Acta Mechanica Solida Sinica

, Volume 23, Issue 4, pp 336–352

# Multiple Parallel Symmetric Permeable Model-III Cracks in a Piezoelectric/Piezomagnetic Composite Material Plane

Article

## Abstract

In this paper, the interactions of multiple parallel symmetric and permeable finite length cracks in a piezoelectric/piezomagnetic material plane subjected to anti-plane shear stress loading are studied by the Schmidt method. The problem is formulated through Fourier transform into dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. To solve the dual integral equations, the displacement jumps across the crack surfaces are directly expanded as a series of Jacobi polynomials. Finally, the relation between the electric field, the magnetic flux field and the stress field near the crack tips is obtained. The results show that the stress, the electric displacement and the magnetic flux intensity factors at the crack tips depend on the length and spacing of the cracks. It is also revealed that the crack shielding effect presents in piezoelectric/piezomagnetic materials.

## Key words

piezoelectric/piezomagnetic composites multiple parallel symmetric cracks crack shielding effect mechanics of solids

## References

1. [1]
Wu, T.L. and Huang, J.H., Closed-form solutions for the magnetoelectric coupling coefficients in fibrous composites with piezoelectric and piezomagnetic phases. International Journal of Solids and Structures, 2000, 37(21): 2981–3009.
2. [2]
Song, Z.F. and Sih, G.C., Crack initiation behavior in magnetoelectroelastic composite under in-plane deformation. Theoretical and Applied Fracture Mechanics, 2003, 39(3): 189–207.
3. [3]
Sih, G.C. and Song, Z.F., Magnetic and electric poling effects associated with crack growth in BaTiO3-CoFe2O4 composite. Theoretical and Applied Fracture Mechanics, 2003, 39(3): 209–227.
4. [4]
Wang, B.L. and Mai, Y.W., Crack tip field in piezoelectric/piezomagnetic media. European Journal of Mechanics A/Solid, 2003, 22(3): 591–602.
5. [5]
Gao, C.F., Kessler, H. and Balke, H., Crack problems in magnetoelectroelastic solids. Part I: exact solution of a crack. International Journal of Engineering Science, 2003, 41(8): 969–981.
6. [6]
Gao, C.F., Kessler, H. and Balke, H., Crack problems in magnetoelectroelastic solids. Part II: general solution of collinear cracks. International Journal of Engineering Science, 2003, 41(8): 983–994.
7. [7]
Gao, C.F., Tong, P. and Zhang, T.Y., Interfacial crack problems in magneto-electroelastic solids. International Journal of Engineering Science, 2003, 41(18): 2105–2121.
8. [8]
Spyropoulos, C.P., Sih, G.C. and Song, Z.F., Magnetoelectroelastic composite with poling parallel to plane of line crack under out-of-plane deformation. Theoretical and Applied Fracture Mechanics, 2003, 39(3): 281–289.
9. [9]
Van Suchtelen, J., Product properties: a new application of composite materials. Phillips Research Reports, 1972, 27(1): 28–37.Google Scholar
10. [10]
Liu, J.X., Liu, X.L. and Zhao, Y.B., Green’s functions for anisotropic magnetoelectroelastic solids with an elliptical cavity or a crack. International Journal of Engineering Science, 2001, 39(11): 1405–1418.
11. [11]
Wang, B.L. and Mai, Y.W., Fracture of piezoelectromagnetic materials. Mechanics Research Communications, 2004, 31(1): 65–73.
12. [12]
Harshe, G. and Dougherty, J.P., Newnham RE, Theoretical modeling of 3-0/0-3 magnetoelectric composites. International Journal of Applied Electromagnetics in Materials, 1993, 4(2): 161–171.Google Scholar
13. [13]
Avellaneda, M. and Harshe, G., Magnetoelectric effect in piezoelectric/magnetostrictive multiplayer (2-2) composites. Journal of Intelligent material Systems and Structures, 1994, 5(2): 501–513.
