Infinitesimal strain theory

 ( Physical Quantities ) 

In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material 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 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. 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.

$$\backslash mathbf\; E\; =\; \backslash frac\{1\}\{2\}\; \backslash left(\backslash nabla\_\{\backslash mathbf\; X\}\backslash mathbf\; u\; +\; (\backslash nabla\_\{\backslash mathbf\; X\}\backslash mathbf\; u)^T\; +\; (\backslash nabla\_\{\backslash mathbf\; X\}\backslash mathbf\; u)^T\backslash nabla\_\{\backslash mathbf\; X\}\backslash mathbf\; u\backslash right)\backslash approx\; \backslash frac\{1\}\{2\}\backslash left(\backslash nabla\_\{\backslash mathbf\; X\}\backslash mathbf\; u\; +\; (\backslash nabla\_\{\backslash mathbf\; X\}\backslash mathbf\; u)^T\backslash right)$$ or $$E\_\{KL\}=\; \backslash frac\{1\}\{2\}\; \backslash left(\backslash frac\{\backslash partial\; U\_K\}\{\backslash partial\; X\_L\}\; +\backslash frac\{\backslash partial\; U\_L\}\{\backslash partial\; X\_K\}+\; \backslash frac\{\backslash partial\; U\_M\}\{\backslash partial\; X\_K\}\; \backslash frac\{\backslash partial\; U\_M\}\{\backslash partial\; X\_L\}\backslash right)\backslash approx\; \backslash frac\{1\}\{2\}\backslash left(\backslash frac\{\backslash partial\; U\_K\}\{\backslash partial\; X\_L\}+\backslash frac\{\backslash partial\; U\_L\}\{\backslash partial\; X\_K\}\backslash right)$$ and $$\backslash mathbf\; e\; =\backslash frac\{1\}\{2\}\; \backslash left(\backslash nabla\_\{\backslash mathbf\; x\}\backslash mathbf\; u\; +\; (\backslash nabla\_\{\backslash mathbf\; x\}\backslash mathbf\; u)^T\; -\; \backslash nabla\_\{\backslash mathbf\; x\}\backslash mathbf\; u(\backslash nabla\_\{\backslash mathbf\; x\}\backslash mathbf\; u)^T\backslash right)\backslash approx\; \backslash frac\{1\}\{2\}\backslash left(\backslash nabla\_\{\backslash mathbf\; x\}\backslash mathbf\; u\; +\; (\backslash nabla\_\{\backslash mathbf\; x\}\backslash mathbf\; u)^T\backslash right)$$ or $$e\_\{rs\}=\backslash frac\{1\}\{2\}\; \backslash left(\backslash frac\{\backslash partial\; u\_r\}\{\backslash partial\; x\_s\}\; +\backslash frac\{\backslash partial\; u\_s\}\{\backslash partial\; x\_r\}\; -\backslash frac\{\backslash partial\; u\_k\}\{\backslash partial\; x\_r\}\; \backslash frac\{\backslash partial\; u\_k\}\{\backslash partial\; x\_s\}\backslash right)\backslash approx\; \backslash frac\{1\}\{2\}\backslash left(\backslash frac\{\backslash partial\; u\_r\}\{\backslash partial\; x\_s\}\; +\backslash frac\{\backslash partial\; u\_s\}\{\backslash partial\; x\_r\}\backslash right)$$

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 $$\backslash mathbf\; E\; \backslash approx\; \backslash mathbf\; e\; \backslash approx\; \backslash boldsymbol\; \backslash varepsilon\; =\; \backslash frac\{1\}\{2\}\backslash left((\backslash nabla\backslash mathbf\; u)^T\; +\; \backslash nabla\backslash mathbf\; u\backslash right)$$ or $$E\_\{KL\}\backslash approx\; e\_\{rs\}\backslash approx\backslash varepsilon\_\{ij\}\; =\; \backslash frac\{1\}\{2\}\; \backslash left(u\_\{i,j\}+u\_\{j,i\}\backslash right)$$ where $\backslash varepsilon\_\{ij\}$ are the components of the infinitesimal strain tensor $\backslash boldsymbol\; \backslash varepsilon$, also called Cauchy's strain tensor, linear strain tensor, or small strain tensor.

