Wei Liu

Franck Morel Professeur

Salima Bouvier

Vincent Grolleau

Dominique Guines

Identiication of strainrate dependent hardening sensitivity of metallic sheets under in-plane biaxial loading

Identification of strainrate dependent hardening sensitivity of metallic sheets under in-plane biaxial loading

Résumé

Les procédés de mise en forme des tôles métalliques sont largement utilisés dans l'industrie mécanique. La simulation numérique des opérations de mise en forme nécessite une caractérisation précise des modèles de comportement rhéologique des matériaux. Dans de nombreuses opérations de mise en forme des tôles métalliques telle que l'emboutissage, l'hydroformage, …, de grandes déformations et des vitesses de déformations dites intermédiaires peuvent être atteintes sous des états biaxiaux de déformation ou de contrainte. L'objectif de ce travail est de montrer le potentiel de l'essai de traction bi-axiale pour caractériser l'écrouissage des tôles métalliques pour de grandes déformations et dans une gamme de vitesse de déformation dite intermédiaire. A partir de simulations numériques, une forme optimale d'éprouvette en croix, permettant d'atteindre 30% de déformation plastique équivalente dans la zone centrale de l'éprouvette sous un chargement équibiaxial, a été proposée.

Par la suite, des essais quasi-statiques et dynamiques de traction bi-axiale ont été réalisés sur la forme d'éprouvette proposée à partir d'une machine dédiée d'essais servo-hydraulique à quatre vérins. Dans un premier temps, le matériau choisi est un alliage d'aluminium AA5086 ne présentant pas de dépendance à la vitesse de déformation. Les déformations expérimentales sont déterminées à partir de la technique de corrélation d'images. L'écrouissage isotrope de différents modèles est identiié à partir d'une procédure inverse basée sur une modélisation éléments inis de l'essai de traction biaxiale. Trois critères de plasticité (Mises, Hill 48 The objective of this work is to show the potential of the biaxial in-plane tensile test to characterize the hardening behaviour of metal sheets up to large strain levels. By numerical investigation, an optimal cruciform shape is designed to obtain large equivalent plastic strain, up to 30%, at the central zone under equi-biaxial strain path.

As expected, the initial cracks of tested specimens are always observed at the central zone.

Then, quasi-static and dynamic biaxial tensile tests on in-plane cross specimens have been performed on a dedicated servo-hydraulic machine. These biaxial tensile tests have been carried out on aluminium alloy AA5086 to validate the identification methodology of hardening behaviour under biaxial loading. This alloy has been chosen since its hardening behaviour is not dependent on the strain rate. Digital Image Correlation (DIC) technique is used for strain measurement. The parameters of isotropic hardening models are identified by inverse analysis based on the finite element model of the biaxial tensile test. Three yield criteria of Mises, Hill48 and Bron and Besson are compared for the parameter identification of different hardening laws. It is shown that the hardening law identified by biaxial test is precise only if an appropriate yield function is preliminarily determined. The biaxial flow stress curve identified with Bron and Besson yield function have been found in good agreement with the experimental flow stress curve obtained from uniaxial tensile tests. For biaxial tests at intermediate strain rates, damping layers are adopted to reduce oscillations on force versus time curves. The comparison of flow stress curves, identified from quasi-static and dynamic biaxial in-plane tensile tests on the non strain rate-dependent material AA5086, validates the identification methodology of strainrate dependent hardening models.

Finally, the proposed methodology is applied to the hardening characterization of a strain-rate dependent Dual Phase steel DP600 at room temperature. Identified biaxial flow stress curves have been compared with uniaxial ones for different strain rates ( [START_REF] Kim | Dynamic tensile test and specimen design of auto-body steel sheet at the intermediate strain rate[END_REF] .................................... Fig. 1-13 Dynamic testing device with long bar [START_REF] Gilat | A New Compression Intermediate Strain Rate Testing Apparatus[END_REF] .................................................... Fig. 1-14 A simplified schematic of the modified servo-hydraulic machine [START_REF] Othman | A modified servohydraulic machine for testing at intermediate strain rates[END_REF] ......... Fig. 1-15 Position of strain gauges on specimen for local force measurement [START_REF] Wood | Strain rate testing of metallic materials and their modeling for use in CAE based automotive crash simulation tools[END_REF] ...... Fig. 1-16 Comparisons of force measured by local force transducer and DLC [START_REF] Wood | Strain rate testing of metallic materials and their modeling for use in CAE based automotive crash simulation tools[END_REF] ..... Fig. 1-17 Dynamic tensile testing device with damping [START_REF] Fitoussi | Experimental methodology for high strain-rates tensile behaviour analysis of polymer matrix composites[END_REF] ..

Tables

XIII

Tables

General introduction

Sheet metal forming processes are widely adopted to produce panels, tubes, profiled parts in industry. Numerical simulation can be used to predict the defects of parts and optimize the forming process. An accurate constitutive model of material is critically demanded for numerical simulation. For phenomenological modelling of sheet metals, many aspects should be considered: anisotropy of rolled sheet metals, strain path, large strain level, strain rate, temperature, … Many constitutive models have been proposed to characterize the plastic behaviour of sheet metals.

Various conventional quasi-static and dynamic tests have been suggested to identify the parameters of material models with some specific limitations for each of them. For example, the deformation level of standardised uniaxial tensile test until necking is not enough large for forming application or the range of strain paths deserved in a bulge test is rather limited. Because the loading ratio along two axes can be online controlled, the in-plane biaxial tests on cruciform specimen can be carried out to realize various proportional or nonpropotional strain paths. The biaxial tests on cruciform specimen have been applied for characterization of yield loci, identification of hardening models and determination of Forming Limit Curves (FLC). The shape design of cruciform specimen is a key issue for tests. Many cruciform shapes have been proposed.

A dynamic biaxial tensile testing machine with four independent servo-hydraulic actuators have been developed in LGCGM (Laboratoire de Génie Civil et Génie Mécanique) of INSA de Rennes. Various linear or nonlinear strain paths can be realised by this biaxial machine. The deformation fields of specimen can be calculated by Digital Image Correlation (DIC).

In this work, a cruciform shape is designed to obtain large strains at the central zone of specimen under quasi-equibiaxial tensile path. Quasi-static and dynamic biaxial tensile tests are performed to identify strain-rate dependent hardening behaviour from quasi-static to intermediate ( 10 - s ) strain rates. Due to a little strain rate sensitivity at room temperature, Aluminium Alloy AA5086 sheet is firstly tested to validate the identification methodology of hardening laws under dynamic biaxial tensile loadings. Then, this methodology is applied to Dual Phase DP600 steel to identify different strain-rate dependent hardening laws.

The main contents of this work are presented in four chapters as follows:

In chapter 1, the literature about constitutive models and mechanical tests is reviewed. Various hardening laws have been suggested with thermal softening effects and strain rate sensitivity. Several conventional tests under different strain paths have been proposed to identify the parameters of material models. Meanwhile, dynamic testing methods have been especially focused on.

In chapter 2, an optimal cruciform shape is designed by numerical investigations. The dynamic biaxial testing machine with four independent servo-hydraulic actuators is briefly introduced. From literature, various cruciform shapes have been designed for biaxial tensile tests. The effects of notches at the intersections, slits in each arm and thickness reduction at the central zone on the equivalent plastic strain level and distribution are investigated by Finite Element (FE) simulations. An optimal shape of cruciform specimen is designed to obtain large strain level at the central zone under biaxial tensile strain path.

In chapter 3, the identification methodology of hardening behaviour under dynamic biaxial loadings is validated by tests on aluminium alloy AA5086. The proposed shape of cruciform specimen is experimentally validated to obtain large strains. The parameters of isotropic hardening laws are identified by inverse analysis based on a FE model of the test. The biaxial flow stress curves identified by biaxial tensile tests are compared with flow stress curves from uniaxial tensile tests. By comparison of major and minor principal strains, the advanced anisotropic yield function of Bron and Besson 2004 has been further validated for parameter identification of hardening laws.

In chapter 4, the proposed biaxial tensile test and parameter identification strategy are applied to identify the strain-rate dependent hardening model of Dual Phase DP600 steel. Different strain-rate dependent hardening models are compared to describe the hardening behaviour. Quasi-static and dynamic biaxial tensile tests on DP600 steel are performed at different loading velocities. Flow stress curves from uniaxial and biaxial tests are compared in order to show the benefits of the proposed methodology.

Finally, a summary of contributions of this work is given and some perspectives for further research on this subject are considered.

Introduction

For many industrial fields such as automobile and aeronautical industry, sheet metal forming processes are of great importance to produce panels, tubes, profile parts and so on [START_REF] Islamoff | Metal forming techniques[END_REF] . Sheet metal forming process consists in making use of the plasticity of metal to obtain the desired shapes by applying loads. The main defects of deformed parts during sheet metal forming processes include: crack, wrinkle, springback and surface deflections [2] [3] . For prediction of failures and optimization of sheet metal forming processes, numerical simulation is widely adopted instead of experimental trial-anderror method to reduce the time and cost of development [4] . An accurate material model is critically demanded for finite element (FE) simulation in the ranges of strain level and strain rate that cover the whole process.

Metal plasticity is fundamentally associated with the nucleation and migration of dislocations in the crystals and polycrystals as the underlying basis for microstructure rearrangement or evolution during plastic flow [START_REF] David | A perspective on trends in multiscale plasticity[END_REF] . Up to date, the dislocation slip theory is the most well-known mechanism for plastic deformation during metal forming processes [START_REF] Boogaard | Thermally enhanced forming of aluminium sheet: modelling and experiments[END_REF] . As shown in Fig. 1-1, although quantifying plastic deformation at microstructure scale is playing an increasing role, the simulation at micro-scale needs very large storage and is very time-consuming. Especially, it is not practical for simulation of sheet metal forming processes on the basis of physically based model [START_REF] Rabahallah | Modélisation de l'anisotropie plastique et application à la mise en forme des tôles métalliques[END_REF] . Therefore, phenomenological modelling of material behaviour remains predominant for numerical simulation of sheet metal forming processes [START_REF] Domkin | Constitutive models based on dislocation density: Formulation and implementation into finite element codes[END_REF] . For characterization of elastoplastic behaviour, experimental techniques and specimens should be carefully designed to carry out the tests at the conditions met in sheet metal forming processes.

Influence of deformation mode

Strain state and path

In the principal strain space ( )

3 2 1 , , ε ε ε with the assumption of 3 2 1 ε ε ε ≥ ≥ , 1
ε and 2 ε are named as the major and minor principal strains in the plane of sheet metal. In the principal stress space ( )

3 2 1 , , σ σ σ
, the stress along thickness direction is neglected

( 0 3 = σ
), because thickness is much smaller than length and width. Then, sheet metal is supposed to deform under plane stress state.

During deformation, the strain state (defined by the ratio of minor and major principal strains ) may keep unchanged or changed. Strain path is usually defined to describe the evolution of the strain state. Due to the geometrical constraints and boundary conditions during sheet metal forming processes, regions of the blank are usually deformed under various strain states and paths [9] [10] . A drawing process of a cylinder part is taken as an example in Fig. 12. Stress and strain states, strain path for different regions (A, B, C) are described in Fig. 1-3 and Fig. 1234.

1 σ 2 σ 1 ε 2 ε 1 σ 2 σ 1 ε 1 σ 2 σ 1 ε 2 ε A B C 2 ε 1 σ 2 σ 1 ε 1 σ 2 σ 1 ε 2 ε A B 1 σ 2 σ 1 ε 2 ε 1 ε C Plane strain
Biaxial tension Punch Die Binder Blank Strain paths may be proportional or non-proportional during deformation [START_REF] Holmberg | Evaluation of sheet metal formability by tensile tests[END_REF] . As it is known, plastic deformation is strain path dependent. To study the effect of strain path changes, the two-stage strain path change tests have been adopted. As an example, a large specimen of uniaxial tension test is firstly deformed, then a smaller specimen is cut from the deformed specimen at an angle, and finally the smaller specimen is uniaxially tested [12] [13] . To study the forming limits under nonlinear strain paths, a two-stage continuous forming process (Benchmark 1 of Numisheet 2014) is studied for a draw/reverse draw panel with a significant strain path change. For these two experimental processes, the strain path can not be changed online.

Strain rate

During sheet metal forming processes, the deformation of material occurs in the time interval dt , and strain rate εɺ is defined by the strain increment per unit time:

dt dε ε = ɺ Eq. 1-1
The strain rates of typical sheet metal forming processes in the automotive industry are approximately s 1 10 . For most autobody parts, stamping is a basic process for forming shallow parts in a press by stretching the sheet over a punch and die set.

With an increasing demand of forming lightweight and more complicated components in automotive and aerospace industry, hydroforming is developed as a set of internal high pressure forming processes by using liquid as the forming media [START_REF] Lang | Hydroforming highlights: sheet hydroforming and tube hydroforming[END_REF] [15] [START_REF] Oh | Applications of hydroforming processes to automobile parts[END_REF] . For manufacturing small series and single products, incremental sheet forming (ISF) is developed as a family of sheet forming processes where the deformation is highly localized and the final shape is determined by three dimensional movement of some tool part without the need of a die [START_REF] Emmens | The technology of Incremental Sheet Forming-A brief review of the history[END_REF] .

Nowadays, high speed metal forming (HSMF) technologies have been developed with higher strain rates ( s s 4 2 10 10 -). For example, the electromagnetic metal forming (EMF) [18] [19] uses pulse magnetic fields to apply forces to the tubular or sheet metal workpieces, of which the material is high electrical conductive. Due to the extremely high velocities in comparison with the conventional quasi-static forming processes, the forming limits can be extended depending on materials [START_REF] Chu | Determination of the flow curve at high strain rates using electromagnetic punch stretching[END_REF] . The approximate forming velocity and strain rate ranges are compared between the high speed forming and conventional processes, as shown in Table 1-1.

For metallic alloys, the elastoplastic behaviour at high strain rates can be very different from the one under quasi-static state. In this case, they are called as ratedependent materials. Otherwise, they are called as rate-independent ones. Therefore, it is necessary to investigate the elastoplastic properties of materials at various strain rates. To study the strain rate sensitivity, the strain rate jump test with abrupt changes of the strain rate can be performed to remove specimen-to-specimen variations. Sung et al. [START_REF] Sung | A plastic constitutive equation incorporating strain, strain-rate,and temperature[END_REF] have adopted a strain rate jump-down test from a higher strain rate to lower strain rate (in Fig. 12345).

Mathematical modelling of elastoplastic behavior

Constitutive relations

The phenomenogical modelling of elastoplastic behaviour are developed on the basis of thermodynamics. The plastic deformation of material is dissipative and irreversible, and the internal variables should be reasonably chosen to represent the evolution of plastic behaviour. In general, the constitutive relations of elastoplasticity include the following aspects [23] [24] :

(1) A decomposition of strain increment ij dε into an elastic, reversible part

e ij
dε and an irreversible, plastic part

p ij dε : p ij e ij ij d d d ε ε ε + = Eq. 1-3
(2) A elastic law which governs the relation between the stress increment ij dσ and elastic strain increment e ij dε , for which the elasticity law is often adopted as follows:

e kl ijkl ij d C d ε σ = Eq. 1-4
where ijkl C is the elastic tensor.

(3) A yield function f which governs the onset of plastic deformation, for which the yield function should be verified as the following relation:

( ) 0 , = q f ij σ Eq. 1-5
where q is a set of internal variables.

(4) A flow rule which governs the plastic flow and the determination of plastic strain increment p ij dε , for which the associated flow rule is often used for many cases of metal plasticity:

ij p ij f d d σ λ ε ∂ ∂ = Eq. 1-6
where the plastic flow is assumed to occur along the normal direction of the yield function and the scale is defined by the increment of plastic multiplier λ d .

(5) A hardening model which governs the evolution of yield function.

During the plastic deformation of material, the size, position and shape of subsequent yield locus may change. For simplicity, the shape of yield locus ψ is often supposed to be unchangeable during plastic flowing and the subsequent yield criterion can be expressed as follows:

( )

0 = - - σ α σ ψ ij ij
Eq. 1-7

Here, the size of yield locus is controlled by the equivalent stress σ and the position of yield in stress space is determined by the back stress ij and σ is changed, the material model is supposed to be isotropic-kinematical combined hardening, as shown in Fig. 123456. [START_REF] Teaca | Caractérisation expérimental et modélisation de la déformation plastique des tôles métalliques[END_REF] Fig. 123456Hardening models

For the monotonic deformation without any abrupt change of strain path, the isotropic hardening model is usually adopted. It is assumed that the shape of yield locus is not distorted and the position does not move. The isotropic hardening behaviour is characterized only by the evolution of σ . To characterize strain path dependency under nonlinear strain path, the position of subsequent yield locus is assumed to move in the stress space, especially for the Bauschinger phenomenon of reverse loading.

Yield criterion

In the plane stress space, the yield criterion is defined to separate the elastic and plastic states of material. The convexity of yield surface should be assured [START_REF] Phillips | On the concept of the yield surface[END_REF] . Due to the crystallographic structure and the characteristics of rolling process, sheet metals usually exhibit a significant anisotropy of mechanical behaviour [START_REF] Banabic | Sheet metal forming processes: constitutive modelling and numerical simulation[END_REF] . It is necessary to consider the anisotropic yield criteria. For the rolled sheet metals, the orientation is usually defined: rolling direction (RD), transverse direction (TD) and normal direction α

+ = Isotropic hardening Kinematical hardening 2 σ 1 σ 2 σ 1 σ 2 σ 1 σ α
Combined hardening (ND). The variation of plastic behaviour along the direction with an angle θ from RD is assessed by a quantity called Lankford parameter or anisotropic coefficient, which is defined as the ratios of true strains along the width and thickness directions of the tensile testing specimens, as shown in Fig. 1234567.

Mises yield criterion

Based on the observation that a hydrostatic pressure cannot cause plastic yielding, the conclusion that only the elastic energy of distortion influences the transition from elastic to plastic was proposed by von Mises (1913) :

( ) ( ) ( ) ( ) [ ]

Hill 48 yield criterion

For the Hill 48 yield criterion, the axis 1 is parallel to RD, the axis 2 is parallel to TD and the axis 3 is parallel to ND. The yield criterion is presented as follows:

( ) ( ) ( ) ( ) ( ) 2
σ σ σ σ σ σ ψ N H F H H G ij + - + + + = Eq. 1-12
The parameters F , G , H , N of Hill 48 yield criterion for sheet metals can be determined by three anisotropic coefficients 0 r , 45 r and 90 r as follows:

( ) Due to the simplicity of mathematical formulation and parameter identification, Hill 48 yield criterion is widely used in practice. Unfortunately, because the mathematical expression is limited and the number of experimental information for parameter identification is only a few, there are also some important drawbacks of Hill 48 yield criterion. For some materials, especially aluminium alloys, Hill 48 yield surface is located inside the Mises yield surface, while the experimental yield surface is located outside Mises yield surface. This phenomenon is called as the anomalous behaviour and was observed by Woodthrope and Pearce. Only four ears can be predicted by Hill 48 yield criterion for the axisymmetric deep drawing process [START_REF] Yoon | Prediction of six or eight ears in a drawn cup based on a new anisotropic yield function[END_REF] .

0 90 0 1 r r r F + = ; 0 1 1 r G + = ; 0 0 1 r r H + = ; ( )( ) ( ) 0

Yld2000-2d yield criterion

Barlat et al. [START_REF] Barlat | Plane stress yield function for aluminum alloy sheets-part 1: theory[END_REF] have introduced a plane stress yield function (Yld2000-2d) using two linear transformations as follows:

( ) (

) a ij 1 2 1 φ φ σ ψ ′ ′ + ′ = Eq. 1-14
where a is an exponent based on the crystallographic structure of material and a should be 6 and 8 for BBC and FCC materials respectively.

X X 2 1 ′ - ′ = ′ φ Eq. 1-15 a a X X X X 2 1 1 2 2 2 ′ ′ + ′ ′ + ′ ′ + ′ ′ = ′ ′ φ Eq. 1-
ij ij ij ij L T C s C X σ σ ′ = ′ = ′ = ′ Eq. 1-17 ij ij ij ij L T C s C X σ σ ′ ′ = ′ ′ = ′ ′ = ′ ′ Eq. 1-18
Finally, the coefficients of L′ and L ′ ′ can be expressed as follows: where all the independent coefficients k α can be determined with eight material characteristics.

