平面断裂力学问题的超奇异积分方程方法

【作者】 万强；

【导师】 乐金朝； 王复明；

【作者基本信息】 郑州大学 ， 水工结构工程， 2004， 硕士

【摘要】 本文使用超奇异积分方程方法，系统地研究了二维均质、双相材料中的单裂纹、和多裂纹问题，获得了一些理论成果和数值结果。本文的主要工作包括以下几个方面： 1．在已有研究成果的基础上，对双相材料平面在集中力作用下的弹性力学基本解进行了详细推导，给出了问题的位移和应力场的显式表达式。此解可以在边界元法中直接使用，为使用边界元—边界积分方程方法研究平面夹杂和双相材料平面力学问题提供了基础。 2．根据裂纹前沿的奇性应力场，得到了裂纹端点应力强度因子的计算公式。使用主部分析方法，研究了裂纹端点处未知函数的性态指数。使用有限部积分的原理，建立了超奇异积分方程的数值求解方法，并给出了相应的计算公式。在此基础上，得到了基于裂纹面上的位移间断计算裂纹端点应力强度因子的具体表达式。 3．利用均质平面弹性力学基本解和双相材料平面弹性力学基本解，采用超奇异积分方程方法，分别对均质和双相材料平面体两种情况下任意角度单裂纹问题及多裂纹问题进行了详细研究。根据有限部积分的原理，获得了问题的超奇异积分方程组，其中的未知函数是裂纹面上的位移间断函数，并为其建立了相应的数值计算方法，并给出了裂纹端点应力强度因子的计算公式。 4．为了验证该方法的可靠性，本文作了考核性计算，结果表明，该方法是成功的。本文对几种典型的裂纹问题(如均质材料共线裂纹，双相材料斜裂纹，双相材料平行裂纹等)进行了数值计算，并对裂纹之间、裂纹与界面之间的相互干扰问题进行了较为系统的分析，得到了部分有价值的结论。其中一些结果是第一次报道。 本文的研究工作，为平面断裂力学特别是复合材料平面断裂力学的研究提供了一条新的有效途径。更多还原

【Abstract】 The use of composite materials as structural components has inspired considerable research on the effect of flaws and imperfections on the structural strength of composites. The research usually involves two-dimensional modeling. In this dissertation, the two-dimensional crack problems in composite materials are investigated systematically by use of the hypersingular integral equation method. Some theoretical conclusions and a great deal of numerical results are obtained. The main contributions of the dissertation are as follows:1. The elasticity point-force solutions for perfectly bonded elastic half-plane in literatures are further studied. The compact formulae of displacements and stresses are given which can be used in the boundary element method.2. According to the singular stress filed, the formula of stress intensity factor at crack terminal is acquired. The nature of the unknown solutions of the hypersigular integral equations is studied by the main-part analysis method of singular integral equations, and the singular index of the unknown solutions is obtained. The finite-part integral is used to discretize the hypersingular equations as a set of linear algebraic equations, where the singular integrals of various types are specially treated and the specific calculating formula are given. Based on this, the formulae determining the stress intensity factors with the displacement discontinuities on the crack surface are proposed.3. Based on the fundamental solution of homogeneous and bimaterial plane, the hypersingular integral equation method is used to investigate the problems of an arbitrary angle crack and multiply cracks respectively .Both of the problems are reduced to a system of one-dimensional hypersingular integral equations with theory of finite-part integral. the unknown function is displacement discontinuities. and their numerical method are founded for them. the formulae of the stress intensity factors is given.4. Some examples are shown that the numerical approach is reliable. Some typical examples of planar crack problems (such as homogeneous material colinear cracks, bimaterial oblique crack, bimaterial parallel cracks etc.) are calculated, and the effect between cracks or between interface and cracks is analyzed systematically. some worthy results are obtained, part of which are not available in the literature. Therefore, a new efficient path to research plane crack problems especially composite materials plane fracture mechanicals is found.更多还原