Myriam Maxence

Takwa Ahmed Matthew

De mon point de vue la thèse n'est pas un travail solitaire. Je souhaite remercier un grand nombre de personnes avec lesquelles j'ai pu collaborer et vivre au cours de ces années d'études.

À Saint-Gobain Recherche qui m'a confié un sujet très varié avec de nombreuses pistes expérimentales et scientifiques à traiter. Aux fructueux échanges avec l'équipe d'encadrement, Xavier, René, et les collègues Benoît, Keyvan, Jérémie... À Richard, Grégorie, Maxime, Quentin, David... les membres de l'équipe de Chantereine qui m'ont accueilli chaleureusement lors des essais sur le banc cannon.

Au LMPS et l'épanouissement scientifique que j'ai pu ressentir. À François pour ma première chance. À Stéphane et Bastien, mes encadrants qui ont supervisé entièrement le projet de thèse. Ils ont su me diriger vers les points essentiels à traiter, avec complémentarité et régularité. Aux membres des centres d'essai et de gestion, facilitateurs de l'ombre. À Rémi et Xavier qui ont grandement aidé de par leur expérience, leur expertise, et donné de leur personne pour la réalisation du dispositif blowpipe. À Xuyang pour le temps consacré à la stéréo-vision et la corrélation d'images. À Mahmoud et Sofiane, les "Trouaeducs". À Marcello, Maxime, Cédric pour notre passion commune. À Roxane,

Definition and application

Many interpretations of glass can be distinguished in the ages. The first scientific consensus about its definition was proposed by Jerzy Zarzycki: "A glass is a non-crystalline solid (i.e. amorphous) presenting the phenomenon of glass transition" however the definition of common sense often obscures it: glass is a hard material or alloy, brittle and transparent to visible light, made of silicon oxide and fluxes [START_REF] Zanotto | The glassy state of matter: Its definition and ultimate fate[END_REF]. More precisely, the glass of glazing is obtained mixing silica with other atoms: sodium (e.g. soda-lime glass) and boron (e.g. borosilicate glass) for the glazing in the table 1.1, but also inoculants such as metal oxides for glass-ceramics. This insertion modifies some covalent bonds of the structure into ionic bonds. [START_REF] Seward | High temperature glass melt property database for process modeling[END_REF].

Soda-lime

Glass appeared quite early in man's civilization. Over the different periods, it has never ceased evolving and finding an ever-increasing number of applications. Today, it is a material with well-developed technology, as evidenced by the evolution of its technical and mechanical characteristics: structural part, resistance to impact, temperature, UV, integration of technologies... Its amorphous state coupled with zero porosity makes it an ideal material in many cases.

Today, glass is a part of our daily lives as it is in larger projects. For example, in the building sector, one can mention the Louvre pyramid, the glass floors of the Eiffel Tower, and the Grand Canyon Skywalk (glass floor built at 1200 m height, 24 m from the edge of the cliff supporting 74 tons).

Glass grades and manufacturing processes

Glass is a material that has made many appearances throughout the ages. Its utilization started in the 2 nd millennial B.C. in mosaics found in Mesopotamia and then in ancient Egypt. The glazes were formed in vertical furnaces, changing the raw materials (sands, natron) to vary their tints. Glass vessels that appeared in the 1 st millennial B.C. were formed through the deposition of the hot, melted, and viscous glass on a clay core. Later in the 1 st millennial A.D., the natron additives were replaced with plant ashes, and the use of antimony could make the glass colorless [Freestone 2021].

The blowpipe technique, the first notable glass-making technique whose concept is still utilized in modern industries, was invented in the 1 st century B.C. in the Eastern Mediterranean region. Murano glass-makers modeled glass for several medieval centuries using this method: the craftsman takes a molten glass ball from a silica furnace and blows it into a tube to shape it. The construction of sleeves made it possible to create one meter long glass plates. At the end of the 17 th century, the table casting technique appeared in France. The molten glass is deposited on a metal plate and then distributed over its entire surface. The rolling process was invented in 1920 in Germany. Max Bicheroux's idea was to compress the glass between two rollers, quickly installed in the factories because it allowed the molten glass to be poured and pulled continuously. The finished product strip is then cut at regular intervals to form large glass plates. This process will be followed by casting in molds for hollow parts such as glass bottles and jars, and other weaving and centrifugation techniques for the glass wool manufacturing.

FIG. 1.1 -Scheme of float glass manufacturing.

In the case of glass panels for building and automotive applications, the float process has been the most widely used by flat glass manufacturers since the 1960s. The principle schematized in Fig 1 .1 consists in casting the molten glass on a liquid tin bath. Rollers then draw the glass ribbon, which slowly cools to relax the stresses inside the plate. The thickness of the glass is then controlled by the feed rate of the glass, i.e. the rotation speed of the rollers (i.e. RADS in Fig 1 .1).

Mechanical properties

Conversely to silica glass, the ionic and covalent bonds in the soda-lime glass allow little room for plasticity and densification. As a result, it exhibits elastic and brittle behavior at room temperature without permanent deformation. Even if the intrinsic tensile strength of glass is good, it is always weakened by surface defects (scratches, indentations, notches) and behaves brittle. Often, cracks start in regions with a higher stress concentration. The table 1.2 shows some mechanical properties of soda-lime and borosilicate glasses.

1.1. Glass

Ch. 1. Introduction

The first crack is an indelible impression left by the indentation of the projectile near the glass surface. The glass is subjected to tensile stress and shear in different directions in this small area. When the kinetic energy of the impact is higher, a crushed zone with tiny glass fragments is observed. A plastic zone is created in the amorphous material [START_REF] Baret | Extremal model for amorphous media plasticity[END_REF]. Then comes the concentric circular crack, orthoradial or ring cracks, initiated from radial stresses. The Hertz cone crack starts with a circular shallow crack that remains stable for a while, and then a second instability gives way to the cone. For plates, the bending may prevent this second stage and favors the initiation of another set of circular cracks instead. Then, radial cracks extending straight from the impact location can be observed. The presence of orthoradial stresses forms them. A star-like appearance on the indented plane is observed when these cracks are formed. The median cracks are supposedly circular and tangent to the free surface.

Its geometry goes from a circle to a semi-circle with the same depth, but the semi-circle has a center on the surface. The median crack may turn into radial when the indenter is released. Finally, lateral cracks are created beneath the glass surface, forming circles parallel to the indented plane.

In dynamic loading conditions, the greater the kinetic energy or speed of the projectile, the more radial cracks will get formed. These cracks are numerous during high-velocity bullet impact on glass armor [START_REF] Bless | Impact damage in layered glass[END_REF]. Some new shapes appear, like the fan cracks or the bow-tie cracks. In-depth cracks also evolve vertically in the thickness of the glass. A comminution zone may appear during an impact. This area is located under the tip of an indenter as it continues its way into the already cracked glass producing local secondary cracks and, therefore, small fragments.

Fracture mechanic

Glass is representative of a brittle material. It can be deformed in the elastic domain with small deformations and breaks when critical stress is reached. The consideration of critical stress allowed the engineers of the 19 th century to dimension glass structures. The cracking phenomenon only began to be analyzed by Griffith in the 1920s. The author developed a criterion for crack extension in a 2D plate under known loading, based on the theorems of mechanics and thermodynamics [START_REF] Lawn | Fracture of Brittle Solids[END_REF]]. After the first law of thermodynamics, the total of energy, U, of the considered system is

U = U M + U S + U k
with a surface creation contribution U S , and a mechanical energy U M = U E +U A split into elastic energy stored in the media, U E , and potential energy from the applied forces, U A . The inertia term U k added to the initial static problem is non-negligible as soon as dynamic loading of the plate is considered. At equilibrium, the Griffith energy-balance concept gives In a linear elastic fracture mechanics problem, considering plain stress, Irwin introduced the planar stress intensity factors to express the stresses near the crack tip, K I and K II

dU dc = 0
σ rr = K I √ 2πr f Irr (θ) + K II √ 2πr f IIrr (θ) σ θθ = K I √ 2πr f Iθθ (θ) + K II √ 2πr f IIθθ (θ) σ rθ = K I √ 2πr f Irθ (θ) + K II √ 2πr f IIrθ (θ)
and an anti-planar stress intensity factor, K III

σ rz = K III √ 2πr f IIIrz (θ) σ θz = K III √ 2πr f IIIθz (θ)
where f i (θ) are non geometrically nor mechanically dependent functions. K I , K II and K III vary with the applied loading, the crack tip size and geometrical parameters. The three modes, I, II and III, correspond to the fracture modes, respectively, an opening crack mode, a planar shear mode, and an anti-planar shear mode. Irwin used a new material parameter to define a crack propagation criterion. The fracture toughness corresponds to a critical value of the stress intensity factor. As a remark, in plain strain, the stress component σ zz is not null.

In Linear Elastic Fracture Mechanics, one can write Irwin's equation and link the energy release rate G with the stress intensity factors K i for a simple or mixed-mode opening crack in its initial direction,

G = 1 -ν 2 E K 2 I + K 2 II + 1 + ν E K 2

III

with ν the Poisson ratio and E the Young's modulus. The critical energy released rate, or Griffith energy, is also known for soda-lime glass: G c ∼ 10 J/m 2 [START_REF] Freund | Dynamic Fracture Mechanics[END_REF]].

Concerning the fast crack propagation in mode I opening, one may express the crack propa-

υ = c R 1 - K Ic K I n f or K I ≥ K Ic , υ ≤ υ max
with the Rayleigh waves celerity c R = 3300 m/s, the maximum crack opening celerity υ max = 1550 m/s and n = 2 [START_REF] Freund | Dynamic Fracture Mechanics[END_REF]].

Dealing with glass reliability is a significant issue for glass industries. The intrinsic strength of glass material is linked to its structure, its chemical composition, and the energy required for atomic bond cleavage [START_REF] Hühn | Ab Initio energetics of Si-O bond cleavage[END_REF][START_REF] Wondraczek | Advancing the Mechanical Performance of Glasses: Perspectives and Challenges[END_REF]]. Experimentally, the strength of a glass sample highly depends on extrinsic factors [Kurkjian and Gupta 2020]: the environment, the presence of surface flaws, and defects. All surface defects affect the material variability in lifetime and strength and must be determined and controlled at best. The typical probabilistic distribution to model the presence of the flaws is obtained from Weibull theory [START_REF] Weibull | A statistical distribution function of wide applicability[END_REF]], which links the probability of failure, P, and the failure stress, σ f P = 1e -(σ f /σ 0 ) m with a Weibull modulus m and σ 0 the size dependent scaling stress.

Laminated glass 1.2.1 Origin and history

The brittleness of glass is a problem in many applications, particularly in ensuring the safety of its users. For example, in the case of a windshield, if the glass is an optimal choice for structural aspects and provides good visibility, it breaks in the form of large chips in a crash. Therefore, combining the glass with a soft, adhesive material that can deform in large strain regimes to retain the deadly fragments ejection during an impact is made in most applications. Laminated glass is a composite composed of two or more glass plies separated by polymeric interlayers that ensure user safety and is nowadays widely used as glazing.

Sometimes clumsiness may lead to great things. Edouard Benedictus illustrated this concept by dropping a glass jar containing a celluloid solution on the floor of his laboratory. Instead of splintering, the pot cracked, but the fragments remained together. This unexpected discovery allowed the french chemist to file the patent for Triplex glass in 1909. Safety glass quickly met with success in the automotive and aeronautical industries, ensuring the safety of passengers in a vehicle, as shown in Fig 1 .4. The significant role of the interlayer is to maintain the cohesion between glass shards and to dissipate a higher amount of energy during a choc.

Today, laminated glass is becoming more and more widespread in the building industry for several reasons. The first is the possibility of constructing large glass windows, such as a high-rise building, while ensuring the safety of the users by preventing a quick and easy perforation of the laminate. In addition to excellent mechanical properties, nanocoatings on the different surfaces of the composite allow the material adaptation to various applications. One example is the low-emissivity for thermal management. Another advantage is the possibility of integrating technology in the interlayer, thus using the glass as a screen displaying text and images. Finally, according to the objective of reaching carbon neutrality in 2050, glass seems to be an appreciated choice: the use of cullets, i.e. recycled glass debris from the float, in the manufacture of glass is increasingly widespread. Also, hydrogen or biogas can replace natural gas or coal, for example (e.g. [Saint-Gobain 2022]).

The manufacturing process involves laminating a sandwich composed of glass plies or panels connected to thermoplastic interlayers. The composite is produced in an autoclave with specific temperature and pressure conditions for a defined period, varying with desired adhesion strength. There, chemical bonds are formed between the polymer hydroxyl groups and the glass silanol groups, facilitated at moderate temperatures (∼100°C) .

The polymeric interlayer has an enormous influence on the global performance of a laminate during a dynamic loading. The polymeric interlayer is sensitive to temperature variation near its glass transition temperature, T g . A slight change in room temperature during an experiment with a laminate might lead to significant variation in the mechanical properties of the polymer. Zhang et al. highlighted this characteristic by performing impact tests with low-velocity loading, changing the interlayer material, and varying the room temperature between -30°C and 50°C.

In addition, to identify the differences between the response of the polymer under a blunt impact loading with an incident energy of 15 J, the authors highlighted the brittle behavior of the SGP (the ionoplast SentryGlas ® Plus) layers, forming large fragments at -30° [START_REF] Zhang | Temperature effects on the low velocity impact response of laminated glass with different types of interlayer materials[END_REF]].

The choice of a thermoplastic interlayer depends on the desired characteristics of the laminated glass. Some are well-known and used in different applications. For instance, thermoplastic polyurethane (TPU) and polycarbonate (PC) are used in aeronautics. Ethylene-vinyl acetate (EVA) and polyvinyl butyral (PVB) are cheaper and mainly used in building and automotive industries. The latter will be further considered in this section. The PVB presents good dissipating properties and is the most common interlayer in laminated glass because of its transparency, large dissipating capacities, large elastic strain regimes, good adhesion to the glass, and has a reasonable price: app. [START_REF] Cleary | Characterization and quantication of in-service windshield fracture mechanisms and evaluation of laminate sharp impact resistance as a function of construction[END_REF].4 e/m 2 for a 0.76 mm foil [START_REF] Martín | Polymeric interlayer materials for laminated glass: A review[END_REF]].

Polymer structure

It seems appropriate to look at the properties of the polymeric interlayer before visiting the whole composite behavior. Polyvinyl butyral (PVB) is a thermoplastic polymer whose macromolecules are essentially composed of three different monomers: vinyl butyral, vinyl alcohol, and vinyl acetate, whose chemical formulas are presented in

Polymer mechanical behavior

The PVB non-linear hyperelastic behavior in tension might be assessed regarding Fig 1 .6. Still, when the stress is released, it is indeed a visco-elastic behavior which is highlighted by the unloading path. It can also be seen that, with faster loading, the material stiffens and dissipates more energy. Its ductility also reduces with the increase of the strain rate. These phenomena are highlighted by authors like Zhang, that have conducted tensile experiments on PVB, some of which at very high strain rates, up to 1360 s -1 [START_REF] Zhang | The mechanical properties of Polyvinyl Butyral (PVB) at high strain rates[END_REF]].

Some mechanical properties of PVB are represented at ambient temperature in table 1.3 [S-LEC ® 2020; Saflex ® 2020; Trosifol ® 2020]. Because of the different possible formulations of this polymer, the PVB characteristics may vary significantly. However the table provides values for some of the PVB characteristics that give a good overview. The large strains and the ultimate tensile engineer strength then correspond to the elongation obtained by Fourton in Fig 1 .6.

When used close to its glass transition temperature, PVB presents non-linear, visco-elastic behavior. Also, visco-plasticity can be observed via tensile tests [Elzière 2016]. The visco-elastic behavior of PVB in a dynamic regime can be described using a generalized Maxwell rheological model (assuming small strains and a constant temperature). It gives the polymer shear modulus G(t) as a relaxation function expressed via its Prony series. Considering the PVB as isotropic, G(t) = E(t)/2(1 + ν) ∼ E(t)/3. It is modeled with N branches in parallel, presenting a spring with a modulus G i in series with a damper that has a relaxation time

PVB -Mechanical properties

τ i G(t) =G ∞ + N i=1 G i e -t/τ i G 0 =G ∞ + N i=1 G i (1.1)
One defines the instantaneous modulus G 0 as the immediate response of the material and G ∞ as the long-term elasticity [Elzière 2016]. Writing g i = G i /G 0 the real and loss moduli, directly sensible to pulsation ω in the complex domain, are

G ′ (ω) =G ∞ + G 0 N i=1 g i τ 2 i ω 2 1 + τ 2 i ω 2 G ′′ (ω) =G ∞ + G 0 N i=1 g i τ i ω 1 + τ 2 i ω 2 (1.
2)

The parameters of the PVB Prony series are reported in table 1.4.

g i = G i /G 0 τ i (s) g i = G i /G 0 τ i (

Glass-PVB interface adhesion

The adhesion between the PVB interlayer and the glass plies is a key parameter for the impact response of laminates. Too poor or too strong adhesion may lead to a loss in terms of performances of the composites [Elzière 2016]. Some easy to set-up tests are commonly used in industry. To cite for instance the pummel test (i.e. cohesive failure test), which is very dependent on the operator and its hammer, or the peeling test (i.e. mode I delamination of the polymer from the glass). However, none provide the mode II failure activation during a dynamic loading on laminates.

A simple test, firstly called "tension adhesion test" by Sha et al. [START_REF] Sha | Analysis of adhesion and interface debonding in laminated safety glass[END_REF]] then named TCT test, consists of a tensile test on a pre-cracked laminated glass specimen. One can measure the adhesion force versus the large strain of the polymer. The major energy is dissipated at the polymeric bridges (i.e. location where glass shards are interconnected via polymer ligaments). Two main phenomena are involved: the visco-elastic dissipation in the bulk of the PVB, and the delamination occurring near the cracks [START_REF] Galuppi | A homogenized analysis à la Hashin for cracked laminates under equi-biaxial stress. Applications to laminated glass[END_REF]. Hence, the TCT has become the most-used experiment to measure the residual and post-fragmentation behavior of cracked laminated glass [START_REF] Muralidhar | Mechanical behaviour in tension of cracked glass bridged by an elastomeric ligament[END_REF] Concerning the modeling of the interface, one common description of the glass-PVB interface is the cohesive zone model. The technique correlates the adhesion energy with the energy required to delaminate the interface. The Orowan theory is adapted and demonstrated by Needleman from the principle of virtual work [START_REF] Needleman | An analysis of tensile decohesion along an interface[END_REF]]. The author represents the normal tensile stress or force versus strain or normal displacement and assesses the adhesive energy of the interface. The cohesive zone developed by Needleman is a continuous model based on the description of void initiation and growth, in which the unbounded zone expands with delamination. This model fits well with the PVB, whose yield strength at ambient temperature is much higher in practice than the interface strength between the matrix and the glass [START_REF] Needleman | A continuum model for void nucleation by inclusion debonding[END_REF]. This cohesive zone model is adopted by Gao [START_REF] Gao | Intrinsic cohesive modeling of impact fracture behavior of laminated glass[END_REF] or Fourton [Fourton 2019] for instance, to study and simulate the mixed fracture mode I+II delamination.

Fourton has made the adhesion between the layers vary, grafting different silanes on the glass surface before lamination with PVB, then peeling the samples. The combined measurement of the macroscopic work of fracture and development of TCT tests on such samples have informed the author that adhesion modifies the interface dissipation and increases the crack contribution of polymer [START_REF] Fourton | Adhesion rupture in laminated glass: influence of adhesion on the energy dissipation mechanisms[END_REF]] (i.e. a strong adhesion causes a rapid rupture of the interlayer). The authors also splits the macroscopic work of fracture in the cohesive zone, G m , into a bulk dissipation proportional to the interlayer thickness h and an interface work of separation.

G m = h • Π bulk + 2Γ crack
The bulk dissipation, Π bulk , only has a slight change with the interface level of adhesion; its mean value is Π bulk = 5.2 MJ/m 3 . Thus, the crack contribution, Γ crack , is proportional to the separation energy term of the total fracture energy, Γ 0 . Fourton summarizes

Γ crack ∼ (Γ 0 ) 1 Π bulk ∼ (Γ 0 ) 0

Composite assembly process

There are several ways of manufacturing laminated glass. The first step is choosing the geometry of the final composite: the shape does not necessarily have to be flat (e.g. windscreens, curved panels), or the laminate can be composed of several polymer layers (e.g. armored glass). The thickness of the plies will vary according to the glazing application: armored glass being much thicker than a standard windshield, for instance.

Next comes the choice of material. Different types of glass and interlayers are available. A typical stack is chosen for the tests: two soda-lime glass plies joined with a PVB interlayer. Glass sharp edges are polished with an abrasive band to manipulate the plates easily during the making process. The float glass used for the impact loading presented in the manuscript will be the Saint-Gobain Glass Planilux ® and its clearer successor, the Planiclear ® . The PVB plies used in the composites are RB41 and RB11 from the supplier Eastman Saflex ® . After sizing the two glass plies and the PVB foils, assembly is carried out in a clean room to avoid the insertion of dust or external particles between the layers and then control the adhesion between the plies. Glass plies are cleaned up with distilled water and soap on both sides. After rinsing, the sandwiched plies are set into a vacuum bag under pressure and temperature for a specific duration.

Numerical models

Modeling glass

Glass subjected to dynamic loading can be modeled as an elastic brittle isotropic material, characterized by its Young's modulus, Poisson ratio, energy release rate, and tensile failure stress. The challenge remains in choosing a correct failure criterion, the crack propagation description, and their typology.

Easy to interpret, the first criterion corresponds to a critical stress criterion. For instance, Peng et al. defined a glass strength limit, implementing shell element deletion after cracking [START_REF] Peng | Finite element modeling of crash test behavior for windshield laminated glass[END_REF]]. Pyttel el al. used a Rankine criterion that also deletes the concerned elements when the stress overpasses the failure criterion [START_REF] Pyttel | Failure criterion for laminated glass under impact loading and its application in Finite Element simulation[END_REF]]. The two methods are implemented in a FEM solver, with an erosion of the fully damaged elements. One drawback of this technique is the loss of mass in eroded zones, which means a loss of inertia and a violation of the mass conservation in the case of dynamic loads.

Still, other methods exist. For instance, X-FEM splits the former elements with the crack evolution in the media (e.g. [START_REF] Xu | Investigation of dynamic multi-cracking behavior in PVB laminated glass plates[END_REF]). The peridynamic method also represents the glass discontinuity, as the materials are represented via discrete elements of a specific volume and density. Elements interactions are directly described with the nodes neighbors' relative positions and displacements [START_REF] Lai | A non-ordinary state-based peridynamics modeling of fractures in quasi-brittle materials[END_REF]].

The difference between the X-FEM and the peridynamic methods is shown in Fig 1 .7. For the same experiment, the authors are reproducing numerically the cracking pattern obtained after a low-velocity impact on a windshield. (e.g. energy dissipated per unit area of debonding), at 20°C, that is much constant for tensile rates between 10 s -1 and 100 s -1 , G a ∼2500 J/m 2 , but decreases for quasi-static TCT tests, G a ∼300 J/m 2 at 0.1 s -1 .

Other plate models have been used to describe the behavior of laminated glass. Vallabhan's model considers the bending as well as the tension-compression of the glass plies coupled with the shear of the interlayer, which is compared to quasi-static free bending tests of a laminated plate [START_REF] Vallabhan | Analysis of laminated glass units[END_REF]]. The shear of the interlayer is also considered quasi-statically in the Foraboschi model [START_REF] Foraboschi | Analytical model for laminated-glass plate[END_REF]]. The visco-elasticity of the interlayer is visited by Galuppi and Royer-Carfagni [START_REF] Galuppi | Enhanced effective thickness of multi-layered laminated glass[END_REF]. Steady-state delamination and stretching are explored by Fourton [Fourton 2019]. Finally, non-empirical models are also proposed for laminated glass. The high contrast laminate model developed by Viverge is using asymptotic methods, i.e. without assumptions on the shape of the stress and strain fields [START_REF] Viverge | Model of highly contrasted stratified plates : application to laminated glass[END_REF]]. All these models are very sophisticated for a limited payback, as they are not adapted to the description of the impacts presented in the following sections.

Dynamic behavior

After introducing the materials' origin, their behavior, and the composite properties, the focus is made in the following on the laminated glass behavior subjected to dynamic loading.

The behavior of laminated glass is complex. The latter does not react in the same way when it is subjected to quasi-static or dynamic loads. The study of the different damage steps represents a challenge: describing the cracks initiation, propagation, and arrest mechanisms in laminates. Thus the PVB interlayer plays a central role in the energy dissipating process, ensuring the cohesion of the composite and avoiding large fragments from splintering. Its visco-elasticity is a determinant factor for the global efficiency of a laminated glass submitted to an impact.

Describing the response of brittle materials, such as glass, to low-velocity impacts by solid bodies is an important issue because of its significance in terms of structural degradation and integrity. [Nassr et al. 2021]. The influence of the boundary conditions is also at stake. The reflection on the plate edges and the positioning of the plate can change the cracking and the overall response of the laminated glass during an impact. The authors note that the impactor head influences the transient response, i.e. the first period of the impact, but does not significantly affect the stretching and tiring phase that follows or changes the perforation threshold.

