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A coupled FEM–Bernstein approach for computing the J k integrals

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Abstract

The stress intensity factors of mixed mode I–II crack loading are often evaluated numerically via FEM implementation combined with the equivalent domain integral method to derive the J k integrals. Mesh refinement or singular elements are classic techniques for reproducing the sharp stress gradient due to material fracture. In this context, high-order Bernstein shape functions are proposed for a model that couples a radial spectral expansion for the stress field around crack tip with a finite element mesh in the outer regions. Lagrange multipliers are used to enforce the coupling in displacements along the FEM–Bernstein approximations interface, in which non-matching nodal patterns are allowed.

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References

  1. Rice J.R.: A path independent integral and the approximate analysis of strain concentration by notches and cracks. J. Appl. Mech. 35, 379–386 (1968)

    Article  Google Scholar 

  2. Anderson T.L.: Fracture Mechanics Fundamentals and Applications, 2nd edn. CRC Press LLC, Boca Raton (1995)

    MATH  Google Scholar 

  3. Atluri S.N.: Path-independent integrals in finite elasticity and inelasticity, with body forces, inertia, and arbitrary crack-face conditions. Eng. Fract. Mech. 16(3), 341–364 (1982)

    Article  Google Scholar 

  4. Lorenzi, H.G.de : On the energy release rate and the J-integral for 3-D crack configurations. Int. J. Fract. 19(3), 183–193 (1982)

  5. Li F.Z., Shih C.F., Needleman A.: A comparison of methods for calculating energy release rates. Eng. Fract. Mech. 21(2), 405–421 (1985)

    Article  Google Scholar 

  6. Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method. Volume 1: The Basis, 5th edn. Butterworth-Heinemann, London (2000)

  7. Shih C.F., de Lorenzi H.G., German M.D.: Crack extension modeling with singular quadratic isoparametric elements. Int. J. Fract. 12(4), 647–651 (1976)

    Google Scholar 

  8. Dodds Jr, R.H., Vargas, P.M.: Numerical evaluation of domain and contour integrals for nonlinear fracture mechanics: formulation and implementation aspects. Report UILU-ENG-8-2006, University of Illinois, Urbana, IL (1988)

  9. Hibbitt H.D.: Some properties of singular isoparametric elements. Int. J. Numer. Methods Eng. 11(1), 180–184 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  10. Pian, T.H.H., Tong, P., Luk, C.H.: Elastic crack analysis by a finite element hybrid method. In: Proceedings of the Third International Conference on Matrix Methods in Structural Mechanics. Wright-Patterson Air Force Base, Ohio (1971)

  11. Tong P., Pian T.H.H., Lasry S.J.: A hybrid-element approach to crack problems in plane elasticity. Int. J. Numer. Methods Eng. 7(3), 297–308 (1973)

    Article  MATH  Google Scholar 

  12. Atluri S.N., Kobayashi A.S., Nakagaki M.: An assumed displacement hybrid finite element model for linear fracture mechanics. Int. J. Fract. 11(2), 257–271 (1975)

    Article  Google Scholar 

  13. Kuna M., Zwicke M.: A mixed hybrid finite element for three dimensional elastic crack analysis. Int. J. Fract. 45(1), 65–79 (1990)

    Article  Google Scholar 

  14. Cavalcante Neto J.B., Wawrzynek P.A., Carvalho M.T.M., Martha L.F., Ingraffea A.R.: An algorithm for three-dimensional mesh generation for arbitrary regions with cracks. Eng. Comput. 17, 75–91 (2001)

    Article  MATH  Google Scholar 

  15. Yagawa G., Yoshimura S., Soneda N., Nakao K.: Automatic two- and three-dimensional mesh generation based on fuzzy knowledge processing. Comput. Mech. 9(5), 333–346 (1992)

    Article  MATH  Google Scholar 

  16. Tanaka M., Hamada M., Iwata Y.: Computation of a two-dimensional stress intensity factor by the boundary element method. Arch. Appl. Mech. 52(1–2), 95–104 (1982)

    MATH  Google Scholar 

  17. Blandford G.E., Ingraffea A.R., Liggett J.A.: Two-dimensional stress intensity factor computations using the boundary element method. Int. J. Numer. Methods Eng. 17(3), 387–404 (1981)

