Christine Mme

M Raphael Espinosa

M Michel Dupuis

M Franz Arrigoni

Chouly

Mr Monzer

Mr Al Lorenzo Khalil

Mr Taddei

Mr Hassan Jianbo

Fris

Xiao Pan

Zhou Yue

Chen Kui

Liulin Runjia

Zhuo Shengrong

Xie Shiming

Wu Hongjian

Zhou Yang

Chen Xianlong

Liu

Liu meimei, Zhu dan Zhao Hui

Fan Xiujuan

Li Yizhuo

Wang zhoucheng, and many other friends in France and in China.

I strongly appreciate the beautiful encounters in Belfort, which make me

GENERAL INTRODUCTION 1.1/ BACKGROUND

The mechanical behavior of structures under high-speed loading is of main interest in vehicle engineering, military framework, and modern pharmaceuticals. Indeed, the material used in the design whatever is the context has to satisfy different requirements, especially specific criteria on manufacturing such as lightweight and high-strength, or similar properties as simulants of the biological body on medicine or pharmaceuticals. As an example of structures with these kinds of important requirements, body-armors can be cited with a primary need of human body protection and dissipation of energy when impacted by the projectiles, and also a need of lightweight and mobility for human ergonomic aspects. This balance between different development requirements for the designs of such protection devices needs a deep understanding of the behavior of the material under complex physical loading such as ballistic impacts.

The research to investigate the processes and behavior of various materials during the ballistic impacts can be categorized into three aspects: Experiments, theoretic analyses, and numerical simulations. The experiment is the most natural and essential way to observe and understand the physical world, which allows us gathering measurable or visual information via employing specific equipment. It is still one of the most important methods in scientific research. Analytical models consist of developing mathematical equations based on materials parameters allowing calculating physical parameters, such as penetration history, velocity history, deformation ranges in the target. The usages of analytical models require some supplements from experimental measurements and semi-empirical fits, which are developed and employed in a variety of fields because of 2 CHAPTER 1. GENERAL INTRODUCTION its simple and low-cost features. The numerical simulation is the calculation that is run on a computer following a program that implements a mathematical model for a physical system. Owing to the fast development of computer science in recent years, the third way is paid more and more attention. It has been developed and applied in various fields particularly for some complex and huge engineering problems.

The processes of high-velocity impacts often involve various damages and even severe destruction, thereby they have many high and strict requirements for experimental techniques and equipment. Particularly, for some research relevant to biological issues in the fields of medicine or pharmaceuticals, analytical and simulation methods demonstrate considerable significance because of the security and ethics problems in the experiments. The numerical method like the finite element method(FEM) based on element discretization was developed from the late eighteenth century. They integrate the engineering scientific methods and principles of calculus in mathematics, as well as modern computer science techniques. The CAE (computer aidered engineering) software developed based on these numerical methods such as FEM, CFD, and , has become the main tool to investigate complex physical problems in engineering.

1.2/ OBJECTIVES

Investigations on the ballistic behavior of soft tissue materials are significant research to the development of the medicine and pharmaceutical fields. They mainly involve the procedure about the particles or fragmentation penetrating or traveling into the soft materials. For instance, Fig. 1.1 shows a needle-free drug delivery device, which is one of transdermal delivery technologies in the pharmaceutical industry. The micro-scale particles as drug powders are injected into skin with high velocity, and several factors should considered such as the particle diameters, trajectories, and momentum or velocities of projectile in this process. From the insight of mechanical investigation, the behavior of micro-particles is performed and observed by the experimental process like Fig. 1.2. Fig. 1.2 shows an example that a sphere with µm-size impacts into a soft tissue captured by some experimental equipment to understand the ballistics procedure. The size of projectile, the cavity deformation of soft tissue, as well as the trajectory of projectile fragments are important factors to design and evaluate the performance of the drug delivery device. The process of a 13-µm steel sphere penetrating into a simulant of soft tissue called SEBS gel observed by the experiment [2].

The numerical methods are suitable approaches to study these kinds of processes. Although grid-based methods like FEM have been powerful tools in engineering, some numerical limitations still exist. One of the most severe problems is the large distortions of elements when solid structures are extremely deformed. The methods based on the meshing technique are not very suitable for simulating hydrodynamic phenom-CHAPTER 1. GENERAL INTRODUCTION ena like the high velocity impacts which include large deformations, moving material interfaces, deformable boundaries as well as fragmentations, typically the process as shown in Fig. 1.2. The mesh distortion can cause significant errors in these analyses. To avoid it, other numerical concepts have been developed, based on particle discretization, also called meshfree or meshless methods like The Smoothed Particle Hydrodynamics method (SPH).

SPH method is the earliest mehsfree method, proposed initially for the by Gingold & Monaghan in 1977. Compared to traditional grid-based numerical methods, some distinct benefits of the SPH method have been recognized during the various applications:

1. The particle modeling of Lagrangian nature in SPH can concern the time history of the material particles, which allows straightforward handling of very large deformations since the connectivity between particles are generated as part of the computation and can change with time.

2.

The free surfaces, material interfaces, and moving boundaries can all be traced freely in the process of simulation, which is always challenging to many Eulerian methods.

3.

The SPH method is a suitable alternative for non-continuum problems, typically examples involving bio-and nano-engineering at the micro and nanoscale, and astrophysics at astronomic scale, as well as the debris clouds phenomenon in HVIs.

4.

SPH formulations are comparatively simple and easy to be developed or improved as 3D-codes by some algorithms or to be coupled with other numerical methods.

Owing to these advantages, the SPH has been extended to the solid mechanics in recent 30 years. Its potential in simulating extremely large deformations of materials has been identified, but the applications in micro-penetration are still not so much in the literature.

Therefore, this thesis highlights the development of an SPH code and its applications on the soft materials penetrations, particularly, in the micro-scale. Various simulants as human body substitutes such as ballistic gelatin and polymers gels will be investigated under high-velocity impacts by SPH simulations.

3/ THESIS ORGANIZATION

The rest of this thesis is organized as follows:

Chapter 2: This chapter provides a state of the art for the advances in ballistic perforation impact on structures using the SPH method. Though the literature reviewing, the advantages and limitations of the SPH method applied in high-velocity impacts are recognized.

Chapter 3: Although the SPH method has been developed for the application in solid dynamics, the tensile instability is a main obstacle to simulate the material strength with a large deformations. Therefore, the first work is to investigate the stability and accuracy of SPH simulations for solid problem. A SPH code is programmed and implemented into the equations of elastic-plasticity for materials.

Then, the influence on the stability and accuracy derived from the kernel functions is investigated. Combined with this analyses on the types of kernel functions, various problems such as vibrated beam and Taylor rod impacts are simulated by this SPH code.

Chapter 4: After programming a SPH code and validating it on several solid problems, a numerical model consisting of SPH and FEM is developed for the micro penetrations into soft materials. Gelatin is the most widely used soft material as a simulant of biological body, and its ballistic behavior under impact loading by microparticles is investigated by this numerical model. A series of simulations involving the micro-particle (less than 10 µm) impact 10 % gelatin with a wide range of velocities from 200-1500 m/s are performed by this SPH model. The soft material SEBS gel as another simulant of biological body involves several benefits, but its ballistic behavior is paid less attention in the literature. The second highlight of this chapter is to investigate the responses of SEBS gel as a target in the micro-penetrations by the numerical simulations using SPH model. Also, the usability and university of several analytical models for penetrations is observed and discussed in the microscale.

Chapter 5: The work of this thesis is summarised and the future works are discussed in the last chapter. industry. The mechanical process of HVIs is complex and involves the extreme deformations on the materials. Penetration is a general description as to the sequence of impacts, which involves the perforation, embedment, ricochet, as well as the fragments and debris clouds with efficiently high velocity. Perforation means that the projectile passes through the target structure. Embedment means that the projectile is stopped in the target that appears generally in enough thick structures. Ricochet means that the projectile is deflected from the target without perforating. The fragments and debris clouds are the most comment damage in the target structures caused by the projectile which passes out the thin target with very high impact velocity. Therefore, the damage forms or material behavior during the HVIs depend on the configurations of projectile and target, impact conditions like initial velocity, as well as the material properties.

As the mentioned in the general introduction, the methods to investigate the process of HVIs can be categorized to three ways [3,4]: experimental tests, analytical modeling involving semi-empirical data fits and numerical modeling. The numerical simulation is 8CHAPTER 2. A STATE OF THE ART ON IMPACTS TO STRUCTURES SIMULATED BY SPH METHOD widely developed in the literature since today's computational powers allow conducting high-performance calculations such as the finite element method (FEM) and finite difference method (FDM). Particularly, the FEM method as one of the typical grid-based methods is the main numerical tool for engineers because of its apparent advantages like (a) Lagrangian grid fixed on the moving material enables it easy to track and obtain the time history of any field variable; (b) irregular or complicated geometries can be modeled by using an irregular mesh; (c) it has been highly developed by academia and industry for a long time, and become a mature and comparatively commercialized approach. It also becomes a powerful predicting tool for the behavior of mechanical structures under HVI [START_REF] Arias | Numerical simulations of impact behaviour of thin steel plates subjected to cylindrical, conical and hemispherical non-deformable projectiles[END_REF][START_REF] Manes | An experimental-numerical investigation on aluminium tubes subjected to ballistic impact with soft core 7.62 ball projectiles[END_REF][START_REF] Žmind Ák | Finite Element Modelling of High Velocity Impact on Plate Structures[END_REF] in the past decades. However, grid-based methods like FEM provide are suffering the mesh distortion problems when simulating hydrodynamic phenomena like explosion and HVIs which typically include large deformations, moving material interfaces, deformable boundaries as well as fragmentations. As an alternative, the mesh-free methods based on particle discretization, also called meshless methods like The Smoothed Particle Hydrodynamics method (SPH) [START_REF] Gingold | Smoothed particle hydrodynamics -Theory and application to non-spherical stars[END_REF][START_REF] Lucy | A numerical approach to the testing of the fission hypothesis[END_REF], Discrete Element Method (DEM) [START_REF] Ad | Numerical simulations of impacts using a discrete element method[END_REF][START_REF] Antonio | The combined finite-discrete element method[END_REF], Dissipative Particle Dynamics (DPD) [START_REF] Ye | A Comparative Review of Smoothed Particle Hydrodynamics, Dissipative Particle Dynamics and Smoothed Dissipative Particle Dynamics[END_REF], Material Point Method (MPM) [START_REF] Ma | Comparison study of MPM and SPH in modeling hypervelocity impact problems[END_REF] etc, are developed. Some reviews on the mesh-free and particle methods and their differences can be found such as [START_REF] Matthias R Öthlin | Meshless methods for large deformation elastodynamics[END_REF][START_REF] Belytschko | Meshless methods: an overview and recent developments[END_REF][START_REF] Li | Meshfree and particle methods and their applications[END_REF], but the SPH method as the earliest mesh-free method is the highlight in this thesis. It has been initially developed in the astrophysical field [START_REF] Gingold | Smoothed particle hydrodynamics -Theory and application to non-spherical stars[END_REF][START_REF] Lucy | A numerical approach to the testing of the fission hypothesis[END_REF]. Since the nineties, some researchers have adapted this method to solid mechanics in the framework of HVIs [START_REF] Randles | Smoothed Particle Hydrodynamics: Some recent improvements and applications[END_REF][START_REF] Libersky | High Strain Lagrangian Hydrodynamics[END_REF]. This method has been widely improved during the 20 last years and appears to be very promising for solid mechanics for the understanding of large deformed mechanics.

Therefore, the objective of this chapter is to review the simulations for high-velocity impacts by the SPH method in the past 30 years. The structure of this chapter is organized by: 1. Firstly, an introduction about standard SPH and its improvements are represented, respectively.

2.

Then, the application of SPH method in high velocity impacts (HVIs) problems is reviewed in the literature. This section focuses on the impacts consisting of the thin structures.

SPH METHOD AND ITS DEVELOPMENTS

3. The limitations and improvements on impact problems by SPH simulations are reviewed based on the literature.

4.

Following, the deep penetrations in various materials are represented, combined with the simulation approaches.

5.

Finally, the deep penetrations in the biomaterials in the micro-scale is determined as the main investigation in this thesis by SPH simulations.

2.2/ SPH METHOD AND ITS DEVELOPMENTS

2.2.1/ THE ALGORITHM OF SPH METHOD

2.2.1.1/ APPROXIMATION OF VARIABLES

Standard SPH also called conventional SPH initially developed by Gingold et.al [START_REF] Gingold | Smoothed particle hydrodynamics -Theory and application to non-spherical stars[END_REF][START_REF] Lucy | A numerical approach to the testing of the fission hypothesis[END_REF] is regarded as a typical meshfree numerical method based on particle modeling, which mainly consists of two processes, kernel approximation, and particle discretization. The first major process in SPH formulation is the kernel approximation, which approximates the field variables. The nearby domain is determined by the kernel function W. The kernel approximation can be expressed by the following equations,

< f ( x) >= Ω f ( x )W( x -x , h)dΩ (2.1)
where, x is a parameter, and x is a variable. The values of kernel function W depend on the distance of xx . Ω is the kernel support centred at a point x. A function should satisfy several conditions like,

1) Normalization W( x -x , h)dx = 1 (2.2)
Obviously, SPH kernel function W is related with the distance between particles and parameter h. Because h determines the size of impact domain, its value directly influences the accuracy and computational quantity in particle approximation.

10CHAPTER 2. A STATE OF THE ART ON IMPACTS TO STRUCTURES SIMULATED BY SPH METHOD

2) Compact supportness: this condition imposes the following equations,

i f | x -x | ≤ κh, W( x -x ) ≥ 0 else W( x -x ) = 0 (2.3)
Where, the combination between smoothing length h and a scale factor κ determines the spread of the specified smoothing function. κ is one of properties of kernel function.

3) Dirac delta function condition when h → 0,

lim h→0 W( x -x , h) = δ( x -x ) (2.4)
Except for above mathematically obligatory conditions, some properties of kernel function are found to have a considerable weight in this algorithm like smoothness, the position of peak of kernel's shape as well as the features of its derivatives. Otherwise, kernel function also should be dimension-independent concerning its practicability for different problems, which means it can be written by the below form ,

W( x -x , h) = α d K(q), q = | x -x |/h, d = 1, 2, 3 (2.5)
Where, α d is the dimension-dependent normalization constant for particular kernel functions.

Based on above conditions, the kernel function in SPH algorithm can be built systematically, as a study by Liu and his co-workers in [START_REF] Vignjevic | Sph in a total lagrangian formalism[END_REF].

Then, particle discretization process convert the kernel integral of Eq. (2.1) into a volume weighted sum, following by:

< f ( x i ) >= N j=1 m j ρ j f ( x j )W( x i -x j , h) (2.6)
In this summation, the subscript i and j denote particle number, m j and ρ j the mass and the density of particle j respectively. N is the total number of the neighbouring particles of particle i in its compact support which is determined by h. This implies that a summation is carried out over the the domain which depends on the types of kernel function. Taking the derivative of Eq. (2.1) to approximate the derivative of f ( x),

< ∇ • f ( x) >= Ω [∇ • f ( x)] W( x -x , h)d x (2.7)
the right hand side of above equation can be written as:

Ω [∇ • f ( x)]W( x -x , h)d x = Ω ∇ • [ f ( x )W( x -x , h)]d x - Ω f ( x ) • ∇W( x -x , h)d x (2.8)
Through the Green-Gauss theorem, the first term on the right hand side can be transferred into the form over its boundary,

Ω ∇ • [ f ( x )W( x -x , h)]d x = Σ f ( x )W( x -x , h) • ndΣ (2.9)
At the boundary Σ with the unit vector normal , the values of W( xx , h) decease to zero according to its characteristic in Eq. (2.3) , then the term in above equation vanishes.

Therefore, Eq. (2.8) become the form like,

Ω [∇ • f ( x)]W( x -x , h)d x = - Ω f ( x ) • ∇W( x -x , h)d x (2.10)
Following the same step with Eq. (2.6) , the above equation can be discretized by a number of particles to obtain its summation,

- Ω f ( x ) • ∇W( x -x , h)d x ≈ N j=1 m j ρ j f ( x j ) ∂W( x i -x j , h) ∂x (2.11)
With the symmetric nature of the smoothing functions, the approximation of the derivative f ( x) at particle i with the position vector x i has the form like,

< ∇ • f ( x) >= N j=1
m j ρ j f ( x j ) ∂W( x ix j , h) ∂x (2.12) 12CHAPTER 2. A STATE OF THE ART ON IMPACTS TO STRUCTURES SIMULATED BY SPH METHOD

2.2.1.3/ TECHNIQUES IN DERIVING SPH FORMULATIONS

To summary and reorganize the SPH formulation for approximating the function and its derivative, according to last section, for a given particle i , its any variable and derivative can be written as the summation as,

< f ( x i ) >= N j=1 m j ρ j f ( x j )W i j (2.13) < ∇ • f ( x i ) >= N j=1 m j ρ j f ( x j ) • ∆ i W i j (2.14) W i j = W( x i -x j , h) = W(| x i -x j |, h) (2.15)
∇ i W i j = x ix j r i j ∂W i j ∂r i j (2.16) Based on [START_REF] Joe | Smoothed particle hydrodynamics[END_REF] which used a technique to derive these SPH formulations, the following two formats are employed to involve the density inside the gradient operation,

∇ • f ( x) = 1 ρ [∆ • (ρ f ( x)) -f ( x) • ∆ρ] (2.17) ∇ • f ( x) = ρ[∆ • ( f ( x) ρ ) + f ( x) ρ 2 • ∆ρ] (2.18)
Linked with above formats, Eq. (2.14) can be expressed as, Firstly, as one of the most common and earliest techniques to reduce the numerical oscillation, artificial viscosity (AV) term is added to momentum and energy conservation equations in SPH formulations [START_REF] Gingold | Smoothed particle hydrodynamics -Theory and application to non-spherical stars[END_REF][START_REF] Joseph | Shock simulation by the particle method sph[END_REF][START_REF] Jj Monaghan | On the problem of penetration in particle methods[END_REF], given by: i j = -qac i j µ i j + qbµ 2 i j ρi j , v i j • x i j < 0 (2.21)

< ∇ • f ( x i ) >= 1 ρ i          N j=1 m j [ f ( x j ) -f ( x i )] • ∆ i W i j          (2.19) < ∇ • f ( x i ) >= ρ i          N j=1 m j [ f ( x j ) ρ 2 j + f ( x i ) ρ 2 i ] • ∆ i W i j          (2.
µ i j = hi j v i j • x i j | x i j | 2 + (κ hi j ) 2 (2.22)
where qa and qb are constants. The notations ci j , ρi j and hi j are the quantities average of particle i and j. When v i j • x i j is positive, µ i j = 0. Although artificial viscosity has been improved in several modified versions, it is still limited to ensure stability during SPH simulations, especially, during the material with large tension.

Artificial stress term was developed to specifically remove tensile instability in elastic dynamics [START_REF] Joseph | Sph without a tensile instability[END_REF][START_REF] James P Gray | Sph elastic dynamics[END_REF]. It is implemented by adding a strong repulsive force only when particles become too close to each other, resulting in a more stable simulation for elastic problems by SPH, but just for 2D problems. The artificial stress term has the following form,

R αβ i j = (R αβ i + R αβ j ) • f n i j (2.23)
Here, in 2-D problems, for example, when α = x and β = x, the term R xx i has the following form

R xx i = c 2 Rxx i + s 2 Ryy i (2.24) Rxx i =              -ζ σxx i ρ 2 σxx i > 0, 0 σxx i < 0.
(2.25)

In above equations, R αβ i is the component of the artificial stress in the original coordinates,

Rαβ

i is new component of the stress tensor in a rotated frame. c denotes cos θ i and s denotes sin θ i ( θ i being the rotation angle which makes the stress tensor diagonal). ζ is a constant that controls the weight of artificial stress. In addition, in Eq. (2.23) , f n i j is related kernel function, presented by W(r i j )/W(∆x) and the optimum value of n is given 4 by Monaghan in [START_REF] James P Gray | Sph elastic dynamics[END_REF]. 14CHAPTER

A STATE OF THE ART ON IMPACTS TO STRUCTURES SIMULATED BY SPH METHOD

The artificial viscosity term Eq. (2.21) and artificial stress terms Eq. (2.23) are added into stress terms in order to improve SPH numerical stability, as following

σ αβ i ρ 2 i + σ αβ j ρ 2 j - i j δ αβ + R αβ i j (2.26)
In addition, XSPH also called conservation smoothing updates the velocity by an average of the particle velocities in every time step [START_REF] Jj Monaghan | On the problem of penetration in particle methods[END_REF], which is suggested to be applied in tension problems simulated by SPH method to remit particles clumping in the study of [START_REF] Larry D Libersky | Recent improvements in sph modeling of hypervelocity impact[END_REF]. It is following the form like,

vi = v i + α cs j 2m j ρ i + ρ j (v j -v i )W i j (2.27)
where vi is the smoothed velocity modified by an average of the particle velocities in the neighbourhood of particle i.

Corrective SPH (CSPH) proposed by Chen also targeted to address the tensile instability [START_REF] Jk Chen | An improvement for tensile instability in smoothed particle hydrodynamics[END_REF]. The idea of this corrective method is to transfer the kernel estimate concept to the Taylor series expansion. Sugiura and Inutsuka [START_REF] Sugiura | An extension of godunov sph: Application to negative pressure media[END_REF] mitigate the tensile instability using the Godunov SPH method [START_REF] Inutsuka | Reformulation of smoothed particle hydrodynamics with riemann solver[END_REF] that utilizes a Riemann solver and achieves the secondorder accuracy in space. They conducted the linear stability analysis for the equations of the Godunov SPH method, and find that the tensile instability can be suppressed by selecting appropriate interpolation for density distribution in the equation of motion for the Godunov SPH method even in the case of elastic dynamics.

Above several improvements are completely based on the formulations of standard SPH where particles interact through a kernel function defined on the current configuration.

According to the study in [START_REF] Rabczuk | Stable particle methods based on lagrangian kernels[END_REF], it means that particles can enter or exit each other's support domains as the material deforms, which is maybe one of the reasons leading to tensile instability. Total lagrangian SPH developed recently calculated the kernel function on the reference configuration, which appears a good performance in removing tensile instability [START_REF] Juan R Reveles | Development of a total lagrangian sph code for the simulation of solids under dynamioc loading[END_REF][START_REF] Üller | An hourglass control algorithm for lagrangian smooth particle hydrodynamics[END_REF]. The doctoral thesis of Reveles [START_REF] Juan R Reveles | Development of a total lagrangian sph code for the simulation of solids under dynamioc loading[END_REF] developed a total Lagrangian SPH code for solid dynamics and it seems at present one of the most promising approaches to keep the numerical stability when applied SPH to HVI problems with quickly extreme deformations. Then, the pseudo-spring technique was developed particularly for modeling 2.3. HVIS ON THIN STRUCTURES 15 crack growth in impact problems [START_REF] Chakraborty | A pseudo-spring based fracture model for sph simulation of impact dynamics[END_REF][START_REF] Rushdie | A computational model for failure of ductile material under impact[END_REF][START_REF] Rushdie | Pseudo-spring sph simulations on the perforation of metal targets with different damage models[END_REF], combined with SPH formulation (so-called Pseudo-spring SPH). An efficient immediate neighbor interaction was formulated by connecting neighbors through pseudo-springs in this technique. The damage of material is evaluated by the behavior of pseudo-springs, which seems a very interesting approach to investigate the crake propagation during impacts. There are also other forms of SPH improvements such as MLSPH [START_REF] Gary | Moving-least-squares-particle hydrodynamics-i. consistency and stability[END_REF][START_REF] Gary | Moving least-squares particle hydrodynamics ii: conservation and boundaries[END_REF], Godunov SPH [START_REF] Inutsuka | Reformulation of smoothed particle hydrodynamics with riemann solver[END_REF][START_REF] Sugiura | An extension of godunov sph: Application to negative pressure media[END_REF], δ-SPH [START_REF] Molteni | A simple procedure to improve the pressure evaluation in hydrodynamic context using the sph[END_REF], and Gamma-SPH [START_REF] Limido | Modeling of hypervelocity impact experiments using gamma-sph technique[END_REF] The response of structures under high-velocity impact loading is complicated and uncertain, which depends on the material properties and impact conditions including projectile shapes, deformable or rigid projectile, impact velocity, target configuration particularly the The types of damages in HVI from [START_REF] Marvin | The mechanics of penetration of projectiles into targets[END_REF].

At the same time, the shapes of projectiles also determiner the mechanical damages during the HVIs, for example, the research in [START_REF] Rushdie | Pseudo-spring sph simulations on the perforation of metal targets with different damage models[END_REF] which studied the influences on the [START_REF] Børvik | Ballistic penetration of steel plates[END_REF]; b) shows the petaling and ductile hole enlargement after perforation in [START_REF] Manes | An experimental-numerical investigation on aluminium tubes subjected to ballistic impact with soft core 7.62 ball projectiles[END_REF]; c) shows the perforation accompanying with radial fracture in [START_REF] Michel | Hypervelocity impacts on thin brittle targets: Experimental data and SPH simulations[END_REF]; d) shows the debris could during the impact in [START_REF] Andrew | Characteristics of debris clouds produced by hypervelocity impact of aluminum spheres with thin aluminum plates[END_REF].

