Polynomial Estimation of J-integral for Through-thickness Crack in Elastic Perfectly-Plastic Conditions

Norwahida Yusoff, Feizal Yusof


Through the means of finite element analysis, -integral for a straight through-thickness crack has been consistently reported to decrease along the crack front with the maximum and minimum values are at the midplane and the free-surface, respectively. The present study aims to examine the through-thickness profile of -integral and represent it mathematically that it could be used to replace the demanding works of finite element analysis. The -integral profile was numerically examined using finite element analysis of three-dimensional boundary layer formulation that is subjected to a uniform load on the outermost boundary of the model. The results verify that in three-dimensional cracked bodies, although the applied load is uniform, the intensity of deformation in terms of -integral varies along the crack front and the profile has been expressed in a polynomial equation. The analytical solution allows experimentalists to estimate the crack-tip deformation in terms of local  for a known value of  that is applied on a related test specimen.


Finite element; J-integral; Polynomial; Three-dimensional; Through-thickness crack.

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J. R. Rice, A path independent integral and the approximate analysis of strain concentration by notches and cracks, Journal of Applied Mechanics, 35(2), 1968, 379-386.

V. Kumar, M. D. German and C. F. Shih, Engineering Approach For Elastic-Plastic Fracture Analysis, Technical report EPRI-NP-1931, General Electric Co., Schenectady, NY, USA, 1981.

D. M. Parks, The virtual crack extension method for nonlinear material behavior, Computer Methods in Applied Mechanics and Engineering, 12, 1977, 353-364.

D. M. Parks, A toughness derivative finite element technique for determination of crack tip stress intensity factors, International Journal of Fracture, 10, 1974, 487-502.

T. Hellen, On the method of virtual crack extensions, International Journal for Numerical Methods in Engineering, 9(1), 1975, 187-207.

R. Krueger, Virtual crack closure technique: History, approach, and applications, Applied Mechanics Reviews, 57(2), 2004, 109-143.

F. Z. Li, C. F. Shih and A. Needleman, A comparison of methods for calculating energy release rate, Engineering Fracture Mechanics, 21, 1985, 405-421.

C. Shih, B. Moran, and T. Nakamura, Energy release rate along a three-dimensional crack front in a thermally stressed body, International Journal of Fracture, 30(2), 1986, 79-102.

H. DeLorenzi, On the energy release rate and the J-integral for 3-D crack configurations, International Journal of Fracture, 19, 1982, 183-193.

W. Brocks and H. Yuan, Numerical investigations on the significance of J for large stable crack growth, Engineering Fracture Mechanics, 32(3), 1989, 459-468.

W. Brocks, I. Scheider and G.-F. Geesthacht, Numerical Aspects of the Path-Dependence of the J-integral in Incremental Plasticity, Geesthacht: GKSS Forschungszentrum, 2001.

G. Nikishkov and S. Atluri, Calculation of fracture mechanics parameters for an arbitrary three-dimensional crack, by the 'equivalent domain integral' method, International Journal for Numerical Methods in Engineering, 24(9), 1987, 1801-1821.

M. Gosz and B. Moran, An interaction energy integral method for computation of mixed-mode stress intensity factors along non-planar crack fronts in three dimensions, Engineering Fracture Mechanics, 69(3), 2002, 299-319.

T. Nakamura, Three-dimensional stress fields of elastic interface cracks, Journal of Applied Mechanics, 58(4), 1991, 939-946.

T. Nakamura and D. M. Parks, Three-dimensional crack front fields in a thin ductile plate, Journal of the Mechanics and Physics of Solids, 38(6), 1990, 787-812.

D. K. Yi and T. C. Wang, The effect of out-of-plane constraint on the stress fields near the front of a crack in a thin ductile plate, International Journal of Solids and Structures, 190, 2020, 244-257.

E. Wang, W. Zhou and G. Shen, Three-dimensional finite element analysis of crack-tip fields of clamped single-edge tension specimens – Part I: Crack-tip stress fields, Engineering Fracture Mechanics, 116, 2014, 122-143.

Y. Tkach and F. M. Burdekin, A three-dimensional analysis of fracture mechanics test pieces of different geometries – Part 1 Stress-state ahead of the crack tip, International Journal of Pressure Vessels and Piping, 93-94, 2012, 42-50.

F. Yusof and J. W. Hancock, In-plane and out-of-plane constraint effects in three-dimensional elastic perfectly-plastic crack tip fields, Proceedings of the 11th International Conference on Fracture, Turin, 2005, 1-6.

J. R. Rice, Mechanics of crack tip deformation and extension by fatigue, in Fatigue Crack Propagation, J. Grosskreutz, Ed. West Conshohocken, PA: ASTM International, 1967, 247-311.

J. R. Rice, Mathematical analysis in the mechanics of fracture, in Fracture: An Advanced Treatise, H. Liebowitc, Ed. NY, USA: Academic Press, 2, 1968, 191-311.

ABAQUS, ABAQUS v6.12 User's Manual, Rhode Island: Dassault Systèmes Simulia Corp., 2012.

F. Yusof, Three-Dimensional Constraint Based Fracture Mechanics, Ph.D. Dissertation, Dept. Mech. Eng., Glasgow Univ., Glascow, 2006.


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