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2012-05-06T05:41:33Z

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[[Image:Simple shear.PNG|right|frame|Simple shear]]

In [[fluid mechanics]], '''simple shear''' is a special case of [[Deformation_(mechanics)|deformation]] where only one component of [[velocity]] vectors has a non-zero value:

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

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

And the [[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]] and:

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

The [[deformation gradient]] tensor <math>\Gamma</math> for this deformation has only one non-zero 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 [[Strain tensor|pure shear strain]] with the rate of <math>\dot \gamma \over 2</math> and [[rotation]] with the rate of <math>\dot \gamma \over 2</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 & {\dot \gamma \over 2} & 0 \\ {\dot \gamma \over 2} & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \\ \mbox{pure shear} \end{matrix}
+ \begin{matrix} \underbrace \begin{bmatrix} 0 & {\dot \gamma \over 2} & 0 \\ {- { \dot \gamma \over 2}} & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \\ \mbox{solid rotation} \end{matrix} </math>

Important examples of simple shear include [[Laminar flow|laminar]] flow through long channels of constant cross-section ([[Poiseuille flow]]), and elastomeric bearing pads in [[base isolation]] systems to allow critical buildings to survive earthquakes undamaged.

== Simple shear in solid mechanics ==
{{Main|Deformation (mechanics)}}
In solid mechanics, a '''simple shear''' deformation is defined as an [[Deformation (mechanics)#Plane deformation|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.<ref name=Ogden>Ogden, R. W., 1984, '''Non-linear elastic deformations''', Dover.</ref> This deformation is differentiated from a '''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/2013/03/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/2013/03/Comparing-Pure-Shear-and-Simple-Shear.pdf|title=Comparing Simple Shear and Pure Shear|accessdate=12 April 2013}}</ref>

If <math>\mathbf{e}_1</math> is the fixed reference orientation in which line elements do not deform during the deformation and <math>\mathbf{e}_1-\mathbf{e}_2</math> 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>

== See also ==
* [[Deformation (mechanics)]]
* [[Infinitesimal strain theory]]
* [[Finite strain theory]]
* [[Pure shear]]

== References ==
{{reflist}}

{{DEFAULTSORT:Simple Shear}}
[[Category:Fluid mechanics]]

Cross fluid - Revision history

en>Helpful Pixie Bot: ISBNs (Build KC)