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International Journal of Fracture

, Volume 216, Issue 2, pp 185–210 | Cite as

Smooth Generalized/eXtended FEM approximations in the computation of configurational forces in linear elastic fracture mechanics

  • Diego Amadeu F. TorresEmail author
  • Clovis S. de Barcellos
  • Paulo de Tarso R. Mendonça
Orginal Paper
  • 110 Downloads

Abstract

The computation of crack severity parameters in the linear elastic fracture mechanics (LEFM) modeling is strongly dependent on the local quality of the approximated stress fields right at the crack tip vicinity. This work investigates the behavior of extrinsically enriched smooth mesh-based approximations, obtained via \(C^{k}\)-GFEM framework (Duarte et al. in Comput Methods Appl Mech Eng 196:33–56, 2006), in the computation of \(\mathcal {J}\)-integral in both pure mode I and mixed-mode loadings for two-dimensional problems of the LEFM. The method of configurational forces is used for this purpose as shown in Steinmann et al. (Int J Solids Struct 38:5509–5526, 2001), for instance, by performing some adaptations according to Häusler et al. (Int J Numer Methods Eng 85:1522–1542, 2011). As such method provides vector quantities, it is also possible to compute the angle \(\theta _{{\mathrm{ADV}}}\) of probable crack advance. The \(C^{k}\)-GFEM is quite versatile and shares similar features with the standard FEM regarding the domain partition and numerical integration (Mendonça et al. in Finite Elem Anal Des 47:698–717, 2011). The tests were conducted using three-noded triangular element meshes and numerical integrations were performed using only global coordinates. The evaluations combined different schemes of polynomial and discontinuous/singular (Moës et al. in Int J Numer Methods Eng 46:131–150, 1999) enrichments. The use of a smooth partition of unity (PoU) can influence the accuracy of computed crack severity parameters. The configurational forces computation is favored by the smoothness, reducing the dependence on the way the crack severity parameters are evaluated.

Keywords

Smooth generalized/extended FEM Crack modeling Configurational forces Eshelbian mechanics J-integral Angle of crack advance Convergence analysis Linear elastic fracture mechanics Mixed mode Crack severity parameters 

Notes

Acknowledgements

Diego A. F. Torres, Clovis S. de Barcellos and Paulo T. R. Mendonça gratefully acknowledge the financial support provided by the Brazilian government agency National Council for Scientific and Technological Development–CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) for this research, under research Grants 163.461/2012-0, 304.698/2013-0 and 304.702/2013-7, respectively. The authors also gratefully acknowledge the Grant provided by DIRPPG/UTFPR–Câmpus Londrina, through the DIRPPG 02/2017 Official Notice, for the final preparation of the manuscript.

