data-driven sampling; failure; incremental-secant; mean-field homogenization; nonlocal; random field; size objectivity; stochastic; surrogate; Data driven; Data-driven sampling; Incremental-secant; Mean-field homogenizations; Nonlocal; Random fields; Size objectivity; Stochastics; Surrogate; Volume elements; Numerical Analysis; Engineering (all); Applied Mathematics; General Engineering
Abstract :
[en] This paper presents the construction of a mean-field homogenization (MFH) surrogate for nonlinear stochastic multiscale analyses of two-phase composites that allows the material response to be studied up to its failure. The homogenized stochastic behavior of the studied unidirectional composite material is first characterized through full-field simulations on stochastic volume elements (SVEs) of the material microstructure, permitting to capture the effect of the microstructural geometric uncertainties on the material response. Then, in order to conduct the stochastic nonlinear multiscale simulations, the microscale problem is substituted by a pressure-dependent MFH reduced order micromechanical model, that is, a MF-ROM, whose properties are identified by an inverse process from the full-field SVE realizations. Homogenized stress-strain curves can be used for the identification process of the nonlinear range, however, a loss of size objectivity is encountered once the strain softening onset is reached. This work addresses this problematic introducing a calibration of the energy release rate obtained with a nonlocal MFH micromechanical model, allowing to scale the variability found on each SVE failure characteristics to the macroscale. The obtained random effective properties are then used as input of a data-driven stochastic model to generate the complete random fields used to feed the stochastic MF-ROM. To show the consistency of the methodology, two MF-ROM constructed from SVEs of two different sizes are successively considered. The performance of the MF-ROM is then verified against an experimental transverse-compression test and against full-field simulations through nonlocal Stochastic Finite Element Method (SFEM) simulations. The implementation of the energy release rate calibration allows to conduct stochastic studies on the failure characteristics of material samples without the need for costly experimental campaigns, paving the way for more complete and affordable virtual testing.
FRIA - Fonds pour la Formation à la Recherche dans l'Industrie et dans l'Agriculture [BE]
Funding text :
The authors would like to acknowledge the financial support from FRS–FNRS as this publication benefits from the support of the Walloon Region within the framework of a FRIA grant. The authors would also like to acknowledge the supercomputing facilities of the Consortium des Équipements de Calcul Intensif en Federation Wallonie Bruxelles (CÉCI) that were funded by the FRS–FNRS that were made available to us.
This is the peer reviewed version of the following article: “Calleja Vázquez, JM, Wu, L, Nguyen, V-D, Noels, L. A micromechanical mean-field homogenization surrogate for the stochastic multiscale analysis of composite materials failure. Int J Numer Methods Eng. 124 (23), 5200–5262, 2023”, which has been published in final form at 10.1002/nme.7344. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions. This article may not be enhanced, enriched or otherwise transformed into a derivative work, without
express permission from Wiley or by statutory rights under applicable legislation. Copyright notices must not be removed, obscured or modified. The article must be linked to Wiley’s version of record on Wiley Online Library and any embedding, framing or otherwise making available the article or pages thereof by third parties from platforms, services and websites other than Wiley Online Library must be prohibited.
Ghanem R, Spanos P. Stochastic Finite Elements: A Spectral Approach. Springer Verlag; 1991.
Charmpis D, Schuëller G, Pellissetti M. The need for linking micromechanics of materials with stochastic finite elements: a challenge for materials science. Comput Mater Sci. 2007;41(1):27-37. doi:10.1016/j.commatsci.2007.02.014
Ostoja-Starzewski M, Wang X. Stochastic finite elements as a bridge between random material microstructure and global response. Comput Methods Appl Mech Eng. 1999;168(1):35-49. doi:10.1016/S0045-7825(98)00105-4
Noels L. Toward stochastic multiscale methods in continuum solid mechanics. Advances in Applied Mechanics. Vol 55. Elsevier; 2022:1-254.
