Hyperelastic homogenized law for reinforced elastomer at finite strain with edge effects. (English) Zbl 0916.73028

The author deals with the static microstructural effects of periodic hyperelastic composites at finite strain. Using the asymptotic process, he develops a homogenization procedure at finite strain and examines hyperelastic behavior of each constituent. The elastomer is assumed as nearly incompressible. An application of this approach is given using periodically stratified composites. A numerical simulation is then investigated on steel/elastomer composite, and the validity of the models is studied. When each constituent is neo-Hookean, the continuum homogenized model can be determined analytically. Since the classical asymptotic solution is not valid in the neighbourhood of the boundaries, boundary layers are added to the classical asymptotic terms. In this way, the effect of only one boundary and the simultaneous effects of two boundaries (angular boundary) are taken into account.


74E30 Composite and mixture properties
74B20 Nonlinear elasticity
74E05 Inhomogeneity in solid mechanics
Full Text: DOI


[1] Bensoussan, A., Lions, J.-L., Papanicolaou, G.: Asymptotic analysis for periodic structures. Amsterdam: North-Holland 1978. · Zbl 0404.35001
[2] Sanchez-Palencia, E.: Non-homogeneous media and vibration theory. Lecture Notes in Physics, Vol. 127. Berlin Heidelberg New York: Springer 1980. · Zbl 0432.70002
[3] Nguetseng, G.: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal.20, 608-623 (1989). · Zbl 0688.35007
[4] Allaire, G.: Homogenization and two-scale convergence. SIAM J. Math. Anal.23, 1482-1518 (1992). · Zbl 0770.35005
[5] Suquet, P.: Plasticité et homogénéisation. Thèse de Doctorat d’Etat, Université Paris VI 1982.
[6] Suquet, P.: Elements of homogenization for inelastic solid mechanics. In: Homogenization techniques for composite media, pp. 193-278. Lecture Notes in Physics Vol. 272. Berlin Heidelberg New York: Springer 1987. · Zbl 0645.73012
[7] Aboudi, J.: Mechanics of composite materials ? a unified micromechanical approach. Amsterdam: Elsevier 1991. · Zbl 0837.73003
[8] Lene, F.: Contribution à l’étude des matériaux composites et de leur endommagement. Thèse de Doctorat d’Etat, Université Paris VI 1984.
[9] Devries, F., Dumontet, H., Duvaut, G., Lene, F.: Homogenization and damage for composite structures. Int. J. Numer. Meth. Eng.27, 285-298 (1989). · Zbl 0709.73059
[10] Guedes, J. M.: Non linear computational model for composite material using homogenization. Ph. D. Thesis, University of Michigan, 1990.
[11] Guedes, J. M., Kikuchi, N.: Preprocessing and postprocessing for materials based on the homogenization method with adaptative finite element method. Comp. Meth. Appl. Mech. Eng.83, 143-198 (1991). · Zbl 0737.73008
[12] Pruchnicki, E.: Contribution à l’homogénéisation en phases linéaire et non linéaire application au renforcement des sols. Mémoire de thèse, Université de Lille, 1991.
[13] Jansson, S.: Homogenized non linear constitutive properties and local stress concentrations for composites with periodic internal structure. Int. J. Solids Struct.29, 2181-2200 (1992). · Zbl 0825.73426
[14] Ghosh, S., Lee, K., Moorthy, S.: Mutiple 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
[15] Nemat-Nasser, S., Taya, M.: On effective moduli of an elastic body containing y periodically distributed voids. Q. Appl. Math.39, 43-59 (1981). · Zbl 0532.73009
[16] Dumontet, H.: Homogénéisation par développements en séries de Fourier. C. R. Acad. Sci. Paris Ser.II296, 1625-1628 (1983). · Zbl 0522.73013
[17] Walker, K. P., Jordan, E. H., Freed, A. D.: Thermoplastic analysis of fibrous periodic composites using triangular subvolumes. NASA report TM 106076. NASA-Lewis Research Center, Cleveland, OH, 1993.
