Bounds on the effective response for gradient crystal inelasticity based on homogenization and virtual testing
Journal article, 2019

This paper presents the application of variationally consistent selective homogenization applied to a polycrystal with a subscale model of gradient-enhanced crystal inelasticity. Although the full gradient problem is solved on Statistical Volume Elements (SVEs), the resulting macroscale problem has the formal character of a standard local continuum. A semi-dual format of gradient inelasticity is exploited, whereby the unknown global variables are the displacements and the energetic micro-stresses on each slip-system. The corresponding time-discrete variational formulation of the SVE-problem defines a saddle point of an associated incremental potential. Focus is placed on the computation of statistical bounds on the effective energy, based on virtual testing on SVEs and an argument of ergodicity. As it turns out, suitable combinations of Dirichlet and Neumann conditions pertinent to the standard equilibrium and the micro-force balance, respectively, will have to be imposed. Whereas arguments leading to the upper bound are quite straightforward, those leading to the lower bound are significantly more involved; hence, a viable approximation of the lower bound is computed in this paper. Numerical evaluations of the effective strain energy confirm the theoretical predictions. Furthermore, heuristic arguments for the resulting macroscale stress-strain relations are numerically confirmed.

boundary conditions

gradient crystal plasticity

computational homogenization


Kristoffer Carlsson

Chalmers, Industrial and Materials Science, Material and Computational Mechanics

Fredrik Larsson

Chalmers, Industrial and Materials Science, Material and Computational Mechanics

Kenneth Runesson

Chalmers, Industrial and Materials Science, Material and Computational Mechanics

International Journal for Numerical Methods in Engineering

0029-5981 (ISSN) 1097-0207 (eISSN)

Vol. 119 4 281-304

Variationally Consistent Selective Homogenization of Multifield Problems with Application to Gradient-Enhanced Modeling of Polycrystals

Swedish Research Council (VR), 2014-01-01 -- 2016-12-31.

Subject Categories

Applied Mechanics

Computational Mathematics

Control Engineering


C3SE (Chalmers Centre for Computational Science and Engineering)



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