RÉSUMÉ

Pour fournir des informations précieuses pour la conception dŠalgorithmes dŠanalyse de tests non destructif (NDT) robustes et Ąables, la combinaison, de mesures acoustiques résolues en temps, de la corrélation dŠimages numériques (DIC), des mesures en volume ainsi que de lŠanalyse postmortem de surface de rupture appliquées à des expériences de Ąssuration en traction dans le but de localiser la source des emissions acoustiques, de détecter la pointe de Ąssure et de déterminer les complexités et la cinématique de la Ąssure, respectivement. Un dialogue entre lŠactivité acoustique, les variables dŠétat, les mécanismes de rupture et les faciès de rupture postmortem est établie au cours dŠexpériences modèles contrôlées produisant des événements AE de différentes natures. Le Poly(méthacrylate de méthyle) (PMMA), traditionnellement considéré comme la matériau modèle par excellence, est utilisé au regard ses caractéristiques distinctives: la capacité à présenter un mode de défaillance alternatif par le biais de la propagation dŠune seule Ąssure, conduisant au développement de deux motifs distincts de croissance contrôlée de Ąssures. De plus, pour la première fois, un tel comportement de rupture cyclique matériau/structurel est examiné à travers le prisme de la mécanique de la rupture élastique linéaire (LEFM) en utilisant la corrélation dŠimages numériques (DIC) in-situ à haute vitesse (HS) et à ultra-haute vitesse (UHS). Les taux de restitution dŠénergie et les vitesses de Ąssuration au cours des expériences de rupture sont déduits à partir de mesures en champ complet en utilisant lŠexpansion en série de Williams. Les surfaces de rupture postmortem des échantillons ont été systématiquement analysées à lŠaide de la microscopie optique. Ce travail fournit des données expérimentales clés pour une meilleure compréhension dŠun cadre théorique uniĄé des instabilités de Ąssuration.

LIST OF FIGURES

GENERAL INTRODUCTION: BACKGROUND AND MOTIVATIONS

Ensuring the safety and reliability of engineering structures and components is governed by the understanding of the mechanical behavior of materials. Fracture presents a fundamental concept in materials science and engineering involving the study of material failure behavior under the inĆuence of external forces or loads. When materials are subjected to loads, they undergo deformation. Depending on the materialŠs properties and the magnitude of the applied force, this deformation can be elastic (temporary) or plastic (permanent). However, when the applied stress exceeds a critical threshold, materials may no longer deform reversibly but might experience fracture, which is the sudden and irreversible separation of a material into independent pieces. The broad Ąeld of fracture mechanics encompasses the study of various modes of failure, including brittle fracture, ductile fracture, and fatigue.

With the concept being introduced in the Ąrst half of the 19th century by the engineer and jurist W. A. J. Albert [START_REF] Albert | Uber trebseile am Haz. Archive fur Mineralogie[END_REF], material fatigue is still considered one of the most critical failure mechanisms in engineering. In material science, fatigue is described as structural damage or failure under the inĆuence of cyclic loading. More accurately, it is the initiation of micro-cracks in the material, and their cyclic events induced growth, leading to abrupt propagation of a macro crack, which most likely results in structural failure. This phenomenon is especially critical due to its unpredictability, which, due to its occurrence in different Ąelds of technology, leads to a comprehensive cross-domain research interest (e.g. aerospace engineering [START_REF] Bhaumik | Fatigue failure of aircraft components[END_REF], microelectronics [START_REF] Lee | Solder joint fatigue models: review and applicability to chip scale packages[END_REF],

or nanotechnology [START_REF] Suhr | Fatigue resistance of aligned carbon nanotube arrays under cyclic compression[END_REF]) up to this day. In mechanics, three stages are commonly dedicated to the description of the phenomenon of fatigue: (i) microscopic damage development, (ii) macroscopic crack growth, and (iii) Ąnal failure when critical loads are reached. Each stage is inĆuenced by different surface, material, or mechanical properties.

Catastrophic events caused by fatigue failure have been common in the past. Due to the severe extent of damage and the thereby sparked public interest, three events are mostly cited in the literature:

Ů In 1954, design and construction errors on an airplane led to a small fatigue crack in a window opening resulting in a sudden structural failure of the pressure cabin causing the crash of the passenger airplane [START_REF] Withey | Fatigue failure of the de Havilland comet I. en[END_REF].

Ů In 1980, welding defects inducing early fatigue failure of a brace resulting in an overload failure of the brace attachments to the columns caused the capsizing of the semisubmersible drilling platform Alexander L. Kielland [START_REF] Moan | The progressive structural failure of the Alexander L. Kielland platform[END_REF].

Ů In 1992, during inspections overlooked fatigue crack on a bolt of the structural component connecting the jet engine and wing caused the separation of two engines from the wings, resulting in the loss of dispersibility and controllability of the freighter aircraft. The incident led to the crash of the airplane into an apartment building [START_REF] Alwi | Fault Detection and Fault-Tolerant Control Using Sliding Modes Series Title: Advances in Industrial Control[END_REF].

Undetected fatigue cracks have, as seen in the forgone examples, not only disastrous dimensions of deaths but also severe economic damages, underlying the importance of further research and conscientious inspection policies to detect and thereby avoid the catastrophic cracking of engineering structures.

There is a wide variety of critical engineering structures spanning the domains of Civil, Mechanical to Aerospace engineering, where monitoring and early detection are essential to prevent failure leading to possible catastrophic incidents [START_REF] Liu | Structural Health Monitoring: State of the Art and Perspectives[END_REF]. In the Ąeld of mechanical engineering, the understanding of fracture behaviors, their initiation, and propagation is vital. • Fatigue tests need to be interrupted to obtain radiographs

• The user needs to be experienced

• Specimens and test specimens need to be small to Ąt within the required space and capture a high image resolution Ultrasound [START_REF] Shull | Nondestructive evaluation: theory, techniques, and applications[END_REF][START_REF] Meola | Recent Advances in Non Destructive Inspection[END_REF] Use of ultrasonic waves into a specimen and measuring the wave response

• Can detect the presence of cracks without direct access to the region of interest

• Detects internal and surface cracks

• Ultrasound probes can be designed for complex geometries

• Determine the cycles to crack initiation

• Approximate crack location needs to be known

• Difficult to inspect materials with low sound transmission

• There needs to be access to the surface to transmit the ultrasound

• Transducer needs to be in contact with the surface or through a Ćuid

• Cracks planar and parallel to the ultrasound wave propagation may go undetected Thermography [START_REF] Hung | Review and comparison of shearography and active thermography for nondestructive evaluation[END_REF][START_REF] Ibarra-Castanedo | Nondestructive testing with thermography[END_REF] An infrared camera is used to observe the temperature change in a specimen

• A contactless technique • Detect crack initiation due to temperature increase at the crack tip

• Determine the crack location

• Crack lengths can be measured

• Determine the cycles to crack initiation

• Thermal visual • Cracks can be scanned over a large area

• Temperature rise needs to be high enough at the crack tip to be detected subjected to mechanical loads lead to the occurrence of stress redistribution. While most energy is contributed to the separation of atomic bonds, a portion of the release of elastic stress energy is found in the form of dissipated heat [START_REF] Döll | Kinetics of crack tip craze zone before and during fracture[END_REF] and elastic waves (acoustic emissions) [START_REF] Bonamy | Interaction of Shear Waves and Propagating Cracks[END_REF]. The transient elastic waves, as surface motion, are captured by a transducer translating the mechanical to electrical energy through the process of piezoelectricity [START_REF] Arnau | Piezoelectric Transducers and Applications[END_REF]. After pre-ampliĄcation, in the Ąeld of seismology, where they are considered as repeated stress releases occurring at a defect [START_REF] Matsuzawa | Repeating earthquakes and quasi-static slip on the plate boundary east off northern Honshu, Japan. en[END_REF]. Hence, the concept of multiplets goes beyond material fatigue and might be present in all sorts of quasi-static and dynamic fracture events. However, to broaden the understanding of such multiplets (i.e. nearly identical waveforms as the signature of a unique source [START_REF] Deschanel | Acoustic emission multiplets as early warnings of fatigue failure in metallic materials[END_REF]), one needs to understand their connection to different source mechanisms. Hence, model experiments need to be developed to produce controlled AE events with increasing complexity being driven by different physics. In addition to its quasi-brittle nature and amorphous structure, Poly(methyl methacrylate) (PMMA), although conventionally deemed an exemplary model material due to its extensive existing literature, presents a distinctive feature: the capability to exhibit an alternative mode of failure through a single crack propagation, leading to the development of two distinct controlled crack growth patterns [START_REF] Ravi-Chandar | On the mechanics and mechanisms of crack growth in polymeric materials[END_REF]. Figure 1 shows the phases of a single crack propagation on a TDCB PMMA specimen subjected to mode I loading conditions with a constant loading rate of 0.5 mm.s -1 .

Figure 1 Ű Marks on the crack surface visualizing the phases of intermittent/alternate crack propagation on a PMMA specimen produced at GeM laboratory (Ecole Centrale de Nantes, France).

Alternate cracking can be separated into two different crack growth types: quasi-static and dynamic. While these phases vary depending on the geometry and the applied loading rate, such controlled multi-mechanisms-related AE responses are expected to produce a better signal-to-noise ratio than fatigue cracking. Hence, fracture processes in PMMA present a promising model case for the investigation of source mechanisms. However, it does not only depict a suitable tool in the context of the Ąrst scientiĄc motivation, the phenomenon of alternate cracking itself is far from being entirely understood. Thus, the second scientiĄc motivation of this work is found in attempting to improve the physical understanding of the alternate recurrence of quasi-static and dynamic crack propagation in PMMA.

Project organization

The early detection of fatigue crack growth by the method of acoustic emissions (AE) was deĄned as the goal for the consortium project Še-WARNINGSŠ. The Agence Nationale de la Recherche (ANR) project Ű under the grant ANR-19-CE42-0012 Ű brought together different entities pooling their expertise. According to the speciĄc expertise, Ąve entity-related work packages were elaborated as shown in Figure 2. Furthermore, experiments are challenged by a simple numerical twin to improve the understanding of their observations.

Contribution GeM

Chapter 1

STATE OF THE ART

Alternate (stick-slip) cracking

A crucial parameter to distinguish among fracture phenomena during experimental investigations of crack propagation is the crack velocity, as it is strongly related to underlying fracture mechanisms. In brittle materials, a wide range of velocities can already be explored by relatively minor variations in loading conditions. Within this span of observable velocities, two independent groups of interestŮslow (quasi-static) and fast (dynamic) crackingŮare covered in the literature. Indeed, at signiĄcant constant extension rates, cracks typically propagate at very high velocities (about 0.5 times the Rayleigh wave speed) [37Ű39]. In that case, cracks are considered unstable, which means that no additional load is required to signiĄcantly propagate the crack, eventually resulting in total failure of the sample. On the contrary, with a sufficiently slow loading rate, the crack propagates continuously stable with velocities at least two orders of magnitude slower [40Ű42]. Between these two regimes, the phenomenon of alternate cracking [START_REF] Ravi-Chandar | On the mechanics and mechanisms of crack growth in polymeric materials[END_REF][START_REF] Hattali | Low Velocity Surface Fracture Patterns in Brittle Material: A Newly Evidenced Mechanical Instability[END_REF] is observed for some materials. Between these two propagation types, a range of velocities is suggested that the crack cannot exploit. This causes a gap between the two well-deĄned phases of quasi-static and dynamic propagation that is commonly described as ŚforbiddenŠ velocity regime. Due to experimental (crack velocity mismatch of about two orders of magnitude) and theoretical considerations (apparent theoretical mismatch regarding the mechanisms), these two regimes have been investigated individually up to now. As a consequence, the ŚforbiddenŠ velocity regime has only been approached from both sides without clearly quantifying energies and velocities involved during transitions.

Basic theoretical background

Beforehand, different terminologies are found in literature discussing this fracture phenomenon. The most commonly used expression is stick-slip, due to the theoretical proximity with observations during the frictional phenomenon of sliding [START_REF] Feeny | A Historical Review on Dry Friction and Stick-Slip Phenomena[END_REF]. In this work, however, alternate cracking is used to specify and separate between sliding and fracture-related observa- This kinetic law can be subdivided into three regions. Region I depicts a quasi-static propagation phase in which the Griffith criterion [START_REF] Griffith | The phenomena of rupture and Ćow in solids[END_REF] should apply. However, when varying the crack velocity over decades, it has been observed experimentally that the critical energy release rate depends on the crack velocity. Consequently, in this region, the crack speed is driven by its relationship with the critical energy release rate that varies between G c for in-Ąnitely low crack speeds and avalanche threshold G a at the transition with Region II (A -→ B).

Extreme crack tip accelerations take place during the following Region II with a supposedly "softening" branch causing the jump from Region I to the dynamic Region III (B -→ C). Hereinafter, the crack Ąnds itself in the dynamic region where the generic KanninenŠs relationship

(K ID = K IC /(1 -(v/c R ) m ) with the dynamic SIF K ID , static SIF K IC , crack tip velocity v,
Rayleigh wave speed c R and material constant m) of dynamic fracture [START_REF] Kanninen | Advanced fracture mechanics Oxford engineering science series 15[END_REF] can be applied, and crack tip speeds are restricted by the dynamic limiting speed of Rayleigh wave speed (c R ). However, during the high-velocity branch (Region III), crack tip speeds in alternate cracking slow down after a certain propagation length to a point of arrest. This arrest marks the beginning of the second jump back to the slow velocity branch. For this, authors in existing literature generally propose the crack to follow along the reference points D to A. All this, due to the reiterating behavior, explains the appearance of a hysteresis implying a ŚforbiddenŠ velocity zone (between v B and v D ).

A typical example of a fracture surface linked to such cracking is shown in Figure 1.2. Here, the two particular surface markings related to quasi-static and dynamic propagating cracks can be observed.

notch tip quasi-static quasi-static dynamic

Existing literature

While Region I and Region III are being well deĄned and investigated independently, insuf-Ącient experimental data and few modeling attempts are provided in the literature to fuse the two regions with a proper description of the ŚforbiddenŠ Region II. In the following, by focusing on the most recent modeling attempt and presented experimental methods, existing literature on alternate cracking will be presented brieĆy.

Most recently, [START_REF] Vincent-Dospital | How heat controls fracture: the thermodynamics of creeping and avalanching cracks[END_REF] provided a predictive model of the fracturing behavior in PMMA based on the theoretical concept of thermal softening. In this model, the failure process is thermally activated and combined with the dissipation and diffusion of heat in the process zone surrounding the crack tip. Covering the vast range of different crack tip speeds found in PMMA fracturing, their thermodynamic approach showed good compliance with the experimental data of quasistatic and dynamic fracturing experiments. Furthermore, the two well-deĄned ŚstableŠ regimes were connected by the model through a continuous description of the ŚsofteningŠ branch. Noteworthily, no experimental data of alternate cracking was considered. Notice, for terminology purposes, that this uniĄed model introduces the concept of sub-critical failure. Indeed, contrary to the discussion made around Figure 1.1, where velocity domains were classiĄed through a discontinuous transition from classical Griffith theory to KanninenŠs type rate-dependent relationship, the authors propose that the actual critical energy release rate is reached at high velocity (∼165 m.s -1 ), while initiation and propagation at lower speed are supposedly sub-critical.

To this day, different methods have been used to experimentally investigate the phenomenon of alternate cracking. In the work of [START_REF] Ravi-Chandar | On the mechanics and mechanisms of crack growth in polymeric materials[END_REF], PMMA and Homalite-100 fracturing of single-edgenotched (SEN) and compact tension (CT) specimens were investigated by focusing on the temporal evolution of the stress intensity factor. Stress intensity factor (SIF) data has been obtained through the method of caustics [START_REF] Beinert | Experimental evaluation of stress concentration and intensity factors[END_REF] using a camera system at 12 kfps over a duration of one minute. Crack length and K I measurements were retrieved during quasi-static propagation phases of alternate cracking experiments, while the experimental setup did not support dynamic phase measurements.

[43] investigated the phenomenon in PMMA, by focusing on crack tip speeds and fracture surfaces, using a high-resolution camera (maximal temporal and spatial resolution of 396 fps and 21 Mpix, respectively) and an in-house image processing tool. The system setup only allowed for the investigation of quasi-static crack propagation phases. Thus, only the lower limit of the ŚforbiddenŠ velocity regime, i.e. v a , was investigated and found to be 3.5 cm.s -1 and 4.0 cm.s -1 for the two observed dynamic bursts, respectively. However, with an interframe of approx. 2.5 ms, it has to be expected that phase transitional velocities were not captured accurately.

[49] contributed an in-depth investigation of cracking in PMMA, dedicating a chapter of his Ph.D. manuscript to the transition phase from dynamic to quasi-static propagations. During the experiments, load cell, and clip gauge have been used to obtain force and displacement data, respectively. Crack tip positions have been obtained by using a high-speed camera system (temporal and spatial resolution of 48 kfps and ∼ 175 kpix, respectively) and a homemade algorithm.

By combining these results, SIF and thereby fracture energies have been derived through a numerical (FE) based compliance chart relying on load and displacement data. However, difficulties in this setup arise in the high-velocity regime, where the accuracy of load and displacement measurements can not be assured. Furthermore, the chosen sample geometry had the unique purpose of studying dynamic to quasi-static crack propagation transitions. TDCB specimens with the notch being blunted by a circular hole allow for the initiation of a dynamically propagating crack while using low tensile extension rates (∼ 2.5 µm.s -1 ). Here, with respect to the theoretical concept of fracture in PMMA (Figure 1.1), the crack Ąnds its way back to the master curve far below the value of v a by following the path D-→A, which can be explained by the relatively low extension rate. After the arrival of the initiated dynamic crack, by assuming an inĄnite sample length, the crack would have propagated entirely quasi-static in the wake.

Hence, the presented experiments are a matter of quasi-static experiments introduced by dynamic crack propagation.

Under the scope of providing a model of the fracturing behavior in PMMA, [START_REF] Vincent-Dospital | How heat controls fracture: the thermodynamics of creeping and avalanching cracks[END_REF] performed wedge-splitting fracture experiments on PMMA samples over a wide range of quasi-static driving speeds (1.6 µm.s -1 to 1.2 mm.s -1 ). Force data were monitored in real-time by the load cell, while the propagating crack was followed by a high-resolution camera. In this case, for quasistatic propagation measurements, the temporal resolution of 10 fps provided crack length and subsequently velocity measurements by post-treating the images with the in-house crack tip detection algorithm. On the other hand, fast crack propagations were detected, through an oscilloscope, by the successive rupture of parallel 500 µm large chromium or gold lines on the surface. Numerical charts based on FE analysis supported the determination of mode I related stress intensity factor K I . Energy release rates then were retrieved under plane stress assumption.

In summary, the literature shows that the investigation of Region II has been conducted by experimental procedures only suitable for either of the two other enclosing regions. Crack speeds are usually obtained from camera systems (either high-speed or high-resolution) and tracking algorithms. K I and subsequently energy release rates are mostly obtained from numerical compliance methods or analytical charts, which are limited due to low accuracies in high loading rate cases. Note that Region II is classically referred to as the ŚforbiddenŠ velocity regime, but the model proposed in [START_REF] Vincent-Dospital | How heat controls fracture: the thermodynamics of creeping and avalanching cracks[END_REF] suggests that it could be reached if the simulation or the experiment can be conducted by controlling the crack speeds. However, this is impossible to achieve neither numerically nor experimentally, with the energy release rate being governed by the applied load as the controlled parameter.

Acoustic emission analysis

In the realm of non-destructive testing (NDT) and structural health monitoring, Acoustic Emission (AE) analysis stands as a valuable and versatile technique. With its ability to continuously monitor in real-time, a unique window into material and structural integrity is offered to proactively address issues, prevent catastrophic failures, and ensure the reliability of critical assets. Nonetheless, the effective utilization of AE is not without its scientiĄc challenges. These challenges, rooted in the physics of acoustic emissions and the intricacies of material behavior, must be addressed to unlock the full potential of this technique. In this section, the classical AE analysis setup is introduced, its theoretical foundation is presented, and its application in damage identiĄcation is brieĆy discussed.

Setup for acoustic emission analysis

The equipment setup of an AE device typically consists of several key components designed Ů Calibration Equipment: To ensure the accuracy of AE measurements, calibration equipment may be used periodically to calibrate the sensors and the entire system. This ensures that the system is capable of detecting and quantifying acoustic emissions accurately. The pencil lead break procedure [START_REF] Scruby | An introduction to acoustic emission[END_REF] can be considered to be the most common calibration method. Furthermore, experimental setups for the analysis of AE signals might additionally consist of real-time monitoring software (visualizing the AE data as it is collected), signal processing and analysis software (includes tools for Ąltering, event location, event classiĄcation, and statistical analysis), or environmental sensors (monitoring of environmental conditions that could affect AE measurements).

Basic theory of AE

AE is grounded in the fundamental principles of elastic wave propagation within materials.

When a material undergoes stress or experiences structural changes, it responds by deforming.

The deformation involves temporary or permanent changes in the materialŠs microstructure.

During deformation, stress concentrations can occur Ű especially in regions where the material is inhomogeneous or where there are structural defects or discontinuities Ű leading to various types of defects within the material. As these defects evolve or propagate, they release energy in the form of dissipated heat [START_REF] Döll | Kinetics of crack tip craze zone before and during fracture[END_REF] and elastic waves (acoustic emissions) [START_REF] Bonamy | Interaction of Shear Waves and Propagating Cracks[END_REF]. The latter travels through the material, carrying information about the evolving stress and structural changes. Specialized transducers are utilized to capture and record these acoustic emissions. Most classically, AE systems are equipped with piezoelectric sensors, which take advantage of the unique property of piezoelectric materials [START_REF] Arnau | Piezoelectric Transducers and Applications[END_REF] to generate electrical charges when subjected to mechanical stress or pressure changes. When an acoustic wave impinges on a piezoelectric sensor, it generates a voltage signal proportional to the acoustic energy received.

Other sensor types are Fibre Optic sensors [START_REF] Willberry | Structural Health Monitoring Using Fibre Optic Acoustic Emission Sensors[END_REF], where optical Ąbers and a sensing element respond to mechanical vibrations or strain that are caused by acoustic waves. When an acoustic wave interacts with the sensing element, it induces strain or displacements in the optical Ąber, leading to changes in the ĄberŠs optical properties, such as phase or intensity. These changes in the optical properties of the Ąber are then detected and converted into electrical signals using photodetectors or other optical measurement techniques. MEMS (Micro-Electro-Mechanical Systems) sensors [START_REF] Ozevin | MEMS Acoustic Emission Sensors[END_REF] present another alternative. They consist of microscale mechanical structures responding to various physical phenomena, such as acceleration, pressure, temperature, or elastic waves. The interaction of the elastic waves with these structures induces mechanical motion or vibrations, which are converted into electrical signals using integrated microscale sensors.

Eventually, the signals produced by each sensor solution are digitized and analyzed.

AE in damage identiĄcation

Globally, AE analysis can be subdivided into three branches of application: localization, lifetime prediction, and damage analysis. The aim of this Ph.D. work is solely related to AE as a technique for damage analysis, with the AE localization method presenting a means of support. In damage analysis, AE signals are highly dependent on the type of source mechanism. Different physical or chemical phenomena have been identiĄed in the literature as potential AE source mechanisms. D. G. Aggelis [START_REF] Aggelis | ClassiĄcation of cracking mode in concrete by acoustic emission parameters[END_REF] used AE for the classiĄcation of cracking modes during four-point bending experiments on different concrete mixture beams. ClassiĄcation of the cracks according to their mode was proposed through different AE descriptor relationships, e.g. average frequency vs. RA-value (rise time over amplitude). Regarding reversible processes, Wadley and Mehrabian [START_REF] Wadley | Acoustic emission for materials processing: a review[END_REF] reviewed existing literature concerning the AE activity from phase transitions (e.g. melting or solidiĄcation). In the context of fabrication processes, Jayakumar et al. [START_REF] Jayakumar | A review of the application of acoustic emission techniques for monitoring forming and grinding processes[END_REF] reviewed the application of AE monitoring during various forming and grinding pro-cesses. Furthermore, Cao et al. [START_REF] Cao | Chemical acoustic emissions from gas evolution processes recorded by a piezoelectric transducer[END_REF] identiĄed and classiĄed the AE responses of various gas evolution processes.

