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Extraction of Stress Intensity Factors by Using the P-Version Finite Element Method and Contour Integral Method

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Abstract

The stress intensity factors (SIFs) for two-dimensional cracks are extracted using the p-version finite element method (P-FEM) and the contour integral method. Several numerical experiments, e.g., crack initiating from the edge of a circular hole under an unidirectional uniform tension and two equal-length, unequal-length hole–edge cracks, respectively, at a rectangular plate, an inclined centered crack under uniaxial tension at a square plate and a pipeline crack model, are used to demonstrate the accuracy and effectiveness of the approaches. SIFs are presented for the effects of various crack lengths and length–width ratio. Numerical results are analyzed and compared with reference solutions and results obtained by the Voronoi cell finite element method, boundary element method, high-order extended finite element method (high-order XFEM) and commercial finite element software ABAQUS in the available literature. Numerical results are in good agreement with the benchmark problems and show faster convergence rate, higher accuracy and better numerical stability.

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References

  1. Babuška I, Miller A. The post-processing approach in the finite element method-part 2: the calculation of stress intensity factors. Int J Numer Methods Eng. 1984;20:1111–29.

    Article  Google Scholar 

  2. Maiti SK. Finite element computation of crack closure integrals and stress intensity factors. Eng Fract Mech. 1992;41:339–48.

    Article  Google Scholar 

  3. Belytschko T, Black T. Elastic crack growth in finite elements with minimal remeshing. Int J Numer Methods Eng. 1999;45:601–20.

    Article  Google Scholar 

  4. Portela A, Aliabadi MH. The dual boundary element method: effective implementation for crack problems. Int J Numer Methods Eng. 1992;33(6):1269–87.

    Article  Google Scholar 

  5. Lecampion B, Detournay E. An implicit algorithm for the propagation of a hydraulic fracture with a fluid lag. Comput Methods Appl Mech Eng. 2007;196:4863–80.

    Article  MathSciNet  Google Scholar 

  6. Yan XQ. A boundary element analysis for stress intensity factors of multiple circular arc cracks in a plane elasticity plate. Appl Math Model. 2010;34:2722–37.

    Article  MathSciNet  Google Scholar 

  7. Rabczuk T, Belytschko T. Cracking particles: a simplified meshfree method for arbitrary evolving cracks. Int J Numer Methods Eng. 2004;61:2316–43.

    Article  Google Scholar 

  8. Duflot M, Hung ND. A meshless method with enriched weight functions for fatigue crack growth. Int J Numer Methods Eng. 2004;59:1945–61.

    Article  Google Scholar 

  9. Duflot M. A meshless method with enriched weight functions for three-dimensional crack propagation. Int J Numer Methods Eng. 2006;65:1970–2006.

    Article  Google Scholar 

  10. Daux C, Moës N, Dolbow J, Sukumar N, Belytschko T. Arbitrary branched and intersecting cracks with the extended finite element method. Int J Numer Methods Eng. 2000;48:1741–60.

    Article  Google Scholar 

  11. Liu XY, Xiao QZ, Karihaloo BL. XFEM for direct evaluation of mixed mode SIFs in homogeneous and bi-materials. Int J Numer Methods Eng. 2004;59(8):1103–18.

    Article  Google Scholar 

  12. Pereira JP, Duarte CA. The contour integral method for loaded cracks. Commun Numer Meth Eng. 2006;22(5):421–32.

    Article  MathSciNet  Google Scholar 

  13. Fries TP, Belytschko T. The extended/generalized finite element method: an overview of the method and its applications. Int J Numer Methods Eng. 2010;84:253–304.

    Article  MathSciNet  Google Scholar 

  14. Bowie OL. Analysis of an infinite plate containing radial cracks originating at the boundary of an internal circular hole. J Math Phys. 1956;35:60–71.