 \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}
     

  \varepsilon_{yx} & \varepsilon_{yy} & \varepsilon_{yz} \\
  \varepsilon_{zx} & \varepsilon_{zy} & \varepsilon_{zz} \\
 \end{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} 
     

Furthermore, since the deformation gradient can be expressed as $\backslash boldsymbol\{F\}\; =\; \backslash boldsymbol\{\backslash nabla\}\backslash mathbf\{u\}\; +\; \backslash boldsymbol\{I\}$ where $\backslash boldsymbol\{I\}$ is the second-order identity tensor, we have $$\backslash boldsymbol\backslash varepsilon\; =\; \backslash frac\{1\}\{2\}\; \backslash left(\backslash boldsymbol\{F\}^T+\backslash boldsymbol\{F\}\backslash right)-\backslash boldsymbol\{I\}$$

Also, from the general expression for the Lagrangian and Eulerian finite strain tensors we have

For very small displacement gradients, i.e., $\backslash |\backslash nabla\; \backslash mathbf\; u\backslash |\; \backslash ll\; 1$, we have $$\backslash overline\; \{ab\}\; \backslash approx\; dx\; +\; \backslash frac\{\backslash partial\; u\_x\}\{\backslash partial\; x\}\; dx$$

The normal strain in the $x$-direction of the rectangular element is defined by $$\backslash varepsilon\_x\; =\; \backslash frac\{\backslash overline\; \{ab\}-\backslash overline\; \{AB\}\}\{\backslash overline\; \{AB\}\}$$ and knowing that $\backslash overline\; \{AB\}=\; dx$, we have $$\backslash varepsilon\_x\; =\; \backslash frac\{\backslash partial\; u\_x\}\{\backslash partial\; x\}$$

Similarly, the normal strain in the and becomes $$\backslash varepsilon\_y\; =\; \backslash frac\{\backslash partial\; u\_y\}\{\backslash partial\; y\}\; \backslash quad\; ,\; \backslash qquad\; \backslash varepsilon\_z\; =\; \backslash frac\{\backslash partial\; u\_z\}\{\backslash partial\; z\}$$

The engineering shear strain, or the change in angle between two originally orthogonal material lines, in this case line $\backslash overline\; \{AC\}$ and $\backslash overline\; \{AB\}$, is defined as $$\backslash gamma\_\{xy\}=\; \backslash alpha\; +\; \backslash beta$$

From the geometry of Figure 1 we have $$\backslash tan\; \backslash alpha\; =\; \backslash frac\{\backslash dfrac\{\backslash partial\; u\_y\}\{\backslash partial\; x\}dx\}\{dx\; +\; \backslash dfrac\{\backslash partial\; u\_x\}\{\backslash partial\; x\}\; dx\}\; =\; \backslash frac\{\backslash dfrac\{\backslash partial\; u\_y\}\{\backslash partial\; x\}\}\{1+\backslash dfrac\{\backslash partial\; u\_x\}\{\backslash partial\; x\}\}\; \backslash quad\; ,\; \backslash qquad\; \backslash tan\; \backslash beta=\backslash frac\{\backslash dfrac\{\backslash partial\; u\_x\}\{\backslash partial\; y\}\; dy\}\{dy+\backslash dfrac\{\backslash partial\; u\_y\}\{\backslash partial\; y\}\; dy\}=\backslash frac\{\backslash dfrac\{\backslash partial\; u\_x\}\{\backslash partial\; y\}\}\{1+\backslash dfrac\{\backslash partial\; u\_y\}\{\backslash partial\; y\}\}$$

