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A comparison of homogenization and standard mechanics analyses for periodic porous composites

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Abstract

Composite material elastic behavior has been studied using many approaches, all of which are based on the concept of a Representative Volume Element (RVE). Most methods accurately estimate effective elastic properties when the ratio of the RVE size to the global structural dimensions, denoted here as ν, goes to zero. However, many composites are locally periodic with finite ν. The purpose of this paper was to compare homogenization and standard mechanics RVE based analyses for periodic porous composites with finite ν. Both methods were implemented using a displacement based finite element formulation. For one-dimensional analyses of composite bars the two methods were equivalent. Howver, for two- and three-dimensional analyses the methods were quite different due to the fact that the local RVE stress and strain state was not determined uniquely by the applied boundary conditions. For two-dimensional analyses of porous periodic composites the effective material properties predicted by standard mechanics approaches using multiple cell RVEs converged to the homogenization predictions using one cell. In addition, homogenization estimates of local strain energy density were within 30% of direct analyses while standard mechanics approaches generally differed from direct analyses by more than 70%. These results suggest that homogenization theory is preferable over standard mechanics of materials approaches for periodic composites even when the material is only locally periodic and ν is finite.

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References

  • Accorsi, M. L. (1988): A method for modelling microstructural material discontinuities in a finite element analysis. Int. J. Num. Meth. Eng. 36, 2187–2197

    Google Scholar 

  • Accorsi, M. L.; Nemat-Nasser, S. (1986): Bounds on the overall elastic and instantaneous elastoplastic moduli of periodic composites. Mech. Materials 5, 209–220

    Google Scholar 

  • Babuska, I. (1976a): Homogenization approach in engineering. Proc. 2nd Int. Symposium on Comp. Meth. in App. Science and Eng. 137–153. Berlin, Heidelberg, New York: Springer

    Google Scholar 

  • Babuska, I. (1976b): Solution of interface problems by homogenization. I. SIAM J. Math. Anal. 7, 603–634

    Google Scholar 

  • Babuska, I. (1976c): Solution of interface problems by homogenization. II. SIAM J. Math. Anal. 7, 635–645

    Google Scholar 

  • Babuska, I. (1976d): Solution of interface problems by homogenization. III. SIAM J. Math. Anal. 7, 923–937

    Google Scholar 

  • Bakhvalov, N.; Panasenko, G. (1989): Homogenization: averaging processes in periodic media. Dordrecht, The Netherlands: Kluwer Academic Publishers

    Google Scholar 

  • Beaupre, G.; Hayes, W. (1985): Finite element analysis of a three-dimensional open-celled model for trabecular bone. J. Biomech. Eng. 107, 249–256

    Google Scholar 

  • Bensoussan, A.; Lions, J.; Papanicolaou, G. (1978): Asymptotic Analysis for Periodic Structures. Amsterdam: North-Holland

    Google Scholar 

  • Benveniste, Y. (1987): A new approach to the application of Mori-Tanaka's theory in composite materials. Mech. Materials 6, 147–157

    Google Scholar 

  • Bourgat, J. (1977): Numerical experiments of the homogenization method for operators with periodic coefficients. In: Dold, A.; Eckmann, B. (eds): Lecture Notes in Mathematics 704, pp. 330–356. Berlin, Heidelberg, New York: Springer

    Google Scholar 

  • Budiansky, B. (1965): On the elastic moduli of some heteorogeneous materials. J. Mech. Phys. Solids 13, 223–227

    Google Scholar 

  • Christenson, R. (1986): Mechanics of low density materials. J. Mech. Phys. Solids 34, 563–578

    Google Scholar 

  • Cleary, M. P.; Chen, I. W.; Lee, S. M. (1980): Self-consistent techniques for heterogeneous solids. ASCE J. Eng. Mech. 106, 861–887

    Google Scholar 

  • Dumontet, H. (1985): Boundary layer stresses in elastic composites. In: Ladeveze, P. (ed): Local Effects in the Analysis of Structures. Amsterdam: Elsevier 115–232

    Google Scholar 

  • Duvaut, G. (1976): Homogeneization et materiaux composite. In: Ciarlet, P.; Rouseau, M. (eds): Theoretical and Applied Mechanics. Amsterdam: North-Holland

