Acta Mechanica

, Volume 230, Issue 10, pp 3593–3612 | Cite as

A nonlocal fracture criterion and its effect on the mesh dependency of GraFEA

  • Parisa Khodabakhshi
  • J. N. ReddyEmail author
  • Arun Srinivasa
Original Paper


Recently, Khodabakhshi et al. (Meccanica 51(12):3129–3147, 2016. presented a new method (by the name GraFEA) capable of studying fracture based on edge breakage within a classical FEA scheme which combines the best features of FEA and bond-breakage methods in a single framework. In this study, an attempt is made to investigate the mesh dependency of GraFEA by a set of numerical examples, and it is shown that using a local fracture criterion for edge failure will yield mesh-dependent results, as is already well known. A physically motivated nonlocal fracture criterion is implemented along with the edge breakage model, and its efficacy in eliminating the mesh sensitivity is investigated. The nonlocal criterion introduces a length scale into the problem. It is shown that by increasing the magnitude of the length scale parameter from zero, the damage pattern moves from localized fracture to diffuse damage pattern, yet with complete material separation (fracture) across a certain plane. It is shown by numerical results that, as expected, the introduction of the nonlocal fracture criterion eliminates the issue of mesh sensitivity, and thus predictions of the approximate crack paths and damage zone can be done within the classical FEA framework without the need for special formulations.



The first two authors gratefully acknowledge the support of the present research by the Oscar S. Wyatt Endowed Chair.