14. [14]
Nan, C.W., Magnetoelectric effect in composites of piezoelectric and piezomagnetic phases. Physical Review B, 1994, 50(7): 6082–6088.
15. [15]
Benveniste, Y., Magnetoelectric effect in fibrous composites with piezoelectric and magnetostrictive phases. Physical Review B, 1995, 51(9): 16424–16427.
16. [16]
Huang, J.H. and Kuo, W.S., Analysis of piezoelectric/piezomagnetic composite materials containing ellipsoidal inclusions. Journal of Applied Physics, 1997, 81(3): 1378–1386.
17. [17]
Li, J.Y., Magnetoelectroelastic multi-inclusion and inhomogeneity problems and their applications in composite materials. International Journal of Engineering Science, 2000, 38(18): 1993–2011.
18. [18]
Zhang, P.W., Zhou, Z.G. and Wu, L.Z., Basic solutions of two parallel Mode-I permeable cracks or four parallel Mode-I permeable cracks in magnetoelectroelastic composite materials. Philosophical Magazine, 2007, 87(22): 1–34.
19. [19]
Zhou, Z.G. and Wang, B., Two parallel symmetry permeable cracks in functionally graded piezoelectric/piezomagnetic materials under anti-plane shear loading. International Journal of Solids and Structures, 2004, 41(16–17): 4407–4422.
20. [20]
Zhou, Z.G. and Wang, B., The scattering of the harmonic anti-plane shear stress waves by two collinear interface cracks between two dissimilar functionally graded piezoelectric/piezomagnetic material half infinite planes. Proceedings of the Institution of Mechanical Engineers, Part C, Journal of Mechanical Engineering Science, 2006, 220(2): 137–148.
21. [21]
Zhou, Z.G., Wu, L.Z. and Wang, B., The dynamic behavior of two collinear interface cracks in magneto-electro-elastic composites. European Journal of Mechanics A/Solids, 2005, 24(2): 253–262.
22. [22]
Wang, B.L. and Han, J.C., Multiple cracking of magnetoelectroelastic materials in coupling thermo-electro-magneto-mechanical loading environments. Computational Materials Science, 2007, 39(2): 291–304.
23. [23]
Tian, W.Y. and Gabbert, U., Multiple crack interaction problem in magnetoelectroelastic solids. European Journal of Mechanics A/Solid, 2004, 23(3): 599–614.
24. [24]
Morse, P.M. and Feshbach, H., Methods of Theoretical Physics. Vol.1, McGraw-Hill, New York, 1958, 926.Google Scholar
25. [25]
Soh, A.K., Fang, D.N. and Lee, K.L., Analysis of a bi-piezoelectric ceramic layer with an interfacial crack subjected to anti-plane shear and in-plane electric loading. European Journal of Mechanics A/Solid, 2000, 19(5): 961–977.
26. [26]
Gradshteyn, I.S. and Ryzhik, I.M., Table of Integrals, Series and Products, Academic Press, New York, 1980, 1035–1037.
27. [27]
Erdelyi, A., Tables of Integral Transforms, Vol.1, McGraw-Hill, New York, 1954, 34–89.Google Scholar
28. [28]
Itou, S., Three dimensional waves propagation in a cracked elastic solid. Journal of Applied Mechanics, 1978, 45, 807–811.
29. [29]
Zhou, Z.G., Du, S.Y. and Wang, B., Dynamic behavior of two parallel symmetric permeable cracks in a piezoelectric material strip. Acta Mechanica Solida Sinica, 2002, 15(3): 294–302.Google Scholar
30. [30]
Zhou, Z.G., Du, S.Y. and Wang, B., Analysis of the dynamic behavior a Griffith permeable crack in piezoelectric materials use of non-local theory. Acta Mechanica Solida Sinica, 2003, 16(1): 52–60.Google Scholar
31. [31]
Tong, Z.H., Jiang, C.P., Lo, S.H. and Cheung, Y.K., A closed form solution to the antiplane problem of doubly periodic cracks of unequal size in piezoelectric materials. Mechanics of Materials, 2006, 38(3): 269–286.
32. [32]
Ratwani, M. and Gupta, G.D., Interaction between parallel cracks in layered composites. International Journal of Solids and Structures, 1974, 10(6): 701–708.