View source for Elasticity tensor

The '''elasticity tensor''' is a fourth-rank [[tensor]] describing the [[Hooke's law|stress-strain relation]] in
a [[Linear elasticity|linear elastic]] material.{{sfn|Thorne|Blandford|2017|p=580}}{{sfn|Itin|Hehl|2013}} Other names are '''elastic modulus tensor''' and '''stiffness tensor'''. Common symbols include <math>\mathbf{C}</math> and <math>\mathbf{Y}</math>.

The defining equation can be written as
:<math>
T^{ij} = C^{ijkl} E_{kl}
</math>
where <math>T^{ij}</math> and <math>E_{kl}</math> are the components of the [[Cauchy stress tensor]] and [[infinitesimal strain tensor]], and <math>C^{ijkl}</math> are the components of the elasticity tensor. Summation over repeated indices is implied.<ref group=note name=n1>Here, upper and lower indices denote [[Covariance and contravariance of vectors|contravariant and covariant components]], respectively, though the distinction can be ignored for [[Cartesian coordinates]]. As a result, some references represent components using only lower indices.</ref> This relationship can be interpreted as a generalization of [[Hooke's law]] to a 3D [[Continuum mechanics|continuum]].

A general fourth-rank tensor <math>\mathbf{F}</math> in 3D has 3<sup>4</sup> = 81 independent components <math>F_{ijkl}</math>, but the elasticity tensor has at most 21 independent components.{{sfn|Thorne|Blandford|2017|p=581}} 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]].{{sfn|Thorne|Blandford|2017|p=581}}

== Definition ==
The most general linear relation between two second-rank tensors <math>\mathbf{T}, \mathbf{E}</math> is
:<math>
T^{ij} = C^{ijkl} E_{kl}
</math>
where <math>C^{ijkl}</math> are the components of a fourth-rank tensor <math>\mathbf{C}</math>.{{sfn|Thorne|Blandford|2017|p=580}}<ref group=note name=n1/> The elasticity tensor is defined as <math>\mathbf{C}</math> for the case where <math>\mathbf{T}</math> and <math>\mathbf{E}</math> are the stress and strain tensors, respectively.

The '''compliance tensor''' <math>\mathbf{K}</math> is defined from the inverse stress-strain relation:
:<math>
E^{ij} = K^{ijkl} T_{kl}
</math>
The two are related by
:<math>
K_{ijpq} C^{pqkl} = \frac{1}{2} \left(\delta_{i}^{k} \delta_{j}^{l} + \delta_{i}^{l} \delta_{j}^{k} \right)
</math>
where <math>\delta_{n}^{m}</math> is the [[Kronecker delta]].{{sfn|Hill|1965}}{{sfn|Cowin|1989}}<ref group=note>Combining the forward and inverse stress-strain relations gives {{math|''E''<sub>''ij''</sub> {{=}} ''K''<sub>''ijpq''</sub> ''C''<sup>''pqkl''</sup>''E''<sub>''kl''</sub>}}.
Due to the minor symmetries {{math|''C''<sup>''pqkl''</sup> {{=}} ''C''<sup>''qpkl''</sup>}} and {{math|''C''<sup>''pqkl''</sup> {{=}} ''C''<sup>''pqlk''</sup>}}, this equation does not uniquely determine {{math|''K''<sub>''ijpq''</sub> ''C''<sup>''pqkl''</sup>}}. In fact, {{math|''K''<sub>''ijpq''</sub> ''C''<sup>''pqkl'' </sup>{{=}}'' a'' δ<sup>''k''</sup><sub>''i''</sub>δ<sup>''l''</sup><sub>''j''</sub> + (1 − ''a'') δ<sup>''l''</sup><sub>''i''</sub>δ<sup>''k''</sup><sub>''j''</sub>}} is a solution for any {{math|0 ≤ ''a'' ≤ 1}}. However, only  {{math|''a'' {{=}} 1/2}} preserves the minor symmetries of '''K''', so this is the correct solution from a physical standpoint.</ref>

Unless otherwise noted, this article assumes <math>\mathbf{C}</math> is defined from the stress-strain relation of a linear elastic material, in the limit of small strain.

