Sufficient Symmetry Conditions for Isotropy of the Elastic Moduli Tensor

Abstract

Symmetry conditions are found that assure isotropy of the fourth rank tensor of elastic moduli. Crystallography provides the answer to this problem in the two-dimensional context, namely one axis of three-fold symmetry assures the isotropy of properties in the plane normal to the axis. The present work provides the answer in the three-dimensional problem: 6 axes of five-fold symmetry are sufficient to give isotropy of the elastic moduli. An important restriction must accompany the present result. The derivation is given in the special form appropriate to low density materials which have a microstructure that transmits load according to the axial deformation of a space network of material distributed into micro-struts. The corresponding fiber composite idealization is that of a fiber dominated system, it therefore follows that if the fibers take the 6 specific orientations in three-space then isotropy is obtained.