Axial translation of a rigid disc inclusion embedded in a penny-shaped crack in a transversely isotropic solid

Document Type : Article

Authors

1 School of Civil Engineering, College of Engineering, University of Tehran, Tehran, P.O. Box 11155-4563, Iran

2 School of Engineering Science, College of Engineering, University of Tehran, Tehran, P.O. Box 11155-4563, Iran

Abstract

In this paper, an analytical solution for the axisymmetric interaction of a rigid disc inclusion embedded in bonded
contact with the surfaces of a penny-shaped crack and a transversely isotropic medium is investigated. By using a
method of potential functions and treating dual and triple integral equations, the mixed boundary value problem is
written in the form of two coupled integral equations, which are amenable to numerical treatments. The axial sti ness
of the inclusion and the shearing stress intensity factor at the tip of the penny-shaped crack for di erent degrees of
material anisotropy are illustrated graphically. Useful limiting cases such as a rigid disc inclusion in an uncracked
medium and in a completely cracked solid are recovered

Keywords

Main Subjects

References

References:
1. Selvadurai, A.P.S. "Mechanics of a rigid circular disc bonded to a cracked elastic half-space", Int. J. Solids Struct., 39(24), pp. 6035-6053 (2002).
2. Menshykov, O.V., Menshykov, V.A., and Guz, I.A. "The contact problem for an open penny-shaped crack under normally incident tension-compression wave", Eng. Fract. Mech., 75(5), pp. 1114-1126 (2008).
3. Selvadurai, A.P.S. "A contact problem for a smooth rigid disc inclusion in a penny-shaped crack", Z. angew. Math. Phys. (ZAMP), 45(1), pp. 166-173 (1994).
4. Sneddon, I.N. "The distribution of stress in the neighbourhood of a crack in an elastic solid", Proc. Math. Phys. Eng. Sci., 187, pp. 229-260 (1946).
5. Collins, W. "Some axially symmetric stress distributions in elastic solids containing penny-shaped cracks. i. Cracks in an infinite solid and a thick plate", Proc. Math. Phys. Eng. Sci., 266, pp. 359-386 (1962).
6. Sneddon, I.N. and Tweed, J. "The stress intensity factor for a penny-shaped crack in an elastic body under the action of symmetric body forces", Int. J.Fract. Mech., 3(4), pp. 291-299  (1967).
7. Sneddon, I.N. and Lowengrub, M., Crack Problems in the Classical Theory of Elasticity, John Wiley & Sons, 221 pages (1969).
8. Kassir, M. and Sih, G.C., Three-dimensional crack problems: A new selection of crack solutions in threedimensional elasticity, Noordhoof International Publishing, Leyden (1975).
9. Sih, G.C. "A review of the three-dimensional stress problem for a cracked plate", Int. J. Fract. Mech., 7(1), pp. 39-61 (1971).
10. Erdogan, F. and Arin, K. "Penny-shaped interface crack between an elastic layer and a half space", Int. J. Eng. Sci., 10(2), pp. 115-125 (1972).
11. Rahman, M. "A penny-shaped crack under timeharmonic torsional body forces", Eng. Fract. Mech., 49(4), pp. 599-610 (1994).
12. Selvadurai, A.P.S., Singh, B.M., and Au, M. "Axial loading of a rigid disc inclusion with a debonded region", Int. J. Solids Struct., 25(7), pp. 783-795 (1989).
13. Selvadurai, A.P.S. "On the axisymmetric loading of an annular crack by a disk inclusion", J. Eng. Math., 46(3-4), pp. 377-393 (2003).
14. Vrbik, J., Singh, B.M., Rokne, J., and Dhaliwal, R.S. "On the expansion of a penny-shaped crack by a rigid circular disc inclusion in a thick plate", Z. Angew. Math. Mech. (ZAMM), 84(8), pp. 538-550 (2004).
15. Eskandari, M., Moeini-Ardakani, S.S., and Shodja, H.M. "Axisymmetric contact of a rigid inclusion embedded at the interface of a piezoelectric bimaterial", Quart. J. Mech. Appl. Math., 62(3), pp. 281-295 (2009).
16. Shodja, H.M., Moeini-Ardakani, S.S., and Eskandari, M. "Axisymmetric problem of energetically consistent interacting annular and penny-shaped cracks in piezoelectric materials", J. Appl. Mech., 78(2), pp. 021010- 10 (2011).
17. Eskandari, M., Moeini-Ardakani, S.S., and Shodja, H.M. "An energetically consistent annular crack in a piezoelectric medium", Eng. Fract. Mech, 77(5), pp. 819-831 (2010).
