NOTICE: this is the author’s version of a work that was accepted for publication in Composite Structures. 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 Composite Structures 189C, 2018, 206-227, doi:10.1016/j.compstruct.2018.01.051
All documents in ORBi are protected by a user license.
[en] In this work, the elastic response of unidirectional fiber (UD) reinforced composites is studied in a stochastic multiscale way. First, the micro-structure of UD carbon fiber reinforced composites is statistically studied based on SEM images of its cross-section and an algorithm to generate numerical micro-structures with an equivalent random distribution of fibers is developed. In particular, based on the images spatial analysis, the empirical statistical descriptors are considered as dependent variables and represented using the copula framework, allowing generating micro-structure realizations used as Stochastic Volume Elements (SVEs). Second, a stochastic scale transition is conducted through the homogenization of SVEs. With a view to the use of the resulting meso-scale random field in structural stochastic analyzes, the homogenization is performed in two steps in order to respect the statistical content from the micro-meter-long
SVEs to the millimeter-long structural finite elements. To this end, the computational homogenization is applied in a hierarchy model: i) Micro-structure generator produces Small SVEs (SSVEs) which are homogenized; ii) Big SVEs (BSVEs) are constructed from the SSVEs. Finally, it is shown on simple illustrative examples that the scatter of the (homogenized) stress distribution in a composite ply can be simulated by means of the developed methodology.
Research center :
A&M - Aérospatiale et Mécanique - ULiège
Disciplines :
Materials science & engineering
Author, co-author :
Wu, Ling ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > Computational & Multiscale Mechanics of Materials (CM3)
Chung, Chi Nghia; Johannes Kepler University Linz - JKU > Institute of Polymer Product Engineering
Major, Zoltan; Johannes Kepler University Linz - JKU > Institute of Polymer Product Engineering
Adam, Laurent; e-Xstream Engineering
Noels, Ludovic ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > Computational & Multiscale Mechanics of Materials (CM3)
Language :
English
Title :
From SEM images to elastic responses: a stochastic multiscale analysis of UD fiber reinforced composites
FP7 - 291826 - M-ERA.NET - From materials science and engineering to innovation for Europe.
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 [BE]
Ghanem, R., Spanos, P., Stochastic finite elements: a spectral approach. 1991, Springer Verlag.
Lemaitre, O., Knio, O., Spectral methods for uncertainty quantification – with applications to computational fluid dynamics. 2010, Springer.
Stefanou, G., The stochastic finite element method: past, present and future. Comput Methods Appl Mech Eng 0045-7825, 198(9-12), 2009, 9–1051.
Alzebdeh, K., Ostoja-Starzewski, M., Micromechanically based stochastic finite elements: length scales and anisotropy. Probab Eng Mech 0266-8920, 11(4), 1996, 205–214, 10.1016/0266-8920(96)00015-X Third International Stochastic Structural Dynamics Conference.
Ostoja-Starzewski, M., Wang, X., Stochastic finite elements as a bridge between random material microstructure and global response. Comput Methods Appl Mech Eng 0045-7825, 168(1-4), 1999, 35–49, 10.1016/S0045-7825(98)00105-4.
Mehrez, L., Moens, D., Vandepitte, D., Stochastic identification of composite material properties from limited experimental databases, part I: experimental database construction. Mech Syst Signal Process 27 (2012), 471–483, 10.1016/j.ymssp.2011.09.004 http://www.sciencedirect.com/science/article/pii/S0888327011003724.
Mehrez, L., Doostan, A., Moens, D., Vandepitte, D., Stochastic identification of composite material properties from limited experimental databases, Part II: uncertainty modelling. Mech Syst Signal Process 27 (2012), 484–4498, 10.1016/j.ymssp.2011.09.001 http://www.sciencedirect.com/science/article/pii/S0888327011003694.