More importantly, the damage types considerably rely on the target properties like ductile plates, brittle plates, soft structures as well as advanced composite plates [START_REF] Chen | High-velocity impact of graphite/epoxy composite laminates[END_REF][START_REF] Shintate | Numerical simulation of hypervelocity impacts of a projectile on laminated composite plate targets by means of improved sph method[END_REF][START_REF] Yashiro | Characterization of high-velocity impact damage in CFRP laminates: Part II -prediction by smoothed particle hydrodynamics[END_REF][START_REF] Nikolas A Nordendale | A comparison of finite element analysis to smooth particle hydrodynamics for application to projectile impact on cementitious material[END_REF][START_REF] Monzer | Sph-based method to simulate penetrating impact mechanics into ballistic gelatin: Toward an understanding of the perforation of human tissue[END_REF][START_REF] Frissane | Sph modeling of high velocity impact into ballistic gelatin. development of an axis-symmetrical formulation[END_REF]. Particularly, for the brittle thin plates like the study [START_REF] Michel | Hypervelocity impacts on thin brittle targets: Experimental data and SPH simulations[END_REF], the ejecta and debris environment was emphasized to be investigated during HVIs. Ceramic as a typical brittle material has been widely used as a target material in the various engineering environment. The first application dedicated to Hight Velocity Impacts on structures by the use of the SPH method seems to appear in the early 90s from one side in [START_REF] Stellingwerf | Smooth particle hydrodynamics[END_REF] and from the other side in [START_REF] Cloutman | An evaluation of smoothed particle hydrodynamics[END_REF]. A few years earlier, Johnson et al. [START_REF] Johnson | Dynamic Lagrangian computations for solids, with variable nodal connectivity for severe distortions[END_REF] had introduced a particle method named NARBOR nodes to model violent impacts. This grid-less method uses the idea of neighbor particle interactions to define simplified strain and stress fields in a Lagrangian frame. This procedure based on free-nodes allows variable nodes connectivities and thus to deal with severe distortions. This first approach had given the inspiration that particle methods can offer interesting tools to investigate such complex situations.

In 1990, at the Next Free-Lagrange Conference, both Stellingwerf [START_REF] Stellingwerf | Smooth particle hydrodynamics[END_REF] and Cloutman [START_REF] Cloutman | An evaluation of smoothed particle hydrodynamics[END_REF] had exhibited a very similar work. Cloutman [START_REF] Cloutman | An evaluation of smoothed particle hydrodynamics[END_REF] introduced a very common standard formulation based on the work in [START_REF] Joseph | Shock simulation by the particle method sph[END_REF][START_REF] Joseph | An introduction to sph[END_REF], which discretized the Euler's equations associated with a Tillotson EOS to model shocks, expansion, and change of phase of metals, such as Iron in this case. As firstly introduced in [START_REF] Joseph | Shock simulation by the particle method sph[END_REF], in the presence of shock waves to prevent inter-penetration and spurious oscillations, an artificial viscosity had been introduced in the momentum and energy equations. The numerical test introduced in this study corresponded to the case of an Iron disk of 0.6 cm thickness and a 2.1 cm radius impacted at 6 km/s by an Iron sphere. The radius of the projectile was not given but seemed to be very close to the target thickness of 0.6 cm. The two bodies were discretized with 1984 particles for the target and 192 particles for the projectiles, distributed randomly. This work did not show any strong validation, although it compared, without showing any support of this, the SPH result with a Finite Difference result. Only some shape differences and dimensions were displayed, which suggested that the SPH procedure could offer a "realistic" solution for HVI problems at that age.

Subsequently, the authors in [START_REF] Cloutman | An evaluation of smoothed particle hydrodynamics[END_REF] provided the encouraging results by the SPH method in applied impacts. And more worthy, this study pointed out the important limitations such as: running time where he suggested the use of parallel computation; reducing shocks smoothing and inter-penetration; the implementation of physical phenomena such as surface tension, viscosity, and material strength; treatment of boundary conditions;

optimal kernel and h-value because he used a Gaussian kernel. On the other side, the authors in [START_REF] Stellingwerf | Smooth particle hydrodynamics[END_REF] presented the code-named SPHC which seemed to be very similar to the one proposed in [START_REF] Cloutman | An evaluation of smoothed particle hydrodynamics[END_REF] at the difference that the SPHC code used two-dimensional

Cartesian modeling and employed a Mie-Gruneisen EOS. The validation test consisted of an aluminum cylinder impacting an Aluminum plate at a speed of 7 km/s. The dimensions of the projectile and the plate were not given but should be in the same order as the previous test. Very few information was given except for the fact that the whole model was discretized with 500 particles. Again, no validation was presented, only the shape of the deformed target was discussed. These two works are the very beginning of HVI SPH modeling and even if no true validation is presented, both could conclude that the SPH method is very appealing for its simplicity and its capability to model important material distortions.

In the SPH models provided by both Cloutman [START_REF] Cloutman | An evaluation of smoothed particle hydrodynamics[END_REF] and Stellingwerf [START_REF] Stellingwerf | Smooth particle hydrodynamics[END_REF] at that time, they used the same standard formulation by considering only the Euler's equations with symmetric SPH discretization including an artificial viscosity. Particularly, at that age, the density evolution was not governed by the continuity equation but instead by the density summation:

ρ h = j W h i j m j . ( 2

2.3.2.2/ DEVELOPMENTS OF SPH CODES IN HVIS

As one of the pioneers in developing the SPH modeling impact problems, Petschek et al. [START_REF] Petschek | Cylindrical Smoothed Particle Hydrodynamics[END_REF] developed a 3D standard SPH code involving the artificial viscosity by Monaghan [START_REF] Jj Monaghan | On the problem of penetration in particle methods[END_REF] and simulated a 3-g copper disk (11.18 mm diameter x 3.45 mm thick) impact on a 2.87 mm thick aluminum bumper plate at 5.55 km/s. Although a predetermined fixed value was given to describe the plastic flow regime during this high impact process, an encouraging result obtained by this code was that the maximum width and length of the debris clouds after impact were in agreement with experiments. Also, Stellingwerf is one of the pioneers in this field. At the beginning of the use of SPH method, Stellingwerf and his co-workers made a review of this computational technique in [START_REF] Stellingwerf | Impact modeling with smooth particle hydrodynamics[END_REF], and illustrated the great potential of this SPH method by the number of different simulations: a spherical aluminum projectile with radius 0.475 cm impacting a thin aluminum sheet with thickness 0.0381 cm at a velocity of 6.75 km/s, and including a series of impact simulations with disk projectiles radius of 0.95 cm impacting a 0.127 thick target at 10 km/s. The same group of researchers studied the variations in crater volume caused in the copper sphere impacts copper target with 4.0/3.0 cm thick at initial velocities from 6.0 km/s to 24.5 km/s using four numerical codes including SPH, MESA, EPIC and CALE codes in [START_REF] Ca Wingate | Models of high velocity impact phenomena[END_REF]. The investigation on the effect of a change in the yield strength in the material model was done and good agreement was found between these codes. In 1993 Except for EPIC software, SPH method was also incorporated into the other public numerical codes like AUTODYN-2D hydrocode and PAM-SHOCK 3D computer code. Some researchers employed these codes to study the impacting process on the thin structures [START_REF] Hayhurst | Cylindrically symmetric sph simulations of hypervelocity impacts on thin plates[END_REF][START_REF] Faraud | SPH simulations of debris impacts using two different computer codes[END_REF]. For example, in 1999, Faraud et al. investigated the debris cloud of the projectile impact on an all-aluminum triple wall system by the SPH method in [START_REF] Faraud | SPH simulations of debris impacts using two different computer codes[END_REF]. The authors mainly compared the performances of two SPH formulations which had been incorporated into the PAM-SHOCK 3D computer code and the AUTODYN 2D hydrocode. Different constitutive laws were tested (Johnson-cook, Steinberg-Guinan), as well as different equations of states (Tillotson, Shock, Sesame). This study showed that both the SPH codes achieved a good representation of the debris cloud expansion before the impact on the 2nd BS, but the simulations of the impact on the 2nd BS and BW could not agree with the experimental observations. Both the PAM-SHOCK 3D computer code and the AUTODYN 2D hydrocode existed severe problems in energy and momentum conservation. Therefore, this study showed that at that age, the SPH technique was not sufficiently mature to support the development of complex debris shielding systems. At same time, Hiermaier et al. [START_REF] Hiermaier | Computational simulation of the hypervelocity impact of al-spheres on thin plates of different materials[END_REF] and Groenenboom et al [START_REF] Paul | Numerical simulation of 2D and 3D hypervelocity impact using the SPH option in PAM-SHOCK™[END_REF] also developed SPH codes and applied it to simulate HVI problem on thin structures. The simulations concluded a 0.5 cm diameters Al projectiles impact 0.15 to 0.4 cm thick plates at 5.75 to 6.85 km/s in [START_REF] Hiermaier | Computational simulation of the hypervelocity impact of al-spheres on thin plates of different materials[END_REF].

Groenenboom et al. [START_REF] Paul | Numerical simulation of 2D and 3D hypervelocity impact using the SPH option in PAM-SHOCK™[END_REF] conducted HVI both in 3D and 2D plane strain, investigating the role of a constant or variable smoothing length in SPH kernels, as well as the type of equation of state.

Actually, the SPH method obtained less interest and attention in the application of solid dynamics until the 21st century. The incorporation of SPH formulations to LS-DYNA commercial software provides much convenience to simulate HVI problems using this approach. For example, Michel et al. [START_REF] Michel | Hypervelocity impacts on thin brittle targets: Experimental data and SPH simulations[END_REF] HB armor steel against 7.62 mm 54R B32 API hardened steel core ammunition [START_REF] Kılıc | Ballistic resistance of high hardness armor steels against 7.62 mm armor piercing ammunition[END_REF].

Johnson-Cook constitutive relations for both strength and failure models were developed in these tests. It is worthy to notice that, as concluded by authors in this study, the use of SPH formulation did not predict spall and fragmentation behavior more accurately as expected, even involving a finer particle size in the SPH discretization. What is worse, even a hybrid SPH modeling used to decrease numerical cost, still, SPH is too costly when compared with the Lagrangian formulation.

A series of experiments about cylindrical Lexan projectiles impact on steel target plates with a velocity range of 4.5-6.0 km/s were performed in the study of [START_REF] Brendan | Modeling Plastic Deformation of Steel Plates in Hypervelocity Impact Experiments[END_REF]. The authors also develped numerical models to these impact by SPH in LS-DYNA Pseudo-spring SPH was developed initially by Amit and chakraborty et al. in [START_REF] Chakraborty | A pseudo-spring based fracture model for sph simulation of impact dynamics[END_REF][START_REF] Chakraborty | Crack propagation in bi-material system via pseudo-spring smoothed particle hydrodynamics[END_REF].

The main concept of the pseudo spring strategy can be represented by Fig. 2.5, in which SPH interaction is restricted only to the nearest neighboring particles through connecting pseudo-springs. This network of springs is supposed between all the interacting particles, and they do not provide any extra stiffness to the system. The function of these springs is to collect all the information about damage state of the material and hence a damage evolution law defines degradation of the inter-neighbor spring forces. The interactions between particles are dominated by a parameter f , defined as f i j = 1-D i j , where D i j is the accumulated damage in the pseudo-spring between the ij particles pair. D i j is defined as D i j = 0.5(D i + D j ), where D i and D j are the damage parameters at the ith and jth particles respectively. Therefore, by evaluating the interaction factor f i j when it reaches the critical value, this spring breaks. The breaking of spring denotes the formation of a crack and the corresponding disconnected particles represent the material body on either side of this newly formed crack surfaces. Therefore, this pseudo spring technique has a fundamental strength in modeling crack propagation. The same group of researchers furtherly applied this model to study crack propagation in the bi-material interface in [START_REF] Chakraborty | Crack propagation in bi-material system via pseudo-spring smoothed particle hydrodynamics[END_REF],

and to study ballistic sensitivity of pre-notch in the metallic beam in [START_REF] Chakraborty | Prognosis for ballistic sensitivity of pre-notch in metallic beam through mesh-less computation reflecting material damage[END_REF].

In 2017, Islam et al. used this pseudo spring strategy combined with SPH to study the failure of ductile material under impact loading in [START_REF] Rushdie | A computational model for failure of ductile material under impact[END_REF]. A series of impacts of a cylindrical projectile on plates with 8 mm and 10 mm were simulated in this study, the material both of projectiles and targets being Weldox 460 E steel. As well known, ductile material failure is much more complex including large plastic deformation, heat generation due to plastic work, material instability due to localization of plastic deformation, and the growth of microvoids leading to damage. The attempt of this study implied that this SPH based on the pseudo spring method is a very promising approach to model the complex damages in impacts. In the same year, Islam et.al. employed the pseudo-spring SPH to investigate the damage in ceramic and ceramic-metal composite structures under high impacts in [START_REF] Chakraborty | A computational framework for modelling impact induced damage in ceramic and ceramic-metal composite structures[END_REF].

The ceramics are typically brittle materials that involve abrupt failure and cracks, which is easier to be captured and modeled by use of the pseudo spring concept. Afterward, A Total Lagrangian SPH method was improved combined with Pseudo-spring contact in [START_REF] Rushdie | A total lagrangian sph method for modelling damage and failure in solids[END_REF] (KGC)) to model the impacts of an aluminum sphere penetrating an aluminum plate in two-and three-dimensional spaces [START_REF] Zhang | Smoothed particle hydrodynamics with kernel gradient correction for modeling high velocity impact in two-and three-dimensional spaces[END_REF]. The KGC technique can mathematically improve the accuracy of the gradient of the kernel functions, which has been implemented to the SPH schemes both of 2D and 3D but only modeling the incompressible fluid flows.

The authors firstly extended it to investigate the HVI process. In the KGC technique, a modified or corrected kernel gradient is obtained by multiplying the original kernel gradient with a local reversible matrix, and the detailed formulations can be found in [START_REF] Shao | An improved sph method for modeling liquid sloshing dynamics[END_REF][START_REF] Zhang | Smoothed particle hydrodynamics with kernel gradient correction for modeling high velocity impact in two-and three-dimensional spaces[END_REF]. The Johnson-Cook model and the Mie-Gruneisen eos were applied in the simulations. The simulations for 2D and 3D HVIs of 1.0 cm diameter aluminum cylinder onto a 0.4 cm thick aluminum plate at 6180 m/s and 4119 m/s were carried out. All the simulation results illustrated that the SPH method with KGC can be effective in modeling HVI problems and can produce more accurate results than the conventional SPH methods without kernel gradient correction. The followed work about the application of the SPH-KGC coupled method was provided in [START_REF] Zhang | A new formula for predicting the crater size of a target plate produced by hypervelocity impact[END_REF]. They mainly investigated the sizes of the craters produced by HVI at different initial impact velocities and the variation of the crater size over the impact velocity. A series of 3D HVIs in which the aluminum sphere impacts the aluminum plate with a thickness of 0.4 cm at different velocities from 2 to 11 km/s (with an interval of 0.5 km/s) were performed and observed in this study. The two-stage phenomenon (the varying stage and steady stage during the increase of the crater size versus impact velocity) was accurately predicted by this new formula.

Limido et al. [START_REF] Limido | Modeling of hypervelocity impact experiments using gamma-sph technique[END_REF] proposed an improved SPH called Gamma-SPH to reduce standard SPH drawbacks like lack of interpolation completeness, tensile instability, and the existence of spurious pressure, as well as high computational time. The main concept of this improvement is based on a centered formulation for the pressure gradient term in

HVIS ON THIN STRUCTURES

the momentum equation and on the introduction of a stabilization term proportional to the pressure difference between two neighbor particles. The stability of this new formulation is ensured and theoretically proved under a CFL-like condition. The simulations of 9.53mm diameter aluminum sphere impact aluminum plates with 0.8 mm and 4.039 mm thick at 6.7 km/s velocity were conducted. Oblique impact consisting of a 3 mm aluminum sphere against to 2mm thick aluminum plate at 32°tilted and 4059 m/s velocity was also simulated by this Gamma-SPH. The debris clouds for both normal and oblique impacts observed in simulation and experimental results were in good agreement.

An alternative scheme called γ-SPH-ALE was deveoped to model the HVI cases and investigate the debris clouds in [START_REF] Coll | An accurate sph scheme for dynamic fragmentation modelling[END_REF]. Indeed, γ-SPH-ALE is the combination of SPH-ALE formulation [START_REF] Vila | On particle weighted methods and smooth particle hydrodynamics[END_REF] and FV low-Mach scheme [START_REF] Lavalle | A numerical reduced model for thin liquid films sheared by a gas flow[END_REF]. The simulations of aluminum spheres impact aluminum plates with different thick at 6.7 km/s were performed by this SPH scheme.

An oblique impact with a 3 mm-diameter sphere to the plate with 2 mm thick at velocity 4050 m/s was also deducted by γ-SPH-AL method. An elastic perfectly plastic material model combined with the Mie-Gr üneisen equation of state was applied to the material of both projectiles and targets. The simulation results revealed that the proposed scheme increases both stability and accuracy and reduces the computation time compared to classical solvers.

Zhang et al. also investigated the damaging effect of target plates induced by highvelocity impact using a decoupled finite particle method (DFPM) in [START_REF] Zhang | Predicting the damage on a target plate produced by hypervelocity impact using a decoupled finite particle method[END_REF]. DFPM was a recently developed corrective SPH method by Liu and his co-workers in [START_REF] Liu | Restoring particle consistency in smoothed particle hydrodynamics[END_REF][START_REF] Zhang | A finite particle method with particle shifting technique for modeling particulate flows with thermal convection[END_REF]. DFPM does not need to solve redundant and challenging pointwise corrective matrix equations when approximating a field variable and its derivatives. The authors made a comparison on the formulations of different SPH improvements like FPM [START_REF] Zhang | A finite particle method with particle shifting technique for modeling particulate flows with thermal convection[END_REF], CSPM [START_REF] Jk Chen | A generalized smoothed particle hydrodynamics method for nonlinear dynamic problems[END_REF] and KGC [START_REF] Zhang | Smoothed particle hydrodynamics with kernel gradient correction for modeling high velocity impact in two-and three-dimensional spaces[END_REF], and the point can be concluded that DFPM indeed has its advantages on mathematical accuracy and implementations. Different from these formulations of SPH versions like [START_REF] Chen | High-velocity impact of graphite/epoxy composite laminates[END_REF]. In recent years, some studies [START_REF] Yashiro | Characterization of high-velocity impact damage in CFRP laminates: Part II -prediction by smoothed particle hydrodynamics[END_REF][START_REF] Nikolas A Nordendale | A comparison of finite element analysis to smooth particle hydrodynamics for application to projectile impact on cementitious material[END_REF] also employed a similar conception to model the laminated structures as the target in impacts. Except for developing the SPH model for laminated structures, the main contribution in [START_REF] Shintate | Numerical simulation of hypervelocity impacts of a projectile on laminated composite plate targets by means of improved sph method[END_REF] is to develop an improved SPH method with new particle generation and particle merger techniques. The new particle generation technique means generating new particles if the area with no particles exceeds a certain value or the distance between the adjacent particles exceeds a certain value, following two rules. Because the new particle generation technique to prevent the numerical fractures causes the overabundance of particles. Another particle merger technique was presented in simulation in which two close particles of a triangular element are merged into one particle if the distance between the two particles attains a certain value. Fig. 2.9 displays a comparison with an HVI example between the results by standard SPH and this improved SPH. This improved SPH was proved not only a promising approach to decrease the numerical fracture during HVI problems involving the deformation with a large tensile strain by a conventional SPH, but also a robust and effective way to model the laminated plates.

Carbon fiber reinforced plastics (CFRPs) have good specific strength, specific modulus, and fatigue properties compared with conventional metals, so they have been applied to primary load-bearing structures such as wings and a fuselage in the latest airplanes. Yashiro et.al investigated mechanisms of the extension of high-velocity impact damage in CFRP laminates by experiments in [START_REF] Yashiro | Characterization of high-velocity impact damage in cfrp laminates: Part i-experiment[END_REF] and numerical simulations with the SPH method in [START_REF] Yashiro | Characterization of high-velocity impact damage in CFRP laminates: Part II -prediction by smoothed particle hydrodynamics[END_REF]. Fig. (Johnson and Holmquist (1994)) was used to describe material behaviors of Al2O3 ceramic, which includes pressure, strain rate, damage, and bulking effects. The simulation results showed that the JH-2 model is suitable and attractive for modeling the high-velocity impact of ceramic-type materials. In this study, two methods including a penalty contact method (PCM) and particle approximate method (PAM) were employed to handle interactions between the projectile and the target. The simulation results illustrated that PCM and PAM induced insignificant differences in modeling the high-velocity impact of the deformable projectile on the ceramic tile. The sensitivity of SPH numerical parameters like artificial viscosity, smoothing length, and particle spacing on simulation results was also investigated.

Another new study in [START_REF] Scazzosi | Fe coupled to sph numerical model for the simulation of high-velocity impact on ceramic based ballistic shields[END_REF] is to develop an FE-SPH coupled model to investigate the responses of HVIs with the Al 2 o 3 tiles target and multi-layered targets with Al2O3 front layer and AA6061-T6 backing layer. In this model, the FE belonging to the ceramic part was converted into SPH particles when the element conversion criterion was met.

Two projectiles were used: 7.62x51 P80 and 12.7x99 AP in the simulations. This study concluded that the FE model can not obtain accurately simulating the transmission of the projectile decelerating force to the rear plate with a consequently incorrect prediction of the residual velocity and damage morphology in multi-layered targets. In contrast, the SPH model combined with FEM elements is more suited for the simulation of the response of ceramic materials. Fig. 2.10 is the numerical result of impact multi-layered target, from which it can be seen that the crack is the principal damage for brittle ceramic plate. Indeed, as the above description of the new types of SPH schemes, 'pseudospring' SPH seems the most promising approach to simulate the ceramic damages under impact conditions because of its natural property to capture the crack transmission by the behavior of pseudo-spring [START_REF] Rushdie | Ballistic performance of ceramic and ceramic-metal composite plates with jh1, jh2 and jhb material models[END_REF].

2.4/ LIMITATIONS AND IMPROVEMENTS

Some advantages of the SPH method applied to simulate the dynamic solids have been well recognized, and from the literature review in the last section, it can be seen that the SPH method also has become one of the most important approaches to investigate the responses of high-velocity impacts. However, the pioneers warned about numerical weaknesses in the use of this mathematical formulation for solid mechanics. The shortcomings of SPH algorithm like estimated accuracy of neighbors, interfaces for large density discontinuity, high time-consuming, and the particles instabilities become more visible in modeling large solid structures. Particularly, the instabilities due to tensile behavior are inevitable in simulating solid dynamic problems like HVIs.

In the earlier 1990s, some techniques including conservation smoothing in [START_REF] Wen | Stabilizing sph with conservative smoothing[END_REF] in 1994, implementation of kernel renormalization in [START_REF] Gordon | Normalized smoothing functions for sph impact computations[END_REF][START_REF] Gordon R Johnson | Sph for high velocity impact computations[END_REF] in 1996, more accurate computation on the boundary in [START_REF] Campbell | Some new algorithms for boundary value problems in smooth particle hydrodynamics[END_REF][START_REF] Takeda | Numerical simulation of viscous flow by smoothed particle hydrodynamics[END_REF] had been developed to improve the SPH performances. The details are provided by the earlier review articles in [START_REF] Randles | Smoothed Particle Hydrodynamics: Some recent improvements and applications[END_REF][START_REF] Libersky | Recent improvements in SPH modeling of hypervelocity impact[END_REF]. With the motivation of simulating HVI problems that involve more complicated and uncertain physical phenomenon, some improvements for SPH formulations are still under research combined with the mechanical properties in HVIs. Besides, the fracture models merged into SPH formulas also should be developed to improve the realism of complete fragmentation during HVI, as claimed in [START_REF] Libersky | Recent improvements in SPH modeling of hypervelocity impact[END_REF]. This section would show some limitations and improvement of SPH 34CHAPTER 2. A STATE OF THE ART ON IMPACTS TO STRUCTURES SIMULATED BY SPH METHOD method applied in simulating HVI problems.

2.4.1/ ACCURACY, INSTABILITY

Firstly, numerical accuracy is the primary consideration that relays on numerical stability to some degree, but numerical instability is still the inevitable problem for most numerical algorithms. In terms of SPH simulating HVI problems, the particles often suffer the severe clumping phenomenon when the materials are under extremely large deformations. Some analyses for the stability of the SPH method has been done by Belytschko and his co-workers based on the features of the particle method with Eulerian and Lagrangian kernels in [START_REF] Belytschko | Stability analysis of particle methods with corrected derivatives[END_REF][START_REF] Rabczuk | Stable particle methods based on lagrangian kernels[END_REF][START_REF] Belytschko | A unified stability analysis of meshless particle methods[END_REF]. Several types of instabilities including tensile instability were identified, and they claimed that tensile instability is a severe limitation to the SPH application in solid dynamics.

2.4.1.1/ ARTIFICIAL VISCOSITY

Artificial viscosity is the most common technique to mitigate the numerical oscillations, also to reduce tensile instability for some problems. The earliest standard SPH also involves an artificial viscosity proposed by Monaghan et al. [START_REF] Joseph | Shock simulation by the particle method sph[END_REF][START_REF] Jj Monaghan | On the problem of penetration in particle methods[END_REF]. Nevertheless, the performances of this artificial viscosity depend on the choice of parameters (α, β), and it easily makes the system over-dissipative if unsuitable parameters are used. Shaw et al. [START_REF] Shaw | Heuristic acceleration correction algorithm for use in sph computations in impact mechanics[END_REF] discussed this point in detail and introduced a modified method to avoid choosing unsuitable parameters. There are also some studies focusing on the effects of artificial viscosity on the simulation results for impact problems. The smoothing length (h) of kernel function and distance (d) between particles are also important parameters that can significantly affect the stability of simulation results. The study by Mehra et al. in [START_REF] Mehra | High velocity impact of metal sphere on thin metallic plates: A comparative smooth particle hydrodynamics study[END_REF] considered this point and also investigated the influence of four SPH schemes including three types of artificial viscosity (MAV, BAL, and MON in Tab.2.2) and a contact SPH scheme (CON) for the simulations of impact problems.

They conducted a series of simulations on HVIs with variable values of the ratio h/d. The diameters of craters and shapes of the debris clouds during the impacts achieved in the simulations were observed and compared with experiments. The authors discovered that both BAL and MON viscosities had a good performance for hypervelocity impacts, but the MAV viscosity and CON scheme can not obtain the correct debris cloud shape as measured even adjusted a wide range of h/d. However, it is worthy to notice that the CON algorithm (also called the Godunov-type scheme) with its attractive quality of being free from arbitrary parameters, performed well at moderate impact velocities, not suffering from the numerical fracture. So, there is still no existing artificial viscosity term which is capable to keep generalized stability for SPH simulations, especially for the HVI problems with complex physical responses. When particles are under tensile stress state or sufficiently close to one another, the motion of the particles becomes unstable, which may lead to particle clumpings representing as large tension exhibit 'numerical fracture' or artificial void in structure. This situation becomes a more significant concern in impact and penetration problems since the elastic dynamics involve considerable attractive forces. In 2012, Liu and his co-workers in [START_REF] Liu | Perforation of steel and aluminum targets using a modified johnson-cook material model[END_REF] used the coupled SPH-FEM (SFM) to simulate high velocity perforation of steel and aluminum alloy plates of different thicknesses impacted by steel projectiles with various geometries, such as blunt, conical and ogival noses. The main contribution in this study is that a modified Johnson-Cook (MJC) model focusing on strain rate and adiabatic heating effects was adopted for metals, and material properties for Weldox 460E steel and AA5083-H116 aluminum plates were determined in the simulations. The authors claimed that for thin plates, due to the inherent tensile instability problem associated with the SFM approach, the agreement between numerial and experimental results was not obtained.

