Editing Infinitesimal strain theory (section)

==Infinitesimal strain tensor==

For infinitesimal deformations of a [[Continuum mechanics|continuum body]], in which the [[displacement gradient tensor]] (2nd order tensor) is small compared to unity, i.e. <math>\|\nabla \mathbf u\| \ll 1 </math>, 
it is possible to perform a geometric linearization of any one of the finite strain tensors used in finite strain theory, e.g. the ''[[Lagrangian finite strain tensor]]'' <math>\mathbf E</math>, and the ''[[Eulerian finite strain tensor]]'' <math>\mathbf e</math>. In such a linearization, the non-linear or second-order terms of the finite strain tensor are neglected. Thus we have

<math display="block">\mathbf E = \frac{1}{2} \left(\nabla_{\mathbf X}\mathbf u + (\nabla_{\mathbf X}\mathbf u)^T + (\nabla_{\mathbf X}\mathbf u)^T\nabla_{\mathbf X}\mathbf u\right)\approx \frac{1}{2}\left(\nabla_{\mathbf X}\mathbf u + (\nabla_{\mathbf X}\mathbf u)^T\right)</math>
or
<math display="block">E_{KL}= \frac{1}{2} \left(\frac{\partial U_K}{\partial X_L} +\frac{\partial U_L}{\partial X_K}+ \frac{\partial U_M}{\partial X_K} \frac{\partial U_M}{\partial X_L}\right)\approx \frac{1}{2}\left(\frac{\partial U_K}{\partial X_L}+\frac{\partial U_L}{\partial X_K}\right)</math>
and
<math display="block">\mathbf e =\frac{1}{2} \left(\nabla_{\mathbf x}\mathbf u + (\nabla_{\mathbf x}\mathbf u)^T - \nabla_{\mathbf x}\mathbf u(\nabla_{\mathbf x}\mathbf u)^T\right)\approx \frac{1}{2}\left(\nabla_{\mathbf x}\mathbf u + (\nabla_{\mathbf x}\mathbf u)^T\right)</math>
or
<math display="block">e_{rs}=\frac{1}{2} \left(\frac{\partial u_r}{\partial x_s} +\frac{\partial u_s}{\partial x_r} -\frac{\partial u_k}{\partial x_r} \frac{\partial u_k}{\partial x_s}\right)\approx \frac{1}{2}\left(\frac{\partial u_r}{\partial x_s} +\frac{\partial u_s}{\partial x_r}\right)</math>

This linearization implies that the Lagrangian description and the Eulerian description are approximately the same as there is little difference in the material and spatial coordinates of a given material point in the continuum. Therefore, the [[material displacement gradient tensor]] components and the [[spatial displacement gradient tensor]] components are approximately equal. Thus we have
<math display="block">\mathbf E \approx \mathbf e \approx \boldsymbol \varepsilon = \frac{1}{2}\left((\nabla\mathbf u)^T + \nabla\mathbf u\right) </math>
or
<math display="block"> E_{KL}\approx e_{rs}\approx\varepsilon_{ij} = \frac{1}{2} \left(u_{i,j}+u_{j,i}\right)</math>
where <math>\varepsilon_{ij}</math> are the components of the ''infinitesimal strain tensor'' <math>\boldsymbol \varepsilon</math>, also called ''Cauchy's strain tensor'', ''linear strain tensor'', or ''small strain tensor''.