    Article  MATH  Google Scholar 

  18. Ariza M.P., Sáez A., Domínguez J.: A singular element for three-dimensional fracture mechanics analysis. Eng. Anal. Bound. Elements 20, 275–285 (1997)

    Article  Google Scholar 

  19. Liu G.R.: Mesh Free Methods Moving Beyond the Finite element Method. CRC Press, Boca Raton (2003)

    MATH  Google Scholar 

  20. Fleming M., Chu Y.A., Moran B., Belytschko T.: Enriched element-free Galerkin methods for crack tip fields. Int. J. Numer. Methods Eng. 40, 1483–1504 (1997)

    Article  MathSciNet  Google Scholar 

  21. Rao B.N., Rahman S.: An enriched meshless method for non-linear fracture mechanics.. Int. J. Numer. Methods Eng. 59, 197–223 (2004)

    Article  MATH  Google Scholar 

  22. Belytschko T., Krongauz Y., Organ D., Fleming M., Krysl P.: Meshless methods: an overview and recent developments. Comput. Methods Appl. Mech. Eng. 139(1-4), 3–47 (1996)

    Article  MATH  Google Scholar 

  23. Krysl P., Belytschko T.: Element-free Galerkin method: convergence of the continuous and discontinuous shape functions. Comput. Methods Appl. Mech. Eng. 148(3-4), 257–277 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  24. Moës N., Dolbow J., Belytschko T.: A finite element method for crack growth without remeshing. Int. J. Numer. Methods Eng. 46, 131–150 (1999)

    Article  MATH  Google Scholar 

  25. Belytschko T., Moës N., Usui S., Parimi C.: Arbitrary discontinuities in finite elements. Int. J. Numer. Methods Eng. 50, 993–1013 (2001)

    Article  MATH  Google Scholar 

  26. Sukumar N., Moran B., Black T., Belytschko T.: An element-free Galerkin method for three-dimensional fracture mechanics. Comput. Mech. 20, 170–175 (1997)

    Article  MATH  Google Scholar 

  27. Rao B.N., Rahman S.: A coupled meshless-finite element method for fracture analysis of cracks. Int. J. Press. Vessels Pip. 78, 647–657 (2001)

    Article  Google Scholar 

  28. Berger J.R., Karageorghis A., Martin P.A.: Stress intensity factor computation using the method of fundamental solutions: mixed-mode problems. Int. J. Numer. Methods Eng. 1, 1–13 (2005)

    Google Scholar 

  29. Ventura G., Xu J.X., Belytschko T.: A vector level set method and new discontinuity approximations for crack growth by EFG. Int. J. Numer. Methods Eng. 54, 923–944 (2002)

    Article  MATH  Google Scholar 

  30. Geubelle P.H., Rice J.R.: A spectral method for three-dimensional elastodynamic fracture problems. J. Mech. Phys. Solids 43(11), 1791–1824 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  31. Cochard A., Rice J.R.: A spectral method for numerical elastodynamic fracture analysis without spatial replication of the rupture event. J. Mech. Phys. Solids 45(8), 1393–1418 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  32. Woo K.S., Lee C.G.: p-Version finite element approximations of stress intensity factors for cracked plates including shear deformation. Eng. Fract. Mech. 52(3), 493–502 (1995)

    Article  MathSciNet  Google Scholar 

  33. Rahulkumar P., Saigal S., Yunus S.: Singular p-version finite elements for stress intensity factor computations. Int. J. Numer. Methods Eng. 40(6), 1091–1114 (1997)

    Article  MATH  Google Scholar 

  34. Lorenz, G.G.: Bernstein Polynomials, 2nd edn. Chelsea Publishing Company, New York (1986)

  35. Fries, T., Matthies H.: Classification and Overview of Meshfree Methods. Institute of Scientific Computing. Technical University Braunschweig, Brunswick, Germany (2004)

  36. Doha, E.H., Bhrawy, A.H., Saker M.A.: On the Derivatives of Bernstein Polynomials: An Application for the Solution of High Even-Order Differential Equations. Hindawi Publishing Corporation Boundary Value Problems (2011)

  37. Valencia Ó.F., Gómez-Escalonilla F.J., Garijo D., López J.: Bernstein polynomials in EFGM. Proc. IMechE Part C J. Mech. Eng. Sci. 225(8), 1808–1815 (2011)

