Simple Shear

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In fluid mechanics, **simple shear** is a special case of deformation where only one component of velocity vectors has a non-zero value:

<math> V_x=f(x,y)</math>

<math> V_y=V_z=0</math>

And the gradient of velocity is constant and perpendicular to the velocity itself:

<math>frac = dot gamma </math>,

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

<math>frac = frac = 0 </math>

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

<math>Gamma = begin 0 & & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 end</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 underbrace begin 0 & & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 end\ mboxend =begin underbrace begin 0 & & 0 \ & 0 & 0 \ 0 & 0 & 0 end \ mbox end+ begin underbrace begin 0 & & 0 \ & 0 & 0 \ 0 & 0 & 0 end \ mbox end </math>

Important examples of simple shear include laminar flow through long channels of constant cross-section (Poiseuille flow), and elastomeric bearing pads in base isolation systems to allow critical buildings to survive earthquakes undamaged.

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<math> V_x=f(x,y)</math>

<math> V_y=V_z=0</math>

And the gradient of velocity is constant and perpendicular to the velocity itself:

<math>frac = dot gamma </math>,

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

<math>frac = frac = 0 </math>

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

<math>Gamma = begin 0 & & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 end</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 underbrace begin 0 & & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 end\ mboxend =begin underbrace begin 0 & & 0 \ & 0 & 0 \ 0 & 0 & 0 end \ mbox end+ begin underbrace begin 0 & & 0 \ & 0 & 0 \ 0 & 0 & 0 end \ mbox end </math>

Important examples of simple shear include laminar flow through long channels of constant cross-section (Poiseuille flow), and elastomeric bearing pads in base isolation systems to allow critical buildings to survive earthquakes undamaged.

Read More

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