Analysis of Elastic-Plastic J Integrals for 3-Dimensional Cracks Using Finite Element Alternating Method

유한요소 교호법을 이용한 삼차원 균열의 탄소성 J 적분 해석

  • 박재학 (충북대학교 공과대학 안전공학과)
  • Published : 2009.02.01


SGBEM(Symmetric Galerkin Boundary Element Method)-FEM alternating method has been proposed by Nikishkov, Park and Atluri. In the proposed method, arbitrarily shaped three-dimensional crack problems can be solved by alternating between the crack solution in an infinite body and the finite element solution without a crack. In the previous study, the SGBEM-FEM alternating method was extended further in order to solve elastic-plastic crack problems and to obtain elastic-plastic stress fields. For the elastic-plastic analysis the algorithm developed by Nikishkov et al. is used after modification. In the algorithm, the initial stress method is used to obtain elastic-plastic stress and strain fields. In this paper, elastic-plastic J integrals for three-dimensional cracks are obtained using the method. For that purpose, accurate values of displacement gradients and stresses are necessary on an integration path. In order to improve the accuracy of stress near crack surfaces, coordinate transformation and partitioning of integration domain are used. The coordinate transformation produces a transformation Jacobian, which cancels the singularity of the integrand. Using the developed program, simple three-dimensional crack problems are solved and elastic and elastic-plastic J integrals are obtained. The obtained J integrals are compared with the values obtained using a handbook solution. It is noted that J integrals obtained from the alternating method are close to the values from the handbook.


Finite Element Alternating Method;3-Dimensional Crack;Elastic-plastic Analysis;J Integral;Symmetric Galerkin Boundary Element Method


  1. Li, S. and Mear, M.E., "Singularity-reduced Integral Equations for Displacement Discontinuities in Three Dimensional Linear Elastic Media," Int. J. Fracture, Vol. 93, pp. 87-114
  2. Park, Jai Hak and Park, Sang Yun, 2007, “Elastic-plastic Analysis of a 3-Dimensional Inner Crack Using Finite Element Alternating Method”, Transactions of the KSME A, Vol. 31, No 10, pp. 1009-1016
  3. Nayak, G.C. and Zienkiewicz, O.C., 1972, "Elasto-plastic Stress Analysis Generalization for Various Constitutive Relations Including Strain Softening," Int. J. Numer. Methods in Engng, Vol. 5, pp. 113-135
  4. Atluri, S.N., 1997, Structural Integrity and Durability, Tech Science Press, Forsyth
  5. Andra, H., 1998, "Integration of Singular Integrals for the Galerkin-type Boundary Element Method in 3D Elasticity," Com. Methods Appl. Mech. Engng, Vol. 157, pp. 239-249
  6. Erichsen, S. and Sauter, S.A., 1998, "Efficient Automatic Quadrature in 3-d Galerkin BEM," Com. Methods Appl. Mech. Engng, 157, pp. 215-224
  7. Frangi, A., Novati, G., Springhetti, R. and Rovizzi, M., 2000, "Fracture Mechanics in 3D by the Symmetric Galerkin Boundary Element Method," VIII Conf. on Numerical Methods in Continuum Mechanics, Liptovsky Jan, Slovak Republik
  8. Zahoor, A., 1991, Ductile Fracture Handbook, EPRI
  9. Nikishkov, G.P., Park, J.H. and Atluri, S.N., 2001, "SGBEM-FEM Alternating Method for Analyzing 3D Non-planar Cracks and Their Growth in Structural Components," Comp. Modeling in Engng & Sci., Vol. 2, No. 3, pp. 401-422
  10. Nikishkov, G.P. and Atluri, S.N., 1994, "Analytical-Numerical Alternating Method for Elastic-Plastic Analysis of Cracks," Comput. Mech., Vol. 13, No. 6, pp. 427-442

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