Understanding Stress Tensor in MTW Ex. 5.4

[TerryW]

I've been working on Ex 5.4 in MTW. The maths is fairly straight forward, but I don't really understand what is going on!

In part (b) what are the 'forces' pushing the volume through a distance? If they are forces, they must produce an acceleration but we have a constant velocity. Are these forces producing the stress in the volume?

What would be an example of a 'stressed' medium as used in this example?

If m^jk is 'inertial mass per unit volume' equation 5.51 will sum to produce the total momentum in the direction of v, not one of the components.

Can anyone shed any light?


Regards


TerryW

 

[Bill_K]

What would be an example of a 'stressed' medium as used in this example?

Anything where the spatial components of the stress tensor are nonzero. Could be a gas under pressure. Could be a squeezed rubber ball.

In part (b) what are the 'forces' pushing the volume through a distance? If they are forces, they must produce an acceleration but we have a constant velocity. Are these forces producing the stress in the volume?

The forces, for example, could be the forces acting on this element of the medium by neighboring elements. Whether the velocity is constant depends on whether the forces are balanced.

If m^jk is 'inertial mass per unit volume' equation 5.51 will sum to produce the total momentum in the direction of v, not one of the components.

It will be the three spatial components of a vector, maybe not in the same direction as v.

 

[TerryW]

Problem sorted

Thanks for your reply. Your comments on the statement

"If m^jk is 'inertial mass per unit volume' equation 5.51 will sum to produce the total momentum in the direction of v, not one of the components."

didn't really help me, but I took another look at it and I re-wrote the expression for T⁰¹ in terms of the stress tensor in the volume's rest frame:

T⁰¹ = (T^0'0' + T^1'1')v¹ + T^1'2'v² + T^1'3'v³

and it then becomes much clearer.

For a unit volume, T⁰¹ is the x-component of momentum which is made up from

1. the contributions from the x-component of momentum arising from the rest mass moving at v¹ + the x-component of flux of x-momentum x v^x

2. the contribution from the y-component of the flux of x-momentum x v^y and

3. the contribution from the z-component of the flux of x-momentum x v^z

The rationale for the contributions 2 and 3 come from the equation preceding (5.16) on page 139.

Regards


TerryW