Abstract
This paper investigates the influence of crack geometry, crack-face and loading conditions, and the permittivity of a medium inside the crack gap on intensity factors of planar and non-planar cracks in linear piezoelectric media. A weakly singular boundary integral equation method together with the near-front approximation is adopted to accurately determine the intensity factors. Obtained results indicate that the non-flat crack surface, the electric field, and the permittivity of a medium inside the crack gap play a crucial role on the behavior of intensity factors. The mode-I stress intensity factors (KI) for two representative non-planar cracks under different crack-face conditions are found significantly different and they possess both upper and lower bounds. In addition, KI for impermeable and semi-permeable non-planar cracks treated depends strongly on the electric field whereas those of impermeable, permeable, and semi-permeable penny-shaped cracks are identical and independent of the electric field. The stress/electric intensity factors predicted by permeable and energetically consistent models are, respectively, independent of and dependent on the electric field for the penny-shaped crack and the two representative non-planar cracks. Also, the permittivity of a medium inside the crack gap strongly affects the intensity factors for all crack configurations considered except for KI of the semi-permeable penny-shaped crack.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Kuna M. Fracture mechanics of piezoelectric materials—where are we right now? Engineering Fracture Mechanics, 2010, 77(2): 309–326
Sladek J, Sladek V, Wünsche M, Zhang C. Effects of electric field and strain gradients on cracks in piezoelectric solids. European Journal of Mechanics. A, Solids, 2018, 71: 187–198
Sladek J, Sladek V, Stanak P, Zhang C, Tan C L. Fracture mechanics analysis of size-dependent piezoelectric solids. International Journal of Solids and Structures, 2017, 113–114: 1–9
Ghasemi H, Park H S, Rabczuk T. A multi-material level set-based topology optimization of flexoelectric composites. Computer Methods in Applied Mechanics and Engineering, 2018, 332: 47–62
Hamdia K M, Ghasemi H, Zhuang X, Alajlan N, Rabczuk T. Sensitivity and uncertainly analysis for flexoelectric nanostructures. Computer Methods in Applied Mechanics and Engineering, 2018, 337: 95–109
Thai T Q, Rabczuk T, Zhuang X. A large deformation isogeometric approach for flexoelectricity and soft materials. Computer Methods in Applied Mechanics and Engineering, 2018, 341: 718–739
Nguyen B H, Nanthakumar S S, Zhuang X, Wriggers P, Jiang X, Rabczuk T. Dynamic flexoelectric effect on piezoelectric nanostructures. European Journal of Mechanics. A, Solids, 2018, 71: 404–409
Ghasemi H, Park H S, Rabczuk T. A level-set based IGA formulation for topology optimization of flexoelectric materials. Computer Methods in Applied Mechanics and Engineering, 2017, 313: 239–258
Nanthakumar S S, Lahmer T, Zhuang X, Zi G, Rabczuk T. Detection of material interfaces using a regularized level set method in piezoelectric structures. Inverse Problems in Science and Engineering, 2016, 24(1): 153–176
Mishra R K. A review on fracture mechanics in piezoelectric structures. In: Proceedings of Materials Today. Amsterdam: Elsevier, 2018, 5407–5413
Parton V Z. Fracture mechanics of piezoelectric materials. Acta Astronautica, 1976, 3(9–10): 671–683
Deeg W F. The analysis of dislocation, crack and inclusion problems in piezoelectric solids. Dissertation for the Doctoral Degree. Palo Alto: Standford University, 1980
Hao T H, Shen Z Y. A new electric boundary condition of electric fracture mechanics and its applications. Engineering Fracture Mechanics, 1994, 47(6): 793–802
Landis C M. Energetically consistent boundary conditions for electromechanical fracture. International Journal of Solids and Structures, 2004, 41(22–23): 6291–6315
Rungamornrat J, Phongtinnaboot W, Wijeyewickrema A C. Analysis of cracks in 3D piezoelectric media with various electrical boundary conditions. International Journal of Fracture, 2015, 192(2): 133–153
