International Journal of Fracture

, Volume 203, Issue 1–2, pp 183–209 | Cite as

Ductile failure modeling and simulations using coupled FE–EFG approach

  • A. S. Shedbale
  • I. V. Singh
  • B. K. Mishra
  • Kamal Sharma


In the present work, a ductile fracture model has been employed to predict the failure of tensile specimen using coupled finite element–element free Galerkin (FE–EFG) approach. The fracture strain as a function of stress triaxiality has been evaluated by analyzing the notched tensile specimens. In the coupled approach, a small portion of the domain, where severe plastic deformation is expected, is modeled by EFG method whereas the rest of the domain is modeled by FEM to exploit the advantages of both the methods. A ramp function has been used in the interface region to maintain the continuity between FE and EFG domains. The nonlinear material behavior is modeled by von-Mises yield criterion and Hollomon’s power law. An implicit return mapping algorithm is employed for stress equilibrium in the plasticity model. The effect of geometric nonlinearity as a result of large deformation is captured by updated Lagrangian approach. The coupled approach is used to study the fracture behavior of two different cracked specimens in order to highlight its capabilities.


Ductile fracture Coupled FE–EFG approach Ramp function Hollomon’s power law Large deformation 


\(\mathbf{a}_J \)

Degrees of freedom associated with Heaviside function


Body force vector per unit volume

\(\mathbf{b}_K^\alpha \)

Degrees of freedom associated with asymptotic functions


Matrix of shape functions gradient


Elastic constitutive matrix


Elasto–plastic constitutive matrix


Green–Lagrange strain tensor


Deformation gradient tensor


Matrix of shape functions derivatives


Heaviside function


Second order unit tensor


Material tangent stiffness matrix


Geometric stiffness matrix

\(\mathbf{K}_T \)

Total tangent stiffness matrix


Strength coefficient

\(\mathbf{M}_{{\varvec{\upsigma }} } \)

Matrix of Cauchy stress components

\(N_I \)

Standard finite elements shape functions

\(\bar{{N}}_I \)

Interface elements shape functions


Hardening exponent


Radial distance from the crack tip


Ramp function


Second Piola–Kirchhoff stress tensor

\(T_S \)

Stress triaxiality


Surface traction vector


Displacement vector

\(\mathbf{u}_I \)

Nodal parameter associated with node I

\(\Delta {\varvec{\upvarepsilon }}\)

Incremental total strain tensor

\(\Delta {\varvec{\upvarepsilon }}^\mathrm{pl}\)

Incremental plastic strain tensor

\(\bar{{\varepsilon }}\)

Effective plastic strain

\(\varepsilon _f \)

Fracture strain

\(\phi _\alpha \left( \mathbf{x} \right) \)

Crack tip asymptotic functions

\(\Delta \gamma \)

Plastic multiplier

\(\varpi \)

Accumulated damage

\(\theta \)

Polar angle with respect to crack tip

\(\xi , \eta \)

Coordinates of parent element

\({{\varvec{\upsigma }} }\)

Cauchy stress tensor

\({\varvec{\sigma }'}\)

Deviatoric stress tensor

\(\sigma _{eq} \)

von-Mises equivalent stress

\(\sigma _m \)

Mean normal stress

\(\varsigma _n (\mathbf{x})\)

Normal level set function

\(\varsigma _t (\mathbf{x})\)

Tangential level set function

\(\psi _I \)

Element free Galerkin shape functions

\(\Phi \)

Yield function

\(\Omega \)

Total domain

\(\Omega ^\mathrm{FE}\)

Finite element sub-domain

\(\Omega ^\mathrm{EFG}\)

Element free Galerkin sub-domain

\(\Omega ^\mathrm{IE}\)

Interface sub-domain

\(\Psi _N \)

State variable at node

\(\Psi _G \)

State variable at Gauss point



This research work is supported by Bhabha Atomic Research Centre (BARC), Mumbai, India.


