Skip to main content
SpringerLink
Log in

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

  • Published:
Acta Mechanica Solida Sinica Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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.

    Book  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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. 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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  13. Faal, R.T. and Fariborz, S.J., Stress analysis of orthotropic planes weakened by cracks. Applied Mathematical Modelling, 2007, 31(6): 1133–1148.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Chapter  Google Scholar 

  19. Karihaloo, B.L. and Wang, J., On the solution of doubly periodic array of cracks. Mechanics of Materials, 1997, 26(4): 209–212.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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.

    Article  Google Scholar 

  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. 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.

    Article  Google Scholar 

  24. Hwu, C., Collinear cracks in anisotropic bodies. International Journal of Fracture, 1991, 52(4): 239–256.

    Google Scholar 

  25. Hu, Y.T. and Zhao, X.H., Collinear periodic cracks in an anisotropic medium. International Journal of Fracture, 1996, 76(3): 207–219.

    Article  Google Scholar 

  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. 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.

    Article  Google Scholar 

  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. 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.

    Article  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Solid Mechanics, Beijing University of Aeronautics and Astronautics, 100191, Beijing, China

    Junhua Xiao & Chiping Jiang

  2. Department of Engineering Mechanics, Yanshan University, 066004, Qinhuangdao, China

    Junhua Xiao

Authors
  1. Junhua Xiao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chiping Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chiping Jiang.

Additional information

Project supported by the National Natural Science Foundation of China (No.10672008).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xiao, J., Jiang, C. Exact solution for orthotropic materials weakened by doubly periodic cracks of unequal size under antiplane shear. Acta Mech. Solida Sin. 22, 53–63 (2009). https://doi.org/10.1016/S0894-9166(09)60090-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1016/S0894-9166(09)60090-2

Key words

Search

Navigation