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

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.