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J-integral evaluation for 2D mixed-mode crack problems employing a meshfree stabilized conforming nodal integration method

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Abstract

Two-dimensional (2D) in-plane mixed-mode fracture mechanics problems are analyzed employing an efficient meshfree Galerkin method based on stabilized conforming nodal integration (SCNI). In this setting, the reproducing kernel function as meshfree interpolant is taken, while employing the SCNI for numerical integration of stiffness matrix in the Galerkin formulation. The strain components are smoothed and stabilized employing Gauss divergence theorem. The path-independent integral (J-integral) is solved based on the nodal integration by summing the smoothed physical quantities and the segments of the contour integrals. In addition, mixed-mode stress intensity factors (SIFs) are extracted from the J-integral by decomposing the displacement and stress fields into symmetric and antisymmetric parts. The advantages and features of the present formulation and discretization in evaluation of the J-integral of in-plane 2D fracture problems are demonstrated through several representative numerical examples. The mixed-mode SIFs are evaluated and compared with reference solutions. The obtained results reveal high accuracy and good performance of the proposed meshfree method in the analysis of 2D fracture problems.

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References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Liu WK, Jun S, Zhang YF (1995) Reproducing kernel particle methods. Int J Numer Methods Fluid 20:1081–1106

    Article  MathSciNet  MATH  Google Scholar 

  3. Atluri SN, Zhu T (1998) A new meshless local Petrov–Galerkin (MLPG) approach in computational mechanics. Comput Mech 22:117–127

    Article  MathSciNet  MATH  Google Scholar 

  4. Belytschko T, Black T (1999) Elastic crack growth in finite elements with minimal remeshing. Int J Numer Methods Eng 45:601–620

    Article  MathSciNet  MATH  Google Scholar 

  5. Melenk JM, Babuška I (1996) The partition of unity finite element method: basic theory and applications. Comput Methods Appl Mech Eng 139:289–314

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Singh IV, Mishra BK, Pant M (2011) An enrichment based new criterion for the simulation of multiple interacting cracks using element free Galerkin method. Int J Fract 167:157–171

    Article  MATH  Google Scholar 

  8. Liu ZL, Menouillard T, Belytschko T (2011) An XFEM/Spectral element method for dynamic crack propagation. Int J Fract 169:183–198

    Article  MATH  Google Scholar 

  9. Sladek J, Sladek V, Krahulec S, Zhang CH, Wünsche M (2013) Crack analysis in decagonal quasicrystals by the MLPG. Int J Fract 181:115–126

    Article  Google Scholar 

  10. Liu P, Yu T, Bui QT, Zhang C (2013) Transient dynamic crack analysis in non-homogeneous functionally graded piezoelectric materials by the X-FEM. Comput Mater Sci 69:542–558

    Article  Google Scholar 

  11. Nguyen NT, Bui QT, Zhang C, Truong TT (2014) Crack growth modeling in elastic solids by the extended meshfree Galerkin radial point interpolation method. Eng Anal Bound Elem 44:87–97

    Article  MathSciNet  MATH  Google Scholar 

  12. Kang Z, Bui QT, Nguyen DD, Saitoh T, Hirose S (2015) An extended consecutive-interpolation quadrilateral element (XCQ4) applied to linear elastic fracture mechanics. Acta Mech 226:3991–4015

    Article  MathSciNet  MATH  Google Scholar 

  13. Bui QT (2015) Extended isogeometric dynamic and static fracture analysis for cracks in piezoelectric materials using NURBS. Comput Methods Appl Mech Eng 295:470–509

    Article  MathSciNet  Google Scholar 

  14. Tanaka S, Okada H, Okazawa S, Fujikubo M (2013) Fracture mechanics analysis using the wavelet Galerkin method and extended finite element method. Int J Numer Methods Eng 93:1082–1108

    Article  MathSciNet  Google Scholar 

  15. Tanaka S, Suzuki H, Ueda S, Sannomaru S (2015) An extended wavelet Galerkin method with a high-order B-spline for 2D crack problems. Acta Mech 226:2159–2175

    Article  MathSciNet  MATH  Google Scholar 

  16. Tanaka S, Sannomaru S, Imachi M, Hagihara S, Okazawa S, Okada H (2015) Analysis of dynamic stress concentration problems employing spline-based wavelet Galerkin method. Eng Anal Bound Elem 58:129–139

    Article  MathSciNet  Google Scholar 

  17. Tanaka S, Okada H, Okazawa S (2012) A wavelet Galerkin method employing B-spline bases for solid mechanics problems without the use of a fictitious domain. Comput Mech 50:35–48

    Article  MathSciNet  MATH  Google Scholar 

  18. Chen JS, Wu CT, Yoon S, You Y (2001) A stabilized conforming nodal integration for Galerkin meshfree methods. Int J Numer Methods Eng 50:435–466

    Article  MATH  Google Scholar 

  19. Chen JS, Yoon S, Wu CT (2002) Non-linear version of stabilized conforming nodal integration for Galerkin mesh-free methods. Int J Numer Methods Eng 53:2587–2615

    Article  MATH  Google Scholar 

  20. Wang D, Chen JS (2004) Locking-free stabilized conforming nodal integration for meshfree Mindlin-Reissner plate formulation. Comput Methods Appl Mech Eng 193:1065–1083

    Article  MATH  Google Scholar 

  21. Wang D, Chen JS (2006) A locking-free meshfree curved beam formulation with the stabilized conforming nodal integration. Comput Mech 39:83–90