                          - - =                 ′ ′ ′ ′ ′
                                - - - - - - - =                 ′ ′ ′ ′ ′ ′ ′ ′ ′ ′
It is validated on binary Al-Mg alloy sheet metal sample that the predicted yield surface using Yld2000-2d is in excellent agreement with both the experimental and polycrystal yield surfaces. The yield function Yld2000-2d was also applied to the aluminium alloy sheet sample, AA2090-T3 and AA6022-T4. The coefficients are determined by three uniaxial tensile tests along the rolling, diagonal and transversal directions, the bulge test and the through-thickness disk compression [START_REF] Barlat | Plane stress yield function for aluminum alloy sheets-part 1: theory[END_REF] .

Yld2004-18p and Yld2004-13p yield criteria

Barlat et al. [START_REF] Barlat | Linear transformation-based anisotropic yield functions[END_REF] have presented the general terms of anisotropic yield functions based on linear transformations of the stress deviator and discussed the methods to determine yield surface experimentally, including uniaxial tensile tests, hydraulic bulge test, disk compression test, crystal plasticity with microstructure modelling and so on. Two yield functions, Yld2004-18p and Yld2004-13p, have been proposed to describe the anisotropic behaviour of metallic alloys for a full stress state (3D).

The Yld2004-18p yield criterion is proposed with 18 material parameters as follows:

( )

a a a a a a a a a a ij S S S S S S S S S S S S S S S S S S 1 3 3 2 3 1 3 3 2 2 2 1 2 3 1 2 1 1 1 ~4 1       ′ ′ - ′ + ′ ′ - ′ + ′ ′ - ′ + ′ ′ - ′ +       ′ ′ - ′ + ′ ′ - ′ + ′ ′ - ′ + ′ ′ - ′ + ′ ′ - ′ = σ ψ Eq. 1-21
The Yld2004-13p yield criterion is proposed with 13 material parameters as follows:

( ) where the symmetric tensor C contains 18 and 13 material parameters, respectively. The Yld2004-18p and Yld2004-13p yield criteria were applied to a mildly anisotropic 6111-T4 and a strongly anisotropic 2090-T3 aluminium alloy sheet samples. The first one with 18 parameters describe accurately the anisotropic behaviour. The second one with 13 parameters can reasonably describe the main trends [START_REF] Barlat | Linear transformation-based anisotropic yield functions[END_REF] .

a a a a a a a a a a ij S S S S S S S S S S S S 1 3 2 1 3 2 1 2 1 2 1 2 1 ~2 1       ′ ′ + ′ ′ + ′ ′ +             ′ ′ + ′ ′ + ′ ′ - ′ - ′ + ′ - ′ + ′ - ′ = σ ψ

BBC 2005 yield criterion

Banabic et al. [START_REF] Banabic | An improved analytical description of orthotropy in metallic sheets[END_REF] have proposed a BBC 2005 yield criterion for the orthotropic sheet materials in the plane stress space (2D) as follows:

( ) ( ) ( ) ( ) ( ) [ ] k k k k ij a a a 2 1 2 2 2 2 1 Λ ⋅ - + Ψ - Γ ⋅ + Ψ + Γ ⋅ = σ ψ Eq. 1-24
The terms Γ , Ψ and Λ are defined as follows: The parameters L , M , N , P , Q , R , S , T can be determined by the error minimization calibration method, and eight experimental data should be provided. Those data includes: three directional yield stresses obtained from uniaxial tests along the rolling, diagonal and transversal directions; three anisotropic coefficients 0 r , 45 r , 90 r ; the equibiaxial yield stress obtained by cross tensile test or bulge test and the equibiaxial anisotropy coefficient. BBC2005 can also be deduced from Yld2000-2d, and they are not fundamentally different. BBC2005 yield criterion has been validated for characterization of the yield locus of aluminium alloy AA6181-T4 sheet [START_REF] Banabic | An improved analytical description of orthotropy in metallic sheets[END_REF] .

Bron and Besson 2004 yield criterion

In order to represent complex yield surfaces, Bron and Besson (2004) [START_REF] Bron | A yield function for anisotropic materials: Application to aluminum alloys[END_REF] have proposed a phenomenological yield function with 16 parameters as follows:

( ) ( )

a K k a k k ij 1 1       = ∑ = σ α σ ψ Eq. 1-28
Where the K functions k σ are convex with respect to the definition of a yield surface.

Here, only two functions are used ( 2 = K

) and they are defined by: ( )

k b k k 1 φ σ = Eq. 1-29       - + - + - = 1 1 1 1 2 1 1 1 1 1 3 1 3 1 2 1 2 1 b b b S S S S S S φ Eq. 1-30       + + + =
s : ij k ij k ij L s σ = Eq. 1-32 ( ) ( ) ( )                     + - - - + - - - + = k k k k k k k k k k k k k k k k ij c c c c c c c c c c c c c c c L 6 5 4 2 1 1 2 1 2 1 3 2 3 3 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 3 0 0 0 3 3 3 0 0 0 3 3 3
Eq. 1-33

The shape of yield surface is influenced by 4 parameters: a , 1 b , 2 b and 1

α ( 1 2 1 α α - =
). The anisotropy is only controlled by 12 parameters

2 1 6 1 - = - = k i c
. When

1 1 = α and 1 1 = i c
, it becomes Mises yield function.

Bron and Besson 2004 yield function was applied to an aluminium alloys 2024-T4.

The uniaxial tests of smooth tensile samples and U-notched samples with two different notches along the rolling, transverse and diagonal directions are performed for parameter identification by inverse analysis. This yield function was also applied to other sheets of binary Al-Mg, 6022-T4, 2090-T3 and 7075-T351. It was found to be very accurate to describe the plastic anisotropy of these various aluminium sheets [START_REF] Bron | A yield function for anisotropic materials: Application to aluminum alloys[END_REF] .

Hardening law

On the micro-scale, when the material is deformed, dislocations will be generated and annihilated, and the texture evolution will occur. It is usually assumed that the concept of dislocation density links the flow stress to the underlying microstructure evolution.

The flow stress σ can be calculated as follows [START_REF] Lemaitre | Mécanique des matériaux solides[END_REF] :

( )

0 0 δ δ α σ σ - + = Gb Eq. 1-34
Where 0 σ is initial yield stress corresponding to the initial density 0 δ of dislocation, α is a material coefficient, G is the transversal elastic modulus, b is the Burgers vector and δ is the current dislocation density.

Strain hardening

On the macro-scale, if the material is deformed under monotonic strain path without the thermal and strain-rate effect, the equivalent plastic strain p ε is usually chosen to represent the dislocation density δ . In the similar way of Eq. 1-34, the hardening law can be expressed by a one-internal-variable model as following:

( ) p ε σ σ Η 0 + = Eq. 1-35
Where 0 σ is initial yield stress and ( ) p H ε represents the strain hardening effect. Several mathematical formulas of hardening laws have been widely used as follows: Ludwick law is an unsaturated one:

n p Kε σ σ + = 0 Eq. 1-36
Voce law is a saturated one:

(

)

) exp( 1 0 p n K ε σ σ - - + = Eq. 1-37
Because it is found to become saturated too fast sometimes, the Voce law has been modified to decrease the saturating speed, such as Hockett-Sherby law:

( )

) exp( 1 0 m p n K ε σ σ - - + = Eq. 1-38
A generalized Voce law [START_REF] Gavrus | Constitutive equation for description of metallic materials behavior during static and dynamic loadings taking into account important gradients of plastic deformation[END_REF] has also been suggested as follows:

( ) α ε σ σ 1 0 ) exp( 1 p n K - - + = Eq. 1-39
The value of α varies with the type of crystallographic system:

Influence of temperature and strain rate

When the material is deformed at various temperature and strain rate, the hardening law should take three items into account: strain hardening ( )

p ε H , thermal softening ( )
T T and strain-rate sensitivity ( ) [36] . The thermal softening effect ( ) T T and strain-rate sensitivity term ( ) ε ɺ

ε ɺ V [35]
V can be manipulated additively or multiplicatively with the basic formulations of strain hardening law. Generally speaking, according to the coupling effect, four types of flow stress curve [START_REF] Paul | Tensile flow behavior of ultra low carbon, low carbon and micro alloyed steel sheets for auto application under low to intermediate strain rate[END_REF] can be presented in Fig. 12345678. The examples are given as follows:

1) Type-1: initial yield stress 0 σ is not influenced by the temperature and strain rate, but the strain hardening behaviour ( )

p ε H is influenced by temperature ( )
T T and strain rate ( ) ε ɺ V . A simple multiplicative form is given as follows:

( ) ( ) ( ) ε ε σ σ ɺ V T H 0 T p + = Eq. 1-40
The flow stress curves have the same initial point and different hardening rate. The Zerilli-Armstrong model (1987) for FCC [START_REF] Zerilli | Dislocation mechanics based constitutive relations for material dynamics calculations[END_REF] is an example of Type-1.

2) Type-2: initial yield stress 0 σ is influenced by the temperature ( ) T T and strain rate ( ) ε ɺ

V , but the strain hardening behaviour ( ) p H ε is not influenced by the temperature and strain rate. A simple multiplicative form is given as follows:

( ) ( ) ( )

p T ε ε σ σ H V T 0 + = ɺ
Eq. 1-41

It appears that the flow stress curves move upwards or downwards with the same hardening rate. The examples includes: Zerilli-Armstrong model (1987) for BCC, Modified Khan-Huang model ( 2009) proposed by H. Yu et al [START_REF] Yu | Constitutive model on the description of plastic behavior of DP600 steel at strain rate from 10 -4 to 10 3 s -1[END_REF] .

3) Type-3: both initial yield stress 0 σ and strain hardening behaviour ( )

( ) ( ) ( ) ( ) ε ε σ σ ɺ V T H 0 T p + = Eq. 1-42
The examples can be given by Johnson-Cook model (1983), Khan-Huang model (1992) [START_REF] Khan | Experimental and theoretical study of mechanical behavior of 1100 aluminum in the strain rate range 10 -5 -10 4 s -1[END_REF] . 4) Type-4: both initial yield stress 0 σ and strain hardening behaviour ( )

p ε H are
influenced by the temperature ( )

T i T
and strain rate ( )

ε ɺ i V
, but the influences are different. A simple multiplicative form is given as follows:

( ) ( ) ( ) ( ) ( ) ε ε ε σ σ ɺ ɺ 2 2 1 1 0 V T H V T T T p + = Eq. 1-43
This type is a generalised form of other three types. There are many examples, such as Lin-Wagoner model (1987) [START_REF] Lin | Aw experimental investigation of deformation induced heating during tensile testing[END_REF] , Zhao model (1997) [START_REF] Zhao | A constitutive model for metals over a large range of strain rates: Identification for mild-steel and aluminum sheets[END_REF] , Khan-Liang model (1999) [START_REF] Khan | Behaviors of three BCC metal over a wide range of strain rates and temperatures: experiments and modeling[END_REF] , Rusinek-Klepaczko (2007) [START_REF] Rusinek | Constitutive relations in 3-D for a wide range of strain rates and temperatures -Application to mild steels[END_REF] , H/V model (2010) [START_REF] Sung | A plastic constitutive equation incorporating strain, strain-rate, and temperature[END_REF] , Paul model (2012) [START_REF] Paul | Predicting the flow behavior of metals under different strain rate and temperature through phenomenological modeling[END_REF] . [START_REF] Paul | Tensile flow behavior of ultra low carbon, low carbon and micro alloyed steel sheets for auto application under low to intermediate strain rate[END_REF] 1.

Fig. 1-8 Schematic representation of four types stress-strain curves

Simple models

The temperature sensitivity ( ) T T is traditionally described as follows:

Power model:

( ) β         = 0 T T T T Eq. 1-44
where T , 0 T are the current and reference temperatures, β is a material constant.

Exponential model:

( )       = T T β exp T Eq. 1-45
Johnson-Cook model:

ε ɺ T ε ɺ T ε ɺ T ε ɺ T σ p ε σ p ε σ p ε σ p ε Type-1 Type-2 Type-3 Type-4 ( ) m m T T T T T T         - - - = 0 0 1 Eq. 1-46
where m T are the melting temperature of material and m is a material constant.

The strain-rate sensitivity ( )

ε ɺ
V is traditionally described as follows:

Power model:

( ) m         = 0 V ε ε ε ɺ ɺ ɺ Eq. 1-47
where ε ɺ , 0 ε ɺ are the current and reference strain rates, m is a material parameter.

Cowper-Symonds model:

( ) m C 1 1 V         + = ε ε ɺ ɺ Eq. 1-48
where C , m are the material parameters Johnson-Cook model:

( ) 0 ln 1 V ε ε ε ɺ ɺ ɺ C + = Eq. 1-49
The hyperbolic arcsine model [START_REF] Diot | Caractérisation expérimentale et numérique du comportement dynamique des matériaux[END_REF] :

( )                +         ≈                 = large is when ln 1 small is when 2 2 arcsinh V 0 0 0 ε ε ε ε ε ε ε ε ε ɺ ɺ ɺ ɺ ɺ ɺ ɺ ɺ ɺ m e e m

Eq. 1-50

The power model is suitable for low strain rates, while the Cowper-Symonds and Johnson-cook models are used for high strain rate. Although the Johnson-Cook model is widely adopted for high strain rate model due to the simplicity, it cannot reasonably describe the experimental flow stresses when the strain rate changes from quasi-static strain rate ( s / 10 3 -) to high strain rate ( s / 10 3 ). The hyperbolic arcsine model is proposed to transit very well between quasi-static and high strain rates.

Integrated models

• Zerilli-Armstrong model (1987) [START_REF] Zerilli | Dislocation mechanics based constitutive relations for material dynamics calculations[END_REF] :

( ) ( ) n p p C T C T C C C ε ε ε σ σ 5 4 3 2 1 0 ln exp + + - + + = ɺ
Eq. 1-51

for FCC ( 0

5 1 = = C C ): [ ] ε ε σ σ ɺ ln exp 4 3 2 0 T C T C C p + - + = Eq. 1-52 for BCC( 0 2 = C ):
( )

n p C T C T C C ε ε σ σ 5 4 3 1 0 ln exp + + - + = ɺ
Eq. 1-53

Where 0

σ , 1 C , 2 C , 3 C , 4
C and 5 C are the material constants. Based on the results of high speed impact tests, this model has been applied for characterization of OHFC copper (FCC) and Armco iron (BCC).

• Lin-Wagoner model (1987) [START_REF] Lin | Aw experimental investigation of deformation induced heating during tensile testing[END_REF] :

( ) ( ) ( ) [ ] β ε ε ε σ                 - + - = 0 0 0 2 1 exp 1 T T T T C C B A m p p ɺ ɺ Eq. 1-54 Where A , B , 1 C , 2
C , m and β are the material parameters, 0 ε ɺ and 0

T are the reference strain rate and temperature. This model was applied to characterize Armco interstitial-free (IF) steel and stainless steel type 310SS.

• Khan-Huang model (1992) [START_REF] Khan | Experimental and theoretical study of mechanical behavior of 1100 aluminum in the strain rate range 10 -5 -10 4 s -1[END_REF] :

( ) [ ] ( ) ( ) 1 0 0 ln ln 1 exp - ∞       - - - + = ε ε ε α ε σ σ ɺ ɺ p p a E Eq. 1-55
Where 0 σ , a , D , ∞ E , m and α are the material parameters and the reference strain rate s

6 0 10 = ε ɺ
. This model is applied to a quite strain rate sensitive material of 1100-0 aluminium over the strain rates ε ɺ from • Zhao model (1997) [START_REF] Zhao | A constitutive model for metals over a large range of strain rates: Identification for mild-steel and aluminum sheets[END_REF] :

( ) ( ) T E D C B k m p n p ∆ -       +         - + + = µ ε ε ε ε ε σ σ 1 ln 0 0 ɺ ɺ ɺ Eq. 1-56
Where 0 σ , B , C , D , E , k , m , n and 0 ε ɺ are the material parameters, and T ∆ is the temperature difference with respect to the room temperature. In this model, the influence of temperature on the initial yield stress and strain hardening are the same, while the influence of the strain rate on the initial yield stress and strain hardening is different. This model is applied to characterize the plastic behaviour of mild steel and commercial aluminium sheets.

• Khan-Liang model (1999) [START_REF] Khan | Behaviors of three BCC metal over a wide range of strain rates and temperatures: experiments and modeling[END_REF] :

                - - -                 - + = m m C n n T T T T B 0 0 0 0 1 ln ln 1 0 1 ε ε ε ε σ σ ɺ ɺ ɺ Eq. 1-57 Where 0 σ , B , C , 0 n , 1
n , m are the material parameters, the reference strain rate s / 10 6 0 = ε ɺ . In this model, the influence of temperature on the initial yield stress and strain hardening are the same. The model has been adopted to describe three BBC metals, tantalum, tantalum alloy and AerMet 100 steel over a wide range of strains (15%), strain rates ( s 4 6 10 10 --) and temperatures (25-315°C).

• Modified Khan-Huang model ( 2009) proposed by H. Yu [START_REF] Yu | Constitutive model on the description of plastic behavior of DP600 steel at strain rate from 10 -4 to 10 3 s -1[END_REF] : • H/V model (2010) [START_REF] Sung | A plastic constitutive equation incorporating strain, strain-rate, and temperature[END_REF] :

( ) p p m p a E D ε α ε ε ε σ σ - - +                 + = ∞ exp
( ) ( ) ( ) ( ) T h g T f T f p p ⋅ ⋅ = = ε ε ε ε σ ɺ ɺ , , ,

Eq. 1-59

The strain hardening ( )

T f p ,
ε is given by a linear combination of Voce and power law as follows:

( ) ( ) ( ) ( ) V H p f T f T T f ⋅ - + = α α ε 1 , Eq. 1-60 ( ) ( ) ( ) ( )      - - = = - - = p HV HV HV V n p HV H B A V f H f T T T HV ε ε α α α exp 1 0 2 1
Eq. 1-61

( ) ( ) ( ) 0 1 2 log 2 0 ε ε γ γ ε ε ε ɺ ɺ ɺ ɺ ɺ ⋅ +         = g Eq. 1-62 ( ) ( ) 0 1 T T T h - - = β Eq. 1-63 Where 1 γ , 2
γ and β are material constants, and 0 ε ɺ is the reference strain rate.

The H/V model has been adopted for modelling of DP590, DP780 and DP980 steels in the uniform strain range at 25, 50 and 100°C by the strain rate jump tests and the isothermal tensile tests.

• Paul model (2012) [START_REF] Paul | Predicting the flow behavior of metals under different strain rate and temperature through phenomenological modeling[END_REF] :

( ) ( ) ( ) [ ] ( ) (
)

0 0 0 0 0 1 ln 1 exp 1 ln exp T T G H C B T T k A p p - -                 - - - + +         - -         = ε ε ε β ε ε ε σ σ ɺ ɺ ɺ ɺ Eq. 1-64
Where 0 σ , A , B , C , β , k , H , G are the material parameters and 0 T is room temperature. This model is validated on mild steel ES, DP600 steel and TRIP700 steel over a wide strain, strain rate and temperature range.

Mechanical tests of sheet metal

To investigate phenomenological modelling of materials for numerical simulation of sheet metal forming processes, many key issues should be considered, such as: plastic anisotropy of rolled sheet metals, multiaxial loading conditions, large plastic strain, temperature and strain rate effects, strain path changes, and so on. The experimental techniques and specimen design should be proposed to consider specific conditions met in the practical sheet metal forming processes. In this section, the tests under different proportional deformation paths and dynamic conditions will be focused on.

First of all, four traditional tests of sheet metal (uniaxial tensile test, bulge test, plane strain test and simple shear test) are usually carried out to obtain different deformation modes and strain paths [START_REF] Bruschi | Testing and modelling of material behaviour and formability in sheet metal forming[END_REF] [49] [50] [START_REF] Chen | Material Modeling and Correlative Mechanical Testing on AHSS Sheet Forming Simulation[END_REF] . These four tests are usually adopted for characterization of plastic behaviour at four representative points in the principal stress space, as shown in Fig. 123456789.

