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Applied Physics 2026
电弹性耦合偏微分方程组带型裂纹问题
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Abstract:
根据压电体中扩展的介质裂纹模型，本文讨论了压电带型机电荷载下的Griffith裂纹问题，在裂纹边界条件假设下，得到一组耦合偏微分方程组。运用傅立叶变换，将电弹性耦合偏微分方程组问题转化为求解第二类Fredholm类积分方程组。采用Lobatto-Chebyshev配置法，得到了非线性代数方程，并对其进行数值求解。通过数值计算分析了不同边界条件下断裂参量的变化规律并和已有实验结果进行了比较。
Based on the extended dielectric crack model in piezoelectric materials, this study examines the Griffith crack problem in piezoelectric strip structures under electromechanical loading. By assuming specific crack boundary conditions, a system of coupled partial differential equations governing the electroelastic field is formulated. Applying Fourier transforms to convert the coupled partial differential equations into a system of the second kind Fredholm integral equations. The Lobatto-Chebyshev collocation method is used to form a nonlinear system of algebraic equations, which is solved by proposing an algorithm. Numerical results are carried out to show the variations of the fracture parameters of concern on the physical properties of the dielectric inside the crack under the two boundary conditions. the formats.
| [1] | Ishihara, M. and Noda, N. (2005) Control of Thermal Stress Intensity Factor in a Piezothermoelastic Semi-Infinite Body with an Edge Crack. European Journal of Mechanics A/Solids, 24, 417-426. https://doi.org/10.1016/j.euromechsol.2005.01.009 |
| [2] | Schneider, G.A., Felten, F. and McMeeking, R.M. (2003) The Electrical Potential Difference across Cracks in PZT Measured by Kelvin Probe Microscopy and the Implications for Fracture. Acta Materialia, 51, 2235-2241. https://doi.org/10.1016/s1359-6454(03)00027-2 |
| [3] | Zhang, T.Y., Zhao, M.H. and Gao, C.F. (2005) The Strip Dielectric Breakdown Model. International Journal of Fracture, 132, 311-327. https://doi.org/10.1007/s10704-005-2054-8 |
| [4] | Zhu, T. and Yang, W. (1997) Toughness Variation of Ferroelectrics by Polarization Switch under Non-Uniform Electric Field. Acta Materialia, 45, 4695-4702. https://doi.org/10.1016/s1359-6454(97)00123-7 |
| [5] | Ricoeur, A. and Kuna, M. (2009) Electrostatic Tractions at Crack Faces and Their Influence on the Fracture Mechanics of Piezoelectrics. International Journal of Fracture, 157, 3-12. https://doi.org/10.1007/s10704-009-9321-z |
| [6] | Zikun, W. (1994) Penny-Shaped Crack in Transversely Isotropic Piezoelectric Materials. Acta Mechanica Sinica, 10, 49-60. https://doi.org/10.1007/bf02487657 |
| [7] | Zhong, X.C. (2011) Fracture Analysis of a Piezoelectric Layer with a Penny-Shaped and Energetically Consistent Crack. Acta Mechanica, 223, 331-345. https://doi.org/10.1007/s00707-011-0565-0 |
| [8] | Li, X.F. and Lee, K.Y. (2004) Effects of Electric Field on Crack Growth for a Penny-Shaped Dielectric Crack in a Piezoelectric Layer. Journal of the Mechanics and Physics of Solids, 52, 2079-2100. https://doi.org/10.1016/j.jmps.2004.02.012 |
| [9] | Guo, J.H., Lu, Z.X., Han, H.T., et al. (2010) The Behavior of Two Non-Symmetrical Permeable Cracks Emanating from an Elliptical Hole in a Piezoelectric Solid. European Journal of Mechanics A/Solids, 29, 654-663. https://doi.org/10.1016/j.euromechsol.2010.01.001 |
| [10] | Guo, J.H., Lu, Z.X., Han, H.T., et al. (2009) Exact Solutions for Anti-Plane Problem of Two Asymmetrical Edge Cracks Emanating from an Elliptical Hole in a Piezoelectric Material. International Journal of Solids and Structures, 46, 3799-3809. https://doi.org/10.1016/j.ijsolstr.2009.07.002 |
| [11] | Zhong, X.C. and Zhang, K.S. (2010) Electroelastic Analysis of an Electrically Dielectric Griffith Crack in a Piezoelectric Layer. International Journal of Engineering Science, 48, 612-623. https://doi.org/10.1016/j.ijengsci.2010.02.002 |
| [12] | Ueda, S. (2006) The Crack Problem in Piezoelectric Strip under Thermoelectric Loading. Journal of Thermal Stresses, 29, 295-316. https://doi.org/10.1080/01495730500360450 |
| [13] | Chen, Z.T. (2006) Dynamic Fracture Mechanics Study of an Electrically Impermeable Mode III Crack in a Transversely Isotropic Piezoelectric Material under Pure Electric Load. International Journal of Fracture, 141, 395-402. https://doi.org/10.1007/s10704-006-9003-z |
| [14] | Zhong, X.C. (2011) Fracture Analysis of a Piezoelectric Layer with a Penny-Shaped and Energetically Consistent Crack. Acta Mechanica, 223, 331-345. https://doi.org/10.1007/s00707-011-0565-0 |
| [15] | Nowacki, W. (1978) Some General Theorems of Thermopiezoelectricity. Journal of Thermal Stresses, 1, 171-182. https://doi.org/10.1080/01495737808926940 |
