Cohesive element; Crack; Generalized continua; Higher-order continua; Micromorphic continua; Regularization; Second gradient; Strain localization; Transition; Cracks propagation; Higher order continuum; Micromorphic continuum; Mode I crack; Regularisation; Strain localizations; Analysis; Engineering (all); Computer Graphics and Computer-Aided Design; Applied Mathematics; General Engineering
Abstract :
[en] Modeling strain localization with a second gradient model can become problematic at the late stages of softening response, when the second gradient terms become significant compared to the first gradient terms. This is particularly true for mode I crack problems where an unrealistic spreading of the localized zone can be encountered. To deal with these limitations, a novel second gradient interface element for mode I crack propagation problems is introduced. It is shown that the model is able to correctly reproduce all the different phases up to failure; the adherence phase, strain localization, the transition from localized strains to cohesive zones and full crack opening.
Disciplines :
Civil engineering
Author, co-author :
Jouan, Gwendal; École Centrale de Nantes, Université de Nantes, CNRS Institut de Recherche en Génie Civil et Mécanique (GeM), UMR 6183, Nantes, cedex 3, France ; ArGEnCo Department, University of Liège, Liège, Belgium
Kotronis, Panagiotis ; École Centrale de Nantes, Université de Nantes, CNRS Institut de Recherche en Génie Civil et Mécanique (GeM), UMR 6183, Nantes, cedex 3, France
Caillerie, Denis ; University Grenoble Alpes, CNRS, Grenoble INP, Grenoble, France
Collin, Frédéric ; Université de Liège - ULiège > Urban and Environmental Engineering
Language :
English
Title :
A second gradient cohesive element for mode I crack propagation
Peerlings, R., Massart, T., Geers, M., A thermodynamically motivated implicit gradient damage framework and its application to brick masonry cracking. Comput. Methods Appl. Mech. Eng. 193:30 (2004), 3403–3417.
Jouan, G., Kotronis, P., Collin, F., Using a second gradient model to simulate the behaviour of concrete structural elements. Finite Elem. Anal. Des. 90 (2014), 50–60, 10.1016/j.finel.2014.06.002 URL http://www.sciencedirect.com/science/article/pii/S0168874X14001085.
Comi, C., Mariani, S., Perego, U., An extended FE strategy for transition from continuum damage to mode I cohesive crack propagation. Int. J. Numer. Anal. Methods Geomech. 31:2 (2007), 213–238.
Jirasek, M., Zimmermann, T., Embedded crack model. Part II: Combination with smeared cracks. Int. J. Numer. Methods Eng. 50:6 (2001), 1291–1305.
Cuvilliez, S., Feyel, F., Lorentz, E., Michel-Ponnelle, S., A finite element approach coupling a continuous gradient damage model and a cohesive zone model within the framework of quasi-brittle failure. Comput. Methods Appl. Mech. Eng. 237 (2012), 244–259.
Germain, P., The method of virtual power in continuum mechanics. Part 2: Microstructure. SIAM J. Appl. Math. 25:3 (1973), 556–575, 10.1137/0125053.
Mindlin, R., Eshel, N., On first strain-gradient theories in linear elasticity. Int. J. Solids Struct. 4:1 (1968), 109–124, 10.1016/0020-7683(68)90036-X.
Mindlin, R., Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1:4 (1965), 417–438, 10.1016/0020-7683(65)90006-5.
Lorentz, E., A mixed interface finite element for cohesive zone models. Comput. Methods Appl. Mech. Eng. 198:2 (2008), 302–317.
Bruggi, M., Venini, P., Modeling cohesive crack growth via a truly-mixed formulation. Comput. Methods Appl. Mech. Eng. 198:47 (2009), 3836–3851.
Wells, G., Sluys, L., A new method for modelling cohesive cracks using finite elements. Int. J. Numer. Methods Eng. 50:12 (2001), 2667–2682.
