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Three-dimensional Green's functions of steady-state motion in anisotropic half-spaces and bimaterials

机译：各向异性半空间和双材料中稳态运动的三维格林函数

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摘要

Three-dimensional Green's functions (GFs) of steady-state motion in linear anisotropic elastic half-space and bimaterials are derived within the framework of generalized Stroh formalism and two-dimensional Fourier transforms. The present study is limited to the subsonic case where the sextic equation has six complex eigenvalues. If the source and field points reside in the same material, the GF is expressed in two parts: a singular part that corresponds to the infinite-space GF, and a complementary part that corresponds to the reflective effects of the interface in the bimaterial case and of the free surface in the half-space case. The singular part in the physical domain is calculated analytically by applying the Radon transform and the residue theorem. If the source and field points reside in different materials (in the bimaterial case), the GF is a one-term solution. The physical counterparts of the complementary part in the half-space case and of the one-term solution in the bimaterial case are derived as a one-dimensional integral by analytically carrying out the integration along the radial direction in the Fourier-inverse transform. When the source and field points are both on the interface in the bimaterial case or on the surface in the half-space case, singularities appear in the Fourier-inverse transform of the GF. The singularities are treated explicitly using a method proposed recently by the authors. Numerical examples are presented to demonstrate the effects of wave velocity on the stress fields, which may be of interest in various engineering problems of steady-state motion. Furthermore, these GFs are required in the steady-state boundary-integral-equation formulation of anisotropic elasticity.
机译：线性各向异性弹性半空间和双材料中的稳态运动的三维格林函数（GFs）是在广义Stroh形式主义和二维傅立叶变换的框架内得出的。本研究仅限于亚音速情况下，其中性方程具有六个复杂的特征值。如果源点和场点位于同一材料中，则GF表示为两部分：与无限空间GF相对应的奇异部分，与在双材料情况下界面的反射效应相对应的互补部分；以及半空间情况下的自由表面的角度。通过应用Radon变换和残差定理可解析地计算出物理域中的奇异部分。如果源点和场点位于不同的材料中（在双材料情况下），则GF是一个单项解决方案。通过在傅立叶逆变换中沿径向方向进行积分，可以将半空间情况下的互补部分和双材料情况下的一阶解的物理对应物导出为一维积分。当源点和场点在双材料情况下都在界面上或在半空间情况下都在表面上时，奇异性就会出现在GF的傅里叶逆变换中。使用作者最近提出的方法明确处理奇点。数值算例表明了波速对应力场的影响，这可能是稳态运动的各种工程问题中令人关注的问题。此外，各向异性弹性的稳态边界积分方程式中需要这些GF。

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