Propagation of polarization in elastodynamics with residual stress and travel times

On uniqueness in the inverse problem for transversely isotropic elastic media with a disjoint wave mode

“…The principal example of elastic media with GWP studied in [15] is isotropic elastodynamics with residual stress. Hansen and Uhlmann [16] establish similar results for isotropic elastodynamics with residual stress in the presence of caustics.…”

“…In particular, they show that the travel time of the wave through the object is uniquely determined by the associated DN map. Uniqueness also holds for isotropic elastodynamics [12], [13], [14], and, up to the natural obstruction, for isotropic media with residual stress [15], [16]. Hansen and Uhlmann [16] use a microlocal analytic approach to the study of reflection of singularities to prove that the scattering relation at the surface is determined by the Dirichlet-to-Neumann map, which extends Rachele's result [15] to the case that caustics are present.…”

“…Uniqueness also holds for isotropic elastodynamics [12], [13], [14], and, up to the natural obstruction, for isotropic media with residual stress [15], [16]. Hansen and Uhlmann [16] use a microlocal analytic approach to the study of reflection of singularities to prove that the scattering relation at the surface is determined by the Dirichlet-to-Neumann map, which extends Rachele's result [15] to the case that caustics are present. In addition, Robertson shows that the Dirichlet-to-Neumann map uniquely determines the residual stress tensor and one of the Lamé parameters at the boundary [17], and establishes uniqueness in the interior [18] for the linearized problem using Man's model [19] for residual stress.…”

“…For example, Hansen and Uhlmann decouple the equations near the boundary in studying propagation of polarization in elastodynamics with residual stress [16]. Here, we adapt the approach taken by Stolk and de Hoop, who treat seismic inverse scattering for anisotropic media in [3], under the assumption that the multiplicity of some eigenvalues is constant, at least microlocally.…”

Uniqueness and Stability of Determining the Residual Stress by One Measurement

“…There are other results concerning the determination of the residual stress by infinitely many boundary measurements, i.e. by the Dirichlet-to-Neumann map, we refer to Hansen and Uhlmann (2003), Rachele (2003), and Robertson (1977Robertson ( , 1988.…”