[note 1] This relationship can be interpreted as a generalization of Hooke's law to a 3D continuum.
[3] This fact follows from the symmetry of the stress and strain tensors, together with the requirement that the stress derives from an elastic energy potential.
For isotropic materials, the elasticity tensor has just two independent components, which can be chosen to be the bulk modulus and shear modulus.
[3] The most general linear relation between two second-rank tensors
is defined from the stress-strain relation of a linear elastic material, in the limit of small strain.
is the metric tensor in the reference frame of the material.
[6][7] In an orthonormal Cartesian coordinate basis, there is no distinction between upper and lower indices, and the metric tensor can be replaced with the Kronecker delta: Substituting the first equation into the stress-strain relation and summing over repeated indices gives where
is the bulk modulus, and are the components of the shear tensor
The elasticity tensor of a cubic crystal has components where
are scalars; because they are coordinate-independent, they are intrinsic material constants.
Thus, a crystal with cubic symmetry is described by three independent elastic constants.
[9] In an orthonormal Cartesian coordinate basis, there is no distinction between upper and lower indices, and
[10] The number of independent elastic constants for several of these is given in table 1.
[9] The elasticity tensor has several symmetries that follow directly from its defining equation
[11][2] The symmetry of the stress and strain tensors implies that Usually, one also assumes that the stress derives from an elastic energy potential
must be symmetric under interchange of the first and second pairs of indices: The symmetries listed above reduce the number of independent components from 81 to 21.
If a material has additional symmetries, then this number is further reduced.
generally acquire different values under a change of basis.
Nevertheless, for certain types of transformations, there are specific combinations of components, called invariants, that remain unchanged.
Invariants are defined with respect to a given set of transformations, formally known as a group operation.
For example, an invariant with respect to the group of proper orthogonal transformations, called SO(3), is a quantity that remains constant under arbitrary 3D rotations.
They are also complete, in the sense that there are no additional independent linear or quadratic invariants.
This decomposition is irreducible, in the sense of being invariant under rotations, and is an important tool in the conceptual development of continuum mechanics.
In addition, it is not irreducible, so it is not invariant under linear transformations such as rotations.
[2] An irreducible representation can be built by considering the notion of a totally symmetric tensor, which is invariant under the interchange of any two indices.
, this sum reduces to The difference is an asymmetric tensor (not antisymmetric).
[2][15] However, this decomposition is not irreducible with respect to the group of rotations SO(3).
into two: See Itin (2020)[15] for explicit expressions in terms of the components of
This representation decomposes the space of elasticity tensors into a direct sum of subspaces: with dimensions These subspaces are each isomorphic to a harmonic tensor space
is the space of 3D, totally symmetric, traceless tensors of rank