[S-SS07] Seismic wave propagation: Theory and Application

[SSS07-P19] Spectral formulation of the 3D elastodynamic boundary integral equations for a bi-material interface

*Avinash Gupta¹, Ranjith Kunnath¹ (1.Mahindra University )

Keywords:Friction, Wave propogation, Boundary Integral Equation Method, Fracture Plane, Interfacial Waves

In this work, a spectral scheme for 3D elastodynamic fracture problems at a planar interface is presented. In this scheme, the boundary integral equations between field quantities (i.e., the tractions and resulting displacement discontinuities) along the crack faces are used. Key benefit of this technique is its numerically efficiency as field quantities are required to be evaluated on the planar interface rather than throughout the entire domain (i.e., an overall reduction in dimension by one). In the spectral method, spatial convolution is replaced with a spectral domain multiplication operation, resulting in greater computational efficiency. The transformation between the spatial domain and the spectral domain is performed by the FFT (Fast Fourier Transform) algorithm. In the literature, Geubelle and Rice [1995] were the first to introduce the 3D spectral representation of Budiansky and Rice’s [1979] approach. In their approach, displacement history at the planar interface is used in the time-convolution. Later, 3D formulation for a bi-material interface was proposed by Breitenfeld and Geubelle [1998] where they had adopted the “independent formulation” over “combined formulation” due to its greater numerical stability. Recently, a spectral formulation of Kostrov [1966] was proposed by Ranjith [2015] for 2D in-plane elasticity. This formulation involves performing of the time-convolution using the traction history. A benefit of the current approach is that, unlike Breitenfeld and Geubelle [1998], whose convolution kernels had to be obtained numerically, the convolution kernels for a bi-material interface are obtained in closed form. Since evaluating the convolution integrals comprises the majority of the computational effort required to solve such problem, the proposed scheme is computationally more effective than Breitenfeld and Geubelle's [1998] method. In the current work, a spectral form of the boundary integral equation method for 3D elastodynamic fracture problems at a bi-material interface is developed following Ranjith’s [2015] methodology. The convolution kernels are validated through comparison with existing solutions of the 3D Lamb problem. The proposed scheme is illustrated by simulating frictional rupture propagation along a bi-material interface.