[18] Moulinec, H., Suquet, P.: A fast numerical method for computing the linear and nonlinear mechanical properties of composites. C. R. Acad. Sci. Paris Ser.II 318, 1417-1423 (1994). · Zbl 0799.73077
[19] Pruchnicki, E.: Homogenized nonlinear constitutive laws using Fourier series expansion. Int. J. Solids Struct.35, 1895-1913 (1998). · Zbl 0935.74058
[20] Hill, R.: The essential structure of constitutive laws for metal composites and polycrystals. J. Mech. Phys. Solids15, 79-95 (1967).
[21] Hill, R.: On constitutive macro-variables for heterogeneous solids at finite strain. Proc. R. Soc. London Ser.A 326, 131-147 (1972). · Zbl 0229.73004
[22] Hill, R.: On macroscopic effects of heterogeneity in elastoplastic media at finite strain. Math. Proc. Camb. Phil. Soc.95, 481-494 (1984). · Zbl 0553.73025
[23] Hill, R., Rice, J. R.: Elastic potentials and the structure of inelastic constitutive laws. SIAM J. Appl. Math.25, 448-461 (1973). · Zbl 0275.73028
[24] Ogden, R. W.: Local and overall extremal properties of time ? independent materials and non-linear elastic comparison materials. Int. J. Solids Struct.12, 147-158 (1976). · Zbl 0324.73005
[25] Ogden, R. W.: Extremum principles in non linear elasticity and their application to composites ? I. Int. J. Solids Struct.14, 265-282 (1978). · Zbl 0384.73022
[26] Ponte Castenada, P.: The overall constitutive behaviour of non linearly elastic composites. Proc. R. Soc. London Ser.A 422, 147-171 (1989). · Zbl 0673.73005
[27] Nemat-Nasser, S., Obata, M.: Rate-dependent, finite elasto-plastic deformation of polycristals. Proc. R. Soc. London Ser.A 407, 343-375 (1986).
[28] Nemat-Nasser, S., Iwakuma, T.: Elastic-plastic composites at finite strains. Int. J. Solids Struct.21, 55-65 (1985). · Zbl 0554.73003
[29] Marcellini, P., Sbordone, C.: Sur quelques questions de G-convergence et d’homogénéisation non linéaire. C. R. Acad. Sci. Paris Ser.A 284, 535-537 (1977). · Zbl 0353.49015
[30] Müller, S.: Homogenization of non convex integral functionals and cellular elastic materials. Arch. Rat. Mech. Anal.99, 189-212 (1987). · Zbl 0629.73009
[31] Abeyaratne, R., Triantafyllidis, N.: An investigation of localization in a porous elastic material using homogenization theory. J. Appl. Mech.51, 481-486 (1984).
[32] Geymonat, G., Müller, S., Triantafyllidis, N.: Quelques remarques sur l’homogénéisation des matériaux élastiques non linéaires. C. R. Acad. Sci. Paris Ser.I311, 911-916 (1990). · Zbl 0722.73004
[33] Geymonat, G., Müller, S., Triantafyllidis, N.: Homogenization of non linearly elastic materials, microscopic bifurcation and macroscopic loss of rank-one convexity. Arch. Rat. Mech. Anal.122, 231-290 (1993). · Zbl 0801.73008
[34] Triantafyllidis, N., Maker, B. N.: On the comparison between microscopic and macroscopic instability mechanisms in a class of fiber-reinforced composites. J. Appl. Mech.52, 794-800 (1985). · Zbl 0586.73112
[35] Bakhvalov, N., Panasenko, G.: Homogenisation: Averaging processes in periodic media. Studies in Mathematics and its Applications, vol.36. Dordrecht: Kluwer 1989. · Zbl 0692.73012
[36] Pruchnicki, E.: Loi hyperélastique homogénéisée pour les structures composites à matrice élastomère en grandes déformations. In: Multiple scale analyses and coupled physical systems. Ponts & Chaussées St Venant Symposium (Salençon, J., ed.), pp. 275-282 Paris: Publishers 1997.
