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Failure-oriented multi-scale variational formulation: micro-structures with nucleation and evolution of softening bands. (English) Zbl 1286.74009

Summary: This contribution presents the theoretical foundations of a Failure-Oriented Multi-scale variational Formulation (FOMF) for modeling heterogeneous softening-based materials undergoing strain localization phenomena. The multi-scale model considers two coupled mechanical problems at different physical length scales, denoted as macro and micro scales, respectively. Every point, at the macro scale, is linked to a Representative Volume Element (RVE), and its constitutive response emerges from a consistent homogenization of the micro-mechanical problem.
At the macroscopic level, the initially continuum medium admits the nucleation and evolution of cohesive cracks due to progressive strain localization phenomena taking place at the microscopic level and caused by shear bands, damage or any other possible failure mechanism. A cohesive crack is introduced in the macro model once a specific macroscopic failure criterion is fulfilled.
The novelty of the present Failure-Oriented Multi-scale Formulation is based on a proper kinematical information transference from the macro-to-micro scales during the complete loading history, even in those points where macro cracks evolve. In fact, the proposed FOMF includes two multi-scale sub-models consistently coupled:
(i) a Classical Multi-scale Model (ClaMM) valid for the stable macro-scale constitutive response.
(ii) A novel Cohesive Multi-scale Model (CohMM) valid, once a macro-discontinuity surface is nucleated, for modeling the macro-crack evolution.
When a macro-crack is activated, two important kinematical assumptions are introduced: (i) a change in the rule that defines how the increments of generalized macro-strains are inserted into the micro-scale and (ii) the Kinematical Admissibility concept, from where proper Strain Homogenization Procedures are obtained. Then, as a consequence of the Hill-Mandel Variational Principle and the proposed kinematical assumptions, the FOMF provides an adequate homogenization formula for the stresses in the continuum part of the body, as well as, for the traction acting on the macro-discontinuity surface.
The assumed macro-to-micro mechanism of kinematical coupling defines a specific admissible RVE-displacement space, which is obtained by incorporating additional boundary conditions, Non-Standard Boundary Conditions (NSBC), in the new model. A consequence of introducing these Non-Standard Boundary Conditions is that they guarantee the existence of a physically admissible RVE-size, a concept that we call through the paper “objectivity” of the homogenized constitutive response.
Several numerical examples are presented showing the objectivity of the formulation, as well as, the capabilities of the new multi-scale approach to model material failure problems.

MSC:

