Keywords: Mode II, High confinement, Pre-stressed cell, R30A7 concrete, High Performance Concrete, Saturation ratio, High strain-rate, Giga, Split-Hopkinson Pressure Bar iii Mode II, Fort confinement, Pré-confinement, Béton R30A7, Béton haute performance, Degré de saturation, vitesse de déformation élevée, Giga, Système aux barres de Hopkinson

I debt my gratitude to a lot of people who helped me to accomplish the PhD thesis.

First, I would like to thank my supervisors Pascal Forquin and Dominique Saletti

Since it is necessary to protect massive structures from these severe loadings, different studies have emerged to understand the behavior of concrete under extreme conditions.

Experimental methods are necessary in order to better understand real phenomena and validate analytical solutions [START_REF] Ravichandran | Modeling constitutive behavior of particulate composites undergoing damage[END_REF], [START_REF] Carol | Normal/Shear cracking model: Application to discrete crack analysis[END_REF]) and numerical models [START_REF] Pontiroli | Predicting concrete behaviour from quasi-static loading to hypervelocity impact[END_REF], [START_REF] Forquin | A probablistic damage model of the dynamic fragmentation process in brittle materials[END_REF]).

Triaxial compression tests have been performed for several decades on concrete [START_REF] Schmidt | Experimental and theoretical investigation of the high-pressure behavior of concrete[END_REF] and [START_REF] Warren | Penetration into low-strength (23 MPa) concrete : target characterization and simulations[END_REF]). The concrete specimen is subjected to a hydrostatic pressure in a first stage then an axial stress is applied. A transition from a brittle to a ductile behavior, which is a characteristic of cohesive materials, was observed. A study [START_REF] Gabet | Triaxial behaviour of concrete under high stresses: Influence of the loading path on compaction and limit states[END_REF]) on 'real concrete' specimens allowed applying a pressure up to 0.85 GPa and a maximum volumetric strain of about 12% was reached.

However, triaxial tests are difficult to perform under dynamic conditions because they require impermeability between the fluid and the specimen. Another method was used to characterize the confined behavior of concrete over a wide range of strain-rates: the quasi-oedometric compression test. This method was proposed by Bazant et al. [1986], [START_REF] Burlion | Compaction des bétons: éléments de modeélisation et caractérisation expérimentale[END_REF] and [START_REF] Gatuingt | Prévision de la rupture des ouvrages en béton sollicités en dynamique rapide[END_REF] and fully revisited by [START_REF] Forquin | An experimental method of measuring the confined compression strength of geomaterials[END_REF] under static conditions and by Forquin et al. [2008c], Forquin et al. [2010] and [START_REF] Piotrowska | Experimental Investigation of the Confined Behavior of Dry and Wet High-Strength Concrete: Quasi Static Versus Dynamic Loading[END_REF] in dynamic (SHPB) testing conditions. A maximum pressure of 800 MPa was reached with a volumetric strain of 15 %.

Since shear fracturing in mode II was also observed in impact tests, it is important to understand the concrete behavior under such loading (Forquin et al. [2008c], Forquin et al.

[2010] and [START_REF] Lukić | Experimental characterization of the punch through shear strength of an ultra-high performance concrete[END_REF]). The Punch through shear test is one of the techniques that exist in the literature to study the shear behavior of concrete in which high deformations are reached inside concrete sample. These tests were conducted on concrete with a low level of pressure (around 8 MPa) [START_REF] Montenegro | Characterization of concrete in mixed mode fracture under confined conditions[END_REF]) or on rocks with a hydrostatic pressure up to 70 MPa [START_REF] Backers | Fracture Toughness Determination and Micromechanics of Rock Under Mode I and Mode II Loading[END_REF]). Moreover, punch-through shear testing technique was developed and performed in static and dynamic conditions with a passive confining pressure which varies during the test [START_REF] Forquin | Influence of strain-rate and confining pressure on the shear strength of concrete[END_REF]) and constitutes a main drawback of the method.

In the present work, we are concerned in studying the confined shear behavior under low and high rates of loading, this behavior has not yet been completely studied despite being of a real concern during impact of concrete structures. The objective of the work is to complete the tests existing in the literature to investigate the behavior of concrete subjected to high confining pressures with high deformation levels and over a wide range of strain-rates.

For this purpose, two types of tests were developed. The first test is a triaxial punchthrough shear test, it consists of applying in a first phase a hydrostatic pressure to a predetermined level, this pressure is maintained constant while the axial load increases at a constant loading rate before material failure in the second phase. The second test consists of confining the concrete sample with a pre-stressed metallic cell prior to applying the shear loading.The numerical simulations done for the design of each test will be described then the test procedures and the different results will be presented.

In the first chapter, a bibliographic study will be devoted to recapitulate the tests describing the behavior of concrete structures subjected to impact. Different experimental methods will be presented, the compression tests, the triaxial tests, the quasi-oedometric compression tests and the punching-through shear tests.

In chapter two, quasi static tests performed by applying a hydrostatic pressure and then a shearing to the concrete sample are presented. These tests are done with the press Giga in which the loading stamps were modified in order to be able to apply shearing to concrete samples. These tests will be performed on real concrete with the largest aggregates size of 8 mm.

In the third chapter, a new technique based on pre-stressing concrete samples with a metallic ring is presented. This ring is instrumented with hoop gages and is deformed elastically by applying a compressive load which is released after inserting the specimen.

The ring applies a confinement to the sample while retrieving its initial position. After applying the confinement, a normal hydraulic press (Schenck) is used to perform static shear testing and a Split Hopkinson Pressure bar setup is used to perform dynamic testing. This study was performed on two different types of concrete, an ordinary concrete R30A7 and a high performance concrete (HPC). Also, two sets of R30A7 specimens have been studied, dry and wet samples. Wet samples are placed in water and dry specimens are oven dried at 60 • C until their mass stabilization.

In the last chapter, final conclusions of this PhD work are drawn by comparing the results of this work to the ones that can be found in the literature. The perspectives raised by this work will be also proposed.

Chapter 1

State of the art 1.1 Experimental characterization of concrete

Concrete is the most used material in civil engineering due to its low cost and durability.

It is widely used in structures such as bridges, dams, bunkers or nuclear power plants.

When subjected to an impact, high mean stresses and high strain rates occur inside concrete targets. Adequate experimental techniques are generally used to understand the concrete mechanical response and model it as accurately as possible.

In order to answer the problematic of the thesis, a state of the art of the existing experiments is presented in this chapter. The mechanical behavior of concrete will be presented for different loading path: simple compression, quasi-oedometric compression, triaxial and shear loading. Results for different moisture contents and concrete compositions over a wide range of strain-rates will be discussed.

Simple Compression tests under static and dynamic loading

Simple compression test is one of the classical tests done on concrete to measure its compressive strength, the main property of concrete. The test is conducted on cylindrical specimens aged of at least 28 days, the two surfaces of the sample are rectified to obtain two flat and parallel surfaces and avoid the frictional resistance in contact with the compression plates. The measurement of the axial strain of the sample can be obtained by means of different sensors: LVDT (Linear Variable Displacement Transducer), extensometers or strain gauges. A part of this pulse is reflected back in the bar and the remaining part is transmitted through the specimen into the transmitted bar. Strain-gauges, glued on the bars, allow the knowledge of forces and particle velocities at both faces of the specimen. [START_REF] Bischoff | Impact behavior of plain concrete loaded in uniaxial compression[END_REF] gathered different tests done on concrete and found that there is an increase of the compressive strength at high strain rate (Figure 1.3). This relative increase might be produced because of two different modes: the first mode corresponds to a strain rate less than 10/s and can be explained by the presence of water in the pores of concrete which exhibits the so-called Stefan effect causing the strengthening in concrete [START_REF] Toutlemonde | Résistance au choc des structures en béton: du comportement du materiau au calcul des ouvrages[END_REF]). The second mode for which the strain rate is higher than 10/s is related, at least in part, to a structural effect. The inertia generates radial constraints which act like a confining pressure and thus increase concrete strength [START_REF] Rossi | A physical phenomenon which can explain the material behaviour of concrete under high strain rates[END_REF], [START_REF] Bailly | Une modélisation d'un matériau fragile avec prise en compte d'effets dynamiques[END_REF]).

Confined compression behavior of geomaterials

During impact problems, concrete is subjected to high confinement compressive stresses and is compacted mechanically [START_REF] Burlion | Compaction des bétons: éléments de modeélisation et caractérisation expérimentale[END_REF] and [START_REF] Burlion | Compaction and tensile damage in concrete: constitutive modelling and application to dynamics[END_REF]). During compaction process, there is a damage of the cementitious matrix and closing of the porosity accompanied by an irreversible decrease of the material volume and an evolution of the mechanical properties. In order to highlight the importance of lateral pressure on the behavior of concrete, several experimental techniques have been developed.

Triaxial tests have shown their interest to characterize the confined behavior of concrete in quasi-static loading but they can not be easily adapted to the characterization of concrete in dynamics. Another common characterization method, which uses a different loading path for studying the confined behavior of concrete, is the quasi-odometric test.

Both tests will be presented hereafter.

Quasi-oedometric compression tests

The standard triaxial test does not closely simulate the conditions of concrete in structures designed to resist explosions during impact. In these structures, concrete is confined by heavy reinforcement in three directions, which causes the lateral normal stresses to increase along with the axial stress. The behavior of concrete in these structures is better simulated by confined compression tests (Bazant et al. [1986]). Therefore, the strength of concrete can be thought to be pressure dependent and strain independent as a first approximation. This assumption is necessary to deduce the confined behavior of concrete materials from a single quasi-oedometric compression test [START_REF] Forquin | An experimental method of measuring the confined compression strength of geomaterials[END_REF]). et al. [1986] carried out confined compression tests on specimens of hardened Portland cement paste and mortar. Samples were tightly fit into a cylindrical cavity in a pressure vessel which prevents lateral deformation and then loaded axially by a hard piston. A significant improvement of the test was obtained by considering a steel cell which allows the application of pressure of several hundreds of MPa in one hand and measuring the confinement pressure in the other hand [START_REF] Burlion | Compaction des bétons: éléments de modeélisation et caractérisation expérimentale[END_REF] and [START_REF] Burlion | Experimental analysis of compaction of concrete and mortar[END_REF]). Figure 1.4 shows a description of the apparatus. The pressure applied by concrete on the inner part of the steel tube was supposed uniform and was obtained by the measurement of the transverse strains on the external face of the cell assumed to be elastic. The gap between the specimen and the cell is filled with epoxy chrysor C6120 and a Teflon coating is applied on the inner surface of the specimen in order to minimize friction and ensure a correction of any possible defects of cylindricality, parallelism and coaxiality of the surfaces of the specimen. Several assumptions were made: the interface product is incompressible and the friction between specimen and steel is negligible. A pressure of 600 MPa has been reached on concrete samples of 50 mm diameter and 100 mm height. Tests on mortar with different W/C ratio were done and they showed that the higher the porosity, the larger the irreversible volumetric strain which represents the compaction process. The author also compared the results of quasi oedometric compression tests with true hydrostatic tests on mortar specimens (W/C=0.5), a much lower compaction process is observed in this case and it was concluded that the hydrostatic response of the mortar depends on the applied deviatoric stresses (Figure 1.5). The axial contraction of the specimen and the barrel deformation of the vessel were not taken into account in the analysis of [START_REF] Burlion | Experimental analysis of compaction of concrete and mortar[END_REF]. [START_REF] Forquin | Endommagement et fissuration de matériaux fragiles sous impact balistique[END_REF] made an improvement on the measurement technique and took into consideration these two subjects. He considered a relationship between the orthoradial deformations and the internal pressure taking into account that the confining pressure is not distributed over the entire height of the cell but on a portion that corresponds to the height of the specimen. A new processing method was introduced to make possible the use of a confining cell that deforms with non-purely elastic (elasto-plastic) behavior. A validation of the testing procedure was done experimentally and numerically on aluminum [START_REF] Forquin | An experimental method of measuring the confined compression strength of geomaterials[END_REF]) and applied to different concrete microstructures (Forquin et al. [2008a]).

Static tests

Bazant

More detailed results will be presented in the following paragraph. The results obtained for different strain-rates are close to each other which justifies the conclusion obtained by [START_REF] Gatuingt | Prévision de la rupture des ouvrages en béton sollicités en dynamique rapide[END_REF] that the behavior of concrete is weakly influenced by the strain-rate. It was noticed that for a given pressure, the level of compaction is greater for dried specimens. Moreover, one can clearly see on Figure 1.10 that the deviatoric stress of dry concrete keeps increasing with the increase of pressure while wet concrete has a limited strength even under high hydrostatic pressure. An increase of pore-pressure in the micro-structure is the cause of the limited strength in saturated samples according to tests and results obtained with micro-concrete MB50 (Forquin et al. [2010]). Similar behavior was seen for dynamic tests (Figure 1.11). A concentration of deviatoric stress was observed for saturated concrete which also has a stiffer hydrostatic response. It was concluded that the saturation ratio has a big influence on the response of the concrete R30A7 while the strain rate has a limited (but non-zero) influence on its response. The same testing technique was conducted on high strength concrete (fc28=80 MPa) [START_REF] Piotrowska | Experimental Investigation of the Confined Behavior of Dry and Wet High-Strength Concrete: Quasi Static Versus Dynamic Loading[END_REF]) and compared to ordinary one. In both static and dynamic conditions, dry specimens of the two concretes reach approximately same deviatoric stress while for saturated specimens, the deviatoric stress is much greater in high strength concrete (Figure 1.12 and Figure 1.13).

