# Thermally induced stress

1. Jul 25, 2012

### Hassan2

When a bar is heated evenly, and it expands freely, they say it has zero induced stress. How about the stored energy? Isn't some energy stored in the process of expansion? Can this energy be calculated the same way the elastic energy is calculated? If so, why we state that it has zero stress?

Thanks.

2. Jul 26, 2012

### titaniumpen

Like you said, the bar is allowed to expand freely. However, if it isn't (e.g. stuck between two pillars) the bar cannot expand at all. There's a force acting on the bar, and that force causes thermal stress.

3. Jul 26, 2012

### Studiot

That's a good question and here is what I hope is a good answer.

When you heat something that is free to expand you put in themal energy not work.

This energy input is not calculated by the same equations as elastic strain energy.
Think about this, there is zero resisting force so any work that equals the force times displacement (expansion) equals zero.

So where does the energy go?
It raises the temperature of the body.
You calculate this energy input as the specific heat times the temperature change.

To further understand this at a molecular level.

Start with the solid lattice at equilibrium. That means the intermolecular forces are balanced.
The lattice energy is the potential energy held in all these balance internal forces.

Elastic strain energy is extra potential energy held in the matrix of the material as a result of applying an external force and straining it. It represents the non equilibrium increase in average separation between the molecules. That is in a strained solid there exists a restoring force due to the external applied force, because the matrix is no longer in equilibrium without them.

Thermal energy is the kinetic energy of the molecules. In the case of solids it is vibrational energy - they vibrate further and faster as you heat them.
These vibrations are all about the equilibrium position of the molecules.
When the solid expands due to thermal vibrations the lattice or matrix is always still in equilibrium. There are no 'restoring forces'. This equilibrium without restoring forces is maintained even though the molecules are now further apart.

You should not confuse the elastic 'restoring forces' with the restoring forces associated with the simple harmonic motion of vibration. They are separate forces although they stem from the same cause viz the intermolecular forces.

If you think about liquids and gasses, the separation between kinetic and potential energy is even more distinct.
The elastic potential energy is measured by fluid pressure and again the heat energy is measured by the specific heat times temperature change.

Last edited: Jul 26, 2012