A nonlocal damage model for plain concrete consistent with cohesive fracture

A rate-independent damage constitutive law is proposed to describe the fracture of plain concrete under tensile loading. Here, the target scale is the individual crack. In order to deal with localised damage, the model is inherently nonlocal: the gradient of the damage field is explicitly involved in the constitutive equations; it is parameterised by a nonlocal length scale which is interpreted as the width of the process zone. The model is defined so that its predictions are close to those of a cohesive law for vanishing nonlocal length scales. Therefore, the current model is plainly consistent with cohesive zone model analyses: the nonlocal length scale appears as a small parameter which does not need any specific identification. And four parameters—among which the tensile strength and the fracture energy—enable to adjust the softening cohesive response. Besides, a special attention has been paid to the shape of the initial damage surface and to the relation between damage and stiffness. The damage surface takes into account not only the contrast between tensile and compressive strengths but also experimental evidences regarding its shape in multiaxial tension. And the damage–stiffness relation is defined so as to describe important phenomena such as the stiffness recovery with crack closure and the sustainability of compressive loads by damaged structures. Finally, several comparisons with experimental data (global force/opening responses, size dependency, curved crack paths, crack opening profiles) enable to validate qualitatively and quantitatively the pertinence of the constitutive law in 2D and 3D.

[1]  J. Mazars.  A description of micro- and macroscale damage of concrete structures , 1986 .

[2]  Christian Miehe,et al.  Thermodynamically consistent phase‐field models of fracture: Variational principles and multi‐field FE implementations , 2010 .

[3]  Hans Muhlhaus,et al.  A variational principle for gradient plasticity , 1991 .

[4]  Donald E. Carlson,et al.  The derivative of a tensor-valued function of a tensor , 1986 .

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

[6]  Alain Millard,et al.  Regularised crack behaviour effects on continuum modelling of quasi-brittle materials under cyclic loading , 2015 .

[7]  Paul Steinmann,et al.  Modeling three‐dimensional crack propagation—A comparison of crack path tracking strategies , 2008 .

[8]  Ahmed Loukili,et al.  Crack propagation and size effect in concrete using a non-local damage model , 2013 .

[9]  R. de Borst,et al.  Wave propagation and localization in a rate-dependent cracked medium-model formulation and one-dimensional examples , 1992 .

[10]  Kyrylo Kazymyrenko,et al.  Modelling large crack propagation: from gradient damage to cohesive zone models , 2012, International Journal of Fracture.

[11]  Gilles A. Francfort,et al.  Revisiting brittle fracture as an energy minimization problem , 1998 .

[12]  Sang-keun Lee,et al.  Biaxial behavior of plain concrete of nuclear containment building , 2004 .

[13]  Milan Jirásek,et al.  Numerical aspects of the crack band approach , 2012 .

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

[15]  N. Chevaugeon,et al.  A level set based model for damage growth: The thick level set approach , 2011 .

[16]  B. Bourdin,et al.  The Variational Approach to Fracture , 2008 .

[17]  Marc Louis Maurice François,et al.  A new yield criterion for the concrete materials , 2008 .

[18]  Pierre Badel,et al.  Application of some anisotropic damage model to the prediction of the failure of some complex industrial concrete structure , 2007 .

[19]  P. Petersson.  Crack growth and development of fracture zones in plain concrete and similar materials , 1981 .

[20]  Z. Bažant,et al.  Crack band theory for fracture of concrete , 1983 .

[21]  A gradient approach to localization of deformation 227 materials , 2004 .

[22]  A. Needleman.  Material rate dependence and mesh sensitivity in localization problems , 1988 .

[23]  Alain Combescure,et al.  Damage model with delay effect Analytical and numerical studies of the evolution of the characteristic damage length , 2003 .

[24]  Frédéric Feyel,et al.  A finite element approach coupling a continuous gradient damage model and a cohesive zone model within the framework of quasi-brittle failure , 2012 .

[25]  Jean-Jacques Marigo,et al.  Initiation of a periodic array of cracks in the thermal shock problem: a gradient damage modeling , 2014 .

[26]  Wam Marcel Brekelmans,et al.  Comparison of nonlocal approaches in continuum damage mechanics , 1995 .

[27]  René Chambon,et al.  A simplified second gradient model for dilatant materials: Theory and numerical implementation , 2008 .

[28]  Stéphane Andrieux,et al.  Analysis of non-local models through energetic formulations , 2003 .

[29]  Claudia Comi,et al.  Computational modelling of gradient‐enhanced damage in quasi‐brittle materials , 1999 .

[30]  Z. Bažant,et al.  Nonlocal Continuum Damage, Localization Instability and Convergence , 1988 .

[31]  Eric Lorentz,et al.  Gradient damage models: Toward full-scale computations , 2011 .

[32]  L. J. Sluys,et al.  From continuous to discontinuous failure in a gradient-enhanced continuum damage model , 2003 .

[33]  P. Steinmann,et al.  Theoretical and computational aspects of a thermodynamically consistent framework for geometrically linear gradient damage , 2001 .

[34]  G. I. Barenblatt.  The formation of equilibrium cracks during brittle fracture. General ideas and hypotheses. Axially-symmetric cracks , 1959 .

[35]  Antonio Huerta,et al.  Extraction of a crack opening from a continuous approach using regularized damage models , 2008 .

[36]  B. Bourdin,et al.  Numerical experiments in revisited brittle fracture , 2000 .

[37]  R. Borst,et al.  Localisation in a Cosserat continuum under static and dynamic loading conditions , 1991 .

[38]  J. Rots.  Computational modeling of concrete fracture , 1988 .

[39]  P. M. Naghdi,et al.  On continuum thermodynamics , 1972 .