Vaughan T, McCarthy C. A combined experimental–numerical approach for generating statistically equivalent fibre distributions for high strength laminated composite materials. Compos Sci Technol. 2010;70(2):291-297. doi:10.1016/j.compscitech.2009.10.020
Melro A, Camanho P, Pinho S. Generation of random distribution of fibres in long-fibre reinforced composites. Compos Sci Technol. 2008;68(9):2092-2102. doi:10.1016/j.compscitech.2008.03.013
Wu L, Chung CN, Major Z, Adam L, Noels L. From SEM images to elastic responses: a stochastic multiscale analysis of UD fiber reinforced composites. Compos Struct. 2018;189:206-227. doi:10.1016/j.compstruct.2018.01.051
Blacklock M, Bale H, Begley M, Cox B. Generating virtual textile composite specimens using statistical data from micro-computed tomography: 1D tow representations for the binary model. J Mech Phys Solids. 2012;60(3):451-470. doi:10.1016/j.jmps.2011.11.010
Tal D, Fish J. Generating a statistically equivalent representative volume element with discrete defects. Compos Struct. 2016;153:791-803. doi:10.1016/j.compstruct.2016.06.077
Vanaerschot A, Cox BN, Lomov SV, Vandepitte D. Stochastic multi-scale modelling of textile composites based on internal geometry variability. Comput Struct. 2013;122:55-64. doi:10.1016/j.compstruc.2012.10.026
Gupta A, Cecen A, Goyal S, Singh AK, Kalidindi SR. Structure-property linkages using a data science approach: application to a non-metallic inclusion/steel composite system. Acta Mater. 2015;91:239-254. doi:10.1016/j.actamat.2015.02.045
Kanouté P, Boso D, Chaboche J, Schrefler B. Multiscale methods for composites: a review. Arch Comput Methods Eng. 2009;16:31-75.
Matouš K, Geers MG, Kouznetsova VG, Gillman A. A review of predictive nonlinear theories for multiscale modeling of heterogeneous materials. J Comput Phys. 2017;330:192-220. doi:10.1016/j.jcp.2016.10.070
Feyel F. Multiscale FE2 elastoviscoplastic analysis of composite structures. Comput Mater Sci. 1999;16:344-354.
Miehe C. Strain-driven homogenization of inelastic microstructures and composites based on an incremental variational formulation. Int J Numer Methods Eng. 2002;55(11):1285-1322. doi:10.1002/nme.515
Pivovarov D, Mergheim J, Willner K, Steinmann P. Stochastic local FEM for computational homogenization of heterogeneous materials exhibiting large plastic deformations. Comput Mech. 2022;69:467-488. doi:10.1007/s00466-021-02099-x
Kouznetsova V, Brekelmans W, Baaijens F. An approach to micro-macro modeling of heterogeneous materials. Comput Mech. 2001;27(1):37-48.
Kouznetsova V, Geers M, Brekelmans W. Multi-scale second-order computational homogenization of muli-phase materials: a nested finite element solution strategy. Comput Methods Appl Mech Eng. 2004;193:5525-5550.
Pingaro M, Reccia E, Trovalusci P, Masiani R. Fast statistical homogenization procedure (FSHP) for particle random composites using virtual element method. Comput Mech. 2019;64(1):197-210.
Pingaro M, Reccia E, Trovalusci P. Homogenization of random porous materials with low-order virtual elements. ASCE-ASME J Risk Uncert Eng Syst Part B Mech Eng. 2019;5(3):030905. doi:10.1115/1.4043475
Pingaro M, De Bellis M, Reccia E, Trovalusci P, Sadowski T. Fast statistical homogenization procedure for estimation of effective properties of ceramic matrix composites (CMC) with random microstructure. Compos Struct. 2023;304:116265. doi:10.1016/j.compstruct.2022.116265
McDowell DL. A perspective on trends in multiscale plasticity. Int J Plastic. 2010;26(9):1280-1309. doi:10.1016/j.ijplas.2010.02.008
Ostoja-Starzewski M, Du X, Khisaeva ZF, Li W. Comparisons of the size of the representative volume element in elastic, plastic, thermoelastic, and permeable random microstructures. Int J Multiscale Comput Eng. 2007;5(2):73-82.
