基于新型裂尖杂交元的压电材料断裂力学研究

提出了一种裂尖邻域杂交元模型，将其与标准杂交应力元结合来求解压电材料裂纹尖 端的奇性电弹场和断裂参数的数值解．裂纹尖端杂交元的建立步骤为：1) 利用高次内插有限元特征法求解特征问题，得到反映裂尖奇异性电弹场状况的特 征值和特征角分布函数；2) 利用广义Hellinger-Reissner变分泛函以及特征问题的解来建立裂尖邻域杂交元模型．该 方法求解电弹场时，摒弃了传统有限元方法中裂尖奇异性场需要借助解析解的做法，也避免 了单纯有限元方法中需要在裂尖端部进行高密度单元划分．采用PZT5板中心裂纹问题 作为考核例，数值结果显示了良好的精确性．作为进一步应用，求解了含中心界面裂纹 的PZT4-PZT5两相压电材料的应力强度因子和电位移强度因子．所有的算例都考虑 了3种裂纹面电边界条件．

Fracture mechanics researches on piezoelectric materials based on a novel crack-tip hybrid finite element method

Singular electro-elastic fields surrounding crack-tips of piezoelectric materials can be expressed as \Sigma = \beta r^\lambda F(\theta ), in which (r,\theta) is the polar coordinate system whose origin is set at the singular point; $\la$ is the eigenvalue; F(\theta) is the characteristic angular variation function; \beta is a coefficient to be determined. The authors have developed a new {\it ad doc} finite element method to solve eigenvalues $\la$ and characteristic angular variation functions F(\theta) in paper 20]. To solve all the singular electro-elastic fields，coefficient $\beta $ should be determined. In this paper a new super crack-tip hybrid element model together with an assumed hybrid stress finite element model is developed to solve the singular electro-elastic fields near the crack-tip of piezoelectric materials. The procedure is as follows: 1) an {\it ad doc} one dimensional finite element method is developed to determine the characteristic problems; 2) The numerical results of step 1 are substituted into the generalized Hellinger-Reissner variational functional, and then a finite element formulation of the super crack-tip element is derived. This new model has two obvious advantages: One is to use numerical solutions but not analytical solutions, the other is to avoid mesh refinement near the crack-tip. To verify efficiency and accuracy of the present model, a benchmark example on the singular electro-elastic fields, stress intensity factors and electric displacement intensity factors for a central crack in an infinite PZT5 panel is given. Interfacial crack problem of PZT4-PZT5 panel is also considered as a further application of the new model. In all examples, three kinds of electric boundary conditions 13], i.e., impermeable boundary condition, permeable boundary condition and conducting boundary condition on the crack surfaces, are considered. This model can be used in more complicated fracture problems, such as piezoelectric wedges, piezoelectric junctions or other complex geometries.