# Pure shear

In mechanics and geology, pure shear is a three-dimensional homogeneous flattening of a body.[1] It is an example of irrotational strain in which body is elongated in one direction while being shortened perpendicularly. For soft materials, such as rubber, a strain state of pure shear is often used for characterizing hyperelastic and fracture mechanical behaviour.[2] Pure shear is differentiated from simple shear in that pure shear involves no rigid body rotation. [3][4]

If ${\displaystyle \lambda }$ is the stretch ratio applied to the material, then the deformation gradient in pure shear can be expressed as[5]

${\displaystyle {\boldsymbol {F}}={\begin{bmatrix}\lambda &0&0\\0&1&0\\0&0&1/\lambda \end{bmatrix}}.}$

The linear elastic stress-strain law for the case of pure shear is:

${\displaystyle {\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{12}\end{bmatrix}}\,=\,{\begin{bmatrix}\sigma \\\nu \sigma \\0\end{bmatrix}}\,=\,{\frac {E}{1-\nu ^{2}}}{\begin{bmatrix}1&\nu &0\\\nu &1&0\\0&0&{\frac {1-\nu }{2}}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{11}\\0\\0\end{bmatrix}}}$