Salmi M, Auslender F, Bornert M, Fogli M. Apparent and effective mechanical properties of linear matrix-inclusion random composites: improved bounds for the effective behavior. Int J Solids Struct. 2012;49(10):1195-1211. doi:10.1016/j.ijsolstr.2012.01.018
Trovalusci P, Ostoja-Starzewski M, De Bellis ML, Murrali A. Scale-dependent homogenization of random composites as micropolar continua. Eur J Mech A/Solids. 2015;49:396-407. doi:10.1016/j.euromechsol.2014.08.010
Reccia E, De Bellis ML, Trovalusci P, Masiani R. Sensitivity to material contrast in homogenization of random particle composites as micropolar continua. Compos Part B Eng. 2018;136:39-45. doi:10.1016/j.compositesb.2017.10.017
Ostoja-Starzewski M. Scale effects in plasticity of random media: status and challenges. Int J Plastic. 2005;21(6):1119-1160. doi:10.1016/j.ijplas.2004.06.008
Hachemi A, Chen M, Chen G, Weichert D. Limit state of structures made of heterogeneous materials. Int J Plastic. 2014;63:124-137. doi:10.1016/j.ijplas.2014.03.019
Lucas V, Golinval JC, Paquay S, Nguyen VD, Noels L, Wu L. A stochastic computational multiscale approach; application to MEMS resonators. Comput Methods Appl Mech Eng. 2015;294:141-167. doi:10.1016/j.cma.2015.05.019
Yvonnet J, He QC. The reduced model multiscale method (R3M) for the non-linear homogenization of hyperelastic media at finite strains. J Comput Phys. 2007;223(1):341-368. doi:10.1016/j.jcp.2006.09.019
Hernández J, Oliver J, Huespe A, Caicedo M, Cante J. High-performance model reduction techniques in computational multiscale homogenization. Comput Methods Appl Mech Eng. 2014;276:149-189. doi:10.1016/j.cma.2014.03.011
Soldner D, Brands B, Zabihyan R, Steinmann P, Mergheim J. A numerical study of different projection-based model reduction techniques applied to computational homogenisation. Comput Mech. 2017;60(613):625. doi:10.1007/s00466-017-1428-x
Michel JC, Suquet P. Non-uniform transformation field analysis: a reduced model for multiscale non-linear problems in solid mechanics. Comput Exper Methods Struct. 2009;159-206. doi:10.1142/9781848163089_0004
Michel JC, Suquet P. A model-reduction approach in micromechanics of materials preserving the variational structure of constitutive relations. J Mech Phys Solids. 2016;90:254-285. doi:10.1016/j.jmps.2016.02.005
Liu Z, Bessa M, Liu WK. Self-consistent clustering analysis: an efficient multi-scale scheme for inelastic heterogeneous materials. Comput Methods Appl Mech Eng. 2016;306:319-341. doi:10.1016/j.cma.2016.04.004
Pivovarov D, Steinmann P, Willner K. Acceleration of the spectral stochastic FEM using POD and element based discrete empirical approximation for a micromechanical model of heterogeneous materials with random geometry. Comput Methods Appl Mech Eng. 2020;360(1):112689. doi:10.1016/j.cma.2019.112689
Clément A, Soize C, Yvonnet J. Computational nonlinear stochastic homogenization using a nonconcurrent multiscale approach for hyperelastic heterogeneous microstructures analysis. Int J Numer Methods Eng. 2012;91(8):799-824. doi:10.1002/nme.4293
Yvonnet J, Monteiro E, He QC. Computational homogenization method and reduced database model for hyperelastic heterogeneous structures. Int J Multiscale Comput Eng. 2013;11(3):201-225.
Hashemi MS, Safdari M, Sheidaei A. A supervised machine learning approach for accelerating the design of particulate composites: application to thermal conductivity. Comput Mater Sci. 2021;197:110664. doi:10.48550/ARXIV.2010.00041
Lu X, Yvonnet J, Papadopoulos L, Kalogeris I, Papadopoulos V. A stochastic FE2 data-driven method for nonlinear multiscale modeling. Materials. 2021;14(11):2875. doi:10.3390/ma14112875
Lu X, Glovanis D, Yvonnet J, Papadopoulos V, Detrez F. A data-driven computational homogenization method based on neural networks for the nonlinear anisotropic electrical response of graphene/polymer nanocomposites. Comput Mech. 2019;64:307-321. doi:10.1007/s00466-018-1643-0
Fritzen F, Fernández M, Larsson F. On-the-fly adaptivity for nonlinear two scale simulations using artificial neural networks and reduced order modeling. Front Mater. 2019;6:75. doi:10.3389/fmats.2019.00075
Liu Z, Wu C, Koishi M. A deep material network for multiscale topology learning and accelerated nonlinear modeling of heterogeneous materials. Comput Methods Appl Mech Eng. 2019;345:1138-1168. doi:10.1016/j.cma.2018.09.020
Huang T, Liu Z, Wu C, Chen W. Microstructure-guided deep material network for rapid nonlinear material modeling and uncertainty quantification. Comput Methods Appl Mech Eng. 2022;398:115197. doi:10.1016/j.cma.2022.115197
Wu L, Adam L, Noels L. A micromechanics-based inverse study for stochastic order reduction of elastic UD fiber reinforced composites analyses. Int J Numer Methods Eng. 2018;115(12):1430-1456. doi:10.1002/nme.5903
Eshelby JD, Peierls RE. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc R Soc London Series A Math Phys Sci. 1957;241(1226):376-396. doi:10.1098/rspa.1957.0133
Mori T, Tanaka K. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall. 1973;21(5):571-574. doi:10.1016/0001-6160(73)90064-3
Benveniste Y. A new approach to the application of Mori-Tanaka's theory in composite materials. Mech Mater. 1987;6(2):147-157. doi:10.1016/0167-6636(87)90005-6
Kröner E. Berechnung der elastischen Konstanten des Vielkristalls aus den Konstanten des Einkristalls. Zeitschrift für Physik. 1958;151:504-518.