<math display="block">\begin{align}
\varepsilon_{ij} &= \frac{1}{2}\left(u_{i,j}+u_{j,i}\right)  \\
&=
\begin{bmatrix}
\varepsilon_{11} & \varepsilon_{12} & \varepsilon_{13} \\
\varepsilon_{21} & \varepsilon_{22} & \varepsilon_{23} \\
\varepsilon_{31} & \varepsilon_{32} & \varepsilon_{33} \\
\end{bmatrix} \\
&=
\begin{bmatrix}
  \frac{\partial u_1}{\partial x_1} & \frac{1}{2} \left(\frac{\partial u_1}{\partial x_2}+\frac{\partial u_2}{\partial x_1}\right) & \frac{1}{2} \left(\frac{\partial u_1}{\partial x_3}+\frac{\partial u_3}{\partial x_1}\right) \\
   \frac{1}{2} \left(\frac{\partial u_2}{\partial x_1}+\frac{\partial u_1}{\partial x_2}\right) & \frac{\partial u_2}{\partial x_2} & \frac{1}{2} \left(\frac{\partial u_2}{\partial x_3}+\frac{\partial u_3}{\partial x_2}\right) \\
   \frac{1}{2} \left(\frac{\partial u_3}{\partial x_1}+\frac{\partial u_1}{\partial x_3}\right) & \frac{1}{2} \left(\frac{\partial u_3}{\partial x_2}+\frac{\partial u_2}{\partial x_3}\right) & \frac{\partial u_3}{\partial x_3} \\
  \end{bmatrix}
\end{align} </math>
or using different notation:
<math display="block">\begin{bmatrix}
\varepsilon_{xx} & \varepsilon_{xy} & \varepsilon_{xz} \\
   \varepsilon_{yx} & \varepsilon_{yy} & \varepsilon_{yz} \\
   \varepsilon_{zx} & \varepsilon_{zy} & \varepsilon_{zz} \\
  \end{bmatrix}
=
\begin{bmatrix}
  \frac{\partial u_x}{\partial x} & \frac{1}{2} \left(\frac{\partial u_x}{\partial y}+\frac{\partial u_y}{\partial x}\right) & \frac{1}{2} \left(\frac{\partial u_x}{\partial z}+\frac{\partial u_z}{\partial x}\right) \\
   \frac{1}{2} \left(\frac{\partial u_y}{\partial x}+\frac{\partial u_x}{\partial y}\right) & \frac{\partial u_y}{\partial y} & \frac{1}{2} \left(\frac{\partial u_y}{\partial z}+\frac{\partial u_z}{\partial y}\right) \\
   \frac{1}{2} \left(\frac{\partial u_z}{\partial x}+\frac{\partial u_x}{\partial z}\right) & \frac{1}{2} \left(\frac{\partial u_z}{\partial y}+\frac{\partial u_y}{\partial z}\right) & \frac{\partial u_z}{\partial z} \\
  \end{bmatrix} </math>

Furthermore, since the [[deformation gradient]] can be expressed as <math>\boldsymbol{F} = \boldsymbol{\nabla}\mathbf{u} + \boldsymbol{I}</math> where <math>\boldsymbol{I}</math> is the second-order identity tensor, we have
<math display="block">\boldsymbol\varepsilon = \frac{1}{2} \left(\boldsymbol{F}^T+\boldsymbol{F}\right)-\boldsymbol{I}</math>

Also, from the [[Finite strain theory#Seth-Hill family of generalized strain tensors|general expression]] for the Lagrangian and Eulerian finite strain tensors we have
<math display="block"> \begin{align}
\mathbf E_{(m)}& =\frac{1}{2m} (\mathbf U^{2m}-\boldsymbol{I}) = \frac{1}{2m} [(\boldsymbol{F}^T\boldsymbol{F})^m - \boldsymbol{I}] \approx \frac{1}{2m} [\{\boldsymbol{\nabla}\mathbf{u}+(\boldsymbol{\nabla}\mathbf{u})^T + \boldsymbol{I}\}^m - \boldsymbol{I}]\approx \boldsymbol{\varepsilon}\\
\mathbf e_{(m)}& = \frac{1}{2m} (\mathbf V^{2m}-\boldsymbol{I})= \frac{1}{2m} [(\boldsymbol{F}\boldsymbol{F}^T)^m - \boldsymbol{I}]\approx \boldsymbol{\varepsilon}
\end{align} </math>

===Geometric derivation===
[[Image:2D geometric strain.svg|class=skin-invert-image|400px|right|thumb|Figure 1. Two-dimensional geometric deformation of an infinitesimal material element.]]
Consider a two-dimensional deformation of an infinitesimal rectangular material element with dimensions <math>dx</math> by <math>dy</math> (Figure 1), which after deformation, takes the form of a rhombus. From the geometry of Figure 1 we have

