Hervé Laurent

M Laurent Dubar

M Laurent

Langlois They

Renault Valérie

Ibrahima, Véronique, Chengkai, Mathieu, Dominique Nicolas Thomas

Jean-Baptiste B Have

Mathieu, Fadik Jeanne Philippe

Francis Gilbert

Keywords: High-Speed joining, High strain-rate, Ductile damage, Numerical simulation, Material modelling

for help in the material characterization tests and for the manufacturing of samples and set-up for the nailing campaigns. A special thank is also given to Marie-Francoise for her tremendous kindness, solving daily-life problems of CEMEF PhD students.

I want to

General Introduction

Context

Environmental policies as well as growing stresses on fuel supply tend to be significant parameters in the design of future mechanical structures. Aerospace, railway, automotive industries have to evolve according to these new constraints without loss in the product quality. Hence, the automotive industry is challenged to find technological solutions to reduce global vehicle CO 2 emission per km. Restrictions are slightly different from one country to another, but it has became a worldwide concern which implies drastic changes in the manufacturing process.

Among the solutions which have been explored by car manufacturers to reduce carbon dioxide emissions, the most relevant one consists in a multi-materials structure using lightweight materials (Figure 1). On one hand, materials with lower density would be promoted, such as alumimiun alloys, magnesium alloys, composites, for vehicle components which do not require high mechanical strengths.

On the other hand, Advanced High Strength Steels (AHSS) would be added to stiffen parts subjected to very high stresses, and balance global loss of rigidity due to local changes from steel to softer materials. Use of metals with very high mechanical properties, such as AHSS, enables thinner parts than usual, which consequently induces once again mass savings.

Joining dissimilar materials

Materials with different chemical compositions and extended mechanical properties usually require joining solutions aside from traditional processes. Meschut et al. (2014a) showed that spot-welding incompatibility might appear between structural parts made of different materials. As consequences, the joining processes based on plastic deformation are investigated to fulfill both industrial requirements -limited preparation of the joining area, reliability, high industrial pace -and technical requirements -flexibility of the process parameters, wide range of join-able material configurations, high joint strengths. Among these processes, clinching, riveting and self-pierce riveting (SPR) have been used for long time in the industry for sheet metal applications on aluminium alloys and low strength steels but [START_REF] Mori | Joining by plastic deformation[END_REF] highlighted that it reached a limit with very high strength steels and low ductility materials. Thus, it has become a first order issue to adapt processes for materials with advanced properties. Optimization were conducted on several parameters affecting join-ability, such as works of [START_REF] Abe | Joining of high strength steel and aluminium alloy sheets by mechanical clinching with dies for control of metal flow[END_REF] in which die shape is optimized to control plastic flow and delay crack occurrence, the works of [START_REF] Huang | Dissimilar joining of aluminum alloy and stainless steel thin sheets by thermally assisted plastic deformation[END_REF] which propose thermally-assisted joining process to increase metal formability and the works of [START_REF] Meschut | Hybrid Technologies for Joining Ultra-high-strength Boron Steels with Aluminum Alloys for Lightweight Car Body Structures[END_REF] which suggest to boost rivet material property to enable boron steel join-ability. In addition to these investigated parameters, [START_REF] Jäckel | Effects of Increased Tool Velocity on Mechanical Joining of Steel and Aluminum Sheet Metals[END_REF] revealed that an increasing tool velocity would improve blind rivet join-ability in a 22MnB5/6016 aluminium material superposition and get a cleaner sheet-rivet fracture interface.

From these considerations, several industrially-based research programs have tried to develop innovative joining solutions for automotive car bodies using material sensitivities to temperature and velocity to extend the range of process join-ability. Martinsen

High-speed nailing process

The process under investigation in the present work is one these promising technologies used for multimaterials applications to join highly dissimilar materials. It is called High-Speed Nailing (HSN) and works at relatively high tool velocity (20 to 39 m/s) to extend the materials perforation-ability and therefore the range of materials join-ability. It consists in a pneumatic-driven piston used to punch a nail into several material layers. The founding principles of nailing as an industrial joining process have been established in the works Torsten [START_REF] Draht | Entwicklung des Bolzensetzens für Blech-Profil-Verbindungen im Fahrzeugbau[END_REF] in which both nailing device technology and nailed-joint properties have been investigated. [START_REF] Hußmann | Weiterentwicklung des Bolzensetzens für innovative Fahrzeugkonzepte des automobilen Leichtbaus[END_REF] refined nailed-joint industrial applicability with mechanical tests on mono and multi-points samples taking into account different sheet materials stiffness, inter-sheets glue, corrosion and also experiments on fatigue bench. The investigations regarding nailed-joint validity have been continued in the works of [START_REF] Flüggen | Einseitiges Fügen von Blech-Profil-Konstruktionen mit lösbaren und nicht lösbaren Hilfsfügeelementen[END_REF] through applications on aluminium profiles and composites. Results from local bending measurements in the nail surrounding area, joint strengths and also failure modes are discussed.

As the joining technology has become more understandable by researchers and engineers, it has been adapted to industrial applications with high production rates by Böllhoff Group, as presented in Figure 2. Nailing device has been set on a robot with a limited joining cycle time (∆t cycle ≈ 1 s).

However car manufacturers remain skeptical since the joining mechanisms are not clearly understood and joint strength can not be predicted yet by engineering simulations.

Numerical simulation of fastener insertion stage

Engineers in charge of joining applications keep claiming for quick and reliable tools to facilitate the decision-making process at the design stage of structures. It should significantly reduce the time and cost of experiments required for the calibration of the joining process settings and the evaluation of optimum joint strength. Indeed most of the preliminary tests could be done numerically through two main simulations: (1) the joining stage simulation and (2) the joint testing simulation. This approach has already been proven on clinching process, in the works of [START_REF] Lee | Design of hole-clinching process for joining of dissimilar materials -Al6061-T4 alloy with DP780 steel, hot-pressed 22mnb5 steel, and carbon fiber reinforced plastic[END_REF], or self-piercing riveting in the works of [START_REF] Porcaro | Self-piercing riveting process: An experimental and numerical investigation[END_REF] and also [START_REF] Fayolle | Etude de la modélisation de la pose et de la tenue mécanique des assemblages par déformation plastique[END_REF], with good results obtained on the prediction of sheets reaction curve to the fastener insertion as well as in-service joint strength. However the new generation of joining process implies the perforation of every layers before the nailed-joint is completed.

Dynamics, damage growth and sheets fracture are additional phenomena to take into account for the simulation of such joining processes. Nowadays it has become a topic of interest for researchers and engineers since it drives the design of multi-materials structures in the automotive, railway, aeronautical and aerospace industries.

Objectives and outlines

The present work results from the partnership of Renault, the department of innovative joining (Technocentre, Guyancourt) and the Center for Material Forming of Mines ParisTech (CEMEF, Sophia-Antipolis) supported by the French National Agency for Research-based Technology (ANRT). This project aims to investigate the high-speed nailing process for mixed-materials applications in the automotive industry and the capability of finite element simulations to evaluate the feasibility and the mechanical features of nailed-joint at the design stage of vehicle development.

Selected materials Three materials have been selected for investigation: (1) a 2.5 mm-thick cast aluminium alloy (AlSi10MnMg(T6)) which is inexpensive and highly convenient in the industry for parts with complex geometry; (2) a 1.5 mm-thick dual-phase steel with high ductility (DP780) and ( 3 Facilities Every nailing tests have been performed using the exact same automated device as used on car production lines in Bremen (Germany) and in South Fremont (USA) on a platform dedicated to joining investigation at IRT M2P laboratory (Metz, France). The RIVTAC ® device is carried on a robot and materials to join are clamped into a special set-up. Nailed-joints are mechanically tested in Renault facilities whereas the materials characterization tests are conducted at CEMEF and LAMIH laboratory (Valenciennes, France).

Observations Several observations are developed throughout this document as markers of the progression of knowledge on this joining process. These ones are based on experimental and numerical considerations and are countered and related to a dominant topic of interest. At the end of the manuscript is presented the "Summary of Observations" which depicts information gathered on the following themes: Joining process settings, Nail insertion stage, Deformation and fracture modes, Sheets temperature increase and Nailed-joint properties.

This document consists in five chapters. Chapter 1 is dedicated to introduce the high-speed nailing process through its components, the joining steps, the main phenomena involved in the nail insertion stage and the basis of nailed-joint cohesion mechanisms. Chapter 2 presents at first the experimental set-up designed for the joining tests. Then preliminary nailing tests are conducted without any consideration to the joint in a large range of process parameters settings to investigate the nailing device capability (joining pressure and hold-down pressure). The nail insertion stage is explored through its sensitivity to materials superposition, joining velocity, nail material and shape. An insight into the mechanisms involved at joining stage is also given through tests conducted with high-speed camera and infra-red camera. Nailing tests are finally processed under clamping conditions on multi-materials superposition to investigate at the joining stage, the sheets reaction, the sheets fracture mode, the cross-section and the final joint features through cross-tension and shear-tension experiments. Chapter 3 introduces the material modelling issues encountered in joining applications. Constitutive and damage laws are proposed to account for plasticity and ductile fracture under dynamic loading. In Chapter 4 these models are calibrated on our materials through compression tests on the nail-shaft in quasi-static conditions and various tension and compression tests on sheet specimens at strain-rates up to 6000 s -1 . Good agreement is obtained with the coupled-damage approach chosen to model tension tests until fracture on a wide range of stress-states. Chapter 5 gathered the process and materials data discussed in the previous chapters to build a finite element model of the nail insertion stage under FORGE ® software. Despite local gap in the joint characteristic dimensions, good agreement is obtained from simulation on the joining kinematics, sheets fracture and deformation modes. Simulation capture the main phenomena observed at the joining stage for every materials superposition, including non-joinable configuration at which nail fractures after impact. Sensitivity analysis to the variation of mesh size, friction, loads, boundary conditions and material formulation are conducted to ensure the consistency of the current model. Finally, results of the nail insertion stage are used as input of the joint testing simulation run in cross-tension and shear-tension configuration.

CHAPTER 1

High-Speed Nailing: Overview of a Joining Process

Introduction

The present work does not investigate every possible configuration of nail insertion processes or nail geometries. It focuses on a particular joining technology designed for applications in automotive industry, in production lines, on parts of vehicle structure made of materials with advanced mechanical properties. Basically, it enables metal-metal joining -total thickness from 3 to 6 mm -in a very short cycle time (0, 9 s ≤ ∆t cycle ≤ 1, 2 s). This method is highly convenient for industrial paces since only one side access and no pre-hole are needed. It consists in a nail punched at high velocity (20 to 39 m/s) into sheets. After full penetration, nail head is fully seated on the upper layer and nail shaft is stuck into the lower one, therefore connecting both layers.

Nail setting device

Nailing device used in the present thesis is commercially known as RIVTAC ® . It has been developed

by Böllhoff Group1 and launched on the market in 2009. In its first version, it was a hand operated tool. But defects quickly appeared: tests from different car manufacturers reached similar conclusions.

Joint was not reproducible since it closely depends on the operator reaction force applied on the tool when the nail strikes the sheets. A second version was developed to correct problems of operator dependency on joint quality and nail re-loading process. This new design became attractive as it could be easily integrated in automated production line, driven by a robot, to process up to 180 joints/min.

Every nailing mentioned in the present manuscript have been done at IRT M2P Laboratory 2 , located in Metz (France), where a platform is dedicated to joining investigations. The automated nailing device inspected here is called "Magazine-Feed" in a version made in 2014. For conveniency, nail-setting device and its main components are presented under a scheme-form in Figure 1.1. It will make easier further study on joining kinematics and avoid unnecessary technical considerations.

Nail

A joint is created between sheets by means of a "connector" component: a rivet, a screw, a nail. . .

In the particular case of nailing, fastener is designed according the two main functions it has to fulfill:

• Perforation and penetration into layers

• Mechanical cohesion at the nail-sheets interface These functions were investigated by Torsten [START_REF] Draht | Entwicklung des Bolzensetzens für Blech-Profil-Verbindungen im Fahrzeugbau[END_REF] which suggests technical solutions to nail insertion process and nail geometry. It leads to a patent by [START_REF] Draht | Method for establishing connection and a nail for this purpose[END_REF] and a first industrial production by Böllhoff one year later. Nail final geometry, chosen for industrial application under the reference FK16-8-3-R07, includes the following features (see Figure 1.2):

• Ogival-shape nose, for penetration To obtain the mechanical properties required to join a wide range of materials, quenching is performed on the nail and followed by a homogenization heat treatment. Thus hardness is supposed to be uniform inside and outside the nail. Table 1

Joining piston

Joining piston, also called plunger, is accelerated in the main joining chamber by air pressure flow up to impact velocity. Mass is around 130 g -whereas nail mass is 1,2 g. Consequently, 99% of impact energy comes from piston kinematics. In other word, piston role is to punch nail into layers to connect.

Hold-down piston

Hold-down piston is a pneumatic component of the automatic nailing tool which evolves in its own pressurized chamber, called hold-down piston chamber, which is separated from the main chamber. It ensures two main functions:

• Maintain contact between headpiece (barrel) and upper sheet;

• Control backward reaction of the tool.

Indeed, Figure 1.1 highlights that hold-down piston is mechanically connected to the setting tool.

Any motion of the setting-tool in the vertical axis implies motion of hold-down piston into its own chamber. Consequently, a suitable set of pressures surrounding the hold-down piston is able to trigger setting-tool motion on the vertical axis. Thus joining can be done even if joining direction is opposite to gravity one, with a contact force which can be set by operators. Reciprocally, backward motion of the setting-tool after nail impact can be damped and controlled by air pressures inside the hold-down chamber. To achieve these functions, a double-acting piston has been chosen by engineers.

Buffer

If inappropriate kinetic energy has been chosen for joiningi.e. a suitable air pressure to be injected into the main chamber -, piston might not be stopped after full nail insertion into the sheets. It keeps moving downward until it has reached bottom of the chamber. To dissipate this energy excess, a buffer is located at the bottom of the joining chamber. In the current version of the setting device, it is made of elastomer and is shaped to contain an important air volume. It has a mixed damper-spring behavior:

buffer deformation tends to damp piston motion but also propel it upwards as a spring. Hence a state of equilibrium is reached when piston oscillations have stopped.

Sensors

Two default sensors are integrated into the setting device: force sensor and piston-displacement sensor.

They both provide important data to the monitoring process chain. The piezo-electrical force sensor is located on the upper part of the setting device. It is generally used to record sheets reaction in dynamic applications from changes in polarization. Coupled to a data unit, it covers force variations such [12.5; 112000±0.5 N] with a time-response in the range [0.033; 0.33 ms]. Thus very fast phenomena can be detected, especially impact. Piston-displacement sensor is located right under the buffer and is based on magneto-resistivity. It tracks piston marks displacement with a 100 µm-precision and capture its motion all along the descent until possible velocities up to 100 m/s.

Setting parameters

Before every joint to be processed, two parameters have to be set: (1) the joining pressure, which drives piston motion and therefore nail insertion path; (2) the hold-down pressure, which controls the contact force applied on sheets by means of the setting-device barrel. RMK: α is the angle, depicted in (b), between joining and gravity directions.

Joining pressure

In its current version, the setting device allows joining pressure to be set in the range 3 to 8 bars. This is possible thanks to a separate tank unit which adjust pressure to the one desired for joining. A servo-valve is dedicated to control the air flowing from the tank to the main chamber, pushing piston downwards. This servomechanism uses pressure measurement inside the main chamber to correct air flow and obtain a measured pressure to be as close as possible to the referent pressure. Pressure sensor and servomechanism are both gathered in the servo-valve which measures and adjusts joining pressure as set. However its reliability to maintain constant joining pressure all along the piston motion is questionable since the response-time is 0.2 s.

As consequence, pressure in the joining chamber should decrease due to two main phenomena: (1) the chamber volume is increased because piston slides downwards and therefore air pressure decreases as well;

(2) the air flowing into the chamber is not adjusted because response-time is bigger than piston descent duration (4.5 ms≤ ∆t piston-descent ≤ 8 ms).

On the basis of measurements discussed in Chapter 2, joining piston velocity can be evaluated at impact time, i.e. when nail contacts the upper layer. Results obtained with different initial joining pressures are plotted in Figure 1.3a. It can be noticed that piston impact velocity -nail impact velocity as well -evolves almost linearly with the joining pressure initially set. It makes this process really convenient to handle for those in charge of the joining pressure evaluation: increasing from 3 bars to 4

bars the joining pressure affects in equal proportion the velocity than if it would be done from 6 to 7 bars. Hence impact velocity set in the range 20 to 39 m/s generates a joining kinetic energy at impact ranging 20 to 100 J.

Observation 1.1: Joining piston velocity

Piston impact velocity evolves almost linearly with the joining pressure.

Hold-down pressure

The range of possible hold-down pressure is 1 to 5 bars. As previously mentioned, hold-down pressure is used to control contact conditions between setting tool and sheets. It implies two pressure components, each one in a side of the piston. Basically hold-down force is the result of force balance acting both side of the piston. Assuming a setting device mass of 13 kg and pressure balance between both chambers, the sheets contacting force can be computed from the force equilibrium principle. As it can be noticed in Figure 1.3b, if joining direction is toward the gravity one (α = 0°), the hold-down force obtained from calculations is about 450 N for 1 bar and 1800 N for 5 bars. However if joining direction is opposite to the gravity one (α = 180°), hold-down force would be lower than in previous configuration: it would reach 200 N for 1 bar and 1550 N for 5 bars.

Observation 1.2: Hold-down force (1)

1. Nail setting device generates a contact force on the sheet called Hold-down force.

2. This force amplitude increases with the Hold-down pressure.

3. If joining is conducted towards the gravity direction, the Hold-down force is significantly bigger than if it was in the opposite direction. contact with the nail head all along this step. It prevents the nail from gravity and makes that only piston force is driving nail descent into the sheets. Three forces are involved in the piston acceleration -therefore its kinematics -and can be listed as follows:

• Air injection (force acting downward )

• Friction at the piston-piston chamber interface (force acting upward )

• Friction at the nail head-headpiece blades interface (force acting upward ) • Air injection (force acting downward )

• Friction at the piston-piston chamber interface (force acting upward )

• Piston's material reaction to compression (force acting upward )

• Nail's material reaction to compression (force acting upward )

• Sheets reaction to perforation (force acting upward )

• Sheets reaction to bending (force acting upward )

Understanding and modelling the phenomena involved into this step will be the main target of this PhD work and will be discussed in depth later.

Stage 3. Piston rebound

The energy excess after sheets have been fully perforated necessary implies some piston rebounds.

The impact with the buffer occurs after a measured piston displacement of 133.6 mm and remaining energy can not be dissipated instantaneously into the buffer. Consequently the piston is propelled upwards (see Figure 1.5d). However air pressure in the joining chamber prevents piston from going too far. The energy is slowly dissipated into air compression while piston is moving. When energy has been fully consumed, piston is static and only subjected to air pressure acting downwards. Piston is consequently driven back to the buffer for a second impact. Depending on the energy excess after joining, this pattern can be reproduced several times until new static equilibrium could be obtained and process completed. On the basis of similar considerations than in the two previous stages, the set of forces involved into this step can be listed as follows:

• Air injection (force acting downward )

• Friction at the piston-piston chamber interface (force acting upward )

• Buffer reaction to compression (force acting upward )

Joint quality indicators

Experiments conducted on a wide range of materials and process setting configurations have revealed naturally, test after test, that some influent parameters exist, and could improve or decrease the joint cohesion. Hence it is possible to associate a joint statement to the proper scale of a parameter. For example, interlock is a dimension used to qualify rivet connection inside the sheets. It can be related to joint quality considerations since tests have highlight without any doubt that interlock affects joint strengths. Similar relationships could be found for High-Speed Nailing applications. Such relationship between joint quality statement and measurement will be later called "joint indicator" and based on in-process data, geometries or mechanical tests.

Process monitoring

Nailing tool is dedicated to be used on production lines. Hence, joint quality must be ideally evaluated after each cycle without any operator intervention. A message has to be returned to the command center: OK if joint satisfies quality requirements and NOK if it does not. The two sensors previously mentioned are involved into the monitoring process: force and piston-displacement sensors.

Draht and Kolac (2012) have detailed in a patent the methodology implemented in the current version of the setting tool. It is stated that a joint is considered OK if it fulfills the following set of conditions:

• Piston displacement curve -which somehow indicates nail penetration state -reaches a maximum followed with at least a minimum before a second maximum (see Figure 1.6a);

• Piston doesn't impact the buffer located at the bottom of the joining chamber.

An additional control step, mentioned in the same patent, consists in a force contactor located under the buffer. If impact with buffer occurs, contactor will capture it and return a NOK joint statement.

As far as I know, it is not implemented in the current device used at IRT M2P.

From a practical point of view, joint statement returned after joining needs to be previously configured by operator. A joint which had been considered mechanically and visually optimized will be used as a reference for following ones. The operator sets a window in the maximum displacement region reached by the piston: if maximum recorded position is in the defined range, joint is stated OK. 1. Nail insertion is driven by joining piston kinematics therefore nail penetration depth is somehow related to piston displacement Z(t).

Non-Valid

Valid Non-Valid Z Z 1 t Z Z 2 t Z Z 3 t (
2. An under-estimated joining pressure leads to partial nail insertion.

3. An over-estimated joining pressure leads to an excessive sheet-materials deformation in the area surrounding the nail shaft.

4. Between those states exists an optimum joining pressure which enables full nail penetration and avoid decohesion in the sheets-nail shaft contact area.

Visual inspection

Because High-Speed Nailing is an energy-based process, it seems reasonable to suggest that a joint with optimized features is produced under optimized joining energy. Lack of energy would not enable nail insertion whereas an energy excess would over-deform material layers. These joining configurations are represented in Figure 1.6b. It is only common sense but it drives the whole joint control methodology.

Methodology

As formulated in Draht and Kolac (2012) joint is visually considered OK if it fulfills the following set of conditions:

• Nail's head is fully seated on the upper-sheet without excessive sheets bending;

• Tightening torque is bigger than reference torque: sheets cannot freely rotate around the nail under manual checks.

It is also mentioned that joint control process could be done using a torque wrench calibrated from a torque reference. When torque would become higher than reference, tightening would be stopped to preserve the joint. However the current nail head geometry is circular, without grip, and therefore unadapted to torque tightening.

Nail grip control

In addition to excessive sheets bending around the impact location, another indication of inappropriate joining energy could be found from visual inspection. If it is possible to access the joint from below, it should be noticed that contact interface is limited between bottom sheet and nail. Although made from strong material, sheet could deform so much that nail could not get locked into the sheets after penetration but keeps bending materials at the interface. It produces this failure pattern called "petalling" -which can be noticed on the right picture of Figure 1.6b -which in this particular case has been claimed by [START_REF] Steinig | RIVTAC® : The Innovative High Speed Joining Process[END_REF] to not be good for joint cohesion.

Joint cross-section

These geometrical indicators of joint quality can be defined more accurately through analysis of a nailed-joint cross-section. Figure 1.7 presents an example of cross-section which could be assumed to be made of {Steel/Aluminium} layers superposition. Seven dimensions arise from the considerations of the nailed-joint geometry and can be defined as follows:

• h ∈ [0; 14 mm]: Residual height between nail head and the upper-sheet (red);

• l 1 : Grip length of the upper sheet at the interface with the nail (blue);

• l 2 : Grip length of the lower sheet at the interface with the nail (green);

• g: Gap between upper and lower sheets (purple);

• n 1 ∈ ]0; 6 mm]: Total length of the nail ogival-tip (orange);

• n 2 ∈ ]0; 3.15 mm]: Diameter of the nail ogival-tip (orange);

• n 3 ∈ ]0; 6 mm]: Height of the nail ogival-tip (orange);

The first dimension h states if a joint can be either considered deficient h > 0 or optimized h = 0.

Two other main dimensions l 1 and l 2 define nail-sheets interface and nail grip conditions as well. The larger is the grip length of the lower sheet l 2 , the bigger is supposed to be the joint strength. Gap between upper and lower sheet g can be useful to investigate bonding area if glue is inserted at the interface. Then nail dimensions n 1 , n 2 and n 3 attest how crushed the nail is after impact. Stiff layers superposition usually induce very large nail deformation with the ogival length n 1 which tends to decrease and the nail-tip diameter n 2 to increase. Those notations and dimensions will be used in further sections to treat nailed-joint in various materials superposition and compare with geometries obtained from simulations.

Mechanical testing

Metal joining in automotive industry was and is still dominated by welding standards. They have been established to qualify mechanical properties of spot welding joints with multiple layers superposition.

From one country to another, specimens dimensions have been set to different dimensions but the chosen testing methodology remains equivalent. Mechanical tests are conducted as close as possible to in-service loading conditions. For this purpose, two tests are commonly used to characterize joint strengths in the two main loading directions: perpendicular to the joint in shear-tensile configuration (see Figure 1.8a) and co-linear to the joint in cross-tension configuration (see Figure

Shear-tensile test

Shear-tension means that loading direction is perpendicular to nailed-joint principal orientation given by the nail shaft axis. Thus both sheets are moving in opposite directions; nail playing a role of intermediary, therefore transmitting loads through its interface. As consequence, stress is distributed into the whole nail volume with an amplitude higher than it should be into the sheets. Because stresses are more diffuse into sheets than the in nail, joint mechanical response in this direction is driven, at first order, by nail material properties, nail size and contact interactions between the nail and the sheets.

Cross-tension test

Cross-tension means that loading direction is co-linear to the nail shaft axis. While lower sheet is moving in one direction, the upper one goes to the opposite. If joining stage is considered valid, nail head lies on the upper-sheet and the shaft is stucked into material layers. The upper-sheet displacement is consequently pulling-out nail head whereas nail shaft is gripped into the lower one. An example of cross-tension test curve can be seen in Figure 1.6c (Center) in the case of an optimized joint whereas incomplete joining would induce an early failure (see Figure 1.6c (Left)) and excessive joining, the destruction of cohesion state at the interface (see Figure 1.6c (Right)). In this loading direction, joint weakest area can be find in the surroundings of the nail ridges-lower sheet interface where loads are transmitted between sheets through a limited volume.

Observation 1.4: Nailed-joint strengths (1)

1. Joint response in shear tensile configuration is mostly related to the nail shaft shear resistance.

2. Joint response in cross-tension configuration is mostly related to the nail ridges grip on the lower sheet. Despite the hierarchical considerations between joint strengths in these different material configurations, this graph highlights that each configuration experiences a plateau where, whatever is the joining pressure chosen to process the joint, maximum cross-tension is unchanged. This plateau might be bigger in superposition which have lower-sheet material with high ductility or extended thickness. Thus superposition involving layers of given materials and thickness can be used for industrial applications only if plateau is large enough to accept slight variations of input joining pressure. This is necessary to ensure that even in manufacturing lines working at very high pace, small variations of input joining pressure would not affect the joint to be processed out of the quality requirements.

Max. Cross-tension force

Joining pressure 1. Cross-tension strength evolution with respect to input joining pressure is assumed to start at zero, increase until a plateau with a maximum value is reached and decrease back to zero.

2. Both span and amplitude of the optimized cross-tension strength plateau depends mostly on the material behavior of the connected layers.

Summary of Chapter 1

High-Speed Nailing device has been presented through its main components: nail, joining piston/plunger, hold-down piston and buffer. Nail design and material property are related to its main functions, i.e.

sheets perforation and sheets cohesion. Nail setting device parameters are twofold: joining pressure and hold-down pressure. They can both be interpreted into physical-based variables much more convenient to handle process capacity and limitation. Hence, joining pressure sets in the range of 3 to 8 bars would accelerate piston from impact velocity of 20 to 39 m/s. Hold-down pressure can be set from 1 to 5 bars and adapt sheets pretension from 0.2 to 1.5 kN.

A joining cycle is then described and the main joining stages are detailed under the following sequence:

Piston acceleration; Nail insertion into the sheets; Piston rebounds. Piston motion is described from ignition of the joining process to the time joint is completed and influent forces are identified. High-Speed Nailing is consequently an energy-based joining process which has to be set properly by operators in order to avoid either lack of energy, due to limited impact velocity, or excess of energy, which would induce piston oscillations until remaining energy is fully dissipated.

Joint quality is controlled through different approaches: monitoring piston position across time; inspection of nail insertion depth and sheets failure pattern; mechanical tests. Process monitoring uses reference curve to identify possible defects occurring in the joining stage. It enables a joint quality evaluation even if we have no access to the joint for visual inspection. Mechanical tests remain necessary to qualify joint strengths and are usually limited to cross-tension and shear-tensile configurations. Due to the configuration of nail-sheets interlock, cross-tension strength is most likely weaker than shear-tensile strength. This explains why charts in form of "Cross-tension strength vs. Joining pressure" are widely estimated among engineers to qualify joint features before any integration into industrial applications.

Résumé en français

Le procédé de clouage grande-vitesse est présenté au travers de ses principaux composants: le clou, le

Introduction

Despite the general understanding of High-Speed Nailing process, discussed in Chapter 1, the present technology remains poorly understood by engineers and researchers as well. We considered that the modelling outcome of the nail insertion stage, targeted in the present work, could not be separated from a good understanding of the process and its involved phenomena.

Experimental set-up designs

To ensure that valuable information are obtained from experimental nailing investigations, special setup have been designed and manufactured. One has been used for High-speed camera and Infra-Red camera experiments (Figure 2.2), an other one for nailing under fully-clamped conditions (Figure 2.1).

Camera set-up

This set-up is designed to access the joining location from beside and below. It is made from an aluminium tube, 170 mm of diameter and 20 mm-thick, fixed to a plate for stability. The access from beside is protected by a safety pane if nail or sheets projections occur. Recording from below is possible if a mirror is put under the sheets. Set-up description is presented in Figure 2

High-Speed camera configuration

Camera set-up is used to capture nail insertion at very high frame rate. Minimization of the processing window has allowed to record the joining stage at 100 000 frames/sec. Assuming that joining duration does not exceed 1 ms, an average of 100 frames is obtained to sequence nail penetration path into the sheets. Figure 2.3a shows experimental set-up with high-speed camera positioned as close as possible to the joining location, surrounded by two spotlights. Sheets are clamped between two plates of the set-up upper part using four bolts. Four clamping tools are also used to maintain set-up position on the table all along the experiments. Finally, setting device headpiece has been partially cut, on 180°a ngle, as shown in Figure 2.4a, to ensure that camera would capture nail motion from impact to the moment joint is completed. Thus camera has a free access to nail penetration through one view from beside (Figure 2.4b) and another one from below (Figure 2.4c). Camera can not record at the same time the images taken from below and beside but tests were repeated in identical conditions to get those data separately.

Infra-Red camera configuration

IR camera used for analysis is a SC7600 (FLIR) with a 25 mm-diameter lens. Set-up configuration is presented in Figure 2.3b with camera positioned as close as possible to the joining location. It targets a mirror located under the sheets with an angle which has prealably set to 45°. This mirror is adapted to IR wave lengths and access the joining location from below, as presented in Figure 2.4c. Furthermore, lower sheet has been painted in mat black in order to increase surface emissivity and improve thermal measurement accuracy.

Sheets-clamping set-up

Sheets-clamping set-up has been designed to ensure that boundary conditions of the sheets are controlled. It is made of two main parts: upper-plate and base. The base is a cylindrical steel made frame on which sheets are placed. In addition to its 40 kg-mass, a groove has been integrated into the set-up design to improve stability on the working table using clamping-tools, as noticed in camera set-up of Figure 2.3. The upper plate is also cylindrical and has a central hole to enable the setting headpiece to access the upper-sheet and insert the nail. Clamping effect is achieved when bolts are tightened to connect both parts, as can be noticed in Figure 2.1. Contact area between the sheets and the set-up parts is controlled through rings located above and below the sheets. Thus clamping force is concentrated on a circular area and transmitted to a load cell located right under the lower ring.

It measures clamping force and in-process maximum joining force. Sheets clamping can be set up to a maximum force of 82 kN. However most experiments were conducted with a force which has been initially set around 40 kN, therefore local stress in contact area reaches at least 77 MPa. Three rings of different internal diameter have been machined: 28 mm, 30 mm and 34 mm. External diameter is set to 50 mm but contact area is bounded on one side by ring internal diameter and on the other side by specimen external free border (38 mm-width). Thus sheets reaction can be changed according to the clamping diameter; it enables to investigate sheets stiffness effect on the nail insertion path.

Investigations on Joining force and Hold-down force

The nail setting device has been already presented in Chapter 1 in terms of components and main joining stages. However lot of information are missing at this point to build a numerical model of the high-speed nailing process. Two parameters have been mentioned to drive conditions in which joint is going to be processed: the joining pressure and the hold-down pressure. Both of these parameters are investigated in the following to understand how it could affect joining and be modelled afterwards.

Joining force: Analysis of the piston kinematics

As previously mentioned, nailing device is an energy-based joining device in which a piston is accelerated until a suitable impact velocity is reached and promotes nail insertion into the layers. Hence the joining pressure parameter drives nail insertion through impact velocity which can be set in a range 20 to 39 m/s. This information reveals in what condition impact occurs and how settings could be adapted to slight changes in materials thicknesses or grades. However it does not state for the evolution of force applied to piston while nail insertion has begun. To quantify this force evolution, we assumed that piston kinematics is driven by the same set a force -materials response exceptedwhatever there are layers to connect or not. In other words, nailing tests conducted without sheets enable an estimation of the forces involved in the joining stage. The following campaign will present nailing tests conducted on the whole range of joining pressures -without sheets -and enables to draw the capabilities of the nailing device and gives afterwards an estimation of the sheets reaction.

Capability of the nail setting device

Experimental set-up Twenty-six nailing tests have been performed with joining pressure increased from 3 bars to 8 bars (increment = + 0.2 bar) and no sheet at the joining location. As detailed in Figure 2.5b which presents the experimental methodology, nail is punched out the barrel to begin a free flight. It is collected in a foam located 10 cm below the area the sheets are supposed to be. Two stiff metal sheets, separated by 12 mm, have been preliminary positioned in the joining location to maintain contact with the headpiece but without being an obstacle to the nail descent. Figure 2.5a shows how sheets have been positioned

to avoid any contact with the nail.

Piston displacement curves without layers to join

In configuration detailed in Figure 2.5, piston displacement curve is not affected by sheets reaction.

Piston kinematics is driven by a set of forces listed in section 1.4.1 regarding piston acceleration stage, section 1.4.2 regarding nail penetration and section 1.4.3 regarding piston oscillations, nail and sheets reactions excepted. From the knowledge of piston displacement at impact with sheets (Z impact = 119.5 mm) and piston displacement at which contact with buffer starts (Z buf f er = 133.6mm), three stages can identified in Figure 2.6a: Piston acceleration, Joining stage and Contacting buffer. Hence piston travel range in which joining is supposed to happen is pictured by a red line, started at impact and continued until maximum piston displacement is reached. Indeed piston keep moving downward -in the joining direction -even after contact with buffer has occurred. As presented in Figure 2.6b, piston displacement can be reset at impact to focus on the joining stage. All nailings are plotted, from highpressure with high slope to low-pressure with low slope. It can be noticed that: (1) piston displacement over time is almost linear in the "joining range" (area marked n°2 in Figure 2.6b) whereas it is highly non-linear in the "contacting buffer range" (area marked n°3 in Figure 2.6b). The buffer reaction to compression F buf f. affects the piston motion linearity ; (2) piston maximum displacement inside the buffer increases with the joining pressure.

Joining force formulation

The equilibrium principle, written in (2.1), describes piston motion at every moment, from impact to maximal piston displacement, as a function of these forces:

m pn • --→ γ(t) = -------→ F air inj. (t) + ------→ F f rict.1 (t) + ------→ F f rict.2 (t) + -----→ F buf f. (t) (2.1) = ---→ F eq (t) (2.2)
An equivalent force -→ F eq is used to express the resulting force applied to joining piston under one term.

As shown in (2.1), it results from every possible force components, noted:

-----→ F air inj. : Air pressure force from the joining chamber ----→ F f rict.1 : Friction force at the Piston-Piston's chamber interface ----→ F f rict.2 : Friction force at the Nail head-Headpiece blades interface ---→ F buf f. : Buffer reaction to compression Mass m pn is the sum of piston and nail mass and summarized the global mass subjected to motion.

Piston acceleration is noted -→ γ under vectorial form but in the present case it can be reduced into scalar form γ since motion is constraint on a single axis, toward joining direction, as:

γ(t) = d 2 z(t) dt 2 (2.3)
Where z is piston displacement in the joining travel range, fitted from experimental data with a 4 thdegree polynomial formulation and its parameters B 1 , B 2 , B 3 , B 4 , B 5 . Thus piston displacement z, velocity v, acceleration γ are respectively expressed as:

z(t) = B 1 t 4 + B 2 t 3 + B 3 t 2 + B 4 t + B 5 (2.4) v(t) = 4B 1 t 3 + 3B 2 t 2 + 2B 3 t + B 4 (2.5) γ(t) = 12B 1 t 2 + 6B 2 t + 2B 3 (2.6) Polynomial coefficients B 1 , B 2 , B 3 , B 4 , B 5
are identified and given in Table A.1. Consequently, proper calculation of the joining force F eq applied on piston can be done using (2.6) and plotted with respect to time or displacement, as shown in Figure 2.7a. It can be noticed that bigger is the joining pressure higher is the force pushing downward the piston. At impact, pushing force is about 700 N for 8 bars, 300 N for 7 bars, 230 N for 6 and 5 bars, 80 N for 4 bars and almost negligible for 3 bars. It means that nail penetration is not only made by impact velocity but also by force balance which keeps pushing downward while joint is being processed. This force is maximum at impact and decreases with piston motion. There is a displacement threshold at which force is not pushing piston anymore. As presented in Figure 2.7b, this point appears earlier at low pressure -low impact velocity -than high pressurehigh impact velocity. In addition to piston and nail inertia, joining pressure inserted in the main chamber induces a force which keep pushing the joining piston all along its descent.

