Bounds on shear moduli for orthotropic elastic composites

Composite and porous materials often appear in nature. Many composites may be considered orthotropic such as wood or bone. The elastic behavior of these composites under shear stresses is characterized by three independent shear moduli. We consider the totality of orthotropic composites made from two isotropic linearly elastic components in fixed proportion. For a prescribed triple of shear stresses we find optimal bounds on the strongest and weakest orthotropic composites. Mathematically this problem is one of constrained optimization. The set of constraints are related to the convex hull of a surface in three dimensions. For given values of the component elasticities the bounds are computed numerically