Preface

Marc BERNACKI1 and Samuel FOREST2

1CNRS UMR 7635, Centre for Material Forming (CEMEF), MINES Paris, PSL Research University, Sophia Antipolis, France

2CNRS UMR 7633, Centre des Matériaux, MINES Paris, PSL Research University, Evry, France

Digital materials have now become an integral part of design methods for industrial components and structures. Their mechanical properties can be predicted from a description of the microstructure and behavior laws of the constituents. This topic is now well established through the concept of integrated computational materials engineering (ICME).

This book covers a wide range of material properties from transport phenomena to the mechanics of materials and microstructure changes in physical metallurgy. It is not, however, an exhaustive review of numerical simulation methods for these phenomena, but rather a current look at the continuum simulation of elasticity, viscoplasticity, phase changes and grain boundary migration in metals, alloys and composites.

The elementary mechanisms of deformation and damage in materials involve complex atomic processes, which are now well explored by appropriate numerical simulations such as molecular dynamics. In contrast, this book shows how these mechanisms can be integrated into a continuum approach to the physics and mechanics of materials at the mesoscopic scale. It therefore focuses on the mechanics of continuous media and the continuum thermodynamics of irreversible processes. These models take into account the heterogeneities of materials, namely, their polycrystalline and/or composite nature, and constitute an intermediate step toward the establishment of effective laws for engineers in the calculation of structures and manufacturing processes.

The scientific topics selected in the book focus on the homogenization theory developed for periodic or random composite materials. The mean field approach is based on an idealization of the microstructure and provides estimates of the mean stresses and strains in the various phases present. These models can be fed by the full-field approach, where the representative elementary volume is the key concept. Plasticity is also given a significant place in this book, particularly at the grain scale, thanks to crystal plasticity in its discrete form, namely, the dynamics of dislocations, or its continuous form, which evaluates the quantities of slip and strain hardening associated with each slip system. Another strong point is the treatment of moving surfaces within the microstructure of materials, whether for phase changes, grain boundary migration and recrystallization. Statistical methods are used extensively to take account of the random nature of the distribution of constituents, dislocations and cracks, and also to exploit the large volumes of data produced by these simulations.

The book is structured into two parts. The first three chapters are devoted to mean-field and full-field approaches to establish the constitutive laws of elastoviscoplastic heterogeneous materials. The next four chapters focus on the numerical simulation of microstructure evolution, with particular attention paid to the modeling of the motion of interfaces (grain boundaries, moving interfaces for phase changes, etc.) and a whole range of numerical front-tracking or front-capturing methods that are now well established or in full development.

The first chapter is a summary presentation by Sylvain Queyreau of the methods and results of discrete dislocation dynamics (DDD) simulations. It begins with a description of the simulation technique, which is based on the elastic theory of dislocations in crystals, as well as on information relating to the core of dislocations and dislocation interaction rules derived from molecular dynamics analyses and microscopic observations. The method is capable of making predictions about the behavior of the single crystal, in particular the strain hardening laws. Recent developments address the response of polycrystals to complex loading, particularly cyclic loading conditions.

In the second chapter, Dominique Jeulin immediately places the question of the representative volume element (RVE) at the heart of the debate on the construction of the effective behavior of heterogeneous materials. He promotes statistical analysis of the morphology of phase distribution within the microstructure and the construction of random morphological models on which to base simulations of physical and mechanical properties. He proposes a precise statistical definition of the size of the RVE according to the precision required for the targeted property estimates. It emphasizes the biases associated with the choice of boundary conditions applied to the boundaries of the domain and stresses the importance of the concept of integral range for the definition of the RVE. The approach is highly effective for predicting linear material properties such as thermal and electrical conductivity, permeability and elasticity, but also acoustics and wave propagation. The statistical definition of the RVE, initially developed for linear properties, is extended here to the nonlinear case by using the space average of the local energy. Finally, the approach is heuristically extended to the elastoplastic behavior of heterogeneous materials, in particular metallic polycrystals and porous media.

The micromechanics of materials is addressed in Chapter 3 within the framework of the mean field approach. In a discipline whose history, outlined in the introduction, has been particularly rich over the last 60 years, the chapter focuses on the thorny issue of homogenization in the case of elastoviscoplastic behavior of the phases involved. The translated field method is highlighted and compared with other models available in the literature. It is first applied to the case of two-phase composites and then to metallic polycrystals. The second part of the chapter concerns the pragmatic approach developed by Cailletaud and Pilvin, which has proven its worth in the prediction of complex loading paths, particularly cyclic loading. It is extended here to the case of large deformations of metallic polycrystals in order to predict the evolution of the crystallographic texture and internal stresses for different deformation paths encountered particularly during metal forming.

The second part of the book, devoted to the simulation of evolving microstructures during thermomechanical treatments, begins with a presentation of the Vertex method in Chapter 4. The migration of grain boundaries and the growth of new grains are modeled by tracking grain boundaries, triple lines and vertices whose movements are dictated by surface energy, grain boundary curvature and, possibly, the energy stored in the grains in the form of dislocation densities assumed homogeneous per grain. Topological changes in the microstructure represent a major challenge in the tracking of these geometric objects within representative elementary volumes. Robust methods are now available and have proved their worth in simulating numerous metallurgical phenomena such as dynamic recrystallization or grain growth with second-phase particles and anisotropic properties of grain boundaries.

Predicting microstructural evolution during complex transformation processes, including mechanical, thermal and metallurgical aspects, is now possible thanks to the thermodynamics of continuous media and irreversible processes. This energy-based approach combines mechanical contributions (elasticity, viscoplasticity and strain hardening), chemical contributions (multi-constituent diffusion) and phase changes (oxidation, diffusive and displacive transformations). Electromagnetic coupling is also possible. The implementation of this holistic modeling is enhanced by the phase field method, which enables the kinetics and dynamics of interface motion to be calculated by formulating coupled free energy potentials and substituting diffuse interfaces for ideal surfaces. This is the subject of Chapter 5 by Ingo Steinbach and Oleg Schchyglo. The theoretical part of the chapter presents the multi-phase and multi-component formulation, which enables realistic alloy compositions to be addressed and the evolution of phase diagrams to be taken into account in detail. The phase field method excels in predicting changes in phase morphology, both coalescence and separation, but is often unsuitable for nucleation processes. Illustrations concern multi-component solidification, martensitic transformation and recrystallization phenomena.

The level-set method is described in Chapter 6 for applications to diffusive phase transformations (Oswald ripening and precipitate coalescence), grain growth and static or dynamic recrystallization in polycrystals. The driving pressures for interface migration are described in detail, as is the finite element numerical implementation. The crucial ingredient of the underlying model is the interface mobility law linking its velocity to the driving pressures of curvature and the stored energy jump in the case of recrystallization, or the Gibbs free energy jump in the case of diffusive phase changes. The technical subtleties of implementing these methods in finite element codes are explained, and the applications concern both 2D and 3D simulations, with original methods for meshing grain boundaries and interfaces. Once again, particular attention is paid to the treatment of multiple junctions. Large-scale computations of polycrystalline aggregates are used to study the influence of grain boundary types on their growth in relation to the anisotropy of surface energies.

The last chapter...