2. This force amplitude is maximum at impact and decreases while the nail is being inserted.

3. At impact this force is about 700 N for 8 bars, 300 N for 7 bars, 100 N for 4 bars and almost negligible for 3 bars.

Piston loading capacity formulated as a surface

From the considerations of both Figure 2.7a and Figure 2.7b, joining force applied to piston appears to affect nail insertion only when it is positive. Negative force would imply that piston motion is promoted in the opposite direction than the joining one, therefore loosing contact with the nail. A convenient approach to present force evolution applied to piston along the joining path could be a 3D surface. Planar coordinates are distributed along x and y axis, respectively allocated to Impact velocity and Piston displacement whereas z axis represents Joining force F eq . Piston loading surface is drawn in Figure 2.8. This surface will be really convenient when piston kinematics modelling will be introduced in Chapter 5. Thus force applied to piston can be set for any given impact velocity and piston displacement. Along the piston path, this force brings additional joining work to kinetic energy already stated by impact conditions. , from (2.8), can be found in Appendix A.

Sheets reaction in the nail insertion stage

When joint is being processed, piston kinematics is driven by the set of forces listed in section 1.4.3.

It includes "Joining force" F eq previously discussed which remains even if there are no sheets to join.

According to (2.1), piston kinematics under joining -with the sheets properly set -is expressed:

m pn • --→ γ(t) = ---→ F eq (t) + ---------→ F pist comp. (t) + ---------→ F nail comp. (t) + -----→ F perf. (t) + -----→ F bend. (t) (2.7) = ---→ F eq (t) + -----→ F reac. (t) (2.8)
and force components noted:

------→ F pist comp. : Piston reaction to compression -------→ F nail comp. : Nail reaction to compression ---→ F perf. : Sheets reaction to perforation ---→ F bend. : Sheets reaction to bending ---→

F reac. = ---→ F perf. + ---→ F bend. + -------→ F nail comp.
Piston reaction to compression F pist comp. is assumed negligible regarding sheets components since

(1) its large diameter prevents materials from high amplitude local stress ; (2) it is made from stiff tempered material and (3) it is free to move in joining direction therefore rigid body motion is more likely preferred than plastic deformations. Thus a set of reaction force components can be gathered under the term F reac. and calculated from the consideration of equation (2.8). Details of this calculation process can be find in Appendix A with an example of computed sheet reaction curve in Figure 2.9.

Observation 2.2: Sheets reaction to the nail insertion (1) Due to differences between the joining force and the sheets reaction (F eq <<< F reac ), it can be assumed that piston kinematics is almost entirely assigned by sheets reaction to nail penetration.

Hold-down force: Experiments on multi-materials configurations

As mentioned in Chapter 1, the hold-down pressure is a parameter which controls the contact force between nailing device headpiece and the joining area. It results from the pressure balance in the hold-down chamber and gravity. The pressure value which has been set by the operator is certainly not the pressure or stress which can be found at the interface between the tool and the upper layer.

It is consequently an important data for joining process modelling as well as a better understanding of joining forces. This issue is investigated in the present section with joining tests conducted on multi-materials superposition and different hold-down pressure.

Experimental set-up

In these experiments, the instrumented sheets-clamping set-up has been used to ensure identical boundary conditions between tests. An amplifier Quantum MX840B (HBM) is used to collect electrical data from the nail setting device and the instrumented frame. The signal from the setting device ranges [21.6; 26.4V] (sensitivity = 1 V/bar) and attests the pressure amplitude inside the hold-down chamber.

The signal from the instrumented frame is generated by a piezoelectric sensor CFW/140kN (HBM) located under the joining location (see Figure 2.1) which is amplified through a CMD600 (HBM) before it can be treated. From the moment joining sequence starts to the moment it ends, both hold-down pressure and sheets reaction will be measured which will reveal how is the pressure evolution and therefore how does it affect the sheets contact force.

To investigate these two parameters, thirteen joining configuration have been chosen and are listed in 1. Before joining starts, the tool contacts the upper sheet and hold-down pressure is adjusted.

2. Hold-down force is constant whatever is the materials superposition (steel, cast alu ...).

Hold-down pressure evolution

Tests #1 to #5 were performed with identical clamping diameter d = 30 mm with only hold-down pressure changed from 1 to 5 bars (increment = + 1 bar). Hold-down pressure is measured all along the preparation time in which nailing device is moved to joining location and contact force with the sheets adjusted. These evolution are plot for different pressure settings in Figure 2.10a. For every curve, a peak C i can be noticed which attests that contact has happened. Sheets reaction have temporarily increased the pressure in the hold-down chamber until balance has readjusted to the appropriate one.

Then a peak in the negative direction appears N i which shows that nail insertion stage also affects pressure in the hold-down chamber. Indeed sheets reaction to impact is a force acting upwards which tends to compress the air remaining in the upper part of the joining chamber. It triggers setting device motion, sliding upwards along a vertical rail pictures in the diagram Figure 1.1 with little balls. Since the setting tool is connected to the hold-down piston, the piston is moved upwards which therefore tends to increase pressure in the upper part of the chamber. Pressure balance in the hold-down chamber globally decreases as it can be noticed in Figure 2.10a.

Sheets loading force

The preparation joining stage starts when contact between the headpiece/barrel and upper-sheet occurs. Thus contact force is captured by the piezo-electric sensor located under the sheets. The same set of colors have been used to distinguish hold-down pressure with different amplitude. As shown in

v r = a(v p i -v p b ) 1 r
(2.9) in which v i , v r and v b are respectively the initial projectile velocity, the residual projectile velocity and the ballistic limit velocity. This is a simplified perforation model based on conservation laws. It assumes that impact momentum is fully transferred to the plug, so in the original form a = m p /(m p + m pl )

with m p and m pl the masses of projectile and plug, and r a curve-fitting parameter. This model has been used and extended to more complex impact applications in the works of [START_REF] Forrestal | Perforation equations for conical and ogival nose rigid projectiles into aluminum target plates[END_REF] Investigations on Terminal Ballistics cover a wide range of impact velocity, from 50 m/s to 12 000 m/s. [START_REF] Zukas | Impact dynamics: theory and experiment[END_REF] highlighted that target undergoes totally different failure mechanisms whether impact occurs at low-velocity or high-velocity. A criterion called the Johnson's damage number δ, from the works of [START_REF] Johnson | Contact mechanics[END_REF], is widely used to classify impact severity according to the range of impact velocity v, projectile density ρ and target yield stress σ Y . Table 2.3 presents variation range of δ-parameter according impact velocity. It can be noticed that high-velocity is a range at which inertial forces are dominant regarding material cohesion forces. At low-velocity, material response is elastic. In between, material exhibits plastic behavior until impact velocity of 100 to 1000 m/s at which hydrodynamic material behavior tends to appear.

The interested joining process implies velocities from 20 to 39 m/s with a material response which is mostly plastic and fracture patterns -petalling, shear plugging, ductile hole enlargement -which are typically associated in the literature rather to low velocity impact than high regime. If the projectile and target parameters are assumed to be made of steel (ρ = 7800 kg.m -3 ; σ Y = 1000 MPa) and impact velocity taken as v = 39 m/s, sheets response regime is estimated to be fully plastic because δ ≈ 10 -2

(see Table 2 # 20 1. Superpositions made of steel and/or aluminium layers fail mainly under three modes: petalling, shear plugging or mixed petalling-shear plugging.

(d) t = ms (Impact) (
2. Dominant petalling mode occurs for {Cast alu./Steel} superposition, i.e. when nail ogival shape is preserved and steel sheet fails by ductile hole enlargment.

3. Dominant shearing mode occurs for {Steel/Cast alu.} superposition, i.e. when nail tip is flattened after impact with the steel layer which promotes the area down the tip -called a plug -to be separated from each sheet. Plug shape is even bigger if nail tip is flat. which is mechanically deposit on the surface whereas H3 nail is coated with an Zinc-Nikel mixture which is deposit by electroplating. It would be reasonable to suggest that only coating is involved in the projection flow which is ejected after impact.

Effect of the joining pressure

As expected, joining pressure affects impact velocity and consequently nail penetration depth inside the material layers. In {Steel/Steel} superposition, this assumption is confirmed in all tests where pressure is increased gradually from 5 bars to 7 bars. For example, in test #1

Effect of the nail features

Tests #1 to #12 have covered two nail geometries -Ogival and Flat nail-tip -and two nail materials -H3 and H5 -at 5 bars, 6 bars and 7 bars. For every pressure setting, nail geometry and material effect on sheets reaction can be compared. Figure 2.27 presents these results with square markers on curves allocated to experiments with flat nail-tip, without markers for ogival nail-tip ones, continuous lines for H3 materials ones and discontinuous line for H5. It appears from these graph that force response slope and amplitude is higher in flat nail-tip confugrations than in ogival ones. It makes sense because sheets materials volume involved in the penetration process is bigger and contact area extended when a flat nail-tip is used. Reciprocally ogival nail-tip has a reduced contact area which implicates a smaller material volume in the mechanism of penetration which therefore induces a softer structural response. This phenomenon can also be observed in Figure 2.22 and Figure 2.24 in mixed materials configurations. Another point to notice is that nail material H3 always produces a sheets reaction to penetration which is slightly higher than H5 's. This can be related to intrinsic material property of each nail. As indicated in Table 1.1, H5 material hardness is higher than H3 which suggests that H5 nail is less subjected to plastic deformation. Thus in a joining stage triggered with a given impact velocity, kinetic energy is going to be mainly dissipated into plastic deformation work of the sheets and the nail for perforation. This distribution should be different if nail material is hard or soft. If nail material is hard, as in the H5 case, a limited plastic work is assumed for the nail and balance suggests that plastic work dissipated into the sheets will be high. If nail material is soft, as in the H3 case, plastic work is distributed into the nail and the sheets and suggests that plastic work dissipated into the sheets is lower than in the previous configuration, with the hard nail. The amount effectively dissipated into the sheets will be spent faster than in the identical configuration with harder nail. Thus piston displacement reached is smaller and therefore the force amplitude has to be higher displacement curve and nail penetration picture attest that nail progression is slightly higher with H5 nail than with H3 's.

Effect of the materials superposition

For a given nail, Figure 2.28 shows how materials superposition reacts to the nail insertion. Whatever the nail is, {Steel/Steel} response is higher than {Steel/Cast alu.} and {Cast alu./Steel} which are almost identical. Only {Cast alu./Steel} maximum force is delayed of about 1.5 mm because steel reaction appears later as second layer than first layer. Global sheets reaction to penetration seems to be a composition of every individual sheet reaction.

Observation 2.6: Sheets reaction to the nail insertion (2)

1. Sheets response to nail penetration is inherent to the sheets and nail materials, geometries... , i.e.

nail-sheets interactions parameters.

2. The maximum sheets force amplitude is affected by materials superposition: it is bigger in {Steel/Steel} superposition than in {Steel/Cast alu.} superposition.

3. The maximum sheets force amplitude is affected by nail features: it is bigger with a flat nail-tip than an ogival one or with a hard nail material than a soft one.

Limitations

Evaluation of local temperature increase

Nail insertion at high velocity into metal sheets naturally suggests that thermal increase could play a role in the joining stage. Plastic work is dissipating throughout the material all along nail impact and penetration stages. Hence the areas which have been plastically deformed become sources of thermal increase with a temperature evolution driven by heat transfer mechanisms. It is a general concern of dynamic forming but the following section will introduce some concepts which will be used later on.

Heat exchange considerations in dynamic forming processes

In dynamic forming processes as in classical ones, large plastic strains are induced into the material to get the desired final geometry. It requires an energy provided by a master piece (forge-hammer, punch ...) which can be directly assessed to the work absorbed for material transformations. The amount of energy which is actually used by a material volume to be plastically deformed per unit time is called plastic work density Ẇ and written as:

Ẇ = σ ij εij = σ Y ε (2.10)
in which σ ij is the Cauchy stress-tensor, εij is the strain-rate tensor, σ Y is the material yield stress and ε is the equivalent strain-rate. It is assumed that most of the energy actually used for metal forming is dissipated into heat. [START_REF] Felder | Effet thermique de la mise en forme -Theorie et phénomènes volumiques[END_REF] has presented a review in which thermal exchange in cold forming processes are investigated. Some studies have tried to quantify the part of plastic work χ which is actually changed into heat. [START_REF] Farren | The heat developed during plastic extension of metals[END_REF] have inserted thermocouples into specimens made of different materials and subjected to tensile tests. It has been found that χ is about 90% for aluminium, copper and mild steel. Then [START_REF] Taylor | The latent energy remaining in metals after cold working[END_REF] in torsion tests on copper specimens have established that the amount of energy which is stored into the material after forming

(1 -χ) -as residual stress, cristal defects ... -is about 9% when the equivalent strain is lower than 2 and decreases when it reaches 3. It has also been shown by [START_REF] Schmidt | A thermal balance method and mechanical investigation for evaluating machinability[END_REF] in cold working experiments on metal tube that total work effectively dissipated into heat after operation is in the range of 0.97 to 1. This explains why the Taylor-Quinney coefficient χ is generally taken as a constant value χ = 0.9 for most materials even if underlying sensitivites to strain or strain-rate can be supposed.

Plastic work is not instantaneously changed into internal heat flow Ė. Competition between different heat transfer mechanisms occurs but could be limited as a first approximation to conduction flow Qcond. between masterpiece and material interface. Thus the evolution of temperature distribution of a material volume which is getting deformed can be expressed as:

Ė = Qcond. + χ Ẇ (2.11) ρC p dT dt = ∂ ∂x j (λ ∂T ∂x i ) + χ σ ij εij (2.12)
As proposed by [START_REF] Felder | Effet thermique de la mise en forme -Theorie et phénomènes volumiques[END_REF], a characteristic dimension can be set to every term of the equation (2.12) to estimate the temperature increase material could experience under forming: ∆t = l/V the characteristic time function of the characteristic length l and characteristic velocity V , τ the thickness of the area and ∆T an estimation of temperature increase.

ρC p dT dt ≈ ρC p ∆T ∆t (2.13) ∂ ∂x j (λ ∂T ∂x i ) ≈ λ ∆T (τ /2) 2
(2.14)

An adimensional parameter called Fourier number F o can be used to quantify competition between heat flow which localizes and heat flow which is pumped out the material volume as the ratio of (2.13) and (2.14).

F o = τ 2 4Ω∆t ≡ V τ 2 4lΩ (2.15)
with the diffusivity Ω = λ/ρC p . Hence Fourier number F o suggests the dominant mechanism which drives the heat transfer. Three main cases are possible:

       F o 1 ⇒ Ė Qcond. Adiabaticity F o 1 ⇒ Ė Qcond.
Equivalence between absorbed heat and pumped-out heat

F o 1 ⇒ Ė Qcond. Conduction is dominant
Forming processes working with high-velocities, i.e. with high V -value in (2.15), can be stated as adiabatic if Fourier number is high enough to consider that conduction can be neglected. Hence conduction term Qcond. in (2.12) can be neglected and energy balance equation becomes:

ρC p dT dt = χ σ ij εij (2.16)
Temperature can therefore be calculated as a function of the plastic work and time.

T (t) -T (t 0 ) = t t 0 χ ρC p σ Y dε pl (t) (2.17) ∆T ad ≈ T d ∆ε (2.18)
with ∆T ad the estimated temperature increase under adiabatic condition which is driven by an amplitude plastic strain ∆ε. The characteristic heat T d represents the temperature increase of a proper material per unit deformation magnitude.

T d = χσ Y ρC p (2.19)
It depends on the material thermal properties and its initial temperature T i and can be estimated at different temperature as presented in In addition to the obvious camera limitations, another phenomenon might affect the accuracy of temperature measurement: sheet material emissivity . It is defined as the ratio of radiated flow from a real surface at a given temperature on the radiated flow from a black body surface at identical temperature.

They are respectively called radiance of the real surface element L m and L°radiance of reference black body surface element with unit W.m -2 .sr -1 and related through (2.20):

= L(T ) L°(T ) (2.20)
Whatever is the temperature T , the radiance of a black body L°is maximum and involves that emissivity of a real surface ranges ∈ [0; 1]. Emissivity and therefore radiance are properties which are not easy to obtain from a real material surface. That is why experiments have been done with the lower sheet painted in mat black. This is really convenient since it allows to quantify surface temperature with emissivity = 1 as if it behaves as a black body. Graph presented in Figure 2.30 highlights how emissivity uncertainty can be related to temperature measurement uncertainty. Indeed, continuous curve states behavior of black body ( = 1) and discontinuous line one example of a real body behavior ( = 0.7). As expected, the radiated flow per unit area per unit solid angle L which is plotted against the vertical axis is different between the two body when a particular temperature is considered, plotted on the horizontal axis. For given flow radiating from the material surface and then catch by sensors, temperature calculation obtained with an emissivity = 1 or = 0.7 will be totally different. The one calculated with the right emissivity would give real surface temperature T r whereas the one calculated with reference black body emissivity would give an apparent temperature T a . As drawn in Figure 2.30, T r would be solution to black body radiation if the heat flow would be higher than the one we have considered just before. Moreover it can be noticed that the higher is the surface temperature, the higher will be the gap between emissivity. In other words, uncertainty on temperature increases with temperature. This is the main issue of these experiments in which perforation occurs on our surface of interest and measured only one time along the joining stage. Emissivity can be assumed to be = 1 at the beginning but it quickly changes as ductile fracture spreads inside the lower sheet and black painting is removed.

Pictures shot in Figure 2.29 with hot points can be either attributed to nail ogival tip which has been heated by friction or lower sheet petals' whose interface has been highly deformed by perforation. After joint has been processed both of these areas are far from mat black color which suggests that real peak temperature T r is larger than apparent temperature T a measured in Figure 2.29d.

Observation 2.7: Sheets temperature increase (1)

1. In the joining stage, the steel sheet is under adiabatic condition (F o ≫ 1) whereas the aluminium one is not (F o < 1).

2. Temperature measurement ensures that steel must exceed locally T > 450°C. The investigations conducted with the high-speed camera in section 2.4 have revealed that nail can be assumed to act as a projectile and the sheets as a target and therefore their interactions could be modified through changes in nail material and geometry or even sheets materials and superposition.

It has also been proved through several tests that structural response to impact -usually presented under a Force-Piston displacement curve -is inherent to the set of parameters which characterizes the nail-sheet interaction (nail-tip geometry, materials, superposition ...), i.e. the structural response is assumed constant within a defined set of impact loading conditions.

However this reaction might change with the sheets boundary conditions. That is why we have chosen to use the sheets clamping set-up with clamping diameter d = 28 mm and d = 34 mm to investigate how sheets reaction could be influenced by local stiffness of the joining area; but also to provide a set of tests in which nailing sensivities keep being investigated through several materials configuration and loading conditions. But at first, the effects of clamping force on sheets reaction has to be preliminary verified before any test on sensitivity is run. This has been done in Experimental campaign n°1 followed with Experimental campaign n°2 on sensitivity to stiffness variation and finally Experimental campaign n°3 focus on nailed-joint reproducibility in terms of sheets reaction but also in terms of characteristic dimensions observed in joint cross-sections.

Experimental campaign n°1: Sheets clamping force

To ensure full clamping of the layers along the joining stage, eighteen joining tests have been done on {Steel/Cast alu.} superposition with three different clamping force f ∈ {26; 36; 46 kN}. It respectively produces normal stress at the interface such as σ f N ∈ {61; 84; 107 MPa}. Joining pressure is gradually increased from 5 bars, in which nail is not fully inserted, to 5.7 bars, in which energy excess tends to destroy the nail-sheets interface in charge of joint cohesion. Joining pressure P = 5.3 bars has been considered the optimized pressure between all tests from the discussions of section 1.5.2. Figure 2.31 presents the "Force vs. Piston displacement" curves of these configurations with increasing joining pressure with color range from black to orange and increasing clamping force from discontinuous to continuous line.

The set of curves of similar joining pressure -same color -is not significantly affected by clamping forces. Orange curves and green curves are gathered and follow the same trend but slight differences can be notice between black curves. Clamping force seems to have no significant effect on maximum force response because curves of same color follow almost the same trend. However it can be notice that maximum piston displacement is different between tests conducted at different joining pressurewhatever is the clamping force. It is maximum for tests performed at P = 5. Joining tests conducted with increasing joining pressure result in increasing piston displacement maximum values, i.e. increasing nail insertion depths (see Observation 1.3.1). excess. This methodology has already been pictured in Figure 1.9 in which joint strength starts from zero force, increases with joining pressure until a plateau is reached and then decreases back to zero.

Experimental campaign n°2: Stiffness of the joining area

Because joining pressure can not be calculated prior to processing the joint, it is found through an iterative procedure. A minimum of five tests is usually needed to find the appropriate joining pressure:

the one which attests that a plateau has been reached. As presented in Figure 1.6b, it is assumed to be the first trial in which nail head has contacted the upper-sheet.

These tests fulfill four main experimental targets:

(1) to compare sheets reaction curves in different configurations;

(2) to observe nail-sheet interface from cross-sections analysis;

(3) to produce nailed-joint thereafter tested mechanically in cross or shear-tensile configuration;

(4) to get database for further comparisons with simulation.

Target ( 1) and ( 2) will be treated in this section whereas (3) in the further section 2.6. Another aspect of {Steel/Cast alu.} joining superposition is the fracture mode, presented in Figure 2.33, which can be assumed to be a mix petalling-shear plugging mode. Aluminium lower-sheet fails with radial cracks (see Figure 2.33a) with a plug has been previously ejected by a nail flattened after impact with the steel upper-sheet. The competition between shear plugging and crack propagation mechanisms is obvious from the consideration of plug shape presented in Figure 2.33b. A crack seems to have spread radially in two main directions of the aluminium sheet bottom part and propagating upwards. Aluminium cylindrical plug shape also suggests that fracture has been promoted downwards by localized through-thickness cracks pushed by above steel plug. This later can be seen in Figure 2 Since steel ductility is higher than aluminium's, this configuration presents a dominant petalling fracture mode. Figure 2.36a shows large petals surrounding the nail shaft and a small plug which has not grown due to ogival nail-tip. Nail color is lighter after penetrating aluminium layer than before. It can be noticed on Figure 2.36b from craved marks on the ogival tip as well as an aluminium groove which flows up towards the nail head.

Such scraping phenomenon involved in the nail penetration kinematics can also be observed in the blue frame Figure 2.37. This occurs because aluminium is firstly bent by impact and then deformed downwards by nail descent. The interface extends so much that it becomes larger than the initial layer thickness. Materials are gathered at the boundary with lower sheet and creates a bulk which is gradually sheared and its components carried out of this region by the nail ridges. In addition to this phenomenon, aluminium volume is conserved since shear plugging fracture is not promoted by nail geometry or stiffness conditions. This explains why a significant gap remains between both sheets even after joint has being processed. Nail-steel sheet interface seems flat with no significant interlock with the ridges. As consequences, cross-tension joint strength obtained in this configuration would be only based on friction forces localized in a very small region of ridge tips, as it can be noticed in the red frame Figure 2.37.

Observation 2.12: Sheets fracture modes (3) About {Cast alu./Steel} superposition 1. As noticed for upper steel made sheet in Observation 2.10.1, the nail-aluminium interface is sheared all along nail descent: sheared pieces are found inside the ridges and on the ogival tip.

2. A steel plug has been formed but the radial cracks triggered petalling sheet failure before the plug separation could happen. 2.39 matches the so-called "asymmetrical principal shear crack" mode. 1. Nail-tip is flattened by impact with a hard material (steel and AHSS).

2. Materials superposition affects impact toughness: the ogival nail-tip is preserved when steel is the lower layer whereas it gets flat when it is used as upper layer.

3. Material strength affects impact toughness: whatever is the position of AHSS layer in the aluminium-AHSS superposition, nail-tip will be crushed by impact.

Observation 2.17: Nail fracture mode (1)

1. Nail penetration capability is limited by its brittleness; it can not be extended by an increasing impact velocity.

2. Nail-tip fracture occurs after impacting a stiff material superposition. Fracture mode is similar to the so-called "asymmetrical principal shear" mode.

Characteristic dimensions variability

The source of variability which has been revealed in sheet reaction curves does also affect nailed-joint main dimensions. Nail penetration can slightly change from one test to another but contact interface between the sheets and the nail can also change. Seven characteristic dimensions have been mentioned in section 1.5.2 and listed h, l 1 , l 2 , g, n 2.44a but with a smaller influence. This nailed-joint asymmetry can be accounted to the lower-sheet fracture mode therefore driven by material inherent ductility. The bigger is ductility, the larger will be l 2 -value and the smaller will be l 2 -standard deviation.

Nail shape deformation is quantified through dimensions n 1 , n 2 and n 3 respectively allocated to nail ogival length, nail-tip diameter and nail-tip diameter depth. These dimensions are not assumed to be important in the joint cohesion mechanism but they might be useful to quantify effect of impact on the nail and use it for further comparisons with numerical simulation of the joining process. Nail 1. Nail-Upper sheet contact distance l 1 variability is negligible (less than 0.15 mm).

length
2. Nail-Lower sheet contact distance l 2 is ranging 116%-122% of initial sheet thickness for steel but only 88%-94% for aluminium. l 2 -variability is bigger for aluminium -0.73 mm -than for steel -0.17 mm. From Observation 2.12.2 and Observation 2.11.2, these differences can be assigned to the materials inherent ductility.

Nailed-joint strengths

Once nail has been inserted into the sheets, the only interest for industrial applications is in-service joint mechanical properties. It has already been mentioned in section 1.5.3 that nailed-joint is subjected in automotive industry to the same qualification process than the one used for rivet or welding point.

It has to respect the same standards and is usually characterized with the same set of tests. For the moment, in Renault joint qualification process it is only limited to these two tests: shear tensile and cross-tension tests. One characterizes joint response perpendicularly to the nail shaft axis whereas the other one characterizes joint response towards the nail shaft axis. It assumes that the joint is characterized at least by these two tests which can explain any other test conducted at intermediate loading direction to the nail shaft axis.

To understand how joint mechanical properties can be assessed to sheets materials joined in specific conditions -nail shaft material, sheets materials or even joining pressure -mechanical testing set-up are introduced at first and then tests results are presented under the form of force-displacement curves and joint fracture modes. They will be therefore used as references for further comparisons with results obtained from numerical simulations.

Mechanical testing set-up

Nailed-joint characterization has been done at Technocentre Renault (Guyancourt, France) with testing devices used on 38 mm x 125 mm sheet samples joined with welding points, rivets, flow drill screws and nails as well. These tests are conducted on two different testing devices and clamping tools.

Shear-tensile configuration

Every shear-tensile tests are performed on INSTRON 5582 testing device at constant velocity 10 mm/min. A load cell located in the lower part of the testing device measures force up to maximum 100 kN (deviation about 0.05% of the measured force). The displacement deviation is estimated to be +/-0.02 mm1 . Set-up conditions can be observed in Figure 2.45a and joint sample in position in Figure 2.45b. The appropriate material layer is added into each jaws to ensure that loading direction lies on a line which starts at one jaw and finish at the other. This knack is pictured in Figure 2.45c and enables joint to be wedged in both upper and lower jaws before test can start.

Cross-tension configuration

Cross-tension tests are conducted on INSTRON 5500R testing device at constant velocity 10 mm/min.

The load cell used in this device is adapted to force measurement up to 100 kN 1 . Set-up conditions can be observed in Figure 2.46a and clamping tool on

Cross-tension strength

As mentionned in Observation 1.5, joint load carrying capability evolves according to the joining pressure. Optimum setting is supposed to be found when a plateau is reached (see Figure 1.9).

Optimized joint strength plateau

Based on Experimental campaign n°2 database, joint samples performed on multi-materials superposi- 1. Optimized cross-tension strength amplitude is proportional to the lower-sheet material yield stress.

2. The plateau span -range of joining pressure in which cross-tension force remains maximum -is proportional to the lower-sheet material ductility.

Joint strength variability

After cross-tension strength plateau has been identified in multi-materials configurations, the joint For further comparisons with simulation results, some dimensions related to sheets bending can be extracted from the joint cross-sections. Scheme presented in Figure 2.55 pictures typical joint geometry after cross-tension failure and its two main dimensions: b 1 is the relative gap between upper and lower sheet before joint decohesion occurs; b 2 is the lower sheet reverse bending displacement. 1.15 ± 0.16 0.88 ± 0.14 The maximum force amplitudes can also be compared between these configurations: {Steel/Cast alu.} has significantly lower force values than the other two. Aluminium considered as the lower layer might not resist as well as steel to the nail twist. The same reason could also justify why the AHSS-based joint has a bigger force response than the steel one.

Shear-tensile strength

Fracture modes

In addition to force-displacement curves, indications on joint geometry can be useful to understand the cohesion mechanism in shear-tensile configuration. Joints are finally considered under the form of cross-tension and shear-tensile samples and tested for various multi-materials superpositions. Finally an explanation of the cohesion mechanisms in these two tests is proposed on the basis of force-displacement curves and post-mortem joint cross-sections.

Résumé en français

Introduction

It is generally assumed that in forming processes as well as joining processes using high tool velocities, materials undergo large plastic strain, high strain-rate and high temperature. Dynamic-sensitive models enable the prediction of yield stress and fracture occurrence and can be used in finite element simulations for complex structural analysis. The yield prediction is modelled using the so-called yield-function whereas sensitivities to the yield stress are accounted by a constitutive law. Fracture is generally assumed to be the final event of critical damage accumulation into the material and is modelled through damage laws.

This chapter presents plasticity and damage formulations adapted to high-speed forming applications on sheet materials. The stress-state parameters are firstly introduced before going through modelling considerations. Then plasticity is discussed with the yield function and constitutive laws. Finally an overview of the damage formulations is presented with coupled-damage models and uncoupled-ones with several propositions of ductile fracture criterion.

Stress-state parameters

To account for metal plasticity and damage models three invariants ((3.1), (3.2) and (3.3)) of the stress tensor σ need to be introduced:

p = -σ h = - 1 3 (σ 1 + σ 2 + σ 3 ) = - I 1 3 (3.1) q = σ = 1 2 [(σ 1 -σ 2 ) 2 + (σ 2 -σ 3 ) 2 + (σ 3 -σ 1 ) 2 ] = 3J 2 (3.2) r = 27 2 (σ 1 -σ h )(σ 2 -σ h )(σ 3 -σ h ) 1 3 = 27 2 J 3 1 3 (3.3)
where σ 1 ≥ σ 2 ≥ σ 3 are the ordered principal stresses. The stress-state can be uniquely described by the three invariant p, q, r -also expressed I 1 , J 2 , J 3 -or by two dimensionless parameters, the stress-triaxialility σ * in (3.4) and the Lode parameter µ σ in (3.5) defined as :

σ * = σ h σ (3.4) µ σ = 2σ 2 -σ 1 -σ 3 σ 1 -σ 3 (3.5)
where σ is the equivalent stress and σ h the hydrostatic stress. The Lode parameter µ σ is also used under the form of the Lode angle θ L or Lode angle parameter θ (-1 ≥ θ ≥ 1) respectively defined in (3.6) and (3.7).

θ L = arctan 1 √ 3 µ σ (3.6) θ = -6 π θ L (3.7)
Normality implies that the expression of this function is limited to the following conditions:

• Φ(σ) < 0 : elastic state

Yield functions Φ(σ) max (|σ 1 -σ 2 |; |σ 2 -σ 3 |; |σ 3 -σ 1 |) -σ Y ≤ 0 (3.12) 1 3 (σ 11 -σ 22 ) 2 + (σ 22 -σ 33 ) 2 + (σ 33 -σ 11 ) 2 + 2σ 2 12 + 2σ 2 13 + +2σ 2 23 -σ 2 Y ≤ 0 (3.13) F 1 (σ 11 -σ 22 ) 2 + F 2 (σ 22 -σ 33 ) 2 + F 3 (σ 33 -σ 11 ) 2 + 2F 4 σ 2 12 + 2F 5 σ 2 13 + +2F 6 σ 2 23 -σ 2 Y ≤ 0 (3.14) σ σ Y 2 + 2q 1 Φ * cosh - 3q 2 2 p σ Y -1 -q 3 Φ * 2 ≤ 0 (3.15)
Table 3.1: Some of the most popular yield functions applied to metals with σ Y the material yield stress and F 1...6 the coefficients of Hill function and q 1 , q 2 , q 3 are material constants and Φ * the effective void volume fraction of the GTN model used in [START_REF] Tvergaard | Analysis of the cup-cone fracture in a roundtensile bar[END_REF].

C Σ O Φ(σ, σ Y (ε 1 )) = 0 (Tresca) Φ(σ, σ Y (ε 1 )) = 0 (Von-Mises) σ 2 σ 1 Φ(σ, σ Y (ε 2 )) = 0 (Von-Mises) Φ(σ, σ Y (ε 3 )) = 0 (Von-Mises) ε1 ≥ ε2 ≥ ε3 Φ(σ) < 0: Elastic

Constitutive laws

A perfect material structure should imply a constant yield stress without any sensitivity to the plastic strain. But real materials have defects and the yield stress evolves with physical parameters. This relationship is called a constitutive law. The ones adapted to dynamic applications are generally classified inside one of these two categories: phenomenological or thermally-activated models.

Phenomenological models

The phenomenological model tries to predict material yield stress according to evolution of state variables: in most cases it is limited to effective plastic strain εpl , strain-rate εpl and temperature T .

A mathematical formulation relates somehow the yield stress to state-variables which are consistent with the targeted phenomena to model:

σ Y = f (ε pl , εpl , T ) (3.16)
Sensitivities are generally assessed to the yield stress using multiplicative terms:

σ Y = k 1 (ε pl ) k 2 ( εpl ) k 3 (T ) (3.17)
Several mathematical form can be given to k 1 , k 2 and k 3 term (see Table 3. Noteworthy extensions have been attempted by several authors to account for strain-hardening dependency to temperature effect [START_REF] Bäker | Finite element simulation of high-speed cutting forces[END_REF], yield saturation phenomenon under very high strain-rates or even to the yield stress sensitivity to the Lode-angle [START_REF] Chocron | A new plasticity and failure model for ballistic application[END_REF]. Anyhow these limitations can not be overcome without physical-based constitutive laws.

Phenomenological laws:

σ Y = k 1 (ε pl ) k 2 ( εpl ) k 3 (T ) k 1 (ε pl ) k 2 ( εpl ) k 3 (T ) λ 1 ε λ 2 pl (3.18) ε β 1 pl (3.19) 1 -θ 1 (T -T 0 ) (3.20) λ 1 (ε 0 + εpl ) λ 2 (3.21) εpl ε0 β 1 (3.22) exp θ 1 T (3.23) λ 1 εpl ε0 λ 2 (3.24) 1 + εpl ε0 β 1 (3.25) 1- T -T 0 T m -T 0 θ 1 (3.26) λ 1 1 + εpl ε0 λ 2 (3.27) 1 + εpl β 2 β 1
(3.28) -

λ 1 + λ 2 ε λ 3 pl (3.29) 1 + β 1 ln εpl ε0 (3.30) - λ 1 +λ 2 (1-exp(-λ 3 εpl )) (3.31) 1+β 1 ln εpl ε0 +β 2 ln 2 εpl ε0 (3.32) - Table 3
.2: Some of the most popular phenomenological-based constitutive laws with strain, strain-rate and temperature dependencies with λ 1,2,3 and ε0 the coefficients of term k 1 , β 1,2 and ε0 the coefficients of term k 2 and θ 1 , T 0 and T m the coefficients of term k 3 .

Thermally-activated models

Zerilli and Armstrong (1988) proposed a formulation (ZA) (see Table 3. rate such as the one from Follansbee and Kocks (1988) which focus on the mechanism which drives the increasing yield stress sensitivity to the strain-rate on copper. An other significant formulation which is valid on a wide range of strain-rates -from 10 -4 s -1 to 10 12 s -1 -is the one from [START_REF] Preston | Model of plastic deformation for extreme loading conditions[END_REF] in which yield stress saturation at high strain-rates or high temperature is well predicted but calibration has to be done for eleven material parameters.

Damage

An interesting model which can be qualified as "hybrid" between the two previous categories is the Rusinek-Klepaczko (RK) model (see Table 3.4). It is a phenomenological law but partially based on the theory of dislocations and it has already been used in impact and perforation simulations (Kpenyigba et al., 2015). The total stress is decomposed into an athermal component k 4 and thermally-activated one k 5 modelled under the Arrhenius form.

Zerilli-Armstrong: Rusinek-Klepaczko: The Johnson-Cook law has been somewhat criticized in the literature (Børvik et al., 2003a), both due to its empirical origin and to the fact that the work-hardening is independent of temperature and strain rate but it remains the most convenient for parameters calibration and gives good results as long as the range of the loading conditions is kept close to the testing conditions.

σ Y = k G + k θ (ε pl , εpl , T ) k G k θ (ε pl , εpl , T ) BCC σ G + ζ 1 ζ -1 2 2 α 1 exp[-α 3 T + α 4 T ln( εpl )] + α 5 ε α 6 pl (3.33) FCC σ G + ζ 1 ζ -1 2 2 α 2 εpl exp[-α 3 T + α 4 T ln( εpl )] (3.34)
σ Y = E(T ) [k 4 (ε pl , εpl , T ) + k 5 ( εpl , T )] E(T ) k 4 (ε pl , εpl , T ) 1 -T Tm exp γ 1 1 -Tm T γ 2 T Tm log εmax εpl -γ 3 (ε 0 + εpl ) γ 4 1-γ 5 T Tm log εpl εmin k 5 ( εpl , T ) γ 6 1 -γ 7 T T m log εmax εpl γ 8 (3.35)

Damage

Damage is usually known as a micro-scale mechanism which pilots material intrinsic ductility (the amount of plastic deformation that material undergoes before a crack is produced). The global fracture is therefore the final event of damage accumulation process inside the material. It is driven by voids nucleation, growth and coalescence inside the matrix -or close to inclusions -under positive stresstriaxiality and shear-band formation under low stress-triaxiality. Therefore the damage evolution changes the geometry at the micro-scale -and reduces the loads transmitting area -which affects the load carrying capability at the macro-scale. This phenomenon is pictured on the stress-strain curve on

Φ * = 0 if Φ < Φ c Φ c + Φ * u -Φc Φ f -Φc (Φ -Φ c ) if Φ ≥ Φ c (3.36)
where Φ c is the Φ-value at which void coalescence is triggered, Φ f the Φ-value at fracture and Φ * u = ± q 2 1 -q 3 /q 3 with a void evolution rate expressed through a void coalescence component and a void growth component.