To understand the laminated glass behavior under low-velocity impacts, one must qualify the existing impact phenomena and quantify their contribution to the total dissipated energy. It is essential to establish a correlation between the experimental parameters of the dynamic loading and the measured dissipating phenomena. Experimental parameters are choices the experimenter has made before a test, e.g. the projectile mass, its blunt or sharp geometry, ply thickness, material aging, adhesion, glass chemical composition and PVB formulation. However, the different stored or dissipated energies that have to be measured are roughly represented in Fig 1 .9: the crack formation in the glass plies, the interface decohesion, the PVB visco-elastic dissipation, the inertia of the sample and the projectile, the energy lost in the test device or stored as elastic energy. Knowing the proportion of energy dissipated by each phenomenon and the influence of the parameters mentioned above provide keys for enhancing laminated glass performance and safety.

The glass company must submit a certain number of normalized tests that the new products must pass to be approved for sale. Manufacturers aim to evaluate the objective and representative performance of laminates submitted to an application case, like a stone impact on a windshield or a building break-in. Such tests may vary according to the requirements. Two European norms for the building industry can be mentioned: The data accumulated from many repetition of the cannon and blowpipe-like experiments has already shed some light on several laminated glass parameters. Examples include the size of the glass plies and interlayer, their adhesion, and the relationship between the number of chips and the impact energy dissipated.

The whole study of the dynamic behavior of laminated glass will be divided into three problem areas. The first study concerns the propagation and arrest of cracking in the glass from the earliest moments of an impact. The second part concerns glass fragmentation, the evolution of the crack density or number of shards created with the dissipated impact energy. The third section focuses on the post-impact residual resistance and residual strength of laminated glass.

The objective is to depict the composite behavior of pre-fragmented glass, i.e. to head towards a model of cracked laminated glass.

Propagation and arrest during dynamic indentation

Observing crack initiation, propagation and arrest help to anticipate the final crack typologies on laminates. One generally seeks to predict the evolution of a crack created by an impact. This phase firstly differs according to the indent geometry (Vickers, Berkovitch, Rockwell). If the latter is blunt (Fig 1 .10a), a cone crack followed by median cracks appears in the glass. The stored energy is initially more significant in the thin glass ply, and the many radial cracks rapidly appear at failure. If the indent is sharp ( This section mainly concerns the sharp indent impact in the spirit of gravel pitting. Well described in the litterature [START_REF] Dörkop | Investigation of the mechanism of stone impact on laminated glass windscreens[END_REF][START_REF] Lawn | Fracture of Brittle Solids[END_REF]], the formation of the star crack under these conditions can be divided into three stages. A first initiation phase through dynamic indentation of the glass having a localized defect on the impacted surface. A second phase is then initiated and corresponds to a sequenced in-depth progression of the crack up to the plate median plane. The third phase is unstable because the star-shaped crack forms and spreads very quickly, around 1100 m/s according to ultra-fast camera observations. Indeed, the crack reaches a zone of tensile stress due to plate bending, and propagation to the bottom of the first ply and lateral extension proceed very fast.

The propagation and arrest steps of a star crack formation are more complicated phenomena because of the existing competition between dissipating phenomena: crack opening energy, bending of the plates, and local stiffness decrease.

The study will focus on the design of a blowpipe test device to represent such a dynamic indentation, despite the conventional aspect of the test compared to a real graveling. The impact energy of the present kind of loading goes from 30 to 100 mJ. This device is highly instrumented to measure phenomena (gauges, high-speed camera). Observing through-thickness or in 3/4 view at 1 to 5 Mfps is possible with the high-speed camera Shimadzu ® HPV-X and allows to follow the damage evolution in the laminate. Concerning the parameters, the use of a chemically tempered glass or borosilicate glass will have to be reconsidered (e.g. [START_REF] Kang | Crack nucleation and growth during dynamic indentation of chemically-strengthened glass[END_REF]). Indeed, the strong superficial compression stresses prevent the nucleation of the cracking at a very shallow depth but, on the contrary, favor it at a sufficient depth (then highly unstable), which makes the formation of the star crack much more challenging to observe.

Fragmentation and energy dissipation during an impact

The laminated glass fragmentation study is in line with the reflection initiated by Nourry [Nourry 2005]. The device he developed allows to link impactor speed and kinetic energy dissipated during the impact with the glass damage, as shown in Some properties have a noticeable influence: impacting the atmosphere or tin side of float glass, PVB foil thickness, sample temperature during the tests, initial presence of surface scratches at the glass surface, glass ply adhesion with the interlayer, thickness difference between the two glass plies...

New cannon tests will be performed with well-controlled experimental conditions (e.g. controlled temperature via a thermostat set at 23°C) and richer instrumentation: use of high-speed cameras and stereo-digital image correlation (S-DIC) to monitor the laminated glass deformation, a telemeter to measure the impactor movement and optical sensors to capture the projectile velocity. The impact energy of the present kind of loading goes from 30 to 400 J. The experimental campaign will focus on the crack appearance on the laminated glass. At first, radial cracks are formed, quickly followed by orthoradial cracks throughout the test. The fragmentation pattern, which varies with the properties and the projectile velocity, is a critical factor in energy dissipation and will be highly reviewed via instrumented cannon experiments. The stretching of the PVB and glass-PVB delamination will be further analyzed regarding the live fragmentation evolution. This final part will be based on the described TCT model [Elzière 2016].

Residual resistance

Repeated impacts are historically linked to anti-infraction tests in buildings. To experimentally understand the behavior of a fragmented laminate, many tests can be set up. For example, a first impact can be performed, and the glass can be loaded, dynamically or statically, on different areas to observe the link between cracking and dissipated energy. We can also repeat the impact on the same point of our laminated glass. This second choice is made in the article Seshadri has been one of the pioneers that describe the interconnection between two fragments, connected by an interlayer polymeric bridge [Seshadri et al. 2002]. The author have looked at the delamination and the particular behavior of the areas between two glass fragments, which are considered here as two substrates connected by the very deformable polymer interlayer. In Fig 1 .12 the decohesion zone near these interfaces is schematized. Tensile stress of such sample introduces rate-dependent large deformations, stretching of the interlayer, and decohesion starting from the initial notch, followed by tearing. Conditions like ambient temperature or local heating during impact also play a role in the global response. The large strain rheology of PVB and its adhesion with glass are extensively studied in the theses of Elziere [Elzière 2016] and Fourton [Fourton 2019]. In addition to describing the stretching stage of PVB during an impact, with a simple test such as the TCT test, they also model the cohesive zone between glass and PVB in different ways. The importance of adhesion is also highlighted, and it is shown that the optimum performance of the laminate during impact is highly dependent on this property. The cannon device will host multi-impact tests to compare the performances between uncracked and already damaged laminated glass. A model will be proposed to measure the delamination at the crack edges of the fragmented laminated glass plate.

Experimental data processing: methods

After introducing the three problem areas, some experimental methods will be described in the current section.

Instrumented tests are based on experimental techniques whose concepts have already been introduced in the literature. Only the well-established theory, i.e. state-of-the-art, is introduced in these two sections. Any further developments will be explained in the bulk of the manuscript.

First, the strain gauge measurements and the study of the uniaxial elastic wave propagation performed to determine the force and velocity in the indenter during the blowpipe experiments will be explained. The theory is well-established in the community of dynamicists, especially in the analysis of Hopkinson bars experiments [START_REF] Kolsky | An investigation of the mechanical properties of materials at very high rates of loading[END_REF]].

Then, the stereo-digital image correlation technique (S-DIC) and the computer vision needed to analyze the laminated glass movement during cannon experiments will be introduced.

Hopkinson-Kolsky bar theory

A Kolsky-Hopkinson bar test uses long entrance and exit bars as a mechanical waveguide. The uniaxial description of the movement is based on the small diameter size compared to the total length of the bar. The set-up is also instrumented with strain gauges that deform with the elastic wave passage. Wheatstone bridges are used to detect the change in resistance of the gauges, which is converted into measured strain.

1.4. Experimental data processing: methods

Ch. 1. Introduction

Elastic wave propagation

In this section, small strain assumption is made:

|∂ x u x | ≪ 1.
The uniaxial wave propagation hypothesis in the indenter cylindrical body gives the following momentum equation on the cylinder axis, i.e. x-axis

∂σ ∂x = ρ ∂ 2 u x ∂t 2 (1.3)
where σ = σ xx is the axial stress, u x the displacement along x, t the time and ρ the density.

To perform a Hopkinson bar experiment analysis, the indenter is considered to remain elastic through the whole test, so that the elastic behavior σ = E∂ x u x , with E the Young's modulus, is introduced in eq (1.

3)

∂ 2 u x ∂t 2 = c 2 ∂ 2 u x ∂x 2 (1.4)
with c = E/ρ the elastic tension-compression waves celerity in the indenter.

The general solution of equation (1.4) is

u x (x, t) = f (x -ct) + g(x + ct) (1.5)
where f represents a wave propagating from left to right and g from right to left [START_REF] Meyers | Dynamic behavior of materials[END_REF]].

The instantaneous material velocity in the cross section is

v x (x, t) = ∂u x (x, t) ∂t = c -f ′ (x -ct) + g ′ (x + ct) (1.6)

Lagrange diagram

The Lagrange diagram allows the determination of the 1D velocity and stress in a bar from the gauge strain measurements, inspired from the Kolsky-Hopkinson bar set-up [START_REF] Hopkinson | A method of measuring the pressure produced in the detonation of high explosives or by the impact of bullets[END_REF][START_REF] Kolsky | An investigation of the mechanical properties of materials at very high rates of loading[END_REF]]. In the diagram, constant values are represented by straight lines of absolute slope c. The impedance z can be introduced in the equation

d(σ + zv x ) dt = ∂(σ + zv x ) ∂t + ∂(σ + zv x ) ∂x dx dt = 0 (1.7)
leading to z = ∓ √ Eρ in both ways dx/dt = ±c. Therefore, the stress and velocity are first computed at the gauge locations, considering the system at rest initially: σ(x, t = 0) = 0 and

v x (x, t = 0) = 0.
To measure the stress and velocity at the indenter, at a distance ℓ from the closer gauges, the waves information is transported at the cone bottom where the 1D propagation hypothesis is still valid. The stress σ(x = ℓ, t) and velocity v x (x = ℓ, t) can be expressed for all t > (L + ℓ) /c σ(0, t + ℓ/c) + zv x (0, t + ℓ/c) = σ(ℓ, t) + zv x (ℓ, t)

σ(0, t -ℓ/c) -zv x (0, t -ℓ/c) = σ(ℓ, t) -zv x (ℓ, t) (1.8)
L being the x-distance between the two sets of strain gauges.

The Lagrange diagram used to analyze the blowpipe experiments is represented in Fig A .2.

Digital Image Correlation Principle and stereo-digital image correlation (S-DIC) formulation

Digital Image Correlation (DIC) most often relies on the assumption of gray level conservation between a reference image f and a deformed image g, so that for each pixel x

f (x, t = 0) = g (x + u(x), t) (1.9)
where t is the time and u the pixel displacement. However, this equation is never strictly obeyed due to the presence of acquisition noise, dependence on the camera performances, and hence the displacement field is computed from the minimization of the quadratic difference between the two members of Eq. (1.9).

W(u) = ∥ f (.) -g(. + u)∥ 2 (1.10)
calculated on all pixels of the region of interest (ROI).

The philosophy is similar for global S-DIC with multiple view system, but minimization is performed for the 3D displacements and considering all the N c cameras at once [START_REF] Beaubier | CAD-based Calibration of a 3D DIC System[END_REF]Dufour et al. 2015b;[START_REF] Berny | High-temperature tests for ceramic matrix composites : from full-field regularised measurements to thermomechanical parameter identification[END_REF]. Using the parametric model with world coordinates as the reference, any point X moves at time t to a new position in 3D, X ′ = X + U(X). Based on the calibration step, this relation is expressed for all camera images, labeled by c, as

x ′c = x c + u c = [Π] c X ′ where x c = [Π] c X. With [Π]
a projection matrix explained in the next section. Thus a residual field is defined as

ϱ c (X, U(X)) = g c (x ′c ) -f c (x c ) (1.11)
and hence the 3D displacement field is computed from the minimization of

W(U) = N c c=1 X∈ROI ϱ c (X, U(X)) 2 (1.12)
The global displacement is decomposed over the mesh using the finite element shape functions, θ i (X), as

U = i U i θ i (X),
H c i j = X∈ROI ∂u c ∂U i • ∇ f c (x c ) ∂u c ∂U j • ∇ f c (x c ) b c i = X∈ROI ∂u c ∂U i • ∇ f c (x c ) f c (x c ) -g c (x ′c ) (1.13)
The minimization problem finally consists of successive corrections of the displacement field, δU = U -Ũ, from a system assembled from the sum over all cameras of their contribution to the global residual

N c c=1 [H] c • δU = N c c=1 b c (1.14)
1.4. Experimental data processing: methods

Ch. 1. Introduction

It is worth emphasizing that the measured displacement is directly the 3D nodal displacement field, minimizing the camera contribution (i.e. residuals for each camera) simultaneously.

As S-DIC matrices are ill-conditioned, some tricks such as the Tikhonov regularization can be used to solve the minimization problem [START_REF] Tikhonov | Solutions of ill-posed problems[END_REF]. The minimization problem becomes

        N c c=1 [H] c + λ[I]         • δU = N c c=1 b c + λ (U 0 -U) (1.15)
with [I] the identity matrix, U the current corrected displacement and U 0 the estimated displacement for this iteration. The weight λ associated to the Tikhonov regularization is chosen as

λ = 10 -2 • λ max (1.16)
with λ max the larger eigenvalue of the S-DIC Hessian matrix N c c=1 [H] c . Even if using a trick such as the Tikhonov regularization improves the computation and the problem conditioning, the solution is not satisfying as the enforced constraint is incorrect.

Stereo vision

Stereo vision consists in registering multiple images from different cameras. To correctly measure the physical shape and kinematics of objects, the calibration of stereo-camera system (i.e. identifying the projection matrix [Π]), is crucial to bridge the real 3D World coordinates The projection matrix, [Π], can be decomposed into an intrinsic matrix, [K] which characterizes the intrinsic parameters of camera (optical center, focal distance and skewness) and an extrinsic projection matrix, [RT] which gives the relative position of the camera and 3D object (translation and rotation). s is a scale factor.

The projection matrix [Π] of each camera is commonly measured via a calibration object [START_REF] Besnard | Caractérisation et quantification de surfaces par stéréocorrélation pour des essais mécaniques du quasi statique à la dynamique ultra-rapide[END_REF]Dufour et al. 2015b;[START_REF] Berny | High-temperature tests for ceramic matrix composites : from full-field regularised measurements to thermomechanical parameter identification[END_REF]]. The technique consists in monitoring this well-known object, whose geometry and size have been independently measured and stored as a 3D-mesh file, expressed with the world coordinates of the problem. A pre-calibration phase is implemented to provide a first determination of the projection matrices.

Then comparing the gray level monitored by each camera at predefined integration points location, the second phase of calibration provides a refined determination of the projection matrices [START_REF] Berny | High-temperature tests for ceramic matrix composites : from full-field regularised measurements to thermomechanical parameter identification[END_REF][START_REF] Beaubier | CAD-based Calibration of a 3D DIC System[END_REF]].

Chapter 2

Star-crack on laminated glass

Gravel pitting context

Resisting small impacts like graveling is at stake for laminated glass glazing. A sharp glass indentation is rarely avoided during an impact, and sometimes a crack appears. Many parameters may drive this breakage, including the projectile mass and speed, its geometry, or the laminated glass characteristics (adhesion of the plies, geometries, thicknesses) [Grant et al. 1998]. Therefore, minimizing the defect size during low-velocity impacts is a priority for windshield manufacturers since locally damaged parts on the external glass layer can be repaired if the crack extension is not too large. The existing norms in the domain concern the repair conditions of a windshield (e.g. norm NF R19-601 [Afnor, French Norm 2011]) that depend on the critical crack length and location on the windshield. Also thee are no defined construction standards for glass manufacturers, they have created their resistance criteria to highlight their product performances, with, for instance, a series of repetitive indentation tests characterized by empirical damage criteria. Various cracks may be observed during graveling, such as Hertzian cone or star cracks. The latter are critical defects as they may grow under quasi-static sub-critical loads or vibrations: e.g. glass residual stresses, aerodynamic load, car body deformations, and thermal stresses due to temperature variations. This damage geometry is first constituted of a local sub-millimetric indentation caused by the sharp indenter on the glass surface, then extended by branching selected radial cracks propagated due to orthoradial stress in the glass ply. The poly-vinyl butyral (PVB) layer and the inner glass ply -which remains undamaged at the end of a typical gravel impact -play a role in the final crack geometry.

The behavior of thin glass or laminated glass plates under the low-velocity impact of small projectiles has been reported in the publications [Behr et al. 1993 Cleary et al. 2020]. Authors analyzed the live crack formation, performed postmortem analyses, and designed specific devices to reproduce such impacts in a controlled way. It can already be observed that the projectile impact angle and velocity are important parameters for the star crack appearance on the laminated glass sample. For an identical loading, a star crack is formed for thin plies, but a conical mark appears on the surface of thick plies. Examining the angle of incidence during the test reveals that only the normal components are involved in the damage [Grant et al. 1998;Cleary et al. 2020]. Such methods can be extended to many different glass compositions as ion-exchanged glass or borosilicate glass. The median crack evolution in strengthened glass can also be at stake because the ply breakage in the whole thickness cannot be avoided when it reaches a critical depth. This phenomenon has been observed via X-ray observations [START_REF] Kang | Crack nucleation and growth during dynamic indentation of chemically-strengthened glass[END_REF]].

However, few measurements are acquired directly on the indenter itself except for its speed. In the present study, a new device named blowpipe is introduced to replicate conditions met for stone impacts, leading to similar crack patterns. This device is designed to measure both force and velocity, with a high temporal resolution, during the impact on the laminated glass sample. The measurement principle is inspired by the Kolsky-Hopkinson method [START_REF] Hopkinson | A method of measuring the pressure produced in the detonation of high explosives or by the impact of bullets[END_REF]; [START_REF] Kolsky | An investigation of the mechanical properties of materials at very high rates of loading[END_REF], with strain gauges positioned on an input bar. This input bar, inserted between the tested glass and the impactor, transfers the dynamic loading and is therefore used as the indentor. As it remains elastic under graveling impact, the gauge strain measurements can be used to estimate the contact force. Additionally, a synchronized high-speed camera is used to monitor the star crack formation on the laminated glass: the Shimadzu ® HPV-X2 camera is used at an acquisition rate between 1 to 5 Mfps, with a fixed definition of 400×250 pixels. A movie of 256 frames is registered during a test. The resulting spatial resolution will be approximately 20 µm/px.

A long blowpipe

A first gravel resistance test is carried out with a projectile striking a flat laminated plate of about 300×300 mm 2 [Kishta 2018a[Kishta ,b, 2019]. The 3.2 g projectile is dropped from a specified height, from 50 cm to 2 m, led through a vertically suspended pipe. The impact energy varies with the projectile's initial height, giving the theoretical impact velocity at the end of its stroke. Many of those experiments were carried out to see the influence of different parameters: projectile height, glass initial damage, and glass ply thickness.

Finally, with such a device, it is possible to analyze and classify the influence of the laminate properties, considering a significant standard deviation in the results. The different tests are not highly instrumented and thus not quantitative. Indeed, the projectile velocity, overestimated due to the friction in the pipe, is not measured. Moreover, the impact angle of the indenter is often not normal to the glass surface and varies with the pipe wall, making the observations difficult. So, it is impossible to detect differences in cracking scenarios between a free bending sample and a sample laid on a rigid support. Some high-rate films have been recorded during an impact. The 5 Mfps movie is triggered with an accelerometer placed on the glass surface. Two viewpoints have been chosen to identify the initiation and the crack propagation: an edge vision to follow the in-depth propagation of the damage and a 3/4-view that enables live monitoring of the star-crack formation. Firstly, cracks begin to propagate through the thickness of the impacted ply. Secondly, an unstable propagation happens once the crack tip reaches the middle of the first glass ply and a centimeter-length star crack pops at 1100 m/s around 15 µs after impact. This crack propagation velocity is measured with the high-speed camera with a good uncertainty due to the good temporal resolution of the movie.

The crack initiation is well documented via those videos and is consistent with other observations, for instance, the crack in-depth evolution observed by Chai and Ravichandran [Chai and Ravichandran 2009]. However, the impact conditions are not quantitatively well described. The conditions of the arrest of the star-crack are not fully understood and measured. The role of the PVB interlayer and its adhesion to the glass are not known, neither is the bending of the plate, nor is the potential local PVB-glass delamination. These primary questions are still pending after the long blowpipe experiments. So, the main goal of the present study is to develop an instrumented test device, including indenter motion and force measurements. Following the crack evolution in the thickness and 3/4-view is also a plus that will validate the first conclusions of the former tests. Moreover, a shorter device design is chosen to facilitate the impact tests and their repeatability.

A shorter (enhanced) blowpipe

The device aims to reproduce the star crack and not to make a census of all possible graveling tests.

FIG. 2.1 -Blowpipe prototype

The short blowpipe prototype is represented in Fig 2 .1a. It has been designed to fire a 3 g projectile at 5 m/s. The structure is cut from a 40×40 mm 2 section aluminum rail. A support is made from profile bars, making the adjustment of the vertical or horizontal position of the blowpipe possible. The barrel is clamped in a bore. When adjusted to the correct height, the tube allows the extremity of the indenter to be held over 2 to 3 mm. A pin, the projectile, and the spring are then placed in the tube. The loading is applied in the form of elastic energy in the spring, varying the stroke of the screw. As a remark, spring-aided systems have already been used by other authors for impact tests with balls [Sun et al. 2005b]. A shot is made when the pin is pulled out. Thanks to the large mass of the bench, the spring and projectile inertia are well absorbed during the test.

In Fig 2 .1b, One can see the instrumented indenter, inserted at the pipe exit. Gauge red cables are long enough (∼100 mm) to adjust the position of the indenter with the desired impacted location on the glass ply. Impact simulations are performed on a laminated glass (Fig 2 .3) and on void (i.e. no sample at the tip of the indenter, Fig 2.4). The purple curves correspond to the measurements made via the two virtual gauges and the Lagrange diagram, using the wave equation (1.4). High frequency components are present in the Lagrange diagram, however the low frequency components (green curves) of the Lagrange-measured force and speed (purple curves) are quite matching the reference (nodal fields computed at the cone bottom, black curves).

A direct low-pass filtering of the gauge signals is also possible (i.e. before the Lagrange procedure). The resulting force and velocity, in 6 is considered a dispersion of the dynamic waves in the bar. Equations introducing the Poisson effect are described in appendix A. Therefore, using this method, a physics-based filter penalizes the higher frequencies in the Lagrange diagram, and the obtained signal is closer to the reference. The model gives the best results and will be used on noisy experimental data.

The preliminary simulations of the blowpipe test justify using strain gauges to measure the speed and the force at the indenter-glass contact.

Gravel impacts are then performed, synchronizing these measurements with an ultra-fast camera monitoring the crack formation.

Blowpipe prototype 2.2.1 First experiments

First tests are carried out with a Rockwell indenter. The camera is triggered with an accelerometer detecting surface waves on the glass. In the surface wave acceleration on the glass surface. The lack of accuracy and the impossibility of synchronizing the camera images relatively to the gauges curves led us to the decision of deciding for another means to trigger the camera. In the following, a gauge signal threshold is used to start the camera shooting and to ensure a perfect synchronization of the devices.

Indenter body bending

In Fig 2 .8a are localized the four strain gauges (principle further explained in section A.4) glued to the indenter at two longitudinal positions L and ℓ. Each gauge is connected with a quarter

ε i = ∆e i U i • f • G (R 1 + R 4 ) 2 R 1 R 4 = 4∆e i U i • f • G (with R 1 = R 4 )
with U i the generator voltage, the gauge factor f and the gain G. 9d are similar and validate that the average of the two axisymmetric gauges makes it possible to suppress measured bending induced by an off-centered impact between the projectile and the fixed indenter, expressing the mean signals:

ε upp. = ε 1 + ε 2 2 & ε low. = ε 3 + ε 4 2
Due to dispersion, the velocity of the bending waves is much lower than the traction/compression ones. On the right of Fig 2.9, the bending signal is detected way after the first passage of the compression wave on the lower gauges. This difference is not significant at first, but it can affect the mean signal analysis, thus the force and velocity measurements, at the end of the impact (from ∼35 µs).

Synchronization of gauges and camera

As mentioned earlier, a good synchronization between strain gauges and camera images is essential. Indeed, the objective is to observe the opening and arrest mechanisms of the star cracks and, in parallel, to measure quantities at the base of the indenter cone via the gauges signal. Fig 2 .10 shows the different periods at stake. Here the formation of the star-shaped crack takes place over 50 µs. The camera triggers off when the first gauge measures a descending signal at a threshold of -150 mV, corresponding to a compression strain measurement of about 10 -5 . The level was decided to be low enough to ensure the trigger function and high enough to be insensitive to the permanent signal noise (i.e. oscillations around 0 V with a small amplitude: under 0.1 V). The time interval between the strain gauge data points is 0. 2 MHz movie is not an issue in the capture of cracking due to the camera (Shimadzu ® HPV-X) ability to record the previous frames when receiving the trigger. However, a 5 MHz film has a fixed duration of 51.2 µs and misses the first moments of impact testing: elastic deformation and dynamic indentation. Still, it is possible to capture the crack opening and arrest.

In the experiments presented in section 2.3.2, performed with the final device, this issue is corrected by using an oscilloscope as an external trigger for the gauge signal recording device and camera frame.

Impacts with Rockwell indenter

This section deals with impact tests performed with a Rockwell (i.e. spherical) indenter. The curvature radius of the tip is close to 200 µm. The indenter body diameter at the gauge locations is 5 mm, and the indenter cone is 120°. The elastic properties of the indenter, used in the Lagrange diagram, are given in table 2.2.

