Geometrical modeling and numerical analysis of edge dislocation

Abstract

In this study, we conduct modeling and numerical analysis of a straight edge dislocation in a three-dimensional geometrically nonlinear elastic medium. Based on the fundamental framework of geometrical elasto-plasticity, kinematics of the continuum is represented by the reference R, intermediate B and current S states and local deformations are described by the multiplicative decomposition of the deformation gradient; F = F_e·F_p. Here, the reference and current states are Euclidean submanifolds while the intermediate state B is represented by Weitzenböck manifold with non-zero torsion in the affine connection. Following to the equivalence of torsion and dislocation density tensor through Hodge star operation, a plastic deformation gradient F_p is obtained from the Cartan first structure equation using the homotopy operator. The current state S is determined so that it minimizes the strain energy functional. We solve the variational problem by using isogeometric analysis; Galerkin method with non-uniform B-spline basis function. The present analysis shows that all stress components agree quantitatively well with those of the Volterra dislocation model. Around the dislocation core, stress fields are non-singular on the contrary to the Volterra dislocation. In addition, we found that all stress components distribute asymmetrically due to the geometrical nonlinearity of the model.