A numerical methodology for rupture propagation at bi-material interfaces with rate- and state-dependent friction

Frictional sliding at bi-material interfaces (when contacting bodies possess different elastic properties) is important in context of earthquake dynamics. Dissimilarity in elastic materials across the interface give rises to complex rupture propagation phenomena and instabilities, compared to the case when the material is similar across the interface. This is due to the coupling between the normal stress and interfacial slip, which is absent in the homogenous case. In the literature, various numerical schemes have been proposed but still many aspects of bi-material ruptures are not well-understood such as the rupture mode, velocity selection and stability. The present work proposes a new numerical scheme to study frictional rupture at a bi-material interface governed by a rate- and state-dependent friction law. It uses a spectral form of the boundary integral equation method (BIEM) as derived in Ranjith (2015, 2022), to evaluate the field quantities at the interface. The BIEM approach computes elastodynamic convolution of traction over its temporal history at the interface only, without need to calculate in regions away from interface, making it numerically efficient, compared to other conventional approaches. In prior work, an alternative spectral form of BIEM was used by Breitenfeld and Geubelle (1998) for 2D in-plane elasticity and Morrissey and Geubelle (1997) for 2D antiplane elasticity. In their approach, time-convolution is performed of the displacement history at the interface. An advantage of Ranjith’s 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. Conversion between real space and spectral domain is done by the Fast Fourier Transform (FFT). Rupture propagation is studied for both in-plane and antiplane frictional sliding at a bi-material interface by coupling the BIEM with a rate- and state-dependent friction law. Such a friction law is known to be suitable for a bi-material interface because it gives rise to well-posed problems (Rice et al., 2001). In earlier studies, an alternative numerical scheme for rate- and state-dependent friction was proposed by Lapusta et al. (2001) to study earthquake sequences on a fault. The disadvantage of their approach is that convolution kernels need to be evaluated multiple times for higher order accuracy. In the present work, a simpler numerical scheme is proposed for bi-material interfaces following a rate- and state-dependent friction law which is computationally more efficient.