CT. Herakovich/Mechanics Research Communications 41(2012)1-20 Table 2 Developments in micromechanics. Author(s) ountry 837 Strain energy density defined -21 elastic constants Great Britain Germany Determination of the elastic field of an ellipsoidal inclusion and related problems rediction of Elastic Constants of Multiphase Materials lashin Variational Approach to the Theory of The Elastic Behavior of Multiphase Materials ashin and shtrikman USA and israel heory of Mechanical Behavior of Heterogeneous Media USA trengthened Materials-L. Elastic Behaviour The el uli of Fiber-Reinforced Materials Hashin and Rosen Reinforcement of metals Kelly and Davies Great Britain Tsai, Halpin and Pagano Theory of Fiber Reinforced Materials On the Effective Moduli of Composite Materials: Slender rigid Inclusions at Dilute Concentrations Russel and acrivos 975 A Theory of Elasticity with Microstructure for Directionally Reinforced Composites Fiber Composites w ropic Constituents Mechanics of Composite Materials: A Unified Micromechanical Approach Micromechanics: overall properties of heterogeneous materials Nemat-Nasser and hori USA The development of micromechanics models for predicting the of a fiber core surrounded by a matrix annulus, such that the effective properties of composites experienced a flurry of activity size of the cylinders varies as needed to fill the entire volume of eginning in the late 1950s and extending through the 1960s and material. The ratio of fiber radius to cylinder radius is held constant to the 1970s. The earlier works were concerned with the pre- throughout, thereby maintaining a constant fiber volume fraction diction of effective properties for materials(both solids and fluids) in each cylinder Four of the necessary five effective properties for a consisting of inclusions in a carrier material (see Hashin, 1964). transversely isotropic composite can be determined using the CCa Paul (1960) and Hill (1963)used energy approaches to obtain model. upper and lower bounds on elastic moduli of heterogeneous mate- In the following from Christensen(1979), with subscripts f and rials consisting of inclusions in a matrix. In general, the inclusions m indicating fiber and matrix respectively, V fiber volume fraction, were of arbitrary shape but both authors did make reference to E axial modulus, v Poissons ratio, u shear modulus, k bulk modulus, fiber-like inclusions. It is noteworthy that they both showed that and the fiber and matrix are taken to be transversely isotropic, the the voigt and reuss approximations are upper and lower bounds CCa model provides expressions for four of the effective properties on moduli Hashin and Shtrikman(1963)presented a variational in terms of the phase properties and fiber volume fraction. approach to derive upper and lower bounds for the effective elas- Effective axial modulus. tic moduli of quasi-isotropic and quasi-homogeneous multiphase materials of arbitrary phase geometry. Hill(1964)addressed the v(1-v Xu-Vm) elastic mechanical properties of fiber-strengthened materials In E1 (1972)Russel and Acrivos considered the effective modulus of com- osites with slender rigid inclusions at dilute concentrations. Effective axial Poisson's ratio: Hashin and Rosen(1964)employed a co cylinder assem- blage(CCA)model to develop upper and lower bounds as well V12=(1-v)Vm+ Vry as specific expressions for(some of) the effective elastic mod- uli of transversely isotropic composites. Their model consists of v(1-VXy-m)(m/(km+m/3)-m(k+H/3) an assemblage of concentric cylinders, each cylinder consisting +(-V)m)/(k+/3)+Vm/(km+m/3)+1 (5) a)Carbon/Epoxy b)Silicon Carbide/Titanium Fig. 5. Fiber and matrix photomicrographsC.T. Herakovich / Mechanics Research Communications 41 (2012) 1–20 5 Table 2 Developments in micromechanics. Year Development Author(s) Country 1837 Strain energy density defined – 21 elastic constants George Green Great Britain 1887/1889 Uniform strain modulus prediction W. Voigt Germany 1929 Uniform stress modulus prediction A. Reuss Germany 1957 Determination of the elastic field