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Simple shear

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Simple shear is a special case of deformation of a fluid 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 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 $\dot \gamma \over 2$ :

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