Uniqueness in Inverse Problems for Elastic Media with Residual Stress

Abstract

We consider dynamic inverse problems for bounded, three-dimensional elastic media with residual stress. In fact, we consider more general (nonclassical) linear hyperelastic media with the property that wave propagation occurs along the geodesics of Riemannian metrics. We show that the travel times of wave propagation through the object, together with entry and exit locations and directions, are determined by the Dirichlet-to-Neumann map in the absence of caustics and when the characteristic cones do not intersect. It follows by a boundary rigidity result of Lassas et al. (Lassas, M., Sharafutdinov, V., Uhlmann, G. ([2003]). Semiglobal boundary rigidity for Riemannian metrics. Math. Ann. 325(4):767–793.) that, under certain conditions, the geometry of the wave paths through the medium is determined up to the pullback by diffeomorphisms that fix the boundary. We also prove that the Dirichlet-to-Neumann map determines the six independent parameters at the boundary that describe the material properties of elastic media with residual stress.

Acknowledgment

The author would like to express thanks and appreciation to Todd Quinto for his valuable input and many helpful conversations.