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Periodic limited permeable cracks in magneto-electro-elastic media

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Abstract

A plane strain problem for a periodic set of the limited electrically and magnetically permeable cracks is considered in this work. The tensile mechanical stress, magnetic induction and electrical displacement are applied at infinity. Using the presentations of electro-magneto-mechanical components via sectionally analytic functions, the formulated problem is reduced to a matrix problem of linear relationships with the associated conditions at infinity. The exact analytical solution of this problem is found. A system of two transcendental equations for the determination of the electric displacement and magnetic induction in the crack regions is found. The solution of this system can be easily found by standard procedures. Formulae for the stresses, magnetic induction and electric displacement vectors, elastic displacements, magnetic and electric potential jumps at the interface as well as the intensity factors at the crack tips are presented in the form of simple analytical formulas. As a limiting case, a single limited permeable crack in magneto-electro-elastic media is studied as well. A comparison with the associated results for a periodic set of limited electrically and magnetically permeable cracks is performed.

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Abbreviations

ERR:

Energy release rate

IF:

Intensity factor

SIF:

Stress intensity factor

V f :

The volume fraction of piezoelectric BaTiO3 in BaTiO3–CoFe2O4 composite

References

  1. Parton V.Z., Kudryavtsev B.A.: Electromagnetoelasticity. Gordon and Breach Science Publishers, New York (1988)

    Google Scholar 

  2. Hao T.H., Shen Z.Y.: A new electric boundary condition of electric fracture mechanics and its applications. Fract. Mech 47, 793–802 (1994). doi:10.1016/0013-7944(94)90059-0

    Article  Google Scholar 

  3. McMeeking, R.: Crack tip energy release rate for a piezoelectric compact tension specimen. Eng Fract Mech (1999). doi:10.1016/S0013-7944(99)00068-5

  4. Gruebner O., Kamlah M., Munz D.: Finite element analysis of cracks in piezoelectric materials taking into account the permittivity of the crack medium. Eng. Fract. Mech. 70, 1399–1413 (2003). doi:10.1016/S0013-7944(02)00117-0

    Article  Google Scholar 

  5. Wang B.-L., Mai Y.-W.: On the electrical boundary conditions on the crack surfaces in piezoelectric ceramics. Int. J. Eng. Sci. 41, 633–652 (2003). doi:10.1016/S0020-7225(02)00149-0

    Article  Google Scholar 

  6. Lapusta Y., Loboda V.: Electro-mechanical yielding for a limited permeable crack in an interlayer between piezoelectric materials. Mech. Res. Commun. 36, 183–192 (2009). doi:10.1016/j.mechrescom.2008.09.001

    Article  MATH  MathSciNet  Google Scholar 

  7. Loboda V., Lapusta Y., Sheveleva A.: Limited permeable crack in an interlayer between piezoelectric materials with different zones of electrical saturation and mechanical yielding. Int. J. Solids Struct. 47, 1795–1806 (2010). doi:10.1016/j.ijsolstr.2010.03.015

    Article  MATH  Google Scholar 

  8. Govorukha V.B., Loboda V.V., Kamlah M.: On the influence of the electric permeability on an interface crack in a piezoelectric bimaterial compound. Int. J. Solids Struct. 43, 1979–1990 (2006). doi:10.1016/j.ijsolstr.2005.04.009

    Article  MATH  Google Scholar 

  9. Li Q., Chen Y.: Solution for a semi-permeable interface crack in elastic dielectric/piezoelectric bimaterials. ASME J. Appl. Mech. 75, 011010-1-13 (2008). doi:10.1115/1.2745397

    Google Scholar 

  10. Gorbatikh L., Kachanov M.: A simple technique for constructing the full stress and displacement fields in elastic plates with multiple cracks. Eng. Fract. Mech. 66, 51–63 (2000)

    Article  Google Scholar 

  11. Lapusta Y., Henaff-Gardin C.: An analytical model for periodic α°-layer cracking in composite laminates. Int. J. Fract. 102, L73–76 (2000)

