# Documentation

## Structural Mechanics — Plane Strain

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 is called plane strain. You can solve plane strain problems with Partial Differential Equation Toolbox™ software by setting the application mode to Structural Mechanics, Plane Strain. The stress-strain relation is only slightly different from the plane stress case, and the same set of material parameters is used. The application interfaces are identical for the two structural mechanics modes.

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.