Journal of Integral Equations and Applications

Solvability of a volume integral equation formulation for anisotropic elastodynamic scattering

Marc Bonnet

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This article investigates the solvability of volume integral equations arising in elastodynamic scattering by penetrable obstacles. The elasticity tensor and mass density are allowed to be smoothly heterogeneous inside the obstacle and may be discontinuous across the background-obstacle interface, the background elastic material being homogeneous. Both materials may be anisotropic, within certain limitations for the background medium. The volume integral equation associated with this problem is first derived, relying on known properties of the background fundamental tensor. To avoid difficulties associated with existing radiation conditions for anisotropic elastic media, we also propose a definition of the radiating character of transmission solutions. The unique solvability of the volume integral equation (and of the scattering problem) is established. For the important special case of isotropic background properties, our definition of a radiating solution is found to be equivalent to the Sommerfeld-Kupradze radiation conditions. Moreover, solvability for anisotropic elastostatics, directly related to known results on the equivalent inclusion method, is recovered as a by-product.

Article information

J. Integral Equations Applications Volume 28, Number 2 (2016), 169-203.

First available in Project Euclid: 1 July 2016

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Zentralblatt MATH identifier

Primary: 35J15: Second-order elliptic equations 45F15: Systems of singular linear integral equations 65R20: Integral equations 74J20: Wave scattering

Volume integral equation elastodynamics anisotropy scattering


Bonnet, Marc. Solvability of a volume integral equation formulation for anisotropic elastodynamic scattering. J. Integral Equations Applications 28 (2016), no. 2, 169--203. doi:10.1216/JIE-2016-28-2-169.

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