Finite element analysis of geometrically necessary dislocations in crystal plasticity

Abstract

SUMMARY We present a finite element method for the analysis of ductile crystals whose energy depends on the density of geometrically necessary dislocations (GNDs). We specifically focus on models in which the energy of the GNDs is assumed to be proportional to the total variation of the slip strains. In particular, the GND energy is homogeneous of degree one in the slip strains. Such models indeed arise from rigorous multiscale analysis as the macroscopic limit of discrete dislocation models or from phenomenological considerations such as a line‐tension approximation for the dislocation self‐energy. The incorporation of internal variable gradients into the free energy of the system renders the constitutive model non‐local. We show that an equivalent free‐energy functional, which does not depend on internal variable gradients, can be obtained by exploiting the variational definition of the total variation. The reformulation of the free energy comes at the expense of auxiliary variational problems, which can be efficiently solved using finite element approximations. The addition of surface terms in the formulation of the free energy results in additional boundary conditions for the internal variables. The proposed framework is verified by way of numerical convergence tests, and simulations of three‐dimensional problems are presented to showcase its applicability. A performance analysis shows that the proposed framework solves strain‐gradient plasticity problems in computing times of the order of local plasticity simulations, making it a promising tool for non‐local crystal plasticity three‐dimensional large‐scale simulations. Copyright © 2012 John Wiley & Sons, Ltd.

Fourier‐based numerical approximation of the Weertman equation for moving dislocations

“…[13][14][15][16] Quite generally, methods to compute the shape of static or moving cores encompass variational approaches and involve finite-element and/or phase-field-type implementations. 13,14,[17][18][19][20] Yet, in spite of such a wealth of enrichments of the PN model and its associated numerical methods of solution, the 1-dimensional Weertman equation-a comparatively simpler extension-has not been investigated as thoroughly, while the specific problem of determining the allowed velocities of steadily moving dislocations for general force laws F ′ remains an open question of major practical interest. 7 For this reason, the present work focuses on solving the simplest, scalar, and 1-dimensional case.…”

A Gibbs-potential-based framework for ideal plasticity of crystalline solids treated as a material flow through an adjustable crystal lattice space and its application to three-dimensional micropillar compression

“…Most of the methods that have been developed so far adopt a Lagrangian description of the continuum problem. In the context of the present paper we specifically highlight the following references focused on the application of CPFEM to micropillar compression: [8,9,12,10,26].…”

The special‐linear update: An application of differential manifold theory to the update of isochoric plasticity flow rules

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