Bounds and estimates of overall moduli of composites with periodic microstructure

Abstract Bounds on the overall moduli of a broad class of composites with periodic microstructure are obtained by generalized Hashin—Shtrikman variational principles. Piecewise constant eigenstrains and eigenstresses are used, and exact, computable bounds are developed. The formulation is valid for composites comprised of an anisotropic (or isotropic) matrix with an arbitrary number of periodically distributed anisotropic (or isotropic) inhomogeneities. Examples of two-phase particulate, whisker, and fiber-reinforced composites are considered for illustration. Finally, an estimate of the overall moduli, based on the selection of the effective medium as the reference material, is proposed for periodic microstructure.

[1]  J. Mackenzie,et al.  The Elastic Constants of a Solid containing Spherical Holes , 1950 .

[2]  Pedro Ponte Castañeda.  The effective mechanical properties of nonlinear isotropic composites , 1991 .

[3]  David J. Jeffrey,et al.  The Rheological Properties of Suspensions of Rigid Particles , 1976 .

[4]  G. Broughton,et al.  Agglomeration and Viscosity in Dilute Suspensions , 1938 .

[5]  T. Iwakuma,et al.  Composites with periodic microstructure , 1983 .

[6]  A. Fröhlich,et al.  Theory of the rheological properties of dispersions , 1946, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[7]  S. Shtrikman,et al.  A variational approach to the theory of the elastic behaviour of polycrystals , 1962 .

[8]  S. Shtrikman,et al.  On some variational principles in anisotropic and nonhomogeneous elasticity , 1962 .

[9]  S. H. Maron,et al.  Rheology of synthetic latex: V. Flow behavior of low-temperature GR-S latex , 1955 .

[10]  Zvi Hashin,et al.  On elastic behaviour of fibre reinforced materials of arbitrary transverse phase geometry , 1965 .

[11]  J. Willis,et al.  Variational Principles for Inhomogeneous Non-linear Media , 1985 .

[12]  J. Willis,et al.  Variational and Related Methods for the Overall Properties of Composites , 1981 .

[13]  R. Luebbers,et al.  Viscosity of suspensions of spherical and other isodimensional particles in liquids , 1957 .

[14]  J. Willis.  Bounds and self-consistent estimates for the overall properties of anisotropic composites , 1977 .

[15]  H. Eilers.  Die Viskosität von Emulsionen hochviskoser Stoffe als Funktion der Konzentration , 1941 .

[16]  S. Shtrikman,et al.  Note on a variational approach to the theory of composite elastic materials , 1961 .

[17]  M. Taya,et al.  On effective moduli of an elastic body containing periodically distributed voids , 1981 .

[18]  J. D. Eshelby.  The determination of the elastic field of an ellipsoidal inclusion, and related problems , 1957, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[19]  L. Walpole.  On bounds for the overall elastic moduli of inhomogeneous systems—I , 1966 .

[20]  Frank L. Saunders,et al.  Rheological properties of monodisperse latex systems I. Concentration dependence of relative viscosity , 1961 .

[21]  V. Vand.  Viscosity of solutions and suspensions; experimental determination of the viscosity-concentration function of spherical suspensions. , 1948, The Journal of physical and colloid chemistry.

[22]  E. Kröner.  Berechnung der elastischen Konstanten des Vielkristalls aus den Konstanten des Einkristalls , 1958 .

[23]  James V. Robinson.  The Viscosity of Suspensions of Spheres. , 1949 .

[24]  S. H. Maron,et al.  Rheology of synthetic latex. II. Concentration dependence of flow in type V GR-S latex , 1951 .

[25]  R. D. Geckler,et al.  The Rheology of Suspensions , 1954 .

[26]  J. Willis,et al.  On methods for bounding the overall properties of nonlinear composites: Correction and addition , 1992 .

[27]  S. G. Ward,et al.  Studies of the viscosity and sedimentation of suspensions Part 1. - The viscosity of suspension of spherical particles , 1959 .

[28]  R. Hill.  The Elastic Behaviour of a Crystalline Aggregate , 1952 .

[29]  S. Shtrikman,et al.  A variational approach to the theory of the elastic behaviour of multiphase materials , 1963 .

[30]  M. Taya,et al.  On effective moduli of an elastic body containing periodically distributed voids: comments and corrections , 1985 .

[31]  L. Walpole,et al.  On the overall elastic moduli of composite materials , 1969 .

[32]  Salvatore Torquato,et al.  Random Heterogeneous Media: Microstructure and Improved Bounds on Effective Properties , 1991 .

[33]  A. Reuss,et al.  Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle . , 1929 .

[34]  Sia Nemat-Nasser,et al.  On composites with periodic structure , 1982 .

[35]  P. Suquet.  On bounds for the overall potential of power law materials containing voids with an arbitrary shape , 1992 .

[36]  Richard M. Christensen,et al.  A critical evaluation for a class of micro-mechanics models , 1990 .

[37]  T. Iwakuma,et al.  Cavity growth and grain boundary sliding in polycrystalline solids , 1986 .

[38]  Sia Nemat-Nasser,et al.  Solids with periodically distributed cracks , 1993 .

[39]  J. Willis,et al.  On methods for bounding the overall properties of nonlinear composites , 1991 .

[40]  S. G. Mason,et al.  THE VISCOSITY OF SUSPENSIONS OF SPHERES: A NOTE ON THE PARTICLE INTERACTION COEFFICIENT , 1954 .

[41]  L. Walpole.  Elastic Behavior of Composite Materials: Theoretical Foundations , 1981 .

[42]  Zvi Hashin,et al.  The Elastic Moduli of Heterogeneous Materials , 1962 .

[43]  David G. Thomas.  Transport characteristics of suspension: VIII. A note on the viscosity of Newtonian suspensions of uniform spherical particles , 1965 .

[44]  Z. Hashin,et al.  The Elastic Moduli of Fiber-Reinforced Materials , 1964 .

[45]  J. Willis.  Elasticity Theory of Composites , 1982 .

[46]  Sia Nemat-Nasser,et al.  Bounds on the overall elastic and instantaneous elastoplastic moduli of periodic composites , 1986 .