The appearance of these source activities is generally identiĄed Ű supported by methods such as parameter clustering [START_REF] Li | Cluster analysis of acoustic emission signals for 2D and 3D woven glass/epoxy composites[END_REF], machine learning [START_REF] Almeida | Identifying damage mechanisms of composites by acoustic emission and supervised machine learning[END_REF][START_REF] Ciaburro | Machine-Learning-Based Methods for Acoustic Emission Testing: A Review[END_REF], waveform processing [START_REF] Arumugam | Effect of fuzzy C means technique in failure mode discrimination of glass/epoxy laminates using acoustic emission monitoring[END_REF], Waveform cross-correlation [START_REF] Deschanel | Acoustic emission multiplets as early warnings of fatigue failure in metallic materials[END_REF][START_REF] De La Selle | Acoustic emission classiĄcation by cross-correlation in[END_REF], moment tensor inversion [START_REF]Acoustic Emission Testing en[END_REF], b-value analysis [START_REF] Sagar | An experimental study on cracking evolution in concrete and cement mortar by the b-value analysis of acoustic emission technique[END_REF] and wavelet analysis [START_REF] Sung | Impact Monitoring of Smart Composite Laminates Using Neural Network and Wavelet Analysis[END_REF] Ű through the detection of similar waveform characteristics of repetitive signals. However, the complexity around the accurate interpretation of the AE activity by the user prevails in each case.

To reduce the need for interpretation, AE is often extended by fractography. With a great abundance of information on fracture processes being hidden on the fracture surface, the methods of fractography focus on the determination of failure causes and thereby on the identiĄcation of fracture source mechanisms. By establishing links to known loading patterns, state variables, and fracture mechanics variables, the causes of crack initiation and growth can be revealed. Furthermore, experimental investigations diverge more and more from the classical experimental setups with an individual application of NDT methods. Nowadays, the combination of different NDT methods has become more prominent. These setups can provide a more robust and adequate understanding of the phenomenon of interest, due to the particular set of advantages and limitations of each NDT technique (see Table 1) [START_REF] Kong | A review of non-destructive testing techniques for the in-situ investigation of fretting fatigue cracks[END_REF][START_REF] Wang | Non-destructive testing and evaluation of composite materials/structures: A state-of-the-art review[END_REF]. AE and Digital Image Correlation (DIC) is one of the most common combinations of NDT methods. The combination of AE, DIC, and fractography bears great potential in investigating fracture source mechanisms, which, however, is not fully exploited in the literature.

Andraju et al. [START_REF] Andraju | Damage characterization of CFRP laminates using acoustic emission and digital image correlation: Clustering, damage identiĄcation and clas-siĄcation[END_REF] identiĄed intra/inter-laminar damage modes in CFRP laminates under tensile loading by using AE and DIC. AE data has been classiĄed by the unsupervised K-means clustering algorithm taking solely the amplitude and peak frequency into consideration. DIC displacement and strain Ąelds provided an understanding of the damage evolution. Eventually, the identiĄed AE clusters have been assigned to damage modes based on the pre-existing understanding of postmortem damage mode features. However, damage mode classiĄcation and the source identiĄcation on the postmortem samples still relied strongly on preexisting knowledge and the usersŠ understanding of the investigated phenomenon.

Such multi-NDT instrument investigation for thick hybrid (carbon/glass Ąber reinforced) composite structures under Ćexural loading was presented in Gul et al. [START_REF] Gul | An experimental investigation on damage mechanisms of thick hybrid composite structures under Ćexural loading using multi-instrument measurements[END_REF]. AE acquisition was used for the identiĄcation of failure types, with DIC monitoring the evolution of transverse and shear strains on the through-thickness surface for additional understanding. The observations of the combination of NDTs were conĄrmed by failure modes identiĄcation through fractography. However, proper interpretation of observations presented itself as crucial for establishing accurate links between measurements and source activities. Shrama et al. [START_REF] Shrama | Fatigue crack monitoring in mild steel specimens using acoustic emission and digital image correlation[END_REF] performed AE and DIC monitoring on the uniaxial fatigue cracking of mildsteel specimens. Various failure mechanisms have been identiĄed by AE. DIC and fractography helped the understanding of the detected AE signals and veriĄed the presence of the identiĄed source activities, respectively. In this study, the combination of the two NDT techniques reduced the need for assumptions in the interpretation of the AE activity. However, interpretation-free signal-to-source activity relationships exceeded the investigation potential.

Xu et al. [START_REF] Xu | InĆuence of graphite content on fracture behavior of MgOŰC refractories based on wedge splitting test with digital image correlation method and acoustic emission[END_REF] monitored wedge splitting test on MgOŰC refractories with different graphite content. Their investigation focused on the effect of graphite content on the fracture behavior of the MgOŰC refractories. In doing so, damage mechanisms were identiĄed based on AE signals, while DIC was used to visualize deformation and identify the crack propagation. Eventually, AE and fracture source activities were only quantitatively evidenced along the crack path through the microscopic fractographic analysis.

With the majority of weight in distinguishing among various source mechanisms being carried by AE, DIC measurements are classically only seen as support to simplify the understanding of observations through the visualization of displacement and strain Ąelds. Fractography is applied independently, where evidence of the expected source activities is quantitatively searched for over the entire postmortem fracture surface. Hence, the link between source activity and fracture Ąngerprints is solely established by interpretation and empirical evidence. Thus, pre-existing knowledge of the expected phenomenon is fundamental for the AE identiĄcation of source activities where validation still remains challenging. In the following, the aim is to identify source mechanisms without the need for interpretation. A stage of zero pre-existing knowledge is pretended to enforce the need to reconstruct the entire chain of signal detection by tracing signals back in space and time to their event of initiation. For that purpose, it seems useful to approach AE analysis through reverse engineering. Due to direct visible access to the Ąngerprint of the source activity on the fracture surface, cracking phenomena seem to present a suitable activity of interest for such investigation. The complete picture of the link between AE activities, state variables, fracture mechanisms, and Ąngerprints on the post-mortem fracture surface might provide valuable information for the design of robust and reliable NDT algorithms and modelling of AE signals. To achieve such a purpose, various complexities of mode I controlled experiments on a transparent, fragile, homogeneous material will be used. The idea is to step aside from the ultimate goal (fatigue in metals) and conĄrm or contradict if it is possible to establish an unambiguous link between AE signals (captured far from the source), and a fracture mechanism. Remark 1 Results presented in this chapter are published in [START_REF] Heinzmann | Investigation of fracture source mechanisms through full-Ąeld imaging and acoustic emission[END_REF].

Part I

Investigation of fracture source mechanisms through full-Ąeld imaging and acoustic emission

Sample speciĄcations, test conĄguration, and experimental setup 2.2.1 Material, sample geometry, and AE sensor placement

TDCB-shaped samples were obtained from 8 mm thick molded PMMA plates (Plexiglas®) through laser cutting (Class-4 laser system). To make intended in-volume recordings possible, only transparent PMMA plates were considered. Due to its ideal quasi-brittle behavior and the presence of different cracking behavior at varying loading conditions, PMMA is a material of great interest in fracture mechanics research and an ideal model material for this investigation.

For details on material properties and their mechanical behavior, the authors refer to [START_REF] Ali | A Review of the Properties and Applications of Poly (Methyl Methacrylate) (PMMA). en[END_REF]. The tapered double-cantilever beam (TDCB) shape was chosen to have a constant specimen com- A cut through the sample in the position of the pin holes is presented in Figure 2.2 for the unloaded and loaded case. Due to clearance angles, the radius of the pin holes differs depending on the sample surface side. In the unloaded case, no pressure is put on the pinhole walls, with the pins only being in touch with the smaller radius side of the pin holes. However, as soon as the tensile load rises, pins are pressed on the upper/lower wall forcing the wall to align with the pin. Thus, the opposite alignment of the pins and the pinhole walls with the propagating crack creates one sample face being under tension while the other is under compression.

With the objective of this part being the investigation of AE and crack growths relationship, this imperfect mode I cracking case has been considered a good candidate with the perfect mode I crack propagation case serving as a reference. In the continuation of the work, this type of cracking will be referred to as complex, while referring to the reference case as smooth.

Notice that caution is required regarding sample preparation to achieve good repeatability. Indeed, PMMA fracture behavior is very sensitive to temperature [START_REF] Atkins | Time-temperature dependent fracture toughness of PMMA: Part 1[END_REF][START_REF] Atkins | Time-temperature dependent fracture toughness of PMMA: Part 2[END_REF], but also to any signiĄcant interaction with organic solvents [START_REF] Wang | Ethanol-induced crack healing in poly(methyl methacrylate)[END_REF]. In that context, ethanol (eventually used for surface cleaning) should be avoided, and solvent-free paints should be preferred for DIC patterning of the sample surface.

Loading and test conĄguration

Experiments were conducted with an ElectroPuls E10 000 Instron device with a load cell

3). The tool is

designed to trigger a user-selected action when the monitored signal falls below or rises above a pre-deĄned value (event value). 25 N was selected for the event value, which generally led to slow propagating cracks of about 5-10 mm.

Intended crack propagation.

A second loading stage was introduced to release the intended crack propagation from this fresh pre-crack with an extension rate of 0.07 mm.s -1 .

Event value

Experimental setup and methods

The investigated crack propagation was simultaneously monitored by an AE system (MIS-TRAS Express-8) and a camera system with two high-resolution cameras (Viework VP-50MX-

Experimental methods

Fracture parameters identiĄcation and crack tip detection through Digital Image Correlation (DIC)

The foundation for the detection of the crack tip and retrieval of fracture mechanics variables is found in DIC by providing full-Ąeld displacement measurements from sequential images. By following the principle of conservation of brightness between a reference (f ) and a deformed image (g), i.e. the equation of optical Ćow, DIC presents an ill-posed non-linear inverse problem:

f (X) = g(X + u(X)) (2.1)
with u(X) being the sought displacement Ąeld. To bypass this ill-posed problem, the pixel displacement has to be parameterized using shape functions. Here, FE discretization has been used [START_REF] Besnard | ŞFinite-ElementŤ Displacement Fields Analysis from Digital Images: Application to PortevinŰLe Châtelier Bands[END_REF]. Eventually, the problem can be linearized and solved iteratively in a least-squares sense. A median regularization is used to mitigate the detrimental impact of noise and smaller elements.

By post-treating the displacement Ąeld, variables relevant for fracture mechanics analysis (e.g.

SIFs and crack tip positions) were derived through WilliamsŠ series expansion. Traditionally, in the case of a semi-inĄnite linear elastic isotropic media, stress and displacement Ąelds around the crack tip depend on SIF, the distance to the crack tip (r), and the angle (θ) in a polar reference system attached to the crack tip [START_REF] Williams | On the Stress Distribution at the Base of a Stationary Crack[END_REF]. Thus, by knowing the displacement Ąeld through DIC and projecting it onto the analytical solution, different fracture mechanics parameters can be derived through a non-linear inverse problem [74Ű76]. The displacement Ąeld around the crack tip is written as follows:

u(r, θ) = i=I,II ∞ n=-∞ A n i r n/2 g n i (θ) (2.2)
with the distance to the crack tip r, the angle θ in a polar reference system attached to the crack tip, WilliamsŠ coefficients A and base function g. The base function g n i (θ) takes the following form: Notice that the projection zone is deĄned by R min and R max . On the one hand, the asymptotic behavior near the crack tip of the Ąelds computed by the super-singular terms (n <0) does not provide any physical meaning and is therefore classically neglected. However, when the crack tip position is sought, these super-singular functions appear on the required basis (n=-1 being, up to a scaling factor, the derivative of n=1 with respect to the assumed crack tip position). To not induce biases due to truncation, terms for n down to -3 are considered. Hence, data at a distance to the crack tip smaller than R min are discarded to maintain reasonable conditioning of the least-squares problem. On the other hand, R max Ű deĄning the projection zone size externally Ű has to be small enough to avoid inĆuences of the free boundary, but large enough to still include a sufficient amount of mesh points in the domain. The right size of the projection zone, i.e. deĄning R min and R max , will be identiĄed through a parametric study for each experimental campaign. Furthermore, by using a pre-deĄned crack path, the super-singular term n = -1 is used to estimate the position to the equivalent elastic crack tip along this path [START_REF] Réthoré | Optimal and noise-robust extraction of Fracture Mechanics parameters from kinematic measurements[END_REF].

g n I (θ) = 1 2µ  (κ + n/2 + (-1) n )cos[(n/2)θ] -(n/2)cos[(n/2 -2)θ] (κ -n/2 -(-1) n )sin[(n/2)θ] + (n/2)sin[(n/2 -2)θ] (et,en) (2.3) g n II (θ) = 1 2µ  -(κ + n/2 -(-1) n )sin[(n/2)θ] + (n/2)sin[(n/2 -2)θ] (κ -n/2 -(-1) n )cos[(n/2)θ] + (n/2)cos[(n/2 -2)θ] (et,en) (2.
It is worthwhile mentioning, that the quasi-static form (not in function of the velocity) of the displacement Ąeld around the crack tip is used (even for dynamic crack instabilities) in line with previous works showing that marginal errors can be assumed for velocities below 0.5c R [START_REF] Lee | Quasi-static and dynamic fracture of graphite/epoxy composites: An optical study of loading-rate effects[END_REF]. Eventually, the outcome can be used to derive crack velocities and energy release rates.

The second, under the assumption of plane stress, will be computed in the context of this work through Equation 2.5.

G = K 1 2 /E (2.

Crack front identiĄcation

Collecting information from within the volume, one would need CT scan techniques [START_REF] Crandall | CT scanning and Ćow measurements of shale fractures after multiple shearing events[END_REF] for the investigation of opaque materials. However, with PMMA existing as a transparent material, capturing crack front kinematics is feasible with standard imaging techniques. To do so, the camera system has to be placed with an inclination to the sample surface causing experimental challenges in providing sufficient light and sharpness over the full sample length and thickness.

Thus, lens and light settings were deĄned on a test sample, in which the crack propagated to about half of the sample length. However, already tiny deviations from the crack path of this example case led to strong local variations and losses in sharpness and light with respect to each newly propagating crack. Hence, a compromised lens setting and light positions had to be used to facilitate the recovery of the crack front kinematics for any possible crack path. For post-processing, a crack front extraction algorithm was built along the following steps:

1. Normalization: To begin with, images were normalized to correct for light variations between frames.

ZOI:

A zone of interest (ZOI) around the crack front was deĄned, within which the extraction was performed.

Revealing advancement:

To reveal the advance of the crack front, preceding images were subtracted from the current one. To increase robustness, not only one preceding image but the average of a pre-deĄned number of preceding images was taken into consideration.

Binarization:

A pixel threshold, based on the mean and standard deviation (std) of the newly created image (step 3), was deĄned. This threshold was used to binarize the image.

Erosion and dilation:

The binary image was eroded (to remove unwanted pixels) and dilated (to recover continuous objects). It is worthwhile mentioning that the difficulties in recording sharp images of the in-volume crack propagation resulted in the algorithm not being able to clearly identify and extract the crack front edges.

Assessment of crack tip detection from DIC

In literature, using DIC displacement Ąelds and WilliamsŠ series expansion has become a classical method for the identiĄcation of the crack tip position [75, 76, 82Ű85]. However, while a numerical investigation with a curved crack front in [START_REF] Réthoré | Optimal and noise-robust extraction of Fracture Mechanics parameters from kinematic measurements[END_REF] estimated the apparent elastic crack tip position systematically in front of its free surface appearance, it has not been explored experimentally. Here, we have tried to elucidate this question experimentally through in-volume The uncertainty of the Špix2mŠ value has been evaluated to be approx. 1.27×10 -4 pixel as the FFT sampling was chosen to be the horizontal pixel size of the image (7856 pixel). First approximation of the crack tip localization uncertainty (σ T (a)) can be evaluated through the following equation: By taking all the above into consideration, the following points can be observed:

σ T (a) = (a × σ pix2m + σ notch ) × pix2m (2.
Ů Evaluated uncertainties are lower than the in-depth crack front variations.

Ů Despite the in-depth crack front complexity, the crack tip is systematically detected at the position of the crack front on the DIC face.

Hence, the DIC-based estimation of the crack tip position is suggested to be the estimation of the crack tip at the surface and not an in-depth average. The presented experiment, with a highly heterogeneous crack front made of multiple ligaments progressing and nucleating at different rates shown by bright white lines in Figure 2.5(c) and (d), eventually shows that DIC-based crack tip detection does not fully render such in-volume complexities.

Acoustic emission(AE)

AE analysis was performed with four resonant piezoelectric sensors (Medium Frequency resonant Miniature sensors Nano30 by MISTRAS). They have been mounted with Kapton tape and super glue. AE data were obtained by the AEwin streaming option to improve the optimal recovery of AE activity by taking advantage of the short experiment duration (less than 5 s).

Hence, waveforms and signal descriptors were computed, through in-house Matlab scripts during post-processing. The AE responses were enhanced by a pre-ampliĄer with a gain of 40 dB, while background noise Ąltering through an imposed threshold was unnecessary since AE data was retrieved by streaming. The system was calibrated and tested for appropriate coupling before each test through the pencil lead break procedure [START_REF] Scruby | An introduction to acoustic emission[END_REF].

Assessment of AE source localization

The method of AE source localization is a widespread addition to standard AE analysis.

With the use of multiple AE transducers, depending on the intended dimension of observa-

Methodology

For the presented work, an inverse problem for AE source localization Ű treating wave velocity as unknown Ű is proposed using four AE transducers. While this problem presents itself as ill-posed [START_REF] Zhou | A New Algebraic Solution for Acoustic Emission Source Localization without Premeasuring Wave Velocity[END_REF], it can easily be built for multiple sensors (here, n = 4). Based on proportional relationships of distances and travel times between source and sensors, 2D source coordinates can be expressed by the nonlinear governing equation, with the assumed AE source location (x s , y s ) and sensor positions (x i , y i ) for ∀i in [1,n]. The equation takes the following form:

(x i -x s ) 2 + (y i -y s ) 2 -v 2 (t i -t s ) 2 = 0, ∀i ∈ [1, n] (2.7)
where t i and t s are the arrival time of the signal at each sensor and the time the signal has been released at the source, respectively. To linearize such non-linear and coupled equation, variables x s , y s , t s and v are substituted, respectively, by x o + dx, y o + dy, t o + dt, and v o + dv.

These linearized contributions of Equation 2.7 then take the following form:

(x i -x s ) 2 = (x i -x o ) 2 + 2dx(x i -x o ) + dx 2 (2.8) (y i -y s ) 2 = (y i -y o ) 2 + 2dy(y i -y o ) + dy 2 (2.9) -v 2 (t i -t s ) 2 = -(v 2 o + 2dvv o + dv 2 )[(t i -t o ) 2 + 2dt(t i -t o ) + dt 2 ] (2.10)
where x o , y o , t o , and v o are the initial guesses with the current increments dx, dy, dt, and dv.

After neglecting squared and weakly coupled terms, Equation 2.7 extended by Equations 2.8, 2.9 and 2.10 leads to a four equation four unknown linear system.

L = 2 x i -x o y i -y o -v 2 o (t i -t o ) v o (t i -t o ) 2 (2.11) b = (x i -x o ) 2 + (y i -y o ) 2 -v 2 o (t i -t o ) 2 (2.12)
The system is eventually solved in a least squares minimization process as follows: Firstly, the non-linear problem is highly sensitive to the initial guess. To highlight this sensitivity, localization error due to the initial guess has been mapped for various random source positions.

L ⊺ L = L ⊺ b (2.
For each random position of the AE source, the AE localization was performed for a grid of 50x50 initial guesses. As an illustration, the initial guess error map is presented in Figure 2.6 for four random AE source positions. σ denotes the distance between the located source position and its real location. The Figure reveals that for some initial guesses (yellow areas), the algorithm may not only poorly converge but even diverge which highlights the strong initial guess dependency. In practice, the cost function is not always convex depending on the source location, so no convergence may be achieved leading to areas of ŠforbiddenŠ initial guesses. Furthermore, no particular initial guess area leading to good localization (σ =∼1 mm) for any potential source position was detected within the Ąeld of interest. Hence, an evaluation to Ąnd the most suitable initial guess has to be performed before each source localization run, i.e. any new sought-for AE signal requires its own optimal initial guess identiĄcation. This can easily be achieved a priori by knowing the crack tip position from DIC and using the proposed numerical twin being presented in this section.

Secondly, to shift the model scenario closer to a real case, main-order experimental biases are introduced to the process of localization. A random normal distributed error was drawn and appropriately scaled for each term of interest. The target term with their standard deviation of the imposed uncertainty can be found in sampling) to stress-test the algorithm. Notice that the signal-to-noise ratio will vary depending on the source position being more critical when the distance between the source and one of the sensors reduces. Indeed, signals need on average 10 µs to propagate through the Ąeld of view, which presents, on average, a signal-to-noise ratio of 90% on the TOA. However, when the source goes closer to one of the sensors since the uncertainty is kept at 1 µs, the signal-to-noise ratio of the crack front (±600µm), the 2D AE localization will not provide valuable information for discriminating from which part of the front the AE has been emitted. Notice that the sensibility of the method to in-volume locations of the source has been evaluated (not presented here), and showed no particular additional bias on in-plane localization, other than slight variations in the velocity identiĄcation.

Waveform classiĄcation methods

Next to the rather standard method of parametric clustering [START_REF] Guo | Fracture process zone characteristics and identiĄcation of the microfracture phases in recycled concrete[END_REF][START_REF] Guo | IdentiĄcation of fatigue damage modes for carbon Ąber/epoxy composites using acoustic emission monitoring under fully reversed loading[END_REF], plenty of analytical methods are proposed in the literature (e.g. waveform cross-correlation [START_REF] Deschanel | Acoustic emission multiplets as early warnings of fatigue failure in metallic materials[END_REF][START_REF] De La Selle | Acoustic emission classiĄcation by cross-correlation in[END_REF], wavelet analysis [START_REF] Sung | Impact Monitoring of Smart Composite Laminates Using Neural Network and Wavelet Analysis[END_REF], b-value analysis [START_REF] Sagar | An experimental study on cracking evolution in concrete and cement mortar by the b-value analysis of acoustic emission technique[END_REF], moment tensor inversion [START_REF]Acoustic Emission Testing en[END_REF]) for the classiĄcation of AE signals and their characteristics. For this investigation, the more speciĄc method of multiplet iden-tiĄcation through waveform cross-correlation [START_REF] Deschanel | Acoustic emission multiplets as early warnings of fatigue failure in metallic materials[END_REF][START_REF] De La Selle | Acoustic emission classiĄcation by cross-correlation in[END_REF] will be pursued in addition to the method of parametric clustering of AE signals in the principle component (PC) space.

In AE analysis, 16 AE waveform descriptors are computed by the AE system. They are used as waveform descriptors to deĄne the characteristics of each AE hit. The descriptors are deĄned as follows:

Ů Amplitude -highest voltage in the AE waveform, expressed on the dB AE amplitude scale. Ů Frequency Centroid -the center of mass of the power spectrum graph.

Ů Peak Frequency -the point where the power spectrum is greatest.

Threshold

Rise time Counts Energy

Implusive Coda Duration Time Amplitude

Principle component analysis (PCA)

In literature and industry, different methods are being used to distinguish among sets of AE responses. Here, the Mistras group software Noesis has been taken as a reference and replicated in Matlab. The following methodology has then been pursued, presented on example data:

Correlation matrix of waveform descriptors

Here, a Pearson [100] pairwise linear correlation between each pair of columns, i.e. waveform descriptors, has been performed as follows:

ρ(a, b) = n i=1 (X a,i -X a )(X b,i -X b ) [ n i=1 (X a,i -X a ) 2 n i=1 (X b,i -X b ) 2 ] (2.14) with X k = n i=1 X k,i n , k = [a, b] (2.15)
where X a and X b are columns in the parametric matrix, while n is the length of the column.

Dendrogram plot of correlation matrix

The degree of correlation between the features of the data set has then been displayed in the form of a dendrogram. With the most correlated features joined together at the top, the degree of correlation decreases towards the bottom of the graph. This has been done to narrow down the number of descriptors to be used for clustering and hence neglect descriptors with similar tendencies. The dendrogram plot is presented in Figure 2.9.

Descriptors that have high coefficients of correlation (again, 1.0 being the maximum) are linked at their value of correlation and can thereby be identiĄed.

Figure 2.9 Ű Correlation matrix of waveform descriptors presented as a dendrogram.

In this example, a very strong correlation is found between Energy and Absolute Energy, which would allow the user to neglect one of them to reduce complexity without losing information. The remaining descriptors are then used for the continuation of the analysis.

Principle components analysis

The principle component analysis [101] has been applied to increase understanding of the AE observation. In the Ąrst step, the most relevant principle components have to be identiĄed.

Waveform cross-correlation

As for the foundation of AE analysis, the method relies on the assumption that similar waveforms are most likely related to the same source mechanism. The identiĄcation of multiplets directly compares signal characteristics by cross-correlating normalized waveforms to deĄne highly correlated groups of signals, i.e. multiplets (nearly identical waveforms as the signature of a unique source [START_REF] Deschanel | Acoustic emission multiplets as early warnings of fatigue failure in metallic materials[END_REF]). Each signal has been tested for similarity with respect to all other signals. In detail, the similarity of two signals (W F i and W F j ) over the time sequence was computed by means of cross-correlation as follows:

R W F i ,W F j (τ ) = (W F i (t) -W F i )(W F j (t + τ ) -W F j ) σ W F i σ W F j (2.16)
with τ and σ being the time delay and standard deviation of W F i and W F j , respectively. Due to normalization, a factor of 1.0 represents a maximal correlation, which is obtained by correlating a waveform by itself. The maximum of the normalized correlation sequence thus quantiĄes the similarity of the two signals and is placed in a correlation matrix.

