A mathematical strip-saturation model for a piezoelectric plane weakened by two collinear equal cracks

Abstract

A plane problem for a poled transversely isotropic piezoelectric plane cut along two equal collinear straight cracks is considered. It is assumed that the electrical yielding occurs at the continuations of the cracks due to the applied mechanical and electrical loadings. We model these crack continuations as the zones with constant cohesive saturation limit electrical displacement. The Stroh formalism and a complex variable technique are adopted to obtain the analytic solution of the problem. Closed-form expressions are derived for the developed saturation zone length, the crack opening displacement, the crack opening potential drop, the stress intensity factors, and the energy release rate. A qualitative numerical case study is presented for ceramics PZT-4, PZT-5H, and BaTiO₃ to study the effects of various parameters as follows: developed saturation zone length and prescribed load, stress intensity factor, energy release rate, and crack opening displacement on crack growth resistance. The energy release rate and the stress intensity factor variations are investigated with respect to the inter-crack distance. The results obtained are presented graphically and discussed.

Keywords

Crack opening displacement crack opening potential drop energy release rate saturation zone Stroh formalism

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References

Suo

Kuo

Barnett

. Fracture mechanics for piezoelectric ceramics. Mech Phy Solids 1992; 40: 739–765.

Pak

. Linear electro-elastic fracture mechanics of piezoelectric materials. Int Fract 1992; 54: 79–100.

Wang

. Analysis of strip electric saturation model of crack problem in piezoelectric materials. Int Solids Struct 2000; 37: 6031–6049.

Zhang

Gao

. Fracture behaviour of piezoelectric materials. Theor Appl Fract Mech 2004; 41: 339–379.

Gao

Wang

. Generalized 2-D problem of piezoelectric media containing collinear cracks. Acta Mech Sinica 1999; 15: 235–244.

. Closed form solution for a piezoelectric strip with two collinear cracks normal to the strip boundaries. Eur Mech A/Solids 2002; 21: 981–989.

Singh

Rokne

Dhaliwal

. Closed form solution for two anti-plane collinear cracks in magneto-electro-elastic layer. Eur Mech A/Solids 2009; 28: 599–609.

Lee

. Two collinear unequal cracks in a poled piezoelectric plane. Int Fract 2010; 165: 47–60.

Gao

Zhang

Tong

. Local and global energy release rates for an electrically yielded crack in a piezoelectric ceramic. Mech Phy Solids 1997; 45: 491–510.

10.

Shen

Nishioka

Kuang

. Nonlinear electromechanical interfacial fracture for piezoelectric materials. Mech Mater 2000; 32: 57–64.

11.

Bhargava

Saxena

. Solution for a cracked piezoelectric plate subjected to variable load on plastic zones under mode-I loading. Mater Process Tech 2005; 164–165: 1495–1499.

12.

Loboda

Lapusta

Govorukha

. Mechanical and electrical yielding for an electrically insulated crack in an interlayer between piezoelectric materials. Int Eng Sci 2008; 46: 260–272.

13.

Bhargava

Jangid

. Strip-saturation model for piezoelectric plane weakened by two collinear cracks with coalesced interior zones. Appl Math Model 2012; 37: 4093–4102.

14.

Gao

Husler

Balke

. Periodic permeable interface cracks in piezoelectric materials. Int Solids Struct 2004; 41: 323–335.

15.

Barnett

Lothe

. Dislocation and line charges in anisotropic piezoelectric insulators. Phys Stat Sol (b) 1975; 67: 105–111.

16.

Muskhelishvili

. Some basic problems of the mathematical theory of elasticity. The Netherlands: Nordhoff, 1963.

17.

Gao

Noda

Zhang

. Dielectric breakdown model for a conductive crack and electrode in piezoelectric materials. Int Eng Sci 2006; 44: 256–272.

18.

. On the crack-tip stress singularity of interfacial cracks in transversely isotropic piezoelectric biomaterials. Int Solids Struct 2003; 40: 7499–7511.

19.

Loboda

Lapusta

Sheveleva

. Limited permeable crack in an interlayer between piezoelectric materials with different zones of electrical saturation and mechanical yielding. Int Solids Struct 2010; 47: 1795–1806.