Simple shear

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Simple shear
Simple shear

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}


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