Experimental investigation of perfect and imperfect mode I cracking cases

The Only limited AE activity is detected during the reference case experiment, which suggests that elastic waves, potentially released during this cracking, are below the experimental sensit-ivity of the AE sensors. For the presented results, the average noise level is found to be 3.5 mV (31 dB) for all experiments. However, no clear link between the AE activity along the test and the velocity variations of the complex cracking is found at this stage. However, with only a peak amplitude of 1 V, the third one (appearing at approx. 4.1 s shortly before the second HA signal) does deviate from the majority of the lower amplitude signals (average peak amplitude of approx. 0.01 V) but poses are rather small signal when compared to the HA signals (close to 10 V). Thus, this signal presents are speciĄc case, that will not be associated with the HA signals.

Investigation of complex cracking cases

AE response of local dynamic instabilities

To identify the origin of the high amplitude (HA) signals, the position of their potential source activity needs to be spatially located on the fracture surface. As previously discussed, the With both signals being located within the zone of uncertainty, it is conĄrmed that the AE signals have been emitted from or near the crack tip. Moreover, the uncertainty, as introduced in Section 2.3.4, is a clear upper bound. Indeed, localization is found at a distance of ∼500 µm and ∼900 µm of the DIC apparent crack tip which even approximates the ±650um of the maximal in-volume crack front complexity. Thus, a deeper analysis of the fracture surface in the spatial vicinity of the crack front for the two considered instants can be pursued. ing the discussion in Section 2.2.1, it can be concluded that these variations in fracture surface roughness do appear on the side being under compression due to the imperfect loading condition. Based on preexisting knowledge on PMMA fracturing reported in the literature, these smooth triangular zones, visible in fracture surface zooms (Figure 2.17 (b) and (c)), can be associated with unstable dynamic crack propagation. Indeed, the recurrence of dynamic and quasi-static crack propagation with their Ąngerprint on the fracture surface has already been observed in many studies [START_REF] Bonamy | Interaction of Shear Waves and Propagating Cracks[END_REF][START_REF] Ravi-Chandar | On the mechanics and mechanisms of crack growth in polymeric materials[END_REF][START_REF] Hattali | Low Velocity Surface Fracture Patterns in Brittle Material: A Newly Evidenced Mechanical Instability[END_REF][START_REF] Vasudevan | Deciphering triangular fracture patterns in PMMA : how crack fragments in mixed mode loading[END_REF]. Such dynamic instabilities, i.e. cracks suddenly jumping over signiĄcant distances at average speeds of approx. 100 m.s -1 , might be induced by thermovisco-elastic effects through rising temperatures at crack tip [START_REF] Vincent-Dospital | How heat controls fracture: the thermodynamics of creeping and avalanching cracks[END_REF]. This phenomenon will be analyzed in depth later on in Part II, where velocity and energy release rate measurements will be provided to contribute to the incomplete understanding of this phenomenon.

Through the identiĄcation of local instabilities of the crack front, crack velocity measurements can be re-evaluated. For this, the crack front was computed for each frame by the crack front extraction algorithm (Section 2.3.2). Before deriving the temporal evolution of the crack front to obtain the velocities, the crack front displacements were smoothed over time by a second-order polynomial Savitzky-Golay Ąlter over a window of three frames. To summarize, while it seems visually obvious looking at Figure 2.19(a) that a correlation exists between AE and heterogeneity of the crack front velocity (i.e. gradient of velocity within the front), the front-by-front correlation between either std or mean velocity and the energy was not found. The count above the threshold was found to be the best variable, however, the correlation remains scattered. This might be explained by the local uncertainty on velocity (due to the low frame rate) or the fact that many AE signals can be produced during the same captured front (AE temporal resolution 1 MHz compared to 62.5 Hz for the cameras). On the one hand, with almost no AE activity (see Figure 2.14(a)), purely mode I cracking experiments in PMMA produce a surface with symmetric, wave-like ligaments (see also [START_REF] Hattali | Low Velocity Surface Fracture Patterns in Brittle Material: A Newly Evidenced Mechanical Instability[END_REF][START_REF] Vasudevan | Deciphering triangular fracture patterns in PMMA : how crack fragments in mixed mode loading[END_REF]). These waves are found almost parallel to the crack propagation direction with a tendency to the boundaries while showing topographical variations of approx. ±70 µm. In addition, tiny regularly spaced waves (every ∼50-100 µm), perpendicular to the crack propagation, are observed. These waves might be associated with small crack advances as observed during fatigue experiments [104]. On the other hand, deep and unstructured scratches on a rather Ćat fracture surface are observed during complex cracking cases, where strong AE activity is detected (see Indeed, waveforms associated with the two multiplets were shown to be highly similar over For granular materials, the separation of AE hits in the space of average frequency and RA-value has been correlated to tensile (above diagonal) and shear (below diagonal) fracture modes [START_REF] Aggelis | ClassiĄcation of cracking mode in concrete by acoustic emission parameters[END_REF]105]. However, with limited information on the source activity in the case of PMMA,

ClassiĄcation of remaining AE signals

Multiplets vs fracture surface Ąngerprints

AE localization results are presented in Figure 2.24 to conĄrm the link between AE signals and crack tip activity. For both multiplets, no repeating event can be unambiguously identiĄed that is solely linked to their expected position of waveform formation. We clearly see that fronts are pulled back at the axial ligaments (shown in Figure 2.20(b)), with an intense pull back on the main central one (as inferred in Figure 2.18(c)). The latter covers the entire specimen which probably makes the unique feature identiĄcation even more complex.

Nevertheless, Figure 2.19 has shown that the link between AE and the crack front may be more connected to the actual kinematic of the crack front than to its post-mortem characteristics. To take advantage of the observations in Figure 2. [START_REF] Harris | Continuous monitoring of fatigue-crack growth by acoustic-emission techniques: Purpose of this investigation was to further explore the relationship between crack-growth characteristics and acoustic-emission variables such as instrumentation gain and sensor frequency. The threshold conditions for crack detection were also investigated[END_REF] This section, while presenting a side discussion, is a perfect example of potential misinterpretation when dealing with partial AE data. Indeed, it is shown that a large amount of highly similar signals Ű coming from the crack tip Ű can not be systematically attributed, at least from post-mortem fracture surface features, to a particular mechanism or behavior. While discriminating fracture phenomena on signal levels has been found to be straightforward, classifying within signals of similar amplitudes has been found potentially misleading.

Conclusion

A model lab experiment under mode I dominating opening mode Ű with slight front/back opening asymmetry leading to complex cracking behavior and high AE activity Ű has been de- Chapter 3

AE RESPONSE OF CONTROLLED GLOBAL INSTABILITIES

Introduction

After the local instability phenomenon has been observed in complex cracking experiments, the interest will shift to the investigation of these instabilities in a controlled experiment with perfect mode I loading conditions. By increasing the scale of these instabilities and isolating them, the understanding of the local phenomenon might be increased. In Section [START_REF] Scheibert | Brittle-Quasibrittle Transition in Dynamic Fracture: An Energetic Signature[END_REF]. On the contrary, with a sufficiently slow loading rate, the crack propagates continuously stable without any instabilities. Here, a quasi-static-like wavy pattern is produced by the propagating crack in the case of pure mode I loading conditions (see Figure 2.20(a)). Within these two regimes, the phenomenon of alternate (stick-slip) cracking [START_REF] Ravi-Chandar | On the mechanics and mechanisms of crack growth in polymeric materials[END_REF][START_REF] Hattali | Low Velocity Surface Fracture Patterns in Brittle Material: A Newly Evidenced Mechanical Instability[END_REF] is observed. Even though the crack is driven by a constant extension rate, the two unique cracking behaviors can be found within one fracturing.

In this section, the experimental adaptions to investigate such cracking experiments are introduced, followed by a discussion of the captured AE activity and the link to its local phenomenon discussed in Section 2. However, the physical phenomenon of alternate cracking will be discussed in detail in Part II.

Experimental program

Experiments have been performed during two independent experimental campaigns, called M1 and M2. Setup, material, and loading conditions have been kept consistent while using different sample thicknesses. 5 mm and 8 mm thick plates have been used for the specimen preparation of the M1 and M2 campaigns, respectively.

Material and geometry of the specimen

Sample geometry and sensor placing were kept constant from Section 2. However, with the intention of producing controlled pure mode I alternate cracking experiments, pin holes have been manually drilled to avoid the laser beam (conical shape) related clearance angles (∼1.5°), which would induce the transverse loads, i.e. potential sample bending. The detailed sample geometry and sensor coordinates can be found in Figure 2.1.

Loading and test conĄguration

All experiments were performed using an Instron all-electric dynamic and fatigue testing system (ElectroPuls E10000) with a load cell of 1 kN maximum loading capacity in a temperature-controlled (21 °C) room. Beforehand, preliminary experiments were conducted to identify the alternate cracking domain within the experimental environment. With respect to the used material and geometry, solely quasi-static propagation has consistently been found below extension rates of 0.1 mm.s -1 . First dynamic bursts have been observed between 0.1 mm.s -1 and 0.5 mm.s -1 with strong variability from one test to another. Following this domain of inconsistency, alternate cracking was systematically determined in a repeatable manner. Entirely dynamic cracks have been found for extension rates above 20 mm.s -1 . Extension rates close to the lower end of the identiĄed alternate cracking domain have been used to provoke higher repetition of crack arrests while maintaining reasonable quasi-static lengths.

The different second-stage loading conditions (see Section 2.2.2) related to the experiments are found in Table 3.1.

Experimental setup

Alternate cracking is a phenomenon, where two entirely different cracking behaviors are to be found during solely one fracturing process. Hence, for relevant data acquisition during the dynamic crack propagation phases, the experiments are in need of high-speed (HS) imaging.

Here, images of the cracking sample were captured using a Photron FASTCAM Mini AX camera system with an acquisition speed of 40 kfps. Except for the replacement of the DIC-facing highresolution camera with the high-speed camera, no additional changes were made with respect to the experimental setup presented in Section 2.2.3. However, it is worthwhile mentioning that with the expected crack velocities during alternate cracking experiments, no information was expected to be gained from the in-volume camera due to its low temporal resolution. It was solely used to obtain presentable images of the fracture surface within the volume (see Fig-

ure 1.2).
Camera hardware parameters for the HS camera system are presented in Table 3.2.

Post-processing

To get derivatives such as the crack velocity, raw crack tip measurements had to be Ąltered accordingly. The need for high-speed imaging to obtain a sufficient amount of data points during dynamic crack propagation led to comparable huge sets of data for the quasi-static propagation phases. With a difference of tens (dynamic) to thousands (quasi-static) of data points between the two regimes, the data set had to be separated and post-Ąltered by different kernel sizes. The post-Ąltering has been performed by using a Savitzky Golay Ąlter with a second-order polynomial. Depending on the regime affiliation, a Ąltering window size of 7.275 ms (291 frames) or 75 µs (3 frames) has been applied for quasi-static and dynamic regimes, respectively. The relatively strong and rather insigniĄcant kernel size for, respectively, the quasi-static and dynamic phases were deemed to be Ątting to not drastically inĆuence crack speed evolution in the quasistatic parts or decrease the accuracy of the dynamic phase measurements.

Due to camera resolution, used lens, and the clearance angle of the sample cut, the uncertainty on the notch tip localization has been evaluated to be about σ notch = 1.5 pix. The uncertainty of the Špix2mŠ value has been evaluated to be about σ pix2m = 1.12 ×10 -3 pix as the FFT sampling was chosen to be the horizontal pixel size of the image (896 pix). Following Equation 2.6, a maximal error of approx. 160 µm and 140 µm have to be expected for the considered Ąeld of view (FOV) for experiments related to campaigns M1 and M2, respectively.

AE activity

Before focusing on the AE activity produced by the alternate fracture, Figure 3.1 presents the fracture surface of experiment M2-T15. Again, velocity or stress intensity factor measurements related to this experiment are found in Part II. Amplitude and normalized crack length are presented as a function of the recording time (1 s). However, only the Ąrst 400 ms are presented, since no activity is recorded thereupon.

Here, dynamic crack propagation (instability) is deĄned by AE signals with peak amplitudes of 10 V (100 dB) Ű as observed during experiments in Chapter 2 Ű and vertical jumps in the temporal evolution of the normalized crack length. Furthermore, a zoom is presented focusing on the second dynamic instability. Here, two things can be observed:

Ů 30 µs delay. A delay between the beginning of the vertical jump of the crack length and the high amplitude AE activity is detected, while only approx. 10 µs can be attributed to the wave travel time from the activity to the AE sensor. This can be explained by a synchronization offset between the camera and the AE system that has been found during preliminary experiments. This offset has been quantiĄed experimentally to be approx. 1400±20 µs.

Hence, the 1400 µs offset has been corrected but it remains a ±20 µs uncertainty. Since the dynamic propagation phase is about 200 µs, AE seems to be released at the beginning of the phase.

To additionally verify the accuracy of the synchronization of the measurements, AE localization was performed on the two dynamic instabilities. The localization results are presented in Figure 3.3. and ∼151 µm is going way below the characteristic size of the macroscopic scratch-like ligaments, the unambiguous and systematic identiĄcation of similar fracture surface Ąngerprints for LA signals with the naked eye becomes very difficult. It seems to be partly contradictory to our Ąrst statement that the AE activity difference between smooth and complex cracking might be related to macro features on the surface. In line with Figure 2.19, it seems much more connected to tiny heterogeneities within the front, the latter being obviously the consequence of the observed macro-scratch. However, while more advanced image analysis tools may help, no obvious solution has been found.

Conclusion

In Remark 2 Results presented in this chapter are published in [START_REF] Heinzmann | Experimental investigation of the alternate recurrence of quasi-static and dynamic crack propagation in PMMA[END_REF].

Alternate cracking under high temporal and moderate spatial resolution

PMMA samples were subjected to the loading conditions introduced in Section 3.2.2 and followed by a high-speed camera system. About 40,000 successive images over a duration of one second were obtained for each experiment. The images were used for DIC analysis and subsequently for the retrieval of fracture mechanic parameters. Oscillations in the crack length of the quasi-static rest phases always appear closely following the dynamic bursts. These Ćuctuations can be explained by system vibrations due to historydependent effects induced by brutal, instantaneous dynamic bursts. It leads to oscillations of the crack lips (opening and closing) causing the crack tip detection to record backward moving cracks. Furthermore, as they become more prominent when the crack approaches the free end of the sample, the vanishing resistivity of the entire sample seems to have an enhancing effect on the phenomenon.

For the sake of clarity, two experiments will be prioritized for the continuation of the discussion. For this, M1-T2 and M2-T15 have been selected as representatives of the two experimental campaigns as the tests with the most quasi-static rest repetitions within their experimental campaign.

Crack speed variations

To further differentiate among the crack propagation types observed during alternate cracking, crack speeds have been derived from the crack length. Here, cracking phases with biased velocity measurements are marked in red. Globally, two groups of propagation phases can be observed in the Ągure. Cracks with rest bands fall all within the lower velocity range, while the others are found above. It appears that the limit average velocity separating these two can be allocated at approx. 60 m.s -1 . Indeed, with sufficient temporal resolution to capture velocity variations during dynamic propagation, rest band decelerations could have been singled out. This would most certainly cause the upward shift in average velocity during these phases with the option of excluding rest bands. By taking all bursts into consideration, a relation of increasing average velocity with increasing burst length can be observed. A speciĄc case is found during the Ąrst dynamic burst of M2-T15. The particular surface roughening is observed, even though no rest band is entirely formed. Nevertheless, crack deceleration inĆuencing the velocity measurement has to be expected. Thus, this burst will be considered biased and included in the rest band-affected bursts. With an average velocity of ∼60 m.s -1 , the dynamic burst seems to be at the velocity threshold for the formation of rest bands. Hence, if dynamic bursts affected by rest bands (biased velocity measurements) are neglected, a ŚforbiddenŠ velocity regime of ∼36 cm.s -1 to ∼36 m.s -1 can be implied taking all conducted experiments into consideration. This is expressed in Figure 4.6, showing velocities with respect to the normalized crack length for all conducted experiments excluding dynamic phases that are considered to be biased. Notice, with average velocities above the identiĄed threshold value (∼60 m.s -1 ) for the formation of rest bands, instantaneous velocity measurements at the beginning and the end of rest band free dynamic bursts can fall below the threshold and thereby present the upper bound of the ŚforbiddenŠ velocity regime. The next chapter will provide a dedicated analysis of the rest band by increasing the temporal resolution of our imaging system.

Connections to the fracture surface

In the following, more attention is given to the relationship between experimental data and fracture surfaces. Microscopic images have been obtained by an optical microscope (Nikon) with a resolution of 0.67 µm.pix -1 . K1 and velocity data as a function of the crack length are presented in phase with their corresponding fracture surface in Figure 4.7. Red color numbers again indicate the presence of rest bands falsifying the velocity measure-ment of this particular dynamic burst. Hence, if dynamic bursts with rest band occurrences are neglected, a trend with a growing relationship of dynamic velocity and distance e can be suggested. While Ąrst Wallner lines seem to appear after 193±78 µm for mean dynamic crack speeds of 85±5 m.s -1 , they Ąrst appear after 490±125 µm for mean crack tip speeds above 100 m.s -1 .

It seems to show that depending on the velocity, a critical distance is required (with an almost linear trend) to reach quasi-static propagations.

Energy release rate vs crack speed

Energy release rates have been computed from K1 to reproduce the kinetic law of materials

showing alternate cracking (Section 1.1.1). Again, a detailed presentation of the hysteresis will only be presented for two tests (here, M1-T2 and M2-T19). For clearer visualization, only the crack propagation phases directly associated with the alternate cracking are presented. This means, only the phases surrounded by dynamic bursts are taken into consideration with the hysteresis always starting with the Ąrst dynamic burst. In each of the two subĄgures, presented in Figure 4.9, two related graphs are found: the temporal evolution of the crack length (a(t))

and the energy release rate as a function of the crack tip speed (G(v)). Each individual phase of cracking is colored concordant over the two plots. Also, triangular and square markers indicate the beginning or the end of a phase, respectively. During this propagation phase, the energy release rate drops from ∼520 J.m -2 to ∼195 J.m -2 , while the crack speed undergoes an increasing, stabilizing, and decreasing phase with velocities around 85 m.s -1 , 120 m.s -1 and 70 m.s -1 , respectively. Subsequently, no data points are acquired until the Ąrst points in the quasi-static region, indicating the Ąrst jump between the two accessible regions. By arriving at the Ąrst quasi-static propagation phase (dark blue), G rises from ∼240 J.m -2 to ∼375 J.m -2 , while the crack speed continues to decrease from 5 cm.s -1 to 3 cm.s -1 . After this, while G continues to grow (∼375 J.m -2 to ∼420 J.m -2 ), v Ąnds itself increasing from the low point to around 15 cm.s -1 . With G decreasing after reaching its peak value (∼420 J.m -2 ), the crack continues to accelerate to 23 cm.s -1 before jumping back to the dynamic region. Globally, this repeats itself until full crack arrest. However, the last two dynamic branches (orange and light red) have to be taken with caution since the measurements are inĆuenced by the presence of rest bands. Furthermore, the seeming separation of the last quasi-static phase (dark red) into two arms can be explained by the post-treatment done on the crack length to remove post-dynamic oscillation leading to zero velocities for a short duration. Additionally, it is worthwhile mentioning that the value of critical fracture energy (G), introducing the ŚforbiddenŠ region II, is 430±30 J.m -2 for all quasi-static phases. Test M2-T19 (Figure 4.9(b)) generally follows this description with the difference of a long quasi-static rest resulting in more data points and stronger velocity oscillations during this phase.

B C D A' (a)
To conclude the results of this section, all experimental data (excluding phases with biased velocity measurements) is presented in the same (v,G) space (Figure 4.10). Additionally, an entirely quasi-static experiment (reference experiment used for the discussion in Chapter 2), that has been subjected to an extension rate (0.07 mm.s -1 ) close to the threshold of the observed appearance of alternate cracking, was added. To place the experimental results into the context of the literature, experimental data points from [START_REF] Vincent-Dospital | How heat controls fracture: the thermodynamics of creeping and avalanching cracks[END_REF] and [START_REF] Scheibert | Brittle-Quasibrittle Transition in Dynamic Fracture: An Energetic Signature[END_REF] (quantitatively extracted using the online software [118]) were furthermore included. While our entirely quasi-static experiment (see gray points on Figure 4.10) seems to align well with the quasi-static experiments conducted in [START_REF] Vincent-Dospital | How heat controls fracture: the thermodynamics of creeping and avalanching cracks[END_REF], alternate cracking results do seem to add the missing experimental data to adequately cover the transition between quasi-static and dynamic crack propagation in the full spectrum of PMMA fracturing. Starting with the dynamic branch, experimental data points found in this study Ąt well with the trend of existing literature.

However, the last value of G (before jumping to the quasi-static phase) is systematically found below any G value of the quasi-static regime, and therefore below the expected value of G c . In detail, velocity measurements and energy release rates introducing the jump from dynamic to quasi-static propagation are found to be 67±31 m.s -1 and 236±40 J.m -2 , respectively. During quasi-static phases, the crack grows at an almost constant speed. Observed variations during quasi-static propagation can be explained partly by physical and partly by post-dynamic effects.

Avalanche values for energy release rate and velocity marking the beginning of the ŚforbiddenŠ region II are found to be 497±92 J.m -2 and 0.19±0.15 m.s -1 , respectively. Taking both sides into consideration, region II, i.e. ŚforbiddenŠ velocity regime, is thereby found to be more narrow than presented in the literature. Nevertheless, a brutal switch is seen from the rate-dependent dynamic branch to the quasi-static regime with only, as a transition, the deceleration distance e.

General variations in the energy release rate are found to be in line with the literature. In the quasi-static regime, the material strength increases with increasing strain rates due to the viscosity of PMMA. The resulting increase of G in the quasi-static regime is supported by the experimental data of Vincent-Dospital et al. [START_REF] Vincent-Dospital | How heat controls fracture: the thermodynamics of creeping and avalanching cracks[END_REF]. The increase in the energy release rate in the dynamic regime can be explained by the increase of the process zone around the crack tip, which is in line with KanninenŠs theory [START_REF] Kanninen | Advanced fracture mechanics Oxford engineering science series 15[END_REF]. However, differing energy release rates for dynamic crack initiation and arrest Ű where the SIF at dynamic crack initiation KID ini has been found to be higher than at dynamic crack arrest KID a Ű have also been discussed in Grégoire et al. [119].

Furthermore, inĆuences on the fracture surface of the variations of the dynamic crack propagation and the increase of surface roughness due to the appearance of micro-cracking have also been discussed in the literature [START_REF] Ravi-Chandar | On the mechanics and mechanisms of crack growth in polymeric materials[END_REF]120,121]. This is in line with the presented observations, where conical shapes on the fracture surface (micro-cracking) are found at the beginning of the dynamic propagation (Figure 4.4, detail 1) with high levels of G. At the same time, they vanish as G decreases to the point of dynamic crack propagation arrest.

Conclusion

For this section, alternate cracking in PMMA has been observed by high-speed imaging and investigated with the help of DIC and post-mortem analysis. Quasi-static and dynamic crack propagations linked to this speciĄc fracture phenomenon have been investigated by a compromised experimental setup. The setup however allowed for the investigation of the two cracking types within one fracture process. Velocity and SIF measurements obtained through DIC and WilliamsŠ series expansion have been tied to microscopic images to highlight the regimedependent behavior of alternate cracking. By doing so, it was possible to Ąnd relationships through the synchronization of experimental devices. The main observations can be summed as follows:

Ů ŚForbiddenŠ velocity regime. The literature described ŚforbiddenŠ velocity regime was not approached from one of the sides, but studied through holistic alternate cracking experiments with highspeed imaging. Here, the ŚforbiddenŠ regime has been identiĄed by quasi-static and dynamic velocity threshold measurements of ∼36 cm.s -1 and ∼ 36 m.s -1 , respectively. Notice that the limit values stem from the lowest/highest measured velocity of all observed phases.

Ů Relationship of mean dynamic velocity with burst length. Dynamic crack speeds and their burst lengths have been registered and used to establish a relationship. By doing so, it was shown that higher velocities result in longer dynamic bursts.

