Regularized formulation of the variational brittle fracture with unilateral contact: numerical experiments. (English) Zbl 1426.74257

Summary: This paper presents a modified regularized formulation of the Ambrosio-Tortorelli type to introduce the crack non-interpenetration condition in the variational approach to fracture mechanics proposed by G. A. Francfort and J. J. Marigo [J. Mech. Phys. Solids 46, No. 8, 1319–1342 (1998; Zbl 0966.74060)]. We focus on the linear elastic case where the contact condition appears as a local unilateral constraint on the displacement jump at the crack surfaces. The regularized model is obtained by splitting the strain energy in a spherical and a deviatoric parts and accounting for the sign of the local volume change. The numerical implementation is based on a standard finite element discretization and on the adaptation of an alternate minimization algorithm used in previous works. The new regularization avoids crack interpenetration and predicts asymmetric results in traction and in compression. Even though we do not exhibit any gamma-convergence proof toward the desired limit behavior, we illustrate through several numerical case studies the pertinence of the new model in comparison to other approaches.


74R10 Brittle fracture
74M15 Contact in solid mechanics
74S05 Finite element methods applied to problems in solid mechanics


Zbl 0966.74060
Full Text: DOI


[1] Alart, P., Méthode de Newton généralisée en mécanique du contact, J. math. pures appl., 76, 83-108, (1997) · Zbl 0868.49021
[2] Allaire, G., Jouve, F., Van Goethem, N., 2007. A level set method for the numerical simulation of damage evolution. Internal Report 629, CMAP, Ecole Polytechnique. · Zbl 1419.74250
[3] Ambrosio, L.; Fusco, N.; Pallara, D., Functions of bounded variation and free discontinuity problems. Oxford mathematical monographs, (2000), Oxford Science Publications Oxford
[4] Ambrosio, L.; Tortorelli, V., Approximation of functionals depending on jumps by elliptic functional via gamma-convergence, Commun. pure appl. math., 43, 999-1036, (1990) · Zbl 0722.49020
[5] Aranson, I.; Kalatsky, V.; Vinokur, V., Continuum field description of crack propagation, Phys. rev. lett., 85, 118-121, (2000)
[6] Babuska, I.; Melenk, J., The partition of unity method, Int. J. numer. methods eng., 40, 727-758, (1997) · Zbl 0949.65117
[7] Badel, P.; Godard, V.; Leblond, J.B., Application of some anisotropic damage model to the prediction of the failure of some complex industrial concrete structure, Int. J. solids struct., 44, 5848-5874, (2007) · Zbl 1178.74153
[8] Bellettini, G.; Coscia, A., Discrete approximation of a free discontinuity problem, Numer. funct. anal. optim., 15, 201-224, (1994) · Zbl 0806.49002
[9] Benallal, A.; Marigo, J.-J., Bifurcation and stability issues in gradient theories with softening, Modelling simulation mater. sci. eng., 15, S283-S295, (2007)
[10] Bourdin, B., Numerical implementation of the variational formulation of quasi-static brittle fracture, Interfaces free boundaries, 9, 411-430, (2007) · Zbl 1130.74040
[11] Bourdin, B.; Francfort, G.; Marigo, J.-J., Numerical experiments in revisited brittle fracture, J. mech. phys. solids, 48, 797-826, (2000) · Zbl 0995.74057
[12] Bourdin, B.; Francfort, G.; Marigo, J.-J., The variational approach to fracture, J. elasticity, 91, 5-148, (2008) · Zbl 1176.74018
[13] Braides, A., 1998. Approximation of free-discontinuity problems. In: Lecture Notes in Mathematics, vol. 1694. Springer, Berlin. · Zbl 0909.49001
[14] Camacho, G.; Ortiz, M., Computational modelling of impact damage in brittle materials, Int. J. solids struct., 33, 2899-2938, (1996) · Zbl 0929.74101
