NOTICE: this is the author’s version of a work that was accepted for publication in Computer Methods in Applied Mechanics and Engineering. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Computer Methods in Applied Mechanics and Engineering, Vol. 233-236, pages 164 - 179, 2012, DOI: 10.1016/j.cma.2012.04.011
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[en] In this work, a gradient-enhanced homogenization procedure is proposed for fiber reinforced materials. In this approach, the fiber is assumed to remain linear elastic while the matrix material is modeled as elasto-plastic coupled with a damage law described by a non-local constitutive model. Toward this end, the mean-field homogenization is based on the knowledge of the macroscopic deformation tensors, internal variables and their gradients, which are applied
to a micro-structural representative volume element (RVE). The macro-stress is then obtained from a homogenization procedure. The methodology holds for 2-phase composites with moderate fiber volume ratios, and for which, at the RVE size, the matrix can be considered as homogeneous isotropic and the ellipsoidal fibers can be considered as homogeneous transversely isotropic. Under these assumptions, the method is successfully applied to simulate the damage process occurring in unidirectional carbon-fiber reinforced epoxy composites submitted to different loading conditions.
FP7 - 235303 - MATERA+ - ERA-NET Plus on Materials Research
Name of the research project :
SIMUCOMP no 1017232 (CT-EUC 2010-10-12)
Funders :
The research has been funded by the Walloon Region under the agreement SIMUCOMP no 1017232 (CT-EUC 2010-10-12) in the context of the ERA-NET +, Matera + framework. CE - Commission Européenne
Kanouté P., Boso D., Chaboche J., Schrefler B. Multiscale methods for composites: a review. Arch. Comput Methods Engrg. 2009, 16:31-75. 1134-3060, 10.1007/s11831-008-9028-8.
Eshelby J.D. The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc. Roy. Soc. Lond Ser. A 1957, 241(1226):376-396. 00804630.
Mori T., Tanaka K. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metallur. 1973, 21(5):571-574. cited by (since 1996) 1814.
Benveniste Y. A new approach to the application of Mori-Tanaka's theory in composite materials. Mech. Mater. 1987, 6(2):147-157. 0167-6636, 10.1016/0167-6636(87)90005-6.
Budiansky B. On the elastic moduli of some heterogeneous materials. J. Mech. Phys. Solids 1965, 13(4):223-227. 0022-5096, 10.1016/0022-5096(65)90011-6.
Kröner E. Berechnung der elastischen Konstanten des Vielkristalls aus den Konstanten des Einkristalls. Zeitschrift für Phys. A Hadrons Nuclei 1958, 151:504-518. 0939-7922.
Hershey A. The elasticity of an isotropic aggregate of anisotropic cubic crystals. J. Appl. Mech. Trans. ASME 1954, 21(3):236-240.
Hill R. A self-consistent mechanics of composite materials. J. Mech. Phys. Solids 1965, 13(4):213-222. 0022-5096, 10.1016/0022-5096(65)90010-4.
Talbot D.R.S., Willis J.R. Variational principles for inhomogeneous non-linear media. IMA J. Appl. Math. 1985, 35(1):39-54. 10.1093/imamat/35.1.39.
Talbot D.R.S., Willis J.R. Bounds and self-consistent estimates for the overall properties of nonlinear composites. IMA J. Appl. Math. 1987, 39(3):215-240. 10.1093/imamat/39.3.215.
Ponte Castañeda P. The effective mechanical properties of nonlinear isotropic composites. J. Mech. Phys. Solids 1991, 39(1):45-71. 0022-5096, 10.1016/0022-5096(91)90030-R.
Ponte Castañeda P. A new variational principle and its application to nonlinear heterogeneous systems. SIAM J. Appl. Math. 1992, 52(5):1321-1341. 00361399.
Talbot D., Willis J. Some simple explicit bounds for the overall behaviour of nonlinear composites. Int. J. Solids Struct. 1992, 29(14-15):1981-1987. 0020-7683, 10.1016/0020-7683(92)90188-Y.
Hill R. Continuum micro-mechanics of elastoplastic polycrystals. J. Mech. Phys. Solids 1965, 13(2):89-101. 0022-5096, 10.1016/0022-5096(65)90023-2.
Pettermann H.E., Plankensteiner A.F., Böhm H.J., Rammerstorfer F.G. A thermo-elasto-plastic constitutive law for inhomogeneous materials based on an incremental Mori-Tanaka approach. Comput. Struct. 1999, 71(2):197-214. 0045-7949, 10.1016/S0045-7949(98).
Doghri I., Ouaar A. Homogenization of two-phase elasto-plastic composite materials and structures: study of tangent operators cyclic plasticity and numerical algorithms. Int. J. Solids Struct. 2003, 40(7):1681-1712. 0020-7683, 10.1016/S0020-7683(03)00013-1.
