Physics: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]

The deformation gradient for pure shear is given by:

$\displaystyle{ F = \begin{bmatrix}1&\gamma&0 \\\gamma&1&0\\0&0&1\end{bmatrix} }$

Note that this gives a Green-Lagrange strain of:

$\displaystyle{ E = \frac{1}{2}\begin{bmatrix}\gamma^2&2\gamma&0\\2\gamma&\gamma^2&0\\0&0&0\end{bmatrix} }$

Here there is no rotation occurring, which can be seen from the equal off-diagonal components of the strain tensor. The linear approximation to the Green-Lagrange strain shows that the small strain tensor is:

$\displaystyle{ \epsilon = \frac{1}{2}\begin{bmatrix}0&2\gamma&0\\2\gamma&0&0\\0&0&0\end{bmatrix} }$

which has only shearing components.