<math display="block">\begin{align}
\overline {ab} &= \sqrt{\left(dx+\frac{\partial u_x}{\partial x}dx \right)^2 + \left( \frac{\partial u_y}{\partial x}dx \right)^2} \\
&= dx\sqrt{1+2\frac{\partial u_x}{\partial x}+\left(\frac{\partial u_x}{\partial x}\right)^2 + \left(\frac{\partial u_y}{\partial x}\right)^2} \\
\end{align}</math>

For very small displacement gradients, i.e., <math>\|\nabla \mathbf u\| \ll 1 </math>, we have
<math display="block">\overline {ab} \approx dx + \frac{\partial u_x}{\partial x} dx</math>

The [[Deformation (mechanics)#Strain measures|normal strain]] in the <math>x</math>-direction of the rectangular element is defined by
<math display="block">\varepsilon_x = \frac{\overline {ab}-\overline {AB}}{\overline {AB}}</math>
and knowing that <math>\overline {AB}= dx</math>, we have
<math display="block">\varepsilon_x = \frac{\partial u_x}{\partial x}</math>

Similarly, the normal strain in the {{nowrap|<math>y</math>-direction,}} and {{nowrap|<math>z</math>-direction,}} becomes
<math display="block">\varepsilon_y = \frac{\partial u_y}{\partial y} \quad , \qquad  \varepsilon_z = \frac{\partial u_z}{\partial z}</math>

The [[Deformation (mechanics)#Strain measures|engineering shear strain]], or the change in angle between two originally orthogonal material lines, in this case line <math>\overline {AC}</math> and <math>\overline {AB}</math>, is defined as
<math display="block">\gamma_{xy}= \alpha + \beta</math>

From the geometry of Figure 1 we have
<math display="block">\tan \alpha = \frac{\dfrac{\partial u_y}{\partial x}dx}{dx + \dfrac{\partial u_x}{\partial x} dx} = \frac{\dfrac{\partial u_y}{\partial x}}{1+\dfrac{\partial u_x}{\partial x}} \quad , \qquad \tan \beta=\frac{\dfrac{\partial u_x}{\partial y} dy}{dy+\dfrac{\partial u_y}{\partial y} dy}=\frac{\dfrac{\partial u_x}{\partial y}}{1+\dfrac{\partial u_y}{\partial y}}</math>

For small rotations, i.e., <math>\alpha</math> and <math>\beta</math> are <math>\ll 1</math> we have
<math display="block">\tan \alpha \approx \alpha \quad , \qquad \tan \beta \approx \beta</math>
and, again, for small displacement gradients, we have
<math display="block">\alpha=\frac{\partial u_y}{\partial x} \quad , \qquad \beta=\frac{\partial u_x}{\partial y}</math>
thus
<math display="block">\gamma_{xy}= \alpha + \beta = \frac{\partial u_y}{\partial x} + \frac{\partial u_x}{\partial y}</math>
By interchanging <math>x</math> and <math>y</math> and <math>u_x</math> and <math>u_y</math>, it can be shown that <math>\gamma_{xy} = \gamma_{yx}</math>.

Similarly, for the <math>y</math>-<math>z</math> and <math>x</math>-<math>z</math> planes, we have
<math display="block">\gamma_{yz} = \gamma_{zy} = \frac{\partial u_y}{\partial z} + \frac{\partial u_z}{\partial y} \quad , \qquad \gamma_{zx} = \gamma_{xz} = \frac{\partial u_z}{\partial x} + \frac{\partial u_x}{\partial z}</math>

It can be seen that the tensorial shear strain components of the infinitesimal strain tensor can then be expressed using the engineering strain definition, {{nowrap|<math>\gamma</math>,}} as
<math display="block"> \begin{bmatrix}
\varepsilon_{xx} & \varepsilon_{xy} & \varepsilon_{xz} \\
\varepsilon_{yx} & \varepsilon_{yy} & \varepsilon_{yz} \\
\varepsilon_{zx} & \varepsilon_{zy} & \varepsilon_{zz} \\
\end{bmatrix} = \begin{bmatrix}
\varepsilon_{xx} & \gamma_{xy}/2 & \gamma_{xz}/2 \\
\gamma_{yx}/2 & \varepsilon_{yy} & \gamma_{yz}/2 \\
\gamma_{zx}/2 & \gamma_{zy}/2 & \varepsilon_{zz} \\
\end{bmatrix}</math>

