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# Simple shear

### Additional recommended knowledge

Simple shear is a special case of deformation of a fluid where only one component of velocity vectors has a non-zero value:

Vx = f(x,y)

Vy = Vz = 0

And the gradient of velocity is perpendicular to it: $\frac {\partial V_x} {\partial y} = \dot \gamma$,

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 Γ 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 $\dot \gamma \over 2$: $\Gamma = \begin{matrix} \underbrace \begin{bmatrix} 0 & {\dot \gamma} & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \\ \mbox{simple shear}\end{matrix} = \begin{matrix} \underbrace \begin{bmatrix} 0 & {\dot \gamma \over 2} & 0 \\ {\dot \gamma \over 2} & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \\ \mbox{pure shear} \end{matrix} + \begin{matrix} \underbrace \begin{bmatrix} 0 & {\dot \gamma \over 2} & 0 \\ {- { \dot \gamma \over 2}} & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \\ \mbox{solid rotation} \end{matrix}$

An important example of simple shear is laminar flow through long channels of constant cross-section (Poiseuille flow).

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Simple_shear". A list of authors is available in Wikipedia.
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