3/4 view

The excellent synchronization between the camera and the deformation gauges makes it possible to follow star cracking and discern different stages during the impact. In Fig 2 .11, the first observation to be emphasized is a peak in force and velocity at the beginning of impact. If the velocity and displacement signals show a double load, the latter is more difficult to observe on the force curve that presents multiple peaks. The initial positive force (in Fig 2 .11) is caused by small defects in the gauge position on the bar. It has been corrected in the final device results in section 2.3.2, making small adjustments of the gauge location on the indenter body. At the same time, the force measured here seems slightly too high compared to test dimensions: greater than anticipated in the elastic simulations. However, the energy (i.e. The force times velocity integrated over time) transmitted to the laminated glass during the impact is of the same order of magnitude as the theoretical energy initially loaded into the spring: approximately 80 mJ. The velocity peak magnitude is also expected. Some of the difference between the 50 mJ on the curve is due to energy loss caused by friction in the pipe and to the kinetic energy remaining elsewhere (e.g. projectile, bench parts).

It is possible to follow the star crack branches length evolution during the impact with the help of the camera. The use of monovision makes it challenging to measure the crack evolution outside the focal plane. Therefore, only the standard lengths (i.e. current lengths divided by their final value) of the four visible cracks are shown in Fig 2 .13. The fast crack opening during phase 2 is highlighted with the observed crack length evolution occurring at 20 µs. In phase 3, two plateaus are observable at 73% and 60% of the standard length.

Side view

The 3/4-view film makes it difficult to observe cracking in the thickness of the external glass ply and, more generally, any cracking that does not appear on the impacted side of the glass. The camera installation shown in Fig 2 .14 allows the steps of crack formation through the glass ply thickness to be followed.

The idea of this tracking in the thickness of the glass is to reflect the light into the lens as the crack forms. The two 400 W spotlights are initially directed towards the indenter but perpendicular to the camera. This gives a dark image at the beginning of the test, except for a reflection on the metal part of the indenter, as observed in Fig 2 .15a. The aim behind this throughthickness view is to show the crack via light reflection. The two 400 W light spots are initially directed towards the indenter but perpendicular to the camera. The first image of the test is, therefore, completely dark. However, the metal of the indenter reflects a small fraction of the light in the camera sensor (see Fig A dashed red line represents the plane between glass and air that behaves as a mirror. A dashed green line separates glass and PVB. Orange lines represent the reflections of the same crack, and the blue lines highlight the indenter and its reflection. One may notice three distinct crack visualization zones: the upper zone is a reflection on the external layer of the external glass ply, in the middle is the direct observation, and at the bottom occurs a reflection on the interface with the PVB layer.

The triboluminescence phenomenon appears at time t=22.2 µs and disappears after 1.4 µs. This brief flash of light and the luminous dot at the indenter tip, captured in A two-step crack opening is interpreted from the two global increases in gray levels. A peak at 22.8 µs is due to ephemeral light emission during fracture of the glass sample. The literature reports similar emissions of photons, electrons, and ions with ceramic fracture [START_REF] Dickinson | The emission of electrons and positive ions from fracture of materials[END_REF][START_REF] Lawn | Fracture of Brittle Solids[END_REF][START_REF] Kawaguchi | Fractoluminescence in minerals[END_REF]]. The second peak at 26 µs is the light reflection due to a permanent crack in the glass that remains in the following images. Finally, this side-view validates the rebound phase happening at around 40 µs and the secondary crack growth phase between 40 and 60 µs. The indenter stops sinking into the glass at the end of phase 4 (∼55 µs), which is consistent with the measured decreasing light intensity. It signals the end of the crack formation.

Reproducibility

Excellent reproducibility of the impact tests performed with the blowpipe test prototype is essential to understand the difference between the different crack patterns. Velocity and force curves are plotted for six successful tests in Fig 2 .18. The overall trends are relatively similar but not identical. It can be noted the use of the same laminated glass sample, the same spring, and indenter, therefore all experimental conditions were chosen to be identical. The observed slight variability is attributed to the friction and the chosen impact location on a laminated glass plate. The imposed velocity of the projectile is also not perfectly tuned with the prototype but a better control is achieved, via a cell force, with the final device. Those repeatable experiments allow us to draw parallels between tests and to create links between the different observations made with the camera.

The prototype allows reliable velocity measurements of the indenter movement. However, concerning the force, positive peaks can be measured at the beginning of the test (Fig 2 .18), corresponding to noise and non-physical signal, which renders the force measurement reliability weaker than the velocity even if its averaged signal remains consistent (Fig 2 .3). The noise level on the force signal will be further discussed at the end of the next section.

Final blowpipe device

Many information introduced in this section is taken from [Le Gourriérec et al. 2022].

Experimental set-up

In this section, several features are common to the previous prototype. These are still worth presenting because of the different details and enhancements provided by the final blowpipe device.

A test using the final blowpipe device is fully detailed in Appendix B.

Experimental blowpipe device

The experimental device is inspired by a car manufacturer test for gravel impact resistance on windshields. Many impacts are performed on a large laminated glass plate, and the critical size of the defect and its occurrence regarding a large number of impacts determines whether the laminates pass an empirical criteria.

These tests use cylindrical steel-made projectiles with a cone-end. The cone tip has a 200 µm curvature radius and is made of synthetic diamond. The projectile velocity range is 3 to 8 m/s. on the sample surface. An aluminum frame supports the 15 cm long pipe. Inside the tube, the impactor is initially retained via two balls. Then a spring is compressed between the impactor and a steel spacer, ending with a cell force. The cell force gives the initial static force in the spring. A compressed air intake pushes the balls aside during the release phase and clears the way out to the impactor. After a 5 cm stroke, the impactor hits the indenter, only guided on 3 mm at the pipe extremity to avoid undue friction from the tube. The steel-made projectile mimics the gravel mass and velocity. The stored elastic energy in the spring is converted into kinetic energy, and the cell force measurement then enables to estimate the impactor velocity.

As shown in

A PMMA frame, whose dimensions are identical to the laminated glass plate, supports the sample. The width of the sides is 5 mm, which enables the plate to bend under the indenter during an impact. The choice of PMMA as the frame material does not interfere with the global compliance of the bending laminate during the first tens of microseconds of the experiment.

The key property is that the surface of the laminate, opposite to the indenter, must be a free surface to help the star-crack to be formed.

The crack evolution is fully monitored via a Shimadzu ® HPV-X2 camera at a 1 Mfps minimum frame rate. Its focus on the known impact location can be finely tuned before the experiment. Moreover, the fixed indenter is instrumented with two pairs of strain gauges. Gauges of the same pair are glued on both sides of the cylinder. Two sites can be distinguished: the upper site (labeled 'A' in

Synchronization between the high-speed camera and the gauges

The trigger adjustment requires a high accuracy in dynamic experiments, as high-speed phenomena are occurring and need to be apprehended. On the final device, the Shimadzu ® cam-era is synchronized with a 0.1 µs uncertainty with the descending signal on an upper site gauge at a level of -0.2 V. Once this level reached, a 5 V step is sent to trig the camera. It is also plotted over the gauge signals and allows for the synchronization of the camera and the strain gauges through the definition of the same time origin. The excellent synchronization between the measurement devices is shown in Fig 2 .20. The local standard deviation of the gray level under the diamond tip of the indenter, corresponding to the median crack evolution in the thickness of the glass ply, is correlated to the tip displacement measured via the strain gauges. This gray-level evolution can be measured thanks to side lighting that is reflected in the camera lens with the occurrence of the crack, as further explained in section 2.3.2.

Impact velocity range

Tests on 1.6 mm thick soda-lime glass plies are carried out to determine a link between the first peak velocity of the indenter extremity and the initial compression force in the spring measured by the piezoelectric cell force. The results are plotted in Fig 2 .21. A linear interpolation is proposed, validated thanks to the experimental data, allowing one to predict the peak velocity from the spring loading. The results are strongly dependent on the impacted sample response to the dynamic loading and may also vary with the friction in the tube, the fast crack formation on glass right under the indenter tip, or the initial position of the indenter on the glass surface. However, the bench is deemed reliable, with results matching the linear regression.

Reproducibility

The superimposition of the force and velocity measurements at the bottom of the indenter cone is shown in Due to the measurement technique via the Lagrange diagram, the stress measured close to the indenter is quite noisy. The signal rippling sometimes leads to a local null force value or even a small tension load due to wave propagation in the steel bar. These instants are linked to crack growth, hence a local gray level variation under the indenter measured by the camera (as will be discussed further down).

Three velocity peaks are clearly measured at the cone bottom on the velocity data. Their width corresponds to the necessary duration for the elastic waves to propagate along the 60 mm long indenter body: 12 µs. The maximum value for each peak decreases, so the indenter globally slows down as expected from the energy dissipation in the damage of the indentation.

The choice of representing only the three first velocity peaks (i.e. 60 µs) is justified experimentally, as the radial growth of the branches of a newly formed star crack on a laminated glass sample becomes negligible after these three periods.

Experiments and results

Monitoring a star crack formation on laminated glass

An instrumented impact is performed on a laminated glass plate.

Through-thickness observation on a free-bending sample

Through-thickness measurement is also performed to estimate the crack depth evolution. In this experiment, a free-bending laminated glass sample movement is captured, and the crack growth under the indenter tip is monitored. c) and corresponds to a bright zone under the indenter. Until reaching the apex of the first velocity peak, the crack grows in depth up to 0.8 mm, which is half the impacted glass ply thickness. Then, a fast crack growth occurs: it propagates through the whole glass thickness, and its radial growth rate approaches 1000 m/s (cf the orange curve in Fig 2 .25b). This fast propagation phase of the crack may only take place when the crack tip has reached up to half the glass ply thickness beyond which the bending of the laminated induces tensile stress. This is in accordance with the already known effect of the impacted glass ply thickness on the gravel impact resistance [Grant et al. 1998]. During the decreasing phase of the second peak, slower secondary crack growth is also observed (Fig 2 .26(d-f)).

A previous study [Kishta 2018a] reported the observation of crack growth through the thickness of the stack with the same camera. On the first frame in

Force measurement uncertainties

The excellent reproducibility of the force and velocity signals is illustrated in Fig 2 .22. The force changes slightly from one impact to another in the same testing configuration. If the mean force imposed on the structure looks repeatable, the accuracy of the peak still needs to be discussed.

Therefore, impacts on polystyrene foams are performed using the blowpipe device. Such impact allows uncertainty measurement as the stiffness and impedance contrasts are very high between the steel body of the indenter and the foam. The transmitted force should be null or close to zero during the test.

The force and velocity of three tests on polystyrene, performed with an initial static load in the spring, F cell , of 30 N, are reported in Fig 2 .28. Velocities are represented on the bottom row. It shows the very accurate and reproducible measurement of the velocity peaks that may be found in a dynamic indentation test on laminated glass with the blowpipe device. However, the raw force signal is noisier. The mean force is obtained with a mean glide filter applied on 24 µs. It oscillates slightly around the 0 N. The peaks amplitude may rise to at most 200 N in the duration of interest of these particular experimental cases.

The oscillations and the high amplitude of the reflected waves during an impact on the glass surface are partly due to the impedance ratio between glass and steel: Z steel /Z glass = 3.

The low level of transmitted force is also explained by the conical shape of the Rockwell indenter that allows Hertzian contact during the first microseconds of an experiment: the designed angle of 130°close to the diamond tip provokes spurious elastic waves reflection at the cone surface. The mean force on 24 µs is plotted (red line, --), showing more minor oscillations around 0 N.

The stress imposed on the glass surface is locally high, allowing crack initiation. Still, the force signal reported at the bar end remains challenging to interpret and illustrates the large amplitude of the reflection waves during a test on laminated glass.

Considering these results, it was chosen in the model to simulate the whole indenter body and to apply the measured force signal at the contact section with the projectile (i.e. at the top of the indenter). The amplitude of the first impulse reaching at least 2.5 kN (Fig 2 .30), the relative error regarding the determined force uncertainty is decreased at 8%, versus 90 to 100% at the cone bottom.

Dynamic indentation model

The imposed force and velocity on the glass sample are determined from strain history measurements by using a Lagrange diagram. These measurements are representative of the indentation behavior during the experiment. Hence, it is not yet possible to conclude about the damage zone behavior as the measured signal does not only depend on the response of the impacted sample but is also sensitive to the boundary conditions, the sample geometry, the material, or the damage evolution. In order to identify the behavior of a locally damaged zone at the glass surface, the use of dynamic simulations is proposed to interpret the measurements as a response of the local damage created on the glass ply.

The idea, illustrated in glass surface. This element is tiny compared to the whole sample (3) and the indenter (1), so it has virtually no inertia, which implies that the measured force at the end of the input bar ( 1) is also locally applied to the element (2) and to the plate (3).

U e , an extrapolation of the laminated glass displacement at the initial contact point between the indenter and the glass surface, is calculated via a visco-elastic (linear) simulation reproducing the diamond tip and the glass plate with the effort boundary conditions of the experiment. By comparing the numerical result to the experimental displacement at the cone bottom U x , one can evaluate the indentation depth δ = U x -U e , i.e. the effective displacement discontinuity across the damage region. The behavior of the non-linear zone (2) is then finally captured by the evolution of the force at the indenter F as a function of the indentation of glass δ.

So, the objective is to identify an effective behavior of the local damaged zone, comparing the experimental response of a sample to the simulated response of a visco-elastic laminated glass.

The same philosophy was pioneered by Lawn [START_REF] Lawn | Fracture of Brittle Solids[END_REF]], where the author considered the overall quasi-static response of the indented glass sample as a superposition of the reversible elastic behavior of the set-up and the non-linear local plastic zone created in a radial/median crack system.

The first step is the making of a visco-elastic dynamic simulation to obtain the damage-free response under the same imposed loading. To do so, the set-up geometry is meshed and further studied in the Abaqus ® CAE solver via a dynamic explicit scheme, introducing its steelmade cylindrical body down to the diamond indenter. The tip is in contact with the glass surface, modeled as an elastic material, respecting the normal incidence between its revolution axis and the plate surface.

The different parts and their characteristics are reported in Table B.1. It contains the data used in the simulations for the steel-made indenter and its diamond tip, the glass and PVB material. The parts geometry replicates the experimental device. Therefore, the steel-made body cylinder is 60 mm long and has a 2.5 mm radius. The indenter ends with a 200 µm curvature radius diamond set initially on the glass surface. The layer thicknesses of the 100×100 mm 2 plates are respectively 1.6, 0.76, and 1.1 mm from top to bottom. The free bending is ensured by a It is then possible to identify an homogenized behavior of the locally non-linear indented glass by plotting the experimental force, which should be identical in the simulation and the experiment (neglecting the size and inertia of the small volume responsible for the non-linear behavior) versus indentation δ (see Fig 2.31b). This curve is an average obtained by simulating eight impact tests on laminated glass with initial speeds ranging from 5 to 8 m/s. The geometry of the stack is unchanged. Only the mean measured force is used because the higher frequencies of the force signal are induced by the dynamic effects coupled with the device geometry.

Parameters

Glass

Fig 2 .31a shows the obtained result after considering a local confined damage zone under the indenter tip. The damage behavior is described during the loading up to the maximum force value with a plastic law, introducing its characteristic length, which is approximately the width of the crater created on the glass surface: 600 µm. Indeed, the plasticity of the damage zone is performed using the direct F vs δ law (Fig 2 .31b) and a characteristic volume length υ, which is for a large part arbitrary, but needed to speak the language of a constitutive law for a volume element. Therefore, the stress σ = F/υ 2 and the plastic strain ε p = δ/υ are defined to comply with the plastic law used in the 3D simulation.

Of course, this model only describes the loading phase and cannot be used to characterize the discharge. The displacement obtained at the bottom of the cone with the plastic model (U ep ) matches the experimental results (U x ), meaning the model allows to replicate the correct experiment kinematics.

Comments on the indentation zone

The indented zone is challenging to analyze. The multiple fissurations at the indenter tip after an impact can be seen on a SEM image (Fig 2 .32). The topology shows a complex comminution zone with many cracks in the spherical newly created crater at the glass surface. Two starcracks are expending outside the main damage, up to a few millimeters long. The circular flake around the indentation is present at the surface of the crack, caused by crack propagation just beneath the glass surface. These are not major defects. In Observing these two images makes the 1D approach, developed to characterize the damage volume behavior under dynamic indentation, elementary. It undoubtedly can be improved and complexified, but describing the laminated glass under gravel pitting does not necessarily need a too fine description of the complex phenomenology occurring at the end of an indenter. Therefore, the non-linear local behavior described in this section is one of the best compromises that can be made to represent this kind of impact.

Conclusion

This chapter has introduced a newly designed device to analyze the star crack formation via dynamic indentation on laminated glass. The input force and velocity are measured close to the damaged surface of the sample and synchronized with ultra-high-speed camera imaging, with one tenth of a microsecond uncertainty. Depending on the camera location, one may measure the median crack in-depth propagation or the radial branches appearance. The difference between the experiment and the linear visco-elasto-dynamic simulation leads to an adequate unidimensional elastic-plastic description of the confined damage for such dynamic indentation.

Even if the device geometry is quite different from a true gravel impact configuration, the indenter large-scale geometry and stiffness, the projectile initial velocity, mass, and size can be considered to be close to real gravel dimensions and impact velocities. Moreover, the uncertainty in the force measurement can be managed by reducing the projectile and bar impedance to be comparable to the glass impedance. But it requires a change in material or cross-section, which would denature the test.

Thanks to these quantitative measurements on the star crack formation, it is possible to determine the behavior of the locally damaged laminated glass and potentially see the influence of various parameters, such as the ply geometry or the material, on the damage size and dissipated energy.

From an experimental perspective, it is also possible to add a local measurement of the deflection on the opposite side of the sample to directly assess the experimental indentation δ and compare it with the visco-elastic simulation results. A device improvement in terms of higher range of initial indenter velocity (more than 10 m/s) can be planned by increasing the dimensions of the load spring or adapting the compressed air arrival directly on the projectile section.

The short-term use of the device today can be adapted to other studies: a comparative study of varying glass shades, PVB formulation... Alternatively, one could change the indenter shape or material to test carbide tools indentation via a controlled and instrumented experiment, for instance.

The final objective of this study is to predict the star-crack length during a gravel impact. The next move using the blowpipe device should be the use of an elasto-plastic simulation, plotting the stress state in the elastic glass plate, then comparing it with the glass stress at break during dynamic loading. The results could be compared with an enhanced crack propagation monitoring to ensure the crack propagation and arrest are well described in the simulations. Else, it is also possible to initiate a radial crack in the simulation and study its propagation during an impact while still modeling the non-linear zone at the indented location.

Chapter 3

Fragmentation under impact

Context

Characterizing and quantifying the mechanical response of laminated glass during a lowvelocity impact, especially structural degradation and integrity, is a critical issue for laminated glass manufacturers. Such impacts can be made by people hit by a car, people falling from their height, or hand tools during a burglary, for instance. Each loading case has a velocity in the range of 10 m/s.

Many experimental studies have been carried out to fully understand the mechanics of laminated glass under impact, e.g. the semi-rigid or head-form impact [Yu et Describing the response on the whole fragmented laminated glass plate is rarely performed in the case of rigid-body impact. These experimental works have brought significant characteristics of the dynamic behavior of laminated glass through a few key quantities. However, these measurements do not provide a global quantitative picture of the whole plate. In particular, because of the very heterogeneous nature of fragmentation and deformation, it is tough to validate numerical modeling from such observations. The present study aims to introduce an instrumented test that provides a more detailed description, global and well-resolved both spatially and temporally.

Ball-drop experiment

The present study focuses on the low-velocity rigid body impact on laminated glass in the context of the anti-break-in as described in standard EN 356 [Norme Européenne NF EN 356 2000], corresponding to a series of three successive 4 kg steel ball drops from a height in the range of 1.5 m to 9 m, on a 1 m 2 laminate. This standard focuses on the prevention of perforation, which is also of utmost importance for protecting people against falls.

Several steps are observed after an impact between the ball and the laminated glass sample [Nourry 2005]. First comes the formation of the radial cracks, expanding from the impact point to the edges, and the first glass fragments. Secondly, the laminated glass plate crack density increases with bending. The number of radial cracks increases slightly, but orthoradial cracks are created at each loading step, up to perforation. Sometimes, in the central region where the impactor hits the surface, a dense fragmentation takes place,leading to the formation of a cohesionless powder (communution zone). Thirdly, PVB is stretched, and delamination of the glass fragments occurs. The two latter phenomena are coupled and represent the majority of dissipated energy during an impact on the laminated glass.

Adhesion in steady-state regime

During an impact on the laminated glass, the glass fragmentation dissipates a small amount of energy, which is directly linked to the total crack area created on the glass plates. This dissipation reaches 1 to 2 percent of the initial kinetic energy of the steel ball [Nourry 2005]. Thus, this phenomenon will not help to stop the projectile.

However, at each created crack location, the delamination and stretching of the interlayer occur. The adhesion strength between the layers becomes an essential parameter for the impact resistance: the higher the adhesion, the more energy is needed to separate PVB from glass. But in practice, too much adhesion prevents the PVB from being stressed over a larger area.

The localized stress initiates a tearing of the polymer layer and finally dissipates a small portion of the initial energy of the projectile. Conversely, a lower adhesive strength allows the stretching of the PVB bulk over a more significant distance, activating visco-elastic dissipation in the interlayer [Fourton 2019]. In the latter reference, the author studied the coupling between the dissipation at the glass PVB interface and in the polymer bulk via TCT tests, changing the imposed tensile velocity and interlayer thickness.

In the literature, models such as a dissipation via cohesive interface simulation expressed with a power law [START_REF] Landis | Crack velocity dependent toughness in rate dependent materials[END_REF]] can be found. The associated steady-state model with Fourton work uses a visco-plastic description of the energy dissipation at the crack tip. It links the crack opening velocity with the plastic strain rate. This plastic strain rate εp i j is given in J 2 -flow theory by

εp i j = εp 3 2 σ ′ i j σ with σ ′ i j
ẇp = σ i j εp i j
Understanding the author model is of interest to describe a ball drop. At the crack tip, the dissipation mechanisms are similar to those described by the author. Using tracking with a fast camera during the test, may lead to a quantitative estimation of the total area of decohesion. In that case, one may deduce the amount of incident kinetic energy dissipated in the breakage of the glass, then in decohesion by linking the advance of the breakage in the cohesive zone to the cracking area correlated to the energy dissipated in the bulk of the material. This leads to a good performance evaluation of the material under impact.

Cannon device

Due to the large size of the laminated glass sample and the use of a spherical steel ball as proposed in the EN 356 norm, it is quite complicated to perform quantitative measurements of all the phenomena that occur during a ball drop test, whether or not the sample has been perforated being the only required output for the standard specifications.

Initially developed by Nourry [Nourry 2005], the cannon test device aims at reproducing hard impacts (with blunt and rigid objects) on laminated glass in a controlled way, representative of serviceability or ultimate limit state via a low-velocity/large-mass impact.

The device allows testing of different specimen geometries and materials to compare their impact performances quantitatively, with good reproducibility, exploring a large range of impact velocities from 2 to 15 m/s.

The impacted samples are square-shaped plates of 300×300 mm 2 : this surface is about ten times smaller than that of ball-drop tests and corresponds to the size of the damaged surface during standard EN 356 impacts. Here, the fastening of the plate is well-controlled and reproducible: two steel rings, polished with a curvature radius of 1 mm in contact with the glass to avoid any indentation and tightened by four screws at 10 N•m. Additionally, Decourcelle highlighted that mechanical wave propagation on the glass sample, provoking the first orthoradial cracks, is not significantly dependent on the selected sample size (from 300 mm to 1 m side length) [Decourcelle 2011], further legitimating the use of smaller samples than those used for standard EN 356 impacts. A telemeter measures the hemisphere displacement at 30,000 Hz, with a 3 µm uncertainty. These high rate measurements allow the evaluation of axial accelerations and velocities through finite differences. The kinetic energy can be estimated assuming the impactor to be perfectly rigid during the test.

During a typical fragmentation test with the cannon, the half-sphere apex impacts the outer glass ply with an initial velocity v 0 (from 2-15 m/s) and at a normal incidence. In the first microseconds after impact, the laminate behaves elastically for the most part. In contrast, a small zone of the first ply right under the impactor head is comminuted under intense multiaxial loading. Depending on the impact velocity and the sample strength, the first radial cracks may appear from the plate center within a few microseconds to a few hundreds of microseconds. As the plate bends, the radial cracks extend and sometimes multiply, while orthoradial cracks are also formed. More and more cracks are created during the whole impact duration. When the fragmentation is well-advanced, the stretching (and possibly tearing out) of the PVB interlayer dissipates a large amount of the impact energy as the impactor stroke increases. This dissipation occurs at the PVB bridge locations that connect all the glass fragments [Seshadri et al. 2002]. These steps are quite similar to the ball-drop experiments.

From an industrial point of view, repetitive tests were carried out on sample series to detect the influence of different parameters: the ply thickness and adhesion. Many conclusions were reported, but the studies did not focus on the fragmentation that occurs during such impact nor on the response of the fragmented composite submitted to stretching, tearing, and delamination. The principal goal of the new experiments performed in the present study is to replicate such experiments, while measuring glass fragmentation, (total crack length, fragment density and geometry) and finally estimating the total dissipated impact energy.

