Application of homogenization theory and limit analysis to the evaluation of the macroscopic strength properties of fiber reinforced composite materials

The macroscopic strength domain of a composite material reinforced by long, parallel fibers is, in general, unknown but for its theoretical definition, In this note it is shown how a homogenization technique applied to yield design theory allows the derivation of two domains (in the space of the macroscopic stresses) which are a lower and an upper bound to the composite strength domain. The dependence of these domains on the fiber content and on the shape of the fiber array is pointed out. Analytical equations for the approximate uniaxial macroscopic strength of composites with Drucker-Prager or von Mises type matrix are derived. For more complex stress conditions, the relevant strength domains are numerically evaluated as well. The discrepancy between the two bounds is in many cases relatively small. In particular, the two bounds yield the same value for the uniaxial strength of the composite along the fiber direction, which by consequence is exactly determined.