Acta Mechanica Solida Sinica

, Volume 22, Issue 1, pp 53–63

# Exact solution for orthotropic materials weakened by doubly periodic cracks of unequal size under antiplane shear

Article

## Abstract

Orthotropic materials weakened by a doubly periodic array of cracks under far-field antiplane shear are investigated, where the fundamental cell contains four cracks of unequal size. By applying the mapping technique, the elliptical function theory and the theory of analytical function boundary value problems, a closed form solution of the whole-field stress is obtained. The exact formulae for the stress intensity factor at the crack tip and the effective antiplane shear modulus of the cracked orthotropic material are derived. A comparison with the finite element method shows the efficiency and accuracy of the present method. Several illustrative examples are provided, and an interesting phenomenon is observed, that is, the stress intensity factor and the dimensionless effective modulus are independent of the material property for a doubly periodic cracked isotropic material, but depend strongly on the material property for the doubly periodic cracked orthotropic material. Such a phenomenon for antiplane problems is similar to that for in-plane problems. The present solution can provide benchmark results for other numerical and approximate methods.

## Key words

orthotropic material a doubly periodic array of cracks antiplane shear boundary value problem stress intensity factor effective modulus

## References

1. [1]
Sih, G.C. and Chen, E.P., Cracks in Composite Materials: A Compilation of Stress Solutions for Composite Systems with Cracks. The Hague: Martinus Nijhoff Publishers, 1981.
2. [2]
Liu, C.D., Analytical solution for orthotropic composite plate containing a mode I crack along principle axis. International Journal of Fracture, 1996, 76(1): 21–38.
3. [3]
Liu, C.D., Analytical stress around mode II crack parallel to principle axis in orthotropic composite plate. Engineering Fracture Mechanics, 1996, 54(6): 791–803.
4. [4]
Federici, L., Piva, A. and Viola, E., Crack edge displacement and elastic constant determination for an orthotropic material. Theoretical and Applied Fracture Mechanics, 1999, 31(3): 173–187.
5. [5]
Charalambides, P.G. and Zhang, W.B., An energy method for calculating the stress intensities in orthotropic bimaterial fracture. International Journal of Fracture, 1996, 76(2): 97–120.
6. [6]
Ozturk, M. and Erdogan, F., Mode I crack problem in an inhomogeneous orthotropic medium. International Journal of Engineering Science, 1997, 35(9): 869–883.
7. [7]
Zhao, C. and Huang, P.Y., Fracture criterion of I-II mixed mode crack for orthotropic plate. Acta Mechanica Solida Sinica, 2002, 23(4): 426–430 (in Chinese).Google Scholar
8. [8]
Nobile, L., Piva, A. and Viola, E., On the inclined crack problem in an orthotropic medium under biaxial loading. Engineering Fracture Mechanics, 2004, 71(4-6): 529–546.
9. [9]
Gruescu, C., Monchiet, V. and Kondo, D., Eshelby tensor for a crack in an orthotropic elastic medium. Comptes Rendus Mecanique, 2005, 333(6): 467–473.
10. [10]
Zhou, Z.G. and Wang, B., Investigation of the behavior of a Griffith crack at the interface between two dissimilar orthotropic elastic half-planes for the opening crack mode. Applied Mathematics and Mechanics (English Edition), 2004, 25(7): 730–740.
11. [11]
Qian, W. and Sun, C.T., Methods for calculating stress intensity factors for interfacial cracks between two orthotropic solids. International Journal of Solids and Structures, 1998, 35(25): 3317–3330.
12. [12]
Chen, D.H., Stress intensity factor for a crack normal to an interface between two orthotropic materials. International Journal of Fracture, 1997, 88(1): 19–39.
13. [13]
Faal, R.T. and Fariborz, S.J., Stress analysis of orthotropic planes weakened by cracks. Applied Mathematical Modelling, 2007, 31(6): 1133–1148.
14. [14]
Mukherjee, S. and Das, S., Interaction of three interfacial Griffith cracks between bonded dissimilar orthotropic half planes. International Journal of Solids and Structures, 2007, 44(17): 5437–5446.
15. [15]
Chandra, A., Hu, K.X. and Huang, Y., A hybrid BEM formulation for multiple cracks in orthotropic elastic components. Computers and Structures, 1995, 56(5): 785–797.
16. [16]
Delameter, W.R., Herrman, G. and Barnett, D.M., Weakening of an elastic solid by a rectangular array of cracks. Journal of Applied Mechanics, 1975, 42(1): 74–80.
17. [17]
Delameter, W.R., Herrmann, G. and Barnett, D.M., Erratum on ‘Weakening of an elastic solid by a rectangular array of cracks’. Journal of Applied Mechanics, 1977, 44: 190.
18. [18]
Hori, H. and Sahasakmontri, K., Mechanical properties of cracked solids: validity of the selfconsistent method. In: Micromechanics and Inhomogeneity, New York: Springer-Verlag, 1990, 137–159.
19. [19]
Karihaloo, B.L. and Wang, J., On the solution of doubly periodic array of cracks. Mechanics of Materials, 1997, 26(4): 209–212.
20. [20]
Wang, J., Fang, J. and Karihaloo, B.L., Asymptotics of multiple crack interactions and prediction of effective modulus. International Journal of Solids and Structures, 2000, 37(31): 4261–4273.
21. [21]
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(4): 269–286.
22. [22]
Lavrentieff, M.A. and Shabat, B.V., Methods of Functions of a Complex Variable. Shi, X.L., Xia, D.Z. and Lv, N.G. trans. Beijing: Higher Education Press, 2006 (in Chinese).Google Scholar
23. [23]
Li, X.F., Closed-form solution for two collinear mode-III cracks in an orthotropic elastic strip of finite width. Mechanics Research Communications, 2003, 30(4): 365–370.
24. [24]
Hwu, C., Collinear cracks in anisotropic bodies. International Journal of Fracture, 1991, 52(4): 239–256.Google Scholar
25. [25]
Hu, Y.T. and Zhao, X.H., Collinear periodic cracks in an anisotropic medium. International Journal of Fracture, 1996, 76(3): 207–219.
26. [26]
Jiang, C.P. and Liu, Y.W., Antiplane problems of collinear cracks between dissimilar anisotropic materials. Acta Mechanica Solida Sinica, 1994, 15(4): 327–332 (in Chinese).Google Scholar
27. [27]
Xia, Z.H., Zhang, Y.F. and Ellyin, F., A unified periodical boundary conditions for representative volume elements of composites and applications. International Journal of Solids and Structures, 2003, 40(8): 1907–1921.
28. [28]
Pi, Z., A finite element analysis of doubly periodic crack problems. Master’s Degree Thesis, Beijing: Beijing University of Aeronautics and Astronautics, 2007 (in Chinese).Google Scholar
29. [29]
Zheng, Q.S. and Hwang, K.C., Two-dimensional elastic compliances of materials with holes and microcracks. Proceedings of the Royal Society of London A, 1997, 453(1957): 353–364.
30. [30]
Zheng, Q.S. and Hwang, K.C., Reduced dependence of defect compliance on matrix and inclusion elastic properties in two-dimensional elasticity. Proceedings of the Royal Society of London A, 1996, 452(1954): 2493–2507.