Simple shear

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

:$V\_x=f(x,y)$

:$V\_y=V\_z=0$

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

:$\backslash frac\; \{\backslash partial\; V\_x\}\; \{\backslash partial\; y\}\; =\; \backslash dot\; \backslash gamma$,

where $\backslash dot\; \backslash gamma$ is the shear rate and:

:$\backslash frac\; \{\backslash partial\; V\_x\}\; \{\backslash partial\; x\}\; =\; \backslash frac\; \{\backslash partial\; V\_x\}\; \{\backslash partial\; z\}\; =\; 0$

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

:

Simple shear with the rate $\backslash dot\; \backslash gamma$ is the combination of pure shear strain with the rate of {{sfrac|2}}$\backslash dot\; \backslash gamma$ and rotation with the rate of {{sfrac|2}}$\backslash dot\; \backslash gamma$:

:$\backslash Gamma\; =$

\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}

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.

{{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.{{cite book|last=Ogden|first=R. W.|date=1984|title=Non-Linear Elastic Deformations|publisher=Dover|ISBN=9780486696485}} This deformation is differentiated from a pure shear by virtue of the presence of a rigid rotation of the material.{{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}}{{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}} When rubber deforms under simple shear, its stress-strain behavior is approximately linear.{{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}} A rod under torsion is a practical example for a body under simple shear.{{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}}

If e₁ is the fixed reference orientation in which line elements do not deform during the deformation and e₁ − e₂ is the plane of deformation, then the deformation gradient in simple shear can be expressed as

:

We can also write the deformation gradient as

:$\backslash boldsymbol\{F\}\; =\; \backslash boldsymbol\{\backslash mathit\{1\}\}\; +\; \backslash gamma\backslash mathbf\{e\}\_1\backslash otimes\backslash mathbf\{e\}\_2.$

In linear elasticity, shear stress, denoted $\backslash tau$, is related to shear strain, denoted $\backslash gamma$, by the following equation:{{cite web|url=http://www.eformulae.com/engineering/strength_materials.php#pureshear|title=Strength of Materials|work=Eformulae.com|accessdate=24 December 2011}}

$\backslash tau\; =\; \backslash gamma\; G\backslash ,$

where $G$ is the shear modulus of the material, given by

$G\; =\; \backslash frac\{E\}\{2(1+\backslash nu)\}$

Here $E$ is Young's modulus and $\backslash nu$ is Poisson's ratio. Combining gives

$\backslash tau\; =\; \backslash frac\{\backslash gamma\; E\}\{2(1+\backslash nu)\}$

• Deformation (mechanics)
• Infinitesimal strain theory
• Finite strain theory
• Pure shear

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Category:Fluid mechanics

Category:Continuum mechanics