Interest in full-field measurements

Measurements of the impactor motion allow a better understanding of the behavior of laminated glass under a blunt impact. However, adding direct measurements of the plate motion during an impact will shed new light on the glass fragments kinematics. ) can be flexibly deformed to fit seamlessly with the real deformed 3D shape through the minimization of the quadratic norm of correlation residuals (image difference after correcting for perspective and shape), allowing for shape characterization and kinematic field measurement of complex 3D objects, even for large deformation configuration.

Finding the correspondence between the 3D coordinates of known points in the specimen and their projected 2D coordinates in retinal images captured by stereo cameras (2 cameras or more) is crucial for accurate 3D shape reconstruction. This is performed in a so-called "calibration stage", which involves identifying the projection matrix to bridge the 3D coordinates of points in the lab frame to their 2D projection in the camera image plane.

Guideline

To analyze the cannon in the spirit of TCT, it is proposed to measure the 3D displacement field as well as the velocity of the fragments (giving the driving velocity that locally dictates the PVB delamination and stretching) under the projectile shock using S-DIC technique. A stereo camera system is set up with two ultra-high-speed cameras. The back surface of the glass is monitored, measuring the glass shard formation and kinematics. Direct measurements on the sample are compared to the impactor kinematics. A model is then proposed to evaluate the delamination of the PVB-glass interfaces and determine the location that is the most likely to tear during a secondary impact.

The residual resistance of an impacted and damaged laminated glass is discussed in the final section, where a repetition of successive shots on damaged laminated glass samples is presented and discussed. First, centered impact tests are performed on pre-cracked samples to compare the influence of the cracking pattern on the projectile kinetic energy dissipation. Then, non-centered impacts on larger laminated glass plates are studied to analyze the behavior of fragmented and partially delaminated samples under secondary dynamic loads.

First cannon experiments

Laminated glass samples

The laminated glass sample used in this experiment is composed of two 4 mm thick Planiclear ® soda-lime glass plies and a 0.76 mm PVB interlayer Saflex ® RB41. The laminates are assembled in a clean room, degassed using an oven, and placed in an autoclave, where the temperature is set to about 130°C and pressure to 13 bars for several hours. The laminates are assembled in a clean room, degassed, and put through an autoclave which is represented in perature is increased in the chamber, and rises up to 80°C. The cooling phase extends over 3 hours, under atmospheric pressure. The incident angle between the camera axes and the cannon principal axis varies on each side from 30°to 45°according to the remaining space around the large test bench.

Set-up & equipement

Observing the back surface of the laminated glass at high temporal resolution has two motivations: First, the measurement of surface deformation, and second the characterization of glass plies fragmentation. Unfortunately, these two goals are mutually exclusive. Indeed, the kinematics of the back surface can be obtained from S-DIC, by depositing first a surface marking with a speckle pattern. In turn, the formation of cracks is mostly hidden due to the opaque paint presence. To circumvent this difficulty, a black and white speckle pattern is applied on the back surface of the laminated glass sample, covering the right-bottom part of the glass, providing sufficient Gray Level (GL) contrast for S-DIC measurements (Fig 3 .5). The left-top part is left transparent, but a thin tissue paper can be interleaved between the projectile and the sample to observe the formation of crack patterns on a more uniform background without adding spurious interference with the background scene. The interest of painting only one-half of the plate is to compare the evolving cracking pattern of the laminated glass and the deformed surface measured via S-DIC. Access to these cracks evolution gives information on the time duration of the elastic deformation phase of the sample, the formation of the radial cracks, the orthoradial ones, the beginning of the tearing phase, and the perforation step, for instance. The S-DIC measurements can also be interpolated on the non-speckled area of the plate. Therefore measurement of the evolution of the cracks via the residuals is made possible on the deformed surface.

The trigger solution uses the same photoelectric sensor that detects the projectile presence a few milliseconds before its contact with the impactor, with a 40 kHz sampling rate. This triggering method is also used in the tests presented in section 3.3.

Impactor movement

Telemeter measurements are performed to access the impactor half-sphere displacement. The telemeter is placed at 90°with respect to the axis of the test. The laser beam created by the source is deflected by a mirror, whose angular deflection close to 45°has been tested to be as accurate as possible. The beam is reflected on the face of the half-sphere and returns to the receiver using the same corner mirror. The impactor stroke is deduced from a different height measurement of the incident beam on the sensor. Thus, the speed of the half-sphere is deduced by centered finite differences of the obtained displacement, averaged on five consecutive points to filter out high frequencies. From this speed measurement, and assuming a rigid body movement, the impactor kinetic energy can be deduced from its velocity V as E c = mV 2 /2, with a constant impactor mass m=4.34 kg, whose value is close to that of the ball used in norm EN 356. The impactor acceleration is also estimated via finite differences. With these 1D curves, it is possible to comment on the telemeter accuracy. In Fig FIG. 3.6 -Superimposed telemeter measurements for an initial impactor velocity close to 5 m/s. Just after the contact between the glass and the half-sphere, a divergence of the curves is observed. The kinetic energy is plotted conserving the sign of the velocity curves.

at the location of the cracks, combined with the interlayer stretching and tearing between the glass fragments [Elzière 2016]. The tests last from 16 ms to 20 ms, with a maximum stroke after the contact with the plate varying from 32 mm to 42 mm.

Sometimes the second plateau is not observed: e.g. tests 13 and 14 in Fig 3 .6. They correspond to the samples with the higher crack density formed after the elastic deformation phase of the composite plate.

Even with a numeric induced noise caused by the two successive derivations of the telemeter acquisition, the acceleration curves highlight two events. A first negative peak, before 5 ms, is associated with the contact between the sphere and the glass surface. The second widespread peak of lower amplitude starts around 6.5 ms and remains, along with the experiment, over 10 ms and is synchronized with the plate finite strain and delamination regime.

Field measurements

The main objective of this section is the kinematic field measurement via S-DIC, whose principle is further detailed in section 1. The mean displacement field is interpolated on 5 mm long radial elements to finally get radial and axial displacement fields. The radial range of the interpolation is between 20 mm and 120 mm.

An extrapolation method is proposed using the radial displacement evolution for known locations, i.e. from 20 mm to 120 mm. Boundary conditions are enforced: the axial symmetry implies a null displacement derivative at R=0 mm, and a null displacement along the edge at R=140 mm. Plus, the amplitude of the instantaneous deformed surface is corrected thanks to the telemeter curve that measures the displacement at R=0 mm. A fourth-order polynomial is used for the extrapolation between 0 mm and 150 mm (the plate maximum radius). The displacement and velocity extrapolation are represented in Fig 3 .8, where the displacement curves correspond to the deformed back surface for each time step. From the estimation of the velocity norm in the laminated glass plate, it is possible to categorize the influence of the plate kinetic energy regarding the impact energy. The velocity is considered homogeneous through the plate thickness. So the sample kinetic energy can be formulated as an integral over the glass surface

E lami k = h g ρ g + h pvb ρ pvb 2 x ∥V(x)∥ 2 dx (3.1)
with ρ the density of the glass (g) or the PVB, h the total thickness of the material and ∥V(x)∥ the velocity norm at point x. This kinetic energy is displayed in Fig 3 .9a.

The telemeter measurement of the impactor kinetic energy (i.e. E telemeter ) can be subtracted from its initial 55.7 J value (i.e. E plateau , the mean plateau value). This energy corresponds to the energy stored and dissipated during the test: E lami = E plateau -E telemeter . This energy is displayed with two abscissa: time (Fig 3 .9b) and the impactor penetration (Fig 3 .9c). The first gap between the two energy plateaus is associated with the kinetic energy transfer from the impactor to the plate. Therefore, subtracting the kinetic energy from the signal gives a smoother energy evolution during the test (plain blue curve).

Axisymmetric description of the experiment

Calibration of the Stereo Vision Camera System

Before introducing an enhanced version of the prior axisymmetric measurement, new calibrations have been performed to localize the projection in the world coordinate system using a well-known calibration target.

The general principle of computer vision is introduced in section 1.4.2.

In this study, the calibration target is a dihedral ("open-book") which has two square faces of

T = N n=1 (x n -[Π]X n ) 2 (3.2)
with respect to Π. Initialization is validated by projecting the 3D mesh over the 2D images for each camera (Fig 3 .10d-e.).

New axisymmetric S-DIC method

As introduced in section 1.4.2, the S-DIC problem is ill-conditioned. So, the ideal solution may be to help the problem formulation using mechanical modeling of the laminated glass response. However, the fragmentation of the glass plies, the delamination of the PVB, and its extension involving significant geometric non-linearities (large displacements and rotations) make this route quite challenging. Alternatively, the axisymmetry of the set-up is a very strong constraint which may be used for regularization without any prior assumption about the mechanical model of the monitored sample. For global S-DIC, the T3-mesh has 4 mm length elements superimposed on the reference frames in Fig 3 .11. There is a total of 2366 elements, each of them possesses 32 integration points so that all gray level information on the images is considered during S-DIC calculation.

Due to the geometry of the whole set-up, the problem is axisymmetric, even if this symmetry is expected to be broken locally in the long run as the laminated glass is fragmented and finally torn apart.

A global axisymmetric description is assumed next for the minimization problem. It is a noninvasive technique that constrains the unknown displacement field (3D vector field over a 2D surface) to a much simpler one (2D displacement field over a 1D radial profile). Such a regularization can be used with B-splines, for instance [START_REF] Chapelier | Free-Form Deformation Digital Image Correlation (FFD-DIC): A non-invasive spline regularization for arbitrary finite element measurements[END_REF]], but a piece-wise linear description is adopted in the following. Also, it can be seen as a reduction of the degrees of freedom of the initial mesh nodal displacements.

Expressed in cylindrical coordinates, these new shape functions impose displacement shape between two images

U(X) = [U X (X, Y, Z), U Y (X, Y, Z), U Z (X, Y, Z)]
as a purely radial and out-of-plane displacement

U ax (r) = [U r (r), 0, U Z (r)]
The orthoradial displacement and the dependence on the angular position are estimated to be globally null, which remains a reasonable assumption as long as no tear opens out in the interlayer (a phenomenon, local at first, that propagates and breaks the axisymmetry of the deformation even at a global scale). Radial and axial displacements depend on the radial distance r between the impact location and the nodal position. The FE mesh center is localized at the impact initial position, i.e. the center of the laminated glass plate, with a 200 µm uncertainty.

The radius is directly

r i = X 2 i + Y 2 i , the angle α i = arctan(Y i /X i ).
The transformation from cartesian to cylindrical coordinates is written as a matrix [P], combining the smaller rotation matrices P i for each node i

P i =                 cos α i -sin α i 0 sin α i cos α i 0 0 0 1                 (3.3) such as U(X i ) = P i U ax (r i ) (3.4)
The axisymmetric description will be forced via the introduction of 1D-linear shape functions (lines in Fig 3 .12), limiting the displacement measurement along r and Z directions. The 2Dmesh current nodal positions (circles in Fig 3 .12) are represented on the new shape functions. 3.12 -1D shape functions q i (r) ensuring displacement continuity, in straight lines. Interpolated T3-nodes (Fig 3 .11) as circles on the new shape functions. The plot illustrates 15 mm-long elements for better readability, while 4 mm elements were actually used in the axisymmetric S-DIC measurements.

The 1D shape functions are noted q i (r) and filled in a second transfer matrix [Q] such as the new Hessian and right-hand terms become

[H] ax = ([P][Q]) T •         N c c=1 [H] c         • [P][Q] b ax = ([P][Q]) T •         N c c=1 b c         (3.5)
and the minimization problem

[H] ax • δU ax = b ax (3.6)
The global 3D-displacement increment δU is updated via the two transfer matrices and the nodal axisymmetric displacement δU ax

δU = [P][Q] • δU ax (3.7)
This reduced problem is far better conditioned than the original S-DIC one (decreasing the degrees of freedom from 7098 to 84). Therefore no additional regularization (e.g. Tikhonov) is needed to perform axisymmetric S-DIC.

Results and discussion

Global S-DIC and axisymmetric S-DIC have been performed on the painted deformed surface of the laminated glass during the first three milliseconds following impact, or a total of 90 frames per camera. In A comminuted zone, centered at the impact point, appears as a bright white zone in Fig 3 .13. The formation of radial and orthoradial cracks could be observed during the experiment. First, radial cracks propagate from the impact point to the edge so that fragment sizes increase with the distance from the center. Then orthoradial cracks appear, as can already be seen in Fig 3 .13. As time passes, the density of both radial and orthoradial cracks increases, leading to smaller and smaller fragments.

This growing crack number also increases the deformability of the fragmented laminated glass plate, whose glass shards are still connected by interlayer polymeric bridges that sustain large strains.

The geometry of the deformed plate surface could at first be approximated by a rounded cone of increasing height. This information could also be read on the displacement fields (e-h) for the three figures 3.14, 3.16 and 3.18.

The sample shape evolution and fragment orientation induce light reflection, with severe gray level changes on the captured images. Specular reflections are even observed in some images (see, for instance, the white spot at the top of Fig 3 .17a), making the analysis more and more difficult for the S-DIC software.

A brightness change, not being accounted for by the S-DIC procedure, is finally present in the residual field (a-d). In particular, a contrast in the residuals field across a radial boundary may be interpreted as a localized bending along a radial crack, making some of them apparent. However, they could hardly be perceived in the raw images. Accessing the final crack pattern by an examination of the post-mortem sample in a fragmentation experiment is not easy as many non-reversible phenomena occur: large stretching then tearing of the interlayer, delamination between the PVB and the glass shards, comminution at the center of the plate that loses smaller fragments sometimes crushed into glass powder. Therefore the glass damage is hard to qualify, evolving throughout the test period. Three pictures have been taken at the end of this experiment. As can be seen, the PVB layer has been torn apart, and glass fragments are missing close to the impact location (Fig 3 .19). Out of the torn zone, the laminated glass retains a crater-like or cone-like shape at the end of the experiment, which emphasizes a residual irreversible deformation of the plate.

Thanks to a good spatial resolution, many cracks can be observed in these three images, but the damage temporal evolution is not assessed via post-mortem imaging. Therefore, live monitoring of the crack formation method is proposed.

Due to the global axisymmetry of the measured displacement, it is possible to reproduce the deformation of the non-speckled glass surface on a semi-circular mesh, then analyze the residuals between a reference image and the experiment images. Some residuals of the non-speckled Moreover, an axisymmetric interpretation of the displacement fields is represented in Fig 3 .22, projecting the global S-DIC obtained fields on radial shape functions (legend 'gl'). The out-ofplane displacement is therefore filtered, and a good radial approximation of the displacement can be obtained, reducing the influence of the ill-behaved measurement points. The extrapolated displacement also matches the telemeter curve (black straight line), which can be interpreted as the displacement along the axis (R=0). It fits the axisymmetric S-DIC results (legend 'ax'). For large radius and time, the global and axisymmetric results are moving away from each other showing the difference between a reliable axisymmetric code and the presently limited global S-DIC procedure. It is also validated by the presence of orthoradial stress at the beginning of the test, creating the radial cracks. The radial displacement is higher for smaller radii, up to a characteristic size approximately equal to the size of the half-sphere (i.e. 50 mm radius).

Due to the better calibration than the one performed during the first wave of cannon experiments, the calibration of the stereo-vision system is much better, and does not need a further rescaling.

Energy dissipation models on pre-cracked samples submitted to tensile loads are already performed with a tensile speed that may vary from quasi-static up to 10 m/s [Del Linz et al. 2017] (up to ∼70 s -1 ), which covers the range of strain rates experienced by PVB in the cannon tests (locally up to ∼30 s -1 ).

The link between the interlayer stretching and its delamination from the glass shards is well documented. The gap between a fragmentation experiment on the cannon and a Through Crack Tensile (TCT ) test is narrowing thanks to the full-field and the cracking pattern measurements presented here. It becomes a question of integrating the energy dissipated in TCT at the proper strain rate over all the area created between the glass fragments. Those data are of interest to the experimenter to determine the behavior of different areas of the laminate rather than relying only on a local measurement via strain gauges or impactor measurement, which do not give the whole plate kinematics. Each laminated glass fragment has its specific movement during an impact that can be interpreted as a local TCT test. In a nutshell, these measure-ments, coupled with a correct corresponding model of the fragmented laminated glass plate, will allow the evaluation of the local impact energy dissipation at the glass fragment edges and, after summation over all cracks, the assessment of the total energy absorbed by the broken laminated glass plate.

Delamination model

Developing a model on the laminated glass under impact is challenging due to the numerous radial and orthoradial cracks on the glass plies. Setting up a model requires using appropriate hypotheses. An axial symmetry in the cracking pattern and in the kinematic fields is assumed, and, the rich knowledge acquired from TCT tests in the literature is used to simplify the model design.

Problem definition Through Crack Tensile (TCT ) tests: a starting point

The concept of the TCT test, already described in the introduction, is based on a tensile test on a pre-notched sample of laminated glass. Due to the presence of the crack, the elastic deformation concentrates in the polymer layer. In addition to the interlayer stretching, symmetrical delamination may occur depending on the tensile test imposed speed, δ, of the clamps (see scheme in glass and PVB (see Fig 3.24b). The delamination occurs in a certain velocity range [Elzière 2016]. An essential characteristic of the TCT can be seen on the force versus time curve: as the imposed speed of the test δ is constant, a limit force is reached. This state of the test is called the steady-state regime, as the force plateau is reached, and the steady-state stretch, λ a , that connects the glass shards displacement, δ, and the delamination fronts advancement, a, can be expressed as λ a = 1 + δ/2a.

Locally as shown in Fig 3 .24b, the same situation is encountered all along the experiment at different positions of the crack front a. The load needed to advance the crack is thus a characteristic of the decohesion that only depends on the delamination velocity ȧ, and nothing else. During a time increment dt, the crack front advances by da = ȧ • dt. This added length of PVB ligament was unstretched prior to delamination, but after it is stretched to λ a • da, so that the two ends of the sample move apart by the extra length (λ a -1) • da on each side of the mid section, and hence dδ = 2(λ a -1) • da, or 2(λ a -1) • ȧ = δ.

The central part of the PVB is subjected to a tensile stress depending on ȧ, needed for the decohesion of the PVB. Hence, the stretch λ a is the quasi-static response of the PVB to a macroscopic tensile stress σ. Finally, a time integration may be trivially performed to arrive at The dissipation mechanisms are all localized at the delamination front, therefore in steady-state, the delaminated PVB 'far' from the crack front is uniformly stretched up to λ a . The steady-state force plateau versus the tensile test speed is reported in Fig 3 .26a. The stress is therefore limited in the cracked laminated glass, as delamination should occur at the steady-state stress.

2(λ(σ(ȧ)) -1) • ȧ = δ.
The potential tearing that may occur during a test should be characterized by a specific criterion rates using a high-speed hydraulic testing machine [38,39] . In their data, the stretch does not reach a steady value: it increases until rupture of the interlayer, so that we cannot define a consistent G m value. We chose to compare the "adherence energy"i.e. force at the plateau F ss normalized by the sample width b 0 -in order to combine their data with those of Elzière in figure I.9.

At low velocities, undulation of the crack fronts leads-though not systematicallyto an arrest of the delamination. The stretch increases continuously, until rupture of the interlayer. The empirical "unstable limit" is indicated by the black straight line in figure I.8.

The adherence energy increases with the tensile velocity ˙ : naively, one could think that the higher the velocity, the better the impact performance since more energy can be dissipated. However, the interlayer also bears a higher stress, limited by the strength and toughness of the polymer: the work of fracture is limited by the rupture of the interlayer at high velocities. In the TCTT, the transition from the steady-state regime to the "rupture" regime occurred around 100 mm • s 1 at 20 C.

F ss /b 0 [kJ m -2 ] 5° C 20° C 50° C
Elziere et al.

0° C 10° C 20° C 30° C 40° C 50° C 60° C
Samieian et al.

20° C

Del Linz et al.

[5]

[37]

[39]

Figure I.9 Influence of traction velocity and temperature on the delamination force measured with the TCT-Test (PVB thickness = 0.76 mm) Solid line: empirical limit between unstable and steady delamination behaviors. Dashed line: empirical limit from steady behavior to rupture of the interlayer. [5,37,39] (a) In Fig 3 .26d, the adhesion limit stress σ adh is plotted versus the delamination velocity ȧ. A crack velocity limit appears for high stress values.

Finally the hyperelastic behavior of the PVB, adapted from the visco-elastic behavior measured by Fourton [Fourton 2019], is represented in Fig 3 .26c for two strain rates: 1 s -1 and 0.1 s -1 . The true stress is obtained assuming volume conservation (i.e. Poisson's ratio of 0.5). Hyperelasticity is used here because the discharge is not considered yet. The effective PVB behavior implies a non reversible behavior of the interlayer, but is not at stake here.

Connection between TCT and cannon tests

The idea of the model is to consider the cannon experiment, with its multiple cracks appearing during the test, as a system of many simultaneous TCT experiments. Writing L the distance between two cracks, the relation between the TCT displacement, δ, and the macroscopic apparent stretch λ is

λ = (δ + L)/L (3.8)
The stress plateau, noted σ adh , corresponds to the adhesion limit strength between glass and PVB. An effective constitutive law λ(σ) can therefore be proposed introducing crack density contrary to more standard cases.

Due to the limited resolution for the live crack measurement, a choice is made to separate the different zones of the fragmented plate. The different zones are represented in Fig 3 .27. Larger shards are located far from the impact location (Z3). A circular zone with a larger orthoradial crack density is also visible for median radius (ZO2). The last zone is composed of finer glass fragments (Z2). In the cannon test, the comminuted zone (Z1), located under the impactor head, is assumed to bring only a minor contribution to the globally dissipated energy, but it cannot be measured with the device. The macroscopic strain in this zone is imposed by the impactor head and hence is soon saturated in the experiment.

The model does not consider the deformation of the laminated plate occurring at a radius smaller than 35 mm. S-DIC indicates that U r continuously increases at r ∼20 mm. Thus some dissipation is expected there. But the crack evolution is hardly monitored in this area. Therefore this small area, regarding the plate size, is not studied.

After the crack formation, the laminated glass structure is weakened and becomes heterogeneous [START_REF] Bermbach | Experimental investigation of energy dissipation mechanisms in laminated safety glass for combined blast-temperature loading scenarios[END_REF]]. There are three main distinct zone categories on the plate. The first one corresponds to zones of perfect adhesion between the glass and PVB, representing the plate stiffer zones. The second category includes the PVB ligament (i.e. without glass shards) that is stretched to a specific value. Finally, delamination and visco-elastic dissipation in the interlayer is at stake at the delamination fronts. Defining different area leads to the consideration of crack densities in the radial ρ r (r) and orthoradial ρ θ (r) directions which play the role of 1/L needed to describe the previously defined apparent stretch λ.

For the radial direction, the circular crack number (N cc ) has to be considered between radii r 1 and r 2 delimiting the disk of the selected zone. So,

ρ r (r) = N cc r 2 -r 1 (3.9)
For the orthoradial direction, a radial crack number (N cr ) is considered, depending on the radial position r ρ θ (r) = N cr 2πr (3.10)

A simplified membrane kinematics of the fragmented plate

The deformation of the fragmented laminated glass plate can be assimilated to a large strain hyperelastic membrane composed of stretchable PVB bridges between the stiffer glass shards. Dynamic effects (e.g. plate inertia, kinetic energy) are neglected.

In the current model, the inter-crack distance is postulated from the observations.

The apparent macroscopic stretch is deduced from the particular displacements U = (U r (r), 0, U z (r)). It can be expressed, along the radius of the membrane

λ r = 1 + U r,r 2 + U z,r 2 (3.11)
and, along the orthoradial direction

λ θ = 1 + U r /r (3.12)
From each side of a crack, the relative displacement of the two glass fragments, (i.e. the effective displacement δ e f f , see (3.8)) is

δ e f f r = (λ r -1)/ρ r (3.13)
in the radial direction, and,

δ e f f θ = (λ θ -1)/ρ θ (3.14)
in the orthoradial direction. These effective values relate the cannon (locally) and the TCT mechanics.

That leads to two local delamination lengths a,

a r = λ r -1 λ r a -1 1 2ρ r (3.15)
in the radial direction, with λ r a = 1 + δ e f f r /2a r , and,

a θ = λ θ -1 λ θ a -1 1 2ρ θ (3.16)
in the orthoradial direction, with λ θ a = 1 + δ e f f θ /2a θ . In the axial symmetry hypothesis, the force balance is simplified (see e.g. [START_REF] Chevaugeon | Contribution à l'étude des membranes hyperélastiques en grandes déformations[END_REF]).

σ θ = r • σ r,r + σ r (3.17)
with σ θ the orthoradial stress and σ r the radial stress, in the cylindrical coordinates system (r,θ,z).

Macroscopic behavior law

The behavior law Ψ of the PVB is introduced as

λ a = 1 + Ψ(σ) (3.18)
with λ a the steady-state stretch and σ the stress. This law describes the uniaxial visco-elastic and large strain behavior of the polymer. Neglecting the viscosity as a first assumption transforms Ψ into a scalar function (i.e. local in time). That means a unique hyperelastic law can describe the behavior of the polymer in large strain regimes during the imposed monotonous load. The function Ψ is represented in Fig 3 .26c, for stretch rates of 0.1 and 1 s -1 [Fourton 2019].

The second phenomenon to consider is the rupture of adhesion (i.e. delamination) between the glass and the PVB. The law φ that links the delamination front velocity ȧ and the adhesive strength σ adh is

σ adh = φ(ȧ) for ȧ ≥ 0 (3.19)
The function φ is represented in Fig 3 .26d [Elzière 2016].

Using the membrane kinematics, the behavior leads to

λ =1 + 2ρaΨ (φ(ȧ)) σ =φ(ȧ) (3.20)
as σ = σ adh the stress imposed by the adhesion phenomenon.

