Plasticity and Beyond

Instead of developing a macroscopic phenomenological constitutive law, it is possible to attach a representative volume element (RVE) of the microstructure at each point of the macrostructure; this results in a two-scale modeling scheme. A discrete version of this scheme performing finite element (FE) discretizations of the boundary value problems on both scales, the macro- and the micro-scale, is denoted as the FE²-method or as the multilevel finite element method. The main advantage of this procedure is based on the fact that we do not have to define a macroscopic phenomenological constitutive law; this is replaced by suitable averages of stress measures and deformation tensors over the microstructure.

Furthermore, remarks concerning stability problems on both scales as well as their interactions are given and representative numerical examples for elasto-plastic microstructures are discussed.

The analysis and simulation of microstructures in solids has gained crucial importance, virtue of the influence of all microstructural characteristics on a material’s macroscopic, mechanical behavior. In particular, the arrangement of dislocations and other lattice defects to particular structures and patterns on the microscale as well as the resultant inhomogeneous distribution of localized strain results in a highly altered stress-strain response. Energetic models predicting the mechanical properties are commonly based on thermodynamic variational principles. Modeling the material response in finite-strain crystal plasticity very often results in a nonconvex variational problem so that the minimizing deformation fields are no longer continuous but exhibit small-scale fluctuations related to probability distributions of deformation gradients to be calculated via energy relaxation. This results in fine structures which can be interpreted as the observed microstructures.

This manuscript is supposed to give an overview of the available methods and results in this field. We start by discussing the underlying variational principles for inelastic materials, derive evolution equations for internal variables, and introduce the concept of condensed energy. As a mathematical prerequisite we review the variational calculus of nonconvex potentials and the notion of relaxation. We use these instruments in order to study the initiation of plastic microstructures. Here we focus on a model of single-slip crystal plasticity. Afterward we move on to model the evolution of microstructures. We introduce the concept of essential microstructures and the corresponding relaxed energies and dissipation potentials, and derive evolution equations for microstructure parameters. We then present a numerical scheme by means of which the microstructure development can be computed, and show numerical results for particular examples in single- and double-slip plasticity. We discuss the influence of hardening and of slip system orientations in the present model.

Continuum crystal plasticity models are extended to incorporate the effect of the dislocation density tensor on material hardening. The approach is based on generalized continuum mechanics including strain gradient plasticity, Cosserat and micromorphic media. The applications deal with the effect of precipitate size in two–phase single crystals and to the Hall-Petch grain size effect in polycrystals. Some links between the micromorphic approach and phase field models are established. A coupling between phase field approach and elastoviscoplasticity constitutive equations is then presented and applied to the prediction of the influence of viscoplasticity on the kinetics of diffusive precipitate growth and morphology changes.

Deformation substructures control plastic, creep, fatigue and fracture properties of ductile crystalline solids. The key ingredient of a substructure is a spontaneously formed dislocation arrangement – dislocation structure. The present notes provide 5 different, complementary points of view which present the dislocation structure formation as a multi scale phenomenon: (i) The basic concepts of dislocation theory and plasticity of single crystals and polycrystals (Section 2). (ii) A ”gallery” of commented pictures of dislocation structures as seen by a transmission electron microscope (Section 3). (iii) Discrete dislocation dynamics (Section 4). (iv) An attempt to formulate statistics of dislocations as a transition from discrete dislocation dynamics to continuum crystal plasticity capable of modeling dislocation structure formation (Section 5). (v) Two continuous models of dislocation structure formation: one dimensional model simulating a formation of vein structure and its transformation into a ladder structure of a persistent slip band (Section 6.1), and a model of misoriented dislocation cells (Section 6.2).

An extended crystal plasticity theory that accounts for the length-scale effects in plastic strain gradient fields is presented. First, foundations and kinematics of crystal plasticity theory is reviewed. Then, experimental evidences for the size-effects in small-sized bent single crystals are presented. Total amounts of apparent strain hardening, which were experimentally observed, are decomposed into isotropic and kinematic hardening components. Physically-based models are formulated to describe the size-dependent isotropic and kinematic hardening behaviors, utilizing possible micromechanical information with respect to dislocations and their motions. Roles of the geometrically necessary dislocations (GNDs) in strain hardening behavior are studied in detail. Furthermore, some aspects of numerical computations of the extended size-dependent crystal plasticity theory are presented. The developed theory involves extra boundary conditions for crystallographic slips and/or the GND densities. Effects of these extra boundary conditions are demonstrated through numerical simulations for some basic boundary value problems. Finally, a phenomenological strain gradient plasticity theory is revisited, based on the knowledge from the present size-dependent crystal plasticity theory.