Mixed boundary conditions for FFT-based homogenization at finite strains. (English) Zbl 1359.74356

Summary: In this article we introduce a Lippmann-Schwinger formulation for the unit cell problem of periodic homogenization of elasticity at finite strains incorporating arbitrary mixed boundary conditions. Such problems occur frequently, for instance when validating computational results with tensile tests, where the deformation gradient in loading direction is fixed, as is the stress in the corresponding orthogonal plane. Previous Lippmann-Schwinger formulations involving mixed boundary can only describe tensile tests where the vector of applied force is proportional to a coordinate direction. Utilizing suitable orthogonal projectors we develop a Lippmann-Schwinger framework for arbitrary mixed boundary conditions. The resulting fixed point and Newton-Krylov algorithms preserve the positive characteristics of existing FFT-algorithms. We demonstrate the power of the proposed methods with a series of numerical examples, including continuous fiber reinforced laminates and a complex nonwoven structure of a long fiber reinforced thermoplastic, resulting in a speed-up of some computations by a factor of 1000.


74Q05 Homogenization in equilibrium problems of solid mechanics
35Q74 PDEs in connection with mechanics of deformable solids
74E30 Composite and mixture properties
45G10 Other nonlinear integral equations
65T50 Numerical methods for discrete and fast Fourier transforms
74B20 Nonlinear elasticity


Full Text: DOI


[1] Advani, S; Tucker, C, The use of tensors to describe and predict fiber orientation in short fiber composites, J Rheol, 31, 751-784, (1987)
[2] Altendorf, H; Jeulin, D; Willot, F, Influence of the fiber geometry on the macroscopic elastic and thermal properties, Int J Solids Struct, 51, 3807-3822, (2014)
[3] Anderson, Y; Mikelsons, M; Tamuzh, V; Tarashch, I, Fatigue failure of laminated carbon-fiber-reinforced plastic, Mech Compos Mater, 27, 58-62, (1991)
[4] Andrä, H; Combaret, N; Dvorkin, J; Glatt, E; Han, J; Kabel, M; Keehm, Y; Krzikalla, F; Lee, M; Madonna, C; Marsh, M; Mukerji, T; Saenger, E; Sain, R; Saxena, N; Ricker, S; Wiegmann, A; Zhan, X, Digital rock physics benchmarks—part II: computing effective properties, Comput Geosci, 50, 33-43, (2013)
[5] Andrä H, Gurka M, Kabel M, Nissle S, Redenbach C, Schladitz K, Wirjadi O (2014) Geometric and mechanical modeling of fiber-reinforced composites. In: Bernard D, Buffière JY, Pollock T, Poulsen HF, Rollett A, Uchic M (eds.) Proceedings of the 2nd international congress on 3D materials science (3DMS), Wiley, pp 35-40. http://eu.wiley.com/WileyCDA/WileyTitle/productCd-111894545X.html
[6] ASTM International (2013) Standard Test Method for In-Plane Shear Response of Polymer Matrix Composite Materials by Tensile Test of a +/- 45 \(^{∘ }\) Laminate. ASTM International, West Conshohocken. www.astm.org/Standards/D3518.htm · Zbl 0124.37604
[7] Ball, J, Convexity conditions and existence theorems in nonlinear elasticity, Arch Ration Mech Anal, 63, 337-403, (1976) · Zbl 0368.73040
[8] Barequet, G; Har-Peled, S, Efficiently approximating the minimum-volume bounding box of a point set in three dimensions, J Algorithms, 38, 91-109, (2001) · Zbl 0969.68166
[9] Bonet J, Wood R (1997) Nonlinear continuum mechanics for finite element analysis. Cambridge University Press, Cambridge · Zbl 0891.73001