Considering steady-state with infinite delamination, and an hyperelastic behavior of the polymer, the law becomes

ȧ =φ -1 (σ) λ =1 + 2ρΨ(σ) t 0 φ -1 (σ(τ)) dτ (3.21)

Partial conclusion on the delamination model

The present section links the cannon and the TCT tests. The well-documented theory developed thanks to the tensile tests can be used to simulate a plate bending experiment, adding information on the evolution of the crack pattern.

The model is supposed to describe the delamination evolution versus time, which is quite interesting for multi-impact tests. But it still needs some improvements in the variety of the TCT data used (e.g. delamination rate range, temperatures).

The choice made in the next section is to look at the behavior of a triangular laminated glass fragment with no prior supposition of the crack density.

Triangular fragment model

The overall behavior of a cracked laminated glass plate loaded with a rigid sphere seems challenging. Apart from the materials, elastic-brittle soda-lime-glass, and visco-elastic PVB, the adhesion between the layers plays a significant role in kinetic energy dissipation and in the plate global out-of-plane movement.

In this section is proposed an elastoplastic model of a laminated glass fragment under impact. The loading conditions are imposed from the experimental telemeter measurement: the displacement of the sphere is imposed in the simulation. The goal is to describe the plate kinematics and to model the energy dissipation phenomena.

Half-fragment model

The studied geometry is simplified in the model, assuming angular symmetries.

First, the glass cracking does not dissipate much impact energy during the test. Nourry observed that most of the radial cracks are the first to pop out on the glass plies [Nourry 2005], before the circular cracks for a large part of the cannon tests. After the experiments, the number of radial cracks is measured between 100 to 250, considering an impact energy of 50 J. Also, the radial cracks are formed in the early stage of the test, when the impactor kinetic energy has not dropped significantly yet.

From these observations, the model geometry is based on a triangular laminated glass fragment, represented in Fig 3 .28. Thus at the beginning of the simulation, the radial cracks are already formed. Due to the plane symmetry, each half of a triangular fragment behaves in the same way. Therefore, instead of representing the whole fragments, only their half separated by the radial crack can be simulated with proper symmetry conditions.

Elziere has shown that for constant delamination velocity, the measured steady-state stress between the two glass fragments is constant. Thus, the model can again be simplified, considering that the stretching phase of the experiment occurs at a constant stress (i.e. steady-state regime) at the radial crack location, by imposing such condition: this steady-state stress is estimated between 10 to 15 N/mm at 25°C [Elzière 2016].

It is now possible to simulate one half of a triangular fragment, applying the chosen steady-state force at the radial crack location. This simplification is illustrated in Fig 3 .29.

Homogenization

An orthoradial crack formation is considered instantaneous through the glass ply thickness, regarding the total duration of an experiment. Furthermore, once a crack has appeared on one of the two glass plies, the second ply has more chance of being cracked similarly because of the global bending of the plate. Therefore, as there is a uniformity of the behavior in thickness, it becomes convenient at to first model a plate with shell elements to simplify the problem. Besides the homogenization, the assumptions are: orthoradial cracks are perfectly circular cracks. The glass plies have the same thickness.

Elasticity

To model the three layers of the laminated glass plate, a transverse isotropic model is defined in table 3.1. The transverse direction corresponds to the stacking direction, i.e. through the plate thickness. This direction corresponds to index 3 in the table. The stacking is symmetrical, and the proportions of material i = {Glass ( g ), PVB ( pvb )} are defined with

υ i = h i /h total
where h i is the i-ply thickness, and h total = i h i the total composite plate thickness. In the 3.1 -Transverse isotropic model of the homogenized laminated glass fragment. Direction 3 is the stacking direction (i.e. through the thickness). table 3.1, E is the Young's modulus, G the shear modulus, ν the Poisson's ratio, ρ the density. The visco-elastic coefficient of the Prony series, g i and τ i are detailed in table 1.4.

Laminated glass fragment Homogenized fragment

Glass isotropic: E g , ν g , ρ g E 11 = E 22 = υ g E g + υ pvb E pvb PVB isotropic: E pvb , ν pvb , ρ pvb E 33 = E g E pvb υ g E g +υ pvb E pvb visco-elastic: g i , τ i G 12 = υ g G g + υ pvb G pvb G 13 = G 23 = G g G pvb υ g G g +υ pvb G pvb Table
The homogenization of the bending behavior of the plate is not yet considered as the PVB layer is at first supposed to be very thin.

Plasticity

With the transverse isotropic model, it is possible to reproduce the behavior of a nonfragmented laminated glass. However, at orthoradial crack locations, the creation of some new degrees of freedom has to be considered. Therefore, a plastic behavior is implemented when the strain at break is reached for the soda-lime glass ply, near 2.8 × 10 -3 . The polymer bridge created at a crack location will be simulated via an isotropic plastic zone, whose local stiffness is E crack = υ pvb E pvb . This behavior is schematized on figure 3.30. Using plasticity allows to describe the behavior during the loading phase, but can not predict the unloading behavior of the plate.

Boundary conditions

The first model is driven by displacement. The displacement measured by the telemeter is imposed on the half sphere, supposed to be perfectly rigid (Fig 3 .31b). All planar movements are unauthorized on the apex of a triangular fragment (i.e. lower radius). All rotations and displacements are null at the clamped location, at a radius of 140 mm from the plate center. The symmetry of the half fragment in the (r, z)-plan is represented, retaining movements inside the symmetry plane. Finally, an orthoradial 2D stress of 15 N/mm is imposed throughout the test at the radial crack location, assuming the effective PVB delamination. The boundary conditions are illustrated in Fig 3 .31.

The glass parameters are taken from the soda-lime glass column in Table 1.2. The steel-made half-sphere parameters are reported in Table 3

Results

To compare with the experimental measurements, the simulated out-of-plane displacement and velocity are shown in Fig 3 .32. The displacement has a good trend. The velocity field synchronized in Fig 3 .33b, the blue and red curves show a similar trend, but the energy of the simulation is 20 J higher than expected, corresponding to the simulated elastic energy stored in the plate (purple curve). Considering the clamped edges and the hard contact defined between the glass and the more rigid steel-made half sphere, provoking high frequencies in the velocity signal in Fig 3 .32, the first model is gives correct results. It reproduces the right kinematic of the impacted plate and successfully predicts the first circular crack radius of 72 mm. Additionally, it is manageable to implement, imposing the measured displacement of the sphere or its initial velocity, and runs in a few minutes.

The drawbacks of the simulation are multiple, though. First, an energy gap exists between the experiment and the simulation as the stored elastic energy in the laminate converges but does not dissipate. The plastic model does not reproduce well the unloading of the glass elastic energy after the crack appearance. This leads to two main consequences: an overestimation of the impact energy and a circular crack growth due to the lack of free surface creation. However, the experiments highlight that the stored elastic energy in the fragmented composite can be neglected and that the glass fragments movement can be seen as purely rigid compared to the interlayer stretching.

Experimentally, the contact with the sphere is softened by the local comminution of the glass. A softer contact between the plate and sphere could avoid the abrupt forth and back movement that adds genuine vibrations to the velocity curve.

Following some new ideas can help the enhancement of the simulation. First, the plate characteristics should be better set, paying attention to bending, using quadratic elements in the model or skipping the homogenization phase. The plasticity model at orthoradial cracks is also to be revisited. Indeed, the uniform dissipation occuring on the plate is not representative of the actual discrete places of dissipation, and should be revisited. In this regard, a subroutine can be used to stop the simulation when the strain at break is reached, to create 'manually' a discontinuity in the glass parts (i.e. crack), then continue the simulation by imposing null stress on the newly created free surface.

Multi-impact & residual resistance

After an impact on a laminated glass plate, the broken sample appears resistant to new loads. The plate resists quasi-static in-plane compression because of the remaining glass fragments on the adhesive layer. However, the interlayer stiffness mainly governs the tension. A fragmented and already deformed laminate can absorb impact energy during secondary impacts, as long as delamination between glass and PVB can occur and no tearing of the interlayer takes place. This behavior is already considered in the standards that include three successive ball-drops on the laminated glass plate [Norme Européenne NF EN 356 2000].

The influence of the distance between two successive impact locations and the damage history of the fragmented and partially delaminated sample are critical parameters for the residual resistance characterization of broken glazing in the context of multi-impact (e.g. anti-break-in standard tests).

Second impact on pre-cracked laminated glass

In this section, the behavior of pre-cracked samples is at stake. Tests are performed using the cannon, and the initial impactor kinetic energy is roughly 130 J. This value corresponds to a ball-drop fall of 3 m high, a typical value used in the anti-break-in normative tests. The samples are PVB-laminated glass plates, with two glass sheets of 4 mm thickness, interleaved by a 0.76 mm PVB foil. Table 3.3 -Initial number of cracks and comminution zone size after a first impact.

In Fig 3 .34 are shown five laminated glass samples pre-impacted at various energies. The stroke of the impactor has been limited to 1 cm (with a stop) after its contact with the glass surface, avoiding global deformation of the plates and delamination at the newly formed crack locations.

The impact energy of the first shots are reported in table 3.3. The crack numbers and the comminution diameter are also counted and reported. Various patterns are presented here,e.g. the radial cracks have successive branching along the radius returning a 'pine needle' aspect observed after the impacts with the lower velocity. The orthoradial cracks are not entirely circular, so the number in the table corresponds to their maximum number along a radial direction on the plate. Radial cracking patterns are not equivalent for the two glass plies. For instance, in Fig 3 .34e, the radial crack number is estimated only on the second glass sheet. For all tests, the impactor kinetic energy is entirely dissipated in less than 15 ms after impact.

No tears are observed. The deformed surface can be approximated at first with cones of 140 mm bottom radii and heights equivalent to the impactor stroke. For the five tests, the number of cracks did not increase much in the second impact: in Fig 3 .34 the initial damage represent ∼80% of the final crack patterns.

In Fig 3 .35b, a steeper slope appears after 4 ms, at a radius of 40 mm. The same phenomenon is observed in Fig 3 .35c at 40 mm radius. It seems that the deformation is larger for the comminution zone, whose displacement is closer to the impactor head after some time. It is worth noting that the initial glass fragments are mostly detached, broken or fully delaminated.

Locally, the plate stiffness becomes similar to that of the interlayer. In these two cases, the deformed surface can be approximated with two embedded cones, with a joint at 40 mm radius.

The orthoradial cracks also play a role in the deformed shape of the plate. This is illustrated in Fig 3 .35e, where the surface has a higher curvature radius around 50 mm (i.e. the half-sphere radius) radial position. A localized bending concentrates the deformation, lowering the overall dissipation of the plate. This leads to a premature high stretch of the polymer and tearing, as the stiffness is greatly reduced.

The initial cracks have influenced the shape of the deformed surface. However, it does not much interfere with the impact performance of the laminated glass as no delamination had occurred during the first impact.

Multi-non-centered impacts

The previous section is quite informative on the influence of the pre-existing crack pattern before a second impact. But the influence of the distance between successive impact locations has yet to be explored. Using the cannon device instrumentation to analyze multi-impact experiments can shed new light on the behavior of the fragmented laminated glass plates subjected to a second and a third impacts. 3.36 -Scheme of the three sample sizes (dashed green). From smaller to larger areas: 300×300 mm 2 cannon samples, 560×560 mm 2 cannon new samples, 900×1100 mm 2 ball-drop samples. The device axis and positioning of the plates are represented.

Reworked

The size of the frame support is enlarged to 500×500 mm 2 to impact 560×560 mm 2 plates possibly two or three times. A bigger calibration target of the same dihedral object is used to deal with the larger amplitude of out-of-plane movements and the wider sensor size.

The tests are all performed at a 130 J kinetic energy, i.e. a 7.8 m/s initial velocity. The laminated glass sample still comprises two 4 mm thick glass plies interleaved with 0.76 mm thick PVB foils.

Square sample: First impact

Because of the manufacturing process of float glass, the bending strength of a sheet of glass is dissymmetric. Indeed, the "tin" face, i.e. the surface of glass that was in contact with the tin bath, has a higher density of surface defects (or micro-defects), and hence the glass ply breaks easily when bent with the tin surface in tension. In contrast the "air" face is much resistant to rupture under flexure when it is in tension. That means with a tin side in tension and air side in compression, the crack apparition in the early tens of microseconds of the first impact is facilitated. In Regarding the full-field measurements from 1 to 2 ms after the contact between the glass and the impactor, the plate edge does not deform significantly compared to the indentation area. This section can be defined by a distance to the impactor of about 100 mm. From this distance and a specific duration, the fragmented laminated glass dissipation of the impactor kinetic energy becomes minor.

The laminated glass stops the impactor, and no tear is observed at the end of the test duration (i.e. ∼25 ms including the unloading phase).

S-DIC measurements are performed thanks to the lateral cameras images (Fig 3 .38). On the non-speckled part of the surface, the crack pattern in the low-density fragmented area is observed at 4.5 ms (Fig 3 .38c-d). Referring to the energy curve of the impactor at 4.5 ms, 40% of the impact energy is transferred to the laminate (see 'shot 1' in Fig 3 .41a). The degrees of freedom are reduced using the test axisymmetry (up to 70 mm radius). A cone shape is observed up to 100 mm radius. The constraints are relaxed from 70 mm radius, giving different shape functions for each angular zones. The sections, defined every π/24, have independent axisymmetric kinematics. The orthoradial displacement is still null. These new shape functions allow the ellipse shape to appear: i.e. extended circles for positive Y-values are observed in Fig 3 .38e-h.

This behavior is confirmed by the difference in motion between the bottom left and the upper left corners that do not show identical out-of-plane displacements.

At first, circular and elliptical zones describe the displacement field. Still, at the end of the experiment, the ellipses have formed, and an implementation of the orthoradial displacement appears to be unavoidable to describe the surface kinematics adequately and avoid the apparition of radial marks on the displacement fields in Fig 3 .38e-h. However, regarding the evolution of the residual in Fig 3 .39, the displacement of the ROI is globally well described.

The curves in Fig 3 .39a show a rapid increase of the residuals from 0 to 5 ms caused by the cracks and the large strain regime. The different dynamic from 5 ms is mainly caused by brightness and contrast changes in the different zones of the mesh. The residual field in Fig 3.39b shows the apparition of the large residual areas (in black and red) interpreted as a change in lighting condition. The smaller dots that have appeared are explained by the presence of the cracks in the speckled areas. Radial cracks can be observed, even for higher radius values. The non-converged local regions are mostly contained in the first 80 mm radius, which is the limit of the fully axisymmetric area. The kinematic is very restrictive in this zone to

Square sample: Second & third impacts

During successive ball-drop experiments, the second steel ball fall is performed at 13 cm from the initial impact location [Decourcelle 2011], and the third drop is made at 13 cm from the two previous locations. Thanks to the design of the experiments, successive impacts are performed: a second shot at 14.14 cm from the first impact, and a third shot at 20 cm and 14.14 cm from the first and second impacts, respectively. By comparing the first and final images, it is confirmed that no cracks are formed far from the secondary impact locations (e.g. non-speckled area of . Some fragments still appear under the impactor head and close to the frame support, but none in the finely fragmented areas. During the tearing of the interlayer, no cracks are formed in the glass plies. The tear locations are shown in orange for previous radial damage and in green for previous orthoradial one (see Fig 3 .40b-c). These locations are the major structural defects on the fragmented and delaminated glass plate. They all originate from the first impact cracks. During the third impact, the ball goes through the laminated glass plate, and large glass fragments remain on the sample far from the three impact locations. The second and third shots show smoother contact between the impactor and the fragmented glass because no rapid energy drop is measured in those cases. The energy versus time (or displacement) is similar between the last two tests. The plate stops the impactor during the second shot. However, a tear occurs at 10 ms (or 67 mm of impactor penetration) and causes a slope change during the third shot.

Resistance to three impacts

During this second experimental campaign, the glass surfaces in contact with the tin bath are put in compression during the laminated glass plate bending. The cracks are more difficult to initiate, i.e. the elastic energy stored before the crack apparition is much higher with this configuration. Small glass fragments are created during the late breakage.

The sample cracking patterns are shown in under the impactor head, but no new cracks are formed on the plate edges. Due to the small glass fragments size, the plate has locally conformed to the shape of the sphere. The impact energy is fully dissipated in the three cases, and no tearing of the interlayer is observed at the end of each shot.

The strokes measured before a tear appearance of the second and third impacts are also more significant than those of the previous experiment (see Fig 3.43). Comparing the two series of three impacts, the finer fragments best line up to the impactor shape, and the global plate can dissipate much more impact energy.

Discussion on multi-impacts

Many conclusions can be made based on the study of the many impacts performed on fragmented laminated glass plates.

The first one concerns the crack density influence on the plate residual behavior during secondary impacts. With a fine crack density, the damage is concentrated close to the impact location that becomes a structural weak point of the fragmented composite. So, in the case of a second shot on a very finely fragmented laminated glass plate, if its impact location is close to the weak point, the steel ball should go easily through the sample. On the other hand, far from this weak point, the projectile is more prone to be arrested as the polymer bridges created during the previous impact(s) can conform to the projectile shape and dissipate a much larger amount of its kinetic energy.

When the fragments are bigger, delamination may occur far from the impact zone and weaken the structure on a wider area. Moreover, as many large fragments remain, the weak zone is concentrated at the fewer polymer bridges (i.e. radial and circular cracks location), where tearing is easily initiated and further propagated during secondary impacts.

The apparition of a circular zone (also visible in Nourry's ball-drop experiments [Nourry 2005]) is a consequence of the ability of the fragmented laminated glass to conform to the geometry of the indentor head. So, the boundary condition influence is reduced concerning the global impact performance of the broken laminated glass plate. Here a circular, finely fragmented zone (of 100 mm radius) is formed even if the contact point is not centered on the plate. This newly formed zone becomes the weak point of a structure that can still resist plural impacts.

This result is significant and needs to be further studied because it seems universal for all low-velocity impacts on laminated glass (e.g. anti-break-in with steel ball, head-impact on windshields, sphere-ball pitting, automotive crash). For instance, the circular pattern is also observed in head-impact on automotive windshield studies (e.g. [START_REF] Shahriari | Prediction of vehicle impact speed based on the postcracking behavior of automotive PVB laminated glass: Analytical modeling and numerical cohesive zone modeling[END_REF]). The link between the cracking pattern and the characteristics of a projectile still has to be made. Also, the influence of the large fragments outside this circular, densely cracked, area, and how they participate in a projectile kinetic energy dissipation need to be discussed.

Conclusion

Instrumented and quantitative experiments are performed with the cannon device to analyze the laminated glass plate behavior under hard-body blunt impact. S-DIC analyses are carried out from high-speed camera images to extract the 3D shape variation and the corresponding kinematic information of the laminated glass during the fragmentation test concerning the measured projectile movement.

A newly developed axisymmetric description of the displacement field in S-DIC, which can be seen as a relevant geometric regularization devoid of material-based mechanical assumptions, provides very good convergence properties and remarkable robustness to the analysis. The obtained performances are much greater with the help of this regularization which avoids the divergence of some nodes of the mesh and allows the formation of the cracks to be measured on the deformed, non-speckled part of the laminated glass. Also, the use of a calibration object provides a reliable quantitative estimate of the displacements, and no rescaling of the results are therefore needed to match the telemeter measurements.

The data treatment presented here gives access to the shape evolution of the laminated glass during the impact, yielding the geometrical link between the energy dissipation close to the cracks (energy per delaminated area, which can be measured on TCT experiments) and the global energy dissipated (the diminution of kinetic energy lost by the impactor). This link enables to build of a model that computes the energy that the laminated glass can absorb as a function of material parameters only, such as the glass-interlayer adherence or its largedeformation stress-strain behavior. It could be interesting to perform new TCT experiments to refine the model parameters, especially the relationship between the steady-state stretch λ a and the imposed velocity δ in higher strain rate regimes. Adapting the simulation with new boundary conditions is at stake, considering the large strain regime and the out-of-plane movement. An enhanced model also can integrate the influence of the PVB layer thickness (here fixed at 0.76 mm) and the shearing of the polymer, which is fully neglected with the membrane approximation.

Knowing the mean crack delamination per zone is quite interesting to evaluate the resistance to multi-impact. The macroscopic influence of the cracking pattern is explored in the previous section. Still, it is worth combining the measurement of the crack evolution and the local delamination to estimate the structural damage evolution on the laminated glass plate and therefore know its residual resistance before the tearing of the interlayer.

The next step is to expand the membrane model (without the axial symmetry hypothesis) to analyze multiple, non-centered impacts. A comparison between the sample simulation and the full-field measurements via S-DIC on the same sample can help adjust the model to real test cases. Also, the unloading phase needs to be described, as the elastic and visco-elastic energy stored in the impacted plate is quite significant. Finally, the adhesion recovery needs to be explored by introducing a time-dependent damage variable, considering the duration between two impacts on the same laminated glass plate.

Chapter 4

Conclusion & perspectives

In this work, new highly-instrumented experiments have helped to apprehend the laminated glass behavior under various dynamic loads. The gravel pitting, i.e. the dynamic indentation, has been revisited and the star-crack formation documented. The ball-drop test, i.e. fragmentation during hard-body low-velocity impact representative of an anti-break-in experiment, has been adapted to analyze the influence of glass fragmentation on dissipated energy, and to test its residual resistance to multi-impact.

Based on this work, new leads are proposed for future studies.

Dynamic indentation tests

The blowpipe experiments have revisited the dynamic indentation tests, by performing reliable velocity and force measurements. The addition of a high-speed camera has allowed the synchronized monitoring of the star-crack formation. The device allowed the star-crack formation monitoring and quantitative measurements have been performed. The reliability of the device has been presented, and the uncertainties of force measurements at the indenter are discussed: the signal being very sensitive to noise, the use of low frequencies and of the force measured far from the glass is a choice allowing to exploit numerical simulations representative of the stress and velocity at stake during a blowpipe test.

The focus was made on developing an original testing device for crack formation experiments.

A more extensive series of tests can be performed to achieve a better definition of the indented zone mechanical behavior. Especially performing tests with a Vickers indenter may be interesting as this geometry is very well documented in the literature.

During the blowpipe tests, the crack evolution is performed via sudden jerks caused by the vibration of the impacted plate and the indenter. Therefore, comparing the indentation law (i.e. force F versus indentation δ) obtained during the dynamic indentation tests with a quasi-static indentation, using the same indenter, can give more information on the dynamic effects on the laminated glass behavior.

The main objective of those experiments was to predict the star-crack formation on the laminated glass regarding the impact parameters: gravel shape and mass, its normal to the glass velocity, the sample material, and geometry. Adding parameters such as surface condition or temperature during testing also provides a confidence interval for the model based on the fitted parameters. Furthermore, changing the glass composition using borosilicate glass can help compare the indented zone behavior.

Dynamic indentation model

A model has been designed by reproducing the test boundary conditions. A model involving the difference of behavior between the tests and a viscous elastic plate allowed to characterize, in a residual way, the behavior under the indenter during the formation of the star crack network. This unidimensional point of view, can represent the loss of stiffness under the indenter during a test, and explains the very complex and hardly described indentation zone.

The model can be enhanced by changing the most influential parameters. For instance, the indenter geometry should not change the local damage behavior, the glass ply thickness may have a minor influence on this behavior, and finally, the change of glass formulation should lead to a new law.

Even if the unloading phase has not been considered in the model, the amount of elastic energy stored in the local area is meager regarding the global elastic energy stored in the bending laminated glass plate. However, the stiffness decrease may impact the indentation zone behavior during the second and third force peaks and should be explored.

With a finely modeled behavior under the tip, it becomes possible to use X-FEM in the current elastic part of the glass, to make the crack branches evolve during an impact and, therefore, predict the final crack length. A criterion may be defined to describe the number of branches of the star crack as a function of the impact velocity (in the spirit of [START_REF] Vandenberghe | Star-shaped crack pattern of broken windows[END_REF]) and propagate them from the boundary between the damaged zone and the elastic zone.

Glass fragmentation

Many conclusions have been reached thanks to the instrumentation of the cannon device, especially the new S-DIC measurements of the deformed surface of a fragmented laminated glass plate during an impact. The influence of glass fragmentation on energy dissipation has been explored. Even if the cracking pattern does not influence the direct impact on an unbroken laminated glass plate, it becomes significant for the future dynamic loads as it will dictate which parts of the structure are more damaged than others.

The triangular fragment model can be improved to describe the circular crack initiation during the test and predict the location of the orthoradial cracks that accounts for the finer fragmented zone and the outer plate area, beyond the first crack.

S-DIC enhancement

Some new explorations concerning the S-DIC analyses may be done. To study the whole test duration, the release of some degrees of freedom of the regularized code can help describe the slight orthoradial movement. Mesh evolution can be implemented during the test, depending on the observed movement in certain zones. Using smaller elements than the smallest fragments can help better describe the joint apparition at the crack location. Indeed, with more refined elements (at most two times smaller than the fragment size), the slope breaks of the deformed surface would only correspond to a new crack, considering rigid glass fragments. A multiscale approach can also be considered [START_REF] Hild | Multiscale DIC Applied to Pantographic Structures[END_REF]].

Dealing with the brightness and contrast changes is also at stake, even if this issue can be partly resolved using iterative S-DIC (i.e. using as reference frame the previous image).

A change of instrumentation (e.g. higher magnification, higher image definition) may be needed in further studies to increase the number of integration points in the finer fragmentation areas.