    Article  Google Scholar 

  12. Lapusta Y., Wagner W.: On various material and fibre-matrix interface models in the near-surface instability problems for fibrous composites. Compos. Part A Appl. Sci. Manuf. 32, 413–423 (2001)

    Article  Google Scholar 

  13. Yasniy P., Marushchak P., Lapusta Y.: Experimental study of crack growth in a bimetal under fatigue and fatigue-creep conditions. Int. J. Fract. 139(3-4), 545–552 (2006)

    Article  MATH  Google Scholar 

  14. Kudryavtsev, B., Rakitin, V.: Periodic set of cracks in the boundary of a piezoelectric and a rigid conductor. Isv. AN SSSR. Mechanika Tvordogo Tela No 2, 121–129 (1976) (in Russian)

  15. Gao C.-F., Kessler H., Balke H.: Crack problems in magnetoelectroelastic solids. Part II: general solution of collinear cracks. Int. J. Eng. Sci. 41, 969–981 (2003). doi:10.1016/S0020-7225(02)00324-5

    Article  MATH  MathSciNet  Google Scholar 

  16. Loboda, V., Kozinov, S.: Periodic set of the interface cracks with limited electric permeability. IUTAM Bookseries. IUTAM Symp. Multiscale Model. Fatigue Damage Fract. Smart Mater. 24, 175–187 (2011)

    Article  Google Scholar 

  17. Kozinov S., Loboda V., Lapusta Y.: Periodic set of limited electrically permeable interface cracks with contact zones. Mech. Res. Commun. 48, 32–41 (2013). doi:10.1016/j.mechrescom.2012.12.002

    Article  Google Scholar 

  18. Yue Y., Wan Y.: Multilayered piezomagnetic/piezoelectric composite with periodic interfacial Yoffe-type cracks under magnetic or electric field. Acta Mech. 225(7), 2133–2150 (2014). doi:10.1007/s00707-013-1032-x

    Article  MATH  Google Scholar 

  19. Zhou Z.-G., Wang B., Sun Y.-G.: Two collinear interface cracks in magneto-electro-elastic composites. Int. J. Eng. Sci. 42, 1155–1167 (2004). doi:10.1016/j.ijengsci.2004.01.005

    Article  MATH  Google Scholar 

  20. Wang B.-L., Mai Y.-W.: Applicability of the crack-face electromagnetic boundary conditions for fracture of magnetoelectroelastic materials. Int. J. Solids Struct. 44, 387–398 (2007). doi:10.1016/j.ijsolstr.2006.04.028

    Article  MATH  Google Scholar 

  21. Zhong X.-C.: Closed-form solutions for two collinear dielectric cracks in a magnetoelectroelastic solid. Appl. Math. Model. 35, 2930–2944 (2011). doi:10.1016/j.apm.2010.12.010

    Article  MATH  MathSciNet  Google Scholar 

  22. Sih G.C., Song Z.F.: Magnetic and electric poling effects associated with crack growth in BaTiO3–CoFe2O4 composite. Theor. Appl. Fract. Mech. 39, 209–227 (2003). doi:10.1016/S0167-8442(03)00003-X

    Article  Google Scholar 

  23. Herrmann K.P., Loboda V.V., Khodanen T.V.: An interface crack with contact zones in a piezoelectric/piezomagnetic bimaterial. Arch. Appl. Mech. 80(6), 651–670 (2010). doi:10.1016/j.ijsolstr.2005.02.003

    Article  MATH  Google Scholar 

  24. Gao C.F., Tong P., Zhang T.Y.: Interfacial crack problems in magneto-electroelastic solids. Int. J. Eng. Sci. 41, 2105–2121 (2003). doi:10.1016/S0020-7225(03)00206-4

    Article  Google Scholar 

  25. Gakhov F.: Boundary Value Problems. Pergamon Press, Oxford (1966)

    MATH  Google Scholar 

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Viun, O., Labesse-Jied, F., Moutou-Pitti, R. et al. Periodic limited permeable cracks in magneto-electro-elastic media. Acta Mech 226, 2225–2233 (2015). https://doi.org/10.1007/s00707-014-1296-9

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  • DOI: https://doi.org/10.1007/s00707-014-1296-9

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