    Article  MathSciNet  Google Scholar 

  15. Newman JC. An improved method of collocation for the stress analysis of cracked plates with various shaped boundaries. NASA TN D-6376, 1971.

  16. Tweed J, Rooke DP. The distribution of stress near the tip of a radial crack at the edge of a circular hole. Int J Eng Sci. 1973;11(11):85–95.

    Article  Google Scholar 

  17. Nisitani N, Isida M. Simple procedure for calculating \(K_ {I}\) of a notch with a crack of arbitrary size and its application to nonpropagating fatigue crack. In: Proc joint JSME-SESA conf on experimental mechanics, part I. 1982. p. 150–5.

  18. Lai J, Schijve J. The stress intensity factor and stress concentration for a finite plate with a single crack emanating from a hole. Eng Fract Mech. 1990;36(4):619–30.

    Article  Google Scholar 

  19. Sollero P, Aliabadi MH, Rooke DP. Anisotropic analysis of cracks emanating from circular holes in composite laminates using the boundary element method. Eng Frac Mech. 1994;49:213–24.

    Article  Google Scholar 

  20. Yan XQ. An efficient and accurate numerical method of stress intensity factors calculation of a branched crack. ASME J Appl Mech. 2005;72(3):330.

    Article  Google Scholar 

  21. Yan XQ. An empirical formula for stress intensity factors of cracks emanating from a circular hole in a rectangular plate in tension. Eng Fail Anal. 2007;14:935–40.

    Article  Google Scholar 

  22. Han N, Guo R. Two new Voronoi cell finite element models for fracture simulation in porous material under inner pressure. Eng Fract Mech. 2019;211:478–94.

    Article  Google Scholar 

  23. Lan MY, Waisman H, Harari I. A high-order extended finite element method for extraction of mixed-mode strain energy release rates in arbitrary crack setting. Int J Numer Methods Eng. 2013;96(12):787–812.

    Article  Google Scholar 

  24. Song G, Waisman H, Lan MY, Harari I. Extraction of stress intensity factors from Irwin’s integral using high-order XFEM on triangular meshes. Int J Numer Methods Eng. 2015;102(12):528–50.

    Article  MathSciNet  Google Scholar 

  25. Murakami Y. Stress intensity factors handbook. New York: Pergamon Press; 1987.

    Google Scholar 

  26. Chinese aeronautical establishment. Stress intensity factors handbook. Revised ed. Beijing: Science Press; 1993.

  27. Babuška I, Szabó BA, Katz IN. The p-version of the finite element method. SIAM J Numer Anal. 1981;18(3):515–45.

    Article  MathSciNet  Google Scholar 

  28. Gui W, Babuška I. The h, p and h-p versions of the finite element method in 1 dimension. Part 1: The error analysis of the p-version. Numerische Mathematik. 1986;49(6):577-612.

  29. Babuška I, Suri M. The optimal convergence rate of the p-version of the finite element method. SIAM J Numer Anal. 1987;24(4):750–76.

    Article  MathSciNet  Google Scholar 

  30. Guo BQ. Approximation theory of the p-version of the finite element method in three dimensions, Part 2: Convergence of the p-version of FEM. SIAM J Numer Anal. 2009;47(4):2578–611.

    Article  MathSciNet  Google Scholar 

  31. Guo BQ, Zhang JM. Stable and compatible polynomial extensions in three dimensions and applications to the p and h-p finite element method. SIAM J Numer Anal. 2009;47(2):1195–225.

    Article  MathSciNet  Google Scholar 

  32. Symposium Nineteenth. Szabó BA, Babuška I. Computation of the amplitude of stress singular terms for cracks and reentrant corners, In T. A. Cruse, editor, Fracture Mechanics. ASTM STP. 1988;969:101–24.

    Google Scholar 

  33. Wowk D, Gamble K, Underhill R. Influence of p-method finite element parameters on predictions of crack front Geometry. Finite Elem Anal Des. 2013;73:1–10.

