This is the submitted version of the paper "A damage to crack transition model accounting for stress triaxiality formulated in a hybrid non-local implicit discontinuous Galerkin - cohesive band model framework, International Journal for Numerical Methods in Engineering VOL 115, PAGE 1430-1456, 10.1002/nme.5903" which has been published in final form on URL https://doi.org/10.1002/nme.5903
All documents in ORBi are protected by a user license.
Multiscale; Stochastic problems; Order reduction; Mean-Field Homogenization; Composites
Abstract :
[en] This research develops a stochastic mean-field-homogenization (MFH) process that is used as Reduced Order Model (ROM) to carry out a statistical multiscale analysis on unidirectional (UD) fiber reinforced composites. First full-field simulations of UD Stochastic Volume Elements (SVEs), whose statistical description is obtained from SEM images, are conducted to define statistical meso-scale apparent properties.
A stochastic Mori-Tanaka MFH model is then developed through an inverse stochastic identification process performed on the apparent elastic properties obtained by full-field simulations. As a result, a random vector of the effective elastic properties of phases and micro-structure information of the Mori-Tanaka model is inferred. In order to conduct Stochastic Finite Element Method (SFEM) analyzes, a generator of this random vector is then constructed using the copula method, allowing predicting the statistical response of a composite ply under bending. The statistical dependence of the random vector entries is shown to be respected by the generator. Although this work is limited to the elastic response, we believe that the stochastic Mori-Tanaka model can be extended to nonlinear behaviors in order to conduct efficient stochastic
multiscale simulations.
H2020 - 685451 - M-ERA.NET 2 - ERA-NET for materials research and innovation
Name of the research project :
The research has been funded by the Walloon Region under the agreement no 1410246 - STOMMMAC (CT-INT2013-03-28) in the context of the M-ERA.NET Joint Call 2014.
Funders :
Service public de Wallonie : Direction générale opérationnelle de l'économie, de l'emploi et de la recherche - DG06 CE - Commission Européenne
Kanouté P, Boso DP, Chaboche JL, Schrefler BA. Multiscale methods for composites: a review. Arch Comput Methods Eng. 2009;16(1):31-75. https://doi.org/10.1007/s11831-008-9028-8
Noels L, Wu L, Adam L, et al. Effective Properties. Handbook of Software Solutions for ICME. Weinheim, Germany: Wiley-VCH Verlag GmbH & Co KGaA; 2016:433-485. http://doi.org/10.1002/9783527693566.ch6
Matouš K, Geers MGD, Kouznetsova VG, Gillman A. A review of predictive nonlinear theories for multiscale modeling of heterogeneous materials. J Comput Phys. 2017;330:192-220. https://doi.org/10.1016/j.jcp.2016.10.070
Feyel F. Multiscale FE2 elastoviscoplastic analysis of composite structures. Comput Mater Sci. 1999;16(1-4):344-354. http://www.sciencedirect.com/science/article/pii/S0927025699000774
Michel JC, Moulinec H, Suquet P. Effective properties of composite materials with periodic microstructure: a computational approach. Comput Methods Appl Mech Eng. 1999;172(1-4):109-143. http://www.sciencedirect.com/science/article/pii/S0045782598002278
Terada K, Hori M, Kyoya T, Kikuchi N. Simulation of the multi-scale convergence in computational homogenization approaches. Int J Solids Struct. 2000;37(16):2285-2311. http://www.sciencedirect.com/science/article/pii/S0020768398003412
Kouznetsova V, Brekelmans WAM, Baaijens FPT. An approach to micro-macro modeling of heterogeneous materials. Comput Mech. 2001;27(1):37-48. https://doi.org/10.1007/s004660000212
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. http://doi.org/10.1002/nme.515
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. http://www.sciencedirect.com/science/article/pii/S0021999106004402
Hernández JA, Oliver J, Huespe AE, Caicedo MA, Cante JC. High-performance model reduction techniques in computational multiscale homogenization. Comput Methods Appl Mech Eng. 2014;276:149-189. http://www.sciencedirect.com/science/article/pii/S0045782514000978
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(4):613-625. https://doi.org/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. In: Galvanetto U, Aliabadi MHF, eds. Multiscale Modelling in Solid Mechanics: Computational Approaches. London, UK: Imperial College Press; 2009:159-206. https://hal.archives-ouvertes.fr/hal-00367772
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. http://www.sciencedirect.com/science/article/pii/S0022509616300928
Wirtz D, Karajan N, Haasdonk B. Surrogate modeling of multiscale models using kernel methods. Int J Numer Methods Eng. 2015;101(1):1-28. http://doi.org/10.1002/nme.4767
Perrin G, Soize C, Duhamel D, Funfschilling C. Identification of polynomial chaos representations in high dimension from a set of realizations. SIAM J Sci Comput. 2012;34(6):A2917-A2945. https://doi.org/10.1137/11084950X
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(1-4):35-49. http://doi.org/10.1016/S0045-7825(98)00105-4
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 Computat 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. http://www.sciencedirect.com/science/article/pii/S0020768312000340
Trovalusci P, De Bellis ML, Ostoja-Starzewski M, Murrali A. Particulate random composites homogenized as micropolar materials. Meccanica. 2014;49(11):2719-2727. https://doi.org/10.1007/s11012-014-0031-x
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. http://doi.org/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. http://www.sciencedirect.com/science/article/pii/S1359836817325088