For small rotations, i.e., $\backslash alpha$ and $\backslash beta$ are $\backslash ll\; 1$ we have $$\backslash tan\; \backslash alpha\; \backslash approx\; \backslash alpha\; \backslash quad\; ,\; \backslash qquad\; \backslash tan\; \backslash beta\; \backslash approx\; \backslash beta$$ and, again, for small displacement gradients, we have $$\backslash alpha=\backslash frac\{\backslash partial\; u\_y\}\{\backslash partial\; x\}\; \backslash quad\; ,\; \backslash qquad\; \backslash beta=\backslash frac\{\backslash partial\; u\_x\}\{\backslash partial\; y\}$$ thus $$\backslash gamma\_\{xy\}=\; \backslash alpha\; +\; \backslash beta\; =\; \backslash frac\{\backslash partial\; u\_y\}\{\backslash partial\; x\}\; +\; \backslash frac\{\backslash partial\; u\_x\}\{\backslash partial\; y\}$$ By interchanging $x$ and $y$ and $u\_x$ and $u\_y$, it can be shown that $\backslash gamma\_\{xy\}\; =\; \backslash gamma\_\{yx\}$.

Similarly, for the $y$-$z$ and $x$-$z$ planes, we have $$\backslash gamma\_\{yz\}\; =\; \backslash gamma\_\{zy\}\; =\; \backslash frac\{\backslash partial\; u\_y\}\{\backslash partial\; z\}\; +\; \backslash frac\{\backslash partial\; u\_z\}\{\backslash partial\; y\}\; \backslash quad\; ,\; \backslash qquad\; \backslash gamma\_\{zx\}\; =\; \backslash gamma\_\{xz\}\; =\; \backslash frac\{\backslash partial\; u\_z\}\{\backslash partial\; x\}\; +\; \backslash frac\{\backslash partial\; u\_x\}\{\backslash partial\; z\}$$

It can be seen that the tensorial shear strain components of the infinitesimal strain tensor can then be expressed using the engineering strain definition, as

For infinitesimal strains then we have $$d\backslash mathbf\{x\}^2\; -\; d\backslash mathbf\{X\}^2\; =\; d\backslash mathbf\; X\; \backslash cdot\; 2\backslash mathbf\{\backslash boldsymbol\; \backslash varepsilon\}\; \backslash cdot\; d\backslash mathbf\; X\; \backslash quad\backslash text\{or\}\backslash quad\; (dx)^2\; -\; (dX)^2\; =\; 2\backslash varepsilon\_\{KL\}\backslash ,dX\_K\backslash ,dX\_L$$

Dividing by $(dX)^2$ we have $$\backslash frac\{dx-dX\}\{dX\}\backslash frac\{dx+dX\}\{dX\}=2\backslash varepsilon\_\{ij\}\backslash frac\{dX\_i\}\{dX\}\backslash frac\{dX\_j\}\{dX\}$$

For small deformations we assume that $dx\; \backslash approx\; dX$, thus the second term of the left hand side becomes: $\backslash frac\{dx+dX\}\{dX\}\; \backslash approx\; 2$.

Then we have $$\backslash frac\{dx-dX\}\{dX\}\; =\; \backslash varepsilon\_\{ij\}N\_iN\_j\; =\; \backslash mathbf\; N\; \backslash cdot\; \backslash boldsymbol\; \backslash varepsilon\; \backslash cdot\; \backslash mathbf\; N$$ where $N\_i=\backslash frac\{dX\_i\}\{dX\}$, is the unit vector in the direction of $d\backslash mathbf\; X$, and the left-hand-side expression is the normal strain $e\_\{(\backslash mathbf\; N)\}$ in the direction of $\backslash mathbf\; N$. For the particular case of $\backslash mathbf\; N$ in the $X\_1$ direction, i.e., $\backslash mathbf\; N\; =\; \backslash mathbf\; I\_1$, we have $$e\_\{(\backslash mathbf\; I\_1)\}=\backslash mathbf\; I\_1\; \backslash cdot\; \backslash boldsymbol\; \backslash varepsilon\; \backslash cdot\; \backslash mathbf\; I\_1\; =\; \backslash varepsilon\_\{11\}.$$

Similarly, for $\backslash mathbf\; N=\backslash mathbf\; I\_2$ and $\backslash mathbf\; N=\backslash mathbf\; I\_3$ we can find the normal strains $\backslash varepsilon\_\{22\}$ and $\backslash varepsilon\_\{33\}$, respectively. Therefore, the diagonal elements of the infinitesimal strain tensor are the normal strains in the coordinate directions.

