# Simple shear

**Simple shear** is a deformation in which parallel planes in a material remain parallel and maintain a constant distance, while translating relative to each other.

## In fluid mechanics

In fluid mechanics, **simple shear** is a special case of deformation where only one component of velocity vectors has a non-zero value:

And the gradient of velocity is constant and perpendicular to the velocity itself:

- ,

where is the shear rate and:

The displacement gradient tensor Γ for this deformation has only one nonzero term:

Simple shear with the rate is the combination of pure shear strain with the rate of 1/2 and rotation with the rate of 1/2:

The mathematical model representing simple shear is a shear mapping restricted to the physical limits. It is an elementary linear transformation represented by a matrix. The model may represent laminar flow velocity at varying depths of a long channel with constant cross-section. Limited shear deformation is also used in vibration control, for instance base isolation of buildings for limiting earthquake damage.

## In solid mechanics

In solid mechanics, a **simple shear** deformation is defined as an isochoric plane deformation in which there are a set of line elements with a given reference orientation that do not change length and orientation during the deformation.[1] This deformation is differentiated from a pure shear by virtue of the presence of a rigid rotation of the material.[2][3] When rubber deforms under simple shear, its stress-strain behavior is approximately linear.[4] A rod under torsion is a practical example for a body under simple shear.[5]

If **e**_{1} is the fixed reference orientation in which line elements do not deform during the deformation and **e**_{1} − **e**_{2} is the plane of deformation, then the deformation gradient in simple shear can be expressed as

We can also write the deformation gradient as

### Simple shear stress–strain relation

In linear elasticity, shear stress, denoted , is related to shear strain, denoted , by the following equation:[6]

where is the shear modulus of the material, given by

Here is Young's modulus and is Poisson's ratio. Combining gives

## References

- Ogden, R. W. (1984).
*Non-Linear Elastic Deformations*. Dover. ISBN 9780486696485. - "Where do the Pure and Shear come from in the Pure Shear test?" (PDF). Retrieved 12 April 2013.
- "Comparing Simple Shear and Pure Shear" (PDF). Retrieved 12 April 2013.
- Yeoh, O. H. (1990). "Characterization of elastic properties of carbon-black-filled rubber vulcanizates".
*Rubber Chemistry and Technology*.**63**(5): 792–805. doi:10.5254/1.3538289. - Roylance, David. "SHEAR AND TORSION" (PDF).
*mit.edu*. MIT. Retrieved 17 February 2018. - "Strength of Materials".
*Eformulae.com*. Retrieved 24 December 2011.