References

  1. Anderson TL (2005) Fracture Mechanics: fundamentals and applications, 3rd edn. CRC Press, Boca RatonCrossRefGoogle Scholar
  2. Babuska I, Banerjee U (2012) Stable generalized finite element method (SGFEM). Comput Methods Appl Mech Eng 201–204:91–111CrossRefGoogle Scholar
  3. Babuška I, Melenk JM (1997) The partition of unity method. Int J Numer Methods Eng 40:727–758CrossRefGoogle Scholar
  4. Babuška I, Whiteman JR, Strouboulis T (2011) Finite elements: an introduction to the method and error estimation. Oxford University Press, New YorkGoogle Scholar
  5. Banks-Sills L, Sherman D (1992) On the computation of stress intensity factors for three-dimensional geometries by means of the stiffness derivative and J-integral methods. Int J Fract 53:1–20Google Scholar
  6. Barsoum RS (1974) Application of quadratic isoparametric element in linear fracture mechanics. Int J Fract 10:603–605CrossRefGoogle Scholar
  7. Béchet E, Minnebo H, Moës N, Burgardt B (2005) Improved implementation and robustness study of the XFEM for stress analysis around cracks. Int J Numer Methods Eng 64:1033–1056CrossRefGoogle Scholar
  8. Belytschko T, Black T (1999) Elastic crack growth in finite elements with minimal remeshing. Int J Numer Methods Eng 45:601–620CrossRefGoogle Scholar
  9. Belytschko T, LU YY, Gu L (1994) Element-free Galerkin method. Int J Numer Methods Eng 37:229–256CrossRefGoogle Scholar
  10. Belytschko T, Moës N, Usui S, Parimi C (2001) Arbitrary discontinuities in finite elements. Int J Numer Methods Eng 50:993–1013CrossRefGoogle Scholar
  11. Belytschko T, Gracie R, Ventura G (2009) A review of extended / generalized finite element methods for material modeling. Model Simul Mater Sci Eng 17:043001CrossRefGoogle Scholar
  12. Boresi AP, Chong KP, Lee JD (2011) Elasticity in engineering mechanics, 3rd edn. Wiley, HobokenGoogle Scholar
  13. Braun M (2005) Structural optimization by material forces, In: Steinmann P, Maugin GA, Mechanics of material forces. Advances in mechanics and mathematics, vol 11. p 211–218, Springer, BerlinGoogle Scholar
  14. Braun M (1997) Configurational forces induced by finite-element discretizations. Proc Estonian Acad Sci Phys Math 46:24–31Google Scholar
  15. Braun M (2007) Configurational forces in discrete elastic systems. Arch Appl Mech 77:85–93CrossRefGoogle Scholar
  16. Brebbia CA, Telles JCF, Wrobel LC (1984) Boundary element techniques: theory and applications in engineering. Springer, BerlinCrossRefGoogle Scholar
  17. Chahine E, Laborde P, Renard Y (2008) Crack tip enrichment in the XFEM using a cutoff function. Int J Numer Methods Eng 75:629–646CrossRefGoogle Scholar
  18. Chang JH, Wu DJ (2007) Stress intensity factor computation along a non-planar curved crack in three dimensions. Int J Solids Struct 44:371–386CrossRefGoogle Scholar
  19. Cherepanov G (1967) Rasprostranenie trechin v sploshnoi srede. Prikladnaja Matematika i Mekhanica 31:478–488Google Scholar
  20. Chessa J, Wang H, Belytschko T (2003) On the construction of blending elements for local partition of unity enriched finite elements. Int J Numer Methods Eng 57:1015–1038CrossRefGoogle Scholar
  21. de Barcellos CS, Mendonça PTR, Duarte CA (2009) A \(C^{k}\) continuous generalized finite element formulations applied to laminated Kirchhoff plate model. Comput Mech 44:377–393CrossRefGoogle Scholar
  22. deLorenzi HG (1982) On the energy release rate and the J-integral for 3-D crack configurations. Int J Fract 19:183–193CrossRefGoogle Scholar
  23. deLorenzi HG (1985) Energy release rate calculations by the finite element method. Eng Fract Mech 21:129–143CrossRefGoogle Scholar
  24. Denzer R, Barth FJ, Steinmann P (2003) Studies in elastic fracture mechanics based on the material force method. Int J Numer Methods Eng 58:1817–1835CrossRefGoogle Scholar
  25. Duarte CA, Migliano DQ, Baker EB (2005) A technique to combine meshfree- and finite element-based partition of unity approximations. Technical Report, Department of Civil and Environmental Engineering. University of Illinois at Urbana-ChampaignGoogle Scholar
  26. Duarte CA, Babuška I (2002) Mesh-independent p-orthotropic enrichment using the generalized finite element method. Int J Numer Methods Eng 55:1477–1492CrossRefGoogle Scholar
  27. Duarte CA, Babuška I, Oden JT (2000) Generalized finite element method for three-dimensional structural mechanics problems. Comput Struct 77:215–232CrossRefGoogle Scholar