Φ = Φnucleation + Φgrowth (3.37)
with Φgrowth = (1 -Φ)trace( εpl ) and Φnucleation often expressed as a gaussian curve: 2014). However it is mostly unused for industrial applications since a large number of parameters have to be calibrated from mechanical tests whereas others have to be measured from microscopical analysis.

Φnucleation = Φ N s N √ 2π exp - 1

Continuum damage mechanics

On the basis of continuum mechanics, Lemaitre and Chaboche (1994) proposed a theory in which damage growth Ḋ is driven by the total elastic work w el dissipated into the material (sum of distortion energy and hydrostatic energy). A driving force Y is assumed and called Energy density release rate:

Y = 1 2 dw el dD σ,T = hσ 2 2E(1 -hD) 2 R v (3.39)
with a constant stress σ and temperature T , E and ν are respectively Young modulus and Poisson ratio, h the parameter accounting for micro-closure effect and the function of stress-triaxiality

R v = 2 3 (1 + ν) + 3(1 -2ν)σ 2 .
From the consideration of a damage dissipative potential, evolution rate is: In the present work, an additional formulation of the equivalent strain threshold εD is used (3.42).

Ḋ = 0 if εpl < εD or σ * < -1 3 Y S b εpl if εpl ≥ εD (3.
The exponential form (3.41) has been extended to four different threshold values εD1 , εD2 , εD3 , εD4 which have to be calibrated from experiments and are assessed to the range of stress-triaxiality σ * 1 , σ * 2 , σ * 3 , σ * 4 . This is pictured as a locus in Figure 3.3. For conveniency, parameters are calculated as

k 12 = -ln (ε D2 /ε D1 )/(σ * 2 -σ * 1 ), k 23 = -ln (ε D3 /ε D2 )/(σ * 3 -σ * 2 ) and k 34 = -ln (ε D4 /ε D3 )/(σ * 4 -σ * 3 ). εD =                  εD1 if σ * < σ * 1 εD1 exp(-k 12 σ * ) if σ * 1 ≤ σ * < σ * 2 εD2 exp(-k 23 σ * ) if σ * 2 ≤ σ * < σ * 3 εD3 exp(-k 34 σ * ) if σ * 3 ≤ σ * < σ * 4 εD4 if σ * ≥ σ * 4 ( 3

Phenomenologically-based fracture criteria

The phenomenologically-based fracture model, called PB model hereafter, is based on the consideration that damage scalar variable D (0 ≤ D ≤ 1) can be simply expressed as:

D(ε pl ) = dε pl εf (σ * , µ σ ) c (3.43)
where εf is the effective plastic strain at fracture and c is a non-linear damage evolution exponent. For the same reasons than in Lemaitre formulation, damage growth is driven by a stress-state condition (σ * < -1/3) and an effective plastic strain threshold (ε pl ≥ εD ) as expressed in (3.44):

Ḋ =    0 if εpl < εD or σ * < -1 3 c 1 εf εpl εf c-1 if εpl ≥ εD (3.44)
This formulation is really convenient since damage can be related to any desired physical dependancies and modelled throught the function -also called fracture locusεf = f (σ * , µ σ , εpl , T, . . .). Assuming a constant damage accumulation threshold (ε D = εD0 ), this model requires at least 3 parameters to identify: εD0 , c and the coupling parameter β.

Coupled-damage formulation

Assuming material isotropy, a temperature-dependant and strain-rate dependent Von-Mises yield function (3.45) is considered appropriate to account for material behavior under impact.

Φ = σ(σ) -w(D) σ Y (ε pl , εpl , T ) (3.45)
where σ Y is the yield stress and w(D) is the weakening function in order to take into account damage softening effect. A convenient interpretation of damage uses the ratio of damaged area S D to the total surface S T such as: D = S D /S T . The effective stress tensor σ is used as the one which should be applied to an undamaged material, in order to get the same elastic strain tensor as the one obtained from the damaged material under actual stress σ. It is called strain equivalence principle and is expressed as in (3.46).

σij = σ ij 1 -D (3.46)
where σij and σ ij are respectively components of the effective stress tensor and of the actual stress tensor. The weakening function w(D) -presented in (3.47) -is based on this principle and affects macro-mechanical behavior in a different way if material is rather in a dominant compression stressstate (σ * < 0) than in a tension stress-state (σ * ≥ 0) :

w(D) = (1 -hD β ) if σ * < 0 (1 -D β ) if σ * ≥ 0 (3.47)
where h is a material parameter (0 ≤ h ≤ 1) which accounts for micro-crack closure effect. Lemaitre Both approaches will use the same coupling formulation -accounting for micro-crack closure effect and damage softening -but damage will be calculated according to the inherent model formulation.

In the case of PB-model, the calculation of the damage variable D can be done on the basis of a large number of formulations taken from literature which will be properly introduced in the next section.

Fracture loci

Fracture locus formulations are based on the consideration that damage -modelled with a scalar variable D (3.43) -is driven by effective plastic strain and accumulated inside the material up to fracture, i.e. when εpl = εf . Either it is coupled with the yield stress to get a softening effect or it is uncoupled and damage variable will have no effect on plasticity until damage reaches a critical value D c . When this value is reached, a meso-crack is created and fracture occurs inside the material.

Several formulations have been proposed into the literature to model the effective plastic strain at fracture as a function of stress, strain, strain-rate and temperature components. The choice of an appropriate fracture criterion is always balanced between the accuracy of the formulation to capture experimental data and the number of parameters which have to be calibrated.

Damage

Fracture locus: εf = f (σ * , µ σ , εpl , T )

CFS ϕ 1 (3.48) BFS +∞ if σ * < -1/3 ϕ 1 if σ * ≥ -1/3 (3.49) ECL ϕ 3 σ ϕ 1 (3σ * 3 + µ 2 σ -3 -µ σ ) + 6 3 3 + µ 2 σ -ϕ 2 (3.50) CL ϕ 3 σ 3 3 + µ 2 σ 3σ * 3 + µ 2 σ + 3 -µ σ (3.51) JC (ϕ 1 + ϕ 2 exp(-ϕ 3 σ * )) 1 + ϕ 4 ln εpl ε0 1 + ϕ 5 T -T 0 T m -T 0 (3.52) EJC ϕ 6 + (1 -ϕ 6 ) 6|θ L | π ϕ 7 ε JC f (3.53) XW ϕ 1 exp(-ϕ 2 σ * ) -[ϕ 1 exp(-ϕ 2 σ * ) -ϕ 3 exp(-ϕ 4 σ * )] 1 -cos π 2 (1 -θ) ϕ 5 1 ϕ 5 (3.54) BW 1 2 ε (+) f + ε (-) f -ε 0 f µ 2 σ + 1 2 ε (+) f + ε (-) f µ σ + ε 0 f (3.55) MMC A ϕ 2 1 + ϕ 2
WKS D c (1 -ϕ 1 σ * ) ϕ 2 2 - sin(| θ|π/6) cos[(1 + | θ|)π/6] -ϕ 3 (3.58) GUR 1 ϕ 1 ln Φ f (1 -Φ 0 ) Φ 0 (1 -Φ f ) (3.59) NH 1 ϕ 1 + ϕ 2 ln Φ f (-ϕ 1 f 0 + ϕ 1 + ϕ 2 ) f 0 (-ϕ 1 Φ f + ϕ 1 + ϕ 2 ) (3.

60)

Table 3.5: Some of the most popular fracture criteria with ϕ 1,2,3,4,5,6,7 the main coefficients and D c the critical damage parameter.

RMK 1 : ε (+) f = ϕ 1 exp(-ϕ 2 σ * ), ε 0 f = ϕ 3 exp(-ϕ 4 σ * ) and ε (-) f
= ϕ 5 exp(-ϕ 6 σ * ) are three curve functions of the BW criterion. RMK 2 : A and n are parameters of the Ludwig plastic law, for MMC and MSS criteria. RMK 3 : ϕ 1 = 3 2 q 1 q 2 sinh 3 2 q 2 σ * , ϕ 2 = ϕ w 1 -cos 2 π 2 (1 -θ) with q 1 , q 2 , ϕ w are material parameters and Φ 0 and Φ f respectively the void volume fraction of the undamaged and damaged material state for GUR and NH criteria. In the following are presented the general criteria which have been applied to impact, sheet perforation or joining processes from the simpler with only one parameter to the more complicated ones with seven parameters (see Table 3.5).

The simplest criterion is the Constant Fracture Strain (CFS) and is presented in (3.48). It assumes that fracture strain is independent of the loading conditions. It has been extended to the Bounded Fracture Strain (BFS) to account for the cut-off value for fracture under negative stress-triaxiality (3.49). The consistency of this cut-off value has been shown in [START_REF] Bao | On the cut-off value of negative triaxiality for fracture[END_REF] Both CL and ECL criterion can be found under the strain-based formulation in Table 3.5, (3.51) and

(3.50).

ϕ 3 = ϕ 1 σ 1 σ + (1 -ϕ 1 ) σ 1 -σ 3 σ
= ϕ 4 = ϕ 2 . εf = 1 2 ε (+) f + ε (-) f -ε 0 f µ 2 σ + 1 2 ε (+) f + ε (-) f µ σ + ε 0 f (3.65) with        ε (+) f = ϕ 1 exp(-ϕ 2 σ * ) ε 0 f = ϕ 3 exp(-ϕ 4 σ * ) ε (-) f = ϕ 5 exp(-ϕ 6 σ * ) (3.66)
In the category of stress-based fracture criteria, Bai and Wierzbicki (2010) have reformulated the wellknown Mohr-Coulomb criterion in order to get it written in the space of stress-triaxiality and Lode angle (3.67). In the case of an isotropic material behavior modelled with a power hardening law, the set of parameters is taken as ϕ s θ = 0, ϕ ax θ = 1 which leads to the simplified version of the Modified Mohr-Coulomb (MMC) criterion, presented in (3.56). It can be reduced to a single-parameter criterion, called Maximum Shear Stress (MSS) criterion if ϕ 1 = 0 (3.57) .

εf = A ϕ 2 [1 -ϕ σ * (σ * -σ * 0 )] ϕ s θ + √ 3 2- √ 3 (ϕ ax θ -ϕ s θ ) sec θπ 6 - 1
. . .

εf = D c (1 -ϕ 1 σ * ) ϕ 2 (2 -η) -ϕ 3 (3.68)
with η, a function of the deviatoric principal stress components s 1 ≥ s 2 ≥ s 3 .

η = max s 1 s 2 , s 2 s 3 (3.69)
This formulation is simplified into the one proposed in Table 3. For most traditional joining process simulations, the main concern is to properly model metal flow inside an upper and a lower die without considerations on damage. In these cases an uncoupled fracture criterion was sufficient to account for crack occurrence. But in the new generation of joining processes, such as High-Speed Nailing, Spin-Blind Riveting or even Flow-Drilling Screw, access is made from one side and metal flow is not constrained by a lower die. Every layers have to be perforated before being joined. Thus sheets failure modelling becomes mandatory to get realistic sheets reaction as well as final interface geometry and an appropriate damage evolution law must be chosen. To handle these modelling issues, we have chosen to exclude the Gurson damage model since it has too many parameters to handle and requires microscopical analysis for the parameters calibration. For conveniency, the LEM and PB coupled-based approaches will be used to simulate material softening induced by damage. Several fracture strain formulations will be tested as damage driving force for the PB-models, from the CL-criterion with one parameters to the BW6P-criterion with 6 parameters, and will be compared to results obtained with the LEM. These models have to be calibrated from a suitable set of tests which are presented in the next chapter before being applied to high-speed nailing simulations in Chapter 5.

Summary of Chapter 3

In this chapter, plasticity and damage formulations are introduced for applications which involve impact, sheet perforations and dynamic loading. As a preliminary, three invariant of the stress tensor are reminded and the stress-triaxiality and Lode angle parameters are presented. This is a necessary step since materials behavior are always expressed as a function of the stress tensor invariant. Then plasticity models are introduced with the decomposition of the material response into a hydrostatic part and a deviatoric part. It has been stated that the hydrostatic stress component should be described with a state equation for solid-solid impact velocities which range 0.5 to 2 km/s. Below this velocity, material behavior is only driven by the traditional flow theory of plasticity. Several yield surface formulations and constitutive laws are proposed accounting for strain, strain-rate and temperature dependencies.

Three main approaches are presented to model damage growth and fracture: porous solid plasticity, continuum damage mechanics and phenomenological-based criterion. These models can be either coupled or not to material yield stress to account for damage softening leading to fracture. Thus the coupling equations are presented for Lemaitre and Phenomenologically-based approaches and finally a review is given on the fracture loci adapted to dynamic issues and applied to joining processes.

Résumé en français

Dans ce chapitre sont présentés les modèles de comportement matériaux soumis à des chargements dynamiques tels que l'impact, la perforation de tôles et l'assemblage à grande-vitesse. Dans un premier temps, un rappel est fait sur les invariants du tenseur de contrainte, nécessaires par la suite pour l'établissement de modèles prédictifs du comportement en déformation et à rupture des matériaux.

La réponse du matériau se décompose en une partie hydrostatique et une partie déviatorique. La prise en compte d'une équation d'état est nécessaire lorsque des vitesses d'impact très élevées sont rencontrées, entre 0.5 et 2 km/s. En dessous du seuil de 500 m/s, la réponse du matériau est régie par la traditionelle loi d'écoulement plastique. Plusieurs formulations de surface de plasticité sont proposées ainsi qu'un large nombre de lois de comportements sensibles à la déformation, vitesse de déformation et température.

Trois approches sont abordées pour la modélisation de l'endommagement et de la rupture de tôles: la

Introduction

To achieve complex mechanical simulations, the material constitutive and damage laws have to be adapted to the phenomena to model and the parameters calibrated from mechanical tests performed in loading conditions as close as possible to the conditions of application. Several models have been presented in the previous chapter to treat plasticity and damage for impact and dynamic loads. The present chapter tackles the calibration of these models for sheets materials and nail shaft material and prepare to the simulation of the nailing process.

Three sheet materials have been selected: a 1.

Material testing

The calibration of constitutive and damage models, identified in the previous chapter, requires experimental data obtained from mechanical tests. They are conducted on metal sheets and nail for various specimen geometries, loading conditions and strain rates. These tests are numerous and are detailed in the following sections. process has no significant effect on this material yield surface. However to quantify this yield-stress anisotropy, a ratio r α can be defined at each angle α to the rolling direction and for a given state of equivalent plastic strain εpl . It is expressed as:

r α = σ α Y σ 0°Y εpl (4.1)
where σ 0°Y is the reference yield stress at 0°and σ α Y the yield stress at the investigated angle. These ratio have been evaluated from εpl = 0.02 to εpl = 0.1. Results are gathered in Table 4.3. Even if r-value slightly changes with the plastic strain, no significant difference can be noticed and it remains close to 1.

No investigation regarding cast aluminium anisotropy was possible because of incompatibility between ingot sizes and specimen -specimen length is 120 mm whereas ingot width is 90 mm. However no reason has been found to legitimate anisotropy in the molted material since (i) no particular direction is promoted during the forming step and (ii) no significant irregularity in the microstructure would withstand the final heat treatment. Consequently, it has been assumed that anisotropy can be neglected and isotropy is considered consistent for the rest of this work for the mechanical behavior of cast aluminium, steel and AHSS.

Notched tension

Bridgman (1952) was the first to demonstrate, later verified in the works of [START_REF] Bao | On fracture locus in the equivalent strain and stress triaxiality space[END_REF] and a tremendous amount of authors, that notched specimens subjected to tension induce inside the material thickness a stress-triaxiality magnitude which can be related to the notched radius.

However, as in the present case only sheet materials are concerned, possible stress-states are limited to configurations which satisfy plane stress condition, which means that evolution of Lode angle parameter θ can not be considered apart from the stress-triaxiality σ * , and is related as in the following (4.2).

θ = 1 - 2 π cos -1 - 27 2 σ * σ * 2 - 1 3 (4.2)
Thus three specimens have been designed for tensile tests and are presented in Figure 4.1: one for uniaxial loading called UT0 -different from standard uniaxial specimen -and two notched specimens NT7 and NT2, with respectively a notch radius of 7 mm and 2 mm. Mechanical tests were conducted at velocity 1 mm/min to get -for UT0 specimen only -a mean strain rate until necking of 1.10 -4 s -1 .

Tests were repeated 3 times for every specimen geometry for cast aluminium, steel and AHSS.

In-plane shear

In addition to uniaxial and notched specimens, in-plane shear specimens have been designed to complete the range of investigated stress-states at fracture. Butterfly specimens were used, indicated by SH notation and its geometry detailed in Figure 4.1. Tests have been repeated 3 times for cast aluminium, steel and AHSS at velocity 10 mm/min. Results from tensile tests pictured in Figure 4.3 revealed that good repeatibility is obtained for steel and AHSS samples but cast aluminium exhibits bigger discrepancy. Indeed microstructural defects might promote loss of ductility and precocious crack occurrence.

Effective plastic strain at fracture

To access the instantaneous die-displacement as well as surface strain field, a DIC optical measurement has been used for these tests. A speckle pattern was sprayed over specimens surface and virtual extensometers -pictured on Figure 4.4 with red dots -were positioned on specimens to measure real jaws displacements. In-plane principal strains (Hencky strains) are calculated from the displacement field obtained from the picture sequence and the effective strain is calculated afterward, through (4.3), as:

ε = 2 3 (ε 2 1 + ε 2 2 + ε 2 3 ) (4.3)
where ε 1 , ε 2 are in-plane principal strains and ε 3 is the strain component in the thickness direction.

Plastic incompressibility is assumed and elastic strains are considered negligible regarding total strains, i.e. the relationship εpl = ε is considered valid. As consequences, the effective strain observed in the crack onset area is considered as the effective plastic strain at fracture εf . The identification of crack path is possible when two pictures are considered: the one just before the crack has appeared and the one just after the crack has spread. Hence, sensors have been positioned on the crack path at three different spots for every specimen geometry: s 11 , s 0 , s 12 for respectively the first border, the center and the second border of the specimen zone where crack occurs (see Figure 4.4). Effective plastic strain at fracture εf is therefore calculated for each specimen and each tested material at the sensors location.

Results are gathered in Table 4.4 and will be further used for fracture locus identification. The strain field computation from DIC images taken during the mechanical test uses speckles tracking function and displacement field calculation. Three user-parameters have to be adapted to the specimen size and speckle pattern to get an appropriate strain field computation: the substep, the step and the filtersize. Speckles are not actually tracked, the substep is a geometrical space frame in which a unique speckle pattern is defined. Thus a substep size of 9 means that speckles pattern to track are limited to a mesh size of 9 x 9 pixels. The step size is a parameter which defines the range of the tracking zone. For example, a step size of 2 means that the 9 x 9 pixels area is tracked for every 2 pixels. It is generally chosen as 1/4 of the substep size to keep under control the overlap between the different substep meshes. The filter size corresponds to the number of data points used to interpolate the deformation. Therefore the smoothing area dimension is {step x filter size} pixels.

To avoid a strong underestimation of the strain values or a noisy response in the strain field, some recommendations have to be followed to choose the appropriate set of parameters [START_REF] Schreier | Image Correlation for Shape, Motion and Deformation Measurements[END_REF].

The choice of a suitable combination of step and filtersize is crucial [START_REF] Candau | On the use of a fourcameras stereovision system to characterize large 3d deformation in elastomers[END_REF].

Dynamic compression

To quantify strain-rate hardening and thermal softening beyond 50 s -1 , an Hopkinson bar test has been used. [START_REF] Roth | Experimental and numerical investigation of ductile fracture initiation at low, intermediate and high strain rates[END_REF], [START_REF] Jeunechamps | Simulation numérique, à l'aide d'algorithmes thermomécaniques implicites, de matériaux endommageables pouvant subir de grandes vitesses de déformation[END_REF] and also [START_REF] Gary | Comportement des métaux à grande vitesse de déformation: modélisation. Techniques de l'ingénieur[END_REF] proposed details on this test and the main principles to obtain sample stress-strain path from sensors located on both bars. Twenty-five tests have been performed at LAMIH Laboratory, based in Valenciennes (France), on the same cylinder specimen geometry C0 used for quasi-static compression tests. Impact velocity of the input Hopkinson bar was set to load the specimen at five different strain-rates : 400 s -1 , 800 s -1 , 2000 s -1 , 3200 s -1

and 6000 s -1 . Each configuration has been repeated five times.

Nail material characterization

As previously mentioned, the nail fastener is forged from non-alloyed carbon steel C60. Due to the manufacturing process, final properties of the nail material are far beyond the raw material ones.

To quantify its properties, mechanical compression tests have been performed on specimens directly extracted from the nail cylindrical shaft. Three cylindrical shaft specimens respectively 5.13 mm, 5.13 mm and 5.6 mm high (shaft diameter = 3.05mm) are subjected to compression loading until a final relative displacement of 30%, 40% and 50% of the initial shaft length is reached. Due to funds constraint, Hopkinson test is excluded for the nail-shaft specimen and only quasistatic conditions are considered. Compression velocity has been set to 3 mm/min for an average strain-rate of 3.10 -2 s -1 .

Despite lubrification of both upper and lower dies, nail cylinders got deformed as a so called "barrel effect" '(see Figure 4.9b). Such shape shows that friction is not negligible and induces an heterogeneous strain field distribution. Figure 4.9b shows the three experimental curves with a very similar mechanical response.

Material models calibration

Appropriate material models have been identified and detailed in Chapter 3 and characterization tests have been performed on our materials. The next step is the calibration of material models according to experiments. The parameters identification procedure is shortly introduced before the calibration of constitutive and damage laws is presented.

Inverse analysis procedure

Optimization strategies are used to find the best set of material law parameters and obtain a mechanical response from simulation as close as possible to the experimental one. This calibration procedure is enabled by the use of a metamodel-assisted evolutionary algorithm developed at CEMEF Laboratory [START_REF] Roux | Assemblage mécanique: stratégies d'optimisation des procédés et d'identification des comportements mécaniques des matériaux[END_REF] and called MOOPI (MOdular software dedicated to Optimization and Parameters Identification). Coupled with FORGE ® finite element software, the optimization can be done directly from the results of tensile test simulations. Thus the discrepancy between force-displacement data from simulation and experiment is evaluated by the way of a cost function Ψ which quantifies the average error between curves in the least square sense (4.4).

Ψ(x) = j i=1 (y exp i -y num i (x)) 2 m i=1 (y exp i ) 2 (4.4)
with j the number of experimental points, y exp i the experimental value at i and y num i the numerical value at i. The set of parameters giving the lowest cost function will be promoted and will affect the generation of new parameters. The function has to be minimized until optimization stops, either because the convergency criterion has been reached or the number of simulations is maximum.

Tensile test simulations

To account for the accuracy of material models and predict strain localization and fracture occurrence in the tensile tests, finite element models have been built on the four geometries with boundary conditions taken from experiments. Due to symmetry, only 1/8 of the UT0, NT7 and NT2 specimen geometry is modelled and only 1/2 of SH-specimen. Mesh size is minimum at the specimen center -set to 0.10 mm for every specimen and tested material -and get coarser out the area of interest. Since the coupled damage approach chosen in the present work is mesh sensitive, calibrated material laws will be use with identical mesh-size of 0.10 mm. These models are presented in Figure 4.10 for UT0's, Figure 4.11 for NT7's, Figure 4.12 for NT2's and Figure 4.13 for SH's.

Constitutive law calibration

On the basis of constitutive laws presented in Chapter 3, it has been chosen for conveniency to use a

Johnson-Cook based formulation to model yield stress evolution of our materials according to strain, strain-rate and thermal sensitivities (see Table 3.2). It is expressed under a multiplicative form:

σ Y (ε pl , εpl , T ) = k 1 (ε pl ) k 2 ( εpl ) k 3 (T ) (4.5)
with the strain hardening term k 1 (4.6):

k 1 (ε pl ) = Λ k S (ε pl ) + (1 -Λ) k V (ε pl ) (4.6)
with k S and k V which are respectively the strain hardening Swift and Voce formulations introduced in Table 3.2 and Λ the balance parameter.

k S (ε pl ) = λ 1 (ε pl + ε0 ) λ 2 k V (ε pl ) = λ 3 + λ 4 (1 -exp (-λ 5 εpl )) (4.7)
Strain-rate hardening is modelled through a Johnson-Cook based term k 2 (4.8):

k 2 ( εpl ) = 1 + β eq ln εpl ε0 (4.8)
with β eq = β 1 εpl ε0

+ β 2 such that if β 1 = 0, it reduces to the original Johnson-Cook formulation.

Finally, the thermal softening effect is modelled through the term k 3 (4.9):

k 3 (T ) = 1 - T -T 0 T m -T 0 θ 1 (4.9)
Strain hardening Strain-rate hardening Thermal softening

λ 1 λ 2 λ 3 λ 4 λ 5 Λ ε0 β 1 β 2 ε0 Tm θ 1 (MPa) (-) (MPa) (-) (-) (-) (-) (-) (-) (-) (C°) (-) Steel

Sheet materials

The yield stress sensitivities to strain, strain-rate and temperature are added using the multiplicative formulation presented in (4.5). Hence the calibration of every sensitivity parameters could be done independently: first with the parameters related to strain hardening, then to strain-rate hardening and finally to thermal softening.

Strain hardening

To state for material strain hardening, only λ i -parameters -and ε0 as well -have to be set. It can be done through a traditional parameter identification of the experimental stress-strain curve to fit with an analytical curve. Assuming plastic incompressibilityε 1 + ε 2 + ε 3 = 0 -and transverse isotropyε 2 = ε 3 -the longitudinal true stress component σ 1 until necking can be expressed as:

σ 1 = F/(A 0 exp(-ε 1 )
) with F the force measurement, A 0 the initial area of the specimen and the longitudinal strain ε 1 = ln(l c /l 0 ) function of the current gauge length l c and the initial gauge length l 0 . Under uniaxial loading, the particularity of the stress tensor is that only one component remains, which means that the equivalent stress can be written σ = σ Y = σ 1 . Parameters identification is straightforward with this methodology for strain-hardening law with few parameters, such as Swift's or Ludwig's, but it becomes more complex for laws with much more parameters to identify.

In the present work, we found the use of FE simulation coupled to the optimization algorithm using MOOPI software and FORGE ® more convenient for the parameters identification. It enables more flexibility in the choice of the simulation taken as reference and provide additional data for the cost function evaluation. For steel and cast aluminium, only the UT0 force-displacement curve is used for the identification process whereas three curves have been necessary for AHSS's: UT0, NT7 and NT2.

Results of optimization are presented in Table B.1 and details are given about optimization parameters such as the initial and final domain size, the choice of the best simulation and cost function obtained inside the investigated domain of optimization. The best results are plotted in Figure 4.16 with the associated set of parameters given in Table 4.5.

Both steel and AHSS models are in good agreement with experiment whereas some discrepancies can be noticed for the cast aluminium one. It has been shown previously that tensile tests on cast aluminium are complicated due to low ductility and the existence of defects at the microscale. The cost function could not be minimized beyond the value obtained with simulation #250 (see Table B.1).

Consequently we considered that constitutive law for cast aluminium could not be improved and will be taken as it is illustrated in Figure 4.16b.

Strain-rate hardening

A finite element model of Hopkinson compression test has been run in two different loading configurations: mean strain-rate of 400 s -1 and 6000 s -1 . The optimization is applied on parameters β 1 and β 2 and the best set of parameters is validated through a third reference curve taken from a test conducted at 2000 s -1 . Results are plot on the stress-strain curves presented in Figure 4.14 and ensure good RMK : PB-model uses the BW4P criterion for the fracture strain formulation εf .

Damage law calibration

In the present work we have chosen to treat ductile damage with coupled-based approaches: Lemaitre (LEM) and Phenomenologically-based (PB). Both formulations will be calibrated on the basis of several experimental tests and therefore attest the consistency of damage models to predict material ductility on a wide range of stress-states. alloys) and the strain threshold values εDi corresponding to the current tested stress-state σ * i . For instance, the optimization conducted on cast aluminium damage law used the NT2 force-displacement curve as a reference for parameters identification. Since the average stress-state of this specimen strain localization area is σ * 4 = 0.58 the associated strain threshold triggering damage accumulation to calibrate is εD4 . Then few simulations have to be run to calibrate the other parameters εD1 , εD2 and εD3 and reconstruct the strain threshold presented in Figure 3.3. All details on the conditions and results of optimization conducted on steel, cast aluminium and AHSS can be find in Table B.1 and the best set of parameters are gathered in Table 4.7 for steel, Table 4.8 for cast aluminium and Table 4.9 for AHSS. As it can be noticed on force-displacement curves presented in Figure 4.16 good agreement is obtained from simulations computed with LEM-model except for cast aluminium whose the abrupt decreasing slope is almost impossible to capture.

Fracture locus calibration

The fracture locus is generally assumed to be a really convenient way to predict the onset of fracture in a material as a function of stress-state invariants (σ * , µ σ ) and thermo-mechanical indicators such as temperature T or strain-rate εpl . Several formulations have been introduced in Chapter 3 and gathered in Table 3.5. Assuming a proportional loading path leading to fracture, the tensile test simulation computed on steel LEM-model was used to quantify the mean stress-state of sensors located at different spots of the crack path. These sensors were positioned in the core thickness or at the specimen surface, at the center or at the border of the area of interest. The evolution of stress-triaxiality and lode parameter is plot for every specimen geometry in Figure 4.17a and Figure 4.17b. The average value of each test is written on the graph, close to the corresponding curve, and used to express the mean stress-state of each specimen under the form of a "stress-triaxiality vs. lode parameter" graph (see This graph highlights that sensors located at the specimen center -middle of length and width -are subjected to plane stress whereas the ones at the corner are out of this locus. It can also be noticed that a wide range of stress-state to fracture is covered by the tested specimen geometries, from σ * ≈ 0 with SH specimen to σ * ≈ 0.57 with NT2's. Hence it becomes possible to assign a mean stress-state to the measurement of effective plastic strain at fracture presented in Table 4.4 and therefore calibrate the fracture loci on these data-points. -c). From the consideration of the works of [START_REF] Barsoum | Micromechanical analysis on the influence of the Lode parameter on void growth and coalescence[END_REF], [START_REF] Cao | Models for ductile damage and fracture prediction in cold bulk metal forming processes: a review[END_REF] and also [START_REF] Cao | A model for ductile damage prediction at low stress triaxialities incorporating void shape change and void rotation[END_REF] the HST-t fracture strain is assumed to be 1/2 of the HST-c's and therefore promote asymmetry of the fracture locus. RMK 3 : The set of data-points defines which serie is taken as reference for calibration. RMK 4 : Informations regarding fracture loci can be found in Table 3.5.

On the consideration of a decreasing fracture strain with the stress-triaxiality, selected values for both Generalized tension and Generalized compression points -HST-t and HST-c -had to be lower than experiments for NT2 specimen. Moreover HST-tension values have been chosen to be 1/2 of the HSTcompression fracture strain to promote fracture locus asymmetry regarding the lode parameter axis. This ratio of two is not arbitrary; it is consistent with results from Cao (2017) on micro-mechanical representative cell in which strain to localization for X100 steel in HST-compression is about twice the HST-tension one. Similar ratio on ductility is considered valid according to the works of [START_REF] Barsoum | Micromechanical analysis on the influence of the Lode parameter on void growth and coalescence[END_REF] and also [START_REF] Cao | A model for ductile damage prediction at low stress triaxialities incorporating void shape change and void rotation[END_REF] in the range of stress-triaxiality σ * ≤ 1.5. This gap becomes negligible under very high stress-triaxility σ * > 1.5 and fracture locus appears symmetrical regarding the lode parameter axis.

An appropriate set of data-points has been selected for each fracture locus calibration in order to adapt the database to the number of parameters of the law. Thus the BW4P criterion required six data-points for calibration whereas the JC required only three (see Table 4.6). Examples of Bai-Wierzbicki symmetrical (BW4P) and asymmetrical (BW6P) fracture loci calibrated on steel are shown in Figure 4.18. The difference of fracture strain prediction is obvious considering the shape of these different fracture loci but it is even bigger when fracture surfaces are compared. For instance, the JC lode-independent fracture locus plot in 4.7 for steel, in Table 4.8 for cast aluminium and in Table 4.9 for AHSS. Steel. Fracture loci parameters B.1 and the best set of parameters are gathered in Table 4.7

ϕ 1 ϕ 2 ϕ 3 ϕ 4 ϕ 5 ϕ 6 ϕ 7 q 1 q 2 f 0 f f ϕw Dc ECL 0.
for steel, in Table 4.8 for cast aluminium and in Table 4.9 for AHSS. Good agreement is obtained on computed force-displacement curves for every specimen geometry on steel and AHSS. However, the same discrepancy noticed with the LEM-model can be noticed here with the PB-model for cast aluminium which exhibits significant difference on force-displacement curves. Since this material ductility is small, it is complex to model the abrupt softening of the mechanical response related to the damage accumulation and therefore no attempt to have a better results will be done hereafter because it would require a deep modification of the damage formulation and therefore it could be complicated to use it on industrial applications.

The counterpart of a calibration performed on a single locus (BW4P) is that computations with other fracture criteria might induce a different material response in the tensile test simulations. This effect is pictured in Figure 4.20 for steel, in Figure B.9 for cast aluminium and in Figure B.10 for AHSS.

Significant differences can be noticed between PB-models applied to steel but it has almost no effect for cast aluminium and AHSS (see Appendix B). These considerations will have to be taken into account when such formulations will be applied to high-speed nailing simulations.

Summary of Chapter 4

On the basis of constitutive and damage laws introduced in Chapter 3, the present chapter describes the characterization procedure to calibrate models from experimental tests on our materials. The aim of these tests conducted on sheet materials -steel, cast aluminium and AHSS -are twofold: (1) evaluate the yield stress evolution according to strain, strain-rate and temperature and (2) promote fracture under a wide range of stress-states. Hence, uniaxial tensile tests have been performed under quasistatic conditions to investigate yield stress anisotropy and strain hardening effect. Compression tests have been conducted at different strain rates to investigate yield stress sensitivity to intense dynamic loading. Notched and in-plane shear specimens were tested with the support of a DIC optical measurement set-up to provide accurate details of the crack onset area. For the nail, a compression set-up has been used to test shaft mechanical behavior under quasistatic conditions. All these tests were therefore modelled into FORGE ® software and coupled with an optimization software called MOOPI for the parameters identification procedure.

The constitutive model parameters have been calibrated from tensile tests for the strain hardening part and from Hopkinson compression tests for the strain rate and thermal part. Lemaitre damage model is then calibrated using a set of coupled tensile test simulations -uniaxial, notched and in-plane shear -to cover a wide range of stress-states. Fracture loci introduced in Chapter 3 are calibrated using a Matlab ® optimization routine and the local strain data obtained from experimental sheet fracture tests.

These loci are used as the basis of the calibration procedure of a second coupled-damage approach: the Phenomenologically-based. Good agreement is globally obtained with these two damage approaches in terms of force-displacement curves despite the cast aluminium lack of ductility which makes the calibration a very complex procedure and the perspectives of improvement very limited.

Résumé en français

Dans la continuité du chapitre 3 qui présentait les lois matériaux pour traiter de la plasticité et de l'endommagement, le présent chapitre a pour objectif la description du protocole d'identification des paramètres de ces lois sur la base d'essais mécaniques. Ces essais ont deux objectifs principaux:

(

Introduction

The high-speed nailing process and the main features assessed to the joint have been experimentally investigated all along the Chapter 1 and 2. The constitutive and damage material models introduced in Chapter 3 have been experimentally tested on sheet materials and calibrated in Chapter 4. Consequently with these information it has become possible to build a numerical model of the high-speed nailing process on multi-materials configuration made of steel (DP780), cast aluminium (AlSi10MnMgT6) and AHSS (DP1180).

The present chapter is dedicated to the exploration of the high-speed nailing process through simulations of the nail insertion stage and the nailed-joint testing stage. The target of the nail insertion stage simulation is threefold: (i) evaluate the validity of the simulation and quantify the discrepancy with experiment; (ii) investigate the model sensitivity to variations in the boundary conditions, loading conditions and even to parameters of the material damage models; (iii) make use of the finite element simulation to get an insight into the mechanisms of nail penetration and improve our understanding on this joining process.

The target of the nailed-joint testing simulation consists also in three main points: (i) propose a numerical methodology to get results from the nail insertion simulation as an input of the joint testing simulation; (ii) evaluate the discrepancy with experiment and the main model sensitivities; (iii) suggest the way of model improvements for a better prediction of the joint features.

High-speed nailing of multi-materials superposition

To depict dissimilar materials configuration which are consistent with multi-materials structures expected in the automotive industry, two main configuration of interests have been chosen: {Steel/Cast alu.} and {Cast alu./Steel}. Piston kinematics, deformation modes and fracture modes are discussed on these cases and the model sensitivity to mesh size, friction conditions, loading and boundary conditions are investigated. The significant influence of material model parameters on the nail insertion efficiency is also highlighted in this section. Finally, nailing configurations are extended to {Steel/Steel}, {Cast alu./Cast alu.}, {Cast alu./AHSS.} and {AHSS/Cast alu.} to estimate the model capability to capture a large range of sheets response, even if it implies the nail fracture after impact.

