Gradient theory for plasticity via homogenization of discrete dislocations
By Adriana Garroni, Giovanni Leoni, Marcello Ponsiglione. 2008
161 citations recorded by the source corpus.
We deduce a macroscopic strain gradient theory for plasticity from a model of discrete dislocations. We restrict our analysis to the case of a cylindrical symmetry for the crystal under study, so that the mathematical formulation will involve a two-dimensional variational problem. The dislocations are introduced as point topological defects of the strain fields, for which we compute the elastic energy stored outside the so-called core region.We show that the G-limit of this energy (suitably rescaled), as the core radius tends to zero and the number of dislocations tends to infinity, takes the form E = fO (W(se) + f (Curl se)) dx, where e represents the elastic part of the macroscopic strain, and Curl se represents the geometrically necessary dislocation density. The plastic energy density f is defined explicitly through an asymptotic cell formula, depending only on the elastic tensor and the class of the admissible Burgers vectors, accounting for the crystalline structure. It turns out to be positively 1-homogeneous, so that concentration on lines is permitted, accounting for the presence of pattern formations observed in crystals such as dislocation walls.
Subfields: Computational Theory and Mathematics, Materials Chemistry, Mechanics of Materials
View Paper: https://www.semanticscholar.org/paper/20165628a5d7cf978e7e4e8e5b5e4355b1daf449
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In Computational Theory and Mathematics
In Materials Chemistry
In Mechanics of Materials
Citations in the corpus
- Semi-classical states for the Choquard equation (2013, methodology): “at theorem 3 is equivalent to theorem 1. Theorem 4 was used to obtain a generalized Korn type inequality in the derivation of a strain gradient theory for plasticity by homogenization of dislocations [47]. 1.3. Estimates for circulation integrals. Theorems 1 and 3 are equivalent to the following geo”
- Limiting Bourgain–Brezis estimates for systems of linear differential equations: Theme and variations (2014, methodology): “4 was used to obtain a generalized Korn-type inequality in the derivation of a strain gradient theory for plasticity by homogenization of dislocations [47].”