  1. Areias P, Dias-da-Costa D, Sargado JM, Rabczuk T (2013a) Element-wise algorithm for modeling ductile fracture with the Rousselier yield function. Comput Mech 52:1429–1443CrossRefGoogle Scholar
  2. Areias P, Rabczuk T, Dias-da-Costa D (2013b) Element-wise fracture algorithm based on rotation of edges. Eng Fract Mech 110:113–137CrossRefGoogle Scholar
  3. Areias P, Rabczuk T (2013) Finite strain fracture of plates and shells with configurational forces and edge rotations. Int J Numer Methods Eng 94:1099–1122CrossRefGoogle Scholar
  4. Areias P, Rabczuk T, Camanho PP (2013c) Initially rigid cohesive laws and fracture based on edge rotations. Comput Mech 52:931–947CrossRefGoogle Scholar
  5. Areias P, Rabczuk T, Camanho PP (2014) Finite strain fracture of 2D problems with injected anisotropic softening elements. Theor Appl Fract Mech 72:50–63CrossRefGoogle Scholar
  6. Areias P, Msekh MA, Rabczuk T (2016) Damage and fracture algorithm using the screened Poisson equation and local remeshing. Eng Fract Mech 158:116–143CrossRefGoogle Scholar
  7. Belytschko T, Lu YY, Gu L (1994a) Element-free Galerkin methods. Int J Numer Methods Eng 37:229–256CrossRefGoogle Scholar
  8. Belytschko T, Gu L, Lu YY (1994b) Fracture and crack growth by element free Galerkin methods. Model Simul Mater Sci Eng 2:519–534CrossRefGoogle Scholar
  9. Belytschko T, Organ D, Hegen D (1995) A coupled finite element-element free Galerkin method. Comput Mech 17:186–195CrossRefGoogle Scholar
  10. Belytschko T, Black T (1999) Elastic crack growth in finite elements with minimal remeshing. Int J Numer Methods Eng 45:601–620CrossRefGoogle Scholar
  11. Bhardwaj G, Singh IV, Mishra BK, Bui TQ (2015) Numerical simulation of functionally graded cracked plates using NURBS based XIGA under different loads and boundary conditions. Compos Struct 126:347–359CrossRefGoogle Scholar
  12. Bordas S, Rabczuk T, Zi G (2008) Three-dimensional crack initiation, propagation, branching and junction in non-linear materials by an extended meshfree method without asymptotic enrichment. Eng Fract Mech 75:943–960CrossRefGoogle Scholar
  13. Chen C, Mangasarian OL (1996) A class of smoothing functions for nonlinear and mixed complementarity problems. Comput Optim Appl 5:97–138CrossRefGoogle Scholar
  14. Chen CR, Kolednik O, Scheider I, Siegmund T, Tatschl A, Fischer FD (2003) On the determination of the cohesive zone parameters for the modelling of microductile crack growth in thick specimens. Int J Fract 120:517–536CrossRefGoogle Scholar
  15. Chen L, Rabczuk T, Bordas SPA, Liu GR, Zeng KY, Kerfriden P (2012) Extended finite element method with edge-based strain smoothing (ESm-XFEM) for linear elastic crack growth. Comp Meth Appl Mech Eng 212:250–265CrossRefGoogle Scholar
  16. Cheung S, Luxmoore AR (2003) A finite element analysis of stable crack growth in an aluminium alloy. Eng Fract Mech 70:1153–1169CrossRefGoogle Scholar
  17. Elguedj T, Gravouil A, Combescure A (2007) A mixed augmented Lagrangian-extended finite element method for modelling elastic–plastic fatigue crack growth with unilateral contact. Int J Numer Methods Eng 71:1569–1597CrossRefGoogle Scholar
  18. Gurson AL (1977) Continuum theory of ductile rupture by void nucleation and growth: part I-yield criteria and flow rules for porous ductile media. J Eng Mater Technol 99:2–15CrossRefGoogle Scholar
  19. Hegen D (1996) Element-free Galerkin methods in combination with finite element approaches. Comp Meth Appl Mech Eng 135:143–166CrossRefGoogle Scholar
  20. Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comp Meth Appl Mech Eng 194:4135–4195CrossRefGoogle Scholar