    Article  MATH  Google Scholar 

  22. Wang D, Sun Y (2011) A Galerkin meshfree method with stabilized conforming nodal integration for geometrically nonlinear analysis of shear deformable plates. Int J Comput Meth 8:685

    Article  MathSciNet  MATH  Google Scholar 

  23. Sadamoto S, Tanaka S, Okazawa S (2013) Elastic large deflection analysis of plates subjected to uniaxial thrust using meshfree Mindlin-Reissner formulation. Comput Mech 52:1313–1330

    Article  MathSciNet  MATH  Google Scholar 

  24. Organ D, Fleming M, Terry T, Belytschko T (1996) Continuous meshless approximations for nonconvex bodies by diffraction and transparency. Comput Mech 18:225–235

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Joyot P, Trunzler J, Chinesta F (2005) Enriched reproducing kernel approximation: reproducing functions with discontinuous derivatives. Lect Notes Comput Sci Eng 43:93–107

    Article  MathSciNet  MATH  Google Scholar 

  27. Rao BN, Rahman S (2003) Mesh-free analysis of cracks in isotropic functionally graded materials. Eng Fract Mech 70:1–27

    Article  Google Scholar 

  28. Wang D, Chen JS (2008) A Hermite reproducing kernel approximation for thin-plate analysis with sub-domain stabilized conforming integration. Int J Numer Methods Eng 74:368–390

    Article  MATH  Google Scholar 

  29. Wang D, Lin Z (2010) Free vibration analysis of thin plates using Hermite reproducing kernel Galerkin meshfree method with sub-domain stabilized conforming integration. Comput Mech 46:703–719

    Article  MathSciNet  MATH  Google Scholar 

  30. Wang D, Lin Z (2011) Dispersion and transient analyses of Hermite reproducing kernel Galerkin meshfree method with sub-domain stabilized conforming integration for thin beam and plate structures. Comput Mech 48:47–63

    Article  MathSciNet  MATH  Google Scholar 

  31. Tanaka S, Sadamoto S, Okazawa S (2012) Nonlinear thin-plate bending analyses using the Hermite reproducing kernel approximation. Int J Comput Methods 9:1240012

    Article  MathSciNet  Google Scholar 

  32. Nikishkov GP, Atluri SN (1987) Calculation of fracture mechanics parameters for an arbitrary three-dimensional crack, by the “equivalent domain integral” method. Int J Numer Methods Eng 24:1801–1821

    Article  MATH  Google Scholar 

  33. Raju IS, Shivakumar KN (1990) An equivalent domain integral method in the two-dimensional analysis of mixed mode crack problems. Eng Fract Mech 37:707–725

    Article  Google Scholar 

  34. Tanaka S, Suzuki H, Sadamoto S, Imachi M, Bui QT (2015) Analysis of cracked shear deformable plates by an effective meshfree plate formulation. Eng Fract Mech 144:142–157

    Article  Google Scholar 

  35. Ishikawa H (1980) A finite element analysis of stress intensity factors for combined tensile and shear loading by only a virtual crack extension. Int J Fract 16:R243–246

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  37. Bowie OL (1972) Solutions of plane crack problems by mapping technique (Chapter 1). In: Sih GC (ed) Mechanics of fracture. nternational Publishing, Leyden, pp 1–55

    Google Scholar 

  38. MSC.Marc 2005r3, User’s Guide

  39. Yau J, Wang S, Corten H (1980) A mixed-mode crack analysis of isotropic solids using conservation laws of elasticity. J Appl Mech 47:335–341

    Article  MATH  Google Scholar 

  40. Lim IL, Johnston IW, Choi SK (1993) Stress intensity factors for semi-circular specimens under three-point bending. Eng Fract Mech 44:363–382

    Article  Google Scholar 

Download references

Acknowledgments

This research was partially supported by JSPS KAKENHI Grant-in-Aid for Scientific Research of (A)(15H02328), (B)(15H04212) and (C)(15K06632). This work was performed under the Cooperative Research Program of the Joining and Welding Research Institute, Osaka University. Tinh Quoc Bui gratefully acknowledges the support from the Grant-in-Aid for Scientific Research (No. 26-04055) - JSPS. TT Yu gratefully acknowledges the supports of the National Natural Science Foundation of China (Grant No. 51179063) and the National Sci-Tech Support Plan of China (Grant No. 2015BAB07B10).

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Authors and Affiliations

  1. Graduate School of Engineering, Hiroshima University, 4-1, Kagamiyama 1-chome, Higashi-Hiroshima, 739-8527, Japan

    Satoyuki Tanaka, Hirotaka Suzuki, Shota Sadamoto & Shogo Sannomaru

  2. Department of Engineering Mechanics, Hohai University, Nanjing, China

    Tiantang Yu

  3. Department of Mechanical and Environmental Informatics, Tokyo Institute of Technology, Tokyo, Japan

    Tinh Quoc Bui

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  1. Satoyuki Tanaka
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  2. Hirotaka Suzuki
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  3. Shota Sadamoto
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  4. Shogo Sannomaru
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  5. Tiantang Yu
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  6. Tinh Quoc Bui
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Correspondence to Satoyuki Tanaka.

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Tanaka, S., Suzuki, H., Sadamoto, S. et al. J-integral evaluation for 2D mixed-mode crack problems employing a meshfree stabilized conforming nodal integration method. Comput Mech 58, 185–198 (2016). https://doi.org/10.1007/s00466-016-1288-9

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  • DOI: https://doi.org/10.1007/s00466-016-1288-9

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