Therefore, as introduced in section 2.2, most of the attention to improve the SPH application in sold dynamics is paid on the tensile instability problem.

Artificial stress also called Monaghan stress was the earliest technique to be particularized for solving tensile problems in solid dynamics in [START_REF] Joseph | Sph without a tensile instability[END_REF][START_REF] James P Gray | Sph elastic dynamics[END_REF]. A strong repulsive force is added to momentum and energy conservation equations, only when particles become too close to each other. It has been validated in impact problems by Monaghan's research, but only performed for low-velocity impact problems. According to the study by Mehra et al. [START_REF] Vishal Mehra | Tensile Instability and Artificial Stresses in Impact Problems in SPH[END_REF], artificial stress could reduce the void size caused by particle tensile instability but unable to completely remove it when the impact velocity is high. Mehra et al. deducted simulations consisting of a steel sphere impact a 2 mm thick Aluminium plate at 3.1 km/s by SPH code combined with artificial stress and Godunov-type scheme (CON)

in [START_REF] Vishal Mehra | Tensile Instability and Artificial Stresses in Impact Problems in SPH[END_REF]. The numerical results showed that the Godunov-type scheme (CON) performed better than artificial stress in terms of mitigating tensile instability. But this observation is only by one impact. As the authors claimed, more studies with different impact scenarios should be considered to identify the presence or absence of tensile instability in the Godunov-type SPH. In fact, Godunov SPH method was developed to remove the tensile instability initially applied in fluid dynamics, which uses a Riemann solver and achieves the second-order accuracy in space in [START_REF] Inutsuka | Reformulation of smoothed particle hydrodynamics with riemann solver[END_REF][START_REF] Sugiura | An extension of godunov sph: Application to negative pressure media[END_REF]. In 2017, Sugiura et al. [START_REF] Sugiura | An extension of godunov sph ii: Application to elastic dynamics[END_REF] extended the Godunov SPH method to elastic dynamics by incorporating deviatoric stress tensor that represents the stress for shear deformation or anisotropic compression. The authors validated this algorithm in several classic examples such as the collision of rubber rings, oscillation of plate, as well as the impact problems. But, the impact example was still at moderate impact velocities ( 3.1 km/s), and its performance is still unknown at highvelocity impacts.

There are also some new SPH schemes developed recently involving removing the tensible instability such as Total Lagrangian SPH [START_REF] Juan R Reveles | Development of a total lagrangian sph code for the simulation of solids under dynamioc loading[END_REF][START_REF] Üller | An hourglass control algorithm for lagrangian smooth particle hydrodynamics[END_REF], decoupled finite particle method (DFPM) [START_REF] Liu | Restoring particle consistency in smoothed particle hydrodynamics[END_REF][START_REF] Zhang | A finite particle method with particle shifting technique for modeling particulate flows with thermal convection[END_REF], and Gamma-SPH [START_REF] Limido | Modeling of hypervelocity impact experiments using gamma-sph technique[END_REF], etc, but no adequate examples applied in HVI problems, Tab.2.3 showing these several approaches corresponded to impacts.

Table 2.3: The techniques incorporated to SPH formulations to mitigate tensile instability.

Ref

Name Description [START_REF] Joseph | Sph without a tensile instability[END_REF][START_REF] James P Gray | Sph elastic dynamics[END_REF] Artificial stress Capable for 2D and low-velocity impact in [23] [111, 112, 28] Godunov-type scheme (CON)

Capable for moderate/high-velocity impact (up to 3.1 km/s) in [107, 109, 110] [30, 31] Total Lagrangian SPH No much application in HVIs [START_REF] Liu | Restoring particle consistency in smoothed particle hydrodynamics[END_REF][START_REF] Zhang | A finite particle method with particle shifting technique for modeling particulate flows with thermal convection[END_REF] (DFPM) Capable for High-velocity impact (up to 6 km/s) in [START_REF] Zhang | Predicting the damage on a target plate produced by hypervelocity impact using a decoupled finite particle method[END_REF] 2.4.2/ TIME-CONSUMING, COUPLED SPH AND CONTACT ALGORITHMS Essentially, in SPH formulas, the approximation process is determined based on a contemporary domain of arbitrarily distributed particles, so neighbor searching is one of the most important steps in SPH to seek the nearest neighbor particles of a particle to define nodal connectivity. This directly leads to the SPH process time-consuming, especially for larger-scale and 3D problems requiring tremendous particles. To address this problem, two ways have been considered in the literature. One is employing the CPU/GPU computer algorithms, and the other is by coupling the FEM element model. A few studies like [START_REF] Crespo | Gpus, a new tool of acceleration in cfd: efficiency and reliability on smoothed particle hydrodynamics methods[END_REF][START_REF] Alexis H Érault | Sph on gpu with cuda[END_REF][START_REF] Frissane | 3d smooth particle hydrodynamics modeling for high velocity penetrating impact using gpu: Application to a blunt projectile penetrating thin steel plates[END_REF] have achieved the SPH formulation incorporated with GPU algorithms.

Particularly, one work by Frissane et al. in [START_REF] Frissane | 3d smooth particle hydrodynamics modeling for high velocity penetrating impact using gpu: Application to a blunt projectile penetrating thin steel plates[END_REF] deserving to be mentioned is that a 3D 38CHAPTER 2. A STATE OF THE ART ON IMPACTS TO STRUCTURES SIMULATED BY SPH METHOD SPH code linked with the GPU algorithm was developed to model the process of highvelocity impacts of thin metal plates struck by a blunt projectile. The simulation results

showed that this 3D SPH GPU numerical code was able to accurately predict the response of HVI problems, more importantly, costing a restrained amount on calculating memory and time.

The SPH has a strong ability to solve dynamic problems with large deformations without mesh distortions while it is not as good as the FEM in terms of computational time and boundary conditions. Besides, FEM has been developed for many years and has been a very mature technique involved in different commercial software. With this concern, the coupled SPH-FEM method can effectively use the strengths of these two methods for the simulations of large deformation problems. Some studies focusing on the coupling algorithms also have been developed in the past years [START_REF] Parshikov | Improvements in SPH method by means of interparticle contact algorithm and analysis of perforation tests at moderate projectile velocities[END_REF][START_REF] Seo | Application of an improved contact algorithm for penetration analysis in SPH[END_REF][START_REF] Parshikov | Smoothed Particle Hydrodynamics Using Interparticle Contact Algorithms[END_REF][START_REF] Sw Attaway | Coupling of smooth particle hydrodynamics with the finite element method[END_REF][START_REF] Sauer | Adaptive Kopplung des netzfreien SPH-Verfahrens mit finiten Elementen zur Berechnung von Impaktvorgaengen[END_REF][START_REF] Campbell | A contact algorithm for smoothed particle hydrodynamics[END_REF].

The coupled ways between SPH and FEM methods are mainly identified into two kinds:

one way is converting techniques from FE elements to the SPH particles depending on the physical states of materials; the other is modeling different domains of the structure with SPH and FE respectively based on the interface contact algorithms. Fig. 2.11 shows these interactions between SPH particles and FEM elements. The left one is from the study of [START_REF] Gr Johnson | A combined particle-element method for high-velocity impact computations[END_REF], which displays that all the structure is modeled by FEM elements at the beginning and then the elements convert to SPH particles as the material deforms increasingly. The right form is provided in [START_REF] Swaddiwudhipong | High velocity penetration/perforation using coupled smooth particle hydrodynamics-finite element method[END_REF], where the center part of the target plate existing large deformation is modeled with SPH particles and the other part with finite elements. Also, if the projectiles are deformed largely such as using brittle materials, they are modeled using SPH particles while elements for rigid or small performed projectiles.

An approach to couple FE-SPH methods is based on a Lagrangian mesh whose elements are converted into SPH particles when a conversion variable (strain, stress, or any state variable) reaches a critical value. This idea was proposed and applied to simulate impact problems initially by Johnson et al. [START_REF] Johnson | Linking of Lagrangian particle methods to standard finite element methods for high velocity impact computations[END_REF][START_REF] Gordon | Conversion of 3d distorted elements into meshless particles during dynamic deformation[END_REF][START_REF] Gr Johnson | A combined particle-element method for high-velocity impact computations[END_REF]. In the study of [START_REF] Johnson | Linking of Lagrangian particle methods to standard finite element methods for high velocity impact computations[END_REF], two different converting schemes were used: SPH nodes sliding on a standard finite element mesh and SPH nodes automatically generated from a standard FE mesh. The simulations were carried out by this coupled method involving rigidly pointed and rounded nose projectile impacting aluminum targets, as well as, tungsten penetrator impacting steel target. The same researcher group improved the converting technique based on setting mass nodes Figure 2.11: The FE-SPH coupled models: Left is the elements converting to SPH particles as the material deforms increasingly from [START_REF] Gr Johnson | A combined particle-element method for high-velocity impact computations[END_REF] and Right is modeling different domains of the structure with SPH and FE, respectively, from [START_REF] Swaddiwudhipong | High velocity penetration/perforation using coupled smooth particle hydrodynamics-finite element method[END_REF].

and stress points in [START_REF] Gr Johnson | A combined particle-element method for high-velocity impact computations[END_REF]. A new coupled FE-SPH method called combined particleelement method (CPEM) was developed to model the process of HVI examples in this study. The oblique impact consisting of a steel rod impact 4340 steel plate at 2 km/s was conducted by this CPEM. Subsequently, Rodriguez et.al. also applied this CPEM method modeling 4mm-thick plate of aluminum impacted by the projectiles with different nose shapes (conical, hemispherical, blunt) in [START_REF] Rodriguez-Millan | Perforation mechanics of 2024 aluminium protective plates subjected to impact by different nose shapes of projectiles[END_REF]. It is proved that the coupled method can reduce the time-consuming compared single SPH modeling, as well as using the advantages of both FEM and SPH, but the implementation seems complex.

On the other hand, various contact algorithms are developed for calculating the contact interactions between SPH particles and FEM elements on the interfaces of FE-SPH domains. A contact algorithm based on a contact potential to link the SPH codes and FEM solver in DYNA3D (developed at Lawrence Livermore National Laboratories) is proposed in [START_REF] De Vuyst | Coupling between meshless and finite element methods[END_REF]. In this algorithm, FE nodes were treated as particles, then a particle to particle contact approach was applied, allowing frictionless sliding between FE and particles.

40CHAPTER 2. A STATE OF THE ART ON IMPACTS TO STRUCTURES SIMULATED BY SPH METHOD

The authors validated this FE-SPH coupled method by the simulations for three different impact problems: a plate impact, water impact, and rod penetration. The case of an ogive-nose high strength steel projectile impacting an aluminum plate under an angle of 30 • was simulated, illustrating interesting results. As another example, an SPH-FEM coupling algorithm also was developed by Zhang et al. [START_REF] Zhang | Coupling of smoothed particle hydrodynamics and finite element method for impact dynamics simulation[END_REF], which adopts background particles in the position of FE nodes, considering the attachment and contact between SPH particles and finite elements. The perforation of a cylindrical Arne tool steel projectile impacting a plate Weldox 460 E steel target was simulated in 3D at a series of impact velocities. The coupled computational model of viscoplasticity and ductile damage, combined with Gruneisen EOS was used for the target plate. Considering the comparatively small deformation that happened in the projectile, an elastic material model was used for the projectile material. All the simulation results were agreed with experiments with acceptable accuracy in terms of ballistic limit velocity. Besides, the numerical results of coupling the SPH-FEM method demonstrated higher spatial accuracy and efficiency than that of SPH only and LS-DYNA. What is worthy to notice that the plastic deformation of the projectile may become severe with higher impact velocity, and this absorbs a lot of the initial kinetic energy. More suitable constitutive models should be chosen for the projectile material with a higher velocity impact, as proposed in [START_REF] Zhang | Coupling of smoothed particle hydrodynamics and finite element method for impact dynamics simulation[END_REF].

The benefits of the coupled FE-SPH method applied to impact problems are obviously recognized. Numbers of simulations have been implemented in this way such as [START_REF] Leonard E Schwer | Perforation of metal plates: laboratory experiments and numerical simulations[END_REF][START_REF] Swaddiwudhipong | High velocity penetration/perforation using coupled smooth particle hydrodynamics-finite element method[END_REF][START_REF] Zhang | Fragment identification and statistics method of hypervelocity impact sph simulation[END_REF][START_REF] Zhang | SPH-FEM simulation of shaped-charge jet penetration into double hull: A comparison study for steel and SPS[END_REF]. A hybrid method composed of FE analysis and SPH is used to analyze the perforation of steel material and aluminum plates with varying thickness by different projectiles in [START_REF] Swaddiwudhipong | High velocity penetration/perforation using coupled smooth particle hydrodynamics-finite element method[END_REF]. However, the authors claimed that at a lower range of impact velocities, failure behavior in a thin target was not observed correctly mainly due to the tensile instability problem inherent in the SPH method. FE solutions are in better agreement and may be adopted for this range of impact velocities. Zhang and co-workers applied a FEM-SPH-FEM technique in LS-DYNA combined with fragment identification and statistics methods to investigate the debris cloud in HVI [START_REF] Zhang | Fragment identification and statistics method of hypervelocity impact sph simulation[END_REF]. The case of d=9.52 mm sphere impact plate with 2.2mm thickness at 6640m/s was simulated in this study. The quantity, the size, the mass, and the velocity of a specific fragment were computed and investigated with these statistics on the binary graph, obtaining good simulation results.

A coupled SPH-FE model is also used to simulate a metal-jet penetration into a double hull made of different materials -steel and SPS (Sandwich Plate System) in [START_REF] Zhang | SPH-FEM simulation of shaped-charge jet penetration into double hull: A comparison study for steel and SPS[END_REF]. The 2.5. MECHANICAL CHARACTERIZATION OF SOFT MATERIALS UNDER IMPACT 41 shaped charge was modeled with SPH and the double-hull structure was modeled with FEM; A penalty method was used to treat the interaction between metal-jet particles and target elements during the entire process. The simulation results were compared with experiments and analyzed to acquire a better understanding of the process of a metal-jet penetration into a double-hull structure. More applications of coupled FE-SPH simulations for HVI problems can be found in the appendix table.

2.5/ MECHANICAL CHARACTERIZATION OF SOFT MATERIALS UN-DER IMPACT

A considerable advance on the application of SPH simulating HVIs problems on thin structures has been achieved in the past decades. The research to investigate the behavior of soft materials or bio-materials under impact loading is also paid more and more attention because of the requirements and developments in the fields such as protections on wound ballistics, traumatic injuries, and some pharmaceutical techniques [START_REF] Wilson | The role of biomechanics in understanding dance movement: a review[END_REF][START_REF] John D Finan | Biomechanical simulation of traumatic brain injury in the rat[END_REF]. Numerous research, particularly the experiments, can be found to investigate these similar behaviors of Fig. 2.12 for human parts and organs under dynamic loading and impacted by bullets [START_REF] Bond | Limitations of hairless mouse skin as a model for in vitro permeation studies through human skin: hydration damage[END_REF][START_REF] Maiden | Historical overview of wound ballistics research[END_REF]. The experiments on some live tissues were carried out to observe and understand the ballistic wounds and their bio-mechanical properties. For instance, organs and tissues from fresh human cadavers have been used [START_REF] Bir | The evaluation of blunt ballistic impacts of the thorax[END_REF][START_REF] Nicholas R Maiden | Unpredictable tensile strength biomechanics may limit thawed cadaver use for simulant research[END_REF], and some animals including horses, cattle, goats, sheep, dogs, and pigs also have been used

extensively to conduct trauma studies [START_REF] Schantz | Aspects on the choice of experimental animals when reproducing missile trauma[END_REF]. However, they are being less used today because of the serious ethical issues [140].

Some soft materials as biofidelic human tissue simulants are developed in recent years such as epoxy resin [START_REF] Ak Dabrowska | Materials used to simulate physical properties of human skin[END_REF], synthetic gel [START_REF] Bracq | Experimental study of the strain rate dependence of a synthetic gel for ballistic blunt trauma assessment[END_REF] or ballistic gelatin [START_REF] Martin | The wound profile: a visual method for quantifying gunshot wound components[END_REF][START_REF] Ds Cronin | Ballistic gelatin characterization and constitutive modeling[END_REF]. Particularly, ballistic gelatin has been regarded as one of the most important simulants as real biological tissues. Gelatin blocks are prepared by dissolving the calibrated amount of usually 250A bloom gelatin powder into warm water, possibly adding preservative and pouring the solution into molds where it first cools off and then solidifies. It was firstly used in the ballistic experiment in [START_REF] Harvey | Mechanism of wounding[END_REF] and was found that similar penetration depths were produced in gelatin compared to those measured in soft tissue. At present, two common gelatin formulations (10% and 20% Ballistic Gelatin) are employed in many experiments to understand the behaviors of soft tissues such as [START_REF] Martin | Ordnance gelatin for ballistic studies. detrimental effect of excess heat used in gelatin preparation[END_REF][START_REF] Gareth | On the dynamic behavior of three readily available soft tissue simulants[END_REF][START_REF] Jm Wilgeroth | The shock response of a rendered porcine fat[END_REF][START_REF] Gj Appleby-Thomas | The high strain-rate behaviour of selected tissue analogues[END_REF], which proved that the gelatin simulated close enough soft tissues by comparing the behaviors between gelatin and porcine or chicken meats by experimental tests.

In order to observe the ballistic behavior of gelatin for the understanding of the live tissues, various tests have been carried out to characterize its mechanical properties. Compression tests at different loadings and strain rates up to 1/s using a Polymer Split Hopkinson Pressure Bar (PSHPB) have been conducted by Cronin et al. [START_REF] Ds Cronin | High rate characterization of low impedance materials using a polymeric split hopkinson pressure bar[END_REF][START_REF] Ds Cronin | Characterization of 10% ballistic gelatin to evaluate temperature, aging and strain rate effects[END_REF]. The mechanical properties of 10% ballistic gelatin were calibrated using a compression test apparatus with temperature-controlled platens to maintain the sample temperature at a fixed level.

Penetration testing was carried out using a standard impact test to assess the effect of aging. The primary conclusion of this study was to show the importance of temperature and aging time to achieve a suitable and consistent response from ballistic gelatin. The pression experiments on gelatin specimens at strain rates spanning two orders of magnitude (0.001-0.1s -1 ) and observed a nonlinear load-displacement history and strong strain rate-dependence [START_REF] Ravikumar | A constitutive model for ballistic gelatin at surgical strain rates[END_REF].

Therefore, the mechanical responses of ballistic gelatin are quite different associated with the loading speeds and impact conditions. While the quasi-static and low-velocity behavior (less than 30 m/s) can be described through some hyperelastic models [START_REF] Ds Cronin | Dynamic characterization and simulation of ballistic gelatin[END_REF][START_REF] Ravikumar | A constitutive model for ballistic gelatin at surgical strain rates[END_REF]. As far, the hyperelastic Mooney-Rivlin model is regarded as the most promising constitutive law for the gelatin deformations [START_REF] Liu | Behavior of a novel iterative deconvolution algorithm for system identification[END_REF][START_REF] Srinivasan | Wave propagation in ballistic gelatine[END_REF]. In the study of [START_REF] Srinivasan | Wave propagation in ballistic gelatine[END_REF], the wave propagation behavior in ballistic gelatin was investigated by imparting an impulse motion into a long cylindrical gelatin specimen and then capturing the transient deformation using a high-speed digital camera. The high strain rate behavior of gelatin was considered in the experiments using a polymer split Hopkinson pressure bar (PSHPB). The hyperelastic Mooney-Rivlin model was employed to analyze the responses of gelatin. Kwon and Subhash investigated the mechanical behavior of ballistic gelatin under quasi-static and dynamic compressive loading [START_REF] Kwon | Compressive strain rate sensitivity of ballistic gelatin[END_REF]. The dramatic increase in the strength of gelatin at high rates was discovered in these experiments. The compressive strength increased from 3kPa at a strain rate of around 0.0013/s to 6MPa at a strain rate of around 3200/s. This phenomenon was discussed and a conclusion that ballistic gelatin is highly sensitive to the strain rate was provided in this study. According to the literature [START_REF] Ralph | Motion of spherical particles in a bingham plastic[END_REF]2], the ballistic gelatin behaviors under high-velocity loading also should be described through hydrodynamic theory, as explained for Fig. 2.12. Therefore, the mechanical process of penetrating soft materials is complex and not easy to be understood only by experimental observations.

Gelatin has been widely used in experiments and numerical simulations to understand the ballistic response of soft tissues. At the same time, new homogeneous synthetic simulants are also being developed in order to avoid the instabilities of material properties of gelatin caused by the environments and aging [START_REF] Humphrey | Ballistics and anatomical modelling-a review[END_REF]. Some alternative tissue simulants like Perma-Gel, Associ-ating Gelatine's (PAG), or transparent gel candleis (a clear syn-44CHAPTER [START_REF] Shen | Numerical modeling of a human tissue surrogate sebs gel under high velocity impacts: investigation of the effect of the strain rate in an elasto-hydrodynamic law[END_REF], an elastohydrodynamic constitutive law of SEBS gel combined with the effect of the strain rate was also proposed. The related experiments were conducted with both targets and projectiles of several centimeters dimension. This research demonstrates that the elasticity of SEBS gel material is non-linear and depending on the strain rate. Actually, the synthetic polymer SEBS gel is paid not enough attention at present. Considering it involves the similar mechanical properties with ballistic gelatin but better practical performances, this thesis will provide an investigation for this material in following chapter.

2.5.1.2/ ANALYTICAL INVESTIGATIONS

In the field of penetrations in soft tissues, the factors to be investigated include the various material properties, the trajectory of the projectile, the pressure wave in the soft material, the temporary and permanent cavities, as well as the yawing, tumbling and fragmenting depending on the states of projectiles. analyzed by the model in [START_REF] Liu | A spherical cavity expansion model of large elastic deformation and its application to ballistic gelatin penetration problems[END_REF]. The research investigated the cavity expansion in the soft material called Polyvinyl Alcohol Hydrogels by elastic solution and experiments in [START_REF] Nafo | Experimental and analytical investigation of the cavity expansion method for mechanical characterization of soft materials[END_REF].

In 2018, by combination with parameters obtained from the motion model of rifle bullet in [START_REF] Susu | A new motion model of rifle bullet penetration into ballistic gelatin[END_REF], cavity dynamics in gelatin was studied and a motion model of temporary cavity in gelatin penetrations of rifle bullets was proposed by Liu et al. [START_REF] Liu | Cavity dynamics in 10 wt% gelatin penetration of rifle bullet[END_REF]. The maximum cavity wall radius, radial movement along the penetration trajectory and temporary cavity profiles were investigated by this analytical model. It is significant for the trauma assessment to develop an appropriate cavitation model in gelatin for penetrations of rifle bullet. where [START_REF] Liu | A spherical cavity expansion model of large elastic deformation and its application to ballistic gelatin penetration problems[END_REF]. [START_REF] Dyckmans | Numerical and experimental study of the impact of small caliber projectiles on ballistic soap[END_REF], and also to study deformations induced by the penetrations of ABS (Acrylonitrile-Butadiene-Styrene) plastic spheres into gelatin at velocities up to 160 m/s [START_REF] Koene | Experimental and numerical study of the impact of spherical projectiles on ballistic gelatin at velocities up to 160m/s[END_REF].

In the investigation of ballistic responses in the soft tissues, the simulations are carried out predominately based on the gelatin and validated also by existing experiments related to gelatin blocks such as [START_REF] Md Minisi | Gelatin impact modeling[END_REF][START_REF] Koene | Experimental and numerical study of the impact of spherical projectiles on ballistic gelatin at velocities up to 160m/s[END_REF][START_REF] Wen | Impact of steel spheres on ballistic gelatin at moderate velocities[END_REF][START_REF] Awoukeng-Goumtcha | Investigations of impact biomechanics for penetrating ballistic cases[END_REF]. Most of these simulations were performed by the numerical methods based on meshing. Minisi et al. [START_REF] Md Minisi | Gelatin impact modeling[END_REF] simulated the projectile-gelatin interaction at high impact velocities with the 10% gelatin modeled as a hydrodynamic material with the Mie-Gr üneisen equation of state at high strain rates and a Mooney-Rivlin material at lows train rates. Worthy to be noticed, a study in 2013 by Wen et al. [START_REF] Wen | Impact of steel spheres on ballistic gelatin at moderate velocities[END_REF] successfully simulated the trajectory of projectiles, the time history of the temporary cavity, and the pressure in the 10% gelatin by LS-DYNA software with finite element method. The authors studied experimentally and computationally the penetrations of a steel sphere into ballistic gelatin at a moderately high speed (up to 1630 m/s). They modeled the gelatin materials using an isotropic and homogeneous elastic-plastic constitutive law combined with a polynomial equation of state, which provided good results compared with experiments. The same research group also performed the simulations using FE method to study the penetration of a rifle bullet into a block of ballistic gelatin [START_REF] Wen | Rifle bullet penetration into ballistic gelatin[END_REF]. The effect of the angle of attack of the rifle bullet for 48CHAPTER 2. A STATE OF THE ART ON IMPACTS TO STRUCTURES SIMULATED BY SPH METHOD the penetration process were highlighted in these simulations. Fig. 2.15 shows the FEM model for this soft penetration by Wen et al. in [START_REF] Wen | Impact of steel spheres on ballistic gelatin at moderate velocities[END_REF][START_REF] Wen | Rifle bullet penetration into ballistic gelatin[END_REF].

