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Research Papers

Proposal of Extended Boundary Integral Equation Method for Rupture Dynamics Interacting With Medium Interfaces

[+] Author and Article Information
Nobuki Kame1

Earthquake Research Institute,  The University of Tokyo, Tokyo, Japan 113-0032kame@eri.u-tokyo.ac.jp

Tetsuya Kusakabe

Earthquake Research Institute,  The University of Tokyo, Tokyo, Japan 113-0032

1

Corresponding author.

J. Appl. Mech 79(3), 031017 (Apr 05, 2012) (8 pages) doi:10.1115/1.4005899 History: Received August 01, 2011; Revised December 02, 2011; Posted February 13, 2012; Published April 04, 2012; Online April 05, 2012

The boundary integral equation method (BIEM) has been applied to the analysis of rupture propagation of nonplanar faults in an unbounded homogeneous elastic medium. Here, we propose an extended BIEM (XBIEM) that is applicable in an inhomogeneous bounded medium consisting of homogeneous sub-regions. In the formulation of the XBIEM, the interfaces of the sub-regions are regarded as extended boundaries upon which boundary integral equations are additionally derived. This has been originally known as a multiregion approach in the analysis of seismic wave propagation in the frequency domain and it is employed here for rupture dynamics interacting with medium interfaces in time domain. All of the boundary integral equations are fully coupled by imposing boundary conditions on the extended boundaries and then numerically solved after spatiotemporal discretization. This paper gives the explicit expressions of discretized stress kernels for anti-plane nonplanar problems and the numerical method for the implementation of the XBIEM, which are validated in two representative planar fault problems.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 1

Schematic illustration of the configuration of a crack and homogeneous sub-regions. A crack surface Γ is in a homogeneous volume V that is bounded by a medium surface S. Another volume V’ with a surface S’ is adjoining. The two surfaces S and S′ are identical except that their normal vectors n and n′ are in the opposite direction.

Grahic Jump Location
Figure 2

Configuration of a bimaterial consisting of two homogeneous half-spaces (x2  > 0 for V +  and x2  < 0 for V −  ) bounded by surfaces S ±  adjoining on the x1 -axis. Each medium is characterized by the rigidity μ ±  and the shear wave velocity β ±  . In the validation tests shown in Figs.  34, an initial crack with length 2a is assumed on the x1 -axis.

Grahic Jump Location
Figure 3

Analytic (solid line) and numerical solution (dotted line) for the 101th element of an instantaneous crack discretized into 402 elements. The computation is done with normalized quantities: traction Tnrml  = Tσ, length xnrml  = x/a, time tnrml=t/(α/β) and displacement velocity u·nrml=u·/(βΔσ/μ), where Δσ is the stress drop, a is the crack half-length, μ is the medium rigidity, and β is the shear wave velocity of the medium. A time step is assumed as Δt = Δs/(2β).

Grahic Jump Location
Figure 4

Spatiotemporal evolution of the crack tip position spontaneously propagating along the bimaterial interface for different material contrasts r = 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0. The material contrast is characterized by r = β −  /β +  , the ratio of the lower medium shear wave velocity to the upper velocity. Unit quantities for the computation are based on those in the upper medium. The initial crack with length 2a is discretized by 40 elements, and the unit time a/β +  is discretized by 40 time steps. The interface is represented by 800 elements, which is large enough to eliminate the diffracted wave from the artificial edges of the computational domain coming back to the crack tip. The predicted upper limit rupture velocities for r = 0.5 (vr /β +   = 0.5) and r = 1.0(vr /β +   = 1.0) are plotted for reference.

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