Griffith, A.A., VI. The phenomena of rupture and flow in solids. Phil. Trans. R. Soc. Lond. A 221:582–593 (1921), 163–198.
Bernard, P.-E., Moës, N., Chevaugeon, N., Damage growth modeling using the Thick Level Set (TLS) approach: Efficient discretization for quasi-static loadings. Comput. Methods Appl. Mech. Eng. 233 (2012), 11–27.
Chambon, R., Caillerie, D., Hassan, N.E., One-dimensional localisation studied with a second grade model. Eur. J. Mech. A 17:4 (1998), 637–656, 10.1016/S0997-7538(99)80026-6.
Chambon, R., Caillerie, D., Matsuchima, T., Plastic continuum with microstructure, local second gradient theories for geomaterials: localization studies. Int. J. Solids Struct. 38:46–47 (2001), 8503–8527, 10.1016/S0020-7683(01)00057-9.
Chambon, R., Moullet, J., Uniqueness studies in boundary value problems involving some second gradient models. Comput. Methods Appl. Mech. Eng. 193:27 (2004), 2771–2796.
Matsushima, T., Chambon, R., Caillerie, D., Second gradient models as a particular case of microstructured models: a large strain finite elements analysis. C. R. Acad. Sci. IIB 328:2 (2000), 179–186.
Matsushima, T., Chambon, R., Caillerie, D., Large strain finite element analysis of a local second gradient model: application to localization. Int. J. Numer. Methods Eng. 54:4 (2002), 499–521.
Bésuelle, P., Chambon, R., Collin, F., Switching deformation modes in post-localization solutions with a quasibrittle material. J. Mater. Struct. 1 (2006), 1115–1134.
Kotronis, P., Chambon, R., Mazars, J., Collin, F., Local second gradient models and damage mechanics: application to concrete. 11th International Conference on Fracture, 2005, ICF, Turin, Italy, 20–25 cd, paper 5712.
Soufflet, M., Jouan, G., Kotronis, P., Collin, F., Using a penalty term to deal with spurious oscillations in second gradient finite elements. Int. J. Damage Mech. 28:3 (2019), 346–366.
Kotronis, P., Al Holo, S., Bésuelle, P., Chambon, R., Shear softening and localization: Modelling the evolution of the width of the shear zone. Acta Geotech. 3:2 (2008), 85–97, 10.1007/s11440-008-0061-4.
Sieffert, Y., Al Holo, S., Chambon, R., Loss of uniqueness of numerical solutions of the borehole problem modelled with enhanced media. Int. J. Solids Struct. 46:17 (2009), 3173–3197.
Fernandes, R., Chavant, C., Chambon, R., A simplified second gradient model for dilatant materials: Theory and numerical implementation. Int. J. Solids Struct. 45:20 (2008), 5289–5307, 10.1016/j.ijsolstr.2008.05.032 URL http://www.sciencedirect.com/science/article/pii/S0020768308002205.
Collin, F., Chambon, R., Charlier, R., A finite element method for poro mechanical modelling of geotechnical problems using local second gradient models. Int. J. Numer. Methods Eng. 65:11 (2006), 1749–1772.
Rolshoven, S., Nonlocal plasticity models for localized failure. (Ph.D. thesis), 2003, École Polytechnique fédérale de Lausanne.
Kotronis, P., Stratégies de Modélisation de Structures en Béton Soumises à des Chargements Sévères. (Ph.D. thesis), 2008, Université Joseph-Fourier-Grenoble I http://tel.archives-ouvertes.fr/tel-00350461.
Jouan, G., Modélisation numérique de la localisation des déformations dans le béton avec un modèle de second gradient. (Ph.D. thesis), 2014, Ecole Centrale de Nantes, Université de Liège (cotutelle) http://bictel.ulg.ac.be/ETD-db/collection/available/ULgetd-11042014-112001/.