Chaboche J., Kanouté P., Roos A. On the capabilities of mean-field approaches for the description of plasticity in metal matrix composites. Int. J. Plasticity 2005, 21(7):1409-1434. 0749-6419, 10.1016/j.ijplas.2004.07.001.
Doghri I., Brassart L., Adam L., Gérard J.S. A second-moment incremental formulation for the mean-field homogenization of elasto-plastic composites. Int. J. Plasticity 2011, 27(3):352-371. 0749-6419, 10.1016/j.ijplas.2010.06.004.
Kouznetsova V., Geers M.G.D., Brekelmans W.A.M. Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme. Int. J. Numer. Methods Engrg. 2002, 54(8):1235-1260.
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. 2004, 193(48-51):5525-5550. Advances in Computational Plasticity. 0045-7825, 10.1016/j.cma.2003.12.073.
De Borst R. Simulation of strain localization: a reappraisal of the Cosserat continuum. Engrg. Comput. 1991, 8(4):317-332. 0264-4401.
Bazant Z.P., Belytchko T.B., Chang T.P. Continuum theory for strain-softening. J. Engrg. Mech. - ASCE 1984, 110(12):1666-1692. 10.1061/ASCE)0733-9399(1984)110:12(1666.
Zbib H.M., Aifantis E.C. A gradient-dependent flow theory of plasticity: application to metal and soil instabilities. Appl. Mech. Rev. 1989, 42(11S):S295-S304. 10.1115/1.3152403.
De Borst R., Sluys L., Mühlhaus H.-B., Pamin J. Fundamental issues in finite element analyses of localization of deformation. Engrg. Comput. 1993, 10(2):99-121. 0264-4401.
Peerlings R., de Borst R., Brekelmans W., de Vree J. Gradient-enhanced damage for quasi-brittle materials. Int. J. Numer. Methods Engrg. 1996, 39:3391-3403.
M. Geers, Experimental Analysis and Computational Modelling of Damage and Fracture, Ph.D. Thesis, University of Technology, Eindhoven (Netherlands), 1997.
Peerlings R., de Borst R., Brekelmans W., Geers M. Gradient-enhanced damage modelling of concrete fracture. Mech. Cohesive-Frictional Mat. 1998, 3:323-342.
Liu X., Hu G. A continuum micromechanical theory of overall plasticity for particulate composites including particle size effect. Int. J. Plasticity 2005, 21(4):777-799. 0749-6419, 10.1016/j.ijplas.2004.04.014.
Dascalu C. A two-scale damage model with material length. Comptes Rendus Mécan. 2009, 337(9-10):645-652. 1631-0721, 10.1016/j.crme.2009.09.008.
Knockaert R., Doghri I. Nonlocal constitutive models with gradients of internal variables derived from a micro/macro homogenization procedure. Comput. Methods Appl. Mech. Engrg. 1999, 174(1-2):121-136. 0045-7825, 10.1016/S0045-7825(98)00282-5.
Massart T., Peerlings R., Geers M. A dissipation-based control method for the multi-scale modelling of quasi-brittle materials. C.R. Mec. 2005, 333:521-527.
Massart T., Peerlings R., Geers M. An enhanced multi-scale approach for masonry wall computations. Int. J. Numer. Methods Engrg. 2007, 69(5):1022-1059.
Coenen E., Kouznetsova V., Geers M. Enabling microstructure-based damage and localization analyses and upscaling. Model. Simul. Mater. Sci. Engrg. 2011, 19(7):074008.
Coenen E., Kouznetsova V., Geers M. Novel boundary conditions for strain localization analyses in microstructural volume elements. Int. J. Numer. Methods Engrg. 2012, 90(1):1-21. http://dx.doi.org/10.1002/nme.3298.
Peerlings R., Geers M., de Borst R., Brekelmans W. A critical comparison of nonlocal and gradient-enhanced softening continua. Int. J. Solids Struct. 2001, 38:7723-7746.
Segurado J., Llorca J. A numerical approximation to the elastic properties of sphere-reinforced composites. J. Mech. Phys. Solids 2002, 50(10):2107-2121. 0022-5096.
Berveiller M., Zaoui A. An extension of the self-consistent scheme to plastically-flowing polycrystals. J. Mech. Phys. Solids 1978, 26(5-6):325-344. 0022-5096, 10.1016/0022-5096(78)90003-0.
Weng G.J. The overall elastoplastic stress-strain relations of dual-phase metals. J. Mech. Phys. Solids 1990, 38(3):419-441. 0022-5096, 10.1016/0022-5096(90)90007-Q.
Ponte Castañeda P. Exact second-order estimates for the effective mechanical properties of nonlinear composite materials. J. Mech. Phys. Solids 1996, 44(6):827-862. 0022-5096, 10.1016/0022-5096(96)00015-4.
Lahellec N., Suquet P. On the effective behavior of nonlinear inelastic composites: I. Incremental variational principles. J. Mech. Phys. Solids 2007, 55(9):1932-1963. 0022-5096, 10.1016/j.jmps.2007.02.003.