Hill R. A self-consistent mechanics of composite materials. J Mech Phys Solids. 1965;13(4):213-222. doi:10.1016/0022-5096(65)90010-4
Hill R. Continuum micro-mechanics of elastoplastic polycrystals. J Mech Phys Solids. 1965;13(2):89-101. doi:10.1016/0022-5096(65)90023-2
Talbot D, Willis JR. Variational principles for inhomogeneous non-linear media. IMA J Appl Math. 1985;35(1):39-54. doi:10.1093/imamat/35.1.39
Wu L, Nguyen VD, Adam L, Noels L. An inverse micro-mechanical analysis toward the stochastic homogenization of nonlinear random composites. Comput Methods Appl Mech Eng. 2019;348:97-138. doi:10.1016/j.cma.2019.01.016
Calleja Vázquez JM, Wu L, Nguyen VD, Noels L. An incremental-secant mean-field homogenization model enhanced with a non-associated pressure-dependent plasticity model. Int J Numer Methods Eng. 2022;123(19):4616-4654. doi:10.1002/nme.7048
Verhoosel CV, Remmers JJC, Gutiérrez MA, Borst R. Computational homogenization for adhesive and cohesive failure in quasi-brittle solids. Int J Numer Methods Eng. 2010;83(8-9):1155-1179. doi:10.1002/nme.2854
Nguyen VD, Wu L, Noels L. A micro-mechanical model of reinforced polymer failure with length scale effects and predictive capabilities. Validation on carbon fiber reinforced high-crosslinked RTM6 epoxy resin. Mech Mater. 2019;133:193-213. doi:10.1016/j.mechmat.2019.02.017
Mosby M, Matous K. On mechanics and material length scales of failure in heterogeneous interfaces using a finite strain high performance solver. Model Simul Mater Sci Eng. 2015;23(8):085014. doi:10.1088/0965-0393/23/8/085014
Phu Nguyen V, Lloberas-Valls O, Stroeven M, Johannes SL. On the existence of representative volumes for softening quasi-brittle materials–a failure zone averaging scheme. Comput Methods Appl Mech Eng. 2010;199(45):3028-3038. doi:10.1016/j.cma.2010.06.018
Wu L, Zhang T, Maillard E, Adam L, Martiny P, Noels L. Per-phase spatial correlated damage models of UD fibre reinforced composites using mean-field homogenisation; applications to notched laminate failure and yarn failure of plain woven composites. Comput Struct. 2021;257:106650. doi:10.1016/j.compstruc.2021.106650
Peerlings RHJ, Borst DR, Brekelmans WAM, Vree JHP. Gradient enhanced damage for quasi-brittle materials. Int J Numer Methods Eng. 1996;39(19):3391-3403. doi:10.1002/(SICI)1097-0207(19961015)39:19<3391::AID-NME7>3.0.CO;2-D
Geers M, de Borst R, Brekelmans W, Peerlings R. Strain-based transient-gradient damage model for failure analyses. Comput Methods Appl Mech Eng. 1998;160(1):133-153. doi:10.1016/S0045-7825(98)80011-X
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(44):7723-7746. doi:10.1016/S0020-7683(01)00087-7
Wu L, Sket F, Molina-Aldareguia J, et al. A study of composite laminates failure using an anisotropic gradient-enhanced damage mean-field homogenization model. Compos Struct. 2015;126:246-264. doi:10.1016/j.compstruct.2015.02.070
Nguyen VD, Béchet E, Geuzaine C, Noels L. Imposing periodic boundary condition on arbitrary meshes by polynomial interpolation. Comput Mater Sci. 2012;55:390-406. doi:10.1016/j.commatsci.2011.10.017
Firooz S, Saeb S, Chatzigeorgiou G, Meraghni F, Steinmann P, Javili A. Systematic study of homogenization and the utility of circular simplified representative volume element. Math Mech Solids. 2019;24(9):2961-2985. doi:10.1177/1081286518823834
Khoei AR, Saadat MA. A nonlocal computational homogenization of softening quasi-brittle materials. Int J Numer Methods Eng. 2019;119(8):712-736. doi:10.1002/nme.6070
