内聚力模型裂纹问题分析的解析奇异单元

An Analytical Singular Element to Study the Cohesive Zone Model for Cracks

内聚力模型已经被广泛应用于需要考虑断裂过程区的裂纹问题当中,然而常用的数值方法应用于分析内聚力模型裂纹问题时还存在着一些不足,比如不能准确地给出断裂过程区的长度、需要网格加密等.为了克服这些缺点,论文构造了一个新型的解析奇异单元,并将之应用于基于内聚力模型的裂纹分析当中.首先将虚拟裂纹表面处的内聚力用拉格拉日插值的方法近似表示为多项式形式,而多项式表示的内聚力所对应的特解可以被解析地给出.然后利用一个简单地迭代分析,基于内聚力模型的裂纹问题就可以被模拟出来了.最后,给出二个数值算例来证明论文方法的有效性.

The cohesive zone model is widely used in fracture mechanics. When the fracture process zone （FPZ） in front of the crack tip is too large to be neglected, the nonlinear behavior must be consid- ered. That is to say, in this circumstance the linear fracture mechanics is no longer valid. In order to take into account the nonlinear behavior in FPZ, many fracture models have been proposed, among which, the cohesive zone model （CZM） might be one of the simplest and has been widely used. However, there still remain some problems in the existing numerical methods; for instance, length of the fracture process zone cannot be obtained accurately; dense meshes are required, etc. In order to get over these difficulties, a new analytical singular element is proposed in the present study and further extended into the cohesive zone model for crack propagation problems. In this singular element, the cohesive traction is approximately ex- pressed in the form of polynomial expanding though Lagrange interpolation. The special solution corre- sponding to each expanding term is specified analytically. Each special solution strictly satisfies the re- quirements of both differential equations of interior domain and the corresponding traction expanding terms. The real cohesive traction acting on the cohesive crack surface is thus expressed in a natural and strict way. Then the special solution can be transformed into nodal forces of the present singular element. Assembling the stiffness matrix and nodal force into the global FEM system, the cohesive crack problem can be analyzed. An efficient iteration procedure is also proposed to solve the nonlinear problem. Finally, the cohesive crack propagation under arbitrary external loading can be simulated, and the length of FPZ, crack tip opening displacement （CTOD） and other parameters in the cohesive crack problem can be ob- tained simultaneously. The validity of the present method is illustrated by numerical examples.