Figure 1.12: Comparison of the behaviors of high strength concrete HSC and ordinary concrete OC under static quasi-oedometric compression for saturated samples [START_REF] Piotrowska | Experimental Investigation of the Confined Behavior of Dry and Wet High-Strength Concrete: Quasi Static Versus Dynamic Loading[END_REF])

Because of the low porosity of the high strength concrete, this latter shows a stiffer volumetric behavior. Same conclusion than with the ordinary concrete was deduced for the high strength concrete; their behavior is strongly dependent on the water content and weakly dependent on the strain rate. Triaxial loading with a moderate level of confining pressure have been addressed by many authors [START_REF] Warren | Penetration into low-strength (23 MPa) concrete : target characterization and simulations[END_REF], [START_REF] Sfer | Study of the Behavior of Concrete under Triaxial Compression[END_REF], [START_REF] Williams | Laboratory characterization of sam-35 concrete[END_REF] and [START_REF] Schmidt | Experimental and theoretical investigation of the high-pressure behavior of concrete[END_REF]). These authors have revealed the increase of concrete strength and a transition from brittle to ductile behavior with the confinement (Figure 1.14). during these tests. The results have revealed an influence of the loading path on the compaction behavior which confirms previous studies carried out on mortar specimens [START_REF] Warren | Penetration into low-strength (23 MPa) concrete : target characterization and simulations[END_REF] and [START_REF] Schmidt | Experimental and theoretical investigation of the high-pressure behavior of concrete[END_REF]) and it has also been proved that the deviatoric stress, associated to shear stress increases the compaction process [START_REF] Burlion | Experimental analysis of compaction of concrete and mortar[END_REF]). In other words, for a given mean stress, the higher the deviatoric stress, the more compacted the specimen. This sensitivity to the deviatoric stress can be explained by a granular rearrangement inside the matrix under shear stress.

Influence of saturation ratio

Vu [2007] carried out experiments about the effect of the saturation ratio on the concrete behavior under high confinement. The test results show that the saturation degree has a major influence on the concrete behavior. The behavior of an ordinary concrete is only slightly dependent on the saturation ratio at low confining pressures while under high mean stress, the hydrostatic behavior of very wet or saturated concrete clearly become stiffer than that of dried concrete. The curves of the deviatoric behavior show that the strength of dried concrete strongly increases with the confining pressure whereas it is limited for saturated samples (Figure 1.15). Regarding the limit states, it is found that at a low level of mean stress (below 150 MPa), the states of the dried, wet and saturated samples all lie very close to one another (Figure 1.16) since at such stress levels, concrete behavior is governed by a cohesive character.

The presence of water in the sample does not exert therefore a very significant effect on the limit state.

Conversely, it was shown that for higher confinement levels, the effect of water becomes predominant. The increase in peak deviatoric stress with respect to mean stress remains very low for the saturated samples; this phenomenon may be explained by a pore pressure effect. These observations can be explained by a combination of two phenomena. On the one hand, the loss of cohesion in the cementitious matrix due to high confinement makes the concrete a non-cohesive granular material type (Vu et al. [2009a]). The dried concrete shear strength increases nearly linearly with confining pressure, and this Different density, confining pressure and strain-rate were investigated. They concluded that the pressure has a high influence on the response of the sand studied, the strain-rate has a limited influence and same response was obtained regardless its initial density.

Shear Behavior of geomaterials

In fracture mechanics, cracks or fractures are usually subdivided into three basic propagation modes: namely mode I (opening), mode II (sliding), and mode III(tiring) (Irwin [START_REF] Luong | Tensile and shear strenghts of concrete and rock[END_REF] and [START_REF] Luong | Fracture testing of concrete and rock materials[END_REF] has developed a technique to maximize mode II conditions of crack propagation in rocks. A circular tube is subjected on its internal and external faces to a shear stress parallel to Z axis. To avoid normal stresses induced by sliding on the facets (Reynolds dilatancy effect), four concentric tubular parts were made in the sample as shown in Figure 1. Optimized dimensions of the proposed test [START_REF] Luong | Fracture testing of concrete and rock materials[END_REF])

Triaxial Punch-Through Shear Test

In order to suppress the dilatancy in concrete and to avoid a tensile Mode I fracturing of the sample, some authors applied a confining pressure to the sample prior to shear loading. [START_REF] Backers | Rock fracture toughness testing in Mode II -punch-through shear test[END_REF] and [START_REF] Backers | New data on mode II fracture toughness of rock from the Punch-Through Shear test[END_REF] conducted laboratory tests and finite element modeling on six different types of rock subjected to up to 70 MPa of confining pressure to measure Mode II fracture toughness. The specimen was first subjected to a hydrostatic pressure and then an axial load was applied to create shear stresses in the ligament. They concluded that fracture toughness increases with confining pressure and reaches a constant value at confining pressures higher than 20-35 MPa. The results of two types of rocks are presented in Figure 1.23.

The authors deduced that with the increase of shear stress, primary macroscopic wing fractures develop at about 30% of the maximum stress. They propagate out of the stressed zone and stop. Further elevation of shear stress in limestone results in development of an 'echelon fracture' with an angle depending on the confining pressure. This echelon fracture forms an almost straight fracture (mode II) at the peak of loading.

Marble and granite show a somewhat similar fracture sequence, the energy requirement for the formation of the two types of fractures is different. [START_REF] Montenegro | Characterization of concrete in mixed mode fracture under confined conditions[END_REF] and [START_REF] Montenegro | Experimenal tests and numerical modeling to identify the asymptotic shear-compression mode IIA of concrete fracture[END_REF] used the same method and evaluated the fracture energy associated to mode II fracture in confined and unconfined 1. 4.3.1 Static tests Forquin [2011b] conducted experiments on dry and saturated concrete samples with different stiffnesses of passive confinement cells and with circular notches to avoid selfconfinement of the sample. Since steel is stiffer than aluminum, higher confinement was obtained in the concrete sample an thus higher shear strength was observed (Figure 1.27). Moreover, the author observed higher strength with dry samples than with wet ones (Figure 1.28). More recently, [START_REF] Lukić | Experimental characterization of the punch through shear strength of an ultra-high performance concrete[END_REF] used the same technique to study the shear behavior of Ultra-High Performance Concrete with and without steel fibers under static and dynamic conditions. Four or eight radial notches were performed on the sample to suppress its self confinement and thus deduce radial stresses inside the ligament. They discovered a decrease of shear strength for a higher number of radial notches for both static and dynamic tests. They also obtained higher shear strength of about factor two in dynamic tests in comparison to static ones for Ultra-High Performance Concrete and Fibered Ultra-High Performance Concrete. Moreover, a higher shear strength was obtained for Fibered UHPC (Figure 1.30).

By comparing shear stress as a function of radial stress (Figure 1.31), it was deduced that at high strain-rates the increase of shear strength results from a higher radial stress. The authors stated that this difference of shear strength might be induced by the adjournment of tensile fracturing and an alteration of crack propagation prior to mode II cracking. Fibered Ultra-High performance concrete [START_REF] Lukić | Experimental characterization of the punch through shear strength of an ultra-high performance concrete[END_REF])

The split Hopkinson pressure bar technique was also used to study dynamic shear failure of different materials such as rocks, steel or polymers. [START_REF] Yao | A dynamic punch-through shear method for determining dynamic Mode II fracture toughness of rocks[END_REF] used the same geometry suggested by [START_REF] Backers | ISRM Suggested Method for the Determination of Mode II Fracture Toughness[END_REF] to determine the dynamic fracture toughness of Fangshan marble and concluded that it increases with the loading rate (Figure 1.32). They also used the micro CT to analyze fracture pattern and modes and noticed that the main fracture developed at peak force is the mode II fracture.

Conclusion

Several experimental tests exist in the literature to improve the understanding and modeling of the confined and shear behavior of geomaterials subjected to impact as presented in the bibliographic review. Table 1.1 summarizes most of these tests.

Compression tests on concrete showed the strong influence of strain-rate that mainly results from structural effect observed in dynamic tests. Results through triaxial and quasi-oedometric compression tests showed the important influence of confinement on the deviatoric strength of concrete. The confinement applied was up to 850 MPa and the axial and volumetric deformations reach often a value around 15%. The strength of dried concrete increases with confinement but stays constant under a certain level of confinement for saturated sample. Thus, the presence of water plays a big role on the confined behavior of concrete.

Concrete shear analysis showed the importance of confinement in increasing its strength and reducing the dilatancy effect. Experiments conducted on dry and saturated concrete samples showed a higher strength with dry samples than in wet ones. Furthermore, both concrete showed a very small strain-rate sensitivity.

In previous punching-through shear test, the confinement applied is either quite small in the order of 4 MPa in [START_REF] Montenegro | Characterization of concrete in mixed mode fracture under confined conditions[END_REF]'s study or applied to rocks characterized by a higher strength than concrete [START_REF] Backers | Rock fracture toughness testing in Mode II -punch-through shear test[END_REF]) or applied using a passive confinement pressure [START_REF] Forquin | Influence of strain-rate and confining pressure on the shear strength of concrete[END_REF]). The values of the nominal deformation of the punch-through shear (PTS) tests in Table 1.1 were estimated from the axial displacement applied to the specimen during shearing. The aim of the present study is to perform punch-through shear tests with high levels of confinement over a wide range of strain-rates. Thus, the confined static and dynamic behavior of concrete will be investigated using a high capacity triaxial press or a prestressed metallic ring which allow having an almost constant confinement stress during shearing. The study includes R30A7 concrete in its dry and saturated states since it appeared that the saturation ratio plays a major role in the response of concrete. It also includes tests on high performance concrete to investigate the effect of the concrete composition on the shear behavior of concrete.

Chapter 2

Quasi-static Punching Through

Shear test under high confinement

In this chapter, punch-through shear tests with high level of confinement are presented for the investigation of shear behavior of concrete. It was shown in the existing literature [START_REF] Montenegro | Experimenal tests and numerical modeling to identify the asymptotic shear-compression mode IIA of concrete fracture[END_REF], [START_REF] Backers | New data on mode II fracture toughness of rock from the Punch-Through Shear test[END_REF]) that confining pressure suppresses the growth of tensile mode I cracks, consequently it is neccesary to apply a sufficient high level of confinement to limit mode I and obtain a pure mode II fracturing. In the present work, PTS tests have been conducted with a high-capacity triaxial press able to generate a high confining pressure. The test consists of cylindrical samples with two cylindrical notches in the top and bottom faces, the outer diameter of the top notch and the inner diameter of the bottom notch coincide to form a cylindrical shear ligament. In a first step, the concrete sample is subjected to a hydraulic pressure. Then in a second step, an axial load is applied to the central part of the specimen to produce shearing in the ligament.

Firstly, the hydraulic press used for the tests is introduced. New tools are designed to make the press able to apply shearing to the tested samples. Thus, numerical simulations done with FEM code (Abaqus) to study the influence of different geometries and optimize the sample's dimensions are presented. Afterwards, validation tests on PMMA and tests on wet R30A7 concrete will be shown.

Experimental setup: Triaxial press Giga

The tests have been conducted with a high-capacity triaxial press 'Giga' that originally allows loading a cylindrical concrete specimen 7 cm in diameter and 14 cm long.

Design of the test using numerical simulations

In order to determine the optimal sample geometry, numerical simulations have been conducted with the finite element code Abaqus/explicit. The test consists of a cylindrical specimen with two cylindrical notches drilled in the bottom and top surfaces. The notches were made in such way that the inner diameter of the lower notch and the outer where q is the equivalent Mises stress, P the hydrostatic pressure and c the cohesion of the material related to the input data as

c = (1 - 1 3 tan β)σ c (2.2)
The hardening flow of the material is described in terms of the dilation angle ψ. With

Influence of rings

Because the notches are drilled into the specimen, the top and bottom inner parts remain unconfined in the radial direction during the test. The axial stress applied on this zone being larger than the compressive strength of concrete, this zone could be damaged under compression before reaching any shear deformation. Consequently, two small steel rings were designed to prevent the radial deformation of the unconfined notch region. Figure 2.6 shows the difference in plastic strain in the sample without and with the two rings under a confining pressure of 100 MPa.

The numerical simulations clearly show that without the use of rings (Figure 2.6(a)), the deformation occurs mainly in the part where the vertical load is applied. When the rings are used (Figure 2.6(b)), the plastic strain is localized in the ligament and thus the objective of the test to produce shear stresses can be reached.

Mesh sensitivity

The mesh has a small influence on the shear and radial stresses and a significant effect on the shear deformation in the ligament. Figure 2.7 illustrates three sizes of mesh studied: 0.15 mm, 1.5 mm and 3 mm (with a notch width equal to 3 mm) and for these three configurations, the shear deformation in the ligament is checked (Figure 2.8). For a very 20 mm which leads to a total sample length equal to 30, 35 and 40 mm. The results revealed that the shear and radial stress were the same within the three samples. For a higher upper notch length, higher tensile stresses were noticed near the upper ring, for this reason the upper notch should not be longer than 15 mm.

Influence of notch diameter and self-confinement

Simulations with different notch diameters D lig ranging from 25 mm to 45 mm were performed. Due to the stiffness of the concrete sample, an orthoradial self-confinement can occur. In the literature [START_REF] Forquin | Influence of strain-rate and confining pressure on the shear strength of concrete[END_REF] and [START_REF] Forquin | A testing technique to characterise the shear behaviour of concrete at high strain-rates[END_REF]), four radial notches at 90 • were performed to avoid this self-confinement. In that case, the mean radial stress in the ligament can be deduced from the hydrostatic pressure P applied to the sample knowing the area of the lateral surface S of the specimen and the area of the fracture surface S lig .

