December  2014, 9(4): 575-597. doi: 10.3934/nhm.2014.9.575

A review of non local continuum damage: Modelling of failure?

1. 

Laboratoire des Fluides Complexes et leurs Réservoirs - UMR 5150, Université de Pau et des Pays de l'Adour, Allée du Parc Montaury, Anglet, 64600, France, France

Received  October 2014 Revised  October 2014 Published  December 2014

Failure of quasi-brittle materials such as concrete needs a proper description of strain softening due to progressive micro-cracking and the introduction of an internal length in the constitutive model in order to achieve non zero energy dissipation. This paper reviews the main results obtained with the non local damage model, which has been among the precursors of such models. In most cases up to now, the internal length has been considered as a constant. There is today a consensus that it should not be the case as models possess severe shortcomings such as incorrect averaging near the boundaries of the solid considered and non local transmission across non convex boundaries. An interaction-based model in which the weight function is constructed from the analysis of interaction has been proposed. It avoids empirical descriptions of the evolution of the internal length. This model is also recalled and further documented. Additional results dealing with spalling failure are discussed. Finally, it is pointed out that this model provides an asymptotic description of complete failure, which is consistent with fracture mechanics.
Citation: Gilles Pijaudier-Cabot, David Grégoire. A review of non local continuum damage: Modelling of failure?. Networks & Heterogeneous Media, 2014, 9 (4) : 575-597. doi: 10.3934/nhm.2014.9.575
References:
[1]

Z. P. Bažant, Instability, ductility, and size effect in strain-softening concrete,, Journal of the Engineering Mechanics Division, 102 (1976), 331. Google Scholar

[2]

Z. P. Bažant, Nonlocal damage theory based on micromechanics of crack interactions,, Journal of Engineering Mechanics, 120 (1994), 593. doi: 10.1061/(ASCE)0733-9399(1994)120:3(593). Google Scholar

[3]

Z. P. Bažant and M. Jirasek, Nonlocal integral formulations of plasticity and damage: survey of progress,, Journal of Engineering Mechanics, 128 (2002), 1119. doi: 10.1061/(ASCE)0733-9399(2002)128:11(1119). Google Scholar

[4]

Z. P. Bažant and G. Pijaudier-Cabot, Nonlocal continuum damage localization instability and convergence,, Journal of Applied Mechanics, 55 (1988), 287. doi: 10.1115/1.3173674. Google Scholar

[5]

Z. P. Bažant, J.-L. Le and C. G. Hoover, Nonlocal boundary layer (nbl) model: overcoming boundary condition problems in strength statistics and fracture analysis of quasibrittle materials,, in Fracture Mechanics of Concrete and Concrete Structures-Recent Advances in Fracture Mechanics of Concrete (ed. B.-H. Oh), (2010), 135. Google Scholar

[6]

A. Benallal, R. Billardon and G. Geymonat, Bifurcation and localization in rate-independent materials. Some general considerations,, in Bifurcation and Stability of Dissipative Systems (ed. Q. S. Nguyen), 327 (1993), 1. doi: 10.1007/978-3-7091-2712-4_1. Google Scholar

[7]

B. Bourdin, G. A. Francfort and J.-J. Marigo, Numerical experiments in revisited brittle fracture,, Journal of the Mechanics and Physics of Solids, 48 (2000), 797. doi: 10.1016/S0022-5096(99)00028-9. Google Scholar

[8]

B. Bourdin, G. A. Francfort and J.-J. Marigo, The variational approach to fracture,, Journal of Elasticity, 91 (2008), 5. doi: 10.1007/s10659-007-9107-3. Google Scholar

[9]

R. De Borst, Simulation of strain localization: A reappraisal of the Cosserat continuum,, Engineering computations, 8 (1991), 317. doi: 10.1108/eb023842. Google Scholar

[10]

R. De Borst and H.-B. Mühlhaus, Gradient-dependent plasticity: Formulation and algorithmic aspects,, International Journal for Numerical Methods in Engineering, 35 (1992), 521. doi: 10.1002/nme.1620350307. Google Scholar

[11]

F. Dufour, G. Legrain, G. Pijaudier-Cabot and A. Huerta, Estimation of crack opening from a two-dimensional continuum-based finite element computation,, International Journal for Numerical and Analytical Methods in Geomechanics, 36 (2012), 1813. doi: 10.1002/nag.1097. Google Scholar

[12]

