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 引用本文: 王立安, 赵建昌, 杨华中. 饱和多孔地基与矩形板动力相互作用的非轴对称混合边值问题[J]. 力学学报, 2020, 52(4): 1189-1198.
Wang Li'an, Zhao Jianchang, Yang Huazhong. NON-AXISYMMETRIC MIXED BOUNDARY VALUE PROBLEM FOR DYNAMIC INTERACTION BETWEEN SATURATED POROUS FOUNDATION AND RECTANGULAR PLATE[J]. Chinese Journal of Theoretical and Applied Mechanics, 2020, 52(4): 1189-1198.
 Citation: Wang Li'an, Zhao Jianchang, Yang Huazhong. NON-AXISYMMETRIC MIXED BOUNDARY VALUE PROBLEM FOR DYNAMIC INTERACTION BETWEEN SATURATED POROUS FOUNDATION AND RECTANGULAR PLATE[J]. Chinese Journal of Theoretical and Applied Mechanics, 2020, 52(4): 1189-1198.

## NON-AXISYMMETRIC MIXED BOUNDARY VALUE PROBLEM FOR DYNAMIC INTERACTION BETWEEN SATURATED POROUS FOUNDATION AND RECTANGULAR PLATE

• 摘要: 在同一界面的不同区域具有多种边界条件, 称之为混合边界, 这是一个熟知的力学问题. 对这类问题进行精确分析时, 必须要进行混合边值问题的求解. 而对于一般的三维非轴对称情形, 混合边值问题的求解往往存在数学困难. 本文利用Hilbert定理和双重Fourier变换, 给出了一种求解三维非轴对称混合边值问题的解析方法, 利用该方法对具有混合透水边界的饱和多孔地基上矩形板的振动弯曲进行了解析研究(板与地基接触面为不透水边界, 其余为透水边界). 首先, 基于Kirchhoff理论和Biot多孔介质理论建立矩形板与饱和多孔地基的动力控制方程, 进行耦合求解. 针对板土接触面和非接触面的混合边值问题, 采用双重Fourier变换构造出两对二维对偶积分方程, 以接触应力和接触面孔隙压力为基本未知量, 用Jacobi正交多项式将未知量展开, 再利用Schmidt法对二维对偶积分方程完成求解, 最终推导出板土系统在动力作用下的位移和应力解析式. 通过将本文计算模型退化为单一弹性地基, 与已有研究结果进行对比, 验证了本文方法的正确性和有效性. 最后, 通过数值算例, 对饱和多孔地基上矩形板的动力响应及参数影响做出分析和讨论. 此外, 本文提出的解析法具有一般性, 可广泛应用于复杂接触问题和多场耦合问题的求解.

Abstract: Multiple boundary conditions exist in different areas of the same boundary surface are called mixed boundary, which is a well-known mechanical problem. It is necessary to solve the mixed boundary value problem, when to accurately analyze such problems. For the general 3D non-axisymmetric situation, there are often mathematical difficulties in solving the mixed boundary value problem. In this paper, an analytical method for solving three-dimensional non-axisymmetric mixed boundary value problems is presented by using Hilbert's theorem and double Fourier transform. Basted on this method, the coupled vibration problem for a rectangular plate resting on a saturated porous half-space with mixed permeable boundary is studied. Firstly, the dynamic governing equations of rectangular plates and saturated porous foundations are established based on Kirchhoff theory and Biot's porous medium theory. The operator equation is decoupled by double Fourier transforms to obtain the general solution of the rectangular plate and the foundation. The mixed boundary value problem is converted to two pairs of two-dimensional dual integral equations, with the contact surface stress and pore pressure as the basic unknowns, and to be solved by the Schmidt's method. The displacement and internal force analytical equations of the coupled vibration of the plate-foundation system are obtained. Finally, numerical examples are given to analyze and discuss the vibration response and parameters of the rectangular substrate on the saturated half space.

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