Summary
The paper investigates a numerical procedure for the computation of the overall macroscopic elasticity moduli of linear composite materials with periodic micro-structure. We consider a homogenized macro-continuum with locally attached representative micro-structure which characterizes a representative cell of a composite. The deformation of the micro-structure is assumed to be coupled with the local deformation at a typical point of the macro-continuum by three alternative constraints of the microscopic fluctuation field. The underlying key approach is a finite element discretization of the boundary value problem for the fluctuation field on the micro-structure of the composite. This results into a distinct closed-form representation of the overall elasticity moduli in terms of a Taylor-type upper bound term and a characteristic softening term which depends on global fluctuation stiffness matrices of the discretized micro-structure. With this representation in hand, overall moduli of periodic composites can be computed in a straightforward manner for a given finite element discretization of the micro-structure. We demonstrate the concept for three types of periodic composites and compare the results with well-known analytical estimates.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aboudi, J.: Mechanics of composite materials—A unified micromechanical approach. Amsterdam: Elsevier 1991.
Bennett, J. G., Haberman, K. S.: An alternative unified approach to the micro-mechanical analysis of composite materials. J. Comp. Materials30, 1732–1747 (1996).
Bensoussan, A., Lions, J. L., Papanicolaou, G.: Asymptotic analysis for periodic structures. Studies in Mathematics and its Applications. North-Holland 1978.
Benveniste, Y.: A new approach to the application of Mori-Tanaka's theory in composite materials. J. Mech. Phys. Solids11, 357–372 (1987).
Budianski, B.: On the elastic moduli of some heterogeneous materials. Studies in mathematics and its applications. J. Mech Phys. Solids13, 223–227 (1965).
Eshelby, J. D.: The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. London, Ser. A241, 376–396 (1957).
Guedes, J. M., Kikuchi, N.: Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods. Comp. Meth. Appl. Mech. Engng83, 143–198 (1990).
Hashin, Z.: Analysis of composite materials—A survey. J. Appl. Mech50, 481–505 (1983).
Hill, R.: Elastic properties of reinforced solids: Some theoretical principles. J. Mech. Phys. Solids11, 357–372 (1963).
Hill, R.: A self-consistent mechanics of composite materials. J. Mech. Phys. Solids13, 213–222 (1965).
Hill, R.: On constitutive variables for heterogeneous solids at finite strain. Proc. R. Soc. London, Ser. A11, 131–147 (1972).
Kröner, E.: Elastic moduli of perfect disordered composite materials. J. Mech. Phys. Solids15, 319–340 (1967).
Mori, T., Tanaka, K.: Average stress in a matrix and average energy of materials with misfitting inclusions. Acta Metall.21, 571–574 (1973).
Nemat-Nasser, S., Hori, M.: Micromechanics: overall properties of heterogeneous materials. Appl. Math. Mech.37. North-Holland 1993.
Reuss, A.: Berechnung der Fließgrenze von Mischkristallen aufgrund der Plastizitätsbedingung für Einkristalle. Z. Angew. Math. Mech.9, 49–58 (1929).
Sanchez-Palenzia, E.: Nonhomogeneous media and vibration theory. Lecture Notes in Physics127, 193–278 (1980).
Suquet, P. M.: Elements of homogenization for inelastic solid mechanics. In: Homogenization techniques for composite materials (Sanchez-Palenzia, E.; Zaoui, A., eds). Lecture Notes in Physics272, 193–278 (1987).
Swan, C. C.: Techniques for stress-and strain-controlled homogenization of inelastic periodic composites. Comp. Meth. Appl. Mech. Engng117, 249–267 (1994).
Swan, C. C., Cakmak, A. S.: Homogenization and effective elastoplasticity models for periodic composites. Comm. Num. Meth. Engng10, 257–265 (1994).
Voigt, W.: Lehrbuch der Kristallphysik. Leipzig: Teubner 1889.
Weng, G. J.: The theoretical connection between Mori-Tanaka's theory and the Hashin-Shtrikman-Walpole bounds. Int. J. Engng Sci.28, 1111–1120 (1990).
Willis, J. R.: Variational and related methods for the overall properties of composites. Adv. Appl. Mech.21, pp. 1–78. Academic Press 1981.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Miehe, C., Schröder, J. & Bayreuther, C. On the homogenization analysis of composite materials based on discretized fluctuations on the micro-structure. Acta Mechanica 155, 1–16 (2002). https://doi.org/10.1007/BF01170836
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01170836