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## Laser-matter interaction theory

The interaction mechanisms between matter and electromagnetic waves depend on the waves frequency range. The electromagnetic spectrum (Fig. 2.1) extends from low frequencies used for modern radio communication to high frequencies encoun-tered in nuclear decay, e.g. gamma radiation. The high-power lasers which are the concern of this dissertation emit radiations between infrared and ultraviolet.

The interaction between a laser beam and matter is based on the fast photon energy change into thermal and/or kinetic energy in the first atomic layers of mat-ter. When a material is irradiated with lasers, the laser energy will first be trans-formed into electronic excitation energy and then transferred to lattices of the ma-terial through collisions between the electrons and lattices. The deposition of laser energy will produce a series of effects, such as temperature rise, gasification and ion-ization. The physical processes of laser-matter interaction (Fig. 2.2) depend mainly on the laser intensity IL = EL , where EL is the pulse energy, τL is the Full Dura- τL SL tion at Half Maximum (FDHM) and SL is the energy deposition surface. The present work will focus on laser radiations with laser density above 1012 W cm−2, leading to plasma generation and ablation of matter.

**Generation of shock wave by laser ablation**

Materials subjected to laser irradiation (Fig. 2.3) will absorb the incident laser energy [78]. In the case of normal incidence, the coefficient of energy absorption A is given by Equation (2.1): (nL + 1)2 + k2L A = 4 nL (2.1)

where nL and kL are respectively the real and imaginary parts of the refraction com-plex index (n˜L = n L + ik L). Otherwise, the influence of the incidence angle and the polarization of the laser radiation must be taken into account in the computation of A. The laser energy is firstly absorbed by the free electrons in a small depth lskin determined by Equation (2.2): 2 π kL lskin = λ (2.2).

**Shock wave propagation in materials**

The laser-induced pressure will make the irradiated material under compressive stresses which will propagate in all directions [173, 104, 61]. When the amplitude of the stress waves greatly exceeds the dynamic strength of the material, the shear stresses can be neglected in comparison with the hydrostatic component of the stress tensor. One therefore can consider a high pressure state traveling into the material which can be assumed, in a first approximation, to have no shear resistance (i.e. the shear modulus is zero, µ = 0). Under this assumption, its state is completely characterized by three thermodynamic parameters: pressure P, density ρ (or specific volume v = ρ−1) and internal energy E (or temperature T). The propagation of a high pressure state into a material can lead to the formation of a shock wave. To simplify the understanding of this concept, an ideal gas will be first considered.

For an ideal gas, the associated equation of state in the case of an isentropic process can be written: PVγ = const where V is the volume occupied by the gas, γ = Cp Cv Cv are the specific heat at constant pressure and the specific heat at constant volume, respectively. Derivation of Equation (2.5) leads to: ∂P = −γ P (2.6) ∂V V.

**Hydrodynamic treatment**

Shock waves are characterized by a steep front and require a state of uniaxial strain (no considerable lateral flow of material) which allows the buildup of the hydro-static component of stress to high levels. When the last (hydrostatic component) far exceeds the dynamic flow strength, the material behaves as a fluid. Therefore, the Rankine-Hugoniot conservation equations for fluids [145, 143, 144] can be applied to calculate the shock wave parameters. This is valid taking into account several assumptions:

• the shock is a discontinuous surface and has no apparent thickness.

• the material behaves as a fluid (µ = 0); the theory is, therefore, restricted to high pressures.

• body forces (such as gravitational) and heat conduction at the shock front are negligible.

• there is no elastoplastic behavior.

• material does not undergo phase transformation.

Taking into account these assumptions, the shock wave equations can easily be ob-tained by considering a small region Ωs with cross section A immediately ahead of and behind the shock front (Fig. 2.6).

Ahead of the front (initial state), the pressure is P0, the density is ρ0 and the energy is E0; behind it, they are P, ρ and E, respectively. The velocity of the particles ahead of and behind the shock front, which is moving at a velocity of Vs (shock velocity), are (Vp) P0 = V0 and Vp, respectively. The apparent velocity of the shock front is (Vs − V0), because it is moving into a region of particle velocity Vp = V0. At the same time, the material leaving the shock front is moving at a velocity Vs − Vp. Using this, the equations of mass, momentum and energy conservation can be derived.

### Representative curves for shock waves

Hugoniot curve: The relationship between P and ρ (or v = ρ−1) is usually known as Hugoniot equation. This equation is defined as the locus of all shocked states in the (P, ρ) or (P, v) frame and essentially describes the material properties (Fig. 2.8a). The straight line in Figure 2.8a relaying the (P0, v0) and (P1, v1) states is the Rayleigh line and refers to the shock state at P1. It is very important to realize that when pressure is increased in a shock front, it does not follow the P − v curve. Rather, it jumps discontinuously from P0 to P1. The slope of this line is proportional to the square of the shock velocity (Vs).

