Editing Infinitesimal strain theory (section)

=== Strain transformation rules ===
If we choose an [[orthonormal basis|orthonormal coordinate system]] (<math>\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3</math>) we can write the tensor in terms of components with respect to those base vectors as
<math display="block"> \boldsymbol{\varepsilon} = \sum_{i=1}^3 \sum_{j=1}^3 \varepsilon_{ij} \mathbf{e}_i\otimes\mathbf{e}_j </math>
In matrix form,
<math display="block">\underline{\underline{\boldsymbol{\varepsilon}}} = \begin{bmatrix}
  \varepsilon_{11} & \varepsilon_{12} & \varepsilon_{13} \\
  \varepsilon_{12} & \varepsilon_{22} & \varepsilon_{23} \\
  \varepsilon_{13} & \varepsilon_{23} & \varepsilon_{33} 
\end{bmatrix} </math>
We can easily choose to use another orthonormal coordinate system (<math>\hat{\mathbf{e}}_1,\hat{\mathbf{e}}_2,\hat{\mathbf{e}}_3</math>) instead. In that case the components of the tensor are different, say
<math display="block">
   \boldsymbol{\varepsilon} = \sum_{i=1}^3 \sum_{j=1}^3 \hat{\varepsilon}_{ij} \hat{\mathbf{e}}_i\otimes\hat{\mathbf{e}}_j \quad \implies \quad \underline{\underline{\hat{\boldsymbol{\varepsilon}}}} = \begin{bmatrix}
     \hat{\varepsilon}_{11} & \hat{\varepsilon}_{12} & \hat{\varepsilon}_{13} \\
     \hat{\varepsilon}_{12} & \hat{\varepsilon}_{22} & \hat{\varepsilon}_{23} \\
     \hat{\varepsilon}_{13} & \hat{\varepsilon}_{23} & \hat{\varepsilon}_{33}
\end{bmatrix}
</math>
The components of the strain in the two coordinate systems are related by
<math display="block"> \hat{\varepsilon}_{ij} = \ell_{ip}~\ell_{jq}~\varepsilon_{pq} </math>
where the [[Einstein summation convention]] for repeated indices has been used and <math>\ell_{ij} = \hat{\mathbf{e}}_i\cdot{\mathbf{e}}_j</math>. In matrix form
<math display="block">
   \underline{\underline{\hat{\boldsymbol{\varepsilon}}}} = \underline{\underline{\mathbf{L}}} ~\underline{\underline{\boldsymbol{\varepsilon}}}~ \underline{\underline{\mathbf{L}}}^T
 </math>
or
<math display="block">
\begin{bmatrix}
  \hat{\varepsilon}_{11} & \hat{\varepsilon}_{12} & \hat{\varepsilon}_{13} \\
  \hat{\varepsilon}_{21} & \hat{\varepsilon}_{22} & \hat{\varepsilon}_{23} \\
  \hat{\varepsilon}_{31} & \hat{\varepsilon}_{32} & \hat{\varepsilon}_{33}
\end{bmatrix}
= \begin{bmatrix}
  \ell_{11} & \ell_{12} & \ell_{13} \\
  \ell_{21} & \ell_{22} & \ell_{23} \\
  \ell_{31} & \ell_{32} & \ell_{33}
\end{bmatrix}
\begin{bmatrix}
  \varepsilon_{11} & \varepsilon_{12} & \varepsilon_{13} \\
  \varepsilon_{21} & \varepsilon_{22} & \varepsilon_{23} \\
  \varepsilon_{31} & \varepsilon_{32} & \varepsilon_{33}
\end{bmatrix}
\begin{bmatrix}
\ell_{11} & \ell_{12} & \ell_{13} \\
\ell_{21} & \ell_{22} & \ell_{23} \\
\ell_{31} & \ell_{32} & \ell_{33}
\end{bmatrix}^T </math>