    Google Scholar 

  • Eshelby, J. D. (1957): The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. London A 241, 376–396

    Google Scholar 

  • Fung, Y. (1965): Foundations of Solid Mechanics, Englewood Cliffs: Prentice-Hall

    Google Scholar 

  • Gibson, L.; Ashby, M. (1982): The mechanics of three-dimensional cellular materials. Proc. R. Soc. London A 382, 43–59

    Google Scholar 

  • Gibson, L.; Ashby, M. (1988): Cellular Solids: Structure and Properties. Oxford: Pergamon Press

    Google Scholar 

  • Guedes, J. (1990): Nonlinear computational models for composite materials using homogenization. Ph.D. Dissertation, Ann Arbor: The University of Michigan

  • Guedes, J.; Kikuchi, N. (1990): Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element solutions. Com. Mech. App. Meth. Eng. 83, 143–198

    Google Scholar 

  • Hashin, Z. (1983): Analysis of composite materials—a survey. J. App. Mech. 50, 481–505

    Google Scholar 

  • Hashin, Z.; Shtrikman, S. (1962): A variational approach to the theory of the elastic behavior of polycrystals. J. Mech. Phys. Solids 10, 343–352

    Google Scholar 

  • Hashin, Z.; Shtrikman, S. (1963): A variational approach to the theory of the elastic behavior of multiphase materials. J. Mech. Phys. Solids 11, 127–140

    Google Scholar 

  • Hill, R. (1963): Elastic properties of reinforced solids: some theoretical principles. J. Mech. Phys. Solids 11, 357–372

    Google Scholar 

  • Hill, R. (1965): A self consistent mechanics of composite materials. J. Mech. Phys. Solids 13, 213–222

    Google Scholar 

  • Huber, A.; Gibson, L. (1988): Anisotropy of foams. J. Mat. Sci. 23, 3031–3040

    Google Scholar 

  • Iremonger, M. J.; Lawler, J. P. (1980): Relationship between modulus and density for high-density closed cell thermoplastic foams. J. App. Polymer Sci. 25, 809–819

    Google Scholar 

  • Ishai, O.; Cohen, L. J. (1967): Elastic properties of filled and porous epoxy composites. Int. J. Mech. Sci. 9, 539–546

    Google Scholar 

  • Iwakuma, T.; Nemat-Nasser, S. (1983): Composites with periodic microstructure. Comput. Struct. 16, 13–19

    Google Scholar 

  • Kanakkanatt, S. V. (1973): Mechanical anisotropy of open-cell foams. J. Cell. Plast. 9, 50–53

    Google Scholar 

  • Keller, J. B. (1977): Effective behavior of heterogenous media. Proc. Symposium on Statistical Mechanics and Statistical Methods, Plenum Press, 631–644

  • Kikuchi, N. (1986): Finite Element Methods in Mechanics. Cambridge: Cambridge University Press

    Google Scholar 

  • Kikuchi, N. (in preparation): Variational methods and optimization in mechanics. Englewood Cliffs: Prentice-Hall

  • Ko, W. (1965): Deformations of foamed elastomers. J. Cell. Plast. 1, 45–50

    Google Scholar 

  • Kroner, E. (1977): Bounds for effective elastic moduli of disordered materials. J. Mech. Phys. Solids 25, 137–155

    Google Scholar 

  • Larsen, E. W. (1975): Neutron transport and diffusion in inhomogeneous media. I. J. Math. Phys. 16, 1421–1427

    Google Scholar 

  • Lederman, J. M. (1971): The prediction of the tensile properties of flexible foams. J. App. Polymer Sci. 15, 693–703

    Google Scholar 

  • Lene, F.; Leguillon, D. (1982): Homogenized constitutive law for a partially cohesive composite material. Int. J. Solids Struct. 5, 443–458

    Google Scholar 

  • Lions, J. (1981): Some methods in the Mathematical Analysis Analysis of Systems and their Control. New York: Gordon and Breach Science Publishers

    Google Scholar 

  • Lions, J. L. (1980): Asymptotic expansions in perforated media with a periodic structure. Rocky Mountain J. Math. 10, 125–140

    Google Scholar 

  • Mori, T.; Tanaka, K. (1973): Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall. 231, 571–574

    Google Scholar 

  • Mura, T. (1982): Micromechanics of defectsin solids. The Hague: Martinus Nijhoff