  1. 1.
    Khodabakhshi, P., Reddy, J.N., Srinivasa, A.: GraFEA: a graph-based finite element approach for the study of damage and fracture in brittle materials. Meccanica 51(12), 3129–3147 (2016). MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Xu, X.P., Needleman, A.: Numerical simulations of fast crack growth in brittle solids. J. Mech. Phys. Solids 42(9), 1397–1434 (1994)zbMATHCrossRefGoogle Scholar
  3. 3.
    Camacho, G.T., Ortiz, M.: Computational modelling of impact damage in brittle materials. Int. J. Solids Struct. 33(20–22), 2899–2938 (1996)zbMATHCrossRefGoogle Scholar
  4. 4.
    Rabczuk, T., Belytschko, T.: Cracking particles: a simplified meshfree method for arbitrary evolving cracks. Int. J. Numer. Methods Eng. 61(13), 2316–2343 (2004)zbMATHCrossRefGoogle Scholar
  5. 5.
    Belytschko, T., Black, T.: Elastic crack growth in finite elements with minimal remeshing. Int. J. Numer. Methods Eng. 45(5), 601–620 (1999).<601::AID-NME598>3.0.CO;2-S zbMATHCrossRefGoogle Scholar
  6. 6.
    Moës, N., Dolbow, J., Belytschko, T.: A finite element method for crack growth without remeshing. Int. J. Numer. Methods Eng. 46(1), 131–150 (1999).<131::AID-NME726>3.0.CO;2-J MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Moës, N., Belytschko, T.: Extended finite element method for cohesive crack growth. Eng. Fract. Mech. 69(7), 813–833 (2002). CrossRefGoogle Scholar
  8. 8.
    Pijaudier-Cabot, G., Bažant, Z.P.: Nonlocal damage theory. J. Eng. Mech. 113(10), 1512–1533 (1987)zbMATHCrossRefGoogle Scholar
  9. 9.
    Bažant, Z.P., Pijaudier-Cabot, G.: Nonlocal continuum damage, localization instability and convergence. J. Appl. Mech. 55(2), 287–293 (1988)zbMATHCrossRefGoogle Scholar
  10. 10.
    Jirsek, M., Zimmermann, T.: Rotating crack model with transition to scalar damage. J. Eng. Mech. 124(3), 277–284 (1998). CrossRefGoogle Scholar
  11. 11.
    Bažant, Z.P.: Instability, ductility and size effect in strain-softening concrete. J. Eng. Mech. Div. ASCE 102, 331–344 (1975)Google Scholar
  12. 12.
    Bažant, Z.P., Cedolin, L.: Finite element modeling of crack band propagation. J. Struct. Eng. 109(1), 69–92 (1983)CrossRefGoogle Scholar
  13. 13.
    Bažant, Z.P., Oh, B.H.: Crack band theory for fracture of concrete. Matér. Constr. 16(3), 155–177 (1983). CrossRefGoogle Scholar
  14. 14.
    Bažant, Z.P., Jirásek, M.: Nonlocal integral formulations of plasticity and damage: survey of progress. J. Eng. Mech. 128(11), 1119–1149 (2002)CrossRefGoogle Scholar
  15. 15.
    Needleman, A.: Material rate dependence and mesh sensitivity in localization problems. Comput. Methods Appl. Mech. Eng. 67(1), 69–85 (1988). zbMATHCrossRefGoogle Scholar
  16. 16.
    Triantafyllidis, N., Aifantis, E.C.: A gradient approach to localization of deformation. I. Hyperelastic materials. J. Elast. 16(3), 225–237 (1986). MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Lasry, D., Belytschko, T.: Localization limiters in transient problems. Int. J. Solids Struct. 24(6), 581–597 (1988). zbMATHCrossRefGoogle Scholar
  18. 18.
    Peerlings, R.H.J., Geers, M.G.D., De Borst, R., Brekelmans, W.A.M.: A critical comparison of nonlocal and gradient-enhanced softening continua. Int. J. Solids Struct. 38(44), 7723–7746 (2001)zbMATHCrossRefGoogle Scholar
  19. 19.
    Peerlings, R., De Borst, R., Brekelmans, W., Geers, M.: Localisation issues in local and nonlocal continuum approaches to fracture. Eur. J. Mech. A Solids 21(2), 175–189 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Bažant, Z.P., Belytschko, T.B., Ta-Peng, C.: Continuum theory for strain-softening. J. Eng. Mech. 110(12), 1666–1692 (1984). CrossRefGoogle Scholar
  21. 21.
    Belytschko, T., Bažant, Z.P., Yul-Woong, H., Ta-Peng, C.: Strain-softening materials and finite-element solutions. Comput. Struct. 23(2), 163–180 (1986). CrossRefGoogle Scholar
  22. 22.
    Jirásek, M.: Nonlocal models for damage and fracture: comparison of approaches. Int. J. Solids Struct. 35(31–32), 4133–4145 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Challamel, N., Wang, C.M.: The small length scale effect for a non-local cantilever beam: a paradox solved. Nanotechnology 19(34), 345703 (2008)CrossRefGoogle Scholar
  24. 24.
    Challamel, N.: Variational formulation of gradient or/and nonlocal higher-order shear elasticity beams. Compos. Struct. 105(Supplement C), 351–368 (2013). CrossRefGoogle Scholar
  25. 25.
