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Simple shear

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Translation which preserves parallelism
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 ) {\displaystyle V_{x}=f(x,y)}
V y = V z = 0 {\displaystyle V_{y}=V_{z}=0}

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

∂ V x ∂ y = γ ˙ {\displaystyle {\frac {\partial V_{x}}{\partial y}}={\dot {\gamma }}} ,

where γ ˙ {\displaystyle {\dot {\gamma }}} is the shear rate and:

∂ V x ∂ x = ∂ V x ∂ z = 0 {\displaystyle {\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:

Γ = [ 0 γ ˙ 0 0 0 0 0 0 0 ] {\displaystyle \Gamma ={\begin{bmatrix}0&{\dot {\gamma }}&0\\0&0&0\\0&0&0\end{bmatrix}}}

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

Γ = [ 0 γ ˙ 0 0 0 0 0 0 0 ] ⏟ simple shear = [ 0 1 2 γ ˙ 0 1 2 γ ˙ 0 0 0 0 0 ] ⏟ pure shear + [ 0 1 2 γ ˙ 0 − 1 2 γ ˙ 0 0 0 0 0 ] ⏟ solid rotation {\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&{{\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

Main article: 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. 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

F = [ 1 γ 0 0 1 0 0 0 1 ] . {\displaystyle {\boldsymbol {F}}={\begin{bmatrix}1&\gamma &0\\0&1&0\\0&0&1\end{bmatrix}}.}

We can also write the deformation gradient as

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

Simple shear stress–strain relation

In linear elasticity, shear stress, denoted τ {\displaystyle \tau } , is related to shear strain, denoted γ {\displaystyle \gamma } , by the following equation:

τ = γ G {\displaystyle \tau =\gamma G\,}

where G {\displaystyle G} is the shear modulus of the material, given by

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

Here E {\displaystyle E} is Young's modulus and ν {\displaystyle \nu } is Poisson's ratio. Combining gives

τ = γ E 2 ( 1 + ν ) {\displaystyle \tau ={\frac {\gamma E}{2(1+\nu )}}}

See also

References

  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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