Lower and upper bounds to the macroscopic strength domain of a fiber reinforced composite material

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 macroscopic stresses) that 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.