[10] Bonnet, G, Effective properties of elastic periodic composite media with fibers, J Mech Phys Solids, 55, 881-899, (2007) · Zbl 1170.74042
[11] Brun M, Lopez-Pamies O, Castañeda PP (2007) Homogenization estimates for fiber-reinforced elastomers with periodic microstructures. Int J Solids Struct 44(18-19):5953-5979. doi:10.1016/j.ijsolstr.2007.02.003 · Zbl 1038.74605
[12] Castañeda, PP, Exact second-order estimates for the effective mechanical properties of nonlinear composite materials, J Mech Phys Solids, 44, 827-862, (1996) · Zbl 1054.74708
[13] Chen, L; Chen, J; Lebensohn, R; Ji, Y; Heo, T; Bhattacharyya, S; Chang, K; Mathaudhu, S; Liu, Z; Chen, LQ, An integrated fast Fourier transform-based phase-field and crystal plasticity approach to model recrystallization of three dimensional polycrystals, Comput Methods Appl Mech Eng, 285, 829-848, (2015) · Zbl 1423.74712
[14] Eisenlohr, P; Diehl, M; Lebensohn, R; Roters, F, A spectral method solution to crystal elasto-viscoplasticity at finite strains, Int J Plast, 46, 37-53, (2013)
[15] Flaig C, Arbenz P (2011) A scalable memory efficient multigrid solver for micro-finite element analyses based on ct images. Parallel Comput 37(12):846-854 doi:10.1016/j.parco.2011.08.001. http://www.sciencedirect.com/science/article/pii/S0167819111001037. 6th International workshop on parallel matrix algorithms and applications (PMAA’10)
[16] Fliegener S Micromechanical finite element modeling of long fiber reinforced thermoplastics. Ph.D. thesis, Karlsruhe Institute of Technology (KIT) (to appear)
[17] Fliegener, S; Luke, M; Gumbsch, P, 3D microstructure modeling of long fiber reinforced thermoplastics, Compos Sci Technol, 104, 136-145, (2014)
[18] Gélébart, L; Mondon-Cancel, R, Non-linear extension of FFT-based methods accelerated by conjugate gradients to evaluate the mechanical behavior of composite materials, Comput Mater Sci, 77, 430-439, (2013)
[19] Geymonat, G; Müller, S; Triantafyllidis, N, Homogenization of nonlinearly elastic materials, microscopic bifurcation and macroscopic loss of rank-one convexity, Arch Ration Mech Anal, 122, 231-290, (1993) · Zbl 0801.73008
[20] Henning, F; Ernst, H; Brüssel, R; Geiger, O; Krause, W, LFTs for automotive applications, Reinf Plast, 49, 24-33, (2005)
[21] Herrmann, K; Müller, W; Neumann, S, Linear and elastic-plastic fracture mechanics revisited by use of Fourier transforms—theory and application, Comput Mater Sci, 16, 186-196, (1999)
[22] HEXCEL: 3501-6 Epoxy matrix—high strength, damage-resistant, structural epoxy matrix. http://www.hexcel.com/Resources/DataSheets/Prepreg-Data-Sheets/3501-6_eu.pdf · Zbl 1405.74012
[23] Hill, R, On constitutive macro-variables for heterogeneous solids at finite strain, Proc R Soc Lond A: Math Phys Eng Sci, 326, 131-147, (1972) · Zbl 0229.73004
[24] Hoffmann S (2012) Computational homogenization of short fiber reinforced thermoplastic materials. Ph.D. thesis, University Kaiserslautern, LTM
[25] Kabel, M; Böhlke, T; Schneider, M, Efficient fixed point and Newton-Krylov solvers for FFT-based homogenization of elasticity at large deformations, Comput Mech, 54, 1497-1514, (2014) · Zbl 1309.74013
[26] Kanit, T; Forest, S; Galliet, I; Mounoury, V; Jeulin, D, Determination of the size of the representative volume element for random composites: statistical and numerical approach, Int J Solids Struct, 40, 3647-3679, (2003) · Zbl 1038.74605