Figure 2.15: (a) is the sphere projectile in [START_REF] Wen | Impact of steel spheres on ballistic gelatin at moderate velocities[END_REF]; (b) is the rifle bullet projectile in [START_REF] Wen | Rifle bullet penetration into ballistic gelatin[END_REF]. (c) is the FE model for gelatin block in both of [START_REF] Wen | Impact of steel spheres on ballistic gelatin at moderate velocities[END_REF] and [START_REF] Wen | Rifle bullet penetration into ballistic gelatin[END_REF].

Also, in 2014, a FE model was developed to investigate the ballistic behavior of 20% gelatin under high-velocity impacts [START_REF] Awoukeng-Goumtcha | Investigations of impact biomechanics for penetrating ballistic cases[END_REF]. The gelatin was modeled using an elastoplastic hydrodynamic constitutive law, similar to the one used in the study of Wen et al. [START_REF] Wen | Impact of steel spheres on ballistic gelatin at moderate velocities[END_REF]. A series of simulations were performed by this model for the penetrations involving different sphere projectiles of (2.38, 4.76, 6.3 mm diameters) impacting with a large range of velocities from 230 to 2229 m/s. The penetrating histories were simulated and validated by experimental and analytical results. Zhang et al. [START_REF] Zhang | The experimental and numerical study of indirect effect of a rifle bullet on the bone[END_REF] also used a FEM model to investigate the process of a 56-type 7.62-mm rifle bullet penetrating the gelatinbone composite target. In recent, Chen et al. [START_REF] Chen | The adaptive finite element material point method for simulation of projectiles penetrating into ballistic gelatin at high velocities[END_REF] developed an adaptive finite element material point method (AFEMP) that includes the advantages of both the finite element method (FEM) and the material point method (MPM) to model the penetrations into ballistic gelatin. By the comparisons with the experimental results, the accuracy of the AFEMP program was verified, and the temporary cavity phenomenon in the gelatin target can be better described than the finite element method according to this study.

Although the numerical methods based on meshing techniques have been used to simulate the penetrations in the soft materials by a few research, the mesh distortion problem is still risque and requires extra work when simulating the extremely large deformations that happened in the deep penetrations. As introduced previously, the particle numerical methods like the SPH method is more suitable to simulate this kind of problems. The previous review on the advance of HVIs on thin structures simulated by the SPH method proves this point. However, for the soft penetrations that happened in the thick structure, there is less research on the applications of particle methods.

In 2015, Taddei et al. [START_REF] Taddei | Smoothed particle hydrodynamics formulation for penetrating impacts on ballistic gelatine[END_REF] developed an original numerical model combined SPH particles and FE elements by software HyperWorks with Radioss to investigate the ballistic response of 20% gelatin under high-velocity impacts. The configuration of this model is shown in Fig. 2.16, in which the domain of impact center is discretized by SPH particles and the far domain is by elements. This SPH model simulated a series of penetrations in the gelatin impacted by rigid projectiles and achieved a good agreement with experimental observations. Then, this numerical model was used to simulate the penetrations into 10% gelatin under high-velocity impacts [START_REF] Monzer | Sph-based method to simulate penetrating impact mechanics into ballistic gelatin: Toward an understanding of the perforation of human tissue[END_REF]. In this study, the multi-layered gelatin was the target block impacted by a rigid sphere, which was also stimulated by this numerical model. A good agreement was obtained between SPH simulation and analytical results.

Frissane et.al developed an axisymmetrical SPH code to simulate the penetration process into gelatin [START_REF] Frissane | Sph modeling of high velocity impact into ballistic gelatin. development of an axis-symmetrical formulation[END_REF]. The same penetration cases with the study of [START_REF] Taddei | Smoothed particle hydrodynamics formulation for penetrating impacts on ballistic gelatine[END_REF] were simulated by this code and a better accuracy between the numerical and experimental results was achieved.

50CHAPTER 2. A STATE OF THE ART ON IMPACTS TO STRUCTURES SIMULATED BY SPH METHOD

Compared with the wide usage of the SPH method in simulating perforation on thin structures, there are only a few studies on the application of the SPH method to simulate the deep penetrations into soft materials in the literature. To the author's knowledge, except for ballistic gelatin, there is almost no numerical investigation of another simulant of soft tissues called polymer gels under high-velocity impacts. Therefore, expanding the application of the SPH method to the investigations on deep penetrations is an interesting and original work.

2.5.2/ BALLISTIC PENETRATIONS IN THE MICRO-SCALE

To understand the penetration process in the soft materials is significant work, which has been paid attention via experimental, analytical, and various numerical methods.

However, most of them are investigated under the macro-scale. During the past few years, interest in the development of transdermal delivery technologies of pharmaceuticals has been renewed by the promise of needle-free drug delivery via penetration of high-speed drug-carrying projectiles into the viable epidermis and dermis [START_REF] Hiren | Penetration enhancers for transdermal drug delivery system: A review[END_REF][START_REF] Shital H Bariya | Microneedles: an emerging transdermal drug delivery system[END_REF]. This kind of technique involves a similar penetration process that happened in the soft tissues but in the micro-scale. The sizes of the transdermal powders are on the µm-scale or smaller.

The physical process and mechanical behavior show completely different properties when the penetrations transmit from the macro-scale to the micro-scale. This trend has been validated in the observations via several experiments and analyses [START_REF] Thomas | A ballistic study of micro-particle penetration to the oral mucosa[END_REF][START_REF] Veysset | High-velocity micro-particle impact on gelatin and synthetic hydrogel[END_REF].

In terms of the experiments, the investigation of the deformation of soft materials in extreme dynamic environments is achieved by a novel laser-induced particle impact platform Numerical simulation is an efficient alternative to investigate the complex physical problem of high-velocity microparticle penetration in soft tissue, and very few work seems to exist in the literature on this specific point. Among those works, Guha and Shear in [START_REF] Ra Guha | Ballistic impact of single particles into gelatin: experiments and modeling with application to transdermal pharmaceutical delivery[END_REF] developed a finite element model of the impact and penetration of a 2 m gold particle into the human dermis. They validated this numerical model with macro-scale steel ballgelatin system by using a dimensional analysis and then investigated the performance of transdermal power injection technology. They found an elastic modulus of 2.25 MPa for skin stimulants, which is one of the key material parameters for impact in soft tissue studies. However, this model is based on a scaled-up system, which means the numerical computation is not directly used on the micro-scale structure.

As the author's knowledge, there is no more study to investigate the penetrations of soft materials in the micro-scale by the simulation approaches. The main limits in simulating such phenomena research lie in two points: one is the complex properties of soft tissues, which are still under the developments. As known for several simulants of the soft tissues such gelatin and SEBS gel, their mechanical properties involve hyper-elasticity, 52CHAPTER 2. A STATE OF THE ART ON IMPACTS TO STRUCTURES SIMULATED BY SPH METHOD viscous-elasticity, the high sensitivity to the strain rate, as well as high depending on the loading types. What's more, when the deformation of the soft materials happens in the micro-scale, the mechanical behavior becomes more questionable; The other is that the large deformations under quick and extreme impact loading make the simulation methods based on the meshing techniques no longer sufficient. The micro-scale condition also requires some specific treatments from the numerical models. Taking into account the successful attempts of the SPH method to model the deep penetrations in the gelatin materials [START_REF] Taddei | Smoothed particle hydrodynamics formulation for penetrating impacts on ballistic gelatine[END_REF][START_REF] Frissane | Sph modeling of high velocity impact into ballistic gelatin. development of an axis-symmetrical formulation[END_REF], this thesis provides an original investigation about the ballistic behaviors of soft materials under micro impact loading by developing the numerical model using the SPH method. Two different simulant materials involving 10% gelatin and 40 vol % SEBS gel are simulated as the target materials and their mechanical properties under micro impacts are observed and discussed.

2.6/ CONCLUSION

This chapter reviews the developments of SPH method and its application in the field of HVIs problems. Although SPH method is known to involve some numerical drawbacks like numerical instabilities, boundary inaccuracy, and time-consuming, particularly in solid impact problems, a state of the art in this chapter witnessed the great success of the SPH method applied to diverse impact problems, mainly showing as:

1. As far, different versions of SPH formulations have been developed to improve the comprehensive performance of SPH simulations, like CSPM, Godunov SPH, total

Lagrangian SPH, and pseudo-springs SPH, etc. Besides, most of them have been applied to impact problems.

The SPH formulation involves great compatibility with other different algorithms and

then is developed to some coupled numerical methods like Gamma-SPH, FE-SPH, γ-SPH-ALE, etc.

3.

The most of fragmentation and debris clouds phenomenons in HVIs on thin structures are predicted well by the SPH method thanks to its natural particle property.

4.

The application of the SPH method has been extended to a diversity of HVI problems, not limited to metallic projectile-target systems. It comes from the develop- During the literature review, a trend is found that the target materials in impact systems at present demonstrates a large variety on types of materials such as various composite laminates, brittle materials like ceramics, advanced high-strength cementitious materials, as well as multilayered structures consisting of completely different materials except for the metallic materials. Moreover, the SPH method has also been extended to simulate the ballistic response of soft tissue materials in recent years like [START_REF] Monzer | Sph-based method to simulate penetrating impact mechanics into ballistic gelatin: Toward an understanding of the perforation of human tissue[END_REF][START_REF] Frissane | Sph modeling of high velocity impact into ballistic gelatin. development of an axis-symmetrical formulation[END_REF], particularly, to the bird-strick problem involving the SPH modeling of birds body [START_REF] Zhang | Effect of arbitrary yaw/pitch angle in bird strike numerical simulation using sph method[END_REF][START_REF] Zhou | Bird-striking damage of rotating laminates using sph-cdm method[END_REF][START_REF] Zhou | Experimental and numerical investigation of high velocity soft impact loading on aircraft materials[END_REF]. This trend can be foretold with the high requirements of the modern industry. Therefore, to investigate the HVIs by SPH method in further work still should pay many effects like,

1.

Tensile instability is still an important factor to affect the simulation accuracy. Although lots of improvements have been developed, the performance of them still depends on the impact conditions like low or high impact velocity, the types of materials. Total Lagrangian SPH seems a comparatively stable scheme, but it needs more applications to different HVI systems to validate its performance.

2.

The behavior of crack propagation as one of the most important damages during impacts, very depends on the types of materials and target structures, particularly for some multilayered structures or laminated composites. There is no much simulation in this situation. To the best of the authors ' knowledge, Pseudo-spring SPH seems the most promising scheme for this.

3.

The investigations to fragmentation and debris clouds during HVIs are still on the qualitative analysis. The quantitative analysis is necessary, but it is a challenging work for SPH application. By reviewing the application of the SPH method in the literature, it can be seen that the SPH has become one of most important simulation approaches in solid dynamics because of its great potentials in simulating extremely large deformations. In terms of the essence of the SPH, it uses a kernel function for numerical approximations. Some literature demonstrate mathematically that the types of the kernel function directly influence the stability and overall accuracy of SPH simulation. However, less attention is paid to the influences derived from kernels particularly on simulating dynamic solids problems by SPH. In addition, some research introduce that there are strong relations between SPH kernels and tensile instability which is one of shortcomings in SPH method, a kind of particles clumping phenomenon probably leading to some nonphysical cracks when the solid material is largely stretched or compressed. At present, no method exists to completely avoid these instabilities, although a few corrections like Artificial Viscosity, Artificial Stress, Corrective SPH (CSPH) or the Godunov-type SPH have been proposed. Therefore, the objective in this chapter is that the performances of SPH method with different kernels is studied, including classical types and several new ones proposed recently.

Combined with some corrected techniques,a 2D SPH code is developed and introduced in this chapter, and the suitability of these kernels in SPH method is discussed in solid dynamic problems like bending deformation of elastic beam and some tests about elasticplastic deformation in impact problems. The structure of this chapter is organized by:

1. Introduce the role of kernel function in SPH algorithm, involving the relations between the different types of kernel function and numerical instability and accuracy.

2.

The formulation of Corrective SPH method (SPM) is represented, which is a technique to correct the calculation of kernel function and also is implemented in the SPH code.

3.

The procedure of implementation the SPH code for solids is introduced.

4.

The validation to the SPH code is carried out by numerical examples like bending deformation of elastic beam and elastic-plastic deformation in rod impacts.

5.

The discussion to the numerical results and different performances of different types of kernel function in solid problems.

3.2/ THE ROLE OF KERNEL FUNCTION

3.2.1/ THE TYPES OF KERNEL FUNCTION

At present, series of B-spline functions are classical and traditional SPH kernels in most applications [START_REF] Vignjevic | A treatment of zero-energy modes in the smoothed particle hydrodynamics method[END_REF][START_REF] Liu | Smoothed particle hydrodynamics: a meshfree particle method[END_REF], for examples cubic spline and quintic functions shown in Fig. 3.1.

It also lists several existing SPH kernels, like the Gaussian function, Wendland fucntion and some new types (Double cosine function and Hyperbolic shaped). Fig. 3.2 shows the shapes of these six kernel functions and their derivatives. Gingold and Monaghan in their original paper [START_REF] Gingold | Smoothed particle hydrodynamics -Theory and application to non-spherical stars[END_REF] firstly used the Gaussian function as a SPH kernel because it is sufficiently smooth even for high orders of derivatives. However, the Gaussian function doesn't exactly satisfy the compactness condition, as it never goes to zero theoretically with the variable |r| increasing. As a study by Liu and his co-workers in [START_REF] Mb | [END_REF], the kernel function in SPH algorithm can be built systematically. For instance, type-5 kernel in Fig. 3.1 is constructed based on the cosine functions which is characterized by having enough smoothness in any order of derivatives, thereby a better accuracy can be acquired as SPH kernel [START_REF] Yang | A new kernel function for sph with applications to free surface flows[END_REF]. In addition, a hyperbolic shaped kernel function (type-6 kernel in Fig. 3.1) is also constructed and used in the SPH formulas to simulate viscous liquid drop problems in [START_REF] Yang | Smoothed particle hydrodynamics modeling of viscous liquid drop without tensile instability[END_REF]. This function possesses non-negative second derivatives which is regarded to remove the pairing instability in compressive regimes. As a classical research discussing the numerical instability when applied SPH method in solids, Swegle [START_REF] Jw Swegle | Smoothed particle hydrodynamics stability analysis[END_REF] proposed an unstable growth condition by analyzing the particles clumping phenomenon in stretched material (so called tensile instability or pairing instability) when using B-spline kernel in SPH algorithm. This condition can be written as following,

W " T ≥ 0 (3.1)
where, W " is the second derivative of the kernel W and T is stress state, which is negative in compression and positive in tension. This condition means that the instability of SPH algorithm depends on the sign of the product of the total stress and the second derivative of the kernel function. In 2012, Dehnen in [START_REF] Dehnen | Improving convergence in smoothed particle hydrodynamics simulations without pairing instability[END_REF] discovered that the Wendland kernels have the similar shapes for derivatives with traditional B-spline kernels, but without providing pairing instability even with a large number of neighbour particles. By linear stability analysis, they concluded that non-negativity of the Fourier transform of SPH kernel functions is a necessary Thanks to these analyses, the influences from kernel types are paid attentions in research how to decrease the instability in SPH applications. Yang in [START_REF] Yang | Smoothed particle hydrodynamics modeling of viscous liquid drop without tensile instability[END_REF] proposed the new types of kernels (type-6 kernel in Fig. 3.1) in order to remove the tensile instability and proved these ones can achieve more stable simulations in flow problems, but for solid dynamic problems where exists more severe particles disordering, there is no much particularly practical investigation from the view on kernels.

3.2.3/ THE ACCURACY VIA KERNEL TYPES

It is difficult to estimate the numerical errors of SPH algorithm just from kernels properties, since the algorithm accuracy depends on a number of factors, and also is influenced by the types of problems. Sigalotti (2019) in [START_REF] Di | A new insight into the consistency of the sph interpolation formula[END_REF] developed a general error analysis to simultaneous treatment of both the kernel and particle approximation errors for arbitrary particle distributions. This error analysis using the Poisson summation formula can achieve a functional dependence of the error bounds on SPH interpolation parameters, like the types of kernels, smoothing length h, and the number of neighbours. A term A w is derived in this research as an estimate which involves the parameters from particular kernels as following,

A n w = 2 n α d B n k n n (3.2)
in which, α d and k, respectively, is the dimension-dependent constant like Fig. Wendland function seems a promising alternative as SPH kernel, but it is not often used in some practical simulations. This chapter is intended to highlight how these kinds of kernels in list of Fig. 3.1 influence the instability and accuracy and look for the most suitable kernel when SPH simulating solid dynamics.

3.2.4/ CORRECTIVE KERNEL PROCESS

Based on the formulation of Standard SPH, corrective SPH (CSPH) proposed by Chen in [START_REF] Jk Chen | An improvement for tensile instability in smoothed particle hydrodynamics[END_REF][START_REF] Jk Chen | A generalized smoothed particle hydrodynamics method for nonlinear dynamic problems[END_REF] is implemented in this code. The idea of this corrective method is to transfer the kernel estimate concept to the Taylor series expansion, which has been proved that this transposition is very effective addressing tensile instability. Besides, it also helps to improve boundary deficiency problems which is an another major obstacle for the standard SPH.

Here, the formulation of CSPH is represented in 2D directly. The Taylor series expansion for a 2D function is employed and it is cut after the linear element.

f (x , y ) = f (x i , y i ) + (x -x i ) ∂ f ∂x |(x i , y i ) + (y -y i ) ∂ f ∂y |(x i , y i ) (3.3)
based on above format, the expansion is multiplied by ∂ ∂x W, ∂ ∂y W, then integrated over the whole domain,following as,

∂W ∂x f (x , y )dx = ∂W ∂x f (x i , y i )dx+ (x -x i ) ∂W ∂x ∂ f ∂x |(x i , y i )dx + (y -y i ) ∂W ∂x ∂ f ∂y |(x i , y i )dx (3.4) 3.2. THE ROLE OF KERNEL FUNCTION 61 ∂W ∂y f (x , y )dx = ∂W ∂y f (x i , y i )dx+ (x -x i ) ∂W ∂y ∂ f ∂x |(x i , y i )dx + (y -y i ) ∂W ∂y ∂ f ∂y |(x i , y i )dx (3.5)
Reorganizing these terms,

∂W ∂x ( f (x , y )-f (x i , y i ))dx = ∂ f ∂x |(x i , y i ) (x -x i ) ∂W ∂x dx + ∂ f ∂y |(x i , y i ) (y -y i ) ∂W ∂x dx (3.6) ∂W ∂y ( f (x , y )-f (x i , y i ))dx = ∂ f ∂x |(x i , y i ) (x -x i ) ∂W ∂y dx + ∂ f ∂y |(x i , y i ) (y -y i ) ∂W ∂y dx (3.7)
Discretizing employing the usual nodal integration,

N j=1 ∂W ∂x ( f (x j , y j ) -f (x i , y i )) m j ρ j = ∂ f ∂x |(x i , y i ) N j=1 (x j -x i ) ∂W ∂x m j ρ j + ∂ f ∂y |(x i , y i ) N j=1 (y j -y i ) ∂W ∂x m j ρ j (3.8) N j=1 ∂W ∂y ( f (x j , y j ) -f (x i , y i )) m j ρ j = ∂ f ∂x |(x i , y i ) N j=1 (x j -x i ) ∂W ∂y m j ρ j + ∂ f ∂y |(x i , y i ) N j=1 (y j -y i ) ∂W ∂y m j ρ j (3.9)
Writing this in compact matrix form reveals that the CSPM corrector,

< ∇ • f (x i , y i ) > i = N j=1 ( f j -f i )∇W i j m j ρ j B (3.10) B =          N j=1 (x j -x i ) ∇W i j m j ρ j          (3.

3.3/ THE PROCEDURE OF SPH CODE IMPLEMENTATION

This section mainly introduces the programming implementation of a SPH code, which is based on the standard SPH scheme combined with some techniques involving CSPH, artificial viscosity and stress, as well as XSPH.

3.3.1/ THE SPH FORMULATIONS FOR SOLIDS

3.3.1.1/ SPH FORMULATIONS FOR NAVIER-STOKES EQUATIONS

Firstly, the procedure of deriving conversation equations in solids into SPH formulations is presented. In an Eulerian referential, the conversation laws take the following form:

1. Conservation of mass:

Dρ Dt = -ρ∆ • v (3.12)
2. Conservation of momentum:

ρ D v Dt = -∆ • σ (3.13)

Conservation of energy:

De) Dt = - σ ρ : ∆ ⊗ v (3.14)
The density ρ, specific internal energy e, the velocity v, and the stress tensor component σ are the function of the spatial coordinates x and time t. Here, σ is the Cauchy stress tensor.

The summation for density can be written by,

ρ( x i ) = N j=1 m j W( x i -x j , h) (3.15)
The above expression is simple and used at the initial development of SPH method, but it is not able to ensure the correct calculation at the free boundary. Therefore, the approximation for the rate of change of density is carried out to instead of Eq. (3.15) . To approximate the equation of Conservation of mass (Eq. (3.12) ) by using the formulas of SPH discretization processes, Eq. (3.12) can be written as,

THE PROCEDURE OF SPH CODE IMPLEMENTATION

63

Ω Dρ Dt W( x -x , h)dΩ = - Ω ρ ∂ v ∂x W( x -x , h)dΩ (3.16)
According to the formats in chapter 2, the derivative of a vector field v can be approximated as,

< ∂ v ∂x >= N j=1 m j ρ j v j • ∂W( x i -x j , h) ∂x i (3.17)
Then, the conservation of mass equation can be discretized by,

< Dρ i Dt >= ρ i N j=1 m j ρ j (v α i -v α j ) ∂W( x i -x j , h) ∂x α i (3.18)
To approximate the equation of Conservation of mass (Eq. (3.13) ) by using the formulas of SPH discretization processes, Eq. (3.13) can be written as,

Ω D v Dt W( x -x , h)dΩ = - Ω 1 ρ ∂ σ ∂x W( x -x , h)dΩ (3.19)
The following expression can be obtained to the SPH discritized forms on the equations on conversation of momentum and energy,

< Dv α i Dt >= - N j=1 m j ( σ αβ i ρ 2 i + σ αβ j ρ 2 j ) ∂W( x i -x j , h) ∂x β i (3.20) de α i dt = - σ αβ i ρ ∂v α i ∂x β i = - 1 2 N j=1 m j (v α i -v α j )( σ αβ i ρ 2 i + σ αβ j ρ 2 j ) • ∂W i j ∂x β i (3.21)

3.3.1.2/ THE FORMULATIONS ON ELASTIC-PLASTICITY

In solids, the stress tensor in Eq. (3.20) and Eq. (3.20) can be written as following,

σ αβ = -Pδ αβ + S αβ (3.22) 64CHAPTER 3

. THEORETICAL POINTS: DEVELOPMENT OF A SPH CODE TO DYNAMIC SOLIDS

where, P is pressure and S αβ is the deviatoric stress tensor. In this paper, for elastic problem, the pressure is computed using a simple Equation of State (EoS):

P = K( ρ ρ 0 -1) (3.23)
where, K is the bulk modulus. For the elastic-plastic problem, the Mie-Gruneisen EOS [START_REF] Libersky | High Strain Lagrangian Hydrodynamics[END_REF] is used,

P = (1 - 1 2 Γη)P H + Γρe (3.24)
where,

P H =              a 0 η + b 0 η 2 + c 0 η 3 η > 0, a 0 η 3 η < 0. (3.25) η = ρ ρ 0 -1 (3.26) a 0 = ρ 0 C 2 , b 0 = a 0 [1 + 2(S -1)], c 0 = a 0 [2(S -1) + 3(S -1) 2 ] (3.27)
Here, S and C are respectively the linear shock-velocity and the particle-velocity parameters to describe the Hugoniot fit and Γ is the Gruneisen parameter.

The deviatoric stress tensor S αβ can be calculated by its derivative,

dS αβ dt = 2µ( εαβ - 1 3 δ αβ εγγ ) + S αγ R βγ + S βγ R αγ (3.28) εαβ = 1 2 ( ∂v α ∂x β + ∂v β ∂x α ) (3.29) R αβ = 1 2 ( ∂v α ∂x β - ∂v β ∂x α ) (3.30)
Combined with the SPH formulations, the equations (3.28) - (3.30) can be described as summation forms of particles by following formula:

( ∂v α ∂x β ) i = - N j=1 m j ρ j (v α i -v α j ) ∂W i j ∂x β i (3.31)
For plastic problems, the yield stress σ y is constant in perfectly-plastic behavior law while in Johnson-Cook law, the yield stress should be updated by following equation:

σ y = (a + bε n p )(1 + clin( ε ε0 ))(1 -T * m ) (3.32)
where, the parameters a, b, c, n are constants related to material properties. ε p is the plastic strain and ε is plastic strain rate. T * = T -T room T M -T room , where T M is the fictitious melting temperature of the material, T being the current temperature, T room being the room temperature and m being the thermal softening exponent.

In the case study of this work, Von Mises yield criterion is applied. The second stress invariant J 2 = 0.5S αβ S αβ is checked in every time step. If

f = min( σ y √ 3J 2 , 1) (3.33)
The plastic strain ε αβ p can be calculated by the increment form,

ε αβ p = 1 -f 2µ S αβ (3.34)
The equivalent strain ε e is defined as ε e = √ 0.5ε αβ ε αβ , which includs both elastic and plastic deformation. 1. Initialization. This part includes the input of the initial configuration of the problem geometry, total number information of the discretized particles, particle distance, boundary conditions, material properties, time step and other simulation control parameters.

3.3.2/ THE STRUCTURE OF SPH CODE

2.

Discretization. This setup process can be implemented in some commercial tools like CAD/CAE when the geometry is complex.

3. Time integration. The loop time integration algorithms like Leapfrog, Predictorcorrector and Runge-kutta are some common used methods, which may have different advantages and disadvantages. The loop contains the major modules in SPH simulation, consisting of:

1) Nearest neighboring particle searching (NNPS), which also can be achieved by several different algorithms. Because this step is very time-consuming when the problem includes a large number of particles, and it should be calculated in every time step. So, The efficiency of the specific approach often determiner the time-consuming of the whole simulation.

2)

Calculating the smoothing function.

3) Density estimation. Here, there are two ways for estimation, one using summation density approach, the other calculating density change rate (P124 book liu).

4)

Calculating the internal forces arising from the particle interactions. The pressure is obtained from the density and energy through the equation of state depending on the physical problem.

5)

Calculating the external forces if necessary.

6) Corrective kernel function

7)

Calculating the artificial viscosity and stress terms.

