Fracture Parameters for Cracked Cylincal Shells

Document Type : Research Paper


1 Laboratoire de Mécanique Appliquée, Université des Sciences et de la Technologie d’Oran , Algeria

2 Laboratoire de Recherche des Technologies Industrielles, Université Ibn Khaldoun de Tiaret, Algeria

3 Université d'Oran 2 Mohamed Ben Ahmed, Algeria


In this paper, 2D boundary element stress analysis is carried out to obtain the T-stress for multiple internal edge cracks in thick-walled cylinders for a wide range of cylinder radius ratios and relative crack depth. The T-stress, together with the stress intensity factor K, provides amore reliable two-parameter prediction of fracture in linear elastic fracture mechanics. T-stress weight functions are then derived from the T-stress solutions for two reference load conditions corresponding to the cases when the cracked cylinder is subject to a uniform and to a linear applied stress variation on the crack faces. The derived weight functions are then verified for several non-linear load conditions. Using the BEM results as reference T-stress solutions; the T-stress weight functions for thick-walled cylinder have also been derived. Excellent agreements between the BEM results and weight function predictions are obtained. The weight functions derived are suitable for obtaining T-stress solutions for the corresponding cracked thick-walled cylinder under any complex stress fields. Results of the study show that the two dimensional BEM analysis, together with weight function method, can be used to provide a quick and accurate estimate of T-stress for 2-D crack problems.


