Spectral formulation of the 3D elastodynamic boundary integral equations for a bi-material interface

We present a numerical scheme to study 3D fracture problems at a planar interface. This scheme is based on the spectral representation of the boundary integral equation method which involves the evaluation of elastodynamic convolutions at the interface. The advantage of this method is that it is numerically efficient as it calculates the field quantities only on the fracture plane rather than in the entire domain. In the current approach, spatial convolution is replaced by multiplication in the spectral domain which increase the computational efficiency. In the literature, Geubelle and Rice [1995] first introduced the 3D spectral representation of the formulation of Budiansky and Rice [1979]. In their approach, the time-convolution is performed of the displacement history at the interface. Later, 3D formulation for a bi-material interface was proposed by Breitenfeld and Geubelle [1998]. Recently, a spectral form of the Kostrov [1966] was proposed by Ranjith [2015] for 2D in-plane problems. In this approach, time-convolution is performed of the traction history at the interface. An advantage of this approach is that the convolution kernels for a bi-material interface can be expressed in closed form, whereas Breitenfeld and Geubelle [1998] had to obtain their convolution kernels numerically. In the present work, convolution kernels for 3D elastodynamic fracture problems at a bi-material interface are derived following the approach of Ranjith [2015].