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:

And the gradient of velocity is perpendicular to it:

<math>\frac {\partial V_x} {\partial y} = \dot \gamma </math>,

where <math>\dot \gamma </math> is the shear rate and:

<math>\frac {\partial V_x} {\partial x} = \frac {\partial V_x} {\partial z} = 0 </math>

The deformation gradient tensor <math>\Gamma</math> for this deformation has only one non-zero term:

<math>\Gamma = \begin{bmatrix} 0 & {\dot \gamma} & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}</math>

Simple shear with the rate <math>\dot \gamma</math> is the combination of pure shear strain with the rate of <math>\dot \gamma \over 2</math> and rotation with the rate of <math>\dot \gamma \over 2</math>:

<math>\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} </math>

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".