74A45 Theories of fracture and damage
74M25 Micromechanics of solids
74N15 Analysis of microstructure in solids
74Q20 Bounds on effective properties in solid mechanics
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[1] Barenblatt, G., The mathematical theory of equilibrium of cracks in brittle fracture, Adv. Appl. Mech., 7, 55-129 (1962)
[2] Bazant, Z.; Planas, J., Fracture and Size Effect in Concrete and Other Quasibrittle Materials (1998), CRC Press: CRC Press Boca Raton, FL
[3] Belytschko, T.; Chen, H.; Xu, J.; Zi, G., Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment, Int. J. Numer. Methods Engrg., 58, 1873-1905 (2003) · Zbl 1032.74662
[4] Belytschko, T.; Loehnert, S.; Song, J. H., Multiscale aggregating discontinuities: a method for circumventing loss of material stability, Int. J. Numer. Methods Engrg., 73, 869-894 (2008) · Zbl 1195.74008
[5] Christensen, R.; Lou, K., Solutions for effective shear properties in three phase sphere and cylinder models, J. Mech. Phys. Solids, 27, 315-330 (1979) · Zbl 0419.73007
[7] Dugdale, D., Yielding of steel sheets containing slits, J. Mech. Phys. Solids, 8, 100-108 (1960)
[8] Eshelby, J., The determination of the field of an ellipsoidal inclusion and related problems, Proc. R. Soc. Lond. A, 241, 376-396 (1957) · Zbl 0079.39606
[9] Feyel, F.; Chaboche, J., \(Fe^2\) multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials, Comput. Methods Appl. Mech. Engrg., 183, 309-330 (2000) · Zbl 0993.74062
[10] Fish, J.; Yu, Q.; Shek, K., Computational damage mechanics for composite materials based on mathematical homogenization, Int. J. Numer. Methods Engrg., 45, 1657-1679 (1999) · Zbl 0949.74057
[11] Gitman, I.; Askes, H.; Sluys, L., Representative volume: existence and size determination, Eng. Fract. Mech., 74, 2518-2534 (2007)
[12] Giusti, S.; Blanco, P. J.; de Souza Neto, E.; Feijóo, R. A., An assessment of the Gurson yield criterion by a computational multi-scale approach, Eng. Comput.: Int. J. Comput.-Aid. Eng. Softw., 26, 3, 281-301 (2009) · Zbl 1257.74151
[13] Hashin, Z.; Shtrikman, S., A variational approach to the theory of the elastic behaviour of multiphase materials, J. Mech. Phys. Solids, 11, 127-140 (1963) · Zbl 0108.36902
[14] Hill, R., A self-consistent mechanics of composite materials, J. Mech. Phys. Solids, 13, 213-222 (1965)
[15] Hillerborg, A.; Modeer, M.; Petersson, P., Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements, Cement Concrete Res., 6, 163-168 (1976)
[17] Kouznetsova, V.; Geers, M.; Brekelmans, W., Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme, Int. J. Numer. Methods Engrg., 54, 1235-1260 (2002) · Zbl 1058.74070
[18] Kouznetsova, V.; Geers, M.; Brekelmans, W., Multi-scale second-order computational homogenization of multi-phase materials: a nested finite element solution strategy, Comput. Methods Appl. Mech. Engrg., 193, 5525-5550 (2002) · Zbl 1112.74469
[19] Kulkarni, M.; Geubelle, P.; Matous, K., Multi-scale modeling of heterogeneous adhesives: effect of particle decohesion, Mech. Mater., 41, 573-583 (2009)
[20] Matous, K.; Kulkarni, M.; Geubelle, P., Multiscale cohesive failure modeling of heterogeneous adhesives, J. Mech. Phys. Solids, 56, 1511-1533 (2008) · Zbl 1171.74433
[21] Michel, J.; Moulinec, H.; Suquet, P., Effective properties of composite materials with periodic microstructure: a computational approach, Comput. Methods Appl. Mech. Engrg., 172, 109-143 (1999) · Zbl 0964.74054
[22] Miehe, C.; Koch, A., Computational micro-to-macro transition of discretized microstructures undergoing small strain, Arch. Appl. Mech., 72, 300-317 (2002) · Zbl 1032.74010
[23] Miehe, C.; Schroder, J.; Becker, M., Computational homogenization analysis in finite elasticity: material and structural instabilities on the micro- and macro-scales of periodics composites and their interaction, Comput. Methods Appl. Mech. Engrg., 191, 4971-5005 (2002) · Zbl 1099.74517
[24] Mori, T.; Tanaka, K., Average stress in the matrix and average energy of materials with misfitting inclusions, Acta Metall., 21, 571-574 (1973)
[25] Nemat-Nasser, S.; Hori, M., Micromechanics: Overall Properties of Heterogeneous Materials (1999), Elsevier · Zbl 0924.73006
[26] Nguyen, V.; Valls, O.; Stroeven, M.; Sluys, L., On the existence of representative volumes for softening quasi-brittle materials - a failure zone averaging scheme, Comput. Methods Appl. Mech. Engrg., 199, 3028-3038 (2010) · Zbl 1231.74372
[27] Oliver, J.; Huespe, A. E., Theoretical and computational issues in modelling material failure in strong discontinuity scenarios, Comput. Methods Appl. Mech. Engrg., 193, 2987-3014 (2004) · Zbl 1067.74505
[28] Oliver, J.; Huespe, A. E.; Cante, J.; Díaz, G., On the numerical resolution of the discontinuous material bifurcation problem, Int. J. Numer. Methods Engrg., 83, 786-804 (2010) · Zbl 1197.74194
[29] Perić, D.; de Souza Neto, E.; Feijóo, R. A.; Partovi, M.; Molina, A. C., On micro-to-macro transitions for multi-scale analysis of non-linear heterogeneous materials: unified variational basis and finite element implementation, Int. J. Numer. Methods Engrg., 87, 149-170 (2011) · Zbl 1242.74087
[30] Rice, J., The localization of plastic deformation, (Koiter, W., Theoretical and Applied Mechanics, 14th IUTAM Congress (1976), Amsterdam: Amsterdam North-Holland), 207-220
[32] Rudnicki, J.; Rice, J., Condition for the localization of deformations in pressure sensitive dilatant materials, J. Mech. Phys. Solids, 23, 371-394 (1975)
[33] Runesson, K.; Ottosen, N.; Peric, D., Discontinuous bifurcations of elastic-plastic solutions at plane stress and plane strain, Int. J. Plasticity, 7, 99-121 (1991) · Zbl 0761.73035
[35] Sanchez-Palencia, E., Non-homogeneous media and vibration theory, Lecture Notes in Physics, vol. 127 (1980), Springer-Verlag: Springer-Verlag Berlin · Zbl 0432.70002
[36] Simo, J.; Hughes, T., Computational Inelasticity (1998), Springer-Verlag · Zbl 0934.74003
[37] Simo, J.; Ju, J., Strain- and stress-based continuum damage models - Part II: Formulation, Int. J. Solids Struct., 23, 841-869 (1987) · Zbl 0634.73107
[38] Simo, J.; Oliver, J., A new approach to the analysis and simulation of strain softening in solids, (Bazant, Z.; Bittnar, Z.; Jirásek, M.; Mazars, J., Fracture and Damage in Quasi-Brittle Structures (1994), E & FN Spon), 25-39
[39] Song, J. H.; Belytschko, T., Multiscale aggregating discontinuities method for micro-macro failure of composites, Compos. Part B, 40, 417-426 (2009)
[41] de Souza Neto, E.; Feijóo, R. A., On the equivalence between spatial and material volume averaging of stress in large strain multi-scale solid constitutive models, Mech. Mater., 40, 803-811 (2008)
[42] Verhoosel, C.; Remmers, J.; Gutiérrez, M.; de Borst, R., Computational homogenization for adhesive and cohesive failure in quasi-brittle solids, Int. J. Numer. Methods Engrg., 83, 1155-1179 (2010) · Zbl 1197.74139
[43] Willis, J., Variational and related methods for the overall properties of composites, Adv. Appl. Mech., 21, 1-78 (1981) · Zbl 0476.73053
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