The representation of the linear elastic symmetries by sets of vectors

The objective of this paper is to show that the different forms of the elasticity or compliance tensors that represent the different linear elastic material symmetries may be obtained by assuming that the strain energy depends on a set of vectors, a different specific set for each of the eight linear elastic material symmetries. This connection permits a geometrically intuitive view of the elastic symmetries as a companion to the group theoretic crystallographic viewpoint. It also permits each of the eight linear elastic symmetries to be viewed as characterized by sets of material vectors. Finally, it provides a conceptual basis for the direct and evolving calculation of the changes in symmetry as a result of deformation. This work was motivated by the use of vectors to represent collagen directions in development of constitutive equations for soft tissue deformation.