Ů Observation of rest bands. In line with the previous point, probably for dynamic propagation phases containing lower levels of energy, dynamic bursts show intermittent behavior with extremely small rest bands. An average velocity threshold for the potential occurrence of rest bands has been identiĄed at ∼60 m.s is not smooth after all. In addition, the path that the crack follows to revert to the master curve in the quasi-static regime differs from the observations in [START_REF] Vasudevan | Deciphering triangular fracture patterns in PMMA : how crack fragments in mixed mode loading[END_REF]. However, in their work, very slow extension rates have been used resulting in quasi-static experiments initiated by a dynamic crack. No alternate recurrence of the two propagation types was observed. Regarding quasi-static to dynamic transitions, even with a temporal resolution of 40 kfps, it was not possible to experimentally explore the smooth transition during the ŚsofteningŠ branch predicted by the thermodynamic model of [START_REF] Vincent-Dospital | How heat controls fracture: the thermodynamics of creeping and avalanching cracks[END_REF]. This, however, could potentially be explained by a lack of temporal resolution or the feasibility of exploring this regime due to the way the experiments were conducted. To do so, an experimental conĄguration enabling decreasing K1 values with increasing crack velocities is needed.

Beyond the methodological contribution of investigating alternate cracking through high-speed DIC, the experimental investigation of alternate cracking was addressed by focusing systematically on crack velocities, SIFs, and their relation to the fracture surface. Like this, transitions from dynamic to quasi-static and quasi-static to dynamic were studied. Hence, a compromised experimental conĄguration was proposed, based on high-speed full-Ąeld measurements (DIC), and on asymptotic solutions of kinematic Ąelds to get not only FE-independent and chart-free estimations of the SIFs but also sub-pixel estimations of crack tip positions. Additionally, the contribution of this chapter can be summed by the attempt to provide experimental data for improved estimation of the hysteretic fracturing behavior described by the loop following paths B, C, D, and AŠ in Figure 1.1.

tests to investigate dynamic crack propagating in glass [123], or inducing rupture initiation by short pressure pulses from a wireŠs rapid sublimation [124]), the challenge in this experimental campaign arises from the fact that the alternate cracking behavior is not actively controlled. The spatial and temporal appearance of the event of interest (dynamic crack propagation phase with the transition from/to quasi-static propagation phases during alternate cracking) can not be predicted. PMMA, paint, solvents, notch reproducibility, and setup alignment introduce some variations in loading, material properties (toughness), and initiation resulting in the variation of the quasi-static phases. Indeed, variation in the dynamic phase was not critical considering the speed of crack (>100 m.s -1 ), but 1 mm more on quasi-static phases will shift everything causing Ćash-light (see section Section 5.2.4 for lighting challenges) to be off by some milliseconds. Hence, if the event can be triggered (see Section 5.2.4 for triggering challenges) the camera will record at the right moment, but images will be captured in the dark. Despite a lot of caution, only 1 test was a full success having both transitions whereas two others were darker and missed one of the two transitions. 15 more experiments were totally off, i.e. either the crack was outside of the small FOV or the light was delayed.

The Section will present the complex experimental setup and methodology allowing for the investigation of a dynamic phase during alternate cracking with extreme temporal resolution and a discussion tackling the points raised above.

Experimental setup speciĄcations

Material and geometry of the specimen

For the following experimental campaign, apart from blunting the notch tip with a circular hole to induce dynamic crack initiation [START_REF] Vasudevan | Deciphering triangular fracture patterns in PMMA : how crack fragments in mixed mode loading[END_REF], material and sample geometry were unchanged with respect to experiments presented before (see Section 3.2.1). With the slight change in geometry, decreased stress concentration in the notch tip will enable the crack to directly start in the dynamic phase (Region III in Figure 1.1).

Furthermore, new acoustic emission (AE) transducer positions were chosen based on their spe-ciĄc purpose of system triggering and AE analysis, respectively. Sample geometry, as well as AE sensor positions, are presented in Figure 5.1.

Loading and test conĄguration

As in the previously discussed campaigns, experiments were performed on an ElectroPuls E10 000 Instron device. However, here a 10 kN maximum loading capacity load was mounted.

Due to the speciĄc sample geometry, only relatively low extension rates of 0.025 mm.s -1 were needed to provoke dynamically initiated alternate cracking.

Experimental setup

With the aim of investigating alternate cracking with extreme temporal resolution, one is in need of ultra-high-speed (UHS) imaging techniques and proĄcient DIC competencies. Here, images were acquired using the UHS-HR Cordin Model 580 at 470k fps with a spatial resolution of 3296 × 2472 pixels. While less complex UHS technologies exist regarding their experimental application, none offers comparable spatial at such high temporal resolution (see A. Vinel et al. [125] for an in-depth comparison of different UHS technologies). However, with the Cordin Model 580 being a multi-sensor rotating mirror ultra-high-speed camera, postprocessing of such images is not straightforward. For this, the state-of-the-art calibration methodology was developed by A. Vinel et al. [125], which can be summarized as follows:

Calibration stage 1. Calibration shots: A set of calibration shots are taken from a static sample. With this post-processing methodology, an accuracy of 0.5±0.2 pixels (including algorithmic, camera variability, and experimental bias) can be achieved.

Before introducing the experimental setup, it seems reasonable to introduce the different timescales that have to be dealt with regarding the experiments. Like this, the need for such a complex experimental setup is justiĄed. Table 5.1 summarizes these timescales: In addition, light is not instantaneous (rise time of ∼200 µs), thus a precursor event is needed to provide light during the recording of the event of interest. Here, sufficient lighting was provided by two Pro-10 Profoto Ćashes (2 × 2400J for 1 ms). Two piezoelectric transducers (Medium Frequency resonant Miniature sensors Nano30 by MISTRAS), positioned as visualized in Figure 5.1, were used to capture AE activities for the purpose of triggering and standard AE analysis. The Ąrst AE transducer (S1 in Figure 5.1) was directly connected to an oscilloscope and a delay/trigger signal generator (R&D vision), without any pre-ampliĄcation, while the second transducer (S2 in Figure 5.1) was classically connected through a pre-ampliĄer to the AE system. No AE analysis was intended for the signals recovered with the oscilloscope and delay/trigger signal generator, their sole duty was for the triggering of the camera and the light/AE streaming, respectively. Eventually, all devices were attached back to the oscilloscope to simplify data synchronization in the wake. The experimental setup is detailed as a sketch in

Experimental methodology

A two-dynamic phase conĄguration was chosen, where the Ąrst one is needed just for triggering the light. Indeed, a lot of work has been done on Ąnding repeatability in terms of fracture onset, loading, AE precursor, and triggering by conductive stripes, but no reliable alternative was found. Thus, two clear dynamic events are needed. To begin with, it is again worthwhile mentioning that the complexity of this experimental campaign arises from the fact that the event of interest is not actively controlled. This means, there is no certainty in the proper prediction of the temporal or spatial location of the sought-for activity. With the need to Ąt the recording length (∼167 µs) within a window of perfect light conditions (∼1000 µs), when the appearance of the second dynamic event (trigger for the camera) will already shift by ∼1200 µs for a 100 µm variation of the preceding quasi-static propagation (assuming an average quasi-static crack speed of 8.5cm.s -1 ) poses an extreme experimental challenge. Furthermore, with the FOV being placed in the position of the expected dynamic event based on preliminary experiments, no guarantee existed that the event of interest would actually happen within the anticipated window.

Triggering of the different devices was fully reliant on sensor S1 recovering AE signals being produced by dynamic instabilities (see Section 3.3). However, the signal was split into two different controller devices each being responsible for an individual set of responder devices.

Controller devices:

Oscilloscope: The oscilloscope was solely responsible for triggering the camera system. To do so, a two-stage triggering procedure was established: (1) the Ąrst threshold crossing (Ąrst dynamic instability) armed the oscilloscope, while (2) the second threshold crossing (second dynamic instability) triggered the camera.

Delay/trigger signal generator (R&D vision):

The delay/trigger signal generator was tuned to react based on the Ąrst threshold crossing (dynamic instability). Two independent signals were transferred to trigger the light and the AE streaming. However, while the AE streaming was instantly triggered, a delay of 5.5 ms (identiĄed based on preliminary experiments) was added to the triggering of the light system. The latter was used to account for the expected duration of the quasi-static phase in between.

Responder devices:

Light system (Flash): After being triggered by the delay/trigger signal generator, the Ćashlights are in need of approx. 200 µs to reach its full power, while they are able to provide full intensity for about 1 ms.

AE:

Being instantly triggered, the AE streaming was set to record for 100 ms.

Camera (Cordin):

Taking opportunity of the Cordin Model 580 system allowing for a pretrigger of up to 3/4 of its recording length, approx. 46 µs of the 167 µs recording length were captured before receiving the trigger signal.

The entire experimental methodology is summarized in Figure 5.3. Here, the output of the oscilloscope capturing the different starting time stamps of all responder devices is superposed with the raw streaming output. The green signal presents the AE activity recovered by sensor S2 (AE system), while the black signal presents the AE activity received at sensor S1 (oscilloscope). The red box indicates the time of perfect light conditions (∼1000 µs) that has been activated 5.5 ms after the Ąrst threshold crossing, while the tiny blue box shows the recording length (∼167 µs). The fracture surface is presented as a reference indicating the expected positions of signal release on the fracture surface.

UHS rotating mirror cameras and light: A small digression

As previously discussed, providing sufficient lighting for UHS recording poses one speciĄc difficulty when working with a multi-sensor rotating mirror UHS camera system. In the classical framework, the needed quantity of light is provided by Ćashlights. However, a certain rise time is necessary for Ćashlights to reach their full intensity. To account for this, an event of interest precursor is needed to serve as a trigger. In the conducted experiments, this has been achieved by initiating the alternate cracking experiment with a dynamic propagation phase and its emission of elastic waves. However, in the process of Ąnding more reliable and convenient solutions to simplify the experimental setup, alternative solutions were investigated, namely constant or pulse light sources.

Pulsed laser solutions allow to decrease rise time in the order of some nanosecond, but even the best solution on the market (UHS cavitar system 280 W @ 640 nm Ű tested on site) was not powerful enough to properly light the sample. The current technology can pulse up to 5 MHz, but only during a cumulated pulse duration of 32 µs. In practice, this means that only 400 ns pulse per frame (80 frames) can be produced. Only an FOV in the order of 1 cm 2 can be sufficiently lighted by using such a small exposure period. The presented information thus shows that it does not seem feasible to obtain enough continuous light to record with intended frame rates of above 400 kfps over a large enough surface area for our experiments. Notice that such a limit would be pushed to a couple of Mfps for single and small sensor UHS technologies such as Shimadzu and/or Kirana due to signiĄcantly larger pixel surfaces ( 20 times) and the absence of multiple layers of optical systems (e.g. beam splitter, mirrors) contributing to the waste light. Hence, at the

Camera hardware and DIC parameters

Camera hardware and DIC parameters regarding the UHS experiments are presented in where {u} and {ü} represent the displacement and acceleration at time t, while {F } is the applied force.

[M ] and [K] are the mass and stiffness matrix, respectively. The time integration of displacements and velocities is achieved using an explicit Newmark scheme.

For improved understanding, Figure 5.5 shows the twin nodes around the crack tip at different integration steps of the dynamic simulation stage. The displacement magnitude of the static preload, at fracture onset, is two to three orders higher than the amplitude of the elastic wave. Thus, to visualize the elastic wave propagation, the static preloading will be graphically removed by subtracting the entire displacement Ąeld of the Ąrst integration step.

The horizontal and vertical displacement Ąelds of the same time step as in Parameters used to perform the simulation are listed in Table 5.4.

Number of nodes 9224

Virtual Image Deformation (VID)

One of the major difficulties working with multi-sensor ultra high-speed cameras (Cordin-580) is soothing the inĆuence of the distortion induced by the complex sensor apparatus. The distortion has to be taken into account during the process of virtually deforming images. This process can be summarized in the following equation: By capturing images of a static sample, U F e vanishes from Equation 5.3 and the distortion can be retrieved using DIC. Nevertheless, U Di can slightly vary from one shot to another (e.g. due to vibrations) leading to a collection of distortion Ąelds, which was used by Vinel et al. [125] to create a statistical model U Dstat (see Section 5.2.3). For the presented procedure, synthetically created images will be distorted using a random U Di (introducing the uncertainty in real experiments). In contrast, the DIC procedure will take advantage of this statistical model U Dstat .

x = X + U F e (X) + U Di (X + U F e (X)) (5.
Two further points have to be considered for the process of VID. First, to get the full specimen into the Cordin-580 FOV with full resolution (2472 × 3296 pixels), a virtual lens with a magni-Ącation of about 0.2 would be required. However, capturing signals with peak displacements of ± 6µm horizontally and ± 20µm vertically (see Figure 5.7) will be challenging with the anticipated measurement uncertainty. Secondly, to put difficulties associated with the evaluation of displacements at the crack lips and tip aside (a continuous mesh can not accurately capture discontinuous displacement Ąelds), the FOV was shifted away from the crack. With the investigation interest being the identiĄcation of experimental setup parameters and the feasibility of capturing the propagation of elastic waves, the ROI was positioned with the crack tip Ąnding itself slightly outside of the FOV. Figure 5.8 shows the three different ROIs that were investigated.

Eventually, the idea is to deform images with the simulated displacement plus the distortion induced by the optics in the camera. These distortions, being polynomial, a sufficiently Ąne sampling is required to interpolate the reference image into a virtually deformed one. Thus, the mesh is reĄned to obtain the same mesh size (in pixels) for any magniĄcation. In detail, any virtual increase of the magniĄcation, i.e. zooming into the mesh, results in an increase in mesh size compared to the Ąeld of view, i.e. pixel size. Hence, as the mesh becomes coarser and coarser, an increase in interpolation errors on the synthetically deformed images, due to the displacement from the simulation always being obtained with the initial mesh and just being interpolated on a Ąner one, has to be expected. Therefore, to bypass these restrictions, the mesh was reĄned prior to the procedure of virtual image deformation.

As a reference image for the VID process, a synthetically created speckle pattern image was used [125], which can be seen in Figure 5.9. From an experimental point of view, this is mandatory due to the fact that the Cordin-580

will not be able to provide undistorted images. So contrary to the standard DIC, the correlation will be performed between the camera images and either a synthetically created reference image or a scan of the sample surface (see Section 5.2.3).

Parameters used for the process of DIC, following the methodology introduced in Section 5.2.3, are presented in Table 5.5. Based on the foreseen, the most promising magniĄcation to choose is 1.0. It is worth noticing that there might be some magniĄcation slightly below 1.0, which still gives reasonable results, and that the magniĄcation threshold needed to make the signal overcome the Ćoor of distortion is somewhere between the two magniĄcations (0.5 -1.0). This means the magniĄcation of 1.0 is chosen rather conservative. Notice that the discussed conditions (magniĄcation of 1 with a temporal resolution of 4 Mfps) present the maximal potential of the experimental setup. While results seem promising, a compromise Ű both spatially and temporally Ű had to be made for experimental repeatability issues discussed before. Furthermore, the unique but sudden release of one mode might only give an order of magnitude of the released energy compared to that released during a typical dynamic burst in alternate cracking experiments. More insights are gained by a numerical replication of a dynamic crack propagation phase, roughly resembling the presented experiment, which will be explored in Section 5.4.

UHS investigation of dynamic phases during alternate cracking

PMMA samples were subjected to the loading conditions introduced in Section 5.2.2 and followed by a UHS camera system (Cordin Model 580). Successive images were recorded throughout 167 µs (i.e. 470 kfps) and with a magniĄcation of 0.65. DIC was performed on the images and subsequently fracture mechanic parameters were retrieved by WilliamsŠ series projection. Even for the only successful experiment, one can see by looking at Figure 5.11(a) that, while the temporal synchronization was on point, the beginning of the dynamic crack propagation was almost missed spatially. Thus, underlining the complexity of the experiment.

Connections to the fracture surface

The following section will present a continuation to Section 4.2.2. Again, microscopic images were obtained by an optical microscope (Nikon). Figure 5.12 presents the fracture surface and a detailed view with a resolution of the optical microscope of 0.67 µm.pix -1 and 0.34 µm.pix -1 , respectively. Again, due to manual stitching of the microscopic image, light differences are visible indicating the sub-frames in Figure 5.12(a). In Figure 5.12(a), the two quasi-static phases enclosing the dynamic phase of interest can be observed on the left and right sides of the image.

In between, one fully evolved rest band appears, while a second zone of macroscopic surface roughening is observed between the rest band and the second quasi-static phase. Similar to experiment M2-T15 in Section 4.2.1, the fracture surface here indicates a reduction of crack velocity, while not reaching the state of a fully developed rest band. Furthermore, taking Figure 5.12(b) into consideration, higher velocities have to be expected during this dynamic propagation in comparison to the experiments discussed in Chapter 4 due to the density and continuous presence of conical shapes on the fracture surface (i.e. indicating the appearance of micro-cracking [START_REF] Ravi-Chandar | On the mechanics and mechanisms of crack growth in polymeric materials[END_REF]120,121]).

To underline this observation, Figure 5.13 presents the crack velocity and mode I related stress intensity factor (K1) as a function of the crack length in phase with the fracture surface topography. A STIL 3D surface proĄlometer took topographic images with the linear grid scanning mode and a step of 6 µm. where dynamic propagation phases with rest bands were all found to have average velocities below ∼60 m.s -1 (see Section 4.2.1). It also demonstrates the fact that, if a higher temporal resolution can be achieved, even lower apparent (average) dynamic speed phases do locally propagate at more than 100 m.s -1 .

Taking the topography measurements into consideration, one can Ąnd the Ąrst interruption perfectly in line with the observed rest band, while the phase of reduced crack tip speeds is found along the slight increase of surface roughness. Thus, the assumption of the observed rest bands being the Ąngerprint of quasi-static crack propagation is evidenced. Additionally, the phase of increased surface roughness (around 5.5-6.5 mm) proves that even smaller crack velocity variations can be identiĄed by just observing the fracture surface.

By looking at the spatial evolution of K1 during this dynamic propagation, the main drop in K1 is found directly with the initiation of the dynamic crack propagation with a stabilization until the Ąrst rest band. Similar to the K1 behavior of the quasi-static phases in Chapther 4 (see

Interaction of crack tip and elastic waves

In the process of atomic bond separation during the fracture process, energy is partially released through heat dissipation [START_REF] Döll | Kinetics of crack tip craze zone before and during fracture[END_REF] and acoustic emission [START_REF] Bonamy | Interaction of Shear Waves and Propagating Cracks[END_REF]. While the contribution of energy corresponding to the release of elastic waves is considered to be rather small (below 5% Attempting to Ąnd an answer to the intermittent behavior of the crack during dynamic cracking phases, fracture surface roughness is compared with velocity and T-stress estimation from the WilliamsŠ series projection. The surface roughness was computed with the following equation over a 500 µm slice (at approx. y=6 mm) of the fracture surface topography in Figure 5.13:

Ra = 1 l l 0 |z(x)| dx (5.4)
With l, x, and z being the slice length, width, and altitude, respectively. Computing the surface roughness over a slice rather than over the entire sample thickness was done to avoid misleading expansion of Ra due to the crack curvature. Furthermore, the position of the slice was chosen as close as possible to the DIC face. However, due to the preparation of the sample for the microscopic image, the fracture surface was slightly compromised by cleaning the spray paint (scratches close to the DIC face in Figure 5.12(a)). Thus, reliable roughness measurements were not possible closer to the DIC face. Figure 5.14 shows the surface roughness along the entire slice. the crackŠs arrest or slow down. In particular, an increase of T means in practice that tension stress waves (probably coming from rebounds on free edges) are successively facing the crack tip. Evidence for these observations will be discussed in Section 5.4. Color differentiation is presented for the part before (red) and after (black) the rest band.

Energy release rate

During the Ąrst part, G decreases from ∼170 J.m -2 to ∼120 J.m -2 before increasing again to ∼140 J.m -2 . The velocity increases from ∼6 m.s -1 to ∼215 m.s -1 and back to ∼2.5 m.s -1 before reaching the rest band. For the second part, a Ąrst phase of increasing G (∼125 J.m -2 to ∼143 J.m -2 ) with decreasing v (∼14 m.s -1 to ∼2 m.s -1 ) is observed. After that, a small gap in the curve is seen due to spurious negative velocities during the quasi-static rest phase. It continues with a strong velocity increase (∼9 m.s -1 to ∼145 m.s -1 ) and a slight decrease in G (∼133 J.m -2 to ∼125 J.m -2 ). Afterward, continuous growth of G can be observed with a short stabilization before the second quasi-static phase (∼125 J.m -2 to ∼186 J.m -2 ). However, the phase of reduced crack tip speeds (slowing down to ∼50 m.s -1 ) surrounded by two velocity peaks (∼176 m.s -1 and ∼200 m.s -1 ) can be seen in v. Eventually, after the last crack speed peak, the recording stops at a v of ∼15 m.s -1 at the beginning of the second quasi-static phase.

To bring the observations into the context of the kinetic law of alternate cracking, UHS data has been added to the data presented in Figure 4.10. Thus, Figure 5.17 combines quasi-static and HS experiments with the presented UHS experiment. While increased temporal sampling for the dynamic phase during an alternate cracking experiment provides higher velocity measurements in the peak and further decreases the window of the ŠforbiddenŚ velocity regime on the lower end, energy release rates are found only on the lower end compared to the HS measurements (see Section 4.2.3). This can potentially be explained by PMMA being a rate-dependent material [134] (variation of YoungŠs modulus E from the quasi-static reference at 2.9±0.1 GPa up to 6 GPa at higher strain rates [135]). In the case of rate-dependent materials during dynamic transient loading, E becomes heterogeneous.

High strain rates are found in the vicinity of the crack tip with low strain rates everywhere else.

With the appearance of waves, E potentially becomes heterogeneous everywhere, and transient in time. In such a case, the crack may propagate in a strongly varying E domain. Hence, the following has to be taken into consideration:

1. The approach of assuming a constant E for the WilliamsŠ series projection is not entirely suited to render such measurements. This also applies to the experiments in Chapter 4, however, sufficient temporal resolutions did not exist to capture wave-induced variations in the kinematic Ąeld. Hence, a potential misestimation of the energy release rate by choos-ing E to be close to quasi-static loading rates (3.5 GPa for all presented experiments) has to be expected.

Discussion based on a numerical replication of the experiment

Building on the basic simulation introduced in Chapter 5.2.6, the following section will present a rough numerical replication of the discussed experiment to improve the understanding of the observations. The initial simulation has been shifted closer to the experimental case by small adjustments.

Here, the predeĄned crack has been introduced with its tip at the Ąnal position of the experimental crack tip (obtained via DIC + WilliamsŠ series expansion on the experimental data).

Instead of clamping solely one node (at the tip of the pre-deĄned crack path) during the static pre-load phase, multiple nodes are held together along the approximate crack path of the experiment (∼8.5 mm, see Figure 5.16). Figure 5.18 shows two time instances of the simulation:

(1) before the opening of the Ąrst twin node and (2) after the opening of the Ąrst twin node. Following the temporal evolution of the experimental crack propagation, the nodes were released progressively toward the crack tip. Notice that no propagation law (or initiation) was included, but the crack advance was imposed by the position of the experimental tip at each instant. To smooth the process of the node opening, the mesh has been reĄned by two (element sizes of 50 pixel reduced to 25 pixel around the pre-deĄned crack path). Due to the explicit time integration scheme, a Ąner mesh requires a smaller integration step. Hence, the integration step was also divided in half (0.025 µs to 0.0125 µs). The decrease in time sampling will result in a temporal resolution of 8 Mfps of the simulation. To achieve a fair comparison with the experimental data, simulation results were down-scaled to the temporal resolution of the experiment (2.13 µs, i.e. 470 kfps) in the wake. This was done, for simplicity purposes, by averaging in time the displacement and velocity data of the simulations over the set of integration steps presenting the experimental time step. Furthermore, by changing from a displacement to a load-controlled simulation, the application of the load has been changed from the nodes at the left boundary of the sample to the nodes within the area of the pins. Finally, to improve the experimental resembling of the simulation, the fracture toughness (K1 c ) has been reduced to 0.82 MPa.m -0.5 being the experimentally measured K1 level at the onset of the dynamic crack propagation. Eventually, the displacement data, at the temporal resolution of the experiment, was used in the process of WilliamsŠ series projection. Figure 5.19 presents the experimental crack length, the smoothed crack length that has been imposed on the simulation, and the crack length detected by the WilliamsŠ series in the wake.

Wave propagation

First, let us focus on the transient wave propagation process within the sample. In order to depict the waves emitted from the alternating cracking process on a graph representing space and time, it is necessary to combine the two spatial dimensions (x and y). of elastic waves stops. Following this phase of arrest, the release of elastic waves continues with the crack tip continuing to propagate dynamically from ∼80 µs onward. During the phase of decreased crack speed (∼90 µs to ∼110 µs), the release of elastic waves is evidently reduced before the last fully dynamic crack propagation goes off and the release of elastic waves is restored.

