Analysis and Computation of Microstructure in Finite Plasticity

Slip processes are soft modes of deformation, characteristic of a variety of layered materials. The layers can be at the atomic scale, as in the plastic deformation of crystalline lattices, or on a macroscopic scale, as in stacks of cards or sheets of paper and geological strata. The characteristic deformation processes involve sliding of the layers over one another, leading to a shear deformation with a specific orientation. If the forcing is not parallel to the layers, complex microstructures may form, which have a remarkable similarity over different systems and often consist of alternating shears of different sign. We review here recent results on the detailed analysis of slip processes in crystal plasticity based on the theory of relaxation, discuss the general variational framework for these microstructures, and compare with available experimental results in different systems. We then address the situation in which slip in several different directions may coexist in the same system, as frequently observed in plastically deformed crystals.

In this chapter we investigate the variational modeling of the evolution of inelastic microstructures by the example of finite crystal plasticity with one active slip system. For this purpose we describe the microstructures by laminates of first order.We propose an analytical partial relaxation of an incompressible neo-Hookean energy formulation, keeping the internal variables and geometric microstructure parameters fixed, thus approximating the relaxed energy by an upper bound of the rank-one-convex hull. Based on the minimization of a Lagrange functional, consisting of the sum of rate of energy and dissipation potential, we derive an incremental strategy to model the time-continuous evolution of the laminate microstructure. Special attention is given to the three distinct cases of microstructure evolution, initiation, rotation, and continuous change. We compare a rate-independent approach with another one that employs viscous regularization which has certain advantages concerning the numerical implementation. Simple shear and tension/compression tests will be shown to demonstrate the differences between both approaches and to show the physical implications of the models introduced.

This work summarizes recent results on the formulation and numerical implementation of gradient plasticity based on incremental variational potentials as outlined in a recent sequence of work [Mie14, MMH14, MWA14, MAM13]. We focus on variational gradient crystal plasticity and outline a formulation and finite element implementation of micromechanically-motivated multiplicative gradient plasticity for single crystals. In order to partially overcome the complexity of full multislip scenarios, we suggest a new viscous regularized formulation of rate-independent crystal plasticity, that exploits in a systematic manner the longand short-range nature of the involved variables. To this end, we outline a multifield scenario, where the macro-deformation and the plastic slips on crystallographic systems are the primary fields. We then define a long-range state related to the primary fields and in addition a short-range plastic state for further variables describing the plastic state. The evolution of the short-range state is fully determined by the evolution of the long-range state, which is systematically exploited in the algorithmic treatment. The model problem under consideration accounts in a canonical format for basic effects related to statistically stored and geometrically necessary dislocation flow, yielding micro-force balances including non-convex cross-hardening, kinematic hardening and size effects. Further key ingredients of the proposed algorithmic formulation are geometrically exact updates of the short-range state and a distinct regularization of the rate-independent dissipation function that preserves the range of the elastic domain. The model capability and algorithmic performance is shown in a first multislip scenario in an fcc crystal. A second example presents the prediction of formation and evolution of laminate microstructure.

In a first part we consider evolutionary systems given as generalized gradient systems and discuss various variational principles that can be used to construct solutions for a given system or to derive the limit dynamics for multiscale problems via the theory of evolutionary Gamma-convergence. On the one hand we consider a family of viscous gradient system with quadratic dissipation potentials and a wiggly energy landscape that converge to a rate-independent system. On the other hand we show how the concept of Balanced-Viscosity solution arise in the vanishing-viscosity limit.

As applications we discuss, first, the evolution of laminate microstructures in finite-strain elastoplasticity and, second, a two-phase model for shape-memory materials, where H-measures are used to construct the mutual recovery sequences needed in the existence theory.