Models: circular crack initiation and membrane deformation

The membrane model based on the experimental measurement of the cracking pattern evolution allowed assessing the live delamination length on the surface. This parameter was easily measured during quasi-static TCT tests in the literature, but is very difficult to apprehend during ball-drop experiments. Therefore, the model introduced in this work provides a way to use the extensive tensile test data in a simulation of cannon tests.

Considering finite strain, especially the dominant out-of-plane movement during the test, should complete the model. This evolution is already proposed in Fourton's perpectives [Fourton 2019] and can be addressed using the membrane kinematics [START_REF] Chevaugeon | Contribution à l'étude des membranes hyperélastiques en grandes déformations[END_REF]].

The model generalization to non-axisymmetric impact can be a plus in determining the evolution of the delamination far from the impact areas of the plate. Indeed, it should help to characterize the area of a plate where the impact energy is dissipated during a ball-drop experiment.

One perspective for the model is predicting the crack length instead of imposing the number of radial and orthoradial cracks. It may be possible to avoid using the crack densities and use the deformed surface, with a suited criterion for crack formation which is more easily accessible. It can be performed using the measured strain and strain rate regimes that will be linked to a theoretical crack density. So, the delamination and the dissipated energy could directly be deduced from the deformed state for each image.

Towards a damage description of the structure

The multi-impact instrumented tests have scratched the surface of a much larger topic: the residual resistance of broken laminated glass. The new geometry introduced on the cannon allows repetitive tests to be performed, then compared. During successive impacts, there is competition between the fragmentation density, the evolution of glass-PVB delamination, and the residual strength of the fragmented plate. If the glass fragments are small, the whole structure will be less resistant to a second impact, regardless of the location of the impacted plate. If the fragments are large, there is larger delamination near the previous impact, making that area more susceptible to tearing than the rest of the structure during another loading.

As tearing of the interlayer can happen outside the comminution zone, it is possible to formulate the crack field as an anisotropic "damage" model.

Giving the damage field on the already impacted laminate will help to know the limited damage value of a laminated glass subjected to several impacts and anticipate the residual behavior of the fragmented laminated glass. One should work out the impact energy dissipation capacity of any damaged region. This involves carrying in parallel 1. a theoretical analysis of the appropriate damage propagation and associated dissipation based on the microscopic picture of the crack density and delamination 2. and a new impact test campaign, exploring the influence of the distance between the first and second shots, and comparing the final delamination (or damage) fields.

The impactor velocity (i.e. kinetic energy) may also be changed to access of wide range of experimental conditions for validation.

Presently, the influence of a tear implies estimating the laminated glass plate as fully broken.

It would be interesting to look at the effect of a pre-tear at a certain distance from the second impact and see if its length evolves and if the laminate can still stop a projectile.

Assuming that u is not dependent on r, and in r = 0, the radial displacement due to inertia is null

u r (r, x, t) = -νr ∂u(x, t) ∂x (A.11) the speed v is then v = -νr ∂ 2 u(x, t) ∂t∂x e r + ∂u(x, t) ∂t e x (A.12)
Since the bar is free to move in the radial direction, the force balance on the volume Ω of the bar of section S , defined between x 0 and x 1 and the principle of virtual powers [START_REF] Dhatt | Une présentation de la méthode des éléments finis[END_REF] is then expressed as

Ω ρ ∂ 2 u i ∂t 2 v * i dV + Ω σ i j ε i j (v * i )dV = 0 (A.13)
With v * the trial field, a kinematically admissible velocity, similar to v. Verifying equation for every v * (x, t), and writing I the cylinder's second moment of area

I = S r 2 dS (A.14) one gets S ρ ∂ 2 u(x, t) ∂t 2 -ρν 2 I ∂ 4 u(x, t) ∂x 2 ∂t 2 -S ∂σ(x, t) ∂x = 0 (A.15)
which leads to Love equation [START_REF] Meyers | Dynamic behavior of materials[END_REF]]

∂ 2 u(x, t) ∂t 2 = c 2 ∂ 2 u(x, t) ∂x 2 + ν 2 I S ∂ 4 u(x, t) ∂x 2 ∂t 2 (A.16)
The differential equation is still a 1D-equation but now includes the radial displacement effects on the longitudinal displacement over the bar. The general solution of this equation is

u(x, t) = ES ρν 2 I 0
Re F(ω)e i(ξ(ω)x-ωt) + G(ω)e i(ξ(ω)x+ωt) dω (A.17) F(ω) and G(ω) are two complex functions expressed thanks to the boundary and initial values of the experiment. The wave number ξ(ω) is expressed as follows

ξ(ω) = ± ρω 2 E -ρν 2 I S ω 2 (A.18)

A.2.2 Correction of the harmonic wave dispersion

During a Split Hopkinson Pressure Bar test, strains are measured punctually on the bar via strain gauges. The wave shifting consists of inserting the dispersion between two measuring points, for instance, between the gauges A and B separated by distance L = x Bx A (with

x A < x B , cf Fig A.3), at fixed pulsation ω ε * B (ω) = ε * A (ω)e iξ(ω)L
Zhao and Gary also solved the dispersion in the time domain using a shift function f shi f t evaluated via the Fast Fourier Transform (FFT) [START_REF] Zhao | A new method for the separation of waves. Application to the SHPB technique for an unlimited duration of measurement[END_REF]]

f shi f t ε j asc,des (t) = FFT -1 e iξ(ω)L FFT ε j asc,des (t)
With the ascending and descending terms of the strain component j of A or B on the bar. The wave velocity is now frequency-dependent, i.e. the wave propagation time between the gauges will depend on the frequency. Therefore, the celerity C(ω) of the wave is

C(ω) = ω ξ(ω) = ± E 1 -ρν 2 I ES ω 2 ρ (A.19)
The weakness of this method lies in the fact that a multi-frequency signal is projected within a very limited time interval in the Fourier domain. It leads to a lack of accuracy in the contributions of each frequency, which will be highly averaged.

A.3 Elastic waves: bending and shear waves propagation

In this section, the 1D propagation of bending and shear waves without any other axial displacements is assumed. The considered bending occurs in the (x, y) plan. First, one may express equilibrium of an uni-dimensional element dx, represented on the scheme A.1. The hypothesis of small strain and small strain velocity is made. Which, in the circular beam, directly leads to

∂M(x, t) ∂x + Q(x, t) = Iρ ∂ 2 θ(x, t) ∂t 2 ∂Q(x, t) ∂x = ρS ∂ 2 u y (x, t) ∂t 2 (A.20)
The elastic equation in the case of a Timoshenko beam theory [START_REF] Timoshenko | Theory of bending, torsion and buckling of thin-walled members of open cross section[END_REF]] also gives

M(x, t) = yσ b xx dS = EI ∂θ(x, t) ∂x Q(x, t) = σ b xy dS = GS ∂u y (x, t) ∂x -θ(x, t) (A.21)
With G the shear modulus. Finally, the complete bending equation can be expressed as

ρS ∂ 2 u y (x, t) ∂t 2 = Iρ ∂ 3 θ(x, t) ∂x∂t 2 -EI ∂ 3 θ(x, t) ∂x 3 (A.22)

A.3.1 Neglecting lateral inertia

In this subsection, the lateral inertia term is ignored:

∂ 2 u y (x,t) ∂t 2 = 0. Due to the initial conditions Q(x, t = 0) =0 M(x, t = 0) =0
and the free edges of the indenter, Q(x = R, t) = 0, one may neglect the transverse force during the experiment. The bending equation is then expressed with rotational terms

Iρ ∂ 2 θ(x, t) ∂t 2 -EI ∂ 2 θ(x, t) ∂x 2 = 0 (A.23)
This equation is very similar to equation (A.2), with c = E/ρ the traction-compression wave propagation celerity. This wave creates a momentum along the bar, which is a good approximation of the beam bending in the first microseconds of an impact but is rapidly underestimated due to radial inertia.

A.3.2 Including lateral inertia

Here, one considers equation (A.22), dealing with rotational and lateral inertia. The substitute of the deflection terms leads to a fourth-order equation

∂ 2 θ(x, t) ∂t 2 + Iρ GS ∂ 4 θ(x, t) ∂t 4 + EI ρS ∂ 4 θ(x, t) ∂x 4 - I S 1 + E G ∂ 4 θ(x, t) ∂t 2 ∂x 2 = 0 (A.24)
Which complex equation is resumed to

-ω 2 + Iρ GS ω 4 + EI ρS ξ(ω) 4 - I S 1 + E G ω 2 ξ(ω) 2 = 0 (A.25)
With a wave number ξ(ω) which depends on the pulsation ω.

ξ(ω) = ± ω c 1 + 1 2 + ν ± 1 2 + ν 2 + c R 2 2 ω 2 (A.26) For t > (L + ℓ)/C(ω) σ ω (0, t + ℓ/C(ω)) + Z(ω)v ω (0, t + ℓ/C(ω)) = σ ω (ℓ, t) + Z(ω)v ω (ℓ, t) σ ω (0, t -ℓ/C(ω)) -Z(ω)v ω (0, t -ℓ/C(ω)) = σ ω (ℓ, t) -Z(ω)v ω (ℓ, t) (A.31)
Combining the equations one gets

v ω (ℓ, t) = 1 2Z(ω) [σ ω (0, t + ℓ/C(ω)) -σ ω (0, t -ℓ/C(ω))] + 1 2 [v ω (0, t + ℓ/C(ω)) + v ω (0, t -ℓ/C(ω))] σ ω (ℓ, t) = 1 2 [σ ω (0, t + ℓ/C(ω)) + σ ω (0, t -ℓ/C(ω))] + Z(ω) 2 [v ω (0, t + ℓ/C(ω)) -v ω (0, t -ℓ/C(ω))] (

A.4.3 Bending and shear waves Lagrange diagram

In the same way, Lagrange diagrams for the pure shear and the bending problem can be designed. One may find the impedance Z(ω) of any harmonic of pulsation ω, such as

d(M ω + Z(ω) θω ) dt = ∂(M ω + Z(ω) θω ) ∂t + ∂(M ω + Z(ω) θω ) ∂x dx dt = 0 (A.33)
with the rotation velocity θω = ∂θ ω /∂t. Considering the Timoshenko beam theory for purely elastic problem, the momentum M ω and rotation speed θω in complex forms are

M ω = -jξ(ω)EIθ θω = -jωθ
So, the two characteristic lines will depend on impedance Z(ω)

dx dt = C(ω) & Z(ω) = - EIξ(ω) ω dx dt = -C(ω) & Z(ω) = EIξ(ω) ω
The momentum and rotation speed created during an uncentered impact is measurable with the two axisymmetrical gauge difference.

This measured momentum is not dependent on the traction-compression movement as it is treated separately in the equation. One may improve the measured forces and momentum coupling the traction-compression and the bending phenomena.

In the blowpipe experiments, the momentum was measured to ensure the bending did not interfere with the purely axial loading during the whole test duration. It was verified that the bending contribution was not a problem for the force and velocity measurement in the end, as bending included and bending-neglected calculations have given the same results on the force and velocity.

B.2 Banc d'essai

Le banc d'essai blowpipe permet de réaliser des mesures quantitatives de force et de vitesse lors de la formation d'une fissure en étoile sur du verre feuilleté.

Il est instrumenté à l'aide :

• Des jauges de déformation collées sur la barre cylindrique qui permettent de détecter le passage des ondes élastique dans son corps cylindrique. A partir de ces signaux et dans le cadre d'une description uniaxiale d'un tel essai, on déduit la déformation et la contrainte aux positions des jauges. A l'aide d'un diagramme de Lagrange, on transporte ces informations pour remonter aux vitesse et force à l'extrémité de la barre où se trouve l'indenteur.

• De la caméra ultra-rapide Shimadzu ® "HPV-X2", filmant 256 images à une fréquence de 1, 2 ou 5 MHz. Celle-ci est utilisée pour suivre visuellement la formation de fissure. On peut ainsi quantifier la longueur d'une fissure au cours de l'essai.

• D'une mesure de force dans le ressort à l'aide d'un capteur piézoélectrique qui permet de régler la force dans le ressort de compression avant le tir. Cela permet de prévoir approximativement la vitesse d'impact du projectile. Par un transport de cette information, on remonte à l'état de contrainte et de vitesse en tout point de l'indenteur, et en particulier la base du cône (proche du verre) et la surface impactée (au sommet).

On peut résumer la démarche de la sorte:

1. La mesure de déformation donne directement la contrainte aux positions des jauges (loi de Hooke pour la barre).

2. Connaissant l'état initial du système, ici au repose et sans pré-charge, on mesure l'amplitude des ondes antérogrades et rétrogrades au droit de chaque jauge.

B.3 Dépouillement complet d'un essai

Un essai d'impact, réalisé avec le dispositif, est présenté.

La barre est initialement au repos et sans pré-charge. Celle-ci ne présente pas de variation de section.

Le cadre de 5 mm de côté maintient la plaque de 100 mm×100 mm de verre feuilleté en son bord (cf support de la Figure On mesure en premier la force, F = πr 2 σ (avec une moyenne glissante sur 1 µs), et la vitesse.

On peut déjà noter le signal très bruité de la force comparé à celui de la vitesse. La formation de la fissure arrive autour de 0 µs, après la mesure des deux premiers pics de force et du premier pic de vitesse, puis s'en suit une phase de chargement plus lente au cours de laquelle la fissure croît.

Les mesures intégrées en temps de l'énergie transmise au verre E = t F • V, de l'impulsion I = t F et du déplacement U = t V permettent d'afficher les tendances deux signaux précédents sans trop de bruit. On remarque l'avancée en escalier, ou par à-coups, de l'indenteur.

Il est également possible de remonter à la force de l'impact, mesurée sur la section de contact entre le projectile et la barre. Ce signal est donné par la courbe noire de la Figure B.6.

Les trois premières courbes correspondent au signal au sommet de l'indenteur. On trace en noir le signal brut, un filtrage moyen sur 1 µs en orange et le signal du premier pic de force (en jaune). Ce dernier est utilisé comme signal de force en entrée dans la simulation présentée au chapitre suivant. Le pic de force de l'impact atteint 1.7 kN. Les oscillations qui suivent permettent de définir une incertitude sur la mesure de force de 100 à 150 N.

Les signaux sont comparés avec la mesure de force en base de cône (courbe violette), proche de la surface du verre. La différence de mesure de force montre que l'essentiel de l'onde est réfléchie au niveau de l'indenteur de part l'impédance très différente entre le verre et l'acier ou le diamant.

La fissure finale obtenue est visible sur la Figure B.7. On remarque de larges fissures en forme de pétale qui se situent juste sous la surface extérieur du verre.

Le contact sphérique de l'indenteur créé une zone comminutée (partie blanche) circulaire, dont le diamètre mesure 600 µm.

On remarque à 9h un départ de fissure radiale. (i.e.adhésion parfaite) puisque l'endommagement est localisé dans la première moitié du pli de verre extérieur. Un cadre de 5 mm de côté est tracé sur la face intérieure du pli de verre non impactée, pour simuler les conditions d'appui simple du composite sur le cadre en PMMA.

B.4 Simulation de cet essai

Les paramètres matériaux utilisés sont renseignés Table B.1.

Le maillage est présenté en Figure B.9. Il est constitué d'hexaèdres. Le maillage le plus fin se situe au lieu du contact sphérique indenteur/verre, où il ne dépasse pas 20 µm (i.e.10% du rayon de courbure de la pointe). Au niveau de la barre, le maillage mesure 1 mm. Dans la zone élastique du verre feuilleté, à plus de 2 cm de l'impact, un maillage grossier de 5 mm est utilisé. Ce sont les premiers instants de l'indentation qui nous intéressent ici car c'est l'intervalle de temps dans lequel la fissure se forme, la phase de propagation secondaire n'étant pas étudiée ici. On considère donc une durée de la simulation jusqu'à la fin du premier pic de vitesse (i.e.à 10 µs).

On compare les résultats issus de la simulation élastique de l'essai avec ceux expérimentaux de la section B.3.

Les déformations sont comparées dans la Figure B.10. On remarque une excellente concordance entre les signaux en trait plein de l'essai et ceux en pointillé de la simulation. La différence la plus notable est celle de la jauge aval, principalement après la phase de contact avec le verre qui est très différente entre l'essai et la simulation élastique. L'écart maximum mesuré est de 10 -5 pour la déformation.

On peut également mesurer avec un diagramme de Lagrange les données de force, vitesse, impulsion, déplacement et énergie transmise en Il est clair que la mesure de force est ici la plus sensible pour l'essai sarbacane. Le moindre changement sur le banc d'essai provoque d'importantes différences sur la force. Cependant, les tendances des courbes de force présentent des amplitudes et évolutions comparables jusqu'à la fin du premier pic de vitesse. On peut aussi noter au regard du signal que l'inertie du cône de l'indenteur est négligeable et n'apparaît pas sur les courbes. Donc les mesures effectuées en base de cône sont très proche de la sollicitation réelle de l'indenteur sur le verre ce qui valide le choix de la géométrie de la pointe de l'indenteur.

A la vue des courbes, on remarque que les mesures de vitesse et déplacement sont bien reproduite dans la simulation élastique, tandis que la force, l'impulsion et l'énergie semblent être des données moins raisonnables.

La modélisation élastique linéaire pose dès lors problème. Il faut donc corriger cet écart en décrivant au mieux le contact verre-indenteur, ce qui est proposé dans la suite.

B.4.3 Simulation élasto-plastique

A la suite de plusieurs simulations élastiques d'essais différents, un modèle est proposé pour décrire le comportement non-linéaire du petit volume de la zone indentée (cf section 2.3.3 du manuscrit).

Dans cette partie, l'idée est de considérer l'essai avec trois éléments en série :

1. L'indenteur et son corps cylindrique qui restent élastiques.

2. La plaque de verre feuilleté restant dans sa globalité visco-élastique.

B.5 Conclusion sur le montage et les simulations

Le banc blowpipe permet de former des fissures sur le verre feuilleté, tout en quantifiant les grandeurs imposées à l'échantillon indenté. répétabilité des essais, modélisation de la mécanique non-linéaire de l'indentation dynamique par un comportement plastique équivalent qui rend compte assez fidèlement des mesures. Cela alors que l'indentation dynamique du verre est compliquée à décrire de part le caractère aléatoire de l'initiation et qui influe sur la propagation de la fissure, donc de sa géométrie finale.

Un compromis est fait sur le design du banc d'essai au sens où l'on ne peut pas ajuster l'impédance de la barre portant l'indenteur, car on souhaite répliquer une condition expérimentale représentative du gravillonnage. Par conséquent, l'essentiel de la quantité de mouvement transmise à la barre par le projectile est réfléchi à l'extrémité proche du verre et très peu est transmis au vitrage. Il en résulte des allers-retours d'ondes élastiques dans la barre complexes, et la force estimée sur l'indenteur est très faible et donc soumise à un bruit relatif très important. Inversement, la vitesse et le déplacement estimés à partir des mêmes mesures de jauge montrent un bruit beaucoup plus faible et donc des mesures plus fiables. Les autres mesures semblent quant à elles bien plus satisfaisantes. De plus, les données mesurées expérimentalement sont validées numériquement.

Le modèle du comportement de la zone indentée rend compte raisonnablement des mesures. Il présente cependant encore quelques lacunes pour

• d'une part, décrire les phases de décharge lors des à-coups imposés par l'indenteur au verre. En particulier il n'y a pas de composante visco-élastique ou visco-plastique attachée à cette sollicitation dynamique.

• d'autre part, mais il peut s'agir là d'un second modèle esclave du premier, décrire une phase de propagation des fissures dans le verre et se focaliser sur un critère d'arrêt, et donc fournir une prédiction de la taille du défaut final.

déformation macroscopique de la plaque. Cette dernière est due à l'étirement et au délaminage de l'intercalaire, phénomènes responsables de la majeure partie de la dissipation d'énergie lors d'un tel chargement [16,[START_REF] Elzière | Laminated glass : dynamic rupture of adhesion[END_REF]. De tels impacts représentants des situations de la vie courante : accident de voiture, chute de corps... peuvent être reproduits en laboratoire à l'aide de montages instrumentés (e.g. [START_REF] Del Linz | Delamination properties of laminated glass windows subject to blast loading[END_REF][START_REF] Mohagheghian | Quasi-static bending and low velocity impact performance of monolithic and laminated glass windows employing chemically strengthened glass[END_REF]14]). Avec celui développé dans la thèse de Nourry [16], il est possible de réaliser des essais très instrumentés, représentatifs d'une chute de bille. L'étude se concentre sur l'apparition de la ssuration et la déformation du composite. Une mesure de la déformée face arrière du verre et de l'évolution de l'énergie cinétique de l'impacteur au cours de l'essai sont mis en relation pour relier la performance, déformation du composite et la ssuration des plis de verre.

Comme décrit dans la norme EN 356 [15], le verre feuilleté est capable d'arrêter un projectile lors de chargements successifs, même après ssuration. Après réalisation de plusieurs impacts sur des échantillons de diérentes tailles, l'inuence des ssures survenant au premier impact sera discutée vis-à-vis de la résistance du feuilleté fragmenté, mesurée lors des tirs suivants. Une modication de la géométrie du banc permettra également de faire varier le point d'impact entre deux tirs successifs. L'objectif principal est de créer la ssuration puis de la qualier avant de charger à nouveau le verre. En eet, les zones plus ou moins endommagées ne se comporteront pas de la même manière lors d'essais successifs. Un grand nombre d'essais a été réalisé, mais une large partie de l'exploitation des résultats reste à faire cependant. La synchronisation de cette mesure avec une caméra rapide a permis de mettre en relation l'avancée de la formation de la ssure et les sollicitations appliquées sur le matériau.

Le modèle empirique associé aux essais permet de mesurer la pénétration de l'indenteur dans le verre et d'en déterminer une loi de comportement eective de la zone endommagée au cours de l'impact.

Le point de vue adopté ici est de considérer trois éléments en série : l'indenteur, dont les paramètres matériaux sont connus, la partie non-linéaire qui englobe la pointe en diamant et la ssure nouvellement créée, inconnue, puis le reste de la plaque feuilletée en exion, connue.

Les conditions aux limites d'appuis du composite et des positions relatives des éléments sont maîtrisées. La force de l'impact est appliquée au sommet de l'impacteur. On mesure les champs Concernant les essais de fragmentation, une implémentation des corrections de brillance et de contraste serait un plus pour rendre l'analyse de la déformée de la plaque et de la mesure de la ssuration plus robustes. Les modèles de ssuration et de dissipation doivent évoluer : éviter d'imposer une symétrie axiale, passer en grandes déformations, prédire la densité de ssure... Le chargement sous impacts multiples a été seulement survolé, peut être largement approfondi pour

  FIG. 1.3 -Coordinate system at Irwin slit crack tip.

1. 2 .

 2 FIG. 1.4 -Triplex Safety Glass Co's advertising poster, 1923 [Triplex Co. 1923].

Fig 1 . 5 .

 15 FIG. 1.5 -PVB monomers [Elzière et al. 2019].

FIG. 1 . 6 -

 16 FIG. 1.6 -Tensile test of PVB at ambient temperature (T∼20°C). Engineering stress plot versus elongation for different strain rates [Fourton 2019].

  FIG.1.7 -Authors numerically reproducing some cracking pattern after the same impact experiment on laminated glass.

FIG. 1 . 10 -

 110 FIG. 1.10 -Blunt (a) or sharp (b) projectiles used to impact semi-infinite monolithic glass [Chai and Ravichandran 2009]. (a) represents a Hertz cone crack, and (b) represents a median crack.

  FIG. 1.11 -Evolution of the impactor kinetic energy after contact with the glazing during impact tests and associated cracking profiles (at T=17°C). Impact speed is 9.2 m/s for initial kinetic energy of 170 J [Nourry 2005].

1. 3 .

 3 FIG.1.12 -Model of a cracked laminate[Seshadri et al. 2002].

  (a) Support. (b) Pipe & instrumentation.

Feasibility study via

  simulations (a) Abaqus set-up. (b) Indenter tip.

5 FIG. 2 . 3 -

 523 FIG. 2.3 -Blowpipe simulation on laminated glass.

  Fig 2.5, are closer to the reference. The explicit code used for the numerical tests is sensitive to noise and adds high-frequency artifacts to the signals measured via Lagrange diagram.

5 FIG. 2 . 4 - 3 FIG. 2 . 5 - 3 FIG. 2 . 6 -

 524325326 FIG.2.4 -Blowpipe simulation on void. A null low-frequency force is globally measured, except at the end where it is slightly increasing due to the short duration of the strain signals.

  FIG. 2.7 -Star-cracking on laminated glass sample. (a-c) High-speed imaging (at 5 Mfps) and (d) crack size measurement. Time origin is given by the increase of the detected acceleration (i.e. trigger) signal during the contact.

  FIG. 2.8 -(a) Gauges glued on the indenter and (b) their electric diagram.

  R 1 is the gauge nominal resistance of 120 Ω. The Wheatstone bridge and the 120Ω resistances, R i are shown in Fig 2.8b. Both locations have two diametrically opposed strain gauges to correct bending measured during an off-centered impact. In Fig 2.9a strain measurements from the two upper gauges, ε 1 and ε 2 close to the impact zone are shown, together with the two lower ones, ε 3 and ε 4 close to the base of the cone. In contrast, in Fig 2.9b similar measurements for an off-centered impact with one signal in tension and a second in compression on the two upper gauges are shown. However, the average signals of both experiments displayed in Fig 2.9c and Fig 2.

  FIG. 2.9 -(a)-(c) Centered and (b)-(d) Off-centered strain measurements. (c) and (d) are the mean signal of upper or lower site.

