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在Biot饱和两相介质理论的基础上,对二维Biot波动方程进行水平坐标的Fourier变换和相应的变量代换,得到一组以土骨架位移分量与孔压为变量的二阶常微分方程。同时,对轴对称条件下的Biot波动方程进行径向坐标的Hankel变换,得到一组与二维情况相同的常微分方程。为求解该常微分方程组,运用薄层法原理在竖向进行离散求解,最终得到饱和层状地基的动力Lamb问题的基本解答,给出频域内土骨架位移和孔压的显示表达。此外,针对薄层法仅能在有限深度内求解的不足,根据旁轴近似原理,对饱和半空间频率-波数域内的刚度矩阵进行二阶Taylor级数展开,得到适合于薄层法底层边界的旁轴近似解答。最后,通过计算实例验证本文解答。
On the basis of Biot saturated two-phase medium theory, the two-dimensional Biot wave equation is transformed by Fourier transform in horizontal coordinates and corresponding variable substitution, and a group of second-order ordinary differential equations with soil frame displacement and pore pressure as variables is obtained. At the same time, the Hankel transform of radial coordinate is applied to the Biot wave equation under the axisymmetric condition to obtain a set of ordinary differential equations with the same two-dimensional case. In order to solve this system of ordinary differential equations, the method of thin-layer method is used to discretize vertically. Finally, the fundamental solution to the dynamic Lamb problem in saturated layered soils is obtained, and the displacements and pore pressures of soil skeleton in the frequency domain are given. In addition, aiming at the problem that the thin-layer method can only be solved within the finite depth, according to the paraxial approximation principle, the second-order Taylor series expansion of the stiffness matrix in the frequency-wavenumber domain of the saturation half-space is obtained, Paraxial approximation solution. Finally, verify the solution of this paper by calculating examples.