[1] Rice J.R., 1968, Path-independent integral and the approximate analysis of strain concentration by notches and cracks, Journal of Applied Mechanics 35(2): 379-386.
[2] Anderson T.L., 1995, Fracture Mechanics: Fundamentals and Applications, Boca Raton, CRC Press.
[3] Williams J.G., Ewing P.D., 1972, Fracture under complex stress—the angled crack problem, International Journal of Fracture 8(4): 416-441.
[4] Ueda Y., Ikeda K., Yao T., Aoki M., 1983, Characteristics of brittle failure under general combined modes including those under bi-axial tensile loads, Engineering Fracture Mechanics 18(6):1131-1158.
[5] Smith D.J., Ayatollahi M.R., Pavier M.J., 2001, The role of T-stress in brittle fracture for linear elastic materials under mixed-mode loading, Fatigue & Fracture of Engineering Materials & Structures 24(2):137-150.
[6] Cotterell B., Rice J.R., 1980, Slightly curved or kinked cracks, International Journal of Fracture 16(2):155-169.
[7] Du Z-Z., Hancock J.W., 1991, The effect of non-singular stresses on crack-tip constraint, Journal of the Mechanics and Physics of Solids 39(3): 555-567.
[8] O’Dowd N.P., Shih C.F., Dodds Jr R.H., 1995, The role of geometry and crack growth on constraint and implications for ductile/brittle fracture, In: Constraint effects in fracture theory and applications, American Society for Testing and Materials 2:134-159.
[9] Larsson S.G., Carlson A.J., 1973, Influence of non-singular stress terms and specimen geometry on small-scale yielding at crack tips in elastic–plastic materials, Journal of the Mechanics and Physics of Solids 21(4): 263-277.
[10] Leevers P.S., Radon J.C.D., 1982, Inherent stress biaxiality in various fracture specimen, International Journal of Fracture 19(4): 311-325.
[11] Cardew G.E., Goldthorpe M.R., Howard I.C., Kfouri A.P., 1985, Fundamentals of Deformation and Fracture, Eshelby Memorial Symposium Sheffield.
[12] Kfouri A.P., 1986, Some evaluations of the elastic T-term using Eshelby’s method, International Journal of Fracture 30(4): 301-315.
[13] Sham T.L., 1991, The determination of the elastic T-term using higher-order weight functions, International Journal of Fracture 48(2):81-102.
[14] Wang Y-Y., Parks D.M., 1992, Evaluation of the elastic T-stress in surface cracked plates using the line-spring method, International Journal of Fracture 56(1): 25-40.
[15] Chen C.S., Krause R., Pettit R.G., Banks-Sills L., Ingraffea A.R., 2001, Numerical assessment of T-stress computation using a p-version finite element method, International Journal of Fracture 107(2):177-199.
[16] Sladek J., Sladek V., Fedelinski P., 1997, Contour integrals for mixed-mode crack analysis: effect of nonsingular terms, Theoretical and Applied Fracture Mechanics 27:115-127.
[17] Nakamura T., Parks D.M., 1992, Determination of T-stress along three dimensional crack fronts using an interaction integral method, International Journal of Solids and Structures 29(13):1597-1611.
[18] Fett T., 2002, T -Stress Solutions and Stress Intensity Factors for 1-D Cracks, Dusseldorf, VDI Verlag.
[19] Wang X., 2002, Elastic T-stress for cracks in test specimens subjected to non-uniform stress distributions, Engineering Fracture Mechanics 69: 1339-1352.
[20] Zhao L.G., Chen Y.H.,1996, On the elastic T -term of a main crack induced by near tip microcracks, International Journal of Fracture 82: 363-379.
[21] Williams M.L., 1957, On the stress distribution at the base of a stationary crack, Journal of Applied Mechanics 24:109-114.
[22] Rice J.R., 1974, Limitations to the small scale yielding approximation for crack tip plasticity, Journal of the Mechanics and Physics of Solids 22:17-26.
[23] Suresh S., 1991, Fatigue of Materials, Cambridge University Press.
[24] Sladek J., Sladek V., 2000, Evaluation of the elastic T -stress in three-dimensional crack problems using an integral formula, International Journal of Fracture 101: 47-52.
[25] Rooke D.P., Cright D.J., 1 976, Compendium of Stress Intensity Factors, Willingon Press.
[26] Tada H., Paris P.C., Mn G.R., 1984, The Strew Analysir of Crarks Handbook , Paris Productions.
[27] Chen V.Z., 2000, Closed form solutions of T-stress in plate elasticity crack problems, International Journal of Solid Structure 37: 1629-1637.
[28] Tan C.L., 1987, The Boundary Element Method: A Short Course, Carleton University, OMawa, Ontdo.
[29] Aliabadi M.W., Rooke D.P., 1991, Numerical Fracture Mechanics, Kluwer Aeademic Publishers, Boston.
[30] Tan C.L., Wang X., 2003, The use of quarter-point crack tip elements for T-stress determination in boundary element method (BEM) analysis, Engineering Fracture Mechanics 70: 2247-2252.
[31] Betegon C., Hancock J.W., 1991, Two-parameter characterization of elastic-plastic crack tip field, Journal of Applied Mechanics 58: 104-110.
[32] 0'Dowd N.P., Shih C.F., 1991, Family of crack tip fields characterized by a triaxiality parameter-I : Structure of fields, Journal of the Mechanics and Physics of Solids 24: 989-1015.
[33] Wang Y.Y., 1993, On the Two-Parameter Characterization of Elastic-Plastic Crack Front Fields in Surface Cracked Plates, In: Hackett E.M., Schwalbe K.M., Dodds R.H., Editors.
[34] Sahli A., Rahmani O., 2009, Stress intensity factor solutions for two-dimensional elastostatic problems by the hypersingular boundary integral equation, Journal of Strain Analysis 44(4): 235-247.
[35] Yamada U., Ezawa Y., Nishiguchi I., 1979, Recommendations on singularity or crack tip elements, International Journal of Mechanical Engineering 14: 1525-1544.
[36] Blackbum W.S., 1977, The Mathematics of Finite Elements and Applications, Brunel University.
[37] Akin J.E., 1976, The generation of elements with singularities, International Journal of Mechanical Engineering 10: 1249-1259.
[38] Henshell R.D., Shaw K.G., 1975, Crack-tip finite elements are unnecessary, International Journal of Mechanical Engineering 9: 495-507.
[39] Barsoum R.S., 1976, On the use of isoparametric finite elements in linear fracture mechanics, International Journal of Mechanical Engineering 10: 25-37.
[40] Buecker H.F., 1989, A novel principle for the computation of stress intensity factor, Zeitschrift für Angewandte Mathematik und Mechanik 50: 129-146.
[41] Kfouri A.P., 1986, Some evaluations of the elastic T-stress using Eshelby's method, International Journal of Fracture 20: 301-315.
[42] Fett T., 1997, A Green's function for T-stress in an edge cracked rectangular plate, Engineering Fracture Mechanics 57: 365-373.
[43] Hooton D.G., Sherry A.H., Sanderson D.J., Ainsworth R.A., 1998, Application of R6 constraint methods using weight function for T-stress, ASME Pressure Vessel Piping Conference 365: 37-43.
[44] Andrasic C.P., Parker A.P., 1984, Dimensionless stress intensity factors for cracked thick cylinders under polynomial crack face loadings, Engineering Fracture Mechanics 19: 187-193.
[45] Wu X.R., Carlsson A.J., 1991, Weight Functions and Stress Intensity Factor Solutions, Pergamon Press.