Park S B, Sun C T. Effect of electric field on fracture of piezoelectric ceramics. International Journal of Fracture, 1993, 70(3): 203–216
Pan E. A BEM analysis of fracture mechanics in 2D anisotropic piezoelectric solids. Engineering Analysis with Boundary Elements, 1999, 23(1): 67–76
Chen W Q, Shioya T. Fundamental solution for a penny-shaped crack in a piezoelectric medium. Journal of the Mechanics and Physics of Solids, 1999, 47(7): 1459–1475
Chen W Q, Shioya T. Complete and exact solutions of a penny-shaped crack in a piezoelectric solid: Antisymmetric shear loadings. International Journal of Solids and Structures, 2000, 37(18): 2603–2619
Chen W Q, Shioya T, Ding H J. A penny-shaped crack in piezoelectrics: Resolved. International Journal of Fracture, 2000, 105(1): 49–56
Xu X L, Rajapakse R K N D. A theoretical study of branched cracks in piezoelectrics. Acta Materialia, 2000, 48(8): 1865–1882
Davi G, Milazzo A. Multidomain boundary integral formulation for piezoelectric materials fracture mechanics. International Journal of Solids and Structures, 2001, 38(40–41): 7065–7078
Hou P F, Ding H J, Guan F L. Point forces and point charge applied to a circular crack in a transversely isotropic piezoelectric space. Theoretical and Applied Fracture Mechanics, 2001, 36(3): 245–262
Rajapakse R K N D, Xu X L. Boundary element modeling of cracks in piezoelectric solids. Engineering Analysis with Boundary Elements, 2001, 25(9): 771–781
Xu X L, Rajapakse R K N D. On a plane crack in piezoelectric solids. International Journal of Solids and Structures, 2001, 38(42–43): 7643–7658
Wang X D, Jiang L Y. Fracture behaviour of cracks in piezoelectric media with electromechanically coupled boundary conditions. Proceeding of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2002, 458(2026): 2545–2560
Huang Z, Kuang Z B. A mixed electric boundary value problem for a two-dimensional piezoelectric crack. International Journal of Solids and Structures, 2003, 40(6): 1433–1453
Wang B L, Mai Y W. On the electrical boundary conditions on the crack surfaces in piezoelectric ceramics. International Journal of Engineering Science, 2003, 41(6): 633–652
Chen M C. Application of finite-part integrals to three-dimensional fracture problems for piezoelectric media Part I: Hypersingular integral equation and theoretical analysis. International Journal of Fracture, 2003, 121(3–4): 133–148
Chen M C. Application of finite-part integrals to three-dimensional fracture problems for piezoelectric media Part II: Numerical analysis. International Journal of Fracture, 2003, 121(3–4): 149–161
Wang X D, Jiang L Y. The nonlinear fracture behaviour of an arbitrarily oriented dielectric crack in piezoelectric materials. Acta Mechanica, 2004, 172(3–4): 195–210
Li X F, Lee K Y. Three-dimensional electroelastic analysis of a piezoelectric material with a penny-shaped dielectric crack. ASME Journal of Applied Mechanics, 2004, 71(6): 866–878
Chen W Q, Lim C W. 3D point force solution for a permeable penny-shaped crack embedded in an infinite transversely isotropic piezoelectric medium. International Journal of Fracture, 2005, 131(3): 231–246
Groh U, Kuna M. Efficient boundary element analysis of cracks in 2D piezoelectric structures. International Journal of Solids and Structures, 2005, 42(8): 2399–2416
Chiang C R, Weng G J. Nonlinear behavior and critical state of a penny-shaped dielectric crack in a piezoelectric solid. ASME Journal of Applied Mechanics, 2007, 74(5): 852–860
Ou Z C, Chen Y H. Re-examination of the PKHS crack model in piezoelectric materials. European Journal of Mechanics. A, Solids, 2007, 26(4): 659–675
Qin T Y, Yu Y S, Noda N A. Finite-part integral and boundary element method to solve three-dimensional crack problems in piezoelectric materials. International Journal of Solids and Structures, 2007, 44(14–15): 4770–4783
Wippler K, Kuna M. Crack analyses in three-dimensional piezoelectric structures by the BEM. Computational Materials Science, 2007, 39(1): 261–266
Li Q, Ricoeur A, Kuna M. Coulomb traction on a penny-shaped crack in a three-dimensional piezoelectric body. Archive of Applied Mechanics, 2011, 81(6): 685–700
