f Coordinate-free representation of the elastic tensor

The elastic stiffness tensor relates stress to strain (both tensors of rank two). It is of rank foor and thus has 81 components. The symmetry of both stress and strain and the existence of an elastic potential (the independence of the elastic energy from the strain-history) result in at most 21 of these being independent. Material symmetry reduces the number of independent stiffnesses further (monoclinic 12, orthorhombic 9, trigonal 6, tetragonal 6, hexagonal 5, cubic 3, isotropic 2). However, for this reduction to be valid the stiffnesses must be referred to a coordinate system aligned with the axes of symmetry. With reference to an arbitrary coordinate system all media - with the exception of isotropic media, where every direction is an axis of symmetry - are characterized by 21 non-vanishing elastic stiffnesses, from which the material symmetry is far from obvious. Once the inherent symmetry is detected, the coordinate system can be rotated into one of the preferential coordinate system. This rotation (e .g., by means of the Bond relations) is relatively complicated, since for a tensor of rank four the corresponding expressions are of up to fourth order in the cosines and sines of the angles of rotation.