On the elastic boundary value problem of dislocations in bounded crystals

A new formulation of the elastic boundary value problem of dislocations in bounded crystals is developed. This formulation is based on the ansatz that the stress field of dislocations in bounded domains can be constructed as the sum of a contribution corresponding to the classical infinite-domain solution plus a correction that is determined here from a mathematically well posed problem. The formulation of the elastic boundary value problem given here ensures that the equilibrium of the overall stress field is rigorously satisfied, specifically when dislocations intersect the boundary. The implications of this new formalism for dislocation dynamics simulation are discussed for the cases of bounded crystals and crystal volumes representative of uniformly loaded infinite crystals. An approximate computational solution of the elastic boundary value problem is presented based on the concept of virtual dislocations and the use of a non-singular form of the infinite-domain solution of the dislocation stress field. This computational solution addresses the issues of singularity and global equilibrium of the boundary traction associated with the corrective field. Sample results are presented for the internal stress in bounded crystals containing 3D dislocation configurations produced using the dislocation dynamics simulation method. The results illustrate the statistical character of the internal elastic field.