Energy needed to increase a volume

[Gh778]

A recipient (cube) of 1m³ is filled of small spheres, there are for example 1000³ spheres inside the recipient. There are also 1000³ elastics that attract the spheres to the bottom. The elastic are always vertical. One elastic for each sphere. One end of the elastic is fixed on a sphere and the other end of the elastic is fixed (but can slide) on the bottom.

I can move in lateral the bottom end of the elastic to have always the elastic vertical. The force of the elastic is supposed constant, just to simplify the calculations, so if the length of the elastic is 0.2m, or 1 m or 1.2 m for example, the force is always 1 N. There is no mass, it is a geometric study, I don't need it and there is no friction.

I increase the volume of each sphere of 20%, only the height of the recipient can change (not the lateral walls), so the height passes from 1 m to 1.2 m. I need for that the energy 550*1000²*0.2*1*1*k, with k the sphere packing and 550 is the mean pressure, at start the height is 1m so the pressure is 0.5 and at final the pressure is 0.6, the mean is at 0.55.

The problem is the potential energy in the elastic, I have less: 0.1*1000³*k. Where I'm wrong ? I must take the integral of the pressure ? I took 1000³ spheres but it is possible in theory to take more just to simplify the calculations and to think with the law of pressure.

 

[.Scott]

I haven't followed everything you posted - but I can contradict one item:
Since the tensions on the elastics remain constant, the pressure will also remain constant.

The only addition to the energy of this system will be the work exerted to lengthen those elastics.
In other words, the energy per unit length stored in the elastics remains constant, but the total length of the elastics has increased.

 

[Gh778]

ok I understood thanks !