Improved Voigt and Reuss Formulas with the Poisson Effect.

The Poisson effect, measured by the Poisson's ratio, plays an important role in the regulation of the elastic properties of composite materials, but it is not considered in the conventional Voigt (iso-strain) and Reuss (iso-stress) formulas, which explains why the formulas are inaccurate even if the iso-strain or the iso-stress conditions are satisfied. To consider the Poisson effect, we derived a set of new formulas based on the governing equations of elasticity. The obtained formulas show greater mathematical complexity. To further understand how the Poisson effect influences composite elastic properties, two types of finite element models (FEM) were constructed to simulate the situations where the Poisson effect does or does not have an influence. The results show that the conventional Voigt and Reuss formulas are special cases of the newly derived ones. The Poisson effect induces secondary strains and stresses into the phase materials, which demands more strain energy to achieve the same deformation in the primary (loading) direction, and thus increases composite stiffness; the magnitude of the increase is dependent on the contrast of phase properties. The findings may have significant impact on the study of the emerging nanocomposites and functionally graded materials, where the conventional Voigt and Reuss formulas have wide applications.