 \varepsilon_{11} & \varepsilon_{12} & \varepsilon_{13} \\
 \varepsilon_{12} & \varepsilon_{22} & \varepsilon_{23} \\
 \varepsilon_{13} & \varepsilon_{23} & \varepsilon_{33}
     

  \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}
     

  \underline{\underline{\hat{\boldsymbol{\varepsilon}}}} = \underline{\underline{\mathbf{L}}} ~\underline{\underline{\boldsymbol{\varepsilon}}}~ \underline{\underline{\mathbf{L}}}^T
     
     

 \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}
     

 \ell_{11} & \ell_{12} & \ell_{13} \\
 \ell_{21} & \ell_{22} & \ell_{23} \\
 \ell_{31} & \ell_{32} & \ell_{33}
     

 \varepsilon_{11} & \varepsilon_{12} & \varepsilon_{13} \\
 \varepsilon_{21} & \varepsilon_{22} & \varepsilon_{23} \\
 \varepsilon_{31} & \varepsilon_{32} & \varepsilon_{33}
     

  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})
     

  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})
     

  \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
     
     

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 $$(\backslash underline\{\backslash underline\{\backslash boldsymbol\{\backslash varepsilon\}\}\}\; -\; \backslash varepsilon\_i~\backslash underline\{\backslash underline\{\backslash mathbf\{I\}\}\})~\backslash mathbf\{n\}\_i\; =\; \backslash underline\{\backslash mathbf\{0\}\}$$ This system of equations is equivalent to finding the vector $\backslash mathbf\{n\}\_i$ along which the strain tensor becomes a pure stretch with no shear component.

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

1. a mean strain tensor or volumetric strain tensor or spherical strain tensor, $\backslash varepsilon\_M\backslash delta\_\{ij\}$, related to dilation or volume change; and
2. a deviatoric component called the strain deviator tensor, $\backslash varepsilon\text{'}\_\{ij\}$, related to distortion.

$$\backslash varepsilon\_\{ij\}=\; \backslash varepsilon\text{'}\_\{ij\}\; +\; \backslash varepsilon\_M\backslash delta\_\{ij\}$$ where $\backslash varepsilon\_M$ is the mean strain given by $$\backslash varepsilon\_M\; =\; \backslash frac\{\backslash varepsilon\_\{kk\}\}\{3\}\; =\; \backslash frac\{\backslash varepsilon\_\{11\}\; +\; \backslash varepsilon\_\{22\}\; +\; \backslash varepsilon\_\{33\}\}\{3\}\; =\; \backslash tfrac\{1\}\{3\}I^e\_1$$

The deviatoric strain tensor can be obtained by subtracting the mean strain tensor from the infinitesimal strain tensor:

  \varepsilon'_{11} & \varepsilon'_{12} & \varepsilon'_{13} \\
  \varepsilon'_{21} & \varepsilon'_{22} & \varepsilon'_{23} \\
  \varepsilon'_{31} & \varepsilon'_{32} & \varepsilon'_{33} \\
 \end{bmatrix}
     

  \varepsilon_{11} & \varepsilon_{12} & \varepsilon_{13} \\
  \varepsilon_{21} & \varepsilon_{22} & \varepsilon_{23} \\
  \varepsilon_{31} & \varepsilon_{32} & \varepsilon_{33} \\
 \end{bmatrix}
     

  \varepsilon_M & 0 & 0 \\
  0 & \varepsilon_M & 0 \\
  0 & 0 & \varepsilon_M \\
 \end{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} \\
     

The normal strain on an octahedral plane is given by $$\backslash varepsilon\_\{\backslash mathrm\{oct\}\}\; =\; \backslash tfrac\{1\}\{3\}(\backslash varepsilon\_1\; +\; \backslash varepsilon\_2\; +\; \backslash varepsilon\_3)$$