Vanaerschot, A., Cox, B.N., Lomov, S.V., Vandepitte, D., Stochastic framework for quantifying the geometrical variability of laminated textile composites using micro-computed tomography. Compos Part A: Appl Sci Manuf 1359-835X, 44, 2013, 122–131, 10.1016/j.compositesa.2012.08.020 http://www.sciencedirect.com/science/article/pii/S1359835X12002606.
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 0022-5096, 60(3), 2012, 451–470 http://www.sciencedirect.com/science/article/pii/S0022509611002225.
Tal, D., Fish, J., Generating a statistically equivalent representative volume element with discrete defects. Compos Struct 153 (2016), 791–803, 10.1016/j.compstruct.2016.06.077 http://www.sciencedirect.com/science/article/pii/S0263822316310790.
Vaughan, T., McCarthy, C., A combined experimental-numerical approach for generating statistically equivalent fibre distributions for high strength laminated composite materials. Compos Sci Technol 0266-3538, 70(2), 2010, 291–297 http://www.sciencedirect.com/science/article/pii/S0266353809003832.
Hojo, M., Mizuno, M., Hobbiebrunken, T., Adachi, T., Tanaka, M., Ha, S.K., Effect of fiber array irregularities on microscopic interfacial normal stress states of transversely loaded UD-CFRP from viewpoint of failure initiation. Compos Sci Technol 0266-3538, 69(11), 2009, 1726–1734, 10.1016/j.compscitech.2008.08.032 http://www.sciencedirect.com/science/article/pii/S0266353808003060, Experimental Techniques and Design in Composite Materials (ETDCM8) with Regular Papers.
Vanaerschot, A., Cox, B.N., Lomov, S.V., Vandepitte, D., Stochastic multi-scale modelling of textile composites based on internal geometry variability. Comput Struct 0045-7949, 122, 2013, 55–64, 10.1016/j.compstruc.2012.10.026 http://www.sciencedirect.com/science/article/pii/S0045794912002647, computational Fluid and Solid Mechanics 2013.
Melro, A., Camanho, P., Pinho, S., Generation of random distribution of fibres in long-fibre reinforced composites. Compos Sci Technol 0266-3538, 68(9), 2008, 2092–2102, 10.1016/j.compscitech.2008.03.013 http://www.sciencedirect.com/science/article/pii/S0266353808001048.
Gupta, A., Cecen, A., Goyal, S., Singh, A.K., Kalidindi, S.R., Structure-property linkages using a data science approach: application to a non-metallic inclusion, steel composite system. Acta Mater 91 (2015), 239–254, 10.1016/j.actamat.2015.02.045 http://www.sciencedirect.com/science/article/pii/S1359645415001603.
Charmpis, D., Schuler, G., Pellissetti, M., The need for linking micromechanics of materials with stochastic finite elements: a challenge for materials science. Comput Mater Sci 0927-0256, 41(1), 2007, 27–37, 10.1016/j.commatsci.2007.02.014 http://www.sciencedirect.com/science/article/pii/S0927025607000602.
Kanouté, P., Boso, D.P., Chaboche, J.L., Schrefler, B.A., Multiscale methods for composites: a review. Arch Comput Methods Eng 1886-1784, 16(1), 2009, 31–75, 10.1007/s11831-008-9028-8.
Noels, L., Wu, L., Adam, L., Seyfarth, J., Soni, G., Segurado, J., Laschet, G., Chen, G., Lesueur, M., Lobos, M., Bhlke, T., Reiter, T., Oberpeilsteiner, S., Salaberger, D., Weichert, D., Broeckmann, C., Effective Properties. Handbook of software solutions for ICME, 2016, Wiley-VCH Verlag GmbH & Co. KGaA 9783527693566, 433–485, 10.1002/9783527693566.ch6.
Feyel, F., Multiscale FE2 elastoviscoplastic analysis of composite structures. Comput Mater Sci 0927-0256, 16(1), 1999, 344–354, 10.1016/S0927-0256(99)00077-4.