  28. Duarte CA, Hamzeh ON, Liszka TJ, Tworzydlo WW (2001) A generalized finite element method for the simulation of three-dimensional dynamic crack propagation. Comput Methods Appl Mech Eng 190:2227–2262CrossRefGoogle Scholar
  29. Duarte CA, Kim D-J, Quaresma DM (2006) Arbitrarily smooth generalized finite element approximations. Comput Methods Appl Mech Eng 196:33–56CrossRefGoogle Scholar
  30. Edwards HC (1996) \(C^{\infty }\) finite element basis functions. Technical Report, TICAM Report, The University of Texas at Austin, p 96-45Google Scholar
  31. Eischen JW (1987) An improved method for computing the \(J_{2}\) integral. Eng Fract Mech 26:691–700CrossRefGoogle Scholar
  32. Eshelby JD (1951) The force on an elastic singularity. Philos Trans R Soc Lond A 244:87–112CrossRefGoogle Scholar
  33. Eshelby JD (1975) The elastic energy-momentum tensor. J Elast 5:321–335CrossRefGoogle Scholar
  34. Freitas A, Torres DAF, Mendonça PTR (2015) Comparative analysis of \(C^{k}\)- and \(C^{0}\)-GFEM applied to two-dimensional problems of confined plasticity. Lat Am J Solids Struct 12(5):861–882CrossRefGoogle Scholar
  35. Fries T-P, Belytschko T (2006) The intrinsic XFEM: a method for arbitrary discontinuities without additional unknowns. Int J Numer Methods Eng 68:1358–1385CrossRefGoogle Scholar
  36. Fries TP, Belytschko T (2010) The extended / generalized finite element method: an overview of the method and its application. Int J Numer Methods Eng 84:253–304Google Scholar
  37. Giner E, Fuenmayor FJ, Besa AJ, Tur M (2002) An implementation of the stiffness derivative method as a discrete analytical sensitivity analysis and its application to mixed mode in LEFM. Eng Fract Mech 69:2051–2071CrossRefGoogle Scholar
  38. Glaser J, Steinmann P (2006) On material forces within the extended finite element method. In: Benallal A, Botsis J, Fleck NA et al (eds) Proceedings of the sixth European solid mechanics conference ESMC, August 2006, Budapest, HungaryGoogle Scholar
  39. Glaser J, Steinmann P (2007) Material force method within the framework of the XFEM - distribution of nodal material forces. In: Proceedings in applied mathematics and mechanics, Sixth international congress on industrial applied mathematics (ICIAM07) and GAMM annual meeting, vol 7. Zrich, p 4030017-4030018Google Scholar
  40. Gross D, Mueller R, Kolling S (2002) Configurational forces - morphology evolution and finite elements. Mech Res Commun 29:529–536CrossRefGoogle Scholar
  41. Gupta V, Duarte CA, Babuška I, Banerjee U (2013) A stable and optimally convergent generalized FEM (SGFEM) for linear elastic fracture mechanics. Comput Methods Appl Mech Eng 266:23–39CrossRefGoogle Scholar
  42. Gurtin ME (2000) Configurational forces as basic concepts of continuum physics. Springer, New YorkGoogle Scholar
  43. Häusler SM, Lindhorst K, Horst P (2011) Combination of the material force concept and the extended finite element method for mixed mode crack growth simulations. Int J Numer Methods Eng 85:1522–1542CrossRefGoogle Scholar
  44. Heintz P, Larsson F, Hansbo P, Runesson K (2004) Adaptive strategies and error control for computing material forces in fracture mechanics. Int J Numer Methods Eng 60:1287–1299CrossRefGoogle Scholar
  45. Hellen TK (1975) On the method of virtual crack extensions. Int J Numer Methods Eng 9:187–207CrossRefGoogle Scholar
  46. Hwang CG, Wawrzynek PA, Tayebi AK, Ingraffea AR (1998) On the virtual crack extension method for calculation of the rates or energy release rate. Eng Fract Mech 59:521–542CrossRefGoogle Scholar
  47. Irwin GR (1956) Onset of fast crack propagation in high strength steel and aluminum alloys, Research Report, Department of the Navy, Office of Naval Research - Washington, D.CGoogle Scholar
  48. Irwin GR (1957) Analysis of stress and strains near the end of a crack traversing a plate. J Appl Mech 24:361–364Google Scholar
  49. Ishikawa H, Kitagawa H, Okamura H (1979) \(J\)-integral of a mixed-mode crack and its application. In: Miller K, Smith R (eds) Proceedings of the 3rd international conference of mechanical behaviors of materials. ICM 3, vol 3. The Netherlands, p 447–455Google Scholar
  50. Kienzler R, Herrmann G (2000) Mechanics in material space with applications to defect and fracture mechanics. Springer, BerlinGoogle Scholar
  51. Kim D-J, Pereira JP, Duarte CA (2010) Analysis of three-dimensional fracture mechanics problems: a two-scale approach using coarse generalized FEM meshes. Int J Numer Methods Eng 81:335–365Google Scholar
