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:

V_{x}=f(x,y)

V_{y}=V_{z}=0

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

{\frac {\partial V_{x}}{\partial y}}={\dot {\gamma }}

,

where ${\dot {\gamma }}$ is the shear rate and:

{\frac {\partial V_{x}}{\partial x}}={\frac {\partial V_{x}}{\partial z}}=0

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

\Gamma ={\begin{bmatrix}0&{\dot {\gamma }}&0\\0&0&0\\0&0&0\end{bmatrix}}

Simple shear with the rate ${\dot {\gamma }}$ is the combination of pure shear strain with the rate of ⁠1/2⁠ ${\dot {\gamma }}$ and rotation with the rate of ⁠1/2⁠ ${\dot {\gamma }}$ :

\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&{{\tfrac {1}{2}}{\dot {\gamma }}}&0\\{{\tfrac {1}{2}}{\dot {\gamma }}}&0&0\\0&0&0\end{bmatrix}} \\{\mbox{pure shear}}\end{matrix}}+{\begin{matrix}\underbrace {\begin{bmatrix}0&{{\tfrac {1}{2}}{\dot {\gamma }}}&0\\{-{{\tfrac {1}{2}}{\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.

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.^[1] This deformation is differentiated from a pure shear by virtue of the presence of a rigid rotation of the material.^[2]^[3] When rubber deforms under simple shear, its stress-strain behavior is approximately linear.^[4] A rod under torsion is a practical example for a body under simple shear.^[5]

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

{\boldsymbol {F}}={\begin{bmatrix}1&\gamma &0\\0&1&0\\0&0&1\end{bmatrix}}.

We can also write the deformation gradient as

{\boldsymbol {F}}={\boldsymbol {\mathit {1}}}+\gamma \mathbf {e} _{1}\otimes \mathbf {e} _{2}.

Simple shear stress–strain relation

In linear elasticity, shear stress, denoted $\tau$ , is related to shear strain, denoted $\gamma$ , by the following equation:^[6]

$\tau =\gamma G\,$

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

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

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

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

References

^ Ogden, R. W. (1984). Non-Linear Elastic Deformations. Dover. ISBN 9780486696485.
^ "Where do the Pure and Shear come from in the Pure Shear test?" (PDF). Retrieved 12 April 2013.
^ "Comparing Simple Shear and Pure Shear" (PDF). Retrieved 12 April 2013.
^ Yeoh, O. H. (1990). "Characterization of elastic properties of carbon-black-filled rubber vulcanizates". Rubber Chemistry and Technology. 63 (5): 792–805. doi:10.5254/1.3538289.
^ Roylance, David. "SHEAR AND TORSION" (PDF). mit.edu. MIT. Retrieved 17 February 2018.
^ "Strength of Materials". Eformulae.com. Retrieved 24 December 2011.

[Ogden-1] Ogden, R. W. (1984). Non-Linear Elastic Deformations. Dover. ISBN 9780486696485.

[2] "Where do the Pure and Shear come from in the Pure Shear test?" (PDF). Retrieved 12 April 2013.

[3] "Comparing Simple Shear and Pure Shear" (PDF). Retrieved 12 April 2013.

[4] Yeoh, O. H. (1990). "Characterization of elastic properties of carbon-black-filled rubber vulcanizates". Rubber Chemistry and Technology. 63 (5): 792–805. doi:10.5254/1.3538289.

[5] Roylance, David. "SHEAR AND TORSION" (PDF). mit.edu. MIT. Retrieved 17 February 2018.

[6] "Strength of Materials". Eformulae.com. Retrieved 24 December 2011.

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