Out-of-Plane Tension of Thin Two-Layered Plates of Identically Oriented Hexagonal Crystals

In the framework of the theory of anisotropic elasticity, a theoretical analysis of the out-of-plane extension of thin two-layered plates of hexagonal crystals is carried out. The six-fold axes of all pairs of crystals are assumed to be perpendicular to the plane of the plates. Formulae for effective Young’s modulus and effective Poisson’s ratio are obtained. It is shown that in most cases effective Young’s moduli exceed Reuss’s average, and thus the rule of mixtures is violated. Equality of these characteristics is possible if the ratios of Young’s moduli and the ratios of Poisson’s ratios of crystal pairs are the same. Effective Poisson’s ratio may be greater or less than the corresponding Reuss’s average. In addition, it was found that effective Young’s modulus can surpass Young’s moduli of both crystals forming a two-layered plate. Effective Poisson’s ratio can be both larger and smaller than Poisson’s ratios of the initial pair of crystals. The general theoretical conclusions are illustrated by numerical estimates using the experimental values of the elastic constants of the known hexagonal crystals.