F. Dufour, G. Pijaudier-cabot, M. Choinska and A. Huerta, Extraction of a crack opening from a continuous approach using regularized damage models,, Computers and Concrete, 5 (2008), 375. doi: 10.12989/cac.2008.5.4.375. Google Scholar

[13]

M. Geers, R. De Borst, W. Brekelmans and R. Peerlings, Strain-based transient-gradient damage model for failure analyses,, Computer Methods in Applied Mechanics and Engineering, 160 (1998), 133. doi: 10.1016/S0045-7825(98)80011-X. Google Scholar

[14]

C. Giry, F. Dufour and J. Mazars, Stress-based nonlocal damage model,, International Journal of Solids and Structures, 48 (2011), 3431. doi: 10.1016/j.ijsolstr.2011.08.012. Google Scholar

[15]

P. Grassl, D. Xenos, M. Jirásek and M. Horák, Evaluation of nonlocal approaches for modelling fracture near nonconvex boundaries,, International Journal of Solids and Structures, 51 (2014), 3239. doi: 10.1016/j.ijsolstr.2014.05.023. Google Scholar

[16]

D. Grégoire, H. Maigre and A. Combescure, New experimental and numerical techniques to study the arrest and the restart of a crack under impact in transparent materials,, International Journal of Solids and Structures, 46 (2009), 3480. doi: 10.1016/j.ijsolstr.2009.06.003. Google Scholar

[17]

D. Grégoire, H. Maigre, J. Réthoré and A. Combescure, Dynamic crack propagation under mixed-mode loading - Comparison between experiments and X-FEM simulations,, International Journal of Solids and Structures, 44 (2007), 6517. doi: 10.1016/j.ijsolstr.2007.02.044. Google Scholar

[18]

D. Grégoire, L. B. Rojas-Solano, V. Lefort, P. Grassl, J. Saliba, J.-P. Regoin, A. Loukili and G. Pijaudier-Cabot, Mesoscale Analysis of Failure in Quasi-Brittle Materials: Comparison Between Lattice Model and Acoustic Emission Data,, {International Journal of Numerical and Analytical Methods in Geomechanics}, (2014). Google Scholar

[19]

D. Grégoire, L. B. Rojas-Solano and G. Pijaudier-cabot, Continuum to discontinuum transition during failure in non-local damage models,, International Journal for Multiscale Computational Engineering, 10 (2012), 567. doi: 10.1615/IntJMultCompEng.2012003061. Google Scholar

[20]

D. Grégoire, L. B. Rojas-Solano and G. Pijaudier-Cabot, Failure and size effect for notched and unnotched concrete beams,, International Journal for Numerical and Analytical Methods in Geomechanics, 37 (2013), 1434. doi: 10.1002/nag.2180. Google Scholar

[21]

J. Hadamard, Leçons Sur la Propagation des Ondes, (French) [Lectures on wave propagation],, Hermann, (1903). Google Scholar

[22]

M. Hadamard, Les problèmes aux limites dans la théorie des équations aux dérivées partielles, (French) [Boundary value problems in the theory of partial differential equations],, Journal de Physique Théorique et Appliquée, 6 (1907), 202. Google Scholar

[23]

R. Hill, A general theory of uniqueness and stability in elastic-plastic solids,, Journal of the Mechanics and Physics of Solids, 6 (1958), 236. doi: 10.1016/0022-5096(58)90029-2. Google Scholar

[24]

R. Hill, Some basic principles in the mechanics of solids without a natural time,, Journal of the Mechanics and Physics of Solids, 7 (1959), 209. doi: 10.1016/0022-5096(59)90007-9. Google Scholar

[25]

M. Jirásek and B. Patzák, Consistent tangent stiffness for nonlocal damage models,, Computers & structures, 80 (2002), 1279. doi: 10.1016/S0045-7949(02)00078-0. Google Scholar

[26]

M. Jirásek, S. Rolshoven and P. Grassl, Size effect on fracture energy induced by non-locality,, International Journal for Numerical and Analytical Methods in Geomechanics, 28 (2004), 653. doi: 10.1002/nag.364. Google Scholar

[27]

D. D. Joseph, M. Renardy and J.-C. Saut, Hyperbolicity and change of type in the flow of viscoelastic fluid,, in Analysis and Thermomechanics (eds. B. D. Coleman, 87 (1985), 213. doi: 10.1007/BF00250725. Google Scholar

[28]

H. Kolsky, An investigation of the mechanical properties of material at a very high rate of loading,, Proceedings of the Physical Society, B62 (1949), 676. doi: 10.1088/0370-1301/62/11/302. Google Scholar

