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 and 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), 331-344.

[2]

Z. P. Bažant, Nonlocal damage theory based on micromechanics of crack interactions, Journal of Engineering Mechanics, 120 (1994), 593-617. doi: 10.1061/(ASCE)0733-9399(1994)120:3(593).

[3]

Z. P. Bažant and M. Jirasek, Nonlocal integral formulations of plasticity and damage: survey of progress, Journal of Engineering Mechanics, 128 (2002), 1119-1149. doi: 10.1061/(ASCE)0733-9399(2002)128:11(1119).

[4]

Z. P. Bažant and G. Pijaudier-Cabot, Nonlocal continuum damage localization instability and convergence, Journal of Applied Mechanics, 55 (1988), 287-294. doi: 10.1115/1.3173674.

[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), Korea Concrete Institute, (2010), 135-143.

[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), Springer Vienna, 327 (1993), 1-44. doi: 10.1007/978-3-7091-2712-4_1.

[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), 797-826. doi: 10.1016/S0022-5096(99)00028-9.

[8]

B. Bourdin, G. A. Francfort and J.-J. Marigo, The variational approach to fracture, Journal of Elasticity, 91 (2008), 5-148. doi: 10.1007/s10659-007-9107-3.

[9]

R. De Borst, Simulation of strain localization: A reappraisal of the Cosserat continuum, Engineering computations, 8 (1991), 317-332. doi: 10.1108/eb023842.

[10]

R. De Borst and H.-B. Mühlhaus, Gradient-dependent plasticity: Formulation and algorithmic aspects, International Journal for Numerical Methods in Engineering, 35 (1992), 521-539. doi: 10.1002/nme.1620350307.

[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), 1813-1830. doi: 10.1002/nag.1097.

[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), 375-388. doi: 10.12989/cac.2008.5.4.375.

[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), 133-153. doi: 10.1016/S0045-7825(98)80011-X.

[14]

C. Giry, F. Dufour and J. Mazars, Stress-based nonlocal damage model, International Journal of Solids and Structures, 48 (2011), 3431-3443. doi: 10.1016/j.ijsolstr.2011.08.012.

[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), 3239-3251. doi: 10.1016/j.ijsolstr.2014.05.023.

[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), 3480-3491. doi: 10.1016/j.ijsolstr.2009.06.003.

[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), 6517-6534. doi: 10.1016/j.ijsolstr.2007.02.044.

[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}, in review, 2014.

[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), 567-580. doi: 10.1615/IntJMultCompEng.2012003061.

[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), 1434-1452. doi: 10.1002/nag.2180.

[21]

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

[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), 202-241.

[23]

R. Hill, A general theory of uniqueness and stability in elastic-plastic solids, Journal of the Mechanics and Physics of Solids, 6 (1958), 236-249. doi: 10.1016/0022-5096(58)90029-2.

[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), 209-225. doi: 10.1016/0022-5096(59)90007-9.

[25]

M. Jirásek and B. Patzák, Consistent tangent stiffness for nonlocal damage models, Computers & structures, 80 (2002), 1279-1293. doi: 10.1016/S0045-7949(02)00078-0.

[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), 653-670. doi: 10.1002/nag.364.

[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, M. Feinberg and J. Serrin ), Springer Berlin Heidelberg, 87 (1985), 213-251. doi: 10.1007/BF00250725.

[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), 676-700. doi: 10.1088/0370-1301/62/11/302.

[29]

A. Krayani, G. Pijaudier-Cabot and F. Dufour, Boundary effect on weight function in nonlocal damage model, Engineering Fracture Mechanics, 76 (2009), 2217-2231. doi: 10.1016/j.engfracmech.2009.07.007.

[30]

J. Mazars, A description of micro-and macroscale damage of concrete structures, Engineering Fracture Mechanics, 25 (1986), 729-737. doi: 10.1016/0013-7944(86)90036-6.

[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, Université Paris VI, 1984.