of an ellipsoidal inclusion and related problems Eshelby Great Britain 1960 Prediction of Elastic Constants of Multiphase Materials Paul USA 1962 The Elastic Moduli of Heterogeneous Materials Hashin USA 1963 Elastic Properties of Reinforced solids: some theoretical principles Hill Great Britain 1963 Variational Approach to the Theory of The Elastic Behavior of Multiphase Materials Hashin and Shtrikman USA and Israel 1964 Theory of Mechanical Behavior of Heterogeneous Media Hashin USA 1964 Theory of Mechanical Properties of Fiber-Strengthened Materials – I. Elastic Behaviour Hill USA 1964 The Elastic Moduli of Fiber-Reinforced Materials Hashin and Rosen USA 1965 The Principle of the Fiber Reinforcement of Metals Kelly and Davies Great Britain 1967 Modern Composite Materials Broutman and Krock USA 1968 Composite Materials Workshop Tsai, Halpin and Pagano USA 1972 Theory of Fiber Reinforced Materials Hashin USA 1972 On the Effective Moduli of Composite Materials: Slender Rigid Inclusions at Dilute Concentrations Russel and Acrivos USA 1975 A Theory of Elasticity with Microstructure for Directionally Reinforced Composites Achenbach USA 1979 Analysis of Properties of Fiber Composites with Anisotropic Constituents Hashin Israel 1991 Mechanics of Composite Materials: A Unified Micromechanical Approach Aboudi Israel 1993 Micromechanics: overall properties of heterogeneous materials Nemat-Nasser and Hori USA The development of micromechanics models for predicting the effective properties of composites experienced a flurry of activity beginning in the late 1950s and extending through the 1960s and into the 1970s. The earlier works were concerned with the pre￾diction of effective properties for materials (both solids and fluids) consisting of inclusions in a carrier material (see Hashin, 1964). Paul (1960) and Hill (1963) used energy approaches to obtain upper and lower bounds on elastic moduli of heterogeneous mate￾rials consisting of inclusions in a matrix. In general, the inclusions were of arbitrary shape, but both authors did make reference to fiber-like inclusions. It is noteworthy that they both showed that the Voigt and Reuss approximations are upper and lower bounds on moduli. Hashin and Shtrikman (1963) presented a variational approach to derive upper and lower bounds for the effective elas￾tic moduli of quasi-isotropic and quasi-homogeneous multiphase materials of arbitrary phase geometry. Hill (1964) addressed the elastic mechanical properties of fiber-strengthened materials. In (1972) Russel andAcrivos considered the effective modulus of com￾posites with slender rigid inclusions at dilute concentrations. HashinandRosen(1964) employeda concentric cylinder assem￾blage (CCA) model to develop upper and lower bounds as well as specific expressions for (some of) the effective elastic mod￾uli of transversely isotropic composites. Their model consists of an assemblage of concentric cylinders, each cylinder consisting of a fiber core surrounded by a matrix annulus, such that the size of the cylinders varies as needed to fill the entire volume of material. The ratio of fiber radius to cylinder radius is held constant throughout, thereby maintaining a constant fiber volume fraction in each cylinder. Four of the necessary five effective properties for a transversely isotropic composite can be determined using the CCA model. In the following from Christensen (1979), with subscripts f and m indicating fiber and matrix respectively, V fiber volume fraction, E axial modulus,  Poisson’s ratio,  shear modulus, k bulk modulus, and the fiber and matrix are taken to be transversely isotropic, the CCA model provides expressions for four of the effective properties in terms of the phase properties and fiber volume fraction. Effective axial modulus: E∗ 1 = Vf Ef + (1 − Vf )Em + 4Vf (1 − Vf )(vf − vm) 2m ((1 − Vf )m)/(kf + f /3) + Vf m (4) Effective axial Poisson’s ratio: 12 = (1 − Vf )m + Vf f + Vf (1 − Vf )(f − m)  m/(km + m/3) − m/(kf + f /3) ((1 − Vf )m)/(kf + f /3) + Vf m/(km + m/3) + 1 (5) Fig. 5. Fiber and matrix photomicrographs