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 引用本文: 平学成, 陈梦成, 谢基龙, 李 强. 基于新型裂尖杂交元的压电材料断裂力学研究[J]. 力学学报, 2006, 38(3): 407-413.
Fracture mechanics researches on piezoelectric materials based on a novel crack-tip hybrid finite element method[J]. Chinese Journal of Theoretical and Applied Mechanics, 2006, 38(3): 407-413.
 Citation: Fracture mechanics researches on piezoelectric materials based on a novel crack-tip hybrid finite element method[J]. Chinese Journal of Theoretical and Applied Mechanics, 2006, 38(3): 407-413.

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

• 摘要: 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．

Abstract: Singular electro-elastic fields surrounding crack-tips ofpiezoelectric materials can be expressed as \Sigma = \beta r^\lambdaF(\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 tobe determined. The authors have developed a new \it ad doc finite element methodto solve eigenvalues \la and characteristic angular variation functionsF(\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 hybridelement model together with an assumed hybrid stress finite element model isdeveloped to solve the singular electro-elastic fields near the crack-tip ofpiezoelectric materials. The procedure is as follows: 1) an \it ad doc one dimensional finite element method is developedto determine the characteristic problems; 2) The numerical results of step 1are substituted into the generalized Hellinger-Reissner variationalfunctional, and then a finite element formulation of the super crack-tipelement is derived. This new model has two obvious advantages: One is to usenumerical solutions but not analytical solutions, the other is to avoid meshrefinement near the crack-tip. To verify efficiency and accuracyof the present model, a benchmark example on the singular electro-elasticfields, stress intensity factors and electric displacement intensity factorsfor a central crack in an infinite PZT5 panel is given. Interfacial crackproblem of PZT4-PZT5 panel is also considered as a further application ofthe new model. In all examples, three kinds of electric boundaryconditions 13, i.e., impermeable boundary condition, permeable boundarycondition and conducting boundary condition on the crack surfaces, areconsidered. This model can be used in more complicated fractureproblems, such as piezoelectric wedges, piezoelectric junctions or othercomplex geometries.

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