Editing Infinitesimal strain theory

{{Short description|Mathematical model for describing material deformation under stress}}
{{More footnotes needed|date=August 2023}}
{{Continuum mechanics|cTopic=solid}}
In [[continuum mechanics]], the '''infinitesimal strain theory''' is a mathematical approach to the description of the [[deformation (mechanics)|deformation]] of a solid body in which the [[Displacement (vector)|displacements]] of the material [[particle]]s are assumed to be much smaller (indeed, [[infinitesimally]] smaller) than any relevant dimension of the body; so that its geometry and the constitutive properties of the material (such as [[density]] and [[stiffness]]) at each point of space can be assumed to be unchanged by the deformation.

With this assumption, the equations of continuum mechanics are considerably simplified. This approach may also be called '''small deformation theory''', '''small displacement theory''', or '''small displacement-gradient theory'''. It is contrasted with the [[finite strain theory]] where the opposite assumption is made.

The infinitesimal strain theory is commonly adopted in civil and mechanical engineering for the [[stress analysis]] of structures built from relatively stiff [[elasticity (physics)|elastic]] materials like [[concrete]] and [[steel]], since a common goal in the design of such structures is to minimize their deformation under typical [[Structural load|loads]]. However, this approximation demands caution in the case of thin flexible bodies, such as rods, plates, and shells which are susceptible to significant rotations, thus making the results unreliable.<ref>{{Cite book |last=Boresi, Arthur P. (Arthur Peter), 1924– |title=Advanced mechanics of materials |date=2003 |publisher=John Wiley & Sons |others=Schmidt, Richard J. (Richard Joseph), 1954– |isbn=1601199228 |edition=6th |location=New York |page=62 |oclc=430194205}}</ref>

==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>

==Compatibility equations==
{{Main|Compatibility (mechanics)}}
For prescribed strain components <math>\varepsilon_{ij}</math> the strain tensor equation <math>u_{i,j}+u_{j,i}= 2 \varepsilon_{ij}</math> represents a system of six differential equations for the determination of three displacements components <math>u_i</math>, giving an over-determined system. Thus, a solution does not generally exist for an arbitrary choice of strain components. Therefore, some restrictions, named ''compatibility equations'', are imposed upon the strain components. With the addition of the three compatibility equations the number of independent equations are reduced to three, matching the number of unknown displacement components. These constraints on the strain tensor were discovered by [[Adhémar Jean Claude Barré de Saint-Venant|Saint-Venant]], and are called the "[[Saint-Venant's compatibility condition|Saint Venant compatibility equations]]".

The compatibility functions serve to assure a single-valued continuous displacement function <math>u_i</math>. If the elastic medium is visualised as a set of infinitesimal cubes in the unstrained state, after the medium is strained, an arbitrary strain tensor may not yield a situation in which the distorted cubes still fit together without overlapping.

In index notation, the compatibility equations are expressed as
<math display="block">\varepsilon_{ij,km}+\varepsilon_{km,ij}-\varepsilon_{ik,jm}-\varepsilon_{jm,ik}=0</math>

In engineering notation,
* <math>\frac{\partial^2 \epsilon_x}{\partial y^2} + \frac{\partial^2 \epsilon_y}{\partial x^2} = 2 \frac{\partial^2 \epsilon_{xy}}{\partial x \partial y}</math>
* <math>\frac{\partial^2 \epsilon_y}{\partial z^2} + \frac{\partial^2 \epsilon_z}{\partial y^2} = 2 \frac{\partial^2 \epsilon_{yz}}{\partial y \partial z}</math>
* <math>\frac{\partial^2 \epsilon_x}{\partial z^2} + \frac{\partial^2 \epsilon_z}{\partial x^2} = 2 \frac{\partial^2 \epsilon_{zx}}{\partial z \partial x}</math>
* <math>\frac{\partial^2 \epsilon_x}{\partial y \partial z} = \frac{\partial}{\partial x} \left ( -\frac{\partial \epsilon_{yz}}{\partial x} + \frac{\partial \epsilon_{zx}}{\partial y} + \frac{\partial \epsilon_{xy}}{\partial z}\right)</math>
* <math>\frac{\partial^2 \epsilon_y}{\partial z \partial x} = \frac{\partial}{\partial y} \left ( \frac{\partial \epsilon_{yz}}{\partial x} - \frac{\partial \epsilon_{zx}}{\partial y} + \frac{\partial \epsilon_{xy}}{\partial z}\right)</math>
* <math>\frac{\partial^2 \epsilon_z}{\partial x \partial y} = \frac{\partial}{\partial z} \left ( \frac{\partial \epsilon_{yz}}{\partial x} + \frac{\partial \epsilon_{zx}}{\partial y} - \frac{\partial \epsilon_{xy}}{\partial z}\right)</math>

