Abstract
Based on an interface deformable piezoelectric bi-layer beam model, a bonded piezoelectric bi-material beam with an interface crack perpendicular to the poling axis is analyzed within the framework of the theory of linear piezoelectricity. The layer-wise approximations of both the elastic displacements and electric potential are employed, and each sub-layer is modeled as a single linearly elastic Timoshenko beam perfectly bonded together through a deformable interface. Using the impermeable crack assumption, the closed form solutions for the energy release rate (ERR) and crack energy density (CED) are derived for the layered piezoelectric beam subjected to combined uniformly distributed electromechanical loading. Based on superposition principle, both the ERR and CED and their components are all reduced to the functions of the crack tip loading parameters. Loading dependence of the total CED with respect to the applied electric field is manifested with the analytical results, showing that there is a transformation from an even dependence to an odd dependence for the normalized CED when the applied mechanical loading increases. Compared with the commonly used equivalent single layer model, the proposed analysis augments the crack driving force by alleviating the stress concentration along the interface and thus increases the loading parameters at the crack tip. The proposed model provides improved solutions for fracture analysis of piezoelectric layered structures and sheds light on the loading dependence of the fracture parameters (i.e., the ERR and CED) with respect to the applied electromechanical loadings.
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References
Balke H, Drescher J, Kemmer G (1998) Investigation of mechanical strain energy release rate with respect to a fracture criterion for piezoelectric ceramics. Int J Fract 89: L59–L64
Chen Y-H, Hasebe N (2005) Current understanding on fracture behaviors of ferroelectric/piezoelectric materials. J Intell Mater Syst Struct 16: 673–687. doi:10.1177/1045389X05054330
Chen FL, and Qiao PZ, (2009) Electro-mechanical behavior of interface deformable piezoelectric bi-layer beams. ASCE J Eng Mech (accepted)
Du TB, Liu M, Seghi S, Hsia KJ, Economy J, Shang JK (2001) Piezoelectric actuation of crack growth along polymer-metal interfaces in adhesive bonds. J Mater Res 16: 2885–2892. doi:10.1557/JMR.2001.0397
Fang DN, Liu B, Hwang KC (2000) Energy analysis on fracture of ferroelectric ceramics. Int J Fract 100: 401–408. doi:10.1023/A:1018740911313
Fang D, Zhang Z-K, Soh AK, Lee KL (2004) Fracture criteria of piezoelectric ceramics with defects. Mech Mater 36: 917–928. doi:10.1016/j.mechmat.2003.08.011
Fulton CC, Gao H (2001) Effect of local polarization switching on piezoelectric fracture. J Mech Phys Solids 49: 927–952. doi:10.1016/S0022-5096(00)00049-1
Gao H, Zhang T-Y, Tong P (1997) Local and global energy release rates for an electrically yielded crack in a piezoelectric ceramic. J Mech Phys Solids 45: 491–510. doi:10.1016/S0022-5096(96)00108-1
Gao C-F, Zhao M, Tong P, Zhang T-Y (2004) The energy release rate and the J-integral of an electrically insulated crack in a piezoelectric material. Int J Eng Sci 42: 2175–2192. doi:10.1016/j.ijengsci.2004.08.007
Hao T-H, Shen Zi-Yuan (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)90243-7
Hao TH, Gong X, Suo Z (1996) Fracture mechanics for the design of ceramic multilayer actuators. J Mech Phys Solids 44: 23–48. doi:10.1016/0022-5096(95)00068-2
Jiang LZ, Sun CT (2001) Analysis of indentation cracking in piezoceramics. Int J Solids Struct 38: 1903–1918. doi:10.1016/S0020-7683(00)00142-6
Kumar S, Singh RN (1996) Crack propagation in piezoelectric materials under combined mechanical and electrical loadings. Acta Mater 44: 173–200
Li Q, Chen YH (2008) Solution for a semi-permeable interface crack between two dissimilar piezoelectric material. J Appl Mech 74: 833–844. doi:10.1115/1.2711232
Liu M, Hsia KJ (2003) Interfacial cracks between piezoelectric and elastic materials under in-plane electric loading. J Mech Phys Solids 51: 921–944. doi:10.1016/S0022-5096(02)00120-5
