Shear Stress Problems: Calculating Lateral Movement

[aeroengine123321]

Homework Statement

The bottom surface (8 cm x 12 cm) of a rectangular block of cheese (3 cm thick) is clamped in a cheese grater. The grating mechanism moving across the top surface of the cheese, applies a lateral force of 20 N. The shear modulus, G, of the cheese is 3.7 kPa. Assuming the grater applies the force uniformly to the upper surface, estimate the lateral movement of the upper surface with respect to lower surface?

Homework Equations

How would I calculate the lateral movement of the upper surface in relation to the lower surface?

The Attempt at a Solution

 

[haruspex]

How would I calculate the lateral movement of the upper surface in relation to the lower surface?

What standard equations can you quote regarding shear modulus, shear stress and shear strain?

 

[aeroengine123321]

I've worked out the shear strain but I'm not sure how to get the lateral movement from there

 

[haruspex]

I've worked out the shear strain but I'm not sure how to get the lateral movement from there

The shear strain is an angle, right? From there it is just geometry. You know the thickness of the block. Draw a diagram of it in its strained (parallelogram) shape.

 

[aeroengine123321]

Right, so I can just do sin(shear strain) and then multiply it by the height of the block?

 

[haruspex]

Right, so I can just do sin(shear strain) and then multiply it by the height of the block?

Yes.

 

[aeroengine123321]

Thank you

 

[aeroengine123321]

Would it not be tan(shear strain) then multiplied by the thickness? Because when I draw a diagram of it, the thickness (3cm) is the adjacent while the length I'm looking for is the opposite. Is this correct or am I going wrong with these geometries?

 

[haruspex]

Would it not be tan(shear strain) then multiplied by the thickness? Because when I draw a diagram of it, the thickness (3cm) is the adjacent while the length I'm looking for is the opposite. Is this correct or am I going wrong with these geometries?

I'm sure these things are only valid for small strains anyway, so sine versus tan versus angle in radians makes no difference.
If you treat the block as constant volume then the height doesn't change, so tan would be right. If you treat it as a framework of rigid rods initially at right angles embedded in a compressible matrix then the rods have fixed length, so sine would be right.