Finite element model

A 3D model of the nail insertion stage has been built with FORGE ® software on a limited slice of 10°.

Although a slight asymmetry of the petalling failure process has been observed in few cross-sections presented in Chapter 2, it appears that the main other mechanisms encountered in the experimentssheets bending, shearing, nail crushing -seem highly symmetrical. Thus we have considered that using symmetry planes would be a convenient way to reduce computational time without apparent loss in the model consistency. This model is pictured on Figure 5.1 in the configuration of {Steel/Cast alu.} superposition.

Clamping conditions

Joining Force = f (Time) Piston Nail mass initial velocity Mesh sizes (from impact area to outside) 0.1; 0. To compute the nail insertion simulation, several parameters have to be preliminary set:

• damage sensitivity to the mesh size, the impact area subjected to damage has been set to the identical mesh size as the one used for calibration on tensile specimen, i.e 0.1 mm. An elastic mechanical behavior has been assigned to piston -with Young Modulus E piston = 200000 GPa -in order to consider it as a deformable object and its motion resulting from a mechanical computation.

Numerical parameters are numerous and have to be adapted to the specific conditions of computation. The mesh size is the main one. It is set to 0.1 mm at the impact area and get coarser out of this zone: 0.2 mm, 0.5 mm (in the sheet bending zone) and 1.0 mm (between the clamps). Mesh size in the nail core is 0.3 mm whereas it is 0.1 mm in nail-tip and ridges area. A 0.5 mm-high cylindrical shape is considered to account for the piston and is meshed with element 0.06 mm. Due to nail-tip crushing after impact and shear-plugging of the sheets, remeshing is mandatory at every increment. FORGE ® solver uses an implicit integration scheme, the time step is adapted according to convergency criterion and therefore it has not be to be set by user before simulation.

Boundary conditions are straightforward since a clamping set-up has been used in the experiments.

Different ring diameter could be set to the clamping zone d = {28; 30; 34 mm}. Bilateral sticking condition is used at the upper sheet-upper ring interface and at the lower sheet-lower ring interface.

Loading conditions consist in two main conditions: (1) an initial velocity applied to piston and nail to state for impact; (2) an equivalent joining force F eq applied to the piston upper surface to state for the set of force which keeps pushing the piston downwards during the penetration stage. For sake of simplicity, the piston is modelled with a 0.5 mm high cylinder whose density is set to ρ piston = 6 385 846.5 kg.m -3 , the equivalent density of the 130 g piston total mass assessed to this small volume.

On the basis of the equivalent joining force surface, presented in Figure 2.8, a vertical force serie is calculated according to a serie of piston vertical position. Thus force is a user-defined parameter which can be adapted to represent the nailing device input conditions from 3 bars of joining pressure to 8 bars. To simplify the loading conditions, the hold-down pressure is not considered here.

Contact is accounted between solids with a penalty method using a classification of interactions from master to slave. Hence it has been considered that nail should be the master; the upper sheet is a slave regarding the nail; the lower sheet is a slave regarding the upper sheet.

Friction defines how loads are transmitted between solids. For every interface between solids is assigned a frictional contact law: a Coulomb limited Tresca law is assumed for nail-upper sheet, nail-lower sheet, upper sheet-lower sheet and a bilateral sticking condition for the sheet-clamping ring interface. Default values for nail-steel, nail-cast alu and steel-cast alu interfaces are respectively {m = 0.8 ; µ = 0.4}, {m = 0} and {m = 0}.

Heat transfer is limited in this application to conduction. Observation 2.7.1 highlighted that steel sheet is in adiabatic conditions during the joining process whereas cast aluminium is not. Thus the interface coefficient h φ is assumed to zero for nail-steel and cast aluminium-steel contact whereas a medium transfer value is assumed for nail-cast aluminium and clamped ring-cast aluminium contact, i.e. h φ = 10 000 W.m -2 .K -1 .

Fracture modelling mostly uses kill-element technique to model a crack in materials. When a critical damage value computed in the element is reached, this one is deleted. It creates mesh discontinuities and therefore it is consistent with physical observations. However the mesh size must be fine enough to avoid important volume loss in the material.

In the present work, we have considered that element deletion only based on the consideration of a critical damage value would over-estimate crack locations inside the material. Indeed in simulation damage could be maximum in a large region of the mesh but a real crack would necessarily start at one location and propagate through a single path. Hence the damage variable can be considered as one condition which triggers crack occurrence in simulation, but not the only one. Other conditions can be added to the critical damage value to get a more realistic crack path prediction. We have chosen to use coupled-damage formulation to model material softening induced by damage accumulation. It promotes strain localization and therefore it corroborates what has been observed in plenty of experimental tensile tests conducted on metals. Since the strain localization band is generally assumed to be the area where crack is propagating, we have found really convenient to use the excessive mesh distortion as an additional condition to the kill-element procedure. It would be based on the consideration of damage and mesh-quality. Details of this calculation procedure can be find in Appendix C "Qualitybased damage condition for kill-element".

Configurations of interest

According to nailing experiments conducted in Chapter 2 on steel and cast aluminium, two hybrid material configuration have been selected as references for the simulation of the nail insertion stage:

• {Steel/Cast alu.} joined at P = 5.3 bars (Table 2.7: Experimental campaign n°2 -#3).

• {Cast alu./Steel} joined at P = 5.0 bars (Table 2.8: Experimental campaign n°2 -#21).

Although these two configurations are used for most analysis, three other configurations are also introduced since it will be useful to evaluate the model reliability for various material superposition:

• {Cast alu./AHSS} joined at P = 6.3 bars (Table 2.10: Experimental campaign n°2 -#41).

• {Steel/Steel} joined at P = 6.5 bars (Table 2.11: Experimental campaign n°3 -#6).

• {Cast alu./Cast alu.} joined at P = 4.5 bars (Table 2.11: Experimental campaign n°3 -#23).

Investigation on the nail insertion stage

In the present section, results obtained from {Steel/Cast alu.} and {Cast alu./Steel} joining simulation computed with LEM material models are discussed and the main mechanisms involved in the penetration process are highlighted.

Joining piston kinematics

The piston motion has been used to track nail path history throughout the experimental tests and is going to be used hereafter to evaluate numerical simulation discrepancy. Its kinematics can be investigated under three variables: piston displacement z, piston velocity and force. Figure 5.2 presents results from {Steel/Cast alu.} and {Cast alu./Steel} simulation with respectively discontinuous purple and discontinuous blue lines, stopped at maximum displacement value and error-bars plot on experimental curves from the dispersion measurement in Experimental campaign n°3. Thus it can be noticed that good agreement is obtained between simulations and experiments even if slight variation can be observed in both configurations with a piston position under-estimated in Figure 5.2a and overestimated in Figure 5.2b. Curves from simulation diverge at almost the same time after impact in both simulation: marker 2 at t = 0.35 ms. According to details pictured in Figure 5.3, the sheets perforation process just end at this moment and the nail ridges descent starts. Piston velocity prediction is better in {Steel/Cast alu.} configuration than in the {Cast alu./Steel}'s, which can also be noticed on the force-piston displacement curves. Indeed the maximum force amplitude -at marker 1 -is well predicted in Figure 5.2e whereas a significant gap of almost 1 kN remains in Figure 5.2f.

An insight into the penetration mechanism is pictured in Figure 5.3 where the evolution of characteristic joint dimensions z, h, y and n 1 can be evaluate according to four nail penetration states: 1) maximum force amplitude before upper sheet failure occurs, 2) lower sheet failure, 3) end of the hole expansion stage and 4) nail penetration end. The comparison of Figure 5.3a and Figure 5.3b shows that nail-tip crushing (n 1 -value) is almost negligible if cast aluminium is the upper layer whereas it grows very fast if it is made of steel. Sheet bending distance (y-value) grows also faster in {Steel/Cast alu.} configuration than in {Cast alu./Steel}'s because upper sheet fracture can occur only after the maximum bending has been reached. This is not the case for {Cast alu./Steel} configuration because nail instantaneously penetrates the upper sheet without resistance and bending is delayed until nail contacts the steel layer.

In both case, it can be noticed that piston displacement z distribution into the three deformation mechanisms (h; y; n 1 ) is different if the upper layer is made of steel or of cast aluminium.

Sheets fracture modes

The difference of deformation modes between {Steel/Cast alu.} and {Cast alu./Steel} superposition should find an explanation from a local point of view. Indeed it has been noticed that penetration efficiency is better in {Cast alu./Steel} superposition than {Steel/Cast alu.}'s with lower deformation inside the nail. Therefore only the particular sheets reaction to impacti.e. the fracture modesshould explain these differences. Figure 5.4 shows the main steps of the penetration for both material superposition with plastic strain-rate contour plot from blue to red.

The nail penetrates inside cast aluminium positioned as upper layer without any strain localization.

The ogival nail shape pushes beside the aluminium until it contacts the second steel made layer.

However in the opposite {Steel/Cast alu.} configuration, the nail-tip got flattened just after impact with the upper layer. The sheets got significantly bent and then it triggers strain localization in both layers: the lower one is stretched at the center bottom part whereas the upper one exhibits shearplugging. A plug is separated from the upper sheet in the {Steel/Cast alu.} configuration at t = 0.21 ms (see Figure 5.4e) which therefore induces shear-plugging of the second layer. Since the aluminium material has been moved without resistance in {Cast alu./Steel} configuration, steel layer loading properly starts when the first contact occurs at t = 0.14 ms (see Figure 5.4c). Then a competition appears between strain localization in the bottom part and in through-thickness until a small plug is separated from the second sheet at t = 0.31 ms (see Figure 5.4g).

The contour color-scale attests that material subjected to nail impact and perforation undergoes very high-strain rates: it is about 20 000 s -1 in steel and cast aluminium shearing zone -even beyond in the ligament where crack occurs -and about 10 000 s -1 at the nail ridge-sheet interface. 0 2 000 4 000 6 000 8 000 10 000 12 000 14 000 16 000 18 000 20 000 εpl (s

RMK 1:

The average stress-triaxiality σ * avg and lode parameter µ σ.avg are computed from stress-states values weighted with the effective plastic strain increments. RMK 2: Symbols and colors are assigned to sensors located in a particular zone: triangle for fracture zone; square for nail ridge-sheet interface zone; blue for upper sheet; red for lower sheet.

In (e), small symbols are allocated to {Steel/Cast alu.} and big symbols to {Cast alu./Steel}.

To investigate the mechanisms of nail penetration, sensors have been attached to nodes of the sheets meshes and record the stress-state exhibited by materials at these locations. As it can be noticed in Figure 5.5a and Figure 5.5b, triangles are the symbols assigned to sensors located in both upper and lower sheet fracture zone and squares are the ones assigned to sensors located at the nail ridges-sheet interface area. The stress-state evolution during the joining stage is investigated using two indicators: the stress-triaxiality σ * and the Lode parameter µ σ .

It can be noticed that, except for {Cast alu./Steel} configuration (see Figure 5.5b) which presents stable values, stress-triaxiality can change during the perforation stage from dominant compressionstate (σ * < 0) to dominant tension-state (σ * > 0). For conveniency, the average stress-triaxiality σ * avg has been computed on two periods of the joining process: the perforation and the nail descent stages.

Thus we can observe that the strain localization area in the lower sheet of {Cast alu./Steel} exhibits an almost constant stress-triaxiality (σ * avg = 0.628) up to fracture. The upper sheet remains under an highly compression stress-state during the whole perforation steps (σ * avg = -1.350) which confirms that damage could not grow inside the material in such conditions. On the contrary, sheet materials of the {Steel/Cast alu.} configuration (see Figure 5.5a) undergo totally different stress-states in their strain localization area: the steel sheet bending promotes through-thickness tension with an average stresstriaxiality to fracture σ * avg = 0.467. It implies that damage has to grow and propagate throughout the material in order to complete the plug separation from the sheet. It is one explanation which justifies why nail penetration is slowed down when steel is position as the upper layer. The second layer, made of cast aluminium, is in a compression-state until fracture of the upper layer occurs which causes the change of stress-state into tension-dominant (σ * avg = 0.827). Consequently the damage accumulation process definitely begins when the upper layer has failed. Therefore it is a second argument for a better nail penetration efficiency when cast aluminium is positioned as the upper layer.

A convenient approach to discuss the fracture conditions of metal sheets in the two superpositions consists in a diagram where the average Lode parameter µ σ.avg is on the x-axis and the average stresstriaxiality σ * avg is on the y-axis. Figure 5.5e presents these histories for upper and lower sheet points and the reader has to refer to the particular notation to differentiate points from each others. It becomes straightforward that lower-sheet fails mostly under bi-axial tension -close to plane-stress conditions -, whatever is the material, whereas the upper layer made of steel would fail close to uniaxial tension and the cast aluminium one in a compression dominant state. 2. Joining simulation attests that the average stress-state exhibits by sensors located in the lower sheet fracture zone is close to plane-stress and plane strain loci, i.e. σ * avg ≈ 0.6 and µ σ.avg ≈ 0. 3. The upper sheet fracture zone history is subjected to a lower hydrostatic stress than the lower one: σ * avg ≈ 0.45 and µ σ.avg ≈ 0.1 for steel and σ * avg ≈ -1.3 and µ σ.avg ≈ 0.9 for cast aluminium.

Thermo-mechanical state variables

To depict the phenomena involved in the nail insertion stage, Figure 5.6 presents the effective plastic strain εpl , temperature (T ; T ad ) and Lemaitre damage variable D at the final state of sheets joining process. As noticed on Figure 5.6a, the maximum value is limited to εpl = 1. Nail-tip and the surrounding area of sheets have undergone significant deformation which can locally even go beyond this value.

Damage growth is somehow driven by plastic strain and it shows that the lower part of sheets where strain is important is more likely affected by damage than the upper part. Indeed, lower-sheets and steel upper-sheet are red on the lower part where tension was dominant whereas the cast aluminium upper sheet reveals an almost fully-blue interface preserved from damage by compression. only the plastic work (2.17) and T is solved considering heat transfer computation (section 5.2.1). In any case the cast aluminium layer goes beyond 200°C whereas the steel layer reaches locally more than 1000°C under adiabatic assumption and 500°C with normal computation method -conduction pumped out the heat from nail-steel sheet interface. Moreover the adiabatic calculation suggests that melting temperature reached at the steel interface is unlikely. Material could locally experiences temperature close to melting temperature on a very thin layer of steel at the interface -not captured at the current model scale -but it should remain between the bounds given by the two methods proposed here.

Observation 5.3: Sheets temperature increase (3)

1. From the consideration of Observation 2.7.2, good agreement is obtained on maximum temperature computed in {Cast alu./Steel} simulations -T max = 435°C -and {Steel/Cast alu.} -T max = 454°C.

2.

The simulation inaccuracy at the local scale to model sheet material interface getting sheared down by nail ridge descent legitimates a maximum temperature increase larger than the one currently predicted by simulation.

Model sensitivity I: numerical parameters

Main results of the high-speed nailing simulation in both cast aluminium-steel superposition have been discussed in the previous sections. The joining piston kinematics, the distribution of piston displacement among three main deformation modes, the evolution of thermo-mechanical state variables during the joining stage and the fracture modes have been investigated to get an insight into the mechanisms of nail insertion. However the model sensitivity to slight parameters variation has to be evaluated at first before it could be stated consistent and reliable. That is why we have chosen to study in the following section the model response to mesh size and friction parameter variation.

Mesh size

Previous simulations were performed with a mesh size in the bending zone set to 0.5 mm. Due to the mesh-sensitive damage formulation of the sheets, mesh size is kept constant to 0.1 mm in the impact area and only changed in the bending zone. Figure 5.7 presents the models used for analysis with three other mesh size: 0.2 mm, 0.3 mm and 0.75 mm. and amplitude; the second set is composed of simulations run with mesh size set to 0.2 mm and 0.3 mm with lowest slope, amplitude and nail penetration distance h. It shows that the bigger is the mesh size, the bigger is the sheet bending distance y and the lower is the maximum force amplitude as well. It is consistent with the Observation 2.15.1 which implies that material superposition with a 0.2 mm-mesh size is stiffer with a maximum sheets reaction bigger than with a 0.5 mm-mesh size. Similar effect has been noticed in {Cast alu./Steel} superposition where the model with a mesh size set to 0.2 mm has the lowest sheet bending distance y at every state of nail penetrationi.e. the stiffest configurationand therefore has also the biggest sheets reaction.

Friction parameters

The friction has to be modelled properly to account for the force balance which happens at the interface between objects. It is controlled by a Coulomb-limited Tresca law which uses two coefficient: m and µ. As detailed in section 5.2.1, three interfaces have been targeted to have a significant effect on the nail insertion stage: the nail-cast aluminium, the nail-steel and the cast aluminium-steel. aluminium to the nail in Figure 5.9g and Figure 5.9h and aluminium is brought downwards. However it seems unrealistic since it had already been implied by experimental cross-section analysis that nailcast aluminium interface should have limited friction interactions (see Observation 2.13.1). No plug is formed, aluminium is pushed apart, inter-sheets gap and bulge formation attest that frictionless conditions can be legitimately applied to the nail-cast aluminium interface.

The second point to notice deals with failure modes resulting from high interlayers friction conditions in Figure 5.9c and Figure 5.9d. The lower sheet made of cast aluminium seems to be not subjected to the tearing phenomenon which growth from the bottom part to dig through the material thickness.

This phenomenon can be observed on (1) piston displacement is stopped after the force plateau (Observation 2.15.2);

(2) force does not keep increasing after this plateau (Observation 2.15.2);

(3) nail penetration distance is maximum (14 mm).

Clamping diameter

The experimental nailing campaign n°2 conducted on mixed steel-cast aluminium superpositions in- suggest that simulation might not be accurate enough for the moment to capture short difference of piston kinematics obtained with such a small clamping diameter variation.

Model sensitivity III: damage

One of the main modelling issue of plastic deformation-based joining processes is the choice of a suitable set of constitutive and damage laws. Fracture mode also becomes a concern for such processes to enable the prediction of important volume loss, for instance due to important shear plugging, which would have a significant effect on the final joint features. For these reasons we have chosen in the present section to investigate how different aspects of the damage formulation could affect the piston kinematics of joining simulations. Among the material damage formulations tested here, it can be noticed on

RMK:

The average stress-triaxiality σ * avg and lode parameter µ σ.avg are computed from stress-states values weighted with the effective plastic strain increments.

First point to notice, steel and cast aluminium sheet points experienced very distinct stress-state histories whatever is the damage formulation. Most of the steel fracture points are located in a dominant tension region, close to the plane stress-locus, i.e. where stress-triaxiality is in the range 0.3 < σ * avg (steel) < 0.7 and Lode parameter 0.1 < µ σ.avg (steel) < 0.4 whereas the cast aluminium points are more scattered out of the plane-stress conditions, either in a very high-stress triaxiality region σ * avg (cast alu.) ≈ 0.8 or close to uniaxial tension 0.2 < σ * avg (cast alu.) < 0.4. Most of the fracture points allocated to cast aluminium positioned as upper sheet (small stars) can not be shown on Figure 5.15a due to excessive stress-triaxiality values σ * avg (cast alu.) ≈ -1.4. This compression stress-state prevents the cast aluminium from damage growth and ductile failure. Hence the fracture inclination of the sheets in every tested configuration can be summed up as follows:

(1) steel positioned as the upper sheet fails due to a stress-state which goes from tension to shear, far from the plane-stress condition; (2) cast aluminium positioned as the upper sheet undergoes high compression stress-state, beyond the cut-off value (σ * < -1/3) discussed in Chapter 3 which prevents from any crack initiation and propagation. Consequently damage accumulation effect is limited and therefore the effect of damage formulation on the joining kinematics can be considered negligible as well; (3) steel positioned as the lower sheet is mostly subjected to bi-axial tension state (σ * ≈ 0.66) and lies on the plane-stress locus; (4) cast aluminium positioned as the lower sheet exhibits either a very high tension state until fracture (σ * avg (cast alu.) ≈ 0.8), out of the plane-stress conditions, or a mixed shear-tension state (0.2 < σ * avg (cast alu.) < 0.4).

Using the mean-stress state to fracture (σ * avg , µ σ.avg ) extracted from sensors in the nailing simulation, the fracture strain associated to upper and lower sheet fracture zone are plotted on every calibrated loci (see for instance Consequently the disparity in joining kinematics can not be fully allocated to inappropriate choice of damage formulation but it should result from other aspect of the damage computation method.

Material ductility is controlled by damage evolution law but the limit of damage growth is controlled by another parameter called critical damage parameter D c . According to joining simulations applied to cast aluminium-steel configurations, two factors of the damage computation method affect joining piston kinematics and therefore nail insertion depth:

1. Damage formulation controls the damage growth. Thus one criterion can give better result than another.

2. Critical damage parameter controls ductility. It affects directly the amount of plastic work before fracture occurs and therefore the nail penetration depth.

Applications to multi-materials configurations

The validity of high-speed nailing simulation has been tested and evaluated in the two main configurations of interest: {Steel/Cast alu.} and {Cast alu./Steel}. However in real industrial conditions, this joining process could be applied to a wide range of materials, thicknesses and superposition. The present section shows the main results of simulation run in the configurations investigated in Chapter 2 made of cast aluminium, steel and AHSS sheets and the one which cannot be joined due to excessive materials stiffness, the {AHSS/Cast alu.}'s.

Piston kinematics

Results obtained from simulation are expressed in terms of force-piston displacement curves in Fig- under-estimated in the others.

Cross-sections

Another aspect to evaluate the validity of simulation consists in the cross-sections comparison presented in Figure 5.18. Most of the phenomena are captured by simulation: nail-tip crushing in Figure 5.18a and Figure 5.18b, inter-sheets gap and sheets fracture mode. However significant differences can be noticed on the simulated sheet bending in the area close to the picture boundaries, especially for {Cast alu./Steel} (see Figure 5.18c). To quantify these geometrical discrepancies, seven measurements are performed on the final joint shape as introduced in Figure 1.7 and are presented in RMK: Penetration is not completed in Figure 5.18a therefore simulation is taken at the same state (not final state) of penetration to enable the comparison with experiment.

The gap on penetration distance h is zero in every cases, i.e. nail is fully inserted into the sheets. The nail related dimensions n 1 , n 2 and n 3 show that nail-tip crushing is captured even though discrepancies can be noticed between experiment and simulation. The gap obtained on n 1 -value might be due to inappropriate experimental cutting angle to get the final cross-section or inaccurate cutting position.

Hence the nail-tip length is bigger or smaller than if the cutting was perfectly processed.

However the values of interests are the interface nail-upper sheet contact distance l 1 and the interface nail-lower sheet contact distance l 2 . Those ones are important in the consideration of loads transmitting area from the sheets to the nail in further nailed-joint tests. For upper sheet made of steel or cast aluminium the gap is the same and about 13%. However it is 8% for a lower sheet made of steel and 15 % for a lower sheet made of cast aluminium. This gap is bigger for cast aluminium as a lower sheet since brittleness and fracture mechanisms are unpredictable in our numerical simulation and therefore the contact distance cannot be accurately modelled. This issue is obvious when Figure 5.18b and Figure 5.18d are considered. Asymmetry of the contact distance and asymmetry of the cracks in both sides of the nail are impossible to capture in current finite element simulation.

Observation 5.8: Joint cross-section (6)

1. The gap between experimental and numerical nail-upper sheet contact distance l 1 is the same whatever is the material superposition; about 13%.

2. The gap between experimental and numerical nail-lower sheet contact distance l 2 is bigger if made of cast aluminium (15%) than steel (8%). The cast aluminium brittleness induces unpredictable crack occurrence, accurate fracture mode and therefore contact length.

Limitation of the joining process

Among the configuration investigated in Chapter 2, the mixed cast aluminium-AHSS materials superposition are interesting since they are touching the limit of current joining process capability. Indeed an excessive stiffness of the joining location makes problematic the nail insertion into the sheets. This effect on nail deformation and fracture has been widely discussed in Observation 2.16 and Observation 2.17. Consequently it makes these configuration particularly adapted to test the simulation reliability to predict either the nail excessive deformation -the {Cast alu./AHSS} -or the nail fracture after impact -the {AHSS/Cast alu.}. 2.10).

RMK:

The evolution of piston displacement over time can be found in the Appendix D.

{AHSS/Cast alu.} Simulation has been run in the exact same numerical conditions than {Cast alu./AHSS}'s; except the superposition and the loading conditions which are changed: force serie is set to P = 6.0 bars and velocity to v = 33 m/s as in Experimental campaign n°2, test #37 (Table 2.9).

As expected by experiment, nail penetration failed in simulation because of the upper sheet extended stiffness. Figure 5.20 reveals that nail-tip undergoes very large deformation, depicted at four different time after impact: 0.006 ms, 0.04 ms, 0.10 ms and 0.16 ms. Due to a lack of stiffness, the nail could not generate strain localization zone in the upper and lower sheet impact area. Therefore the joining energy has been mostly dissipated into nail plastic deformation and sheets bending deformation.

It has been noticed in Observation 2.17.2 that nail's brittleness implies material could not actually withstand such strain amplitude observed in simulation without failure. The current nail material formulation does not take into account damage mechanisms since no experimental set-up have been found for the model validation. However the present simulation could be used to investigate some of the mechanisms responsible of crack occurrence at the nail-tip.

In addition to geometrical considerations, Figure 5.20 also draws the stress-triaxiality σ * evolution during the deformation stage from the compression cut-off value (σ * = -0.33) to uniaxial tensionstate (σ * = +0.33). It shows that a compression dominant area is created in both side of the impact contact zone, in the nail-tip and in the upper sheet. While nail's body keeps moving downwards, the compression zone is preserved and surrounded by two others area of interests: (1) a tension-dominant zone in the contact area when material parts are pushed against the upper sheet and stretch out to enlarge the tip; (2) a shear-tension zone in the nail core which is above the compression zone and have to feed the enlargement zone with a constant material flow. The balance between the three zones implies that damage is prevented from growth in the impact compression zone but is promoted in the outside contact zone and in the core. The blue contour -the area where stress-triaxiality is bigger than cut-off value -testifies that damage could legitimately grow in the two region which have been pointed out before. Moreover the orientation between these two region suggests that a crack propagating in between would be really close to a 45°angle. Thus we can assume that a crack created in one of the two zone would definitely propagate at 45°, driven by these preferential stress-states, as the so-called asymmetrical principal shear fracture mode observed in the experiment presented in Figure 2.39.

-0. RMK: Loading conditions are taken from Experimental campaign n°2, test #37 (Table 2.9).

Observation 5.9: Nail deformation (2)

Simulation captures the nail deformation sensitivity to impact discussed in Observation 2.16: nail-tip is more affected by impact with AHSS than steel and even more if it is located as the upper layer of dissimilar materials superposition. RMK: Simulation is run with one symmetry axis to enable non-orthogonal impact (+2°) and promote asymmetrical fracture occurrence.

One way to validate the assumption of the nail fracture mode process consists in a simulation where nail impact is not orthogonal to the upper sheet. Thus the mesh is extended to a 180°slice, with one symmetry plane. The yaw angle has been set to 2°as it can be noticed on Figure 5.21a to slightly promote a preferential nail deformation direction after impact. A Cockroft-Latham fracture locus has been chosen since only one parameter has to be set (ϕ 3 ) and it implicitly accounts for decreasing fracture strain with stress-triaxiality. Therefore it is adapted to develop damage in regions where tension are dominant. For sake of simplicity, we assumed ϕ 3 = 40 MPa for the nail's material fracture locus and the simulation run until t = 0.2 ms (Figure 5.21b).

The result is straightforward: friction sticks the flattened nail-tip to the upper layer. As the nail is moving downwards on a vertical line, an asymmetry is promoted in the nail main axis and shaft is bending. The two regions of tension-dominant states inside the shaft core and on the outside become asymmetrical as well and damage is growing. Finally a fully-damaged border zone (red contour) is drawn at 45°and is in good agreement with the fracture pattern obtained in experiment (Figure 5.21c).

Observation 5.10: Nail fracture mode (2)

1. Impact with stiff configuration generates into the nail three regions with different stress-states:

(i) a compression zone right under the impact area; (ii) a tension zone at the nail-sheet contact area;

(iii) a shear-tension zone in the nail core.

2. The only locations where stress-triaxiality admits damage growth (σ * > -0.33) are the nail-sheet contact area and the nail core.

3. A crack generated in one these locations would inevitably propagate towards the other one, driven at 45°by preferential stress conditions, as obtained in experiment and discussed in Observation 2.17.

Nailed-joint mechanical tests

High-speed nailing simulations have been run on different material superpositions and the validity has been tested in various loading, boundary, mesh size and friction conditions. However joining is the very first stage of the joint design process. Once a joint is completed, it must be tested to quantify what in-service mechanical strengths could be expected. The present chapter is dedicated to investigate through finite element simulation the properties of nailed-joint processed in {Steel/Cast alu.} and {Cast alu./Steel} configurations. At first, the sheets relaxation stage is simulated before cross-tension and shear-tension model could be tested. Then results are compared to experiments from the consideration of force-displacement curves, global as well as local deformation modes observed from post-mortem cross-sections. Finally, a sensitivity analysis is conducted on cross-tension simulationthe most important configuration -to highlight which parameters must be carefully chosen to run joint testing simulation.

Finite element models

The results obtained from the nail insertion stage simulation are used as input of the nailed-joint characterization tests. Nail and sheets meshes and their associated field (stress, strain, temperature ...) are imported and modified to be part of the joint testing simulations.

Mesh transformation

After the nail insertion stage is completed, mesh modifications are necessary to prepare the relaxation stage: piston and ejected plugs from both sheet meshes have to be removed using the red plans presented in Figure 5.22a. This mesh will be used without other modification to simulate sheets springback effect until a new equilibrium is reached. After the relaxation stage is completed, final meshes and fields are modified for the nailed-joint testing simulations: cross-tension and shear-tension tests. The "mirror" application, available in FORGE ® , duplicates nail and sheets meshes around the vertical axis to get 1/4 of the real cross-tension joint (see Figure 5.22b) and 1/2 of the shear-tension one (see Figure 5.22c).

The final preparation step consists in using appropriate cut-plans to reach realistic sample geometries.

High-speed nailing simulation

Piston mesh is removed using this plan Sheets' plug are removed using this plan 1.

2.

Nailed-joint relaxation (a)

Mesh extension from 10° to 90°

1.

Cross-tension Nailed-joint relaxation

Mesh cut with red and blue plans 2.

(b)

Mesh extension from 10° to 180° 1.

Shear-tension Nailed-joint relaxation

Relaxation

This simulation is run with the exact same mesh as the one of the high-speed nailing simulation at final increment. As presented in Figure 5.22a and Figure 5.23, piston motion is no more considered and boundary conditions are changed. The relaxation duration is set to ∆t relaxation = 0.2 s and both sheets are stuck to circumferential planes. Material models are unchanged and heat transfer is not considered. Only friction is modified and set to sticking condition at the nail-lower sheet interface and nail-upper sheet interface.

Clamping conditions Mesh sizes (from impact area to outside) 0.1; 0.2; 0.5; 1.0 mm Relaxation time = 0.2 s

Configurations of interest

In the continuity of investigations conducted on high-speed nailing simulation, nailed-joint testing simulations are computed on {Steel/Cast alu.} and {Cast alu./Steel} superposition. Cross-tension and shear-tension tests have been performed on different joints, therefore different references must be assigned to these configurations.

Purple and light blue curves presented in Figure 2.52 are taken as cross-tension references:

• {Steel/Cast alu.} joined at P = 5.7 bars (Table 2.11: Experimental campaign n°3).

• {Cast alu./Steel} joined at P = 5.3 bars (Table 2.11: Experimental campaign n°3).

Purple and light blue curves with small circles, in Figure 2.56, are taken as shear-tension references:

• {Steel/Cast alu.} joined at P = 5.4 bars (Table 2.7: Experimental campaign n°2 -#5).

• {Cast alu./Steel} joined at P = 5.3 bars (Table 2.8: Experimental campaign n°2 -#22).

These configurations enable the comparison between experimental and numerical force-displacement curves and fracture modes.

Relaxation

The relaxation stage simulation is mandatory to reach a new static equilibrium. Conditions of computation have been detailed in section 5.3.1. To read properly the joint behavior throughout the tests, the force-displacement curves presented in Despite the discrepancy with experiment which will be discussed in a while, these curves enable to capture the main features of the joint response in cross-tension tests. Indeed, it can be noticed that, whatever is the materials superposition, joint response follow the same trend until a displacement of 1 mm is reached. Figure 5.29c and Figure 5.29d depict the joint geometry and it can be noticed that at this displacement, lower sheet is almost horizontal. The whole parts, nail and sheets, are moved down and starting to bend until this displacement value is reached. That is why joint response is identical in both superpositions. However, after 1 mm, a divergence appears between the reaction of {Steel/Cast alu.} and {Cast alu./Steel} joints. Lower sheet clamped part creates a reverse bending which induces an horizontal reaction towards the nail. Therefore a bigger displacement set to the lower sheet will induce a bigger reverse bending and a bigger reaction towards the nail. It improves nail interlock but the balance between vertical and horizontal force component is changed. Hence the joint reaction will be driven by the lower sheet stiffness: on one hand the reverse bending is pushing the sheet against the shaft and on the other hand, interface with the nail is going to be sheared since the nail is static and the lower sheet is moved downwards. Thus the divergence between force-displacement curves observed at 1 mm is explained by the transition between these two stages.

Simulations are stopped after a displacement of about 4 mm to preserve the mesh at the interface from excessive distortion. Moreover it has not been necessary to extend the simulation beyond this value since the main phenomena are already captured and the discrepancy with experiment is noticed.

This gap could be explained in the experiment by local sliding at the interface between sheets and dies: joint response in fully-clamped conditions would be the one obtained in simulation, with a large curve slope, whereas real boundary conditions introduce micro-sliding at the interface, which affects the curve slope but not the force amplitude experienced inside the joint.

If this assumption is considered as a valid explanation for the discrepancy between experimental and numerical force-displacement curves, the joint deformation modes observed in Figure 5.29 attest the simulation consistency to predict the dominant mechanisms discussed in Observation 2.22. The comparisons between final sheets bending at the last increment of simulation presented in Figure 5.31 reveals good agreement on the global geometries; except for cast aluminium as a upper layer which could not be simulated as far as in the experiment, up to crack occurrence.

Figure 5.30 also confirms the considerations made from cross-section analysis in Observation 2.21: the lower sheet reverse bending is well captured by simulation and tend to push the sheet against the nail shaft. This phenomenon can be noticed in Figure 5.30a where cast aluminium flows into the ridge but could not be reached in the opposite {Cast alu./Steel} superposition. Indeed Figure 5.31b shows a steel sheet which is not as bent as in the experiment and therefore with a stress at the interface which is not high enough to trigger material flow into the ridges. While the lower sheet is pushed towards the nail by bending, the upper sheet is pulled out the nail interface by bending as well. This explains why the upper layer in {Cast alu./Steel} configuration is gradually loosing contact with the nail until the load transmitting area reaches a critical limit and brings the whole joint into failure. In both simulations the computation is limited by mesh excessive distortion in the cast aluminium sheet. However the force-displacement curves obtained from {Cast alu./Steel} simulations diverge from the very beginning (Figure 5.34b). Every curves associated to simulation computed without remeshing are superposed, whatever is the time step (f r = 0). The other ones follow a distinct path: the larger is remeshing frequency, the softer is the joint response. The larger is the time step, the softer is the joint response. Remeshing frequencies and time step affect significantly the {Cast alu./Steel} cross-tension simulation; not the {Steel/Cast alu.}. Even the first elastic stage in which joint response is supposed to be linear -and common between the two configurations -is changed. Discontinuous dash curves lay on the linear part until the remeshing occurs and modifies the curve slope. Nevertheless this gap could not be assigned to cast aluminium upper sheet bending since it has barely start when curves are diverging. This gap has to be allocated to accumulated errors on the respective positions of nail ridges' nodes and sheets' throughout the remeshing steps and amplified by an extended time step.

Model sensitivity III: friction parameters

The friction conditions at the nail-lower sheet interface have been initially set to sticking. But reality implies that vertical joint response is driven by the normal stress at the nail-lower sheet interface and therefore subjected to friction. The choice of the friction model parameters could locally affect their interactions and modify globally the joint response. This is why friction parameters are changed to m = {0.90; 0.60; 0.30; 0.05} and µ = 1/2 m in {Steel/Cast alu.} and {Cast alu./Steel} simulations.

Figure 5.35 focuses on the very beginning of the cross-tension simulations (D p ≤ 0.6 mm). As it can be noticed in both configurations, the larger is the friction parameter value, the longer joint response curves lays on the sticking curve. However every curves leave quickly the sticking trend and contact is definitely lost between the nail and the lower sheet. Consequently, friction has to be maximum to model shearing at the nail-lower interface without loosing contact.

Nail insertion stage

The high-speed nailing process is based on two main components: the nail and the joining piston. The effect of nail insertion depth on joint quality has been investigated all along this work. In addition to visual data, process monitoring data obtained from piston kinematics enables an estimation of sheets reaction to the nail insertion. These topics are gathered in the following themes: nail insertion depth and sheets reaction to the nail insertion.

Nail insertion depth 1.1 Nail insertion is driven by joining piston kinematics therefore nail penetration depth is somehow related to piston displacement Z(t).