[29]

A. Krayani, G. Pijaudier-Cabot and F. Dufour, Boundary effect on weight function in nonlocal damage model,, Engineering Fracture Mechanics, 76 (2009), 2217. doi: 10.1016/j.engfracmech.2009.07.007. Google Scholar

[30]

J. Mazars, A description of micro-and macroscale damage of concrete structures,, Engineering Fracture Mechanics, 25 (1986), 729. doi: 10.1016/0013-7944(86)90036-6. Google Scholar

[31]

J. Mazars, Application de la Mécanique de L'endommagement au Comportement non Linéaire et à la rupture du Béton de Structure, (French) [Application of mechanical damage to the nonlinear behavior and breakage of the concrete structure],, PhD thesis, (1984). Google Scholar

[32]

A. Needleman, Material rate dependence and mesh sensitivity in localization problems,, Computer Methods in Applied Mechanics and Engineering, 67 (1988), 69. doi: 10.1016/0045-7825(88)90069-2. Google Scholar

[33]

M. Negri, A non-local approximation of free discontinuity problems in sbv and sbd,, Calculus of Variations and Partial Differential Equations, 25 (2006), 33. doi: 10.1007/s00526-005-0356-3. Google Scholar

[34]

R. Peerlings, R. De Borst, W. Brekelmans, J. De Vree and I. Spee, Some observations on localisation in non-local and gradient damage models,, European Journal of Mechanics - A/Solids, 15 (1996), 937. Google Scholar

[35]

R. Peerlings, M. Geers, R. de Borst and W. Brekelmans, A critical comparison of nonlocal and gradient-enhanced softening continua,, International Journal of Solids and Structures, 38 (2001), 7723. doi: 10.1016/S0020-7683(01)00087-7. Google Scholar

[36]

K. Pham and J.-J. Marigo, Approche variationnelle de l'endommagement: II. Les modèles à gradient, (French) [The variational approach to damage: II. The gradient damage models],, Comptes Rendus Mécanique, 338 (2010), 199. doi: 10.1016/j.crme.2010.03.012. Google Scholar

[37]

G. Pijaudier-Cabot and Y. Berthaud, Effets des interactions dans l'endommagement d'un milieu fragile. Formulation non locale, (French) [Damage and interactions in a microcracked medium. Non local formulation],, Comptes rendus de l'Académie des sciences-Série 2, 310 (1990), 1577. Google Scholar

[38]

G. Pijaudier-Cabot and Z. P. Bažant, Nonlocal Damage Theory,, Journal of Engineering Mechanics, 113 (1987), 1512. doi: 10.1061/(ASCE)0733-9399(1987)113:10(1512). Google Scholar

[39]

G. Pijaudier-Cabot and A. Benallal, Strain localization and bifurcation in a nonlocal continuum,, International Journal of Solids and Structures, 30 (1993), 1761. doi: 10.1016/0020-7683(93)90232-V. Google Scholar

[40]

G. Pijaudier-Cabot and L. Bode, Localization of damage in a nonlocal continuum,, Mechanics Research Communications, 19 (1992), 145. doi: 10.1016/0093-6413(92)90039-D. Google Scholar

[41]

G. Pijaudier-Cabot, L. Bodé and A. Huerta, Arbitrary Lagrangian-Eulerian finite element analysis of strain localization in transient problems,, International Journal for Numerical Methods in Engineering, 38 (1995), 4171. doi: 10.1002/nme.1620382406. Google Scholar

[42]

G. Pijaudier-Cabot and F. Dufour, Non local damage model. Boundary and evolving boundary effects,, European Journal of Environmental and Civil engineering, 14 (2010), 729. doi: 10.1080/19648189.2010.9693260. Google Scholar

[43]

G. Pijaudier-Cabot, K. Haidar and J.-F. Dubé, Non-local damage model with evolving internal length,, International Journal for Numerical and Analytical Methods in Geomechanics, 28 (2004), 633. doi: 10.1002/nag.367. Google Scholar

[44]

G. Pijaudier-Cabot and A. Huerta, Finite element analysis of bifurcation in nonlocal strain softening solids,, Computer methods in applied mechanics and engineering, 90 (1991), 905. doi: 10.1016/0045-7825(91)90190-H. Google Scholar

[45]

J. R. Rice, The Localization of Plastic Deformation,, Division of Engineering, (1976). Google Scholar

[46]

