# What are the assumptions made when analysing a stretched elastic string from a thermodynamic aspect?

Consider a stretched elastic string. Then conducting a thermodynamic analysis of the elastic string. The approach is very similar to that used for $(P, V, T)$ systems, with the pressure and volume state variables replaced, respectively, by the string's tension J and its length L $(J,L,T)$.

The thermodynamic work can be given as $\delta W=-PdV$ for a $(P,V,T)$ system, analogous to this the work done on stretching an elastic string can be given as $\delta W=+JdL$

What assumptions would one make in both systems to arrive at the corresponding work equations above?

For the former case, I thought about the assumptions made when considering an ideal gas, such as:

• Gas molecules obey newtonian mechanics.
• Volume occupied by each molecule is considered negligible compared to the volume occupied by the entire gas. (i.e. point like gas atoms)
• Collisions are elastic.
• There exists no forces between the atoms/molecules constituting the ideal gas. (Except during collisions)

However I have no idea if this is helpful or not and if it is, how I can relate the following information to the latter system.

Pressure P is a compressive stress, and tension is related to tensile stress. So for the gas, the reversible work done by the surroundings on the gas is -PdV. And, for the elastic string, the reversible work done by the surroundings on the string is TdL. This last equation is the same as force times displacement. For the gas, it can be worked into force-displacement form by writing $$PdV=(PA)\frac{dV}{A}=(PA)dL$$where dL is the displacement of the piston, A is the cross sectional area of the cylinder and piston, and PA is the force acting on the piston.