Lei J, Wang H, Zhang C, Bui T, Garcia-Sanchez F. Comparison of several BEM-based approaches in evaluating crack-tip field intensity factors in piezoelectric materials. International Journal of Fracture, 2014, 189(1): 111–120
Lei J, Zhang C, Garcia-Sanchez F. BEM analysis of electrically limited permeable cracks considering Coulomb tractions in piezoelectric materials. Engineering Analysis with Boundary Elements, 2015, 54: 28–38
Lei J, Yun L, Zhang C. An interaction integral and a modified crack closure integral for evaluating piezoelectric crack-tip fracture parameters in BEM. Engineering Analysis with Boundary Elements, 2017, 79: 88–97
Lei J, Zhang C. A simplified evaluation of the mechanical energy release rate of kinked cracks in piezoelectric materials using the boundary element method. Engineering Fracture Mechanics, 2018, 188: 36–57
Xu C H, Zhou Z H, Leung A Y T, Xu X S, Luo X W. The finite element discretized symplectic method for direct computation of SIF of piezoelectric materials. Engineering Fracture Mechanics, 2016, 162: 21–37
Hao T. Multiple collinear cracks in a piezoelectric material. International Journal of Solids and Structures, 2001, 38(50–51): 9201–9208
Denda M, Mansukh M. Upper and lower bounds analysis of electric induction intensity factors for multiple piezoelectric cracks by the BEM. Engineering Analysis with Boundary Elements, 2005, 29(6): 533–550
Sanz J A, Ariza M P, Dominguez J. Three-dimensional BEM for piezoelectric fracture analysis. Engineering Analysis with Boundary Elements, 2005, 29(6): 586–596
Rungamornrat J, Mear M E. Analysis of fractures in 3D piezoelectric media by a weakly singular integral equation method. International Journal of Fracture, 2008, 151(1): 1–27
Solis M, Sanz J A, Ariza M P, Dominguez J. Analysis of cracked piezoelectric solids by a mixed three-dimensional BE approach. Engineering Analysis with Boundary Elements, 2009, 33(3): 271–282
Li Q, Chen Y H. Why traction-free? Piezoelectric crack and coulombic traction. Archive of Applied Mechanics, 2008, 78(7): 559–573
Motola Y, Banks-Sills L. M-integral for calculating intensity factors of cracked piezoelectric materials using the exact boundary conditions. ASME Journal of Applied Mechanics, 2008, 76(1): 011004
Phongtinnaboot W, Rungamornrat J, Chintanapakdee C. Modeling of cracks in 3D piezoelectric finite media by weakly singular SGBEM. Engineering Analysis with Boundary Elements, 2011, 35(3): 319–329
Martin P A, Rizzo F J. Hypersingular integrals: How smooth must the density be? International Journal for Numerical Methods in Engineering, 1996, 39(4): 687–704
Pan E. A BEM analysis of fracture mechanics in 2D anisotropic piezoelectric solids. Engineering Analysis with Boundary Elements, 1999, 23(1): 67–76
Chen M C. Application of finite-part integrals to three-dimensional fracture problems for piezoelectric media Part I: Hypersingular integral equation and theoretical analysis. International Journal of Fracture, 2003, 121(3–4): 133–148
Chen M C. Application of finite-part integrals to three-dimensional fracture problems for piezoelectric media Part II: numerical analysis. International Journal of Fracture, 2003, 121(3–4): 149–161
Qin T Y, Noda N A. Application of hypersingular integral equation method to a three-dimensional crack in piezoelectric materials. JSME International Journal. Series A, Solid Mechanics and Material Engineering, 2004, 47(2): 173–180
Li S, Mear M E, Xiao L. Symmetric weak-form integral equation method for three-dimensional fracture analysis. Computer Methods in Applied Mechanics and Engineering, 1998, 151(3–4): 435–459
Acknowledgements
The authors gratefully acknowledge the financial support provided by Thailand Research Fund (Grant Nos. TRG5880100 and RSA5980032).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rungamornrat, J., Chansavang, B., Phongtinnaboot, W. et al. Investigation of Generalized SIFs of cracks in 3D piezoelectric media under various crack-face conditions. Front. Struct. Civ. Eng. 14, 280–298 (2020). https://doi.org/10.1007/s11709-019-0586-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11709-019-0586-7