Dislocation Microstructures and the Effective Behavior of Single Crystals

Abstract

We consider single-crystal plasticity in the limiting case of infinite latent hardening, which signifies that the crystal must deform in single slip at all material points. This requirement introduces a nonconvex constraint, and thereby induces the formation of fine-scale structures. We restrict attention throughout to linearized kinematics and deformation theory of plasticity, which is appropriate for monotonic proportional loading and confers the boundary value problem of plasticity a well-defined variational structure analogous to elasticity. We first study a scale-invariant (local) problem. We show that, by developing microstructures in the form of sequential laminates of finite depth, crystals can beat the single-slip constraint, i. e., the macroscopic (relaxed) constitutive behavior is indistinguishable from multislip ideal plasticity. In a second step, we include dislocation line energies, and hence a lengthscale, into the model. Different regimes lead to several possible types of microstructure patterns. We present constructions which achieve the various optimal scaling laws, and discuss the relation with experimentally known scalings, such as the Hall-Petch law.

Nonlinear regularization operators as derived from the micromorphic approach to gradient elasticity, viscoplasticity and damage

“…It turns out that the size effects predicted based on a quadratic potential are not consistent with results from physical metallurgy [63,67]. The singular case m = 1 provides consistent scaling laws for the yield stress as a function of channel width in laminate microstructures [47,50]. In particular, the singular character of the model results in a size-dependent abrupt increase of the apparent yield stress.…”

“…Motivations for nonlinear potentials with respect to gradient of the extra-degrees of freedom stem from recent strain gradient crystal plasticity models making use of rank one, power law or logarithmic functions of the gradient terms for better description of dislocation behaviour [47][48][49][50]. Non-quadratic potentials are seldom in the phase field community.…”

Homogenization of the Prager model in one-dimensional plasticity

“…It is therefore an analysis of the time-dependent problem. Regarding the interesting field of variational problems in plasticity we mention [19] as a general reference, [4] for a homogenization result, and [7] for an analysis of microstructures that appear as a consequence of non-convexity.…”

Some recent results on the convergence of damage to fracture

“…Many fine properties and results holding in the BV framework have a counterpart for BD (see [4]), but in general BD functions remain less well understood. In particular the study of lower semicontinuity and relaxation in BD is still in its beginnings (see [10,29,34,26,12,13,21,9]) and integral representation results have been established only in some special cases (see [25]). Compactness and approximation results with more regular functions have been obtained in [10,14,15,23,31].…”