[15] Cazes, F.; Coret, M.; Combescure, A.; Gravouil, A., A thermodynamic method for the construction of a cohesive law from a nonlocal damage model, Int. J. solids struct., 46, 1476-1490, (2009) · Zbl 1236.74013
[16] Chambolle, A., A density result in two-dimensional linearized elasticity, and applications, Arch. ration. mech. anal., 167, 211-233, (2003) · Zbl 1030.74007
[17] Chambolle, A., An approximation result for special functions with bounded deformation, J. math. pures appl., 83, 929-954, (2004) · Zbl 1084.49038
[18] Ciarlet, P.; Nečas, J., Injectivity and self-contact in nonlinear elasticity, Arch. ration. mech. anal., 97, 3, 171-188, (1987) · Zbl 0628.73043
[19] Coleman, T.; Yuying, L., A reflective Newton method for minimizing a quadratic function subject to bounds on some of the variables, SIAM J. optim., 6, 4, 1040-1058, (1996) · Zbl 0861.65053
[20] Comi, C., A non-local model with tension and compression damage mechanisms, Eur. J. mech. A/solids, 20, 1-22, (2001) · Zbl 0982.74005
[21] Dal Maso, G.; Francfort, G.; Toader, R., Quasistatic crack growth in nonlinear elasticity, Arch. ration. mech. anal., 176, 165-225, (2005) · Zbl 1064.74150
[22] Dal Maso, G.; Toader, R., A model for the quasi-static growth of brittle fractures: existence and approximation results, Arch. ration. mech. anal., 162, 101-135, (2002) · Zbl 1042.74002
[23] de Borst, R.; Remmers, J.; Needleman, A.; Abellan, M.-A., Discrete vs smeared crack models for concrete fracture: bridging the gap, Int. J. numer. methods eng., 28, 583-607, (2004) · Zbl 1086.74044
[24] Del Piero, G.; Lancioni, G.; March, R., A variational model for fracture mechanics: numerical experiments, J. mech. phys. solids, 55, 2513-2537, (2007) · Zbl 1166.74413
[25] Francfort, G.; Larsen, C., Existence and convergence for quasi-static evolution in brittle fracture, Commun. pure appl. math., 56, 10, 1465-1500, (2003) · Zbl 1068.74056
[26] Francfort, G.; Marigo, J.-J., Revisiting brittle fracture as an energy minimization problem, J. mech. phys. solids, 46, 8, 1319-1342, (1998) · Zbl 0966.74060
[27] Giacomini, A., Ambrosio – tortorelli approximation of quasi-static evolution of brittle fractures, Calc. var. partial differential equations, 22, 129-172, (2005) · Zbl 1068.35189
[28] Giacomini, A.; Ponsiglione, M., Non-interpenetration of matter for SBV deformations of hyperelastic brittle materials, Proc. R. soc. Edinburgh sect. A, 138, 5, 1019-1041, (2008) · Zbl 1151.74034
[29] Glowinski, R.; Le Tallec, P., Numerical solution of problems in incompressible finite elasticity by augmented Lagrangian methods (i): two-dimensional and axisymmetric problems, SIAM J. appl. math., 42, 400-425, (1982) · Zbl 0506.73045
[30] Hakim, V.; Karma, A., Laws of crack motion and phase-field models of fracture, J. mech. phys. solids, 57, 342-368, (2009) · Zbl 1421.74089
[31] Hughes, J.T., The finite element method: linear static and dynamic finite element analysis, (2000), Dover Publications Mineola, NY · Zbl 1191.74002
[32] Ingraffea, A.; Saouma, V., Numerical modelling of discrete crack propagation in reinforced and plain concrete, (), 171-225
[33] Jirasek, M., Nonlocal models for damage and fracture: comparison of approaches, Int. J. solids struct., 35, 4133-4145, (1998) · Zbl 0930.74054
[34] Lancioni, G.; Royer-Carfagni, G., The variational approach to fracture mechanics. A practical application to the French panthéon in Paris, J. elasticity, 95, 1-30, (2009) · Zbl 1166.74029