== Special cases ==

===Plane strain===
[[Image:Plane strain.svg|class=skin-invert-image|500px|right|thumb|Plane strain state in a continuum.]]
In real engineering components, [[Stress (physics)|stress]] (and strain) are 3-D [[tensor]]s but in prismatic structures such as a long metal billet, the length of the structure is much greater than the other two dimensions. The strains associated with length, i.e., the normal strain <math>\varepsilon_{33}</math> and the shear strains <math>\varepsilon_{13}</math> and <math>\varepsilon_{23}</math> (if the length is the 3-direction) are constrained by nearby material and are small compared to the ''cross-sectional strains''. Plane strain is then an acceptable approximation. The [[strain tensor]] for plane strain is written as:
<math display="block">\underline{\underline{\boldsymbol{\varepsilon}}} = \begin{bmatrix}
\varepsilon_{11} & \varepsilon_{12} & 0 \\
\varepsilon_{21} & \varepsilon_{22} & 0 \\
     0      &     0       & 0
\end{bmatrix}</math>
in which the double underline indicates a second order [[tensor]]. This strain state is called ''plane strain''. The corresponding stress tensor is:
<math display="block">\underline{\underline{\boldsymbol{\sigma}}} = \begin{bmatrix}
\sigma_{11} & \sigma_{12} & 0 \\
\sigma_{21} & \sigma_{22} & 0 \\
     0      &     0       & \sigma_{33}
\end{bmatrix}</math>
in which the non-zero <math>\sigma_{33}</math> is needed to maintain the constraint <math>\epsilon_{33} = 0</math>. This stress term can be temporarily removed from the analysis to leave only the in-plane terms, effectively reducing the 3-D problem to a much simpler 2-D problem.

=== Antiplane strain ===
{{main|Antiplane shear}}
Antiplane strain is another special state of strain that can occur in a body, for instance in a region close to a [[screw dislocation]]. The [[strain tensor]] for antiplane strain is given by
<math display="block">\underline{\underline{\boldsymbol{\varepsilon}}} = \begin{bmatrix}
0 & 0 & \varepsilon_{13} \\
0 & 0 & \varepsilon_{23}\\
\varepsilon_{13} & \varepsilon_{23} & 0
\end{bmatrix}</math>

== Relation to infinitesimal rotation tensor {{anchor|Infinitesimal rotation tensor}} ==
{{see also|Spin tensor (mechanics)}}

The infinitesimal strain tensor is defined as
<math display="block"> \boldsymbol{\varepsilon} = \frac{1}{2} [\boldsymbol{\nabla}\mathbf{u} + (\boldsymbol{\nabla}\mathbf{u})^T]</math>
Therefore the displacement gradient can be expressed as
<math display="block"> \boldsymbol{\nabla}\mathbf{u} = \boldsymbol{\varepsilon} + \boldsymbol{W}</math>
where
<math display="block"> \boldsymbol{W} := \frac{1}{2} [\boldsymbol{\nabla}\mathbf{u} - (\boldsymbol{\nabla}\mathbf{u})^T]</math>
The quantity <math>\boldsymbol{W}</math> is the '''infinitesimal rotation tensor''' or '''infinitesimal angular displacement tensor''' (related to the ''[[infinitesimal rotation matrix]]''). This tensor is [[skew symmetric]]. For infinitesimal deformations the scalar components of <math>\boldsymbol{W}</math> satisfy the condition <math>|W_{ij}| \ll 1</math>. Note that the displacement gradient is small only if {{em|both}} the strain tensor and the rotation tensor are infinitesimal.

