Overall behavior of two-dimensional periodic composites

The overall properties of a binary elastic periodic fiber-reinforced composite are studied here for a cell periodicity of square type. Exact formulae are obtained for the effective stiffnesses, which give closed-form expressions for composites with isotropic components including ones for empty and rigid fibers. The new formulae are simple and relatively easy to compute. Examples show the dependences of the stiffnesses as a function of fiber volume fraction up to the percolation limit. The specific example of glass fibers in epoxy yields new curves, which correct those displayed before by Meguid and Kalamkarov. Comparison with experimental data is very good. Bruno, Hill and Hashin’s bounds are compared with the exact solution. In most cases, the latter is very close to a bound in a given interval. A useful fact to know, where the easy formula afforded by the bound is advantageous. Plots of effective properties are also given for values of the shear moduli ratio of the two media. The overall parameters in the cases of empty and rigid fibers are also shown. The exact formulae explicitly display Avellaneda and Schwarts’s microstructural parameters, which have a physical meaning, and provide formulae for them. The equations easily lead to Hill’s universal relations