While the relationship between the release of elastic waves and the increase of dynamic crack propagation is evidenced with the strongest releases at the times of peak crack tip speed, clear bands of rebounding waves (opposite orientation) can be observed. Two main bands are found rebounding from the notch tip (NT) and the top boundary (TB) to eventually interact with the crack at ∼90 µs and ∼130 µs in line with the slow-down of the crack tip and the full arrest at the beginning of the quasi-static phase, respectively. It is worth noticing that no rebounding of waves that potentially causes the Ąrst arrest between ∼40 µs and ∼60 µs is observed. However, considering theoretical wave speeds of ∼2134 m.s -1 for P-waves (see also To improve the separation of the two body waves, the Ąrst stress invariant (I1) and second invariant of deviatoric stress (J2) Ű respectively representative for compressive and shear wave propagation Ű were computed. This has been done following the theory of linear elasticity. The derivative of the displacements with respect to their spatial coordinates yielded strains, while stresses were determined through the application of HookeŠs Law, which relates stress to strain in linear elastic materials consistently with what is adopted in the numerical simulation. The time and space presentation of the two stress components will be presented in full temporal resolution of the simulation (8 Mfps) to improve the visibility of the waves.

compressive waves. By identifying the time the Ąrst compressive wave has been released and the time the wave reaches the notch tip, a wave speed of ∼2048 m.s -1 can be measured. Similarly, in the case of the second invariant of the deviatoric stress (J2), the two time instances result in a measured shear wave speed of ∼1104 m.s -1 . Hence, with almost similar theoretical wave speeds (see Table 5.7), the presence of P and S-waves is evidenced. It is worth mentioning that, due to the measurement of wave speeds relative to crack tip speed, not the exact same wave speeds are found.

Experimental and numerical AE signal comparison

First of all, it is worth mentioning, that the discussion in this section has strongly been in- 1. In-plane strains. By the derivative of the displacement Ąeld with respect to spatial coordinates.

2. Out-of-plane strains. By using the equations of elasticity under the plane-stress assumption, out-of-plane strains are computed through in-plane strains (ε x and ε y ) and PoissonŠs ratio (ν):

ε z = -ν × (ε x + ε y ) (5.7)
.

3. Out-of-plane displacements. Using the material thickness (t=8 mm), out-of-plane displacements (u z ) were estimated through the out-of-plane strains:

u z = 0.5 × t × ε z ( 5

K1 and T-stress: Experiment and simulation comparison

The experimental results are validated by comparing the experimental K1 and T-stress with their simulation counterparts. Apart from the fact that T-stress results stay almost constant in both, experiment and simulation, the two peaks in T-stress Ű indicating the interaction between waves and crack tip Ű are not presented in the simulation. This is strange since crack tip velocity variation and branching have been clearly observed in previous works (e.g. [START_REF] Eid | InĆuence of rate-dependent damage phase-Ąeld on the limiting crack-tip velocity in dynamic fracture[END_REF]133]). However, in the hypothesis of T variation induced by wave/crack tip interaction, a perfect synchronization of the elastic wave with the crack tip advance would be required. Here, the crack tip history is prescribed but only a weak control on wave propagation exists. Thus, the analysis remains rather qualitative showing the possibility that the rebounding waves can reach the crack tip approximately around the right time. For better wave propagation modeling, the following ingredients should be considered in the numerical model: viscoelastic behavior, 3D effects, and wave rebounds on the pins (inĄnitely stiff compared to PMMA). It is also worth mentioning that the Ąrst interaction occurs with waves rebounding on the notch. In the simulation, however, a ŠsharpŠ notch Ű as used during most of the other experimental campaigns Ű was used and not the blunted notch shape, which might also cause deviations.

Conclusion

In this chapter, a dynamic crack propagation phase during an alternate cracking experiment in PMMA was observed by ultra-high-speed imaging and investigated with the help of DIC and post-mortem analysis. Extreme temporal sampling allowed for the detailed measurement of velocity and SIF variations while linking them to particular fracture surface observations. A numerical twin of the experiment was created to support the understanding of the observations.

The main observations can be summed as follows:

Ů ŚForbiddenŠ velocity regime. The upper limiting velocity has been observed to be ∼2.

CONCLUSION & PERSPECTIVES

Conclusion

This comprehensive study delved into the intricate world of acoustic emissions (AE) and fracture mechanics. The research was built around these two topics by using one to support the understanding of the other. The scientiĄc contribution has been divided into two parts:

Investigation of fracture source mechanisms through full-Ąeld imaging and acoustic emission.

Part I

Alternate cracking in PMMA: Investigation of the mechanism.

Part II

For the Ąrst chapter, our investigation involved conducting model experiments on PMMA under both perfect and imperfect mode I loading conditions. In cases of stable but imperfect crack propagation, we observed heterogeneous crack fronts during the latter scenario, resulting in a high density of AE emissions with localized dynamic instabilities. These experiments served as a valuable means to bridge the gap in our understanding of the relationship between fracture mechanisms and their corresponding AE signatures. We used the proper pure mode I experiment as a baseline for comparison, and in this section, we discussed the complexities involved in correlating AE signals with local fracture regimes. Through this multi-modal approach, we aimed to gain a more comprehensive and nuanced understanding of the link between AE signals and their source activity. The following points have been addressed from a technical point of view:

• The ill-posed problem of AE localization without prior knowledge of wave speed, highlighting the sensitivity to initial guess parameters.

• An image analysis procedure has been proposed to capture the position and the kinematics of a complex crack front using in-volume recording in the case of a transparent material connecting macroscopic AE activity with crack front behavior.

• The combination of in-volume observations and apparent crack tip position from LEFM theory and DIC, has shown that the apparent crack tip position is closely related to the surface crack front position and does not reĆect crack front complexity by a through thickness homogenization process.

From a physical point of view, the following points have been discussed:

• A clear link has been established between acoustic emission (AE) activity and the position of the crack front. This connection has been demonstrated through two methods: the detection of the apparent crack tip using DIC/WilliamsŠ series expansion and AE localization.

• Differences observed between the pure mode I experiment and the perturbed mode I experiments were attributed to the link between AE emissions and the heterogeneity in crack front kinematics.

• Various types of AE signals have been detected, including those with high amplitudes (HA) as well as a predominant category of signals with lower amplitudes (LA). Within the LA signals, there are two primary multiplets, which are sets of signals featuring highly correlated impulsive waveform segments.

• A deĄnitive connection between post-mortem fracture surface characteristics, transient crack front movements, multiplets, or a systematic identiĄcation of source mechanisms has not been established. This complexity arises from the challenge of classifying AE signals that exhibit similar amplitudes, especially in a blind non-destructive testing (NDT) approach. A proposal to link AE signals to localized crack front peak velocities, which aligns with the analysis of high-amplitude (HA) signals and macroscopic observations of AE/velocity heterogeneity has been put forward. However, the conclusions drawn from this proposition remained unclear.

While instabilities have been observed locally, the following chapter aimed at their production in a controlled manner. By taking the phenomenon of alternate cracking in PMMA as a tool to produce two types of crack propagation in a single fracture experiment, the investigation focused solely on the AE responses of global instabilities to improve the understanding of the observed local instabilities. The following points were tackled:

• Investigation of large-scale instabilities triggered by the stick-slip phenomenon in PMMA, leading to signiĄcantly higher AE signals.

• Observation of a linear relationship between AE absolute energy of high-amplitude signals and the log of the fracture growth area.

• Suggestion that lower amplitude emissions may be induced by localized small crack growth rather than macroscopic crack advancement.

The Ąrst part of this work focused on AE by using fracture as a tool to produce AE signals.

The opposite applies to the second part, where AE became a means for the investigation of the fracture phenomenon of alternate cracking in PMMA. During the Ąrst chapter of Part II, the following observations can be concluded:

• Examined the "forbidden" velocity regime in the kinetic law of PMMA fracture and iden-tiĄed threshold velocities.

• Explored the relationship between dynamic crack speeds and burst lengths.

• Observed rest bands in dynamic propagation phases, indicating additional intermittent behavior of dynamic propagating cracks.

• DeĄned a critical deceleration distance between dynamic and quasi-static propagation phases.

• Observed particular surface markings, such as Wallner lines and nucleation phases.

• Improved the sampling of the kinetic law of PMMA fracturing, shedding light on the decelerating (transition from dynamic to quasi-static) propagation phase.

In the Ąnal chapter, we addressed open questions and limitations encountered during previous chapters. The access to Ultra-High-Speed (UHS) camera technology allowed us to explore unprecedented ranges of temporal resolution during alternate cracking experiments. However, challenges were found in the unpredictable nature of alternate cracking. Factors like material properties and setup variations introduced uncertainties. We discussed our complex experimental setup and methodology for capturing dynamic phases during alternate cracking, emphasizing the difficulty of precise timing due to the unpredictable nature of these events. The following was observed:

• Further investigated the "forbidden" velocity regime, narrowing the gap between the stable regimes.

• Discovered a clear relationship between crack tip speed and fracture surface roughness.

• Examined rest bands as evidence of intermittent crack behavior during dynamic propagation phases.

• Proposed that crack arrests during dynamic phases result from interactions with selfinduced elastic waves.

• Understanding of experimental observations was improved by a numerical twin.

These Ąndings collectively contribute to a deeper understanding of AE emissions and fracture mechanics in PMMA. The research offers valuable insights into crack propagation behavior, AE signal characteristics, and fracture surface roughness, providing valuable knowledge for future studies in material science and fracture mechanics.

Perspectives

The work in this thesis has contributed to improving the understanding of AE emissions As discussed in the core of the text, we can observe that the same source event can produce a signiĄcantly different signal on each sensor. While all the analysis in the main work has been done on Sensor 2, it shows that performing the analysis on other sensors may lead to slightly different classiĄcation and multiplets detection results. In the present case, Sensor 1 and Sensor 2 are systematically behind the crack front, while Sensor 3 and Sensor 4 are in front. We observe that the main difference, in terms of impulsive response (see to its distinctive feature: the capability to exhibit an alternative mode of failure through a single crack propagation, leading to the development of two distinct controlled crack growth patterns. Furthermore, for the first time, such material/structural cyclic fracture behavior is examined through the lens of linear elastic fracture mechanics (LEFM) by using in-situ High-Speed (HS) and Ultra-High-Speed (UHS) DIC. Energy release rates and crack velocities during fracture experiments are derived from full-field measurements using Williams' series expansion. Fracture surfaces of post-mortem samples have been systematically analyzed using optical microscopy. This work provides key experimental data regarding the improved understanding of a unified theoretical framework of crack instabilities.

52 2. 7 97 4. 4 98 4. 5

 527974985 Marks on the crack surface visualizing the phases of intermittent/alternate crack propagation on a PMMA specimen produced at GeM laboratory (Ecole Centrale de Nantes, France). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Organizational chart of the e-WARNINGs project extracted from the project proposal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Kinetic law (G c -log v curve) for materials showing alternate cracking (stick-slip) behavior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Crack surface recorded with an inclined high-resolution camera through the bulk of a transparent PMMA specimen visualizing the two phases of alternate cracking. 2.1 Sample geometry with the AE sensor positions and their coordinates (red) with respect to the notch tip [mm]. The radius of the notch tip is deĄned by the laser beam diameter of approximately 200 µm. . . . . . . . . . . . . . . . . . . . . . . . 2.2 Schematic representation of complex loading condition. . . . . . . . . . . . . . . . 2.3 Example of stage change method (Static Crack Detection). . . . . . . . . . . . . . 2.4 (a) Sketch of the experimental setup and (b) picture of an unbroken sample mounted in the mechanical device. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Accuracy in crack tip detection by WilliamsŠ series for Ąrst ((a) and (c)) and last ((b) and (d)) frame of the experiment T13. (a) and (b) shows the crack tip detection on the DIC face, with the white rectangle marking the FOV. (c) and (d) shows the in-volume recording, where the DIC crack tip detection is vertically prolonged from the surface equivalent elastic crack tip position by the dashed red line and the crack front (obtained by the crack front extraction algorithm) is highlighted in green. The DIC face is found at the upper side of the crack surface in (c) and (d), while the maximal potential uncertainty in the equivalent crack tip detection related to the two frames is found to be approx. 40 and 45 µm, respectively. 2.6 Map representing the sensitivity of the initial guess (50×50 grid) on the localization error for four random AE source positions. σ denotes the distance between the located source position and its real location (red star). Sensors are presented in scale of the real sensor base area with the area of interest being approx. 40 × 40 mm (see Figure 2.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Median absolute deviation (MAD), i.e. mean identiĄcation uncertainty discarding outliers, on source localization depending on the source position (20×20 grid) for (a) x-direction and (b) y-direction. Sensors are presented in scale of the real sensor base area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.8 Visualization of some AE descriptors. . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.9 Correlation matrix of waveform descriptors presented as a dendrogram. . . . . . 57 2.10 Evaluation of most inĆuential principle component axes. . . . . . . . . . . . . . . 58 2.11 Feature contributions to the (a) Ąrst and (b) second principle component axis. . 58 2.12 Vertical displacement Ąeld [mm] and corresponding WilliamsŠ series projection [mm] within the domain surrounding the crack-tip (deĄned by R min and R max ) of the Ąrst and last recorded image for experiment T13. The used WilliamsŠ series parameters are listed in Table 2.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.13 (a) Normalized crack length [mm] as a function of recorded time [s] and (b) velocity [mm/s] as a function of the normalized crack length [mm]. . . . . . . . 60 2.14 Raw AE streaming data (Amplitude [V] in log-scale vs. Streaming time [s]) of experiments (a) Reference, (b) T5 and (c) T13. . . . . . . . . . . . . . . . . . . . . 61 2.15 HA signals and an example of an LA signal, contributions to the two most dominant principle component axes (PC1 and PC2) and all AE signals presented in the space of Ąrst and second principal components. . . . . . . . . . . . . . . . . . 63 2.16 AE localization of the HA signals of experiment T13. . . . . . . . . . . . . . . . . 64 2.17(a) Cumulative absolute energy (CAE) [aJ] of AE signals as a function of the crack tip position on the fracture surface with the crack front complexity for the two signals related to the dynamic bursts. (b) and (c) shows a detailed view of the particular fracture surface markings associated with dynamic crack propagation. 65 2.18 For experiment T13, (a) and (b) present, as a function of the recording time [s], DIC determined crack tip speed [mm.s -1 ] and the absolute voltage signal of the AE streaming, respectively, while crack front velocities [mm.s -1 ] along the sample thickness are shown in (c). Notice, that white areas at the borders of the velocity map stem from the algorithm not being able to properly detect the edge of the crack front within the volume. . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.19 (a) AE streaming [dB] and the standard deviation (σ) of the velocity map [mm.s -1 ] as a function of the recording time [s] for experiment T13. (b) and (c) present the interframe (16 ms) threshold (26 dB) crossings (counts) as a function of the mean (v) and standard deviation (σ(v)) of the crack front velocity [mm.s -1 ]. 2.20 Microscopic image of fracture surface for (a) reference and (b) complex (T5) cracking case. Crack propagation is from left to right. . . . . . . . . . . . . . . . . 2.21 Normalized waveforms associated to (a) multiplet 1 and (c) multiplet 2, with their corresponding histogram of correlation coefficients ((b) and (d)). . . . . . . 2.22 (a) Multiplets are presented in the space of the Ąrst and second principal component axis of parameters computed from normalized AE signals, while (b) and (c) show the parametric contributions to the Ąrst and second principal component axis, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.23 Average frequency [kHz] as a function of the RA-value (Rise Time over Amplitude) [s.V -1 ]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.24 AE localization of (a) Multiplet 1 and (b) Multiplets 2. . . . . . . . . . . . . . . . 2.25 Crack front complexities related to waveforms associated with (a) Multiplet 1 and (b) Multiplet 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.26 (a) Fracture surface crop over which the crack propagated during the experimental time and (b) corresponding crack front velocities [mm.s -1 ] presented on the extracted crack fronts from the inclined images. . . . . . . . . . . . . . . . . . 3.1 Fracture surface section of the alternate cracking experiment M2-T15. . . . . . . . 3.2 Normalized crack length [-] and amplitude [V] as a function of the recording time for the alternate cracking experiment M2-T15. . . . . . . . . . . . . . . . . . . 3.3 AE localization results for the two dynamic instabilities of experiment M2-T15. . 3.4 The AE absolute energy [aJ] as a function of the burst area (log-scale) for dynamic propagating cracks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Entire displacement Ąeld in y-direction [mm] and its corresponding WilliamsŠ series projection [mm] on the process zone surrounding the crack-tip (deĄned by R min and R max ) of the Ąrst ((a) and (c)) and last recorded image ((b) and (d)) for experiment M2-T15. The used WilliamsŠ series parameters are listed in Table 3.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Normalized crack length (1.0 = 63 mm) as a function of recording time [s] for all Ąve experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Crack tip speed (v) [m.s -1 ] as a function of the normalized crack length (a) [mm]. Black bars present the phase-dependent mean velocity [m.s -1 ], while their value is presented as numbers next to the bar with its standard deviation (std). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example fracture surface showing different particular surface observations. Details 1, 2, and 3 show the nucleation phase, rest band (RB), and the dynamic to quasi-static deceleration phase with the appearance of Wallner lines, respectively. Mean velocity [m.s -1 ] as a function of the burst length [mm] for the dynamic phases of all conducted experiments. The chronological order of the different dynamic bursts for each individual test is given by the numbers next to the data points. Red colored numbers indicate the appearance of at least one rest band during this particular dynamic burst, while the error bars give the variation (std) of the velocities for each dynamic phase. . . . . . . . . . . . . . . . . . . . . . . . . 99 4.6 Velocity [m.s -1 ] as a function of the normalized crack length [mm] for all conducted experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.7 Velocity [m.s -1 ] and K1 [MPa √ m] as a function of crack length [mm] in phase with the microscopic image of the fracture surface for the experiments: (a) M1-T2 and (b) M2-T15. A spatial matching error between DIC and microscope measurements of ∼160 µm and ∼140 µm has to be considered for M1-T2 and M2-T15, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.8 (a) Mean velocity during dynamic crack propagation [m.s -1 ] as a function of e-length (i.e. distance from dynamic micro roughening to Wallner lines) [µm] of the subsequent DYN to QS transition phase for all experiments. Error bars present the measurements on the two free edges of the fracture surface over the sample thickness. The chronological order of the different dynamic bursts for each individual test is given by the numbers next to the data. Red-colored numbers indicate the appearance of at least one rest band during this particular dynamic burst. (b) Example of the measured distance e on the fracture surface. . . . . . . . . . . 103 4.9 Energy release rate (G) [J.m -2 ] as a function of velocity (v) [m.s -1 ] for the alternate cracking phases with its corresponding temporal evolution of the crack for (a) M1-T2 and (b) M2-T19. Reference to the phases of the theoretical kinetic law (see Figure 1.1) are added for better comprehension. . . . . . . . . . . . . . . 105 4.10 Energy release rate (G) [J.m -2 ] as a function of velocity (v) [m.s -1 ] for the alternate cracking and one entirely quasi-static experiment. . . . . . . . . . . . . . . 106 5.1 Sample geometry and sensor placement . . . . . . . . . . . . . . . . . . . . . . . . 113 5.2 Sketch of experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Experimental trigger concept indicating the different deceive activities with the fracture surface as a reference. Only a Ąfth of the entire streaming (100 ms) is presented here since no activity was recorded afterward. . . . . . . . . . . . . . . 5.4 (a) Luminous Ćux [Mlux] as a function of CCD gain (CDS gain of +3db) at different acquisition speeds, with images taken at different CCD gains and acquisitions speeds: (b) 36% and 80 fps, (c) 75% and 160 fps and (d) 100% and 250 fps. . . . 5.5 Twin nodes close to the crack tip at three simulation times steps. Displacements have been ampliĄed by a factor of 20 to improve the visualization of the node separation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 U simu [µm]: (a) horizontal, (b) vertical at t = 19.5 µs. . . . . . . . . . . . . . . . . 5.7 U simu [µm]: (a) horizontal, (b) vertical at t = 19.5 µs. . . . . . . . . . . . . . . . . 5.8 FOV with different magniĄcations (G). . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Reference image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Strain Ąelds [mm.m -1 ] of the three different levels of magniĄcation (G) at three different integration steps. Elastic waves are emitted from the crack tip, which Ąnds itself slightly outside of the FOV in the bottom right corner. . . . . . . . . . 5.11 Entire displacement Ąeld in y-direction [mm] and its corresponding WilliamsŠ series projection [mm] on the process zone surrounding the crack-tip (deĄned by R min and R max ) for 4 time steps (0.0 µs, 42.22 µs, 94.99 µs, and 166.75 µs). The used WilliamsŠ series parameters are listed in Table 5.3. . . . . . . . . . . . . . . . 5.12 (a) Fracture surface Section of the observed dynamic crack propagation with the DIC face on top and (b) Detail 1 of the transition from quasi-static to dynamic propagation highlighting the nucleation phase with the presence of microcracks. Notice that detail 1 is found in the bottom right corner of (a), while QS1, QS2, and RB stand for the quasi-static phase 1, quasi-static phase 2, and rest band, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.13 K1 [MPa.m -0.5 ] and velocity (v) [m.s -1 ] as a function of the crack length in line with the topography measurement of the fracture surface. Again, the DIC face is on top of the fracture surface topography. . . . . . . . . . . . . . . . . . . . . . . . 5.14 Surface roughness (Ra) [µm] of the 500 µm slice of the fracture surface topography at round y = 6 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.15 Crack tip speed [m.s -1 ] and T-stress [MPa] as a function of the crack length in line with the roughness measurement (Ra) [µm] of the 500 µm slice of the fracture surface topography at around y = 6 mm. . . . . . . . . . . . . . . . . . . . . . 5.16 Energy release rate (G) [J.m -2 ] as a function of velocity (v) [m.s -1 ] with its corresponding temporal evolution of the crack. The dynamic phase is split into the part before (red) and the part after (black) the rest band. Triangular and rectangular markers indicate the beginning and the end of a phase, respectively. . 135 5.17 Energy release rate (G) [J.m -2 ] as a function of velocity (v) [m.s -1 ] for all alternate experiments recorded with the HS and UHS camera. . . . . . . . . . . . . 136 5.18 Twin nodes, clamped twin nodes along over the experimental crack length and the crack tip at two different simulation times steps. Displacements have been ampliĄed for by a factor of 80 to improve the visualization of the node separation. 138 5.19 Temporal evolution of the experimental, imposed simulation, and recovered simulation crack length [mm]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.20 Velocity magnitude [m.s -1 ] at different time steps of the simulation. The mesh is deformed according to displacement. Static pre-load has been removed and residual dynamic displacement has been ampliĄed with a factor of 80 for improved visualization of the crack opening. . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 5.21 Ring discretization (dr = 0.5 mm) at the Ąrst time step. LB, BB, TB, and RB stand for the left, bottom, top, and right boundary, respectively, marking the closest boundary points (for potential wave interaction) from the crack tip in each direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.22 Average velocity magnitude [m.s -1 ] over each ring and the detected crack tip speed computed by the time derivative of the crack length (simulation imposed) at each time instance. Red lines indicate the distance to the notch tip (NT), bottom boundary (BB), top boundary (TB) left boundary (LB), and right boundary( RB).143 5.23 At each time instance and average over each ring: (a) The Ąrst stress invariant (I1), where zones of compression and rarefaction are shown in blue and yellow, respectively. A compressive wave speed of ∼2048 m.s -1 is observed. (b) The second invariant of deviatoric stress (J2), where a shear wave speed of ∼1104 m.s -1 is observed. Red lines indicate the distance to the notch tip (NT), bottom boundary (BB), top boundary (TB) left boundary (LB), and right boundary (RB). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 5.24 Transfer function of different AE transducers extracted from Hamam et al. [137]. 147 5.25 Out-of-plane velocities and computed AE signals in time and frequency space for experimental and simulation results. . . . . . . . . . . . . . . . . . . . . . . . . . . 148 5.26 Experiment and simulation comparison of the temporal evolution of (a) K1 and (b) T -stress. The comparison starts at the onset of the dynamic crack propagation of the simulation (∼35 µs). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 LIST OF TABLES1 Advantages and limitations of NDT techniques used in the detection of cracks for fatigue extracted from [9] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 List of experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Camera hardware parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 DIC & WilliamsŠ series parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Standard deviation (uncertainty) added to synthetic source position, time of arrival, and Ąrst guess of velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.5 Information on non-normalized waveforms associated with the multiplets. . . . . 3.1 Second loading stage extension rates applied during the experiments. . . . . . . . 3.2 DIC hardware parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 DIC & WilliamsŠ series parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Amount of rest band appearances during the dynamic bursts. ( * ) means speciĄc roughening of the surface observed, which implies the early formation of a rest band that is yet not fully evolved. However, biased velocity measurements still have to be expected. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Experimental campaign-related timescales. . . . . . . . . . . . . . . . . . . . . . . 5.2 DIC hardware parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 DIC & WilliamsŠ series parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Simulation parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 DIC parameters for performing DIC on the synthetic images . . . . . . . . . . . . 5.6 WilliamsŠ series parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 PMMA properties considered . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 AE parameters computed for the signal in Figure A.1 in chronological order of the sensor number. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  electrical signals are used to evaluate the structural integrity of the material. Monitoring and detection of fatigue with AE were Ąrst proposed in the mid-1970s by Harris et al. [19], where fatigue-crack propagation was investigated on aluminum and steel singleedge-notched samples. Since then, with research mostly focusing on AE signals released close to the Ąnal fracture (Stage (iii)), the early detection of fatigue is still a challenge up to this date. This can generally be explained by the dominance of noise with respect to the signal even in controlled laboratory environments. Signals coming from a small number of events in the Ąrst stages of fatigue are easily lost. The separation of signals emitted by fatigue crack growth from the ones originating from noise sources is widely investigated [25Ű28]. Nevertheless, thorough in-service feasibility of the literature-proposed approaches has not been demonstrated yet [29, 30]. Most recently, Deschanel et al. [29, 31] associated AE responses to fatigue crack growth in metallic materials. Highly similar AE signals have been detected during cyclic fatigue tests at similar stress levels. These signals have a unique characterization distinguishable by their similarity in waveform and are thereby assumed to be emitted by the same source. The AE multiplets are understood as signals initiated by slowly propagating cracks (fatigue stage (ii)) over a sequence of loading cycles. Even though the analysis of multiplets poses a promising way for early detection of stable crack propagation, a clear understanding of the link between recovered AE signals and their actual source activity has yet to be established. This is in line with the main limita-tion in AE analysis: the user must ensure the correct interpretation of the recorded AE signals (see

Figure 2 Ű■■

 2 Figure 2 Ű Organizational chart of the e-WARNINGs project extracted from the project proposal.

Figure 1 . 1 Ű

 11 Figure 1.1 Ű Kinetic law (G c -log v curve) for materials showing alternate cracking (stick-slip) behavior.

Figure 1 . 2 Ű

 12 Figure 1.2 Ű Crack surface recorded with an inclined high-resolution camera through the bulk of a transparent PMMA specimen visualizing the two phases of alternate cracking.

  to detect, process, and analyze acoustic signals. The general overview of the equipment setup for a classical AE device is given below: Ů AE Sensors (Transducers): AE sensors are at the heart of the setup. The sensors convert mechanical vibrations or acoustic waves into electrical signals. They are typically piezoelectric or resonant sensors, capable of detecting high-frequency acoustic emissions. The number and placement of sensors depend on the application and the structure being monitored. Ů Pre-AmpliĄers: With signals produced by the AE sensors being often weak, pre-ampliĄers are used to boost the signal-to-noise ratio. The pre-ampliĄers are connected to the sensors and help improve the sensitivity of the AE system. Ů Data Acquisition System (DA): The data acquisition system Ű typically including analogto-digital converters (ADCs) and data storage capabilities Ű digitizes the ampliĄed AE signals, converting them into digital data that can be processed and analyzed by a computer or dedicated AE analysis hardware. Ů Computer or Analysis Hardware: The digital data from the (DA) system is sent to a computer or dedicated analysis hardware for further processing. Specialized AE software is often used to analyze and interpret the data, including identifying signal characteristics. Ů Mounting Hardware: Depending on the application, AE sensors may need to be securely mounted to the structure being monitored. Mounting hardware can include brackets, adhesives, or other attachment mechanisms.

