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:

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 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:

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

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

where \lambda is the principal stretch.

See also

References

  1. ^ Reish, Nathaniel E. & Gary H. Girty. "Definition and Mathematics of Pure Shear". San Diego State University Department of Geological Sciences. Retrieved 24 December 2011.
  2. ^ Yeoh, O. H. (2001). "Analysis of deformation and fracture of 'pure shear'rubber testpiece". Plastics, Rubber and Composites. 30 (8): 389–397. Bibcode:2001PRC....30..389Y. doi:10.1179/146580101101541787. S2CID 136628719
  3. ^ "Where do the Pure and Shear come from in the Pure Shear test?". Retrieved 1 April 2026.
  4. ^ "Comparing Simple Shear and Pure Shear". Retrieved 12 April 2013.
  5. ^ Thiel, Christian; Voss, Jendrik; Martin, Robert J.; Neff, Patrizio (2019-06-01). "Shear, pure and simple". International Journal of Non-Linear Mechanics. 112: 57–72. arXiv:1806.07749. doi:10.1016/j.ijnonlinmec.2018.10.002. ISSN 0020-7462