Fig. 1-9 Representative strain and stress states on yield locus

For material modelling at various strain rates, Field et al. [START_REF] Field | Review of experimental techniques for high rate deformation and shock studies[END_REF] have suggested different experimental techniques: the conventional load frame for creep and stress relaxation, servo-hydraulic system for quasi-static and intermediate strain rates, dropweights, Split Hopkinson Pressure Bars (SHPB), Taylor impact and shock loading by plate impact for high strain rates, as shown in Fig. 12345678910. Because dynamic tests are completed in a short time, the dynamic testing machines and measurement systems are usually different from those for quasi-static tests. materials, which has already been standardised [53] [54] . It can provide the Young's modulus, Poisson's ratio, yield strength, ultimate tensile strength (UTS), uniform elongation and anisotropic coefficients. The uniaxial tensile specimen is shown in Fig. 1-11. When the deformation is uniform before localisation, the true strain, stress and strain rate can be calculated. where G is the gauge length and 0 G is the initial length.

True stress is calculated by: ( )

0 exp S F S F ε σ × = = Eq. 1-66
where F is the loading force, 0 S , S are the initial and current cross sections.

True strain rate is calculated by:

G V dt d = = ε εɺ Eq. 1-67
where V is the loading velocity.

Dynamic uniaxial test

For the dynamic tests at intermediate strain rate range ( s

/ 500 1 . 0 ≤ ≤ ε ɺ
), the servohydraulic system is usually adopted. International Iron and Steel Institute Committee (IISI) [START_REF] Borsutzki | Recommendations for dynamic tensile testing of sheet steels[END_REF] has made recommendations for dynamic tensile testing for sheet steels. Several issues have been discussed such as machine type, specimen, clamping method and measurement systems. For tests at strain rate above s / 10 , the measured force is greatly perturbed by multiple passages of the reflected elastic wave or by the vibratory response of mechanical parts of the experimental device. To solve this problem, various methods have been suggested.

(1) The load cell is shorten to reduce the time for the wave to travel back several times, so that elastic deformation within load cell becomes homogeneous quickly; (2) The load cell is lengthen in order to finish the test before the return back of the reflected elastic wave, in other words, it is based on the SHPB method; (3) Strain gauge is attached on grip section of specimen, where deformation is elastic; (4) A damping method is adopted to improve the impact condition when a sudden loading is applied; (5) The forces applied on specimen are determined by inverse analysis. A numerical model of the test allowing to reproduce the oscillations of force signal is needed.

Kim et al. [56] [57] have developed a high speed tensile testing machine to investigate dynamic tensile behaviour of steel sheets at various strain rates from 003 . 0 to s / 200 . The oscillation phenomenon of the force signal measured by the load cell, called as load ringing, is caused by the vibratory response of the mechanical parts or by the propagation of elastic stress wave. The natural frequency increases as the length of grip decreases. The load cell is attached to the upper grip to measure the force, and the grip has been designed to effectively reduce the load ringing, as shown in Fig. 123456789101112. Meanwhile, because the strain of specimen is calculated from the displacement of the crosshead, it is different from the exact strain at the gauge section. The geometric effects are also studied to compensate the strain discrepancy. [START_REF] Kim | Dynamic tensile test and specimen design of auto-body steel sheet at the intermediate strain rate[END_REF] For testing materials in compression at intermediate strain rate between about 10 and s / 200 , Gilat et al. [START_REF] Gilat | A New Compression Intermediate Strain Rate Testing Apparatus[END_REF] have introduced a new apparatus consisting in a large hydraulic actuator and a long transmitter bar of m 40 length, as shown in Fig. 1-13. Once the specimen is loaded by the actuator that impact directly the free end of the specimen,

Load cell

Upper grip which is deformed between the actuator and the transmitter bar. A compressive wave propagates into the transmitter bar. The amplitude of wave is measured by the strain gages located on the transmitter bar to calculated the force of the specimen as in the SHPB. The transmitter bar is enough long that the experiment will be completed before the reflected wave in the transmitter bar return back to gages, and the measured force will not be disturbed. The strain in the specimen during the tests is measured by 3D Digital Image Correlation (DIC) with high speed cameras.

Fig. 1-13 Dynamic testing device with long bar [START_REF] Gilat | A New Compression Intermediate Strain Rate Testing Apparatus[END_REF] Othman et al. [START_REF] Othman | A modified servohydraulic machine for testing at intermediate strain rates[END_REF] have developed a modified servo-hydraulic machine to assess the mechanical properties of aluminium alloy 2017-T4 at the strain rate of approximately s 100 , as shown in Fig. 1234567891011121314. A long duration Hopkinson bar technique and a wave separation method are adopted for less oscillating force measurement. The force at the bar-attaching device interface is firstly calculated the separation method. Then, the force applied to the specimen is secondly deduced by assuming a rigid mass model of the attaching device. The strain in the specimen is measured by using Digital Speckle Photographs (DSP) which is captured via high-speed video camera. [START_REF] Othman | A modified servohydraulic machine for testing at intermediate strain rates[END_REF] To develop tests at strain rates ranging from quasi-static to s / 500 for application in automotive crash simulation, Wood et al. [START_REF] Wood | Strain rate testing of metallic materials and their modeling for use in CAE based automotive crash simulation tools[END_REF] have presented some recommendations and procedures about the test machine requirements, specimen design, force and strain measurements, data processing and strength hardening constitutive relations. Three types of specimen with different gauged lengths have been proposed for the tests at quasi-static strain rates, low to intermediate strain rates and intermediate to high strain rates. Two methods of force measurement are compared during the dynamic test. The dynamic load cell (DLC) is located between the static grip and the machine frame, which is called as "machine-based force sensor". Meanwhile, a load force transducer is proposed on the specimen using strain gauges, which is called as "local force transducer on specimen", as shown in Fig. 1 [START_REF] Wood | Strain rate testing of metallic materials and their modeling for use in CAE based automotive crash simulation tools[END_REF] Fig. 12345678910111213141516Comparisons of force measured by local force transducer and DLC [START_REF] Wood | Strain rate testing of metallic materials and their modeling for use in CAE based automotive crash simulation tools[END_REF] In order to conduct high speed tensile tests on fibre reinforced polymer composites, Fitoussi et al. [START_REF] Fitoussi | Experimental methodology for high strain-rates tensile behaviour analysis of polymer matrix composites[END_REF] have adopted an experimental device on a servo-hydraulic machine, in Fig. 1-17. The hydraulic jack is accelerated to reach the specific velocity over a straight displacement before the contact of sliding bar and hydraulic jack. A damping joint of a low impedance material is placed between the sliding bar and the hydraulic jack, which attenuated partially the wave effects caused by the dynamic shock. After testing with several geometries and materials, the rubber nitrile was chosen for the damping joint to obtain homogeneous deformation of specimen and minimise the amplitude of shock wave. Boyce et al. [START_REF] Boyce | The dynamic tensile behavior of tough, ultrahigh-strength steels at strain-rates from[END_REF] have employed a servo-hydraulic device to perform tensile tests of tough, ultrahigh-strength steels at strain rates from 0002 . 0 up to s / 200 . To reduce oscillations in the tensile bar and load cell, a custom-designed load cell has been used and it shows about % 50 reduction of the oscillation amplitude. Studying the pulse-shaping materials used in split Hopkinson pressure bar test to smooth the elastic wave, a rubber damper have also been placed in the slack adapter and reduce the initial oscillations by more than % 50 . Xiao [ 63 ] has carried out dynamic tensile tests of four plastic materials using a servo-hydraulic testing machine and a damping layer was also adopted. The single-degree freedom spring-mass model with and without damping was studied to obtain the analytical solutions, which describes the influence of loading rate and the natural frequency of testing system on the magnitude of system ringing and its decay rate. [START_REF] Fitoussi | Experimental methodology for high strain-rates tensile behaviour analysis of polymer matrix composites[END_REF] In order to identify the constitutive model at large strains up to 150% and intermediate strain rates up to s 500 , Diot et al. [64] have carried out uniaxial compressive tests by a servo-hydraulic experimental set-up ( in Fig. 123456789101112131415161718) . The dynamic response of the set-up is experimentally analyzed and a FE model of the whole experimental structure is validated to represent the elastic and inertia effects. To reduce the time consuming for parameter identification, a two-step sequential identification method based on inverse analysis has been suggested: the mathematical expression of the force on the bar/specimen interface is firstly identified on the basis of the previously defined FE model of tools. The material parameters is then identified, taking the identified force as the boundary condition of specimen. Finally, the behaviour of a steel 27MnCr5 has been identified under different loading velocities and temperatures. Zhu et al. [65] [66] have conducted dynamic tensile tests on aluminium alloy AA6061-T6 and engineering materials at intermediate strain rate up to approximately s / 200 using a high-speed servo-hydraulic machine. The predominant frequencies of testing machine were identified by modal analysis, and the effect of loading rate and natural frequency of testing machine on the magnitude of machine ringing was also analyzed by a single degree-of-freedom spring-mass-damper model. Two methods for curve smoothing, which are a low-pass filter with a cut-off frequency and a power-law fitting function, are presented to extract the stress from the original data.

Fig. 1-14 A simplified schematic of the modified servo-hydraulic machine

Split Hopkinson pressure bar

Because the Split Hopkinson Pressure Bar (SHPB) is not only very popular for dynamic uniaxial tests at high strain rate, s 500 ≥ ε ɺ , but also widely used as the basis of other dynamic tests, the theory of SHPB is briefly presented here [67] [68] . It consists in the striker, incident bar and transmit bar. The test is completed in an impulse wave, as shown in Fig. 1-19 [START_REF] Khlif | Adaptation d'un banc d'essai Charpy à la traction rapid des polymères[END_REF] . When the incident bar is impacted by the striker, a compressive impulse has emerged and propagates along the incident bar. The impulse will be recorded by the strain gauge 1 as the incident impulse i ε . When i ε has arrived at the interface of incident bar and specimen, it will reflect and transmit. The reflected impulse will be recorded by the strain gauge 1 as the reflected tensile impulse r ε . The transmit impulse will be measured by the strain gauge 2 as the transmit compressive impulse t ε .

Fig. 1-19 Schematic of Split Hopkinson Pressure Bar and travelling wave diagram

The theory of one-dimensional elastic stress wave propagation is adopted. The specimen is considered to become in equilibrium because the stress wave can travel back several times in the specimen during the test. Therefore, the strain ε , strain rate εɺ and stress σ of specimen can be calculated as follows:

( ) ( ) ( ) ( ) ( ) ( )dt t L C dt t t t L C t t r t i t r ∫ ∫ - = - + - = 0 0 2 ε ε ε ε ε Eq. 1-68 ( ) ( ) ( ) ( ) ( ) ( ) t L C t t t L C t r i t r ε ε ε ε ε 2 - = - + - = ɺ Eq. 1-69 ( ) ( ) ( ) ( ) ( ) ( ) t E S A t t t E S A t t t r i ε ε ε ε σ = + + = 2

Eq. 1-70

Where A is the cross-secrion area of bars, L , S are the length and area of specimen,

ρ E C =
is the velocity of wave in the bars, E is the Young's module of bars and ρ is the density of bars.

Bulge test

Quasi-static bulge test

For the bulge test, the central zone of circular plate is deformed under biaxial tensile state, as shown in Fig. 1-20 [START_REF] Gutscher | Determination of flow stress for sheet metal forming using the viscous pressure bulge (VPB) test[END_REF] . The hydraulic fluid, viscous materials or gas [START_REF] Abu-Farha | Limiting strains of sheet metals obtained by pneumatic stretching at elevated temperatures[END_REF] can be chosen as the pressure medium. Compared with the uniaxial test, bulge test is usually adopted to obtain material behaviour under balanced tensile path, especially for the equivalent stress and equivalent strain curves, which can reach up to much larger deformation level, even two times of that obtained by uniaxial tensile test [72] [73] . The main drawbacks of bulge test are the bending effect and lack of changeable strain paths during tests [START_REF] Atkinson | Accurate determination of biaxial stress-strain relationships from hydraulic bulging tests of sheet metals[END_REF] . The membrane theory is used to determine the stresses and strains.

Fig. 1-20 Bulge test [START_REF] Gutscher | Determination of flow stress for sheet metal forming using the viscous pressure bulge (VPB) test[END_REF] If the material is isotropic and Mises yield criterion is adopted, the equivalent stress σ can be calculated by: ( )

d d t pR 2 2 1 2 1 2 2 2 1 2 2 1 = = = + + - = σ σ σ σ σ σ σ Eq. 1-71
where p is the hydraulic pressure, d R is the radius of dome, d t is the current thickness at the top of dome.

The equivalent strain can be calculated with the assumption of constant volume:

        - = - = = = ⋅ + + = 0 3 2 1 2 1 2 2 2 1 ln 2 2 3 2 t t d ε ε ε ε ε ε ε ε Eq. 1-72
where 0 t is the initial thickness of sheet.

With a fillet c R in the cavity, the radius d R of dome can be calculated by:

( ) ( ) d d c d c c d h h R h R d R 2 2 2 2 - + + = Eq. 1-73
where c d is the diameter of cavity and d h is the dome height.

The thickness d t at the top of dome can be estimated as follows:

( )         + = 2 0 2 1 1 c d d d h t t
Eq. 1-74

The unknown dome height d h is measured during the test.

Dynamic bulge test

For the dynamic biaxial test at high strain rates of up to s / 500 , Grolleau et al. [75] [76] have proposed a dynamic bulge testing device, in Fig. 1-21 . The idea is to design a movable bulge cell in a SHPB system. When the input bar is impacted at a speed, a pressure wave is generated and transmitted to the fluid. Then, the bulge cell and the output bar will move forward while the specimen will also be deformed by the pressure of fluid. During the test, the fluid pressure is calculated by the incoming and reflected waves measured by strain gages on the input bar. It is found that the bars should be made of low impedance materials to achieve a satisfactory pressure measurement accuracy. The effective piston displacement is calculated by the incoming and reflected waves, and the transmitted wave measured on the output bar. The Swift laws combined with Cowper-Symonds model are identified for A16111-T4 and DP450 sheets by inverse analysis. The objective function was defined by the pressure-time history and the effective displacement-time history. Ramezani et al. [START_REF] Ramezani | Combined experimental and numerical analysis of bulge test at high strain rates using split Hopkinson pressure bar apparatus[END_REF] have also developed a similar dynamic bulge test set-up on the basis of SHPB system. This high strain rate bulge test technique have been validated by comparison of the analytical and FE simulation with the experimental results.

Plane strain test

The plane strain test is usually performed by the traditional uniaxial tensile testing machine. The testing specimen should be designed to keep the strain along the width direction zero ( 0 22 = ε ) during deformation. For specimen design of plane strain test, the width of deformed zone is usually much larger than the length, as shown in Fig. 1-22. The main drawback of plane strain test is the influence of the free edges on strain field homogeneity and stress calculation [START_REF] An | A novel and simple method for the measurement of plane strain work hardening[END_REF] . Wagoner et al. [START_REF] Wagoner | An experimental and analytical investigation of in-plane deformation of 2036-T4 aluminum sheet[END_REF] [80] [START_REF] Wagoner | Comparison of plane-strain and tensile work hardening in two sheet steel alloys[END_REF] has compared several types of plane strain specimen geometry to study the effects of dimension on the strain distribution and strain level. Flores et al. [START_REF] Flores | Accurate stress computation in plane strain tensile tests for sheet metal using experimental data[END_REF] have presented a methodology to experimentally identify the evolution of the homogeneous planestrain field region during deformation. A general formula for calculating stress has been given by using experimental data and including the edge effect evolution. Fig. 1-23 Cruciform shape for plane-strain tension [START_REF] Kuwabara | Advances in experiments on metal sheets andtubes in support of constitutive modeling and forming simulations[END_REF] Kuwabara et al. [START_REF] Kuwabara | Advances in experiments on metal sheets andtubes in support of constitutive modeling and forming simulations[END_REF] have developed a novel plane-strain tensile test of sheet metals to measure the stress-strain curves, using a hydraulically servo-controlled biaxial tensile testing machine and a newly-devised cruciform specimen, as shown in Fig. 1-23. The principal strains of the specimen are measured by strain gages, and strain rates can be calculated. By online controlling the strain rates, it has been succeeded in realizing plane strain tensile tests. Tensile direction is parallel to the shorter arms (axis-1). The total strain along the longer arms is kept almost to be zero. The slits are made along longer arms to exclude geometric constraint on the deformation of the gauge section.

Simple shear test

Quasi-static simple shear test

The simple shear test is usually adopted to carry out the cyclic loading tests [START_REF] Miyauchi | Deformation path effect on stress-strain relation in sheet metals[END_REF] [85] [START_REF] An | Development of simple shear test for the measurement of work hardening[END_REF] . American Society for Testing and Materials (ASTM) [START_REF]Standard Test Method for Shear Testing of Thin Aluminum Alloy Products[END_REF] has suggested a procedure to determine shear ultimate strengths, which can be carried out on the traditional uniaxial testing machine. The simple shear testing specimen is given in Fig. 1-24.

Fig. 1-24 Shear testing specimen (ASTM B831-11) [START_REF]Standard Test Method for Shear Testing of Thin Aluminum Alloy Products[END_REF] For many simple shear tests, the experimental device can also be designed in order to impose a parallel displacement of two lateral grips along the opposite directions [START_REF] Yin | An experimental and numerical investigation of different shear test configurations for sheet metal characterization[END_REF] . To carry out the simple shear test at the strain rate from 4 10 -to s / 10 2 , Rusinek et al. [START_REF] Rusinek | Shear testing of a sheet steel at wide range of strain rates and a constitutive relation with strain-rate and temperature dependence of the flow stress[END_REF] have adopted a fast hydraulic machine and designed a double-shear specimen, in Fig. 1-25. The loading force F is measured by the machine load sensor and the mean displacement of the specimen U is calculated by using two linear variable differential transducer (VDTL) displacement gauges. The shear strain, strain rate and stress of the quasi-static shear test can be calculated as follows.

Fig. 1-25 Specimen of double shear test [START_REF] Rusinek | Shear testing of a sheet steel at wide range of strain rates and a constitutive relation with strain-rate and temperature dependence of the flow stress[END_REF] Shear strain is calculated by :

h U = γ Eq. 1-75
Here, h is the width of shear zone.

Shear stress is calculated by:

s A F 2 = τ Eq. 1-76
Here, s A is the cross section of shear zone, gH A s 2 = , g is the thickness of shear zone and H the height of shear zone.

Shear strain rate is calculated by:

h V dt dU h = = 1 γɺ Eq. 1-77
Where V is the loading velocity.

Dynamic simple shear test

For the strain rate higher than s / 10 2 , Rusinek et al. [START_REF] Rusinek | Shear testing of a sheet steel at wide range of strain rates and a constitutive relation with strain-rate and temperature dependence of the flow stress[END_REF] have directly attached the double-shear specimen to the Hopkinson tube and loaded by the direct impact of a projectile, as shown in Fig. 1-26 . At the instant of impact, a transmitted elastic wave starts to propagate in the Hopkinson tube and recorded by the strain gages glued on the Hopkinson tube. With the measured transmitted wave, the displacement of tube can be determined by the analysis of propagation of the elastic wave in bars or tubes, and the force transmitted by the specimen can also be computed. The testing methods have been applied to determine the visco-plastic behaviour of cold rolled sheet steel under a wide range of strain rate. An original visco-plastic constitutive relation of Rusinek and Klepaczko (R-K) model has been identified. Kuwabara et al. [START_REF] Kuwabara | Advanced in experiments on metal sheets and tubes in support of constitutive modeling and forming simulations[END_REF] have developed a combined uniaxial tension-internal pressure testing machine. Various strain paths under large deformation can be realised by controlling the ratios of uniaxial tensile force T by hydraulic cylinders and the internal pressure P by an intensifier, as shown in Fig. 1-28. A spherometer is located on the top of specimen to measure the radius of curvature, φ R , in the axial direction. The axial and circumferential true strain on the outer surface of specimen are measured by strain gauges. The true strains and true stresses along the axial and circumferential directions will be calculated on the midwall. For the tension-internal pressure test of tubular specimen, various biaxial strain paths can be realised, but it is limited for application because it is difficult to manufacture the tubular specimen for sheet metal [START_REF] Hill | An investigation of plastic flow and differential work hardening in orthotropic brass tubes under fluid pressure and axial load[END_REF] [92] [93] [START_REF] Liu | Formability determination of titanium alloy tube for high pressure pneumatic forming at elevated temperature[END_REF] .