  To achieve a holistic understanding of source mechanisms during fracture experiments of classical quasi-fragile material, this chapter proposes a multi-modal experimental setup combining different NDT tools. For AE localization, crack tip detection, and determination of crack front complexities, combined time-resolved AE, DIC, and in-volume measurements with postmortem fracture surface analysis are applied to cracking experiments, respectively. Each technique was considered to bypass the need for blind interpretation, while allowing to tie AE signals to fracture surface patterns, i.e. enabling traceability of any captured AE signal to the spatial position of the elastic waveform formation on the fracture surface. Model experiments on PMMA are performed under perfect and imperfect mode I loading conditions leading to stable crack propagation, where heterogeneous crack fronts during the second case generate a high density of AE emissions with localized dynamic instabilities. Such experiments may help the chain of understanding the connection of fracture mechanisms and their AE signature. Compared with the proper pure mode I experiment as a baseline the section discusses the complexity around the ability to correlate AE signals to local fracture regimes.

Figure 2 . 1 Ű

 21 Figure 2.1 Ű Sample geometry with the AE sensor positions and their coordinates (red) with respect to the notch tip [mm]. The radius of the notch tip is deĄned by the laser beam diameter of approximately 200 µm.

Figure 2 .Figure 2 . 2 Ű

 222 Figure 2.2 presents a schematic description of the impact of the clearance angles during loading.

of 1 1 .

 11 kN maximum loading capacity. The room temperature was controlled at 21°C. To achieve an experimental consistency of the crack line by avoiding inĆuences of the notch tip on the subsequent cracking and the point of initiation for different cracking experiments, a two stages loading procedure was established: Controlled quasi-static crack propagation. During this stage a relatively slow extension rate of 0.01 mm.s -1 was imposed. Stage change has been executed with the help of the Static Break Detector option within the Instron WaveMatrix software (Figure 2.

Figure 2 . 3 Ű

 23 Figure 2.3 Ű Example of stage change method (Static Crack Detection).

M/C 30 )Figure 2 . 4 Ű

 3024 Figure 2.4 Ű (a) Sketch of the experimental setup and (b) picture of an unbroken sample mounted in the mechanical device.

4 )

 4 with µ and κ being the shear modulus and KolossovŠs constant, respectively. KolossovŠs constant under plane stress is κ = (3 -ν)/(1 + ν) with ν being the PoissonŠs ratio. Equation2.2 states, that displacements can be computed for an inĄnite sum of modes. However, limiting the solution to n min = -3 and n max = 7 is sufficient to retain the relevant crack features. For quasi-brittle medium, good agreement of the mechanical Ąelds can be established outside the process zone of the crack for the following WilliamsŠ series coefficient (A n i ) solutions: n = 0, in-plane rigid body translations n = 1, asymptotic terms K I and K II n = 2, T-stress and in-plane body rotations

5 )with K 1

 51 and E being the SIF related to mode I and YoungŠs Modulus, respectively. DIC and WilliamsŠ series projection results have been obtained entirely by accessing the library of the open-source software UFreckles [79].

6 . 7 . 8 . 9 .

 6789 Boundary: Tracing the boundaries of each object to deĄne their pixel coordinates by the MATLAB function bwboundaries, which is based on the Moore-Neighbor tracing algorithm modiĄed by JacobŠs stopping criteria [81]. Exclusion: Excluding tiny objects with a minimum amount of associated pixels. Reconstruction: Rebuilding the crack front mask based on the different objects. Finishing: Recovering a clean crack front by only considering the most advanced position of the rough crack front mask. 10. Smoothing: Applying a moving median Ąlter to reduce the inĆuence of outliers along the crack front.

  recordings of transparent PMMA plates with the back-face inclined camera. Thanks to the temporal synchronization of the devices, each DIC frame had a corresponding in-depth image visualizing the crack surface at this time. By taking the pixel-to-meter conversion values of the two cameras into account, spatial matching was performed by taking the notch tip as a common reference point. However, the comparison of data obtained by techniques with different spatial resolutions is in need of harmonization. The most dominant error sources related to the process of spatial matching are the manual selection of the notch position (σ notch ) and the pixel-to-meter conversion (Špix2mŠ) uncertainty (σ pix2m ). Due to camera resolution, used lens, and the clearance angle of the sample cut, the uncertainty on the notch tip localization has been evaluated to be about σ notch = 20 pix. Pixel-to-meter conversion values have been identiĄed by analyzing ruler images taken before each test, with the help of the Fast Fourier transform (FFT) approach.

6 )Figure 2 . 5 Ű

 625 Figure 2.5 Ű Accuracy in crack tip detection by WilliamsŠ series for Ąrst ((a) and (c)) and last ((b) and (d)) frame of the experiment T13. (a) and (b) shows the crack tip detection on the DIC face, with the white rectangle marking the FOV. (c) and (d) shows the in-volume recording, where the DIC crack tip detection is vertically prolonged from the surface equivalent elastic crack tip position by the dashed red line and the crack front (obtained by the crack front extraction algorithm) is highlighted in green. The DIC face is found at the upper side of the crack surface in (c) and (d), while the maximal potential uncertainty in the equivalent crack tip detection related to the two frames is found to be approx. 40 and 45 µm, respectively.

3 Ů 1 Ů

 31 13) Uncertainty To evaluate the performance of the AE source detection and identify possible weak spots in the method, localization has been performed on synthetic AE signals. To mimic the localization of AE signals, the time of arrival (TOA) of the synthetic AE signals for all four sensors is analytically computed. For this, the following elements were considered: Ů YoungŠs modulus (E) = 3.5 GPa Ů Material density (ρ) = 1200 kg.m -Velocity (v) = E/ρ = 1707.8 m.s -Distance to the sensors i (d i ) = (x i -x s ) 2 + (y i -y s ) 2 , with (x i ,y i ) and (x s ,y s ) being the sensor and source coordinates, respectively. Ů Time of arrival (TOA i ) = d i /v In the following, (a) the inĆuence of the initial guess position on the localization uncertainty for various random source positions and (b) the inĆuence of synthetic AE signals corrupted by expected experimental biases are investigated.

Figure 2 . 6 Ű

 26 Figure 2.6 Ű Map representing the sensitivity of the initial guess (50×50 grid) on the localization error for four random AE source positions. σ denotes the distance between the located source position and its real location (red star). Sensors are presented in scale of the real sensor base area with the area of interest being approx. 40 × 40 mm (see Figure 2.1)

  decreases a lot with the propagation time going closer to the uncertainty Ćoor. Furthermore, identiĄcation of the TOA through cross-correlating waveforms Ű to achieve sub-resolution accuracies Ű was not possible with waveform characteristics changing from one sensor to the other (see Part A). While regular threshold crossing and Ćoating threshold crossing are the most basic methods of TOA detection, more advanced methods are proposed in the literature. Cheng et al.[START_REF] Cheng | Acoustic emission source location using Lamb wave propagation simulation and artiĄcial neural network for I-shaped steel girder[END_REF] compared such methods in the case of I-shaped steel girder. With large differences in peak amplitude, the TOA was determined in the presented work through a dynamic threshold and veriĄed by the Akaike Information Criterion (AIC) method[START_REF] Hensman | Locating acoustic emission sources in complex structures using Gaussian processes[END_REF]. The dynamic threshold was deĄned as twice the standard deviation of the absolute volt amplitude. It was additionally possible to control and potentially adapt the TOA, due to the rather manageable amount of AE hits (being initially generously extracted by a Ąxed threshold).Various AE source locations were tested to evaluate the performance of the localization algorithm with respect to different areas of the Ąeld of interest.

Figure 2 .

 2 [START_REF] Alwi | Fault Detection and Fault-Tolerant Control Using Sliding Modes Series Title: Advances in Industrial Control[END_REF] is showing the uncertainty (in x and y direction) of the source localizations spanning a 20×20 grid.

Figure 2 . 7 Ű

 27 Figure 2.7 Ű Median absolute deviation (MAD), i.e. mean identiĄcation uncertainty discarding outliers, on source localization depending on the source position (20×20 grid) for (a) xdirection and (b) y-direction. Sensors are presented in scale of the real sensor base area.

Ů

  Energy -time integral of the absolute signal voltage. The reported magnitude depends on the value selected for Energy Reference Gain. Proportional to Signal Strength. Ů Counts -number of times the signal crosses the detection threshold. Ů Duration -time from Ąrst to last threshold crossing (µs). Ů RMS -root mean square voltage during a period of time-based on a software programmable time constant, referred to as the input to the signal processing board. Ů ASL -RMS, converted to the dB AE scale (0 dB AE = 1 µV at the sensor, before any amp-liĄcation). Ů Threshold -detection threshold, on the dB AE scale. Ů Rise Time -the time from the Ąrst threshold crossing to the highest voltage point on the waveform (µs). Ů Counts to Peak -number of threshold crossings from Ąrst to highest voltage point on the waveform. Ů Average Frequency -Counts divided by Duration, divided by 1000 (thus, kHz). Note that this is not a spectral domain calculation, but a calculation from time domain features. Ů Reverberation Frequency -(Counts -Counts to Peak) divided by (Duration -Risetime). Ů Initiation Frequency -Counts to Peak divided by Risetime. Ů Signal Strength -time integral of the absolute signal voltage, expressed in pVs (picovoltseconds) referenced to the sensor, before any ampliĄcation. Proportional to Energy. Ů Absolute Energy -time integral of the square of the signal voltage at the sensor before any ampliĄcation, divided by a 10 kΩ impedance and expressed in aJ (attojoules).

Figure 2 . 8 Ű

 28 Figure 2.8 Ű Visualization of some AE descriptors.

Figure 2 . 10 ŰFigure 2 . 11 Ű

 210211 Figure 2.10 Ű Evaluation of most inĆuential principle component axes.

  fracturing of TDCB-shaped PMMA samples was investigated through DIC, in-volume recordings, microscopy, and AE. The time of interest for each experiment was 5 seconds during which, based on a common trigger (see Section 2.2.3), all time-dependent devices were recorded simultaneously. Microscopic images of the fracture surface were obtained by a numeric microscope (Keyence) with a spatial resolution of 2.53 µm.pix -1 . With DIC being a surface measurement tool, images obtained a posteriori by the inclined camera provide in-volume understandings of crack front complexities to account for deviations of the crack tip location measured on the DIC face to the crack front within the volume. Eventually, AE localization is performed to conĄrm that AE signals are actually emitted within the vicinity of the crack front and thereby avoid any misconceptions by potential environmental noise. DIC displacement Ąelds and their corresponding WilliamsŠ series projections are presented in Figure 4.1. As introduced earlier, two propagation scenarios Ű a reference (smooth model crack propagation) and two complex cracking experiments (closer to real-case life cracking scenario) Ű have been investigated from a surface kinematics and acoustic emissions point of view. More precisely, a pure mode I fracture experiment is denoted as a reference, with T5 and T13 presenting complex fracturing experiments. Let us start from a macroscopic perspective. By presenting the three experiments in the space of normalized crack length as a function of experimental time (Figure 2.13(a)) and velocity as a function of the normalized crack length (Figure 2.13(b)), their difference in cracking behavior is highlighted.

Figure 2 . 12 ŰFigure 2 .

 2122 Figure 2.12 Ű Vertical displacement Ąeld [mm] and corresponding WilliamsŠ series projection [mm] within the domain surrounding the crack-tip (deĄned by R min and R max ) of the Ąrst and last recorded image for experiment T13. The used WilliamsŠ series parameters are listed in Table 2.3.

Figure 2 .

 2 Figure 2.14 Ű Raw AE streaming data (Amplitude [V] in log-scale vs. Streaming time [s]) of experiments (a) Reference, (b) T5 and (c) T13.

Firstly,

  AE data will be investigated independently by cropping AE waveforms from the streaming and the subsequent computation of AE descriptors from those waveforms. By solely looking at the AE streaming output of experiments T5 and T13, strongly differing signals can already be identiĄed within the set of AE responses. With peak amplitudes almost reaching 10 V (100 dB) (see Figure 2.14(b) and (c)), these signals present a great variation to the average peak amplitude of approx 0.01 V (40 dB) found for all other signals. To understand and identify their origin, we will focus on experiment T13, where two signals (at 2.1 s and 4.1 s) with a peak amplitude of 10 V are detected. To help analyze and classify the AE activity during this experiment, the Principle Component Analysis (PCA) Ű by taking all AE descriptors into consideration except Time and Threshold Ű has been performed on the entire set of recovered AE signals (see Section 2.3.4 for detailed information on the procedure). Underlining the discrepancy, Figure 2.15 shows both high amplitude (HA) signals in comparison with an example waveform of the lower amplitude (LA) signals, and the entire set of signals being presented in the space of the most dominant principle component axis and their AE descriptor contributions.While the differences in waveform characteristics are easily visible by comparing the two HA signals to the example of an LA signal, it is worthwhile mentioning that similarities comparing the two HA signals are found to be rather small. This becomes further evident by comparing all signals in the space of the principal component axis. Indeed, three signals are found to be relatively far from the main cluster of signals that cover the space from approx. -5 to 5 along PC1 and PC2. Among these signals, the two pre-deĄned high amplitude signals are the ones deviating the most from the rest. Their origin is investigated in detail in the following section.

Figure 2 . 15 Ű

 215 Figure 2.15 Ű HA signals and an example of an LA signal, contributions to the two most dominant principle component axes (PC1 and PC2) and all AE signals presented in the space of Ąrst and second principal components.

Figure 2 .

 2 Figure 2.16 Ű AE localization of the HA signals of experiment T13.

Figure 2 .Figure 2 .

 22 Figure 2.17 shows (a) the spatial evolution of the cumulative absolute energy of the AE activity and the crack front at the instant of the two HA signals on the fracture surface, as well as (b) and (c) detail views of the fracture surface.

Figure 2 .

 2 17 (b) and (c) show that the surface roughness strongly differs locally with triangular areas emerging from the side opposite to the DIC face. Consider-

Figure 2 .Figure 2 . 18 Ű

 2218 Figure 2.18 Ű For experiment T13, (a) and (b) present, as a function of the recording time [s], DIC determined crack tip speed [mm.s -1 ] and the absolute voltage signal of the AE streaming, respectively, while crack front velocities [mm.s -1 ] along the sample thickness are shown in (c).Notice, that white areas at the borders of the velocity map stem from the algorithm not being able to properly detect the edge of the crack front within the volume.

Figure 2 .

 2 Figure 2.18(c) shows, for all x positions, the crack front velocity within the 8 mm thickness. The curvature and complexity of the front at each time are visible by the shape of the successive proĄles. Some strong local acceleration can be observed within the volume. Especially, the two dynamic instabilities are clearly visible on the (bottom) in-volume face (side being under compression) at around 16 mm and 27 mm. The increase in velocity, being already observed in

Figure 2 .Figure 2 .

 22 Figure 2.13(b), is also clearly visible on the (top) DIC face. By comparing Figure 2.18(a) and (b), one can observe that the DIC crack speed increases (Figure 2.18 (a)) at 2.2 s appears, in time, approx. 100 ms after the detection of the local instability in the AE streaming (Figure 2.18 (b)). This can be explained by the spatial position of the dynamic burst along the depth of the sample. Taking Figure 2.18(c) into consideration, local dynamic instabilities can be identiĄed as crack front jumps on the inclined camera face at around 16 mm and 27 mm. Hence, the Ąrst peak in the DIC crack velocity can not be associated directly with the instability itself, but rather with the crack front catching up belated with the strong local advancement. It is worth mentioning, that the DIC-side velocities retrieved from the crack front kinematics do match the apparent crack tip speed captured by DIC. However, to enhance visualization of local velocity variations, the presented velocity map in Figure 2.18(c) is limited to an upper bound of 15 mm.s -1 . Thus, velocity peaks around the local instabilities, and their secondary impact on the opposite side are saturated. Nevertheless, velocities associated with the local instabilities have to be taken with caution, since sudden and strong velocity variations can not be appropriately captured with the temporal resolution of the camera (16 ms inter-frame). Hence, it has to be expected that the dynamically related velocity variations are cropped in amplitude (see actual velocities of instabilities in Part II). In the vicinity of the second local instability (4.1 seconds), Figure 2.18 (b) presents a more homogeneous increase in crack front velocity along the sample thickness covering up potential secondary effects on the DIC face. General observations of the crack front kinematics shown in Figure 2.18(b) present a rather homogeneous crack front speed for about 2 s until the appearance of the Ąrst local instability. Thus, it seems that this local instability produces a disturbance of the crack front behavior resulting in global, successive accelerations and declarations of the crack front with local velocity variations. Finally, Figure 2.19(a) presents, as a function of the recording time, both crack front velocity heterogeneity (computed as the standard deviation (σ) of the velocity along the crack front at every time) and the AE streaming of experiment T13. A clear correlation can be observed. Apart from some AE activity and tiny velocity variations, the majority of AE activity and velocity variations are found after the Ąrst dynamic instability at 2.1 seconds. Particularly around 2.1, 2.9, 3.5, and 4.1 seconds, a good temporal agreement can be observed between the increase in crack front velocity heterogeneity and the density of the AE activity. Underlining this correlation, Figure 2.19(b) and (c) present the inter-frame threshold crossings (counts), i.e. cumulative counts within a window of 16 ms, as a function of the mean and standard deviation of the velocity along the crack front, respectively.

Figure 2 .

 2 19(b) clearly shows a separation of the homogeneous crack front phase with low AE activity (i.e. small number of interframe counts) from the heterogeneous crack front phase with high AE activity (i.e. high number of inter-frame counts). However, the separation of these phases is not only found in time with the appearance of the local instability as disruption of the balanced crack front propagation but also by a crack front mean velocity (v) threshold of about 4 mm.s -1 . With the interframe count as a function of the crack front variations (σ(v)), Figure2.19(c) presents a different angle on the correlation deĄned in Figure2.19(a). A more heterogeneous crack front propagation results in a higher density of AE activity. Again, taking the experimental time into consideration, the Ąrst dynamic instability is found in the transition between the two phases.

Figure 2 . 20 Ű

 220 Figure 2.20 Ű Microscopic image of fracture surface for (a) reference and (b) complex (T5) cracking case. Crack propagation is from left to right.

Figure 2 .Figure 2 . 21 Ű

 2221 Figure 2.14(b) and (c)). At this stage, it seems reasonable to say that the change in AE activity is mainly due to the discrepancy observed at the macro-scale, i.e. scratching marks. In the following, LA signals will be investigated by waveform cross-correlation (see Section 2.3.4) with the aim of identifying AE signals, as in the case of HA signals, being linked

Figure 2 .

 2 Figure 2.22(a) shows all signals, with the two multiplets superimposed in green and red, in the space of the Ąrst and second principal component axis of the normalized AE descriptors, with the contributions to the two principle components shown in Figure 2.22(b) and Figure 2.22(c), respectively. In the present case, due to normalization, some meaningless features have been removed from the analysis: Time and Threshold (as in the previous analysis of HA signals in Section 2.4.2), as well as Energy, Signal Strength, and R-frequency (see Section 2.3.4 for feature details). In Figure 2.22(a), clear multiplet separation is mainly observed along the Ąrst principal axis. Hence, taking Figure 2.22(b) into consideration, differentiation between AE signals associated with the two multiplets is mainly driven by rise time, count, and duration due to the clear horizontal separation. However, it is worth mentioning that a classiĄcation, similar to multiplet, could not be achieved using either k-means or DBSCAN unsupervised classiĄcation methods. Different explanations could be proposed for such observation:

Figure 2 .Figure 2 .

 22 Figure 2.22 Ű (a) Multiplets are presented in the space of the Ąrst and second principal component axis of parameters computed from normalized AE signals, while (b) and (c) show the parametric contributions to the Ąrst and second principal component axis, respectively.

Figure 2 . 23 Ű

 223 Figure 2.23 Ű Average frequency [kHz] as a function of the RA-value (Rise Time over Amplitude) [s.V -1 ].

Figure 2 .

 2 Figure 2.23 allows solely for highlighting the presence of two categories of AE signals, where the two multiplets are only represented in one.

Figure 2 . 24 ŰFigure 2 . 25 Ű

 224225 Figure 2.24 Ű AE localization of (a) Multiplet 1 and (b) Multiplets 2.

Figure 2 .

 2 26(b)) at the spatial position of the dynamic instability along the thickness of the sample. Noteworthy, the location of the dynamic instabilities has only been identiĄed on the DIC face up to this point. Its correct fracture surface location was detected through pre-existing knowledge of fracture surface patterns associated with dynamic instabilities. However, by taking the above into consideration, before made assumptions have been evidenced. With respect to AE signals being related to the multiplets, one can observe that all signals appear during the more ŠresponsiveŠ section of the streaming (Figure 2.19) after the Ąrst dynamic instability. While the crucial information provided by the through-thickness crack front growth rate heterogeneity seems to correlate the signals to local perturbations (scratches/artifacts) observed on the surface, it did not provide an unambiguous explanation for the natural multiplet classiĄcation.

1 . 2 .