Liu M, Hsia KJ, Shang JK (2005) Driving forces for interfacial fatigue crack growth by piezoelectric actuator. J Intell Mater Syst Struct 16: 557–566. doi:10.1177/1045389X05051632
McMeeking RM (1999) Crack tip energy release rate for a piezoelectric compact tension specimen. 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
Nam B-G, Watanabe K (2007) Crack energy density and energy release rate for piezoelectric material. Int J Solids Struct 44: 3904–3919. doi:10.1016/j.ijsolstr.2006.10.029
Nam B-G, Watanabe K (2008) Effect of electric boundary conditions on crack energy density and its derivatives for piezoelectric material. Eng Fract Mech 75: 207–222. doi:10.1016/j.engfracmech.2007.03.022
Nam B-G, Liu R, Tsuchida S, Watanabe K (2007) Applicability of crack energy density to fracture strength evaluation of piezoelectric ceramics. Mater Sci Eng A 449(−451): 343–347. doi:10.1016/j.msea.2005.12.104
Ou ZC, Chen Y-H (2004) Interface crack problem in elastic dielectric/piezoelectric bimaterials. Int J Fract 130: 427–454. doi:10.1023/B:FRAC.0000049502.54417.1c
Ou ZC, Wu X (2003) On the crack-tip stress singularity of interfacial cracks in transversely isotropic piezoelectric bimaterials. Int J Solids Struct 40: 7499–7511. doi:10.1016/j.ijsolstr.2003.08.021
Pak YE (1990) Crack extension force in a piezoelectric material. ASME J Appl Mech 57: 647–653
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
Qiao PZ, Chen FL (2008) An improved adhesively bonded bi-material beam model for plated beams. Eng Struct 30: 1949–1957. doi:10.1016/j.engstruct.2007.12.017
Qiao P, Wang J (2004) Mechanics and fracture of crack tip deformable bi-material interface. Int J Solids Struct 41: 7423–7444. doi:10.1016/j.ijsolstr.2004.06.006
Qiao PZ, Wang J (2005) Novel joint deformation models and their application to delamination fracture analysis. Compos Sci Technol 65: 1826–1939. doi:10.1016/j.compscitech.2005.03.014
Ru CQ (2000) Exact solution for finite electrode layers embedded at the interface of two piezoelectric half-planes. J Mech Phys Solids 48: 693–708. doi:10.1016/S0022-5096(99)00056-3
Ru CQ, Mao X, Epstein M (1998) Electric-field induced interfacial cracking in multilayer electrostrictive actuators. J Mech Phys Solids 46: 1301–1318. doi:10.1016/S0022-5096(98)00038-6
Shen S, Nishioka T, Hu SL (2000) Crack propagation along the interface of piezoelectric bimaterial. Theor Appl Fract Mech 34: 185–203. doi:10.1016/S0167-8442(00)00035-5
Shindo Y, Yoshida M, Narita F, Horiguchi K (2004) Electroelastic field concentrations ahead of electrodes in multilayer piezoelectric actuators: experiment and finite element simulation. J Mech Phys Solids 52: 1109–1124. doi:10.1016/j.jmps.2003.09.017
Sosa H (1992) On the fracture mechanics of piezoelectric solids. Int J Solids Struct 29: 2613–2622. doi:10.1016/0020-7683(92)90225-I
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
Wang TC, Han XL (1999) Fracture mechanics of piezoelectric materials. Int J Fract 98: 15–35. doi:10.1023/A:1018656606554
Wang J, Qiao P (2004a) Interface crack between two shear deformable elastic layers. J Mech Phys Solids 52: 891–905. doi:10.1016/S0022-5096(03)00121-2
Wang J, Qiao PZ (2004b) Novel beam analysis of end-notched flexure specimen for mode-II fracture. Eng Fract Mech 71: 219–231. doi:10.1016/S0013-7944(03)00096-1
Wang J, Qiao PZ (2005) Analysis of beam-type fracture specimens with crack-tip deformation. Int J Fract 132: 223–248. doi:10.1007/s10704-005-2181-2
Wang H, Singh RN (1997) Crack propagation in piezoelectric ceramics: effect of applied electric fields. J Appl Phys 81: 7471–7479. doi:10.1063/1.365290
Yang W, Suo Z (1994) Cracking in ceramic actuators caused by electrostriction. J Mech Phys Solids 42: 649–663. doi:10.1016/0022-5096(94)90056-6
Yang W, Zhu T (1998) Switch-toughnessing of ferroelectrics subjected to electric fields. J Mech Phys Solids 46: 291–311. doi:10.1016/S0022-5096(97)00062-8
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Qiao, P., Chen, F. Interface crack between two interface deformable piezoelectric layers. Int J Fract 156, 185–201 (2009). https://doi.org/10.1007/s10704-009-9359-y
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DOI: https://doi.org/10.1007/s10704-009-9359-y