σ rr(no self-confinement) .S lig = P.S (2.5)

where

S lig = π.D lig .H lig (2.6)
and

S = π.D.H (2.7)
If it was possible to assume none self-confinement, the above equation could be used and a value of radial stress in the ligament could be calculated and plotted in Figure 2.12 (dotted green curve). On the other hand, the true radial stress averaged in the ligament at 1.5 mm of displacement (peak of shear stress) can be calculated directly from the numerical simulations (dashed red curve) at 100 MPa of confinement. The influence of the confining pressure is discussed in section 2.5.

The ratio between the radial stress calculated from equation (2.5) and the numerical one obtained by Abaqus is plot in black in Figure 2.12. It is observed that the smaller the notch diameter is, the higher the ratio between stresses is. Thus for a small notch diameter, there is more self-confinement in the peripheral part of the concrete sample.

Moreover, for a large notch diameter, a bending of the peripheral part was noticed because of the reduction of the peripheral thickness. Thus, as a compromise between both cases, the intermediate notch diameter of 35 mm was chosen.

in length and two caps of 110 and 150 mm (Figure 2.15). A space of 20 mm was left inside the caps which allows the piston to slide and apply a displacement to the concrete specimen.

Stress and strain calculation

For all the tests done, first a hydrostatic loading, which consists of generating a confining pressure around the sample, is applied in a linear manner with a pressure rate of 100 MPa/minute (i.e. 1.67 MPa/s). When the pressure reaches the constant value required, the displacement of the piston starts with a rate of 20 µm/s.

The pressure sensor, installed close to the entrance of the confinement cell, determines the pressure within the cell with a measurement accuracy of 1% over the operating range. The force sensor, positioned between the lower loading head and the piston (cf.

Figure 2.1(b)), is a metal body that deforms elastically over the machine loading range.

The force transmitted by the sensor is deduced from its strain. The shear stress is then calculated from this axial force using equation (2.8).

σ shear = F axial S lig (2.8)
with S lig given in equation (2.6).

Each part of the LVDT sensor is fixed on a loading head. The shortening generated by the deformation of the caps is considered negligeable compared to the displacement of the concrete sample. Therefore, it is considered that the LVDT sensor allows to calculate directly the displacement of the central part of the specimen. The nominal strain is thus calculated using equation (2.9).

ǫ nominal = U LVDT 2e (2.9)
Moreover, the nominal strain-rate can then be deduced:

ǫnominal = V 2e (2.10)
Since V = 20µm/s and e = 4 mm, the corresponding nominal strain rate is equal to 2.5x10 -3 /s.

Since it is difficult to measure radial stresses in the ligament, an estimation of this stress was elaborated from the numerical simulations. The Drucker-Prager model, used for the concrete, was described in section 2.2.1.3. Different confining pressures were applied to the numerical model and for each pressure, the average of the radial stress is calculated in the ligament at 1.5 mm of vertical displacement (which corresponds to the peak of shear stresses). Three values of the dilation angle describing the flow rule has been studied (5, 10 and 15 • ), a variation of 10% of the true radial stress in the ligament was observed (Figure 2.17).

From the other hand, if one considers that the self confinement in the concrete sample doesn't happen, the radial stress calculated from the confining pressure multiplied by the surface ratio (equation (2.5)) is also determined. The specimen is then wrapped within a multilayer protective membrane in order to prevent the confining fluid from infiltrating the sample (Figure 2.20).

Validation tests on PMMA

The polymethylmethacrylate (PMMA) is a homogeneous and non porous material that has been studied by many authors due to its testing ease, simplicity of specimen preparation, and possible field applications. In addition, it is a transparent material which makes it easier to investigate fracturing mode inside the samples. For these reasons, PMMA is a good material candidate to proceed to validation tests.

Brief literature review

The polymethylmethacrylate, PMMA, is a material of major interest because it is likely to be used in aeronautics, protective structures and armor systems thus it is important to evaluate its behavior under confinement to ensure a reliable performance. Many authors studied the mechanical behavior of PMMA subjected to compression under static or dynamic loadings. [START_REF] Rittel | Dynamic flow and failure of confined polymethylmethacrylate[END_REF] studied the pressure-sensitivity of PMMA at various strain rates, they confined cylindrical specimens with a tightly fit metal sleeve and subjected them to compression tests. A brittle to ductile failure mode transition appeared in the PMMA as confining pressure increases. The radial cracking and fragmentation process were observed in brittle failure, and a truncated conical plug in the ductile regime. Forquin et al. [2012] applied the same technique developed for concrete to PMMA specimens [START_REF] Forquin | Role of porosity in controlling the mechanical and impact behaviours of cement-based materials[END_REF]), these latter were enclosed in a brass or aluminum cell and then subjected to axial compression either with a normal testing machine or with the Split Hopkinson bar device. For quasi-static loading, the maximum strength is not affected by confining pressure since same results were obtained for aluminum and brass cells. An elasto-plastic behavior was observed and an inclined crack was propagating. A brittle failure occurred beyond a certain level of loading corresponding to the propagation of the crack. Under dynamic loading, an elastic-brittle behavior is observed and a mode II fracture pattern composed of one or two conical fracture surfaces are initiated from the corners of the specimen.

The outcome of the above results is that confinement in brittle polymers causes a failure mode transition from brittle to ductile which is identical to that observed in concretes or ceramics. Thus it would be interesting to study the PMMA behavior under confined mode II conditions.

Mechanical properties

The compressive strength of PMMA was obtained by conducting simple compression tests done on specimens 14 mm in length and 10 mm in diameter. The other mechanical properties of the PMMA are provided by the manufacturer and listed in

Testing program

Four tests were conducted on PMMA specimens (with a ligament length of 15 mm) : the first one PTSS2 was performed with the Schenck press, a normal hydraulic press without a confinement. The 3 others were performed with the Giga press. PTSG1 was conducted with a confining pressure of 50 MPa, the test is stopped when failure occurs and the force starts decreasing. PTSG2 was conducted with the same level of confining pressure (50 MPa) but the test was stopped after 3 mm of displacement in order to check the sample state before failure. PTSG3 was carried out with 30 MPa of confinement. For the unconfined test, after the elastic part, a small softening occurs when the nominal strain reaches 40%. Afterwards, at 70% of strain, a brittle behavior is observed. When a lateral confinement is applied, a clear change in the stress-strain behavior of the material is observed. A similar trend is observed for the elastic part, however a hardening effect is observed for confined tests. For a confinement of 30 MPa, the failure occurs at 80% of of strain (PTSG2) that no cracks were observed inside the ligament (Figure 2.22(d)).

Tests on specimens with a ligament length of 20mm were performed. Since a higher force is applied on the central part of the specimen, the unconfined compression strength of PMMA (equal to 105 MPa) was exceeded and this part swells and was damaged under compression (Figure 2.23). Consequently, this configuration was not taken into consideration.

From the above results, one can conclude that the new testing method developed was able to generate shear fracturing inside the samples tested, this mode II fracture occurred at failure. Moreover, the sample geometry with a ligament length equal to 15 mm is more preferable than the 20 mm in order to avoid the damage of the central part. Finally, the increase of confinement level in the shear ligament of the samples lead to a higher shear strength.

Influence of rings

Preliminary tests have been conducted on R30A7 concrete, one with the ring around the notches and the other without them. The confining pressure applied was 50 MPa.

It was observed after the tests that the central part of concrete specimens confined with the rings was not damaged under compression on one hand and that the rings diameter remained constant on the other hand. These observations confirmed that the rings remained in the elastic deformation regime and allowed the confinement of the central part of the specimen. Thus, using the ring was successful and permitted to reach the objective of the test. A deviation in the slope of PTSG1 test was reported, this problem might occur if there is no perfect contact between the sample and the steel cap before starting the test. For the PTSG3, an initial linear behavior was followed by a strain softening at 20%.

The maximum shear stress reached for a confining pressure of 50 MPa was 78 MPa while on the other hand the specimen subjected to a confining pressure of 100 MPa exhibited higher shear stress of about 95 MPa. Thus, higher shear strength is observed for higher confinement pressure, this can be mainly due to a higher radial stresses generated in the shear ligament.

Post-mortem observation

The specimens subjected to punch-through shear testing are shown in Figure 2.27 and 2.28. The proposed test seems capable of producing a controlled mode II fracture as predicted in numerical simulations. From the viewpoint of post-mortem specimen observation, the lower central part was not damaged, it was punched inside the specimen and the upper central notch was pushed outside the specimen. For the specimen subjected to 50 MPa, one radial crack was observed after the test. This could be explained that even at confinements of 50 MPa the dilatant behavior of concrete is not absolutely suppressed.

(a) (b) 'Giga' has been used to apply the high level of confinement and two steel rings were glued around the unconfined part of the specimen to prevent it to be damaged under compression. Experimental tests conducted on R30A7 concrete showed that the use of the rings was advantageous and allowed orienting the crack parallel to the loading direction as demonstrated in numerical simulations. Two different ligament lengths were chosen to be studied experimentally. The same force was obtained for both tests and higher shear stresses were deduced for a smaller shear surface. This might be explained by the fact that the development of cracking on the edges of the notches is the phenomenon which leads to the initiation of failure in concrete. Different levels of confining pressure were applied to study the influence of confinement on the shear behavior of concrete. It was concluded that for higher confinement, higher force was applied to the specimen before failure and thus higher shear strength is obtained in the shear ligament. This difference confirms the pressure sensitivity of concrete under shear loading. The tests were stopped at around 3 mm of displacement, otherwise the ring would touch the internal part of the specimen. The displacement at the peak of shear stress was around 1.5 mm leading to a nominal strain around 20%. Consequently, the objective of the present work to apply a high confinement pressure and obtain a high deformation level in a concrete sample was achieved.

Chapter 3

2D static and dynamic Punching

Through Shear test

The objective of this part of the study is to develop an experimental method which allows to study the behavior of concrete subjected to a certain confinement under Mode II fracture propagation in static and dynamic conditions. This should supply the development of a testing method able to generate a pre-confinement in concrete prior to testing it under shear. The first part of the chapter will focus on the numerical simulations done for dimensioning the test and in the second part, the experimental procedure and experimental results are presented.

Design of the test using numerical and analytical methods

This section gives a detailed description of the simulations done on Abaqus in order to optimize the test configuration. In parallel, some analytical calculations were conducted to compare them with the simulations. The design of the confinement cell is first presented and the optimization of the concrete sample geometry is then described.

Confinement procedure

As detailed in Chapter 1, in dynamic punching through shear tests existing in the literature, the confinement applied was a passive confinement (Forquin [2011a], [START_REF] Forquin | A testing technique to characterise the shear behaviour of concrete at high strain-rates[END_REF] and [START_REF] Lukić | Experimental characterization of the punch through shear strength of an ultra-high performance concrete[END_REF]). These tests consist of inserting the concrete specimen inside a confinement ring which allows inducing a confinement pressure due where α is a constant, A = bt is the area of the cross section of the ring, E its young modulus and G its modulus of rigidity [START_REF] Srinath | Stresses in a circular ring. Comparison of theory with experiment[END_REF]).

The total strain energy in an element dS = Rdθ of the ring is equal to:

dU = [ M 2 2AEeR + N 2 2AE - M N AER + α V 2 2AG ]Rdθ (3.4)
By replacing the values of the moment, the normal force and the shear in the above equation, it comes that:

U = π 2 0 [ (M 1 -P 2 Rcosθ) 2 2AEeR + (-P 2 cosθ) 2 2AE - (M 1 -P 2 Rcosθ)(-P 2 cosθ) AER + α P 2 sinθ 2 2AG ]Rdθ (3.5)
According to Castigliano's theorem [START_REF] Prasad | Design studies of a square ring shaped force sensor[END_REF]), the moment M 1 should be such that dU/dM 1 = 0, then, equation (3.5) leads to:

π 2 0 ( M 1 e - P R 2e cosθ + P 2 cosθ)Rdθ = 0 M 1 π 2e - P R 2e + P 2 = 0 M 1 = P π (R -e) M = P π (R -e) - P 2 Rcosθ (3.6)
The normal stresses at any cross section are the sum of the stresses due to bending moment and normal force. Let σ θ be the normal stress, σ θ 1 and σ θ 2 the bending and direct stresses component. They are equal to [START_REF] Srinath | Stresses in a circular ring. Comparison of theory with experiment[END_REF]):

σ θ 1 = M y Ae(r n -y) σ θ 2 = - P bt (3.7)
r n being the radial ordinate for the neutral axis and y being the distance of any point from the neutral axis of the ring cross section along a radius.

The total normal stress is then equal to:

σ θ = y Ae(r n -y) [ P π (R -e) - P 2
σ θ ≈ 3P R tb 2 ( 2 π -cosθ) - P bt (3.9)
The maximum radial stress occurs at the position where the load is applied (θ=π/2) and thus

σ θ ≈ 6P R πtb 2 (3.10)
This equation is useful in obtaining an approximation of the hoop stresses and strains in a circular ring. For more precise values of the stress and strain distribution along the modified shape of the ring, finite element analysis has been used.