 12 signed and studied through a multiple-point view perspective. With the aim of linking source mechanisms of AE signals to post-mortem fracture surface characteristics, the combination of devices allowed for a holistic investigation of the fracturing process. DIC, in-volume recordings, and the combination of multiple AE sensor was used for crack tip detection, identiĄcation of crack front complexities/kinematics, and AE source localization, respectively. The analysis relies on both, automatic crack tip detection using DIC and Williams series expansion, as well as AE source localization.The present work provides (1) a methodological/numerical and (2) physical contribution that is summarized in the following: On the methodological side, it has been shown that:Ů The ill-posed problem of AE localization, without a priori knowledge of the wave speed, using multiple AE sensors is highly sensitive to the initial guess when solving it with classical Newton-Raphson algorithms. This issue can be partly overcome by designing an optimization procedure for the initial guess. It has been done to Ąnd for each crack tip position the best initial guess parameters for convergence using a simple numerical twin. Eventually, for the sensor position considered in this work (not optimized), a localization uncertainty lower than 2 mm can be achieved for most of the crack propagation.Ů An image analysis procedure has been proposed to capture the position and the kinematics of a complex crack front using in-volume recording in the case of a transparent material. It allows for capturing local variation of the crack front kinematics, which has been found highly valuable for connecting macroscopic AE activity and crack front behavior. Ů The combination of in-volume observations and apparent crack tip position from LEFM theory and DIC, has shown that the apparent crack tip position is closely related to the surface crack front position and does not reĆect crack front complexity by a through thickness homogenization process. This point remained unclear in the literature associated with crack tip detection using WilliamŠs series expansion and DIC. On the physical interpretation side, the following points have been observed: Ů A systematic connection between AE activity and crack front position has been demonstrated using both apparent crack tip detection, through DIC/WilliamsŠ series expansion, and AE localization. Ů Combining AE streaming and in-volume crack front kinematics data, a strong correlation between the amount and intensity of acoustic emission and heterogeneity of the crack front kinematics has been observed. Indeed, while homogeneous crack front velocity does not produce AE signals, at least high enough to leave the noise Ćoor, once the crack front kinematics become heterogeneous, strong AE activity occurs. It explains the difference in terms of AE activity observed between pure mode I and perturbed mode I experiment in PMMA. This observation is highly valuable since it demonstrates that post-mortem analysis could be misleading in term of AE source mechanism analysis since it does not necessarily reĆects the transient kinematics of the front. Ů Different families of AE signals have been observed: High amplitude (HA), a dominant class of lower amplitude (LA) signals, and among them two main multiplets, i.e. a set of signals having highly correlated impulsive waveform parts. Ů Two important multiplets have been observed composed of 15 and 8 successive AE signals with highly correlated impulsive waveform parts. While their impulsive parts clearly characterize them, it has been found difficult to Ąnd a similar classiĄcation using standard procedures such as k-means or DBSCAN clustering algorithms of the waveforms based on AE descriptors in their dominant PC space. Indeed most of the LA signals share close characteristics, and considering the entire waveform instead of only the impulsive part makes classiĄcation difficult. Ů No clear relationship between post-mortem fracture surface features, transient crack front kinematics, and multiplets, or systematical identiĄcation of source mechanisms has been found, which reĆects the complexity of classifying (especially in a blind NDT approach) AE signals sharing similar AE amplitudes. A proposition has been made, linking AE to localized crack front peak velocities in line with the HA signal analysis and AE/velocity heterogeneity observations done macroscopically. However, conclusions remain vague. This section has demonstrated that combining different investigation tools Ű surface and involume imagine, AE and post-mortem analysis Ű AE sources and the crack front can be undoubtedly connected for AE signals with extreme characteristics such as dynamic instabilities. However, the classiĄcation of LA signals, dominant during quasi-static crack propagation, remains highly difficult to connect unambiguously to a particular source mechanism even for highly correlated waveforms successively reappearing along 15 mm of crack propagation. We do believe that such lab experiments may help the design of the NDT algorithm discriminating in real-time different AE sources.

Figure 3 . 1 Ű

 31 Figure 3.1 Ű Fracture surface section of the alternate cracking experiment M2-T15.

Figure 3 . 2 Ű

 32 Figure 3.2 Ű Normalized crack length [-] and amplitude [V] as a function of the recording time for the alternate cracking experiment M2-T15.

Figure 3 . 3 Ű

 33 Figure 3.3 Ű AE localization results for the two dynamic instabilities of experiment M2-T15.

Figure 3 . 4 Ű

 34 Figure 3.4 Ű The AE absolute energy [aJ] as a function of the burst area (log-scale) for dynamic propagating cracks.

Figure 3 .

 3 Figure 3.4 hence suggesting a linear relationship between the AE absolute energy and the log of the crack propagation area produced dynamic cracks. Results are presented for one sensor, while no attenuation is expected due to small travel distances. However, it is worthwhile mentioning that the established relationship does not reĆect the complete fracture energy. Retrieved data solely relies on the punctual measurement of the AE transducer. Thus, AE absolute energy measurements Ű computed as the time integral of the squared voltage signal Ű bear a proportionate and directional bias, while also relying on the intrinsic transfer (potentially complex) function of the sensor. In an attempt to understand the observation discussed in Section 2.4.4, where no clear macroscopic systematic Ąngerprint is observed at each crack front position associated with multiplets emissions, an extrapolation of Figure 3.2(b) is proposed. Indeed, following the observed linear relationship between AE absolute energies and the log of the growth fractured area in the case of dynamic instabilities (HA signals either macro or localized) and assuming similar mechanisms but at a much smaller scale, an expected crack surface growth associated to multiplet signals can be estimated. With a difference in the dimension of millions between the AE absolute energy of LA signals (below 235 aJ, see Table 2.5) and HA signals (above 2×10 6 aJ), propagation

  With the Ąrst part using alternate cracking as a model experiment for the investigation of source mechanisms and their AE activity, this part will explore alternate cracking from a fracture mechanics point of view. In this section, the compromised experimental design (see Section 3.2.3) allows for the investigation of the reoccurring appearance of quasi-static and dynamic crack propagation during one fracture process. Better estimations of the ŚforbiddenŠ velocity regime, considering in-process viscous effects (history-dependent), can be accomplished. Improved velocity measurements through high temporal and moderate spatial resolution can be achieved with up-to-date high-speed imaging. Energy release rate estimations beneĄt from direct measurements of crack tip positions and linear elastic fracture mechanic (LEFM) parameters through full-Ąeld measurements using digital image correlation (DIC) and WilliamsŠ series expansion.

  Five tests have been taken into con-sideration coming from two different experimental campaigns (M1 and M2). A representative for all experiments, full-Ąeld displacement measurements (in the y-direction) for the Ąrst and last recorded frame superimposed on the appropriate image of experiment M2-T15 are presented in Figure 4.1(a) and Figure 4.1(b), respectively. In the same conĄguration, corresponding WilliamsŠ series projection areas are presented in Figure 4.1(c) and Figure 4.1(d).

Figure 4 . 1 Ű

 41 Figure 4.1 Ű Entire displacement Ąeld in y-direction [mm] and its corresponding WilliamsŠ series projection [mm] on the process zone surrounding the crack-tip (deĄned by R min and R max ) of the Ąrst ((a) and (c)) and last recorded image ((b) and (d)) for experiment M2-T15. The used WilliamsŠ series parameters are listed in Table 3.3.

Figure 4 . 2 Ű

 42 Figure 4.2 Ű Normalized crack length (1.0 = 63 mm) as a function of recording time [s] for all Ąve experiments.

Figure 4 .Figure 4 . 3 Ű

 443 Figure 4.3 Ű Crack tip speed (v) [m.s -1 ] as a function of the normalized crack length (a) [mm]. Black bars present the phase-dependent mean velocity [m.s -1 ], while their value is presented as numbers next to the bar with its standard deviation (std).

Figure 4 . 4 Ű

 44 Figure 4.4 Ű Example fracture surface showing different particular surface observations. Details 1, 2, and 3 show the nucleation phase, rest band (RB), and the dynamic to quasi-static deceleration phase with the appearance of Wallner lines, respectively.

Figure 4 . 5 provides

 45 a relation between crack speeds and their corresponding propagating length.

Figure 4 . 5 Ű

 45 Figure 4.5 Ű Mean velocity [m.s -1] as a function of the burst length [mm] for the dynamic phases of all conducted experiments. The chronological order of the different dynamic bursts for each individual test is given by the numbers next to the data points. Red colored numbers indicate the appearance of at least one rest band during this particular dynamic burst, while the error bars give the variation (std) of the velocities for each dynamic phase.

Figure 4 . 6 Ű

 46 Figure 4.6 Ű Velocity [m.s -1 ] as a function of the normalized crack length [mm] for all conducted experiments.

Figure 4 . 7 ŰFigure 4 . 8 Ű

 4748 Figure 4.7 Ű Velocity [m.s -1 ] and K1 [MPa √ m] as a function of crack length [mm] in phase with the microscopic image of the fracture surface for the experiments: (a) M1-T2 and (b) M2-T15. A spatial matching error between DIC and microscope measurements of ∼160 µm and ∼140 µm has to be considered for M1-T2 and M2-T15, respectively.

Figure 4 . 9 Ű

 49 Figure 4.9 Ű Energy release rate (G) [J.m -2 ] as a function of velocity (v) [m.s -1 ] for the alternate cracking phases with its corresponding temporal evolution of the crack for (a) M1-T2 and (b) M2-T19. Reference to the phases of the theoretical kinetic law (see Figure 1.1) are added for better comprehension.

Figure 4 . 10 Ű

 410 Figure 4.10 Ű Energy release rate (G) [J.m -2 ] as a function of velocity (v) [m.s -1 ] for the alternate cracking and one entirely quasi-static experiment.

- 1 .ŮŮ

 1 Deceleration phase occurring during the transition of dynamic to quasistatic propagation. The inĆuence of dynamically propagating cracks arriving at the quasi-static phase on the fracture surface led to the deĄnition of a critical deceleration distance between the two phases. Here, an almost linear trend is found between the mean crack tip speeds of the dynamic burst and the speciĄc deceleration distance. It is found between 115 and 615 µm for mean velocities ranging from 78 to 106 m.s -1 . It points to the fact that the crack tip undergoes a deceleration of about 10 7 m.s -2 . Particular surface markings. Two particular PMMA-related fracture surface patterns, Wallner lines, and nucleation phases have been observed. The second one, only appearing at the beginning of the dynamic bursts, suggests crack speed phases beyond the velocities measured during the experiments. This can be explained by the limits of the chosen interframe. Ů Improved sampling of the kinetic law of PMMA fracturing in the vicinity of instability. The kinetic law for fracturing in PMMA was established by supplementing the results of this work with experimental data from existing literature. Here, experimental data of the interaction of reoccurring dynamic and quasi-static phases during alternate cracking experiments was provided. For the Ąrst time, direct velocity and SIF measurements have been experimentally identiĄed for both phases during the same alternate cracking experiments. Results led to a proposed adjustment of the classical description of alternate cracking regarding the transition from dynamic to quasi-static propagation (path along reference points B-C-D-AŠ in Figure 1.1.1). On one hand, in view of the chosen extension rates to induce alternate recurrence of the transitions, different mechanisms could have been activated. On the other hand, the transition of dynamic to quasi-static propagation with extreme decelerations (potentially being outside of the framework of [47])

Figure 5 . 1 Ű

 51 Figure 5.1 Ű Sample geometry and sensor placement

2 . 3 . 1 . 3 . 4 . 5 .

 231345 Distortion model: Distortion modeling is performed on the calibration images based on Zernike polynomials. Camera model: A statistical camera model is built upon multiple calibration shots. Experiment stage Extrinsic parameter correction: A static image is being used to update the camera model. 2. DIC: DIC is performed on the experimental images. Distortion deconvolution: DIC results are being corrected for the distortions with the before-deĄned camera model. Scanning artifacts: With reference images being acquired by a standard scanner, scanning artifacts (classically vertical periodic displacement bands related to scanner screw step size) have to be suppressed in the wake. Effective kinematic data: Distortion-free kinematic data can be retrieved.

Figure 5 Figure 5 . 2 Ű

 552 Figure 5.2.

Figure 5 . 3 Ű

 53 Figure 5.3 Ű Experimental trigger concept indicating the different deceive activities with the fracture surface as a reference. Only a Ąfth of the entire streaming (100 ms) is presented here since no activity was recorded afterward.

  Continuous light systems by RELInc provide focus LED rings up to 10 Mlux. To understand the applicability of such equipment in the context of our camera, a sample was continuously lighted by the model light of the two Ćashes (i.e. the included continuous light in Ćashes used to set up experiments Ű max. ∼100 klux). The relationship between luminous Ćux and camera gains was established, by sweeping over the CCD and CDS gain of the camera for a different level of luminous Ćux (from 100 klux to 2.7 klux, measured using a Speedmaster Sekonic lux meter), while using a large and constant exposure time (>1 ms) and keeping the image histogram constant (same mean value and full 14 bit dynamic histogram coverage). Upscaling the values for shorter exposure times (associated with 80, 160, and 250 kfps) enabled the determination of reachable image qualities and limits in terms of achievable frame rates. The luminosity as a function of the CCD gain at different acquisition speeds, next to the images taken at the speciĄc CCD gains, are presented in Figure 5.4. A reference line is drawn to indicate the 10 Mlux produced by the RELInc continuous ring light. It is worth mentioning, that the presented data and conclusions are only valid for the Cordin-580 camera system.

Figure 5 .

 5 Figure 5.4(a) shows the amount of light required to take properly exposed images (i.e. histogram covering the entire 14-bit dynamic) at different acquisition speeds. Crossing points between the 10 Mlux line and the curves demonstrate the smallest possible CCD gain to acquire an image lighted by 10 Mlux at the speciĄc acquisition speed. An image taken at each of these crossing points and thus the CCD gain enhanced noise is presented in Figure 5.4 (b), (c), and (d). Hence, even by ignoring the strong noise introduced by a CCD gain of 100, only a maximal acquisition speed of 250 kfps could be reached with the 10 Mlux provided by the RELInc ring light system. Notice, it may be possible to compromise also on the image histogram, i.e. shrinking the data within only the Ąrst 10 bits, which would possibly allow increasing the frame rate to 400 kfps. However, this would strongly affect the DIC results.

Figure 5 . 4 Ű

 54 Figure 5.4 Ű (a) Luminous Ćux [Mlux] as a function of CCD gain (CDS gain of +3db) at different acquisition speeds, with images taken at different CCD gains and acquisitions speeds: (b) 36% and 80 fps, (c) 75% and 160 fps and (d) 100% and 250 fps.

Figure 5 . 5 Ű

 55 Figure 5.5 Ű Twin nodes close to the crack tip at three simulation times steps. Displacements have been ampliĄed by a factor of 20 to improve the visualization of the node separation.

Figure 5 . 6 Ű

 56 Figure 5.6 Ű U simu [µm]: (a) horizontal, (b) vertical at t = 19.5 µs.

Figure 5 .Figure 5 . 7 Ű

 557 Figure 5.7 Ű U simu [µm]: (a) horizontal, (b) vertical at t = 19.5 µs.

3 )

 3 With X and x being the initial and deformed value at the nodal position, U F e and U Di being the simulation displacement Ąeld and a speciĄc distortion, respectively. The speciĄc distortion Ąeld is taken out of a data collection of distortion Ąelds obtained from real experiments[125].

Figure 5 . 8 Ű

 58 Figure 5.8 Ű FOV with different magniĄcations (G).

Figure 5 . 9 Ű

 59 Figure 5.9 Ű Reference image.

Figure 5 .

 5 Figure 5.10 Ű Strain Ąelds [mm.m -1 ] of the three different levels of magniĄcation (G) at three different integration steps. Elastic waves are emitted from the crack tip, which Ąnds itself slightly outside of the FOV in the bottom right corner.

Figure 5 .Ů MagniĄcation of 1 . 5 :

 515 Figure 5.10 shows the strain Ąeld in the horizontal direction for a magniĄcation of 0.5, 1.0, and 1.5 in the Ąrst, second, and third rows, with the results obtained from images 32, 65, and 78 in the three columns, respectively.From the strain Ąelds visualized in Figure5.10, the following observations can be made: Ů MagniĄcation of 1.5: The propagation of the strain band is clearly visible from the bottom right corner (crack tip) to the top left. Nevertheless, while the visibility is good, the phenomenon can only be seen for a short duration, due to its small Ąeld of view.

Remark 3

 3 The energy being released at the onset of the dynamic propagation Ű provoked in the present case by the special sample design of the blunted notch tip Ű might cause strong out-of-plane motions of the sample (shaking of the system and the sample). Such out-of-plane motion is critical for retrieving fracture mechanics parameters through DIC and WilliamsŠ series projection. Varying extraction areas (see Section 2.3.1) will inĆuence the accuracy of the results, i.e. artiĄcially inĆate or shrink the fracture mechanics variables. Hence, a modiĄed version of the WilliamsŠ series projection algorithm Ű identifying both LEFM parameters and homogeneous dilatation and shear being induced by out-of-plane motion Ű was considered for the present experiment. Furthermore, alternate cracking experiments inChapter 4 have been validated by the modiĄed WilliamsŠ series algorithm. Full-Ąeld displacement measurements (in the y-direction) for the Ąrst and last recorded frame are presented in Figure 5.11(a) and Figure 5.11(b), respectively. In the same conĄguration, corresponding WilliamsŠ series projection areas are presented in Figure 5.11(c) and Figure 5.11(d).

Figure 5 . 11 Ű

 511 Figure 5.11 Ű Entire displacement Ąeld in y-direction [mm] and its corresponding WilliamsŠ series projection [mm] on the process zone surrounding the crack-tip (deĄned by R min and R max ) for 4 time steps (0.0 µs, 42.22 µs, 94.99 µs, and 166.75 µs). The used WilliamsŠ series parameters are listed in Table 5.3.

Figure 5 .

 5 Figure 5.12 Ű (a) Fracture surface Section of the observed dynamic crack propagation with the DIC face on top and (b) Detail 1 of the transition from quasi-static to dynamic propagation highlighting the nucleation phase with the presence of microcracks. Notice that detail 1 is found in the bottom right corner of (a), while QS1, QS2, and RB stand for the quasi-static phase 1, quasi-static phase 2, and rest band, respectively.

Figure 5 .

 5 Figure 5.13 Ű K1 [MPa.m -0.5 ] and velocity (v) [m.s -1 ] as a function of the crack length in line with the topography measurement of the fracture surface. Again, the DIC face is on top of the fracture surface topography. 131

Figure 4 . 7 ,

 47 Figure 4.7, K1 shows a slight increase during the rest band. However, apart from a small dip at around 5.5 mm (in line with the start of the crack slow-down during the phase of increased surface roughness), K1 continues to steadily rise until reaching the second quasi-static phase where Ű again as observed during alternate cracking experiments in Chapther 4 Ű a steep increase in K1 is observed. Eventually, almost the same K1 level of the Ąrst quasi-static phase is reached in the second quasi-static phase.

[

  126Ű128]), elastic wave-induced interference of the crack front dynamics have been reported in the literature[START_REF] Bonamy | Interaction of Shear Waves and Propagating Cracks[END_REF] 126, 129]. Moreover, Ąngerprints of their interaction have been observed on the fracture surface[126, 130]. With non-singular stresses, depending on the geometry and loading conditions of a sample, receiving more attention in the literature recently regarding their role in brittle fracture [131], T-stress (i.e. being the tensile stress parallel to the crack) variations have been linked to the crack tip interacting with elastic stress waves in[132, 133].

Figure 5 . 14 Ű

 514 Figure 5.14 Ű Surface roughness (Ra) [µm] of the 500 µm slice of the fracture surface topography at round y = 6 mm.

Figure 5 .

 5 Figure 5.15 presents the surface roughness, crack tip speed (v), and the T-stress all in function of the crack length.

Figure 5 . 15 Ű

 515 Figure 5.15 Ű Crack tip speed [m.s -1 ] and T-stress [MPa] as a function of the crack length in line with the roughness measurement (Ra) [µm] of the 500 µm slice of the fracture surface topography at around y = 6 mm.

K1Figure 4 . 9 ,

 49 Figure 4.9, crack length measurements are provided for a better understanding of the presented data. Notice that only the phases of dynamic crack propagation are considered. This has been

Figure 5 . 16 Ű

 516 Figure 5.16 Ű Energy release rate (G) [J.m -2 ] as a function of velocity (v) [m.s -1 ] with its corresponding temporal evolution of the crack. The dynamic phase is split into the part before (red) and the part after (black) the rest band. Triangular and rectangular markers indicate the beginning and the end of a phase, respectively.

Figure 5 . 17 Ű

 517 Figure 5.17 Ű Energy release rate (G) [J.m -2 ] as a function of velocity (v) [m.s -1 ] for all alternate experiments recorded with the HS and UHS camera.

Figure 5 . 18 Ű

 518 Figure 5.18 Ű Twin nodes, clamped twin nodes along over the experimental crack length and the crack tip at two different simulation times steps. Displacements have been ampliĄed for by a factor of 80 to improve the visualization of the node separation.

Figure 5 . 19 Ű

 519 Figure 5.19 Ű Temporal evolution of the experimental, imposed simulation, and recovered simulation crack length [mm].

Figure 5 . 20 presents

 520 the in-plane velocity magnitude at different time instances of the simulation.

Figure 5 . 20 Ű

 520 Figure 5.20 Ű Velocity magnitude [m.s -1 ] at different time steps of the simulation. The mesh is deformed according to displacement. Static pre-load has been removed and residual dynamic displacement has been ampliĄed with a factor of 80 for improved visualization of the crack opening.

Figure 5 .

 5 Figure 5.20(a) shows the sample without any wave propagation since no twin node has been allowed to open yet.Figure 5.20(b) and (c) show the transient process of wave propaga-

Figure 5 .

 5 20(b) and (c) show the transient process of wave propagation towards the boundaries. It is worth noticing, that between the two, the Ąrst interaction with the notch tip can be observed. On the bottom of Figure5.20(d), the Ąrst recurrence of the waves towards the crack after interacting with the boundary becomes visible. Hereinafter, wave propagation, and interactions become gradually less traceable. In Figure5.20(i), presenting the last time instance of the simulation, the transient wave propagation is not recognizable anymore.

Figure 5 . 21 Ű

 521 Figure 5.21 Ű Ring discretization (dr = 0.5 mm) at the Ąrst time step. LB, BB, TB, and RB stand for the left, bottom, top, and right boundary, respectively, marking the closest boundary points (for potential wave interaction) from the crack tip in each direction.

Figure 5 . 22 Ű

 522 Figure 5.22 Ű Average velocity magnitude [m.s -1 ] over each ring and the detected crack tip speed computed by the time derivative of the crack length (simulation imposed) at each time instance. Red lines indicate the distance to the notch tip (NT), bottom boundary (BB), top boundary (TB) left boundary (LB), and right boundary( RB).

  20 µs after being released. This would result in a theoretical crack interaction at ∼50 µs and ∼55µs, considering an emission of the waves at the Ąrst release of nodes (∼30 µs) and the start of the main dynamic crack propagation at ∼35 µs. Hence, crack arrest caused by the interaction of P-waves emitted by the release of the Ąrst twin nodes or the onset of the dynamic crack propagation is theoretically possible but might be visually covered by the release of new waves in Figure5.22.

  spired by [136Ű139]. With the ability to compute out-of-plane velocities from the experimental DIC results and the simulation at experimental (470 kfps) and full (8 Mfps) resolution, AE signals were computed by taking the transfer function of the sensor into consideration. Comparison with AE signals recorded by the AE system is thereby possible. Numerical AE signals were computed from full-Ąeld DIC measurements and simulation displacement results as follows:

.8) . 4 . 5 .

 45 Out-of-plane velocities (v z ). Through the time derivative of out-of-plane displacements (u z ). Out-of-plane velocity over sensor area (V surf ace (ω)). The average out-of-plane velocity (v z ) was computed at the experimental sensor position and over the base area of the Nano30 sensor (∼50 µs) and (2) a particular dip at ∼140 µs. However, considering the frequency domain, only one main frequency (90 kHz) is detected for the AE device signal, while two main frequencies (180 kHz and 480 kHz) can be observed for the simulation at full resolution. At this point, it is worth taking the simulation results with a different mesh size (Ąfth column) into account. It has been observed that the choice of mesh size for the simulation has a strong inĆuence on the signalsŠ frequency content and amplitude. By increasing the mesh size, amplitudes will be increased, while frequencies are decreased. In detail, progressively decreasing the mesh size along the crack, one of the two peaks (480 kHz) vanishes and only the Ąrst one remains (180 kHz) but is ampliĄed with considerably stronger values. It seems to shift closer to the experimental observation. Hence, the choice of mesh size for the initial simulation has to be taken into consideration when discussing AE signals retrieved from simulation data. Moving on to comparing the AE signals computed from the DIC measurements of the experiments (second column) and the simulation at a reduced resolution (fourth column). Due to the low temporal resolution, signal comparison is rather difficult. However, with similar amplitudes, waveform characteristics with two main peaks are observed in both graphs. Yet, the peaks do appear later in the simulation than in the experiment. This can be explained by the rate dependency of PMMA (see discussion in Section 5.3.3) and the choice of YoungŠs modulus for the simulation (with 3.5 GPa being rather on the quasi-static end) that might cause an underestimation of the actual wave speeds present in the experiment. Globally, focusing on the signals in the frequency domain, a similar trend of decreasing frequency content is observed. As in the numerical design section (see Section 5.2.6), 4 Mfps and better magniĄcation would have been required to get an improved comparison. Solutions, e.g. different gains, to avoid saturation of the AE signals have not been found with the available equipment. Furthermore, the signal is not only the emission at the start but the combination of signals being emitted at each re-acceleration of the crack tip with a combination of S and P waves. This highlights challenges in the accurate AE localization of signals.

Figure 5 .

 5 [START_REF] Maillet | Waveform-based selection of acoustic emission events generated by damage in composite materials[END_REF] presents the temporal evolution of K1 and T-stress for both, the experiment and simulation.