1.2 An under-estimated joining pressure leads to partial nail insertion. (1) piston displacement is stopped after the force plateau (see Sheets reaction to the nail insertion 3.2);

(2) force does not keep increasing after this plateau (see Sheets reaction to the nail insertion 3.2);

(3) nail penetration distance is maximum (14 mm). 1.3 Dominant shearing mode occurs for {Steel/Cast alu.} superposition, i.e. when nail tip is flattened after impact with the steel layer which promotes the area down the tip -called a plug -to be separated from each sheet. Plug shape is even bigger if nail tip is flat.

1.4 Mixed petalling-shear plugging mode occurs for {Steel/Steel} superposition in which created plugs are small enough to get hole enlargment effect with also petals created from the cracks propagation in radial directions.

2.0 About {Steel/Cast alu.} superposition:

2.1 Nail-steel sheet interface keeps being sheared after perforation because steel sheared pieces can be found afterwards at the nail-aluminium interface and inside the nail ridges.

2.2

The aluminium plug is formed by the competition between plug-shearing and petalling mechanims: radial cracks initiate the petals formation and a shear crack grows and propagates through-thickness to finally trigger plug separation from the sheet. 5.2 Joining simulation attest that the average stress-state exhibits by sensors located in the lower sheet fracture zone is close to plane-stress and plane strain loci, i.e. σ * avg ≈ 0.6 and µ σ.avg ≈ 0. 5.3 The upper sheet fracture zone history is subjected to a lower hydrostatic stress than the lower one: σ * avg ≈ 0.45 and µ σ.avg ≈ 0.1 for steel and σ * avg ≈ -1.3 and µ σ.avg ≈ 0.9 for cast aluminium. 6.1 Sheets failure modes in {Cast alu./Steel} joining simulations are not affected by variations in the friction conditions at the three interfaces: nail-steel, nail-cast aluminium and steel-cast aluminium.

6.2 Sheets failure mechanisms in {Steel/Cast alu.} configurations are significantly influenced by changes in the friction conditions at the interfaces. Only the model including the following set of friction conditions captures the competition between the tearing and shear-plugging failure mechanisms discussed in Sheets fracture mode 2.2: frictionless at cast aluminium-steel interface, frictionless at nail-cast aluminium interface and high friction at nail-steel interface.

Nail deformation mode 1.1 Nail-tip is flattened by impact with a hard material (steel and AHSS).

1.2 Materials superposition affects impact toughness: the ogival nail-tip is preserved when steel is the lower layer whereas it gets flat when it is used as upper layer.

1.3 Material strength affects impact toughness: whatever is the position of AHSS layer in the aluminium-AHSS superposition, nail-tip will be crushed by impact.

2.0 Simulation captures the nail deformation sensitivity to impact: nail-tip is more affected by impact with AHSS than steel and even more if it is located as the upper layer of dissimilar materials superposition.

Nail fracture mode 1.1 Nail penetration capability is limited by its brittleness; it can not be extended by an increasing impact velocity.

1.2 Nail-tip fracture occurs after impacting a stiff material superposition. Fracture mode is similar to the so-called "asymmetrical principal shear" mode.

2.1 Impact with stiff configuration generates into the nail three region with different stress-states:

(i) a compression zone right under the impact area; (ii) a tension zone at the nail-sheet contact area;

(iii) a shear-tension zone in the nail core.

2.2

The only locations where stress-triaxiality admits damage growth (σ * > -0.33) are the nail-sheet contact area and the nail core.

2.3

A crack generated in one these locations would inevitably propagate towards the other one, driven at 45°by preferential stress conditions, as obtained in experiment.

Sheets temperature increase

1.1 In the joining stage, the steel sheet is under adiabatic condition (F o ≫ 1) whereas the aluminium one is not (F o < 1).

1.2 Temperature measurement ensures that steel must exceed locally T > 450°C.

2.0 Blue-colored ring which can be noticed on steel sheared plug indicates that material should have locally reached high temperature during the nail insertion stage. whatever is the material superposition; about 13%.

6.2

The gap between experimental and numerical nail-lower sheet contact distance l 2 is bigger if made of cast aluminium (15%) than steel (8%). The cast aluminium brittleness induces unpredictable crack occurrence, accurate fracture mode and therefore contact length.

Cross-tension simulations capture global sheets bending as well as local phenomena discussed in

Joint cross-section 4: the upper sheet is pulled out the nail contact interface which decreases the load transmitting area; the lower sheet is pushed on the nail shaft which promotes material flow into the ridges. Only fracture and sheared material could not be modelled yet.

Nailed-joint strengths 1.1 Joint response in shear tensile configuration is mostly related to the nail shaft shear resistance.

1.2 Joint response in cross-tension configuration is mostly related to the nail ridges grip on the lower sheet.

About cohesion mechanism in the cross-tension test:

2.1 As the upper and lower layers are moved apart, nail is static. One side of the lower sheet is taken down whereas the other side is contacting the nail (Joint cross-section 4.1), i.e. one side is slowly bent while the other one is pushed by material reaction against the nail shaft. Therefore material flows into the nail ridges and increases nail interlock (Joint cross-section 4.2). The nail-lower sheet interface gets gradually sheared by their relative velocity with fragments separated from the rest of the sheet which remain stuck into the ridges (Joint cross-section 4.3).

3.0 About cohesion mechanism in the shear-tensile test:

3.1 Nail is moved in opposite directions by the upper and lower sheet. Thus two compression areas can be found in the sheets in both side of the nail (Joint cross-section 5.1). It triggers nail rotation inside the sheets and therefore lower sheet constraints the nail and its stiffness modulates the maximum force the joint can handle. However the nail head position, seated on the upper layer, prevents from an excessive shaft bending. Tension and compression dominant stress-states are growing with the shaft bending until a crack appears and drives the whole joint into failure Joint cross-section 5.2. 4.0 Two stages compose the nailed-joint response to cross-tension: 4.1 Nail, upper sheet and lower sheet react as one, with identical stiffness for {Steel/Cast alu.} and {Cast alu./Steel} joint. This stage is complete when the lower sheet moving part is horizontal.

4.2

As the lower sheet moving part goes below the horizontal line (D p > 1 mm), upper sheet bending keeps increasing and a lower sheet reverse bending is created. The upper sheet bending pulls out material from nail contact whereas the lower sheet reverse bending pushes it into the nail ridges. Nail grip depends on the lower sheet material stiffness which is getting sheared at the interface and pictured by a divergence between {Steel/Cast alu.} and {Cast alu./Steel} force-displacement curves.

5.1 Cross-tension simulations are subjected to mesh inter-penetration issues. It appears in region where kill element were significant at the insertion stage and can be twofold: nail's nodes are projected into the lower sheet's mesh and upper sheet's nodes are projected into the lower sheet's mesh. It causes an arbitrary job abortion.

5.2 Despite too few simulations run to evaluate model sensitivity, the following can already be stated:

(i) mesh-size have no significant effect on the joint response; (ii) remeshing brings noise in the joint response curve; (iii) {Cast alu./Steel} is more affected by remeshing and time step variation than {Steel/Cast alu.}; probably caused by accumulation of nodes re-interpolation errors of both nail and lower sheet meshes which introduce an "artificial sliding" in the force-displacement curve.

6.0 Good agreement is obtained by mixed shear-tension simulations on the set of phenomena discussed in Nailed-joint strengths 3.1:

(i) the two zones plastically deformed in both sides of the nail promoted by nail rotation; (ii) the maximum joint force which is driven by lower sheet stiffness (material features) and constraints the nail rotation; (iii) the onset of head tearing mechanism noticed in {Cast alu./Steel} cross-section.

Optimized joining pressure range 1.1 Cross-tension strength evolution with respect to input joining pressure is assumed to start at zero, increase until a plateau with a maximum value is reached and decrease back to zero.

1.2 Both span and amplitude of the optimized cross-tension strength plateau depends mostly on the material behavior of the connected layers.

2.1 Optimized cross-tension strength amplitude is proportional to the lower-sheet material yield stress.

2.2

The plateau span -range of joining pressure in which cross-tension force remains maximum -is proportional to the lower-sheet material ductility.

Nailing references Fully clamped sheets are joined with increasing joining pressure until optimum pressure is reached in the following configuration: {Steel/Cast alu.}, {Cast alu./Steel} and {Cast alu./AHSS}. Sheets fracture modes, cross-sections and sheet reaction curves are analyzed to highlight the main features of a proper nailed-joint: (i) a limited amount of ejected material to promote a better nail locking inside the sheets; (ii) a nail head which lies on the upper sheet; (iii) a piston motion stopped slightly after full nail penetration is completed. The variability of sheet reaction curves and joint characteristic dimensions is also studied from the repetition of identical tests at optimized joining pressure. The biggest sheets reaction variability appeared just after perforation occurred, in the piston displacement range assigned to the hole expansion and nail descent. The same statement is done from the analysis of joint characteristic dimensions; whatever is the material under consideration, the biggest variability is obtained at the nail-lower sheet contact length l 2 . These variabilities must be allocated to the unpredictability of the fracture mode at the lower sheet, driven by the competition between two dominant fracture mechanisms: tensile tearing and shear-plugging.

Nailed-joint properties In-service joint properties are investigated through conventional crosstension and shear-tension tests in quasistatic conditions on 38 x 125 mm 2 samples. These tests revealed the validity of criteria used to state whether a joining pressure is optimum or not, from the consideration of cross-tension tests performed on joints processed at different joining pressure. The shear-tension strength is limitated by nail shaft shear resistance and lower sheet capability to constraint nail from rotation. The cross-tension strength is driven by the lower sheet material yield stress but joint fracture always starts -whatever if located as upper or lower layer -from a crack which has grown inside the material with the lowest ductility.

Mechanical characterization

Several tests are done on nail shaft (C60) and sheet materials (DP780, DP1180 and AlSi10MnMg(T6)) to meet the dynamic and fracture conditions encountered in the highspeed nailing process. Nail material's strain hardening is investigated through quasistatic compression tests whereas the sheet materials are tested under large strain, high strain-rate and fracture: quasistatic and dynamic hopkinson compression tests up to 6000 s -1 and quasistatic tensile tests on uniaxial, notched and in-plane shear specimens. A digital image correlation system is used to capture the surface strain-field of tensile tests until fracture occurred. It provided important data for the further calibration of the material models: an accurate measurement of the displacement applied at the specimens' boundaries and the effective plastic strain in the crack onset area.

Material models calibration

The yield stress is modelled through a mixed Swift-Voce formulation for the strain hardening term and a Johnson-Cook formulation for the strain-rate hardening and thermal softening terms. A coupled-damage approach is chosen to account for material softening with the damage accumulation. Two evolution laws are considered: the phenomenological-based (PB) and the Lemaitre (LEM). Both constitutive and damage law parameters are calibrated from compression and tensile test simulations carried out in the same conditions as in the experiments and coupled with the optimization algorithm. Good agreement is obtained between experimental and numerical forcedisplacement curves which attests the consistency of the model to predict material behavior under different loading velocities and on a large range of stress-states leading to fracture.

q t = ψ t A t l 2 t1 + l 2 t2 + l 2 t3 (C.1)
where A t is the area of the triangle, l t1 , l t2 , l t3 the edge lengths of the triangle and ψ t = 4 √ 3 is the normalizing coefficient to obtain for an equilateral triangle, a maximum quality q t = 1. As it can be noticed from the consideration of these criteria, any association of dimensions can be assumed valid to evaluate mesh quality of a tetrahedron. Thus we have chosen to consider tetrahedron quality Q 5 according to the worst facet quality among the four triangle quality q t1 , q t2 , q t3 and q t4 (C.6).

Q 5 = min [q t1 , q t2 , q t3 , q t4 ] (C.6)

This quality criterion is computed as a user-subroutine from spatial coordinates of nodes X loc given by FORGE ® solver at every increment (see Algorithm 1).

XVI

Algorithm 1: Mesh quality computation in FORGE ® user-subroutine Input : Node location matrix X loc of size 3 × 4, respectively 3 space dimensions and 4 nodes

Output: Quality value such as 0 > Q > 1

1 Vectors of distance between nodes 2 for i = 1 to 3 do 3 v(i, 1) = X loc (i, 1) -X loc (i, 2); 4 v(i, 2) = X loc (i, 1) -X loc (i, 3); 5 v(i, 3) = X loc (i, 1) -X loc (i, 4); 6 v(i, 4) = X loc (i, 2) -X loc (i, 3); 7 v(i, 5) = X loc (i, 2) -X loc (i, 4); 8 v(i, 6) = X loc (i, 3) -X loc (i, 4); Tetrahedron quality Q = min [q(1), q(2), q(3), q(4)];

XVII

Indicator triggering kill-element

Element deletion is triggered only when a damage indicator value D * is set to 1. If this value is equal to zero, no elements are deleted from the mesh. Algorithm 2 shows how damage and quality criteria are used to compute this damage indicator from the consideration of two main fracture modes observed in experiments: (i) a crack generated below the lower sheet which propagates radially to form petals;

(ii) a through-thickness crack produced in a shear band which separates a plug from the sheet.

fεYF 1 , F 2 , F 3 , F 4 , F 5 CoefficientsFI 1 , J 2 , J 3 Characteristic length l 0 1 Nail-Upper sheet interface contact distance l 2 Sum of piston and nail masses n 1 2 3

 123513012123 Abbreviations

  et al. (2015) and Groche et al. (2014) have proposed overviews of the most popular processes used to join dissimilar materials: Friction Stir Blind Riveting (FSBR), Self-Piercing Punch Riveting (SPPR), Flow-Drilling Screw (FDS)... As stated by Chastel and Passemard (2014), some of these technologies have become really attractive for industrial applications since neither pre-hole nor additional support are required but only one side access for the setting tool.

Figure 1 :

 1 Figure 1: Materials distribution applied to a vehicle car body is one of the biggest challenge of automotive light-weighting. Configuration taken from Renault EOLAB prototype (www.renault.com).

Figure 2 :

 2 Figure 2: Automated RIVTAC ® nailing device adapted to production line at the experimental joining platform of IRT M2P laboratory (www.irt-m2p.eu).

) a 1 .

 1 75 mm-thick very high-strength dual-phase steel with low ductility (DP1180), respectively designated in the following Cast alu., Steel and AHSS. Rectangular specimens of dimensions 38 mm x 125 mm are extracted from materials and used in experiments. It enables a wide range of joining configuration which are representative of industrial cases with highly dissimilar behavior: {Cast alu./Cast alu.}, {Steel/Steel} or mixed superposition.

Contents 1 . 1 9 1. 2 . 2

 11922 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2 Nail setting device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.1 Nail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Joining piston . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.3 Hold-down piston . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.4 Buffer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.5 Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 Setting parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.1 Joining pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.2 Hold-down pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 Main joining steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4.1 Positioning stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4.2 Stage 1. Piston acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4.3 Stage 2. Nail insertion into the sheets . . . . . . . . . . . . . . . . . . . . . . . 14 1.4.4 Stage 3. Piston rebound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.5 Joint quality indicators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5.1 Process monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5.2 Visual inspection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1. 5 . 3 19 1. 5 . 4

 531954 Mechanical testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convenient joinability chart: Joint strength vs. Input joining pressure . . . . . 20 1.6 Summary of Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.7 Résumé en français . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1 . 1 :

 11 Figure 1.1: Diagram of the High-Speed Nailing device and its main components

•Figure 1 . 2 :

 12 Figure 1.2: Nail geometry (Left) and RIVTAC ® H3 nail (Right)

Figure 1 . 3 :

 13 Figure 1.3: Nailing device capabilities on (a) joining pressure and (b) hold-down pressure setting.

1. 4 .Figure 1 . 4 :

 414 Figure 1.4: Positioning process of the setting tool

Figure 1 . 5 :

 15 Figure 1.5: Main joining stages of the High-Speed Nailing process

Figure 1 . 6 :Observation 1 . 3 :

 1613 Figure 1.6: Joint quality indicators in three different joining conditions: Non-valid joining for energy lack (Left); Valid joining (Center); Non-valid joining for energy excess (Right)

3 Figure 1 . 7 :

 317 Figure 1.7: Main dimensions of a nailed-joint cross-section

Figure 1 . 8 :

 18 Figure 1.8: Mechanical tests conducted on double layers nailed-joints in (a) shear-tensile and (b) cross-tension configurations.

1. 5 . 4

 54 Convenient joinability chart: Joint strength vs. Input joining pressureFrom the consideration of mutual loading configurations and material volumes contribution in the mechanical responses, nailed-joint strength is mostly weaker in cross-tension configuration than in the shear-tensile one. That is why main concerns of engineers in charge of joining applications are based on cross-tension strength optimization related to nail setting device parameters. As previously mentioned, joining pressure is the main parameter to pilot nail insertion; it is therefore interesting to use these two notions -Cross-tension strength & Joining pressure -to connect in-service joint properties to process parameters. It takes the form of a graph, presented in Figure1.9, where maximal cross-tension force is plotted against joining pressure. Four typical joining configurations with layers made from different materials can be noticed: {Steel/Steel} superposition in purple, {Steel/Aluminium} superposition in orange, {Aluminium/Steel} superposition in dark blue and {Aluminium/Aluminium} superposition in light blue.

Figure 1 . 9 :

 19 Figure 1.9: Joinability charts in four different joining materials superposition

  piston d'assemblage, le piston de maintien et l'amortisseur. La géométrie du clou ainsi que ses propriétés mécaniques répondent à deux fonctions principales: la perforation des tôles et la cohésion en leur sein. L'assemblage par clouage est configuré, dans la machine étudiée ici, via deux paramètres: la pression d'assemblage et la pression de maintien. Toutes deux peuvent être interprétées au moyen de grandeurs physiques bien plus parlantes pour discuter les capacités et les limitations de la machine d'assemblage. Ainsi la pression d'assemblage qui peut être configurée entre 3 et 8 bars permet d'atteindre des vitesses d'impact du piston entre 20 et 39 m/s. La pression de maintien est, pour sa part, configurable entre 1 et 5 bars et sert à moduler la force exercée par le cloueur sur les tôles dans la plage de 0.2 à 1.5 kN. Un cycle d'assemblage est ensuite décrit et ses étapes principales détaillées. Elles suivent l'enchainement suivant: Accélération du piston; Insertion du clou dans les tôles; Rebonds du piston. Le mouvement du piston est ainsi décrit depuis le déclenchement du cycle jusqu'au retour à l'équilibre du piston. Les forces influentes sont également identifiées pour chaque étape. Il apparait ainsi que le clouage grande-vitesse est un procédé nécéssitant un paramétrage précis par l'opérateur afin d'éviter aussi bien un assemblage réalisé avec une énergie trop faible, due à une vitesse d'impact limitée, qu'excessive, ce qui induirait nécéssairement des oscillations du piston pour son retour à l'équilibre. La qualité d'un assemblage cloué peut être évaluée selon différentes approches: le suivi de la position du piston dans le temps; la vérification de la profondeur de pénétration du clou et du profil de rupture des tôles; les essais mécaniques. Le principe de contrôle par suivi du piston repose sur la comparaison avec une courbe de référence pour permettre la détection d'éventuels défauts. Cela permet l'évaluation automatique des assemblages même lorsqu'ils ne sont pas accessibles à l'inspection visuelle. Enfin, les essais mécaniques restent incontournables pour qualifier un assemblage dans les deux directions principales, à savoir en traction-croix et traction-cisaillement. Etant donné la distance caractéristique de contact entre le clou et les tôles, la tenue mécanique en traction-croix est généralement plus faible que celle en traction-cisaillement. Cela explique pourquoi le graphique "Tenue mécanique en traction-croix vs. Pression d'assemblage" est si apprécié des ingénieurs pour statuer, parmi différentes configurations, celle qui est la plus propice à une intégration dans une structure à échelle industrielle.

Figure 2 . 1 :

 21 Figure 2.1: Sheets clamping set-up design for controlled and instrumented nailing conditions

Figure 2 . 2 :Figure 2 . 3 :Figure 2 . 4 :

 222324 Figure 2.2: Camera set-up design for High-Speed recordings and Infra-red analysis

Gap∅Figure 2 . 5 :RMK 1 :

 251 Figure 2.5: Experimental set-up without sheets to be joined to test piston loading capability. RMK 1: Nail is pictured in red. RMK 2: Headpiece is pictured in blue. RMK 3: Figure (b) is a cross-section of Figure (a) joining area.

  Twenty-six nailing [3; 8 bars]

Figure 2 . 6 :Figure 2 . 7 :

 2627 Figure 2.6: Experimental nailing campaign without sheets in the joining location for the identification of piston displacement range at which joining occurs.

Figure 2 . 8 :

 28 Figure 2.8: Piston loading surface after impact: Joining force F eq = f (Impact velocity, Piston displacement)

Figure 2 . 9 :RMK 1 :

 291 Figure 2.9: Computation steps to obtain sheets reaction F reac. from piston motion in the joining stage.RMK 1: Example taken from {Steel/Cast alu.} superposition joined at P = 5.4 bars. RMK 2: Detailed calculation procedure of sheets reaction component F reac. , from (2.8), can be found in Appendix A.

Figure 2 .

 2 Figure 2.10b on {Steel/Steel} superposition, the larger is the hold-down pressure the larger will be the contact force. At P = 5 bars force is about 1.2 kN whereas it is only 0.4 kN at P = 1 bars. Similar tests are conducted on different materials superposition involving Steel, Cast aluminium, AHSS at constant hold-down pressure P = 1 bars with clamping rings of diameter d ∈ {28; 30; 34 mm}. Results are presented in Figure 2.11 where only the maximum force is plotted. It appears that materials superposition does not change too much the contact force because black markers are almost superimposed. However clamping diameter seems to affect the maximum force experienced by sheets with a bigger value with diameter d = 28 mm and a smaller value when d = 34 mm. It brings new information to the hold-down force calculation presented in Figure 1.3b in which force predicted at P = 5 bars is about 1.8 kN (α = 0) whereas it is measured 1.25 kN in Figure 2.11.

Figure 2 .Figure 2 . 11 :

 2211 Figure 2.10: Hold-down pressure evolution and Hold-down force measured in the preparation joining stage of {Steel/Steel} superposition tested at P ∈ {1; 2; 3; 4; 5 bars}

Figure 2 . 12 :

 212 Figure2.12: Main failure modes in impacted plates[START_REF] Zukas | Impact dynamics: theory and experiment[END_REF]

# 1 ( 3 (Figure 2 .Figure 2 . 14 : 4 (Figure 2 .Figure 2 . 16 :

 13221442216 Figure 2.13: High-speed camera records of {Steel/Steel} superposition joined with an H3 nail and ogival tip at P = 5, 6, 7 bars

9 (Figure 2 . 17 :Figure 2 . 18 :

 9217218 Figure 2.17: High-speed camera records of {Steel/Steel} superposition joined with an H5 nail and ogival tip at P = 5, 6, 7 bars

Figure 2 . 19 :Figure 2 . 20 :

 219220 Figure 2.19: High-speed camera records of {Steel/Steel} superposition joined with an H5 nail and flat tip at P = 5, 6, 7 bars

Figure 2 .Figure 2 . 22 :Figure 2 .Figure 2 . 24 :

 22222224 Figure 2.21: High-speed camera records of {Steel/Cast alu.} superposition joined at P = 5.4 bars with an H3 nail and ogival tip, an H3 nail and flat tip and an H5 nail and ogival tip

# 19 (

 19 a) t = ms (Impact) (b) t = 0.17 ms (Max. force) (c) t = 0.32 ms (Fragmentation)

Figure 2 . 27 :

 227 Figure 2.27: Summary of the sheets reaction to variations of nail parameters (H3 vs. H5 nail and Ogival vs. Flat nail-tip) in {Steel/Steel} superposition joined at 5, 6 and 7 bars

4 .

 4 Mixed petalling-shear plugging mode occurs for {Steel/Steel} superposition in which created plugs are small enough to get hole enlargment effect with also petals created from the cracks propagation in radial directions.The last remark which could be made about these pictures deals with the phenomenon observed in tests #7 Figure2.17b, #9 Figure2.17h and #15 Figure2.21h. There is a projection flow which results from impact with the sheets and happens only when an H5 nail is used. Besides differences of material properties between the two nails, H5 nail is covered with a Zinc-Aluminium based coating

Figure 2 .Figure 2 . 28 :

 2228 Figure 2.28: Summary of multi-materials sheets reaction to variations of nail parameters ({Steel/Steel} superposition joined at P = 7 bars ; {Steel/Cast alu.} superposition joined at P = 5.4 bars ; {Cast alu./Steel} superposition joined at P = 5 bars)

  to satisfy the condition that kinetic energy is fully dissipated into plastic work. This assumption is confirmed in {Steel/Steel} tests #1 Figure 2.13c, #4 Figure 2.15c, #7 Figure 2.17c, #10 Figure 2.19c and is also significant in {Aluminium/Steel} tests #16 Figure 2.23c and #18 Figure 2.23i where piston

Figure 2 . 29 :Figure 2 . 30 :

 229230 Figure 2.29: Surface apparent temperature T a distribution measured from below the {Steel/Steel} superposition with an IR camera at several frame rates

3

 3 bars and minimum at P = 5.0 bars. Tests performed with an excessive joining pressure P = 5.7 bars give an intermediate value between these two configurations. A deeper nail insertion does not benefit from a higher joining pressure. These experiments confirm that the sheets clamping force, beyond f ≥ 26 kN, does not affect force responses and maximum piston displacements out of the bonds observed in Figure 2.31.

Figure 2 . 31 :

 231 Figure 2.31: Sheet reaction in the joining stage of {Steel/Cast alu.} superposition for three different clamping force f and joining pressure P (Hold-down pressure is set to 1 bar)

Fifty-four nailing

  tests have been performed on multi-materials superpositions: {Steel/Cast alu.}, {Cast alu./Steel}, {AHSS/Cast alu.} and {Cast alu./AHSS}. Nail material is set to H3 and tip geometry to ogival. Two different clamping configurations were tested: diameter d = 28 mm and d = 34 mm. Joining pressure was gradually increased from configurations in which the nail is not fully inserted into the sheets to configurations in which joint cohesion has been destroyed by energy

  Figure 2.32: Force-Piston displacement curves of {Steel/Cast alu.} superposition with increasing joining pressure and clamping diameter d

Figure 2 .

 2 Figure 2.38: {AHSS/Cast alu.} superposition joined at P = 6 bars involving nail fracture after impact (test #37 ; Hold-down pressure is set to 1 bar)

  Figure 2.40: Force-Piston displacement curves of {Cast alu./AHSS} superposition with increasing joining pressure and clamping diameter d

Figure 2 .

 2 Figure 2.42: {Cast alu./AHSS} superposition joined at P = 6.3 bars (test #41)

Figure 2 .

 2 46b. Due to the tool geometry, the joint upper sheet is positioned into the lower jaw and therefore the joint lower sheet is positioned into the upper jaw, as presented in Figure2.46c. Before test starts, a hydraulic pressure system is triggered to close the jaws and clamp the sheets.

  Figure 2.45: Set-up configuration to test shear-tensile joint properties on INSTRON 5582 device with a maximum 100 kN force and constant velocity 10 mm/min

Figure 2 .

 2 Figure 2.47: Cross-tension curves of {Steel/Cast alu.} superposition with increasing joining pressure and clamping diameter d (seeTable 2.7)

Figure 2 . 51 :

 251 Figure 2.51: Joinability chart in {Steel/Cast alu.} (seeTable 2.7), {Cast alu./Steel} (see Table 2.8) and {Cast alu./AHSS} superposition (see Table 2.10) with clamping diameter d = 28 mm (small circle) and d = 34 mm (big circle)

Figure 2 . 52 :

 252 Figure 2.52: Cross-tension tests of {Cast alu./Steel} and {Steel/Cast alu.} at optimized joining pressure (respectively 5.3 bars and 5.7 bars)

Figure 2 .Figure 2 . 54 :

 2254 Figure 2.53: Nailed-joint before and after being mechanically tested in cross-tension configuration

  These two dimensions have been measured from seven tests in both {Steel/Cast alu.} and {Cast alu./Steel} superpositions and are summed up in Table2.14. b 1 -value is larger if aluminium is taken as upper layer than steel due to bigger global sheets bending. b 2 -value only provides information on the lower sheet relative bending from two points: one is contacting the nail and upper sheet area and the other one is 5 mm away from the nail axis. Again, aluminium layer has the biggest b 2 -value when it is considered as the lower sheet.

Figure 2 .

 2 Figure 2.55: Bending dimensions measured from joint cross-section after failure

Figure 2 .

 2 Figure 2.56: Shear-tensile curves of optimized joining pressure tests in {Steel/Cast alu.} (test #5 and #13), {Cast alu./Steel} (test #22 and #31) and {Cast alu./AHSS} superposition (test #42 and #51) with clamping diameter d = 28 mm (small circle) and d = 34 mm (big circle)

Figure 2 .

 2 57 presents both preserved and fracture joint. Compared to the cross-tension test, sheets do not undergo important deformation in sheartensile test. Everything happen in the nail surrounding area. That is why the main joint fracture mode, presented in Figure 2.58a, is related to nail shaft shearing, right under the head. This one is shared by 5 over 6 experiments discussed about Figure 2.56. The only one which benefits from an extended ductility is the one presented in Figure 2.58b and in the cross-section Figure 2.58d. In this case, an unusual fracture mode has been promoted: nail shaft is slowly teared out the nail head to reach almost horizontal position when the test is stopped. Despite a bigger displacement at fracture, this mode is rather a reason to be considered unstable than to become a reference for this test. In other words, investigation should focus rather on the fracture pattern experienced by the other five tests, presented in Figure 2.58c, than this later. This mode consists in the nail head separation from the shaft by an horizontal crack. Several details in Figure 2.58c indicate what sequence of events could be the cause of joint failure: (1) right part of the lower sheet perfectly matches nail shaft geometry. Lower sheet was obviously pushed against the nail shaft; (2) nail shaft is bent in the lower sheet tensile direction. Different stress-states should have appeared in both side of the shaft: tension on the right, compression on the left; (3) left part of the upper sheet also testifies that contact with the nail shaft has occurred. The crack has very likely grown in the tension dominant area and has propagated to the other side of the shaft.

Figure 2 .Figure 2 . 58 : 2 .

 22582 Figure 2.57: Nailed-joint before and after being mechanically tested in shear-tensile configuration

2. 7

 7 Summary of Chapter 2Experimental investigations have been conducted to understand(1) in which conditions a joint is processed,(2) what kind of phenomena is involved in the insertion stage and (3) what are the main features finally assessed to the joint. Two set-up have been specially designed for the following experiments: one for full sheet clamping conditions and one for camera access to the joining zone. At first, it has been necessary to clarify notions of joining force and hold-down force -two input parameters to set in the joining devicethrough experimental nailing campaign achieved with no consideration to the nail history. Only the headpiece position adjustment and piston kinematics have been considered here with no interest of the joining stage. Then insertion stage is investigated through high-speed camera recordings in various multi-materials superpositions, nail geometry and material, impact velocity ... This study is completed with Infra-Red camera measurements of the joining area in double steel sheet configuration.Then plenty of joints are processed in fully-clamped conditions to provide valuable data for the evaluation of joining process sensitivities and for further comparisons with simulation results. Sheets reaction curves to the nail insertion are investigated as well as fracture modes and joints cross-sections to understand what is the nailing mechanics and what are the limitations.

  Plusieurs campagnes expérimentales ont été menées pour comprendre (1) dans quelles conditions l'assemblage est réalisé, (2) quels sont les phénomènes physiques qui ont lieu pendant la phase d'assemblage et (3) quelles sont les propriétés finales de la liaison.Pour cela, deux montages ont été spécialement conçus et fabriqués: l'un pour assurer l'encastrement des tôles, l'autre pour permettre l'accès visuel des tôles à une caméra. Dans un premier temps, il s'est avéré nécéssaire de clarifier les notions de pression d'assemblage et pression de maintien -les deux paramètres de pilotage de la machine d'assemblage -par le biais d'essais réalisés sans l'utilisation de clou. Seules les actions portant sur la pièce de tête et le piston ont été considérés, sans attention portée sur la phase effective d'assemblage. La phase de pénétration du clou dans les tôles a ensuite été étudiée dans le cadre d'essais réalisés avec une caméra rapide pour différentes superpositions de matériaux, géométries de clou ou matériaux de clou, vitesses d'impact ... Cette étude a été complétée par des mesures thermiques de la zone d'assemblage effectuées sur une superposition de deux tôles d'aciers.Un grand nombre d'essais ont ensuite été réalisés dans les conditions de tôles encastrées afin d'évaluer les différentes sensibilités du procédé et permettre par la suite la comparaison avec les résultats issus de la simulation numérique. Les courbes de force de réaction des tôles à l'assemblage ainsi que les modes de ruptures et les micro-coupes des points assemblés ont été étudiés pour comprendre quels sont les phénomènes principaux agissant pendant la phase d'insertion du clou dans les tôles et quelles sont les actuelles limitations du procédé.Des essais mécaniques ont ensuite été réalisés sur des éprouvettes multi-matériaux en traction-croix et traction-cisaillement. Sur la base des courbes force-déplacement et des faciès de rupture, des explications sur les principaux mécanismes de cohésion et de rupture sont proposées.

•

  Φ(σ) = 0 : plastic state • Φ(σ) > 0 : impossible Various yield functions can be found into the literature and applied to metals but the main ones can be limited to four. If we assume that stress tensor σ takes the form of (3.11), Tresca, Von-Mises, Hill and Gurson-Tvergaard-Needleman (GTN) yield functions are respectively expressed in Table3.1 as (3.12), (3.13), (3.14) and (3.15):

Figure 3 . 1 :

 31 Figure 3.1: Tresca and Von-Mises yield functions in the plane-stress with isotropic hardening

  2). Strain-hardening term has evolved from the traditional Ludwig formulation (3.18) to the Norton-Hoff (3.21) and Swift ones (3.29). Jeunechamps (2008) and Dehra (2006) have modified the power law form in (3.24) and (3.27) for high-speed applications. Grytten et al. (2009) applied a Voce formulation (3.31) with five coefficients to impact and perforation simulations and get a better curve fitting agreement with increasing plastic strain. Strain-rate hardening is generally modelled with a power law (3.19) and transformed for conveniency into a dimensionless parameter (3.22). This formulation has been extended (3.25) and used in terminal ballistic applications in the works of Dey et al. (2007a) and Yadav et al. (2001). A modified version of this equation, from the works of Cowper and Symonds (1957) (3.28) is also widely used in the community of impact simulations. However the most popular formulation is the one from Johnson and Cook (1983) (JC) since only one parameter has to be calibrated (3.30) and it fits well a wide range of materials behavior. It has been extended to a Huh-Kang quadradic form (3.32) and compared with other formulations in the works of Scholl et al. (2018). It is generally assumed that thermal increase softens the material yield stress and can be modelled with a decreasing function of the temperature. The linear formulation (3.20) is rarely used and in most cases a Arrhenius form (3.23) is preferred. However the Johnson-Cook formulation (3.26) remains dominant and is used in almost every dynamic applications (Dey et al., 2007b).

3 )

 3 based on the motion of dislocations which is different between BCC (3.33) and FCC crystallographic structures(3.34). It has not been too much used for ballistic applications even if it is considered to give a better prediction of high strain rates material's behavior.Dey et al. (2007a) compared the ZA model to the JC constitutive model in the case of a steel plate perforation by a blunt-nose projectile. It was concluded that JC relation predicts considerably better both the ballistic limit velocity and the crack path to fracture than the ZA constitutive model. These differences are supposed to result from a bigger strain-rate softening and thermal softening in the JC formulation than the ZA. However the same reason explains why JC-model was considered particularly unadapted to represent specific material behavior such as the blue brittleness steel effect at high temperature(Børvik et al., 2003a). Despite the physical ground of the ZA model, it remains largely unused in the industry due to the amount of experiments required for calibration: 5 parameters coupling strain, strain-rate and temperature sensitivities with some microstructural informations required for the ZA-model vs. 5 separated parameters without coupling sensitivities for the JC-model. Several other formulations have been proposed to account for material behavior under very high strain-

Figure 3 .Figure 3 . 2 :

 332 Figure 3.2 with and without accounting for damage under tensile test.Three main approaches can be found in the literature to model damage accumulation and fracture:

2 εpl -εN s N 2 εpl ( 3

 223 .38) with εN the plastic strain at maximum nucleation, s N the standard deviation of the gaussian distribution and Φ N the total volume of voids that can be nucleated. Due to its microscopical basis, this approach can predict accurately damage occurence at the macro-scale and has been adapted for dynamic applications by Molinari et al. (2014), blanking applications by Nahshon and Xue (2009) and extended for shear failure in Nahshon and Hutchinson (2008) and even used under the Rousselier for-mulation to clinch-joining by Zhao et al. (

  40) with S and b which are material parameters. In addition to a condition introduced by Wierzbicki et al. (2005) in which a cut-off value is assumed to avoid damage growth under dominant compression stress-state, an other condition, based on the minimum value of effective plastic strain before damage can occur, is introduced in (3.40). Under the formulation proposed in the works of Cao et al. (2015a) and written in (3.41), the effective plastic strain threshold εD can variate with stress-triaxiality as: εD = εD0 exp(-kσ * ) (3.41) where εD0 and k are material parameters. This model requires the identification of 4 parameters (ε D0 , k, S, b) which can be reduced to at least 2 parameters (ε D0 , S) -since damage accumulation threshold can be considered constant with k = 0 and exponent b is usually taken as b = 1 for alumimium alloys and b = 2 for steels, as detailed in Lemaitre and Chaboche (1994).

Figure 3 . 3 :

 33 Figure 3.3: Modified strain threshold formulation εD at which damage accumulation begins

  and Chaboche (1994) have found from tension and compression tests that parameter h equals to 0.2 is assumed valid for most materials. According to this weakening function, the effective stress tensor under compression is reformulated as: σij = σ ij /(1 -hD).The weakening exponent β is a calibration parameter -in the original Lemaitre formulation β = 1 -which enables to extend damage effect on mechanical behavior under a more general formulation. A proper damage model includes the coupling effect presented in (3.45), the associated weakening function w(D) in (3.47) and a damage evolution law. This last one is expressed by equation (3.40) for the Lemaitre formulation (LEM) and by equation (3.44) for the phenomenologically-based one (PB).