FIG. 2 . 10 -

 210 FIG. 2.10 -Trigger setting. The movie duration is varying with the acquisition rate as the image number is limited to 256 with this camera.

  12c. Finally, secondary crack propagation occurs in phase 4: slow growth of the crack branches is observable for 20 µs. The final star shape is obtained in Fig 2.12d. The end of the impact happens around 60 µs. A discharge marks it: the displacement changes trend after a short plateau (Fig 2.11), and some of the remaining elastic energy stored in the glass is restituted to the indenter (decrease of the energy curve in Fig 2.11).

FIG. 2 . 11 -

 211 FIG. 2.11 -Velocity, displacement, force, impulse and energy measurements of the indenter during blowpipe experiment. Four steps can be distinguished with a camera set in 3/4-view: 1-Dynamic indentation & elastic deformation. In-depth propagation. 2-Star-crack opening. 3-Bounce. 4-Secondary crack growth. The red dashed line are set at 21.2 µs, 24.6 µs, 39.8 µs and 60.2 µs.

  FIG. 2.12 -Images of the blowpipe experiment corresponding to the red vertical lines in Fig 2.11.

4 FIG. 2 . 13 -

 4213 FIG. 2.13 -The four crack branches length evolution, percent of their maximum length. Corresponding labels on the right-hand image (cf Fig 2.12d).

FIG. 2 . 14 -

 214 FIG.2.14 -Through-thickness camera setting. The lateral lightning is performed through the external glass ply in the z-direction.

  FIG. 2.15 -First and last images of the side-view 5 Mfps film. Initial view, (a) without mask, (b) with mask. Final image, (c) without mask, (d) with mask.

14 FIG. 2 . 17 -

 14217 FIG. 2.16 -Symmetries observable during the side-view captured blowpipe experiment.

4 FIG. 2 . 18 -

 4218 FIG.2.18 -Superposition of 6 force and velocity measurements at the indenter during experiments with the blowpipe prototype.

FIG. 2 . 19 -

 219 FIG. 2.19 -Blowpipe set-up: a steel-made indenter with diamond Rockwell tip is instrumented with strain gauges as a mini-Hopkinson bar.

Fig 2 .

 2 19b), closer to the contact with the impactor, and the lower site ('B' in Fig 2.19b), close to the cone and the glass surface. The gauges position are shown in Fig 2.19b. The reference position (x=0) is the lower site (labeled 'B'). The upper site (labeled'A') is placed at a distance L ∼ 35 mm from the reference, and the cone bottom is distant from ℓ ∼ 10 mm from 'B'. The strain uncertainty is estimated at 1.6×10 -6 (i.e. 0.7% of the maximum deformation, i.e. a tension of 2.5 mV).

FIG. 2 .

 2 FIG. 2.20 -Standard deviation of the gray level (dashed line) under the indenter tip, together with the measured displacement of the indenter cone (solid line) at the same instant. The gray level change under the indenter is shown in Fig 2.26.

FIG. 2 .

 2 FIG. 2.21 -Peak velocity of the indenter tip (m/s) versus the spring initial load (N).

5 FIG. 2 . 22 -

 5222 FIG. 2.22 -Superimposed force (N) and velocity (m/s) for four impacts on flat soda-lime glass samples. No bending of the sample is authorized, and the initial load in the spring is close to 25 N.

7 FIG. 2 .

 72 FIG.2.23 -Force (□) and Velocity (⃝) measurements at the indenter tip.

  FIG. 2.25 -Through thickness crack evolution. (a) Force (□) and Velocity (⃝) measurements at the indenter tip. (b) Crack depth (□) and in-plane width (⃝).

6 FIG. 2

 62 FIG.2.28 -Force (top) and velocity (bottom) of the indenter measurement during three dynamic indentation experiments on polystyrene foam. The raw signal is displayed in blue (-). The mean force on 24 µs is plotted (red line, --), showing more minor oscillations around 0 N.

  FIG.2.29 -Indenter and laminated glass scheme. The whole blowpipe is composed of three elements in series:(1) The indenter, which is instrumented via strain gauges to calculate stress and velocity in the upper cross-section of the cone. (2) Small local non-linear volume between the indenter and the sample. (3) The laminated glass plate, without the damaged volume (2), which remains visco-elastic during the experiment.

  FIG. 2.30 -Blowpipe simulation in Abaqus ® CAE software. The imposed impact load is represented on the curve and applied on the top surface of the indenter. A sectional view of the bottom of the cone and its contact with the glass surface is represented in the top right-hand corner.

FIG. 2 .

 2 FIG. 2.31 -(a) Displacement and indentation evaluated at the bottom of the cone. (b) Average effective behavior of the local zone under the indenter. The experimental force is plotted versus indentation depth. The identified behavior (b) is implemented as a plastic law (defined from experimental stress vs plastic strain curve) of the local damage zone, allowing the measurement of a new displacement of the cone bottom U ep .

  FIG. 2.32 -SEM Low vacuum observation of the indented zone after a blowpipe experiment on the laminated glass. Two radial branches of the star-crack are particularly visible (at 3:30 and 9:00 on a clock). The indentation zone diameter measures 540 µm. The surface flakes around the indentation are formed during the unloading phase.

FIG. 2 . 33 -

 233 FIG. 2.33 -Conical indentation median plane observation through the glass thickness via an optical microscope. The polarized light underlines the residual stress in the glass ply. The non-elastic zone is 500 µm deep and 1 mm large.

  the deviatoric part of the Cauchy stress tensor, σ the stress criterion (e.g. Von Mises) and εp the equivalent plastic strain rate described by Fourton and Landis et al. by a power law of exponent m εp ε0 + 1 = σ σ y m considering the material yield stress σ y and the initial plastic strain rate ε0 . The plastic strain rate εp i j is directly linked to the plastic energy dissipated (i.e. the bulk contribution of the dissipated energy) during the crack advance

FIG. 3 . 1 -

 31 FIG.3.1 -Scheme of a cannon experiment. The impactor and projectile are guided in the barrel. The impactor head finally hits normally the laminated glass. If the laminated glass is completely pierced, the impactor will be stopped by an arrest cone in the barrel.

  Compared to other techniques, Stereo-Digital Image Correlation [Sutton et al. 2009] (S-DIC) is a non-contact, non-destructive, robust, and precise optical method to access 3D shape and kinematic information (motion and deformation) of large structures [Dufour et al. 2015b; Genovese and Sorgente 2018]. Moreover, by expressing S-DIC into a global Finite Element based formulation, the virtual model (e.g. CAD [Dufour et al. 2015a] or Finite Element mesh [Pierré et al. 2017; Berny 2020; Colantonio et al. 2020]

  A classical Point-to-Point initialization using natural features (e.g. edges and corners) provides an initial guess of the projection matrix. Then, the projection matrices are further refined in the Global CAD/FEbased framework as proposed by [Dufour et al. 2015a; Shao et al. 2016; Berny 2020].S-DIC also enables to fill in the gap between impact tests and more classical material tests. The energy dissipation that occurs during an impact through delamination and stretching of a polymer layer can be estimated using the through-crack tensile (TCT ) test results of the litterature[Seshadri et al. 2002; Elzière 2016; Del Linz et al. 2017; Samieian et al. 2018; Fourton et al. 2020; Ma et al. 2022].

Fig 3 . 3 .FIG. 3 . 3 -

 3333 FIG. 3.3 -Vacuum cycle for 44-2 laminated glass samples.

FIG. 3 . 4 -

 34 FIG. 3.4 -The stereo camera setup: 1 The 400W spotlights. 2 Shimadzu ® HPV-X2. 3 Right-view Photron ® SA-5 (24 mm lens). 4 Left-view Photron ® SA-5 (24 mm lens). 5 Telemeter. 6 Barrel box. 7 Laminated glass sample clamped in a circular window. 8 Impactor half-sphere. Cameras, shown in Fig 3.4, are used to record the details of the dynamic fracture. The first camera (2) is a Shimadzu ® 'HPV-X2' that captures the crack initiation at 100,000 Hz with a 400×250 pixel definition. The camera axis is coincident with the cannon z-axis. Two synchronized (at ±20 µs) Photron ® 'SA-5' cameras(3 & 4) are positioned on each side of the device, monitored at 30,000 Hz. The CMOS sensors ensure a good quality of the images captured with the two lateral cameras, but their definition is limited and decreases as the acquisition rate increases. The total definition of each camera is then 640×376 pixels. Moreover, due to the very short exposure time of the fast cameras, two 400 W spotlights are oriented towards the sample to provide enough light with a negligible PVB heating.

FIG. 3 . 5 -

 35 FIG. 3.5 -Initial view of the two lateral cameras. World coordinate system projected on the images.

(d) Acceleration (10 4 m/s 2 )

 2

4 . 2 .

 42 FIG. 3.7 -Example of surface mesh used for S-DIC measurement, projected on the two camera views.

FIG. 3 . 8 -

 38 FIG. 3.8 -Extrapolated out-of-plane (a) displacement and (b) velocity measured at different times.

  Kinetic energy (J) of the laminated glass plate.

FIG. 3 . 9 -FIG. 3 . 10 -

 39310 FIG. 3.9 -(a) Kinetic energy of the laminated glass. (b)-(c) Energy stored or dissipated in the laminated glass, with (black dashed line) or without (plain blue line) the kinetic energy of the plate. The signal appears more smooth when the contribution of the kinetic energy is removed from the telemeter measurement.

  FIG. 3.11 -Initial 2D mesh projected on the two reference frames. T3 element average edge length is 4 mm.

  FIG.3.12 -1D shape functions q i (r) ensuring displacement continuity, in straight lines. Interpolated T3-nodes (Fig 3.11) as circles on the new shape functions. The plot illustrates 15 mm-long elements for better readability, while 4 mm elements were actually used in the axisymmetric S-DIC measurements.

Fig 3 .

 3 14, 3.16 and 3.18, which corresponds to the 20 th , 50 th and 80 th frames, are shown the residual fields for global (a)-(c) and axisymmetric (b)-(d) S-DIC, expressed in gray levels, and the out-of-plane displacement field for global (e) and axisymmetric (f) S-DIC, in millimeters. The raw images (a) left and (b) right of the corresponding instants are represented in Fig 3.13, 3.16 and 3.18. The residuals and displacements are shown on the undeformed geometry (Z=0 plane). The out-of-plane displacement is also plotted on the current deformed 3D surface for global (g) and axisymmetric (h) S-DIC. Images are 8-bit encoded, meaning that the gray level of each pixel may vary from 0 to 255.

FIG. 3 . 13 -

 313 FIG. 3.13 -Stereo images at t=0.63 ms (i.e. iteration 20).

FIG. 3 . 14 -

 314 FIG. 3.14 -Comparison between global S-DIC (left column) and axi-S-DIC (right column) at t=0.63 ms (i.e. iteration 20). The different rows show respectively: (a-b) residual field for the left camera images; (c-d) residual field for the right camera images; (e-f) Out of plane displacement plotted onto the initial mesh; (g-h) Out of plane displacement plotted onto the deformed mesh.

FIG. 3 . 16 -FIG. 3 . 18 -

 316318 FIG. 3.16 -Comparison between global S-DIC (left column) and axi-S-DIC (right column) at t=1.63 ms (i.e. iteration 50). The different rows show respectively: (a-b) residual field for the left camera images; (c-d) residual field for the right camera images; (e-f) Out of plane displacement plotted onto the initial mesh; (g-h) Out of plane displacement plotted onto the deformed mesh.

FIG. 3 .

 3 FIG. 3.19 -Post-breakage sample observations performed just after the experiment: (a) sample is still set in the device. Flat positions: (b) Back side or monitored side view, (c) Front side or impacted side view.

  FIG.3.22 -Out-of-plane displacement (mm) for some radius ('R', in mm) versus time (ms). Axisymmetric ('ax') and global ('gl') S-DIC are compared with the telemeter measurement.

  Z-displacement (mm).

  r-displacement (mm).

  FIG.3.23 -Full-field measurement with the axisymmetric S-DIC. Fields are represented as a function of the radius, at different instants. The results for R=0 mm and R=140 mm are extrapolated with respect to the boundary conditions.

Fig 3 .FIG. 3 .

 33 FIG. 3.24 -Schemes: (a) TCT test [Fourton 2019], (b) Delamination crack front. During a test, the energy dissipation occurs at the crack front (orange area). Symmetry planes are represented in the figure (dashed lines) as the delamination is supposed to be symmetric in the model.

FIG. 3 . 25 -

 325 FIG. 3.25 -Simplified laminated glass behavior based on observation with TCT tests. Inspired from [Elzière 2016; Fourton 2019].

  This argument is schematized in Fig 3.25.

Fig 3 .

 3 Fig 3.25a corresponds to the hyperelastic behavior of a PVB film submitted to tensile stress. On this curve, the delamination occurs in one point that depends on the delamination velocity. This curve depends on the strain rate applied during the tensile test, as the PVB behaves viscoelastically. In the linear left-hand part in Fig 3.25b, λ a is constant: steady-state regime, up to the maximum reachable delamination length a = L/2. Finally, in Fig 3.25c, the macroscopic behavior, as measured during the TCT tests, is schematized. The adhesion limit is reached very quickly with increasing extension. At the end of the steady-state, no delamination occurs anymore, and the stress increases quickly before the tearing of the interlayer.

FIG. 3 .

 3 FIG. 3.26 -Schemes: (a) TCT data in steady-state regime. Solid line: limit between stable and unstable steady delamination. Dashed line: limit between steady delamination and interlayer tearing [Fourton 2019]. (b) Steady-state stretch, λ a , vs tensile velocity, δ [Elzière 2016]. (c) Hyperelastic behavior at 25°C for two cannon strain rates [Fourton 2019].(d) Limit of adhesion σ adh versus the delamination velocity ȧ, the delamination velocity is limited to an asymptotic value of 33 mm/s [Elzière 2016]. These curves are available for a PVB thickness of 0.76 mm.

FIG. 3 .

 3 FIG. 3.27 -(a) Monitored crack number evolution versus time during a cannon test, (b) Defined zone on the fragmented plate. The cracks are numbered watching the high-speed camera images and the residuals projected on the deformed surface to know their precise localization.

FIG. 3 .

 3 FIG. 3.28 -(a) Image of the test after the radial crack formation. (b) Scheme of the triangle model, with a radial crack as a symmetry plane. The shape of the triangle used in the model is visible in red in the image.

FIG. 3 .

 3 FIG.3.29 -Half-fragment with applied linear force on the PVB bridge at the radial crack location. Symmetry planes (dashed orange lines) are represented at the median radius of the triangular fragment, and the middle of the stretched PVB ligament.

FIG. 3 . 30 -

 330 FIG.3.30 -Stress versus strain curve illustrating the softening at a newly formed polymer bridge, i.e. the orthoradial crack location.

FIG. 3 .

 3 FIG. 3.31 -(a) Boundary conditions on the half-fragment in Abaqus ® CAE 6.14. (b) Measured and imposed displacement of the half sphere.

FIG. 3 .

 3 FIG.3.32 -Nodal (a) displacement and (b) velocity evolution for different radius. The imposed displacement of the sphere in the simulation is also represented.

  FIG.3.33 -Energies (J) extracted from the simulation results. Those data are superimposed to the previous experimental curves in blue dashed line.

5 FIG. 3 . 34 -

 5334 FIG.3.34 -Image of the pre-cracked plates before a second impact test at 130 J.

FIG. 3 .

 3 FIG.3.35 -Measurements obtained during the second impact on pre-fragmented laminated glass samples. (a-e) Samples' shapes are measured via S-DIC and plotted every 1 ms. 5 mm elements are used for these axisymmetric S-DIC measurements. (f) Impactor kinetic energy estimated from the telemeter versus time.

  FIG.3.36 -Scheme of the three sample sizes (dashed green). From smaller to larger areas: 300×300 mm 2 cannon samples, 560×560 mm 2 cannon new samples, 900×1100 mm 2 ball-drop samples. The device axis and positioning of the plates are represented.

Fig 3 .

 3 36 represents the new dimension of the window frame. The axis of the device, i.e. the contact location between the half-sphere and the glass surface, is indicated. It is horizontally centered but positioned vertically at 15 cm from the lower edge and 35 cm from the upper edge. The cameras are installed as shown in Fig 3.4. However, the frame acquisition of the two Photron ® SA-5 cameras is reduced at 20 kHz, and the sensor resolution is increased to 704×520 pixels to monitor a larger area of the new plate. Camera lateral views are shown in Fig 3.38.A Memrecam ® HX-3 replaces the Shimadzu ® in the axis of the barrel. Its acquisition rate is also set at 20 kHz for a definition of 640×640 pixels and synchronized with the lateral cameras, using the same trigger signal. Front view is represented in Fig 3.37.

Fig 3 .

 3 37 is shown the crack pattern evolution during the test. At first, radial and orthoradial cracks slowly grow in number on the two glass plies. Cracks are formed throughout the test duration, up to 15 ms. The final damage pattern is observed in Fig 3.37h. A 40 mm diameter comminuted zone is visible under the impactor. A 200 mm external diameter disk zone shows an area with finer fragments. Above this limit, larger glass fragments are created. For instance, 10 cm long glass shards can be measured at the top. New elliptical cracks are formed at the bottom right to comply with the boundary conditions. Those kind of ellipse cracks are observed in ball-drop tests as well [Nourry 2005; Decourcelle 2011; Elzière 2016].

  FIG. 3.37 -Front view of a cannon test at 130 J. (a) Moment of the impact and time reference. Radial cracks are formed in about ten milliseconds after the impact. (b-g) Time evolution of the crack pattern. (h) A 200 mm diameter circular crack zone is formed around the impact location.

FIG. 3 .

 3 FIG. 3.38 -S-DIC measurements: four out-of-plane displacement fields (mm) are plotted on the deformed mesh, projected on the current images. Right-view on the right column. Left-view on the left.

(a) 2

 2 FIG. 3.40 -(a)-(b) Second and (c)-(d) third impacts. First and last frames are shown. The largest defaults from the first impact are plotted in orange (radial crack) and green (orthoradial crack). During the third impact, tearing of the interlayer appears at the highlighted critical crack location.

FIG. 3 .

 3 FIG. 3.41 -Impactor kinetic energy evolution, (a) versus time, (b) versus displacement, for three successive tests on a laminated glass sample. Tin side of the glass layers in tension. The initial energy drop in the first test is important because the cracks progressively appear. The fragmented plate can accommodate the sphere geometry in the two other cases.

FIG. 3 .

 3 FIG. 3.42 -(a) Pattern obtained during the first ply breakage (at t=2.4 ms). Patterns obtained at the end of the tests: (b) First shot, (c) Second shot, and (d) Third shot. The impact is made at 6 on a clock. A pi/2 plate rotation is made between (b)-(c) and (c)-(d). No tear has appeared.

FIG. 3 .

 3 FIG.3.43 -Impactor kinetic energy evolution, (a) versus time, (b) versus displacement, for three successive tests on a laminated glass sample. Tin side of the glass layers in compression. The initial energy drop in the first test is large because the cracks have appeared lately. The final cracking pattern is obtained quasi-instantaneously (in 2 or 3 images). Plus, only a few cracks are created under the impactor during the last two shots.

  FIG.A.1 -Schemes of a beam submitted to bending, in the (x, y) plan.

  FIG. A.2 -Lagrange diagram with no dispersion, used to analyzed blowpipe experiments.

4 FIG. B. 2 -

 42 FIG. B.2 -Déformation de la jauge amont ('top') en rouge et aval ('bot') en bleu. L'ordre de grandeur est de 10 -4 .

  FIG. B.3 -(a) Contraintes et (b) vitesses aux positions des jauges. En rouge la jauge en amont, en bleu la jauge en aval.

FIG. B. 4 -

 4 FIG. B.4 -Propagation des ondes utilisées dans le diagramme de Lagrange.

FIG. B. 5 -❷

 5 FIG. B.5 -Mesure des grandeurs en base de cône, au plus proche de la surface indentée. On mesure (a) la force et (b) la vitesse. Par intégration temporelle (c) l'impulsion, (d) le déplacement et (e) l'énergie.

FIG. B. 9 -B. 4 . 2

 942 FIG. B.9 -Vue en coupe du maillage au lieu du contact entre l'indenteur et le verre. La partie en vert foncé sous l'indenteur est un cube de côté 600 µm.

  Figure B.11.

4 FIG. B. 10 -

 410 FIG. B.10 -Déformations mesurées dans l'essai (-) et dans la simulation (--) aux positions des jauges.

  FIG. B.11 -Mesures des grandeurs en base de cône. On mesure (a) la force et (b) la vitesse. Par intégration temporelle (c) l'impulsion, (d) le déplacement et (e) l'énergie. Les données issues de l'essai (bleu) et de la simulation élastique (orange) sont représentées.

A

  FIG. B.12 -Mesures de déformation aux jauges (a) et mesures en base de cône. On trace (b) la force, (c) la vitesse, (d) l'impulsion, (e) le déplacement et (f) l'énergie. Les données tracées sont issues de l'essai (bleu) et des simulations élastique (orange) et élasto-plastique (jaune).

FIG. B. 14 -FIG. B. 15 -

 1415 FIG.B.14 -Tracé de la force F en fonction de l'indentation δ.

2Fig. 1 -

 1 Fig. 1 -Banc d'essai blowpipe : Le corps en acier de l'indenteur est instrumenté avec des jauges de déformation tel une barre de Hopkinson miniature. Sa pointe Rockwell en diamant est initialement en contact avec la surface du verre.

  cinématiques et la contrainte dans les simulations. La relation entre la force transmise à la pointe de l'indenteur et sa pénétration dans le verre dénissent une loi de comportement eective de l'ensemble des non-linéarités du problème. La force est mesurée expérimentalement et imposée dans la simulation, tandis que la profondeur d'indentation est évaluée comme la diérence de déplacement mesurée entre l'expérience et la simulation 100% visco-élastique réversible. Cela permet de qualier la non-linéarité et ainsi trouver un comportement 1D du dommage non-linéaire créé. Cette étude ouvre la voie vers une caractérisation des performances du verre feuilleté. En faisant varier les géométries, les matériaux, et les topologies de ssure, il sera possible d'optimiser le choix des matériaux pour mieux résister aux impacts de gravillons.3 Fragmentation lors de l'impact et résistance résiduelle Qualier et quantier la réponse mécanique du verre feuilleté lors de chargement dynamique est important pour la sécurité de passagers dans un véhicule ou de passants sous une verrière. De tels impacts sont facilement rencontrés lors d'un usage courant du composite lors d'accidents (et éventuellement d'eractions). De nombreuses réglementations sont alors mises en vigueur et des essais normatifs, tels que la chute à la bille, doivent être réalisés avant la mise sur le marché d'un nouveau produit. Plutôt que de répliquer directement la norme EN 356 à l'aide

Fig. 2 -

 2 Fig. 2 -Installation pour la stéréo-vision : 1 Spots lumineux de 400W. 2 Shimadzu ® HPV-X2. 3 Photron ® SA-5 (focale de 24 mm) de droite. 4 Photron ® SA-5 de gauche. 5 Télémètre. 6 canon. 7 Verre feuilleté encastré. 8 Demi-sphère de l'impacteur.

  Fig 2) se compose d'un projectile et d'un impacteur au corps cylindrique terminé par une demisphère de rayon 50 mm en acier, reproduisant la géométrie de l'impact EN 356 qui utilise des sphères d'acier ayant le même rayon. L'impacteur et le projectile sont guidés dans le canon, dont l'axe est en incidence normale avec la surface de l'échantillon. A la n de sa course l'impacteur est arrêté par un épaulement. Une mesure du mouvement du plan médian de la demi-sphère est réalisée à l'aide d'un télémètre échantillonné à haute fréquence. L'espace autour du banc d'essai permet également l'installation de plusieurs caméras rapides synchronisées orant diérentes vues de la déformée 3D de l'échantillon. La stéréo-corrélation d'images numériques (S-CIN) est revisitée pour pouvoir mesurer une déformée axisymmétrique de la face arrière du verre feuilleté. Le code est régularisé en amont des itérations sur le déplacement et permet de ltrer localement les déplacements erronés à cause des artefacts sur les variations de niveau de gris : changement de brillance et contraste, zone fortement éclairées, apparition des ssures, ous. l'hypothèse de comportement axisymmétrique permet de mesurer la partie non mouchetée du feuilleté et ainsi mesurer l'évolution du nombre de ssures radiales et orthoradiales en fonction du rayon. Lors d'un essai il est possible de réaliser la mesure dans une base axisymétrique du déplacement hors-plan (Fig 3a), radial (Fig 3b) et de la vitesse (Fig 3c). A l'aide de cette méthode

Fig. 3 -Fig. 4 - 4 Perspectives

 344 Fig.3-Champs mesurés à l'aide de la S-CIN axisymmétrique, représentés en fonction du rayon, pour différents instants. Les résultats à R=0 mm et R=140 mm sont extrapolés à partir des conditions aux limites.

Table 1 .

 1 1 -Composition of soda-lime and borosilicate glass

	Borosilicate

Table 1 .

 1

	Units

3 -PVB properties at temperature T∼20°C [Martín et al. 2020]. 1.2. Laminated glass Ch. 1. Introduction

Table 1 . 4

 14

	s)

-PVB Prony series at 25°C

[Elzière 2016

].

  ; Seshadri et al. 2002; Galuppi and Royer-Carfagni 2016; Dam 2017; D'Ambrosio et al. 2019; Samieian et al. 2019].

Table 1 .

 1 The four different sandwiches used for the tests are reported in table 1.5.

		Internal	Impacted	PVB	Total
		Ply	Ply	plies	thickness
				0.38	3.08
	Thicknesses	1.6	1.1	0.76	3.46
	(mm)			3×0.38=1.14	3.84
				0.76+0.38=1.14	3.84

5 -Samples used for blowpipe experiments. Plates surface is 100×100mm 2 .

