Optimal bounds on the in-plane 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. The authors consider the totality of orthotropic composites made form two isotropic linearly elastic components in fixed proportion. For a prescribed triple of shear stresses, optimal bounds are found on the strongest and weakest orthotropic composites. Mathematically, this problem is one of constrained optimization. the set of constraints is related to the convex hull a surface in three dimensions. For given of the component elasticities, the bounds are computed numerically.