Fig. 1-28 Uniaxial tension-internal pressure testing machine [START_REF] Hakoyama | Biaxial tensile test of high strength steel sheet for large plastic strain range[END_REF] The initial and subsequent yield locus of extruded aluminium alloy A5154-H112 tubes have been investigated under different strain paths, using this tension-internal pressure test of tubular specimen [START_REF] Kuwabara | Anisotropic plastic deformation of extruded aluminum alloy tube under axial forces and internal pressure[END_REF] . The forming limit stresses of steel tubes (JIS STKM11A) subjected to proportional and nonproportional loading paths have also been studied [START_REF] Yoshida | Effect of strain hardening behavior on forming limit stresses of steel tube subjected to non proportional loading paths[END_REF] . With the biaxial tests of cruciform and tubular specimens, the plastic behaviour of high strength steel sheet (JSC590R) have also been investigated for a equivalent strain range of 16 . 0 002 . 0 ≤ ≤ p ε [START_REF] Hakoyama | Biaxial tensile test of high strength steel sheet for large plastic strain range[END_REF] .

Combined test

To characterize sheet material behaviour under multi-axial and non proportional loads, Pijlman et al. [START_REF] Pijlman | Sheet material characterisation by multi-axial experiments[END_REF] have designed a biaxial testing equipment, which can combine the plane strain test with the shear test on only one specimen, as shown in Fig. 1-29 . The upper part can move along the horizontal direction to carry out shear test, and the lower part can move along the vertical direction to perform the plane strain test. Using this biaxial device, the shear and plane strain reference points of yield function can be been determined. Meanwhile, the cyclic shear test and orthogonal tests can also be carried out on this device for material modelling under non-proportional loading paths. Forles et al. [99] [100] have developed a similar experimental device to study the plane strain test and simple shear test. Mohr et al. [101] [102] have also developed a similar dual-actuator system and performed the tests of various combinations of the shear and tensile loadings to determine initial yield function and hardening laws of a stainless steel 301LN under isothermal quasi-static conditions at room temperature .

Fig. 1-29 Combined plane strain-simple shear test [START_REF] Van | Strain path dependency in sheet metal: Experiments and models[END_REF] With the development of digital image correlation (DIC) and inverse analysis, many experiments with heterogeneous deformation have been proposed to identify the constitutive parameters of materials on a single sample [103] [104] [105] [106] [107] . Pottier et al. [108] have developed an out-of-plane testing procedure with a hemispherical punch to apply the prescribed displacement at the centre of sample, in Fig. 1

Conclusion

For FE simulation of sheet metal forming processes, accurate phenomenological models of material are demanded. In order to identify parameters of these models, appropriate experimental databases are needed. These databases must be obtained from experimental tests on sheet metal specimens at conditions encountered in the forming processes. In most sheet metal forming processes such as stamping, hydroforming, … large strains can be reached in intermediate strain rate range (up to few hundred 1 s ). Moreover, formed sheets are usually submitted both to in-plane biaxial loadings and to linear or nonlinear strain paths.

Many experimental techniques have been developed for material characterization, including: uniaxial tensile test, bulge test, plane strain test, simple shear test,… There are several specific limitations for each of these traditional experiment. In order to avoid these main drawbacks, an in-plane biaxial tensile tests of a cruciform specimen is proposed in this work. The potential of this test to identify hardening behaviour up to large strains in the range of intermediate strain rates is investigated at room temperature.

In the following work, an optimal shape of cruciform specimen is proposed to obtain large strains under equibiaxial tensile strain path. Then, the rate-dependent hardening sensitivity of sheet metals is identified by inverse analysis based on FE simulation of biaxial tensile tests.

Introduction

Many biaxial tests using cruciform specimens under various strain paths, temperatures and strain rates [START_REF] Shimamoto | The development of servo dynamic biaxial loading device[END_REF] have been carried out with different goals of material modelling, including: ( ) characterization of yield locus, ( ) identification of hardening model, ( ) determination of Forming Limit Curve (FLC), as shown in Fig. 2-1.

Fig. 2-1 Applications of biaxial tests for material modelling

Up to date, several types of biaxial devices have been developed for biaxial tests on cruciform specimen, including link mechanism attachments and stand-alone machines [START_REF] Hannon | A review of planar biaxial tensile test systems for sheet metal[END_REF] . The experimental devices should be controlled to keep the centre of specimen unmoved [111] and apply the specific loading ratios along two perpendicular directions. In section 2.2, the two main types of biaxial device are presented. Although a newly standardised cruciform shape with seven slits in each arm has been proposed by International Organization for Standardization (ISO) [112] , the maximum deformation at the central zone is very small. Therefore in following work, a review of the typical cruciform shapes is presented in section 2.3, in order to investigate the more efficient geometries allowing to obtain large strains. Then, an optimal cruciform shape is designed for large strains by means of the numerical simulation tool.

Biaxial testing devices

Many link mechanism attachments have been design and equipped in a conventional uniaxial testing machines to convert uniaxial tensile or compressive force into two forces along two perpendicular directions of cruciform specimen (in Fig. 2-2) [113] [114] . Although the attachments are economical, the disadvantage is that the load ratios can not be online changed during tests. Moreover, this kind of machine is not dedicated to dynamic tests. So, stand-alone machines seem more appropriate for such tests. Stand-

p ε σ 1 σ 2 σ 1 ε 2 ε
alone biaxial machines are developed with independent actuators along two directions, by which various linear and nonlinear load paths can be realised (in Fig. 23) [115] [116] .

Application of cruciform specimen

Since 1960s, biaxial tests on flat cross-shaped specimens, with various geometries of the central zone, have already been performed for characterization of yield locus, identification of hardening model and determination of FLC. Several typical shapes and applications of cruciform specimens are reviewed in this section.

Characterization of yield locus

Muller et al. [START_REF] Muller | New experiments for determining yield loci of sheet metal[END_REF] have adopted cruciform specimens with notches at intersections of arms. The shape was optimized by varying the parameters, 1 R and 2 R , to obtain a large zone of homogeneous deformation and high strain level before instability occurs, as shown in Fig. 23456. The maximum strain at the central zone was affected by different materials. The stress components were calculated by dividing the forces through the cross section. The yield point was determined by a sharp increase of temperature due to dissipation of plastic work. Finally, the initial yield locus of a steel (St14 O5) and subsequent yield locus after prestraining of an aluminium alloy (AlMgSi1) have been investigated. Banabic [ 118 ] has used the same specimen to obtained yield points under different stress ratios for an aluminium alloy AA5182-0, and compared with BBC2005 yield criterion. Naka et al. [119] [120] have performed biaxial tensile tests on an aluminum-magnesium alloy sheet at temperatures from 30°C to 300°C and strain rates from s / 10 5 to s / 10 2 to study the effects of temperature and strain rate on evolution of yield loci. A cruciform shape has been suggested with notches at the intersection of arms and two slits in each arm to avoid geometrical constrains on the gauge zone, as shown in Fig. 234567. The maximum plastic strain was about 0.06 at room temperature. The strain and strain rate were calculated by the recorded grid images. The yield stresses was defined by the points where the proportional relation between stress and strain disappeared. Kuwabara et al. [121] [122] [123] have proposed a specimen shape with slits in arms to obtain homogeneous deformation field at the gauge zone, as shown in Fig. 2345678. The maximum equivalent strain of specimen was only 0.04. The strain components were measured by four biaxial strain gauges. The stress components were calculated from dividing the measured forces by current cross sectional areas. The successive contours of plastic work for particular plastic strains in the biaxial stress space have been determined for cold-rolled low-carbon or dual phase steel sheets and compared with various phenomenological or crystallographic yield criteria.

Merklein et al. [124] [125] have designed a new biaxial setup on a punch-load moving perpendicular to the sheet. Biaxial tensile tests have been carried out with a local heat on magnesium alloy AZ31 at the temperature from room temperature to 310°C. The shape with slits of different length has been adopted to assure that the maximum strains occur at the central zone and stress concentration is reduced to a minimum, as shown in Fig. 23456789. The strain fields of specimen were determined by DIC techniques. The stress components were calculated on the basis of experimental forces and cross sections. The stress points on the initial and subsequent yield loci were determined for the corresponding equivalent strains. Teaca et al. [126] [127] have proposed heterogeneous biaxial tensile tests to identify the plastic anisotropy of sheet metal. Two shapes of cruciform specimen were designed to cover the whole stress plane. The first one with hole at the central zone covered the stress states from uniaxial tension (UT) to plane-strain tension (PST). The second one covered the stress states from uniaxial tension (UT) to equibiaxial tension (EBT), as shown in Fig. 2-10. The equivalent strain at the central zone of second specimen is about 0.1. The strain fields were calculated by DIC method. A two-step strategy of parameter identification was performed to determine FMM yield function for ES steel and 1050A aluminium sheets. The parameters of hardening law were determined by uniaxial tensile tests. Then, the parameters of FMM yield function were identified by minimizing the difference between experimental and FE-simulated strain fields of specimen. With the identified parameters, the formation of ears in the cup drawing test is accurately predicted by the finite element simulation. Prates et al. [128] have used a single equibiaxial tensile test of cruciform specimen to simultaneously identify the parameters of Hill 48 yield criterion and Swift isotropic hardening law by mixed numerical-experimental method. A cruciform shape (in Fig. 2-11) was proposed to reproduce heterogeneous deformation with strain paths ranging from uniaxial to equibiaxial tension. The strain fields were obtained by DIC method. Zhang et al. [START_REF] Zhang | Characterization of anisotropic yield criterion with biaxial tension test[END_REF] have adopted unique equibiaxial tensile test of cruciform specimen to identify the parameters of Bron and Besson 2004 yield criterion for AA5086 and DP980 sheets. As shown in Fig. 2-12, the specimen with notches was designed to realise a continuous strain states from 2 . 0

1 2 - = ε ε to 1 1 2 = ε ε
. The strain fields were calculated by DIC techniques. The FE-simulated principal strains and strain paths along diagonal, longitudinal and transverse directions were compared with the experimental results. The identified parameters have been validated by comparison with those determined by uniaxial test, bulge test and simple shear test.

Identification of hardening model

Makinde et al. [130] have presented two different geometries of cruciform specimen to investigate the mechanical behaviour of sheet metals and composite materials under monotonic and cyclic biaxial loading conditions, as shown in Fig. 2345678910111213. One was shown with a circular thickness-reduced central zone. The other was given with a rectangular thickness-reduced central zone and slits in each arms. Green et al. [START_REF] Green | Experimental investigation of the biaxial behaviour of an aluminium sheet[END_REF] have adopted the second shape (as-received thickness, mm 3 .

6

) with a thicknessreduced central zone (final thickness, mm 2954 . 1

) to obtain maximum strain in the central zone and seven parallel slots in each arm to maintain the deformation as uniform as possible in the gauge section. The cruciform specimens were deformed up to equivalent strains of approximately 0.15 in the biaxial stretching zone. The strain at the central zone was measured by biaxial extensometers [132] . Several biaxial tensile tests were carried out for AA1145 sheet under different proportional strain paths. The biaxial flow curves were determined by an iterative procedure based on FE simulation. Gozzi et al. [133] [134] [135] have used biaxial tests of cruciform specimen to investigate the plastic behaviour of extra high strength steel under nonlinear strain paths. Two shapes were designed with smooth transition to reduce stress concentration and three slits in each arm to make deformation uniform, as shown in Fig. 2-14. The stress distribution was improved by the different positions of slots in each arm. The maximum strain at the central zone is about 0.01. The strain at the central zone was measured by strain gauge. The stress was obtained from the forces dividing by the modified cross section. A series of tests have been carried out under two-stage loading paths, including a initial proportional loading path, an unloading and a subsequent proportional reloading path in a new direction. Compared with a simple isotropic hardening model, a kinematical hardening model was validated to predict the response of material subjected to non-monotonic loading. Kulawinski et al. [136] have adopted the second shape of cruciform specimen to carry out biaxial tests on a cast TRIP steel. The strain at the central zone of specimen was measured by biaxial extensometers. A new procedure, called as partial unloading method, was proposed to determine the stress of specimen. The influence of strain path changes on the equivalent stress and equivalent strain were investigated by biaxial sequential tests under two-stage loading paths.

Uemori et al. [137] have adopted the shape of cruciform specimen (in Fig. 234567) to conducted several biaxial tests on high strength steel (HSS) sheet under proportional and non-proportional loading paths. The non-proportional loading paths were realised by the sequential proportional loading paths after initial equibiaxial loading and unloading path. Compared with the isotropic hardening (IH) model, the kinematical hardening model (Yoshida-Uemori model) was validated to capture the Bauschinger effect and describe the translation of loading surface. Merklein et al. [138] [139] have designed a stand-alone biaxial machine to realize an online continuous change of stress state from uniaxial tension to biaxial tension on cruciform specimen. A cruciform shape (as-received thickness, mm 1 ) with seven slits in each arm and the central section reduced by half on one side (final thickness, mm 5 . 0

) is shown in Fig. 2-15. The maximum equivalent strain at the central zone is about 1 . 0 . The strain fields were determined by DIC method. The stress components were calculated by the measured force and current cross sectional area. The isotropic and isotropic-kinematical hardening models have been compared with experimental results in the first quadrant of principal stress space for AA6016.

Determination of forming limit curve

Yu et al. [140] have suggested biaxial tests of cruciform specimen to study forming limits under complex loading paths. An optimal shape (as-received thickness, mm 3 )

with two-step thickness reduction was given by FE simulation for a low-carbon steel sheet, as shown in Fig. 2-16. A first cross-shaped thickness reduction (middle thickness, mm 1 ) was made to transfer the main deformation to central zone. A further circular thickness reduction (final thickness, mm 7 . 0 ) was used to concentrate limit strain at the central zone. The specimen shape has not been experimentally validated. Abu-Farha et al. [143] [144] have carried out biaxial tensile tests on cruciform specimen to study microstructural mechanisms on biaxial deformation. Two cruciform shapes of various as-received thicknesses have been experimentally investigated for lightweight materials (AA5083, Mg AZ31B, TWIP steel) at elevated temperature (300°C) and quasi-static state. The shapes with thickness reduction (final thickness, around mm 1 ) have been designed for large strains up to crack at the central zone of specimen, as shown in Fig. 23456789101112131415161718. For the first one, each arm was tapered and a circular profile was used to define the thickness evolution. The other shape was suggested with notches at the corners and a circular flat-bottomed thickness reduction at the gauge area. Lee et al. [START_REF] Lee | A new method for testing formability in sheet metal forming at biaxial tensile state[END_REF] have adopted biaxial test of cruciform specimen to determine the first quadrant of FLC. A cruciform shape (as-received thickness, mm 6 . 1

) with two-step thickness-reduction was proposed to make initial crack located at the central zone, as shown in Fig. 2 Zidane et al. [146] [147] have adopted biaxial tests of cruciform specimen to determine FLC of AA5086. A cruciform shape (as-received thickness, mm 4 ) with two-step thickness reduction at the central zone and slits in each arm has been proposed, as shown in Fig. 2-20. The first-step thickness-reduced zone is a square one (middle thickness, mm 2 ). For further thickness reduction at the central zone, the thickness of a circular zone was defined by a radius curved profile (final thickness, mm 75 . 0

). The specimen has been validated to realise various strain paths at the central point of specimen by loading different speed ratios. The strain fields of specimen were calculated by DIC. When the equivalent plastic strain increment ratios between a point located inside and outside the necking zone has reached a critical value, the onset of necking is determined. The corresponding major and minor strains of the zone inside the necking zone represents one point of FLC (in Fig. 23456789101112131415161718192021). Using the global forces and displacements along each arm of specimen and local equivalent strains at the central point on the quasi-equibiaxial tensile test, the parameters of Hill 48 yield criterion and Ludwick hardening law have been identified under large strains on the basis of FE simulation. Leotoing et al. [START_REF] Leotoing | Cruciform shape benefits for experimental and numerical evaluation of sheet metal formability[END_REF] have also presented the comparison between experimental and numerical predictions of FLC by using this proposed cruciform shape. In conclusion, very few research about plastic behaviour modelling of sheet metals under large strains (up to the deformation level obtained by standardised uniaxial tensile tests) using biaxial tests on cruciform specimen have been reported. For large strains, shape design of cruciform specimen becomes usually very complex. Very few of them have been validated experimentally except the shape proposed by Zidane, which is based on two-step thickness reduction (from as received thickness mm 4 to final thickness mm 75 . 0

). Nevertheless, the use of this shape for any sheet thickness seems difficult, particularly for small initial thicknesses. Another drawback of the cruciform shape with a central thickness reduction defined by a curved profile is that large deformations are obtained only at the central point, which can lead some difficulties in the identification procedure. At the same time, most biaxial tensile tests on cruciform specimen have been carried out at quasi-static condition.

Numerical investigation of cruciform shapes

First of all, an optimal cruciform shape must be defined and the proposed geometry must verify the following specification:

(1) Large strains must be reached at the central zone, which is deformed under biaxial tensile state; (2) Deformation field at the central zone must be as homogeneous as possible to facilitate the identification procedure; (3) Cruciform specimen should be as easy as possible to be manufactured; (4) Maximum load before necking for each actuator must be less than KN 50 ; (5) For dynamic tests, the initial stiffness of the specimen is limited by capacities of the servo-hydraulic device. Consequently, dimensions of the square central zone of the cross specimen are chosen equal to mm 30 .

In the following work, FE software ABAQUS (static-implicit algorithm) is adopted to simulate equibiaxial tensile tests. The initial thickness mm 2 of sheet is chosen. A quarter FE model is adopted due to symmetries. Same displacements are applied along each arm of cruciform shape. Young's modulus GPa E 69 = and Poisson's ratio 33 . 0 = υ are considered for elasticity. For plasticity, isotropic Mises yield criterion is used and the true stress-plastic strain curve, given in Fig. 2-22, is introduced point by point for the isotropic hardening behaviour. ) field of specimen are investigated. Firstly, a basic cruciform shape, which is the simplest one, is studied to present the problems introduced by cruciform shape design. Then, various cruciform shapes with different geometries are investigated. Finally, an optimal cruciform shape is suggested to obtain large strains at the central zone.

Basic shape

A basic cruciform shape consists in three zones: the central zone, the arms and the intersections, as shown in Fig. 2 Therefore, the central zone is the zone of interest under biaxial tensile state, but the deformation is very small. The arms are the force-transferring zones under uniaxial tensile state. The intersections are sharp geometries, where sever strain concentrations occur. For the basic shape, there are two main problems: strain localisation at the intersection and the necking of arms under uniaxial tensile state. Arm Intersection Center

Notches at intersections

As explained above, notches have been suggested at the intersections to reduce the strain localisation. Here, the distance D from notch to symmetrical axis and the radius R of notches are changed for shape design, as shown in Fig. 2-26.

Fig. 2-26 Shape with notches

The effect of distance D on the equivalent plastic strains of specimens is studied with constant radius of notch, mm R 5 = . The equivalent plastic strains at the notches are higher than those at the centre and arm, as shown in Fig. 2-27. The effect of radius R on the equivalent plastic strains of specimens are studied with constant distance from notch edge to symmetrical axis, mm R D 10 = -. It can be seen that the equivalent plastic strains at the notches are also higher than those at the centre and arm, as shown in Fig. 2 It is concluded that the effect of position and size of notch on the equivalent plastic strain at the central zone of cruciform shape is very small. Sever strain localisations always occur at the notches.

Slits along arms

Slits in each arm are proposed to reduce the transverse rigidity of arms and eliminate geometric constraint on the central zone. Here, the effect of position of slits on the deformation of specimen has been investigated. The number of slits in each arm is constant and equal to seven. The point P is fixed so that the end of outside slit is aligned with the arm edge. The ends of slits in each arm are aligned following a straight line. The inclination of this line is defined by θ , as shown in Fig. 2-31. The width of slits ( mm 5 . 0

) and distance between slits are not changed. It is concluded that the effect of slits on the strain level at the central zone is also very small. But slits in each arm ( °= 90 θ

) allow to obtain a homogeneous deformation field at the central zone.

