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.

For the aforementioned deformation gradient, the eigenvalues of the right Cauchy-Green deformation tensor (${\displaystyle \mathbf {C} =\mathbf {F} ^{T}\mathbf {F} =2\mathbf {E} +\mathbf {I} }$, see Finite strain theory) are ${\displaystyle 1,1+2\gamma +\gamma ^{2}}$ and ${\displaystyle 1-2\gamma +\gamma ^{2}}$. The volume change is given as ${\displaystyle J=\det(F)=1-\gamma ^{2}}$, which is not unity. In literature, a volume preserving formulation for ${\displaystyle F}$ is used to denote pure shear in large deformation^[5]. This is written in the principal coordinate frame as:

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

where ${\displaystyle \lambda }$ is the principal stretch.

See also

• Simple shear
• Squeeze mapping