单层非饱和多孔介质一维瞬态响应半解析解

基于Zienkiewicz提出的非饱和多孔介质波动理论，考虑两相流体和固体颗粒的压缩性以及惯性、黏滞和机械耦合作用，采用半解析的方法获得了一类典型边界条件下单层非饱和多孔介质一维瞬态响应解。首先推导出无量纲化后以位移表示的控制方程，并将其写成矩阵形式；然后，将边界条件齐次化，求解控制方程所对应的特征值问题，得到了满足齐次边界条件的特征值和相对应的特征函数。根据变异系数法并利用特征函数的正交性，得到了一系列仅黏滞耦合的关于时间的二阶常微分方程及相应的初始条件。在此基础上，运用精细时程积分法给出了常微分方程组的数值解。最后，通过若干算例验证了结果的正确性并探讨了单层非饱和多孔介质一维瞬态动力响应的特点。该方法可推广应用于其他典型的边界条件。

A semi-analytical solution for one-dimensional transient response of single-layer unsaturated porous media

Based on the theory of unsaturated porous media proposed by Zienkiewicz, considering the inertia, viscous and mechanical couplings and the compressibility of fluid and solid particles, semi-analytical solutions for one-dimensional transient response of single-layer unsaturated porous media are obtained with the example of a typical boundary condition problem. During the solution procedure, the dimensionless displacement governing equations in matrix form are derived firstly. Then the the nonhomogeneous boundary conditions are transformed into homogeneous boundary conditions and the eigenvalue problems of the governing equations are solved. After that, the variation coefficient method and the orthogonality of the characteristics functions are utilized to obtain a series of ordinary differential equations with their initial conditions. Applying the precise time-integration method, the semi-analytical solutions of the ordinary differential equations are developed. Finally, several numerical examples are provided to prove the correctness of the present solution and investigate the features of one-dimensional transient response of unsaturated single-layer porous media. This method can be extended to any other boundary conditions.