  [START_REF] Bao | On fracture locus in the equivalent strain and stress triaxiality space[END_REF] from compression tests on metal cylinders that fracture does not occur whenσ * < -1/3.Cockroft and Latham (1968) have proposed a criterion (CL) in which ductility is driven by the first principal stress component σ 1 and limited by the critical plastic work at fracture ϕ 3 (3.61). Similar criteria have been developed after considering the equivalent stress (3.62) or the shear stress components (3.63) instead of the maximal principal stress one.

ϕ 3 = (σ 1 -

 31 σ 3 ) dε pl (3.63) This criterion has been applied by Gruben et al. (2012) under a more general formulation (3.64), called Extended Cockroft-Latham criterion (ECL), with parameter ϕ 3 controlling the critical plastic work at fracture, ϕ 1 the relative influence of the major principal stress and the maximum shear stress, and ϕ 2 the amplitude of the stress-state dependency. Parameters ranges are 0 ≤ ϕ 1 ≤ 1, ϕ 2 ≥ 0 and ϕ 3 ≥ 0.

  (1985) proposed a fracture strain criterion (3.52) built as the so-called constitutive law (JC). It is inspired from the works of[START_REF] Rice | On the ductile enlargement of voids in triaxial stress fields[END_REF] for the hydrostatic-stress dependency and from the constitutive law for strain-rate and thermal ones. Chocron et al. (2011) and also Erice and Gálvez (2014) used on dynamic applications an extended version of the JC-criterion (3.53) to include the Lode angle dependency -using the second kind Lode function originally formulated in the works of Xue and Wierzbicki (2008). Several authors have therefore taken advantages of the exponential-function capability to state for decreasing fracture strain with the stress-triaxiality. Xue and Wierzbicki (2008) have presented a criterion (XW) with five parameters which associates two exponential functions to build the fracture locus with Lode angle and stress-triaxiality dependencies (3.54). Another version proposed by Bai and Wierzbicki (2008) (BW) consists in three exponential-curves (3.66) related inside a Lode-angle based function (3.65) which enables the construction of an highly asymmetrical fracture locus (BW6P), regarding the sensitivity to the Lode parameter. Assuming ϕ 5 = ϕ 1 and ϕ 6 = ϕ 2 the BW-criterion is changed into a symmetrical fracture locus (BW4P) and even made independent of the Lode parameter if ϕ 5 = ϕ 3 = ϕ 1 and ϕ 6

  interest used to model sheet spallation under impact (Teng and Wierzbicki, 2006) is the one developed by Wilkins et al. (1980) (WKS) and expressed under the fracture strain form in (3.68).

  5 (3.58) but compared to other criteria, it does not intrinsically account for a cut-off region. Finally, a closed form of the Gurson fracture model is used to built two criteria: the one from the original Gurson formulation (3.59) (GUR) and a modified formulation from Nahshon and Hutchinson (2008) (NH) to account for shear failure (3.60). The GUR model can be obtain from the NH considering ϕ w = 0 The Lemaitre damage model can be transformed into a fracture locus for a convenient comparison between the different criteria. Assuming a proportional loading path, it can be expressed as a fracture locus (3.70). εf = εD + 1 -(1 -hD c ) 2b+1 h(2b + 1) Application to joining processes Numerical simulations of joining processes have proven that sheet materials reaction to the fastener insertion as well as in-service joint strength can be accurately predicted if an appropriate set of material constitutive and damage laws is used. Several modelling strategy can be found throughout articles of the literature and applied to joining issue. Porcaro et al. (2006) and Carandente et al. (2016) have simulated the insertion of a self-piercing rivet into a double aluminium-layers configuration with good agreement on the force-displacement tool history (Figure 3.4). In both configurations the upper-sheet failure was modeled through a geometrical-based criterion which triggers mesh deletion when a given distortion value is reached.

Figure 3 . 4 :

 34 Figure 3.4: Comparison between simulation and experiment on rivet insertion stage with (a) Numerical results; (b) cross-section and (c) force-displacement curve (Porcaro et al., 2006)

Figure 3 . 5 :Figure 3 . 6 :

 3536 Figure 3.5: Rivet joint strength models compared to experiments in terms of (a) joint fracture mode and (b) force-displacement curve (Fayolle, 2009).(Notations: red curve is the experiment; blue curve is the numerical curve with coupled-damage model; green curve is the numerical curve with uncoupled-damage model.)

Figure 4 . 1 :

 41 Figure 4.1: Sheet metal specimens for tensile tests (UT0, NT7, NT2, SH) and compression tests (C0)

Figure 4 . 2 :

 42 Figure 4.2: Force-Displacement curves from tensile tests performed on steel (DP780) and AHSS (DP1180) sheets at different angle to the rolling direction

  Cast alu.

Figure 4 . 3 :

 43 Figure 4.3: Force-Displacement curves from tensile tests performed on steel (DP780), AHSS (DP1180) and cast aluminium (AlSi10MnMgT6) specimens with uniaxial (UT0), notched 7 mm-radius (NT7), notched 2 mm-radius (NT2) and in-plane shear (SH) geometries.

Figure 4 . 4 :

 44 Figure 4.4: Quasi-static tensile test set-up including an optical measurement device for UT0, NT7, NT2 and SH sheet metal specimens. Virtual extensometers and sensors positions are indicated by red and yellow dots and used in the post-processing step to extract local strain field.

Figure 4 . 5 presentsFigure 4 . 5 :

 4545 Figure 4.5: Evolution of the strain component ε 12 for a given substep size obtained from DIC computation process on three sensors s 11 , s 0 , s 12 in the shear localization zone of a cast alu. SH specimen.

4. 2 . 1 . 5 Figure 4 . 6 :Figure 4 . 7 :

 2154647 Figure 4.6: Quasi-static compression set-up for C0 sheet metal specimen with lubricant located on both upper and lower die (lubricant has been removed on the present picture to highlight specimen size and position inside the dies).

Figure 4 .

 4 8a presents the Hopkinson bar set-up in which DP780 steel has been tested, Figure 4.8b reveals how sample contour is tracked during the test and Figure 4.8c shows stress-strain curves obtained at five different strain-rates.

Figure 4 . 8 :

 48 Figure 4.8: Hopkinson compression tests conducted on steel (DP780) C0-specimens at high strain-rates

Figure 4 . 9 :

 49 Figure 4.9: Quasistatic compression tests performed on the nail shaft cylinder.Notations: h i is the initial shaft height and h f the final/compressed shaft height.

Figure 4 .Figure 4 .

 44 Figure 4.10: UT0 model for tensile test on steel (DP780) in the undeformed and deformed state in (a) front view and (b) cross-section view (Colorscale is based on LEM damage amplitude).

Figure 4 .

 4 Figure 4.12: NT2 model used for tensile test on steel (DP780) in the undeformed and deformed state in (a) front view and (b) cross-section view (Colorscale is based on LEM damage amplitude).

Figure 4 .

 4 Figure 4.13: SH model used for in-plane shear test on steel (DP780) in the undeformed and deformed state with (a) the global mesh and (b) local mesh of the shearing zone (Colorscale is based on LEM damage amplitude).

Figure 4 . 15 :Figure 4 . 16 :

 415416 Figure 4.15: Quasistatic compression tests performed on the nail shaft cylinder specimen expressed in term of experimental and numerical (a) force-displacement curves and (b) nail shape contour. Notations: h i is the initial shaft height and h f the final/compressed shaft height.

4. 3 . 4 . 1

 341 Lemaitre coupled approach (LEM) The Lemaitre damage formulation was previously introduced in Chapter 3 through equations (3.39), (3.40), (3.42) with a softening material response modelled by a weakening function expressed in (3.45) and (3.47). The parameters of this law are numerous and have to be calibrated according to experimental results. For conveniency the set of reference tensile tests used for calibration is different for steel, cast aluminium and AHSS. UT0 and NT2 curves have been necessary to calibrate parameters of steel damage model whereas cast aluminium required only the NT2 curve and AHSS the NT7 and NT2 curves. The parameters to calibrate are S, b (generally taken as b = 2 for steel and b = 1 for aluminium

Figure 4 .

 4 Figure 4.17c).

  123

Figure 4 . 17 :

 417 Figure 4.17: Stress-triaxiality (a), Lode parameter (b) evolution and mean stress-state (c) obtained via tensile test simulations on steel (DP780) and extracted from sensors located in the specimens' strain localization area.RMK 1 : For (a) and (b), the sensors located at the specimen surface are pictured with continuous lines and inside the core thickness with discontinuous lines. RMK 2 : For (a) and (b), the average stress-state values are calculated and plot on the graph close to the associated curves. RMK 3 : In (c), plane stress locus is drawn with a discontinuous purple line. RMK 4 : Notations for sensors position associated to symbols on the graph (c) are: big symbol for sensor in the core thickness; small symbol for sensor at the surface; square for sensor located at the center; circle for sensor located at the corner/border. RMK 5 : Simulations are conducted with the Lemaitre formulation properly calibrated (see Figure 4.16a).

Figure 4 .

 4 19a would assign to the material a totally different fracture feature than the BW6P asymmetrical locus plot in Figure 4.19b. All fracture locus surfaces are presented for comparison in Figure B.5 for steel, in Figure B.6 for cast aluminium and in Figure B.7 for AHSS. Parameters properly calibrated have been gathered in Table

Figure 4 .Figure 4 .

 44 Figure 4.18: (a) BW4P and (b) BW6P fracture loci calibrated for steel (DP780)

4. 3 . 4 . 3

 343 Phenomenologically-based coupled approach (PB)The Phenomenologically-based damage formulation was introduced in Chapter 3 as well as their governing equations (3.44), (3.45) and (3.47) to account for material softening due to damage. For the same reasons than Lemaitre damage model calibration, inverse analysis have been conducted on tensile test simulations to find the set of parameters which gives the best agreement with the experimental force-displacement curves. The BW4P fracture strain formulation has been used for the computation of damage evolution and the parameters targeted for calibration are c, β and εDi . Details of the optimization procedure are given in Table

Figure 4 . 20 :

 420 Figure 4.20: Comparisons between steel material models using PB formulation (DP780) and experiment on tensile test conducted on (a) UT0, (b) NT7, (c) NT2 and (d) SH specimens.

  l'évaluation des sensibilités de la contrainte d'écoulement à la déformation plastique, la vitesse de déformation et la température; (2) de générer des états de chargements différents au sein des éprouvettes menant à la rupture. De ce fait, des essais de traction ont été réalisés en conditions quasi-statiques sur des éprouvettes uniaxiales pour évaluer l'anisotropie de la contrainte d'écoulement par rapport au sens de laminage des tôles et le phénomène d'écrouissage. Des essais de compression ont aussi été réalisés à plusieurs vitesses de déformations, quasi-statiques et hopkinsons pour évaluer l'évolution de la contrainte d'écoulement avec la vitesse de déformation. Des éprouvettes entaillées et de cisaillement type "butterfly" ont été testées avec en parallèle un système de stéréo-corrélation d'images mesurant localement les déformations en surface des éprouvettes pour étudier les conditions de rupture sur une large gamme d'états de contraintes. Le clou a quant à lui été caractérisé par compression quasistatique. Tout ces essais ont ensuite été modélisés dans le logiciel éléments finis FORGE ® et couplés avec le module d'optimisation MOOPI pour la procédure de calibration des lois matériaux. Les lois de comportements sont calibrées sur la base des essais de traction pour la dépendance à l'écrouissage et des essais de compression Hopkinson pour celle au durcissement dynamique et à l'adoucissement thermique. Le modèle de Lemaitre est ensuite calibré en utilisant plusieurs simulations de traction -uniaxiale, entaillée et cisaillement -de manière à couvrir une large gamme d'état de contrainte à rupture. Les critères de rupture présentés au Chapitre 3 sont ensuite calibrés par le biais d'une routine Matlab ® et des déformations à rupture obtenues experimentalement. Ces critères ou loci sont ensuite intégrés à la procédure de calibration du second modèle d'endommagement selectionné ici: l'approche phénoménologique couplée. Il s'avère que les deux modèles donnent l'un l'autre d'aussi bon résultats, en accord avec les courbes expérimentales force-déplacement, même si un écart persiste pour l'aluminium de fonderie dont la très faible ductilité implique des complications pour la calibration des modèles sans pouvoir à l'heure actuelle proposer des améliorations concrètes.

Figure 5 . 1 :RMK 1 : 2 : 3 :

 51123 Figure 5.1: Finite element model of the nail insertion stage built with FORGE ® on a 10°circular slice. (a) details model's meshes, loading and boundary conditions. (b) and (c) present sensors location to track and measure characteristic joint dimensions. RMK 1: Example taken in a configuration of {Steel/Cast alu.} superposition. RMK 2: For conveniency, the geometry has been extended on the picture from 10°to 180°. RMK 3: Red dots are sensors assessed to particular nodes of the mesh. Position history of sensor s 1 and s 3 are used respectively to compute piston displacement z and evaluate the sheet bending distance y during the joining stage. The relative distance between sensor s 2 and s 3 computes the effective nail penetration h and the one between sensor s 4 and s 5 , the nail-tip crushing reduction distance n 1 .

  Material constitutive and damage laws have been properly defined and calibrated on steel, cast aluminium, AHSS and nail shaft materials. Results are detailed in Chapter 4 and Appendix B. Due to

Observation 5 . 1 : 1 .Figure 5 . 2 :RMK 1 :

 511521 Figure 5.2: Comparisons of joining piston kinematics obtained from experiments and simulations in {Steel/Cast alu.} configuration (a) (c) (e) and {Cast alu./Steel} configuration (b) (d) (f ). RMK 1: Piston displacement is extracted from sensor s 1 location history (see Figure 5.1) and post-treated to compute piston velocity and force (see Appendix A). RMK 2: Both simulations are computed with LEM material model. RMK 3: Events of the joining stage indicated by a number {1;2;3;4} are detailed in Figure 5.3. RMK 4: Error-bars plot on the experimental curves have been established from the dispersion measured on every material superposition tested in Experimental campaign n°3.

Figure 5 . 3 :

 53 Figure 5.3: Details and distribution of the piston displacement into the three main deformation modes obtained in the joining stage simulation of {Steel/Cast alu.} superposition (a) (c) and {Cast alu./Steel} superposition (b) (d). RMK: Deformation modes are described through variables introduced in Figure 5.1: z exp : Piston displacement (Experiment) z sim : Piston displacement (Simulation) h: Effective nail penetration (Simulation) y: Sheet bending distance (Simulation) n 1 : Nail-tip crushing reduction distance (Simulation)

Figure 5 . 4 :

 54 Figure 5.4: Evolution of the nail penetration state and strain-rate variable εpl in the joining stage simulations computed with LEM material models on {Steel/Cast alu.} (Left side of the axis) and {Cast alu./Steel} (Right side of the axis) configurations.

Figure 5 . 5 :

 55 Figure 5.5: Stress-states (a...d) and average values (e) during the joining stage of sensors located in upper and lower sheet fracture zone and nail-sheet interface zone for {Steel/Cast alu.} superposition (a) (c) and {Cast alu./Steel} superposition (b) (d) computed with LEM material models.

Observation 5 . 2 :

 52 Sheets fracture modes (5) 1. Both {Cast alu./Steel} and {Steel/Cast alu.} fracture modes discussed in Observation 2.5.2 and Observation 2.5.3 are well captured by joining simulations.

Figure 5 . 6 :

 56 Figure 5.6: Comparison of thermo-mechanical scalar variables at the end of joining simulations conducted on mixed cast aluminium-steel configurations: effective plastic strain εpl (a), LEM damage D (b), adiabatic temperature T ad and calculated temperature T (c) (d).

Figure 5 .6c and Figure 5 .

 55 6d respectively show temperature field for {Steel/Cast alu.} and {Cast alu./Steel} configurations computed with two different methods: T ad is calculated considering adiabaticity and therefore

Figure 5 . 7 :

 57 Figure 5.7: Mesh size in the sheets bending zone for {Steel/Cast alu.} superposition set to (a) 0.75 mm, (b) 0.5 mm, (c) 0.3 mm and (d) 0.2 mm.RMK: Mesh size in the impact area is kept constant to 0.1 mm and 0.2 mm in the surrounding.

Figure 5 . 8 :

 58 Figure 5.8: Effect of mesh-size reduction in the bending zone (Figure 5.7) on the joining piston kinematics (a)(b) and sheet bending evolution during the penetration stage (c)(d) in simulations run with LEM material models on {Steel/Cast alu.} (Left) and {Cast alu./Steel} (Right) configurations.

  Figure 5.9d); (3) the nail-steel interface is set to {m = 1 ; µ = 0} (see Figure 5.9e and Figure 5.9f); (4) the nail-cast aluminium interface is set to {m = 1 ; µ = 0} (see Figure 5.9g and Figure 5.9h).

Figure 5 .Figure 5 . 10 : 1 . 5 . 4 : 1 . 2 .Figure 5 . 11 :

 551015412511 Figure 5.10: Comparisons of joining piston kinematics obtained from simulations run with LEM material models and different friction conditions at the nail-steel interface in {Steel/Cast alu.} (Left) and {Cast alu./Steel} (Right) configurations. RMK: Friction conditions at the interface are the following: {m = 0 ; µ = 0} at nail-cast alu. interface. {m = 0 ; µ = 0} at steel-cast alu. interface. {m ; µ} at nail-steel interface.

Figure 5 . 12 :

 512 Figure 5.12: Evolution of the nail penetration distance h with an increasing joining pressure from data presented in Figure5.11 on {Steel/Cast alu.} and {Cast alu./Steel} joining simulations.

1 . 2 .

 12 Joining simulations capture the tendency (see Observation 2.8) of nail insertion depth to be proportional to the joining pressure. Although a gap remains between the optimized joining pressure predicted by simulation and given by experiments -5.0 bars vs 5.3 bars in {Steel/Cast alu.} and 4.4 bars vs 5.0 bars in {Cast alu./Steel} -joining simulations can be run at different joining pressure and the one which verifies the following conditions would be considered the optimum:

Figure 5 .

 5 Figure 5.13. Contrary to results obtained from experiments, simulations do not predict a bigger force amplitude if clamping diameter is set to d = 28 mm than if it is set to d = 34 mm. These results

Figure 5 . 13 :

 513 Figure 5.13: Comparisons between joining piston kinematics of (a) {Steel/Cast alu.} and (b) {Cast alu./Steel} material layers obtained from experiments and simulations with clamping diameter d = 28 mm (small circle) and d = 34 mm (big circle).

5. 2 . 6 . 1 Figure 5 . 14 :

 261514 Figure 5.14: Comparisons between simulations run with seven damage formulations and experiments on the piston kinematics of {Steel/Cast alu.} (Left) and {Cast alu./Steel} (Right) superpositions.

Figure 5 .Figure 5 . 15 :

 5515 Figure 5.15: Average stress-state experienced by sensors located in the (a) upper sheet fracture area and (b) lower sheet fracture area of {Steel/Cast alu.} and {Cast alu./Steel} configurations.

  Figure B.5 and Figure B.6). These values are summed up in Table 5.1 for the six tested fracture criteria. Despite obvious differences in the joining kinematics of {Cast alu./Steel} simulation (see Figure 5.14d), no significant gap can be noticed between BW6P associated steel fracture strain εf (σ * avg , µ σ.avg ) = 0.225 and εf (σ * avg , µ σ.avg ) = 0.243 the CL's. Furthermore no significant difference can be noticed between joining kinematics of {Steel/Cast alu.} simulation computed with BW6P and BW4P criteria even though their respective fracture strain values are very different: εf (σ * avg , µ σ.avg ) = 0.273 (steel) and εf (σ * avg , µ σ.avg ) = 0.109 (cast aluminium) for the BW6P criterion and εf (σ * avg , µ σ.avg ) = 0.442 (steel) and εf (σ * avg , µ σ.avg ) = 0.158 (cast aluminium) for the BW4P.

5. 2 Figure 5 . 16 :

 2516 Figure 5.16: Effect of upper sheet and lower sheet critical damage parameter on the nail penetration distance h (see Figure 5.1) on {Steel/Cast alu.} (a) and {Cast alu./Steel} (b) joining configuration computed with LEM material models.

ure 5 .Figure 5 . 17 :

 5517 Figure 5.17: Comparisons between simulations run with LEM material models and experiments on the piston kinematics of (a) {Steel/Steel}, (b) {Steel/Cast alu.}, (c) {Cast alu./Steel} and (d) {Cast alu./Cast alu.} superposition.

3 :

 3 Figure 5.18: Comparisons between the nailed-joint cross-sections obtained by experiments and simulations on four materials superposition to measure characteristic joint dimensions (details on the notations in Figure 1.7 and the conditions of experimental nailing in Table 2.11).

Figure 5 . 19 :

 519 Figure 5.19: Comparison between experiment and simulation run on {Cast alu./AHSS} nailed-joint in terms of (a) cross-section and (b) piston kinematics (test #41 in Table2.10).

Figure 5 . 20 :

 520 Figure 5.20: Evolution of the nail deformation state and stress-triaxiality σ * in the joining stage simulation of {AHSS/Cast alu.} configuration computed with LEM material models.

Figure 5 . 21 :

 521 Figure 5.21: Example of the application of a CL fracture criterion applied to an {AHSS/Cast alu.} nailing simulation at (a) impact and (b) fracture time and (c) the associated fracture modes.

Figure 5 .

 5 Figure 5.22: Mesh transformation procedure to generate from the result of high-speed nailing simulation (a) the appropriate mesh for the sheets relaxation simulation and from the result of relaxation simulation, the nailed-joint meshes associated to (b) cross-tension and (c) shear-tension simulations.

Figure 5 .RMK 1 :RMK 1 : 3 :

 5113 Figure 5.23: Finite element model of the post-nailing relaxation stage built with FORGE ® on a 10°slice. RMK 1: Example taken in a configuration of {Steel/Cast alu.} superposition. RMK 2: Mesh is preserved as given by the last increment of joining simulation.

Figure 5 .Figure 5 .Figure 5 .Figure 5 . 28 :

 555528 Figure 5.26: Nailed-joint relaxation stage in the {Steel/Cast alu.} configuration (a) at initial time, (b) final time and (c) at three locations of the nail-lower sheet interface.

Figure 5 .

 5 Figure 5.28 must be taken into consideration. Discontinuous purple and light blue curves are respectively assigned to {Steel/Cast alu.} and {Cast alu./Steel} joint response obtained from simulation.

Figure 5 .Figure 5 .RMK 1 : 1 0. 5 s; 2 inc - 1 1.0 s; 0 inc - 1 1.0 s; 20 inc - 1 1.0 s; 2 inc - 1 3.0 s; 0 inc - 1 3.0 s; 20 inc - 1 3Figure 5 . 34 :RMK 1 :

 551151111115341 Figure 5.30: Comparisons between experimental and numerical cross-tension joint cross-section towards Plan A in (a) {Steel/Cast alu.} and (b) {Cast alu./Steel} configurations. RMK: Numerical cross-sections taken at the die displacement D p = 4.7 mm.

Figure 5 . 35 :

 535 Figure 5.35: Evolution of numerical joint response for {Steel/Cast alu.} and {Cast alu./Steel} configuration with the friction parameters (m; µ) changed at the nail-lower sheet interface. RMK: Coulomb parameter µ is set according to Tresca parameter m such as: µ = 1/2 m

5. 3 . 5

 35 Shear-tensionSimulation are run in the conditions detailed in section 5.3.1 on 1/2 of the real shear-tension joint geometry. One symmetry plane is used to reduce computation time. The Figure5.36 presents mechanical response of the joint in {Steel/Cast alu.} and {Cast alu./Steel} superpositions.Despite the fact that simulations could not be run as far as in the experiments, the global curve trend has been caught. Each simulation follows its own trend according to the lower sheet stiffness which constraint nail rotation as discussed in Observation 2.24. The {Cast alu./Steel} superposition maximum force goes beyond the values experienced by {Steel/Cast alu.} joint and confirms the validity of the shear-tension simulation. However the computation cost to run such simulations is substantial even though they are stopped well before strain localization occurs inside the nail shaft.The experimental and numerical joint cross-sections are compared in Figure5.37, respectively at fracture (D p > 3 mm) and at the last increment of simulation (D p ≈ 1.8 mm). The two zones which have been plastically deformed, in both sides of the nail by its rotation (Observation 2.23), are well captured by simulation. The nail shaft upper part is constraint by the large head which lays on the sheet whereas the shaft lower part is brought leftwards. This phenomenon appears on Figure5.37b as the starting point of the nail head tearing mechanism. Yellow contours (σ ≈ 1300 MPa) show good agreement on the nail-head stress localization area where fracture occurs in experiment. But stress amplitude at this location (σ ≈ 1000 MPa) is lower in the {Steel/Cast alu.} configuration (Figure5.37a) than in the {Cast alu./Steel}. Nail rotation is less constrained by a cast aluminium layer than a steel one, which preluded crack occurrence inside the nail shaft as noticed in the experiment (Observation 2.23.2).

Figure 5 . 36 :Figure 5 . 37 :

 536537 Figure 5.36: Comparison between experimental and numerical shear-tension joint response in {Steel/Cast alu.} and {Cast alu./Steel} configurations.

1. 3 1 . 4

 314 An over-estimated joining pressure leads to an excessive sheet-materials deformation in the area surrounding the nail shaft. Between those states exists an optimum joining pressure which enables full nail penetration and avoid decohesion in the sheets-nail shaft contact area.

2. 0 3 . 1

 031 Joining tests conducted with increasing joining pressure result in increasing piston displacement maximum values, i.e. increasing nail insertion depths (see Nail insertion depth 1.1). Piston displacement z is distributed into three main deformation modes: nail-tip reduction n 1 , sheet bending y and the effective nail insertion into the sheets h.

3. 2

 2 When sheets perforation stage ends (marker 3 in Figure5.3) the relative displacements are: {Steel/Cast alu.}: n 1 = 4.1% z ; h = 12.2% z ; h = 87.5% z {Cast alu./Steel}: n 1 = 0.8% z ; h = 7.8% z ; h = 92% z 4.1 Joining simulations capture the tendency (see Nail insertion depth 2.0) of nail insertion depth to be proportional to the joining pressure.

4. 2

 2 Although a gap remains between the optimized joining pressure predicted by simulation and given by experiments -5.0 bars vs 5.3 bars in {Steel/Cast alu.} and 4.4 bars vs 5.0 bars in {Cast alu./Steel} -joining simulations can be run at different joining pressure and the one which verifies the following conditions would be considered the optimum:

5. 0 5 . 2

 052 According to joining simulations applied to cast aluminium-steel configurations, two factors of the damage computation method affect joining piston kinematics and therefore nail insertion depth: 5.1 Damage formulation controls the damage growth. Thus one criterion can give better result than another. Critical damage parameter controls ductility. It affects directly the amount of plastic work before fracture occurs and therefore the nail penetration depth.

2. 0 1 . 2

 012 Sheets subjected to impact can exhibit six different main fracture modes. It results from geometrical, material or velocity-based parameters which control the projectile-target interactions. Sheets fracture mode 1.1 Superposition made of steel and/or aluminium layers fail mainly under three modes: petalling, shear plugging or mixed petalling-shear plugging. Dominant petalling mode occurs for {Cast alu./Steel} superposition, i.e. when nail ogival shape is preserved and steel sheet fails by ductile hole enlargment.

3. 0 3 . 1

 031 About {Cast alu./Steel} superposition: As noticed for upper steel made sheet in Sheets fracture mode 2.1, the nail-aluminium interface is sheared all along nail descent: sheared pieces are found inside the ridges and on the ogival tip.

3. 2

 2 A steel plug has been formed but the radial cracks triggered petalling sheet failure before the plug separation could happen.

3. 3 4 . 1

 341 No shear-plug has been formed from the aluminium sheet perforation.4.0 About {Cast alu./AHSS} superposition: Neither in the {Cast alu./Steel} superposition (see Sheets fracture mode 3.3) nor in this superposition an aluminium plug is formed when it lies as the upper layer.

4. 2 5 . 1

 251 AHSS plug shape indicates rather a dominant shear-plugging mode than a mixed petalling-shear plugging mode experienced by steel layer {Cast alu./Steel} superposition (see Sheets fracture mode 3.2). Both {Cast alu./Steel} and {Steel/Cast alu.} fracture modes discussed in Sheets fracture mode 1.2 and Sheets fracture mode 1.3 are well captured by joining simulations.

3. 1

 1 From the consideration of Sheets temperature increase 1.2, good agreement is obtained on maximum temperature computed in {Cast alu./Steel} simulations -T max = 435°C -and {Steel/Cast alu.} simulations -T max = 454°C.

Figure B. 2 :Figure B. 3 :Figure B. 4 :Figure B. 5 :Figure B. 6 :Figure B. 7 :Figure B. 8 :Figure B. 9 :Figure B. 10 :Figure C. 1 :

 23456789101 Figure B.2: (a) BW4P and (b) BW6P fracture loci calibrated for steel (DP780)

For a 3

 3 tetrahedron, the formulation of the quality criterion can be based on different dimensions: volume, area of triangle facet, edge length. Parthasarathy et al. (1994) proposes a comparisons of tetrahedron quality measures. Klingner and Shewchuk (2008) also mentioned some formulations which have been widely used in literature. An interesting formulation uses a ratio of the volume with respect to the sum of the facet area (C.2).where V h the tetrahedron volume calculated asV h = 1/6 | det( -→ IJ, -→ IK, -→ IL)| with I, J, K, L, the four nodes of the tetrahedron, A i is the area of facet i and ψ = 4.58457.10 -4 the normalizing coefficient.Another formulation using the tetrahedron volume is mentioned in Lo (1997) (C.3). and l hi the length of edge i. A variation of the Q 2 -criterion is discussed in[START_REF] Freitag | Tetrahedral mesh improvement via optimization of the element condition number[END_REF] and takes the form (C.4). ψ = 8.47967. A really similar formulation as the one presented for the triangle is extended to tetrahedron (C.5).

9 Area facet 1 P 5 ;facet 2 P ( 1 ) 5 ;facet 3 P ( 1 ) 5 ;facet 4 P

 152153154 (1) = v(2, 1).v(3, 2) -v(3, 1).v(2, 2); P (2) = v(3, 1).v(1, 2) -v(1, 1).v(3, 2);P (3) = v(1, 1).v(2, 2) -v(2, 1).v(1, 2);S(1) = 0.5 [P(1).P (1) + P (2).P (2) + P (3).P (3)] 0.Area = v(2, 1).v(3, 3) -v(3, 1).v(2, 3); P (2) = v(3, 1).v(1, 3) -v(1, 1).v(3, 3); P (3) = v(1, 1).v(2, 3) -v(2, 1).v(1, 3); S(2) = 0.5 [P (1).P (1) + P (2).P (2) + P (3).P (3)] 0.Area = v(2, 2).v(3, 3) -v(3, 2).v(2, 3); P (2) = v(3, 2).v(1, 3) -v(1, 2).v(3, 3);P (3) = v(1, 2).v(2, 3) -v(2, 2).v(1,3); S(3) = 0.5 [P (1).P (1) + P (2).P (2) + P (3).P (3)] 0.Area (1) = v(2, 4).v(3, 5) -v(3, 4).v(2, 5); P (2) = v(3, 4).v(1, 5) -v(1, 4).v(3, 5);P (3) = v(1, 4).v(2, 5) -v(2, 4).v(1, 5); S(4) = 0.5 [P (1).P (1) + P (2).P (2) + P (3).P (3)] 0.5 ; Edge lengthfor i = 1 to 6 do l(i) = [v(1, i).v(1, i) + v(2, i).v(2, i) + v(3, i).v(3, i)] 0.5 ;Facets quality q(1) = 4.(3) 0.5 .S(1) ; q(2) = 4.(3) 0.5 .S(2); q(3) = 4.(3) 0.5 .S(3) ; q(4) = 4.(3) 0.5 .S(4);

Algorithm 2 : 3 /1 3 x 4 y 5 FRACTURE

 23345 Quality-based damage indicator in FORGE ® user-subroutine Input : 1/ Node location matrix X loc 2/ Mesh quality Q Critical quality Q cr 4/ Effective plastic strain εpl 5/ Critical plastic strain εcr 6/ Damage variable D 7/ Critical damage D cr Output: Damage indicator value such as D * = {0; 1} Mesh quality condition2 if Q < Q cr then min = min [X loc (1, 1), X loc (1, 2), X loc (1, 3), X loc(1, 4)]; min = min [X loc (2, 1), X loc (2, 2), X loc (2, 3), X loc (2, 4)]; MODE N°1: Ductile hole enlargement...

6.Large plastic strain condition 9 if εpl ≥ εcr then 10 DFRACTURE

 910 .. element to delete on the symmetry axis 7 if x min < 0.5 & y min < 0.5 then 8 MODE N°2: Shear-plugging...

14... element to delete in a ligament which connect the ejected plug to the sheet 15 Damage condition 16 else if D ≥ D cr then 17 if εpl ≥ εcr then 18 DFigure D. 1 :

 1617181 Figure D.1: Sheets deformation and fracture modes in the high-speed nailing simulation of {Steel/Cast alu.} superposition with different mesh size in the bending zone at t = 0.27 ms (a) (b) (c) (d) and t = 0.65 ms (e) (f ) (g) (h).

Figure D. 2 :

 2 Figure D.2: Sheets deformation and fracture modes in the high-speed nailing simulation of {Cast alu./Steel} superposition with different mesh size in the bending zone at t = 0.29 ms (a) (b) (c) (d) and t = 0.43 ms (e) (f ) (g) (h).

Figure D. 3 :Figure D. 4 :

 34 Figure D.3: Comparisons between simulations run with LEM material models and experiments on the piston kinematics of (a) {Steel/Steel}, (b) {Steel/Cast alu.}, (c) {Cast alu./Steel} and (d) {Cast alu./Cast alu.} superposition.

Table 1 . 1 :

 11 .1 sums up general features of nail material properties. Nail materials features (RIVTAC ® )

	Mass	Hardness	Raw material	Heat treatment
	1.2 g	450 ± 30 HV10 (H3)	C60 steel	24h 220 o C
	1.2 g	500 ± 30 HV10 (H5)	C60 steel	24h 220 o C

  1.8b). Assuming that joint strength of any intermediate loading direction can be decomposed into a shear and a tensile component, joint qualification is usually limited to these two tests. However Fayolle (2009) and also Hoang et al. (2011) have mentioned that full understanding of joint behavior requires testings at different loading angles. ARCAN set-up is typically used on tensile device to enable joint testing in

several directions. It provides mixed joint failure modes between shear-dominant and tension-dominant modes and can be really convenient to investigate the origin of both fracture mechanisms.

However in the present thesis, our choice was driven by Renault industrial needs so we have chosen to respect European standards formulated in ISO (2013) and ISO (2016) widespread in the automotive industry. Rectangular specimens are machined from metal sheets into dimensions 38 mm x 125 mm and tested on standard industrial testing device. For this reason, only shear-tensile and cross tension tests will be considered here.

Table 2

 2

		.1. Tests are repeated in the different materials superposition and hold-down pressure
	and clamping diameter is modified with ring diameter d ∈ {28; 30; 34 mm}. {Steel/Cast alu.} and
	{Cast alu./Steel} superposition have been repeated eight and nine times but {Steel/Steel} only three
	times. Since high-speed nailing process is mostly dedicated to mix-materials applications, more tests
	have been done in dissimilar superposition than in mono-material ones (except for {AHSS/Cast alu.}
	superposition which can not be joined and therefore tested only three times).
	#	Superposition	Clamping diameter (mm)	Hold-down pressure (bars)	Repetition
	1	{Steel/Steel}	30	1	3
	2	{Steel/Steel}	30	2	3
	3	{Steel/Steel}	30	3	3
	4	{Steel/Steel}	30	4	3
	5	{Steel/Steel}	30	5	3
	6	{Steel/Cast alu.}	28	1	9
	7	{Steel/Cast alu.}	34	1	8
	8	{Cast alu./Steel}	28	1	9
	9	{Cast alu./Steel}	34	1	9
	10	{AHSS/Cast alu.}	28	1	2
	11	{AHSS/Cast alu.}	34	1	1
	12	{Cast alu./AHSS}	28	1	9
	13	{Cast alu./AHSS}	34	1	8
	Table 2.1: Experimental tests conducted with different materials superposition, clamping diameter and
		hold-down pressure settings
	Observation 2.3: Hold-down force (2)

Piston velocity induces materials responses which are not common in the field of joining: strain-rate

	2.4.1.1 Considerations on ballistic-like sheets failure modes
	Terminal Ballistics, also called "Mechanics of Penetration" in the works of Backman and Goldsmith
	(1978), is a perspective which has been considerably investigated since 1970 to improve the predictive
	capability of armor plates. Experiments were performed at a large range of impact velocities. Corbett
	et al. (1996) proposed a review of ballistic experiments and reveal that target behavior always fits one
	of the eight most common perforation modes, whose six mains ones are illustrated on Figure 2.12.
	As impact experiments are globally reproducible, it is generally assumed, as mentionned by Holmen
	et al. (2015), that the energy required to perforate a plate in a given configuration is constant. Thus,
	theorization of perforation mechanics is possible via the identification of influent parameters from both
	the target and the projectile, as listed in Table 2.2.
	Corbett et al. (1996), Børvik et al. (2003b) and Børvik et al. (2005) have investigated how changes
	in the projectile or target main parameters could affect perforation capability. Projectile nose shape
	are modified from ogival to flat, from short to large diameter, target material thickness is increased or
	layers added. Parameters are changed and effects are always quantified in terms of residual projectile
	velocity -i.e. velocity after full penetration of the projectile -and ballistic limit velocity -i.e. transition
	velocity between non-penetration and full penetration state. This is generally expressed under the form
	of Lambert equation (2.9), taken from the works of Recht and Ipson (1962):
	hardening, thermal softening due to temperature increase, fracture... Furthermore, nail geometry is
	far from the one of blind or self-piercing rivet. The cylindrical nail shaft ends with an ogival tip and
	therefore nail insertion mechanisms can be somehow related to sheets perforation mechanisms which
	are only encountered in Terminal Ballistics, i.e. science studying penetration mechanisms of a free-
	flying projectile into targeted materials layers. In the following will be introduced few notions from
	this field which could be useful to explain failure modes obtained in our joining process.
		Geometry	Material	Impact 1 st influence	Impact 2 nd influence
		Diameter	Constitutive law	Mass	Obliquity
	Projectile	Length	Fracture law	Velocity	Yaw
		Nose-shape	-	-	Spin
		Span	Constitutive law	Boundary conditions	-
	Target	Thickness	Fracture law	-	-
		Layers	-	-	-

  .3).

