Complex variable approach in studying modified polarization saturation model in two-dimensional semipermeable piezoelectric media

doi:10.1007/s10483-017-2281-9

Abstract

Abstract:

A modified polarization saturation model is proposed and addressed mathematically using a complex variable approach in two-dimensional (2D) semipermeable piezoelectric media. In this model, an existing polarization saturation (PS) model in 2D piezoelectric media is modified by considering a linearly varying saturated normal electric displacement load in place of a constant normal electric displacement load, applied on a saturated electric zone. A centre cracked infinite 2D piezoelectric domain subject to an arbitrary poling direction and in-plane electromechanical loadings is considered for the analytical and numerical studies. Here, the problem is mathematically modeled as a non-homogeneous Riemann-Hilbert problem in terms of unknown complex potential functions representing electric displacement and stress components. Having solved the Hilbert problem, the solutions to the saturated zone length, the crack opening displacement (COD), the crack opening potential (COP), and the local stress intensity factors (SIFs) are obtained in explicit forms. A numerical study is also presented for the proposed modified model, showing the effects of the saturation condition on the applied electrical loading, the saturation zone length, and the COP. The results of fracture parameters obtained from the proposed model are compared with the existing PS model subject to electrical loading, crack face conditions, and polarization angles.

Key words: fracture parameter, crack-tip field, dynamic propagating, strain softening, complex variable, local stress intensity factor (SIF), piezoelectric, polarization saturation (PS) model

2010 MSC Number:

74R10

S. SINGH, K. SHARMA, R. R. BHARGAVA. Complex variable approach in studying modified polarization saturation model in two-dimensional semipermeable piezoelectric media. Applied Mathematics and Mechanics (English Edition), 2017, 38(11): 1517-1532.

TrendMD

References

[1] Dugdale, D. S. Yielding of steel sheets containing slits. Journal of Mechanics and Physics of Solids, 8, 100-104(1960)
[2] Muskhelishvili, N. I. Some Basic Problems of Mathematical Theory of Elasticity, Noordhoff, Leyden (1975)
[3] Broek, D. Elementary Engineering Fracture Mechanics, 4th ed., Martinus Nijhoff Publishers, Dordrecht (1986)
[4] Gdoutos, E. E. Fracture Mechanics:An Introduction, Kluwer Academic Publishers, Dordrecht (1993)
[5] Parton, V. Z. Fracture mechanics of piezoelectric materials. Acta Astronautica, 3, 671-683(1976)
[6] Pak, Y. E. Linear electro-elastic fracture mechanics of piezoelectric materials. International Journal of Fracture, 54, 79-100(1992)
[7] Suo, Z., Kuo, C. M., Barnett, D. M., and Wills, J. R. Fracture mechanics of piezoelectric ceramics. Journal of Mechanics and Physics of Solids, 40, 739-765(1992)
[8] Park, S. B. and Sun, C. T. Effect of electric field on fracture of piezoelectric ceramics. International Journal of Fracture, 70, 203-216(1995)
[9] Zhang, T. Y., Zhao, M. H., and Tong, P. Fracture of piezoelectric ceramics. Advances in Applied Mechanics, 38, 147-289(2002)
[10] Kuna, M. Finite element analyses of cracks in piezoelectric structures:a survey. Archive of Applied Mechanics, 76, 725-745(2006)
[11] Bhargava, R. R. and Sharma, K. X-FEM simulation for two-unequal-collinear cracks in 2D finite piezoelectric specimen. International Journal of Mechanics and Materials Design, 8, 129-148(2012)
[12] Sharma, K., Bui, T. Q., Zhang, C., and Bhargava, R. R. Analysis of subinterface crack in piezoelectric bimaterials with the extended finite element method. Engineering Fracture Mechanics, 104, 114-139(2013)
[13] Gao, H., Zhang, T. Y., and Tong, P. Local and global energy release rate for an electrically yielded crack in a piezoelectric ceramic. Journal of Mechanics and Physics of Solids, 45, 491-510(1997)
[14] Ru, C. Q. Effect of electrical polarization saturation on stress intensity factors in a piezoelectric ceramic. International Journal of Solids and Structures, 36, 869-883(1999)
[15] Gao, C. F. and Zhang, T. Y. Fracture behaviors of piezoelectric materials. Theoretical and Applied Fracture Mechanics, 41, 339-379(2004)
[16] Zhang, T. Y., Zhao, M. H., and Gao, C. F. The strip dielectric breakdown model. International Journal of Fracture, 132, 311-327(2005)
[17] Loboda, V., Laputsa, Y., and Govorukha, V. Mechanical and electrical yielding for an electrically insulated crack in an interlayer between piezoelectric materials. International Journal of Engineering Science, 46, 260-272(2008)
[18] Fan, C. Y., Zhao, M. H., and Zhou, Y. H. Numerical solution of polarization saturation/dielectric breakdown model in 2D finite piezoelectric media. Journal of Mechanics and Physics of Solids, 57, 1527-1544(2009)
[19] Loboda, V., Laputsa, Y., and Sheveleva, A. Limited permeable crack in an interlayer between piezoelectric materials with different zones of electrical saturation and mechanical yielding. International Journal of Solids and Structures, 47, 1795-1806(2010)
[20] Fan, C. Y., Zhao, Y. F., Zhao, M. H., and Pan, E. Analytical solution of a semi-permeable crack in a 2D piezoelectric medium based on the PS model. Mechanics Research Communications, 40, 34-40(2012)
[21] Linder, C. and Miehe, C. Effect of electric displacement saturation on the hysteretic behavior of ferroelectric ceramics and the initiation and propagation of cracks in piezoelectric ceramics. Journal of Mechanics and Physics of Solids, 60, 882-903(2012)
[22] Chen, H. S., Pei, Y. M., Liu, J. X., and Fang, D. N. Moving polarization saturation crack in ferroelectric solids. European Journal of Mechanics/A Solids, 41, 43-49(2013)
[23] Bhargava, R. R. and Jangid, K. Strip-coalesced interior zone model for two unequal collinear cracks weaking piezoelectric media. Applied Mathematics and Mechanics (English Edition), 35(10), 1249-1260(2014) DOI 10.1007/s10483-014-1890-9
[24] Bhargava, R. R. and Jangid, K. Strip-saturation model for piezoelectric plane weakened by two collinear cracks with coalesced interior zones. Applied Mathematical Modeling, 37, 4093-4102(2013)
[25] Bhargava, R. R. and Hasan, S. Crack opening displacement for two unequal straight cracks with coalesced plastic zones:a modified Dugdale model. Applied Mathematical Modeling, 35, 3788-3796(2011)
[26] Hasan, S. and Akhtar, N. Dugdale model for three equal collinear straight cracks:an analytical approach. Theoretical and Applied Fracture Mechanics, 78, 40-50(2015)
[27] Hasan, S. Application of modified Dugdale model to two pairs of collinear cracks with coalesced yield zones. Applied Mathematical Modeling, 40, 3381-3399(2016)
[28] Hao, T. H. and Shen, Z. Y. A new electric boundary condition of electric fracture mechanics and its applications. Engineering Fracture Mechanics, 47, 793-802(1994)