Physics: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]\displaystyle{ V_x=f(x,y) }[/math]
- [math]\displaystyle{ V_y=V_z=0 }[/math]
And the gradient of velocity is constant and perpendicular to the velocity itself:
- [math]\displaystyle{ \frac {\partial V_x} {\partial y} = \dot \gamma }[/math],
where [math]\displaystyle{ \dot \gamma }[/math] is the shear rate and:
- [math]\displaystyle{ \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]\displaystyle{ \Gamma = \begin{bmatrix} 0 & {\dot \gamma} & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} }[/math]
Simple shear with the rate [math]\displaystyle{ \dot \gamma }[/math] is the combination of pure shear strain with the rate of 1/2[math]\displaystyle{ \dot \gamma }[/math] and rotation with the rate of 1/2[math]\displaystyle{ \dot \gamma }[/math]:
- [math]\displaystyle{ \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
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
- [math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \tau }[/math], is related to shear strain, denoted [math]\displaystyle{ \gamma }[/math], by the following equation:[6]
[math]\displaystyle{ \tau = \gamma G\, }[/math]
where [math]\displaystyle{ G }[/math] is the shear modulus of the material, given by
[math]\displaystyle{ G = \frac{E}{2(1+\nu)} }[/math]
Here [math]\displaystyle{ E }[/math] is Young's modulus and [math]\displaystyle{ \nu }[/math] is Poisson's ratio. Combining gives
[math]\displaystyle{ \tau = \frac{\gamma E}{2(1+\nu)} }[/math]
See also
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?". http://www.endurica.com/wp-content/uploads/2015/06/Pure-Shear-Nomenclature.pdf. Retrieved 12 April 2013.
- ↑ "Comparing Simple Shear and Pure Shear". http://www.endurica.com/wp-content/uploads/2015/06/Comparing-Pure-Shear-and-Simple-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". MIT. http://web.mit.edu/course/3/3.11/www/modules/torsion.pdf. Retrieved 17 February 2018.
- ↑ "Strength of Materials". Eformulae.com. http://www.eformulae.com/engineering/strength_materials.php#pureshear. Retrieved 24 December 2011.
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