  Many researchers have examined quasi-static indentation of glass samples[START_REF] Roesler | Brittle fractures near equilibrium[END_REF][START_REF] Frank | On the theory of Hertzian fracture[END_REF];[START_REF] Evans | Quasi-static solid particle damage in brittle solids-I. Observations analysis and implications[END_REF], by moving a spherical indent on a surface or describing a Hertz cone crack formation in the glass thickness. Concerning dynamic loadings on semi-infinite monolithic glass, Chaudhri and Kurkjian observe the formation of indepth, radial, and orthoradial cracks on glass dynamically indented by steel balls[START_REF] Chaudri | Impact of small steel spheres on the surfaces of "normal" and "anomalous" glasses[END_REF]]. Rapid imaging is performed by Knight et al. to follow the cracking steps of the glass block and thus characterize the crack geometries, including the conical crack typical of Other experiments, such as the hard-body impact on windshields, are widely carried on. Those kinds of experiments are mainly designed to replicate head-form impact experiments. In literature, Pyttel et al. model failure in laminated windshields, without considering the visco-elastic behavior of the PVB interlayer, using a local Rankine maximum stress criterion to describe the crack initiation and growth[START_REF] Pyttel | Failure criterion for laminated glass under impact loading and its application in Finite Element simulation[END_REF]]. Peng et al. seek at the optimum fracture stress in their models that matches best the different simulation patterns of the laminates[START_REF] Peng | Finite element modeling of crash test behavior for windshield laminated glass[END_REF]. They have found a 50 MPa failure stress of glass was the most appropriate value to describe their head-forms impact experiments. Comparing experiments and models, Yu et al. have proposed a homogenized single-layered model for laminated windshields that also works for a triple-layered windshield application[START_REF] Yu | Computation modeling of laminated crack glass windshields subjected to headform impact[END_REF]]. Finally, the tempered laminated windshield is reported to be the safer and lighter windshield compared to classic flat soda-lime glass glazing[START_REF] Li | Dynamic mechanical behavior and pedestrian safety characteristics of toughened laminated windshield[END_REF]].Phenomena are reported during hard-body impacts. For instance, Nassr et al. insisted on the predominance of the flexural waves versus the in-plane tension compression ones, in the glass breakage

Hertz contact

[START_REF] Knight | Impact of small steel spheres on glass surfaces[END_REF]

]. In addition, for thin glass plates, particularly laminated glass, it is advisable to take into account the crack opening mechanisms resulting from the bending of the glass, which is dominant during blunt impact

[Behr et al. 1993

]. The different cracking mechanisms of thin float glass (between 3 and 12 mm) and impact speeds of small spherical projectiles around 10 m/s are listed by

[START_REF] Ball | On the low velocity impact behaviour of glass plates[END_REF]

].

  with nodal amplitudes U i [Aleyaasin and Harrigan 2010; Hild and Roux 2012]. The hessian matrix [H] c relative to each camera, c, and right-hand vector {b} c are defined as

Verifying the gauge measurements with the simulations

		Steel	Diamond	Units
	Density	7800	3500	kg/m 3
	Young's modulus	210	1050	GPa
	Poisson ratio	0.27	0.1	-
	Table 2.1 -Indenter and projectile properties.
	In Fig 2.2a the indenter body diameter is 5 mm, then narrows down to 4.6 mm. The cone
	angle is 130°. The external ply (i.e. impacted ply), PVB foil, and internal ply thicknesses are
	respectively 1.6 mm, 0.76 mm, and 1.1 mm.
	The virtual test starts with an initial velocity of the projectile of -5 m/s. It impacts the fixed inden-
	ter, initially in frictionless contact with the glass. The laminate is free of movement: because of
	its sufficient inertia regarding the initial velocity and the 100 µs experiment duration, it does not
	need any boundary conditions. This was verified by a null displacement of the plate boundary.
	The force and velocity are measured at the base of the cone, accessible in practice via the
	Lagrange diagram (see section 1.4.1).
	Velocity and force signal are reported in Fig 2.3, 2.4, 2.5 and 2.6.
	FIG. 2.2 -(a) and (b) Dynamic explicit analysis with axisymmetrical geometry, on Abaqus ® 6.14, for the blowpipe experiment feasibility. (c) Gauge signals: in red the simulated
	upper gauge signal, in blue the simulated lower gauge signal.
	Preliminary simulations are achieved to verify the reliability of the gauge measurements with
	the device. Gauges locations are shown in Fig 2.2a. The forces and velocities determined from
	the simulated gauge measurements can be compared with those directly computed by the finite
	element software at the bottom of the cone. In addition, it will be possible to look at the gauge
	signals (Fig 2.2c) during an impact test on the void (i.e. without the laminate) to evaluate the
	measurement dispersion and its uncertainty. Finally, two filters are compared: a low-pass filter
	and a dispersive model-based filter.
	The simulation reproduces the new experimental set-up. The axisymmetry along the axis of
	the tube is assumed. The glass is supposed to be elastic (cf soda-lime glass in table 1.2)
	and to have a perfect adhesion with the visco-elastic PVB foil (see tables 1.3 and 1.4). The
	indenter and the projectile are made of steel, and the Rockwell tip, in white in Fig 2.2b, is made
	of diamond (cf table 2.1 for material characteristics).

Table 2 .

 2 2 -Rockwell indenter material properties. mental data, 4 phases are noteworthy during the crack formation (Fig 2.11).At the beginning of the impact (zone 1 in Fig 2.11), the glass plate globally behaves elastically, but a local indentation (of a few microns in length) is made under the indenter tip. The crack propagates through the thickness of the glass ply until it reaches half of its thickness. This propagation is done by bursts since the zone is in slight compression and does not facilitate the formation of cracks that have no choice but to evolve vertically through the thickness. This is also observed in a previous reference[Kishta 2018b]. This step is hard to capture with the

	Density	7648	kg/m 3
	Young's modulus	205	GPa
	Poisson ratio	0.27	-

Given the experi-chosen 3/4-view, but it corresponds to the first images (Fig 2.12a). Then, at time t=21.2 µs, a brutal crack opening occurs in less than 4 µs (between Fig 2.12a and Fig 2.12b). At the same time, the speed of the cone decreases, and the maximum impact energy is reached. The star crack is formed during this quick step. During phase 3, the speed becomes negative with a slight decrease of the displacement, which means the indenter bounces off the glass for a few microseconds. The crack length does not evolve much between Fig 2.12b and Fig 2.

Table 2 .

 2 PVB Prony series (at 25°C) in the time domain, with the amplitudes g i , the characteristic times τ i (in s), and G(t) = G 0 (1 + g i exp(-t/τ i ) the shear modulus: 3 -Constitutive parameters used in the simulations. rigid plane contact of 5 mm from the edges. However, it plays a minor role as the impacts are performed far enough from the sides of the laminated glass plate to avoid overlaying reflected elastic waves during dynamic indentation.The force measured at the top of the input bar, at the contact section hit by the impactor (see force and contact in Fig 2.30), is imposed. The displacement U e of the bottom of the cone is obtained via the simulation (Fig 2.31a). The difference at the cone bottom between the experimentally measured displacement U x and the simulated one U e gives the indentation δ.

			PVB	Steel	Diamond	Damage	Units
	Density	2500	1000	7850	3500	2500	kg/m 3
	Young's modulus	70	1	200	1050	70	GPa
	Poisson ratio	0.23	0.49	0.3	0.1	0.23	-
		(0.10690, 1e-04)	(0.09720, 1e-03)	(0.08748, 1e-02)
	(g i ,τ i )	(0.01940, 1e-01)	(0.00583, 1.0)	(0.00107, 1e01)
		(0.00078, 1e02)	(0.00058, 1e03)	(0.00032, 1e04)

  However, the relative movement of the fragments during the tests is seldom studied, the random cracking pattern of the glass layers being quite tricky to apprehend.In a general context, the experimental tools to characterize the impact response of either impactor or laminated glass can be classified into three major categories:1. Direct measurements on the projectile/impactor via an accelerometer or a cell force[Yu et al. 2017; Li et al. 2019; Wang et al. 2020]. 2. Detection of wave propagation on the glass surface via multiple strain gauges glued at different radial distances from the impact location [Wang et al. 2020; Nassr et al. 2021].

	3. Kinematic field measurements in 2D [Del Linz et al. 2017] or 3D [Mohagheghian et al.
	2017].

al. 2017; Li et al. 2019; Nassr et al. 2021] on windshields. These studies have been coupled to various numerical models [Pyttel et al. 2011; Chen et al. 2016; Lai et al. 2018; Samieian et al. 2019; Shahriari and Saeidi Googarchin 2020; Fernandes et al. 2021; Ma et al. 2022; Kim et al. 2022]. Wang et al. impacted very thick samples with a rigid spherical projectile

[START_REF] Wang | Post-fracture performance of laminated glass panels under consecutive hard body impacts[END_REF][START_REF] Wang | Experimental and analytical study on the pre-crack impact response of thick multi-layered laminated glass under hard body impact[END_REF]

. Measurement techniques used by the authors are direct measurements of the projectile acceleration, combined with high-speed cameras capturing the in-plane cracking of the glass plies. Quantitative measurements are performed, mainly locally, using strain gauges or accelerometers, for instance.

Table 3 .

 3 .2.

	Density	7800	kg/m 3
	Young's modulus	210	GPa
	Poisson ratio	0.3	-

2 -Half-sphere elastic material properties.

  ture et du pare-brise, des multiples géométries de projectiles, multiples vitesses d'impact, états de précontrainte liés au formage, raideur structurale, ... On se propose donc dans la suite de se focaliser sur le phénomène d'apparition de la fissure et le traitement des mesures réalisées, plutôt que la comparaison et l'influence des nombreux autres paramètres du système.

	Le matériau impacté est un verre feuilleté composé d'un pli de poly(butyral vinylique) (PVB)
	intercalé entre deux plis de verre (plat) sodo-calcique. Les épaisseurs des couches sont don-
	nées en Figure B.1 : 1.6 mm pour le verre extérieur, 1.1 mm pour le verre intérieur et 0.76 mm
	pour l'intercalaire.

Table B .

 B Le concept du banc d'essai est présenté en Figure B.1. Un projectile en acier entre en contact avec l'indenteur initialement fixe, dont l'extrémité est un indenteur (Rockwell, ou Vickers, et dans le futur possiblement Berkowich, Knoop, Cube corner, ...) en diamant, posé sur sur la surface du verre. Lors de l'impact avec le projectile, des ondes élastiques se propagent du haut vers le bas de la barre et sont détectées par les jauges de déformation (en rouge). L'indenteur se met en mouvement et sa pointe Rockwell, petite sphère en diamant de 200 µm, indente la surface du verre. Une dizaine de microsecondes après se forme la fissure sur le verre après une phase de croissance suivant uniquement l'épaisseur du pli de verre impacté. Le verre feuilleté repose sur un cadre en PMMA, permettant à la plaque de fléchir sous l'axe de l'indenteur au cours de l'essai. Schéma de principe du montage blowpipe pour l'essai traité. On retrouve la géométrie de l'indenteur et de la plaque de verre feuilleté de 100×100 mm 2 . l'indenteur est en diamant sur une hauteur d'environ 400 µm. le comportement visco-élastique du PVB est décrit par une série de Prony. PVB Prony series (at 25°C) in the time domain, with the amplitudes g i , characteristic times τ i (in s), such that the shear modulus reads G(t) = G 0 (1 + g i exp(-t/τ i ): 1 -Paramètres matériaux des éléments de l'essai aussi utilisés dans les simulations.Le traitement du signal pour la mesure de contrainte (σ) et vitesse (v) est présenté dans les annexes précédentes du manuscrit. En supposant une propagation 1D (suivant x positif et x négatif) des ondes élastiques dans la barre, et un état initial nul en contrainte et vitesse (i.e.état statique initial et pas de précharge), on peut remonter au fur et à mesure de l'avancement de l'essai, aux vitesses et contraintes aux jauges.

	Indenter Laminated glass PMMA Frame support Glass 1.6 mm 0.76 mm Glass 1.1 mm (for free bending experiments) Glass Density 2500 Young's modulus 70 Poisson ratio 0.23 (g i ,τ i ) (0.10690, 1e-04) 60 mm PVB 1000 1 0.49 (0.01940, 1e-01) FIG. B.1 -Parameters PVB (0.00078, 1e02)	20.7 mm 31.2 mm 9.5 mm	Projectile (m=3.2g) Gauges: strain measurement processing from Hopkinson method Rockwell diamond tip (200µm curvature radius) 5 mm Diamond Units 3500 kg/m 3 1050 GPa 0.1 -(0.09720, 1e-03) ∅ 5 mm Steel 7850 200 0.3 (0.08748, 1e-02) (0.00583, 1.0) (0.00107, 1e01) (0.00058, 1e03) (0.00032, 1e04)

Les propriétés matériaux des éléments, également utilisés dans les simulations, sont reportés dans la Table B.1. Le projectile et l'indenteur sont en acier et leurs dimensions sont : 5 mm de diamètre, un projectile de longueur 20.7 mm, un indenteur de longueur 60 mm. La pointe de

Table B .

 B 3. La contrainte et la vitesse peuvent être transportées en tout point de la barre, et en particulier au niveau de l'indenteur.Les grandeurs utiles pour le dépouillement par diagramme de Lagrange sont présentés dans la TableB.2.

	Parameters			Units
	Density	ρ	7850	kg/m 3
	Young's modulus	E	200	GPa
	Cylinder radius	r	2.5	mm
	Inter-gauges distance	L	31.2	mm
	From bottom gauge to cone bottom distance	ℓ	9.5	mm
	From impact location to upper gauge distance	d	17.2	mm

2 -Grandeurs utilisées pour le dépouillement des mesures de jauge de déformation.

  B.1). La charge statique dans le ressort, mesurée à l'aide du capteur piézoélectrique, est initialement de 25.69 N, impliquant un premier pic de vitesse attendu autour de 4.2 m/s (cf Fig. 2.21 du manuscrit). La mesure des jauges est présentée en Figure B.2. L'essai est très rapide et la formation de fissure prend environ 30 µs entre le moment où la jauge amont détecte la première onde incidente et le moment où la fissure en étoile est formée sur le verre.L'origine des temps est choisie à l'aide de la réception du signal de trigger par les boîtiers de commande. C'est pourquoi on mesure du signal utile avant 0 µs.

On peut noter l'écart d'arrivée du signal de τ = L ρ/E = 6.2 µs de la jauge amont à la jauge aval. Les deux signaux sont ensuite utilisés pour les mesures de contraintes aux lieux des

B.4.1 Concept et design

	De nombreux modèles apparaissent dans la littérature concernant l'impact des vitrages feuil-
	letés. Différentes approches sont examinées et les résultats concernant l'évolution de la fissure
	et la réponse du matériau fragile sont comparés [Wang et al. 2017] : Modélisation par éléments
	finis (FEM) [Sun et al. 2005b,a; Peng et al. 2013; Chen et al. 2016] utilisant des modèles de
	zones cohésives [Chen et al. 2015; Gao et al. 2017; Prasongngen et al. 2019], modélisation
	par éléments finis étendus (XFEM) [Xu et al. 2017], les éléments mixtes finis-discrets [Wang
	et al. 2018; Chen et al. 2022], et la péridynamique [Lai et al. 2018; Wu et al. 2020]. Cepen-
	dant, la plupart de ces études se concentrent sur les impacts de tête ou de corps dur et sur la

fragmentation des pare-brises stratifiés. Les énergies cinétiques des projectiles sont donc bien plus importantes dans ces essais : autour de 100 J, alors que celles des impacts gravillon se situent plutôt autour de 100 mJ.

La simulation présentée ci-après est très liée aux essais et concerne plusieurs points :

Remerciements

Appendix A

Hopkinson bars theory

Performing measurements of a dynamic experiment is a real issue by itself. Indeed, due to the high-frequency phenomena to measure, the experimenter must adapt its devices -for instance, high-speed rate cameras, laser trackers, or strain gauges. The scientist will also deal with the increase in the uncertainty measurements and then has to filter the appropriate frequencies when needed. This appendix is an echo to the section 1.4.1. It completes the theory of the propagation of the elastic wave in the bar, introducing the dispersion problem due to the lateral inertia of the bar and the bending and shear waves propagation.

A.1 Long bars 1D equations

The Hopkinson bar device, created in 1913 by Hopkinson [START_REF] Hopkinson | A method of measuring the pressure produced in the detonation of high explosives or by the impact of bullets[END_REF]] and revisited in 1949 by Kolsky [START_REF] Kolsky | An investigation of the mechanical properties of materials at very high rates of loading[END_REF]], allows the dynamic loading in compression, traction, or shear of a sample, i.e. at deformation rates of 10 s -1 to 1500 s -1 . At the beginning of the test, a first incident bar is impacted by the projectile with specified energy. The shock waves propagate through the bar, the material, and finally into the outlet bar. At the interfaces, part of the wave is transmitted, and the other part is reflected.

For a Hopkinson bar experiment analysis, we consider the above-mentioned elastic bars throughout the stress. Adding the hypothesis of uniaxial propagation, and neglecting Poisson effects (i.e. the lateral inertia of the bars), because the length of a bar is much greater than its diameter, one can write the equilibrium equation

with E the Young's modulus, u the movement of a material point on the bar, σ = Eε the stress and ε = ∂u/∂x the longitudinal strain x, ρ the density and c = E/ρ the elastic tension/compression waves velocity in the bar.

The general form of u is

with f , a wave, propagating through x > 0 and g, propagating through x < 0. The speed definition is

When the output bar is long enough to neglect the propagation in x < 0 one can consider g = 0, so σ(x, t) = -ρcv(x, t) (A.5)

A reflection at the interface between the bar and the sample is considered. The strain of the incident wave ε i and the reflected wave ε r is then measured by a gauge glued to the input bar.

The speed of the bar extremity x 1 is computed as

idem for the forces F(x 1 , t)

considering the bar section S .

A.2 Elastic waves: axial propagation and dispersion

Writing a subsection dedicated to dispersion is justified here because the configuration described in chapter 2 is a Hopkinson mini-bar system. Indedd, the small length of the bar makes it difficult to justify the usual 1D hypothesis based on Kolsky-Hopkinson theory [START_REF] Hopkinson | A method of measuring the pressure produced in the detonation of high explosives or by the impact of bullets[END_REF][START_REF] Kolsky | An investigation of the mechanical properties of materials at very high rates of loading[END_REF]], the indenter diameter being only ten times shorter than its length.

A.2.1 Considering the lateral inertia

To deal with a 1D model, the lateral inertia of our mechanical waves in the mini-bar should be considered, also described by Aleyaasin for visco-elastic polymeric bars [START_REF] Aleyaasin | Wave dispersion and attenuation in viscoelastic polymeric bars: Analysing the effect of lateral inertia[END_REF]. A simple hypothesis is to take into account a diameter variation due to Poisson effect [START_REF] Meyers | Dynamic behavior of materials[END_REF]], i.e. a displacement u like u = u r (r, x, t)e r + u(x, t)e x (A.9) Adding Hooke's law

The wave celerity C(ω) of an harmonic is

with R the bar radius. From here, the condition over the wave celerity

with an asymptotic celerity c s = G/ρ, meaning the bending waves celerity due to shear stress are slower than the traction and compression ones.

A.4 Strain gauge measurements A.4.1 Principle

A strain gauge is a sensor that converts mechanical displacement into the change of resistance. The device can be attached directly to the surface of a deformed material to measure strains on a sample. It exists in various sizes and shapes, but the working principle of a foil-type gauge is constant: the resistance of the foil changes as it is subjected to mechanical deformation on the tested sample. This change is proportional to the applied strain. Gauges are connected to a Wheatstone bridge to detect the slight changes in the gauge resistance. The electric potential variation measured with a voltmeter informs on the micro-strain using the gauge factor.

The advantages of this measuring device are its simplicity and low-costs. It can also adapt to many surfaces with a suited coating. However, a simple strain gauge provides only a 1D local measurement of the sample deformation. One must make a hypothesis on the 1D fields to match an object global behavior.

A.4.2 Traction-compression Lagrange diagram

The construction of a Lagrange diagram is justified, with a 1D hypothesis, as it will allow us to measure stresses and velocities at the gauges in the presence of incident and reflected waves in the short indenter. Indeed, waves reflection does not allow for the exploitation of the equation (A.5), only applicable during the first wave passage. So, the principle is based on measuring the stress at two places of our indenter, i.e. at two separate gauges of a length L, in x direction. One names "upper strain gauge" the upstream gauge of abscissa x = -L and "lower strain gauge" the downstream gauge of abscissa x = 0. With a 1D purely elastic and axial behavior of the indenter,

To solve the dispersion problem, Zhao and Gary expressed the sum of the ascending and descending waves of the gauge measurement [START_REF] Zhao | A new method for the separation of waves. Application to the SHPB technique for an unlimited duration of measurement[END_REF]]. However, it is decided here to draw a Lagrange diagram for each considered frequency and follow the characteristic lines. We will suppose the harmonic decomposition of the considered quantities as a discrete sum of the frequency components. So,

From Love's motion equation solution (A.17), given by Pochhammer, one should define the impedance Z(ω) giving

Which leads to

Presently the frequency components equation is solved separately. The initial stress and speed are null in the indenter, C(ω) the elastic wave celerity of the frequency in the material defined equation (A. 19) and Z(ω) = E/C(ω) the impedance. The relations between stresses σ ω (x, t) and speeds v ω (x, t) can be expressed For t ≤ L/C(ω)

To measure the stress and speed at the indenter tip, for instance, at length ℓ, one may consider the data computed at the downstream gauge. The stress σ ω (x = ℓ, t) and speeds v ω (x = ℓ, t) can be expressed as

Appendix B

Dépouillement d'un essai blowpipe

Dans cette annexe, des éléments concernant le banc d'essai blowpipe sont ajoutés. Un essai est entièrement dépouillé. Les résultats de simulations sont donnés. Le modèle de comportement de la zone indentée est implémenté, et les résultats des simulations sont comparés aux données expérimentales.

B.1 Gravillonnage sur le verre

Une fissure peut se propager lors de l'impact entre un gravillon et un pare-brise ; on note par exemple dans cette catégorie les fissures en cône de Hertz ou les fissures en étoile.

Ce sont les fissures en étoile qui représentent le dommage le plus critique car leurs branches peuvent croître au cours du temps lors de nouveaux chargements (statiques, vibratoires, dynamiques, thermiques...).

On trouve dans la littérature des essais sur de fines plaques de verre ou de feuilleté permettant d'étudier différents points lors des essais de gravillonnage (i.e.impact d'un gravillon). Même si le gravillon n'est pas toujours utilisé comme projectile de part ses dimensions non standard, Les auteurs [Behr et al. 1993 ont analysé la formation de fissures sous impact, effectué des analyses post-mortem et conçu des dispositifs spécifiques pour reproduire de tels impacts de manière contrôlée. On peut déjà observer que l'angle et la vitesse d'impact du projectile sont des paramètres importants pour l'apparition de la fissure en étoile sur le verre feuilleté : pour une charge identique, une fissure en étoile se forme pour les plis minces, mais une marque conique apparaît à la surface des plis épais. L'examen de l'angle d'incidence pendant l'essai a révélé que seules les composantes normales sont impliquées dans l'endommagement [Grant et al. 1998;Cleary et al. 2020]. Ces analyses peuvent être étendues à de nombreuses compositions de verre différentes comme le verre borosilicaté. L'évolution de la fissure médiane dans le verre peut également être en jeu car lorsqu'elle atteint une profondeur critique, la rupture du pli dans toute l'épaisseur ne peut alors être évitée. Ceci a été observé aux rayons X [START_REF] Kang | Crack nucleation and growth during dynamic indentation of chemically-strengthened glass[END_REF]].

L'étude des conditions de formation d'une telle fissure représente un vrai challenge : zone d'indentation sub-millimétrique, fissuration en moins de 30 µs, dépendance de l'état de la struc- Abstract : Laminated glass, a composite made of two or more dierent glass sheets, interleaved with thin polymeric layers, is a widely used material for building or automotive applications. Used as glazing, laminated glass is designed to insure the user safety by preventing large glass chip ejection during an impact and by dissipating the kinetic energy. The composite response to dynamic loads is therefore a crucial issue to validate the safety of its use through many standards implying a wide variety of tests.

Resisting small impacts like graveling is also at stake for such glazing. It is proposed to study the star crack formation by designing a new device. The set-up is composed of a spring-loaded projectile that mimics a gravel momentum and a mini-Hopkinson input bar equipped with strain gauges, whose end is an indenter initially in contact with the sample. A synchronized highspeed camera, used to measure the crack propagation, is combined with the indenter force and velocity measured at the indenter tip. A simu-lation analyzes the non-linear local behavior accounting for the sample dynamic bending. This work is a prelude to the crack length prediction on windshields submitted to gravel pitting. Hard-body blunt impact is representative of anti-burglary testing. During a ball-drop test, the glass layers are cracked and a global large strain regime, driven by the newly formed polymeric bridges, is activated. A large amount of impact energy is dissipated thanks to the stretching of the interlayer and delamination. To estimate the composite behavior after glass breakage, stereo-digital image correlation analyses, introducing axial regularization, are carried out from high-speed camera images to extract the kinematic information of the sample motion. Leads are proposed to design a predictive model with regards to the observations. Repetitive impacts are also studied as the crack pattern and the delamination history are crucial to evaluate the laminated glass residual resistance.