===Physical interpretation===
From [[Finite deformation tensor|finite strain theory]] we have
<math display="block">d\mathbf{x}^2 - d\mathbf{X}^2 = d\mathbf X \cdot 2\mathbf E \cdot d\mathbf X \quad\text{or}\quad (dx)^2 - (dX)^2 = 2E_{KL}\,dX_K\,dX_L</math>

For infinitesimal strains then we have
<math display="block">d\mathbf{x}^2 - d\mathbf{X}^2 = d\mathbf X \cdot 2\mathbf{\boldsymbol \varepsilon} \cdot d\mathbf X \quad\text{or}\quad (dx)^2 - (dX)^2 = 2\varepsilon_{KL}\,dX_K\,dX_L</math>

Dividing by <math>(dX)^2</math> we have
<math display="block">\frac{dx-dX}{dX}\frac{dx+dX}{dX}=2\varepsilon_{ij}\frac{dX_i}{dX}\frac{dX_j}{dX}</math>

For small deformations we assume that <math>dx \approx dX</math>, thus the second term of the left hand side becomes: <math>\frac{dx+dX}{dX} \approx 2</math>.

Then we have
<math display="block">\frac{dx-dX}{dX} = \varepsilon_{ij}N_iN_j = \mathbf N \cdot \boldsymbol \varepsilon \cdot \mathbf N</math>
where <math>N_i=\frac{dX_i}{dX}</math>, is the unit vector in the direction of <math>d\mathbf X</math>, and the left-hand-side expression is the [[Deformation (mechanics)#Strain measures|normal strain]] <math>e_{(\mathbf N)}</math> in the direction of <math>\mathbf N</math>. For the particular case of <math>\mathbf N</math> in the <math>X_1</math> direction, i.e., <math>\mathbf N = \mathbf I_1</math>, we have
<math display="block">e_{(\mathbf I_1)}=\mathbf I_1 \cdot \boldsymbol \varepsilon \cdot \mathbf I_1 = \varepsilon_{11}.</math>

Similarly, for <math>\mathbf N=\mathbf I_2</math> and <math>\mathbf N=\mathbf I_3</math> we can find the normal strains <math>\varepsilon_{22}</math> and <math>\varepsilon_{33}</math>, respectively. Therefore, the diagonal elements of the infinitesimal strain tensor are the normal strains in the coordinate directions.

=== 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>

=== Strain invariants ===
Certain operations on the strain tensor give the same result without regard to which orthonormal coordinate system is used to represent the components of strain. The results of these operations are called '''strain invariants'''. The most commonly used strain invariants are
<math display="block"> \begin{align}
   I_1 & = \mathrm{tr}(\boldsymbol{\varepsilon}) \\
   I_2 & = \tfrac{1}{2}\{[\mathrm{tr}(\boldsymbol{\varepsilon})]^2 - \mathrm{tr}(\boldsymbol{\varepsilon}^2)\} \\
   I_3 & = \det(\boldsymbol{\varepsilon})
\end{align} </math>
In terms of components
<math display="block"> \begin{align}
   I_1 & = \varepsilon_{11} + \varepsilon_{22} + \varepsilon_{33} \\
   I_2 & = \varepsilon_{11}\varepsilon_{22} + \varepsilon_{22}\varepsilon_{33} + \varepsilon_{33}\varepsilon_{11} - \varepsilon_{12}^2 - \varepsilon_{23}^2 - \varepsilon_{31}^2 \\
   I_3 & = \varepsilon_{11}(\varepsilon_{22}\varepsilon_{33} - \varepsilon_{23}^2) - \varepsilon_{12}(\varepsilon_{21}\varepsilon_{33}-\varepsilon_{23}\varepsilon_{31}) + \varepsilon_{13}(\varepsilon_{21}\varepsilon_{32}-\varepsilon_{22}\varepsilon_{31})
\end{align} </math>

