无限域分层介质的基本解

A Fundamental Solution for Infinite Layered Media

本文从三维弹性力学最基本的平衡方程和本构关系出发,推导出状态传递微分方程。在求解状态传递微分方程时,建议了一种对指数矩阵进行分解的方法,避免了直接解法可能导致状态变量的发散的问题。引入了无穷远处的状态变量为有限值的条件,推导出上、下无限层表面的位移与应力关系式。再根据状态传递方程,可得出层状介质任意点的应力和位移的值。此结果可直接退化到无限域经典的Kelvin解。

A state transfer matrix differential equation was derived from the three-dimensional equilibrium equations and constitutive equations of a homogeneous, isotropic linear elastic body. The exponential matrix was discomposed in order to avoid the non-convergence in solving the transfer matrix differential e-quations directly. Introducing the condition that the statement variables should be finite value at infinite points, the relations between the displacements and the stresses on the interfaces of infinite body were set up. The displacements and stresses at an arbitrary point of the infinite body ware easily deduced by use of the transfer matrix equations. The results derived here can be degenerated to the classic Kelvin's solution for infinite media.