18. Singh, B.M., Rokne, J., and Dhaliwal, R.S. "Closed form solution for an annular elliptic crack around an elliptic rigid inclusion in an infinite solid", J. Appl. Math. Mech. (ZAMM), 92(11-12), pp. 882-887 (2012).
19. Eskandari-Ghadi, M., Ardeshir-Behrestaghi, A., and Neya, B.N. "Mathematical analysis for an axisymmetric disc-shaped crack in transversely isotropic halfspace", Int. J. Mech. Sci., 68, pp. 171-179 (2013).
20. Fabrikant, V.I. "General at crack located in the plane perpendicular to the planes of isotropy in transversely isotropic body", Acta Mech., 226(10), pp. 3289-3306 (2015).
21. Fabrikant, V.I. "Non-traditional crack problem for transversely-isotropic body", Eur. J. Mech. A-Solid, 30(6), pp. 902-912 (2011).
22. Fabrikant, V.I. "General  at crack arbitrarily located in the transversely-isotropic body", Theor. Appl. Fract. Mech., 82, pp. 69-76 (2016).
23. Fabrikant, V. "Relationship between contact and crack problems for generally anisotropic bodies", Int. J. Eng. Sci., 102, pp. 27-35 (2016).
24. Yang, F. and Zhao, Y.-P. "The effect of a capillary bridge on the crack opening of a penny crack", Soft Matter, 12(5), pp. 1586-1592 (2016).
25. Shahmohamadi, M., Khojasteh, A., and Rahimian, M. "Frictionless contact of a rigid disk with the face of a penny-shaped crack in a transversely isotropic solid", Int. J. Solids Struct., 106, pp. 274-283 (2017).
26. Antipov, Y.A. and Mkhitaryan, S.M. "A crack induced by a thin rigid inclusion partly debonded from the matrix", Quart. J. Mech. Appl. Math., 70(2), pp. 153- 85 (2017).
27. Amiri-Hezaveh, A., Moghaddasi, H., Karimi, P., and Ostoja-Starzewski, M. "Dynamic interaction of plates in an inhomogeneous transversely isotropic space weakened by a crack", J. Appl. Math. Mech. (ZAMM), 97, pp. 491-504 (2017).
28. Eskandari-Ghadi, M., Ardeshir-Behrestaghi, A., and Pak, R.Y. "Bi-material transversely isotropic halfspace containing penny-shaped crack under timeharmonic horizontal loads", Eng. Fract. Mech., 172, pp. 152-180 (2017).
29. Singh, B.M., Rokne, J., and Dhaliwal, R.S. "Thermal stresses in a two-dimensional infinite medium containing a rigid inclusion embedded in a line crack", Theor. Appl. Fract. Mech., 46(2), pp. 148-155 (2006).
30. Lekhnitskii, S., Theory of an Anisotropic Elastic Body, 525. Holden-Day, San Francisco (1963).
31. Rahimian, M., Eskandari-Ghadi, M., Pak, R.Y.S., and Khojasteh, A. "Elastodynamic potential method for transversely isotropic solid", J. Eng. Mech., 133(10), pp. 1134-1145 (2007).
32. Cooke, J. "Triple integral equations", Quart. J. Mech. Appl. Math., 16(2), pp. 193-203 (1963).
33. Katebi, A.A., Khojasteh, A., Rahimian, M., and Pak, R.Y.S. "Axisymmetric interaction of a rigid disc with a transversely isotropic half-space", Int. J. Numer. Anal. Methods Geomech., 34(12), pp. 1211-1236 (2010).
34. Selvadurai, A.P.S. "Axial displacement of a rigid elliptical disc inclusion embedded in a transversely isotropic elastic solid", Mech. Res. Commun., 9(1), pp. 39-45 (1982).
35. Fabrikant, V.I., Mixed Boundary Value Problems of Potential Theory and Their Applications in Engineering, Kluwer Academic Publishers (1991).
36. Mossakovskii, V. "The fundamental mixed problem of the theory of elasticity for a half-space with a circular line separating the boundary conditions", Prikl. Mat. Mekh., 18, pp. 187-196 (1954).
37. Fabrikant, V.I., Applications of Potential Theory in Mechanics: a Selection of New Results, 51, Kluwer Academic Publishers, Dordrecht, Boston (1989).
38. Selvadurai, A.P.S. and Singh, B.M. "Axisymmetric problems for an externally cracked elastic solid-ii. Effect of a penny-shaped inclusion", Int. J. Eng. Sci., 25(11-12), pp. 1477-1490 (1987).
39. Baker, C., The Numerical Treatment of Integral Equations, Clarendon Press (1977).
40. Atkinson, K.E. A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind, Soc. for Industrial and Applied Mathematics (1976).
41. Payton, R.C., Elastic Wave Propagation in Transversely Isotropic Media, Springer Science & Business Media (2012).
Scientia Iranica
Volume 26, Issue 2 - Serial Number 2
Transactions on Civil Engineering (A)
March and April 2019
Pages 678-689

Files

Share

How to cite

Statistics