Pulling/pushing an elastic rod at high speeds

[Terry Bing]

Consider an elastic rod lying on a table. If one end of the rod is pulled/pushed along the length of the rod with speed $v$, the other end will not immediately start moving, because any disturbance takes time to propagate along the rod. To be precise, the other end will move after a time $t=L/c$ where L is the length of the rod and $c=\sqrt{\frac{Young's modulus}{density}}$ is the speed of longitudinal disturbance in the rod. What will happen if we (a) pull one end with $v>c$ (b) push the end with $v>c$. (ignoring relativistic effects)
I imagine that in these cases the expression for speed of longitudinal waves I wrote isn't valid. At least in the compression case, intuitively, I think that as the strain increases, maybe the amount of deformation created by a given stress decreases, i.e Modulus of elasticity itself increases with strain after some point and can go arbitrarily high. So the object starts to behave more like a rigid body? I am just thinking out aloud , because I am not an expert on elasticity, deformations etc. I got these questions as I was reading up on waves and sound.

 

[hilbert2]

Speed of sound is constant only in small deformations, as far as I know. So if you apply a sudden very large force on one end of an elastic body, the deformation will not propagate with the usual speed.

 

[sophiecentaur]

The very finite width of a rod will allow the sides to deform laterally and there will be a surface wave which, I think will propagate slower than the compression wave.

 

[jrmichler]

Suddenly pushing one end of a rod at speed V is the same problem as a rod moving at speed V hitting a rigid barrier. The response is no longer a simple elastic wave at rod speeds well below the speed of sound in the rod. When the forces are large enough, the rod is no longer elastic.

An example from the Abaqus FEA example problem manual: http://130.149.89.49:2080/v6.7/books/exa/default.htm?startat=ch02s01aex67.html. The rod is apparently a copper alloy and is hitting a rigid barrier at 340 m/sec. The end of the rod mushrooms.

Another example is a lead bullet hitting a rigid barrier at at 310 m/sec. The lead splatters similar to a jet of water hitting the barrier. The reason the lead splattering became clear when I calculated the forces as if the lead was a liquid with the density of lead at that velocity. The dynamic force (0.5 * rho * V^2) was about 160,000 PSI, while the tensile strength of the .22 LR bullet was only 2,000 or 3,000 PSI. When the dynamic forces are orders of magnitude larger than the tensile strength of the material, the material starts to behave like a liquid. This was from a dynamic measurements class project where I needed to impact a 30,000 lb load cell, and the impact duration needed to be less than 50 microseconds. After the final presentation, we found that the professor's hobby was target shooting.

Of course, if you define the rod as being made from a hypothetical perfectly elastic material with infinite yield strength, then you will have very large elastic deformation.

 

[Terry Bing]

The very finite width of a rod will allow the sides to deform laterally and there will be a surface wave which, I think will propagate slower than the compression wave.

Suddenly pushing one end of a rod at speed V is the same problem as a rod moving at speed V hitting a rigid barrier. The response is no longer a simple elastic wave at rod speeds well below the speed of sound in the rod. When the forces are large enough, the rod is no longer elastic.

An example from the Abaqus FEA example problem manual: http://130.149.89.49:2080/v6.7/books/exa/default.htm?startat=ch02s01aex67.html. The rod is apparently a copper alloy and is hitting a rigid barrier at 340 m/sec. The end of the rod mushrooms.

Another example is a lead bullet hitting a rigid barrier at at 310 m/sec. The lead splatters similar to a jet of water hitting the barrier. The reason the lead splattering became clear when I calculated the forces as if the lead was a liquid with the density of lead at that velocity. The dynamic force (0.5 * rho * V^2) was about 160,000 PSI, while the tensile strength of the .22 LR bullet was only 2,000 or 3,000 PSI. When the dynamic forces are orders of magnitude larger than the tensile strength of the material, the material starts to behave like a liquid. This was from a dynamic measurements class project where I needed to impact a 30,000 lb load cell, and the impact duration needed to be less than 50 microseconds. After the final presentation, we found that the professor's hobby was target shooting.

Ah. Thanks. Lateral deformation makes sense. The simulation was interesting. In the opposite process of pulling (extension), what will happen? Will the object certainly break?

 

[Terry Bing]

Of course, if you define the rod as being made from a hypothetical perfectly elastic material with infinite yield strength, then you will have very large elastic deformation.

Strange things will happen in case we stick to the perfectly elastic material model. if we push one end of the rod with v=c, it will reach where the other end is, even before the other end starts to move. So the rod is effectively reduced to zero volume. (If v>c, I can't even imagine!) Of course, like you mentioned, this model is not valid in the case of large sudden deformations.