Extension of Stroh's formalism to self-similar problems in two-dimensional elastodynamics

The Smirnov-Soloblev method for a two-dimensional scalar wave equation is generalized to self-similar problems in anisotropic elastodynamics. The resulting formulation resembles and may be regarded as an extension to Stroh's formalism for two-dimensional anisotropic elastostatics. In contrast to the Cagniard-De Hoop method which requires inversion of Laplace transforms, the general solution here is directly expressed in terms of the eigenvalues and eigenvectors of a six-dimensional eigenvalue problem. The eigenvalue problem, although now dependent on the time and position, shares the same analytic structure as that for the static case. The formulation is applied to derive the Green's tensor due to a line impulse in an infinite solid or on the surface of a, semi-infinite medium. Certain explicit results of the Green's tensors are derived for monoclinic or orthotropic materials. Numerical calculations are also performed for silicon.