    Srinivasa, A.R., Reddy, J.N.: An overview of theories of continuum mechanics with nonlocal elastic response and a general framework for conservative and dissipative systems. Appl. Mech. Rev. 69(3), 030802 (18) (2017). CrossRefGoogle Scholar
  26. 26.
    Needleman, A.: Some issues in cohesive surface modeling. In: Procedia IUTAM, Mechanics for the World: Proceedings of the 23rd International Congress of Theoretical and Applied Mechanics, ICTAM2012, vol. 10, pp. 221–246 (2014). CrossRefGoogle Scholar
  27. 27.
    de Borst, R., Remmers, J.J.C., Needleman, A.: Mesh-independent discrete numerical representations of cohesive-zone models. Eng. Fract. Mech. 73(2), 160–177 (2006)CrossRefGoogle Scholar
  28. 28.
    Song, J.-H., Wang, H., Belytschko, T.: A comparative study on finite element methods for dynamic fracture. Comput. Mech. 42(2), 239–250 (2008)zbMATHCrossRefGoogle Scholar
  29. 29.
    Silling, S.A.: Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48(1), 175–209 (2000). MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Silling, S.A., Epton, M., Weckner, O., Xu, J., Askari, E.: Peridynamic states and constitutive modeling. J. Elast. 88(2), 151–184 (2007). MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Seleson, P.D.: Peridynamic multi scale models for the mechanics of materials: constitutive relations, upscaling from atomistic systems, and interface problems. Ph.D. thesis, Florida State University (2010)Google Scholar
  32. 32.
    Henke, S.F., Shanbhag, S.: Mesh sensitivity in peridynamic simulations. Comput. Phys. Commun. 185(1), 181–193 (2014). MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Seleson, P., Parks, M.: On the role of the influence function in the peridynamic theory. Int. J. Multiscale Comput. Eng. 9(6), 689–706 (2011)CrossRefGoogle Scholar
  34. 34.
    Bobaru, F., Zhang, G.: Why do cracks branch? A peridynamic investigation of dynamic brittle fracture. Int. J. Fract. 196(1), 59–98 (2015). CrossRefGoogle Scholar
  35. 35.
    Dipasquale, D., Sarego, G., Zaccariotto, M., Galvanetto, U.: Dependence of crack paths on the orientation of regular 2d peridynamic grids. Eng. Fract. Mech. 160, 248–263 (2016). CrossRefGoogle Scholar
  36. 36.
    Ghajari, M., Iannucci, L., Curtis, P.: A peridynamic material model for the analysis of dynamic crack propagation in orthotropic media. Comput. Methods Appl. Mech. Eng. 276, 431–452 (2014). MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Hu, W., Ha, Y.D., Bobaru, F.: Peridynamic model for dynamic fracture in unidirectional fiber-reinforced composites. Comput. Methods Appl. Mech. Eng. 217–220, 247–261 (2012). MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Seleson, P., Du, Q., Parks, M.L.: On the consistency between nearest-neighbor peridynamic discretizations and discretized classical elasticity models. Comput. Methods Appl. Mech. Eng. 311, 698–722 (2016). MathSciNetCrossRefGoogle Scholar
  39. 39.
    Reddy, J.N., Srinivasa, A.R.: On the force-displacement characteristics of finite elements for elasticity and related problems. Finite Elem. Anal. Des. 104, 35–40 (2015)CrossRefGoogle Scholar
  40. 40.
    Reddy, J.N.: An Introduction to the Finite Element Method. McGraw Hill, New York (2006)Google Scholar
  41. 41.
    Ritchie, R.O., Knott, J.F., Rice, J.R.: On the relationship between critical tensile stress and fracture toughness in mild steel. J. Mech. Phys. Solids 21(6), 395–410 (1973)CrossRefGoogle Scholar
  42. 42.
    Lin, T., Evans, A.G., Ritchie, R.O.: A statistical model of brittle fracture by transgranular cleavage. J. Mech. Phys. Solids 34(5), 477–497 (1986)CrossRefGoogle Scholar
  43. 43.
    Mao, Y., Talamini, B., Anand, L.: Rupture of polymers by chain scission. Extreme Mech. Lett. 13, 17–24 (2017). CrossRefGoogle Scholar
  44. 44.
    Pang, S.-D., Bažant, Z.P., Le, J.-L.: Statistics of strength of ceramics: finite weakest-link model and necessity of zero threshold. Int. J. Fract. 154(1), 131–145 (2008). zbMATHCrossRefGoogle Scholar
  45. 45.
    De Borst, R., Sluys, L.J., Muhlhaus, H.-B., Pamin, J.: Fundamental issues in finite element analyses of localization of deformation. Eng. Comput. 10(2), 99–121 (1993)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Oden Institute for Computational Engineering and SciencesUniversity of Texas at AustinAustinUSA
  2. 2.Department of Mechanical EngineeringTexas A&M UniversityCollege StationUSA

Personalised recommendations