Abstract
An electrically limited permeable crack with a frictionless contact zone at the right crack tip between two semi-infinite piezoelectric spaces under the action of a remote electromechanical loading is considered. Attention is focused on the influence induced from the permittivity of the medium inside the crack gap on the contact zone length and the fracture mechanical parameters. Assuming the electric displacement constant inside the open region of the crack, the problem is reduced to a combined Dirichlet-Riemann and Hilbert boundary value problems which have been solved exactly. Stress and electric displacement intensity factors as well as the crack tip energy release rate are found in a clear analytical form. Furthermore, transcendental equations for the determination of the real contact zone length have been obtained for a general case and for a small contact zone length in an especially simple form. The dependencies of the mentioned values on the intensities of the electromechanical loading are presented in tables and associated diagrams.
Similar content being viewed by others
References
Beom HG, Atluri SN (1996) Near-tip fields and intensity factors for interfacial cracks in dissimilar anisotropic piezoelectric media. Int J Fract 75: 163–183
Beom HG (2003) Permeable cracks between two dissimilar piezoelectric materials. Int J Solids Struct 40: 6669–6679
Comninou M (1977) The interface crack. ASME J Appl Mech 44: 631–636
Deeg WF (1980) The analysis of dislocation, crack and inclusion problem in piezoelectric solids. Ph.D.dissertation, Standford University
Dundurs J, Gautesen AK (1988) An opportunistic analysis of the interface crack. Int J Fract 36: 151–159. doi:10.1007/BF00017793
Dunn ML (1994) The effects of crack face boundary conditions on the fracture mechanics of piezoelectric solids. Eng Fract Mech 48: 25–39. doi:10.1016/0013-7944(94)90140-6
Dunn ML, Taya M (1994) Electroelastic field concentrations in and around inhomogeneities in piezoelectric solids. ASME J Appl Mech 61: 474–475
Gao CF, Wang MZ (2000) Collinear permeable cracks between dissimilar piezoelectric materials. Int J Solids Struct 37: 4969–4986
Govorukha VB, Kamlah M (2005) Investigation of an interface crack with a contact zone in a piezoelectric biomaterial under limited permeable electric boundary conditions. Acta Mech 178: 85–99
Govorukha VB, Loboda VV, Kamlah M (2006) On the influence of the electric permeability on an interface crack in a piezoelectric biomaterial compound. Int J Solids Struct 43: 1979–1990. doi:10.1016/j.ijsolstr.2005.04.009
Gruebner O, Kamlah M, Munz D (2003) Finite element analysis of cracks in piezoelectric materials taking into account the permittivity of the crack medium. Eng Fract Mech 70: 1399–1413. doi:10.1016/S0013-7944(02)00117-0
Hao TH, Shen ZY (1994) A new electric boundary condition of electric fracture mechanics and its applications. Eng Fract Mech 47: 793–802. doi:10.1016/0013-7944(94)90059-0
Herrmann KP, Loboda VV (2000) Fracture-mechanical assessment of electrically permeable interface cracks in piezoelectric bimaterials by considerations of various contact zone models. Arch Appl Mech 70: 127–143. doi:10.1007/s004199900052
Herrmann KP, Loboda VV, Govorukha VB (2001) On contact zone models for an electrically impermeable interface crack in a piezoelectric biomaterial. Int J Fract 111: 203–227. doi:10.1023/A:1012269616735
Kharun IV, Loboda VV (2003) A set of interface cracks with contact zones in a combined tension-shear field. Acta Mech 166: 43–56. doi:10.1007/s00707-003-0044-3
Kemmer G (2000) Berechnung von elektromechanischen Intensitätsparametern bei Rissen in Piezokeramiken. Dissertation, VDI Verlag Düsseldorf Nr. 261 Reihe 18
Landis CM (2004) Energetically consistent boundary conditions for electromechanical fracture. Int J Solids Struct 41: 6291–6315
Lapusta Y, Loboda V (2009) Electro-mechanical yielding for a limited permeable crack in an interlayer between piezoelectric materials. Mech Res Commun 36: 183–192. doi:10.1016/j.mechrescom.2008.09.001
Li Q, Chen YH (2007) Solution for a semi-permeable interface crack between two dissimilar piezoelectric material. ASME J Appl Mech 74: 833–844. doi:10.1115/1.2711232