[35] Liebe, T.; Steinmann, P.; Benallal, A., Theoretical and computational numerical aspects of a thermodynamically consistent framework for geometrically linear gradient damage, Comput. methods appl. mech. eng., 190, 6555-6576, (2001) · Zbl 0991.74010
[36] Lorentz, E.; Andrieux, S., A variational formulation of nonlocal damage models, Int. J. plasticity, 15, 119-138, (1999) · Zbl 1024.74005
[37] Lorentz, E.; Benallal, A., Gradient constitutive relations: numerical aspects and applications to gradient damage, Comput. methods appl. mech. eng., 194, 5191-5220, (2005) · Zbl 1092.74049
[38] Lussardi, L.; Negri, M., Convergence of nonlocal finite element energies for fracture mechanics, Numer. funct. anal. optim., 28, 83-109, (2007) · Zbl 1108.74059
[39] Marconi, V.; Jagla, E., Diffuse interface approach to brittle fracture, Phys. rev. E, 71, 036110, (2005)
[40] Mariani, S.; Perego, U., Extended finite element method for quasi-brittle fracture, Int. J. numer. methods eng., 58, 103-126, (2003) · Zbl 1032.74673
[41] Mazars, J.; Pijaudier-Cabot, G., From damage to fracture mechanics and conversely: a combined approach, Int. J. solids struct., 33, 3327-3342, (1996) · Zbl 0929.74091
[42] Meschke, G.; Dumstorff, P., Energy-based modeling of cohesive and cohesionless cracks via X-FEM, Comput. methods appl. mech. eng., 196, 2338-2357, (2007) · Zbl 1173.74384
[43] Moës, N.; Belytschko, T., Extended finite element method for cohesive crack growth, Eng. fract. mech., 69, 813-833, (2002)
[44] Moës, N.; Dolbow, J.; Belytschko, T., A finite element method for crack growth without remeshing, Int. J. numer. methods eng., 46, 131-150, (1999) · Zbl 0955.74066
[45] Mumford, D.; Shah, J., Optimal approximations by piecewise smooth functions and associated variational problems, Commun. pure appl. math., 42, 577-685, (1989) · Zbl 0691.49036
[46] Naik, R.; Prabhakaran, R., Failure modes for polycarbonate under clearance-fit pin-loading, Polym. eng. sci., 27, 22, 1681-1687, (1987)
[47] Negri, M., The anisotropy introduced by the mesh in the finite element approximation of the mumford – shah functional, Numer. funct. anal. optim., 20, 957-982, (1999) · Zbl 0953.49024
[48] Oliver, J.; Huespe, A.; Pulido, M.; Chaves, E., From continuum mechanics to fracture mechanics: the strong discontinuity approach, Eng. fract. mech., 69, 113-136, (2002)
[49] Ortiz, M.; Pandolfi, A., A class of cohesive finite elements for the simulation of three-dimensional crack propagation, Int. J. numer. methods eng., 44, 1267-1282, (1999) · Zbl 0932.74067
[50] Peerlings, R.; de Borst, R.; Brekelmans, W.; Geers, M., Gradient-enhanced damage modelling of concrete fracture, Mech. cohes.-frict. mater., 3, 323-342, (1998)
[51] Pijaudier-Cabot, G.; Bazant, Z., Nonlocal damage theory, J. eng. mech., 113, 1512-1533, (1987)
[52] Ramtani, S.; Berthaud, Y.; Mazars, J., Orthotropic behavior of concrete with directional aspects: modelling and experiments, Nucl. eng. des., 133, 97-111, (1992)
[53] Raous, M.; Cangèmi, L.; Cocu, M., A consistent model coupling adhesion, friction, and unilateral contact, Comput. methods appl. mech. eng., 177, 383-399, (1999) · Zbl 0949.74008
[54] Remmers, J.; de Borst, R.; Needleman, A., The simulation of dynamic crack propagation using the cohesive segments method, J. mech. phys. solids, 56, 70-92, (2008) · Zbl 1162.74438
[55] Song, J.-H.; Wang, H.; Belytschko, T., A comparative study on finite element methods for dynamic fracture, Comput. mech., 42, 239-250, (2008) · Zbl 1160.74048
[56] Xu, X.-P.; Needleman, A., Numerical simulation of fast crack growth in brittle solids, J. mech. phys. solids, 42, 9, 1397-1434, (1994) · Zbl 0825.73579
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.