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 $$\Gamma$$ 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 $$\dot \gamma \over 2$$:

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

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