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In structural mechanics, the equations relating stress and strain arise from the balance of forces in the material medium.

*Plane stress* is a condition that prevails
in a flat plate in the * x*-

$$\left(\begin{array}{c}{\sigma}_{x}\\ {\sigma}_{y}\\ {\tau}_{xy}\end{array}\right)=\frac{E}{1-{\nu}^{2}}\left(\begin{array}{ccc}1& \nu & 0\\ \nu & 1& 0\\ 0& 0& \frac{1-\nu}{2}\end{array}\right)\left(\begin{array}{c}{\epsilon}_{x}\\ {\epsilon}_{y}\\ {\gamma}_{xy}\end{array}\right)$$

where * σ_{x}* and

The deformation of the material is described by the displacements
in the * x* and

$$\begin{array}{c}{\epsilon}_{x}=\frac{\partial u}{\partial x}\\ {\epsilon}_{y}=\frac{\partial v}{\partial y}\\ {\gamma}_{xy}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\end{array}$$

The balance of force equations are

$$\begin{array}{c}-\frac{\partial {\sigma}_{x}}{\partial x}-\frac{\partial {\tau}_{xy}}{\partial y}={K}_{x}\\ -\frac{\partial {\tau}_{xy}}{\partial x}-\frac{\partial {\sigma}_{y}}{\partial y}={K}_{y}\end{array}$$

where * K_{x}* and

$$-\nabla \xb7\left(c\otimes \nabla u\right)=k$$

where **c** is a rank four tensor,
which can be written as four 2-by-2 matrices *c*_{11}, *c*_{12}, *c*_{21},
and *c*_{22}:

$${c}_{11}=\left(\begin{array}{cc}2G+\mu & 0\\ 0& G\end{array}\right),\text{\hspace{1em}}{c}_{12}=\left(\begin{array}{cc}0& \mu \\ G& 0\end{array}\right),\text{\hspace{1em}}{c}_{21}=\left(\begin{array}{cc}0& G\\ \mu & 0\end{array}\right),\text{\hspace{1em}}{c}_{22}=\left(\begin{array}{cc}G& 0\\ 0& 2G+\mu \end{array}\right)$$

$$G=\frac{E}{2\left(1+\nu \right)},\text{\hspace{1em}}\mu =2G\frac{\nu}{1-\nu},\text{\hspace{1em}}k=\left(\begin{array}{l}{K}_{x}\\ {K}_{y}\end{array}\right)$$

Here * G * is the shear modulus, and

$$-\nabla \cdot (c\otimes \nabla u)=\lambda du,\text{\hspace{1em}}d=\left(\begin{array}{cc}\rho & 0\\ 0& \rho \end{array}\right)$$

*Plane strain* is a deformation state where
there are no displacements in the * z*-direction,
and the displacements in the

The places where the plane strain equations differ from the plane stress equations are:

The

parameter in the*µ*tensor is defined as*c*$$\mu =2G\frac{\nu}{1-2\nu}$$

The von Mises effective stress is computed as

$$\sqrt{\left({\sigma}_{1}^{2}+{\sigma}_{2}^{2}\right)\left({\nu}^{2}-\nu +1\right)+{\sigma}_{1}{\sigma}_{2}\left(2{\nu}^{2}-2\nu -1\right)}$$

Plane strain problems are less common than plane stress problems.
An example is a slice of an underground tunnel that lies along the * z*-axis.
It deforms in essentially plane strain conditions.

[1] Cook, Robert D., David S. Malkus, and Michael E. Plesha. *Concepts
and Applications of Finite Element Analysis*. 3rd edition.
New York, NY: John Wiley & Sons, 1989.

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