L. B. Rojas-Solano, D. Grégoire and G. Pijaudier-cabot, Interaction-based non-local damage model for failure in quasi-brittle materials,, Mechanics Research Communications, 54 (2013), 56. doi: 10.1016/j.mechrescom.2013.09.011. Google Scholar

[47]

A. Simone, H. Askes and L. J. Sluys, Incorrect initiation and propagation of failure in non-local and gradient-enhanced media,, International Journal of Solids and Structures, 41 (2004), 351. doi: 10.1016/j.ijsolstr.2003.09.020. Google Scholar

[48]

L. Sluys and R. De Borst, Wave propagation and localization in a rate-dependent cracked medium-model formulation and one-dimensional examples,, International Journal of Solids and Structures, 29 (1992), 2945. doi: 10.1016/0020-7683(92)90151-I. Google Scholar

show all references

References:
[1]

Z. P. Bažant, Instability, ductility, and size effect in strain-softening concrete,, Journal of the Engineering Mechanics Division, 102 (1976), 331. Google Scholar

[2]

Z. P. Bažant, Nonlocal damage theory based on micromechanics of crack interactions,, Journal of Engineering Mechanics, 120 (1994), 593. doi: 10.1061/(ASCE)0733-9399(1994)120:3(593). Google Scholar

[3]

Z. P. Bažant and M. Jirasek, Nonlocal integral formulations of plasticity and damage: survey of progress,, Journal of Engineering Mechanics, 128 (2002), 1119. doi: 10.1061/(ASCE)0733-9399(2002)128:11(1119). Google Scholar

[4]

Z. P. Bažant and G. Pijaudier-Cabot, Nonlocal continuum damage localization instability and convergence,, Journal of Applied Mechanics, 55 (1988), 287. doi: 10.1115/1.3173674. Google Scholar

[5]

Z. P. Bažant, J.-L. Le and C. G. Hoover, Nonlocal boundary layer (nbl) model: overcoming boundary condition problems in strength statistics and fracture analysis of quasibrittle materials,, in Fracture Mechanics of Concrete and Concrete Structures-Recent Advances in Fracture Mechanics of Concrete (ed. B.-H. Oh), (2010), 135. Google Scholar

[6]

A. Benallal, R. Billardon and G. Geymonat, Bifurcation and localization in rate-independent materials. Some general considerations,, in Bifurcation and Stability of Dissipative Systems (ed. Q. S. Nguyen), 327 (1993), 1. doi: 10.1007/978-3-7091-2712-4_1. Google Scholar

[7]

B. Bourdin, G. A. Francfort and J.-J. Marigo, Numerical experiments in revisited brittle fracture,, Journal of the Mechanics and Physics of Solids, 48 (2000), 797. doi: 10.1016/S0022-5096(99)00028-9. Google Scholar

[8]

B. Bourdin, G. A. Francfort and J.-J. Marigo, The variational approach to fracture,, Journal of Elasticity, 91 (2008), 5. doi: 10.1007/s10659-007-9107-3. Google Scholar

[9]

R. De Borst, Simulation of strain localization: A reappraisal of the Cosserat continuum,, Engineering computations, 8 (1991), 317. doi: 10.1108/eb023842. Google Scholar

[10]

R. De Borst and H.-B. Mühlhaus, Gradient-dependent plasticity: Formulation and algorithmic aspects,, International Journal for Numerical Methods in Engineering, 35 (1992), 521. doi: 10.1002/nme.1620350307. Google Scholar

[11]

F. Dufour, G. Legrain, G. Pijaudier-Cabot and A. Huerta, Estimation of crack opening from a two-dimensional continuum-based finite element computation,, International Journal for Numerical and Analytical Methods in Geomechanics, 36 (2012), 1813. doi: 10.1002/nag.1097. Google Scholar

[12]

F. Dufour, G. Pijaudier-cabot, M. Choinska and A. Huerta, Extraction of a crack opening from a continuous approach using regularized damage models,, Computers and Concrete, 5 (2008), 375. doi: 10.12989/cac.2008.5.4.375. Google Scholar

[13]

M. Geers, R. De Borst, W. Brekelmans and R. Peerlings, Strain-based transient-gradient damage model for failure analyses,, Computer Methods in Applied Mechanics and Engineering, 160 (1998), 133. doi: 10.1016/S0045-7825(98)80011-X. Google Scholar

[14]

C. Giry, F. Dufour and J. Mazars, Stress-based nonlocal damage model,, International Journal of Solids and Structures, 48 (2011), 3431. doi: 10.1016/j.ijsolstr.2011.08.012. Google Scholar

[15]

