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Fiber orientation interpolation for the multiscale analysis of short fiber reinforced composite parts. (English) Zbl 1446.74024
Summary: For short fiber reinforced plastic parts the local fiber orientation has a strong influence on the mechanical properties. To enable multiscale computations using surrogate models we advocate a two-step identification strategy. Firstly, for a number of sample orientations an effective model is derived by numerical methods available in the literature. Secondly, to cover a general orientation state, these effective models are interpolated. In this article we develop a novel and effective strategy to carry out this interpolation. Firstly, taking into account symmetry arguments, we reduce the fiber orientation phase space to a triangle in \({\mathbb {R}}^2\) . For an associated triangulation of this triangle we furnish each node with an surrogate model. Then, we use linear interpolation on the fiber orientation triangle to equip each fiber orientation state with an effective stress. The proposed approach is quite general, and works for any physically nonlinear constitutive law on the micro-scale, as long as surrogate models for single fiber orientation states can be extracted. To demonstrate the capabilities of our scheme we study the viscoelastic creep behavior of short glass fiber reinforced PA66, and use Schapery’s collocation method together with FFT-based computational homogenization to derive single orientation state effective models. We discuss the efficient implementation of our method, and present results of a component scale computation on a benchmark component by using ABAQUS ®.

MSC:
74-10 Mathematical modeling or simulation for problems pertaining to mechanics of deformable solids
74D10 Nonlinear constitutive equations for materials with memory
74A40 Random materials and composite materials
74Q05 Homogenization in equilibrium problems of solid mechanics
74S25 Spectral and related methods applied to problems in solid mechanics
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[1] Abaqus (2014) Abaqus 6.14: analysis user’s manual. Dassault Systemes Simulia Corp, Providence
[2] Aboudi, J, The generalized method of cells and high-fidelity generalized method of cells micromechanical models—a review, Mech Adv Mater Struct, 11, 329-366, (2004)
[3] Advani, SG; Tucker, CL, The use of tensors to describe and predict fiber orientation in short fiber composites, J Rheol, 31, 751-784, (1987)
[4] Andrä, H; Kabel, M; Staub, S; Krizikalla, F; Schulz, V, Numerische homogenisierung für viskoelastische faserverbundwerkstoffe, NAFEMS Online-Mag, 21, 70-83, (2012)
[5] Bakhvalov, NS; Knyazev, AV, Efficient computation of averaged characteristics of composites of a periodic structure of essentially different materials, Sov Math Dokl, 42, 57-62, (1991) · Zbl 0746.73001
[6] Bakhvalov NS, Panasenko GP (1989) Homogenization: averaging processes in periodic media. Kluwer, Dordrecht · Zbl 0692.73012
[7] Bella, P; Otto, F, Corrector estimates for elliptic systems with random periodic coefficients, Multiscale Model Simul, 14, 1434-1462, (2014) · Zbl 1351.35273
[8] Bhat, HA; Subramaniam, S; Pillai, AK; L EK Elangovan, M, Analysis and design of mold for plastic side release buckle using moldflow software, Int J Res Eng Technol, 3, 366-372, (2014)
[9] Bhattacharjee, S; Matouš, K, A nonlinear manifold-based reduced order model for multiscale analysis of heterogeneous hyperelastic materials, J Comput Phys, 313, 635-653, (2016) · Zbl 1349.65611
[10] Bhattacharya, K; Suquet, P, A model problem concerning recoverable strains of shape-memory polycrystals, Proc R Soc Lond A Math Phys Eng Sci, 461, 2797-2816, (2005) · Zbl 1186.74028
[11] Bhattacharya, K; Suquet, PM, A model problem concerning recoverable strains of shape-memory polycrystals, Proc R Soc Lond A Math Phys Eng Sci, 461, 2797-2816, (2005) · Zbl 1186.74028
[12] Box G, Draper N (2007) Response surfaces, mixtures and ridge analyses, 2nd edn. Wiley, Hoboken · Zbl 1267.62006