Thickness reduction

Thickness reduction is suggested to decrease the stiffness of central zone and obtain large deformation. Here, a circular area of thickness reduction is adopted at the central zone. It has been chosen to reduce the thickness only on one side of sheet, which is easier for manufacturing. The radius r and thickness t of thickness-reduced zone are changed, as shown in Fig. 2-35. All the other geometries including width of arm, radius and position of notches are kept constant. The strain level at the notch is still highest.

Fig. 2-36 Effect of radius of thickness reduced zone

The effect of thickness reduction on the equivalent plastic strain of specimen is studied with a constant radius of thickness-reduced zone, 5 = r . When the thickness t decreased, the equivalent plastic level at the central zone increased greatly, as shown in Fig. 2-37. The strain level at the notch is always highest. It is concluded that the deformation level at the central zone can be elevated by using thickness reduction. When the size and thickness of thickness-reduced zone become smaller, the strain level at the central zone is larger. A quasi-homogeneous strain field is observed at the central zone with thickness reduction.

Optimal shape design

A shape with circular thickness reduction at central zone, notches at intersections of arms and slits in each arm is suggested to obtain large strains under equibiaxial tensile path. Here, six free parameters are investigated in the shape of cruciform specimen: the radius r and thickness t of flat central zone, the position D and radius R of notch and the positions 1 S and 2 S of slits, as shown in Fig. 2-40. After a parametric study, the optimal dimensions of shape are given in Table 2- It is concluded that an optimal cruciform shape with thickness reduction, slits and notches has been designed for large strains. The equivalent plastic strain at the central zone reaches about 30 . 0 = p ε .

Conclusion

To realize dynamic biaxial tests on cruciform specimens, servo-hydraulic stand-alone machine is more appropriate than mechanical device uses with conventional uniaxial testing machines which can have complex vibratory responses. So in this work, a stand-alone biaxial tensile testing machine with four independent servo-hydraulic actuators available in the Laboratory GCGM of INSA de Rennes will be used to perform both quasi-static and dynamic biaxial tensile tests. The deformation of specimen will be measured by DIC technique.

In the literature, biaxial tests on various well-defined cruciform specimens have been carried out under linear or nonlinear strain paths for characterization of yield criteria, identification of hardening models and determination of FLC. During previous works in the laboratory, anisotropic yield criterion has been calibrated by using a single biaxial tensile test on notched cruciform specimen, and FLC has also been determined by a set of biaxial tests under linear strain paths performed on a two-step thicknessreduced cruciform specimen.

From literature review and FE numerical investigations, the most sensitive geometric parameters have been studied (notches, slits, thickness reduction). A parametric study has been led to define a set of geometric parameters to obtain large equivalent plastic deformation . This proposed shape could be used for other initial thickness sheet by considering the same thickness reduction ratio.

Introduction

The objective of this chapter is to present a procedure of identification applicable to in-plane biaxial tensile test. This procedure must be validated for static tests and dynamic tests as well. Quasi-static and dynamic biaxial tensile tests will be performed on cruciform specimen defined in chapter 2. Hardening behaviour of sheet metals from quasi-static to intermediate strain rates will be identified up to large strains. The experimental force measurement, which is a main problem for dynamic uniaxial tests, will also become a key issue in dynamic biaxial tensile tests. Therefore, the loading ringing problem will be investigated for dynamic biaxial tests. The experimental forces should be carefully pre-processed for parameter identification. Fortunately, the similar experiences of dynamic uniaxial tensile tests can be adopted for dynamic biaxial tensile tests.

Here, aluminium alloy AA5086 is tested at room temperature. This material has been chosen since it is known to present a very small dependency to strain rate. For biaxial tests, the parameters of hardening law will be identified by inverse analysis. The identified biaxial flow stress curve at quasi-static strain rate will be compared with the uniaxial flow stress curve to validate the procedures of experiment and identification. Then, the identified biaxial flow stress curve at intermediate strain rate will be compared with the curve identified from the quasi-static biaxial tensile test to validate the dynamic biaxial testing procedure.

Uniaxial test of AA5086

The 5XXX series aluminium alloys, with magnesium as the major alloying element, combine a wide range of strength, good forming and welding characteristics, and high resistance to general corrosion. The strength of this alloy can be generally increased with increasing magnesium content and further enhanced by cold work. The 5XXX alloys can be easily cold formed and the formability tends to increase as alloy strength decreases. They are widely used for the cryogenic application, pressure vessels, hulls and superstructures of ships, road transport and general engineering [149] .

Digital image correlation

Measurement methods used for strain determination in sheet metal tests include: circle grid analysis (CGA), strain gauge, extensometers [START_REF] Makinde | Design of a biaxial extensometer for measuring strains in cruciform specimens[END_REF] and so on. Nowadays, with the technical development of high-resolution charge-couple device (CCD) cameras, the Digital Image Correlation (DIC) technique is a more popular strain measurement method [START_REF] Schreier | Image Correlation for Shape, Motion and Deformation Measurements: Basic Concepts, Theory and Applications[END_REF] . It is a non-contact approach. The strain measurement by DIC contains three steps: ( ) generation of speckle pattern on specimen, ( ) image record by camera, ( ) displacement and strain calculation, as shown in Fig. 3-1: In this work, the DIC software CORRELA 2006 [152] , which has been developed by the Laboratoire de Mécanique des Solides of Université de Poitiers, is adopted. Different sets of DIC parameters in CORRELA 2006 are compared to chose the best one for strain calculation. These parameters are given in Table 3-1 . The analysed zone of specimen should be assigned for calculation firstly. Then, the subsets should be defined for correlation. It needs to set four parameters: the length and width ( 1 L , 2 L ) of subset, and the horizontal and vertical distances ( 1 D , 2 D ) between the adjacent subsets, as shown in Fig. 3-2. After the displacements of all the subsets are determined by correlation, four subsets are chosen by setting a constant interval 2 to obtain a parallelogram. The strain will be calculated at the diagonal intersection of the parallelogram, as shown in Fig. 3-3. The accuracy of DIC method for strain measurement is dependent on the quality of speckle pattern and digital image, correlation algorithm [START_REF] Pan | Two-dimensional digital image correlation for inplane displacement and strain measurement: a review[END_REF] .

Experimental result

A dynamic uniaxial tensile testing system has been developed in the Laboratory GCGM of INSA de Rennes, which includes a servo-hydraulic testing machine, controlling system, force sensor and deformation measurement by DIC. The shape of uniaxial tensile testing specimen is shown in Fig. 34. The quasi-static uniaxial test at the loading velocity of s mm 1 and dynamic uniaxial test at the loading velocity of s mm 500 are carried out to obtain the true stress-true strain curves at the quasi-static ( s 02 . 0

) and intermediate ( s 10 ) strain rates, respectively. The main parameters for measurement systems are presented in Table 3-2.

Strain calculation

Before necking of specimen, 4623 images have been captured during the quasi-static test and 602 images have been recorded during the dynamic test. The DIC technique is adopted for strain calculation. The uniformly deformed region is chosen as the zone to be analyzed, as shown in Fig. 345. There are 29 and 5 subsets, 27 and 3 strain calculation points along the longitudinal and transverse directions of the specimen, respectively. The magnification factor from pixel to mm is pixel mm 1 . 0

. Finally, components of strain ε are calculated by average value on gauge zone as follows:

( ) ( )

∑∑ = = × = m i n j t j i n m t 1 1 , , 1 ε ε Eq. 3-1
Where m and n are the total number of strain points along longitudinal and transverse directions. ( ) j i, is the position of strain point and t is the time.

Fig. 3-5 Gauge zone of uniaxial testing specimen

The average true, principal strain curves of specimen are presented in Fig. 3456 For the dynamic uniaxial test, the measured force is oscillating at the beginning due to load ringing phenomenon. Because the oscillation is not behaviour of testing material, RD TD the measured force curve is mathematically pre-processed by the median value filtering method in MATLAB. The experimental force curves are shown in Fig. 34567. S is the initial cross area. The true stress-true strain curves of AA5086 under quasi-static and dynamic uniaxial tension are compared in Fig. 345678.

Fig. 3-8 Static and dynamic uniaxial true stress-true strain curves of AA5086

It can been seen that these curves are almost the same. This confirms that the choice of this alloy based on its strain rate independency property. For large strains, a very small effect of strain rate can be observed, which in this case would be negative strain rate sensitivity at room temperature. The negative strain rate sensitivity at room temperature have also been experimentally observed for AA5182-O [154] by uniaxial tension and compression tests at the strain rates between 1 4 10 - -s and 1 0 10 - s , and for AA6092 [START_REF] Rusinel | Thermo-viscoplastic constitutive relation for aluminium alloys, modeling of negative strain rate sensitivity and viscous drag effects[END_REF] at strain rate level of Firstly, the parameters of well-known Ludwick and Voce models are determined based on the true stress-plastic strain curves. The plastic strain p ε is calculated from the true stress-true strain curve as follows:

E e p σ ε ε ε ε - = - = Eq. 3-2
where E is Young's modulus. The fitting process is completed by MATLAB, and this nonlinear curve-fitting problem is solved in least-squares sense. The error δ of fitting is defined as follows:

( ) ( ) ( ) ∑ = - = n i i p i p fit n 1 2 exp 1 ε σ ε σ δ Eq. 3-3
where fit σ , exp σ are the fit and experimental stress, n is number of experimental points. The identified results of static and dynamic uniaxial tests are compared in Table 3-3 and Table 3-4 .

Parameters

Voce model: The experimental and identified curves are compared in Fig. 3456789, where the identified hardening laws are extrapolated up to equivalent plastic strain of % 34 . The two identified hardening laws are very different when the plastic strain becomes larger than % 19 beyond the uniform deformation of uniaxial tests. It is concluded that the strain level of uniaxial tensile test before necking is not enough to characterize the hardening behaviour of sheet metals for large strain level. Therefore, the experimental strain level should be elevated to determine more accurate hardening models.

( ) ( ) p n K ε σ σ - - + = exp 1 0 0 σ (MPa) K (MPa

Quasi-static biaxial test of AA5086

Experimental result

According to optimal shape in chapter 2, cruciform specimens of AA5086 have been prepared by milling. All the dimension of specimen is given in the Appendix (I). During quasi-static biaxial tensile test, a loading velocity of s mm 1 is applied. The high speed camera is adopted to record images at central zone, as shown in Fig. 3-10. The forces along the two perpendicular directions of the specimen are measured by deformation gauges, which are placed on the load sensors (in Fig. 34567891011). The grips are used to align the loading directions and protect the load sensors from the asymmetric loading due to rupture of specimen. The main parameters of high speed camera and load sensors are defined for quasi-static biaxial tensile test, in Table 3-5 . The random speckle pattern is generated at the central zone of specimen before test, and the initial crack are found to occur along the transverse direction in the circular thickness-reduced zone (in Fig. 3456789101112). There are 382 images before initial crack. For DIC, the circular thickness-reduced zone of specimen is defined as the gauge zone (in Fig. 345678910111213). There are 16 subsets and 14 strain points along each direction of specimen.

The magnification factor from pixel to mm is pixel mm 037 . 0

. The displacement vectors of gauge zone are shown in Fig. 3-14. It can be seen that the position of the central point is nearly kept fixed. ).

The equivalent strain at this central zone is above % 30 . The major and minor principal strains at the central zone are about % 20 and % 14 , respectively. The major principal strain localised near the points (

0 = x , mm y 4 ± =
). The strain state is not uniform due to the localisation of major principal strain at the time s t 5 . 1 = . ( ) ( )

∑∑ = = × = n j m i t j i n m t 1 1 , , 1 ε ε Eq. 3-4
where m , n are the number of points to be averaged. t is the time. Here, the average strains are calculated over the 3 3× points ( mm y x mm 6 . 0 , 6 . 0 ≤ ≤ -

) to trace the principal strain history at the central point.

The equivalent and major, minor principal strain curves at the central point are presented in Fig. 3-16. It is can be seen that the equivalent strain at the central point is above % 35 when s t 6 . 1 =

. Due to little asynchronization between two axes and anisotropy of material, major principal strain curve does not exactly coincide with the minor principal strain. Portevin-Le Chatelier (PLC) effect [156] can be observed on the evolution of strain curves which present steps especially between s t 6 . 0 = and

s t 2 . 1 =
. The PLC effect has also been observed from uniaxial tension and simple shear tests from room temperature to 100°C for aluminium alloy AA5754-O [157] .

Fig. 3-16 Experimental strain curves of quasi-static biaxial testing specimen

The curves of strain path and strain rate at the central point are also shown in Fig. 3-17. The strain rate is calculated as follows: For parameter identification by inverse analysis, the measured force curves along two directions have been synchronised with the strain curves. Then, the force curves are smoothed using a moving average filter in MATLAB. Finally, the experimental force and principal strain are prepared for parameter identification, as shown in Fig. 345678910111213141516171819.

( ) ( ) i m i i m i t t t t - - = + + ε ε ε ɺ , when 2 m i i t t t + + = Eq. 3-5

Procedure of inverse analysis

When the deformation of specimen is inhomogeneous, no analytical model can be adopted for stress determination. The inverse analysis is suggested for identification of material parameters [START_REF] Tarantola | Inverse problem theory and methods for model parameter estimation[END_REF] . In this work, the parameters of hardening models are determined by inverse analysis based on FE simulation of biaxial tensile tests on cruciform specimen. The procedure of parameter identification is shown in Fig. 34567891011121314151617181920. The elastic parameters and the parameters of the yield criterion have been determined previously.

Because the experimental curves of displacement measured by displacement sensors along two axes are not exactly the displacements at the arms of specimen, they are not used for inverse analysis. The experimental curves of force along two axes are applied on the FE model. The experimental and simulated principal strains at the same time point i t and the same position of central point on the specimen are compared to identify hardening behaviour of material. Therefore, the cost function to be minimized is defined by the error δ as follows:

( ) ( ) ( ) ( ) ( ) ∑ ∑ ∑ =         - = 2 1 2 exp 2 exp 2 1 i n j j i n j j i j sim i t t t ε ε ε δ Eq. 3-6
Where n is the total number of time points during simulation; sim

Fig. 3-20 Flowchart of parameter identification

The multi-disciplinary and multi-objective optimization platform of modeFRONTIER is used to perform the inverse analysis. A Workflow of modeFRONTIER should be well defined for each optimization problem. Many optimization algorithms can be chosen to perform inverse analysis [159] [160] . Here, Simplex is chosen for parameter identification in modeFRONTIER. Simplex is a simple optimization algorithm seeking the vector of parameters corresponding to the extreme of a function, searching through the parameter space. The movement of Simplex is given by three operations: Refection, Expansion and Contraction [161] . Like hill climbing algorithms, the Simplex method may stop at local optimization and not converge to the global minimum. To make sure that the global minimum is found, several optimizations are performed with different initial values. In order to cover a N-dimensional space, 1 + N initial sets are needed for Simplex while a first-order optimization algorithm requires more test. Nevertheless, the convergence for Simplex method is less efficient than that for many other algorithms when the parameter number is increased. A best approach could consist in applying a hybrid method to localize approximately the global minimum of the cost function and then converge efficiently with a first-order optimization algorithm.

Optimization

Finite element model

Due to the symmetry, a quarter of the FE model of cruciform specimen is built in ABAQUS. Obviously, the accuracy of simulation is dependent on the finite element model. Here, the influence of mesh size at the central zone is studied for simulation. For the material model, Hooke's law is adopted with Young's modulus ) [START_REF] Zidane | Développement d'un banc d'essai de traction biaxiale pour la caractérisation de la formabilité et du comportement élastoplastique de tôles métalliques[END_REF] and the hardening law

Solid element

The model has been discretized by linear tetrahedral elements of type C3D4, and the element size at the central zone and notches is mm 5 . 0

. There are 66322 elements and the CPU time (Intel Core 3.2GHz, RAM 16Go) is s 4 . 786 for simulation. The equivalent plastic strain field is shown in Fig. 3456789101112131415161718192021. It can be seen that the large deformation is primarily located near the central zone and the deformation outside this zone is very small. Therefore, the mesh at this central zone is further refined. The major and minor principal strain evolutions at the central point for different mesh sizes are also compared in Fig. 3-23. When the deformation is small, the strain is nearly the same. When the deformation becomes larger, the strain is critically affected by the element size. The deformation is larger with smaller element size, because the FE model becomes less rigid with smaller elements.

When the deformation field is inhomogeneous and the deformation level is higher, the mesh size of model C3D4_5 is small enough to capture the deformation localisation. Meanwhile, as the mesh size is refined for accurate simulation, the element number will increase and the FE simulation will become more time-consuming. For inverse Refined zone 0.33 0.58 0.47 Y X analysis, it becomes very hard to accept the time cost because it is necessary to perform a lot of simulation for optimization. The mesh size, element number and CPU time for the models of solid (C3D4_5) and shell elements are compared in In order to examine the accuracy of shell element model, the equivalent plastic strain field is compared with that of solid element (C3D4_5), as shown in Fig. 3-25. The levels and distributions of equivalent plastic strain are almost the same for two models, although the localisation at the edge of central thickness-reduced zone in the solid model is a little more severe than that in the shell model. The major and minor principal strain evolutions at the central point are compared between the two models, as shown in Fig. 3-26. It can be seen that the major and minor principal strains curves are nearly the same between the two models. In conclusion, it will be very beneficial to adopt the shell element for parameter identification by inverse analysis, because it not only fulfil the accuracy, but also greatly reduces the time cost of FE simulation.

Parameter identification

With the pre-processed experimental forces along two arms and principal strain curves at the central point of cruciform specimen, the parameters of hardening law are identified by inverse analysis based on the FE model of shell element. For material behaviour, the elasticity is described by isotropic Hooke's law, and the associated flow rule is used for the plastic behaviour. Because the strain path is nearly proportional during quasi-equibiaxial tensile tests on cruciform specimen, the isotropic hardening model is adopted. The preliminarily-known parameters of yield functions are given.

The hardening law to be identified should be preliminarily defined.

Material models

For AA5086, the elasticity is described by Young's modulus MPa E 73022 = and Poisson's ratio 33 . 0 = υ . Here, three yield criteria of Mises, Hill 48, and Bron and Besson 2004 in 2D stress space are adopted to identify the isotropic hardening law.

In the FE software of ABAQUS, Mises and Hill 48 yield criteria can be chosen directly. The subroutine UHARD has been implemented by FORTRAN code for the user-defined hardening law. For the material model with Bron and Besson 2004 yield criterion, the subroutine UMAT has been implemented by FORTRAN code.

For parameter determination with Hill 48 yield function, the uniaxial tensile tests have been carried out along the rolling, diagonal and transverse directions of AA5086 sheet, For parameter identification of isotropic hardening law, the generalized Voce hardening law for FCC alloys is chosen as follows:

) exp( 1

0 p n K ε σ σ - - + =
Eq. 3-7

Identified results

For parameters identification with Mises and Hill 48 yield criteria, the workflow of modeFRONTIER with UHARD is shown in Appendix (II). For Bron and Besson 2004 yield criterion, the workflow of modeFRONTIER with UMAT is shown in Appendix (III).

In order to study the effect of unknown parameters on the cost function, many parameter samples of hardening law have been adopted for simulation. The samples are uniformly distributed by design of experiment (DOE) based a pseudo random Sobol sequence in modeFRONTIER. The lower and upper bounds of parameters 0 σ , K and n are given in Table 3-9. In the design space, 150 samples have been chosen for forward analysis. The forward analysis based on DOE has performed with the Mises and Hill 48 criterion. Because too large strain may occur in the FE model applied by the experimental forces, the simulations are successful to converge until last only for 127 samples with the Mises yield criterion and for 118 samples with the Hill 48 yield criterion. The effects of parameters K and n on the optimization objectives are shown for the Mises and Hill 48 yield criteria (in Fig. 3-28 and Fig. 3-29). It can be seen that the objective is smaller when the parameters K and n decrease.

