多孔饱和半空间上弹性圆板垂直振动的积分方程

INTEGRAL EQUATIONS OF VERTICAL VIBRATION OFAN ELASTIC CIRCULAR PLATE ON A FLUID-SATURATED POROELASTIC HALF SPACE

应用新的方法求解多孔饱和固体的动力基本方程──Ｂｉｏｔ波动方程．首先把Ｂｉｏｔ波动方程化为仅有土骨架位移和孔隙水压力的偏微分方程组，并用逐次解耦方法（不引入位移势函数）求解此偏微分方程组．然后按混合边值条件建立多孔饱和半空间上弹性图板垂直振动的对偶积分方程，用Ａｂｅｌ变换化对偶积分方程为第二类Ｆrｅｄｈｏｌｍ积分方程．文中考虑两种孔隙流体的表面边界条件：（ａ）半空间表面（包括圆板与半空间的接触面）是全部透水的．（ｂ）半．空间表面与圆板的接触面是不透水的，而其余表面是透水的．文末给出了数值算例．

In this paper the integral equations of vertical vibration of an elastic circular plate on asaturated poroelastic half space are studied. Firstly Blot's wave equations in the freqency domainare reduced to the partial differential equations expressed in terms of the displacement of the solidmatrix and the pore-fluid pressure (u, uz, P). The above equations airs solved by a new methodwhich does not introduce the potelltials. Using integral transform techniques the general solutionsfor Hankel transforms of displacements, stresses and pore-fluid pressure are obtained. Then thedual integral equations of vertical vibration of an elastic circular plate on a fluid-saturated poroushalf space are established according to the mixed boundary value condition. Making use of Abeltransform the dual integral equations are reduced to nedholm integral equation of the secondkind. In the paper the two types of drainage conditions at the surface of the poroelastic half spaceare considered: (a) The surfaCe of the poroelastic half space is assumed to be completely perviousboth within and exterior to the plate; (b) The interface between the plate and the poroelastic halfspace is assumed to be impervious and the exterior region is assumed to be pervious.It should be noted that the numerical results for a permeable plate obtained from presentanalysis which does not introduce the potentials is the same as that obtained using the potentials,and small differences exist between the numerical results for an impermeable plate obtained frompresent analysis and from the potentials.The numerical results presented in the paper indicate that the drainage boundary conditionsat the surface of a saturated poroelastic half space and the dimensionless frequency have importantinfluences on the vertical dynamic displacement of elastic plate. Based on the studies, the followingconclusions can be made:(1) The vertical displacements of elastic plate for condition (a) (pervious under the plate) islarger than that for condition (b) (impervious under the plate).(2) The deflection of an elastic plate is larger at the center than at the edge.(3) The vertical displacements of an elastic plate decrease with increase of frequencies.