[13] Camacho, C; Tucker, CL; Yalvac, S; McGee, R, Stiffness and thermal expansion predictions for hybrid short fiber composites, Polym Compos, 11, 229-239, (1990)
[14] Chung, DH; Kwon, TH, Improved model of orthotropic closure approximation for flow induced fiber orientation, Polym Compos, 22, 636-649, (2001)
[15] Doghri, I; Tinel, L, Micromechanical modeling and computation of elasto-plastic materials reinforced with distributed-orientation fibers, Int J Plast, 21, 1919-1940, (2005) · Zbl 1154.74310
[16] Dvorak, G, Transformation field analysis of inelastic composite materials, Proc R Soc Lond A, 4317, 311-327, (1992) · Zbl 0748.73007
[17] Engqvist, EW; Lao, B; Ren, X; Vanden-Eijnden E, W, Heterogeneous multiscale methods: a review, Commun Comput Phys, 2, 367-450, (2007) · Zbl 1164.65496
[18] FeelMath (2017) FeelMath. Fraunhofer Institute for Industrial Mathematics. http://www.itwm.fraunhofer.de/en/fraunhofer-itwm.html
[19] Findley W, Lai J, Onaran K (1989) Creep and relaxation of nonlinear viscoelastic materials: with an introduction to linear viscoelasticity. Dover, Mineola · Zbl 0345.73034
[20] Fritzen, F; Leuschner, M, Reduced basis hybrid computational homogenization based on a mixed incremental formulation, Comput Methods Appl Mech Eng, 260, 143-154, (2013) · Zbl 1286.74081
[21] Furukawa, T; Yagawa, G, Implicit constitutive modelling for viscoplasticity using neural networks, Int J Numer Methods Eng, 43, 195-219, (1998) · Zbl 0926.74020
[22] GeoDict (2017) GeoDict. Math2Market GmbH, Kaiserslautern, Germany. http://www.geodict.de. Accessed on 4 Jan 2017
[23] Germain, P; Nguyen, QS; Suquet, P, Continuum thermodynamics, J Appl Mech, 50, 1010-1020, (1983) · Zbl 0536.73004
[24] Haj-Ali, R; Pecknold, DA; Ghaboussi, J; Voyiadjis, GZ, Simulated micromechanical models using artificial neural networks, J Eng Mech, 127, 730-738, (2001)
[25] Halphen, B; Nguyen, QS, Sur LES matériaux standard généralisés, J Méc, 14, 39-63, (1975) · Zbl 0308.73017
[26] Hashin, Z, Viscoelastic behavior of heterogeneous media, J Appl Mech Trans ASME, 32, 630-636, (1965)
[27] Hashin, Z, Complex modulis of viscoelastic composites—I. general theory and application to particulate composites, Int J Solids Struct, 6, 539-552, (1970) · Zbl 0219.73041
[28] Hill, R, Elastic properties of reinforced solids: some theoretical principles, J Mech Phys Solids, 11, 357-372, (1963) · Zbl 0114.15804
[29] Jack, DA; Smith, DE, An invariant based fitted closure of the sixth-order orientation tensor for modeling short-fiber suspensions, J Rheol, 49, 1091-1115, (2005)
[30] Janovsky, V; Shaw, MKWS; Whiteman, JR, Numerical methods for treating problems of viscoelastic isotropic solid deformation, J Comput Appl Math, 63, 91-107, (1995) · Zbl 0853.73026
[31] Kennedy PK (2013) Flow analysis of injection molds, 2nd edn. Hanser, Munich
[32] Lai, J; Bakker, A, 3-D schapery representation for non-linear viscoelasticity and finite element implementation, Comput Mech, 18, 182-191, (1996) · Zbl 0865.73014
[33] Le, BA; Yvonnet, J; He, QC, Computational homogenization of nonlinear elastic materials using neural networks, Int J Numer Methods Eng, 104, 1061-1084, (2015) · Zbl 1352.74266
[34] Lefik, M; Schrefler, B, Artificial neural network as an incremental non-linear constitutive model for a finite element code, Comput Methods Appl Mech Eng, 192, 3265-3283, (2003) · Zbl 1054.74731
[35] Matouš, K; Geers, MGD; Kouznetsova, VG; Gillman, A, A review of predictive nonlinear theories for multiscale modeling of heterogeneous materials, J Comput Phys, 330, 192-220, (2017)
[36] Michel, J; Suquet, P, Nonuniform transformation field analysis, Int J Solids Struct, 40, 6937-6955, (2003) · Zbl 1057.74031
[37] Montgomery-Smith, S; He, W; Jack, DA; Smith, DE, Exact tensor closures for the three-dimensional jeffery’s equation, J Fluid Mech, 680, 321-335, (2011) · Zbl 1241.76402
[38] Mosby, M; Matouš, K, Computational homogenization at extreme scales, Extreme Mech Lett, 6, 68-74, (2016)