8)

Updating the particle momentum, energy and density; updating the particle position and velocity;

9)

Averaging the velocity if using conservative smoothing.

3.3.3/ THE TIME INTEGRATION

In this code, the predictor-corrector Leap-Frog (LF) scheme is applied in SPH time integration. Denoted the time step t, and the predictor step is where, r presents the position of the particle. For velocity v, v is acceleration and the corrector step is calculated as following:

v n+1/2 = v n + t • vn (3.35)
r n+1/2 = r n + t • v n + 0.5( t)
v n+1 = v n+1/2 + 0.5 t • (v n+1/2 -vn ) (3.37)
here, vn+1/2 is calculated using the predicted values. The density ρ, energy e and deviatoric stress S are calculated by the same steps with velocity v. In this scheme, the time step is evaluated using the following equation

t = c s h c i + |v i | (3.38)
where c i is the elastic wave speed. c i can be valued as c i = E/ρ, where E is Young's modulus of the material. c s is a coefficient, taken as 0.1 ∼ 0.3.

3.4/ THE APPLICATION OF SPH CODE IN SOLIDS

3.4.1/ ELASTIC BEAM EXAMPLE

In this sample, the plane stress assumption is applied for the slender beam, therefore a two-dimensional bending problem is simulated by SPH method with kernels listed in According to [START_REF] Lin | Nonlinear transient analysis of isotropic and composite shell structures under dynamic loading by SPH method[END_REF], when the loading force F max = 1.75kN, the maximum load is not very large and thereby the beam deflection can be considered as a linear change with load.

The reference deflection can be estimated by the static solution reported by Timoshenko [START_REF] Sp Timoshenko | Mechanics of materials[END_REF] with W analytical = FL 3 /3EI + 6FL/5GA = 33.59mm. The average errors e av with different kernels are listed in Tab.3.4, ranging from 17.62% to 2.25%. deflection W c is non-linear related with loading force, and its analytical deflection is W analysical = 81.1mm at the end of time according to Timoshenko [START_REF] Sp Timoshenko | Mechanics of materials[END_REF] and Lin [START_REF] Lin | Nonlinear transient analysis of isotropic and composite shell structures under dynamic loading by SPH method[END_REF].

Different numerical phenomenon with above case F max = 1.75kN, when the beam bends to nearly maximum deflection W analytical = 81.1mm with a large loading force, numerical instabilities with varying degrees happen when using different kernels, as shown in Fig. 3.8.

Obviously, when using the type-2 and type-3 kernel, the non-physical cracks emerge even increasing the coefficient of artificial stress to 0.9 (Tab.3.3), which directly lead to wrong simulations. When applying type-5 kernel, the particles clump severely and cause the pairing instability in compressed part of the beam. The pairing instability also occurs in the case with type-1 kernel. Although the slightly tensile instability still exists in the fixed end of the beam with type-4 and type-6 kernel, it is not very severe, especially the average errors e av is 4.89% using type-4 kernel.

Fig. 3.9 and 3.10 illustrate the energy time history for the 6 different kernel functions. Because the external force is loaded during the whole of beam deformation with the form in Fig. 3.5, the total energy consisting of kinetic and strain energy continue to increase with time. Then the free end of beam tends to keep swing at the point of the maximum deformation, which means the kinetic energy should gradually decrease to zero and deformed energy to the maximum. As shown in energy plots, the oscillation of the curves in kinetic energy and total strain energy tend to weaken except for the cases in type-3 and type-2 kernels. It is remarkable that kinetic energy increases rapidly and strain energy is opposite to decrease quickly at the end of calculating time when using type-3 and type-2 kernels. Additionally, it is worth noticing that the values of kinetic energy using type-5 kernel are always smaller than using other kernels because of the severe pairing instability observed in Fig. 3.8 (e). At the same time its strain energy stabilizes to higher values than others illustrated in Fig. 3.10. The similar case is also found from blue curve in kinetic energy plots connecting with Fig. 3.8 (a) when using type-1 kernel which shows slight particles disordering. There is no much difference between the cases of type-4 and type-6 kernels in Fig. 3.8 (d) and (f) and their kinetic energy plots in Fig. 3.9. But the lower strain energy in type-4 kernel case can be noticed than using kernel-6 in Fig. 3.10. In addition, Fig. 3.11 also shows that the simulation with type-4 kernel is in agreement with the results of both SPH code in [START_REF] Lin | Nonlinear transient analysis of isotropic and composite shell structures under dynamic loading by SPH method[END_REF] and FEM method in [START_REF] Lin | Nonlinear transient analysis of isotropic and composite shell structures under dynamic loading by SPH method[END_REF] in terms of the distribution Petschek [START_REF] Larry | Smooth particle hydrodynamics with strength of materials[END_REF], using the same geometry and material parameters. The simulations are carried out in 2D. Perfectly-plastic behavior is applied coupled to the Mie-Gruneisen EoS.

Material parameters are provided in Tab.3.5. Results of the simulations are compared with simulations conducted with the EPIC-2 code in [START_REF] Larry | Smooth particle hydrodynamics with strength of materials[END_REF]. The rod is discretized by 2050 SPH particles with an initial space dx of 1.058mm and a smoothing length h of 1.5dx. The computational total time is 50µs, starting when the rod interacts with the rigid surface. Fig. 3.13 compares the rod shapes at the end of the impact using different kernel functions, from which it is observed obviously that the simulation using kernel type-4 is the closest to the result by EPIC-2 code. The simulation using The 3D Taylor impact test consists in a cylindrical rod of 37.97mm long and 7.595mm diameter impacting a rigid surface. The structure is made of 4340 steel material and with a velocity of 181 m/s. The Johnson-Cook constitutive law is applied to the rod. The same examples are also studied at an experimental level by House and his colleagues in [START_REF] House | Estimation of flow stress under high rate plastic deformation[END_REF] and at a numerical level by Batra in [START_REF] Romesh | Modified smoothed particle hydrodynamics (msph) basis functions for meshless methods, and their application to axisymmetric taylor impact test[END_REF] The material parameters are listed in Tab.3.6.

The values of parameters in SPH algorithm of our simulations are illustrated in Tab.3.7. for simulating impact problems in literature [START_REF] Romesh | Modified smoothed particle hydrodynamics (msph) basis functions for meshless methods, and their application to axisymmetric taylor impact test[END_REF]. Fig. 3.17 also shows the comparison on time histories of rod length during the impact between our simulations and MSPH. In terms of this value, there is not much difference caused from variable kernels, and the final length at the end of impact is acquired with a value near 34.3mm using all of six kernels in our simulations, which is matched well with 34.7mm by MSPH and 34.5mm by LS-DYNA FE-code, respectively. All the errors of SPH numerical simulations compared with experimental datas are shown in Fig. 3.18 and 3.19. Fig. 3.18 shows using type-4 kernel the accuracy is the best (4.4%) while the traditional ones type-1 to type-3 (B-spline and Gaussian functions) result in more 10% errors. Although the error on the final length is the largest when using type-4 kernel shown in Fig. 3.19, its value is only about 1.05%

which is a vary satisfactory result.

3.5/ DISCUSSION

In this section, a discussion connecting the simulations in above section with some research on analysis for kernels' performances is conducted hereafter.

3.5.1/ EFFECTS OF KERNEL TYPES ON THE INSTABILITY

Tests in last section show that with the deformation increasing, the severe particles clumping resulting from nonphysical cracks still emerges even using artificial stress combined with CSPH and XSPH. The results also demonstrate the distinguishing performances on particles instability influenced by kernel types. Section 3.2.2 has provided the instability analyses associated with kernel properties and this section would discuss again combining with the numerical results.

Swegle gave stability and instability regimes for B-spline kernel [START_REF] Jw Swegle | Smoothed particle hydrodynamics stability analysis[END_REF], which can be a reference for the situation in example 4.1. Using type-1 and type-3 kernel, both of them as one of B-spline kernel, shows the particle pairing in maximum deflection of beam in Fig. 3.8, especially type-3 kernel causing severe nonphysical cracks. Fig. 3.2 shows the shapes of type-3 kernel which can represent the common properties of B-spline function. According to the point from Eq. (3.1) , some researchers modify the shapes of kernel functions to avoid the particles dropping into the instability regimes of kernels. Yang X in [START_REF] Yang | Smoothed particle hydrodynamics modeling of viscous liquid drop without tensile instability[END_REF] introduced that hyperbolic shaped kernel function (type-6 kernel in Tab.3.1) which possesses non-negative second derivatives can completely remove the pairing instability in compressed regimes. The curve of type-6 kernel has been tested and successfully achieved simulating results in viscous liquid drop problems in [START_REF] Yang | Smoothed particle hydrodynamics modeling of viscous liquid drop without tensile instability[END_REF]. For solid problems, the beam deflection tests in Fig. 3.8 illustrate that using hyperbolic shaped kernel function can indeed achieve more stable simulation in compressed part compared with other kernels. However, because of non-negativity in whole of kernel impact domain, this func- Figure 3.17: The time history of rod length in our simulations using 6 types of kernels, and the simulations by MSPH and LS-DYNA in [START_REF] Romesh | Modified smoothed particle hydrodynamics (msph) basis functions for meshless methods, and their application to axisymmetric taylor impact test[END_REF].

tion is limited into compressed problems and can't ensure the stability for large stretched problems. Additionally, it can be noticed that the type-4 kernel named Wendland function involving similar second derivatives with B-spline kernel (seen Fig. 4.20) can also get smooth and stable results in large deformation both in compressed and stretched parts of bending beam, as shown in Fig. 3.8 and 3.11.

Particles instability of course influences a lot in system energy shown in Fig. 3.9 and 3.10.

The nonphysical cracks due to severe pairing instability can lead to the beam cut and free at the maximum deformation and thereby release partial deformed energy such as Fig. 3.8 (b) and (c). It results in the kinetic energy of free part of beam still increasing with the existing loading in Fig. 3.9, and the total strain energy decreasing quickly in Fig. 3.10.

In the cases without obvious fracture, the values of kinetic energy using type-5 and type-1 kernels are smaller than the ones using kernels type-4 and type-6. At the same time, more pairing instability can be found in particles using type-5 and type-1 kernels and they also show higher strain energy compared to type-4 and type-6 kernels. This might be explained that pairing instability with disordering deformation among particles can consume higher deformed energy and thereby reduces the kinetic energy of whole system. This assumption is in accordance with the work of Maronne and Colagrossi [START_REF] Marrone | Prediction of energy losses in water impacts using incompressible and weakly compressible models[END_REF], in which authors conclude that the numerical cavitation due to tensile instability affects the energy decay, the mechanical energy then remaining higher than other solvers. In addition, although there is no obvious difference between the cases using type-4 and type-6 kernels, illustrated in Fig. 3.8 (d) and (f), the lower strain energy in type-4 kernel case in Fig. 3.10 seems imply more smoothness among particles. Therefore, the same result by our simulating tests is obtained with Dehnen's analysis concluding that Wendland kernel can make the simulations more stable both in stretched and compressed sections.

3.5.2/ EFFECTS OF KERNEL TYPES ON THE ACCURACY

Section 2.4 illustrates an error estimation concerning the parameters of kernels proposed by Sigalotti in [START_REF] Di | A new insight into the consistency of the sph interpolation formula[END_REF], which also demonstrates the positive view on SPH accuracy when using Wendland kernel. The simulations in Taylor tests in section 4 indeed show that using type-4 kernel can get more reliable accuracy.

Based on error estimation in [START_REF] Di | A new insight into the consistency of the sph interpolation formula[END_REF], the values of A w (Tab.3.1) for type-3 kernel is the largest both in 2D and 3D, which is possible to lead to a worse accuracy in SPH particle approximation and then influences the performance in SPH method. The phenomenon in numerical examples in section 4 is matched with this theory in some points: the most severe particles instability in bending beam test (Fig. 3.8) and the largest errors in Taylor tests when using type-3 kernel (Fig. 3.13 and 3.16). In addition, values of A w from type-1 kernel are close to the values from type-2 kernel both in 2D and 3D, but these two kernels have different performances in instability and accuracy (type-1 seems better much than type-2 in section 4 examples), a possible reason is that the compactness of type-2 kernel is not strictly satisfied (Eq. (3.6) ) in spite of its sufficiently smooth even for high orders of derivatives. This discussion doesn't include type-5 and type-6 kernels, because a limited understanding about the universality of this estimated means.

78CHAPTER 3. THEORETICAL POINTS: DEVELOPMENT OF A SPH CODE TO DYNAMIC SOLIDS

Type-5 and type-6 kernels are new types and constructed for specific problems having gotten good simulating results. It also noticed that in solid dynamics, using type-6 kernel has a very good stability like type-4 as illustrated in Fig. 3.8 and a second accuracy only to the case using type-4 kernel in Fig. 3.18, but it completely does not work for 2D

Taylor impact problems shown in Fig. 3.14. Concerning type-5 kernel which is characterized by having enough smoothness in any order of derivatives, it yet cannot satisfy the requirement in accuracy, the example shown in Tab.3.4, and still does not have a good performance in pairing instability in Fig. 3.8. Therefore, it is difficult to find a kernel function perfectly fit to all situations. By observation and discussion, combined numerical examples with analysis research in literature, the conclusion is acquired that the Wendland function is a better and more reliable choose as SPH kernel in solid dynamic problems.

3.6/ CONCLUSION

Kernel function is one of the most important concepts in SPH algorithm and its properties cause unignorable influences for the results of simulations. In addition to some basic conditions linked to the definition of SPH formulations, like normalization, compactness and Dirac delta property et., other mathematical features like the shape of the second derivative of the kernel, the sign of its Fourier transform, as well as several constant parameters also play insignificant roles in SPH overall performances. This chapter compares a few kernels' performances with applications in dynamical solids problems, including several conventional kernels like B-splines function, Gauss function, Wendland functions and two new types (double cosine function and Hyperbolic shaped function).

It is well known that the pairing instability is a serious problem in developing SPH applications in simulating solids. Based on our numerical tests, the choice of suitable kernel function is able to improve the pairing instability in some degree, as well as obtains a better accuracy. Thus, Wendland function appeared to be a good candidate as SPH kernel in simulating elastic-plastic problems. internal organs have been developed and investigated, such as epoxy resin [START_REF] Ak Dabrowska | Materials used to simulate physical properties of human skin[END_REF], ballistic gelatin [START_REF] Martin | The wound profile: a visual method for quantifying gunshot wound components[END_REF][START_REF] Ds Cronin | Ballistic gelatin characterization and constitutive modeling[END_REF], and polymers [START_REF] Bracq | Experimental study of the strain rate dependence of a synthetic gel for ballistic blunt trauma assessment[END_REF][START_REF] Fontenier | On the mechanical characterization and modeling of polymer gel brain substitute under dynamic rotational loading[END_REF]. One way to overcome these issues is to use numerical simulation as an efficient way of investigations, but also include difficulties such as accurate models for materials properties, micro-scale computation, and

5.

A discussion about the suitability and limitations of this numerical model is provided.

4.2/ ANALYTICAL MODELS FOR PENETRATIONS

The analytical model is one of the most important approaches to understand the impact process. The formulations of analytical models incorporate some essential and relevant physical and mechanical theories and parameters. The various models that describe the behavior of a spherical fragment penetrating soft materials have been developed and employed in the literature such as [START_REF] Larry M Sturdivan | A mathematical model of penetration of chunky projectiles in a gelatin tissue simulant[END_REF][START_REF] Wijk | Sphere penetration into gelatine and board[END_REF][START_REF] Mj Lighthill | Boundary layer theory[END_REF][START_REF] Steven | Modeling the penetration behavior of rigid into ballistic gelatin[END_REF][START_REF] Monzer | Sph-based method to simulate penetrating impact mechanics into ballistic gelatin: Toward an understanding of the perforation of human tissue[END_REF]. Generally, the suitability of analytical models depends on the physical conditions, experimental fitting, as well as empirical data. Therefore, it is possible that the analytical models represent the different performances when the penetrations happened in the macro-scale and micro-scale. The Poncelet equation describing the complete trajectory of the projectile is the most widely [START_REF] Steven | Modeling the penetration behavior of rigid into ballistic gelatin[END_REF][START_REF] Veysset | High-velocity micro-particle impact on gelatin and synthetic hydrogel[END_REF] and to some numerical simulations [START_REF] Monzer | Sph-based method to simulate penetrating impact mechanics into ballistic gelatin: Toward an understanding of the perforation of human tissue[END_REF]. The equation is written in the [START_REF] William A Allen | Dynamics of a projectile penetrating sand[END_REF] as,

-M V = BV 2 + C (4.1)
where, B and C are considered as the inertial drag and the strength resistance terms.

Given a initial velocity v 0 , the penetration depth P can be solved as a function of the instantaneous velocity v:

P(t) = M B [lncos( √ BC M (T f -T )) -lncos( √ BC M T f )] (4.2)
where T f is the final time at which the velocity of the projectile decreases to zero, which can be calculated by,

T f = M √ BC tan -1 (v 0 B C ) (4.3)
where M is the mass of the sphere projectile, and B and C defined as being two important material properties in [START_REF] Wijk | Sphere penetration into gelatine and board[END_REF] are defined as,

B = ρ t πd 2 C d 8 (4.4) C = Rπd 2 4 (4.5)
where ρ t and R are, respectively, the density and the strength resistance of the gelatin.

C d is the inertial drag coefficient and d is the diameter of the projectile. Combined with the Eq. (4.3) , the maximum penetration depth P max can be obtained from the Eq. ( In the work by Wijk et al. [START_REF] Wijk | Sphere penetration into gelatine and board[END_REF], the strength resistance R was found to be 0.52 MPa for 10wt% gelatin and 3.5 MPa for 20wt% gelatin. Intuitively, the strength of gelatin material should increase with percentage. As for the drag coefficient C d , its value is recommended to 0.35 at 24 • C and 0.375 at 10 • C in [START_REF] Wijk | Sphere penetration into gelatine and board[END_REF].

In the investigation of micro-particle (silica spheres with 7.38 µm diameter) impact 10wt%

gelatin in [START_REF] Veysset | High-velocity micro-particle impact on gelatin and synthetic hydrogel[END_REF], the the value for C d = 0.4 was taken from [START_REF] Schlichting | Boundary-layer theory[END_REF] and the parameter R was obtained by fitting maximum penetration data. According to this study in [START_REF] Veysset | High-velocity micro-particle impact on gelatin and synthetic hydrogel[END_REF], the strength resistance R was extracted by the following relation,

P Exp max = 2ρ p d 3ρ t C d ln( ρ t C d v 2 0 2R + 1) + d 2 (4.7)
where the P Exp max is the experimental maximum penetration. It can be noticed that the above equation includes a term of d/2 compared with the Eq. (4.6) , which takes into account the particle's 'nose' contribution. A value of R = 21 MPa was obtained by the way of Eq. (4.7) and in turn used to calculate the complete particle trajectory in Eq. (4.2) .

It is worth noticing that, for the same target material, the value of strength resistance R = 21 MPa is more than 40 times on the macro scale (0.52 MPa), which could be explained by the sample strain rate sensitivity that is exhibited from the macro to the micro-scale.

Generally, a higher strain rate (markedly more than 10 6 s -1 ) can lead to higher hardness in impact [START_REF] Hassani | Material hardness at strain rates beyond 106 s-1 via high velocity microparticle impact indentation[END_REF], which manifests by a higher resistance. This observation is the most important result in the applications of analytical models for macro-penetrations and micro- its generalization is still questionable. Indeed, the Poncelet equation involves a constant resistance R whose variation with strain rate is unknown. A review paper [START_REF] Jr | Analytical models for penetration mechanics: a review[END_REF] claims that the Poncelet model is one of the most classic analytical models for penetration mechanics, but the authors confirmed that a better fit to the experimental data is achieved only if very high-velocity impacts are not included. The authors in [START_REF] Jr | Analytical models for penetration mechanics: a review[END_REF] also explained that the Poncelet parameters should change during penetration when the very high-velocity impacts occur.

To confirm this point, Fig. 4.2 shows a series of penetration curves of macro-sized spheres impact into 20wt% gelatin by experiments and the Poncelet equation. The experimental data are extracted from [START_REF] Larry M Sturdivan | A mathematical model of penetration of chunky projectiles in a gelatin tissue simulant[END_REF] and the parameters in the Poncelet equation [1] are C d = 0.33 and R = 3.5 MPa from [START_REF] Wijk | Sphere penetration into gelatine and board[END_REF]. It can be noticed that for macro-scale impacts, a same combination of parameters (C d = 0.33, R = 3.5 MPa) in Poncelet equation demonstrates important discrepancies in the case of d = 2.38 mm projectile and v=2229 m/s initial velocity. An important error emerges in this case compared with the other penetrations involving larger projectile and lower impact velocities. On the contrary, an increase for the drag and resistance terms in Poncelet equation [2] with C d = 0.35 and R = 5 MPa can acquire a better agreement between experimental result and Poncelet curve. The high velocity such as 2229 m/s in the above example leads to a sufficiently high strain rate.

Therefore, this test justifies the tendency to have R increase with strain rate. According to [START_REF] Hassani | Material hardness at strain rates beyond 106 s-1 via high velocity microparticle impact indentation[END_REF], both the decrease in dimension scale and the increase in impact velocity can achieve high strain rates, even beyond 10 6 s -1 , and the high-velocity impact of microparticles typically belongs to this phenomenon. Hence, the applicability and generalization of the Poncelet model should be further researched, particularly in micro-sized penetration with high velocity.

4.2.2/ A COMPARISON ON THREE MODELS

Generally, the suitability of analytical models depends on the physical conditions, experimental fitting, as well as empirical data. Considering the uncertainty of the analytical models in micro-penetrations, an investigation on the performance of three analytical models is performed in this section.

4.2.2.1/ THE FORMULATIONS OF LIU AND MODIFIED-CG MODELS

By the above discussion, it can be seen that the impact strain rate seems a key to the applications of the Poncelet model. A few analytical models considered the effects of impact strain rate such as [START_REF] Steven | Modeling the penetration behavior of rigid into ballistic gelatin[END_REF][START_REF] Liu | A spherical cavity expansion model of large elastic deformation and its application to ballistic gelatin penetration problems[END_REF]. Liu et al. [START_REF] Liu | A spherical cavity expansion model of large elastic deformation and its application to ballistic gelatin penetration problems[END_REF] developed an analytical model involving the dependent-strain-rate resistance and a viscous term to investigate the trajectory of a projectile in 20 wt % gelatin. In the recent study of Veysset et al. [2], a Modified-CG model was developed to model the penetration trajectories in SEBS gel material in the micro-scale. It considers the viscosity term involved in the drag coefficient and was proven to have a better performance than the Poncelet model. Firstly, their formulations are represented here.

• Liu's model

This total force to projectile is decomposed into three parts: an inertial component from the traget material including a constant frag coefficient C d ; a viscous component representing the friction on the surface of the fragment, including a constant coefficient C v ; a resistance owing to the inherent structural characteristics of the target material. It follows the form as,

-M V = F I + F v + F T (4.8)
where, every component can be expressed as,

F I = 1 2 C d A 0 v 2 F v = C v A 0 v F T = A 0 σ 0 [1 + ( ˙ ˙ 0 ) 2 ] (4.9)
where, the resistance part is depending on a strain-rate. The analytical expression on time history and maximum penetration depth can be acquired, seen in [START_REF] Liu | A spherical cavity expansion model of large elastic deformation and its application to ballistic gelatin penetration problems[END_REF].

• Modified-CG model

The Modified Clift-Gauvin model is based on an assumption that the Reynolds number Re < 10 5 during the experimental penetrations, combined with a empirical form for C d from the work of [START_REF] Clift | The motion of particles in turbulent gas-streams[END_REF]. The total force is expressed as,

-M V = 1 2 C C-G d ρ s Av 2 + AR (4.10)
Where, the drag coefficient is followed by, Where, Re = ρ s vD/η involves the dynamic viscosity η. Therefore, viscosity η and resistance R are coefficients required to be fitted from experimental data. Additional, in return, the fitted value of η should satisfy the precondition of Re < 10 5 in this model.

C C-G d = 24

4.2.2.2/ EMPLOYMENTS OF THESE MODELS

The Poncelet and Liu formulas can obtain the analytical solutions for penetration history P(t) and maximum penetration depth P max . Numerical iteration in the Modified-CG model is required to get the penetration trajectory. Tab.4.1 lists these three models including their coefficients required fitting, physical meanings of terms, as well as practical application.

Two kinds of fitting approaches * a) and * b) through experimental data are introduced:

1. * a) : fitting time-penetration By fitting a whole trajectory (Time-penetration curve) to acquire the constant parameters in analytical models, then using these fitted parameters to simulate other trajectories or maximum penetration depths.

* b) : fitting maximum-penetration depths

By fitting a large number of maximum penetration depths from impacts involving various diameters and velocities to acquire the constant parameters in analytical models, then using these fitted parameters to simulate other trajectories or maximum penetration depths.

Table 4.1: The list for three analytical models to penetrations.

Poncelet Liu Modified-CG [163] [168] [2] Resistance R σ 0 f (˙ ) R Drag coefficient C d C d C C-G d Viscosity no C v η in C C-G d Analytical solution P(t), P max P(t), P max No Applied scale macro/micro macro micro Fitting parameters R, C d σ 0 ,C d ,C v R,η
The coefficients in analytical models are extracted via fitting with experimental data. The Fig. 4.4 shows the fitting results for three analytical models via the * a) approach.

In every subfigure, the curve2 with red solid style is acquired by fitting with experimental data, and then values of coefficients in the analytical model are obtained. Next, the other two analytical curves using these fitted coefficients are compared with experimental trajectories as the validations. The values of fitted coefficients for three models are

(C d =0.8708, R=36.3 MPa) in Poncelet, (C d =0.3966, C v =6.8112e-6, σ 0 =23.2 MPa) in Liu's
model, (η=7.8911e-2, R=16 MPa) in Modified-CG, all of them locating in the reasonable range. It can be seen that the Poncelet equation performs badly in modeling the curve1 with a lower velocity and bigger projectile, and both Liu and Modified-CG models can not fit well to the curve3 with a higher velocity and smaller size of projectile. Therefore, it has to be admitted that using only one experimental trajectory to fit the analytical model can not get universal coefficients to keep the accuracy for modeling other penetration trajectories. All of these three analytical models illustrate a similar situation.

