Spectral formulation of the elastodynamic boundary integral equations for bi-material interfaces

A spectral formulation of the plane-strain boundary integral equations for an interface between dissimilar elastic solids is presented. The boundary integral equations can be written in two equivalent forms: (a) the tractions can be written as a space-time convolution of the displacement continuities at the interface (Budiansky and Rice, 1979) (b) the displacement discontinuities can be written as a space-time convolution of the tractions at the interface (Kostrov, 1966). Prior work on spectral formulation of the boundary integral equations has adopted the former as the starting point. The present work has for its basis the latter form based on a space-time convolution of the tractions. Tractions and displacement components are given a spectral representation in the spatial coordinate along the interface. The radiation damping term is then explicitly extracted to avoid singularities in the convolution kernels. With the spectral forms introduced, the space-time convolutions reduce to convolutions in time for each Fourier mode. Due to continuity of tractions at the interface, this leads to a simpler formulation and form of the convolution kernels in comparison to the formulation involving convolutions over the slip and opening history at the bi-material interface. The convolution kernels are validated by studying some model problems to which analytical solutions are known. When coupled with a cohesive law or a friction law at the interface, the formulation proposed here is of wide applicability for studying spontaneous rupture propagation.