== Special cases ==
=== Isotropic ===
For an isotropic material, <math>\mathbf{C}</math> simplifies to
:<math>
C^{ijkl} = \lambda \!\left( X \right) g^{ij} g^{kl} + \mu\!\left( X \right) \left(g^{ik} g^{jl} + g^{il} g^{kj} \right)
</math>
where <math>\lambda</math> and <math>\mu</math> are scalar functions of the material coordinates
<math>X</math>, and <math>\mathbf{g}</math> is the [[metric tensor]] in the reference frame of the material.{{sfn|Marsden|Hughes|1994|p=223}}{{sfn|Hehl|Itin|2002}} 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:
:<math>
C_{ijkl} = \lambda \!\left( X \right) \delta_{ij} \delta_{kl} + \mu\!\left( X \right) \left(\delta_{ik} \delta_{jl} + \delta_{il} \delta_{kj} \right) \quad \text{[Cartesian coordinates]}</math>

Substituting the first equation into the stress-strain relation and summing over repeated indices gives
:<math>
T^{ij} = \lambda \!\left( X \right) \cdot \left(\mathrm{Tr} \, \mathbf{E} \right) g^{ij} + 2 \mu\!\left( X \right) E^{ij}
</math>
where <math>\mathrm{Tr}\, \mathbf{E} \equiv E^i_{\,i}</math> is the trace of <math>\mathbf{E}</math>.
In this form, <math>\mu</math> and <math>\lambda</math> can be identified with the first and second [[Lamé parameters]].
An equivalent expression is
:<math>
T^{ij} = K \!\left( X \right) \cdot \left(\mathrm{Tr}\, \mathbf{E} \right)  g^{ij} + 2 \mu\!\left( X \right)\Sigma^{ij}
</math>
where <math>K = \lambda + (2/3) \mu</math> is the bulk modulus, and 
:<math>
\Sigma^{ij} \equiv E^{ij} - (1/3)\left(\mathrm{Tr}\, \mathbf{E} \right) g^{ij}
</math>
are the components of the shear tensor <math>\mathbf{\Sigma}</math>.

===Cubic crystals===
The elasticity tensor of a [[cubic crystal]] has components
:<math>
\begin{align}
C^{ijkl} &= \lambda g^{ij} g^{kl} + \mu \left(g^{ik} g^{jl} + g^{il} g^{kj} \right) \\ &+ \alpha \left(a^i a^j a^k a^l + b^i b^j b^k b^l + c^i c^j c^k c^l\right)
\end{align}
</math>
where <math>\mathbf{a}</math>, <math>\mathbf{b}</math>, and <math>\mathbf{c}</math> are unit vectors corresponding to the three mutually perpendicular axes of the crystal [[unit cell]].{{sfn|Thomas|1966}} The coefficients <math>\lambda</math>, <math>\mu</math>, and <math>\alpha</math> 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.{{sfn|Landau|Lifshitz|1970}}

In an orthonormal Cartesian coordinate basis, there is no distinction between upper and lower indices, and <math>g^{ij}</math> is the Kronecker delta, so the expression simplifies to
:<math>
\begin{align}
C_{ijkl} &= \lambda \delta_{ij} \delta_{kl} + \mu \left(\delta_{ik} \delta_{jl} + \delta_{il} \delta_{kj} \right) \\ &+ \alpha \left(a_i a_j a_k a_l + b_i b_j b_k b_l + c_i c_j c_k c_l\right)
\end{align}
</math>

=== Other crystal classes ===
There are similar expressions for the components of <math>\mathbf{C}</math> in other crystal symmetry classes.{{sfn|Srinivasan|Nigam|1969}} The number of independent elastic constants for several of these is given in table 1.{{sfn|Landau|Lifshitz|1970}}
{| class="wikitable"
|+ Table 1: Number of independent elastic constants for various crystal symmetry classes.{{sfn|Landau|Lifshitz|1970}}
! Crystal family
! [[Crystallographic point group|Point group]]
! Independent components
|-
| [[Triclinic]]
|
| 21
|-
| [[Monoclinic]]
|
| 13
|-
| [[Orthorhombic]]
|
| 9
|-
| [[Tetragonal]]
| '''C'''<sub>4</sub>, '''S'''<sub>4</sub>, '''C'''<sub>4h</sub>
| 7
|-
| Tetragonal
| '''C'''<sub>4v</sub>, '''D'''<sub>2d</sub>, '''D'''<sub>4</sub>, '''D'''<sub>4h</sub>
| 6 
|-
| [[Rhombohedral]]
| '''C'''<sub>3</sub>, '''S'''<sub>6</sub>
| 7
|-
| Rhombohedral
| '''C'''<sub>3v</sub>, '''D'''<sub>6</sub>, '''D'''<sub>3d</sub>
| 6
|-
| [[Hexagonal]]
| 
| 5
|-
| [[Cubic crystal system|Cubic]]
| 
| 3
|}