[40]  F. Iurlano,et al.  Numerical insight of a variational smeared approach to cohesive fracture , 2017 .

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

[42]  Jean-Jacques Marigo,et al.  From the onset of damage to rupture: construction of responses with damage localization for a general class of gradient damage models , 2013 .

[43]  J. Pamin.  COMPUTATIONAL MODELLING OF LOCALIZED DEFORMATIONS WITH REGULARIZED CONTINUUM MODELS , 2011 .

[44]  Stefano Mariani,et al.  An extended FE strategy for transition from continuum damage to mode I cohesive crack propagation , 2007 .

[45]  Qi-Chang He,et al.  Conewise linear elastic materials , 1994 .

[46]  J. Marigo,et al.  Gradient Damage Models and Their Use to Approximate Brittle Fracture , 2011 .

[47]  Zdeněk P. Bažant,et al.  Comprehensive concrete fracture tests: Description and results , 2013 .

[48]  K. Willam,et al.  Localization within the Framework of Micropolar Elasto-Plasticity , 1991 .

[49]  Jean-Jacques Marigo,et al.  From Gradient Damage Laws to Griffith’s Theory of Crack Propagation , 2013 .

[50]  Ahmed Benallal,et al.  Bifurcation and Localization in Rate-Independent Materials. Some General Considerations , 1993 .

[51]  Quoc Son Nguyen,et al.  Bifurcation and stability in dissipative media (plasticity, friction, fracture) , 1994 .

[52]  Davide Bigoni,et al.  Yield criteria for quasibrittle and frictional materials: A generalization to surfaces with corners , 2009, 0904.3869.

[53]  Jean-Jacques Marigo,et al.  Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments , 2009 .

[54]  D. Borst,et al.  Fundamental issues in finite element analyses of localization of deformation , 1993 .

[55]  Ahmed Benallal,et al.  Bifurcation and stability issues in gradient theories with softening , 2006 .

[56]  H.E.J.G. Schlangen,et al.  Experimental and numerical analysis of fracture processes in concrete : proefschrift , 1993 .

[57]  Nicolas Triantafyllidis,et al.  A gradient approach to localization of deformation. I. Hyperelastic materials , 1986 .

[58]  L. J. Sluys,et al.  Incorrect initiation and propagation of failure in non-local and gradient-enhanced media , 2004 .

[59]  Gilles Pijaudier-Cabot,et al.  Isotropic and anisotropic descriptions of damage in concrete structures , 1999 .

[60]  Rhj Ron Peerlings,et al.  Gradient enhanced damage for quasi-brittle materials , 1996 .

[61]  Zdeněk P. Bažant,et al.  Cohesive crack, size effect, crack band and work-of-fracture models compared to comprehensive concrete fracture tests , 2014, International Journal of Fracture.

[62]  Jacques Besson,et al.  Modeling of crack growth in round bars and plane strain specimens , 2001 .

[63]  G. Piero,et al.  A variational model for fracture mechanics - Numerical experiments , 2007 .

[64]  Mgd Marc Geers,et al.  Localisation issues in local and nonlocal continuum approaches to fracture , 2002 .

[65]  M. Frémond,et al.  Damage, gradient of damage and principle of virtual power , 1996 .

[66]  Kyrylo Kazymyrenko,et al.  Convergence of a gradient damage model toward a cohesive zone model , 2011 .

[67]  N. Burlion,et al.  Damage and localisation in elastic materials with voids , 1996 .

[68]  Umberto Perego,et al.  Fracture energy based bi-dissipative damage model for concrete , 2001 .

[69]  Z. Bažant,et al.  Fracture and Size Effect in Concrete and Other Quasibrittle Materials , 1997 .

[70]  D. Barker,et al.  Fracture Process Zone of Concrete , 1985 .

[71]  Alfredo Edmundo Huespe,et al.  Crack-path field and strain-injection techniques in computational modeling of propagating material failure , 2014 .

[72]  G. Holzapfel,et al.  3 D Crack propagation in unreinforced concrete . A two-step algorithm for tracking 3 D crack paths , 2006 .

[73]  R. Peerlings,et al.  Discrete crack modelling of ductile fracture driven by non‐local softening plasticity , 2006 .

[74]  Anthony Duncan Jefferson,et al.  Three dimensional finite element simulations of fracture tests using the craft concrete model , 2004 .

[75]  Mgd Marc Geers,et al.  Damage and crack modeling in single-edge and double-edge notched concrete beams , 2000 .

[76]  H. Reinhardt.  Fracture Mechanics of an Elastic Softening Material like Concrete , 1984 .

[77]  P. Badel,et al.  A new path‐following constraint for strain‐softening finite element simulations , 2004 .

[78]  Kenneth Runesson,et al.  A thermodynamically consistent theory of gradient-regularized plasticity coupled to damage , 1997 .

[79]  Samuel Forest,et al.  Micromorphic Approach for Gradient Elasticity, Viscoplasticity, and Damage , 2009 .

[80]  Gerhard A. Holzapfel,et al.  3D Crack propagation in unreinforced concrete. A two-step algorithm for tracking 3D crack paths , 2006 .

[81]  E. P. Warnke,et al.  CONSTITUTIVE MODEL FOR THE TRIAXIAL BEHAVIOR OF CONCRETE , 1975 .

[82]  Samuel Forest,et al.  Localization phenomena and regularization methods , 2004 .

[83]  Anthony Duncan Jefferson,et al.  The simulation of crack opening–closing and aggregate interlock behaviour in finite element concrete models , 2015 .

[84]  S. Andrieux,et al.  A variational formulation for nonlocal damage models , 1999 .

[85]  Z. Bažant.  Concrete fracture models: testing and practice , 2002 .