Michel, J., Moulinec, H., Suquet, P., Effective properties of composite materials with periodic microstructure: a computational approach. Comput Methods Appl Mech Eng 0045-7825, 172(1), 1999, 109–143, 10.1016/S0045-7825(98)00227-8 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 0020-7683, 37(16), 2000, 2285–2311, 10.1016/S0020-7683(98)00341-2 http://www.sciencedirect.com/science/article/pii/S0020768398003412.
Kouznetsova, V., Brekelmans, W.A.M., Baaijens, F.P.T., An approach to micro-macro modeling of heterogeneous materials. Comput Mech 1432-0924, 27(1), 2001, 37–48, 10.1007/s004660000212.
Miehe, C., Strain-driven homogenization of inelastic microstructures and composites based on an incremental variational formulation. Int J Numer Methods Eng 1097-0207, 55(11), 2002, 1285–1322, 10.1002/nme.515.
Gusella, V., Cluni, F., Random field and homogenization for masonry with nonperiodic microstructure. J Mech Mater Struct 1559-3959, 1(2), 2006, 357–386, 10.2140/jomms.2006.1.357.
Mariani, S., Martini, R., Ghisi, A., Corigliano, A., Beghi, M., Overall elastic properties of polysilicon films: a statistical investigation of the effects of polycrystal morphology. Int J Multiscale Comput Eng 9:3 (2011), 327–346.
Savvas, D., Stefanou, G., Papadrakakis, M., Determination of RVE size for random composites with local volume fraction variation. Comput Methods Appl Mech Eng 305 (2016), 327–346, 10.1016/j.cma.2016.03.002 http://www.sciencedirect.com/science/article/pii/S0045782516300822.
Yin, X., Chen, W., To, A., McVeigh, C., Liu, W., Statistical volume element method for predicting microstructure-constitutive property relations. Comput Methods Appl Mech Eng 0045-7825, 197(43-44), 2008, 3516–3529, 10.1016/j.cma.2008.01.008.
Liebscher, A., Proppe, C., Redenbach, C., Schwarzer, D., Uncertainty quantification for metal foam structures by means of image analysis. Probab Eng Mech 28 (2012), 143–151, 10.1016/j.probengmech.2011.08.015 http://www.sciencedirect.com/science/article/pii/S0266892011000701, computational Stochastic Mechanics CSM6.
Stefanou, G., Savvas, D., Papadrakakis, M., Stochastic finite element analysis of composite structures based on material microstructure. Compos Struct 132 (2015), 384–392, 10.1016/j.compstruct.2015.05.044 http://www.sciencedirect.com/science/article/pii/S0263822315004183.
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 0045-7825, 2017, 10.1016/j.cma.2017.08.002 http://www.sciencedirect.com/science/article/pii/S0045782517305868.
Wu, L., Lucas, V., Nguyen, V.-D., Golinval, J.-C., Paquay, S., Noels, L., A stochastic multi-scale approach for the modeling of thermo-elastic damping in micro-resonators. Comput Methods Appl Mech Eng 310 (2016), 802–839, 10.1016/j.cma.2016.07.042 http://www.sciencedirect.com/science/article/pii/S0045782516303930.
Trovalusci, P., De Bellis, M.L., Ostoja-Starzewski, M., Murrali, A., Particulate random composites homogenized as micropolar materials. Meccanica 1572-9648, 49(11), 2014, 2719–2727, 10.1007/s11012-014-0031-x.
Trovalusci, P., Ostoja-Starzewski, M., De Bellis, M.L., Murrali, A., Scale-dependent homogenization of random composites as micropolar continua. Eur J Mech – A/Solids 0997-7538, 49(0), 2015, 396–407, 10.1016/j.euromechsol.2014.08.010.
Lucas, V., Golinval, J.-C., Voicu, R.C., Danila, M., Gavrila, R., Mller, R., Dinescu, A., Noels, L., Wu, L., et al. Propagation of material and surface profile uncertainties on MEMS micro-resonators using a stochastic second-order computational multi-scale approach. Int J Numer Methods Eng 1097-0207, 111(1), 2017, 26–68, 10.1002/nme.5452, nme.5452.