	#	Superposition	Nail material	Nail-tip	Joining pressure (bars)
		{Steel/Steel}	H3	Ogival	5
		{Steel/Steel}	H3	Ogival	6
	Observation 2.4: A ballistic-like joining process {Steel/Steel} H3	Ogival	7
	{Steel/Steel} 1. According to Johnson's damage number δ ≈ 10 -2 applied to this joining process, sheets response H3 Flat 5 {Steel/Steel} H3 Flat 6
	to impact is only plastic (no significant dynamic effects). {Steel/Steel} H3	Flat	7
	{Steel/Steel} 2. Sheets subjected to impact can exhibit six different main fracture modes. It results from geomet-H5 Ogival 5 {Steel/Steel} H5 Ogival 6 rical, material or velocity-based parameters which control the projectile-target interactions. {Steel/Steel} H5 Ogival 7
		{Steel/Steel}	H5	Flat	5
		{Steel/Steel}	H5	Flat	6
	2.4.1.2 Experimental nailing in multi-materials superposition {Steel/Steel} H5 Flat	7
		{Steel/Cast alu.}	H3	Ogival	5.4
	Experimental nailing tests have been performed at IRT M2P in the conditions explained in section {Steel/Cast alu.} H3 Flat 5.4
	2.2.1. Access from beside and below enables nail impact on the sheets to be captured by high-speed {Steel/Cast alu.} H5 Ogival 5.4
	{Cast alu./Steel} camera at 100 000 frames per second. Time is reset at the moment nail is impacting the upper-sheet. H3 Ogival 5 {Cast alu./Steel} H3 Flat 5 Three joining pressure have been tested in {Steel/Steel} superposition -5, 6 and 7 bars -and only one {Cast alu./Steel} H5 Ogival 5
	in {Steel/Cast alu.}'s and {Cast alu./Steel}'s, respectively 5.4 bars and 5 bars. The configurations {AHSS/Cast alu.} H3 Ogival 6.3 of projectile parameters are changed to modify Projectile-Target interactions and promote different {AHSS/Cast alu.} H5 Ogival 6.3
	sheets failure mechanisms: Ogival nail-tip and Flat nail-tip are tested for nail materials H3 and H5. {Cast alu./AHSS} H3 Ogival 6.3
	Those configurations are listed as in Table 2.4. Each configuration is presented in a sequence composed {Cast alu./AHSS} H5 Ogival 6.3
	of three pictures taken at impact-time, maximum force-time and minimum force-time. A "Force vs. Table 2.4: Experimental nailing tests conducted with different materials superposition, nail features Time" graph is finally used to plot sheets reaction evolution -as calculated in section 2.3.1.3 -in the configurations and joining pressure settings
	investigated configurations. For conveniency a "Force vs. Piston displacement" graph is also plotted
	to enable comparisons between tests conducted at different velocity. Flat nose is obtained when ogival
	nose has been cut from traditional nail, therefore it is shorter than ogival-nose nail: the total shaft
	length is 8 mm instead of 14 mm.

  It can be noticed in test #19 Figure2.25c that H3 nail shatters after impact with the AHSS upper layer and then it is propelled out of the barrel. If the opposite superposition is considered with cast aluminium as the upper layer (see #21 Figure2.25i), H3 nail does not shatter and full insertion is completed. But the nail-tip has been widely deformed after impact with AHSS-made upper sheet which therefore promotes shear plugging of the lower sheet.Thus one thing can already be noticed from these tests: materials superposition shall be significant in the nail join-ability with easier nail penetration in {Cast alu./AHSS} configuration than in {AHSS/Cast alu.}'s. Tests #20 and #22 were conducted in the same materials configuration than tests #19 and #21 but with a H5 nail. In these cases, nail fracture has not appeared after impact with AHSS positioned in upper layer but it has progressed into the sheets even if flattened nail-tip can testify how tough was impact. Test #22 Figure2.25l also attests that nail exhibits extreme loading conditions when AHSS is the first sheet to be impacted. However H5 nail takes advantage of its extended material properties

Tests #19 #20 #21 #22 have been done on different materials superposition than in the previous tests. AHSS is introduced into the set of tested materials in {AHSS/Cast alu.} and {Cast alu./AHSS} superposition joined at pressure P = 6.3 bars successively with H3 nail and H5 nail. These configurations have been chosen since stiffness of layers superposition is touching the perforation limit that the nail can handle.

to withstand impact with stiff material layers what is far beyond H3 nail eventualities. These results can also be observed through Figure

2

.26 in which sheets reaction are plotted accross time. Only curve from test #19, continuous purple line, should be considered with caution since fracture has spread throughout the nail shaft from 0.2 ms to full fragmentation. It leads to non-realistic piston kinematics and therefore non-realistic reaction curve.

Table 2

 2

	Initial temperature T i (°C)	20	200	400	600	800	1000
	Aluminium	30	25	7	-	-	-
	Duralumin	-	90	30	-	-	-
	Copper	70	45	35	15	8	-
	TiAl6V	-	-	85	45	10	-
	Mild-steel	135	95	95	35	15	20
	Stainless-steel	205	115	80	70	45	25

.5. It can be noticed that temperature increases about four times more in steel than in aluminium per unit of deformation. Moreover T d seems to decrease with temperature.

Table 2 .

 2 Because aluminium does not satisfy adiabatic condition, temperature should not locally exceed 30 °C. However steel can exhibit temperature increase which can be higher than 205 °C. Yield stress of advanced steel and strain hardening effect should increase plastic work transformed into heat and characteristic heat T d as well. It would legitimate temperature increase larger than 205 °C inside steel material and therefore will be investigated through experimental tests in the further section.

	F o	= 6.25
	Steel
	F o

5: Characteristic heat T d estimated for typical metals

[START_REF] Felder | Effet thermique de la mise en forme -Theorie et phénomènes volumiques[END_REF]

2.4.2.2 Estimation of temperature increase in the joining stage On the basis of considerations established in the previous section, Fourier number can be estimated with (2.15) in the joining conditions for aluminium and stainless-steel. Thermal diffusivity is assumed to be Ω Alu = 96 mm 2 .s -1 and Ω Steel = 4 mm 2 .s -1

[START_REF] Felder | Effet thermique de la mise en forme -Theorie et phénomènes volumiques[END_REF]

. The characteristic time is taken as ∆t ≈ 0.1 ms since the maximum force of the nail penetration stage occurs always before 0.2 ms. The thickness of the deformed area is taken as τ = 0.1 mm because nail ridge is 0.1 mm-deep and certainly does affect sheet material on this depth/thickness. Alu = 0.26 Condition of adiabaticity, i.e. F o 1, is satisfied with steel but not with aluminium. As consequence adiabatic temperature increase ∆T ad can legitimately be calculated for steel but would be over-estimated for aluminium. The strain magnitude is taken as ∆ε ≈ 1.0 which is close to the one observed in forging and machining with initial temperature set to T i = 20°C. 2.4.2.3 IR measurements conducted in {Steel/Steel} superposition Set-up conditions and device used for IR measurements have been detailed in section 2.2.1. It consists in two steel layers clamped into a camera set-up and a IR camera targeting a mirror located above the sheets. Joining pressure has been set to 6 bars which gives 32 m/s as impact velocity. Experiments started the camera set on full-frame (640 x 512 pixel 2 ) which allows measurements around 100 frames/s. It was gradually increased to 870 frames/s (160 x 128 pixel 2 ), then 1737 frames/s (64 x 68 pixel 2 ) and finally 2577 frames/s (48 x 28 pixel 2 ), the smallest window the camera authorizes. These tests are summed up in Table 2.6 and results obtained with four different camera settings can be seen in Figure 2.29. Only the peak values detected by camera sensors of the targeted windows are reported in Table 2.6.

It can be noticed that the smaller is the window size, the bigger is the camera frequency and therefore joining stage is covered with much more measurements. In test # 4, camera frequency reached 2.577 frames/ms whereas high-speed camera experiments discussed in section 2.4.1 were done at 100 frames/ms. Since the penetration duration is about 0.4 ms, only one measurement is expected in the joining stage. Then the probability to capture the hottest time with only one picture in the whole joining time becomes quite small. This consideration makes that local temperature increase higher than 450 °C is very likely in {Steel/Steel} superposition.

Table 2 .

 2

6: IR measurements conducted on {Steel/Steel} superposition in the joining stage

Table 2 .

 2 Nine tests were run with clamping diameter d = 28 mm and eight tests with d = 34 mm. Experimental conditions are listed from test #1 to #17 in Table2.7 and sheets reaction curves are plotted in the two graph in Figure2.32. Curves obtained from clamping diameter d = 28 mm are marked with small circles whereas the ones of d = 34 mm are mark with bigger circles. In both graph, curves color-scale starts from black for the lowest joining pressure to red for the largest joining pressure. Green curve is used for optimized configuration. Error bar has been built only for the optimized test repeated three times (#4 #5 #6 for d = 28 mm and #12 #13 #14 for d = 34 mm). It can be noticed that both configuration presents a black curve which stops at smaller piston displacement than the rest. Blue curves also exhibits a precipitated stop compared to the green one and the red curve is out of interest since piston motion does not stop after full penetration. The only thing which distinguishes green curve from the others is the maximum piston displacementz max (28 mm) = 16 mm & z max (34 mm) = 16 mm -and no excessive reaction force at the end of piston travel.

	#	Clamping diameter (mm)	Clamping force (kN)	Joining pressure (bars)
	1	28	46.2	4.8
	2	28	48.5	5.0
	3	28	57.3	5.3
	4	28	56.2	5.4
	5	28	51.3	5.4
	6	28	42.7	5.4
	7	28	52.6	5.5
	8	28	45.4	5.7
	9	28	52.9	6.0
	10	34	41.0	5.0
	11	34	37.7	5.3
	12	34	35.9	5.4
	13	34	47.8	5.4
	14	34	37.9	5.4
	15	34	38.7	5.5
	16	34	39.7	5.7
	17	34	44.2	6.0

2.5.2.1 {Steel/Cast alu.} superposition 7: Experimental campaign n°2 nailing tests conducted on {Steel/Cast alu.} layers superposition with increasing clamping diameter d and joining pressure (Hold-down pressure is set to 1 bar)

  Blue-colored ring which can be noticed on steel sheared plug indicates that material should have locally reached high temperature during the nail insertion stage.

	Observation 2.9: Sheets temperature increase (2)
	Observation 2.10: Sheets fracture modes (2)
	About {Steel/Cast alu.} superposition
	1. Nail-steel sheet interface keeps being sheared after perforation because steel sheared pieces can
	be found afterwards at the nail-aluminium interface and inside the nail ridges.
	Observation 2.11: Joint cross-section (1)
	About {Steel/Cast alu.} superposition
	1. Upper sheet global bending and interface contact length l 1 is significant.
	2. Due to aluminium shear-plugging, the volume loss reduces the nail-aluminium sheet interface
	contact area l 2 .

.33c in two different pictures, one from beside and one from below. Several rings of different colors can be noticed in both pictures and attest the different deformation steps and local temperature increase material has undergone in the penetration stage.

2. The aluminium plug is formed by the competition between plug-shearing and petalling mechanims: radial cracks initiate the petals formation and a shear crack grows and propagates through-thickness to finally trigger plug separation from the sheet.

Joint cross-section in Figure

2

.34 brings additional information about fracture mode and nail-sheet interface in {Steel/Cast alu.} superposition. Test #3 reveals several points which have to be noticed.

First, upper sheet bending is important with a contact length with nail -previously noted l 1 -larger than initial layer thickness. It creates an inter-sheets gap g which is particularly important in this case since nail head is not fully seated on the sheets and consequently the gap has not been reduced yet. Two focuses on nail ridges are presented in the red and the blue frame. The nail-lower sheet interface l 2 is limited in this case because shear plugging has removed a large material volume in the perforation stage. The upper sheet-nail interface is sheared and taken away throughout the nail descent. As noticed in both pictures, it has either got stucked at the nail-lower sheet interface or into the nail ridges. Such phenomenon implies that sheet material at the interface with the nail is more likely brought out by nail ridges than flowing into the ridges to get materials continuity. It is rather likely to get a flat sheared nail-sheet interface with local wavy contact area as located in the red frame upper part of Figure

2

.34.

3. Nail-lower sheet interface contact area is mostly flat, i.e. no indication of sheet material flowing into the nail ridges.

Table 2 .

 2 2.5.2.2 {Cast alu./Steel} superposition Nine tests were run with both clamping diameter d = 28 mm and d = 34 mm. Experimental conditions are listed from test #18 to #35 in Table 2.8 and sheets reaction curves are plotted in the two graph in Figure 2.35. Only four curves per clamping configuration are plotted with the same color-scale convention (black, green, orange and red) as previously introduced. Error bar has been built only for the optimized test and repeated three times (#54 #60 #61 for d = 28 mm and #65 #69 #70 for d = 34 mm).All curves are almost identical with only some slight differences on maximum piston displacement z max and force amplitude at contact time between nail head and upper sheet t pen . In both clamping configurations, black and orange curves -respectively considered as joining with a lack and an excess energy -are almost included in the error bar range. It means that sheets reaction variability is in the range of its sensitivity to the joining pressure: about 0.1 to 0.2 bars in Figure2.35a and more than 0.2 bars in Figure2.35b. This is established from the gap between pressure related to the green curve and the curve with the biggest pressure included in the the error bar.

	#	Clamping diameter (mm)	Clamping force (kN)	Joining pressure (bars)
	18	28	48.2	4.8
	19	28	43.2	4.9
	20	28	44.8	5.0
	21	28	45.4	5.0
	22	28	47.7	5.0
	23	28	38.2	5.1
	24	28	40.3	5.3
	25	28	39.9	5.6
	26	28	40.1	6.0
	27	34	49.6	4.9
	28	34	45.7	5.0
	29	34	50.6	5.1
	30	34	45.4	5.2
	31	34	42.4	5.2
	32	34	44.0	5.2
	33	34	39.7	5.4
	34	34	40.9	5.8
	35	34	37.3	6.0

8: Experimental campaign n°2 nailing tests conducted on {Cast alu./Steel} layers superposition with increasing clamping diameter d and joining pressure (Hold-down pressure is set to 1 bar)

Table 2 . 9 :

 29 Two tests have been conducted in {AHSS/Cast alu.} superposition as listed Table2.9. They are joined with basic ogival H3 nail, the one used in section 2.4 in the high-speed camera investigations. Experimental campaign n°2 nailing tests conducted on {AHSS/Cast alu.} layers superposition with increasing clamping diameter d and joining pressure (Hold-down pressure is set to 1 bar)Contrary to high-speed camera experiment in Figure2.25c, the present headpiece is not cut and sheets are positioned inside the sheets clamping set-up. Thus nail cannot escape piston after impact and has to

	12
	10
	8
	6
	4
	2
	0
	0	2	4	6	8	10	12	14	16
				(a) d = 28 mm
	12
	10
	8
	6
	4
	2
	0
	0	2	4	6	8	10	12	14	16
				(b) d = 34 mm
	12
	10
	8
	6
	4
	2
	0
	0	2	4	6	8	10	12	14	16

3. No shear-plug has been formed from the aluminium sheet perforation. Observation 2.13: Joint cross-section

(2)

About {Cast alu./Steel} superposition 1. Aluminium is pushed apart while nail is penetrating the sheets. Volume conservation and Observation 2.12.3 explain why a significant inter-sheets gap g can exist.

2. As noticed for lower aluminium made sheet in Observation 2.11.3, the nail-steel interface is mostly flat. Some wave-like material bulges can be noticed at the ridge front, pushed downwards by the nail descent but there are no indications of sheet material flowing into the nail ridges.

(c) 28 mm vs. 34 mm Figure 2.35: Force-Piston displacement curves of {Cast alu./Steel} superposition with increasing joining pressure and clamping diameter d 2.5.2.3 {AHSS/Cast alu.} superposition deal with it until maximum displacement is reached. The sheets reaction is stiffer in this configuration than in the camera set-up one: maximum force is 12 kN (Figure 2.38) compared to 10.5 kN in camera set-up configuration (Figure 2.26). Nail-tip undergoes large deformation until a crack is created. The shaft consequently slides along the fracture zone and get bent inside the barrel. This is consistent with what Rakvåg et al. (2014) have shown from taylor impact tests conducted with different materials and velocities. Projectile can exhibit one of the five following main fracture modes: mushrooming, tensile splitting, shear cracking, petalling and fragmentation. From these considerations, the nail fracture mode presented in Figure

Table 2

 2 Nine tests were run with clamping diameter d = 28 mm and eight tests with d = 34 mm. Experimental conditions are listed from test #38 to #54 in Table2.10 and sheets reaction curves are plotted in the two graph in Figure2.40. Despite the curve from test #39 which depicts very different behavior to the others, most curves give similar results. Inherent variability of sheet reaction at optimized pressure (P = 6.3 bars) is again presented by green error bars. As expected, this variability covers the variation of the set of curves obtained at different joining pressures, from 6.1 bars to 6.7 bars in Figure2.40a and from 6.0 bars to 6.8 bars in Figure2.40b. Thus force evolution cannot be exclusively allocated neither to joining pressure variation or intrinsic sheets response variability. The only particularity to notice about green/optimized curve compared to the rest of inappropriate joining curves is that it catches the maximum piston displacement value z max with the minimum force amplitude.Interface with the lower sheet is flat but several material waves can be noticed in front of the ridges in blue frame Figure2.42. The aluminium interface is scrapped down according to the same mechanism explained for {Cast alu./Steel} superposition.

	#	Clamping diameter (mm)	Clamping force (kN)	Joining pressure (bars)
	38	28	38.1	5.9
	39	28	40.5	6.1
	40	28	39.1	6.3
	41	28	33.9	6.3
	42	28	42.2	6.3
	43	28	32.2	6.4
	44	28	39.7	6.7
	45	28	36.9	7.0
	46	28	38.1	7.3
	47	34	38.1	6.0
	48	34	37.1	6.2
	49	34	34.9	6.3
	50	34	43.9	6.3
	51	34	43.4	6.3
	52	34	38.6	6.5
	53	34	33.7	6.8
	54	34	41.9	7.0
	tion showed a nail ogival shape preserved after penetrating steel as lower sheet. But if steel is changed
	into AHSS, nail could not overcome such impact without involving large deformations. This means a
	flat, dark and even twisted nail-tip as suggested in Figure 2.41b which obviously promotes plugging
	of the lower-sheet. However the upper-sheet made of aluminium is not sheared since nail deformation

1 mm Figure 2.39: Nail fracture mode in {AHSS/Cast alu.} joining tests #36 and #37 2.5.2.4 {Cast alu./AHSS} superposition .10: Experimental campaign n°2 nailing tests conducted on {Cast alu./AHSS} layers superposition with increasing clamping diameter d and joining pressure (Hold-down pressure is set to 1 bar) Nail material is almost too soft to enable such perforation process. Flattened nail-tip and plug shape from Figure 2.41a imply that dominant fracture mode is shear plugging. {Cast alu./Steel} superposioccurs only when it has reached the lower-sheet. A cross-section in the appropriate joining configuration (test #41) is proposed in Figure 2.42 to picture internal deformation state. Aluminium made upper-sheet looks similar to cross-section obtained in {Cast alu./Steel} superposition (see Figure 2.37) but AHSS does not bend as steel does. The nail-lower sheet interface length l 2 is smaller in {Cast alu./AHSS} superposition than in {Cast alu./Steel}'s. Observation 2.15: Sheets reaction to the nail insertion (3) 1. Whatever is the tested materials superposition, the maximum sheets force amplitude is bigger in a stiff configuration than in soft one (at optimized joining pressure). 2. Sheets reaction curve obtained from test conducted with the optimized joining pressure can be recognized by two features: (1) piston displacement is stopped slightly after the force plateau has been reached; (2) force amplitude does not keep increasing after this plateau. Observation 2.16: Nail deformation (1)

Table 2 .

 2 2.5.3 Experimental campaign n°3: Nailed-joint reproducibilityThe set-up detailed in section 2.2.2 has been used in the conditions presented in Experimental campaign n°1 and Experimental campaign n°2 with a constant clamping diameter d = 30 mm and a H3 ogival nail. For every investigated materials superposition, an iterative procedure has been used to find the optimized joining pressure. Depending on the configuration, five to seven tests were necessary to obtain the appropriate pressure setting. Then optimized joining configuration is repeated six to seven times in order to evaluate force variability in the joining stage and especially to quantify discrepancies on characteristic joint dimensions observed from cross-section analysis. The conditions of these tests are summed up in Table2.11.

	#	Superposition	Joining pressure (bars)	Repetition
	1. . .6	{Steel/Steel}	6.5		6
	7. . .13	{Steel/Cast alu.}	5.7		7
	14. . .20	{Cast alu./Steel}	5.3		7
	21. . .27	{Cast alu./Cast alu.}	4.5		7
	1. Max. Force (kN)	8.43 ± 0.33	6.21 ± 0.09	6.21 ± 0.13	4.47 ± 0.29
	2. Min. Force (kN)	2.33 ± 0.11	1.99 ± 0.13	1.73 ± 0.18	1.67 ± 0.25
	3. Max. z	(mm) 14.95 ± 0.37	16.04 ± 0.33	13.47 ± 0.61	15.01 ± 0.45

11: Experimental campaign n°3 nailing tests conducted on different materials superposition and repeated several times at the optimized joining pressure (Clamping diameter d = 30 mm ; Hold-down pressure is set to 1 bar) Mean value ± Standard deviation {Steel/Steel} {Steel/Cast alu.} {Cast alu./Steel} {Cast alu./Cast alu.}

Table 2

 2 Steel} superposition is 2.33 kN with a standard deviation of 0.11 kN. This value is consistent with variability range observed for point 1 peak force. Then force tends to increase due to contact between upper sheet and the nail head which happens at z ≈ 12 mm. This state is indicated on the graph by point 3 which pictures the maximum piston displacement z reached at the end of the test.It changes a lot between configuration but maximum standard deviation 0.61 mm is small regarding mean value estimated at 13.47 mm in {Cast alu./Steel} superposition. Figure 2.43: Force-Piston displacement curves of different materials superposition and repeated several times at the optimized joining pressure (Clamping diameter d = 30 mm ; Hold-down pressure is set to 1 bar). Notations {1,2,3} are detailed in Table2.12.

	It can be noticed that reaction curves are not so different between tests run in a same configuration
	and curve shape is globally constant. It is controlled by three inflection points whose value is set with
	a certain variability. From impact (z = 0) to the maximum force value (point indicated by a marker
	1), the slope is mainly constant between tests. The peak force is estimated between tests with a mean
	value and a standard deviation as presented Table 2.12.

.12: Sheets reaction characteristic values obtained at optimized joining pressure from at least six Force-Piston displacement curves (see Table

2

.11 and Figure

2

.43)

Sheets reaction variability

All tests reaction curves are plotted in

Figure 2.43 with identical color and an error bar which has been evaluated from a reference test. Tests considered as reference are the following: #1 for {Steel/Steel} superposition, #8 for {Steel/Cast alu.}, #19 for {Cast alu./Steel} and #22 for {Cast alu./Cast alu.}. In {Steel/Steel} superposition, maximum force is 8.43 kN with a standard derivation of 0.33 kN. It gets smaller in other tested superposition: 0.29 kN in {Cast alu./Cast alu.}'s, 0.13 kN in {Cast alu./Steel}'s and 0.09 in {Steel/Cast alu.}'s. As the force is decreasing from point 1 to point 2, gap between curves increases. But the force drop is limited to the value reached at point 2. The mean value obtained in {Steel/(d) {Cast alu./Cast alu.} (test #21. . .#27) displacement marker (3.) (based on standard deviation).

3. Maximum piston displacement (3.) is different between multi-materials configurations: 14.95 mm for {Steel/Steel}, 16.04 mm for {Steel/Cast alu.}, 13.47 mm for {Cast alu./Steel} and 15.01 mm for {Cast alu./Cast alu.}.

  1 , n 2 and n 3 . From the same set of tests used to quantify sheets reaction variability (see Table2.11), cross-sections have been cut from the nailed-joints and characteristic dimensions measured by microscopical analysis. Results of four materials superposition are gathered in Table2.13 and Figure2.44. {Steel/Steel} cross-section in Figure2.44a reveals that, in some tests, nail penetration into the sheets is not complete. It means that nail head is at a distance h from contacting the upper sheet. It has been measured h = 0.57 mm in test #6 but it is statistically evaluated by a mean value ± standard deviation as h = 0.16 ± 0.23 mm (see Table2.13). Variability on the residual distance h appears significant regarding its value but it is not so much bigger than

variability assessed to any other parameter. The lower variability is found to be 0.03-0.05 for parameters g and n 3 whatever is the superposition. But it can be noticed that the l 1 variability which ranges 0.05 to 0.15 is quite low regarding the rest of the parameters. This dimension quantifies the nail-upper sheet interface distance is not so much affected by superposition and intrinsic variability: if the upper layer is made of steel, l 1 is about 2.25-2.27 mm ; if it is made of aluminium, l 1 is about 4.22-4.33 mm.

Taken into account the initial layer thickness, 1.5 mm for steel and 2.5 mm for aluminium, this mean variability estimated around 0.11 mm from one superposition to another and about 0.15 from one test to another (at identical superposition) attests the stability of l 1 dimensions throughout the tests. Similar conclusion cannot be found from results obtained on l 2 -value which measures the nail-lower sheet interface distance. If steel is taken as lower layer, a certain stability of the result is obtained:

1.74 ± 0.16 in {Steel/Steel} superposition and 1.83 ± 0.20 in {Cast alu./Steel}'s. But mean values and standard deviations are bigger if aluminium is taken as lower layer. Indeed deformation and fracture are often asymmetrical which for instance promotes bigger l 1 -value on the left part of Figure

2

.44d than on the right part. It can also be observed occasionally on {Steel/Steel} configuration as presented in Figure

Table 2 .

 2 n 1 is significantly reduced when steel is part of the layers and even more if it is positioned as the upper layer. A shorter nail length is always balanced by a larger nail-tip diameter: 1.89 mm for

	h	0.16 ± 0.23	0.02 ± 0.05	0	0.05 ± 0.14
	l 1	2.25 ± 0.05	2.27 ± 0.07	4.33 ± 0.15	4.22 ± 0.08
	l 2	1.74 ± 0.16	2.34 ± 0.17	1.83 ± 0.20	2.66 ± 0.73
	g	0.35 ± 0.04	0.17 ± 0.03	0.34 ± 0.07	0.28 ± 0.05
	n 1	4.45 ± 0.10	4.60 ± 0.13	4.69 ± 0.66	5.38 ± 0.32
	n 2	1.89 ± 0.17	1.56 ± 0.35	1.11 ± 0.31	0.64 ± 0.20
	n 3	0.30 ± 0.05	0.23 ± 0.05	0.17 ± 0.03	0.16 ± 0.05

{Steel/Steel} superposition, 1.56 mm for {Steel/Cast alu.}'s, 1.11 mm for {Cast alu./Steel}'s and 0.64 mm for {Cast alu./Cast alu.}'s. Mean value ± Standard deviation Dimensions (mm) {Steel/Steel} {Steel/Cast alu.} {Cast alu./Steel} {Cast alu./Cast alu.} 13: Nailed-joint characteristic dimensions obtained at optimized joining pressure from at least six cross-section measurements (see Table 2.11) 1 mm (a) {Steel/Steel} (test #6) 1 mm (b) {Steel/Cast alu.} (test #11) 1 mm (c) {Cast alu./Steel} (test #14) 1 mm (d) {Cast alu./Cast alu.} (test #23) Figure 2.44: Nailed-joint cross-sections obtained at optimized joining pressure in four materials superposition (see Table 2.11) to measure characteristic joint dimensions (see Table 2.13) Observation 2.19: Joint cross-section (3)

Table 2 . 10 )

 210 First thing to notice from these tests is that hierarchical joint response is globally respected: test with under-estimated joining pressure (black curve) drops the first; test with optimized pressure has the best load-carrying capability (green curve); test with excessive joining pressure (red curve) fails before the optimized one. It confirms the relation -introduced in Chapter 1 -between nail insertion depth and mechanical properties assigned to the joint (see Figure1.6c).

	4.5
	4
	3.5
	3
	2.5
	2
	1.5
	1
	0.5
	0
	0	2	4	6	8	10	12	14	16

5 Figure 2.50: Cross-tension curves of optimized joining pressure tests in {Steel/Cast alu.} (test #4 and #12), {Cast alu./Steel} (test #20 and #30) and {Cast alu./AHSS} superposition (test #40 and #49) with clamping diameter d = 28 mm (small circle) and d = 34 mm (big circle)

Table 2

 2 2.50 in multi-materials optimized configurations and in the joinability chart -Maximum crosstension strength vs. Joining pressure chart -on Figure2.51. In addition to similar results obtained between the two clamping configurations, the last diagram also highlights important joinability chart particularities: {Steel/Cast alu.} plateau is short with low amplitude due to the fracture mode -shear plugging -which induces small nail-sheet contact area; {Cast alu./Steel} plateau is wider and presents larger amplitude than {Steel/Cast alu.}'s because of steel ductility. This material property drives sheet fracture mode and therefore the nail-sheet contact area. {Cast alu./AHSS} plateau is controlled by identical mechanism than {Cast alu./Steel}'s but only the amplitude has been shifted up by the extended AHSS stiffness. Plateau range has not benefited from this material upgrading.

	Observation 2.20: Optimized joining pressure range (2)

.7), {Cast alu./Steel} (see Table 2.8) and {Cast alu./AHSS} superposition (see Table 2.10) with clamping diameter d = 28 mm (small circle) and d = 34 mm (big circle)

The optimized pressure has been chosen by iteration, test after test, from visual inspection of the joint (see section 1.5.2). It appears from these tests that methodology is satisfactory since the green curve is almost always the best one. Only random phenomena related to fracture occurrence could explain a green curve to do not have the best features, as for example in {Cast alu./AHSS} superposition (see Figure

2

.49). No significant differences can also be noticed between tests conducted with clamping diameter d = 28 mm and d = 34 mm which means that stiffness differences can be overcome with an adapted input joining pressure and gives very similar joint strength. These considerations are obvious on Figure

Table 2 .

 2 14: Main sheets bending dimensions after cross-tension tests on {Steel/Cast alu.} and {Cast alu./Steel} superpositions (see Figure2.55 for details)As the upper and lower layers are moved apart, nail is static. One side of the lower sheet is taken down whereas the other side is contacting the nail (Observation 2.21.1), i.e. one side is slowly bent while the other one is pushed by material reaction against the nail shaft. Therefore material flows into the nail ridges and increases nail interlock (Observation 2.21.2). The nail-lower sheet interface gets gradually sheared by their relative velocity with fragments separated from the rest of the sheet which remain stucked into the ridges (Observation 2.21.3).

	Observation 2.21: Joint cross-section (4)
	1. Global as well as local sheets bending (from cross-section) suggest that during a cross-tension
	test, contact area at lower sheet-nail interface has evolved according to bending state.
	2. Although no lower sheet material has been observed into the nail ridges after the joining stage

(see Observation 2.11.3 and Observation 2.13.2), joint failure analysis reveals that it did flow into the nail ridges during cross-tension test.

3. Fragments of sheared materials can be found into almost every ridges located in the nail lower part.

Table 3 . 3 :

 33 Zerilli-Armstrong constitutive laws for BCC and FCC crystallographic structure with σ G , ζ 1 and ζ 2 the parameters of the athermal yield stress part and α 1,2,3,4,5,6 the parameters of the thermally-activated part.

Table 3 . 4 :

 34 Rusinek-Klepaczko constitutive law with εmax and εmin , respectively the maximum and minimum strain-rate limit of the model and γ 1,2,3,4,5,6,7,8 , ε0 and T m the material parameters to calibrate for the athermal stress component k 4 and thermally-activated component k 5 .

Table 4 .

 4 5 mm-thick DP780 steel, a 1.75 mm-thick DP1180 AHSS and a 2.5 mm-thick cast aluminium ingot AlSi10MnMg(T6). Both steels have been manufactured by Arcelor Mittal without surface coating. Aluminium has been casted by BronzeAlu into rectangular ingots of dimensions 90 mm x 190 mm followed by a T6-heat treatment. Chemical composition of the material as well as the main mechanical features can be found respectively in Table 4.1 and Table 4.2.

			(% mass) Steel (1) AHSS (1) Cast alu. (3)	C60 (2)
			C	0.085	0.147	-		0.57-0.65
			Mn	1.857	1.903	0.40-0.80		0.60-0.90
			Mg	-	-	0.10-0.60		-
			Si	0.251	0.209	9.00-11.5		≤ 0.40
			P	0.016	0.019	-		≤ 0.045
			S	0.001	0.001	-		≤ 0.045
			Al	0.027	0.026	-		-
			N	0.0046	0.0044	-		-
			Ti	0.002	-	≤ 0.20		-
			Mo	-	0.003	-		≤ 0.10
			Cr	-	0.191	-		≤ 0.40
			Ni	-	-	-		≤ 0.40
			Cu	-	-	≤ 0.05		-
			Zn	-	-	≤ 0.07		-
			Fe	-	-	≤ 0.25		-
			B	-	0.0002	-		-
			V	-	0.003	-		-
	1: Chemical composition of 1.5 mm-thick DP780 steel sheet, 1.75 mm-thick DP1180 AHSS sheet, 2.5
		mm-thick AlSi10MnMg(T6) cast aluminium ingot and C60 nail raw material
		(1) : Measurement from casting process given by Arcelor Mittal
		(2) : C60 steel composition from standards EN-10027-1
		(3) : AlSi10MnMg cast aluminium composition from standards EN AC-43500
		R p 0,2 (MPa)	R m (MPa)	A e (%)	E (GPa)	ν	ρ (kg/m 3 )
		min	max	min	max	-		-	-	-
	Steel	450	550	780	900	15		194	0.33	7800
	AHSS	900	1100	1180	-	5		200	0.33	7800
	Cast alu.	120	-	200	-	7		77	0.3	2700

Table 4 .

 4

2: Mechanical features of 1.5 mm-thick DP780 steel sheet, 1.75 mm-thick DP1180 AHSS sheet and 2.5 mm-thick AlSi10MnMg(T6) cast aluminium ingot

Table 4 .

 4

		.987 0.985
		0.04 1 0.979 0.977
		0.08 1 0.976 0.975
		0.10 1 0.976 0.975
		0.02 1 1.172 1.105
	AHSS	0.04 1 1.088 1.036 0.08 1 0.997 0.997
		0.10 1 0.990 0.996
	Cast alu.	Not tested

3: Yield stress ratio r α to state for material anisotropy

Table 4 .

 4

			εf (Mean value ± Standard deviation)
			UT0	NT7	NT2	SH
		s 11	0.375 ± 0.251	0.109 ± 0.083	0.131 ± 0.024	0.118 ± 0.032
	Cast alu.	s 0	0.411 ± 0.144	0.161 ± 0.097	0.075 ± 0.015	0.097 ± 0.025
		s 12	0.209 ± 0.128	0.255 ± 0.195	0.115 ± 0.033	0.174 ± 0.068
		s 11	0.368 ± 0.006	0.240 ± 0.031	0.292 ± 0.016	0.224 ± 0.032
	Steel	s 0	0.529 ± 0.018	0.301 ± 0.058	0.230 ± 0.031	0.170 ± 0.042
		s 12	0.356 ± 0.017	0.278 ± 0.036	0.296 ± 0.032	0.264 ± 0.059
		s 11	0.452 ± 0.078	0.218 ± 0.015	0.312 ± 0.012	0.183 ± 0.083
	AHSS	s 0	0.468 ± 0.122	0.315 ± 0.004	0.226 ± 0.005	0.165 ± 0.051
		s 12	0.369 ± 0.090	0.226 ± 0.004	0.340 ± 0.014	0.110 ± 0.053

4:

Effective plastic strain at fracture εf obtained from three DIC plane-strain sensor measurements s 11 , s 0 and s 12 on UT0, NT7, NT2, SH tensile specimens made from cast aluminium (AlSi10MnMgT6), steel (DP780) and AHSS (DP1180)

Table 4

 4 .6 presents what are the data-points for every specimen and tested material. Since Matlab ® optimization algorithm requires as many experimental data as parameters to identify, it has been necessary for several criteria to add "optimization bounds" to the actual set of experimental fracture points. It consists in additional fracture points, acting like "virtual experiments", under High Stress Trixiality (HST) conditions (σ * = 1). Two cases of virtual loading path are considered here: (1) the generalized tension state, written HST-tension (HST-t)(µ σ = -1) in Table4.6 and (2) the generalized compression state, written HST-compression (HST-c)(µ σ = +1).

