一类单层饱和多孔介质问题一维瞬态响应精确解
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摘要
本文基于Biot理论,考虑惯性、粘滞和机械耦合作用,系统研究了上表面作用任意竖向荷载、给定任意孔隙流体压力,下表面给定任意固体骨架位移和相对位移这一类非齐次边界条件下单层饱和可压缩和不可压缩多孔介质的一维瞬态响应问题,得到了考虑和忽略相对位移惯性项时的精确解。
从一维Biot基本方程出发,对可压缩情况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粘滞耦合作用的特征值问题,得到一组关于空间坐标的正交函数基。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,并采用状态空间法求解常微分方程组,得到位移分量,并进而求得总应力和孔隙流体压力。对固体颗粒和流体可压缩的情况,可直接根据物理方程确定总应力和孔隙流体压力。
对固体颗粒和流体均不可压缩的情况,首先从连续方程出发得到固体骨架位移和相对位移必须满足的关系式。通过边界条件齐次化和分离变量法求得固体骨架的位移。对整体平衡方程关于空间坐标积分,并根据边界条件确定总应力,进而求得孔隙流体压力。
当相对位移惯性项不计时,采用考虑相对位移惯性项对应的正交函数基,将控制方程化为一系列相互解耦的关于时间的常微分方程求解。算例验证了本文解法的正确性和适用性。
Based on the Biot Theoiy, the one-dimensional transient response of single-layer fluid-saturated porous media loaded arbitrarily at its top surface are studied thorouly, where the inertial, viscous and mechanical couplings are taken into account. Exact solutions are developed for both the Fully Dynamic Formulation and the Partly Dynamic Formulation.
Firstly, dimensionless governing equations in matrix form for the compressible constituents are derived, with the boundary conditions homogenized. Then, the eigen-value problem for the corresponding nonviscous system is solved to get an orthogonal function base in spatial domain. Applying variation coeffient method and making use of the orthogonality of the base functions, a series of decoupled second-order ordinary differential equations together with its corresponding initial conditions are obtained in time domain. To get the solutions for displacement components, the second-order ordinary differential equations are solved by the state-space method.The total stress and fluid pressure can be determined directly by the constitutive equations.
For the incompressible constituents, the relationship, which the displacement of solid particles and the relative displacement must satisfy, is derived from the continuity equation. By homogenizing the boundary conditions and using the method of separation of variables, the displacement of solid particles can be obtained. The total stress and fluid pressure can be determined by integrating the dynamic equilibrium equation of porous media and using the boundary conditions.
For the Partly Dynamic Formulation, where the inertial term of relative displacement is not considered, the problem can be solved by making use of the same orthogonal function base as what is used in the Fully Dynamic Formulation. Finally, examples are studied to demonstrate the correctness and applicability of the presented solutions.
从一维Biot基本方程出发,对可压缩情况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粘滞耦合作用的特征值问题,得到一组关于空间坐标的正交函数基。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,并采用状态空间法求解常微分方程组,得到位移分量,并进而求得总应力和孔隙流体压力。对固体颗粒和流体可压缩的情况,可直接根据物理方程确定总应力和孔隙流体压力。
对固体颗粒和流体均不可压缩的情况,首先从连续方程出发得到固体骨架位移和相对位移必须满足的关系式。通过边界条件齐次化和分离变量法求得固体骨架的位移。对整体平衡方程关于空间坐标积分,并根据边界条件确定总应力,进而求得孔隙流体压力。
当相对位移惯性项不计时,采用考虑相对位移惯性项对应的正交函数基,将控制方程化为一系列相互解耦的关于时间的常微分方程求解。算例验证了本文解法的正确性和适用性。
Based on the Biot Theoiy, the one-dimensional transient response of single-layer fluid-saturated porous media loaded arbitrarily at its top surface are studied thorouly, where the inertial, viscous and mechanical couplings are taken into account. Exact solutions are developed for both the Fully Dynamic Formulation and the Partly Dynamic Formulation.
Firstly, dimensionless governing equations in matrix form for the compressible constituents are derived, with the boundary conditions homogenized. Then, the eigen-value problem for the corresponding nonviscous system is solved to get an orthogonal function base in spatial domain. Applying variation coeffient method and making use of the orthogonality of the base functions, a series of decoupled second-order ordinary differential equations together with its corresponding initial conditions are obtained in time domain. To get the solutions for displacement components, the second-order ordinary differential equations are solved by the state-space method.The total stress and fluid pressure can be determined directly by the constitutive equations.
For the incompressible constituents, the relationship, which the displacement of solid particles and the relative displacement must satisfy, is derived from the continuity equation. By homogenizing the boundary conditions and using the method of separation of variables, the displacement of solid particles can be obtained. The total stress and fluid pressure can be determined by integrating the dynamic equilibrium equation of porous media and using the boundary conditions.
For the Partly Dynamic Formulation, where the inertial term of relative displacement is not considered, the problem can be solved by making use of the same orthogonal function base as what is used in the Fully Dynamic Formulation. Finally, examples are studied to demonstrate the correctness and applicability of the presented solutions.
引文
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