    Article  Google Scholar 

  34. Garzon J, Duarte CA, Pereira JP. Extraction of stress intensity factors for the simulation of 3-D crack growth with the generalized finite element method. Key Eng Mater. 2013;560:1–36.

    Article  Google Scholar 

  35. Rahulkumar P, Saigal S, Yunus S. Singular p-version finite elements for stress intensity factor computation. Int J Numer Methods Eng. 2015;40(6):1091–114.

    Article  Google Scholar 

  36. Cui X, Li H, Cheng GW, Tang C, Gao X. Contour integral approaches for the evaluation of stress intensity factors using displacement discontinuity method. Eng Anal Bound Elem. 2017;82:119–29.

    Article  MathSciNet  Google Scholar 

  37. Paolini A, Kollmannsberger S, Winter C, Buchschmid M. A high-order finite element model for vibration analysis of cross-laminated timber assemblies. Build Acoust. 2017;24(3):135–58.

    Article  Google Scholar 

  38. Belalia SA. A new analysis of nonlinear free vibration behavior of bi-functionally graded sandwich plates using the p-version of the finite element method. Mechanics of Advanced Materials and Structures. 2017;1–14.

  39. Wu Y, Xing YF, Liu B. Analysis of isotropic and composite laminated plates and shells using a differential quadrature hierarchical finite element method. Compos Struct. 2018;205:11–25.

    Article  Google Scholar 

  40. Rachid Z, Kaddour R, Achache H. Dynamic calculation of a tapered shaft rotor made of composite material. Adv Aircr Spacecr Sci. 2018;5(1):51–71.

    Google Scholar 

  41. Chen H, Xiao YX, Guo RQ. Numerical simulation for concrete aggregate models based on the p-version adaptive FEM method. Eng Mech. 2019;(36 suppl):158-164.

  42. Wowk D, Alousis L, Underhill R. An adaptive remeshing technique for predicting the growth of irregular crack fronts using p-version finite element analysis. Eng Fract Mech. 2019;207:36–47.

    Article  Google Scholar 

  43. Wu Y, Xing YF, Liu B. Hierarchical p-version \(\text{ C}^{1}\) finite elements on quadrilateral and triangular domains with curved boundaries and their applications to Kirchhoff plates. Int J Numer Methods Eng. 2019;119:177–207.

    Article  Google Scholar 

  44. Lai JB, Zhang X, Schijve J. An investigation of a hole-edge crack problem by a combined complex variable and least square method. Eng Fract Mech. 1991;39(4):713–37.

    Article  Google Scholar 

  45. Liu SH, Duan SJ. Analytical solutions of cracks emanating from an elliptic hole in an infinite plate under tension. Chin J Mech Eng. 2014;27(5):1057–63.

    Article  Google Scholar 

  46. Szabó BA, Babuška I. Introduction to finite element analysis. New York: Wiley; 2011.

    Book  Google Scholar 

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Acknowledgements

The authors appreciate the financial support of the National Natural Science Foundation of China (Grant No: 51769011) for this work, and the authors are also deeply grateful to the editors and reviewers for their rigorous work and valuable comments.

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Authors and Affiliations

  1. Department of Engineering Mechanics, Kunming University of Science and Technology, Kunming, 650500, China

    Jianming Zhang & Jun Chen

  2. Department of Water Resources and Hydropower Engineering, Kunming University of Science and Technology, Kunming, 650500, China

    Liang Wu

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  1. Jianming Zhang
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  2. Jun Chen
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  3. Liang Wu
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Correspondence to Jianming Zhang.

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Zhang, J., Chen, J. & Wu, L. Extraction of Stress Intensity Factors by Using the P-Version Finite Element Method and Contour Integral Method. Acta Mech. Solida Sin. 33, 836–850 (2020). https://doi.org/10.1007/s10338-020-00188-7

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  • DOI: https://doi.org/10.1007/s10338-020-00188-7

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