It seems that there exist a band, in which the parameters sets ( K , n ) are more feasible for optimization. For the Simplex optimization of parameter identification, the initial sets of parameters can be chosen from the more feasible band in DOE space. With the initial values given in this more feasible band, convergence by Simplex optimization will be faster and the problem of local optimization can be avoided as possible. Meanwhile, the effects of unknown parameters on the optimization function may be different for different mathematical form of hardening laws. Therefore, the hardening law should be well defined to represent the material behaviour as accurately as possible. If an unsuitable mathematical form is used, the optimized parameters may not be obtained finally. The experimental equivalent plastic strain-equivalent stress curve (in Fig. 3-30) is directly calculated by the experimental deformation and force curves of uniaxial tensile test. It can be clearly seen that the equivalent stress-equivalent plastic strain curve identified with the Mises yield criterion is the lowest, and the equivalent stressequivalent plastic strain curve identified with the Hill 48 yield criterion is lower than the uniaxial flow stress curve, especially for larger deformation, while the equivalent stress-equivalent plastic strain curve identified with the advanced yield criterion of Bron and Besson 2004 nearly coincide with the uniaxial flow stress curve.

In fact, all the equivalent stress-equivalent plastic strain curves for one material should be unique under different linear strain paths at the same temperature and strain rate, if the yield criterion is preliminarily well-defined. It can be concluded that the hardening behaviour of AA5086 sheet under the linear path of biaxial tension can be well characterised by the identified hardening law with Bron and Besson 2004 yield criterion.

Dynamic biaxial test of AA5086

For dynamic biaxial tensile test, the specimen is deformed at a specific velocity. The measured force curves must be treated before introduction into the parameter identification loop due to oscillations on the forces signals.

Load ringing phenomenon

For dynamic biaxial tests, the inertia masses are accelerated along the sliding bars to obtain the specific velocity before the specimen is loaded, then the specimen is impacted suddenly and the velocity is maintained due to the inertia effect of additional mass during the test, as shown in Fig. 3-34. To carry out dynamic biaxial tensile test of cruciform specimen at the speed of s mm 250 and room temperature, the main parameters of measurement system are given in Table 3-11 . The velocities of the four actuators ( 10 V and 20 V along TD, 30 V and 40 V along RD) are measured by the displacement sensors, as shown in Fig. 3 The equivalent, principal strain fields and strain state are presented in Fig. 3 The equivalent and principal strain curves averaged by the nine central points are presented in Fig. 3-39. Compared with those of quasi-static test, the PLC effect seems to disappear at higher strain rates. The order of equivalent strain before ms t 6

= is only For dynamic biaxial tests, the load sensors will ring [164] at one of eigen frequencies when the sliding bars are suddenly impacted by the mass bloc. The measured forces can not be used to characterize the material behaviour of specimen without treatment, because they are the combination of dynamic elastic response of load sensor and the true material behaviour of specimen.

Damping and experimental results

In practice, a damping treatment can be adopted to reduce the level of vibration when the structure is subjected to impacts or other transient forces [START_REF] Horr | Modelling of nonlinear damping characteristics of a viscoelastic structural damper[END_REF] . It consists in a material, which can increase the ability to store mechanical energy and dissipate a portion of energy through hysteresis. Viscoelastic dampers have long been used in the control of vibration in the engineering field [START_REF] Park | Analytical modeling of viscoelastic dampers for structural and vibration control[END_REF] .

In order to improve the quality of measured forces during dynamic biaxial tests, a damping layer of elastomer is placed at the interface between the sliding bar and inertia mass to improve the impact condition, as shown in Fig. 3-44. The evolutions of equivalent, major and minor principal strains averaged by nine points are presented in Fig. 3-50. The order of equivalent strain before ms t 8

= is only 4 10 -. The PLC effect of AA5086 seems to disappear at the higher strain rates. The evolutions of strain path and equivalent strain rate at the central point of specimen are presented in Fig. 3 It is seen that the smoothed force coincide with the measured one along axis Y. The smoothed force along axis X is a little different from the measured one before N 4000 . The smoothed forces and experimental principal strains have also been synchronized, as shown in Fig. 3-54. It can be seen that the strain level for the force of N 4000 is very low and corresponds mainly to an elastic one. So, the influence of smooth process on the strain evolution can be neglected. Therefore, the experimental data presented in Fig. 3-54 will be used for parameter identification.

Parameter identification

With the pre-processed force curves along two directions and the principal strain curves at the central point from the dynamic biaxial tensile test with damping layers, a parameter identification has been performed to characterize the hardening behaviour of AA5086 sheet at intermediate strain rates. Here, the elastic behaviour and the shape of Bron and Besson 2004 yield function are assumed to be independent of strain rates. The elastic parameters and coefficients of Bon and Besson 2004 yield function have been determined respectively by uniaxial and biaxial tensile tests at the quasi-static conditions and room temperature.

The generalised Voce hardening law, which was used for parameter identification by the quasi-static biaxial tensile test, is also adopted. The identified parameters of this hardening law by dynamic biaxial tensile test are given in Table 3-12 and compared with those obtained by quasi-static one. The uniaxial and biaxial flow stress curves identified by quasi-static and dynamic biaxial tests are compared in Fig. 3-55. The uniaxial flow stress curve obtained from dynamic uniaxial test is also presented. It is seen that the identified flow stress curve by dynamic biaxial test is a little lower than the one by quasi-static biaxial test. The relative error between the two biaxial flow stress curves is about 5%. This difference can be explained by the small negative strain rate sensitivity of AA5086, also observed for uniaxial characterizations, and measurement errors.

0 p n K ε σ σ - - + = Objective 0 σ (MPa) K (MPa) n ( ) 1 ε δ ( ) 2 ε δ Quasi-
The simulated principal strain curves are compared with the experimental ones of the dynamic biaxial test, as shown in Fig. 3-56. It can be seen that the simulated major principal strain curve is in a good agreement with the experimental one and the simulated minor principal strain curve is slightly higher than the experimental one after ms t 5

= .

Conclusion

With the experimental and identified results from the quasi-static and dynamic biaxial tensile tests on aluminium alloy AA5086, it can be concluded that:

(1) With the optimal shape of cruciform specimen, biaxial tensile tests on AA5086 sheet under quasi-static and dynamic conditions at room temperature have been carried out. The crack initially occurred at the central zone, so the optimal shape of cruciform specimen is experimentally validated to obtain larger strains.

(2) For parameter identification by biaxial tensile test on cruciform specimen, the global forces along the two directions are applied to the FE model, and the local experimental and simulated principal strains at the central point are compared to optimize the parameters of material model. The generalised Voce hardening law is chosen to characterize the hardening behaviour. This strategy of parameter identification is validated for a quasi-static biaxial tensile test.

( (4) During the dynamic tests at intermediate strain rate, the dynamic response of the mechanical parts located between the specimen and actuator leads to oscillation on the force signal. To reduce these oscillations, an elastomer layer (thickness of mm 3 ) has been used. Comparison of experimental results with and without damping layer shows that the amplitude of oscillations is greatly reduced when using an elastomer. The hardening law under dynamic condition has also been identified with the yield criterion of Bron and Besson 2004, and compared with the one obtained under quasi-static condition, so the dynamic testing procedure has been validated for material characterization at intermediate strain rates.

Identification of strain-rate dependent hardening laws of DP600 by biaxial tests

Introduction

For sheet metal forming processes, elastoplastic behaviour of many materials exhibits strain rate sensitivity. The hardening laws should consider both strain hardening and strain rate effects for rate-dependent materials at room temperature.

In this chapter, uniaxial and biaxial tensile tests under quasi-static and dynamic conditions will be performed and compared on a rate-dependent sheet metal DP600.

The same geometry of specimens, testing machines and strategy of parameter identification, which has been validated for characterization of AA5086, are applied for the characterization of viscoplastic behaviour of a DP600 sheet.

Uniaxial test of DP600

DP600 is a dual phase Advanced High Strength Steel (AHSS), which is designed to offer high strength, ductility and formability. DP steels consist in a ferritic matrix containing a hard martensitic second phase in the form of islands [167] . The material is widely used for body-in-white structural end reinforcement components, such as pillar reinforcements, crash structures, as well as for light weight seat structures [168] . and s / 20 , respectively. The high speed camera is also adopted to record the deformation of specimens. The main parameters of measurement system for each loading speed are presented in Table 4-1. DIC technique is adopted to calculate the true strain fields. The subsets in the gauge zone for each specimen have been shown in Fig. 4-1. The magnification factor from pixel to mm is pixel mm 1 . 0

Experimental result

. Then, the true strains on the gauge zone are averaged for each specimen before the onset of necking. To calculate the true stress-true strain curves, the true stress σ is calculated from dividing the force by current cross section of specimen before diffuse necking. The current cross section is obtained by the initial cross section and true strain ε with the assumption of constant volume. If the measurement frequencies of experimental force and deformation are not the same, the experimental forces are interpolated by the method of piecewise cubic Hermit interpolation in MATLAB.

The true stress-true strain curves of DP600 at different strain rates of s 02 . 0

, s 2 , s 10 and s 20 are presented in Fig. 456. It is seen that the true stress-true strain curve of DP600 becomes higher when the strain rate is elevated. The hardening behaviour of DP600 sheet exhibits positive strain rate sensitivity at intermediate strain rate range and room temperature.

Strain rate sensitivity

For parameter identification of hardening laws, the true stress-plastic strain curves have been calculated from the true stress-true strain curves, as shown in Fig. 4567. The yield points on the true stress-true strain curve are determined by the conventional limit at 0.2% plastic strain. and Voce ( ( ) ( ) ( )

p p n K ε ε - - = exp 1 H
) models are compared for characterization of the strain hardening effect and simple power model is adopted to describe the strain rate sensitivity ( ( ) m ε ε

ɺ ɺ = V
). The adiabatic deformation-induced thermal effect is neglected [169] [170] . As seen in chapter 1, four types of hardening law can be defined in a multiplicative way as follows:

(1) multiplying the strain rate term ( ) εɺ V with the strain-hardening terms ( )

p ε H : ( ) ( ) ε ε σ σ ɺ V H 0 ⋅ + = p Eq. 4-1
(2) multiplying the strain rate term ( ) εɺ V with the initial yield stress 0 σ :

( ) ( ) n p ε ε σ σ H V 0 + ⋅ = ɺ
Eq. 4-2

(3) multiplying the same strain rate term ( ) εɺ V with the initial yield stress 0 σ and strain-hardening terms ( )

p ε H : ( ) ( ) ( ) ε ε σ σ ɺ V H 0 ⋅ + = p Eq. 4-3 (
( ) ( ) ( ) ε ε ε σ σ ɺ ɺ 2 1 0 V H V ⋅ + ⋅ = p Eq. 4-4
To determine the parameters of hardening laws, an iterative procedure is adopted to minimize the errors between the experimental and fitted flowing stress curves, as shown in Fig. 45678. Here, the optimized objective δ is defined as follows:

( ) ( ) ( ) ∑ ∑ ∑                   - = = j n i j i p j i p j i p fit j j n 2 , exp, exp , exp, exp , exp, 1 ε σ ε σ ε σ δ δ Eq. 4-5
Here, j is the curve number of true stress-plastic strain curve at specific strain rate, i is the point number on each curve, fit σ and exp σ are the fitted and experimental stress.

The optimization process is completed by modeFRONTIER, in Appendix (IV). The Bounded BFGS (B-BFGS) algorithm is chosen, which can handle the design variables on the actual bounds in a suitable way [171] . It is an extension of the classical Broyden- for low plastic strain (below % 4 ). The strain hardening effects identified on basis of Ludwick law are overestimated when the plastic strain reaches about % 15 .

The identified results of four rate-dependent hardening laws on the basis of Voce law are presented in

Experimental results of biaxial tests

The experimental procedures of quasi-static and dynamic biaxial tests on cruciform shape, which has been validated in previous chapter (on AA5086), have been applied to characterize viscoplastic behaviour of DP600. Three loading speeds for each actuator are adopted: s mm 02 . The main parameters of high speed camera and load sensors are respectively defined for biaxial tests at different loading speeds, as shown in t=128.52s t=127.2s The parameters of these hardening laws are identified by the uniaxial tensile tests. It is seen that the flow stress curves on the basis of Ludwick law become much higher than those on the basis of Voce law, after an equivalent plastic strain of % 15 . Therefore, the uniaxial tests dose not permit to identify an appropriate hardening law for large strains (higher than % 15 ).

The biaxial flow stress curves identified with two strain-rate dependent hardening laws on the basis of Ludwick and Voce models are also compared in Fig. 4-39. The parameters of these hardening laws are identified based on the FE model with Hill 48 yield criterion. It is seen that the strain rate sensitivities of uniaxial and biaxial flow stress curves are identical. Therefore, the dynamic biaxial tensile tests are validated to identify strain-rate hardening laws. The biaxial flow stress curves are relatively close for each strain rate until the equivalent plastic strain of about % 30 . It clearly shows that the biaxial tensile tests permits a better hardening identification for large strains.

Conclusion

The strain-rate dependent hardening behaviour of DP600 has been investigated by uniaxial tensile tests at different strain rates ( s ). Based on the experimental data, it is found that DP 600 exhibits a positive strain rate sensitivity. Ludwick or Voce laws have been adopted to describe the strain hardening effect and simple power law has been used to represent the strain rate sensitivity. Finally, four multiplicative types of strain-rate dependent hardening laws have been compared to characterize the hardening behaviour of DP600 at room temperature. 10 - s , it is more beneficial of biaxial tensile tests to identify strain-rate dependent hardening behaviour up to large strains.

Conclusions and perspectives

The objective of this thesis was to propose quasi-static and dynamic biaxial tensile tests on flat cruciform specimen to identify strain-rate dependent hardening models of sheet metals from quasi-static to intermediate strain rates. The in-plane biaxial testing procedures and parameter identification strategy have been validated on AA5086 and applied to identify rate-dependent hardening laws for DP600 steel.

The main contributions of this work can be summarized as follows:

A cruciform shape has been successfully designed to obtain large equivalent plastic strains (up to % 30 ) at the central zone of specimen under biaxial tensile strain path. Specific notches at the intersections of arms, slits in each arm and thickness reduction at the central zone are adopted. As expected, the initial cracks of tested specimens are always observed at the central zone.

The parameters of hardening laws are identified by inverse analysis based on a FE model of the test. The experimental forces are applied on the arms of FE model. The simulated principal strains at the central zone are compared with experimental results to optimize the material parameters. By comparison of the identified biaxial flow stress curves with the uniaxial flow stress curves, this parameter identification strategy has been validated. Three yield criteria are compared to identify the parameters of hardening laws for quasi-static biaxial tensile test on AA5086. The biaxial flow stress curve identified with advanced anisotropic yield function of Bron and Besson 2004 coincides with the uniaxial flow stress curve. It is shown that the flow stress curve is precise only if an appropriate yield function is preliminarily chosen. Dynamic biaxial tensile tests on cruciform specimen have been carried out to identify strain-rate hardening behaviour of sheet metals at intermediate strain rates. Damping layers are adopted to reduce loading ringing.

Different strain-rate dependent hardening laws have been identified for DP600 steel by biaxial tensile tests. The biaxial flow stress curves identified on the basis of Ludwick and Voce models are close up to equivalent plastic strains of % 30 for each strain rate. The benefits of the proposed methodology are clearly shown since the hardening behaviour is now accurately known for an equivalent strain level of % 30 .

In the future, several perspectives are suggested as following:

In this work, only the principal strains at the central zone of cruciform specimen under equi-biaxial path are used for parameter identification of hardening law. If some parameters of the yield criterion are preliminarily unknown, more experimental data under different strain paths can be adopted for inverse analysis to identify all the parameters of constitutive models simultaneously. It could be also interesting to evaluate the influence of strain path on the identified results of hardening law.

Many other hardening models in the literature can be used for identification of hardening behaviour. As seen in literature, a mixed type of Ludwick and Voce laws is certainly more precise to identify hardening behaviour up to large strains.

When the deformation level is elevated, the evolution of microstructure and textures in sheet metals may change the parameters of yield functions. Therefore, the effects of strain level on the subsequent yield surface need further study [START_REF] Iadicola | Experimental observations of evolving yield loci in biaxially strained AA5754-O[END_REF] [175] [176] [177] . The effects of strain rate on the initial yield surfaces are also very interesting [178] [179] . The yield functions, which consider the influences of strain level and strain rate, will be more accurate to identify strain-rate dependent hardening models up to large strains.

Because successive nonlinear deformation path can be realised by biaxial tensile tests on the cruciform specimen, advanced hardening models, like kinematical combined models, could be identified by the proposed methodology.

(II) modeFRONTIER flowchart with subroutine UHARD

For the user subroutine UHARD of rate-independent hardening law, the yield stress σ and variation of yield stress with respect to the equivalent plastic strain

  et Bron et Besson) ont été successivement utilisés pour l'identiication des paramètres des lois d'écrouissage. Les résultats obtenus montrent d'une part que la modélisation est très sensible au critère de plasticité choisi, et d'autre part que le critère de Bron et Besson permet d'obtenir une très bonne corrélation entre les courbes d'écrouissage identiiées à partir de l'essai bi-axial et de l'essai uni-axial. Pour les tests dynamiques bi-axiaux, les phénomènes de résonance du dispositif mécanique, générés à l'impact initial de début d'essai et matérialisés par de fortes oscillations du signal d'effort, sont atténués par l'interposition d'un élément en élastomère dans le système d'ancrage de chaque bras de l'éprouvette. Pour inir, la méthodologie d'identiication proposée est appliquée à la caractérisation du comportement viscoplastique d'un acier dual phase DP600. Les courbes d'écrouissage identiiées à partir des essais bi-axiaux ont été comparées à celles obtenues par des essais uni-axiaux pour une gamme de vitesse de déformation allant de 10 - Identification of strain-rate dependent hardening sensitivity of metallic sheets under in-plane biaxial loading Wei LIU En partenariat avec Document protégé par les droits d'auteur

  Figures

1

 1 Fig. 1-1 Physical based and phenomenogical modelling of metals ............................... Fig. 1-2 Drawing of cylinder part .................................................................................. Fig. 1-3 Stress and strain state of cylinder part .............................................................. Fig. 1-4 Strain paths for cylinder part ............................................................................ Fig. 1-5 Strain rate jump-down test [22] .......................................................................... Fig. 1-6 Hardening models .......................................................................................... Fig. 1-7 Uniaxial tensile tests along different directions ............................................. Fig. 1-8 Schematic representation of four types stress-strain curves [37] ..................... Fig. 1-9 Representative strain and stress states on yield locus .................................... Fig. 1-10 Schematic of strain rate ranges and experimental techniques [52] ................ Fig. 1-11 Uniaxial tensile specimen ............................................................................. Fig. 1-12 Shortened upper grip and measured force curve [56] .................................... Fig. 1-13 Dynamic testing device with long bar [58] .................................................... Fig. 1-14 A simplified schematic of the modified servo-hydraulic machine[START_REF] Othman | A modified servohydraulic machine for testing at intermediate strain rates[END_REF] ......... Fig.1-15 Position of strain gauges on specimen for local force measurement[START_REF] Wood | Strain rate testing of metallic materials and their modeling for use in CAE based automotive crash simulation tools[END_REF] ...... Fig.1-16 Comparisons of force measured by local force transducer and DLC[START_REF] Wood | Strain rate testing of metallic materials and their modeling for use in CAE based automotive crash simulation tools[END_REF] ..... Fig.1-17 Dynamic tensile testing device with damping[START_REF] Fitoussi | Experimental methodology for high strain-rates tensile behaviour analysis of polymer matrix composites[END_REF] ........................................ Fig. 1-18 Experimental set-up for dynamic uniaxial compression [64] ......................... Fig. 1-19 Schematic of Split Hopkinson Pressure Bar and travelling wave diagram .. Fig. 1-20 Bulge test [70] ................................................................................................ Fig. 1-21 Dynamic bulge testing device [75] ................................................................. Fig. 1-22 Plane strain specimens [79] [82] ....................................................................... Fig. 1-23 Cruciform shape for plane-strain tension [83] ................................................ Fig. 1-24 Shear testing specimen (ASTM B831-11) [87] .............................................. Fig. 1-25 Specimen of double shear test [89] ................................................................ Fig. 1-26 Experimental setup of direct impact [89] ....................................................... Fig. 1-27 Biaxial tensile tests ....................................................................................... Fig. 1-28 Uniaxial tension-internal pressure testing machine ................................... Fig. 1-29 Combined plane strain-simple shear test [12] ................................................ Fig. 1-30 Experimental tools and sample under heterogeneous deformation [107] ..... Fig. 2-1 Applications of biaxial tests for material modelling ...................................... Fig. 2-2 Link mechanism attachments for biaxial test (FER 1988) ............................. Fig. 2-3 Stand-alone biaxial testing machines (MAK 1992, KUW 1998) ................... Fig. 2-4 Dynamic biaxial testing machine ................................................................... Fig. 2-5 Biaxial testing machine and measurement systems ....................................... Fig. 2-6 Cruciform shape (MUL, 1996) ....................................................................... Fig. 2-7 Cruciform shape (NAK, 2003) ....................................................................... Fig. 2-8 Cruciform Shape (KUW, 1998) ..................................................................... Fig. 2-9 Cruciform shape (MER, 2008) ....................................................................... Fig. 2-10 Cruciform shape of UT/PST (a) and UT/EBT (b) specimen (TEA, 2010) ..