Jirásek, M., Rolshoven, S., Localization properties of strain-softening gradient plasticity models. Part I: Strain-gradient theories. Int. J. Solids Struct. 46:11–12 (2009), 2225–2238.
Simone, A., Askes, H., Sluys, L., Incorrect initiation and propagation of failure in non-local and gradient-enhanced media. Int. J. Solids Struct. 41:2 (2004), 351–363.
Wang, Z., Poh, L.H., A homogenized localizing gradient damage model with micro inertia effect. J. Mech. Phys. Solids 116 (2018), 370–390.
Timofeev, D., Barchiesi, E., Misra, A., Placidi, L., Hemivariational continuum approach for granular solids with damage-induced anisotropy evolution. Math. Mech. Solids 26:5 (2021), 738–770.
Pardoen, B., Levasseur, S., Collin, F., Using local second gradient model and shear strain localisation to model the excavation damaged zone in unsaturated claystone. Rock Mech. Rock Eng. 48:2 (2015), 691–714.
Van den Eijnden, A., Bésuelle, P., Collin, F., Chambon, R., Desrues, J., Modeling the strain localization around an underground gallery with a hydro-mechanical double scale model; effect of anisotropy. Comput. Geotech.z 85 (2017), 384–400.
Collin, F., Kotronis, P., Pardoen, B., Numerical modelling of multiphysics couplings and strain localization. Sulem, J., Stefanou, I., Papamichos, E., Veveakis, M., (eds.) Alert Geomaterials Doctoral School, Modelling of instabilities and bifurcation in Geomechnics, 2016, 247–291 http://alertgeomaterials.eu/publications/.
Cuvilliez, S., Passage d'un modèle d'endommagement continu régularisé a un modèle de fissuration cohésive dans le cadre de la rupture quasi-fragile. (Ph.D. thesis), 2012, Mines ParisTech.
Cazes, F., Construction et implémentation de lois cohésives extrinsèques. (Ph.D. thesis), 2010, INSA de Lyon URL https://www.theses.fr/2010ISAL0075.
Simone, A., Sluys, L., The use of displacement discontinuities in a rate-dependent medium. Comput. Methods Appl. Mech. Eng. 193:27 (2004), 3015–3033.
Simone, A., Wells, G., Sluys, L., From continuous to discontinuous failure in a gradient-enhanced continuum damage model. Comput. Methods Appl. Mech. Eng. 192:41 (2003), 4581–4607.
Forest, S., Sievert, R., Elastoviscoplastic constitutive frameworks for generalized continua. Acta Mech. 160:1–2 (2003), 71–111.
Mindlin, R., Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16:1 (1964), 51–78.
Li, J., A micromechanics-based strain gradient damage model for fracture prediction of brittle materials - Part I: Homogenization methodology and constitutive relations. Int. J. Solids Struct. 48:24 (2011), 3336–3345.
Li, J., Pham, T., Abdelmoula, R., Song, S., Jiang, C., A micromechanics-based strain gradient damage model for fracture prediction of brittle materials – Part II: Damage modeling and numerical simulations. Int. J. Solids Struct. 48:24 (2011), 3346–3358.
Yang, Y., Misra, A., Micromechanics based second gradient continuum theory for shear band modeling in cohesive granular materials following damage elasticity. Int. J. Solids Struct. 49:18 (2012), 2500–2514.
Dell'Isola, F., Sciarra, G., Vidoli, S., Generalized Hooke's law for isotropic second gradient materials. Proc. R. Soc. A 465:2107 (2009), 2177–2196.
Zienkiewicz, O., Taylor, R., The Finite Element Method For Solid And Structural Mechanics. 2005, Butterworth-Heinemann.
Zervos, A., Papanastasiou, P., Vardoulakis, I., A finite element displacement formulation for gradient elastoplasticity. Int. J. Numer. Methods Eng. 50:6 (2001), 1369–1388, 10.1002/1097-0207(20010228)50:6<1369::AID-NME72>3.0.CO;2-K.
