Editing Linear elasticity (section)

== Anisotropic homogeneous media ==

{{Main|Hooke's law}}

For anisotropic media, the stiffness tensor <math> C_{ijkl}</math> is more complicated.  The symmetry of the stress tensor <math>\sigma_{ij}</math> means that there are at most 6 different elements of stress. Similarly, there are at most 6 different elements of the strain tensor <math>\varepsilon_{ij}\,\!</math>.  Hence the fourth-order stiffness tensor <math> C_{ijkl}</math> may be written as a matrix <math>C_{\alpha \beta}</math> (a tensor of second order). [[Voigt notation]] is the standard mapping for tensor indices,
<math display="block"> 
\begin{matrix}
ij & =\\
\Downarrow & \\
\alpha  & =
\end{matrix}

 \begin{matrix}
11 & 22 & 33 & 23,32 & 13,31 & 12,21 \\
\Downarrow & \Downarrow & \Downarrow & \Downarrow & \Downarrow & \Downarrow & \\
1  &2 &  3 &  4 &  5 & 6
\end{matrix}</math>

With this notation, one can write the elasticity matrix for any linearly elastic medium as:
<math display="block"> C_{ijkl}  \Rightarrow C_{\alpha \beta} = \begin{bmatrix}
 C_{11} & C_{12} & C_{13} & C_{14} & C_{15} & C_{16} \\
 C_{12} & C_{22} & C_{23} & C_{24} & C_{25} & C_{26} \\
 C_{13} & C_{23} & C_{33} & C_{34} & C_{35} & C_{36} \\
 C_{14} & C_{24} & C_{34} & C_{44} & C_{45} & C_{46} \\
 C_{15} & C_{25} & C_{35} & C_{45} & C_{55} & C_{56} \\
 C_{16} & C_{26} & C_{36} & C_{46} & C_{56} & C_{66} 
\end{bmatrix}.</math>

As shown, the matrix <math> C_{\alpha \beta}</math> is symmetric, this is a result of the existence of a strain energy density function which satisfies <math>\sigma_{ij} = \frac{\partial W}{\partial\varepsilon_{ij}}</math>.  Hence, there are at most 21 different elements of <math> C_{\alpha \beta}\,\!</math>.

The isotropic special case has 2 independent elements:
<math display="block"> C_{\alpha \beta} = \begin{bmatrix}
 K+4 \mu\ /3  & K-2 \mu\ /3 & K-2 \mu\ /3 & 0 & 0  & 0 \\
 K-2 \mu\ /3  & K+4 \mu\ /3 &  K-2 \mu\ /3 & 0 & 0  & 0 \\
 K-2 \mu\ /3  & K-2 \mu\ /3 & K+4 \mu\ /3 & 0 & 0  & 0 \\
 0  & 0 & 0 & \mu\ & 0  & 0 \\
 0  & 0 & 0 & 0 & \mu\  & 0 \\
 0  & 0 & 0 & 0 & 0  & \mu\ 
\end{bmatrix}.</math>

The simplest anisotropic case, that of cubic symmetry has 3 independent elements:
<math display="block"> C_{\alpha \beta} = \begin{bmatrix}
 C_{11}  &  C_{12} &  C_{12} & 0 & 0  & 0 \\
 C_{12}  &  C_{11} &  C_{12} & 0 & 0  & 0 \\
 C_{12}  & C_{12}  &  C_{11} & 0 & 0  & 0 \\
 0  & 0 & 0 & C_{44} & 0  & 0 \\
 0  & 0 & 0 & 0 & C_{44}  & 0 \\
 0  & 0 & 0 & 0 & 0  & C_{44} 
\end{bmatrix}.</math>

The case of [[transverse isotropy]], also called polar anisotropy, (with a single axis (the 3-axis) of symmetry) has 5 independent elements:
<math display="block"> C_{\alpha \beta} = \begin{bmatrix}
  C_{11}  &  C_{11}-2C_{66} &  C_{13} & 0 & 0  & 0 \\
 C_{11}-2C_{66}  &  C_{11} &  C_{13} & 0 & 0  & 0 \\
  C_{13}  & C_{13}  &  C_{33} & 0 & 0  & 0 \\
 0  & 0 & 0 & C_{44} & 0  & 0 \\
 0  & 0 & 0 & 0 & C_{44}  & 0 \\
 0  & 0 & 0 & 0 & 0  & C_{66} 
\end{bmatrix}.</math>

When the transverse isotropy is weak (i.e. close to isotropy), an alternative parametrization utilizing [[Thomsen parameters]], is convenient for the formulas for wave speeds.

The case of orthotropy (the symmetry of a brick) has 9 independent elements:
<math display="block"> C_{\alpha \beta} = \begin{bmatrix}
 C_{11} & C_{12} & C_{13} & 0 & 0 & 0 \\
 C_{12} & C_{22} & C_{23} & 0 & 0 & 0 \\
 C_{13} & C_{23} & C_{33} & 0 & 0 & 0 \\
 0  & 0 & 0 & C_{44} & 0 & 0 \\
 0  & 0 & 0 & 0 & C_{55} & 0 \\
 0  & 0 & 0 & 0 & 0 & C_{66} 
\end{bmatrix}.</math>

=== Elastodynamics ===
The elastodynamic wave equation for anisotropic media can be expressed as
<math display="block"> (\delta_{kl} \partial_{tt} - A_{kl}[\nabla])\, u_l = \frac{1}{\rho} F_k</math>
where
<math display="block"> A_{kl}[\nabla]=\frac{1}{\rho} \, \partial_i \, C_{iklj} \, \partial_j</math>
is the ''acoustic differential operator'', and <math> \delta_{kl}</math> is [[Kronecker delta]].

==== Plane waves and Christoffel equation ====
A ''plane wave'' has the form
<math display="block"> \mathbf{u}[\mathbf{x}, \, t] = U[\mathbf{k} \cdot  \mathbf{x} - \omega \, t] \, \hat{\mathbf{u}}</math>
with <math>\hat{\mathbf{u}}\,\!</math> of unit length.
It is a solution of the wave equation with zero forcing, if and only if <math> \omega^2 </math> and <math>\hat{\mathbf{u}}</math> constitute an eigenvalue/eigenvector pair of the ''acoustic algebraic operator''
<math display="block"> A_{kl}[\mathbf{k}]=\frac{1}{\rho} \, k_i \, C_{iklj} \, k_j.</math>
This ''propagation condition'' (also known as the '''Christoffel equation''') may be written as
<math display="block">A[\hat{\mathbf{k}}] \, \hat{\mathbf{u}} = c^2 \, \hat{\mathbf{u}}</math>
where
<math>\hat{\mathbf{k}} = \mathbf{k} / \sqrt{\mathbf{k}\cdot\mathbf{k}}</math>
denotes propagation direction and <math>c = \omega / \sqrt{\mathbf{k} \cdot \mathbf{k}}</math> is phase velocity.