径向非均匀球各向异性弹性材料弹性动力学的Stroh形式
Elastodynamics of radially inhomogeneous spherically anisotropic elastic materials in the Stroh formalism
摘要：
本文提出了一种求解径向非均匀球各向异性弹性材料弹性动力学问题的方法，即材料，使得$c_{ijkl}= c_{ijkl}(r)$在球坐标系${r,θ,ϕ}$中。时谐位移场$\mathbf{u}(r,θ,ϕ)$以分离变量形式展开，依赖于由振幅依赖于$r$的矢量球谐函数描述的$θ,ϕ$.证明了只有当球面各向异性被限制为横观各向同性且主轴在径向时，这种分离变量解才是可能的，此时振幅由一阶常微分方程组确定.位移场的限制形式，如$\mathbf{u}(r,θ)$，允许这种类型的分离变量的解决方案，为某些较低的材料对称性。这些结果将矩形和圆柱系统弹性动力学的Stroh形式推广到球坐标系。
英文摘要：
A method is presented for solving elastodynamic problems in radially inhomogeneous elastic materials with spherical anisotropy, i.e.\ materials such that $c_{ijkl}= c_{ijkl}(r)$ in a spherical coordinate system ${r,θ,ϕ}$. The time harmonic displacement field $\mathbf{u}(r,θ,ϕ)$ is expanded in a separation of variables form with dependence on $θ,ϕ$ described by vector spherical harmonics with $r$-dependent amplitudes. It is proved that such separation of variables solution is generally possible only if the spherical anisotropy is restricted to transverse isotropy with the principal axis in the radial direction, in which case the amplitudes are determined by a first-order ordinary differential system. Restricted forms of the displacement field, such as $\mathbf{u}(r,θ)$, admit this type of separation of variables solutions for certain lower material symmetries. These results extend the Stroh formalism of elastodynamics in rectangular and cylindrical systems to spherical coordinates.
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