Simple shear

 ( Fluid Mechanics ) 

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.

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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:

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

$V\_y=V\_z=0$

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 Strain tensor with the rate of $\backslash dot\; \backslash gamma$ and rotation with the rate of $\backslash dot\; \backslash gamma$:

$\backslash Gamma\; =$

\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.

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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.

R. W. Ogden (1984). 9780486696485, Dover. ISBN 9780486696485

This deformation is differentiated from a pure shear by virtue of the presence of a rigid rotation of the material. When rubber deforms under simple shear, its stress-strain behavior is approximately linear. A rod under torsion is a practical example for a body under simple shear.

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

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

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Simple shear stress–strain relation In linear elasticity, shear stress, denoted $\backslash tau$, is related to shear strain, denoted $\backslash gamma$, by the following equation:

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

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

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

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See also

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

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