  52. Kreyszig E (1989) Introductory functional analysis with applications. Wiley, HobokenGoogle Scholar
  53. Kuna M (2013) Finite elements in fracture mechanics: theory, numerics, applications. Springer, BerlinCrossRefGoogle Scholar
  54. Laborde P, Pommier J, Renard Y, Salaun M (2005) High-order extended finite element method for cracked domains. Int J Numer Methods Eng 64:354–381CrossRefGoogle Scholar
  55. Lachat JC, Watson JO (1976) Effective numerical treatment of boundary integral equations: a formulation for three dimensional elastostatic. Int J Numer Methods Eng 10:991–1005CrossRefGoogle Scholar
  56. Larsson R, Fagerström M (2005) A framework for fracture modelling based on the material forces concept with XFEM kinematics. Int J Numer Methods Eng 62:1763–1788CrossRefGoogle Scholar
  57. Li S, Wang G (2008) Introduction to micromechanics and nanomechanics. World Scientific, SingaporeCrossRefGoogle Scholar
  58. Li FZ, Shih CF, Needleman A (1985) A comparison of methods for calculating energy release rates. Eng Fract Mech 21:405–421CrossRefGoogle Scholar
  59. Lin SC, Abel J (1988) Variational approach for a new direct-integration form of the virtual crack extension method. Int J Fract 38:217–235Google Scholar
  60. Materna D, Barthold F-J (2008) On variational sensitivity analysis and configurational mechanics. Comput Mech 41:661–681CrossRefGoogle Scholar
  61. Maugin GA (1993) Material inhomogeneities in elasticity. Chapmann and Hall, LondonCrossRefGoogle Scholar
  62. Maugin GA (1995) Material forces: concepts and applications. Appl Mech Rev 48:213–245CrossRefGoogle Scholar
  63. Mendonça PTR, de Barcellos CS, Torres DAF (2011) Analysis of anisotropic Mindlin plate model by continuous and non-continuous GFEM. Finite Elem Anal Des 47:698–717CrossRefGoogle Scholar
  64. Mendonça PTR, de Barcellos CS, Torres DAF (2013) Robust \(C^{k}/C^{0}\) generalized FEM approximations for higher-order conformity requirements: application to Reddy’s HSDT model for anisotropic laminated plates. Compos Struct 96:332–345CrossRefGoogle Scholar
  65. Miehe C, Gürses E (2007) A robust algorithm for configurational-force-driven brittle crack propagation with r-adaptive mesh alignment. Int J Numer Methods Eng 72:127–155CrossRefGoogle Scholar
  66. Miehe C, Gürses E, Birkle M (2007) A computational framework of configurational-force-driven brittle fracture based on incremental energy minimization. Int J Fract 145:245–259CrossRefGoogle Scholar
  67. Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46:131–150CrossRefGoogle Scholar
  68. Moran B, Shih CF (1987) A general treatment of crack tip contour integrals. Int J Fract 35:295–310CrossRefGoogle Scholar
  69. Mueller R, Maugin GA (2002) On material forces and finite element discretizations. Comput Mech 29:52–60CrossRefGoogle Scholar
  70. Mueller R, Kolling S, Gross D (2002) On configurational forces in the context of the finite element method. Int J Numer Methods Eng 53:1557–1574CrossRefGoogle Scholar
  71. Nishioka T, Atluri N (1984) On the computation of mixed-mode \(K\)-factors for a dynamically propagating crack, using path-independent integrals \(J^{^{\prime }}_{k}\). Eng Fract Mech 20:193–208CrossRefGoogle Scholar
  72. Oden JT, Reddy JN (1976) An introduction to the mathematical theory of finite elements. Wiley, New YorkGoogle Scholar
  73. Parks DM (1974) A stiffness derivative finite element technique for determination of crack tip stress intensity factors. Int J Fract 10:487–502CrossRefGoogle Scholar
  74. Qian G, González-Albuixech VF, Niffenegger M, Giner E (2016) Comparison of \(K_{I}\) calculation methods. Eng Fract Mech 156:52–67CrossRefGoogle Scholar
  75. Rice JR (1968) A path independent integral and the approximate analysis of strain concentration by notches and cracks. J Appl Mech 35:379–386CrossRefGoogle Scholar
  76. Rvachev VL (1982) Theory of \(R\)-functions and some of its applications. Naukova Dumka (in Russian)Google Scholar
  77. Rvachev VL, Sheiko TI (1995) \(R\)-functions in boundary value problems in mechanics. Appl Mech Rev 48:151–188CrossRefGoogle Scholar
  78. Schweitzer MA (2008) Meshfree and generalized finite element methods (Habilitation thesis), R. F. -W. Universität BonnGoogle Scholar
  79. Shapiro V (2007) Semi-analytic geometry with \(R\)-functions. Acta Numerica 16:239–303CrossRefGoogle Scholar