Stefanou G, Savvas D, Papadrakakis M. Stochastic finite element analysis of composite structures based on material microstructure. Compos Struct. 2015;132:384-392. http://www.sciencedirect.com/science/article/pii/S0263822315004183
Savvas D, Stefanou G, Papadrakakis M. Determination of RVE size for random composites with local volume fraction variation. Comput Methods Appl Mech Eng. 2016;305:340-358. http://www.sciencedirect.com/science/article/pii/S0045782516300822
Stefanou G, Savvas D, Papadrakakis M. Stochastic finite element analysis of composite structures based on mesoscale random fields of material properties. Comput Methods Appl Mech Eng. 2017;326. http://www.sciencedirect.com/science/article/pii/S0045782517305868
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;189C:206-227. https://doi.org/10.1016/j.compstruct.2018.01.051
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. http://www.sciencedirect.com/science/article/pii/S0045782515001929
Wu L, Lucas V, Nguyen VD, Golinval JC, Paquay S, Noels L. A stochastic multi-scale approach for the modeling of thermo-elastic damping in micro-resonators. Comput Methods Appl Mech Eng. 2016;310:802-839. http://www.sciencedirect.com/science/article/pii/S0045782516303930
Pivovarov D, Steinmann P. Modified SFEM for computational homogenization of heterogeneous materials with microstructural geometric uncertainties. Comput Mech. 2016;57(1):123-147
Fish J, Wu W. A nonintrusive stochastic multiscale solver. Int J Numer Methods Eng. 2011;88(9):862-879. https://doi.org/10.1002/nme.3201
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. https://doi.org/10.1002/nme.4293
Yvonnet J, Monteiro E, He Q-C. Computational homogenization method and reduced database model for hyperelastic heterogeneous structures. Int J Multiscale Comput Eng. 2013;11(3):201-225
Bessa MA, Bostanabad R, Liu Z, et al. A framework for data-driven analysis of materials under uncertainty: countering the curse of dimensionality. Comput Methods Appl Mech Eng. 2017;320:633-667. http://www.sciencedirect.com/science/article/pii/S0045782516314803
Mori T, Tanaka K. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall. 1973;21(5):571-574
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 Plast. 2013;51:80-102. http://www.sciencedirect.com/science/article/pii/S0749641913001174
Yang S, Yu S, Ryu J, et al. Nonlinear multiscale modeling approach to characterize elastoplastic behavior of CNT/polymer nanocomposites considering the interphase and interfacial imperfection. Int J Plast. 2013;41:124-146. http://www.sciencedirect.com/science/article/pii/S0749641912001404
Hoang TH, Guerich M, Yvonnet J. Determining the size of RVE for nonlinear random composites in an incremental computational homogenization framework. J Eng Mech. 2016;142(5):04016018. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001057
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. https://doi.org/10.1016/j.ijsolstr.2013.07.022
Wu L, Sket F, Molina-Aldareguia JM, et al. A study of composite laminates failure using an anisotropic gradient-enhanced damage mean-field homogenization model. Compos Struct. 2015;126:246-264. http://www.sciencedirect.com/science/article/pii/S0263822315001580
Alzebdeh K, Ostoja-Starzewski M. Micromechanically based stochastic finite elements: length scales and anisotropy. Probab Eng Mech. 1996;11(4):205-214. Part of special issue: Third International Stochastic Structural Dynamics Conference. http://doi.org/10.1016/0266-8920(96)00015-X
Ghanem R, Spanos P. Stochastic Finite Elements: A Spectral Approach. New York, NY: Springer-Verlag New York; 1991
Le Maître OP, Knio OM. Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics. Dordrecht, The Netherlands: Springer Science+Business Media; 2010
Strelen JC, Nassaj F. Analysis and generation of random vectors with copulas. In: Proceedings of the 39th Conference on Winter Simulation: 40 Years! The Best is Yet to Come; 2007; Piscataway, NJ. http://dl.acm.org/citation.cfm?id=1351542.1351639
Miehe C, Koch A. Computational micro-to-macro transitions of discretized microstructures undergoing small strains. Arch Appl Mech. 2002;72(4-5):300-317. https://doi.org/10.1007/s00419-002-0212-2
Ainsworth M. Essential boundary conditions and multi-point constraints in finite element analysis. Comput Methods Appl Mech Eng. 2001;190(48):6323-6339. http://www.sciencedirect.com/science/article/pii/S0045782501002365
Nguyen V-D, 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(3):483-505. https://doi.org/10.1007/s00466-016-1358-z
Perić D, de Souza Neto EA, Feijóo RA, Partovi M, Molina AJC. 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 Eng. 2010;87(1-5):149-170. http://doi.org/10.1002/nme.3014
Schröder J, Labusch M, Keip MA. Algorithmic two-scale transition for magneto-electro-mechanically coupled problems FE2-scheme: localization and homogenization. Comput Methods Appl Mech Eng. 2016;302:253-280. http://www.sciencedirect.com/science/article/pii/S0045782515003242
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. http://www.sciencedirect.com/science/article/pii/S0927025611005866
Hill R. A self-consistent mechanics of composite materials. J Mech Phys Solids. 1965;13(4):213-222. https://doi.org/10.1016/0022-5096(65)90010-4
Eshelby JD. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc Royal Soc Math Phys Eng Sci. 1957;241(1226):376-396
Segurado J, Llorca J. A numerical approximation to the elastic properties of sphere-reinforced composites. J Mech Phys Solids. 2002;50(10):2107-2121
Székely GJ, Rizzo ML, Bakirov NK. Measuring and testing dependence by correlation of distances. Ann Stat. 2007;35(6):2769-2794. http://doi.org/10.1214/009053607000000505
