Séminaire GIOCOMO PO

Date: 23 Mai 14h30 / 15h30

Lieu: Salle verte / 3ème Etage GTL

An anisotropic non-singular theory of dislocations with atomic resolution

Abstract

The singular nature of the elastic fields produced by dislocations presents conceptual challenges and computational difficulties in the implementation of discrete dislocation-based models of plasticity. In this work we consider theoretical and numerical aspects of the non-singular theory of discrete dislocation loops in a particular version of Mindlin’s anisotropic gradient elasticity with up to six independent gradient parameters. The framework models anisotropic materials where there are two sources of anisotropy, namely the bulk material anisotropy and a weak non-local anisotropy relevant at the nanoscale. The Green tensor of this framework, which we derive as part of the work, is non-singular and it rapidly converges to its classical counterpart a few characteristic lengths away from the origin. Therefore, the new Green tensor can be used as a physical regularization of the classical Green tensor. The Green tensor is the basis for deriving a non-singular eigenstrain theory of defects in anisotropic materials, where the non-singular theory of dislocations is obtained as a special case. The fundamental equations of curved dislocation loops in three dimensions are given as non-singular line integrals suitable for numerical implementation using fast one-dimensional quadrature. These include expressions for the interaction energy between two dislocation loops and the line integral form of the generalized solid angle associated with dislocations having a spread core. The six characteristic length scale parameters of the framework are obtained from the components of the rank-six tensor of strain gradient coefficients of Mindlin’s theory. In turn, the components of such tensor are obtained from atomistic calculations. As an extension of the Born-Huang lattice theory of elasticity, we show that the rank-six tensor of strain gradient coefficients admits an explicit local representation in terms of the derivatives of atomistic potentials. By virtue of this explicit representation, the link between atomistic and the simplified theory of gradient elasticity is established, and a non-singular and parameter-free theory of dislocations in anisotropic materials is obtained. Several applications of the theory are presented.