== Properties ==
=== Symmetries ===
The elasticity tensor has several symmetries that follow directly from its defining equation <math>T^{ij} = C^{ijkl} E_{kl}</math>.{{sfn|Thorne|Blandford|2017}}{{sfn|Itin|Hehl|2013|p=}} The symmetry of the stress and strain tensors implies that
:<math>
C_{ijkl} = C_{jikl} \qquad \text{and} \qquad C_{ijkl} = C_{ijlk},
</math>
Usually, one also assumes that the stress derives from an elastic energy potential <math>U</math>:
:<math>
T^{ij} = \frac{\partial U}{\partial E_{ij}}
</math>
which implies
:<math>
C_{ijkl} = \frac{\partial^2 U}{\partial E_{ij}\partial E_{kl}}
</math>
Hence, <math>\mathbf{C}</math> must be symmetric under interchange of the first and second pairs of indices:
:<math>
C_{ijkl} = C_{klij}
</math>
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.{{sfn|Landau|Lifshitz|1970}}

=== Transformations ===
Under rotation, the components <math>C^{ijkl}</math> transform as
:<math>
C'_{ijkl} = R_{ip} R_{jq} R_{kr} R_{ls} C^{pqrs}</math>
where <math>C'_{ijkl}</math> are the covariant components in the rotated basis, and <math>R_{ij}</math>
are the elements of the corresponding [[rotation matrix]]. A similar transformation rule holds for other linear transformations.

=== Invariants ===
The components of <math>\mathbf{C}</math> 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.
	
<math>\mathbf{C}</math> possesses two linear invariants and seven quadratic invariants with respect to SO(3).{{sfn|Norris|2007}} The linear invariants are
:<math>
\begin{align}
L_1 &= C^{ij}_{\,\,\,ij} \\
L_2 &= C^{ii}_{\,\,\,jj}
\end{align}
</math>
and the quadratic invariants are
:<math>
\left\{ L_1^2, \, L_2^2, \, L_1 L_2, \, C_{ijkl} C^{ijkl}, \,C_{iikl} C^{jjkl}, \,C_{iikl} C^{jkjl}, \,C_{kiil} C^{kjjl} \right\}
</math>
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.{{sfn|Norris|2007}}

== Decompositions ==
{{See also | Tensor decomposition}}
A common strategy in tensor analysis is to decompose a tensor into simpler components that can be analyzed separately. For example, the
displacement gradient tensor <math>\mathbf{W} = \mathbf{\nabla} \mathbf{\xi}</math> can be decomposed as
:<math>
\mathbf{W} = \frac{1}{3} \Theta \mathbf{g} + \mathbf{\Sigma} + \mathbf{R}
</math>
where <math>\Theta</math> is a rank-0 tensor (a scalar), equal to the trace of <math>\mathbf{W}</math>; 
<math>\mathbf{\Sigma}</math> is symmetric and trace-free; and <math>\mathbf{R}</math> is antisymmetric.{{sfn|Thorne|Blandford|2017|p=571}} Component-wise,
:<math>
\begin{align}
\Sigma^{ij} \equiv W^{(ij)} &= \frac{1}{2} \left(W^{ij} + W^{ji} \right) - \frac{1}{3} \left(\mathrm{Tr}\, \mathbf{W} \right) g^{ij} \\
R^{ij} \equiv W^{[ij]} &= \frac{1}{2} \left(W^{ij} - W^{ji} \right)
\end{align}
</math>
Here and later, symmeterization and antisymmeterization are denoted by <math>(ij)</math> and <math>[ij]</math>, 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.{{sfn|Thorne|Blandford|2017}}

The elasticity tensor has rank 4, and its decompositions are more complex and varied than those of a rank-2 tensor.{{sfn|Moakher|Norris|2006|pp=221–222}} A few examples are described below.