    Article  Google Scholar 

  38. Bhatti M.I., Bracken P.: Solutions of differential equations in a Bernstein polynomial basis. J. Comput. Appl. Math. 205, 272–280 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  39. Mirkov, N., Rasuo, B.: A Bernstein Polynomial Collocation Method for the Solution of Elliptic Boundary Value Problems. Cornell University Library (2012)

  40. Liu, J., Zheng, Z., Xu, Q.: Bernstein-polynomials-based highly accurate methods for one-dimensional interface problems. J. Appl. Math. doi:10.1155/2012/859315

  41. Garijo D., Valencia Ó.F., Gómez-Escalonilla F.J., López J.: Bernstein–Galerkin approach in elastostatics. Proc. IMechE Part C J. Mech. Eng. Sci. 228(3), 391–404 (2014)

    Article  Google Scholar 

  42. Atkinson K.E.: An Introduction to Numerical Analysis, 2nd edn. Wiley, London (1988)

    Google Scholar 

  43. Raju I.S., Shivakumar K.N.: An equivalent domain integral method in the two-dimensional analysis of mixed mode crack problems. Eng. Fract. Mech. 37(4), 707–725 (1990)

    Article  Google Scholar 

  44. Eischen J.W.: An improved method for computing the J 2 integral. Eng. Fract. Mech. 26(5), 691–700 (1987)

    Article  Google Scholar 

  45. Nikishkov G.P., Atluri S.N.: An equivalent domain integral method for computing crack-tip integral parameters in non-elastic, thermo-mechanical fracture. Eng. Fract. Mech. 26(6), 851–867 (1987)

    Article  Google Scholar 

  46. Shih C.F., Moran B., Nakamura T.: Energy release rate along a three-dimensional crack front in a thermally stressed body. Int. J. Fract. 30(2), 79–102 (1986)

    Google Scholar 

  47. Belytschko T., Lu Y.Y., Gu L.: Element-free Galerkin methods. Int. J. Numer. Methods Eng. 37, 229–256 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  48. Dolbow J., Belytschko T.: An introduction to programming the meshless element free galerkin method. Arch. Comput. Methods Eng. 5(3), 207–241 (1998)

    Article  MathSciNet  Google Scholar 

  49. Kuna M.: Finite Elements in Fracture Mechanics; Theory—Numerics—Applications. Springer, Berlin (2013)

    Book  MATH  Google Scholar 

  50. Spivey M.Z.: Combinatorial sums and finite differences. Discrete Math. 307, 3130–3146 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  51. Wilson, W.K.: Research Report 69-IE7-FMECH-RI. Westinghouse Research Laboratories, Pittsburgh (1969)

  52. Gdoutos E.E.: Fracture Mechanics: An Introduction, 2nd edn. Springer, Berlin (2005)

    Google Scholar 

  53. Rooke, D.P., Cartwright, D.J.: Compendium of Stress Intensity Factors. Stationery Office (1976)

  54. Bowie O.L.: Solutions of plane crack problems by mapping technique. Mech. Fract. I (Methods Anal. Sol. Crack Prob.) 1, 1–55 (1973)

    Article  MathSciNet  Google Scholar 

  55. Nishioka T., Atluri S.N.: On the computation of mixed-mode K-factors for a dynamically propagating crack, using path-independent integrals \({J'_{k}}\). Eng. Fract. Mech. 20(2), 193–208 (1984)

    Article  Google Scholar 

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Authors and Affiliations

  1. Safran Engineering Services (Safran Group), Calle Arnaldo de Vilanova, 5, 28906, Getafe, Madrid, Spain

    Diego Garijo

  2. Airbus Defence and Space, Avda John Lennon s/n, 28906, Getafe, Madrid, Spain

    Francisco Javier Gómez-Escalonilla

  3. FACC AG, Fischerstraße, 9, 4910, Ried im Innkreis, Austria

    Óscar Francisco Valencia

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  1. Diego Garijo
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  2. Francisco Javier Gómez-Escalonilla
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  3. Óscar Francisco Valencia
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Correspondence to Diego Garijo.

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Garijo, D., Gómez-Escalonilla, F.J. & Valencia, Ó.F. A coupled FEM–Bernstein approach for computing the J k integrals. Arch Appl Mech 84, 1883–1902 (2014). https://doi.org/10.1007/s00419-014-0893-3

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  • DOI: https://doi.org/10.1007/s00419-014-0893-3

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