## Journal of Integral Equations and Applications

### Solvability of a volume integral equation formulation for anisotropic elastodynamic scattering

Marc Bonnet

#### Abstract

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

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

Dates
First available in Project Euclid: 1 July 2016

https://projecteuclid.org/euclid.jiea/1467399274

Digital Object Identifier
doi:10.1216/JIE-2016-28-2-169

Mathematical Reviews number (MathSciNet)
MR3518482

Zentralblatt MATH identifier
1383.35226

#### Citation

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. https://projecteuclid.org/euclid.jiea/1467399274

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