[37] Pruchnicki, E.: Sur quelques aspects de la théorie de l’homogénéisation des milieux périodiques. Mémoire d’Habilitation, Université de Lille, 1997.
[38] Salamon, M. D. G.: Elastic moduli of a stratified rock mass. Int. J. Rock. Mech. Min. Sci.5, 519-527 (1968).
[39] Sawicki, A.: Yield condition for layered composites. Int. J. Solids Struct.17, 969-979 (1981). · Zbl 0471.73030
[40] Herrmann, L. R., Welch, K. R., Lim, C. K.: Composite FEM analysis for layered systems. J. Eng. Mech.110, 1284-1302 (1984).
[41] Pruchnicki, E., Shahrour, I.: Loi d’évolution homogénéisée du matériau multicouche à constituants élastoplastiques parfaits. C. R. Acad. Sci. Paris Ser.II 315, 137-142 (1992). · Zbl 0749.73051
[42] Pruchnicki, E., Shahrour, I.: A macroscopic elastoplastic constitutive law for multilayered media: Application to reinforced earth matrial. Int. J. Num. Anal. Meth. Geomech.18, 507-518 (1994). · Zbl 0811.73054
[43] Boutin, C.: Microstructural effects in elastic composites. Int. J. Solids Struc.33, 1025-1051 (1996). · Zbl 0920.73282
[44] Lourenço, P. B.: A matrix formulation for the elastoplastic homogenisation of layered materials. Mech. Cohesive-frictional Mat.1, 273-294 (1996).
[45] Dumontet, H.: Homogénéisation d’un matériau à structure périodique stratifié, de comportement linéaire et non linéaire et viscoélastique. C. R. Acad. Sci. Paris Ser.II 295, 633-636 (1982) · Zbl 0501.73014
[46] Dumontet, H.: Homogénéisation des matériaux stratifiés du type élastique linéaire, non linéaire et viscoélastique. Thèse de doctorat de 3ème cycle. Université de Paris VI 1983.
[47] Dumontet, H.: Homogénéisation et effects de bords dans les matériaux composites. Thèse de doctorat d’état. Université de Paris VI 1990.
[48] Jankovich, E., Leblanc, F., Durand, M., Bercovier, M.: A finite element method for the analysis of rubber parts, experimental and analytical assessment. Comp. Struct.14, 385-391 (1981). · Zbl 0467.73100
[49] Häggblad, B., Sundberg, J. A.: Large strain solutions of rubber components. Comp. Struct.17, 835-843 (1983)
[50] Ciarlet, P. G.: Mathematical elasticity, vol. I. Three-dimensional elasticity. Amsterdam New York Oxford Tokyo: North-Holland 1988. · Zbl 0648.73014
[51] Ogden, R. W.: Large deformation isotropic elasticity ? on the correlation of theory and experiment for incompressible rubber like solids. Proc. R. Soc. London Ser.A 326, 565-584 (1972). · Zbl 0257.73034
[52] Malkus, D. S.: Finite elements with penalties in non linear elasticity. Int. J. Numer. Meth. Eng.16, 121-136 (1980). · Zbl 0441.73103
[53] Bar-Yoseph, P., Avrashi, J.: New variational-asymptotic formulations for interlaminar stress analysis in laminated plates. J. Appl. Math. Phys.37, 305-321 (1986). · Zbl 0598.73064
[54] Dumontet, H.: Study of a boundary layer problem in elastic composite materials. Math. Modell. Numer. Anal.20, 265-286 (1986). · Zbl 0602.73012
[55] Sanchez-Palencia, E.: Boundary layer and edge effects in composite. In: Homogenization techniques for composite media, pp. 121-147. Lecture Notes in Physics, Vol. 272. Berlin Heidelberg New York: Springer 1985.
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.