			Stress-state		εf
			σ * avg	µσ.avg	Steel	Cast alu.	AHSS
		SH	0.072	-0.178	0.264	0.174	0.183
	Experimental	UT0	0.353	-0.910	0.529	0.411	0.468
		NT7	0.487	-0.402	0.301	0.161	0.315
		NT2	0.569	-0.019	0.230	0.075	0.226
	Virtual	HST-tension	1	-1	0.050	0.025	0.050
		HST-compression	1	+1	0.100	0.050	0.100
		Fracture locus calibration. Set of data-points
		Steel			Cast alu.	AHSS
	ECL	SH,NT2,HST-t			SH,NT2,HST-t	SH,NT2,HST-t,HST-c
	CL	SH,UT0,NT2,HST-t,HST-c	SH,UT0,NT2,HST-t,HST-c	SH,UT0,NT2
	JC	UT0,NT7,NT2			UT0,NT7,NT2	UT0,NT7,NT2
	EJC	SH,UT0,NT2,HST-t		SH,UT0,NT2,HST-t	SH,UT0,NT2,HST-t
	BW4P	SH,UT0,NT7,NT2,HST-t,HST-c	SH,UT0,NT7,NT2,HST-t,HST-c	SH,UT0,NT7,NT2,HST-t,HST-c
	BW6P	SH,UT0,NT7,NT2		SH,UT0,NT7,NT2	SH,UT0,NT2,HST-c
	GUR	UT0,NT7,NT2			UT0,NT7,NT2	UT0,NT7,NT2
	NH	SH,UT0,NT7,NT2,HST-t	SH,UT0,NT7,NT2,HST-t	SH,UT0,NT7,NT2,HST-t

Fracture locus calibration. Data-points εf = f (σ * avg , µσ.avg)

Table 4 . 6 :

 46 Summary of the experimental and virtual effective fracture strain data-points used for fracture locus calibration of steel, cast aluminium and AHSS materials.RMK 1 : σ * avg and µ σ.avg are respectively the average stress-triaxiality and lode parameter that material exhibits in the crack onset area. RMK 2 : High Stress Triaxiality (HST) virtual data-points are either tension dominant (HST-t) or compression dominant (HST

Table 4 .

 4 7: Calibrated parameters of the damage models for steel (DP780) (see Table3.5).

	256	2.601	270.7

RMK 1 : Parameters are unit-free expect for S which is expressed in [MPa.s]. RMK 2 : PB-model parameters have been calibrated using BW4P fracture strain formulation εf . RMK 3 : Critical damage D c is the maximum damage value obtained in tensile simulations.

Table 4

 4

						AHSS. Fracture loci parameters
		ϕ 1	ϕ 2	ϕ 3	ϕ 4	ϕ 5	ϕ 6	ϕ 7	q 1	q 2	f 0	f f	ϕw	Dc
	ECL	0.730 1.090 137.72										0.451
	CL			215.8										0.425
	JC	0	1.473	3.233										0.394
	EJC	0	1.155	2.224			0.233	2.872						0.453
	BW4P 1.382 2.776	0.221	9.83e -12	1.382 2.776							0.410
	BW6P 1.609 2.778	0.204	-0.171	0.827 1.291							0.402
	GUR								1.122 1.783	-0.088	-0.619
	NH								0.363 1.557 5.94e -5	8.88e -5	1.548
					AHSS. Damage evolution and coupling parameters
		σ * 1	σ * 2	σ * 3	σ * 4	εD1	εD2	εD3	εD4	b	S	c	β	Dc
	LEM	0	0.33	0.47	0.58	1.8	1	1.51e -4	0.05	2	7.54		1	0.88
	PB	0	0.33	0.47	0.58	1.8	0.125 1.35e -2	0.03			0.118	1.416

.8: Calibrated parameters of the damage models for cast aluminium (AlSi10MnMgT6) (see Table

3

.5).

RMK 1 : Parameters are unit-free expect for S which is expressed in [MPa.s]. RMK 2 : PB-model parameters have been calibrated using BW4P fracture strain formulation εf . RMK 3 : Critical damage D c is the maximum damage value obtained in tensile simulations.

Table 4 . 9 :

 49 Calibrated parameters of the damage models for AHSS (DP1180) (see Table3.5).

RMK 1 : Parameters are unit-free expect for S which is expressed in [MPa.s]. RMK 2 : PB-model parameters have been calibrated using BW4P fracture strain formulation εf . RMK 3 : Critical damage D c is the maximum damage value obtained in tensile simulations.

Table 5 . 1 :

 51 Effective plastic strain at fracture εf for different locus at the mean stress-state to fracture (µ σ.avg , σ * avg ) of the upper and lower sheet in high-speed nailing simulations applied to {Cast alu./Steel} and {Steel/Cast alu.} superpositions (see Figure B.5 and Figure B.6).

Table 5 . 2 :

 52 }, (c) {Cast alu./Steel} and (d) {Cast alu./Cast alu.} superposition. RMK 1: Materials superposition are respectively joined in conditions of test #1, test #12, test #19 and test #22 (see Table 2.11). RMK 2: The evolution of piston displacement over time can be found in the Appendix D.The best agreement with experiment is reached in {Steel/Cast alu.} configuration and the worst in {Cast alu./Cast alu.} configuration. Curve shape is globally respected in every cases, except the {Cast alu./Cast alu.} one in which the noisy piston velocity profile induced a discontinuity in the acceleration curves (see Figure5.17d). Markers located at the three inflection points established in Observation 2.18.1 are used to quantify the discrepancies between sheets reaction from simulation and experiment. Table5.2 presents these results and highlights three main points: Comparisons between sheets reaction characteristic values obtained by experiments and simulations (details on the notations in Figure5.14 and cross-section in Figure5.18).

	• Maximum force is supposed to be identical in mixed cast aluminium-steel configuration but is
	effectively smaller in {Cast alu./Steel} (4.73 kN) than {Steel/Cast alu.} (6.13 kN)
	• Despite the gap with experimental values, maximum force hierarchy between configuration ob-
	tained from simulation is respected
	• Prediction of minimum force value is satisfactory in {Cast alu./Steel} configuration whereas it
	is under-estimated in the others
			{Steel/Steel}	{Steel/Cast alu.}	{Cast alu./Steel}	{Cast alu./Cast alu.}
			Exp	Num	Exp	Num	Exp	Num	Exp	Num
	1. Max. Force	(kN)	8.43 ± 0.33	7.52	6.21 ± 0.09	6.13	6.21 ± 0.13	4.73	4.47 ± 0.29	3.26
	2. Min. Force	(kN)	2.33 ± 0.11	0.99	1.99 ± 0.13	0.73	1.73 ± 0.18	1.71	1.67 ± 0.25	0
	3. Max. z	(mm)	14.95 ± 0.37	15.64	16.04 ± 0.33	15.92	13.47 ± 0.61	15.8	15.01 ± 0.45	14.84
	Observation 5.7: Sheets reaction to the nail insertion (5)
	Joining simulation run on {Steel/Steel}, {Steel/Cast alu.}, {Cast alu./Steel} and {Cast alu./Cast
	alu.} configuration have the following features:
	1. Maximum force amplitude which is supposed to be identical in mixed cast aluminium-steel con-
	figuration is smaller in {Cast alu./Steel} (4.73 kN) than {Steel/Cast alu.} (6.13 kN);
	2. Despite the gap with experimental values, maximum force hierarchy between the four materials
	configuration obtained from simulation is respected;
	3. Prediction of minimum force value is satisfactory in {Cast alu./Steel} configuration whereas it is

Table 5

 5 .3.

	Dimensions	{Steel/Steel}	{Steel/Cast alu.}	{Cast alu./Steel}	{Cast alu./Cast alu.}
	(mm)	Exp	Num	Exp	Num	Exp	Num	Exp	Num
	h	0.16 ± 0.23	0	0.02 ± 0.05	0	0	0	0.05 ± 0.14	0
	l 1	2.25 ± 0.05	1.78	2.27 ± 0.07	1.98	4.33 ± 0.15	3.70	4.22 ± 0.08	3.75
	l 2	1.74 ± 0.16	2.24	2.34 ± 0.17	2.61	1.83 ± 0.20	1.94	2.66 ± 0.73	3.19
	g	0.35 ± 0.04	0.19	0.17 ± 0.03	0.40	0.34 ± 0.07	0.23	0.28 ± 0.05	0.42
	n 1	4.45 ± 0.10	4.86	4.60 ± 0.13	5.33	4.69 ± 0.66	5.72	5.38 ± 0.32	5.85
	n 2	1.89 ± 0.17	1.72	1.56 ± 0.35	1.51	1.11 ± 0.31	1.17	0.64 ± 0.20	1.15
	n 3	0.30 ± 0.05	0.39	0.23 ± 0.05	0.36	0.17 ± 0.03	0.34	0.16 ± 0.05	0.50

Table 5 .

 5

www.boellhoff.com

Table2.2: Influent parameters in Projectile-Target interaction[START_REF] Backman | The mechanics of penetration of projectiles into targets[END_REF]

2.4. Nail insertion stage: Insight into the perforation mechanismsComparisons #19,20,21,22

http://bcodata.whoi.edu/Carrington/Instron_5565_Materials_Testing_Frame_M10-14190-EN(RevA).pdf

5.5. Résumé en françaisla validité de la simulation est confirmée par la comparaison des déformations du clou et des tôles, des modes de rupture, des configurations non-clouables et des dimensions caractéristiques de l'assemblage. L'écart le plus grand entre la simulation et l'expérience concerne la distance à l'interface clou-tôle inférieure. Plus important pour l'aluminium que pour l'acier, cet écart provient du caractère aléatoire du mode de rupture de la tôle inférieure et très fortement dépendant de la ductilité du matériau.Les simulations d'essais de traction-croix et de traction-cisaillement sont réalisées sur la base des résultats obtenus après relaxation des tôles. En dépit d'écarts observés entre les courbes force-déplacement expérimentales et numériques, les courbes de simulations en traction-croix présentent une tendance vraisemblable et permettent l'identification de deux étapes principales: une, linéaire, dans laquelle tous les éléments de la liaison réagissent de manière commune; une seconde, non-linéaire, qui présente une réponse proportionnelle aux propriétés mécaniques de la tôle inférieure. Parmi les sensibilités du modèle étudié, le remaillage et le pas de temps affectent significativement la réponse de l'assemblage en introduisant des erreurs qui s'accumulent sur la position spatiale des noeuds des maillages, et créant un glissement artificiel à l'interface clou-tôle inférieure. Enfin, les simulations de traction-cisaillement donnent des résultats en accord avec l'expérience aussi bien en termes de courbes force-déplacement que sur les mécanismes de déformations au sein des tôles, responsables à terme de la rupture de l'assemblage.

Acknowledgments

First of all, I would like to thank both my academic and industrial supervisors for giving me the opportunity to carry out this PhD thesis between Technocentre Renault and CEMEF laboratory. I would like to express my gratitude to Katia Mocellin who has always supervised my work in a kind and supportive way and let me benefit her experience on leading research projects. I also want to thank Philippe Michel, Lucas Passemard and Mathieu De Metz Nobalt from Renault to help me to catch what are the industrial issues of multi-materials applications in automotive manufacturing and help me to refine the targets of this PhD. I want to sincerely acknowledge the members of my PhD thesis committee, Ms. Patricia Verleysen, M.

Plasticity

The conditions of the material subjected to impact and perforation mechanisms had been formulated in [START_REF] Ortiz | Adaptive lagrangian modelling of ballistic penetration of metallic targets[END_REF] by assuming that the volumetric or dilatational response is governed by a state equation while the shear or deviatoric response obeys to conventional flow theory of plasticity.

The Cauchy stress-tensor σ is decomposed into a deviatoric part s and a volumetric part p with I the identity matrix as (3.8). σ = pI + s (3.8)

Hydrostatic stress

A state equation describes the hydrostatic stress or pressure as a function of density and temperature.

The Mie-Gruneisen formulation (3.9) is generally adopted as in [START_REF] Backman | The mechanics of penetration of projectiles into targets[END_REF] and [START_REF] Ortiz | Computational modelling of impact damage of brittle material[END_REF].

where ξ = J -1, J = det(F * ) = ρ 0 /ρ is the Jacobian of deformation, F * is the transformation gradient, ρ is the current mass density, e the internal energy per unit mass, Γ is the Gruneisen coefficient and K 1 , K 2 and K 3 are material coefficients, obtained from Hugoniot curve which represent the evolution of shock front data for a given material. In the range of impact velocity from 0,5 to 2 km/s, according to [START_REF] Zukas | Impact dynamics: theory and experiment[END_REF], only moderate pressures are generated by solid-solid impact. Thus a linear form of the state equation would be sufficient to treat weak shock as (3.10):

where K is an artificial bulk modulus and ε V the volumetric strain. Below 500 m/s, in the range of "low impact velocities", no use of state equation in the constitutive model is mentioned by leading researchers.

Deviatoric stress and yield function

Metal incompressibility is assumed whatever is the state of stress or strain of the material volume under consideration. Consequently hydrostatic stress has no effect on plasticity and metal flow is only driven by the deviatoric part of the stress-tensor, i.e. the second invariant J 2 and third invariant J 3 .

Hence two configurations are possible for material: either it is in an elastic state or in a plastic state.

The domain of elasticity C is assumed to be a convex and closed part of the stress space Σ with the origin O located inside this domain. According to Hill's principle, the border of the elastic domain defines the plastic state region of the stress space Σ and is expressed by a yield function Φ(σ) pictured in Figure 3.1.

plasticité des milieux poreux, la mécanique de l'endommagement et les modèles phénoménologiques.

Ceux-ci sont présentés tels qu'il puissent être couplés ou découplés au tenseur des contraintes pour représenter la dégradation des propriétés mécaniques avec la croissance de l'endommagement. Plusieurs critères de ruptures rencontrés dans la littérature sur des problématiques de dynamique sont ensuite présentés pour être utilisés indistinctement de manière couplée ou découplée. Enfin une courte présentation est faite sur les modèles rencontrés dans la littérature pour la simulation d'assemblage par déformation plastique et les problématiques principales y sont discutées.

agreement of the Johnson-Cook formulation to capture strain-rate hardening of steel. Since Hopkinson tests have been done only for steel (DP780), parameters β 1 , β 2 are taken from [START_REF] Teng | High Velocity Impact Fracture[END_REF] for cast aluminium -considering a 2024-T351 aluminium alloy -and nail-shaft material -considering a Weldox 460E steel -and from [START_REF] Jeunechamps | Simulation numérique, à l'aide d'algorithmes thermomécaniques implicites, de matériaux endommageables pouvant subir de grandes vitesses de déformation[END_REF] for AHSS (DP1180) -considering an another dual-phase steel (DP800). See Table 4.5 for additional informations.

Thermal softening

Thermal softening is treated in a different way for steel and cast aluminium. Since very high strainrate implies a significant temperature increase (due to adiabaticity), thermal softening can not be dissociated from the dynamic hardening. As consequences we have considered that dynamic-thermal competition on yield stress can be well captured only with two coefficients β 1 and β 2 . For steel, these coefficients have been evaluated from Hopkinson tests and thermal parameter set to θ 1 = 0 whereas for cast aluminium they have been taken in the literature from the works of [START_REF] Teng | High Velocity Impact Fracture[END_REF] and for AHSS they have been taken from Jeunechamps (2008) on another dual-phase steel (DP800).

Nail-shaft material

As mentioned in section 4.2.1.5, a compression set-up has been used to test nail-shaft mechanical behavior. The Voce yield stress formulation was assumed valid and parameters λ 3 , λ 4 and λ 5 were selected for inverse analyis. Former described procedure were used to simulate shaft compression until a satisfying set of parameters is obtained. The friction parameters have been preliminary tested through several simulations to ensure that "barrel effect" is captured independently from material parameters identification. A Coulomb-limited Tresca model was chosen to state for die-specimen friction parameters {m = 0.7; µ = 0.2} with m and µ which are respectively the Tresca and Coulomb coefficients. The best result from optimization is obtained with simulation #111 (see Table B.1) and is presented in Figure 4.15 with the set of parameters detailed in Table 4.5. The first point to notice is that {Cast alu./Steel} superposition does not undergo the diversity of failure modes observed in the {Steel/Cast alu.}'s. Indeed the lower sheet made of steel always fails -whatever are the friction conditions -with a mixed shear-plugging and tearing mode without significant variation in the plug geometry. The failure mode of the upper sheet made of cast aluminium does not change with friction conditions but only the bulge is prevent from growth. The high friction contact sticks the

Cross-tension

The cross-tension simulation is built from results of relaxation simulation (see Figure 5.24). The generation of a new mesh has been necessary due to the mesh transformation presented in Figure 5.22b but with an identical mesh size distribution as the previous one. According to experimental conditions, upper sheet end is clamped while the lower sheet end is moved downwards at constant velocity 1 mm/min. Remeshing procedure is forbidden to avoid that interpolation of node spatial coordinates creates virtual sliding at the nail-lower sheet interface. Finally, material models are unchanged and friction is set to sliding conditions at every interface, except at the nail-lower sheet one which is set to sticking condition.

Shear-tension

The shear-tension simulation is built from results of relaxation simulation (see Figure 5.25). The generation of a new mesh has been necessary due to the mesh transformation presented in Figure 5.22c but with an identical mesh size distribution as the previous one. According to experimental conditions, lower sheet end is clamped while the upper sheet end is moved rightwards at constant velocity 1 mm/min. Remeshing procedure is mandatory to account for the large deformation undergone by both sheets at the interface with the nail. Material models are unchanged and friction is set to sliding conditions at every interface, except at the nail-lower sheet one which is set to {m = 1 ; µ = 1} for a maximum friction conditions without objects to be totally stuck together.

Lower sheet clamped Upper sheet moved Mesh sizes (from impact to outside) 0.1; 0.2; 0.5; 0.7 mm Two stages compose the nailed-joint response to cross-tension:

1. Nail, upper sheet and lower sheet react as one, with identical stiffness for {Steel/Cast alu.} and {Cast alu./Steel} joint. This stage is complete when the lower sheet moving part is horizontal.

2. As the lower sheet moving part goes below the horizontal line (D p > 1 mm), upper sheet bending keeps increasing and a lower sheet reverse bending is created. The upper sheet bending pulls out material from nail contact whereas the lower sheet reverse bending pushes it into the nail ridges.

Nail grip depends on the lower sheet material stiffness which is getting sheared at the interface and pictured by a divergence between {Steel/Cast alu.} and {Cast alu./Steel} force-displacement curves.

Observation 5.12: Joint cross-section (7)

Cross-tension simulations capture global sheets bending as well as local phenomena discussed in Observation 2.21: the upper sheet is pulled out the nail contact interface which decreases the load transmitting area; the lower sheet is pushed on the nail shaft which promotes material flow into the ridges. Only sheared material and fracture could not be modelled yet.

Model sensitivity I: mesh size

The mesh size has been changed in the two main regions of {Steel/Cast alu.} and {Cast alu./Steel} joined sheets: the bending zone and the nail-lower sheet interface. The bending zone is changed from 0.5 mm to 0.3 mm-element while nail-lower sheet interface is kept to 0.1 mm. Then the nail-lower sheet interface is changed from 0.1 to 0.05 mm (damage field is reset to zero) while the mesh size in the bending zone remained set to 0.5 mm. Consequently the joint response is considered equivalent between tests, whatever is the mesh size in the bending zone and at the nail-sheet interface, until the force dropped. Unfortunately this force drop results from the interruption of computation due to mesh issues. Figure 5.33 reveals that mesh inter-penetration becomes critical when a node position is re-interpolated at a wrong location.

In Figure 5.33a, a node from the nail ridge has been projected into the lower sheet mesh, which is definitely going to cause job abortion at short-term. The same pattern has been observed in Figure 5.33b on {Cast alu./Steel} simulation; except that two spots are subjected to mesh inter-penetration. One is from a nail's node projected into the lower sheet mesh, the other one is from a upper sheet's node projected into the lower sheet mesh. These defects are promoted by mesh low quality induced by the nail insertion stage and the kill element procedure. Therefore contact simulation involving broken meshes necessarily end with inter-penetration issues.

In addition to mesh issues which affect both materials configuration, the {Cast alu./Steel} joint response significantly changes with the mesh size conditions. Since the cast aluminium sheet is subjected to a large bending in this material configuration, it would be relevant to partly allocate this variation to the mesh size reduction in the bending zone from 0.5 mm to 0.3 mm. Moreover, the mesh size reduction at the nail-steel interface facilitates mesh adaptation when material flows into the nail ridges which once again affects the joint response and the force-displacement curve.

Information gathered in previous chapters have enabled the construction of a realistic 3D model of the high-speed nailing process under FORGE ® . {Steel/Cast alu.} and {Cast alu./Steel} are chosen as references to investigate the main features of mixed cast aluminium-steel joining simulations. Its validity is evaluated through comparisons between experimental and numerical joining piston kinematics.

Then fracture mode differences observed between both superposition are allocated to the distribution of piston displacement among three main deformation modes: sheets bending, nail-tip crushing and effective nail penetration. Since the nail impact with upper layer is tougher in {Steel/Cast alu.} configuration than in the {Cast alu./Steel}, crushing distance is bigger and therefore the effective nail penetration is reduced as much. An explanation can also be extracted from local analysis of both sheets stress-states in their strain localization zones: lower sheet fracture zone undergoes a loading path with higher stress-triaxility (σ * avg ≈ 0.6) than for the upper sheet (σ * avg ≈ 0.45). Thermomechanical variables such as damage and temperature are also investigated at the final state of nail insertion. It corroborates the IR thermal measurement discussed in Chapter 2 and highlights the zone inside the sheets where damage has spread.

The model sensitivity to mesh-size, friction, loading, boundary conditions and damage parameters is evaluated and the following conclusions can be drawn: (i) mesh-size does not affect the sheets reaction but it significantly changes the nail penetration distance; (ii) friction parameters do not modify {Cast alu./Steel} fracture modes but it promotes dominant shear-plugging in {Steel/Cast alu.} if friction is maximized at every interfaces and rather tensile-tearing if it is minimized. Frictionless conditions at nail-cast aluminium interface and high friction at nail-steel interface give the best results and account for the competition between these two fracture mechanisms; (iii) as observed in experiments, the nail penetration state increases with the joining pressure but sensitivity to the clamping conditions can not be captured by the current model; (iv) none of the tested damage formulation enables a perfect prediction of the joining piston kinematics. Moreover, the best prediction cannot be fully allocated to inherent damage formulation. It is highlighted that the critical damage prevails to control material's ductility and therefore nail penetration distance.

Applied to {Steel/Steel}, {Cast alu./Cast alu.} and mixed cast aluminium-AHSS superposition, the simulation validity is confirmed to account for nail and sheets deformations, fracture modes, nonjoinable configuration and joint characteristic dimensions. The main discrepancy is noticed at the nail-lower sheet contact distance; bigger for cast aluminium than steel due to the unpredictability of the crack location and the fracture mode.

Cross-tension and shear-tension simulations are then computed from the results of a relaxation stage simulation. Despite a gap between experimental and numerical force-displacement curves, a similar trend is captured by cross-tension simulation and two main stages are identified: a linear stage driven by joint components reacting as one; a non-linear stage with the nail grip depending on the lower sheet material stiffness. Among the sensitivity simulation parameters which have been investigated, remeshing frequency and time step introduce errors on the nodes location which accumulate and affect

Summary of Observations

Several observations have been developed throughout this document. It provides key points in the exploration of this joining process and the mechanisms involved in the nail insertion stage and nailedjoint final properties. These observations were countered and sorted per topic of interests in order to highlight the progression of knowledge in the following themes: Joining process settings, Nail insertion stage, Deformation and fracture modes, Sheets temperature increase and Nailed-joint properties.

Joining process settings

Three topics are related to the joining process settings, i.e. the loading conditions which are effectively applied to nail and piston during the joining stage: the joining velocity, the joining force and the hold-down force.

Joining velocity 1.0 Piston impact velocity evolves almost linearly with the joining pressure.

Joining force 1.0 In addition to piston and nail inertia, joining pressure inserted in the main chamber induces a force which keep pushing the joining piston all along its descent.

2.0 This force amplitude is maximum at impact and decreases while the nail is being inserted. 1.2 This force amplitude increases with the Hold-down pressure.

1.3

If joining is conducted towards the gravity direction, the Hold-down force is significantly bigger than if it was in the opposite direction.

2.1 Before joining starts, the tool contacts the upper sheet and hold-down pressure is adjusted.

2.2 Hold-down force is constant whatever is the materials superposition (steel, cast alu ...).

Sheets reaction to the nail insertion 1.0 Due to differences between the joining force and the sheets reaction (F eq <<< F reac ), it can be assumed that piston kinematics is almost entirely assigned by sheets reaction to nail penetration.

2.1 Sheets response to nail penetration is inherent to the sheets and nail materials, geometries... , i.e. nail-sheets interactions parameters.

2.2

The maximum sheets force amplitude is affected by materials superposition: it is bigger in {Steel/Steel} superposition than in {Steel/Cast alu.} superposition.

2.3

The maximum sheets force amplitude is affected by nail features: it is bigger with a flat nail-tip than an ogival one or with a hard nail material than a soft one.

3.1 Whatever is the tested materials superposition, the maximum sheets force amplitude is bigger in a stiff configuration than in soft one (at optimized joining pressure).

3.2 Sheets reaction curve obtained from test conducted with the optimized joining pressure can be recognized by two features: (1) piston displacement is stopped slightly after the force plateau has been reached; (2) force amplitude does not keep increasing after these plateau.

Deformation and fracture modes

The high-speed nailing process is a fracture-based process and therefore the present work has been focused on these issues. The following themes are considered: ballistic-like joining process, sheets fracture mode, nail deformation and fracture mode.

A ballistic-like joining process 1.0 According to Johnson's damage number δ ≈ 10 -2 applied to this joining process, sheets response to impact is only plastic (no significant dynamic effects).

3.2

The simulation inaccuracy at the local scale to model sheet material interface getting sheared down by nail ridge descent legitimates a maximum temperature increase larger than the one currently predicted by simulation.

Nailed-joint properties

Nailed-joint properties have been investigated through different angles: the joint cross-section which are available after the joint is processed and after it has been mechanically tested; the joint strength obtained from tensile tests and the relation between in-process data and in-service final properties.

Joint cross-section 1.0 About {Steel/Cast alu.} superposition:

1.1 Upper sheet global bending and interface contact length l 1 is significant.

1.2 Due to aluminium shear-plugging, the volume loss reduces the nail-aluminium sheet interface contact area l 2 .

1.3 Nail-lower sheet interface contact area is mostly flat, i.e. no indication of sheet material flowing into the nail ridges.

About {Cast alu./Steel} superposition:

2.1 Aluminium is pushed apart while nail is penetrating the sheets. Volume conservation and Sheets fracture mode 3.3 explain why a significant inter-sheets gap g can exist.

2.2

As noticed for lower aluminium made sheet in Joint cross-section 1.3, the nail-steel interface is mostly flat. Some wave-like material bulges can be noticed at the ridge front, pushed downwards by the nail descent but there are no indications of sheet material flowing into the nail ridges.

3.1 Nail-Upper sheet contact distance l 1 variability is negligible (less than 0.15 mm).

3.2 Nail-Lower sheet contact distance l 2 is ranging 116%-122% of initial sheet thickness for steel but only 88%-94% for aluminium. l 2 -variability is bigger for aluminium -0.73 mm -than for steel -0.17 mm.

From Sheets fracture mode 3.2 and Joint cross-section 1.2, these differences can be assigned to the materials inherent ductility.

4.1 Global as well as local sheets bending (from cross-section) suggest that during a cross-tension test, contact area at lower sheet-nail interface has evolved according to bending state.

4.2

Although no lower sheet material has been observed into the nail ridges after the joining stage (see Joint cross-section 1.3 and Joint cross-section 2.2), joint failure analysis reveals that it did flow into the nail ridges during cross-tension test.

4.3

Fragments of sheared materials can be found into almost every ridges located in the nail lower part.

5.1 Two zones have been plastically deformed by the nail contact. These zones are found on both sides of the nail. The first zone is in the upper sheet, the second is in the lower sheet.

5.2

Nail shaft is twisted by the sheets displacement. A crack propagates through the shaft and lead to global joint failure.

6.1

The gap between experimental and numerical nail-upper sheet contact distance l 1 is the same

Conclusions and Outlook

Achievements

This work aims to build a model of the high-speed nailing process applied to dissimilar materials in order to use it at the design stage of multi-materials structures. It would provide an helpful evaluation of the feasibility and the main features of the joint. Experimental nailing campaigns were necessary to understand the joining process principle, supply a database for further comparisons with simulation and captured the phenomena which have to be accounted in the model. The scientific literature was also investigated to find an appropriate set of material models and characterization tests adapted to dynamic and perforation issues. These objectives were reached through the following tasks.

Loading conditions

The joining pressure and hold-down pressure, i.e. the input settings of the nailing device, are investigated through twenty-six nailing tests conducted without sheets at the joining spot. The piston kinematics is analyzed to get a more convenient representation of the loading condition applied to piston for various joining pressure setting: a loading surface (Figure 2.8) draws the airpressure force evolution according to impact velocity and piston displacement.

Nailing phenomena Two set-up are specially designed to (i) ensure the access for camera recordings from beside and below the sheets while joint is being processed and (ii) ensure sheets clamping conditions and prevent the joining stage from uncertainties at the boundaries. Tests conducted with an high-speed camera gave an insight into the sheets fracture mechanisms observed during the nail insertion stage: petalling in {Cast alu./Steel} superposition with the upper layer pushed apart while the lower one is teared at the center; mixed petalling-shear plugging in {Steel/Cast alu.} promoted by the nail-tip which is crushed after impact with the upper layer. Therefore a steel plug is formed under the nail-tip and a competition arises at the lower sheet between through-thickness crack which grows from the upper part and tensile tearing which creates radial cracks at the bottom part. Superposition made of AHSS and cast aluminium were also tested to depict the current limitation of the process: excessive nail deformation if the AHSS is located as second layer, nail fracture if it located as the upper one. Experiments revealed that nail geometry, material properties, superposition, impact velocity and stiffness are all parameters which control the nail-sheets interactions and the joining stage as well. Temperature increase undergone by steel sheets were also investigated with Infra-Red camera recordings up to 2500 frame/s with a maximum surface temperature estimated at 450°C.

Nailing simulation A finite element model of the joining process is built with FORGE ® on a limited slice of 10°to reduce the computation time (∆t CP U = 59 h 1 ). {Steel/Cast alu.} and {Cast alu./Steel} are taken as reference configurations for the joining simulation. It highlighted the sequence of mechanisms which are involved in the nail insertion stage. The piston displacement is transformed into three main phenomena: (i) nail-tip reduction, (ii) sheets bending and (iii) effective nail penetration, i.e. deformation in the impact location and fracture. The same pattern observed in experiment was captured by simulation: {Steel/Cast alu.} implies large deformation at the nail-tip after impact with the steel sheet which therefore restricts the energy available for the nail insertion. {Cast alu./Steel} preserves the nail-tip ogival shape until steel sheet is bent enough to be easily perforated. Thus the simulation accounts for the nail penetration-ability regarding material stiffness and superposition.

Sheets fracture pattern and final joint geometries are also satisfactory for configurations made of steel, cast aluminium and AHSS layers. The process limitations are also captured by simulation through (i) under-estimated joining pressure set as loading conditions which results in partial nail insertion; (ii) excessive stiffness of the materials superposition which results in nail critical deformation and fracture.

Several sensitivity analysis are assessed to the model on mesh-size requirements, friction and material model parameters. These studies highlighted that the main concern of energy-based joining processes are parameters which affect significantly the distribution of the deformation, especially the critical damage parameter D c which controls material ductility.

Nailed-joint testing simulation After the sheets relaxation simulation is completed, cross-tension and shear-tension models are created with the proper boundary and loading conditions. Despite the discrepancies between experimental and numerical force-displacement curves, cross-tension simulations captured the main deformation modes observed in experiment: a first stage in which nail and sheets reactions are united; it explains why joint response is identical at the beginning of {Steel/Cast alu.} and {Cast alu./Steel} tests. A second stage in which a reverse bending of the lower sheet makes the joint response material-dependent to the lower sheet stiffness. The model is also validated by phenomena captured at the nail-sheets interface: the lower sheet bending tends to push materials inside the nail ridges while the upper sheet bending tends to pull it out the contact interface. The stress distribution is affected and induces a preferential localization leading to fracture in the lowest ductile material, i.e. the cast aluminium.

Shear-tension complexity is bigger than cross-tension one since it implies, from the beginning of simulation, intense contact stresses at the nail-sheets interface with possible mesh inter-penetration. Besides this consideration, simulation involves almost two times more elements than in the cross-tension one.

It makes shear-tension average computation time (∆t CP U = 666 h 2 ) 33 times bigger than cross-tension one (∆t CP U = 20 h 3 ). Despite a simulation run on 1.4 mm-total displacement applied to the joint, good agreement is obtained on force-displacement curves and most of the mechanisms observed in experiment are captured: (i) the nail rotation which is dominant since it drives damage growth; (ii) the two zones in upper and lower sheet which are deformed by nail displacement; (iii) the upper sheet free corner which is detached from contact with the lower sheet by nail displacement. 1 Computation run on 9 cores with a Intel(R) Xeon(R) CPUE5-1650v3 3.50GHz; 32 Go RAM 2 Computation run on 16 cores with a Intel(R) Xeon(R) CPUE5-2665 0 2.40GHz; 32 Go RAM 3 Computation run on 16 virtual cores with a Intel(R) Xeon(R) Gold 6148 CPU 2.40GHz, 2394 MHz Suggestions for future studies Numerical models are powerful tools to simulate physical phenomena within a defined framework. It is convenient since laws, rules, coupling can be added step-by-step to get closer to the reality. The counterpart is that it is a endless target and reality always escapes from models. Choices have to be made on whether a phenomenon should be taken into account or not to properly model a problem.

The present work is a result of choices which tried to be as relevant as possible to model the highspeed nailing process. Several other approaches could have been followed and improvements can still be considered. The following suggestions do not pretend to be exhaustive.

Short-term suggestions Shear-tension joint properties are mostly driven by nail ductility. Therefore it would be convenient to perform instrumented compression tests on nail specimens made of different height/diameter ratio on a steel which has experienced similar forging steps. A damage model could be calibrated on these tests through an inverse analysis procedure, applied to high-speed nailing simulation and used in shear-tension simulation to predict crack occurrence in the nail shaft. Strain-rate hardening of the nail material could also be prospected to investigate its effect on the joining piston kinematics obtained from simulation, i.e. the force-piston displacement curve. The mesh-size could be reduced in the impact location to capture with more accuracy the volume of ejected material and the nailsheets interface geometry which is going to affect the load carrying capability of the joint. Finally, an ARCAN set-up could be used to control joint testing boundary and loading conditions in experiment and simulation and pursue the investigation of the nailed-joint features and their predictability.

Mid-term suggestions An interesting approach to move forward a better description of material's fracture mode and ductility consists in the reconstruction of the loading path which is actually experienced by materials under joining conditions. It would depend on stress-state (σ * , µ σ ), strain-rate ( εpl ) and temperature (T ) evolution and take the form of a 4D fracture locus. Every thermo-mechanical state would be associated to a critical plastic strain to fracture. This strategy is rather unlikely since such conditions could not be recreated yet under laboratory conditions. But at least the fracture strain dependency to strain-rate could be evaluated through tension and compression hopkinson set-up coupled with a DIC measurement system and applied to high-speed nailing simulations. It could affects somehow the sheets reaction during the nail insertion stage and the fracture modes as well.

The phenomenological-based damage approach should also be reformed with a distinction between crack formation and crack propagation. Indeed impact loading generally causes damage to spread in a large region of the sheets. Each element hosts a micro-crack which grows independently of the surrounding elements and their proper damage states'. Therefore a crack path is rather assigned to a serie of isolated damaged elements than one element where damage has been initiated and has propagated thoughout the continuum in a preferential direction. Coupled with an adaptative remeshing procedure at the crack-tip location, a propagation criterion would avoid an over-estimated number of elements to delete due to maximum damage value, improve the prediction of the sheet fracture mode and the nail-sheets interface geometry.

APPENDIX A

Analysis of the piston kinematics

I

The piston loading surface has been established from the twenty-six curves discussed before and relates the evolution of the joining force F eq applied to piston according to initial impact velocity and piston displacement (see

Calculation of the sheets reaction

During the joining stage, piston kinematics is driven by the set of forces listed in section 1.4.3. It includes "Joining force" F eq previously discussed which remains even if there are no sheets to join.

Thus piston kinematics subjected to normal joining configuration -with the sheets properly set -is expressed:

with all the force components related to perforation, bending and deformation phenomena gathered in term F reac. . Thus it is possible to obtain sheets reaction F reac. if joining force F eq is substituted from the piston acceleration term γ. However the piston acceleration calculation is not as easy as it seems to be. Lot of sheets reaction changes are expected throughout the nail penetration progress and they should be captured under small velocity variations.

A polynomial formulation is generally adapted to fit a global curve shape but it is not for local slope variations. Consequently we have considered that a polynomial approximation of piston displacement curve could lead to inaccurate estimation of acceleration, and force as well. Hence, the methodology chosen to calculate sheets reaction F reac. from piston displacement z and joining force F eq is presented in the following steps: II 1. Piston displacement values z(i) are interpolated to have 10 times more i-values than obtained from experiment. After interpolation, z(i)-serie is approximately composed of 1000 i-values (dt = 1.10 -6 s).

2. First velocity estimation is obtained from a discrete formulation v a with an interval between values p = 2 :

3. As noticed in Figure A.2a, v a -serie is noisy which is not good to predict accurately the acceleration. A moving-average computation has been chosen to smooth the curve into v b using the filter function of MATLAB ® library. It acts as a filter which slides a window of length l w = 2 along v a -serie, computing averages of the data contained in each window as proposed in the following equation:

Since acceleration γ and joining force F eq are known, the proper sheets reaction F reac. VI