  80. Shepard D (1968) A two-dimensional interpolation function for irregularly-spaced data. In: Proceedings of the 23rd ACM national conference - ACM’68, New York, p 517-524Google Scholar
  81. Shih CF, Moran B, Nakamura T (1986) Energy release rate along a three-dimensional crack front in a thermally stressed body. Int J Fract 30:79–102Google Scholar
  82. Steinmann P (2000) Application of material forces to hyrelastostatic fracture mechanics. part I: continuum mechanics setting. Int J Solids Struct 37:7371–7391CrossRefGoogle Scholar
  83. Steinmann P, Ackermann D, Barth FJ (2001) Application of material forces to hyperelastostatic fracture mechanics. II. computational setting. Int J Solids Struct 38:5509–5526CrossRefGoogle Scholar
  84. Steinmann P, Scherer M, Denzer R (2009) Secret and joy of configurational mechanics: from foundations in continuum mechanics to applications in computational mechanics. J Appl Math Mech 89:614–630Google Scholar
  85. Strouboulis T, Babuska I, Datta DK, Copps K, Gangaraj SK (2000) A posteriori estimation and adaptive control of the errors in the quantity of interest. Part I: a posteriori estimation of the error in the von Mises stress and tye stress intensity factor. Comput Methods Appl Mech Eng 181(2000):261–294CrossRefGoogle Scholar
  86. Sukumar N, Malsch EA (2006) Recent advances in the construction of polygonal finite element interpolants. Arch Comput Methods Eng 13:129–163CrossRefGoogle Scholar
  87. Sukumar N, Tabarraei A (2004) Conforming polygonal finite elements. Int J Numer Methods Eng 82:2045–2066CrossRefGoogle Scholar
  88. Szabó BA (1986) Estimation and control of error based on \(p\)-convergence. In: I. Babuška, Ed., Accuracy estimates and adpative refinements in finite element computations. Wiley Series in Numerical methods in Engineering, p 61–78Google Scholar
  89. Szabó B, Babuška I (2011) Introductions to finite element method: formulation, verification and validation. Wiley series in computational mechanics. Wiley, HobokenCrossRefGoogle Scholar
  90. Szabó BA, Yosibash Z (1996) Superconvergent extraction of flux intensity factors and first derivatives from finite element solutions. Comput Methods Appl Mech Eng 129:349–370CrossRefGoogle Scholar
  91. Talischi C, Paulino GH, Pereira A, Menezes IFM (2010) Polygonal finite elements for topology optimization: a unifying paradigm. Int J Numer Methods Eng 82:671–698Google Scholar
  92. Tarancón JE, Vercher A, Giner E, Fuenmayor FJ (2009) Enhanced blending elements for XFEM applied to linear elastic fracture mechanics. Int J Numer Methods Eng 77:126–148CrossRefGoogle Scholar
  93. Torres DAF, Mendonça PTR (2010) Analysis of piezoelectric laminates by generalized finite element method and mixed layerwise-HSDT models. Smart Mater Struct 19:035004CrossRefGoogle Scholar
  94. Torres DAF, Mendonça PTR, de Barcellos CS (2011) Evaluation and verification of an HSDT-Layerwise generalized finite element formulation for adaptive piezoelectric laminated plates. Comput Methods Appl Mech Eng 200:675–691CrossRefGoogle Scholar
  95. Verron E, Aït-Bachir M, Castaing P (2009) Some new properties of the Eshelby stress tensor. In: P. Steinmann, IUTAM symposium on progress in the theory and numerics of configurational mechanics, vol 17. p 27-35Google Scholar
  96. Waismann H (2010) An analytical stiffness derivative extended finite element technique for extraction of crack tip strain energy release rates. Eng Fract Mech 77:3204–3215CrossRefGoogle Scholar
  97. Wandzura S, Xiao H (2003) Symmetric quadrature rules on a triangle. Comput Math Appl 45:1829–1840CrossRefGoogle Scholar
  98. Westergaard HM (1939) Bearing pressures and cracks. J Appl Mech 6:49–53Google Scholar
  99. Xiao Q, Karihaloo B (2006) Improving the accuracy of XFEM crack tip fields using higher order quadrature and statically admissible stress recovery. Int J Numer Methods Eng 66:1378–1410CrossRefGoogle Scholar
  100. Yosibash Z (2012) Singularities in elliptic boundary value problems and elasticity and their connections with failure initiation. Springer, BerlinCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mechanical EngineeringFederal University of Technology of Paraná - UTFPRLondrinaBrazil
  2. 2.Group of Mechanical Analysis and Design, Department of Mechanical EngineeringFederal University of Santa Catarina - UFSCFlorianópolisBrazil

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