Coenen E, Kouznetsova V, Geers M. Novel boundary conditions for strain localization analyses in microstructural volume elements. Int J Numer Methods Eng. 2012;90(1):1-21. doi:10.1002/nme.3298
Nguyen VD, Wu L, Noels L. Unified treatment of microscopic boundary conditions and efficient algorithms for estimating tangent operators of the homogenized behavior in the computational homogenization method. Comput Mech. 2017;59:483-505. doi:10.1007/s00466-016-1358-z
Bonet J, Burton A. A simple orthotropic, transversely isotropic hyperelastic constitutive equation for large strain computations. Comput Methods Appl Mech Eng. 1998;162(1):151-164. doi:10.1016/S0045-7825(97)00339-3
Wu L, Tjahjanto D, Becker G, Makradi A, Jérusalem A, Noels L. A micro–meso-model of intra-laminar fracture in fiber-reinforced composites based on a discontinuous Galerkin/cohesive zone method. Eng Fract Mech. 2013;104:162-183. doi:10.1016/j.engfracmech.2013.03.018
Nguyen VD, Lani F, Pardoen T, Morelle X, Noels L. A large strain hyperelastic viscoelastic-viscoplastic-damage constitutive model based on a multi-mechanism non-local damage continuum for amorphous glassy polymers. Int J Solids Struct. 2016;96:192-216. doi:10.1016/j.ijsolstr.2016.06.008
Wu L, Noels L, Adam L, Doghri I. A combined incremental-secant mean-field homogenization scheme with per-phase residual strains for elasto-plastic composites. Int J Plastic. 2013;51:80-102. doi:10.1016/j.ijplas.2013.06.006
Berveiller M, Zaoui A. An extension of the self-consistent scheme to plastically-flowing polycrystals. J Mech Phys Solids. 1978;26(5):325-344. doi:10.1016/0022-5096(78)90003-0
Tandon GP, Weng GJ. A theory of particle-reinforced plasticity. J Appl Mech. 1988;55(1):126-135. doi:10.1115/1.3173618
Castaneda PP. The effective mechanical properties of nonlinear isotropic composites. J Mech Phys Solids. 1991;39(1):45-71. doi:10.1016/0022-5096(91)90030-R
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. doi: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. doi:10.1016/j.jmps.2011.09.004
Boudet J, Auslender F, Bornert M, Lapusta Y. An incremental variational formulation for the prediction of the effective work-hardening behavior and field statistics of elasto-(visco)plastic composites. Int J Solids Struct. 2016;83:90-113. doi:10.1016/j.ijsolstr.2016.01.003
Lucchetta A, Auslender F, Bornert M, Kondo D. A double incremental variational procedure for elastoplastic composites with combined isotropic and linear kinematic hardening. Int J Solids Struct. 2019;158:243-267. doi:10.1016/j.ijsolstr.2018.09.012
Molinari A, Canova G, Ahzi S. A self consistent approach of the large deformation polycrystal viscoplasticity. Acta Metall. 1987;35(12):2983-2994. doi:10.1016/0001-6160(87)90297-5
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(6):1203-1227. doi:10.1016/S0022-5096(99)00071-X
Pierard O, Doghri I. An enhanced affine formulation and the corresponding numerical algorithms for the mean-field homogenization of elasto-viscoplastic composites. Int J Plastic. 2006;22(1):131-157. doi:10.1016/j.ijplas.2005.04.001
Doghri I, Adam L, Bilger N. Mean-field homogenization of elasto-viscoplastic composites based on a general incrementally affine linearization method. Int J Plastic. 2010;26(2):219-238. doi:10.1016/j.ijplas.2009.06.003
Miled B, Doghri I, Brassart L, Delannay L. Micromechanical modeling of coupled viscoelastic-viscoplastic composites based on an incrementally affine formulation. Int J Solids Struct. 2013;50(10):1755-1769. doi:10.1016/j.ijsolstr.2013.02.004
Hutchinson JW, Hill R. Elastic-plastic behavior of polycrystalline metals and composites. Proc R Soc London A Math Phys Sci. 1970;319(1537):247-272. doi:10.1098/rspa.1970.0177
