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## Plane Stress and Plane Strain

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-y plane, loaded only in its own plane and without z-direction restraint. Assuming isotropic and isothermal conditions, the stress-strain relation can then be written as follows:

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

where σx and σy are the normal stresses in the x and y directions, and τxy is the shear stress. The material properties are expressed as a combination of E, the elastic modulus or Young's modulus, and ν, Poisson's ratio.

The deformation of the material is described by the displacements in the x and y directions, u and v, from which the strains are defined as

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

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 Kx and Ky are volume forces (body forces). Combining the preceding relations yields the displacement equation:

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

where c is a rank four tensor, which can be written as four 2-by-2 matrices c11, c12, c21, and c22:

`${c}_{11}=\left(\begin{array}{cc}2G+\mu & 0\\ 0& G\end{array}\right),\text{ }{c}_{12}=\left(\begin{array}{cc}0& \mu \\ G& 0\end{array}\right),\text{ }{c}_{21}=\left(\begin{array}{cc}0& G\\ \mu & 0\end{array}\right),\text{ }{c}_{22}=\left(\begin{array}{cc}G& 0\\ 0& 2G+\mu \end{array}\right)$`
`$G=\frac{E}{2\left(1+\nu \right)},\text{ }\mu =2G\frac{\nu }{1-\nu },\text{ }k=\left(\begin{array}{l}{K}_{x}\\ {K}_{y}\end{array}\right)$`

Here G is the shear modulus, and k is a vector of volume forces. In addition, you can describe the plane stress problem as an eigenvalue problem where, ρ is the material density:

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

Plane strain is a deformation state where there are no displacements in the z-direction, and the displacements in the x- and y-directions are functions of x and y but not z. The stress-strain relation is only slightly different from the plane stress case, and the same set of material parameters is used.

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

• The µ parameter in the c tensor is defined as

`$\mu =2G\frac{\nu }{1-2\nu }$`
• 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.

## References

[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.