[27] Kanjarla A, Lebensohn R, Balogh L, Tomé C (2012) Study of internal lattice strain distributions in stainless steel using a full-field elasto-viscoplastic formulation based on fast fourier transforms. Acta Mater 60(6-7):3094-3106. doi:10.1016/j.actamat.2012.02.014 · Zbl 0799.73077
[28] Kaßbohm S, Müller W, Feßler R (2005) Fourier series for computing the response of periodic structures with arbitrary stiffness distribution. Computat Mater Sci 32(3-4):387-391. doi:10.1016/j.commatsci.2004.09.028. http://www.sciencedirect.com/science/article/pii/S0927025604002186. IWCMM
[29] Knowles, J, The finite anti-plane shear field near the tip of a crack for a class of incompressible elastic solids, Int J Fract, 13, 611-639, (1977)
[30] Kobayashi, S, Theory of connections, Ann Mat Pura Appl, 43, 119-194, (1957) · Zbl 0124.37604
[31] Lahellec N, Michel J, Moulinec H, Suquet P (2003) Analysis of inhomogeneous materials at large strains using fast fourier transforms. In: Miehe C (ed) IUTAM symposium on computational mechanics of solid materials at large strains, solid mechanics and its applications. Springer, Netherlands, pp 247-258. doi:10.1007/978-94-017-0297-3_22 · Zbl 1039.74040
[32] Lebensohn R, Idiart M, Castañeda PP (2012) Modeling microstructural effects in dilatational plasticity of polycrystalline materials. Proced IUTAM 3:314-330. doi:10.1016/j.piutam.2012.03.020. http://www.sciencedirect.com/science/article/pii/S2210983812000211. IUTAM symposium on linking scales in computations: from microstructure to macro-scale properties
[33] Lebensohn, R; Idiart, M; Castañeda, PP; Vincent, PG, Dilatational viscoplasticity of polycrystalline solids with intergranular cavities, Philos Mag, 91, 3038-3067, (2011)
[34] Lebensohn R, Kanjarla A, Eisenlohr P (2012) An elasto-viscoplastic formulation based on fast fourier transforms for the prediction of micromechanical fields in polycrystalline materials. Int J Plast 32-33:59-69. doi:10.1016/j.ijplas.2011.12.005
[35] Lebensohn, R; Rollett, A; Suquet, P, Fast Fourier transform-based modeling for the determination of micromechanical fields in polycrystals, JOM, 63, 13-18, (2011)
[36] Lee, SB; Lebensohn, R; Rollett, A, Modeling the viscoplastic micromechanical response of two-phase materials using fast Fourier transforms, Int J Plast, 27, 707-727, (2011) · Zbl 1405.74012
[37] Lefebvre G, Sinclair C, Lebensohn R, Mithieux JD (2012) Accounting for local interactions in the prediction of roping of ferritic stainless steel sheets. Model Simul Mater Sci Eng 20(2):024008. http://stacks.iop.org/0965-0393/20/i=2/a=024008
[38] Li, J; Meng, S; Tian, X; Song, F; Jiang, C, A non-local fracture model for composite laminates and numerical simulations by using the FFT method, Compos Part B: Eng, 43, 961-971, (2012)
[39] Li, J; Tian, XX; Abdelmoula, R, A damage model for crack prediction in brittle and quasi-brittle materials solved by the FFT method, Int J Fract, 173, 135-146, (2012)
[40] Liu B, Raabe D, Roters F, Eisenlohr P, Lebensohn R (2010) Comparison of finite element and fast fourier transform crystal plasticity solvers for texture prediction. Model Simul Mater Sci Eng 18(8):085005. http://stacks.iop.org/0965-0393/18/i=8/a=085005 · Zbl 1295.74006
[41] Monchiet, V; Bonnet, G, Numerical homogenization of nonlinear composites with a polarization-based FFT iterative scheme, Comput Mater Sci, 79, 276-283, (2013)
[42] Moore, E, On the reciprocal of the general algebraic matrix, Bull Am Math Soc, 26, 394-395, (1920)