===Principal strains===
It can be shown that it is possible to find a coordinate system (<math>\mathbf{n}_1,\mathbf{n}_2,\mathbf{n}_3</math>) in which the components of the strain tensor are
<math display="block">
   \underline{\underline{\boldsymbol{\varepsilon}}} = \begin{bmatrix} \varepsilon_{1} & 0 & 0 \\
     0 & \varepsilon_{2} & 0 \\
     0 & 0 & \varepsilon_{3} \end{bmatrix} \quad \implies \quad \boldsymbol{\varepsilon} = \varepsilon_{1} \mathbf{n}_1\otimes\mathbf{n}_1 + \varepsilon_{2} \mathbf{n}_2\otimes\mathbf{n}_2 + \varepsilon_{3} \mathbf{n}_3\otimes\mathbf{n}_3 
 </math>
The components of the strain tensor in the (<math>\mathbf{n}_1,\mathbf{n}_2,\mathbf{n}_3</math>) coordinate system are called the '''principal strains''' and the directions <math>\mathbf{n}_i</math> are called the directions of principal strain. Since there are no shear strain components in this coordinate system, the principal strains represent the maximum and minimum stretches of an elemental volume.

If we are given the components of the strain tensor in an arbitrary orthonormal coordinate system, we can find the principal strains using an [[eigenvalue decomposition]] determined by solving the system of equations
<math display="block"> (\underline{\underline{\boldsymbol{\varepsilon}}} - \varepsilon_i~\underline{\underline{\mathbf{I}}})~\mathbf{n}_i = \underline{\mathbf{0}} </math>
This system of equations is equivalent to finding the vector <math>\mathbf{n}_i</math> along which the strain tensor becomes a pure stretch with no shear component.

===Volumetric strain===
<!-- anchor: [[Volumetric strain]] redirects here -->
The '''volumetric strain''', also called '''bulk strain''', is the relative variation of the volume, as arising from ''[[dilation (physics)|dilation]]'' or ''compression''; it is the [[#Strain invariants|first strain invariant]] or [[trace (matrix)|trace]] of the tensor:
<math display="block">\delta=\frac{\Delta V}{V_0} = I_1 = \varepsilon_{11} + \varepsilon_{22} + \varepsilon_{33}</math>
Actually, if we consider a cube with an edge length ''a'', it is a quasi-cube after the deformation (the variations of the angles do not change the volume) with the dimensions <math>a \cdot (1 + \varepsilon_{11}) \times a \cdot (1 + \varepsilon_{22}) \times a \cdot (1 + \varepsilon_{33})</math> and ''V''<sub>0</sub> = ''a''<sup>3</sup>, thus
<math display="block">\frac{\Delta V}{V_0} = \frac{\left ( 1 + \varepsilon_{11} + \varepsilon_{22} + \varepsilon_{33} + \varepsilon_{11} \cdot \varepsilon_{22} + \varepsilon_{11} \cdot \varepsilon_{33}+ \varepsilon_{22} \cdot \varepsilon_{33} + \varepsilon_{11} \cdot \varepsilon_{22} \cdot \varepsilon_{33} \right ) \cdot a^3 - a^3}{a^3}</math>
as we consider small deformations,
<math display="block">1 \gg \varepsilon_{ii} \gg \varepsilon_{ii} \cdot \varepsilon_{jj} \gg \varepsilon_{11} \cdot \varepsilon_{22} \cdot \varepsilon_{33} </math>
therefore the formula.

[[Image:Approximation volume deformation.png|class=skin-invert-image|400px|<small>Real variation of volume  (top) and the approximated one (bottom): the green drawing shows the estimated volume and the orange drawing the neglected volume</small>]]

In case of pure shear, we can see that there is no change of the volume.

===Strain deviator tensor===
The infinitesimal strain tensor <math>\varepsilon_{ij}</math>, similarly to the [[Cauchy stress tensor]], can be expressed as the sum of two other tensors: 
# a '''mean strain tensor''' or '''volumetric strain tensor''' or '''spherical strain tensor''', <math>\varepsilon_M\delta_{ij}</math>, related to dilation or volume change; and
# a deviatoric component called the '''strain deviator tensor''', <math>\varepsilon'_{ij}</math>, related to distortion.