The analytical models are fitted via the * b) approach using the experimental data in fitting approach can get more stable and universal fitting coefficients for analytical models than the * a) approach. This is because that the * b) fitting approach uses more experimental data including a large diversity on projectile size and impact velocity, while the * a) approach uses the information only from one penetration trajectory. the SPH model, the area located far from the impact center where involves small deformations is discretized by finite elements while SPH particles are distributed in the dominant penetrating part. The projectile is regarded as a rigid sphere and also modeled by finite elements. The contact is based on master surface-slave nodes approach between projectile elements and particles in impact center (TYPE7 which is widely used for HVI problems in the commercial software Radioss.) Interface Type2 in Radioss is used for the modeling of the interaction between the SPH domain and the FEM domain of the target.

So, the link between the FEM elements and SPH particles constructed with a kinematic interface is well defined and provide a correct extension of the gelatin sample. More details of Type7 and Type2 contact algorithms can be seen in [213]. L sph defines the size of the whole SPH domain in the whole model. dx i is the discretization length of the SPH components, and it is increased gradually with the distance to penetrating center ( the same strategy was adopted for macroscale simulation in [START_REF] Taddei | Smoothed particle hydrodynamics formulation for penetrating impacts on ballistic gelatine[END_REF]).

4.3.2/ MATERIAL CONSTITUTIVE LAW

The hyper-elastic or visco-elastic properties of soft materials like gelatin and gels have been recognized, but the motion of the projectile in a soft tissue target can be described by yield-stress fluids according to several studies such as [START_REF] Ralph | Motion of spherical particles in a bingham plastic[END_REF]2]. That means the deformations of soft tissues involve both solid-like behavior and Newtonian fluid-like behavior during the penetrations. The target material represents an elastic-plastic deformation like solids under law-velocity impact and then transition to hydrodynamic movements like fluids when its mechanical stresses exceed a yield strength under higher velocity. Therefore, an elastic-plastic hydrodynamic constitutive law is employed for the target materials. In terms of modeling the solid-like behavior, the linear-elastic plasticity model is used, and the yield-stress in soft materials follows the form,

σ y = A + Bε n p (4.12)
where σ y and ε n p are, respectively, the yield stress and the effective plastic strain. The apparent yield stress A, the hardening parameter B, and the hardening exponent n are the constant values related to the plastic property of materials. About failure, working on ballistic gelatin, Moy et al.in [START_REF] Moy | Tensile deformation and fracture of ballistic gelatin as a function of loading rate[END_REF] showed a very small plasticity domain : There is a notable lack of plastic deformation in the gelatin. The specimens were subjected to large strains and after failure, the ballistic gelatin exhibited little to no plastic deformation.

Knowing that the SEBS gel has a similar behavior, it is assumed that SEBS gel breaks after reaching maximum stress which is close to the yield-stress value.

Besides, the Mie-Gruneisen EOS is applied to describe the non-linear pressure evolution, considering that the hydrodynamic effect becomes predominant upon the impacts into soft tissues at high impact velocities . The equation has the following form:

P =              3 k=0 C k k > 0, C 0 + C 1 < 0.
                                 C 0 = P 0 C 1 = ρ 0 c 2 0 -0.5Γ 0 P 0 C 2 = (2S -1)C 1 C 3 = (S -1)(3S -1)C 1 (4.15)
According to [START_REF] Awoukeng-Goumtcha | Investigations of impact biomechanics for penetrating ballistic cases[END_REF], the C k coefficients depend on the parameters ρ 0 , c 0 , S , E 0 and Γ 0 , which are, respectively, the target materials density, the sound of speed, the slope of the linear Shock Hugoniot function, the initial volumetric internal energy, and the Gruneisen parameter.

The physical problems simulated by this model are based on a series of experiments on 10 wt% gelatin conducted in [START_REF] Veysset | High-velocity micro-particle impact on gelatin and synthetic hydrogel[END_REF] and 40 vol% SEBS gel conducted in [2]. The parameters about these two kinds of soft materials as simulant of soft tissues used in this constitutive law are represented in Tab.4.3. In the numerical model, the projectile is considered as a non-deformable body and modeled with a linear elastic constitutive law and a high value for the Young modulus. The material parameters of the projectiles are shown in Tab.4.4.

Table 4.3: Material parameters of the target materials. The value of Young's modulus underlined will be investigated combined with numerical simulations. The value of apparent yield stress A is equal to the ballistic resistance which is extracted from the study in [START_REF] Veysset | High-velocity micro-particle impact on gelatin and synthetic hydrogel[END_REF] for gelatin and [2] for SEBE gel. Young's modulus E and apparent yield stress A can represent the material resistance to some degree, but there is no direct correlation between them. The value 21 MPa of apparent yield stress A in Tab.4.3 is extracted from the literature [START_REF] Veysset | High-velocity micro-particle impact on gelatin and synthetic hydrogel[END_REF]. Based on this investigation, the target's resistance in micro-particle penetration is raised to 21 MPa compared with the value of 0.52 MPa obtained in macro-scale experiments using the same target. Therefore, the Young's modulus is predicted to a range of 1-100Mpa in the micro-scale, considering an increase on the strength of target materials. For the target of SEBS gel, the similar range for the values of Young's modulus is predicted taking into account a value of 0.12 MPa used as constant Young's modulus in the study of [START_REF] Shen | Numerical modeling of a human tissue surrogate sebs gel under high velocity impacts: investigation of the effect of the strain rate in an elasto-hydrodynamic law[END_REF], as well as a resistance value in the Modified-CG analytical model [2].

4.4/ SIMULATIONS ON BALLISTIC GELATIN

The simulation results are obtained with the commercial software RADIOSS, and validated by the experimental data derived from the laser-induced particle impact test (LIPIT) by Veysset et al. in [START_REF] Veysset | High-velocity micro-particle impact on gelatin and synthetic hydrogel[END_REF]. In these experiments, silica spheres (micro-scale diameter)

were accelerated from 200 to 1500 m/s and penetrated into gelatin sample. Fig. 4.8 shows a typical image sequence of a silica particle (d=7.38 µm) impact on a 10wt% gelatin sample at a velocity of 1290 m/s and its experimental trajectory is extracted from [START_REF] Veysset | High-velocity micro-particle impact on gelatin and synthetic hydrogel[END_REF], as a validation for the following simulations.

4.4.1.2/ ARTIFICIAL VISCOSITY AND XSPH

Artificial viscosity and XSPH (also called Conservative Smoothing) were developed in order to ensure numerical stability of the SPH method, but both of them can generate unavoidable and unphysical energy dissipation. Shaw et al. [START_REF] Shaw | Heuristic acceleration correction algorithm for use in sph computations in impact mechanics[END_REF] showed that it easily makes the system over-dissipative if unsuitable artificial viscosity parameters (qa, qb) are used.

Studies in the literature also indicate that artificial viscosity has an important influence on the penetration history [START_REF] Taddei | Smoothed particle hydrodynamics formulation for penetrating impacts on ballistic gelatine[END_REF][START_REF] Gordon R Johnson | Artificial viscosity effects for sph impact computations[END_REF]. between the numerical and experimental trajectories, (0.5,0.2) is better for the values of (qa, qb) (see Fig. 4.11), which is in accordance with previous studies on macro-scale particles impacts in gelatin [START_REF] Taddei | Smoothed particle hydrodynamics formulation for penetrating impacts on ballistic gelatine[END_REF].

Likewise, we investigate the effects of the XSPH coefficient α cs on penetration trajectories(see the Fig. 4.12). A larger value of α cs leads to a smaller final penetration compared to the experimental result because of more kinetic energy loss among particles. The XSPH term is regarded as an approach to mitigate the tensile instability in solids. What can be noticed in Fig. 4.12 is XSPH coefficient which seems to affect more at the end than at the beginning of impact. It can be explained by the fact that at the end of impact, when the penetration velocity is small, the elastic-plastic behavior is more important and tensile instability happens, which may be the cause of poor accuracy. Then, a value of α cs = 0.015 is chosen as a suitable value for the microparticle penetration problem. . This is because that a value of C d = 0.375 can keep a consistency between maximum penetration length Eq. (4.6) and the whole time history of penetration Eq. (4.2) . It

4.4.2.2/ CAVITY EVALUATION AND DISCUSSION

Another important point in the phenomenon of gelatin perforation by a projectile is the size of the cavity. Although the penetration is a very important parameter to investigate, the size of the cavity in the radial dimension is also of interest. As it can inform, for instance, about the extent of tissue damage following projectile penetration. Fig. 4. [START_REF] Li | Meshfree and particle methods and their applications[END_REF] shows the shapes of the cavity obtained using the different elastic modulis considering the same silica particle impacting the target at 1290 m/s. The shape is exhibited like a slender cone that with the tunnel of time expands in both radial and longitudinal directions. It also can be observed that the cavity tends to collapse more quickly after expanding to the maximum size in the case the target sample with a larger Young's modulus ( more intuitively by the contrast between case 1 and case 4). The behavior of collapse in the temporary cavity is derived from the recovery of elastic deformations of target gelatin. several times in the application of macro-penetrations such as [START_REF] Taddei | Smoothed particle hydrodynamics formulation for penetrating impacts on ballistic gelatine[END_REF][START_REF] Monzer | Sph-based method to simulate penetrating impact mechanics into ballistic gelatin: Toward an understanding of the perforation of human tissue[END_REF]. In this thesis, for the numerical trajectories, especially at the end of penetrations. This result in the micro-scale was not observed before since the standard employment of this corrective procedure does not have such influence in the classical macro-scale [START_REF] James P Gray | Sph elastic dynamics[END_REF]. It is worthy to notice that the values of the XSPH coefficient and Young's modulus seem to represent a uniform effect on the trends of numerical penetrations at the half-section. It can be explained that an increase in Young's modulus leads to the harder target material and then more resistance stops the projectile more quickly. At the same time, an increase in the XSPH coefficient also makes a faster stop of the projectile during the penetrations, which can be surmised from multiple factors.

Firstly, at the end of penetrations, the velocity of the projectile decreases, and the target materials behaves predominately as elastic-plastic deformation, which can lead to particles clumping, and finally lead to tensile instability. Tensile instability is recognized as a severe problem when the solid materials are stretched simulated by the SPH method [START_REF] Joseph | Sph without a tensile instability[END_REF]. The XSPH also called smoothing conservative can mitigate the tensile instability by smoothing the velocity field of the particles in the influence domains controlled by the SPH kernel function, but it produces energy dissipation in the system [START_REF] Wen | Stabilizing sph with conservative smoothing[END_REF]. That is the reason why XSPH coefficient produce larger discrepancies between experimental and numerical results at the end of the penetration than at the beginning. Secondly, the function of the XSPH also can be regarded as adding a kind of friction to the interface between projectile and target material. The retardation force to the projectile coming from the target material consists of inertia force, resistance force, as well as friction part according to the analytical study for penetrations [START_REF] Liu | A spherical cavity expansion model of large elastic deformation and its application to ballistic gelatin penetration problems[END_REF]. Therefore, it is perhaps also a factor that the higher values of the XSPH coefficient provide more friction to make the projectile stop quickly.

4.6.2/ MATERIALS PARAMETERS

The linear-elastic hydrodynamic law is attempted to study the ballistic behavior of gelatin and SEBS gel. The first consideration to use this is that the movement of soft material under high-velocity impact has a fluid hydrodynamic behavior especially at the beginning of penetration, which has been reported in various research like [START_REF] Holzapfel | Numerical prediction of damage in composite structures from soft body impacts[END_REF][START_REF] Wen | Impact of steel spheres on ballistic gelatin at moderate velocities[END_REF]. The second concern is that a macro-scale investigation to SEBS gel by [START_REF] Shen | Numerical modeling of a human tissue surrogate sebs gel under high velocity impacts: investigation of the effect of the strain rate in an elasto-hydrodynamic law[END_REF] demonstrated a small difference between using a linear-elasticity and nonlinear-elasticity depending on the strain rate. Moreover, the experimental and analytical observation for gelatin under micro-impacts [2,[START_REF] Hassani | Material hardness at strain rates beyond 106 s-1 via high velocity microparticle impact indentation[END_REF] discovered a multiplier increase on the strength of target materials when the penetrations involve a higher strain rate, especially the dimension transitions from macro-scale to micro-scale.

Therefore, during the simulations, the initial Young's modulus of ballistic gelatin was predicted in the range of 1-100 MPa referring the value of E = 1 Mpa used for the macro penetrations [START_REF] Monzer | Sph-based method to simulate penetrating impact mechanics into ballistic gelatin: Toward an understanding of the perforation of human tissue[END_REF]. Subsequently, the range of Young's modulus was narrowed down to a more suitable value to predict the general trajectories of penetrations. The final values of Young's modulus for 10 % gelatin are 20-40 MPa, which is matched with this trend.

The same range of 1-100 MPa was for ballistic SEBS gel under micro-impacts compared to 0.12 MPa used in the macro cases [START_REF] Shen | Numerical modeling of a human tissue surrogate sebs gel under high velocity impacts: investigation of the effect of the strain rate in an elasto-hydrodynamic law[END_REF]. The final values of Young's modulus for 40 vol% SEBS gel are 40-50 MPa. According to the investigation of the relation between mechanic properties and penetrations for SEBS gel in the macro-scale [START_REF] Mrozek | The relationship between mechanical properties and ballistic penetration depth in a viscoelastic gel[END_REF], the authors described the penetration depths in SEBS gel using the elastic Froude number involving Young's modulus ranging from 60 to 630 kPa on the macro-scale. Therefore, a range of The other concern is the high strain rate which is identified in the cases involving the micro-scale and the high-velocity impact. It also can be noticed that in the Fig. 4.25, the accuracy of numerical result decreases at the case with a higher impact velocity (close to 900 m/s), while the modified-CG model demonstrates a better fit in this point, as seen in the Fig. 4.5. It can be explained that the viscosity becomes more important in the micropenetration with a higher strain rate, which is taken into account into the modified-CG model [2]. Therefore, a more accurate constitutive law for ballistic gelatin and SEBS gel including the strain rate, viscosity, and even failure criterion should be further developed. deducting the retardation force from different parts including inertia, resistance, and viscosity or friction. But, for the penetrations in the micro-scale involving more complex mechanics, especially, the strain rate higher than 10 6 s -1 , the robustness of their coefficients highly depends on the fitting procedure requiring tremendous experimental data, which weaken the value of avoiding high experimental costs as analytical models.

The main contribution in this chapter is to develop a numerical model based on SPH method to the micro-impacts with high velocities. A large number of simulations were performed for penetrations into two kinds of soft materials involving a variety of projectile sizes from 7-22 µm with velocities from 50-1300 m/s. The numerical trajectories and maximum penetrating depths were predicted well and agreed with the experimental and analytical models. The ranges of Young's modulus for both ballistic gelatin and SEBS gel were extracted based on the agreement between numerical and experimental results.

Then, a discussion about numerical results was provided, including the numerical parameters, physical parameters, as well as the limitations of this model.

CONCLUSIONS, LIMITS, AND PERSPECTIVES 5.1/ CONCLUSIONS

Transdermal drug delivery is one of the most significant technologies in the pharmaceutical industry. This kind of techniques such as a needle-free drug delivery device typically include the process of micro-particles travelling through the soft tissues. The micro-particles size of the drug powders generally is on the range of 1-20 µm, and they penetrate into soft tissues at a high velocity and finally integrate into the soft tissues at some depths. A good understanding on several factors in this process involving the particle diameters, trajectories, and momentum or velocities of projectile, as well as the deformation of soft tissues is required in order to develop this technique and design the relevant device. At the same time, numerical simulation become one of the important approaches to study the engineering problems thanks to the large advance of computer science. Therefore, this thesis developed a numerical model based on a particle method to investigate the ballistic behavior of two kinds of simulant of soft tissues during the micro-penetrations.

Firstly, this thesis provided a state of the art on the advances in ballistic perforation impact on structures. The advantages of SPH method are convinced during this review. To sum up, the main contribution of this thesis is further expanding the application of SPH method to complex dynamic solids. One part focus on the SPH algorithm itself to study the relation between the SPH kernel types and the instability in simulating elastic-plastic solids. The other part is to develop a SPH model for the penetrations of micro-particles into soft materials (gelatin and SEBS gel), and a good investigation is achieved by SPH simulations.

5.2/ LIMITS

This thesis mainly provided an investigation on the ballistic behavior of the two soft materials (gelatin and SEBS gel) under micro-scale impacts via the numerical approaches.

The aim is to improve and develop the transdermal drug delivery techniques which involve the process of the micro-particles penetrating soft tissues. Although the predictions of penetrating parameters like trajectories and maximum depths of the projectiles are obtained by a series of simulations by an SPH model, the investigation is still limited to several points.

Soft materials like gelatin and SEBS gel are regarded as the simulant of living soft tissues to investigate the penetrating process because they have similar behavior. However, the complex environment of living soft tissues is difficult to be defined on artificial materials like gelatin and SEBS gel. The behavior model of soft tissues involves viscosity, hyper-elasticity, temperature sensitivity, hydrodynamic behavior, thermodynamic behavior, phase changes (liquid-solid). Besides, the mechanical properties of these soft tissues under high-velocity impact have been recognized as highly-sensitive to the strain rate in the literature. Unfortunately, there is no existence of an accurate constitutive law for this kind of material including so many complex properties.

Therefore, the employment of behavior law in this numerical simulation is the most challenging work. An elastohydrodynamic law was attempted to model the behavior of the gelatin and SEBS gel, and a simple failure criterion was employed to model their breaking process. Firstly, it is not enough accurate to model real physics, which leads to a limited investigation. Another difficulty is that the quantitative values of materials parameters, particularly, in the micro-scale cases are not confirmed in the literature. There is no much study on the mechanical properties of gelatin and SEBS gel in the micro-scale. The limited knowledge on the soft materials leads to the limited investigation in this numerical work:

It only can acquire a satisfying prediction to the trajectories and maximum penetration depths of projectiles, but the specific deformation of target materials can not be captured accurately. The temporary cavity is another important phenomenon during the penetrations into soft materials, and its evolution with time is strongly linked to the hyper-elasticity and viscosity properties. As an initial attempt in chapter 4, the maximum cavity size in ballistic gelatin is simulated closely to the experimental result, but the time-evolution is hard to model correctly. This limit also leads to the validations of numerical results depending on a fixed set of numerical parameters such as the SPH particle discretization and XSPH coefficient, as shown in the discussion part of chapter 4.

In addition, this numerical model also has some limitations by itself. By developing an SPH code to study the instability of the SPH algorithm, the main discovery in chapter 2 is that the types of kernel functions have considerable influences on the SPH simulation. A more suitable kernel called Wendland function is found to the solid dynamics, a better al- ing. Therefore, a more stable and accurate SPH algorithm is required to get a better understanding of the penetrating process in the micro-scale.

5.3/ FUTURE WORKS

The ballistic behavior of soft tissues under high-velocity impact by micro-particles is investigated via numerical approaches at the first time. Taking into account the conclusion and limitations, several interesting points can be further researched:

1. The first and also the most important study on the development of the behavior laws for soft materials should be paid more attention. Especially for the simulant of soft materials including ballistic gelatin and SEBS gel, the hyper-elasticity, viscosity and dependence on strain-rate should be considered. The hydrodynamic behavior, therm-dynamic behavior, phase changes (liquid-solid) as well as failure model of soft materials under the high-velocity impact loading should be further investigated. There are a few constitutive law involving hyper-elastic or viscous-elastic properties developed for some soft materials in the literature, such as a static hyperelastic constitutive for rubber [START_REF] Yang | A visco-hyperelastic approach to modelling the constitutive behaviour of rubber[END_REF], Visco-hyperelastic model for skeletal muscle [START_REF] Lu | A viscohyperelastic model for skeletal muscle tissue under high strain rates[END_REF], Neo-Hookean model, Ogden model, and Mooney-Rivlin model for soft materials like ballitic gelatin and SEBS gel [START_REF] Kim | A comparison among neo-hookean model, mooney-rivlin model, and ogden model for chloroprene rubber[END_REF][START_REF] Bracq | On the modeling of a visco-hyperelastic polymer gel under blunt ballistic impacts[END_REF][START_REF] Azeem Aslam | Ballistic gelatin lagrange mooney-rivlin material model as a substitute of bird in finite element bird strike case studies[END_REF]. Especially, an elastohydrodynamic constitutive law depending on impact strain rate for SEBS gel was proposed in a recent work [START_REF] Shen | Numerical modeling of a human tissue surrogate sebs gel under high velocity impacts: investigation of the effect of the strain rate in an elasto-hydrodynamic law[END_REF], which is a very promising behavior to improve the numerical investigation on the ballistic soft tissues in the micro-penetrations.

2.

In terms of the SPH algorithm, the tensile instability is still an important barrier, particularly, when it simulates the impact problems involving large deformations of solid Lagrangian SPH should be implemented for the penetration problems, instead of standard SPH algorithm [START_REF] Juan R Reveles | Development of a total lagrangian sph code for the simulation of solids under dynamioc loading[END_REF][START_REF] Üller | An hourglass control algorithm for lagrangian smooth particle hydrodynamics[END_REF]. In addition, the interface between the projectile and target should be modeled accurately maybe considering the surface tension, which is predicted as a key factor to capture the evolution of temporary cavity in the soft materials during penetrations [START_REF] Liu | A spherical cavity expansion model of large elastic deformation and its application to ballistic gelatin penetration problems[END_REF].

3.

To investigate the evolution of the temporary cavity in the gelatin or SEBS gel during penetrations is important to understand the deformations of soft-tissue, which is highly based on a more accurate constitutive law for target materials as mentioned.

Chapter 4.4.2 studied the temporary cavity in the ballistic gelatin, and the numerical results show that the SPH method is a very promising way to model this process, but still requires a more suitable constitutive law involving the hyper-elastic or viscous-elastic properties of soft tissues. In the drug delivery techniques, the human body penetrated by drug particles generally consists of several different soft tissues such as skin, muscle, water, inner organs, etc. The multi-layered target structure demonstrates various behaviors and then influence the trajectory of drug particles.

Therefore, to investigate the penetrating parameters into the multi-layered gelatin, or multi-layered SBES gels, or even mix-layered gelatin and SEBS gels is an important study, which is also under-researched in my group [START_REF] Monzer | Sph-based method to simulate penetrating impact mechanics into ballistic gelatin: Toward an understanding of the perforation of human tissue[END_REF]. In real applications like transdermal drug delivery, the drug particles integrate to the soft tissues gradually during the penetrations, which means the large deformations should also be considered on the projectiles. This is another interesting research in the future. In addition, as experimental observations [START_REF] Veysset | Dynamics of supersonic microparticle impact on elastomers revealed by real-time multi-frame imaging[END_REF], the micro-particles have the rebounding process after arriving at the maximum penetration depths. This phenomenon is never investigated via both analytical and numerical methods, therefore, this point will be possibly studied in the future, or based on an improved numerical model.

142

LIST OF FIGURES

2.11

The FE-SPH coupled models: Left is the elements converting to SPH particles as the material deforms increasingly from [START_REF] Gr Johnson | A combined particle-element method for high-velocity impact computations[END_REF] and Right is modeling different domains of the structure with SPH and FE, respectively, from [START_REF] Swaddiwudhipong | High velocity penetration/perforation using coupled smooth particle hydrodynamics-finite element method[END_REF]. 2.15 (a) is the sphere projectile in [START_REF] Wen | Impact of steel spheres on ballistic gelatin at moderate velocities[END_REF]; (b) is the rifle bullet projectile in [START_REF] Wen | Rifle bullet penetration into ballistic gelatin[END_REF].

(c) is the FE model for gelatin block in both of [START_REF] Wen | Impact of steel spheres on ballistic gelatin at moderate velocities[END_REF] and [START_REF] Wen | Rifle bullet penetration into ballistic gelatin[END_REF]. . . . . . . .

2.16

The SPH model combined with FE elements for soft penetration in [START_REF] Taddei | Smoothed particle hydrodynamics formulation for penetrating impacts on ballistic gelatine[END_REF]. . underlined will be investigated combined with numerical simulations. The value of apparent yield stress A is equal to the ballistic resistance which is extracted from the study in [START_REF] Veysset | High-velocity micro-particle impact on gelatin and synthetic hydrogel[END_REF] for gelatin and [2] for SEBE gel. . . . . . . and AUTODYN 2D hydrocodes [START_REF] Faraud | SPH simulations of debris impacts using two different computer codes[END_REF] • normal impact Keywords: Smoothed Particles Hydrodynamics, high-velocity impact, soft tissues, micro-impacts Abstract: Investigations on the ballistic behaviours of soft tissue materials are significant research in a medical and pharmaceutical framework. They mainly focus on particles penetrating or traveling into the soft materials. For instance, in the needle-free drug delivery, the micro-scale particles as drug powders are injected into skin with high velocity, and several factors should be considered such as the particle diameters, trajectories, and momentum or velocities of projectile in this process. There have been investigated by some experiments, but there is still no study from the numerical insights.

The numerical methods are suitable to study these kinds of processes. Although grid-based methods like FEM have been powerful tools in engineering, the large distortions of elements exist when solid structures are extremely deformed. The methods based on the particle modelling are very suitable for simulating phenomena like the high velocity impacts. As the earliest meshless method, Smoothed Particles Hydrodynamics (SPH) has been applied in solid dynamics because of its great potentials in simulating extremely large deformation and perforation of targets by various projectiles at high velocities. This thesis mainly studies the SPH method by developing a SPH code for solid dynamics, and then propose a numerical model based on SPH method to investigate the penetration process into soft tissues (gelatin and synthetic polymer SEBS gel (styrene-ethylene-butylene-styrene) considered as biofidelic soft tissue simulants) at microscale.

Université Bourgogne Franche-Comté 32, avenue de l'Observatoire 25000 Besançon

3 Figure 1 . 1 :

 311 Figure 1.1: The system of needle-free drug delivery device provided in the [1].

Figure 1 . 2 :

 12 Figure 1.2: The process of a 13-µm steel sphere penetrating into a simulant of soft tissue called SEBS gel observed by the experiment [2].

A

  STATE OF THE ART ON IMPACTS TO STRUCTURES SIMULATED BY SPH METHOD 2.1/ INTRODUCTION High velocity impacts (HVIs) are very important phenomenon in various fields such as vehicle engineering, military framework, medicine techniques, as well as manufacturing

2. 2 . 11 2. 2 . 1 . 2 /

 211212 SPH METHOD AND ITS DEVELOPMENTS APPROXIMATION OF DERIVATIVES

16CHAPTER 2 .

