Bounds on the elastic strain energy density in 3-D bodies with material symmetries

Dealt with is the determination of the orientation of the material symmetry axes at which a 3D body made of aleotropic materials exhibits the stiffest (or possibly, the most flexible) response. As a general rule, this is shown to require co1linearity of principal stresses and principal strains. The two cases of materials with cubic symmetry and transversely isotropic materials are studied in detail. For transversely isotropic materials, optimal solutions can be either ‘trivial’, i.e. featured by full col1inearity of material symmetry axes and principal strains. or ‘quasi-trivial’, where collinearity is only partial1y preserved. For materials with cubic symmetry. possible are also ‘non-trivial’ solutions. featured by equal axial strains along the three symmetry axes. Optimal orientations are analytically derived and the relevant solutions classified: of great importance to this end is a parameter depending only on the elastic material properties.