# Pure shear

In mechanics and geology, pure shear is a three-dimensional homogeneous flattening of a body. 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. Pure shear is differentiated from simple shear in that pure shear involves no rigid body rotation. 

The deformation gradient for pure shear is given by:

$F={\begin{bmatrix}1&\gamma &0\\\gamma &1&0\\0&0&1\end{bmatrix}}$ Note that this gives a Green-Lagrange strain of:

$E={\frac {1}{2}}{\begin{bmatrix}\gamma ^{2}&2\gamma &0\\2\gamma &\gamma ^{2}&0\\0&0&0\end{bmatrix}}$ Here we note that there is no rotation occurring which can be seen from the equal off-diagonal components of the strain tensor. It is also worth noting that if we take the linear approximation to the Green-Lagrange strain we see that the small strain tensor is:$\epsilon ={\frac {1}{2}}{\begin{bmatrix}0&2\gamma &0\\2\gamma &0&0\\0&0&0\end{bmatrix}}$ Which has only shearing components.