[32]

A. Needleman, Material rate dependence and mesh sensitivity in localization problems, Computer Methods in Applied Mechanics and Engineering, 67 (1988), 69-85. doi: 10.1016/0045-7825(88)90069-2.

[33]

M. Negri, A non-local approximation of free discontinuity problems in sbv and sbd, Calculus of Variations and Partial Differential Equations, 25 (2006), 33-62. doi: 10.1007/s00526-005-0356-3.

[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), 937-953.

[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), 7723-7746. doi: 10.1016/S0020-7683(01)00087-7.

[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), 199-206. doi: 10.1016/j.crme.2010.03.012.

[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), 1577-1582.

[38]

G. Pijaudier-Cabot and Z. P. Bažant, Nonlocal Damage Theory, Journal of Engineering Mechanics, 113 (1987), 1512-1533. doi: 10.1061/(ASCE)0733-9399(1987)113:10(1512).

[39]

G. Pijaudier-Cabot and A. Benallal, Strain localization and bifurcation in a nonlocal continuum, International Journal of Solids and Structures, 30 (1993), 1761-1775. doi: 10.1016/0020-7683(93)90232-V.

[40]

G. Pijaudier-Cabot and L. Bode, Localization of damage in a nonlocal continuum, Mechanics Research Communications, 19 (1992), 145-153. doi: 10.1016/0093-6413(92)90039-D.

[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), 4171-4191. doi: 10.1002/nme.1620382406.

[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), 729-749. doi: 10.1080/19648189.2010.9693260.

[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), 633-652. doi: 10.1002/nag.367.

[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), 905-919. doi: 10.1016/0045-7825(91)90190-H.

[45]

J. R. Rice, The Localization of Plastic Deformation, Division of Engineering, Brown University, 1976.

[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), 56-62. doi: 10.1016/j.mechrescom.2013.09.011.

[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), 351-363. doi: 10.1016/j.ijsolstr.2003.09.020.

[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), 2945-2958. doi: 10.1016/0020-7683(92)90151-I.

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), 331-344.

[2]

Z. P. Bažant, Nonlocal damage theory based on micromechanics of crack interactions, Journal of Engineering Mechanics, 120 (1994), 593-617. doi: 10.1061/(ASCE)0733-9399(1994)120:3(593).

[3]

Z. P. Bažant and M. Jirasek, Nonlocal integral formulations of plasticity and damage: survey of progress, Journal of Engineering Mechanics, 128 (2002), 1119-1149. doi: 10.1061/(ASCE)0733-9399(2002)128:11(1119).

[4]

Z. P. Bažant and G. Pijaudier-Cabot, Nonlocal continuum damage localization instability and convergence, Journal of Applied Mechanics, 55 (1988), 287-294. doi: 10.1115/1.3173674.

[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), Korea Concrete Institute, (2010), 135-143.

[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), Springer Vienna, 327 (1993), 1-44. doi: 10.1007/978-3-7091-2712-4_1.

[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), 797-826. doi: 10.1016/S0022-5096(99)00028-9.

[8]

B. Bourdin, G. A. Francfort and J.-J. Marigo, The variational approach to fracture, Journal of Elasticity, 91 (2008), 5-148. doi: 10.1007/s10659-007-9107-3.

[9]

R. De Borst, Simulation of strain localization: A reappraisal of the Cosserat continuum, Engineering computations, 8 (1991), 317-332. doi: 10.1108/eb023842.

[10]

R. De Borst and H.-B. Mühlhaus, Gradient-dependent plasticity: Formulation and algorithmic aspects, International Journal for Numerical Methods in Engineering, 35 (1992), 521-539. doi: 10.1002/nme.1620350307.

[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), 1813-1830. doi: 10.1002/nag.1097.

[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), 375-388. doi: 10.12989/cac.2008.5.4.375.

[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), 133-153. doi: 10.1016/S0045-7825(98)80011-X.