=== The axial vector ===
A skew symmetric second-order tensor has three independent scalar components. These three components are used to define an '''axial vector''', <math>\mathbf{w}</math>, as follows
<math display="block"> W_{ij} = -\epsilon_{ijk}~w_k ~;~~ w_i = -\tfrac{1}{2}~\epsilon_{ijk}~W_{jk} </math>
where <math>\epsilon_{ijk}</math> is the [[permutation symbol]]. In matrix form
<math display="block"> \underline{\underline{\boldsymbol{W}}} = \begin{bmatrix} 0 & -w_3 & w_2 \\ w_3 & 0 & -w_1 \\ -w_2 & w_1 & 0\end{bmatrix} ~;~~ \underline{\mathbf{w}} = \begin{bmatrix} w_1 \\ w_2 \\ w_3 \end{bmatrix} </math>
The axial vector is also called the '''infinitesimal rotation vector'''. The rotation vector is related to the displacement gradient by the relation
<math display="block"> \mathbf{w} = \tfrac{1}{2}~ \boldsymbol{\nabla} \times \mathbf{u} </math>
In index notation
<math display="block"> w_i = \tfrac{1}{2}~\epsilon_{ijk}~u_{k,j} </math>
If <math>\lVert\boldsymbol{W}\rVert \ll 1 </math> and <math>\boldsymbol{\varepsilon} = \boldsymbol{0}</math> then the material undergoes an approximate rigid body rotation of magnitude <math>|\mathbf{w}|</math> around the vector <math>\mathbf{w}</math>.

=== Relation between the strain tensor and the rotation vector ===
Given a continuous, single-valued displacement field <math>\mathbf{u}</math> and the corresponding infinitesimal strain tensor <math>\boldsymbol{\varepsilon}</math>, we have (see [[Tensor derivative (continuum mechanics)]])
<math display="block">\boldsymbol{\nabla}\times\boldsymbol{\varepsilon} = e_{ijk}~\varepsilon_{lj,i}~\mathbf{e}_k\otimes\mathbf{e}_l
 = \tfrac{1}{2}~e_{ijk}~[u_{l,ji} + u_{j,li}]~\mathbf{e}_k\otimes\mathbf{e}_l </math>
Since a change in the order of differentiation does not change the result, <math>u_{l,ji} = u_{l,ij}</math>. Therefore
<math display="block"> e_{ijk} u_{l,ji} = (e_{12k}+e_{21k}) u_{l,12} + (e_{13k}+e_{31k}) u_{l,13} + (e_{23k} + e_{32k}) u_{l,32} = 0 </math>
Also
<math display="block"> \tfrac{1}{2}~e_{ijk}~u_{j,li} = \left(\tfrac{1}{2}~e_{ijk}~u_{j,i}\right)_{,l} = \left(\tfrac{1}{2} ~ e_{kij}~u_{j,i}\right)_{,l} = w_{k,l} </math>
Hence
<math display="block"> \boldsymbol{\nabla} \times \boldsymbol{\varepsilon} = w_{k,l}~\mathbf{e}_k\otimes\mathbf{e}_l = \boldsymbol{\nabla}\mathbf{w} </math>

=== Relation between rotation tensor and rotation vector ===
From an important identity regarding the [[Tensor derivative (continuum mechanics)|curl of a tensor]] we know that for a continuous, single-valued displacement field <math>\mathbf{u}</math>,
<math display="block"> \boldsymbol{\nabla}\times(\boldsymbol{\nabla}\mathbf{u}) = \boldsymbol{0}. </math>
Since <math>\boldsymbol{\nabla}\mathbf{u} = \boldsymbol{\varepsilon} + \boldsymbol{W}</math> we have
<math display="block"> \boldsymbol{\nabla}\times\boldsymbol{W} = -\boldsymbol{\nabla}\times\boldsymbol{\varepsilon} = - \boldsymbol{\nabla} \mathbf{w}. </math>

