Boundary element method in anisotropic media with grain sliding and dislocation dynamics

Abstract

Boundary element method (BEM) is an effective numerical technique to solve complex engineering problems. The fundamental solutions for both isotropic and anisotropic boundary element method are studied as the basic to develop elastostatic boundary integral equations. The numerical implementation of BEM is described in a very detailed fashion. Multi zone BEM is introduced to calculate polycrystal grains structure. The connectivity between grains is modeled with a stiffness spring system. The sliding effect at grain boundaries is simulated by a non linear sliding model. After anisotropy and grain sliding are implemented with BEM, the information on the grain boundaries can be calculated effectively. Inside the grains, the dislocation theory is discussed. For multiple dislocations stress field, two calculation methods are introduced: discrete dislocation method and dislocation density tensor method. For the dislocation density tensor method, the domain integral involved are transformed into boundary integral to save the runtime and be compatible with the BEM formulation. To control the total error and save time, a combined method with those two is developed to calculate multiple dislocations stress field. The new mixed method reduce the run time from the order O(n2) to O(n) and keep the error within 2%. With dislocation stress field, the dislocation dynamics are generated with the dislocation calculation and the superposition with the boundary conditions. The size effect for yielding is shown and satisfies the Hall-Petch law. The results with grain sliding and anisotropy are also shown and analyzed.