    Google Scholar 

  • Murakami, H.; Toledano, A. (1990): A high-order mixture homogenization of bi-laminated composites. J. App. Mech. 57, 388–396

    Google Scholar 

  • Murakami, H.; Maewal, A.; Hegemier, G. A. (1981): A mixture theory with a director for linear elastodynamics of periodically laminated media. Int. J. Solids Struct. 17, 155–173

    Google Scholar 

  • Nemat-Nasser, S.; Taya, M. (1981): On effective moduli of an elastic body containing periodically distributed voids. Quart. J. App. Math. 39, 43–59

    Google Scholar 

  • Nemat-Nasser, S.; Iwakuma, T.; Hejazi, M. (1982): On composites with periodic structure. Mech. Materials 1, 239–267

    Google Scholar 

  • Norris, A. N. (1985): A differential scheme for the effective moduli of composites. Mech. Materials 4, 1–16

    Google Scholar 

  • Norris, A. N. (1989): An examination of the Mori-Tanaka effective medium approximation for multiphase composites. J. App. Mech. 56, 83–88

    Google Scholar 

  • Patel, M.; Finnie, I. (1970): Structural features and mechanical properties of rigid cellular plastics. J. Mater. 5, 909–932

    Google Scholar 

  • Reuss, A. (1929): Berechnung der Fliessgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle. Z. Angew. Math. Mech. 9, 49–58

    Google Scholar 

  • Sanchez-Palencia, E. (1974): Comportements local et macroscopique d'un type de milieux physiques heterogenes. Int. J. Eng. Sci. 12, 331–351

    Google Scholar 

  • Sanchez-Palencia, E. (1980): Non-Homogeneous Media and Vibration Theory. Berlin, Heidelberg, New York: Springer

    Google Scholar 

  • Sanchez-Palencia, E. (1987): Boundary layers and edge effects in composites. In: Sanchez-Palencia, E.; Zaoui, A. (eds): Homogenization techniques for composite media, pp. 194–278. Berlin, Heidelberg, New York: Springer

    Google Scholar 

  • Suquet, P. (1987): Elements of homogenization theory for inelastic solid mechanics. In: Sanchez-Palencia, E.; Zaoui, A. (eds): Homogenization techniques for composite media, pp. 194–278. Berlin, Heidelberg, New York: Springer

    Google Scholar 

  • Voigt, W. (1889): Über die Beziehung zwischen den beiden Elastizitätskonstanten isotroper Körper, Wied. Ann. 38, 573–587

    Google Scholar 

  • Walpole, L. J. (1966a): On bounds for the overall elastic moduli of inhomogeneous systems—I. J. Mech. Phys. Solids 14, 151–162

    Google Scholar 

  • Walpole, L. J. (1966b): On bounds for the overall elastic moduli of inhomogeneous systems—II. J. Mech. Phys. Solids 14, 289–301

    Google Scholar 

  • Warren, W. E.; Kraynik, A. M. (1988): The linear elastic properties of open-cell foams. J. App. Mech. 55, 341–346

    Google Scholar 

  • Willis, J. R. (1977): Bounds and self-consistent estimates for the overall properties of anisotropic composites. J. Mech. Phys. Solids 25, 185–202

    Google Scholar 

  • Zhikov, V. V.; Kozlov, S. M.; Oleinik, O. A.; Kha Ten Ngoan (1979): Averaging and G-convergence of differential operators. Rus. Math. Survey 34, 69–147

    Google Scholar 

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Authors and Affiliations

  1. Orthopaedic Research Laboratories, Section of Orthopaedic Surgery, The University of Michigan, Rm. G-0161, 400 N. Ingalls Bldg., 48109, Ann Arbor, MI, USA

    S. J. Hollister

  2. Computational Mechanics Laboratory, Department of Mechanical Engineering and Applied Mechanics, The University of Michigan, 48109, Ann Arbor, MI, USA

    N. Kikuchi

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  1. S. J. Hollister
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  2. N. Kikuchi
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Communicated by S. N. Atluri, December 31, 1991

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Hollister, S.J., Kikuchi, N. A comparison of homogenization and standard mechanics analyses for periodic porous composites. Computational Mechanics 10, 73–95 (1992). https://doi.org/10.1007/BF00369853

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