  21. Koplik J, Needleman A (1988) Void growth and coalescence in porous plastic solids. Int J Solids Struct 24:835–853CrossRefGoogle Scholar
  22. Krongauz Y, Belytschko T (1996) Enforcement of essential boundary conditions in meshless approximations using finite elements. Comput Meth Appl Mech Eng 131:133–145CrossRefGoogle Scholar
  23. Kumar S, Singh IV, Mishra BK (2014) A multigrid coupled (FE-EFG) approach to simulate fatigue crack growth in heterogeneous materials. Theor Appl Fract Mech 72:121–135CrossRefGoogle Scholar
  24. Kumar S, Singh IV, Mishra BK (2015a) A homogenized XFEM approach to simulate fatigue crack growth problems. Comput Struct 150:1–22CrossRefGoogle Scholar
  25. Kumar S, Shedbale AS, Singh IV, Mishra BK (2015b) Elasto–plastic fatigue crack growth analysis of plane problems in the presence of flaws using XFEM. Front Struct Civ Eng 9:420–440CrossRefGoogle Scholar
  26. Leblond JB, Lazarus V, Karma A (2015) Multiscale cohesive zone model for propagation of segmented crack fronts in mode I + III fracture. Int J Fract 191:167–189CrossRefGoogle Scholar
  27. Liu WK, Karpov EG, Zhang S, Park HS (2004) An introduction to computational nanomechanics and materials. Comput Meth Appl Mech Eng 193:1529–1578CrossRefGoogle Scholar
  28. Liu L, Dong X, Cong-xin L (2009) Adaptive finite element-element-free Galerkin coupling method for bulk metal forming processes. J Zhejiang Uni Sci A 10:353–360CrossRefGoogle Scholar
  29. Lu YY, Belytschko T, Gu L (1994) A new implementation of the element free Galerkin method. Comput Meth Appl Mech Eng 113:397–414CrossRefGoogle Scholar
  30. McClintock FA (1968) A criterion of ductile fracture by the growth of holes. J Appl Mech 35:363–371CrossRefGoogle Scholar
  31. Moes N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46:131–150CrossRefGoogle Scholar
  32. Oh C, Kim N, Kim Y, Baek J, Kim Y, Kim W (2011) A finite element ductile failure simulation method using stress-modified fracture strain model. Eng Fract Mech 78:124–137Google Scholar
  33. Osher S, Sethian JA (1988) Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations. J Comput Phys 79:12–49CrossRefGoogle Scholar
  34. Owen D, Hinton E (1980) Finite elements in plasticity: theory and applications. Pineridge Press, SwanseaGoogle Scholar
  35. Pathak H, Singh A, Singh IV, Brahmankar M (2015) Three-dimensional stochastic quasi-static fatigue crack growth simulations using coupled FE-EFG approach. Comput Struct 160:1–19CrossRefGoogle Scholar
  36. Rabczuk T, Belytschko T (2004) Cracking particles: a simplified meshfree method for arbitrary evolving cracks. Int J Numer Methods Eng 61:2316–2343CrossRefGoogle Scholar
  37. Rabczuk T, Belytschko T (2007) A three-dimensional large deformation meshfree method for arbitrary evolving cracks. Comput Meth Appl Mech Eng 196:2777–2799CrossRefGoogle Scholar
  38. Rabczuk T, Areias PMA, Belytschko T (2007) A simplified mesh-free method for shear bands with cohesive surfaces. Int J Numer Methods Eng 69:993–1021CrossRefGoogle Scholar
  39. Rabczuk T, Samaniego E (2008) Discontinuous modelling of shear bands using adaptive meshfree methods. Comput Meth Appl Mech Eng 197:641–658CrossRefGoogle Scholar
  40. Rabczuk T, Zi G, Bordas S, Nguyen-Xuan H (2008) A geometrically non-linear three-dimensional cohesive crack method for reinforced concrete structures. Eng Fract Mech 75:4740–4758CrossRefGoogle Scholar
  41. Rabczuk T, Zi G, Bordas S, Nguyen-Xuan H (2010) A simple and robust three-dimensional cracking-particle method without enrichment. Comput Methods Appl Mech Eng 199:2437–2455CrossRefGoogle Scholar