Baxter, S.C., Graham, L.L., Characterization of random composites using moving-window technique. J Eng Mech 126:4 (2000), 389–397, 10.1061/(ASCE)0733-9399(2000) 126:4(389), http://ascelibrary.org/doi/abs/10.1061/%28ASCE%290733-9399%282000%29126%3A4%28389%29.
Yin, X., Lee, S., Chen, W., Liu, W.K., Horstemeyer, M.F., Efficient random field uncertainty propagation in design using multiscale analysis. J Mech Des 1050-0472, 131(2), 2009, 021006, 10.1115/1.3042159.
Guilleminot, J., Noshadravan, A., Soize, C., Ghanem, R., A probabilistic model for bounded elasticity tensor random fields with application to polycrystalline microstructures. Comput Methods Appl Mech Eng 0045-7825, 200(17-20), 2011, 1637–1648, 10.1016/j.cma.2011.01.016.
Noshadravan, A., Ghanem, R., Guilleminot, J., Atodaria, I., Peralta, P., Validation of a probabilistic model for mesoscale elasticity tensor or random polycrystals. Int J Uncertainty Quantification 2152-5080, 3(1), 2013, 73–100.
Shinozuka, M., Simulation of multivariate and multidimensional random processes. J Acoust Soc Am 49:1B (1971), 357–368, 10.1121/1.1912338.
Popescu, R., Deodatis, G., Prevost, J., Simulation of homogeneous non-Gaussian stochastic vector fields. Probab Eng Mech 0266-8920, 13(1), 1998, 1–13, 10.1016/S0266-8920(97)00001-5.
Yamazaki, F., Shinozuka, M., Digital generation of non-Gaussian stochastic fields. J Eng Mech 114:7 (1988), 1183–1197.
Deodatis, G., Micaletti, R.C., Simulation of highly skewed non-Gaussian stochastic processes. J Eng Mech 127:12 (2001), 1284–1295.
Maligno, A., Warrior, N., Long, A., Effects of inter-fibre spacing on damage evolution in unidirectional (UD) fibre-reinforced composites. Eur J Mech – A/Solids 0997-7538, 28(4), 2009, 768–776, 10.1016/j.euromechsol.2008.10.009 http://www.sciencedirect.com/science/article/pii/S0997753808001113.
Hobbiebrunken, T., Hojo, M., Jin, K., Ha, S., Influence of non-uniform fiber arrangement on microscopic stress and failure initiation in thermally and transversely loaded CF/epoxy laminated composites. Compos Sci Technol 0266-3538, 68(15), 2008, 3107–3113, 10.1016/j.compscitech.2008.07.006 http://www.sciencedirect.com/science/article/pii/S0266353808002637.
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 0020-7683, 50(24), 2013, 3843–3860, 10.1016/j.ijsolstr.2013.07.022.
Wu, L., Sket, F., Molina-Aldareguia, J., Makradi, A., Adam, L., Doghri, I., Noels, L., A study of composite laminates failure using an anisotropic gradient-enhanced damage mean-field homogenization model. Compos Struct 126 (2015), 246–264, 10.1016/j.compstruct.2015.02.070 http://www.sciencedirect.com/science/article/pii/S0263822315001580.
Bansal, M., Singh, I., Mishra, B., Sharma, K., Khan, I., A stochastic XFEM model for the tensile strength prediction of heterogeneous graphite based on microstructural observations. J Nucl Mater 487 C (2017), 143–157, 10.1016/j.jnucmat.2016.12.045 http://www.sciencedirect.com/science/article/pii/S0022311516306973.
Bansal, M., Singh, I., Mishra, B., Sharma, K., Khan, I., A two-scale stochastic framework for predicting failure strength probability of heterogeneous materials. Compos Struct 179(C) (2017), 294–325, 10.1016/j.compstruct.2017.07.044 http://www.sciencedirect.com/science/article/pii/S0263822317315088.
