Marc Blétry

Non-linear mechanics of materials

Preface by Jean Lemaitre Chapter 1 Introduction 1.1. Model construction 1.2. Applications to models Chapter 2 General concepts 2.1. Formulation of the constitutive equations 2.2. Principle of virtual power 2.3. Thermodyna~nicso f irreversible processes 2.4. Main class of constitutive equations 2.5. Yield criteria 2.6. Numerical methods for nonlinear equations 2.7. Numerical solution of differential equations 2.8. Finite element Chapter 3 Plasticity and 3D viscoplasticity 3.1. Generality 3.2. Formulation of the constitutive equations 3.3. Flow direction associated to the classical criteria 3.4. Expression of some particular constitutive equations in plasticity 3.5. Flow under prescribed strain rate 3.6. Non-associated plasticity 3.7. Nonlinear hardening 3.8. Some classical extensions 3.9. Hardening and recovery in viscoplasticity 3.10. Multimechanism models 3.1 1. Behaviour of porous materials Chapter 4 Introduction to damage mechanics 4.1. Introduction 4.2. Notions and general concepts 4.3. Damage variables and state laws 4.4. State and dissipative couplings 4.5. Damage deactivation 4.6. Damage evolution laws 4.7. Examples of damage models in brittle materials Chapter 5 Microstructural mechanics 5.1. Characteristic lengths and scales in microstructural mechanics 5.2. Some homogenization techniques 5.3. Application to linear elastic heterogeneous materials 5.4. Some examples. applications and extensions 5.5. Homogenization in thermoelasticity 5.6. Nonlinear homogenization 5.7. Computation of RVE 5.8. Homogenization of coarse grain structures Chapter 6 Finite deformations 6.1. Geometry and kinematics of continuum 6.2. Sthenics and statics of the continuum 6.3. Constitutive laws 6.4. Application: Simple glide 6.5. Finite deformations of generalized continua Chapter 7 Nonlinear structural analysis 7.1. The material object 7.2. Examples of implementations of particular models 7.3. Specificities related to finite elements Chapter 8 Strain localization 8.1. Bifurcation modes in elastoplasticity 8.2. Regularization methods Appendix Notation used A.1. Tensors A.2. Vectors, Matrices A.3. Voigt notation

Dislocation dynamics simulations with climb: kinetics of dislocation loop coarsening controlled by bulk diffusion

Dislocation climb mobilities, assuming vacancy bulk diffusion, are derived and implemented in dislocation dynamics simulations to study the coarsening of vacancy prismatic loops in fcc metals. When loops cannot glide, comparison of the simulations with a coarsening model based on the line tension approximation shows good agreement. Dislocation dynamics simulations with both glide and climb are then performed. Allowing for glide of the loops along their prismatic cylinders leads to faster coarsening kinetics, as direct coalescence of the loops is now possible.

Fluctuations, structure factor and polytetrahedra in random packings of sticky hard spheres

Sequentially-built random sphere-packings have been numerically studied in the packing fraction interval 0.329 < γ < 0.586. For that purpose fast running geometrical algorithms have been designed in order to build about 400 aggregates, containing 106 spheres each one, which allowed a careful study of the local fluctuations and an improved accuracy in the calculations of the pair distribution P(r) and structure factors S(Q) of the aggregates. Among various parameters (Voronoi tessellation, contact coordination number distribution,…), fluctuations were quantitatively evaluated by the direct evaluation of the fluctuations of the local sphere number density, which appears to follow a power law. The FWHM of the Voronoi cells volume shows a regular variation over the whole packing fraction range. Dirac peaks appear on the pair correlation function as the packing fraction of the aggregates decreases, indicating the growth of larger and larger regular polytetrahedra, which manifest in two ways on the structure factor, at low and large Q values. These low PF aggregates have a composite structure made of polytetrahedra embedded in a more disordered matrix. Incidentally, the irregularity index of the building tetrahedron appears as a better parameter than the packing fraction to describe various features of the aggregates structure.

3D plasticity and viscoplasticity

This chapter is devoted to 3D plasticity theory. It includes a comprehensive description of the building bricks available to develop constitutive equations in a plastic or viscoplastic framework. Various criteria and hardening rules are introduced. The models account for cyclic loading paths, thermomechanical histories, recovery, extra-hardening, anisotropy, etc. Both single potential and multipotential approaches are investigated. The latter includes models suitable for DS alloys or single crystals. The behaviour of porous material is also investigated.

Introduction to damage mechanics

This chapter introduces the main concepts used in Continuum Damage Mechanics (CDM), a specific theory for coupling constitutive equations with the material deterioration processes that evolve before fracture. Similarities and differences with classical Fracture Mechanics are pointed out.

Nonlinear structural analysis

This chapter first introduces a generic description of material constitutive equations and of the link between finite elements and material laws. Examples of the implementation of specific constitutive models are given (plasticity, viscoplasticity, kinematic hardening, porous materials). Some specific finite element treatments are then reviewed including: incompressibility, periodicity, finite strains and Cosserat elements.

Inelastic constitutive laws at finite deformation

The modern continuum thermomechanics framework is settled for finite deformations of materials. Objective stress and strain measures are defined. Elastoviscoplastic constitutive equations are formulated based on the introduction of the free energy and dissipation potentials. Analytical and numerical results for the simple glide of isotropic or single crystalline materials provide a clear assessment of various classes of finite deformation material models. Finally the theory is extended to generalized continua such as the Cosserat medium.

Strain localization phenomena

Strain localization phenomena such as necking and shear banding very often occur prior to final failure. They can be predicted by a bifurcation analysis of the system of constitutive and equilibrium equations. Notions like stability, acceleration waves and loss of ellipticity are defined. Finite element simulations can be performed to predict the onset of various localization modes. The analysis of the post-localization behavior requires the introduction of regularization methods, such as the mechanics of generalized continua, to restore the well-posedness of the boundary value problem.

Elements of microstructural mechanics

Microstructural mechanics aims at determining the impact of the microstructure of materials on the effective properties and the local fields in heterogeneous materials. Homogenization techniques are designed to derive overall linear and non-linear properties of materials from the knowledge of the properties of individual phases and their arrangement in space. The notion of Representative Volume Element is defined and its size is estimated in the case of random materials. The limits of homogenization are explored when the characteristic wave length of mean field variation becomes close to the size of heterogeneities.