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
CompMech

Abstract

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.

Keywords

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

Nomenclature

\(\mathbf{a}_J \)

Degrees of freedom associated with Heaviside function

\(\mathbf{b}\)

Body force vector per unit volume

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

Degrees of freedom associated with asymptotic functions

\(\mathbf{B}\)

Matrix of shape functions gradient

\(\mathbf{D}\)

Elastic constitutive matrix

\(\mathbf{D}^{\mathrm{ep}}\)

Elasto–plastic constitutive matrix

\(\mathbf{E}\)

Green–Lagrange strain tensor

\(\mathbf{F}\)

Deformation gradient tensor

\(\mathbf{G}\)

Matrix of shape functions derivatives

\(H(\mathbf{x})\)

Heaviside function

I

Second order unit tensor

\(\mathbf{K}^\mathrm{mat}\)

Material tangent stiffness matrix

\(\mathbf{K}^{\mathrm{geo}}\)

Geometric stiffness matrix

\(\mathbf{K}_T \)

Total tangent stiffness matrix

\(\tilde{K}\)

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

\(\hat{{n}}\)

Hardening exponent

\(\tilde{r}\)

Radial distance from the crack tip

\(R(\mathbf{x})\)

Ramp function

\(\mathbf{S}\)

Second Piola–Kirchhoff stress tensor

\(T_S \)

Stress triaxiality

\(\mathbf{t}\)

Surface traction vector

\(\mathbf{u}\)

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

Notes

Acknowledgments

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

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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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