=== M and N tensors ===
This decomposition is obtained by symmeterization and antisymmeterization of the middle two indices:
:<math>
C^{ijkl} = M^{ijkl} + N^{ijkl}
</math>
where
:<math>
\begin{align}
M^{ijkl} \equiv C^{i(jk)l} = \frac{1}{2}\left(C^{ijkl} + C^{ikjl} \right) \\
N^{ijkl} \equiv C^{i[jk]l} = \frac{1}{2}\left(C^{ijkl} - C^{ikjl} \right)
\end{align}
</math>
A disadvantage of this decomposition is that <math>M^{ijkl}</math> and <math>N^{ijkl}</math> do not 
obey all original symmetries of <math>C^{ijkl}</math>, 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.{{sfn|Itin|Hehl|2013|p=}}

=== Irreducible representations ===
{{See also|Young tableau|Irreducible representation}}
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. A totally symmetric tensor <math>\mathbf{S}</math> can be constructed from
<math>\mathbf{C}</math> by summing over all <math>4! = 24</math> [[permutations]] of the indices
:<math>
\begin{align}
S^{ijkl} &= \frac{1}{4!}\sum_{(i,j,k,l)\in S_4} C^{ijkl} \\ &=\frac{1}{4!}\left(C^{ijkl} + C^{jikl}+ C^{ikjl} + \ldots \right)
\end{align}
</math>
where <math>\mathbb{S}_4</math> is the set of all permutations of the four indices.{{sfn|Itin|Hehl|2013|p=}} Owing to the symmetries of <math>C^{ijkl}</math>, this sum reduces to
:<math>
S^{ijkl} = \frac{1}{3}\left(C^{ijkl} + C^{iklj} + C^{iljk} \right)
</math>
The difference
:<math>
 A^{ijkl} \equiv C^{ijkl} - S^{ijkl} = \frac{1}{3}\left(2 C^{ijkl} - C^{ilkj} - C^{iklj} \right)
 </math>
is an asymmetric tensor (''not'' antisymmetric). The decomposition <math>C^{ijkl} = S^{ijkl} + A^{ijkl}</math> can be shown to be unique and irreducible with respect to <math>\mathbb{S}_4</math>. In other words, any additional symmetrization operations on <math>\mathbf{S}</math> or <math>\mathbf{A}</math> 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]] <math>G(3,\mathbb{R})</math>.{{sfn|Itin|Hehl|2013|p=}}{{sfn|Itin|2020}}

However, this decomposition is not irreducible with respect to the group of rotations SO(3). Instead, <math>\mathbf{S}</math> decomposes into three irreducible parts, and <math>\mathbf{A}</math> into two:
:<math>
\begin{align}
C^{ijkl} &= S^{ijkl} + A^{ijkl} \\
&= \left(^{(1)}\!S^{ijkl} + \, ^{(2)}\!S^{ijkl} + \, ^{(3)}\!S^{ijkl} \right) + \, \left(^{(1)}\!A^{ijkl} + ^{(2)}\!A^{ijkl} \right)
\end{align}
</math>
See Itin (2020){{sfn|Itin|2020}} for explicit expressions in terms of the components of <math>\mathbf{C}</math>.

This representation decomposes the space of elasticity tensors into a direct sum of subspaces:
:<math>
\mathcal{C} = \left(^{(1)}\!\mathcal{C} \oplus \, ^{(2)}\!\mathcal{C} \oplus \, ^{(3)}\!\mathcal{C} \right) \oplus \, \left(^{(4)}\!\mathcal{C} \oplus \, ^{(5)}\!\mathcal{C} \right)
</math>
with dimensions
:<math>
21 = (1 \oplus 5 \oplus 9) \oplus (1 \oplus 5)
</math>
These subspaces are each isomorphic to a [[harmonic tensor]] space <math>\mathbb{H}_n(\mathbb{R}^3)</math>.{{sfn|Itin|2020}}{{sfn | Olive | Kolev | Auffray | 2017}} Here, <math>\mathbb{H}_n(\mathbb{R}^3)</math> is the space of 3D, totally symmetric, traceless tensors of rank <math>n</math>. In particular, <math>^{(1)}\!\mathcal{C}</math> and <math>^{(4)}\!\mathcal{C}</math> correspond to <math>\mathbb{H}_1</math>, <math>^{(2)}\!\mathcal{C}</math> and <math>^{(5)}\!\mathcal{C}</math> correspond to <math>\mathbb{H}_2</math>, and <math>^{(3)}\!\mathcal{C}</math> corresponds to <math>\mathbb{H}_4</math>.

== See also ==
* [[Continuum mechanics]]
* [[Solid mechanics]]
* [[Constitutive equation]]
* [[Strength of materials]]
*{{annotated link|List of materials properties#Mechanical properties|List of materials properties § Mechanical properties}}
* [[Representation theory of finite groups]]
* [[Voigt notation]]

==Footnotes==
{{reflist|group=note}}

==References==
{{reflist}}

==Bibliography==
{{refbegin|2|indent=yes}}
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{{refend}}

[[Category:Tensor physical quantities]]
[[Category:Continuum mechanics]]