 2 A STATE OF THE ART ON IMPACTS TO STRUCTURES SIMULATED BY SPH METHOD thickness, and target materials. Backman et al. in [48] classified the most frequent types of damages during HVI for thin or intermediate targets, as shown in Fig.2.1. All of these damages result from the initial compression wave and they often happen with more than one single form in specific impacts, which often involve the elastic-plastic deformation, shear deformation, hydrodynamic process, phase transition, etc. The definition of the thin target means the stress and deformation gradients throughout its thickness do not exist. The mechanical problems like perforation, fragments, and debris clouds principally happen on the thin plates impacted by the high velocities. For example, the perforations accompanying plugging a), ductile hole enlargement b), radial fracture c), as well as fragmentation and debris cloud phenomenons d), as shown in Fig.2.2, have been observed in many experiments about impact thin structures.

Figure 2 . 1 :

 21 Figure 2.1: The types of damages in HVI from [48].

2. 3 .

 3 HVIS ON THIN STRUCTURES 17 perforation of metal targets impacted by different types of bullets, the simulation carried out by SPH framework.

Figure 2 . 2 :

 22 Figure 2.2: The types of damages observed by HVI experiments: a) shows the plugging situation after perforation by[START_REF] Børvik | Ballistic penetration of steel plates[END_REF]; b) shows the petaling and ductile hole enlargement after perforation in[START_REF] Manes | An experimental-numerical investigation on aluminium tubes subjected to ballistic impact with soft core 7.62 ball projectiles[END_REF]; c) shows the perforation accompanying with radial fracture in[START_REF] Michel | Hypervelocity impacts on thin brittle targets: Experimental data and SPH simulations[END_REF]; d) shows the debris could during the impact in[START_REF] Andrew | Characteristics of debris clouds produced by hypervelocity impact of aluminum spheres with thin aluminum plates[END_REF].

. 28 )Figure 2 . 3 :

 2823 Figure 2.3: Illustration of the Stellingwerf's test [63].

  used the SPH model in LS-DYNA to study the damaging and fragmenting processes during HVI on thin brittle targets. The simulation by the SPH model consisted of the steel projectiles on brittle targets (fused silica plates) 22CHAPTER 2. A STATE OF THE ART ON IMPACTS TO STRUCTURES SIMULATED BY SPH METHOD of 2 mm thickness with velocity impact ranging from 1 to 4 km/s. The Johnson Holmquist material model and pressure cut-off criteria were chosen for target material. This SPH model successfully predicted the fragments clouds geometries and velocities. Besides, the perforation hole and spallation zones in the target were obtained in the simulation results, being in a good agreement with experimental observations. The influence of target thickness on damages and matter ejection was also investigated numerically in this study.

Fig. 2 .

 2 Fig.2.4 is the simulation result from [51], showing the spalling phenomenon, the particle clustering representing the macro and micro spalls naturally.

Figure 2 . 4 :

 24 Figure 2.4: Illustration of SPH simulation in [51]: two types of spalls can be identified (macro and micro spalls) represented by SPH particles clusters as fragments.

Figure 2 . 5 :

 25 Figure 2.5: The pseudo-spring model (a) undamaged configuration (b) cracked configuration from [32].

Figure 2 . 6 :

 26 Figure 2.6: Numerical laminated plate model by SPH particle from [54].

Figure 2 . 7 :

 27 Figure 2.7: The SPH model for CFRP laminate from [55].

30CHAPTER 2 .Figure 2 . 8 :

 228 Figure 2.8: The SPH model of the laminated Gr/Ep composite plate (left); debris cloud produced by impact at 4.0 km/s (right) provided in [54].

Figure 2 . 9 :

 29 Figure 2.9: The unphyscial fractures during standard SPH model (left) and improved SPH model (right) in [54].

Figure 2 . 10 :

 210 Figure 2.10: Impact sequence of 7.62x51 P80 against multi-layered target simulated by the numerical models from [62].

Back in 1996 ,

 1996 Johnson et.al. studied the effects of different artificial viscosities for the SPH method in [41]. The types of artificial viscosity developed by different authors are listed in Fig.2.2. Johnson et.al. used five examples elected to describe a wide range of physical problems including plastic flow, wave propagation, erosion, rigid projector impact, and localized shear in this study. They were simulated by standard SPH formulations with six schemes of artificial viscosities, involving MAV by Monaghan al.et. in [8, 21, 22], BAL by Balsara et al. [104], node and bond viscosity [105], as well as three different combinations of parameters in MAV viscosity. By analyzing the simulation results, the authors declared that only changing the coefficients of MAV-type artificial viscosity can 2.4. LIMITATIONS AND IMPROVEMENTS 35 not provide improved results. Additionally, the types of artificial viscosity illustrated different performances for different problems. For example, the MAV-type viscosity showed generally good results in plastic flow, rigid projector impact, and wave propagation problems but appeared severe numerical instability in the other two examples. It is difficult to find that a certain existing artificial viscosity algorithm or set of coefficients was clearly superior for the range of problems. The conclusion of this study was only to remind users should pay much caution when performing SPH computations for specific problems. At the same time, more work was required to develop artificial viscosity algorithms (or other algorithms) to improve the SPH instability problem.

36CHAPTER 2 .

 2 A STATE OF THE ART ON IMPACTS TO STRUCTURES SIMULATED BY SPH METHOD 2.4.1.2/ TENSILE INSTABILITY Actually, in the SPH method, tensile instability is still the most unmanageable problem.

Fig. 2 .

 2 Fig.2.12 is a typical result of soft materials penetrated by the projectile with high velocity. The obvious difference between the biomaterials and metallic materials as an impact target is that the soft and large elastic properties of biomaterials can lead to a deep penetration and a temporary cavity phenomenon in the target materials. The hydrodynamic process involving the extreme deformations and the behavior like fluid is also characterized in biomaterials under impacts in the literature like[START_REF] Shen | Numerical modeling of a human tissue surrogate sebs gel under high velocity impacts: investigation of the effect of the strain rate in an elasto-hydrodynamic law[END_REF][START_REF] Taddei | Smoothed particle hydrodynamics formulation for penetrating impacts on ballistic gelatine[END_REF][START_REF] Monzer | Sph-based method to simulate penetrating impact mechanics into ballistic gelatin: Toward an understanding of the perforation of human tissue[END_REF].

Figure 2 . 12 :

 212 Figure2.12: The process of a bullet penetrating into soft tissue observed by the experiment[START_REF] Beat | Wound ballistics: basics and applications[END_REF].

2. 5 .

 5 MECHANICAL CHARACTERIZATION OF SOFT MATERIALS UNDER IMPACT 43 material properties were estimated using the stress at failure, strain at failure, and material stiffness as characterized by the Neo-Hookean constitutive model. Further research was predicted by authors that a strain-rate dependence and a hyperelastic behavior law should be developed for gelatin material. Ravikumar et.al conducted mechanical com-

Fig. 2 .Figure 2 . 13 :

 2213 Fig.2.12 involving the trajectory, time history of cavity size etc.

2. 5

 5 Figure 2.14: The shapes the temporary cavity caused by (a) steel sphere of 0.45 g (856 m/s); (b) steel cylinder of 0.9 g (1032 m/s); (c) steel rhombus of 0.25 g (651 m/s) and (d) bullet of 4.15 g (738 m/s). The black lines in the figure are used to calculate the cavity volume. where[171].

2. 5 .

 5 Figure 2.16: The SPH model combined with FE elements for soft penetration in [133].

(

  LIPIT) in recent years, which was developed by a research group including J.-H. Lee and D. Veysset et al.[START_REF] Jh Lee | High strain rate deformation of layered nanocomposites nat[END_REF] 187,[START_REF] Veysset | Dynamics of supersonic microparticle impact on elastomers revealed by real-time multi-frame imaging[END_REF][START_REF] Veysset | Molecular influence in high-strain-rate microparticle impact response of poly (urethane urea) elastomers[END_REF] 2]. In the LIPIT platform shown in 2.17, an intense short laser pulse is used to accelerate microparticles to supersonic velocities, and individual particle impact events are observed with a high-frame-rate camera. It can capture real-time in-situ visualization of the deformation response with high accuracy. Fig.1.2 is one of the results obtained by this experimental technique, which provided an observation of the response and deformation of ballistic SEBS gel with various concentrations (10vol%,20vol%,30vol% and 30vol%)[2]. Veysset et.al. used this experimental platform to investigate the ballistic behaviors of gelatin and elastomers under high-velocity impacts in[START_REF] Veysset | High-velocity micro-particle impact on gelatin and synthetic hydrogel[END_REF]. The authors conducted a considerable increase of resistance in the target ma-2.5. MECHANICAL CHARACTERIZATION OF SOFT MATERIALS UNDER IMPACT 51 terials in the micro-scale penetrations via combining with the analytical model. Thanks to these experimental observations, the performance of soft materials under micro-impact loading can be observed and evaluated, but the quantified mechanical properties are still required to be further researched.

Figure 2 . 17 :

 217 Figure 2.17: Particle launch and imaging configuration in the LIPIT platform provided in [188].

2. 6 . CONCLUSION 53 ments

 653 of constitutive laws for different materials, and also the good compatibility of SPH schemes.

4 .

 4 The other important further work is HVIs problems happened in the soft tissues. Developing the SPH model with material constitutive law to investigate the responses of soft tissues, esoterically in the micro-scale, is a interesting work. This study is also the main contribution of this thesis.To sum up, the strong capability of the SPH simulating high-speed velocity impacts has been witnessed in the literature reading. The study to the ballistic behaviors of soft tissues 54CHAPTER 2. A STATE OF THE ART ON IMPACTS TO STRUCTURES SIMULATED BY SPH METHOD under micro-impacts is an significant work, but no much simulating investigations. Therefore, this thesis focuses on the application of the SPH method modeling the penetrating process in the soft materials such as gelatin and polymer gels in the micro-scale. THEORETICAL POINTS: DEVELOPMENT OF A SPH CODE TO DYNAMIC SOLIDS 3.1/ INTRODUCTION

3. 2 . 57 Figure 3 . 1 :

 25731 Figure 3.1: A list for six types of SPH kernels.

Figure 3 . 2 :

 32 Figure 3.2: The shapes of kernel functions and their derivatives.

Fig. 3 .

 3 3 shows the the regions of stability and instability of a cubic B-spline kernel.

Figure 3 . 3 :

 33 Figure 3.3: The stability analysis from the research of Swegle et.al [199]: (a) The particle clumping phenomena in the SPH simulation. (b) The regions of instability and stability of a cubic B-spline kernel.

11 )62CHAPTER 3 .

 113 THEORETICAL POINTS: DEVELOPMENT OF A SPH CODE TO DYNAMIC SOLIDS

√ J 2 exceeds the yield stress σ y / √ 3 ,

 3 the individual stress components are given new values S αβ new = f S αβ , where the value of f is defined by

Fig. 3 .

 3 Fig.3.4 is the flow diagram of a standard SPH combined with these modifications mentioned in above sections. It is applied in elastic-plastic formulations. The SPH method is the typical Lagrangian particle method, which approximates a function and its spatial derivatives through averaging or summation over neighboring particles. Therefore, a step called Nearest neighboring particle searching (NNPS) is one of main calculated modules.During it, the types of kernel function is a key which has an important influence for the instability and accuracy of numerical results. In my code, six different types of kernel function are included in order to investigate their different performances in solid dynamic problems.

3. 3 . 4 .

 34 THE PROCEDURE OF SPH CODE IMPLEMENTATION 67 Output results. When the time step reaches to a prescribed one or at some interval, the resultant information for a computational state is saved to external files for later analyses or post-processing.

Figure 3 . 4 :

 34 Figure 3.4: The flow diagram of SPH code implementation.

Fig. 3 . 1 .

 31 Fig.3.1. The geometrical dimensions of the beam structure are shown in Fig.3.5, where the length L = 100mm, the width t = 10mm and the beam thickness is fixed to 1mm. As illustrated in Fig.3.5, the left edge of the beam is fixed and the other edge is free and has a concentrated load F. The value of the load is a quadratic function of time, starting from 0 to the maximum value F max at time t = 1.5ms. F max is kept constant until to the end of simulation t = 3.0ms (Fig.3.6). All the material parameters are provided in Tab.3.2 and the related parameters in CSPH are given in Tab. 3.3.

3. 4 . 69 Figure 3 . 5 :

 46935 Figure 3.5: The beam model and the left is fixed.

Figure 3 . 6 :

 36 Figure 3.6: The dynamical loading.

Fig. 3 6 Errors 2 Figure 3 . 7 :

 36237 Figure 3.7: the deflection distributions: (a) is simulated by our code with type-1 kernel (m) and (b) is simulated by Lin et al. [202] (mm).

3. 4 . 71 Figure 3 . 8 :

 47138 Figure 3.8: Distributions of σ xx at the maximum deflection with 6 kernels, (a)-(f) matching with type-1 to type-6 kernel, respectively.

72CHAPTER 3 .

 3 Figure 3.9: time history of total kinetic energy.

Figure 3 .Fig. 3 .

 33 Figure 3.10: time history of total strain energy.

3. 4 . 73 Figure 3 . 11 :

 473311 Figure 3.11: Distributions of deflections: (a) is simulated by our code using type-4 kernel (m), (b) is simulated by Lin's SPH code in [202] (mm) and (c) by FEM method in [202] (mm).

74CHAPTER 3 .Figure 3 . 12 :

 3312 Figure 3.12: Model geometric of rob.

Figure 3 . 13 :

 313 Figure 3.13: The shapes of rod at the impact end.

Figure 3 . 14 :

 314 Figure 3.14: The distribution of equivalent strain with type-6 kernel.

Figure 3 . 15 :

 315 Figure 3.15: The distribution of equivalent strain with type-4 kernel.

76CHAPTER 3 .Figure 3 . 16 :

 3316 Figure 3.16: The time history of mushroom radius in our simulations using 6 types of kernels, and the simulations by MSPH and LS-DYNA in [206].

Figure 3 . 18 :

 318 Figure 3.18: The errors of mushroom radius at impacted end compared with experimental data from[205].

Figure 3 . 19 :

 319 Figure 3.19: The errors of rod length at impacted end compared with experimental data from[205].

  Investigating high-speed microparticle impact responses into soft tissue is essential to several fields such as medicine and biology with technological applications like transdermal delivery of pharmaceuticals. The understanding of the physical process involved in such a phenomenon is complex and few experimental tools are available to study highvelocity microparticle penetration in soft tissues. Besides, considering the practical and ethical issues, various simulants of human tissues including skin, muscle, brain, and other

1 . 3 . 4 .

 134 mathematical formulations. Smoothed Particles Hydrodynamics (SPH) has been applied in solid dynamics to simulate extremely large deformation and perforation of targets by various projectiles, as reviewed in the state of the art in the chapter??. After the theoretic study in the code and its validations to several benchmark problems in the chapter 3, this part develops an original numerical model based on SPH to study the dynamical phenomena during micro-scale impact of spheres into soft materials including ballistic gelatin (BG) and SEBS gel. The structure of this chapter is organized by: 80CHAPTER 4. SPH APPLICATION: HIGH-VELOCITY IMPACTS INTO SOFT TISSUES BY MICRO-PARTICL The analytical models for penetrations including the Poncelet equation, Liu's model and Modified-CG model are introduced firstly, which are often as the effective tools to validate the numerical results. The Poncelet model is represented and its coefficients are studied for the micro-penetrations. Then, the suitability and the fitting approaches of the analytical models based on a series of experiments on the micropenetrations are discussed. 2. A 3D numerical model combined with SPH particles and FEM elements is developed using Hyperworks software. The material constitutive law for gelatin and SEBS gel is introduced. Then, the simulations on ballistic behavior of 10 wt% BG impacted by a silica sphere with 7.38 µm diameter with 200-1290 m/s velocity are performed using this 3D numerical model. The perpetrating history of the projectile and the temporary cavity of the gelatin are investigated and validated by experimental results. The second part of the simulations is on ballistic behavior of 40 vol% SEBS gel impacted by a steel sphere with a range of diameters like 10-25 µm diameters. The impact velocities are in a range of 20-1000 m/s. The trajectory of the projectile and the elasticity of the SEBS gel are investigated and discussed based on the comparisons between the numerical and experimental results.

  penetrations. Although the Poncelet model with the parameters (C d = 0.4, R = 21 MPa) was validated by Veysset et al. [167] to model penetration of 7.38-µm diameter silica spheres into 10wt% gelatin. We found that a variation of drag coefficient C d = 0.375 combined with R = 21 MPa shows that the curve by Poncelet equation is also in agreement with experimental result. Fig.4.1 illustrates that there is a small difference at the final penetration between the curves using C d = 0.4 and C d = 0.375 in Poncelet equation (Eq. (4.2) ) to model the trajectory. The combination of (C d = 0.375, R = 21 MPa) can achieve a better fitting than the other, because it keeps the consistency for the whole time history on penetration length. Another point deserving to be discussed is the applicability of the Poncelet model with specific values of parameters (C d , R) in micro-sized penetration. Although the parameters C d = 0.375 or 0.4 and R = 21 MPa have demonstrated their ability to fit well with the impact trajectory of the microparticle (only for d = 7.38 µm) in Veysset's work [167],

84CHAPTER 4 .

 4 Figure 4.1: The curves of silica particle d = 7.38µm impact with 1290 m/s in 10wt% gealtin, compared between experimenal data and the Poncelet model with two combinations of parameters C d and R.

Figure 4 . 2 :

 42 Figure 4.2: The curves of steel spheres impact 20wt% gealtin: the experimental data is from [163] ; the Poncelet equation [1] with C d = 0.33 and R = 3.5 MPa [164], and the Poncelet equation [2] with C d = 0.35 and R = 5.0 MPa.

Fig. 4 .Table 4 . 2 :

 442 Fig.4.3 and Tab.4.2 include the experiments about impact SEBS gel material by the mi-

Fig. 4 .

 4 Fig.4.3 (b). Fig.4.5 shows the fitting results on the normalized maximum penetration depth as a function of impact velocity. The values of (C d =0.8, R=33.7MPa) in Poncelet, (C d =0.8, C v =5.31e-7, σ 0 =15MPa) in Liu's model, and (η =4.5e-2, R=25MPa) in Modified-CG are obtained in the reasonable ranges. Fig.4.6 shows the results using these fitted coefficients to model the penetration trajectories. It can be seen that a general agreement compared to different experimental trajectories is acquired for every model. It implies that the * b) fitting approach can get more stable and universal fitting coefficients for analytical

88CHAPTER 4 .Figure 4 . 3 :

 443 Figure 4.3: The experiments performed on the penetration SEBS gel by steel microparticles in [2]: (a) The experimental images showing the projectile trajectory of Curve3 with particle diameter 13 µm penetration into 40-vol% SEBS gel. (b) Normalized penetration depths as a function of impact velocity, involving various diameter particles marked by colors.

Figure 4 . 4 : 89 Figure 4 . 5 :

 448945 Figure 4.4: Fitting results via * a) approach: (a) Poncelet model (b) Liu's model (c) Modified-CG model. The configurations of curves: Curve1 (v 0 =215 m/s, D=20 µm), Curve2 (v 0 =440 m/s, D=16 µm), Curve3 (v 0 =630 m/s, D=13 µm).

Figure 4 . 6 :

 46 Figure 4.6: Validating the fitted coefficeints from Fig.4.5: (a) Poncelet model (b) Liu's model (c) Modified-CG model. The configurations of curves: Curve1 (v 0 =215 m/s, D=20 µm), Curve2 (v 0 =440 m/s, D=16 µm), Curve3 (v 0 =630 m/s, D=13 µm).

4. 3 . 91 Figure 4 . 7 :

 39147 Figure 4.7: Numerical model combining SPH and finite elements is built in the commercial software Hypermesh: L sph defines the width of SPH colume; SPH part consists of several SPH components with different discretization lengths dx i , and the values of dx i increase with the distance between SPH component and impact center (dx 1 < dx 2 < ... < dx i < ...).

(4. 13 )92CHAPTER 4 .

 134 SPH APPLICATION: HIGH-VELOCITY IMPACTS INTO SOFT TISSUES BY MICRO-PARTICL

4. 4 . 1 /Figure 4 . 8 :

 4148 Figure 4.8: Multi-frame sequence showing a high-velocity impact on 10 wt% gelatin at 1290 m/s and a maximum penetration depth reaching around 200-250 ns after impact, provided in [167].

Figure 4 . 9 :

 49 Figure 4.9: The numerical trajectories with different width L sph of SPH column, compared to experimental result.

Figure 4 . 10 :Fig. 4 .

 4104 Figure 4.10: The numerical trajectories with different discretization length dx 1 of SPH particles, compared to experimental result.

96CHAPTER 4 .

 4 Figure 4.11: Comparison between experimental result and numerical curves using different values of artificial viscocity coefficients (qa, qb).

Figure 4 . 12 :

 412 Figure 4.12: Comparison between experimental result and numerical curves using different values of conservative smoothing α cs .

4. 4 97 Figure 4 .

 4974 Figure 4.13: (a) Comparison between experimental result and numerical curves of four cases simulations in Tab. 4.5. (b) The errors on maximum penetration length of simulation results in four cases compared to experimental result.

Figure 4 . 14 :

 414 Figure 4.14: Normalized maximum as a function of impact velocity for impacts on 10 wt% gelatin with a same projectile (silica sphere with d=7.38 µm). The maximum penetration data obtained by simulations and experiments are fitted with the Poncelet model in [167].

98CHAPTER 4 .Figure 4 . 15 :Fig. 4 .

 44154 Figure 4.15: Penetration velocity vs penetration depth fitting between the Poncelet equation and simulation curves in case 2 at different impact velocities v 0 .

  figures, it can be seen that the numerical curves are also in agreement with the Poncelet equation.

Figure 4 . 16 :

 416 Figure 4.16: The dynamic process of cavity in gelatin sample impact with 1290 m/s; Contours of stress in the gelatin around the temporary cavity and a comparison in three cases of simulations with different Young's modulus E are presented. The result of case 3 ignored involves the similar shape with that in the case 2.

Figure 4 . 17 :

 417 Figure 4.17: The diameter of temporary cavity in gelatin sample in the case 2 simulation with E = 20 MPa at 10 ns, 70 ns, 150 ns, and 300 ns after impact.

100CHAPTER 4 .Figure 4 . 18 :

 4418 Figure 4.18: The measure of cavity diameter in in the numerical simulation from Fig.4.17; the blue area between D1 and D2 means the damaged SPH particles with zero stress.

Figure 4 . 19 :

 419 Figure 4.19: The maximum cavity diameters obtaind by experiment (D exp ) and simulation (D2).

Fig. 4 .

 4 Fig.4.17 shows the cavity diameters positioned at 15 µm to the impact face at different time points in the case 2 simulation with E = 20 MPa. Fig.4.18 presents the approach to measure the cavity diameter in simulation result, denoting that D1 is the size of real empty cavity and D2 is the diameter of cavity including damaged particles area. The values of maximum cavity size obtained by the experiment and simulation (Fig.4.19) are observed and we note that the maximum diameter D2 of the numerical cavity is very close to the experimental data if the cavity considering the damaged particles (the error less than 10%).

4. 5 . 101 Figure 4 . 20 :

 5101420 Figure 4.20: The time history of cavity diameter in gelatin for the experiment and numerical simulation. The experimental data is provided by Veysset et al.[167].

Fig. 4 .

 4 Fig.4.20 displays the time history of cavity diameter in gelatin for the experimental and simulation results. It can be noticed that the cavity in simulation expands very rapidly at the primary stage after impact. After 150 ns, nevertheless, the expanding trend of the numerical cavity is close to the experimental result. The discrepancy may be attributed to the predominantly hydrodynamic behavior at initial penetration, or reasonably no considering the hyperelastic property of the gelatin material in the numerical model. Although the value of the maximum cavity diameter obtained by SPH simulation is in the same order of amplitude with experimental results, the dynamic process of the cavity in gelatin should be more investigated in further research.

Figure 4 . 21 : 103 Figure 4 . 22 :

 421103422 Figure 4.21: The numerical results for simulating the penetration in 40 vol% SEBS gel by D=13µm at 630 m/s: (a) α cs =0.1 (b) α cs =0.08 (c) α cs =0.05; the value of Young's modulus is in the range of 1-150 MPa.

Figure 4 . 24 :

 424 Figure 4.24: The curves of steel particle D = 22 impact with 300 m/s are obtained by SPH model with the coefficients α cs = 0.08, (qa, qb)=(2,1), compared to the experimental data.

Figure 4 . 25 :

 425 Figure 4.25: The normalized maximum penetration depths is the function of the impact velocity, involving a various projectile diameters from 10-22 µm and impact velocities from 50-900 m/s. The darker color is associated with a larger diameter of the projectile in the simulation results. A comparison is represented between numerical and experimental results.

Figure 4 . 26 :

 426 Figure 4.26: The curves of steel particle D = 12 impact with 735 m/s are obtained by SPH model with the coefficients α cs = 0.08, (qa, qb)=(2,1), compared to the experimental data.

Figure 4 . 27 :

 427 Figure 4.27: The curves of steel particle D = 16 impact with 440 m/s are obtained by SPH model with the coefficients α cs = 0.08, (qa, qb)=(2,1), compared to the experimental data.

Figure 4 . 28 :

 428 Figure 4.28: The comparisons between the analytical results by Modified-CG and numerical simulations in various penetrations.

4. 6 . DISCUSSION 107 Figure 4 . 29 :

 6107429 Figure 4.29: The temporary cavity in the SEBS gel during the penetration by D=13µm at 630 m/s: the strain distributions at three time points from the longitudinal direction; the value of Young's model used in this case is 50 MPa.

Fig. 4 .

 4 Fig.4.12 and Fig.4.21 illustrate that the usage of XSPH produces a substantial influence

108CHAPTER 4 .

