Analytical Modelling to Improve the Predictive Capacity of Newtonian Fluid Theories on the Elastic Moduli Ratio (Ec/Em) of Particulate Composites

The limitations of existing 2-phase Newtonian fluid theories in predicting the elastic properties of 2-phase particulate Hookean composites, are addressed by first rewriting their equations in the form of the elastic modulus ratio (Ec/Em) instead of the shear modulus ratio (Gc/Gm), based on the known relationship between shear modulus, elastic modulus and Poisson’s ratio of isotropic solids. The assumptions of incompressibility (thus a Poisson’s ratio of 0.5) or simply equality of the Poisson’s ratios of the filler and matrix are shown to satisfy the known Ec/Em ratio of particulate composites of unity at a filler volume fraction of 0%. General expressions for the Ec/Em ratios are developed by replacing the constants in the equations relating shear stress and strain in 2-phase Newtonian fluids to the equations relating shear stress to strain in Hookean solids, with variables. The resulting expressions are then equated to the known values of the elastic modulus ratios at a filler volume fraction of 100%, to obtain expressions relating the variables to the Poisson’s ratios and elastic modulus ratios. The expressions, together with the assumption of incompressibility of the composite constituents, thus perfect plasticity (Poisson’s ratio of 0.5), are finally used to develop expressions of elastic modulus ratio Ec/Em in terms of the volume fraction and the modulus ratio (Ep/Em) for plotting curves of the Ec/Em ratio against volume fraction of reinforcing filler. These curves are plotted on the same graphs with curves of the lower and upper bounds defined by the Reuss and Voigt rules and comparisons made. The equations are also tested for application to composite constituents that are not perfectly plastic. In both cases, the developed equations are seen to be a great improvement on existing theory in that they give good estimates of the elastic modulus ratio Ec/Em ratio for the full range of particle filler volume fraction between 0 and 100%. This is a big improvement from the case for the original equations, whose utility is limited to very low filler volume fractions.