Brassart L., Stainier L., Doghri I., Delannay L. A variational formulation for the incremental homogenization of elasto-plastic composites. J. Mech. Phys. Solids 2011, 59(12):2455-2475. 0022-5096, 10.1016/j.jmps.2011.09.004.
Lahellec N., Ponte Castañeda P., Suquet P. Variational estimates for the effective response field statistics in thermoelastic composites with intra-phase property fluctuations. Proc. Roy. Soc. A: Math. Phys. Engrg. Sci. 2011, 467(2132):2224-2246. 10.1098/rspa.2010.0609.
Masson R., Bornert M., Suquet P., Zaoui A. An affine formulation for the prediction of the effective properties of nonlinear composites and polycrystals. J. Mech. Phys. Solids 2000, 48:1203-1227. 10.1016/S0022-509(99)00071-X.
Pierard O., Doghri I. Study of various estimates of the macroscopic tangent operator in the incremental homogenization of elastoplastic composites. Int. J. Multiscale Comput. Engrg. 2006, 4(4):521-543. 1543-1649.
Bazant Z.P., Pijaudier-Cabot G., damage Nonlocal continuum localization instability and convergence. J. Appl. Mech. 1988, 55(2):287-293. . 10.1115/1.3173674.
Lemaitre J., Desmorat R. Engineering Damage Mechanics: Ductile, Creep, Fatigue and Brittle Failures 2005, Springer-Verlag, Berlin.
Lemaitre J. Coupled elasto-plasticity and damage constitutive equations. Comput. Methods Appl. Mech. Engrg. 1985, 51(1-3):31-49. 0045-7825, 10.1016/0045-7825(85)90026-X.
Doghri I. Numerical implementation and analysis of a class of metal plasticity models coupled with ductile damage. Int. J. Numer. Methods Engrg. 1995, 38(20):3403-3431. 1097-0207, 10.1002/nme.1620382004.
Geers M.G.D., de Borst R., Brekelmans W.A.M., Peerlings R.H.J. Strain-based transient-gradient damage model for failure analyses. Comput. Methods Appl. Mech. Engrg. 1998, 160:133-153.
Simo J.C., Taylor R.L. Consistent tangent operators for rate-independent elastoplasticity. Comput. Methods Appl. Mech. Engrg. 1985, 48(1):101-118. 0045-7825, 10.1016/0045-7825(85)90070-2.
Dvorak G.J. Transformation field analysis of inelastic composite materials. Proc.: Math. Phys. Sci. 1992, 437(2132). pp. 311-327.
Dvorak G.J., Bahei-El-Din Y.A., Wafa A.M. Implementation of the transformation field analysis for inelastic composite materials. Comput. Mech. 1994, 14(3):201-228.
L. Brassart, Homogenization of elasto-(visco) plastic composites: history-dependent incremental and variational approaches, Ph.D. Thesis, The Universitcatholique de Louvain, Louvain-la-Neuve, Belgium, 2011.
Doghri I. Mechanics of Deformable Solids - Linear, Nonlinear, Analytical and Computational Aspects 2000, Springer-Verlag, Berlin.
Peerlings R., Massart T., Geers M. A thermodynamically motivated implicit gradient damage framework and its application to brick masonry cracking. Comput. Methods Appl. Mech. Engrg. 2004, 193(30):3403-3417. . 0045-7825, 10.1016/j.cma.2003.10.021.
Geuzaine C., Remacle J.-F. Gmsh: a 3-D finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Methods Engrg. 2009, 79(11):1309-1331. .
Gitman I., Askes H., Sluys L. Representative volume: existence and size determination. Engrg. Fract. Mech. 2007, 74(16):2518-2534. . 0013-7944, 10.1016/j.engfracmech.2006.12.021.
Pelissou C., Baccou J., Monerie Y., Perales F. Determination of the size of the representative volume element for random quasi-brittle composites. Int. J. Solids Struct. 2009, 46(14-15):2842-2855. . 0020-7683, 10.1016/j.ijsolstr.2009.03.015.
Nguyen V.P., Lloberas-Valls O., Stroeven M., Sluys L.J.N. On the existence of representative volumes for softening quasi-brittle materials a failure zone averaging scheme. Comput. Methods Appl. Mech. Engrg. 2010, 199(45-48):3028-3038. . 0045-7825, 10.1016/j.cma.2010.06.018.
González C., LLorca J. Multiscale modeling of fracture in fiber-reinforced composites. Acta Mater. 2006, 54(16):4171-4181. 1359-6454.
Geers M., de Borst R., Brekelmans W., Peerlings R. Validation and internal length scale determination for a gradient damage model: application to short glass-fibre-reinforced polypropylene. Int. J. Solids Struct. 1999, 36(17):2557-2583. 0020-7683.