Numerical Simulations

Ring dimensions (b, t and R) were optimized by performing numerical simulations on Abaqus explicit. These simulations aim to obtain the best dimensions for the final geometry which allows the ring to handle a certain diametrical force while remaining in the elastic mode. A 2D model was used; it was drawn taking into consideration the symmetry with respect to X and Y axis. In the following simulations, the material properties used for aluminum and steel are presented in Table 3 After performing several simulations, the final dimensions were chosen as: the mean radius R=57.5 mm (R 1 =50mm and R 2 =65 mm), the width b=15 mm and the thickness t=20 mm.

One example of the calculation done is shown in Figure 3.3. A load of 20 kN was applied to a small plate in interaction with the ring in order to avoid the concentration of stresses and a high distortion on the node where the load is applied. The Mises stress obtained in the ring are shown in Figure 3.3(b).

After calculating Huber-Mises stresses inside the ring, two types of ring will be tested, one made of aluminum and the other made of steel with a yield strength of 485 MPa and 1100 MPa respectively.

Considering the first step of loading, with an applied force F 2 reaching the maximum value:

u max 2 = F max 2 K 2 (3.11) with u max 2 + L ring 2 = L sample 2 (3.12)
Now the sample is introduced and the ring and concrete are put in contact:

u f inal = F f inal 1 K 1 = F f inal 3 K 3 (3.13)
When the force applied F 2 starts to decrease and F 3 begins to increase:

u = u(F 2 ) + u(F 3 ) = F 2 K 2 + F 3 K 3 (3.14)
When the force applied F 2 is totally released, F 2 =0 :

F comp = F 3 = K 3 (u 2 -u 3 ) = K 3 (u max 2 -u 3 ) (3.15)
and since F 3 = K 1 u 1 , we have:

F comp = K 3 [ F max 2 K 2 - F comp K 1 ] F comp = F max 2 K 2 [ 1 1 K 3 + 1 K 1 ] F comp = K 1 K 3 (K 1 + K 3 ) F max 2 K 2 (3.16)
Finally we can write,

F comp = αF applied (3.17) with α = K 1 K 3 K 2 (K 1 + K 3 ) (3.18)
If one considers that K 2 and K 3 have approximately the same values, the above relation can be written:

α = K 1 (K 1 + K 3 ) (3.19)
-Second step: The force applied to the ring is released, and an interaction law between the ring and the sample is defined to create the contact. The force applied on the ring and the one obtained in the contact between the concrete and the ring during the two time steps are shown in Figure 3.9. We can see that in the first step, a force of 36 kN is applied and the force in the concrete is equal to zero. At the end of the second step, when the force applied is totally released a force of 28 kN is transferred to concrete.

Homogeneity of stresses in the ligament

In this section, different geometries of concrete sample were studied in order to obtain a better homogeneity inside the ligament after applying the confinement to the sample.

A brief description of each case studied is presented in Table 3. Only one quarter of the sample is modeled as shown in Figure 3.10, the ring and the plate have been hidden from the graphs for better display of the stresses distribution on the surface of the concrete specimen.

Furthermore, the confinement stresses along the path drawn in Figure 3.12 for the different cases are presented in Figure 3.13 and Figure 3.14. 3.3.

Figure 3.14: Variation of the confinement stress σ xx along the path Z (cf. Figure 3.12) as function of its normalized distance for different cases studied as detailed in Table 3.3.

As shown in Figure 3.14, for the cases c and d where the width (B) of concrete is equal to 30 mm, the variation of confinement stress along the ligament is approximately 3

MPa (8%) whereas for the cases where the width is equal to 40 mm, the confinement varies from 21 MPa at the surface of the specimen to 31 MPa inside the sample (29%).

One can conclude that in order to obtain a better homogeneity of confinement stress in the concrete sample, the width B of concrete should be equal to 30 mm, the height equal to 30 mm with a ligament height equal to 20 mm. This ligament length is equal to the contact length between the ring and the concrete sample (H contact =H lig ). One should also note that for smaller ligament section, higher stresses for the same applied load are obtained. Hence, a width and a height equal to 30 mm are more desirable in terms of stress homogeneity and amplitude.

Strain gauges configuration

Two types of strain gauge configurations were used: quarter-bridge for the ring gauges and full-bridge for the bars. These configurations are based on the concept of a Wheatstone bridge. A description of the Wheatstone bridge as well as the relation between the voltage measured and the gauge deformation are detailed in Appendix B.

Quarter-bridge configuration

An expression for the strain gauge as a function of the measured voltage in a quarter bridge configuration is given by:

ε = 4 K.G.V EX ∆V measured (3.20)

Full-bridge configuration

The gauges used on the Hopkinson bars are used in sets, consisting of a transverse gauge and a longitudinal gauge. At a measuring point there are two sets of diametrically opposed gauges which are mounted as a Full Wheatstone bridge.

Confinement of concrete

As mentioned in section 3.1.1, the confinement consists of pre-stressing a metallic ring by applying a compressive load. The concrete specimen is then inserted inside the ring and the load is released, so a confinement is applied to the concrete sample. Steel sheets are incorporated between the ring and the concrete sample in order to fill the gap between them and thus obtain different levels of confinement.

The relation between the force and gauges deformation being known, one can calculate the compression force transferred to concrete by checking the deformation of the gauges at the end of the unloading phase. For instance, if a certain level of confinement is required, the corresponding deformation can be calculated (

ε Ji = β Ji F 2 , β Ji being known).
Aan example of 11 kN of compression force (8 MPa of confinement) applied with an aluminum ring is plotted in

Quasi-static tests results

Table 3.7 summarizes all static shear tests performed in this study and which will be discussed in the following sections. The table includes the name of each test, the type of the ring used to apply the confinement and the type of material tested. In addition, it includes the strain-rate calculated from equation (2.10), the level of confinement pressure at the peak of the shear stress as well as the peak shear stress in the ligament. The composition of the concrete tested is presented in section 2.

Repeatability of tests

Some tests were performed twice to check the repeatability. For both tests, a good agreement between the two sets of results was noticed. Figure 3.33 shows two tests done on the same concrete with the same confining level. Same shear strength and mean confining stress were obtained for both tests in particular until the peak shear stress was reached.

Influence of confinement

To study the influence of confinement, experiments were carried out with both confining ring, aluminum and steel. A comparison of shear tests performed with dry samples is presented in Figure 3.34.

For the static shear test done with an aluminum ring, the confinement stress is approximately equal to 9 MPa at the beginning of the test and increases to reach 14 MPa at the peak of the shear stress. With the steel ring, the initial confinement is around 20

MPa and increases up to 25 MPa. For these different confining stresses, we can notice a

Based on the strains measured in the bars and according to the one dimensional wave theory, forces and particles velocities at the bar/specimen interfaces can be calculated.

Processing technique

The forces and particles velocities at the specimen faces are calculated by shifting the waves to the contact surfaces assuming that no wave dispersion occurs in slender elastic bars. If ε incident , ε reflected and ε transmitted represent the shifted incident, reflected and transmitted waves respectively, the forces and velocities at the contact surfaces can be calculated as follows:

F input = -S input E bar (ε incident + ε ref lected ) (3.25) F output = -S output E bar ε transmitted (3.26) V input = -C bar (ε incident -ε ref lected ) (3.27) V output = -C bar ε transmitted (3.28)
ε incident , ε transmitted being negative and ε reflected being positive, the shear stress in the ligament is obtained by averaging the input and output forces divided by the section of the ligament:

σ shear = F input + F output 2.S ligament (3.29)
The axial displacement is then deduced by integrating the axial velocity obtained as follows:

V axial = V input -V output (3.30) This processing technique was applied to a series of dynamic experiments performed on different types of concrete specimens. Experimental results are presented hereafter.

Pulse-shaper technique

The use of a pulse shaper technique increases the rising time of the incident wave and reduces the inertial effects which can lead to a force equilibrium at both sides of the concrete sample and a more constant strain-rate during loading. Thus, a thin lead pulse-shaper with the geometry shown in Figure 3.41 is mounted on the impact end of the incident bar. It has an internal diameter of 36 mm and an external of 42 mm with a thickness of 0.8 mm. This system made possible to considerably lengthen the rising time without significantly modifying the total duration of the incident pulse. For these reasons, the following experimental tests could be analyzed by considering equation (3.29). In total, fifteen tests were done with the Hopkinson bar device. Thirteen were successful:

five of them were done on wet R30A7, four were conducted on dry R30A7 and four conducted on High Performance Concrete HPC.

The description of each test is detailed in the Table 3.8. It includes the name of the test, the type of ring used to apply the confinement, the type of concrete tested, the impact velocity measured from the speed barriers, the strain rate in the shear zone, the level of confinement pressure at the peak of the shear stress as well as the shear stress calculated from the average of the input and output forces. The experimental setup used for dynamic tests is presented in Figure 3.42. This crack is a tensile crack which was observed in the studies carried by Backers et al.

[2004] and [START_REF] Yao | A dynamic punch-through shear method for determining dynamic Mode II fracture toughness of rocks[END_REF] on rocks and by [START_REF] Lukić | Experimental characterization of the punch through shear strength of an ultra-high performance concrete[END_REF] on ultra-high performance concrete. These sub-critical cracks are expected to initiate before the peak [START_REF] Watkins | Fracture toughness test for soil-cement samples in mode II[END_REF]), they moves out of the induced shear zone and stops and their influence on the mode II fracture is negligible [START_REF] Backers | ISRM Suggested Method for the Determination of Mode II Fracture Toughness[END_REF]).

Chapter 4

Conclusion

In this chapter, results presented in previous chapters will be compared and discussed in a general frame. Then final conclusions of this work will be drawn and research perspectives will be discussed

General discussion

Results obtained on wet and dry R30A7 concrete samples confined with the Giga press or with the pre-stressed ring are compared with previous punching-through shear tests done with a passive confinement (Forquin [2011a,b], [START_REF] Forquin | A testing technique to characterise the shear behaviour of concrete at high strain-rates[END_REF]). Shear stresses as function of confinement stresses in the ligament are plotted in Figure 4.1.

An increase of shear strength with an increase of confining pressure is considerably reported. This increase of strength was also markedly observed in others experiments on concrete samples like in triaxial tests and quasi-oedometric compression tests. This observation confirms the influence of confinement in increasing the strength of concrete.

It should be noted that it is difficult to measure the confinement in the ligament of PTS samples performed with the Giga press due to a possible self-confinement of the sample.

Thus confining stresses were calculated from numerical simulations conducted with the Drucker-Prager model.

Moreover, results obtained in this study and those obtained in the literature over a wide range of strain-rates are gathered in Figure 4.2. One the one hand, it is shown that the shear strength of dry concrete samples is higher than that of wet ones for each of the tests done whatever the loading rate. One the other hand, there is a trend of increasing strength with strain-rates for the different sets of concrete in the studied range (dry and wet). This observation can be explained by the influence of confinement: the lower the confinement, the higher the sensitivity of concrete strength to strain-rate.

Consequently, it is concluded on the one hand, that the confinement has a major effect on the shear behavior of concrete. On the other hand, the strain-rate sensitivity is small compared to traction or unconfined compression tests.

Conclusion and perspectives

The security of concrete structures used in many applications such as nuclear containments, bunkers or dams is of considerable concern since these structures can be exposed to various extreme loadings during which different compressive, tensile and shear modes might be observed. Various experimental methods have been used to explore the behavior of concrete under different types of loading over a wide range of strain rates. It was shown in the literature that concrete exhibits an increase of strength under confinement (triaxial and quasi-oedometric compression tests) and dynamic loadings (spalling tests). However, in contrast to tensile behavior, the shear behavior of concrete under compression and dynamic loadings has not been carefully studied at a constant confining level.

The aim of the present work is to investigate the quasi-static and dynamic shear behavior of concrete under confined conditions. In previous experiments, concrete samples were either subjected to a small level of confinement [START_REF] Montenegro | Characterization of concrete in mixed mode fracture under confined conditions[END_REF]) or higher level of confinement with a passive metallic ring [START_REF] Forquin | Influence of strain-rate and confining pressure on the shear strength of concrete[END_REF]). In the last method, a change of confinement level occurs during the tests and constitutes a major drawback of this technique. Two methods have been developed to apply confinement to samples before testing them under shear. The first one is based on applying a hydrostatic pressure using a high capacity press and the second one consists of pre-confining the concrete sample using a pre-stressed metallic cell.

In the first chapter of this manuscript, a bibliographic review summarizing the different techniques existing in the literature and which describes the behavior of concrete when subjected to compression or shear at different strain rates is presented. Dynamic simple compression tests showed an increase of strength with strain rate, this increase being mainly structural as it is due to an artificial lateral pressure induced by inertial effects that prevent the sample expansion. Moreover, regarding the material behavior under compression, it was shown that brittle materials are very sensitive to lateral pressure.

An increase of strength with confinement and a transition from quasi-brittle to ductile behavior is revealed. The behavior of concrete under compression is also dependent on the saturation ratio. At high levels of confinement, the presence of water has a major importance: the shear strength of saturated concrete is limited to a maximum value whatever the confining pressure. Sample confined with a passive cell and subjected to dynamic loadings showed a small sensibility to strain-rates. This observation is in contrast to what it is observed in dynamic uniaxial compression tests. This can be explained by the fact that the radial confinement seems to cancel the effect of the radial inertia of the material. Furthermore, static and dynamic experiments conducted considering different water saturation ratios showed a strong influence on the confined behavior of concrete. By reducing the level of effective stress applied to the concrete skeleton, the pore pressure strongly influences the dynamic behavior of confined concrete. Besides, quasi static and dynamic punch-through shear tests done on dry and wet concrete showed a small strain rate sensitivity. It was concluded that the Stephan effect which is supposed to explain the strain rate sensitivity of wet concrete in tension is inoperative under shear loading.