  42. Rajesh KN, Rao BN (2010) Coupled meshfree and fractal finite element method for mixed mode two dimensional crack problems. Int J Numer Methods Eng 84:572–609Google Scholar
  43. Rao BN, Rahman S (2001) A coupled meshless-finite element method for fracture analysis of cracks. Int J Press Vessels Pip 78:647–657CrossRefGoogle Scholar
  44. Reddy JN (2009) An introduction to nonlinear finite element analysis. Oxford University Press, OxfordGoogle Scholar
  45. Rice JR, Tracey DM (1969) On the ductile enlargement of voids in triaxial stress fields. J Mech Phys Solids 17:201–217CrossRefGoogle Scholar
  46. Rousselier G (1987) Ductile fracture models and their potential in local approach of fracture. Nucl Eng Des 105:97–111CrossRefGoogle Scholar
  47. Shedbale AS, Singh IV, Mishra BK (2013) Nonlinear simulation of an embedded crack in the presence of holes and inclusions by XFEM. Proc Eng 64:642–651CrossRefGoogle Scholar
  48. Shedbale AS, Singh IV, Mishra BK, Sharma K (2015) Evaluation of mechanical properties using spherical ball indentation and coupled FE-EFG approach. Mech Adv Mater Struct 23:832–843CrossRefGoogle Scholar
  49. Shedbale AS, Singh IV, Mishra BK (2016) A coupled FE-EFG approach for modeling crack growth in ductile materials. Fatigue Fract Eng Mater Struct. doi: 10.1111/ffe.12423 Google Scholar
  50. 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 167:157–171CrossRefGoogle Scholar
  51. Stolarska M, Chopp DL, Moës N, Belytschko T (2001) Modelling crack growth by level sets in the extended finite element method. Int J Numer Methods Eng 51:943–960CrossRefGoogle Scholar
  52. Sukumar N, Chopp DL, Moran B (2003) Extended finite element method and fast marching method for three-dimensional fatigue crack propagation. Eng Fract Mech 70:29–48Google Scholar
  53. Thomason PF (1990) Ductile fracture of metals. Pergamon Press, OxfordGoogle Scholar
  54. Tvergaard V (1982) On localization in ductile materials containing spherical voids. Int J Fract 18:237–252Google Scholar
  55. Tvergaard V, Needleman A (1984) Analysis of the cup-cone fracture in a round tensile bar. Acta Metall 32:157–169CrossRefGoogle Scholar
  56. Tvergaard V, Hutchinson JW (1992) The relation between crack growth resistance and fracture process parameters in elastic–plastic solids. J Mech Phys Solids 40:1377–1397Google Scholar
  57. Wagner GJ, Liu WK (2003) Coupling of atomistic and continuum simulations using a bridging scale decomposition. J Comput Phys 190:249–274CrossRefGoogle Scholar
  58. Xiangqiao Y (2006) A boundary element modeling of fatigue crack growth in a plane elastic plate. Mech Res Commun 33:470–481CrossRefGoogle Scholar
  59. Xiao SP, Belytschko T (2004) A bridging domain method for coupling continua with molecular dynamics. Comput Methods Appl Mech Eng 193:1645–1669CrossRefGoogle Scholar
  60. Zhang ZL (1995) Explicit consistent tangent moduli with a return mapping algorithm for pressure-dependent elasto–plasticity models. Comput Methods Appl Mech Eng 121:29–44CrossRefGoogle Scholar
  61. Zhuang X, Augarde CE, Mathisen KM (2012) Fracture modeling using meshless methods and level sets in 3D: framework and modeling. Int J Numer Methods Eng 92:969–998CrossRefGoogle Scholar
  62. Zhuang X, Zhu H, Augarde C (2014) An improved meshless Shepard and least squares method possessing the delta property and requiring no singular weight function. Comput Mech 53:343–357CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of Mechanical and Industrial EngineeringIndian Institute of Technology RoorkeeRoorkeeIndia
  2. 2.Reactor Safety DivisionBhabha Atomic Research CenterMumbaiIndia

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