# Simple shear

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{{Short description|Translation which preserves parallelism}}
right|frame|Simple shear
'''Simple shear''' is a [deformation](/source/Deformation_(mechanics)) 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](/source/fluid_mechanics), '''simple shear''' is a special case of [deformation](/source/Deformation_(mechanics)) where only one component of [velocity](/source/velocity) vectors has a non-zero value:

:<math>V_x=f(x,y)</math>

:<math>V_y=V_z=0</math>

And the [gradient](/source/gradient) of velocity is constant and perpendicular to the velocity itself:

:<math>\frac {\partial V_x} {\partial y} = \dot \gamma </math>,

where <math>\dot \gamma </math> is the [shear rate](/source/shear_rate) and:

:<math>\frac {\partial V_x} {\partial x} = \frac {\partial V_x} {\partial z} = 0 </math>

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

:<math>\Gamma = \begin{bmatrix} 0 & {\dot \gamma} & 0 \\  0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}</math>

Simple shear with the rate <math>\dot \gamma</math> is the combination of [pure shear strain](/source/Strain_tensor) with the rate of {{sfrac|2}}<math>\dot \gamma</math> and [rotation](/source/rotation) with the rate of {{sfrac|2}}<math>\dot \gamma</math>:

:<math>\Gamma =
\begin{matrix} \underbrace \begin{bmatrix} 0 & {\dot \gamma} & 0 \\  0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}
\\ \mbox{simple shear}\end{matrix} =
\begin{matrix} \underbrace \begin{bmatrix} 0 & {\tfrac12 \dot \gamma} & 0 \\  {\tfrac12 \dot \gamma} & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \\ \mbox{pure shear} \end{matrix}
+ \begin{matrix} \underbrace \begin{bmatrix} 0 & {\tfrac12 \dot \gamma} & 0 \\  {- { \tfrac12 \dot \gamma}} & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \\ \mbox{solid rotation} \end{matrix} </math>

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

== In solid mechanics ==
{{Main|Deformation (mechanics)}}
In solid mechanics, a '''simple shear''' deformation is defined as an [isochoric plane deformation](/source/Deformation_(mechanics)) in which there are a set of line elements with a given reference orientation that do not change length and orientation during the deformation.<ref name=Ogden>{{cite book|last=Ogden|first=R. W.|date=1984|title=Non-Linear Elastic Deformations|publisher=Dover|ISBN=9780486696485}}</ref>  This deformation is differentiated from a [pure shear](/source/pure_shear) by virtue of the presence of a rigid rotation of the material.<ref>{{cite web|url=http://www.endurica.com/wp-content/uploads/2015/06/Pure-Shear-Nomenclature.pdf|title=Where do the Pure and Shear come from in the Pure Shear test?|accessdate=12 April 2013}}</ref><ref>{{cite web|url=http://www.endurica.com/wp-content/uploads/2015/06/Comparing-Pure-Shear-and-Simple-Shear.pdf|title=Comparing Simple Shear and Pure Shear|accessdate=12 April 2013}}</ref> When rubber deforms under simple shear, its stress-strain behavior is approximately linear.<ref>{{cite journal|last1=Yeoh|first1=O. H.|title=Characterization of elastic properties of carbon-black-filled rubber vulcanizates|journal=Rubber Chemistry and Technology|date=1990|volume=63|issue=5|pages=792–805|doi=10.5254/1.3538289}}</ref> A rod under torsion is a practical example for a body under simple shear.<ref>{{cite web|last1=Roylance|first1=David|title=SHEAR AND TORSION|url=http://web.mit.edu/course/3/3.11/www/modules/torsion.pdf|website=mit.edu|publisher=MIT|accessdate=17 February 2018}}</ref>

If '''e'''<sub>1</sub> is the fixed reference orientation in which line elements do not deform during the deformation and '''e'''<sub>1</sub>&nbsp;−&nbsp;'''e'''<sub>2</sub> is the plane of deformation, then the deformation gradient in simple shear can be expressed as
:<math> \boldsymbol{F} = \begin{bmatrix} 1 & \gamma & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}. </math>
We can also write the deformation gradient as
:<math> \boldsymbol{F} = \boldsymbol{\mathit{1}} + \gamma\mathbf{e}_1\otimes\mathbf{e}_2. </math>

=== Simple shear stress–strain relation ===
In linear elasticity, [shear stress](/source/shear_stress), denoted <math>\tau</math>, is related to [shear strain](/source/shear_strain), denoted <math>\gamma</math>, by the following equation:<ref>{{cite web|url=http://www.eformulae.com/engineering/strength_materials.php#pureshear|title=Strength of Materials|work=Eformulae.com|accessdate=24 December 2011}}</ref>

<math>\tau = \gamma G\,</math>

where <math>G</math> is the [shear modulus](/source/shear_modulus) of the material, given by

<math> G = \frac{E}{2(1+\nu)} </math>

Here <math>E</math> is [Young's modulus](/source/Young's_modulus) and <math>\nu</math> is [Poisson's ratio](/source/Poisson's_ratio).  Combining gives

<math>\tau = \frac{\gamma E}{2(1+\nu)}</math>

== See also ==
* [Deformation (mechanics)](/source/Deformation_(mechanics))
* [Infinitesimal strain theory](/source/Infinitesimal_strain_theory)
* [Finite strain theory](/source/Finite_strain_theory)
* [Pure shear](/source/Pure_shear)

== References ==
{{reflist}}

{{DEFAULTSORT:Simple Shear}}
Category:Fluid mechanics
Category:Continuum mechanics

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Adapted from the Wikipedia article [Simple shear](https://en.wikipedia.org/wiki/Simple_shear) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Simple_shear?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.