Chapter two describes the new experimental technique developed to study the behavior of concrete subjected to high level of confinement. This method consists of applying a high level of hydrostatic pressure to a concrete sample prior to the application of the shear loading. The specimens consists of cylindrical samples with coaxial cylindrical notches on the top and bottom faces leaving a cylindrical ligament. Two steel rings were glued around the cylindrical notches in order to prevent this zone to be damaged under compression. The high capacity triaxial press Giga has been used to apply a confinement up to 100 MPa. It was shown that the confinement has a major influence on the shear behavior of concrete. An increase of shear strength was noticed for higher confinement level.

In chapter three, the method based on the pre-stressing of the concrete sample is investigated. It consists in inserting a concrete specimen into a pre-stressed metallic ring.

When the loading is released, confining stresses are transferred to the ligament of the concrete sample. Then, a vertical displacement is applied to the central part of the sample to produce shear deformation in the ligament. Two types of metallic ring were used (aluminum and steel) in order to study the influence of confinement: higher shear strength was reported with higher confinement which confirms pressure sensitivity of concrete. Dynamic tests performed with the Split Hopkinson Bar technique were performed to analyze the effect of strain-rate on the confined shear behavior of concrete.

It was found that at the same level of confinement, concrete samples subjected to dynamic loadings exhibited higher shear strength than the one subjected to static loadings.

Moreover, both dried and saturated samples have been tested: dry samples exhibited higher shear strength. In addition, tests on high performance concrete were conducted:

the shear strength of high strength concrete strongly exceeds the one of the ordinary concrete.

In this chapter, the results obtained in both methods are discussed and compared with data existing in the literature. By plotting the ratio of dynamic strength to static strength, it was found that, whereas the strain rate has a strong influence on the tensile strength of concrete, this influence decreases with the level of confinement and appeared quite limited under highly confined compression tests.

Finally, the perspectives for this study are first to complete experimental tests done with the Giga press. Tests with different confinement levels and different saturation ratios could be performed to verify the influence of free water on the shear behavior of concrete under very high confinement level.

Furthermore, tests on concretes mixed with different aggregates composition could be considered to study their effect on the concrete shear behavior. It is also interesting to conduct experimental tests with the pre-stressed cell at intermediate strain-rate to investigate the strain-rate sensitivity of concrete shear strength over different ranges of strain-rate. Moreover, micro-tomography could be used during quasi-static testing in order to examine 3D damage evolution and failure mechanism.

On the other hand, since it is difficult to understand all the mechanical phenomena happening inside concrete with the macroscopic modeling, it is necessary to simulate the real microstucture of concrete and analyze the effects of each constituent on the macroscopic response. This is why performing mesoscopic numerical simulations in which the behavior of cement paste and aggregates is differentiated is very important. It would be necessary to use a model capable of simulating numerically the crack propagation in mode II in aggregates and in cement paste. The model must be identified and validated by experiments. In addition, the model should be capable of taking into account the microstructural parameters of the material (free water content, characteristic size of the mesostructure, resistance of the aggregates). The calculations will highlight the influence of the mesostructure on the confined shear behavior of concrete. A preliminary model and results are presented in Appendix C.

Résumé en francais

Introduction

Le béton est le matériau le plus utilisé en génie civil. Au cours de leur durée de vie, les structures en béton armé peuvent être soumises à différentes types de sollicitations qui peuvent être provoquées intentionnellement ou accidentellement comme les impacts ou les séismes. Par exemple, lors de la pénétration d'un projectile dans une dalle en béton, différents modes de rupture peuvent être générés: en particulier pendant la phase de pénétration de l'impacteur, des contraintes de cisaillement et de compression tri-axiale peuvent être observées à l'intérieur de la cible. De plus, due aux effets inertiels, le confinement atteint dans le béton peut être assez important en fonction de la vitesse d'impact et des caractéristiques de la cible et du projectile [START_REF] Bailly | Modélisation de géomatériaux sous sollicitations dynamiques élevées. i : Un tir de pénétration sur cible en béton[END_REF]). D'autre part, dans le cas d'un séisme, si les poteaux en béton armé ne sont pas suffisamment renforcés transversalement, une rupture en cisaillement se produit. D'où la nécessité d'étudier le comportement du béton soumis à une sollicitation de cisaillement confiné.

D'après l'étude bibliographique, des essais de compression tri-axiale ont été réalisés depuis plusieurs décennies sur le béton [START_REF] Schmidt | Experimental and theoretical investigation of the high-pressure behavior of concrete[END_REF], [START_REF] Warren | Penetration into low-strength (23 MPa) concrete : target characterization and simulations[END_REF]).

L'échantillon de béton est soumis à une pression hydrostatique dans un premier temps puis une contrainte axiale est appliquée. Il a été observé qu'une augmentation du confinement augmente la contrainte déviatorique maximale atteinte et le comportement du béton tend vers un comportement plus ductile. Une étude [START_REF] Gabet | Triaxial behaviour of concrete under high stresses: Influence of the loading path on compaction and limit states[END_REF]) sur des éprouvettes en 'béton réel' a permis d'appliquer une pression allant jusqu'à 0,85 GPa et une déformation volumique maximale d'environ 12% a été atteinte dans l'échantillon.

Cependant, les essais tri-axiaux sont difficiles à réaliser dans des conditions dynamiques, ces difficultés sont liés à l'application d'un confinement à l'aide d'un fluide sous haute pression et à l'application d'un confinement axial avant le chargement dynamique. Une autre méthode a été utilisée pour caractériser le comportement confiné du béton sur une large gamme de vitesse de déformation : l'essai de compression quasi-oedométrique.

Cette méthode a été proposée par Bazant et al. [1986], [START_REF] Burlion | Compaction des bétons: éléments de modeélisation et caractérisation expérimentale[END_REF] et [START_REF] Gatuingt | Prévision de la rupture des ouvrages en béton sollicités en dynamique rapide[END_REF] et entièrement revisitée par [START_REF] Forquin | An experimental method of measuring the confined compression strength of geomaterials[END_REF] dans des conditions statiques et par [START_REF] Forquin | Endommagement et fissuration de matériaux fragiles sous impact balistique[END_REF], Forquin et al. [2008c], Forquin et al. [2010] et [START_REF] Piotrowska | Experimental Investigation of the Confined Behavior of Dry and Wet High-Strength Concrete: Quasi Static Versus Dynamic Loading[END_REF] dans des conditions dynamiques (SHPB). Une pression maximale de 800 MPa a été atteinte avec une déformation volumique de 15%.

Puisque la rupture en cisaillement en mode II a également été observée dans les essais d'impact, il est important de comprendre le comportement du béton sous un tel chargement (Forquin et al. [2008c], Forquin et al. [2010] et [START_REF] Lukić | Experimental characterization of the punch through shear strength of an ultra-high performance concrete[END_REF]). L'essai 'Punch Through Shear' est l'une des techniques qui existent dans la littérature pour étudier le comportement en cisaillement du béton dans lequel des déformations élevées sont atteintes à l'intérieur d'un échantillon de béton. Ces essais ont été menés sur du béton à faible pression (environ 8 MPa) [START_REF] Montenegro | Characterization of concrete in mixed mode fracture under confined conditions[END_REF]) ou sur des roches à des pressions hydrostatiques allant jusqu'à 70 MPa [START_REF] Backers | Fracture Toughness Determination and Micromechanics of Rock Under Mode I and Mode II Loading[END_REF]).

Test de cisaillement triaxial

Dans le chapitre deux, des essais quasi statiques effectués en appliquant une pression hydrostatique puis un cisaillement à l'échantillon de béton sont présentés. Ces tests sont réalisés avec la presse Giga dont les enclumes ont été modifiées afin de pouvoir appliquer le chargement de cisaillement sur des échantillons de béton. Des simulations numériques ont été conduites pour dimensionner le test et choisir les dimensions optimales de l'échantillon testé. Le modèle 'Drucker-Prager" qui prend en compte la dilatance du béton et qui prévoit l'augmentation des contraintes de cisaillement avec le confinement a été utilisé pour modéliser l'éprouvette en béton. En choisissant les paramètres du modèle déjà calibré sur des essais de cisaillement avec confinement passif [START_REF] Forquin | Influence of strain-rate and confining pressure on the shear strength of concrete[END_REF]), les contraintes radiales ont été estimées à partir de la pression de

Test de cisaillement avec cellule de confinement

Dans le troisième chapitre, une nouvelle technique est présentée : une cellule métallique précontraint l'échantillon de béton appliquant ainsi une pression de confinement active.

Cette cellule métallique est déformée élastiquement pour pouvoir y insérer un échantillon prismatique. Le retour élastique de l'anneau durant la phase de décharge applique alors une pression de confinement sur l'échantillon. La cellule est instrumentée avec des jauges de déformation qui permettent de calculer les niveaux de contraintes de confinement au cours de l'essai de cisaillement. Une fois le confinement appliqué, un déplacement est imposé sur la partie centrale de l'échantillon pour produire une déformation de cisaillement dans le béton. Une presse hydraulique (Schenck) est utilisée pour effectuer des essais de cisaillement statique et un dispositif de barres de Hopkinson a été utilisé pour réaliser des essais dynamiques avec une barre en entrée et deux barres en sortie.

Une série de simulations numériques a été menée afin de dimensionner les essais et par suite trouver les dimensions optimales des cellules de confinement, de l'échantillon de béton et des barres de Hopkinson.

Deux cellules de confinement métalliques ont été utilisées (en acier ou en aluminium) dans le but de mettre en évidence l'influence du niveau de pression de confinement : une contrainte maximale de cisaillement plus élevée a été rapportée avec un confinement plus élevé, ce qui confirme la sensibilité à la pression du béton sur toute la plage de déformation considérée (1e -4 /s -100/s).

Des échantillons de béton R30A7 secs et humides ont été testés pour étudier l'influence de la teneur en eau. Les échantillons humides sont placés dans l'eau et les échantillons secs sont séchés au four à 60 • C jusqu'à stabilisation de leur masse. Les échantillons secs présentent une résistance au cisaillement plus élevée que les échantillons humides sur la plage de vitesses de déformation considérée.

Un béton haute performance (HPC) a également été étudié et les résultats ont été comparés à ceux obtenus avec le béton ordinaire pour mettre en évidence l'influence de la composition du béton sur sa résistance au cisaillement. La résistance au cisaillement du béton haute performance dépasse fortement celle du béton ordinaire.

Finalement, la comparaison des essais statiques et dynamiques des trois types de béton a permis de conclure qu'il y a une augmentation de la résistance avec la vitesse sur la plage de vitesses de déformation explorée.

D'après les vidéos pris durant les essais statiques, on a pu remarquer une surface de rupture de cisaillement droite reliant les deux entailles de chaque côté de l'éprouvette.

Cette rupture correspond à une rupture en mode II initiée aux alentours du pic de chargement. De même, une analyse par tomographie a été faite sur un échantillon testé sous chargement dynamique pour examiner le plan de rupture à l'intérieur de l'échantillon. Il a été observé que la surface de rupture en cisaillement est verticale à travers tout le volume de l'éprouvette.

Conclusions

Dans le dernier chapitre, les résultats obtenus avec la presse Giga et ceux obtenus avec la cellule de confinement sont comparés avec les résultats des essais conduits par Forquin Les essais ont permis d'aboutir aux conclusions suivantes. D'une part, une augmentation de la résistance au cisaillement avec une augmentation de la pression de confinement est rapportée pour tous les essais de cisaillement réalisés. Une telle augmentation de résistance a également été observée dans d'autres expériences comme pour des essais triaxiaux [START_REF] Vu | Experimental analysis of concrete behavior under high confinement: Effect of the saturation ratio[END_REF]). Les résultats obtenus dans le cas présent confirment l'influence du confinement sur l'augmentation de la résistance du béton. D'autre part, en traçant la résistance au cisaillement en fonction de la vitesse de déformation, une augmentation de cette résistance avec la vitesse de déformation apparaît pour les différents types de béton dans la gamme de vitesses de déformation étudiée. De plus, la résistance au cisaillement des échantillons de béton sec est supérieure à celle des échantillons humides quel que soit la vitesse de déformation.

Afin de comparer l'ordre de grandeur de l'augmentation de la résistance avec la vitesse de déformation, le rapport de la résistance dynamique sur la résistance statique est calculé puis comparé aux rapports obtenus pour des essais de traction (conduits par [START_REF] Erzar | Ecaillage, cratérisation et comportement en traction dynamique de bétons sous impact : approches expérimentales et modélisation[END_REF]), des essais quasi-oedométriques (Forquin et al. [2015a] and Vu et al.