- 1

 1 Fig. 1-1 Physical based and phenomenogical modelling of metals ............................... Fig. 1-2 Drawing of cylinder part .................................................................................. Fig. 1-3 Stress and strain state of cylinder part .............................................................. Fig. 1-4 Strain paths for cylinder part ............................................................................ Fig. 1-5 Strain rate jump-down test [22] .......................................................................... Fig. 1-6 Hardening models .......................................................................................... Fig. 1-7 Uniaxial tensile tests along different directions ............................................. Fig. 1-8 Schematic representation of four types stress-strain curves [37] ..................... Fig. 1-9 Representative strain and stress states on yield locus .................................... Fig. 1-10 Schematic of strain rate ranges and experimental techniques [52] ................ Fig. 1-11 Uniaxial tensile specimen ............................................................................. Fig. 1-12 Shortened upper grip and measured force curve [56] .................................... Fig. 1-13 Dynamic testing device with long bar [58] .................................................... Fig. 1-14 A simplified schematic of the modified servo-hydraulic machine[START_REF] Othman | A modified servohydraulic machine for testing at intermediate strain rates[END_REF] ......... Fig.1-15 Position of strain gauges on specimen for local force measurement[START_REF] Wood | Strain rate testing of metallic materials and their modeling for use in CAE based automotive crash simulation tools[END_REF] ...... Fig.1-16 Comparisons of force measured by local force transducer and DLC[START_REF] Wood | Strain rate testing of metallic materials and their modeling for use in CAE based automotive crash simulation tools[END_REF] ..... Fig.1-17 Dynamic tensile testing device with damping[START_REF] Fitoussi | Experimental methodology for high strain-rates tensile behaviour analysis of polymer matrix composites[END_REF] ........................................ Fig. 1-18 Experimental set-up for dynamic uniaxial compression [64] ......................... Fig. 1-19 Schematic of Split Hopkinson Pressure Bar and travelling wave diagram .. Fig. 1-20 Bulge test [70] ................................................................................................ Fig. 1-21 Dynamic bulge testing device [75] ................................................................. Fig. 1-22 Plane strain specimens [79] [82] ....................................................................... Fig. 1-23 Cruciform shape for plane-strain tension [83] ................................................ Fig. 1-24 Shear testing specimen (ASTM B831-11) [87] .............................................. Fig. 1-25 Specimen of double shear test [89] ................................................................ Fig. 1-26 Experimental setup of direct impact [89] ....................................................... Fig. 1-27 Biaxial tensile tests ....................................................................................... Fig. 1-28 Uniaxial tension-internal pressure testing machine ................................... Fig. 1-29 Combined plane strain-simple shear test [12] ................................................ Fig. 1-30 Experimental tools and sample under heterogeneous deformation [107] ..... Fig. 2-1 Applications of biaxial tests for material modelling ...................................... Fig. 2-2 Link mechanism attachments for biaxial test (FER 1988) ............................. Fig. 2-3 Stand-alone biaxial testing machines (MAK 1992, KUW 1998) ................... Fig. 2-4 Dynamic biaxial testing machine ................................................................... Fig. 2-5 Biaxial testing machine and measurement systems ....................................... Fig. 2-6 Cruciform shape (MUL, 1996) ....................................................................... Fig. 2-7 Cruciform shape (NAK, 2003) ....................................................................... Fig. 2-8 Cruciform Shape (KUW, 1998) ..................................................................... Fig. 2-9 Cruciform shape (MER, 2008) ....................................................................... Fig. 2-10 Cruciform shape of UT/PST (a) and UT/EBT (b) specimen (TEA, 2010) ..

  Fig. 1-1 Physical based and phenomenogical modelling of metals ............................... Fig. 1-2 Drawing of cylinder part .................................................................................. Fig. 1-3 Stress and strain state of cylinder part .............................................................. Fig. 1-4 Strain paths for cylinder part ............................................................................ Fig. 1-5 Strain rate jump-down test [22] .......................................................................... Fig. 1-6 Hardening models .......................................................................................... Fig. 1-7 Uniaxial tensile tests along different directions ............................................. Fig. 1-8 Schematic representation of four types stress-strain curves [37] ..................... Fig. 1-9 Representative strain and stress states on yield locus .................................... Fig. 1-10 Schematic of strain rate ranges and experimental techniques [52] ................ Fig. 1-11 Uniaxial tensile specimen ............................................................................. Fig. 1-12 Shortened upper grip and measured force curve [56] .................................... Fig. 1-13 Dynamic testing device with long bar [58] .................................................... Fig. 1-14 A simplified schematic of the modified servo-hydraulic machine[START_REF] Othman | A modified servohydraulic machine for testing at intermediate strain rates[END_REF] ......... Fig.1-15 Position of strain gauges on specimen for local force measurement[START_REF] Wood | Strain rate testing of metallic materials and their modeling for use in CAE based automotive crash simulation tools[END_REF] ...... Fig.1-16 Comparisons of force measured by local force transducer and DLC[START_REF] Wood | Strain rate testing of metallic materials and their modeling for use in CAE based automotive crash simulation tools[END_REF] ..... Fig.1-17 Dynamic tensile testing device with damping[START_REF] Fitoussi | Experimental methodology for high strain-rates tensile behaviour analysis of polymer matrix composites[END_REF] ........................................ Fig. 1-18 Experimental set-up for dynamic uniaxial compression [64] ......................... Fig. 1-19 Schematic of Split Hopkinson Pressure Bar and travelling wave diagram .. Fig. 1-20 Bulge test [70] ................................................................................................ Fig. 1-21 Dynamic bulge testing device [75] ................................................................. Fig. 1-22 Plane strain specimens [79] [82] ....................................................................... Fig. 1-23 Cruciform shape for plane-strain tension [83] ................................................ Fig. 1-24 Shear testing specimen (ASTM B831-11) [87] .............................................. Fig. 1-25 Specimen of double shear test [89] ................................................................ Fig. 1-26 Experimental setup of direct impact [89] ....................................................... Fig. 1-27 Biaxial tensile tests ....................................................................................... Fig. 1-28 Uniaxial tension-internal pressure testing machine ................................... Fig. 1-29 Combined plane strain-simple shear test [12] ................................................ Fig. 1-30 Experimental tools and sample under heterogeneous deformation [107] ..... Fig. 2-1 Applications of biaxial tests for material modelling ...................................... Fig. 2-2 Link mechanism attachments for biaxial test (FER 1988) ............................. Fig. 2-3 Stand-alone biaxial testing machines (MAK 1992, KUW 1998) ................... Fig. 2-4 Dynamic biaxial testing machine ................................................................... Fig. 2-5 Biaxial testing machine and measurement systems ....................................... Fig. 2-6 Cruciform shape (MUL, 1996) ....................................................................... Fig. 2-7 Cruciform shape (NAK, 2003) ....................................................................... Fig. 2-8 Cruciform Shape (KUW, 1998) ..................................................................... Fig. 2-9 Cruciform shape (MER, 2008) ....................................................................... Fig. 2-10 Cruciform shape of UT/PST (a) and UT/EBT (b) specimen (TEA, 2010) ..

FiguresVIIIFig. 2 -

 2 Figures

FiguresIXFig. 3 - 7

 37 Figures

FiguresXFig. 3 -

 3 Figures

FiguresXIFig. 4 -

 4 Figures

Fig. 1 - 1

 11 Fig. 1-1 Physical based and phenomenogical modelling of metals

Fig. 1 - 2 Fig. 1 - 4

 1214 Fig. 1-2 Drawing of cylinder part Fig. 1-3 Stress and strain state of cylinder part

Bending

2 Fig. 1 - 5

 215 Fig. 1-5 Strain rate jump-down test[START_REF] Sung | A plastic constitutive equation incorporating strain, strain-rate,and temperature[END_REF]

  becomes the isotropic hardening model. When 0 ≠ ij α and σ is constant, it becomes the kinematical hardening model. When 0 ≠ ij α

Fig. 1 - 7

 17 Fig. 1-7 Uniaxial tensile tests along different directions

  yield criterion becomes Mises criterion.

  a

Eq. 1 - 22

 122 In these two yield criteria, the diagonal tensor representation S ~ are defined by the principal values of ij s ~, and the tensor ij s ~ is defined as a linear transformation of the stress deviator ij s .

  values of a modified stress deviator k ij

  are the same. A simple multiplicative form is given as follows:

σ. 3 10 - to s 3 10,

 33 , a , D , ∞ E , m and α are the material parameters and the reference strain rate s This model describes well the hardening behaviour for DP600 steel at the strain rates ε ɺ from s and shows that the strain hardening of DP600 have little dependence on the strain rate.

Fig. 1 -

 1 Fig. 1-10 Schematic of strain rate ranges and experimental techniques[START_REF] Field | Review of experimental techniques for high rate deformation and shock studies[END_REF]

Fig. 1 -

 1 Fig. 1-11 Uniaxial tensile specimen

Fig. 1 -

 1 Fig. 1-12 Shortened upper grip and measured force curve[START_REF] Kim | Dynamic tensile test and specimen design of auto-body steel sheet at the intermediate strain rate[END_REF]

  -15. The measured force by two methods are shown in Fig. 1-16. It can be seen that the force measured by the local force transducer on the specimen is less oscillating that that measured by the DLC.

Fig. 1 -

 1 Fig. 1-15 Position of strain gauges on specimen for local force measurement[START_REF] Wood | Strain rate testing of metallic materials and their modeling for use in CAE based automotive crash simulation tools[END_REF]

Fig. 1 -

 1 Fig. 1-17 Dynamic tensile testing device with damping[START_REF] Fitoussi | Experimental methodology for high strain-rates tensile behaviour analysis of polymer matrix composites[END_REF]

Fig. 1 -

 1 Fig. 1-18 Experimental set-up for dynamic uniaxial compression[64]

Fig. 1 -

 1 Fig. 1-21 Dynamic bulge testing device[START_REF] Grolleau | Biaxial testing of sheet materials at high strain rates using viscoelastic bars[END_REF]

Fig. 1 -

 1 Fig. 1-22 Plane strain specimens[79] [82]

Fig. 1 - 1

 11 Fig. 1-26 Experimental setup of direct impact[START_REF] Rusinek | Shear testing of a sheet steel at wide range of strain rates and a constitutive relation with strain-rate and temperature dependence of the flow stress[END_REF]

Fig. 1 -

 1 Fig. 1-27 Biaxial tensile tests

  -30. With the punch moving, the deformation of sample combines two uniaxial tensile paths, two shear paths and one biaxial tensile paths. The parameters of Hill 48 yield function and Ludwick hardening law have been identified for a commercial pure titanium sheet. The identified material parameters were validated by comparison the top profiles of a deep-drawing cup between experimental and simulated results.

Fig. 1 -

 1 Fig. 1-30 Experimental tools and sample under heterogeneous deformation[107]

Fig. 2 - 2 Fig. 2 - 3 2 .

 22232 Fig. 2-2 Link mechanism attachments for biaxial test (FER 1988)

Fig. 2

 2 Fig. 2-4 Dynamic biaxial testing machine (Loading capacity: KN 50 , Quasi-static velocity: s m 01 . 0 ~ s m 2 ,

Fig. 2 - 6 Fig. 2 - 7

 2627 Fig. 2-6 Cruciform shape (MUL, 1996) Fig. 2-7 Cruciform shape (NAK, 2003)

Fig. 2 - 8 Fig. 2 - 9

 2829 Fig. 2-8 Cruciform Shape (KUW, 1998) Fig. 2-9 Cruciform shape (MER, 2008)

Fig. 2 -

 2 Fig. 2-10 Cruciform shape of UT/PST (a) and UT/EBT (b) specimen (TEA, 2010)

Fig. 2 -

 2 Fig. 2-11 Cruciform shape (PRA, 2014)

Fig. 2 -

 2 Fig. 2-12 Cruciform shape (ZHA, 2014)

Fig. 2 -

 2 Fig. 2-13 Cruciform shapes (MAK, 1992; GRE, 2004)

Fig. 2 -

 2 Fig. 2-14 Cruciform shapes (GOZ, 2004; KUL, 2011)

Fig. 2 -

 2 Fig. 2-15 Cruciform shape (MER, 2013)

Fig. 2 -

 2 Fig. 2-16 Cruciform shape (YU, 2002)

Fig. 2 -

 2 Fig. 2-17 Cruciform shape (TAS, 2008)

Fig. 2 -

 2 Fig. 2-18 Cruciform shapes (ABU, 2009)

  -19. The square thickness-reduced zone (middle thickness, mm 1) with rounded corners is rotated by 45° relative to the arms. The second-step thicknessreduced zone (final thickness,

Fig. 2 -

 2 Fig. 2-19 Cruciform Shapes (LEE, 2015)

Fig. 2 -

 2 Fig. 2-20 Cruciform specimen with further curved thickness reduction

Fig. 2 -.

 2 Fig. 2-22 True stress-plastic strain curve for FE analysis

- 23 .

 23 The model has been discretized by 5362 linear triangular shell elements of type S3. The element size at the central zone has been refined

Fig. 2 -

 2 Fig. 2-23 Basic cruciform shape Fig. 2-24 Mesh of basic shape

Fig. 2 -

 2 Fig. 2-25 Major principal, equivalent plastic strain and strain state of basic shape

Fig. 2 -

 2 Fig. 2-27 Effect of distance D (R=5mm)

2 .

 2 Shape design of cruciform specimen59

Fig. 2 -

 2 Fig. 2-28 Effect of radius (D-R=10mm)

Fig. 2 -

 2 Fig. 2-29 Mesh of shape with notches (D=13mm, R=3mm)

Fig. 2 -

 2 Fig. 2-30 Major principal, equivalent plastic strain and strain state of shape with notches

Fig. 2 -

 2 Fig. 2-31 Shape with slits

Fig. 2 -

 2 Fig. 2-32 Comparison of shapes with slits

Fig. 2 -

 2 Fig. 2-33 Mesh of shape with slits

Fig. 2 -

 2 Fig. 2-34 Major principal, equivalent plastic strain and strain state of shape with slits

Fig. 2 -

 2 Fig. 2-35 Shape with thickness reduction

Fig. 2 -

 2 Fig. 2-37 Effect of thickness of thickness-reduced zone

Fig. 2 -

 2 Fig. 2-38 FE model of shape with thickness reduction and notch ( r=5mm, t=0.5mm)

Fig. 2 -

 2 Fig. 2-39 Major principal, equivalent plastic strain and strain state of shape with thickness reduction and notch

  1.

Fig. 2 A

 2 Fig. 2-40 Optimal shape for large strains

Fig. 2 -

 2 Fig. 2-41 Mesh of optimal shape

Fig. 2 -

 2 Fig. 2-42 Major principal, equivalent plastic strain and strain state of optimal shape

Fig. 2 -

 2 Fig. 2-43 Strain paths of optimal shape

  central zone of cruciform specimen under equibiaxial tensile test. The defined shape with one thickness reduction is proposed from an initial thickness sheet of mm 2 . The final thickness of the thickness-reduced central zone is mm 625 . 0

Fig. 3 - 1

 31 Fig. 3-1 Procedure of strain measurement by DIC

  Fig. 3-2 Subsets for DIC Fig. 3-3 Mode for strain calculation

Fig. 3 - 4

 34 Fig. 3-4 Shape of uniaxial testing specimen

Fig. 3 - 6

 36 Fig. 3-6 Strain curves of static (1mm/s) and dynamic (500mm/s) uniaxial test

Fig. 3 - 7 3 . 2 . 2

 37322 Fig. 3-7 Force curves of static (1mm/s) and dynamic (500mm/s) uniaxial test

Fig. 3 - 9

 39 Fig. 3-9 Extrapolation of static and dynamic uniaxial flowing curve

Fig. 3 -

 3 Fig. 3-10 Cruciform specimen of AA5086

Fig. 3 -

 3 Fig. 3-11 Linkage between specimen and sliding bar

Fig. 3 -Fig. 3 -

 33 Fig. 3-12 Central zone of quasi-static specimen (speckle pattern and initial crack)

Fig. 3 -

 3 Fig. 3-15 Equivalent strain, principal strain field and strain state of quasi-static biaxial specimen

ε 1 ε 2 ε

 2 where m is time interval. The strain path at the central point is nearly proportional (

Fig. 3 -

 3 Fig. 3-17 Strain path and strain rate evolution at central point of quasi-static specimen

Fig. 3 -

 3 Fig. 3-18 Measured force curves of quasi-static biaxial test

Fig. 3 -

 3 Fig. 3-19 Pre-processed experimental force and strain curves of quasi-static biaxial test

ε and exp 2 ε

 2 simulated major and minor principal strains at the central element of FE model; exp 1 are the experimental principal strain at each simulated time point.

  used. The experimental forces are applied to FE models.

For

  FE models with linear tetrahedral element of type C3D4, the refined size at the central zone are varied with mm 3 . The equivalent plastic strain fields are compared in Fig.3-22. It can be seen that the strain level and distribution at the refined zone are greatly influenced by the mesh size. The model C3D4_5 with the smallest mesh size ( mm 1 . 0 ) can capture the deformation localisation at the thickness reduced zone.

Fig. 3 -Fig. 3 -

 33 Fig. 3-21 Zone to be refined (C3D4_1, Size:0.5, Elements:66322, CPU Time: 786.4s)

Fig. 3 - 3 . 3 . 3 . 2

 33332 Fig. 3-23 Comparison of major and minor principal strains

Fig. 3 -

 3 Fig. 3-24 Three zones of specimen

Fig. 3 -

 3 Fig. 3-25 Comparison of equivalent strain fields between solid (a) and shell (b) elements

Fig. 3 -

 3 Fig. 3-26 Comparison of principal strain evolutions between solid (C3D4_5) and shell elements

Fig. 3 -Fig. 3 -

 33 Fig. 3-28 Results of DOE with Mises yield criterion

Fig. 3 -

 3 Fig. 3-30 Comparison of quasi-static flow stress curves of AA5086

  strains with the different identified hardening laws and Mises,Hill 48, and Bron and Besson 2004 yield criteria are also compared with the experimental principal strains in Fig.3-31, Fig.3-32 and Fig.3-33, respectively. It can be seen that the discrepancies of principal strains between the experiments and simulation by the identified hardening laws with Mises, Hill 48 yield criteria are more severe. For the simulation by the identified hardening law and the Bron and Besson 2004 yield criterion, the simulated major principal strain curve nearly coincide with the experimental one, while the simulated minor principal strain curves also coincide well with the experimental one until the strain becomes too large. Therefore, it can be concluded that the yield function of Bron and Besson 2004, which has been identified by a single biaxial tensile test of cruciform specimen, is further validated to describe the yielding anisotropy of AA5086 sheet.