P. Grassl, D. Xenos, M. Jirásek and M. Horák, Evaluation of nonlocal approaches for modelling fracture near nonconvex boundaries,, International Journal of Solids and Structures, 51 (2014), 3239. doi: 10.1016/j.ijsolstr.2014.05.023. Google Scholar

[16]

D. Grégoire, H. Maigre and A. Combescure, New experimental and numerical techniques to study the arrest and the restart of a crack under impact in transparent materials,, International Journal of Solids and Structures, 46 (2009), 3480. doi: 10.1016/j.ijsolstr.2009.06.003. Google Scholar

[17]

D. Grégoire, H. Maigre, J. Réthoré and A. Combescure, Dynamic crack propagation under mixed-mode loading - Comparison between experiments and X-FEM simulations,, International Journal of Solids and Structures, 44 (2007), 6517. doi: 10.1016/j.ijsolstr.2007.02.044. Google Scholar

[18]

D. Grégoire, L. B. Rojas-Solano, V. Lefort, P. Grassl, J. Saliba, J.-P. Regoin, A. Loukili and G. Pijaudier-Cabot, Mesoscale Analysis of Failure in Quasi-Brittle Materials: Comparison Between Lattice Model and Acoustic Emission Data,, {International Journal of Numerical and Analytical Methods in Geomechanics}, (2014). Google Scholar

[19]

D. Grégoire, L. B. Rojas-Solano and G. Pijaudier-cabot, Continuum to discontinuum transition during failure in non-local damage models,, International Journal for Multiscale Computational Engineering, 10 (2012), 567. doi: 10.1615/IntJMultCompEng.2012003061. Google Scholar

[20]

D. Grégoire, L. B. Rojas-Solano and G. Pijaudier-Cabot, Failure and size effect for notched and unnotched concrete beams,, International Journal for Numerical and Analytical Methods in Geomechanics, 37 (2013), 1434. doi: 10.1002/nag.2180. Google Scholar

[21]

J. Hadamard, Leçons Sur la Propagation des Ondes, (French) [Lectures on wave propagation],, Hermann, (1903). Google Scholar

[22]

M. Hadamard, Les problèmes aux limites dans la théorie des équations aux dérivées partielles, (French) [Boundary value problems in the theory of partial differential equations],, Journal de Physique Théorique et Appliquée, 6 (1907), 202. Google Scholar

[23]

R. Hill, A general theory of uniqueness and stability in elastic-plastic solids,, Journal of the Mechanics and Physics of Solids, 6 (1958), 236. doi: 10.1016/0022-5096(58)90029-2. Google Scholar

[24]

R. Hill, Some basic principles in the mechanics of solids without a natural time,, Journal of the Mechanics and Physics of Solids, 7 (1959), 209. doi: 10.1016/0022-5096(59)90007-9. Google Scholar

[25]

M. Jirásek and B. Patzák, Consistent tangent stiffness for nonlocal damage models,, Computers & structures, 80 (2002), 1279. doi: 10.1016/S0045-7949(02)00078-0. Google Scholar

[26]

M. Jirásek, S. Rolshoven and P. Grassl, Size effect on fracture energy induced by non-locality,, International Journal for Numerical and Analytical Methods in Geomechanics, 28 (2004), 653. doi: 10.1002/nag.364. Google Scholar

[27]

D. D. Joseph, M. Renardy and J.-C. Saut, Hyperbolicity and change of type in the flow of viscoelastic fluid,, in Analysis and Thermomechanics (eds. B. D. Coleman, 87 (1985), 213. doi: 10.1007/BF00250725. Google Scholar

[28]

H. Kolsky, An investigation of the mechanical properties of material at a very high rate of loading,, Proceedings of the Physical Society, B62 (1949), 676. doi: 10.1088/0370-1301/62/11/302. Google Scholar

[29]

A. Krayani, G. Pijaudier-Cabot and F. Dufour, Boundary effect on weight function in nonlocal damage model,, Engineering Fracture Mechanics, 76 (2009), 2217. doi: 10.1016/j.engfracmech.2009.07.007. Google Scholar

[30]

J. Mazars, A description of micro-and macroscale damage of concrete structures,, Engineering Fracture Mechanics, 25 (1986), 729. doi: 10.1016/0013-7944(86)90036-6. Google Scholar

[31]

J. Mazars, Application de la Mécanique de L'endommagement au Comportement non Linéaire et à la rupture du Béton de Structure, (French) [Application of mechanical damage to the nonlinear behavior and breakage of the concrete structure],, PhD thesis, (1984). Google Scholar