Olave, M., Vanaerschot, A., Lomov, S.V., Vandepitte, D., Internal geometry variability of two woven composites and related variability of the stiffness. Polym Compos 1548-0569, 33(8), 2012, 1335–1350, 10.1002/pc.22260.
Illian, J., Penttinen, A., Stoyan, H., Stoyan, D., Statistical analysis and modelling of spatial point patterns, vol. 70, 2008, John Wiley & Sons.
Szkely, G.J., Rizzo, M.L., Bakirov, N.K., Measuring and testing dependence by correlation of distances. Ann Statist 35:6 (2007), 2769–2794, 10.1214/009053607000000505.
Strelen, J.C., Nassaj, F., Analysis and generation of random vectors with copulas. Proceedings of the 39th conference on winter simulation: 40 years! the best is yet to come, 2007, WSC '07, IEEE Press, Piscataway, NJ, USA 1-4244-1306-0, 488–496 http://dl.acm.org/citation.cfm?id=1351542.1351639.
Reza Sanei, S.H., Barsotti, E.J., Leonhardt, D., Fertig, R.S.I., Characterization, synthetic generation, and statistical equivalence of composite microstructures. J Compos Mater 51:13 (2017), 1817–1829, 10.1177/0021998316662133.
Miehe, C., Koch, A., Computational micro-to-macro transitions of discretized microstructures undergoing small strains. Arch Appl Mech 0939-1533, 72(4-5), 2002, 300–317, 10.1007/s00419-002-0212-2.
Ainsworth, M., Essential boundary conditions and multi-point constraints in finite element analysis. Comput Methods Appl Mech Eng 0045-7825, 190(48), 2001, 6323–6339, 10.1016/S0045-7825(01)00236-5 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 1432-0924, 59(3), 2017, 483–505, 10.1007/s00466-016-1358-z.
Peric, D., de Souza Neto, E.A., Feijóo, R.A., Partovi, M., Molina, A.J.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 Eng 1097-0207, 87, 2010, 149–170, 10.1002/nme.3014.
Schröder, J., Labusch, M., Keip, M.-A., Algorithmic two-scale transition for magneto-electro-mechanically coupled problems: FE2-scheme: localization and homogenization. Comput Methods Appl Mech Eng 32 (2016), 253–280, 10.1016/j.cma.2015.10.005 http://www.sciencedirect.com/science/article/pii/S0045782515003242.
Nguyen, V.-D., Béchet, E., Geuzaine, C., Noels, L., Imposing periodic boundary condition on arbitrary meshes by polynomial interpolation. Comput Mater Sci 0927-0256, 55, 2012, 390–406, 10.1016/j.commatsci.2011.10.017 http://www.sciencedirect.com/science/article/pii/S0927025611005866.
Shinozuka, M., Deodatis, G., Response variability of stochastic finite element systems. J Eng Mech 114:3 (1988), 499–519, 10.1061/(ASCE)0733-9399(1988) 114:3(499).
Soize, C., Ghanem, R., Data-driven probability concentration and sampling on manifold. J Comput Phys 321 (2016), 242–258, 10.1016/j.jcp.2016.05.044 http://www.sciencedirect.com/science/article/pii/S0021999116301899.
Pivovarov, D., Steinmann, P., Modified SFEM for computational homogenization of heterogeneous materials with microstructural geometric uncertainties. Comput Mech 57:1 (2016), 123–147.
Fish, J., Wu, W., A nonintrusive stochastic multiscale solver. Int J Numer Methods Eng 1097-0207, 88(9), 2011, 862–879, 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 1097-0207, 91(8), 2012, 799–824, 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 1543-1649, 11(3), 2013, 201–225.
Bessa, M., Bostanabad, R., Liu, Z., Hu, A., Apley, D.W., Brinson, C., Chen, W., Liu, W., A framework for data-driven analysis of materials under uncertainty: countering the curse of dimensionality. Comput Methods Appl Mech Eng 320 (2017), 633–667, 10.1016/j.cma.2017.03.037 http://www.sciencedirect.com/science/article/pii/S0045782516314803.
