Propagation of Polarization in Elastodynamics with Residual Stress and Travel Times

Here 0 < ρ ∈ C(Ω) denotes the density, S is the Piola-Kirchhoff stress tensor which obeys the relation SF T = FS where F = I + ∇u is the deformation gradient. Divergence and transpose are taken with respect to the Euclidean metric | · |. The elasticity tensor C maps infinitesimal strain tensors E = (∇u+∇u )/2 to symmetric stress tensors CE. C represents the material properties on the elastic medium. R(x), the residual stress tensor, is a symmetric 3×3-matrix, C on Ω. It satisfies ∇·R = 0. See [Gur84, Sect. 23], [Hog86]. We call Pu = −ρ ∂u/∂t +∇ · ( R +∇uR+ CE ) (2)