#### Phenomena spoiling the propagation of shock waves

Hydrodynamic damping: As shown above, for most materials, the concavity of the Hugoniot curve is positive. Therefore, the compression front will steepen up as it travels through the material because the higher amplitude regions of the front travel faster than the lower amplitude regions. In contrary, the release front will spread out (the release rate decreases) as it travels through the material. The head of the release part (at the maximum pressure) is traveling at a velocity (Vp + C)Pf which is greater than the shock velocity Vs, where Vp and C are the particle velocity and the longitudinal sound velocity at the pressure Pf . As the wave progresses, the release part of the wave overtakes the front. This will reduce the pulse duration to zero. After it is zero, the peak pressure starts to decrease and so the shock velocity (Fig. 2.10). This is generally referred to as “hydrodynamic damping”.

**Table of contents :**

**1 Introduction **

1.1 Motivation and objective research

1.2 Methodology of the work

1.3 Dissertation structure

**2 State of the art **

2.1 Laser-matter interaction theory

2.1.1 Generation of shock wave by laser ablation

2.1.2 Shock wave propagation in materials

2.1.2.1 Hydrodynamic treatment

2.1.2.2 Representative curves for shock waves

2.1.2.3 Permanent deformation and shock wave

2.1.2.4 Phenomena spoiling the propagation of shock waves

2.1.2.5 Shock wave transmission and reflexion

2.1.3 Laser-induced damage in materials

2.2 Mechanical behavior of silica glass

2.2.1 Silica glass response under static hydrostatic compression .

2.2.1.1 Elasticity

2.2.1.2 Permanent deformation: densification

2.2.1.3 Effects of the shear stresses on the silica glass densification

2.2.1.4 Fracture

2.2.2 Silica glass response under shock compression

2.2.2.1 Propagation of compression wave in the region of reversibility

2.2.2.2 Propagation of compression wave in the region of irreversibly

2.2.2.3 Spalling strength

2.2.3 Summary of the silica glass response under shock compression

2.3 Numerical simulation

2.3.1 Continuum methods

2.3.2 Discrete methods

2.3.3 Coupling methods

2.3.4 What class of numerical methods best meets the objectives of this dissertation?

2.4 Conclusion

**3 Choice of the numerical methods **

3.1 Introduction

3.2 Choice of the discrete method

3.2.1 Lattice models

3.2.2 Particle models

3.2.3 Contact dynamics

3.2.4 Classification and choice of the discrete method: DEM

3.3 Choice of the continuum method

3.3.1 Grid-based methods

3.3.1.1 Lagrangian methods

3.3.1.2 Eulerian methods

3.3.1.3 Combined Lagrangian-Eulerian methods

3.3.2 Meshless methods

3.3.2.1 Approximation methods

3.3.2.2 Interpolation methods

3.3.3 Classification and choice of continuum method: CNEM

3.4 Conclusion

**4 Discrete-continuum coupling **

4.1 Introduction

4.2 The discrete element method: DEM

4.2.1 Construction of the DEM domain

4.2.2 Cohesive beam bond model

4.3 The constrained natural element method: CNEM

4.3.1 Natural Neighbor (NN) interpolation

4.3.1.1 Voronoï diagram

4.3.1.2 NN shape functions

4.3.1.3 Support of NN shape functions

4.3.1.4 Properties of NN shape functions

4.3.2 Visibility criterion

4.3.3 Constrained Natural Neighbor (CNN) interpolation

4.3.4 Numerical integration

4.4 Discrete-continuum coupling method: DEM-CNEM

4.4.1 Arlequin approach: brief description

4.4.2 Arlequin approach: application to the DEM-CNEM coupling .

4.4.2.1 DEM formulation

4.4.2.2 CNEM formulation

4.4.2.3 Coupling formulation

4.4.2.4 Global weak formulation

4.4.3 Discretization and spatial integration

4.4.4 Time integration

4.4.5 Algorithmic

4.4.6 Implementation

4.5 Parametric study of the coupling parameters

4.5.1 Influence of the junction parameter l

4.5.2 Influence of the weight functions a

4.5.2.1 Constant weight functions a = ¯a = 0.5

4.5.2.2 Constant weight functions a 6= 0.5

4.5.2.3 Continuous weight functions

4.5.3 Influence of the approximated mediator spaceMhO

4.5.4 Influence of the width of the overlapping zone LO

4.5.5 Dependence between LO and MhO

4.5.6 How to choose the coupling parameters in practice?

4.6 Validation

4.7 Conclusion

**5 Silica glass mechanical behavior modeling **

5.1 Introduction

5.2 Modeling hypotheses

5.3 Beam-based mechanical behavior modeling

5.3.1 Modeling of nonlinear elasticity

5.3.2 Modeling of densification

5.3.3 Static calibration and validation

5.3.4 Dynamic calibration and validation

5.3.5 Discussion

5.4 Virial-stress-based mechanical behavior modeling

5.4.1 Virial stress

5.4.2 Modeling of nonlinear elasticity

5.4.3 Modeling of densification

5.4.4 Application: Plates impact

5.4.5 Discussion

5.5 Brittle fracture modeling

5.5.1 Standard fracture model and its limitations

5.5.2 Fracture model based on the virial stress

5.6 Conclusion

**6 Simulation of LSP processing of silica glass **

6.1 Introduction

6.2 Brief description of the LSP test to be simulated

6.3 Numerical model

6.4 Results

6.5 Conclusion

**7 Conclusion and future work **

**Bibliography**