<math display="block">\varepsilon_{ij}= \varepsilon'_{ij} + \varepsilon_M\delta_{ij}</math>
where <math>\varepsilon_M</math> is the mean strain given by
<math display="block">\varepsilon_M = \frac{\varepsilon_{kk}}{3} = \frac{\varepsilon_{11} + \varepsilon_{22} + \varepsilon_{33}}{3} = \tfrac{1}{3}I^e_1</math>

The deviatoric strain tensor can be obtained by subtracting the mean strain tensor from the infinitesimal strain tensor:
<math display="block">\begin{align}
\ \varepsilon'_{ij} &= \varepsilon_{ij} - \frac{\varepsilon_{kk}}{3}\delta_{ij} \\
\begin{bmatrix}
   \varepsilon'_{11} & \varepsilon'_{12} & \varepsilon'_{13} \\
   \varepsilon'_{21} & \varepsilon'_{22} & \varepsilon'_{23} \\
   \varepsilon'_{31} & \varepsilon'_{32} & \varepsilon'_{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}
   \varepsilon_M & 0 & 0 \\
   0 & \varepsilon_M & 0 \\
   0 & 0 & \varepsilon_M \\
  \end{bmatrix} \\
&=\begin{bmatrix}
   \varepsilon_{11}-\varepsilon_M & \varepsilon_{12} & \varepsilon_{13} \\
   \varepsilon_{21} & \varepsilon_{22}-\varepsilon_M & \varepsilon_{23} \\
   \varepsilon_{31} & \varepsilon_{32} & \varepsilon_{33}-\varepsilon_M \\
  \end{bmatrix} \\
\end{align}</math>

===Octahedral strains===
Let (<math>\mathbf{n}_1, \mathbf{n}_2, \mathbf{n}_3</math>) be the directions of the three principal strains. An '''octahedral plane''' is one whose normal makes equal angles with the three principal directions. The engineering [[shear strain]] on an octahedral plane is called the '''octahedral shear strain''' and is given by
<math display="block"> \gamma_{\mathrm{oct}} = \tfrac{2}{3}\sqrt{(\varepsilon_1-\varepsilon_2)^2 + (\varepsilon_2-\varepsilon_3)^2 + (\varepsilon_3-\varepsilon_1)^2} </math>
where <math>\varepsilon_1, \varepsilon_2, \varepsilon_3</math> are the principal strains.{{citation needed|date=January 2012}}

The [[normal strain]] on an octahedral plane is given by
<math display="block"> \varepsilon_{\mathrm{oct}} = \tfrac{1}{3}(\varepsilon_1 + \varepsilon_2 + \varepsilon_3) </math>
{{citation needed|date=January 2012}}

=== Equivalent strain ===
A scalar quantity called the '''equivalent strain''', or the [[Richard von Mises|von Mises]] equivalent strain, is often used to describe the state of strain in solids. Several definitions of equivalent strain can be found in the literature. A definition that is commonly used in the literature on [[plasticity (physics)|plasticity]] is
<math display="block">
   \varepsilon_{\mathrm{eq}} = \sqrt{\tfrac{2}{3} \boldsymbol{\varepsilon}^{\mathrm{dev}}:\boldsymbol{\varepsilon}^{\mathrm{dev}}}  = \sqrt{\tfrac{2}{3}\varepsilon_{ij}^{\mathrm{dev}}\varepsilon_{ij}^{\mathrm{dev}}} 
  ~;~~ \boldsymbol{\varepsilon}^{\mathrm{dev}} = \boldsymbol{\varepsilon} - \tfrac{1}{3}\mathrm{tr}(\boldsymbol{\varepsilon})~\boldsymbol{I} 
 </math>
This quantity is work conjugate to the equivalent stress defined as
<math display="block"> \sigma_{\mathrm{eq}} = \sqrt{\tfrac{3}{2} \boldsymbol{\sigma}^{\mathrm{dev}}:\boldsymbol{\sigma}^{\mathrm{dev}}} </math>