[39] Moulinec, H; Suquet, P, A fast numerical method for computing the linear and nonlinear mechanical properties of composites, Comptes rendus de l’Académie des sciences Série II, Mécanique, physique, chimie, astronomie, 318, 1417-1423, (1994) · Zbl 0799.73077
[40] Moulinec, H; Suquet, P, A numerical method for computing the overall response of nonlinear composites with complex microstructure, Comput Methods Appl Mech Eng, 157, 69-94, (1998) · Zbl 0954.74079
[41] Ospald, F, Numerical simulation of injection molding using openfoam, Proc Appl Math Mech (PAMM), 14, 673-674, (2014)
[42] Python (2017) Python language reference, version 2.7. Python Software Foundation, Wilmington
[43] Schapery RA (1962) Approximate methods of transform inversion in viscoelastic stress analysis. In: Proceedings of the 4th US national congress on applied mechanics, vol 2, pp 1075-1085
[44] Schneider, M, The sequential addition and migration method to generate representative volume elements for the homogenization of short fiber reinforced plastics, Comput Mech, 59, 247-263, (2017) · Zbl 06764463
[45] Schneider, M; Ospald, F; Kabel, M, Computational homogenization on a staggered grid, Int J Numer Methods Eng, 105, 697-720, (2016)
[46] Steiner K, für Techno-und Wirtschaftsmathematik FI (2012) MISES-FOK: Multiskalenintegrierende Struktureigenschaftssimulation der Faserorientierung für faserverstärkte Kunststoffe im Automobil- und Flugzeugbau: Abschlussbericht zum Forschungsvorhaben 03x0513F im BMBF-Rahmenprogramm WING : Bearbeitungszeitraum vom 01.03.2007 bis 31.12.2011
[47] Suquet, P; Sawchuk, A (ed.); Bianchi, G (ed.), Local and global aspects in the mathematical theory of plasticity, 279-310, (1985), London
[48] Suquet, P; Sanchez-Palencia, E (ed.); Zaoui, A (ed.), Elements of homogenization for inelastic solid mechanics, No. 272, 193-278, (1987), New York
[49] Takano, N; Zako, M; Ohnishi, Y, Macro-micro uncoupled homogenization procedure for microscopic nonlinear behavior analysis of composites, Mater Sci Res Int, 2, 81-86, (1996)
[50] Temizer, I; Wriggers, P, An adaptive method for homogenization in orthotropic nonlinear elasticity, Comput Methods Appl Mech Eng, 35-36, 3409-3423, (2007) · Zbl 1173.74378
[51] Terada, K; Kikuchi, N, Nonlinear homogenization method for practical applications, ASME Appl Mech Div Publ AMD, 212, 1-16, (1995)
[52] Unger, JF; Könke, C, Coupling of scales in multiscale simulation using neural network, Comput Struct, 86, 1994-2003, (2008)
[53] Ward IM (2013) Mechanical properties of solid polymers, 3rd edn. Wiley, Hoboken
[54] Weller, HG; Tabor, HJG; Fureby, C, A tensorial approach to computational continuum mechanics using object-oriented techniques, Comput Phys, 12, 620-631, (1998)
[55] Wirtz, D; Karajan, N; Haasdonk, B, Surrogate modeling of multiscale models using kernel methods, Int J Numer Methods Eng, 101, 1-28, (2015) · Zbl 1352.65144
[56] Woldekidan M (2011) Response modelling of bitumen, bituminous mastic and mortar. Ph.D. thesis, Delft University of Technology
[57] Yang, JL; Zhang, Z; Schlarb, AK; Friedrich, K, On the characterization of tensile creep resistance of polyamide 66 nanocomposites. part I. experimental results and general discussions, Polymer, 47, 2791-2801, (2006)
[58] Yang, JL; Zhang, Z; Schlarb, AK; Friedrich, K, On the characterization of tensile creep resistance of polyamide 66 nanocomposites. part II. modeling and prediction of long-term performance, Polymer, 47, 6745-6758, (2006)
[59] Yvonnet, J; He, QC, The reduced model multiscale method (R3M) for the non-linear homogenization of hyperelastic media at finite strains, J Comput Phys, 223, 341-368, (2007) · Zbl 1163.74048
[60] Yvonnet, J; Monteiro, E; He, Q, Computational homogenization method and reduced database model for hyperelastic heterogeneous structures, Int J Multiscale Comput Eng, 11, 201-225, (2013)
[61] Zaoui, A, Continuum micromechanics: survey, J Eng Mech, 128, 808-816, (2002)
[62] Zeman, J; Vondřejc, J; Novak, J; Marek, I, Accelerating a FFT-based solver for numerical homogenization of periodic media by conjugate gradients, J Comput Phys, 229, 8065-8071, (2010) · Zbl 1197.65191
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