A class of general algorithms for multi-scale analyses of heterogeneous media. (English) Zbl 1001.74095

Summary: We develop a class of computational algorithms for multi-scale analysis. The two-scale modeling scheme for analysis of heterogeneous media with fine periodic microstructures is generalized by using relevant variational statements. Instead of the method of two-scale asymptotic expansion, we employ mathematical results on the generalized convergence in the two-scale variational description. Accordingly, the global-local type computational schemes can be unified with the homogenization procedure for general nonlinear problems. After formulating a problem in linear elastostatics, a problem with local contact condition, and an elastoplastic problem, we present numerical examples along with computational algorithms consistent with two-scale modeling strategy, as well as some direct approaches.


74Q05 Homogenization in equilibrium problems of solid mechanics
74E05 Inhomogeneity in solid mechanics
Full Text: DOI


[1] I. Babuska, Homogenization approach in engineering, in: J.-L. Lions, R. Glowinski (Eds.), Computing Methods in Applied Sciences and Engineering, Lecture Note in Economics and Mathematical Systems, vol. 134, Springer, Berlin, 1976, pp. 137-153 · Zbl 0376.73070
[2] Benssousan, A.; Lions, J.-L.; Papanicoulau, G., Asymtotic analysis for periodic structures, (1978), North-Holland Amsterdam
[3] E. Sanchez-Palencia, Non-homogeneous Media and Vibration Theory, Lecture Notes in Physics, vol. 127, Springer, Berlin, 1980 · Zbl 0432.70002
[4] J.-L. Lions, Some Methods in Mathematical Analysis of Systems and their Control, Kexue Chubanshe Science Press, Beijing, and Gordon and Breach, London, 1981 · Zbl 0542.93034
[5] Kalamkarov, A.L.; Kolpakov, A.G., Analysis, design and optimization of composite structures, (1997), Wiley Chichester, UK · Zbl 0936.74002
[6] J.-L. Lions, Asymptotic calculus of variations, in: Singular Perturbations and Asymptotics, Proc. Adv. Sem., Math. Res. Center, University of Wisconsin, 1980, Madison, Academic Press, New York, 1980, pp. 277-296
[7] G. Duvaut, M. Nuc, A new method of analysis of composite structure, in: Ninth European Rotor Craft Forum, Stresa, Italie, Paper No. 88, 1983
[8] Devries, F.; Léné, F., Homogenization at set macroscopic stress: numerical implementation and application, Rech. aerosp., 1, 34-51, (1987) · Zbl 0615.73011
[9] Paumelle, P.; Hassim, A.; Léné, F., Microstress analysis in woven composite structures, Rech. aerosp., 6, 47-62, (1991)
[10] Léné, F.; Leguillon, D., Homogenized constitutive law for a partially cohesive composite material, Int. J. solids struct., 18, 443-458, (1982) · Zbl 0488.73065
[11] Devries, F.; Dumontet, H.; Duvaut, G.; Léné, F., Homogenization and damage for composite structures, Int. J. numer. meth. eng., 27, 285-298, (1989) · Zbl 0709.73059
[12] Léné, F., Damage constitutive relations for composite materials, Eng. fract. mech., 25, 713-728, (1986)
[13] Guedes, J.M.; Kikuchi, N., Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods, Comput. meth. appl. mech. eng., 83, 143-198, (1990) · Zbl 0737.73008
[14] Hollister, S.J.; Kikuchi, N., Homogenization theory and digitial imaging: a basis for studying the mechanics and design principles of bone tissue, Biotechnol. bioeng., 43, 586-596, (1994)
[15] Ghosh, S.; Lee, K.; Moorthy, S., Multiple scale analysis of heterogeneous elastic structures using homogenization theory and Voronoi cell finite element method, Int. J. solids struct., 32, 27-62, (1995) · Zbl 0865.73060
[16] Fish, J.; Belsky, V., Multigrid method for periodic heterogeneous media. part 1. convergence studies for one-dimensional case, Comput. meth. appl. mech. eng., 126, 1-16, (1995) · Zbl 1067.74574
