层状饱和土Biot固结问题状态空间法

针对饱和多孔介质空间非轴对Biot固结问题，引入状态变量，构造了两组相比独立的状态变量方程，利用Fourier级数和Laplace-Hankel变换，将状态变量方程转换为两组一阶常微分方程组，提出了均质饱和多孔介质空间非轴对称Biot固结问题的传递矩阵，得到以状态变量和传递矩阵乘积的形式表示的均质饱和多孔介质空间非轴对称Biot固结问题的解，利用层间完全接触的条件，可得到N层饱和多孔介质空间非轴对称Biot固结问题的一般解析表达式，文中考虑几种不同的边界条件，分析了两个算例，数值结果表明该方法具有较高的计算精度和良好的计算稳定性。

THE STATE SPACE METHOD OF THE BIOT CONSOLIDATION PROBLEM FOR MULTILAYERED POROUS MEDIA

In this paper, the non-axisymmetric Biot consolidation problem for multilayered porous media is studied. Taking stresses, pore pressure and displacements on layer interfaces as basic unknown functions, two sets of partial differential equations are formulated in the cylindrical coordinates. These partial differential equations are independent each other. Using Fourier expansion, Laplace transforms and Hankel transforms with respect to the circumferential, time and radial coordinates, respectively, the partial differential equations presented are reduced to the ordinary differential equations. By the use of the Cayley-Hamilton theorem, the transfer matrices describing the transfer relation between the state vector of surface and the state vector of arbitrary depth for a finite layer are derived explicitly in the transform space. Using the transfer matrices presented, in the transform space, explicit the solutions of the non-axisymmetric Biot consolidation problems of a semi-infinite saturated porous media are discussed for the permeable and impermeable surface. Considering the continuity condition on layer interfaces, the solutions of the non-axisymmetric Biot consolidation problems for N layers porous media, which the lower surface adheres to half space, permeable and impermeable rough rigid, can be presented in the integral form. The time histories of displacements, stresses and pore pressure can be obtained by solving a linear equation system for discrete values of Laplace-Hankel transform inversions. The two numerical examples are given. The time domain solutions are obtained by applying the numerical scheme proposed by Crump for Laplace inversion and Hankel transform inversion by using direct numerical quadrature. The advantage of the present method is that order of linear algebraic equations is not related to number of layers compared with the conventional method based on the determination of layer arbitrary coefficients. The method presented has high accuracy and efficiency.