A simple J-integral governed bilinear constitutive relation for simulating fracture propagation in quasi-brittle material

The present paper develops a bilinear constitutive relation for material at fracture tip based on the cohesive zone model and the J-integral theory. In this bilinear constitutive relation, the stress linearly decreases rather than suddenly drops down to zero at the post-peak stage. The slope of stress degradation is related to the critical J-integral of material and the element size. By this bilinear constitutive relation, the tensile and mixed fracture propagation can be well simulated for the strain energy release rate can be preserved with element size decreasing. The simulation results are almost free of the element size sensitivity. The present model and its implementation are very simple. For the elements at fracture tip, the derived bilinear constitutive relation is adopted while for the rest elements, the ideal elastic-brittle model is adopted. For the fracture propagation is represented by the deterioration of stiffness through constitutive relation, no separate fracture criterion is needed, which makes the evaluation of the fracture propagation by separate fracture criterion and the mesh modification after each load step unnecessary. The simulation examples of tensile and mixed fracture suggest that the present method is valid and highly efficient.

[1]  J. C. Simo,et al.  An analysis of strong discontinuities induced by strain-softening in rate-independent inelastic solids , 1993 .

[2]  Tian Jian Lu,et al.  On the path dependence of the J-integral in notch problems , 2004 .

[3]  Ronaldo I. Borja,et al.  A finite element model for strain localization analysis of strongly discontinuous fields based on standard Galerkin approximation , 2000 .

[4]  D. S. Dugdale.  Yielding of steel sheets containing slits , 1960 .

[5]  Rena C. Yu,et al.  A comparative study between discrete and continuum models to simulate concrete fracture , 2008 .

[6]  Wei Zhang,et al.  Formulation of a cohesive zone model for a Mode III crack , 2005 .

[7]  G. I. Barenblatt.  THE MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS IN BRITTLE FRACTURE , 1962 .

[8]  P. Gambarova,et al.  Smeared crack analysis for fracture and aggregate interlock in concrete , 1990 .

[9]  Y. R. Rashid,et al.  Ultimate strength analysis of prestressed concrete pressure vessels , 1968 .

[10]  Filippo Berto,et al.  Relationships between J-integral and the strain energy evaluated in a finite volume surrounding the tip of sharp and blunt V-notches , 2007 .

[11]  E. M. Morozov,et al.  Calculation of the energy J-integral for bodies with notches and cracks , 2004 .

[12]  Zhennan Zhang,et al.  Simulation of fracture propagation subjected to compressive and shear stress field using virtual multidimensional internal bonds , 2009 .

[13]  J-integral evaluation for U- and V-blunt notches under Mode I loading and materials obeying a power hardening law , 2007 .

[14]  Z. Bažant,et al.  Blunt Crack Band Propagation in Finite Element Analysis , 1979 .

[15]  F. Berto,et al.  Some Expressions for the Strain Energy in a Finite Volume Surrounding the Root of Blunt V-notches , 2005 .

[16]  A. Needleman,et al.  A cohesive segments method for the simulation of crack growth , 2003 .

[17]  Filippo Berto,et al.  A review of the volume-based strain energy density approach applied to V-notches and welded structures , 2009 .

[18]  Jaime Planas,et al.  Mixed Mode Fracture of Concrete under Proportional and Nonproportional Loading , 1998 .

[19]  A. Hillerborg,et al.  Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements , 1976 .

[20]  René de Borst,et al.  Discrete vs smeared crack models for concrete fracture: bridging the gap , 2004 .

[21]  M. Ortiz,et al.  A finite element method for localized failure analysis , 1987 .

[22]  R. Borst.  Numerical aspects of cohesive-zone models , 2003 .

[23]  G. Zi,et al.  Asymptotic stress intensity factor density profiles for smeared-tip method for cohesive fracture , 2003 .

[24]  F. J. Gómez,et al.  Fracture of U-notched specimens under mixed mode: Experimental results and numerical predictions , 2009 .

[25]  Zdenek P. Bazant,et al.  Rock fracture via strain-softening finite elements , 1984 .

[26]  J. M. Kennedy,et al.  Concrete cracking simulations for nuclear applications , 1984 .

[27]  Jaime Planas,et al.  The cohesive crack concept: application to PMMA at −60 °C , 2005 .

[28]  Paolo Livieri,et al.  Use of J-integral to predict static failures in sharp V-notches and rounded U-notches , 2008 .

[29]  Alberto Carpinteri,et al.  Mixed mode fracture energy of concrete , 1990 .

[30]  G. Sih,et al.  Fracture mechanics applied to engineering problems-strain energy density fracture criterion , 1974 .

[31]  M. Elices,et al.  The cohesive zone model: advantages, limitations and challenges , 2002 .

[32]  M. Elices,et al.  Local strain energy to assess the static failure of U-notches in plates under mixed mode loading , 2007 .

[33]  Zdeněk P. Bažant,et al.  Confinement-shear lattice CSL model for fracture propagation in concrete , 2006 .

[34]  W. Brekelmans,et al.  Reduction of mesh sensitivity in continuum damage mechanics , 1995 .

[35]  T. Belytschko,et al.  A finite element with embedded localization zones , 1988 .

[36]  J. Rice.  A path-independent integral and the approximate analysis of strain , 1968 .

[37]  A. Needleman.  A Continuum Model for Void Nucleation by Inclusion Debonding , 1987 .

[38]  René de Borst,et al.  Some recent issues in computational failure mechanics , 2001 .

[39]  F. J. Gómez,et al.  Generalizations and specializations of cohesive crack models , 2003 .

[40]  Paolo Lazzarin,et al.  A finite-volume-energy based approach to predict the static and fatigue behavior of components with sharp V-shaped notches , 2001 .