Simple shear

{{Short description|Translation which preserves parallelism}} right|frame|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:

:<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 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 with the rate of {{sfrac|2}}<math>\dot \gamma</math> and 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 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 == {{Main|Deformation (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.<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 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'''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 :<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, denoted <math>\tau</math>, is related to 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 of the material, given by

Here <math>E</math> is Young's modulus and <math>\nu</math> is Poisson's ratio. Combining gives

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

== References == {{reflist}}

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