[2009b]) et des essais de cisaillement avec confinement passif (Forquin [2011a] and 1.1 Instrumented sample ready to be tested under simple compression [START_REF] Vu | Caractérisation expérimentale du béton sous fort confinement : influences du degré de saturation et du rapport eau / ciment[END_REF] [START_REF] Bischoff | Impact behavior of plain concrete loaded in uniaxial compression[END_REF]) 1.4 Uniaxial confined compression apparatus [START_REF] Burlion | Experimental analysis of compaction of concrete and mortar[END_REF]) . . . . . . 1.5 Experimental hydrostatic and oedometric tests results on mortar (W/C=0.5) [START_REF] Burlion | Experimental analysis of compaction of concrete and mortar[END_REF]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Geometry of quasi-oedometric tests [START_REF] Gatuingt | Prévision de la rupture des ouvrages en béton sollicités en dynamique rapide[END_REF]) . . . . . . . . . . . 1.7 Stresses versus deformations for QOC dynamic tests [START_REF] Gatuingt | Prévision de la rupture des ouvrages en béton sollicités en dynamique rapide[END_REF]) . . 1.8 Loading cell used for quasi-oedometric tests (Forquin et al. [2008c]) . . . . 1.9 Deviatoric and hydrostatic behavior of MB50 concrete conducted by (Forquin et al. [2008c]) at different strain-rates . . (Forquin et al. [2015a]) . . . . . . . . . . 1.12 Comparison of the behaviors of high strength concrete HSC and ordinary concrete OC under static quasi-oedometric compression for saturated samples [START_REF] Piotrowska | Experimental Investigation of the Confined Behavior of Dry and Wet High-Strength Concrete: Quasi Static Versus Dynamic Loading[END_REF]) . . . . . . . . . . . . . . . 1.13 Comparison of the behaviors of high strength concrete HSC and ordinary concrete OC under dynamic quasi-oedometric compression for saturated samples [START_REF] Piotrowska | Experimental Investigation of the Confined Behavior of Dry and Wet High-Strength Concrete: Quasi Static Versus Dynamic Loading[END_REF]

  discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Conclusion and perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . Résumé en francais A Technical drawings of the new setup of the Giga press B Strain gages configuration B.1 Strain gages configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1.1 Quarter-bridge configuration . . . . . . . . . . . . . . . . . . . . . B.1.2 Full-bridge configuration . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1 . 1 :

 11 Figure1.1: Instrumented sample ready to be tested under simple compression[START_REF] Vu | Caractérisation expérimentale du béton sous fort confinement : influences du degré de saturation et du rapport eau / ciment[END_REF]).

Figure 1 . 2 :

 12 Figure 1.2: A schematic view of the Split-Hopkinson bar apparatus

Figure 1 . 3 :

 13 Figure 1.3: Compressive strength increase versus strain-rate (Bischoff and Perry [1995])

Figure 1 . 4 :

 14 Figure 1.4: Uniaxial confined compression apparatus (Burlion et al. [2001])

Figure 1 . 5 :

 15 Figure 1.5: Experimental hydrostatic and oedometric tests results on mortar (W/C=0.5) (Burlion et al. [2001])

1. 3 . 1 . 2

 312 Dynamic tests Gatuingt [1999] made static and dynamic confined test on samples of micro-concrete MB50 of 50 mm height and 30 mm or 50 mm internal diameter. The geometry of these tests is presented in Figure 1.6.

Figure 1 . 9 :

 19 Figure1.9: Deviatoric and hydrostatic behavior of MB50 concrete conducted by(Forquin et al. [2008c]) at different strain-rates

Figure 1 .

 1 Figure 1.10: Quasi-static tests performed on R30A7 concrete (a) Compaction curve (b) Deviatoric behavior (Forquin et al. [2015a])

Figure 1 .

 1 Figure 1.11: Dynamic quasi-oedometric experiments on R30A7 concrete (a) volumetric behavior (b) deviatoric behavior (Forquin et al. [2015a])

Figure 1 . 13 :

 113 Figure 1.13: Comparison of the behaviors of high strength concrete HSC and ordinary concrete OC under dynamic quasi-oedometric compression for saturated samples (Piotrowska and Forquin [2015])

Figure 1 . 14 :

 114 Figure 1.14: Deviatoric behavior for triaxials tests done on SAM-35 concrete (Poros-ity=20%, Sr=40%) for confining pressures going from 5 MPa to 400 MPa (Williams et al. [2006])

Figure 1 . 15 :

 115 Figure 1.15: Effect of saturation ratio on the deviatoric behavior of concrete subjected to different level of confining pressure (a) Dried concrete Sr=11% (b) Saturated concrete Sr=100% (Vu et al. [2009a])

Figure 1 . 17 :

 117 Figure 1.17: Evolution in volumetric stiffness vs.mean stress during hydrostatic loading (Zingg et al. [2016])

Figure 1 . 18 :

 118 Figure 1.18: Schematic view of the confined Kolsky bar set-up used in triaxial experiments done by Martin et al. [2013]

  22

Figure 1

 1 Figure 1.22: (a) Geometry of the shearing zone of the direct shear test specimen, (b)Optimized dimensions of the proposed test[START_REF] Luong | Fracture testing of concrete and rock materials[END_REF])

Figure 1 Figure 1

 11 Figure 1.27: Quasi-static tests done on dry specimens with (a) aluminum and (b)steel confining cells[START_REF] Forquin | Influence of strain-rate and confining pressure on the shear strength of concrete[END_REF])

Figure 1 .

 1 Figure 1.29: Quasi-static and dynamic tests performed on dry and wet concrete for different strain-rate, (Forquin [2011a])

Figure 1 . 30 :Figure 1 . 31 :

 130131 Figure 1.30: Shear stress Vs Radial stress of quasi-static and dynamic tests done on (a) Ultra-High Performance Concrete and (b) Fibered Ultra-High Performance Concrete (Lukić and Forquin [2015])

Figure 1 .

 1 Figure 1.32: Mode I and Mode II fracture toughness for Fangshan marble at different levels of strain-rate (Yao et al. [2017])

Figure 2 .

 2 Figure 2.1(a) shows a general view of the press. A cross-section of the confining cell is provided inFigure 2.1(b) . This press is able to generate a confining pressure of up to 0.85 GPa and an axial stress reaching 2.3 GPa. The concrete specimen is placed in the confining cell, and the confining fluid, diethylhexyl azelate -a non-volatile organic liquid, is injected into the cell through the upper opening. The cell is then pressurized by means of a multiplying jack. The axial force is generated from a 13 MN jack placed underneath the cell; this force is transmitted to the specimen via a piston that passes through the lower cell plug. An axial displacement sensor located on the machine is used to control the axial jack displacement, while the axial stress applied to the specimen and the pressure inside the cell are monitored by means of force and pressure sensors. The confining pressure and the axial jack displacement are controlled independently which allows different loading paths. A detailed description of the Giga press can be found in[START_REF] Gabet | Comportement triaxial du béton sous fortes contraintes[END_REF] and[START_REF] Vu | Caractérisation expérimentale du béton sous fort confinement : influences du degré de saturation et du rapport eau / ciment[END_REF].

Figure 2

 2 Figure 2.1: (a) General view of the Giga press (b) Cross section of the confining cell

  ψ=10 • , a dilatancy is imposed during plastic deformation. The yield surface of the linear Drucker-Prager model is illustrated in Figure 2.5.

Figure 2 . 5 :

 25 Figure 2.5: Linear Drucker-Prager model: Yield surface in the P-q plane (Forquin and Sallier [2012])

Figure 2 .

 2 16 illustrates a 3D sketch of the new set up. Joints are used between the caps and the piston in order to avoid the infiltration of the confining fluid. Numerical simulations were conducted to study the maximum stress the cap and the piston can sustain and thus choose the adequate material for them. Detailed technical drawings of the new setup developed can be found in Appendix A.

Figure 2 .

 2 Figure 2.15: 2D section of the new shear test set-up

2. 6 . 2

 62 Sample preparationAfter casting concrete in the rectangular molds, cylindrical specimens are drilled then a sample of 40mm height is cut with a saw. Afterwards, the two faces of the samples are ground and circular notches are drilled to the depth needed on the top and bottom surfaces with a diamond drill bit placed in a concrete lathe (Figure2.18).

Figure 2 .

 2 Figure 2.18: (a) Conrete lathe used to make the notches in concrete samples (b) zoom on the sample

Figure 2 .Figure 2 .

 22 Figure 2.19: Concrete sample equipped with a ring

Figure 2 .

 2 Figure 2.21 shows the stress-strain curve of four specimens with and without confinement loaded statically using a hydraulic normal press.

Figure 2 .

 2 Figure 2.21: Results of punching-through shear tests on PMMA specimens for different confining pressure

Figure 2 .Figure 2 .

 22 Figure 2.22: PMMA specimen (a) before testing (b) after test PTSS2 (without confinement) (c) after test PTSG1 (confining pressure P=50 MPa) (d) after test PTSG2 (P=50 MPa, specimen before failure)

Figure 2 .

 2 Figure 2.24: Concrete specimens after PTS testing with a confining pressure of 50 MPa (a) without rings (b) with rings

Figure 2 .

 2 Figure 2.25: Results of quasi-static punching through shear tests performed on R30A7 concrete for a confining pressure of 100 MPa and different ligament lengths

Figure 2 .

 2 Figure2.25 shows a quasi-linear behavior then a decrease of stress after the peak. The average shear stress between the notches increases for a smaller ligament length i.e for a smaller shear surface. For a ligament of 15 mm in length, the stress reaches a value of 120 MPa while for a 20 mm ligament length shear stress is around 97 MPa. This can be explained by the fact that for a smaller ligament length, the radial stresses are higher and thus the shear stresses are higher. It could also be explained by the development of micro-cracks at the edges of the notches which leads to the initiation of failure in the concrete sample.

2. 8 . 4

 84 Influence of confining pressureExperiments with two different confining pressures (50 and 100 MPa) were carried out on R30A7 concrete. Samples with a ligament length of 20 mm were used. Results of these quasi-static PTS tests are reported in Figure2.26. The radial stress estimated in the ligament after 1.5 mm of displacement is around 113 MPa for a lateral confining pressure of 50 MPa and 190 MPa for a lateral pressure of 100 MPa (cf. Figure2.17).

Figure 2 .

 2 Figure 2.26: Results of quasi-static punching through shear test performed on R30A7 concrete for a ligament length of 20 mm at different confinement levels

Figure 2 .

 2 Figure 2.27: Concrete specimen subjected to a confining pressure of 50 MPa

Figure 2 . 28 :

 228 Figure 2.28: Concrete specimen subjected to a confining pressure of 100 MPa

Figure 3 . 9 :

 39 Figure 3.9: Results obtained numerically on Abaqus showing the values of the force applied on a steel ring and the compression force transmitted to the concrete sample

  3. A force of 36 kN is applied on a steel ring and the confinement stresses in the sample at the end of loading for the different cases are presented in Figure 3.11. L B H H lig H sup H inf Case a

Figure 3 .

 3 Figure3.13: Variation of the confinement stress σ xx along the path Y (cf. Figure3.12) as function of its normalized distance for different cases studied as detailed in Table3.3.

Figure 3 .

 3 Figure 3.25: The aluminum ring instrumented with the gauges Four gauges in a quarter-bridge configuration were glued on the rings in order to measure the hoop strain during tests. The gauges used for the aluminum ring are the FCA-1-23, gauges of 1 mm length and which has a temperature compensation of 23. The ones used for the steel are the FCA-2-17 with 2 mm length and a compensation factor of 17. For this gauges, an amplifier gain G=2000 and an excitation voltage of 1V are used.

Figure 3 . 29 :

 329 Figure 3.29: Deformation of the ring (measured by the gauge J 1 ) as function of the applied force

Figure 3 .Figure 3 . 32 :

 3332 Figure 3.32: Schematic representation of the static Punch-through shear tests

Figure 3 .

 3 Figure 3.41: Thin disk of lead used as pulse shaper between the projectile and the incident bar (a) Before dynamic testing and (b) After dynamic testing

Figure 3 . 42 :

 342 Figure 3.42: Experimental device used for dynamic shear tests (a) General view of the setup (b) zoom on a concrete sample with the confinement ring before and after testing

Figure 3 . 50 :

 350 Figure 3.50: Crack pattern for different dynamic tests (a-b) Sheardyn06 (c-d) Shear-dyn08

  confinement appliquée. Un diamètre de 70 mm, une hauteur de 40 mm ont été choisis pour l'échantillon avec un diamètre de 35 mm et une hauteur de 15 ou 20 mm pour les entailles. Afin d'obtenir une surface de rupture cylindrique et non tronconique, le diamètre intérieur de l'entaille circulaire inférieure coïncide avec le diamètre extérieur de l'entaille circulaire supérieure.Les tests sont réalisés sur le R30A7, un béton ordinaire dont les plus gros granulats sont de 8 mm. Des bagues de confinement ont été collées sur les extrémités de la partie centrale de l'échantillon pour éviter son endommagement sous compression et par suite son effondrement durant l'essai. L'influence de la hauteur de l'entaille a été étudiée en testant les deux hauteurs d'entaille 15 et 20 mm et il a été observé que les contraintes de cisaillement diminuent avec la hauteur d'entaille. Cela peut s'expliquer par le fait que les contraintes radiales sont plus élevées pour une plus petite surface de cisaillement d'une part ou par le fait que le développement de micro-fissures sur le fond des entailles mène à la rupture de l'échantillon. En plus, deux niveaux de confinement ont été testés et une augmentation des contraintes de cisaillement avec la pression de confinement appliquée a été observée. Une pression de confinement de l'ordre de 100 MPa a pu être appliquée et une déformation de l'ordre de 30% a été estimée dans l'échantillon.

[

  2011b] et[START_REF] Forquin | A testing technique to characterise the shear behaviour of concrete at high strain-rates[END_REF] avec une cellule de confinement passif. Puis des conclusions finales sont tirées et des perspectives sont proposées.