Fig. 3 -Fig. 3 -Fig. 3 -

 333 Fig. 3-31 Comparison of principal strains between experiments and simulation with Mises

Fig. 3 -

 3 Fig. 3-34 Process of dynamic biaxial tensile test

Fig. 3 -

 3 Fig. 3-35 Velocity of each actuator during dynamic biaxial test

Fig. 3 -

 3 Fig. 3-36 Crack of dynamic specimen Fig. 3-37 Displacement vectors of dynamic specimen

Fig. 3 -

 3 Fig. 3-38 Experimental equivalent and principal strain fields of dynamic biaxial testing specimen

Fig. 3 -

 3 Fig. 3-39 Experimental strain curves of dynamic biaxial test

Fig. 3 - 1 ε 2 εFig. 3 -Fig. 3 -

 31233 Fig. 3-40 Strain path and equivalent strain rate at the central point during dynamic test

Fig. 3 -

 3 Fig. 3-43 First and second modes by numerical simulation

Fig. 3 -

 3 Fig. 3-44 Position of damping layer

Fig. 3 -

 3 Fig. 3-45 Experimental force-displacement curves of damping layer

Fig. 3 -Fig. 3 -

 33 Fig. 3-46 Loaded velocity of dynamic biaxial test with damping

Fig. 3 -

 3 Fig. 3-49 Experimental equivalent and principal strain field of dynamic specimen with damping

Fig. 3 -

 3 Fig. 3-50 Experimental strain curves of dynamic biaxial test with damping

Fig. 3 -

 3 Fig. 3-51 Strain path and strain rate of dynamic biaxial test with damping

Fig. 3 -

 3 Fig. 3-52 Measured force curves of dynamic biaxial test with damping

Fig. 3 -

 3 Fig. 3-53 Smoothed force curves of dynamic biaxial test with damping

Fig. 3 -

 3 Fig. 3-55 Comparison of static and dynamic biaxial flow stress curves

Fig. 3 -

 3 Fig. 3-56 Comparison of experimental and simulated principal strains for dynamic test

)

  In this work, three yield criteria of Mises, Hill 48 and Bron and Besson 2004 are preliminarily defined to identify the parameters of hardening laws for the quasistatic biaxial tensile test. The identified biaxial flow stress curves are also compared with uniaxial flow stress curve. The biaxial flow stress curve identified with the advanced anisotropic yield criterion of Bron and Besson 2004 coincide very well with the uniaxial experimental flow stress curve. It is further validated that Bron and Besson 2004 yield criterion can well describe the anisotropic behaviour of AA5086.

Fig. 4 - 1

 41 Fig. 4-1 Gauge zones of specimen at speeds of 1, 100, 500, 1000 mm/s

1 Fig. 4 - 2 Fig. 4 - 4 Fig. 4 - 5 Fig. 4 - 6

 142444546 Fig. 4-2 Uniaxial test of DP600 (1mm/s) Fig. 4-3 Uniaxial test of DP600 (100mm/s)

Fig. 4 - 7

 47 Fig. 4-7 True stress-plastic strain curves of DP600 by uniaxial tensile tests

  4) multiplying the different strain rate terms ( )

Fig. 4 - 9

 49 Fig. 4-9 Comparison of experimental and identified curves on basis of Ludwick

Fig. 4 -

 4 Fig. 4-10 Comparison of experimental and identified curves on basis of Voce

  specimens of DP600 have been prepared, as shown in Fig. 4-11.

Fig. 4 -

 4 Fig. 4-11 Cruciform specimen of DP600

4. 3 . 1

 31 Loading speed of 0.02mm/s For quasi-static biaxial tensile test at the loading speed of s mm 02 . 0 , the initial crack of specimen can be seen at the central zone, as shown in Fig. 4-12. The deformation fields are calculated by DIC technique. There are 16 subsets and 14 strain points along each direction. The magnification factor is pixel mm 37 . 0 . The displacement vectors are shown in Fig. 4-13. It is seen that the central point is nearly unmoved.

Fig. 4 -Fig. 4 -

 44 Fig. 4-12 Initial crack at 0.02mm/s Fig. 4-13 Displacement vectors at 0.02mm/s

Fig. 4 -

 4 Fig. 4-14 Equivalent and principal strain fields and strain state at 0.02mm/s

Fig. 4 -

 4 Fig. 4-15 Equivalent and principal strain evolutions at the central zone (0.02mm/s)

Fig. 4 -

 4 Fig. 4-16 Strain path and strain rate evolution at the central point (0.02mm/s)

Fig. 4 -

 4 Fig. 4-17 Measured forces along two arms (0.02mm/s)

Fig. 4 - 4 . 3 . 2

 4432 Fig. 4-18 Experimental forces and principal strains for identification (0.02mm/s)

Fig. 4 -

 4 Fig. 4-19 Initial crack at 1mm/s Fig. 4-20 Displacement vectors at 1mm/s

Fig. 4 -Fig. 4 - 2 ε 4 .Fig. 4 -

 44244 Fig. 4-21 Equivalent and principal strain fields and strain state at 1mm/s

Fig. 4 -

 4 Fig. 4-24 Measured forces along two arms (1mm/s)

Fig. 4 -

 4 Fig. 4-25 Experimental forces and principal strains for identification (1mm/s)

Fig. 4 -

 4 Fig. 4-26 Loading velocities of dynamic biaxial test on DP600

Fig. 4 -

 4 Fig. 4-27 Initial crack at 250mm/s Fig. 4-28 Displacement vectors at 250mm/s

Fig. 4 -

 4 Fig. 4-29 Equivalent and principal strain fields and strain state at 250mm/s

Fig. 4 -

 4 Fig. 4-30 Equivalent, principal strain evolution at the central zone (250mm/s)

Fig. 4 -

 4 Fig. 4-31 Strain path and strain rate evolution at the central zone (250mm/s)

ε 1 ε 2 εFig. 4 -

 24 Fig. 4-32 Measured forces along two arms (250mm/s)

Fig. 4 - 4 . 4

 444 Fig. 4-33 Experimental forces and principal strains for identification (250mm/s)

Fig. 4 -Fig. 4 -Fig. 4 -

 444 Fig. 4-35 Comparison of experimental and simulated principal strains (Mises + Ludwick)

Fig. 4 -Fig. 4 -

 44 Fig. 4-38 Strain-rate dependent hardening laws identified by uniaxial tests

  Quasi-static and dynamic biaxial tensile tests on cruciform specimens are performed at different loading speeds of s based on FE model has been applied for parameter identification of hardening laws with strain rate sensitivity. By comparison of uniaxial and biaxial flow stress curves at strain

Where 4 ( 1 - s to 200 1 -s

 411 modeFRONTIER flowchart with subroutine UMAT With the user subroutine of UMAT, the transverse shear stiffness values for shell elements in ABAQUS are defined as follows: G is the shear modulus and t is the thickness of shell. The shear modulus G can be calculated by Young's module E and Poisson's ratio υ as follows: IV) modeFRONTIER flowchart with SRS by uniaxial tensile tests (a) For hardening laws of type 1, 2, 3 (b) For hardening laws of type 4 ; International Journal of Impact Engineering, 36 (2009) 263-271 [63] X. Xiao; Dynamic tensile testing of plastic materials; Polymer Testing, 27 (2008) 164-178 [64] S. Diot, D. Guines, A. Gavrus, E. Ragneau; Two-step procedure for identification of metal behavior from dynamic compression tests; International Journal of Impact Engineering, 34 (2007) 1163-1184

Table 1 -

 1 1 Forming velocity and strain rate ranges of different forming processes ..... Table 2-1 Optimal dimensions in mm ........................................................................ Table 3-1 Main parameters of DIC in CORRELA 2006 ............................................ Table 3-2 Main parameters of measurement system for uniaxial tests of AA5086 .... Table 3-3 Ludwick parameters identified from uniaxial test on AA5086 .................. Table 3-4 Voce parameters identified from uniaxial test on AA5086 ........................ Table 3-5 Main parameters of measurement system for quasi-static biaxial test ....... Table 3-6 Comparison between solid and shell elements ........................................... Table 3-7 Parameters of Hill 48 yield function for AA5086 ...................................... Table 3-8 Parameters of Bron and Besson 2004 yield function for AA5086 ............. Table 3-9 Lower and upper bounds of parameters .....................................................

Table 3 -

 3 10 Identified parameters of hardening laws with different yield criteria ......

Table 3 -

 3 11 Main parameters of measurement system for dynamic biaxial test on

AA5086 .....................................................................................................

Table 3 -

 3

12 Comparison of identified hardening laws by quasi-static and dynamic tests ..........................................................................................................

Table 4 -

 4 1 Main parameters of measurement system for uniaxial tests on DP600 .... Table 4-2 Identified results on basis of Ludwick law by uniaxial tests on DP600 ... Table 4-3 Identified results on basis of Voce law by uniaxial tests on DP600 ........

Table 4 -

 4 4 Main parameters of measurement system for biaxial tests on DP600 ......

Table 4 -

 4 5 Parameters of Hill 48 yield criterion for DP600 ......................................

Table 4 -

 4 6 Identified results of DP600 on basis of Mises and Ludwick ....................

Table 4 -

 4 7 Identified results of DP600 on basis of Hill 48 and Ludwick ..................

Table 4 -

 4 8 Identified results of DP600 on basis of Hill 48 and Voce ........................

	ε	ij	Components of strain tensor
	ε	i	Components of principal strains
	ε	p	Equivalent plastic strain
	ε ɺ		Equivalent strain rate
	σ	ij	Components of stress tensor
	σ	i	Components of principal stresses
	σ		Equivalent stress
	σ	0	Initial yield stress
	ε			True strain in one dimension
	ε	p	Plastic strain in one dimension
	εɺ			True strain rate in one dimension
	σ		True stress in one dimension
	θ r		Anisotropic coefficient or Lankford parameter
	E		Young's modulus
	υ			Poisson's ratio
	t			Time
	T			Temperature
	AA	Aluminium Alloy
	DP	Dual Phase
	BCC	Body Centred Cubic
	FCC	Face Centred Cubic
	HCP	Hexagonal Close Packed
	DIC	Digital Image Correlation
	FE	Finite Element
	FLC	Forming Limit Curve
	PLC	Portevin-Le Châtelier
	SHPB	Split Hopkinson Pressure Bar

XV

Notations

Table 1 -1 Forming velocity and strain rate ranges of different forming processes [21]

 1

	Process	Forming velocity ( s m )	Strain rate ( 1 -s )
	Hydraulic press	0.03	10	0
	Brake press	0.03	10	0
	Mechanical press	0.03-0.73	0 10 -1 10
	Drop hammer	0.24-4.2	1 10 -2 10
	Gas-actuated ram	2.4-82	10	2
	Explosive forming	9-228	2 10 -4 10
	Magnetic forming	27-228	3 10 -4 10
	Electrohydraulic forming	27-228	3 10 -4 10

  The transient response of stress after the jump is minimized by extrapolating the flow curve. A logarithmic strain rate sensitivity index m can be determined by the flow stresses A

	σ and B σ at the corresponding strain rates A εɺ and
	B εɺ as following:

16

	where i X ′ and i X ′ ′ are principal values of the linearly transformed stress tensors ij X ′
	and ij X ′ ′ . ij X ′ and ij X ′ ′ are obtained by linear transformations C′ and C ′ ′ of the
	deviatoric stress tensor ij s . ij s is derived from stress tensor ij σ by the transformation
	matrix T :

Table 3 -1 Main parameters of DIC in CORRELA 2006

 3

	Subset	Size Distance	32 pixels × 32 pixels 16 pixels ×16 pixels
		Method	FFT and quadratic polynomial interpolation
	Correlation	Initial value Displacement	By precedent displacement Between precedent and current images
		Iterative accuracy	0.01 pixel
	Strain	Mode Lagrange	X2 Large deformation,	n	=	2

Table 3 -2 Main parameters of measurement system for uniaxial tests of AA5086

 3

		Camera (FASTCAM-APX RS 250K)	Load sensor
	Test	Acquisition	Resolution	Shutter speed	Acquisition
		rate (fps)	(pixel)	(s)	rate (Hz)
	1mm/s	250	704 ×	128	1/300	1000
	500mm/s	15000	704 ×	128	1/15000	45000

Table 3 -3 Ludwick parameters identified from uniaxial test on AA5086

 3

	Parameters	0 σ (MPa)	Ludwick model: K (MPa)	σ	σ = 0 n +	Kε	p	n	Error δ
	Static	119.6	473.6		0.435			3.75
	Dynamic	132	466.5		0.473			3.73

Table 3 -4 Voce parameters identified from uniaxial test on AA5086

 3

Table 3 -

 3 6. The element sizes of Zone_1 and Zone_2 are the same for both the shell and solid element. It can be seen that the element number and time cost of shell element have been greatly reduced to 4460 and

	s 221 , 9 .

Table 3 -6 Comparison between solid and shell elements

 3

	Solid (C3D4_5)	Shell

Table 3 -9 Lower and upper bounds of parameters

 3

	0 σ (MPa)	K (MPa)	n
	[ 130 , 160	]	[ 200 , 500	]	[ ] 8 , 1

Table 3 -10 Identified parameters of hardening laws with different yield criteria

 3

	Yield criteria	0 σ (MPa) K (MPa)	n	δ	Objective δ ( ) 1 ε ( ) 2 ε δ
	Mises	147.72	329.65	1.352	10.12%	10.88%
	Hill 48	153.62	249.75	3.392	12.46%	12.10%
	B & B	136.90	323.5	2.618	7.19%	16.09%

Table 3 -11 Main parameters of measurement system for dynamic biaxial test on AA5086

 3

		Camera (FASTCAM-APX RS 250K)	Load sensor
	Test	Acquisition	Resolution	Shutter speed	Acquisition
		rate (fps)	(pixel)	(s)	rate (Hz)
	250mm/s	18000	384×320	1/18000	50000

  -35. The loading velocity for each actuator rises from 0 to about

							250	mm	s	during the acceleration
	stage from about ms 4	to ms 9 . It can be seen that the loaded velocity are maintained
	at about	250	mm	s	during the test from ms 9	to ms 16	after acceleration.

  -38. It is seen that the equivalent and principal strains are homogeneous at central zone The stain state is not uniform because of the principal strain localisation at the time

	(	-	2	mm	≤	x	,	y	≤	2	mm	). The equivalent strain at the central zone is about % 30 . The
	equivalent and major principal strain localised near the points (	x	=	0	,	y	=	±	4	mm	).
	t	=	15	.	222	ms	.

  -51. The strain path is about

									ε	2	/ 1 ε	≈	8 . 0	after
	t 12 =	ms	. The strain rate is about	ε ɺ	≈	40	/	s	, and increased quickly after	t 17 =	ms	.

Table 3 -12 Comparison of identified hardening laws by quasi-static and dynamic tests

 3

	Hardening law	1	exp(	)
	Test

Table 4 -1 Main parameters of measurement system for uniaxial tests on DP600

 4

		Camera (FASTCAM-APX RS 250K)	Load sensor
	Test	Acquisition	Resolution	Shutter speed	Acquisition
		rate (fps)	(pixel)	(s)	rate (Hz)
	1mm/s	250	704×128	1/600	250
	100mm/s	3000	704×128	1/5000	9000
	500mm/s	15000	704×128	1/15000	45000
	1m/s	15000	704×128	1/15000

Table 4 -

 4 3. The identified values of initial yield stress 0

Table 4

 4

-4.

Table 4 -4 Main parameters of measurement system for biaxial tests on DP600

 4

		Camera (FASTCAM-APX RS 250K)	Load sensor
	Test	Acquisition	Resolution	Shutter speed	Acquisition
		rate (fps)	(pixel)	(s)	rate (Hz)
	0.02mm/s	50	384×384	1/500	50
	1mm/s	250	1024×1024	1/1000	250
	250mm/s	18000	384×352	1/18000	50000

s -1 à 10 1 s -1 . Le DP600 présente une même sensibilité à la vitesse de déformation quelque soit la sollicitation, uni-axiale ou bi-axiale. Les lois d'écrouissage de Ludwick et de Voce, identiiées jusqu'à 30% de déformation plastique équivalente sur la base de données expérimentales constituées des essais bi-axiaux, sont relativement proches. Les différences observées entre ces courbes d'écrouissage et celles identiiées à partir des essais de traction uni-axiaux montrent tout l'intérêt de l'essai de traction bi-axiale sur éprouvette en croix.

Shape design of cruciform specimen

Acknowledgements

First and foremost, I would like to express my sincere appreciation to my supervisors, Professor Eric RAGNEAU, Dr. Dominique GUINES and Dr. Lionel LETOING, for giving me the opportunity to work in Laboratory of Civil and Mechanical Engineering (LGCGM). Thank you so much for your patience, encouragement and inspiration. I have greatly benefited from your instructions and attitudes for the thesis.

I would like to express my great gratitude to Professor Salima Bouvier (Université Technologique de Compiègne), Professor Vincent Grolleau (ENSI de Bretagne-Sud) and Professor Franck Morel (ENSAM Angers) for your work as members of jury. I would also like to say many thanks to all the colleagues at LGCGM. Thank you so much for your discussion and friendship. I 'd like to express my special appreciation to engineers for specimen preparation and technical support. I could not complete the work without your help. I will cherish my experiences of working and living in INSA de Rennes. I would like to express great acknowledgement to China Scholarship Council (CSC) for the financial support during my work. I would like to acknowledge Northwestern Polytechnical University (NPU) for helping me to apply for studying in France.

Identification methodology of hardening behaviour under in-plane biaxial loading

and three anisotropic coefficients 0 r , 45 r and 90 r are obtained [START_REF] Zhang | Characterization of anisotropic yield criterion with biaxial tension test[END_REF] . The parameters F , G , H , N of Hill 48 yield criterion are calculated by the anisotropic coefficients, in Table 3-7. For the aluminium alloy AA5086, the parameters of Bron and Besson 2004 yield criterion have been calibrated by a single biaxial test of notched cruciform specimen [START_REF] Zhang | Characterization of anisotropic yield criterion with biaxial tension test[END_REF] . The identified parameters are given in Table 3-8. Compared with Mises and Hill48 yield criteria, the identified Bron and Besson 2004 yield criterion has been proved to be more accurate to characterize the anisotropy of AA5086.

The shapes of the three yield functions are compared in 2D principal stress space (in Fig. 3 Fletcher-Goldfarb-Shanno (BFGS) algorithm. The BFGS optimization is a so-called quasi-Newton method. A descent direction is traced out by the Newton step using the gradient information. Therefore, it can achieve fast convergence.

Fig. 4-8 Parameter identification for rate-dependent hardening model of uniaxial tests

The identified results of four rate-dependent hardening laws on the basis of Ludwick law are presented in Table 4-2. It can be seen that the error of identified hardening law Ludwick_4 is smallest. The strain hardening behaviour seems not to be affected by strain rate ( 2 1

m m >>

). The strain rate sensitivity of DP600 can be multiplied only with initial yield stress. The similar conclusion has also been obtained by Yu et al. [START_REF] Yu | Constitutive model on the description of plastic behavior of DP600 steel at strain rate from 10 -4 to 10 3 s -1[END_REF] for DP600 at strain rates from 4 10 -to 1 3 10 - s . The identified hardening models have been compared with the experimental one at each strain rate, as shown in Fig. 456789.

Table 4-2 Identified results on basis of Ludwick law by uniaxial tests on DP600

Type

Hardening law Parameters Error Ludwick_1

identification by inverse analysis based on FE model of biaxial tests. As already presented, the FE model of shell element is adopted.

Material models

For isotropic elasticity of DP600, Young's modulus GPa E 200 = and Poisson's ratio 3 . 0 = υ are considered. The Mises and Hill 48 yield criteria are compared for parameter identification of hardening laws. The parameters of Hill 48 yield criterion have been calculated from three anisotropic coefficients proposed by Ozturk et al. [START_REF] Ozturk | Effects of anisotropic yield functions on prediction of forming limit diagrams of DP600 advanced high strength steel[END_REF] , as shown in Table 4- Isotropic hardening model is considered for FE simulation of biaxial tensile tests at different loading velocities. The parameters of yield criteria are supposed to be independent to strain rate and strain level. The adiabatic deformation-induced thermal effects are neglected for hardening law. The hardening laws of Eq. 4-4 (Type_4) on the basis of Ludwick and Voce laws are adopted as follows:

Identified results

The optimization process has been performed in modeFRONTIER, in Appendix (V). The cost function is defined by the simulated and experimental principal strains, as given in Eq. 3.6. The results identified on the basis of Mises yield criterion and Ludwick law are given in Table 4- Hill 48 yield criterion. It is seen from these identified models that the influence of strain rate on strain hardening term is more important than that on initial yield stress. The simulated principal strains at the central point of specimen have been compared with the experimental curves, as shown in Fig. 4

A . Appendix (I) Drafting of cruciform specimen (V) modeFRONTIER flowchart with SRS by biaxial tensile tests