Hookean composites

The text addresses the limitations of the existing 2-phase Newtonian fluid theories in predicting the elastic properties of 2-phase particulate Hookean composites. The equations are rewritten in the form of the elastic modulus ratio (Ec/Em) instead of the shear modulus ratio (Gc/Gm), based on the relationship between shear modulus, elastic modulus, and Poisson’s ratio of isotropic solids. The assumptions of incompressibility or simply the 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 derived by replacing the constants in the equations relating shear stress and strain in 2-phase Newtonian fluids with variables. The resulting expressions are equated with 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, are used to develop expressions of elastic modulus ratio Ec/Em in terms of the volume fraction and the modulus ratio (Ep/Em) to plot curves of the Ec/Em ratio against the 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 are 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 significant improvement on the existing theory, giving 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 compared to the original equations, whose utility is limited to very low filler volume fractions.

Author(s) Details:

Maina Maringa,
Department of Mechanical and Mechatronic Engineering, Central University of Technology, Private Bag X20539, 20 President Brand Street Westdene, Bloemfontein, 9300, Free State, South Africa.

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