Forquin

  and Sallier [2013]). Ces résultats montrent une influence significative de la vitesse de déformation quand le béton est soumis à de la traction alors qu'une sensibilité modérée est notée lors des essais de cisaillement et une sensibilité limitée est observée lors des essais de compression quasi-oedométrique et cela pour les deux bétons (secs et humides). Cette observation peut s'expliquer par l'influence du confinement : plus le confinement est faible, plus la sensibilité du béton à la vitesse de déformation est élevée. Par conséquent, il est conclu d'une part, que le confinement a un effet majeur sur le comportement au cisaillement du béton et que la résistance du béton sec est plus élevée que le béton humide. D'autre part, la sensibilité à la vitesse de déformation est faible par rapport à celle observée sous chargement de traction. 1 Missile impact effects on concrete target, (a) Penetration, (b) Cone cracking, (c) Spalling, (d) Cracks on (i) proximal face and (ii) distal face, (e) Scabbing, (f) Perforation, and (g) Overall target response. [Li et al. [2005]]

  . . . . . . . . . . . . . . . . . . . 1.10 Quasi-static tests performed on R30A7 concrete (a) Compaction curve (b) Deviatoric behavior (Forquin et al. [2015a]) . . . . . . . . . . . . . . . 1.11 Dynamic quasi-oedometric experiments on R30A7 concrete (a) volumetric behavior (b) deviatoric behavior

  (a) without rings (b) with rings . . . . . . . . . . . . . . . . . . . . . . . . 2.25 Results of quasi-static punching through shear tests performed on R30A7 concrete for a confining pressure of 100 MPa and different ligament lengths 2.26 Results of quasi-static punching through shear test performed on R30A7 concrete for a ligament length of 20 mm at different confinement levels . . 2.27 Concrete specimen subjected to a confining pressure of 50 MPa . . . . . . 2.28 Concrete specimen subjected to a confining pressure of 100 MPa . . . . . 2.29 View from the inside of a concrete specimen subjected to a confining pressure of 100 MPa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 A schematic view of a concrete sample confined with a ring . . . . . . . . 3.2 A circular ring and its quadrant subjected to a compressive force . . . . . 3.3 2D numeircal model for ring dimensioning (a) geometry and loading (b) Mises stress distribution inside the ring for a force of 20 kN . . . . . . . . 3.4 (a) Distribution of hoop strain inside the ring (b) Gauge position . . . . . 3.5 The rigidity of concrete or ring calculated for different cases . . . . . . . . 3.6 Numerical simulations done to calculate the rigidity of the concrete or the ring (Refer to Figure 3.5). . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 3D model for concrete specimen with the ring . . . . . . . . . . . . . . . . 3.8 Dimensions of concrete specimen . . . . . . . . . . . . . . . . . . . . . . . 3.9 Results obtained numerically on Abaqus showing the values of the force applied on a steel ring and the compression force transmitted to the concrete sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 The quarter sample modeled and shown in Figure 3.11 . . . . . . . . . . . 3.11 Distribution of confinement stresses in the concrete sample for different geometries studied detailed inTable 3.3 (H contact =H lig ) . . . . . . . . . . 3.12 Path of nodes used to plot the confinement and shear stresses for different geometries (a) Path Y (b) Path Z . . . . . . . . . . . . . . . . . . . . . . . 3.13 Variation of the confinement stress σ xx along the path Y (cf. Figure 3.12) as function of its normalized distance for different cases studied as detailed in Table 3.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.14 Variation of the confinement stress σ xx along the path Z (cf.Figure 3.12) as function of its normalized distance for different cases studied as detailed in Table3.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.15 Ring geometry and dimensions (in mm) . . . . . . . . . . . . . . . . . . . 3.16 Concrete sample geometry and dimensions (in mm) . . . . . . . . . . . . . 3.17 Loading applied in the static shear loading step . . . . . . . . . . . . . . . 3.18 Shear stresses inside concrete sample under static shear loading after 0.5 mm of displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.19 Shear and confinement stresses in the ligament after 0.5 mm of displacement along the two paths defined in Figure 3.12 . . . . . . . . . . . . . . . 3.20 Mesh used in the case of a ring or a spring . . . . . . . . . . . . . . . . . . 3.21 (a) Numerical 3D mesh for dynamic PTS setup (b) Incident pulse applied to the input bar in dynamic numerical simulations . . . . . . . . . . . . . 3.22 Comparison of the stresses deduced at the specimen/bars interfaces with the stress calculated in the ligament . . . . . . . . . . . . . . . . . . . . . 3.23 Procedure for preparation of concrete samples . . . . . . . . . . . . . . . 3.24 Signal measurement technique . . . . . . . . . . . . . . . . . . . . . . . . . 3.25 The aluminum ring instrumented with the gauges . . . . . . . . . . . . . . 3.26 Layout of gauges on the bars . . . . . . . . . . . . . . . . . . . . . . . . . 3.27 A scheme representing the ring, the position of the gauges and the forces applied to calibrate the gauges . . . . . . . . . . . . . . . . . . . . . . . . 3.28 The configurations of the forces applied for the calibration of the gauges (a)F1 applied (b)F2 applied . . . . . . . . . . . . . . . . . . . . . . . . . . 3.29 Deformation of the ring (measured by the gauge J 1 ) as function of the applied force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.30 Procedure of confinement (a) Method used (b) Steel sheets glued . . . . . 3.31 (a) The evolution of the forces for a confinement of 11 kN (b) A concrete specimen confined with the ring . . . . . . . . . . . . . . . . . . . . . . . . 3.32 Schematic representation of the static Punch-through shear tests . . . . . 3.33 Results of quasi-static shear tests performed on dry R30A7 concrete with a steel ring at the same confining level (a)Shearstat03 (b)Shearstat06 . . . 3.34 Results of quasi-static shear tests performed on dry R30A7concrete with (a) aluminum and (b) steel confining cells . . . . . . . . . . . . . . . . . . 3.35 Results of quasi-static shear tests performed with an aluminum ring on (a) dry and (b) wet R30A7 concrete specimens . . . . . . . . . . . . . . . 3.36 Results of quasi-static shear tests performed with an aluminum ring on (a) OC and (b) HPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.37 (a) Location of pictures taken for 'Shearstat06' test (b) View of the sample before testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.38 Fracturing on the surface of a concrete sample (Shearstat06) at different stages of static loading (Figure 3.37(a)). . . . . . . . . . . . . . . . . . . . 3.39 Crack pattern for different static tests (a) Shearstat01 (b) Shearstat11 (c) Shearstat06 (d) Shearstat12 . . . . . . . . . . . . . . . . . . . . . . . . . . 3.40 Schematic representation of the SPHB system for dynamic Punch-through shear tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.41 Thin disk of lead used as pulse shaper between the projectile and the incident bar (a) Before dynamic testing and (b) After dynamic testing . . 3.42 Experimental device used for dynamic shear tests (a) General view of the setup (b) zoom on a concrete sample with the confinement ring before and after testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.43 Typical signals recorded by strain gauges glued on the bars . . . . . . . . 3.44 Typical stresses obtained in the ligament of the concrete the sample . . . 3.45 Typical elastic shear deformation curve for determining the strain-rate of dynamic tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.46 Results of dynamic shear tests performed on high performance concrete specimens with steel confining cells (a) ShearDyn13 (b)Sheardyn15 . . . . 3.47 Results of dynamic shear tests performed on dry specimens with (a) aluminum and (b) steel confining cells . . . . . . . . . . . . . . . . . . . . . . 3.48 Results of dynamic shear tests performed with an aluminum ring on (a) dry and (b) wet specimens . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.49 Results of dynamic shear tests performed with an aluminum ring on (a) OC and (b) HPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.50 Crack pattern for different dynamic tests (a-b) Sheardyn06 (c-d) Sheardyn08 3.51 CT scans done on BHP specimen tested under dynamic loading (Shear-dyn08) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.52 Quasi-static and dynamic shear tests done on wet R30A7 . . . . . . . . . 3.53 Quasi-static and dynamic shear tests done on dry R30A7 . . . . . . . . . 3.54 Quasi-static and dynamic shear tests done on HPC . . . . . . . . . . . . . 3.55 Quasi-static and dynamic shear tests done on wet R30A7 . . . . . . . . . 3.56 Quasi-static and dynamic shear tests done on dry R30A7 . . . . . . . . . 3.57 Quasi-static and dynamic shear tests done on HPC . . . . . . . . . . . . . 4.1 Results of shear strength with respect to confinement level in the ligament compared to the literature data . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Shear strength with respect to strain rate for different concrete samples in different studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Relative increase of strength with respect to strain rate for wet R30A7 concrete subjected to different types of loading . . . . . . . . . . . . . . . 4.4 Relative increase of strength with respect to strain rate for dry R30A7 concrete subjected to different types of loading . . . . . . . . . . . . . . . A.1 PTS assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 PTS assembly -Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 PTS assembly -Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 PTS assembly -Part III . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5 PTS assembly -Part IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.6 PTS assembly -Part V . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1 The Wheatstone bridge circuit . . . . . . . . . . . . . . . . . . . . . . . . C.1 (a) Heterogeneous concrete sample generated by Abaqus (b) Particle size distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2 Mesh and loading of the mesoscopic model . . . . . . . . . . . . . . . . . .

  Triaxial tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shear Behavior of geomaterials . . . . . . . . . . . . . . . . . . . . . . . . Static tests . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3.2 Dynamic tests . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 Quasi-static Punching Through Shear test under high confinement 2.1 Experimental setup: Triaxial press Giga . . . . . . . . . . . . . . . . . . . 2.2 Design of the test using numerical simulations . . . . . . . . . . . . . . . . 2.2.1 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1.1 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1.2 Loading and boundary conditions . . . . . . . . . . . . . Influence of rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Mesh sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . Influence of concrete composition . . . . . . . . . . . . . 3.3.3.5 Post-mortem observation . . . . . . . . . . . . . . . . . . 3.4 Dynamic tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Split Hopkinson Pressure Bar Technique . . . . . . . . . . . . . . . 3.4.2 Processing technique . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Pulse-shaper technique . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Dynamic tests results . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4.1 Typical dynamic test results . . . . . . . . . . . . . . . . 3.4.4.2 Repeatability of tests . . . . . . . . . . . . . . . . . . . . 3.4.4.3 Influence of confinement . . . . . . . . . . . . . . . . . . . 3.4.4.4 Influence of moisture content . . . . . . . . . . . . . . . . 3.4.4.5 Influence of concrete composition . . . . . . . . . . . . . 3.4.4.6 Post-mortem observation . . . . . . . . . . . . . . . . . . 3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	Contents	vi
	3.3.3.4

Introduction 1 State of the art 1.1 Experimental characterization of concrete . . . . . . . . . . . . . . . . . . 1.2 Simple Compression tests under static and dynamic loading . . . . . . . . 1.3 Confined compression behavior of geomaterials . . . . . . . . . . . . . . . 1.3.1 Quasi-oedometric compression tests . . . . . . . . . . . . . . . . . 1.3.1.1 Static tests . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1.2 Dynamic tests . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 1.3.2.1 Static tests . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2.2 Dynamic tests . . . . . . . . . . . . . . . . . . . . . . . . 1.4 1.4.1 Punch-Through Shear Test concept . . . . . . . . . . . . . . . . . . 1.4.2 Triaxial Punch-Through Shear Test . . . . . . . . . . . . . . . . . 1.4.3 Punch-Through Shear Test with passive confinement . . . . . . . . 1.4.3.1 2.2.1.3 Material parameters . . . . . . . . . . . . . . . . . . . . . 2.2.2

Table 2 . 2

 22 strength at 28 days f c28 =80 MPa. Concrete composition and mechanical properties are presented in Table2.2.

	Concrete mix proportions	OC (R30A7) HPC
	Siliceous gravel D 0.5/8 mm (kg/m 3 )	1008	1008
	Sand Dmax 1.8 mm (kg/m 3 )	838		795.4
	Cement CEM I 52.5 N PM ES CP2 (kg/m 3 )	263		420
	Water (kg/m 3 )	169		140
	Silica fumes (kg/m 3 )	-		46.7
	Superplasticizer (kg/m 3 )	-		4.7
	Mechanical properties	wet	dry
	Density (Kg/m 3 )	2278 2290 2415
	Compressive strength f c28 (MPa)	30	42	80
	Porosity accessible to water	11.8	8.8
	Water/cement ratio	0.64	0.3

: Composition and mechanical properties of ordinary concrete (OC) and high-strength concrete (HPC)

[START_REF] Vu | Experimental analysis of concrete behavior under high confinement: Effect of the saturation ratio[END_REF]

and

[START_REF] Zingg | Influence of cement matrix porosity on the triaxial behaviour of concrete[END_REF]

Table 2

 2

	.3.

Table 2 . 3 :

 23 Mechanical properties of PMMA tested

Table 2 .

 2 4 summarizes punching-through shear tests performed on PMMA specimens.

	Test name Hydraulic press used	Description of the test
	PTSS2	Schenck	P = 0 MPa
	PTSG1	Giga	P = 50 MPa
	PTSG2	Giga	P = 50 MPa (the test is stopped before failure)
	PTSG3	Giga	P = 30 MPa

Table 2 . 4

 24

: Summary of the tests performed on PMMA specimens 2.7.4 PTS tests on PMMA

Table 2 . 5 :

 25 Three tests were conducted on R30A7 concrete specimens and are summarized in Table2.5. Different ligament lengths and different confinement pressures were tested to study their influence on the shear response of concrete. Summary of the tests performed on R30A7 concrete specimens

	2.8 Experimental Results on concrete
	2.8.1 Testing program
	Test name Ligament length Confining pressure
	PTSG1	20 mm	50 MPa
	PTSG2	15 mm	50 MPa
	PTSG3	20 mm	100 MPa

Table 3 .