[17] Taliercio, A.; Sagramoso, P., Uniaxial strength of polymeric-matrix fibrous composites predicted through a homogenization approach, Int. J. solids struct., 32, 2095-2123, (1995) · Zbl 0919.73051
[18] Konke, C., Damage evolution in ductile materials: from micro- to macro-damage, Comput. mech., 15, 497-510, (1995) · Zbl 0825.73535
[19] Pegon, P.; Anthoine, A., Numerical strategies for solving continuum damage problems with softening: application to the homogenization of marsonry, Comput. struct., 64, 623-642, (1995) · Zbl 0919.73004
[20] Luciano, R.; Sacco, E., Homogenization technique and damage model for old masonry material, Int. J. solids struct., 34, 3191-3208, (1997) · Zbl 0942.74610
[21] Lee, K.; Moorthy, S.; Ghosh, S., Multiple scale computational model for damage in composite materials, Comput. meth. appl. mech. eng., 172, 175-201, (1999) · Zbl 0972.74063
[22] Jansson, S., Homogenized nonlinear constitutive properties and local stress concentrations for composites with periodic internal structure, Int. J. solids struct., 29, 2181-2200, (1992) · Zbl 0825.73426
[23] Swan, C.C.; Cakmak, A.S., A hardening orthotropic plasticity model for non-frictional composites: rate formulation and integration algorithm, Int. J. numer. meth. eng., 37, 839-860, (1994) · Zbl 0803.73033
[24] Ghosh, S.; Moorthy, S., Elastic – plastic analysis of arbitrary heterogeneous materials with the Voronoi cell finite element method, Comput. meth. appl. mech. eng., 121, 373-409, (1995) · Zbl 0853.73065
[25] K. Terada, N. Kikuchi, Nonlinear homogenization method for practical applications, in: S. Ghosh, M. Ostoja-Starzewski (Eds.), Computational Methods in Micromechanics, AMSE AMD 212, 1995, pp. 1-16
[26] Fish, J.; Shek, K.; Pandheeradi, M.; Shephard, M., Computational plasticity for composite structures based on mathematical homogenization: theory and practice, Comput. meth. appl. mech. eng., 148, 53-73, (1997) · Zbl 0924.73145
[27] Fish, J.; Shek, K., Finite deformation plasticity for composite structures: computational models and adaptive strategies, Comput. meth. appl. mech. eng., 172, 145-174, (1999) · Zbl 0956.74009
[28] Knockaert, R.; Doghri, I., Nonlocal constitutive models with gradients of internal variables derived from a micro/macro homogenization procedure, Comput. meth. appl. mech. eng., 174, 121-136, (1999) · Zbl 0964.74053
[29] Fish, J.; Yu, Q.; Shek, K., Computational damage mechanics for composite materials based on mathematical homogenization, Int J. numer. meth. eng., 45, 1657-1679, (1999) · Zbl 0949.74057
[30] van der Sluis, O.; Vosbeek, P.H.J.; Schreurs, P.J.G.; Meijer, H.E.H., Homogenization of heterogeneous polymers, Int. J. solids struct., 36, 3193-3214, (1999) · Zbl 0976.74055
[31] E. De Giorgi, Convergence problems for functionals and operators, in: E. De Giorgi, E. Magenes, U. Masco (Eds.), International Meeting on Recent Methods in Nonlinear Analysis, Rome, 1978, Pitagora Editrice, Bologna, 1979, pp. 131-188
[32] Zhikov, V.V.; Kozlov, S.M.; Oleinik, O.A., Homogenization and differential operators and integral functionals, (1995), Springer New York
[33] Dal Maso, G., Introduction to G-convergence, (1993), Birkhäuser Berlin · Zbl 0816.49001
[34] Müller, S., Homogenization of nonconvex integral functionals and cellular elastic materials, Arch. rat. mech. anal., 99, 189-212, (1987) · Zbl 0629.73009
[35] Tartar, L., Nonlinear constitutive relations and homogenization, (), 472-484
[36] Tartar, L., H-measures and small amplitude homogenization, (), 89-99 · Zbl 0790.73009
[37] F. Murat, L. Tartar, H-convergence, in: R.V. Kohn (Ed.), Topics in the Mathematical Modeling of Composite Materials, Progress in Non-linear Differential Equations and their Applications, Birkhäuser, Boston, 1995, p. 19