[32]

A. Needleman, Material rate dependence and mesh sensitivity in localization problems,, Computer Methods in Applied Mechanics and Engineering, 67 (1988), 69. doi: 10.1016/0045-7825(88)90069-2. Google Scholar

[33]

M. Negri, A non-local approximation of free discontinuity problems in sbv and sbd,, Calculus of Variations and Partial Differential Equations, 25 (2006), 33. doi: 10.1007/s00526-005-0356-3. Google Scholar

[34]

R. Peerlings, R. De Borst, W. Brekelmans, J. De Vree and I. Spee, Some observations on localisation in non-local and gradient damage models,, European Journal of Mechanics - A/Solids, 15 (1996), 937. Google Scholar

[35]

R. Peerlings, M. Geers, R. de Borst and W. Brekelmans, A critical comparison of nonlocal and gradient-enhanced softening continua,, International Journal of Solids and Structures, 38 (2001), 7723. doi: 10.1016/S0020-7683(01)00087-7. Google Scholar

[36]

K. Pham and J.-J. Marigo, Approche variationnelle de l'endommagement: II. Les modèles à gradient, (French) [The variational approach to damage: II. The gradient damage models],, Comptes Rendus Mécanique, 338 (2010), 199. doi: 10.1016/j.crme.2010.03.012. Google Scholar

[37]

G. Pijaudier-Cabot and Y. Berthaud, Effets des interactions dans l'endommagement d'un milieu fragile. Formulation non locale, (French) [Damage and interactions in a microcracked medium. Non local formulation],, Comptes rendus de l'Académie des sciences-Série 2, 310 (1990), 1577. Google Scholar

[38]

G. Pijaudier-Cabot and Z. P. Bažant, Nonlocal Damage Theory,, Journal of Engineering Mechanics, 113 (1987), 1512. doi: 10.1061/(ASCE)0733-9399(1987)113:10(1512). Google Scholar

[39]

G. Pijaudier-Cabot and A. Benallal, Strain localization and bifurcation in a nonlocal continuum,, International Journal of Solids and Structures, 30 (1993), 1761. doi: 10.1016/0020-7683(93)90232-V. Google Scholar

[40]

G. Pijaudier-Cabot and L. Bode, Localization of damage in a nonlocal continuum,, Mechanics Research Communications, 19 (1992), 145. doi: 10.1016/0093-6413(92)90039-D. Google Scholar

[41]

G. Pijaudier-Cabot, L. Bodé and A. Huerta, Arbitrary Lagrangian-Eulerian finite element analysis of strain localization in transient problems,, International Journal for Numerical Methods in Engineering, 38 (1995), 4171. doi: 10.1002/nme.1620382406. Google Scholar

[42]

G. Pijaudier-Cabot and F. Dufour, Non local damage model. Boundary and evolving boundary effects,, European Journal of Environmental and Civil engineering, 14 (2010), 729. doi: 10.1080/19648189.2010.9693260. Google Scholar

[43]

G. Pijaudier-Cabot, K. Haidar and J.-F. Dubé, Non-local damage model with evolving internal length,, International Journal for Numerical and Analytical Methods in Geomechanics, 28 (2004), 633. doi: 10.1002/nag.367. Google Scholar

[44]

G. Pijaudier-Cabot and A. Huerta, Finite element analysis of bifurcation in nonlocal strain softening solids,, Computer methods in applied mechanics and engineering, 90 (1991), 905. doi: 10.1016/0045-7825(91)90190-H. Google Scholar

[45]

J. R. Rice, The Localization of Plastic Deformation,, Division of Engineering, (1976). Google Scholar

[46]

L. B. Rojas-Solano, D. Grégoire and G. Pijaudier-cabot, Interaction-based non-local damage model for failure in quasi-brittle materials,, Mechanics Research Communications, 54 (2013), 56. doi: 10.1016/j.mechrescom.2013.09.011. Google Scholar

[47]

A. Simone, H. Askes and L. J. Sluys, Incorrect initiation and propagation of failure in non-local and gradient-enhanced media,, International Journal of Solids and Structures, 41 (2004), 351. doi: 10.1016/j.ijsolstr.2003.09.020. Google Scholar

[48]

L. Sluys and R. De Borst, Wave propagation and localization in a rate-dependent cracked medium-model formulation and one-dimensional examples,, International Journal of Solids and Structures, 29 (1992), 2945. doi: 10.1016/0020-7683(92)90151-I. Google Scholar

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