Derivation of linearized polycrystals from a two-dimensional system of edge dislocations

Fanzon, Silvio; Palombaro, Mariapia; Ponsiglione, Marcello

Abstract

In this paper we show the emergence of polycrystalline structures as a result of elastic energy minimization. For this purpose, we consider a well-known variational model for twodimensional systems of edge dislocations, within the so-called core radius approach, and we derive the \Gamma-limit of the elastic energy functional as the lattice space tends to zero. In the energy regime under investigation, the symmetric and skew part of the strain become decoupled in the limit, the dislocation measure being the curl of the skew part of the strain. The limit energy is given by the sum of a plastic term, acting on the dislocation density, and an elastic term, which depends on the symmetric strains. Minimizers under suitable boundary conditions are piecewise constant antisymmetric strain fields, representing in our model a polycrystal whose grains are mutually rotated by infinitesimal angles. In the energy regime under investigation, the symmetric and skew part of the strain become decoupled in the limit, the dislocation measure being the curl of the skew part of the strain. The limit energy is given by the sum of a plastic term, acting on the dislocation density, and an elastic term, which depends on the symmetric strains. Minimizers under suitable boundary conditions are piecewise constant antisymmetric strain fields, representing in our model a polycrystal whose grains are mutually rotated by infinitesimal angles.

Citation

Fanzon, S., Palombaro, M., & Ponsiglione, M. (2019). Derivation of linearized polycrystals from a two-dimensional system of edge dislocations. SIAM Journal on Mathematical Analysis, 51(5), 3956-3981. https://doi.org/10.1137/18M118726X

• Journal Article Type: Article
• Acceptance Date: Apr 30, 2019
• Online Publication Date: Oct 11, 2019
• Publication Date: Jan 1, 2019
• Deposit Date: May 9, 2023
• Journal: SIAM Journal on Mathematical Analysis
• Print ISSN: 0036-1410
• Electronic ISSN: 1095-7154
• Publisher: Society for Industrial and Applied Mathematics
• Peer Reviewed: Peer Reviewed
• Volume: 51
• Issue: 5
• Pages: 3956-3981
• DOI: https://doi.org/10.1137/18M118726X
• Keywords: Geometric rigidity; Linearization; Polycrystals; Dislocations; Variational methods
• Public URL: https://hull-repository.worktribe.com/output/4271032