A method for calculating the Green's function of spatial dynamic boundary value problems of the anisotropic elasticity theory taking into account asymptotic components

Abstract

The paper proposes a theoretically accurate method for taking into account the asymptotic components of the two-dimensional Fourier integral representing the Green's functions for spatial dynamic boundary value problems of the anisotropic theory of elasticity. The body is a multilayer stack of layers or a multilayer half-space and is excited by a surface harmonic stress source. The expressions for the contribution of the asymptotics include the Fourier series, the coefficients of which contain easily computable integrals from the Bessel functions. Asymptotic expressions for the integrals from the Bessel functions for large values of the parameters are given. Numerical examples are given for a lithium niobate crystal. It is shown that the asymptotics of the symbols of the Green's functions in the anisotropic case can contain real and imaginary components simultaneously, which distinguishes them from the isotropic case. Moreover, the resulting solution is always real. The number of operations for calculating the contribution of asymptotics is several orders of magnitude less than the number of operations for calculating the wave part of the integral. The developed formulas make the direct contour integration method logically complete, theoretically accurate and relatively simple to implement.