Image decomposition method for the analysis of a mixed dislocation in a general multilayer

The elastic solutions for a mixed dislocation in a general multilayer with N dissimilar anisotropic layers are obtained via a generalized image decomposition method. The original problem is decomposed into N homogeneous subproblems with strategically placed continuously distributed image (virtual) dislocations which satisfy the consistency conditions for degenerate N − M (M < N) layer problems. The image dislocations are used to satisfy the interface or free surface conditions, and represent the unknowns of the problem. The resulting singular Cauchy integral equations are transformed into non-singular Fredholm integral equations of the second kind using certain H- and I-integral transforms. The Fredholm integral equations are then solved via the classical Nyström method. The general decomposition and the elimination of all singular integrals yield an exact formulation of the problem; the approximation arises only in the Nyström method. The dislocation mixity and the number of layers dissimilar in thickness and elastic anisotropy can be handled without difficulty, constrained only by the number of linear algebraic equations in the Nyström method for large N. For the numerical study, image forces on a dislocation in two- and three-layer systems are calculated. The accuracy of the results is verified by checking the boundary conditions and by comparison with previous results. The dependence of the image force on the dislocation position and mixity, and on the layer thicknesses and elastic anisotropies, is also illustrated via numerical investigations.