  \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}
     
     

The compatibility functions serve to assure a single-valued continuous displacement function $u\_i$. 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 $$\backslash varepsilon\_\{ij,km\}+\backslash varepsilon\_\{km,ij\}-\backslash varepsilon\_\{ik,jm\}-\backslash varepsilon\_\{jm,ik\}=0$$

In engineering notation,

• $\backslash frac\{\backslash partial^2\; \backslash epsilon\_x\}\{\backslash partial\; y^2\}\; +\; \backslash frac\{\backslash partial^2\; \backslash epsilon\_y\}\{\backslash partial\; x^2\}\; =\; 2\; \backslash frac\{\backslash partial^2\; \backslash epsilon\_\{xy\}\}\{\backslash partial\; x\; \backslash partial\; y\}$
• $\backslash frac\{\backslash partial^2\; \backslash epsilon\_y\}\{\backslash partial\; z^2\}\; +\; \backslash frac\{\backslash partial^2\; \backslash epsilon\_z\}\{\backslash partial\; y^2\}\; =\; 2\; \backslash frac\{\backslash partial^2\; \backslash epsilon\_\{yz\}\}\{\backslash partial\; y\; \backslash partial\; z\}$
• $\backslash frac\{\backslash partial^2\; \backslash epsilon\_x\}\{\backslash partial\; z^2\}\; +\; \backslash frac\{\backslash partial^2\; \backslash epsilon\_z\}\{\backslash partial\; x^2\}\; =\; 2\; \backslash frac\{\backslash partial^2\; \backslash epsilon\_\{zx\}\}\{\backslash partial\; z\; \backslash partial\; x\}$
• $\backslash frac\{\backslash partial^2\; \backslash epsilon\_x\}\{\backslash partial\; y\; \backslash partial\; z\}\; =\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\}\; \backslash left\; (\; -\backslash frac\{\backslash partial\; \backslash epsilon\_\{yz\}\}\{\backslash partial\; x\}\; +\; \backslash frac\{\backslash partial\; \backslash epsilon\_\{zx\}\}\{\backslash partial\; y\}\; +\; \backslash frac\{\backslash partial\; \backslash epsilon\_\{xy\}\}\{\backslash partial\; z\}\backslash right)$
• $\backslash frac\{\backslash partial^2\; \backslash epsilon\_y\}\{\backslash partial\; z\; \backslash partial\; x\}\; =\; \backslash frac\{\backslash partial\}\{\backslash partial\; y\}\; \backslash left\; (\; \backslash frac\{\backslash partial\; \backslash epsilon\_\{yz\}\}\{\backslash partial\; x\}\; -\; \backslash frac\{\backslash partial\; \backslash epsilon\_\{zx\}\}\{\backslash partial\; y\}\; +\; \backslash frac\{\backslash partial\; \backslash epsilon\_\{xy\}\}\{\backslash partial\; z\}\backslash right)$
• $\backslash frac\{\backslash partial^2\; \backslash epsilon\_z\}\{\backslash partial\; x\; \backslash partial\; y\}\; =\; \backslash frac\{\backslash partial\}\{\backslash partial\; z\}\; \backslash left\; (\; \backslash frac\{\backslash partial\; \backslash epsilon\_\{yz\}\}\{\backslash partial\; x\}\; +\; \backslash frac\{\backslash partial\; \backslash epsilon\_\{zx\}\}\{\backslash partial\; y\}\; -\; \backslash frac\{\backslash partial\; \backslash epsilon\_\{xy\}\}\{\backslash partial\; z\}\backslash right)$

    0      &     0       & 0
     

    0      &     0       & \sigma_{33}
     

= \tfrac{1}{2}~e_{ijk}~[u_{l,ji} + u_{j,li}]~\mathbf{e}_k\otimes\mathbf{e}_l 
     

   \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)
     

  \mathbf{u} = u_r~\mathbf{e}_r + u_\theta~\mathbf{e}_\theta + u_\phi~\mathbf{e}_\phi
     
     

   \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)