We consider a variational formulation of gradient elasto-plasticity subject to a class of single-slip side conditions, and show how the nonconvexity effects induced by such conditions can be not only resolved mathematically, but also tested physically. We first show that, for a large class of plastic deformations, a given single-slip condition (specification of Burgers’ vectors and slip planes) can be relaxed by introducing a microstructure through a two-stage process of mollification and lamination. This yields a relaxed side condition which only prescribes certain slip planes, and allows for arbitrary slip directions in these planes. The relaxed model should be a useful tool for simulating macroscopic plastic behavior without the need to resolve arbitrarily fine spatial scales. After deriving the relaxed model, we discuss a partial result on the existence of minimizers. Finally, we apply the relaxed model to a specific physical system, in order to be able to compare the analytical results with experiments. In particular, a rectangular shear sample in which only two slip planes are active is clamped at each end, and is subjected to a prescribed horizontal shear, which requires a certain amount of energy. We show that above some critical aspect ratio the energy is strictly positive and below that aspect ratio it is zero. Moreover, in the respective regimes determined by the aspect ratio, we prove energy scaling bounds, expressed in terms of the amount of prescribed shear, and we show that the scalings as well as the critical aspect ratio change radically if the single-slip condition or the strain gradient penalization is neglected.

This article gives a short description and a slight refinement of recentwork [MSZ15], [SZ12] on the derivation of gradient plasticity models fromdiscrete dislocations models.We focus on an array of parallel edge dislocations. This reduces the problem to a two-dimensional setting. As in the work Garroni, Leoni & Ponsiglione [GLP10] we show that in the regime where the number of dislocation N _ε is of the order log \({1}\over{\varepsilon}\) (where ε is the ratio of the lattice spacing and the macroscopic dimensions of the body) the contributions of the self-energy of the dislocations and their interaction energy balance. Upon suitable rescaling one obtains a continuum limit which contains an elastic energy term and a term which depends on the homogenized dislocation density. The main novelty is that our model allows for microscopic energies which are not quadratic and reflect the invariance under rotations. A key mathematical ingredient is a rigidity estimate in the presence of dislocations which combines the nonlinear Korn inequality of Friesecke, James & Müller [FJM02] and the linear Bourgain & Brezis estimate [BB07] for vector fields with controlled divergence. The main technical improvement of this article compared to [MSZ15] is the removal of the upper bound W(F) ≤ Cdist ²(F,SO(2)) on the stored energy function.

We review aspects of pattern formation in plastically deformed single crystals, in particular as described in the investigation of a copper single crystal shear experiment in [DDMR09]. In this experiment, the specimen showed a band-like microstructure consisting of alternating crystal orientations. Such a formation of microstructure is often linked to a lack of convexity in the free energy describing the system. The specific parameters of the observed bands, namely the relative crystal orientation as well as the normal direction of the band layering, are thus compared to the predictions of the theory of kinematically compatible microstructure oscillating between low-energy states of the non-convex energy. We conclude that this theory is suitable to describe the experimentally observed band-like structure. Furthermore, we link these findings to the models used in studies of relaxation and evolution of microstructure.

In modern engineering, micro-heterogeneous materials are designed to satisfy the needs and challenges in a wide field of technical applications. The effective mechanical behavior of these materials is influenced by the inherent microstructure and therein the interaction and individual behavior of the underlying phases. Computational homogenization approaches, such as the FE² method have been found to be a suitable tool for the consideration of the influences of the microstructure. However, when real microstructures are considered, high computational costs arise from the complex morphology of the microstructure. Statistically similar RVEs (SSRVEs) can be used as an alternative, which are constructed to possess similar statistical properties as the realmicrostructure but are defined by a lower level of complexity. These SSRVEs are obtained from a minimization of differences of statistical measures and mechanical behavior compared with a real microstructure in a staggered optimization scheme, where the inner optimization ensures statistical similarity and the outer optimization problem controls themechanical comparativity of the SSRVE and the real microstructure. The performance of SSRVEs may vary with the utilized statistical measures and the parameterization of the microstructure of the SSRVE.With regard to an efficient construction of SSRVEs, it is necessary to consider statistical measures which can be computed in reasonable time and which provide sufficient information of the real microstructure.Minkowski functionals are analyzed as possible basis for statistical descriptors of microstructures and compared with other well-known statistical measures to investigate the performance. In order to emphasize the general importance of considering microstructural features by more sophisticated measures than basic ones, i.e. volume fraction, an analysis of upper bounds on the error of statistical measures and mechanical response is presented.