**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**30761

##### Numerical Solution for Elliptical Crack with Developing Cusps Subject to Shear Loading

**Authors:**
Nik Mohd Asri Nik Long,
Koo Lee Feng,
Zainidin K. Eshkuvatov,
A. A. Khaldjigitov

**Abstract:**

This paper study the behavior of the solution at the crack edges for an elliptical crack with developing cusps, Ω in the plane elasticity subjected to shear loading. The problem of finding the resulting shear stress can be formulated as a hypersingular integral equation over Ω and it is then transformed into a similar equation over a circular region, D, using conformal mapping. An appropriate collocation points are chosen on the region D to reduce the hypersingular integral equation into a system of linear equations with (2N+1)(N+1) unknown coefficients, which will later be used in the determination of shear stress intensity factors and maximum shear stress intensity. Numerical solution for the considered problem are compared with the existing asymptotic solution, and displayed graphically. Our results give a very good agreement to the existing asymptotic solutions.

**Keywords:**
Conformal Mapping,
stress intensity factors,
Elliptical crack,
hyper singular integral equation,
shear loading

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1071598

**References:**

[1] J. D. Eshelby, "The determination of the elastic field of an ellipsoidal inclusion, and related problem," Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, vol. A241, pp. 376-396, 1957.

[2] M. K. Kassir and G. H. Sih, "Three dimensional stress distribution around an elliptical crack under arbitrary loading," Journal of Applied Mechanics, Transactions ASME, vol. 33, no. 196, pp. 601-611, 1966.

[3] C. M. Segedin, "Some three-dimensional mixed boundary value problems in elasticiy," Department of Aeronautics and Astronautics, University of Washington, Seattle., Tech. Rep., 1967.

[4] F. W. Smith and D. R. Sorensen, "The elliptical crack subjected to nonuniform shear loading," Journal of Applied Mechanics, Transactions ASME, vol. 41, no. 2, pp. 502-506, 1974.

[5] D. R. Sorensen and F. W. Smith, "The semi-elliptical surface crack - a solution by the alternating method," International Journal of Fracture, vol. 12, no. 1, pp. 47-57, 1976.

[6] P. A. Martin, "Orthogonal polynomial solutions for elliptical cracks under shear loadings," Quarterly Journal of Mechanics and Applied Mathematics, vol. 39, no. 4, pp. 519-534, 1986.

[7] P. A. Martin, "Orthogonal polynomial solutions for pressurized elliptical cracks," Quarterly Journal of Mechanics and Applied Mathematics, vol. 39, no. 2, pp. 269-287, 1986.

[8] A. Roy and M. Chatterjee, "Interaction between coplanar elliptic cracks- I. normal loading," International Journal of Solids and Structures, vol. 31, no. 1, pp. 127-144, 1994.

[9] A. Roy and M. Chatterjee, "Interaction between coplanar elliptic cracks- II shear loading," International Journal of Solids and Structures, vol. 36, no. 4, pp. 619-637, 1999.

[10] D. S. Lee, "Stress around an elliptical crack in an elastic plate," Quarterly Journal of Mechanics and Applied Mathematics, vol. 46, no. 3, pp. 487-500, 1993.

[11] Y. Z. Chen, X. Y. Lin, and Z. J. Peng, "Evaluation of stress intensity factors of elliptical crack by using differential-integral equations," International Journal of Fracture, vol. 8, no. 4, pp. R73-R78, 1996.

[12] Y. Z. Chen, X. Y. Lin, and Z. Q. Peng, "Application of differentialintegral equation to elliptical crack problem under shear load," Theoretical and Applied Fracture Mechanics, vol. 27, no. 1, pp. 63-78, 1997.

[13] E. E. Theotokoglou, "Boundary integral equation method to solve embedded planar crack problems under shear loading," Computational Mechanics, vol. 33, no. 5, pp. 327-333, 2004.

[14] N. A. Noda and M. Kagita, "Variations of stress intensity factors of a semi-elliptical surface crack subjected to mode i, ii, iii loading," International Journal of Pressure Vessels and Piping, vol. 81, no. 7, pp. 635-644, 2004.

[15] B. K. Hachi, S. Rechak, M. Haboussi, M. Taghite, Y. Belkacemi, and G. Maurice, "Analysis of elliptical cracks in static and in fatigue by hybridization of green-s functions," in Damage and Fracture Mechanics: Failure Analysis of Engineering Materials and Structures, T. Boukharouba, M. Elboujdaini, and G. Pluvinage, Eds., 2009, pp. 375- 385.

[16] Q. H. Fang, H. P. Song, and Y. W. Liu, "Elastic behaviour of an edge dislocation near a sharp crack emanating from a semi-elliptical blunt crack," Chinese Physics B, vol. 19, no. 1, pp. 016 102-1-8, 2010.

[17] J. T. Guidera and R. W. Lardner, "Penny-shaped cracks," Journal of Elasticity, vol. 5, no. 1, pp. 59-73, 1975.

[18] N. I. Ioakimidis, "Two-dimensional principal value hypersingular integrals for crack problems in three-dimensional elasticity," Acta Mechanica, vol. 82, no. 1-2, pp. 129-134, 1990.

[19] P. A. Martin, "Mapping flat cracks onto penny-shaped cracks: shear loadings," Journal of the Mechanics and Physics of Solids, vol. 43, no. 2, pp. 275-294, 1995.

[20] P. A. Martin, "The discontinuity in the elastostatic displacement vector across a penny-shaped crack under arbitrary loads," Journal of Elasticity, vol. 12, no. 2, pp. 201-218, 1982.

[21] N. M. A. Nik Long, L. F. Koo, and Z. K. Eshkuvatov, "Computation of energy release rates for a nearly circular crack," Mathematical Problem In Engineering, vol. 2011, pp. 1-17, 2011.

[22] T. K. Saha and A. Roy, "Weight function for an elliptic crack in an infinte medium-ii shear loading," International Journal of Fracture, vol. 112, no. 1, pp. 1-21, 2001.

[23] J. R. Rice, "First order variation in elastic fields due to variation in location of a planar crack front," Journal of Applied Mechanics, Transactions ASME, vol. 52, no. 3, pp. 571-579, 1985.

[24] Z. Nehari, Conformal Mapping, 1st ed. New York: McGraw-Hil, 1952.

[25] N. M. Borodachev, "A method of constructing a weight function for a body with a crack," Journal of Applied Mathematics and Mechanics, vol. 62, no. 2, pp. 303-307, 1998.

[26] S. Krenk, "A circular crack under asymmetric loads and some related integral equations," Journal of Applied Mechanics, Transactions ASME, vol. 46, no. 4, pp. 821-826, 1979.

[27] H. Gao and J. R. Rice, "Shear stress intensity factors for a planar crack with slightly curved front," Journal of Applied Mechanics, Transactions ASME, vol. 53, no. 4, pp. 774-778, 1986.

[28] H. Gao, "Nearly circular shear mode cracks," International Journal of Solids and Structures, vol. 24, no. 2, pp. 177-193, 1988.