A generalized approach for solution to image stresses of dislocations

This paper proposes an adaptive meshfree method to calculate the image stresses of the dislocations at the free boundaries and bicrystal interfaces. Based on the superposition method (van der Giessen and Alan Needleman, 1995) originally proposed for single crystal material with free boundaries, a weak formulation is developed for the bicrystal materials containing dislocations, where the interface constraints are imposed using Lagrangian multiplier. The meshfree approximation is introduced to improve the accuracy and smoothness from conventional FEM interpolation functions, and it is straightforward to adaptively enrich meshfree nodes at the surrounding area of dislocations to effectively improve the solution accuracy. The numerical examples demonstrate the effectiveness of the proposed method to calculate the image stress in 2D and 3D problems with various boundary and interface conditions using either the classical elasticity (J.P.Hirth and J.Lothe, 1982) or the singularity-free dislocation theory (M.Lazar, 2012, 2013, 2014; G.Po and M.Lazar, 2014).