[38] S. Spangnolo, Convergence in energy for elliptic operators, in: B. Hubbard (Ed.), Numerical Solution of Partial Differential Equations-III, SYNSPADE 1975, Accademic Press, New York, 1975, pp. 69-498
[39] Pankov, A., G-convergence and homogenization of nonlinear partial differential operators, (1997), Kluwer Academic Publishers Dordrecht · Zbl 0883.35001
[40] Allaire, G., Mathematical approaches and methods, (), 225-250
[41] Allaire, G., Homogenization and two-scale convergence, SIAM J. math. anal., 23, 1482-1518, (1992) · Zbl 0770.35005
[42] P.M. Suquet, Local and global aspects in the mathematical theory of plasticity, in: A. Sawczuk et al. (Eds.), Plasticity Today, Elsevier, London, 1985, pp. 279-309
[43] P.M. Suquet, Elements of homogenization theory for inelastic solid mechanics, in: E. Sanchez-Palencia, A. Zaoui, (Eds.), Homogenization Techniques for Composite Media, Lecture Note on Physics, vol. 272, Springer, Berlin, 1987, pp. 193-278 · Zbl 0645.73012
[44] Suquet, P.M., Overall potentials and external surfaces of power law or ideally plastic composities, J. mech. phys. solids, 41, 981-1002, (1993) · Zbl 0773.73063
[45] Michel, J.C.; Moulinec, H.; Suquet, P.M., Effective properties of composite materials with periodic microstructure: a computational approach, Comput. meth. appl. mech. eng., 172, 109-143, (1999) · Zbl 0964.74054
[46] Costanzo, F.; Boyd, J.G.; Allen, D.H., Micromechanics and homogenization of inelastic composite materials with growing cracks, J. mech. phys. solids, 44, 333-370, (1996) · Zbl 1054.74703
[47] Chung, P.W.; Tamma, K.K., Woven fabric composites – developments in engineering bounds, homogenization and applications, Int J. numer. meth. eng., 45, 1757-1790, (1999) · Zbl 0979.74063
[48] Terada, K.; Hori, M.; Kyoya, T.; Kikuchi, N., Simulation of the multi-scale convergence in computational homogenization approaches, Int. J. solids struct., 37, 2285-2311, (2000) · Zbl 0991.74056
[49] Miehe, C.; Schroder, J.; Schotte, J., Computational homogenization analysis in finite plasticity simulation of texture development in polycrystalline materials, Comput. meth. appl. mech. eng., 171, 387-418, (1999) · Zbl 0982.74068
[50] Washizu, K., Variational methods in elasticity and plasticity, (1982), Pergamon Press Oxford · Zbl 0164.26001
[51] Chacha, D.; Sanchez-Palencia, E., Overall behavior of elastic plates with periodically distributed fissures, Asymptotic anal., 5, 381-396, (1992) · Zbl 0748.73002
[52] Duvaut, G.; Lions, J.-L., Inequalities in mechanics and physics, (1976), Springer New York · Zbl 0331.35002
[53] Kikuchi, N.; Oden, J.T., Contact problem in elasticity: A study of variational inequalities and finite element methods, (1988), SIAM Philadelphia, PA · Zbl 0685.73002
[54] Zavarise, G.; Wriggers, P.; Schrefler, B.A., A method for solving contact problems, Int. J. numer. meth. eng., 42, 473-498, (1998) · Zbl 0914.73048
[55] Johnson, C., A mixed finite element for plasticity, SIAM J. numer. anal., 575-583, (1977) · Zbl 0374.73039
[56] Simo, J.C.; Kennedy, J.G.; Taylor, R.L., Complementary mixed finite element formulations of elastoplasticity, Comput. meths. appl. mech. eng., 74, 177-206, (1988) · Zbl 0687.73064
[57] Simo, J.C.; Hughes, T.J.R., Computational inelasticity, (1998), Springer New York · Zbl 0934.74003
[58] Maugin, G.A., The thermomechanics of plasticity and fracture, (1992), Cambridge University Press Cambridge · Zbl 0753.73001
[59] K. Matsui, K. Terada, K. Yuge, Feasibility and efficiency of a global-local analysis method for nonlinear heterogeneous solids, submitted
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.