Computationally efficient representation for elastostatic and elastodynamic Green's functions for anisotropic solids.

Abstract

A computationally efficient representation of the three-dimensional elastostatic and elastodynamic Green's functions for anisotropic solids is derived by solving the Christoffel equation in terms of delta functions. The representation is also applicable to other wave equations. The method is applied to calculate the transient and the static displacement field due to a point source in infinite and semi-infinite anisotropic cubic solids. For elastodynamic calculations in anisotropic solids, our representation saves the computational time by a factor of about 1000 over the conventional Fourier-Laplace representation. In the elastostatic case, the computational efficiency of our method is much more than the conventional Fourier representation but comparable to the methods of and and Lothe in specific cases. I. INTRQDUCTIGN We derive an integral representation for the elastodynarnic Green's function in terms of a 5 function. In this representation, even in the general anisotropic 3D (three-dimensional} case, only a 1D integration needs to be done numerically. The integration does not involve oscillatory functions or singularities. This reduces the problem of numerical convergence. The CPU time required for calculating the elastodynamic Green's function is only about, ' of that required for the conventional Fourier-Laplace representation. ' Qur technique is also applicable to other wave equations such as the electromagnetic or acoustic wave equation. We also calculate the elastostatic Green's function by taking the static limit of the elastodynamic response of a solid. Qur method is much more efticient than the conventional Fourier representation. However, the computational eKciency of our method is comparable to that of Barnett for an infinite solid and to that of and Lothe for calculations of surface displacements in a semi-infinite solid. The elastodynamic Green's function is useful for calculating physical properties of solids involving longwavelength phonons. Presently, there is a strong interest in wave-form-based ultrasonics for nondestructive characterization of anisotropic materials. These techniques measure the response of a material to an elastic pulse. which is very well modeled in terms of the elastodynamic Green's function. The elastostatic Green's function is useful for calculating stress distribution in solids containing defects and discontinuities. ' In 1attice statics calculations and the computer simulation of lattice defects, the elastostatic Green's function is needed to fix the asymptotic limit of the lattice distortions. ' The Green's function for an isotropic solid is usually' calculated by using the Fourier-I. aplace representation in the wave-vector — frequency space. This is adequate for traditional materials which could be approximated as isotropic. Modern composite materials are highly anisotropic. In this case the conventional Fourier representation requires a 4D integration over oscillatory functions for the elastodynamic Green's function and 3D integration for the elastostatic Green's function. In many applications, we need to calculate the Green's function for bounded solids (containing free surfaces of interfaces). The Fourier-Laplace representation of the Green's function in such cases is poorly convergent and its evaluation is CPU intensive. A powerful technique for calculating the 3D elastostatic Green's function was suggested by Barnett. This technique is very well suited for calculation of the Green's function for infinite solids. For this case, the computational e%ciency of our method is about the same as that of Barnett. An extension of Stroh's formalism has been developed by and Lothe for bounded solids, but it is applicable only to 2D problems. Qur representation retains its simplicity even for 3D bounded solids. For example, our expression for the Green's function for a semi-infinite solid satisfies the free-surface boundary con