Fischer, P., Mergheim, J., Steinmann, P., On the C1 continuous discretization of non-linear gradient elasticity: A comparison of NEM and FEM based on Bernstein–Bézier patches. Int. J. Numer. Methods Eng. 82:10 (2010), 1282–1307.
Caillerie, D., Second gradient linéaire et séries de Fourier. 2013 personal communication, University Grenoble Alpes.
Enakoutsa, K., Leblond, J., Numerical implementation and assessment of the GLPD micromorphic model of ductile rupture. Eur. J. Mech. A 28:3 (2009), 445–460.
Shu, J., King, W., Fleck, N., Finite elements for materials with strain gradient effects. Int. J. Numer. Methods Eng. 44:3 (1999), 373–391.
Boffi, D., Brezzi, F., Fortin, M., et al. Mixed Finite Element Methods And Applications. 2013, Springer.
Gantier, M., Modélisation numérique robuste et fiable de la fissuration des roches et des interfaces. (Ph.D. thesis), 2021, Université Grenoble Alpes.
Placidi, L., Misra, A., Barchiesi, E., Two-dimensional strain gradient damage modeling: a variational approach. Z. Angew. Math. Phys., 69(3), 2018.
Placidi, L., Misra, A., Barchiesi, E., Simulation results for damage with evolving microstructure and growing strain gradient moduli. Contin. Mech. Thermodyn. 31:4 (2019), 1143–1163.
Placidi, L., Barchiesi, E., Energy approach to brittle fracture in strain-gradient modelling. Proc. R. Soc. A, 474(2210), 2018, 20170878.
Mazars, J., Application de la mécanique de l'endommagement au comportement non linéaire et à la rupture du béton de structure. 1984 Thèse de doctorat d’état de l'Université Paris VI.
Bésuelle, P., Implémentation d'un nouveau type d’élément fini dans le code Lagamine pour une classe de lois á longueur interne: Tech. Rep., 2003, Laboratoire Sols, Solides Structures, Grenoble, France, 20 Rapport d'activité.
Cazes, F., Coret, M., Combescure, A., A two-field modified Lagrangian formulation for robust simulations of extrinsic cohesive zone models. Comput. Mech. 51 (2013), 865–884, 10.1007/s00466-012-0763-1.
Radovitzky, R., Seagraves, A., Tupek, M., Noels, L., A scalable 3d fracture and fragmentation algorithm based on a hybrid, discontinuous galerkin, cohesive element method. Comput. Methods Appl. Mech. Eng. 200:1 (2011), 326–344.
Camacho, G., Ortiz, M., Computational modelling of impact damage in brittle materials. Int. J. Solids Struct. 33:20 (1996), 2899–2938.
Souza, F., Allen, D., Modeling failure of heterogeneous viscoelastic solids under dynamic/impact loading due to multiple evolving cracks using a two-way coupled multiscale model. Mech. Time-Dependent Mater. 14:2 (2010), 125–151.
Moės, N., Belytschko, T., Extended finite element method for cohesive crack growth. Eng. Fract. Mech. 69:7 (2002), 813–833.
Dvorkin, E., Cuitiño, A., Gioia, G., Finite elements with displacement interpolated embedded localization lines insensitive to mesh size and distortions. Int. J. Numer. Methods Eng. 30:3 (1990), 541–564.
Jirásek, M., Comparative study on finite elements with embedded discontinuities. Comput. Methods Appl. Mech. Eng. 188:1 (2000), 307–330.
Alam, S., Experimental study and numerical analysis of crack opening in concrete. (Ph.D. thesis), 2011, Ecole Centrale de Nantes (ECN) https://tel.archives-ouvertes.fr/tel-00669877.
Alam, S., Kotronis, P., Loukili, A., Crack propagation and size effect in concrete using a non-local damage model. Eng. Fract. Mech. 109 (2013), 246–261.