Li Q, Chen YH (2008) Solution for a semi-permeable interface crack in elastic dielectric/piezoelectric bimaterials. ASME J Appl Mech 75: 0110101–01101013. doi:10.1115/1.2745397
Li W, McMeeking RM, Landis CM (2008) On the crack face boundary conditions in electromechanical fracture and an experimental protocol for determining energy release rates. Eur J Mech A Solids 27: 285–301
Liu Y, Chen YH (2005) An analytical solution for a cracked piezoelectric plate according to the PKHS electric boundary condition. Acta Mech 180: 233–244. doi:10.1007/s00707-004-0103-4
Ma LF, Chen YH (2001) Weight functions for interface cracks in dissimilar anisotropic piezoelectric materials. Int J Fract 110: 263–279
McMeeking RM (1999) Crack tip energy release rate for a piezoelectric compact tension speciment. Eng Fract Mech 64: 217–244. doi:10.1016/S0013-7944(99)00068-5
McMeeking RM (2001) Towards a fracture mechanics for brittle piezoelectric and dielectric materials. Int J Fract 108: 25–41. doi:10.1023/A:1007652001977
McMeeking RM (2004) The energy release rate for a Griffith crack in a piezoelectric material. Eng Fract Mech 71: 1149–1163. doi:10.1016/S0013-7944(03)00135-8
Muskhelishvili NI (1953) Some basic problems of mathematical theory of elasticity. Noordhoff, Groningen
Nakhmein EL, Nuller BM (1986) Contact between an elastic half-plane and a partly separated stamp. J Appl Math Mech 50: 507–515. doi:10.1016/0021-8928(86)90017-1
Ou ZC, Chen YH (2005) On approach of crack tip energy release rate for a semi-permeable crack when electromechanical loads become very large. Int J Fract 133: 89–105. doi:10.1007/s10704-005-3123-8
Ou ZC, Chen YH (2007) Re-examination of the PKHS crack model in piezoelectric materials. Euro J Mech A Solids 26: 659–675. doi:10.1016/j.euromechsol.2006.09.007
Pak YE (1992) Linear electro-elastic fracture mechanics of piezoelectric materials. Int J Fract 54: 79–100. doi:10.1007/BF00040857
Park SB, Sun CT (1995) Fracture criteria for piezoelectric ceramics. J Am Ceram Soc 78: 1475–1480. doi:10.1111/j.1151-2916.1995.tb08840.x
Parton VZ (1976) Fracture mechanics of piezoelectric materials. Acta Astronaut 3: 671–683. doi:10.1016/0094-5765(76)90105-3
Parton VZ, Kudryavtsev BA (1988) Electromagnetoelasticity. Gordon and Breach Science Publishers, New York
Ricoeur A, Enderlein M, Kuna M (2005) Calculation of the J-integral for limited permeable cracks in piezoelectrics. Arch Appl Mech 74: 536–549. doi:10.1007/s00419-004-0370-5
Ricoeur A, Kuna M (2009) Electrostatic tractions at dielectric interfaces and their implication for crack boundary conditions. Mech Res Commun 36: 330–335
Rogowski B (2007) The limited electrically permeable crack model in linear piezoelasticity. Int J Press Vess Piping 84: 572–581. doi:10.1016/j.ijpvp.2007.04.006
Scherzer M, Kuna M (2004) Combined analytical and numerical solution of 2D interface corner configurations between dissimilar piezoelectric materials. Int J Fract 127: 61–99
Suo Z, Kuo CM, Barnett DM, Willis JR (1992) Fracture mechanics for piezoelectric ceramics. J Mech Phys Solids 40: 739–765. doi:10.1016/0022-5096(92)90002-J
Tian WY, Rajapakse RKND (2006) Fracture parameters of a penny-shaped crack at the interface of a piezoelectric bi-material system. Int J Fract 141: 37–48
Wang BL, Mai YW (2003) On the electrical boundary conditions on the crack surfaces in piezoelectric ceramics. Int J Eng Sci 41: 633–652. doi:10.1016/S0020-7225(02)00149-0
Williams ML (1959) The stresses around a fault or crack in dissimilar media. Bull Seismol Soc Am 49: 199–204
Wippler K, Ricoeur A, Kuna M (2004) Towards the computation of electrically permeable cracks in piezoelectrics. Eng Fract Mech 71: 2567–2587. doi:10.1016/j.engfracmech.2004.03.003
Xu XL, Rajapakse RKND (2001) On plane crack in piezoelectric solids. Int J Solids Struct 38: 7643–7658. doi:10.1016/S0020-7683(01)00029-4
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Govorukha, V., Kamlah, M. On contact zone models for an electrically limited permeable interface crack in a piezoelectric bimaterial. Int J Fract 164, 133–146 (2010). https://doi.org/10.1007/s10704-010-9465-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10704-010-9465-x