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. doi:10.1016/S0020-7683(03)00013-1
Mercier S, Kowalczyk-Gajewska K, Czarnota C. Effective behavior of composites with combined kinematic and isotropic hardening based on additive tangent Mori–Tanaka scheme. Compos Part B Eng. 2019;174:107052. doi:10.1016/j.compositesb.2019.107052
Chaboche J, Kanouté P, Roos A. On the capabilities of mean-field approaches for the description of plasticity in metal matrix composites. Int J Plastic. 2005;21(7):1409-1434. doi:10.1016/j.ijplas.2004.07.001
Wu L, Noels L, Adam L, Doghri I. An implicit-gradient-enhanced incremental-secant mean-field homogenization scheme for elasto-plastic composites with damage. Int J Solids Struct. 2013;50(24):3843-3860. doi:10.1016/j.ijsolstr.2013.07.022
Wu L, Adam L, Doghri I, Noels L. An incremental-secant mean-field homogenization method with second statistical moments for elasto-visco-plastic composite materials. Mech Mater. 2017;114:180-200. doi:10.1016/j.mechmat.2017.08.006
Haddad M, Doghri I, Pierard O. Viscoelastic-viscoplastic polymer composites: development and evaluation of two very dissimilar mean-field homogenization models. Int J Solids Struct. 2022;236-237:111354. doi:10.1016/j.ijsolstr.2021.111354
Wu L, Zulueta K, Major Z, Arriaga A, Noels L. Bayesian inference of non-linear multiscale model parameters accelerated by a deep neural network. Comput Methods Appl Mech Eng. 2020;360:112693. doi:10.1016/j.cma.2019.112693
Soize C, Ghanem R. Data-driven probability concentration and sampling on manifold. J Comput Phys. 2016;321:242-258. doi:10.1016/j.jcp.2016.05.044
Székely GJ, Rizzo ML, Bakirov NK. Measuring and testing dependence by correlation of distances. Ann Stat. 2007;35(6):2769-2794. doi:10.1214/009053607000000505
Chevalier J, Camanho P, Pardoen T. Multi-scale characterization and modelling of the transverse compression response of unidirectional carbon fiber reinforced epoxy. Compos Struct. 2019;209:160-176.
Wu L, Adam L, Noels L. Micro-mechanics and data-driven based reduced order models for multi-scale analyses of woven composites. Compos Struct. 2021;270:114058. doi:10.1016/j.compstruct.2021.114058
Calleja Vázquez JM, Wu L, Nguyen VD, Noels L. Data of “A micromechanical mean-field homogenization surrogate for the stochastic multiscale analysis of composite materials failure. Int J Numer Methods Eng. 2023”. 10.5281/zenodo.7998798
Melro A, Camanho P, Andrade Pires F, Pinho S. Micromechanical analysis of polymer composites reinforced by unidirectional fibres: part I–constitutive modelling. Int J Solids Struct. 2013;50(11):1897-1905. doi:10.1016/j.ijsolstr.2013.02.009
Al-Rub RKA, Tehrani AH, Darabi MK. Application of a large deformation nonlinear-viscoelastic viscoplastic viscodamage constitutive model to polymers and their composites. Int J Damage Mech. 2015;24(2):198-244. doi:10.1177/1056789514527020
Vogler M, Rolfes R, Camanho P. Modeling the inelastic deformation and fracture of polymer composites–part I: plasticity model. Mech Mater. 2013;59:50-64. doi:10.1016/j.mechmat.2012.12.002
Simo JC, Hughes TJ. Computational Inelasticity. Springer Science and Business Media; 2006.
Lemaitre J, Chaboche JL. Mechanics of Solid Materials. Cambridge University Press; 1990.
Morelle C, Chevalier J, Bailly C, Pardoen T, Lani F. Mechanical characterization and modeling of the deformation and failure of the highly crosslinked RTM6 epoxy resin. Mech Time-Depend Mater. 2017;21:419-454. doi:10.1007/s11043-016-9336-6