[43] 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
[44] 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
[45] Müller, S, Homogenization of nonconvex integral functionals and cellular elastic materials, Arch Ration Mech Anal, 99, 189-212, (1987) · Zbl 0629.73009
[46] Müller V, Böhlke T, Kabel M, Andrä H (2015) Homogenization of linear elastic properties of discontinuous reinforced composites—a comparison of mean field and voxel-based methods. Int J Solids Struct. doi:10.1016/j.ijsolstr.2015.02.030. http://www.sciencedirect.com/science/article/pii/S0020768315000761
[47] Ortega, J, The Newton-Kantorovich theorem, Am Math Mon, 75, 658-660, (1968) · Zbl 0183.43004
[48] Ortega J, Rheinboldt W (1970) Iterative solution of nonlinear equations in several variables. Academic Press, New York · Zbl 0241.65046
[49] Penrose, R, A generalized inverse for matrices, Math Proc Camb Philos Soc, 51, 406-413, (1995) · Zbl 0065.24603
[50] Peterson, C; Ehnert, G; Liebold, R; Kühfusz, R; Miracle, D (ed.); Donaldson, S (ed.), Compression molding, No. 21, 516-535, (2001), Netherlands
[51] Prakash A, Lebensohn R (2009) Simulation of micromechanical behavior of polycrystals: finite elements versus fast fourier transforms. Model Simul Mater Sci Eng 17(6):064010. http://stacks.iop.org/0965-0393/17/i=6/a=064010 · Zbl 0183.43004
[52] R&G Faserverbundwerkstoffe GmbH (2009) Faserverbundwerkstoffe Handbuch—composite materials handbook, R&G Faserverbundwerkstoffe GmbH, Waldenbuch
[53] Rollett A, Lebensohn R, Groeber M, Choi Y, Li J, Rohrer G (2010) Stress hot spots in viscoplastic deformation of polycrystals. Model Simul Mater Sci Eng 18(7):074005. http://stacks.iop.org/0965-0393/18/i=7/a=074005 · Zbl 0629.73009
[54] Schneider M, Ospald F, Kabel M (2015) Computational homogenization of elasticity on a staggered grid. Int J Numer Methods Eng. doi:10.1002/nme.5008 · Zbl 0954.74079
[55] Shanthraj P, Eisenlohr P, Diehl M, Roters F (2015) Numerically robust spectral methods for crystal plasticity simulations of heterogeneous materials. Int J Plast 66:31-45. doi:10.1016/j.ijplas.2014.02.006. http://www.sciencedirect.com/science/article/pii/S0749641914000709. Plasticity of textured polycrystals in honor of Prof. Paul Van Houtte · Zbl 0799.73077
[56] Sliseris, J; Andrä, H; Kabel, M; Dix, B; Plinke, B; Wirjadi, O; Frolovs, G, Numerical prediction of the stiffness and strength of medium density fiberboards, Mech Mater, 79, 73-84, (2014)
[57] Spahn, J; Andrä, H; Kabel, M; Müller, R, A multiscale approach for modeling progressive damage of composite materials using fast Fourier transforms, Comput Methods Appl Mech Eng, 268, 871-883, (2014) · Zbl 1295.74006
[58] Suchocki, C, A finite element implementation of knowles stored-energy function: theory, coding and applications, Arch Mech Eng, 58, 319-346, (2011)
[59] Todd R, Allen D, Alting L (1994) Manufacturing processes reference guide. Industrial Press. https://books.google.de/books?id=6x1smAf_PAcC
[60] Vinogradov, V; Milton, G, An accelerated FFT algorithm for thermoelastic and non-linear composites, Int J Numer Methods Eng, 76, 1678-1695, (2008) · Zbl 1195.74302
[61] Willot, F; Gillibert, L; Jeulin, D, Microstructure-induced hotspots in the thermal and elastic responses of granular media, Int J Solids Struct, 50, 1699-1709, (2013)
[62] Zeman, J; Vondřejc, J; Novák, 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.