Numerical Solution for Elliptical Crack with Developing Cusps Subject to Shear Loading
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.
[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.
[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.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:57889", author = "Nik Mohd Asri Nik Long and Koo Lee Feng and Zainidin K. Eshkuvatov and A. A. Khaldjigitov", title = "Numerical Solution for Elliptical Crack with Developing Cusps Subject to Shear Loading", 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 = "Elliptical crack, stress intensity factors, hyper singular integral equation, shear loading, conformal mapping.", volume = "5", number = "10", pages = "1614-5", }
