含孔von Karman板中非线性波散射与边值问题

基于ｖｏｎ Ｋａｒｍａｎ板大挠度弯曲理论，利用小参数摄动法，分析研究了含孔ｖｏｎＫａｒｍａｎ板的非线性波散射与动应力集中问题，其中一类可看成是薄板弯曲波动问题的控制方程。当有单频波入射时，由于弯曲应力与膜应力状态的非线性耦合，孔洞会产生高次谐波散射现象。建立了求解本问题的边界积分方程法，利用积分方程法交替求求这两类问题，最终可获得问题的近似分析解。

NONLINEAR WAVE SCATTERING AND BOUNDARY VALUE PROBLEM OF VON KARMAN'S PLATES WITH A CIRCULAR CAVITY

The problem of elastic wave motion and dynamic stress concentrations in thin plates,because of its technical importance, has been the subject of many investigators. A number of analytic methods were established for the investigation of stress concentrations, among which, the method developed by N.I. Muskhelishivili is prominent. The problem of static stress concentrations on the edge of an arbitrary cutout can be solved by Muskhelishivili's method. Diffractions of flexural waves and dynamic stress concentrations in plates are different from static stress problem.In 60's, Y.H. Pao investigated scattering of elastic wave and dynamic stress concentrations in classical thin plates with a circulax cavity by using special function method and had given numerical results. In 70's, D.K. Liu et al. proposed the complex variable method and conformal mapping technique for solving elastic wave scattering and dynamic stress concentrations in solids. In 40's,W.Z. Chien et al. developed the small parameter perturbation method to analyzing the large deflection of circular plates.In this paper, based on the bending deflections theory of von Karman's plates, using perturbation method and method of harmonic balance, scattering of flexural wave and dynamic stress concentrations in the plate with a circular cavity have been studied.The first equation is treated as a problem of flexural wave motions in thin plates, and the second equation is analogous to a plane elasticity problem. Incident wave of single frequency results in superharmonic wave scattering on the edge of cutouts because of coupling of bending stress with membrane stress. With reduced order of the governing equations, the fundamental solutions of the two equations are given,respectively. A boundary integral method to solve dynamic stress concentrations in pates is proposed. Computational formulas of dynamic stress concentration factors of plates with a cutout are developed. As examples, numerical results are graphically presented.