 3

1: Mechanical properties of material used in numerical simulations

Table 3 . 3 :

 33 Different geometries studied

Table 3 .

 3

	6.
	aluminum	Steel		aluminum	Steel
	α J1 -0.000138 -0.0000451	β J1 0.000139	0.000047
	α J2 0.000212	0.000071	β J2	-0.00014 -0.000048
	α J3 0.000208	0.000021	β J3	-0.00014 -0.000014
	α J4 -0.000057 -0.0000477	β J4	0.00006	0.000051

Table 3 .

 3

6: Coefficients of each gauge used to calculate the confinement pressure

  6.1.

	Test Name	Ring used Concrete Sample Strain rate (/s)	Confinement stress (MPa)	Shear stress (MPa)
	ShearStat01	Alu	R30A7	5x10 -4	19	24
	ShearStat02	Alu	Wet R30A7	5x10 -4	14	18
	ShearStat03	Steel	Dry R30A7	5x10 -4	25	33
	ShearStat04	Steel	Wet BHP	5x10 -4	42	55
	ShearStat05	Steel	Wet R30A7	5x10 -4	20	22
	ShearStat06	Steel	Dry R30A7	5x10 -4	25	33
	ShearStat07	Steel	Wet BHP	5x10 -4	44	65
	ShearStat08	Steel	Wet R30A7	5x10 -4	18	18
	ShearStat09	Alu	Wet R30A7	5x10 -4	11	19
	ShearStat10	Alu	Dry R30A7	5x10 -4	12	25
	ShearStat11	Alu	Wet BHP	5x10 -4	14	37
	ShearStat12	Alu	Wet BHP	5x10 -4	20	42
	ShearStat13	Alu	Dry R30A7	5x10 -4	16	21
	ShearStat14	Alu	Wet R30A7	5x10 -4	15	19

Table 3 .

 3

7: Summary of the different quasi-static shear tests performed

Table 3 .

 3

8: Characteristics of dynamic shear tests done

  De plus, la technique de test de cisaillement 'Punch Through Shear' a été développée et réalisée dans des conditions statiques et dynamiques avec une cellule passive. Durant cet essai, les contraintes radiales varient et dépendent du comportement du matériau testé[START_REF] Forquin | Influence of strain-rate and confining pressure on the shear strength of concrete[END_REF]), ce qui constitue un inconvénient majeur de la méthode.L'objectif du travail de cette thèse est de combler les lacunes des tests existants dans la littérature pour étudier le comportement du béton soumis à des pressions de confinement relativement élevées (de quelques dizaines à quelques centaines de MPa) à des niveaux de déformation élevés et sur une large gamme de vitesses de déformation. Pour cela,

	deux nouvelles méthodes expérimentales dérivées du 'Punch Through Shear test' ont
	été développées pour caractériser le comportement en cisaillement confiné du béton sous
	chargement statique et dynamique.
	Le premier test est un test de cisaillement 'Punch Through Shear' tri-axial qui consiste	à
	appliquer dans une première phase une pression hydrostatique à un niveau prédéterminé.
	Dans une deuxième phase, la pression de confinement étant maintenue constante, une
	force axiale est appliquée sur la partie centrale de l'échantillon pour produire un ci-
	saillement dans le ligament. La deuxième méthode expérimentale consiste à confiner
	l'échantillon de béton avec une cellule métallique précontrainte avant d'appliquer la
	charge de cisaillement. Des simulations numériques réalisées pour la conception de
	chaque test sont décrites puis les procédures de test et les différents résultats sont
	présentés.

  ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 A schematic view of the Split-Hopkinson bar apparatus . . . . . . . . . . 1.3 Compressive strength increase versus strain-rate

  ) . . . . . . . . . . . . . . . . . . . 1.14 Deviatoric behavior for triaxials tests done on SAM-35 concrete (Poros-ity=20%, Sr=40%) for confining pressures going from 5 MPa to 400 MPa[START_REF] Williams | Laboratory characterization of sam-35 concrete[END_REF]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.15 Effect of saturation ratio on the deviatoric behavior of concrete subjected to different level of confining pressure (a) Dried concrete Sr=11% (b) Saturated concrete Sr=100%(Vu et al. [2009a]) . . . . . . . . . . . . . . . 1.16 Limit states of the dry and saturated concrete in the stress invariant space(Vu et al. [2009a]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.17 Evolution in volumetric stiffness vs.mean stress during hydrostatic loading (Zingg et al. [2016]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.18 Schematic view of the confined Kolsky bar set-up used in triaxial experiments done by Martin et al. [2013] . . . . . . . . . . . . . . . . . . . . . . 1.19 Illustration of the different fracture modes . . . . . . . . . . . . . . . . . . 1.20 Mode II fracture toughness testing methods. (a) Antisymmetric Four-Point Bending, (b) Anti-symmetric Four-Point Bending Cube, (c) Punch through Shear, (d) Compression-Shear Cube, (e) Short Beam Compression, (f) Centrally cracked Brazilian test, (g) Three point bending semidisk and (h) Triaxial Compression. [Backers [2004]] . . . . . . . . . . . . . 1.21 (a) Geometry of the Punch-Through Shear test done by Watkins (b) A typical Load displacement curve obtained from experiments (Watkins [1983]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.22 (a) Geometry of the shearing zone of the direct shear test specimen, (b)Optimized dimensions of the proposed test[START_REF] Luong | Fracture testing of concrete and rock materials[END_REF]) . . . . . . . . 1.23 Influence of confining pressure on the stress intensity factor for two types of rocks (a) Carrara Marble (b) Aue Granite[START_REF] Backers | Rock fracture toughness testing in Mode II -punch-through shear test[END_REF]) . . . 1.24 Fracture evaluation in limestone at different stages of loading for a confining pressure of 5MPa[START_REF] Backers | Rock fracture toughness testing in Mode II -punch-through shear test[END_REF]) . . . . . . . . . . . . . . . . 1.25 Shear stress and radial displacement obtained for unconfined and confined tests (Montenegro et al. [2007]) . . . . . . . . . . . . . . . . . . . . . . . . 1.26 Concrete sample confined in a metallic ring and subjected to punchthrough shear tes (Forquin [2011a]) . . . . . . . . . . . . . . . . . . . . . . 1.27 Quasi-static tests done on dry specimens with (a) aluminum and (b) steel confining cells (Forquin [2011b]) . . . . . . . . . . . . . . . . . . . . . . . . 1.28 Quasi-static tests done on (a) dry and (b) wet specimens (Forquin [2011b]) 1.29 Quasi-static and dynamic tests performed on dry and wet concrete for different strain-rate, (Forquin [2011a]) . . . . . . . . . . . . . . . . . . . . 1.30 Shear stress Vs Radial stress of quasi-static and dynamic tests done on (a) Ultra-High Performance Concrete and (b) Fibered Ultra-High Performance Concrete (Lukić and Forquin [2015]) . . . . . . . . . . . . . . . . . 1.31 Shear stress Vs Strain-rate for (a) Ultra-High preformance concrete (b) Fibered Ultra-High performance concrete (Lukić and Forquin [2015]) . . 1.32 Mode I and Mode II fracture toughness for Fangshan marble at different levels of strain-rate (Yao et al. [2017]) . . . . . . . . . . . . . . . . . . . . Finite element geometry and mesh . . . . . . . . . . . . . . . . . . . . . . 2.4 Loading applied to the specimen (a) Confining step (b) Shearing step . . 2.5 Linear Drucker-Prager model: Yield surface in the P-q plane (Forquin and Sallier [2012]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Equivalent plastic strain after 1.3mm of displacement (a) without and (b) with rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Different size of mesh studied (a) 0.15 mm (b) 1.5 mm (c) 3 mm . . . . . 2.8 Shear Logarithmic strain after a displacement of 1 mm for different meshes 2.9 Shear deformation in the ligament . . . . . . . . . . . . . . . . . . . . . . 2.10 Different configurations with different meshes and different element sets in which shear strain is averaged and plotted in Figure 2.11. . . . . . . . . 2.11 Shear strain values for different configurations (Figure 2.10) compared to the calculated nominal strain (equation (2.4)). . . . . . . . . . . . . . . . 2.12 The radial stress given by equation (2.5) and the average true stress obtained by numerical simulations for different notch diameters after 1.5 mm of displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.13 Axial force applied on the central part of the specimen vs axial displacement for different notches lengths. (Numerical simulations) . . . . . . . . 2.14 Geometry of the specimen tested experimentally . . . . . . . . . . . . . . 2.15 2D section of the new shear test set-up . . . . . . . . . . . . . . . . . . . . 2.16 3D sketch of the new shear test set-up . . . . . . . . . . . . . . . . . . . . 2.17 The radial stress given by equation (2.5) and the true average radial stress obtained by the numerical simulations for different confining pressures for a ligament length of 20 mm . . . . . . . . . . . . . . . . . . . . . . . . . . 2.18 (a) Conrete lathe used to make the notches in concrete samples (b) zoom on the sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.19 Concrete sample equipped with a ring . . . . . . . . . . . . . . . . . . . . 2.20 Membrane used for testing samples in the Giga press . . . . . . . . . . . . 2.21 Results of punching-through shear tests on PMMA specimens for different confining pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.22 PMMA specimen (a) before testing (b) after test PTSS2 (without confinement) (c) after test PTSG1 (confining pressure P=50 MPa) (d) after test PTSG2 (P=50 MPa, specimen before failure) . . . . . . . . . . . . . 2.23 PMMA specimen with a ligament length of 20mm, central part damaged under compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.24 Concrete specimens after PTS testing with a confining pressure of 50 MPa

	2.1 (a) General view of the Giga press (b) Cross section of the confining cell .
	2.2 3D and 2D geometry for punching-through shear test specimen . . . . . .
	2.3

Acknowledgements

B.1.1 Quarter-bridge configuration

The quarter-bridge has one active leg, i.e., one leg with a changing resistance. Let's consider R 0 the nominal resistance of this strain gage, the strain-induced change in resistance, ∆R is given by: ∆R

where k is the gage coefficient. The bridge equation can be written to express the strain as function of the measured voltage.

Since the Output voltage is too small (in the range of few mV), the acquisition card has an amplifier which multiplies V 0 by a gain G and which introduces the relation:

where ∆V 0 is the variation of the voltage measured at the end of the chain measurement.

Therefore, from equations (B.2), (B.3) and (B.4) we can derive an expression for the strain gage as a function of the measured voltage:

B.1.2 Full-bridge configuration

In the full-bridge configuration, each of the four gages is considered in the resistances of the bridge. If one considers R L the resistance of the longitudinal gage and R T the one for the transversal gage for each set of gages and assuming that the bars are isotropic, one can write:

The equation (B.1) of the Wheatstone bridge can thus be written;

By combining equations (B. 6) and (B.7) with equation (B.8), we can write:

Finally, by using equation (B.4), the relation between the voltage measured and the strain gage deformation is the following:

Mesoscopic Simulations

A 3D mesoscopic modeling for the numerical analysis of the shear behavior of concrete is proposed. The real micro-structure of concrete is simulated where two different phases are differentiated: the mortar paste which includes the cement past with sand and fine aggregates and the aggregates. An anisotropic damage model is used to simulate the behavior of concrete under shear. It correctly reproduces the damage zones observed on actual samples which can help in the understanding of meso-scale phenomena and their influence on macro-scale behavior.

Because of the geometry of the sample, there is a presence of distorted finite elements during meshing. This should be avoided by choosing the best random distribution and meshing technique. Due to time constraints, this issue could not be developed in this report. A lot of hypothesis were taken into account in order to reduce the computational time and some results are presented to validate the model even though they don't represent the correct values.

C.1 Mesoscopic method principles

Concrete is a multi-scale material which include aggregates, sand, cement and pores. To improve the understanding of the concrete behavior, the properties of these individual components have to be taken into account [START_REF] Wriggers | Mesoscale models for concrete: Homogenisation and damage behaviour[END_REF], [START_REF] Segurado | A numerical approximation to the elastic properties of sphere-reinforced composites[END_REF], [START_REF] Schlangen | Experimental and Numerical Analysis of Micromechanisms of Fracture of Cement-Based Composites[END_REF], [START_REF] Erzar | Experiments and mesoscopic modelling of dynamic testing of concrete[END_REF] and [START_REF] Caballero | A meso-level approach to the 3D numerical analysis of cracking and fracture of concrete materials[END_REF]). Some authors considered the concrete a biphasic material composed of coarse aggregates embedded in mortar [START_REF] Comby-Peyrot | Development and validation of a 3D computational tool to describe concrete behaviour at mesoscale. Application to the alkali-silica reaction[END_REF]). Others authors introduced the interfacial transition zone as a third component [START_REF] Caballero | A meso-level approach to the 3D numerical analysis of cracking and fracture of concrete materials[END_REF]).

These elements were inserted along all potential crack planes in the mesh.

C.2.4 Simplified model

In order to simplify the calculation and reduce the computational time, only aggregates present in the unnotched zone are simulated. For the remaining zone, the elements are assigned the concrete parameters (Figure C.3). Moreover, in the following simulations, only the aggregates having a diameter larger than 5 mm were generated and the total volume of aggregates inside the sample doesn't exceed the 20%. These approximations