压电材料中三维I型断裂力学分析

讨论了不可导通情况下三维横观各向刚性压电材料中受拉伸和电载荷作用的平片裂纹Ⅰ型断裂力学问题．使用自限部分概念，从二维线性压电理论出发，严格得到了一组以裂纹面位移间断和电势间断为未知变量的超奇异积分方程组；应用二维超奇异积分的主部分析法，从理论上分析得到了裂纹前沿应力和电势奇性指数以及应力和电位移奇性场，从而找到了以裂纹面位移间断和电势间断表示的应力和电位移强度因子、能量释放率表达式；为所得到的超奇异积分方程组建立了数值法，并用此计算了若干典型的平片裂纹问题，数值结果令人满意．

THREE-DIMENSIONAL FRACTURE MECHANICS FOR A MODE I CRACK IN PIEZOELECTRIC MEDIA~1)

In recent years,due to its characteristic direct-converse piezoelectric effect,piezoelectric materialshave been extensively used in sensors,actuators,resonators and intelligent structures.However,many man-made piezoelectric materials like piezoelectric ceramic are very brittle easily crack and thus lead to failure.Reliable service lifetime predictions of piezoelectric components demand a complete understanding of theirfracture processes.Most previous studies on crack problems in piezoelectric materials are mainly limited to the two-dimensionalcases.For three-dimensional crack problems,due to the extreme difficulties of mathematics and mechanics,onlyfew theoretical analyses for specially shaped cracks are available.To our best knowledge,numerical researcheson three-dimensional arbitrary-shaped crack problems in piezoelectric materials are very limited up to now.The finite-part integral concept is used to prove rigidly hypersingular integral equations for 3D mode Iplanar crack problems in impermeable case from the three-dimensional linear piezoelectricity theory.Inves-tigations on the singularities,the singular stress and electric displacement fields in the vicinity of the crackare made by the dominant-part analysis of the two-dimensional hypersingular integrals.Thereafter,the stressand electric displacement intensity factor K-fields and the energy release rate G are exactly obtained by thedefinitions similar to those of elasticity.Then,a numerical method for solving hypersingular integral equationsis developed,in which the displacement and eiectric potential differences across the crack surfaces are approx-imated with a product of basic density functions and polynomials.The influences of the collocation pointsand the exponents of the variables in the polynomial on the convergence of numerical results are discussed.Computational experiences state that the selection of exponents in the polynomial with collocation points givendepends on whether the numerical results satisfy final compatible boundary conditions on the crack surface ornot.The stress and electric displacement intensity factors,the total energy release rates and the mechanicalstrain energy release rates for several typical planar cracks are indicated in tables and figures.The numericalresults show that present method yields solutions with high accuracy.