非均匀饱和半空间的Lamb问题

基于Biot多孔介质波动理论，建立孔隙率、密度、剪切模量和渗透系数相互耦合且同时沿深度变化的非均匀饱和半空间模型，引入量化的非均匀梯度表征地基非均匀程度。在柱坐标系下构建以土骨架位移和孔隙压力为基本未知量的三维动力控制方程，采用算子运算和Hankel积分变换求解控制方程，推导出简谐集中力作用下半空间地基振动响应的积分解。将所得结果分别退化到均匀饱和半空间和弹性半空间与经典Lamb解做了对比，验证了结果的正确性。利用已有研究结论，给出孔隙率、密度、剪切模量和渗透系数之间的耦合关系式，代入推导结果进行数值计算。分别对水饱和地基和气饱和（干土）地基的动力响应进行分析，给出两类地基在动力作用下的振动位移和孔隙压力分布，并对非均匀性的影响作出分析。结果表明：4参数沿地基深度耦连变化对地基的动力响应产生一定程度影响，振动位移和孔隙压力在地层中的衰减速度由此加快。由于水的黏滞性远大于气体，所以水饱和地基中的振动衰减更快。非均匀程度越高，耦合效应的影响越明显。

Lamb problem for non-homogeneous saturated half-space

Based on Biot wave theory of porous media, a model was established for non-homogeneous saturated half-space, in which porosity, density, shear modulus and coefficient of permeability were coupled with each other and varied along the depth at the same time. A three-dimensional (3D) dynamic control equation with soil skeleton displacement and pore pressure as the fundamental unknowns is constructed in the cylindrical coordinate system. Operator operation and Hankel integral transformation are used to solving the control equation, and the product decomposition of the vibration response of the half-space under the action of simple harmonic concentrated forces is obtained. The results obtained in this paper are reduced to homogeneous saturated half-space and elastic half-space, respectively, and compared with the classical Lamb solution, the accuracy of results is also verified. Based on the existing results, the coupling relations among porosity, density, shear modulus and coefficient of permeability are given and substituted into the derived results for numerical calculation. The dynamic responses of water-saturated foundation and gas saturated foundation (dry soil) are analyzed respectively. The effect of non-uniform gradient on the calculated results is also investigated. The results show that the coupling of four parameters along the depth of the foundation has a certain influence on the dynamic response of the foundation, and the attenuation rate of vibration displacement and pore pressure in the formation is accelerated. Since the viscosity of water is much higher than that of gas, the vibration attenuation in the water-saturated ground is much faster. The higher the degree of non-uniformity, the more obvious the influence of coupling effect.