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推导了有限矩形区域饱和多孔介质因表面载荷诱发的Biot固结的一个解析解。假设多孔介质为均匀各向同性和线弹性,并被单相流体所饱和;控制方程组采用不可压缩多孔介质模型;孔隙压力场采用狄利克雷边界条件,上下表面位移场符合物理边界,而左右侧面位移场边界条件则由人为特别给定。利用有限正余弦变换和拉普拉斯变换及数值反演获得了物理空间孔隙压力场和位移场的半解析解,其体现为双重级数和的封闭形式。最后以某软黏土层平面应变固结为例,利用有限元分析软件ABAQUS对所给出的解析解进行了验证,同时基于该解析解考察了孔隙压力场和位移场的时空演化规律。所给出的解析解可用于深入分析有限二维饱和多孔介质的流-固耦合力学行为。
An analytical solution of Biot consolidation induced by surface loading in saturated porous media with finite rectangular area is derived. The porous media is assumed to be uniform and linear elastic and saturated by single-phase fluid. The governing equations are of incompressible porous media. The pore pressure field uses the Dirichlet boundary conditions. The displacement fields of the upper and lower surfaces conform to the physical boundary, Lateral displacement field boundary conditions are man-made special given. Semi-analytical solutions of pore pressure field and displacement field in physical space are obtained by using finite cosine transform, Laplace transform and numerical inversion. The semi-analytical solution is represented by the closed form of double series sum. Finally, taking the plane strain consolidation of a soft clay layer as an example, the analytical solution given by ABAQUS is validated. At the same time, the temporal and spatial evolution of pore pressure field and displacement field are investigated. The given analytical solutions can be used to further analyze the fluid-solid interaction mechanics of finite two-dimensional saturated porous media.