Hyperbolicity of Velocity-Stress Equations for Waves in Anisotropic Elastic Solids
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This paper reports mathematical properties of the three-dimensional, first-order, velocity-stress equations for propagating waves in anisotropic, linear elastic solids. The velocity-stress equations are useful for numerical solution. The original equations include the equation of motion and the elasticity relation differentiated by time. The result is a set of nine, first-order partial differential equations (PDEs) of which the velocity and stress components are the unknowns. Cast into a vector-matrix form, the equations can be characterized by three Jacobian matrices. Hyperbolicity of the equations is formally proved by analyzing (i) the spectrum of a linear combination of the three Jacobian matrices, and (ii) the eigenvector matrix for diagonalizing the linearly combined Jacobian matrices. In the three-dimensional space, linearly combined Jacobian matrices are shown to be connected to the classic Christoffel matrix, leading to a simpler derivation for the eigenvalues and eigenvectors. The results in the present paper provide critical information for applying modern numerical methods, originally developed for solving conservation laws, to elastodynamics.
KeywordsHyperbolicity Anisotropy Velocity-stress equations Eigen structure Elasticity
Mathematics Subject Classification (2010)35L02 35L04 35L05 35L65
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