 4 SPH APPLICATION: HIGH-VELOCITY IMPACTS INTO SOFT TISSUES BY MICRO-PARTIC

4. 6 . DISCUSSION 109 40- 50

 610950 MPa extracted for the value of Young's modulus for the SEBS gel under the micro penetrations in this study can be reasonable. As mentioned that this numerical model is capable to predict the trajectory of micropenetrations, but the results are depending on a fixed set of numerical parameters including the discretization density and coefficients of the XSPH. Because the numerical validations are based on these numerical parameters, the extracted values of Young's modulus for both ballistic gelatin and SEBS gel are limited to the exact physic. As known that the mechanical properties of soft tissues become questionable when they are under the micro-impact loading. Although the non-linear elastic or hyper-elastic properties are identified widely in the literature, there is no much study to extract the quantitative values of elasticity (Young's modulus or shear modulus) for soft materials like ballistic gelatin and SEBS gel. Gelatin has become the most common simulant to be investigated to understand the penetrating process in the macro-scale, but the behavior shows a considerable different in the micro-scale. The SEBS gel is recognized as involving a better performance as a simulant of soft tissues compared to the gelatin, unfortunately, its mechanical properties are less investigated in the literature. To the authors' knowledge, there is no failure model developed for SEBS gel up to now, particularly, for the micro-scale condition. In this numerical model, the simplest failure model was attempted for both of gelatin and SEBS gel, which maybe should be further improved to model the breaking process of soft materials during the impacts. The size of temporary cavity is strongly linked to failure criterion in the penetration, as shown in the Fig.4.17 for gelatin and Fig.4.29 for SEBS gel, which also limits the quantitative investigation on the cavity size in this model.

110CHAPTER 4 . 10 %

 410 SPH APPLICATION: HIGH-VELOCITY IMPACTS INTO SOFT TISSUES BY MICRO-PARTIC4.7/ CONCLUSIONThis chapter provided the investigations on ballistic behavior of soft materials including gelatin and 40 vol% by analytical and numerical approaches. Firstly, the performances and fitting coefficients for three analytical models were discussed to model the penetrating trajectories in the micro-scale. Analytical models such as Poncelet, Liu, and Modified-CG models are practical tools to obtain the trajectory thanks to their simple formulations by

5. 3 .

 3 FUTURE WORKS 115 ternative than the conventional kernels like B-spline functions. But the numerical model to the micro-penetration into soft materials is built with the commercial software Hyperworks, and the solving processor is the Radioss which is based on a standard SPH algorithm involving the B-spline kernel. Therefore, the tensile instability should be removed by extra corrective terms like XSPH. That is also one of the factors why the coefficient of XSPH produces a large difference at the end of penetration when the target material behaves predominately as elastic-plastic deformation and more possibly includes particles clump-

CHAPTER 5 .

 5 CONCLUSIONS, LIMITS, AND PERSPECTIVES materials. Based on the work of chapter 2, a 3D SPH code including a more suitable kernel can be developed to model the impact problem. Of course, considering the high accuracy and effectiveness, the more creative SPH formulations like Total

2. 12

 12 The process of a bullet penetrating into soft tissue observed by the experiment [134]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13 The process of a bullet penetrating into soft tissue observed by the experiment [134]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.14 The shapes the temporary cavity caused by (a) steel sphere of 0.45 g (856 m/s); (b) steel cylinder of 0.9 g (1032 m/s); (c) steel rhombus of 0.25 g (651 m/s) and (d) bullet of 4.15 g (738 m/s). The black lines in the figure are used to calculate the cavity volume. where[171]. . . . . . . . . . . . . .

2. 17 Fig. 4 . 1 T

 1741 Fig.4.17; the blue area between D1 and D2 means the damaged SPH particles with zero stress. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4. 4

 4 Material parameters of the projectiles. . . . . . . . . . . . . . . . . . . . . .

  et.al., all of which are developed under the motivation to different SPH applications. Tab.2.1 lists the developments of SPHs in the past forty years, not involving all but including the most several important formats. More detailed reviews about developments of SPH formulations can be found in[START_REF] Liu | Smoothed particle hydrodynamics (sph): an overview and recent developments[END_REF][START_REF] Joseph | Smoothed particle hydrodynamics and its diverse applications[END_REF][START_REF] Matthias R Öthlin | Meshless methods for large deformation elastodynamics[END_REF].

Table 2 .

 2 1: The improvements and developments of SPH method

	Name of methods	Year	Authors and references
	Standard SPH	1977	Gingold and Lucy [8, 9]
	Artificial viscosity	1989	Monaghan[22]
	Axis-symmetry SPH	1996	Johnson etc. [41, 42]
	CSPM	1999	Chen etc. [43, 44]
	MLSPH	1999	Dilts etc. [35, 36]
	Artificial stress	2000	Monaghan and Gray [23, 24]
	Godunov SPH	2002	Inutsuka etc.[28, 27]
	MSPH	2004	Batra etc. [45]
	Total Lagrangian SPH	2006 Vignjevic, Reveles etc. [19, 30]
	FPM	2006	Liu etc. [46]
	δ-SPH	2009	Molteni etc. [37]
	Pseudo-spring SPH	2013	Chakraborty, Amit etc. [32]
	Hourglass for TL SPH formulation 2015	Ganzenmulle etc. [31]
	JST SPH	2016	Lee etc. [47]
	Gamma-SPH	2017	Limido etc. [38]
	2.3/ HVIS ON THIN STRUCTURES
	2.3.1/ HVI PROBLEM DESCRIPTIONS

  rod impact steel plate with 2.6 cm thick at 2 km/s and 4 km/s, and copper rod impact aluminum plate with 12.7 cm thick at 5 km/s. These studies involving the validations between SPH code simulations and experimental or other existing codes results made more and more researchers pay attention to the advantages of the SPH method in the application of HVI problems.

, Johnson et.al incorporated SPH algorithm into a standard Lagrangian code such as EPIC in [70]. It 2.3. HVIS ON THIN STRUCTURES 21 modeled the copper

Table 2 .

 2 2: The types of artificial viscosity incorporated to SPH formulations

	Ref	Abbreviation Parameters description
	[8, 21, 22]	MAV	α, β	The most common type, not stable
	[104]	BAL	α, β	capable for High-velocity impact
	[105]	NBAV	α, β	not stable
	[106]	MON	α

* capable for High-velocity impact

  2. A STATE OF THE ART ON IMPACTS TO STRUCTURES SIMULATED BY SPH METHODthetic medium) can be found in the literature[START_REF] Ali İhsan Uzar | A new ballistic simulant" transparent gel candle"(experimental study). Ulusal travma ve acil cerrahi dergisi= Turkish journal of trauma & emergency surgery[END_REF][START_REF] Moy | Dynamic response of an alternative tissue simulant, physically associating gels (pag)[END_REF][START_REF] Humphrey | Ballistics and anatomical modelling-a review[END_REF]. Particularly, the synthetic polymer SEBS gel (styrene-ethylene-butylene-styrene) has been recognized its practical advantages recently as a sutiable simulant. According to the experimental observations in[START_REF] Mrozek | The relationship between mechanical properties and ballistic penetration depth in a viscoelastic gel[END_REF][START_REF] Bracq | Experimental study of the strain rate dependence of a synthetic gel for ballistic blunt trauma assessment[END_REF][START_REF] Bracq | On the modeling of a visco-hyperelastic polymer gel under blunt ballistic impacts[END_REF], synthetic SEBS gel involves environmental stability, reproducibility, mechanical consistency, and transparency. In the study of[START_REF] Mrozek | The relationship between mechanical properties and ballistic penetration depth in a viscoelastic gel[END_REF] [START_REF] Mrozek | The relationship between mechanical properties and ballistic penetration depth in a viscoelastic gel[END_REF], the authors investigated the mechanical properties of SEBS gels by using tensile, compression, and rheological tests. The conclusions of this study were that the projectile penetration during SEBS gel impacts could be accurately described using mechanical properties like modulus and toughness measured in the quasi-statics strain rate regime. The group of

Bracq et al. highlights the development of constitutive modeling for the synthetic polymer SEBS gel. According to their experimental studies like tensile and compressive tests at diverse strain rates

[START_REF] Bracq | Experimental study of the strain rate dependence of a synthetic gel for ballistic blunt trauma assessment[END_REF][START_REF] Bracq | On the modeling of a visco-hyperelastic polymer gel under blunt ballistic impacts[END_REF]

, a novel strain-rate-dependent hyper-viscoelastic law for the SEBS gel was developed first. This constitutive law was validated by implementing it in a FEM model and acquired an agreement with experimental data. In recent work by

[START_REF] Shen | Numerical modeling of a human tissue surrogate sebs gel under high velocity impacts: investigation of the effect of the strain rate in an elasto-hydrodynamic law[END_REF]

  3.2. THE ROLE OF KERNEL FUNCTION59condition for stability against particles pairing, which is a different insight with Swegle's analysis. Besides, they proved theoretically that Wendland kernel has this property and can keep stable in all the numbers of neighbour particles.

Table 3 .

 3 3.1 and scale factor of smoothing length. n is the dimension and B n = 2π n/2 /Γ(n/2), in which Γ is the Gamma function. According to Di G.Sigalotti, a bigger value of A w means a lower accuracy in SPH particle approximation. They also validated this estimate means by discretizing a simple analytical problem and deducted that Lucy's kernel is a better choice in 2D whereas Wendland C2 is the better one in 3D problems. Tab.3.1 shows the values of A w for 4 different kernel functions from Fig.3.1. The values 1: Numerical values of A w

	types of kernels	a α	a α	k	2D	3D
		2D	3D
	type-1	80/(7π)	16/π	1 45.7	170
	type-2	1/π	1/π 3/2	3	36	162.7
	type-3	3 7 • 7/(478π) 3 7 /(40π) 1 128.1 583.2
	type-4	7/π	21/(2π) 1	28	112
	Both Sigolotti's errors estimation and Walter Dehnen's instability analysis show that

of A w from type-4 kernel is the smallest, which implies that it is able to obtain a better accuracy in SPH particle approximation. The satisfied accuracy in particle approximation which is the major step in SPH formulation can ensure a good simulating result. 60CHAPTER 3. THEORETICAL POINTS: DEVELOPMENT OF A SPH CODE TO DYNAMIC SOLIDS

Table 3 . 2

 32

			: Material parameters
	Material Young's modulus Poisson's ratio	Density	Sound speed
	Steel	210GPa	0.3	7800kg/m 3	5960m/s

Table 3 .

 3

	F max dx	t	h	AV	AS XSPH
	kN mm	s	mm	α, β	ζ	ζ
	1.75	1	1.0e-7 1.2dx 0.1,0.1 0.5	0.5
	17.5	1	1.0e-7 1.4dx 0.5,0.8 0.9	0.5

3: The parameters in CSPH: dx is the initial distance between two particles; t is the time step; AV and AS, respectively, is the artificial viscosity and artificial stress; XSPH is from the literature

[START_REF] Di | A new insight into the consistency of the sph interpolation formula[END_REF]

.

Table 3 .

 3

		5: Material parameters in 2D rob impact.
	Parameters Young's modulus Shear modukus Yield stress	C	S	Γ
	Values	208GPa	80GPa	600MPa	3630m/s 1.80 1.80

Table 3 .

 3

	ρ	µ	K	C	a	b	n	c	m	T M	T room ε0
	kg/m 3 GPa GPa J/KgK GPa MPa				K	K	/s
	7830 89.2 169.1	460	792 510 0.26 0.014 1.03 1293 293	1

6: Material parameters for Jonhnson-Cook law.

Table 3 .

 3 7: the SPH parameters in 3D rob impact.

	dx	t	h	AV	XSPH
	3.88mm 1.0e-8 1.5dx 1.5,1.5	0.1
	Fig.3.16 demonstrates the time histories of the radius of impact mushroom simulated by
	SPH with six kernels listed in Tab.3.1, from which we can see that the result using type-4

kernel is the closest one to the result by MSPH

[START_REF] Romesh | Modified smoothed particle hydrodynamics (msph) basis functions for meshless methods, and their application to axisymmetric taylor impact test[END_REF]

, compared with simulations using other kernels. MSPH has been compared with existing commercial code LS-DYNA and corresponding experiments and been validated that it is a vary effective SPH algorithm

  In the Poncelet equation, the retardation force of the projectile during the impact consists of a constant strength term C and a term proportional to the square of the instant penetration velocity BV 2 , where the strength resistance R, related to the apparent yield stress σ t , is a material-specific constant. This analytical model was initially validated by Sturdivan's experiments in 1978[START_REF] Larry M Sturdivan | A mathematical model of penetration of chunky projectiles in a gelatin tissue simulant[END_REF] for a 20wt% gelatin target. The explicit values of drag coefficient C d and strength resistance R should be changed in specific penetrations.

	82CHAPTER 4. SPH APPLICATION: HIGH-VELOCITY IMPACTS INTO SOFT TISSUES BY MICRO-PARTICL
	P max = P(T f ) =	2ρ p d 3ρ t C d	ln(	ρ t C d v 2 0 2R	+ 1)	(4.6)
	4.2.1.2/ A DISCUSSION FOR THE PARAMETERS
						4.2) ,

Table 4 .

 4 4: Material parameters of the projectiles.

	Parmeters	10 wt% gelatin 40 vol% SEBS gel
	Density	1030 kg/m 3	875 kg/m 3
	Poisson's ratio	0.499	0.495
	Young's modulus	1 -100 MPa	1 -100 MPa
	A	21 MPa	25 MPa
	B	0.01	0.01
	n	0.1	0.1
	c 0	1520 m/s	1520 m/s
	S	1.8	1.87

It should be noticed that, in Tab.4.3, two material parameters are underlined. One is Young's modulus E which is given a value in a range, and an further investigation about the influence on penetration process from this value is provided in simulation results. Both

  CHAPTER 5. CONCLUSIONS, LIMITS, AND PERSPECTIVES thick structures, typically happened in soft materials, there are only a few studies by SPH method. Particularly, for high-velocity impacts into soft materials in the micro-scale, there is no investigations concerning numerical simulations in the literature. Taking into account this vacancy in the applications of SPH simulations, this thesis developed an SPH model to investigate the ballistic process in the soft materials including 10% gelatin and 40 vol% SEBS gel under micro impacts.After confirmed the target and method of this research, the first part in the chapter 2 is to develop a SPH code to study the dynamic solids, aiming to understand the algorithm and procedure of SPH method and its performances in some benchmark problems. This code is implemented for elastic-plastic formulations based on the standard SPH combined with artificial viscosity, artificial stress, XSPH terms , as well as a correction on kernel calculation (CSPH). The role of SPH kernel was paid attentions during programming this code. One of the challenges in this study is to determine the material constitutive law and related material parameters because of the complex mechanical properties of soft materials such as hyper-elasticity, viscous-elasticity, high sensitivity on strain rate, as well as the hydrodynamic process. A elasto-hydrodynamic law using a constant elastic modulus and EOS is employed for both gelatin and SEBS gel. According to the validations of the model against experimental results, the ranges of young's modulus were determined, being 20-40 MPa for 10% ballistic gelatin and 40-50 MPa for 40 vol% SEBS gel, respectively. This result matches with the observations from other experiments and analytical models that the resistance and strength of soft material increase when the penetrations transform from macro-scale to micro-scale. The numerical parameters including artificial viscosity and XSPH coefficients, as well as particle density were studied to achieve a stable numerical model. Then, a series of simulations for gelatin impacted by a silica particle with 7.38 µm with initial velocities ranging from 200

	5.2. LIMITS	113
	micro impacts with high velocities.
	Ballistic gelatin is one of the main simulant of soft tissues, which has been investigated ex-
	perimentally and numerically in the literature in the macro-scale. The synthetic polymer
	SEBS gel (styrene-ethylene-butylene-styrene) is recognized recently as a better simu-
	lant because of its several benefits like environmental stability, reproducibility, mechanical
	consistency, transparency, and biofidelic ability to reproduce human soft tissues behavior.

The property of particle discretizations of SPH method makes it naturally suitable for modeling the structures involving large deformations, particularly for the debris clouds and fragmentation phenomenons during HVIs. The appendix table lists a large number of examples on SPH simulating HVI process. However, in terms of penetrations on 112

The relationship between the inhered shortcoming called tensile instability happened in the SPH simulation for solid problems and its kernel function types was studied. This code involves six kinds of kernel functions like traditional ones (Gaussian function, B-spline functions, Wendland function), and new types constructed for some specific problems (Double cosine function and Hyperbolic shaped). These kernel functions demonstrate a different accuracy and stability for SPH algorithm because of their mathematical properties on shapes, derivatives, smoothness, and influence domains. A series of common examples including elastic beam vibration and Taylor rod impacts were simulated with this code. By comparisons, the most suitable function called Wendland is confirmed to achieve a stable and accurate simulations for elastic-plastic solids. This code also provides an useful preparation including the kernel function and numerical parameters to develop the SPH model to complex ballistic process.

The second contribution of this thesis in the chapter 4 is the development of a numerical model based on SPH particles and FE elements using the commercial software Hyperworks to investigate the ballistic behavior of 10% gelatin and 40 vol% SEBS gel under to 1290 m/s were carried out and good agreements were obtained between numerical and experimental results in terms of trajectories of projectiles. This model also successfully simulated the trajectories of steel spheres with various sizes penetrating 40 vol% SEBS gel by different velocities.

lisation numé rique d'impacts à hautes vitesses par la méthode sans maillage Smoothed Particles Hydrodynamics. Application aux micro-impacts dans des tissus mous. Mots clé s

  : SPH (Smoothed Particles Hydrodynamics), impact à hautes vitesse, tissus mous, pé né trations à micro-é chelle.Ré sumé : L'é tude sur les impacts à hautes vitesses dans les tissus mous constituent un des pans de la recherche en biomé canique des chocs, et peuvent avoir des applications importantes en médecine, que ce soit à l'échelle microscopique ou macroscopique. La perforation d'un projectile dans les tissus mous doit ê tre compris, à l'échelle macroscopique, pour analyser un impact ballistique perforant par exemple, mais é galement à plus petite é chelle, avec l'administration de particules médicamenteuses à hautes vitesses au travers de diffé rents tissus, comme la peau, en prenant en compte diffé rents paramè tres du phé nomè ne, comme la vitesse d'impact ou la taille du projectile. Un des moyens efficace pour comprendre ces phé nomè nes à trè s hautes vitesses est la simulation numé rique. Les modè les numé riques doivent ê tre alors validé s par rapport à des tests expérinentaux permettant de s'assurer de leurs biofidé lité s.Bien que les mé thodes basé es sur un maillage comme la mé thode des é lé ments finis sont des outils puissante, les grandes distorsions des é lé ments tels qu'ils peuvent apparaître lors d'impacts pénétrant gé nè rent des erreurs numé riques importantes. Ainsi, des mé thodes sans maillage basé essur une modé lisation particulaire s'affranchissant de toutes grillessont trè s adapté es pour simuler à hautes vitesses et grandes dé formations. Un des plus anciennes mé thode, la mé thode SPH (Smoothed Particles Hydrodynamics) a é té appliqué e, dans cette thè se à la mé canique du solide en grandes vitesses et grandes dé formations. Cette thè se s'attache à é tudier la mé thode SPH en dé veloppant un code "maison" pour ls impacts perforants ou non, puis propose un modè le numé rique pour é tudier le processus de pé né tration de projectiles dans les tissus mous (la gé latine et le polymè re synthé tique gel SEBS sont considé ré s dans la lité rature comme des substituts de tissus biofidè les) à é chelle micromé trique.

	Projectile-Target system Projectile-Target system Physical problem target behavior law target behavior law Physical problem	• thin Al plates • Johnson-Cook model SPH limitations to debris shielding • Al/Steel plate • Elastic-perfectly plastic • Al plates • Johnson-Cook model shapes of debris clouds, effects of the craters size and debris clouds	• triple wall system • Steinberg-Guinan system • 0.2 cm model • 2-4 mm • Mie-Gr üneisen types of artificial viscosities	• Sesame, Shock, Tillotson • Steinberg-Guinan • Tillotson equation	• Mie-Gruneisen EOS	• steel plate • elasto-plastic model contact algorithm, numerical frac-• HOM equation • AL2O3 ceramic • JH-2 model contact algorithm, fragmentation,	• 6 mm tures • 12.7 mm residual velocity	• fused silica plates • Johnson Holmquist model debris cloud, perforation hole and	• steel plate • elasto-perfectly-plastic contact algorithm, numerical frac-• disposable debris shields • pressure cut-off criteria spallation zones	• 3 mm model tures (DDS)	• 2-10 mm	• Gr/Ep laminate • Al/steel/brass plates • Johnson-Cook model with • Elastic-plastic of single layer damage and debris clouds residual velocities, ballistic limit,	• 1 -8 layers • 3-12 mm EOS perforation	• 1mm	• steel plate • elasto-plastic model contact algorithm, numerical frac-	• Al plate • elastic perfectly plastic • 1 mm • Mie-Gruneisen the contact potential algorithm tures	• 2.63 cm model	• Gruneisen EOS • Al plate • Elastic-plastic model shapes of debris clouds	• 0.8 mm • Mie-Gruneisen equation	• Al plates • Johnson-Cook model with fragment identification and statistics	• 2 mm EOS • steel/Al plates • Johnson-Cook model with method,debris clouds residual and ballistic limit velocities,	• 6-30 mm EOS perforation	• 2 mm plates • elastic-perfectly plastic law the effects of impact velocity and	• Al/Lead/Steel • Weldox 460 E steel • Viscoplasticity and ductile target thickness on penetration coupling algorithm, ballistic limit ve-	• 12 mm damage model depth and crater size locity.	• Mie-Gruneisen equation	• 500 HB armor steel • Johnson-Cook model ballistic limit, spall and fragmenta-	• 9-20mm • Mie-Gruneisen • Al plate • not introduced tion boundary deficiencies, shapes of	• mm-scale formed craters and apertures.	• Steel plate • elastic-plastic law • Steel/Al plates • modified Johnson-Cook element-partile convert and contact residual and ballistic limit velocities	• mm-scale • 6-20 mm (MJC) model algorithms			• CRFP laminate • Elastic-plastic of single layer damage pattern, fiber breaks,	• 1.6 mm • Chang-Chang critera cracks
	projectile projectile	Al sphere Al/Steel spheres, Cu, Al, deformed	deformed			Lead/steel rod, alloy, deformed	deformed	steel sphere, de-	steel cube, de-formed	formed		aluminum sphere, Caliber/steel/Al,	deformed deformed		steel sphere, de-	steel rod with formed	ogive-nose, rigid	Al sphere, de-	formed	Al, deformed	steel rod with var-	ious noses, rigid	Al/Lead/Steel	spheres steel cylinder,	rigid		54R B32 API	hardened steel Al sphere, de-	core ammunition formed	Steel, deformed steel cylinder with	blunt, conical,	ogival noses,	rigid	steel ball, de-	formed
	Impact condition Impact condition	• normal impact • normal impact	• 7 km/s • 3.1 km/s • oblique impact	• 2-12 km/s		• normal impact • normal impact	• 820-840 m/s • 903 m/s	• normal impact	• normal impact • 1-4 km/s	• oblique impact	• 2000 m/s	• normal impact • normal impact	• oblique impact • 70-1050 m/s	• 4.0 km/s	• normal impact	• oblique impact • 200 m/s	• 400 m/s	• normal impact	• 6150 m/s	• normal impact	• 6.64 m/s • normal impact	• 10-400 m/s	• normal impact	• 0.3-3.1 km/s • normal impact	• 180-300 m/s		• normal impact	• 7 km/s • normal impact	• 6.18/6.7 km/s	• oblique impact • normal impact	• 2 km/s • 200-400 m/s			• normal impact	• 200-1200 m/s
	Model & techniques Model & techniques	SPH code [107] Corrective SPH (DFPM)	[92]			2D-SPH code [111] SPH in LS-DYNA (con-	tact model: PCM and	SPH in LS-DYNA [51] PAM) [59]	3D-SPH code [112]			Improved SPH ( new SPH in LS-DYNA [127]	particle generation and	merger techniques) [54]	3D-SPH code [116]	Coupled FE-SPH [125]		SPH and MPM [13]		FE-SPH-FE in LS-DYNA	[128] SPH in LS-DYNA [121]		2D SPH [221]	SPH-FEM coupled [126]			SPH in LS-Dyna [75]	modified SPH [222]		Combined Particle-FE-SPH code [108]	Element Method (CPEM)	[120]		3D SPH code [55]

Titre : Modé Title : Numerical

modeling of impacts at high velocities by a meshfree method Smoothed particles Hydrodynamics. Application to micro-impacts in soft tissues.

ACKNOWLEDGEMENTS

I would like to particularly thank my supervisors, Prof.Sebastien Roth. This research could not have been accomplished without his sustaining support, guidance, patience and encouragement all the way. Especially the words 'step by step' give me much encourage when i hit a bottleneck in the research.

thesis work. I am also grateful to the China Scholarship Council (CSC) for putting faith in my studies and providing the financial support for this research at the Technological University of Belfort Montb éliard (UTBM) in France.

LIST OF FIGURES

1. 1 The system of needle-free drug delivery device provided in the [1]. . . . . . 1.2 The process of a 13-µm steel sphere penetrating into a simulant of soft tissue called SEBS gel observed by the experiment [2]. . . . . . . . . . . .

2.1

The types of damages in HVI from [START_REF] Marvin | The mechanics of penetration of projectiles into targets[END_REF]. . . . . . . . . . . . . . . . . . . . .

2.2

The types of damages observed by HVI experiments: a) shows the plugging situation after perforation by [START_REF] Børvik | Ballistic penetration of steel plates[END_REF]; b) shows the petaling and ductile hole enlargement after perforation in [START_REF] Manes | An experimental-numerical investigation on aluminium tubes subjected to ballistic impact with soft core 7.62 ball projectiles[END_REF]; c) shows the perforation accompanying with radial fracture in [START_REF] Michel | Hypervelocity impacts on thin brittle targets: Experimental data and SPH simulations[END_REF]; d) shows the debris could during the impact in [START_REF] Andrew | Characteristics of debris clouds produced by hypervelocity impact of aluminum spheres with thin aluminum plates[END_REF]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [START_REF] Stellingwerf | Smooth particle hydrodynamics[END_REF]. . . . . . . . . . . . . . . . . . . . . [START_REF] Michel | Hypervelocity impacts on thin brittle targets: Experimental data and SPH simulations[END_REF]: two types of spalls can be identified (macro and micro spalls) represented by SPH particles clusters as fragments.

Illustration of the Stellingwerf's test

Illustration of SPH simulation in