== Strain tensor in non-Cartesian coordinates ==

=== Strain tensor in cylindrical coordinates ===
In [[cylindrical polar coordinates]] (<math>r, \theta, z</math>), the displacement vector can be written as
<math display="block"> \mathbf{u} = u_r~\mathbf{e}_r + u_\theta~\mathbf{e}_\theta + u_z~\mathbf{e}_z </math>
The components of the strain tensor in a cylindrical coordinate system are given by:<ref name=Slaughter>{{cite book |last1=Slaughter |first1=William S. |title=The Linearized Theory of Elasticity |date=2002 |publisher=Springer Science+Business Media |location=New York |isbn=9781461266082 |doi=10.1007/978-1-4612-0093-2}}</ref>
<math display="block">\begin{align}
    \varepsilon_{rr} & = \cfrac{\partial u_r}{\partial r} \\
    \varepsilon_{\theta\theta} & = \cfrac{1}{r}\left(\cfrac{\partial u_\theta}{\partial \theta} + u_r\right) \\
    \varepsilon_{zz} & = \cfrac{\partial u_z}{\partial z} \\
    \varepsilon_{r\theta} & = \cfrac{1}{2} \left(\cfrac{1}{r}\cfrac{\partial u_r}{\partial \theta} + \cfrac{\partial u_\theta}{\partial r} - \cfrac{u_\theta}{r}\right) \\
    \varepsilon_{\theta z} & = \cfrac{1}{2} \left(\cfrac{\partial u_\theta}{\partial z} + \cfrac{1}{r} \cfrac{\partial u_z}{\partial \theta}\right) \\
    \varepsilon_{zr} & = \cfrac{1}{2} \left(\cfrac{\partial u_r}{\partial z} + \cfrac{\partial u_z}{\partial r}\right) 
\end{align}</math>

=== Strain tensor in spherical coordinates ===
[[File:3D Spherical.svg|class=skin-invert-image|thumb|240px|right|Spherical coordinates (''r'', ''&theta;'', ''&phi;'') as commonly used in ''physics'': radial distance ''r'', polar angle ''&theta;'' ([[theta]]), and azimuthal angle ''&phi;'' ([[phi]]). The symbol ''&rho;'' ([[rho]]) is often used instead of ''r''.]]
In [[spherical coordinates]] (<math>r, \theta, \phi</math>), the displacement vector can be written as
<math display="block">
   \mathbf{u} = u_r~\mathbf{e}_r + u_\theta~\mathbf{e}_\theta + u_\phi~\mathbf{e}_\phi
 </math>
The components of the strain tensor in a spherical coordinate system are given by <ref name=Slaughter/>
<math display="block">\begin{align}
    \varepsilon_{rr} & = \cfrac{\partial u_r}{\partial r} \\
    \varepsilon_{\theta\theta} & = \cfrac{1}{r}\left(\cfrac{\partial u_\theta}{\partial \theta} + u_r\right) \\
    \varepsilon_{\phi\phi} & = \cfrac{1}{r\sin\theta}\left(\cfrac{\partial u_\phi}{\partial \phi} + u_r\sin\theta + u_\theta\cos\theta\right)\\
    \varepsilon_{r\theta} & = \cfrac{1}{2}\left(\cfrac{1}{r}\cfrac{\partial u_r}{\partial \theta} + \cfrac{\partial u_\theta}{\partial r}- \cfrac{u_\theta}{r}\right) \\
    \varepsilon_{\theta \phi} & = \cfrac{1}{2r}\left(\cfrac{1}{\sin\theta}\cfrac{\partial u_\theta}{\partial \phi} + \cfrac{\partial u_\phi}{\partial \theta} - u_\phi\cot\theta\right) \\
    \varepsilon_{\phi r} & = \cfrac{1}{2}\left(\cfrac{1}{r\sin\theta}\cfrac{\partial u_r}{\partial \phi} + \cfrac{\partial u_\phi}{\partial r} - \cfrac{u_\phi}{r}\right) 
\end{align} </math>

==See also==
*[[Deformation (mechanics)]]
*[[Compatibility (mechanics)]]
*[[Cauchy stress tensor|Stress tensor]]
*[[Strain gauge]]
*[[Elasticity tensor]]
*[[Stress–strain curve]]
*[[Hooke's law]]
*[[Poisson's ratio]]
*[[Finite strain theory]]
*[[Strain rate]]
*[[Plane stress]]
*[[Digital image correlation]]

== References ==
{{Reflist}}

== External links ==

{{Branches of physics}}
{{Infinitesimals}}

{{DEFAULTSORT:Infinitesimal Strain Theory}}
[[Category:Physical quantities]]
[[Category:Elasticity (physics)]]
[[Category:Materials science]]
[[Category:Solid mechanics]]
[[Category:Mechanics]]