The elasticity tensor is a fourth-rank tensor describing the stress-strain relation in a linear elastic material. Other names are elastic modulus tensor and stiffness tensor.

The defining equation can be written as

are the components of the Cauchy stress tensor and infinitesimal strain tensor are the components of the elasticity tensor. Summation over repeated indices is implied. This relationship can be interpreted as a generalization of Hooke's law to a 3D continuum.

in 3D has 34, but the elasticity tensor has at most 21 independent components. 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.

Definition

is

. are the stress and strain tensors, respectively.

The compliance tensor is defined from the inverse stress-strain relation:

The two are related by

is the Kronecker delta.

is defined from the stress-strain relation of a linear elastic material, in the limit of small strain.

Special cases

Isotropic

simplifies to

is the metric tensor in the reference frame of the material. 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

can be identified with the first and second Lamé parameters. An equivalent expression is

is the bulk modulus, and

.

Cubic crystals

The elasticity tensor of a cubic crystal has components

are unit vectors corresponding to the three mutually perpendicular axes of the crystal unit cell. 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.

is the Kronecker delta, so the expression simplifies to

Other crystal classes

in other crystal symmetry classes. The number of independent elastic constants for several of these is given in table 1.

Properties

Symmetries

. The symmetry of the stress and strain tensors implies that

:

which implies

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.

Transformations

transform as

are the elements of the corresponding rotation matrix. A similar transformation rule holds for other linear transformations.

Invariants

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.

possesses two linear invariants and seven quadratic invariants with respect to SO(3). The linear invariants are

and the quadratic invariants are

These quantities are linearly independent, that is, none can be expressed as a linear combination of the others. They are also complete, in the sense that there are no additional independent linear or quadratic invariants.

Decompositions

can be decomposed as

is antisymmetric. Component-wise,

, respectively. This decomposition is irreducible, in the sense of being invariant under rotations, and is an important tool in the conceptual development of continuum mechanics.

The elasticity tensor has rank 4, and its decompositions are more complex and varied than those of a rank-2 tensor. A few examples are described below.

M and N tensors

This decomposition is obtained by symmeterization and antisymmeterization of the middle two indices:

where

, as they are not symmetric under interchange of the first two indices. In addition, it is not irreducible, so it is not invariant under linear transformations such as rotations.

Irreducible representations

An irreducible representation can be built by considering the notion of a totally symmetric tensor permutations of the indices

is the set of all permutations of the four indices., this sum reduces to

The difference

is an asymmetric tensor (not will either leave it unchanged or evaluate to zero. It is also irreducible with respect to arbitrary linear transformations, that is, the general linear group.

into two:

See Itin (2020).

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..

See also

• Continuum mechanics

• Solid mechanics

• Constitutive equation

• Strength of materials

• Representation theory of finite groups

• Voigt notation

Footnotes

References

Bibliography

• The Feynman Lectures on Physics - The tensor of elasticity