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:
,
where $dot\; gamma$ is the shear rate and:

The deformation gradient tensor $Gamma$ for this deformation has only one non-zero term:

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 $dot\; gamma\; over\; 2$:
$Gamma\; =$
egin{matrix} underbrace egin{bmatrix} 0 & {dot gamma} & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 end{bmatrix}
\ mbox{simple shear}end{matrix} =
egin{matrix} underbrace egin{bmatrix} 0 & {dot gamma over 2} & 0 \ {dot gamma over 2} & 0 & 0 \ 0 & 0 & 0 end{bmatrix} \ mbox{pure shear} end{matrix}
+ egin{matrix} underbrace egin{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).