Figure 5 . 26 Ű

 526 Figure 5.26 Ű Experiment and simulation comparison of the temporal evolution of (a) K1 and (b) T -stress. The comparison starts at the onset of the dynamic crack propagation of the simulation (∼35 µs).

  5 m.s -1 reducing the ŠforbiddenŚ velocity regime further towards a continuous connection of the two stable regimes (Region I and Region III in Figure 1.1). Ů Rest bands. The observation of rest bands has been evidenced, indicating a discontinuous propagation of the crack tip during dynamic cracking phases. Thus, two types of alternate behaviors are identiĄed: alternate (stick-slip) cracking and discontinuous crack propagation during dynamic phases. Ů Elastic wave induced crack arrest. In line with the previous point, it has been suggested that crack arrests during the dynamic phases are caused by the interaction of the crack tip with crack-induced elastic waves. Ů Crack velocity and fracture surface roughness. A clear link between the crack tip speed and the fracture surface roughness has been identiĄed, highlighting that velocity variations can be identiĄed by solely looking at the fracture surface. Ů Numerical twin of experiments The numerical twin was created to support the understanding of experimental observations. Ů Presence of two types of body waves. With the simulation sampling, the emission of two types of waves Ű moving at different speeds Ű are identiĄed. Ů IdentiĄcation of rebounding waves. The assumption of the interaction of crack front and self-emitted elastic waves has been underlined by the identiĄcation of rebounding waves in the simulation data. Potential interactions of rebounding waves and the crack tip have been detected at approximately the right time. Ů Comparison of experimental and numerical AE signals. ArtiĄcial AE signals have been computed from the full-Ąeld DIC measurement and the simulation displacements to compare with the AE signal recorded by the AE device. The dependency of the computed numerical AE signals on the choice of mesh size has been highlighted. However, the lack of temporal resolution and saturation of the AE signals made a proper comparison difficult.

  and fracture mechanics in PMMA with valuable insights into crack propagation behavior, AE signal characteristics, and fracture surface roughness. Yet, further work could be pursued along the following points: Ů Enabling proper wave visualization with UHS recording at increased temporal resolution (>1 Mfps). Exploring the potential to track AE signals back to their source, visualizing the type of wave, and improving the understanding of wave/structure and wave/crack tip interaction leading to a better understanding of the content of AE signals for different fracture mechanisms. Ů DeĆectometry [140] could be explored as a possible alternative. This imaging technique allows the capturing of out-of-plane motion with an amazing resolution (∼10 nm). Such a technique would potentially make the capturing of waves produced by quasi-static crack propagation possible. Thus, a simpliĄed triggering could be applied, since low-amplitude waves are potentially produced continuously. Ů Supporting the UHS experiment results with additional experimental campaigns. Here, the experimental repeatability could be improved. The unpredictable nature of PMMA fracturing has been a major challenge for the repeated success of such experiments. A parametric investigation of the inĆuences on the fracturing of PMMA (e.g. temperature, spray paint types for the DIC speckle pattern, production of the sample, etc.) would allow for the optimization of the experimental program. Ů Improving wave propagation modeling by including e.g. viscoelastic behavior, 3D effects, and wave rebounds on the pins (in-Ąnitely stiff compared to PMMA) in the numerical model. Ů In the spirit of the previous point, continuing the numerical exploration of AE signals, based on an improved simulation, and their comparison to experimental results.

Figure A. 1 (

 1 Figure A.1(b) corresponds to a signal being presented in the core of the manuscript in Figure 2.21 associated with Multiplet 1.

  Figure A.1(b)), is observed between signals in front or behind the source, which potentially evidences a clear directionality of the wave propagating. It underlines the complexity of analysis and classifying, in bulk, crack propagating, branching, and turning with respect to the sensors.

Table 1 Ű

 1 Advantages and limitations of NDT techniques used in the detection of cracks for fatigue extracted from[START_REF] Kong | A review of non-destructive testing techniques for the in-situ investigation of fretting fatigue cracks[END_REF]

	NDT Technique	Ref.	Description	Advantages		Limitations
	Optical Microscopy	[10]	An optical microscope is	• A contactless technique	• Only captures surface
			directed towards the ROI	• Visual		cracks
			or replica and microscopic	• Locate crack initiation	• Well-illuminated Ąeld of
			images are captured of the	• Obtain number of cycles	view is required
			crack	to crack initiation		• Crack initially needs
				• Monitor crack evolution	to be located before
				• Obtain crack length with	measurements	and
				number of cycles		images can be taken
				• Obtain crack path		• Smaller specimens are
				• Simple to use and read-	required to ensure the
				ily available		area of interest is within
				• Tests can be adapted	the Ąeld of view
				for higher temperatures
				with the use of a replica
	SEM	[11, 12] An electron beam is
			scanned over the surface
			of a specimen during a
			test and an image of its
			microstructure is produced
			from the response signals
						• Detects only surface and
						subsurface cracks in
						plane of view
						• Smaller specimens are
						required to ensure area
						of interest is within the
						Ąeld of view of view
	X-Ray Radiograph	[13] An X-ray source penetrates
			specimen onto an X-ray
			imaging detector
				• Determine	crack
				location
				• Identify crack propaga-
				tion modes
				• Monitor crack length
				• Determine the cycles to
				crack initiation

To collect experimental data to support the understanding through simulations, different fracture processes are provoked in laboratories by destructive testing (DT) techniques (e.g. fatigue, tensile (elongation) or torsion experiments), with non-destructive testing (NDT) techniques monitoring material and fracture responses. However, for proper application of NDT techniques, advantages and limitations need to be determined in controlled experimental environments. Apart from Acoustic Emissions, NDT techniques for fatigue failure detection are covering, in general, the following technologies: Optical microscopy, SEM, X-ray radiography, ultrasound, thermography, and direct current potential drop methods. The advantages and limitations of these techniques are given in the following table.

• A contactless technique • Visual • Locate crack initiation • Determine crack location and monitor path • Monitor crack length • Determine the cycles to crack initiation • Identify crack propagation modes • Obtain microstructural features, such as slip lines • Tests can be adapted for high temperature tests • Space constraints of samples due to the test having to be adapted to Ąt within vacuum chamber • Tests need to be interrupted for image capture • A contactless technique • Detects internal and surface cracks • Visual • Can be used for various geometries and materials • Expensive • Large and complex equipment with limited access

Table

the Ąrst scientiĄc motivation of this work.

  ). Through reverse engineering by developing an experimental setup for a 360°perspective discussion on crack-related AE source mechanisms, one could shed light on existing blind spots and trace AE signals back to their origin. Like this, AE signals could be linked to their source mechanism by avoiding the need for blind interpretation, which can be considered During more than half of the fatigue life span, these multiplets have been solely observed during tension, which opens the possibility of analyzing AE signals during controlled laboratory experiments of isolated single crack propagation. Such multiplets have previously been observed

Table 2 .

 2 1 Ű List of experiments.

	Test	Cracking type
	Reference Purely mode I (smooth)
	T5, T13	Predominantly mode I with bending (complex)

Table 2 .

 2

	2 Ű Camera hardware parameters
	Purpose	DIC	In-volume
	Camera	Viework VP-50MX-M/C 30
	Image resolution Acquisition rate	7856 × 1300 pix 62.5 fps
	Lens	TOKINA 100 mm
	Aperture	f/2.8
	Field of view	37.18×6.15 mm 49.75×8.23 mm
	Resolution	4.65 µm	6.17 µm
	Patterning technique Spray paint

Table 2

 2

	.3 Ű DIC & WilliamsŠ series parameters
	DIC
	Software	UFreckles [79]
	Discretization	FE-based global DIC
	Shape function	Bi-linear quadrilateral Lagrange element (Q4P1)
	Element size	95 µm
	Post-Ąltering	Median Ąlter using 1 st neighbour applied to U
	WilliamsŠ series projection
	E	3.5 GPa
	ν	0.32
	R min	127.8 µm
	R max	1301.6 µm
	2D approximation Plane-stress

Table 2

 2

	.4. The potential errors for sensor placement,

source localization. A rather large uncertainty was thus introduced here (about 10 times the AE

Table 2 .

 2 4 Ű Standard deviation (uncertainty) added to synthetic source position, time of arrival, and Ąrst guess of velocity.

	Error	Standard deviation
	Sensor position 500 µm
	TOA	1 µs
	Velocity	5 m.s -1

Table 2 .

 2 the duration of the deĄned impulsive part (40 µs). With an average correlation coefficient of 0.48 among all captured AE signals, average (and lowest) correlation coefficients of 0.88 (and 0.77) and 0.92 (and 0.83) were found among AE waveforms associated with Multiplet 1 and 2, respectively. Furthermore, information on the non-normalized waveforms presented in Figure 2.21 are shown in Table 2.5. 5 Ű Information on non-normalized waveforms associated with the multiplets.

	Multiplet Peak amplitude range	Absolute Energy range (mean)
	1	8.9 mV (39 dB) to 50 mV (54 dB) 7 aJ to 235 aJ (63 aJ)
	2	7.9 mV (38 dB) to 25 mV (48 dB) 4 aJ to 50 aJ (28 aJ)

  for the correlation of AE activity with crack front velocity heterogeneity, Figure 2.26 is proposing an identiĄcation of potential locations of the AE signal origin on the fracture surface based on crack front velocity heterogeneity. For this, not assume the same for LA signals, since the multiplet signal does not necessarily present the highest recorded AE-signals associated with this frame.

	Nevertheless, let us Ąrst focus only on the relation between the fracture surface and the crack
	front kinematics. Interestingly, no particular difference in terms of crack surface features can be
	identiĄed with respect to the two phases of homogeneous (10-15 mm) and heterogeneous (15-
	32 mm) crack front propagation. However, velocity heterogeneity Ű although different in scale Ű
	can be seen in both phases in the vicinity of the artifacts (white marks) on the fracture surface.
	Especially notable are the two local crack front jumps between two consecutive frames at ap-
	prox. 15.5 mm and 27 mm that are associated with the Ąrst and second local dynamic instability,
	respectively. Extending the discussion by the AE waveforms, the two dynamic instabilities Ű be-
	ing already well understood and localized during the previous discussion Ű are well identiĄed
	(asterisk markers colored in magenta in

Table 3 .

 3 1 Ű Second loading stage extension rates applied during the experiments.

	Extension rate [mm.s -1 ] Test
	1.0	M1-T2, M2-T15, M2-T19
	0.5	M1-T3, M1-T6

Table 3

 3

	.2 Ű DIC hardware parameters
	Campaign	M1	M2
	Camera	Photron FASTCAM Mini AX
	Image resolution	896×160 pix
	Acquisition rate	40,000 fps
	Exposure time	25 µs
	Lens	TOKINA 100 mm
	Aperture	f/2.8
	Field of view	74×13 mm	64.5×11.5 mm
	Image scale	1 pix = 83 µm 1 pix = 72 µm
	Patterning technique Spray paint
	3.2.4 DIC and WilliamsŠ series parameters
	All relevant DIC and WilliamsŠ series parameters for the two experimental campaigns are
	listed in Table 3.3.

Table 2 .

 2 5) and HA signals (above 2×10 6 aJ), propagation areas below ∼22.7×10 3 µm 2 maybe expected for LA signals. Considering, only for illustrative purposes, a homogeneous through-thickness crack propagation would result in crack advances of approx. 2.8 µm over the sample thickness of 8 mm for the strongest LA signal. By considering a local crack growth over a square surface, the latter would reach ∼151 µm sides. The area compared to such a feature is symbolized with a black square in Figure 2.20(b) in the left bottom corner as reference. Knowing that the spatial resolution of the microscope is 2.53 µm.pix -1

  this section, macro alternate cracking in PMMA has been produced and observed by high-

	speed imaging and investigated with the help of DIC and AE. Quasi-static and dynamic crack
	propagations linked to this speciĄc fracture phenomenon have been investigated by a comprom-
	ised experimental setup. The setup however allowed for the investigation of the two cracking
	types within one fracture process. Main observations regarding the link between AE activity
	and dynamic instabilities can be summed as follows:
	Ů Large-scale instabilities were provoked by the alternate (stick-
	slip) phenomenon in PMMA producing signiĄcantly higher AE
	signals compared to any other crack front-related AE activity.

Ů A linear relationship between AE absolute energy of HA signals and the log of the fracture growth area has been found considering both: the present macroscopic stick-slip phenomenon and the localized dynamic instabilities observed in Chapter 2. An attempt of extrapolation, from this relationship to LA signals, has been done leading to a potential fracture growth area of about 150 × 150 µm.

This may be evidence that LA emissions (see Section 2.4.3) are rather induced by localized small crack growth than macroscopic crack advance.

Table 4 .

 4

	1 Ű Amount of rest band ap-
	pearances during the dynamic bursts. ( * )
	means speciĄc roughening of the surface
	observed, which implies the early form-
	ation of a rest band that is yet not fully
	evolved. However, biased velocity meas-
	urements still have to be expected.

Table 5 .

 5 1 Ű Experimental campaign-related timescales.

	Activity	Average duration
	Experiment	80 s
	Alternate cracking	40 ms
	Dynamic instability 100 µs
	Wave travel	max. 10 µs
	Camera recording	max. 167 µs
	Light	1 ms

Table 5 .

 5 2 and Table 5.3, respectively.

	Table 5.2 Ű DIC hardware parameters
	Camera	Cordin Model 580
	Image resolution Acquisition rate	3296 × 2472 pix 470,000 fps
	Exposure time	2.12 µs
	Lens	TOKINA 100 mm
	Aperture	f/2.8
	Field of view	28×21 mm
	Image scale	1 pix = 8.5 µm
	Patterning technique Spray paint
	Table 5.3 Ű DIC & WilliamsŠ series parameters
	DIC
	Software	UFreckles [79]
	Discretization	FE-based global DIC
	Shape function	Bi-linear quadrilateral Lagrange element (Q4P1)
	Element (subset) size 0.55 mm (64 pix)
	Post-Ąltering	Median Ąlter using 1 st neighbours applied to U
	WilliamsŠ series projection
	E	3.5 GPa
	ν	0.32
	R min	0.55 mm
	R max	5.5 mm

Table 5 .

 5 4 Ű Simulation parameters.

	Solver	Ufreckles (open-source DIC-FE software [79])
	Material behaviour	elastic
	Density (ρ)	1200 kg.m -3
	YoungŠs modulus (E)	3.5 GPa
	PoissonŠs ratio (ν)	0.32
	Fracture toughness (K1 c )	1.5 MPa.m -0.5
	Crack mode	mode I
	Time (t)	100 µs
	Time step (dt)	0.025 µs
	Element type	T3P1 (3 nodes, polynomial 1st order)
	Mesh size	50-100 pixel
	Image scale	1 pix = 10 µm

Table 5 .

 5

	5 Ű DIC parameters for performing DIC on the synthetic images
	DIC software	Ufreckles [79]
	Mesh size	32 px
	DIC coarse-graining process	5 scales
	Max. iteration	50
	Convergence limit	1e -3
	Regularization type	Tikhonov
	Regularization parameter	2 x mesh size
	Strain post-treatment	MedianFilterCell (Ufreckles library)

  2. Repeating wave propagation through the projection zone might inĆuence the accuracy of the result since an inĄnite domain without boundary effects is assumed for the WilliamsŠ series projection. Furthermore, provoking the alternate behavior by the speciĄc geometry of a blunted notch tip might produce a different cracking Ű similar to a pre-stretched sample, where Eid et al.[START_REF] Eid | InĆuence of rate-dependent damage phase-Ąeld on the limiting crack-tip velocity in dynamic fracture[END_REF] have identiĄed a higher YoungŠs modulus (E up to ∼6.5 GPa) Ű than provoking the alternate behavior classically by a sharp notch tip and a limit extension rate (extension rates of 0.025 mm.s -1 for blunted notch tip samples compared to 1 mm.s -1 for the samples with sharp notch tips). The energy stored in the notch tip due to the geometry is causing the alternate cracking, while the extension rate is far away from supporting this alternate behavior. Eventually, when the energy of the blunted notch is lost, the crack will propagate quasi-statically, i.e. the energy provided to the crack causing alternate cracking solely relies on the energy stored before initiation. Thus, choos-

ing a quasi-static related YoungŠs modulus would not suit the dynamic propagation caused by the geometry. Moreover, in the previous investigation (see Section 4.2.2) G tends to stabilize after a certain length during quasi-static phases. In the present case, however, the amount of energy stored due to the blunted notch does not allow long enough quasi-static regions, which leads to not fully stabilized K1 (during short QS phases) and thereby G values. Due to all the above, a direct G comparison of the two types of alternate cracking experiments might not be straightforward. Even when looking only at Figure

5

.17, results coming from different notch geometries and camera setups overlap quite well.

Table 5 .

 5

	6 Ű WilliamsŠ series parameters
	E	3.5 GPa
	ν	0.32
	R min 0.55 mm
	R max 5.5 mm

Table 5 .

 5

	7 Ű PMMA properties considered
	Bulk modulus (K)	3.2 GPa
	Shear modulus (µ)	1.7 GPa
	Density (ρ)	1200 kg.m -3
	Compressive wave speed (v p ) 2134 m.s -1
	S-waves (v s )	1190 m.s -1

Table 5 .

 5 [START_REF] Alwi | Fault Detection and Fault-Tolerant Control Using Sliding Modes Series Title: Advances in Industrial Control[END_REF]) and a distance of 21 mm to the notch tip, rebounding waves should interact with the crack approx.

Table A .

 A 1 Ű AE parameters computed for the signal in Figure A.1 in chronological order of the sensor number.

	Time	RISE COUN ENER DURA AMP A-FRQ PCNTS THR R-FRQ I-FRQ SIG STR ABS-ENER FRQ-C P-FRQ
	2.4769143 12	27	3	975	53	28	3	31	25	250	2606.29	19.72	1904	106
	2.4769145 12	31	2	975	52	32	3	31	29	250	2169.2	17.21	1449	41
	2.4769161 17	32	2	973	47	33	4	31	29	235	1773.92	8.07	1276	93
	2.4769203 28	27	2	969	48	28	6	31	22	214	1795.97	7.3	1667	1

  Titre : Réponses acoustiques et mécaniques à la propagation stable et instable de fissures dans le PMMA Mot clés : Polymères ; Mécanique de la Rupture ; Corrélation d'Images Numériques ; Emission AcoustiqueRésumé : Pour fournir des informations précieuses pour la conception d'algorithmes d'analyse de tests non destructif (NDT) robustes et fiables, la combinaison, de mesures acoustiques résolues en temps, de la corrélation d'images numériques (DIC), des mesures en volume ainsi que de l'analyse postmortem de surface de rupture appliquées à des expériences de fissuration en traction dans le but de localiser la source des emissions acoustiques, de détecter la pointe de fissure et de déterminer les complexités et la cinématique de la fissure, respectivement. Un dialogue entre l'activité acoustique, les variables d'état, les mécanismes de rupture et les faciès de rupture postmortem est établie au cours d'expériences modèles contrôlées produisant des événements AE de différentes natures. Le Poly(méthacrylate de méthyle) (PMMA), traditionnellement considéré comme la matériau modèle par excellence, est utilisé au regard ses caractéristiques distinctives : la capacité à présen-ter un mode de défaillance alternatif par le biais de la propagation d'une seule fissure, conduisant au développement de deux motifs distincts de croissance contrôlée de fissures. De plus, pour la première fois, un tel comportement de rupture cyclique matériau/structurel est examiné à travers le prisme de la mécanique de la rupture élastique linéaire (LEFM) en utilisant la corrélation d'images numériques (DIC) in-situ à haute vitesse (HS) et à ultra-haute vitesse (UHS). Les taux de restitution d'énergie et les vitesses de fissuration au cours des expériences de rupture sont déduits à partir de mesures en champ complet en utilisant l'expansion en série de Williams. Les surfaces de rupture post-mortem des échantillons ont été systématiquement analysées à l'aide de la microscopie optique. Ce travail fournit des données expérimentales clés pour une meilleure compréhension d'un cadre théorique unifié des instabilités de fissuration. To provide valuable information for the design of robust and reliable nondestructive testing (NDT) algorithms, combined timeresolved Acoustic Emission (AE), Digital Image Correlation (DIC), and in-volume measurements with post-mortem fracture surface analysis are applied to tensile cracking experiments for the purpose of AE localization, crack tip detection, and determination of crack front complexities and kinematics, respectively. The complete picture of the link between AE activities, state variables, fracture mechanisms, and fingerprints on the post-mortem fracture surface is established during controlled model experiments producing AE events of different physical phenomena. Poly(methyl methacrylate) (PMMA), conventionally deemed an exemplary model material, presents an ideal material of interest due

Title: Acoustic and mechanical responses of stable and unstable crack propagation in PMMA Keywords: Polymers; Fracture Mechanics; Digital Image Correlation; Acoustic Emission Abstract:

A.1 Same signal detected at all four sensors for (a) 300 µs and (b) 40 µs (coda) of the signal. The threshold is indicated by the red dashed lines. . . . . . . . . . . . . . . 177

Numerical feasibility study

The numerical imitation of the dynamic crack propagation recording was performed to improve the understanding of the experimental chain and its biases. Experimental-like data are produced by using synthetical data coming from an FE simulation in combination with experimental biases. The chain must include Ąrst-order bias such as camera distortions and their random variations, and limits in terms of resolutions (space and time), sensor noise, and DIC spatial Ąltering effects. The presented section aims to understand the main experimental parameters to play with to achieve the resolution and accuracy needed to record dynamic crack propagation with the Cordin-580 and potentially capture crack-induced elastic waves.

Brief simulation details

At Ąrst, the specimen is loaded with an elastic pre-load (quasi-static simulation, Equation 5.1) until reaching a stress intensity factor equal to the experimental fracture toughness (K1 c ). In the second step, a twin node is released at the tip of a pre-determined crack path introducing the dynamic transient analysis (Equation 5.2). The simulation was done by using the library of Ufreckles [START_REF] Réthoré | UFreckles Language: en[END_REF].

In detail, the following steps were considered for the basic concept of the simulation:

1. Quasi-static preload with unit-applied displacement. In this step, the last pair of twin nodes before the crack tip are clamped together using Lagrange multipliers.

Computation of SIF

Here, the results are also scaled to reach the expected SIF value.

Obtained displacements

The quasi-static displacement was considered as the initial condition for the dynamic simulation (with zero velocity Ąeld).

Dynamic part

The dynamic simulation is launched with no constraint on the twin nodes meaning that the crack is assumed to propagate instantaneously by one element.

A 2D FE model, discretized by an unstructured triangular mesh, was used to solve the problem.

To determine the static pre-loading, the global equation system is as follows:

The Ąnite element equation of motion to solve the dynamic problem of wave propagation can be written in a general form as follows (no damping was taken into consideration):

After visualizing the transient process, it is worth classifying the observed waves. Material waves, commonly referred to as mechanical waves, are waves that depend on a physical medium for their propagation. These waves transmit energy by causing the particles within the medium to oscillate or vibrate, and their distinguishing feature is their capacity to convey energy through a medium without causing a permanent displacement of that medium. Mechanical waves can be separated into the three following main types:

Ů Longitudinal: Within longitudinal waves, the mediumŠs particles shift in alignment with the waveŠs direction of travel. These waves are characterized by alternating regions of compression, where particles are pushed closely together, and rarefaction, where particles are pulled apart.

Example: Sound waves in the air.

Ů Transverse: Transverse waves are generated when the particles within the medium shift in a direction that is perpendicular to the waveŠs propagation. These waves are characterized by the presence of high points called crests and low points referred to as troughs, with the particles oscillating laterally or vertically.

Example: Electromagnetic waves (e.g. light).

Ů Surface: Surface waves are a variety of mechanical waves that travel along the boundary or interface separating two distinct media. These waves display traits of both longitudinal and transverse waves and propagate exclusively along the surface of the medium, without delving into its interior. Rayleigh and love waves present two primary types of surface waves.

Example: Surface waves on the water.

In contrast to surface waves, longitudinal and transverse waves, also referred to as compressive (P-waves) and shear waves (S-waves), both travel within a medium. While typically having smaller amplitudes, they are faster than surface waves. P-waves present the fastest traveling mechanical waves, with S-waves classically traveling at about half of their wave speed. The theoretical wave speed of P-waves (v p ) and S-waves (v s ) can be computed as follows:

with K, µ, and ρ being the bulk modulus, shear modulus, and the material density, respectively. For PMMA, a P-wave is supposed to travel at ∼2134 m.s -1 and an S-wave at ∼1190 m.s -1 by using 3.2 GPa, 1.7 GPa, and 1200 kg.m -3 , for K, µ, and ρ, respectively. Material properties 7. AE response. In the frequency domain, the transfer function was added as follows:

V sensor (ω) = V surf ace (ω) × S sensor (ω) (5.9) with V sensor , V surf ace , and S sensor being the velocity calculated with the considered sensor, the average of out-of-plane velocity in the sensor area, and the sensor transfer function,