不同透水边界下单层饱和多孔介质一维瞬态响应解

Semi-analytical solutions for transient response of one-dimensional saturated single-layer porous media under various permeable boundary conditions

基于Biot饱和多孔介质一维波动控制方程,给出具有普遍意义的一般性透水边界条件、任意初始条件和任意竖向荷载作用下单层饱和多孔介质一维瞬态响应定解模型。首先提出可通过调整相应参数实现采用孔压p和液相相对位移w的线性组合来表示不同的透水边界。之后采用分离变量法求得无阻尼控制方程的特征值和特征函数,利用待定系数法和特征函数的正交性,将原问题转换为求解时域内一系列阻尼项耦合的常微分方程组的初值问题,采用精细时程积分法给出瞬态响应问题的半解析解。同前人研究相比,提出的半解析解具有普遍意义,可精确退化至前人结果,通过与文献不同边界条件的结果对比,验证了解答的正确性。最后,系统分析了不同透水条件下瞬态响应的特点,结果表明半透水条件下响应介于透水和不透水之间,在阶跃荷载下,位移呈现先增大后减小的变化模式,入射波和反射波产生相同相位的孔压。

Based on the one-dimensional basic equations for waves in saturated porous media proposed by Biot,a mathematical model for one-dimensional transient response of single-layer saturated porous media is established under various permeable boundary conditions,arbitrary initial conditions and arbitrary vertical loads.Firstly,the independent parameters fluid pressure p and fluid relative displacement w are proposed to describe the various permeable boundary conditions by linear combination through adjusting the parameters.By utilizing the variable separation method,the eigen-values and the eigen-functions are obtained for undamped governing equations.With the help of undetermined coefficients and orthogonality of eigenfunctions methods,the solution to the problem can be converted to solve the initial value problem of a series of ordinary differential equations.The semi-analytical solutions are approached by using the precise time-integration method.Compared with those of the previous researches,the semi-analytical solutions of this research are more general and can be degenerated into various conditions exactly.Several numerical simulations are carried out to validate the proposed method.Finally,the one-dimensional transient responses of single-layer saturated soil with general boundary conditions under step loads are analyzed.The results demonstrate that the responses of semi-permeable condition are between the permeable and impermeable conditions.The displacements of solid and fluid increase first and then decrease.The incident and reflected waves stimulate the same phase pore pressure.