任意竖向荷载作用下单层饱和多孔介质一维瞬态响应精确解

Exact solution for one-dimensional transient response of single-layer fluid-saturated porous media under arbitrary vertical loadings

基于Biot理论,考虑流体和固体颗粒的压缩性以及惯性、黏滞和机械耦合作用,得到了表面任意竖向荷载作用下单层饱和多孔介质一维瞬态响应的精确解。文章首先导出了以无量纲位移表示的矩阵形式的控制方程,并将边界条件齐次化。采用分离变量法求解不考虑黏滞作用的特征值问题,得到一组关于空间坐标的正交函数基。在此基础上,利用变异系数法和基函数的正交性,得到一系列可以通过状态空间法求解的相互解耦的关于时间的二阶常微分方程组及相应的初始条件。最后通过算例验证本文的正确性。

Based on the Biot＇s Theory,considering the inertial,viscous and mechanical couplings and the compressibility of fluid and solid particles,exact solution for one-dimensional transient response of single-layer fluid-saturated porous media under arbitrary loadings applied on its top surface are developed.Firstly,the dimensionless displacement governing equations in matrix form are derived and the boundary conditions is homogenized.Then,by using the separate variable method the eigen-value problem for the corresponding nonviscous problem is solved to get a series of orthogonal function base with respect to space.After this,by applying the variation coefficient method and making use of the orthogonality of the function base,a series of decoupling second-order ordinary differential equations with respect to time together with their corresponding initial conditions,which can be solved by the state-space method,are obtained.Finally,two examples are given to demonstrate the correctness of the present solution.