Simple shear

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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 deformation gradient tensor \Gamma for this deformation has only one non-zero 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 {{\dot \gamma } \over 2} and rotation with the rate of {{\dot \gamma } \over 2}:

\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&{{\dot \gamma } \over 2}&0\\{{\dot \gamma } \over 2}&0&0\\0&0&0\end{bmatrix}}\\{\mbox{pure shear}}\end{matrix}}+{\begin{matrix}\underbrace {\begin{bmatrix}0&{{\dot \gamma } \over 2}&0\\{-{{\dot \gamma } \over 2}}&0&0\\0&0&0\end{bmatrix}}\\{\mbox{solid rotation}}\end{matrix}}

Important examples of simple shear include laminar flow through long channels of constant cross-section (Poiseuille flow), and elastomeric bearing pads in base isolation systems to allow critical buildings to survive earthquakes undamaged.

Simple shear 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.[1] This deformation is differentiated from a pure shear by virtue of the presence of a rigid rotation of the material.[2][3]

If {\mathbf {e}}_{1} is the fixed reference orientation in which line elements do not deform during the deformation and {\mathbf {e}}_{1}-{\mathbf {e}}_{2} 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}.

See also

References

  1. ↑ Ogden, R. W., 1984, Non-linear elastic deformations, Dover.
  2. ↑ "Where do the Pure and Shear come from in the Pure Shear test?". Retrieved 12 April 2013. 
  3. ↑ "Comparing Simple Shear and Pure Shear". Retrieved 12 April 2013. 
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