Elastic modulus of a two-phase system

Many ordinary solids, such as ceramics, are made up of several phases. Strictly speaking, these are not composite materials, but similar reasoning can be applied to obtain the elastic modulus and other mechanical properties of such systems. Although the equations for a solid composed of several phases with a complex microstructure are frequently unwieldy, simpler equations exist for well-defined geometries. 

A ceramic body composed of two phases, one of which is distributed as particles within the matrix of the other, has a modulus of elasticity given by  

A ceramic body composed of layers aligned parallel to a uniaxial stress, in which the strain is shared equally by the two phases, the iso-strain condition, has an elastic modulus  

This approximation is called the Voigt model, and the value of the elastic modulus is often known as the Voigt bound. The expression is identical to that for a continuous aligned fibre composite under a longitudinal load, and gives the elastic modulus when the load is applied parallel to the sheets. Similarly, if the stress is applied perpendicular to the layers, and an iso-stress condition applies (the Reuss model), the elastic modulus is  

The value of the elastic modulus, often called the Reuss bound, is identical to that for transverse loading on a fibre composite, and gives a value for the elastic modulus normal to the layers. In both of these equations, E, E and Ep are the elastic moduli of the ceramic, matrix and particles, respectively, and Vc (equal to 1.0), Vm and Vp are the corresponding volume fractions. 

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