[14]

C. Giry, F. Dufour and J. Mazars, Stress-based nonlocal damage model, International Journal of Solids and Structures, 48 (2011), 3431-3443. doi: 10.1016/j.ijsolstr.2011.08.012.

[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), 3239-3251. doi: 10.1016/j.ijsolstr.2014.05.023.

[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), 3480-3491. doi: 10.1016/j.ijsolstr.2009.06.003.

[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), 6517-6534. doi: 10.1016/j.ijsolstr.2007.02.044.

[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}, in review, 2014.

[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), 567-580. doi: 10.1615/IntJMultCompEng.2012003061.

[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), 1434-1452. doi: 10.1002/nag.2180.

[21]

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

[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), 202-241.

[23]

R. Hill, A general theory of uniqueness and stability in elastic-plastic solids, Journal of the Mechanics and Physics of Solids, 6 (1958), 236-249. doi: 10.1016/0022-5096(58)90029-2.

[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), 209-225. doi: 10.1016/0022-5096(59)90007-9.

[25]

M. Jirásek and B. Patzák, Consistent tangent stiffness for nonlocal damage models, Computers & structures, 80 (2002), 1279-1293. doi: 10.1016/S0045-7949(02)00078-0.

[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), 653-670. doi: 10.1002/nag.364.

[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, M. Feinberg and J. Serrin ), Springer Berlin Heidelberg, 87 (1985), 213-251. doi: 10.1007/BF00250725.

[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), 676-700. doi: 10.1088/0370-1301/62/11/302.

[29]

A. Krayani, G. Pijaudier-Cabot and F. Dufour, Boundary effect on weight function in nonlocal damage model, Engineering Fracture Mechanics, 76 (2009), 2217-2231. doi: 10.1016/j.engfracmech.2009.07.007.

[30]

J. Mazars, A description of micro-and macroscale damage of concrete structures, Engineering Fracture Mechanics, 25 (1986), 729-737. doi: 10.1016/0013-7944(86)90036-6.

[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, Université Paris VI, 1984.

[32]

A. Needleman, Material rate dependence and mesh sensitivity in localization problems, Computer Methods in Applied Mechanics and Engineering, 67 (1988), 69-85. doi: 10.1016/0045-7825(88)90069-2.

[33]

M. Negri, A non-local approximation of free discontinuity problems in sbv and sbd, Calculus of Variations and Partial Differential Equations, 25 (2006), 33-62. doi: 10.1007/s00526-005-0356-3.

[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), 937-953.

[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), 7723-7746. doi: 10.1016/S0020-7683(01)00087-7.

[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), 199-206. doi: 10.1016/j.crme.2010.03.012.

[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), 1577-1582.

[38]

G. Pijaudier-Cabot and Z. P. Bažant, Nonlocal Damage Theory, Journal of Engineering Mechanics, 113 (1987), 1512-1533. doi: 10.1061/(ASCE)0733-9399(1987)113:10(1512).

[39]

G. Pijaudier-Cabot and A. Benallal, Strain localization and bifurcation in a nonlocal continuum, International Journal of Solids and Structures, 30 (1993), 1761-1775. doi: 10.1016/0020-7683(93)90232-V.

[40]

G. Pijaudier-Cabot and L. Bode, Localization of damage in a nonlocal continuum, Mechanics Research Communications, 19 (1992), 145-153. doi: 10.1016/0093-6413(92)90039-D.

[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), 4171-4191. doi: 10.1002/nme.1620382406.

[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), 729-749. doi: 10.1080/19648189.2010.9693260.

[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), 633-652. doi: 10.1002/nag.367.

[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), 905-919. doi: 10.1016/0045-7825(91)90190-H.

[45]

J. R. Rice, The Localization of Plastic Deformation, Division of Engineering, Brown University, 1976.

[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), 56-62. doi: 10.1016/j.mechrescom.2013.09.011.

[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), 351-363. doi: 10.1016/j.ijsolstr.2003.09.020.

[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), 2945-2958. doi: 10.1016/0020-7683(92)90151-I.

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