功能梯度压电板条中电绝缘型运动裂纹的电弹性场

Electroelastic Field for an Impermeable Moving Crack in a Functionally Graded Piezoelectric Strip

基于三维弹性理论和压电理论,对材料系数按指数函数规律分布的功能梯度压电板条中的反平面运动裂纹问题进行了求解。利用Fourier积分变换方法将电绝缘型运动裂纹问题化为对偶积分方程,并进一步归结为易于求解的第二类Fredholm积分方程。通过渐近分析,获得了裂纹尖端应力、应变、电位移和电场的解析解,给出了裂纹尖端场各个变量的角分布函数,并求得了裂纹尖端场的强度因子,分析了压电材料物性梯度参数、几何尺寸及裂纹运动速度对它们的影响。结果表明,对于电绝缘型裂纹,功能梯度压电板条中运动裂纹尖端附近的各个场变量都具有-1/2阶的奇异性;当裂纹运动速度增大时,裂纹扩展的方向会偏离裂纹面。

Based on the three-dimensional theory of piezoelectric elasticity, the anti-plane moving crack problem was solved for a piezoelectric strip with the material gradient properties being in the form of exponential functions. By using the Fourier transform, the problem involving an impermeable anti-plane moving crack is first reduced to two pairs of dual integral equations and then into Fredholm integral equations of the second kind. The closed forms of the singular stress, strain, electric displacement and electric field are obtained by using asymptotic expansion, and we also got the angular distribution function of the field variables and the intensity factors of relevant quantities near the crack tip. Finally, the influence of material properties gradient, geometrical size, and the crack moving velocity on the intensity factors was studies. The results obtained for impermeable crack show that the field variables near the crack tip in a functionally graded piezoelectric strip all possess the square root singularity; the moving crack has a tendency to deviate from the crack face when the velocity is increased.