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 = egin{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 =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).


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