Free Thermal Expansion and Stress

[RohansK]

This is a question put to me by a colleague of mine which I don't quite know how to answer and has lead to a confusion ( or may be a misconception of physics)

The question is as follows:

When a solid rod is heated it undergoes a Free Thermal Expansion > means that the energy of the atoms increases > leading to a tempreature rise from t1 to t2 > there will be a greater energy the atoms posses at t2 than at t1 > so they overcome the interatomic forces of attraction > their interatomic separation increases > force of attraction between the two atoms is greater at t1 than at t2 > the solid expands linearly FREELY ( ie WITHOUT ANY STRESS INDUCED IIN THE SOLID)

When a solid constrained at one end and loaded at the other end then > the load induces a stress > this overcomes the exsisitng force of attraaction between the atoms > increases the interatomic distance and reach equilibrium when the reactive forces induced equals the load applied ( STRESS IS INDUCED IN THE SOLID)


So What is the difference between the two cases:

When the Interatomic Seperation between the atoms increases in both the cases ( Thermal Expansion and Tensile Loading ) then

Why is Stress not induced in the case of Thermal Expansion when there is an increase in the interatomic separation ( don't just say due to constrain , but explain HOW)

Can anyone explain this citing the DIFFERENCE between the two cases in terms of the interatomic separation and intratomic forces.

 

[Studiot]

Your difficulty lies in your picture of thermal expansion.
Your picture of mechanical loading induced stress is adequate.

Consider the mechanisms by which energy can be stored in a lattice.

The first mechanism is as you describe it for mechanical loading the strain energy is stored as the product of an internal restoring force and the displacement or strain. If you like the bonds are stretched.

Now thermal energy is stored in a different way in the lattice - it does not stretch or strain the bond lengths.
At any given temperature there is a mean bond length which corresponds to the solution of an energy equation. The actual energy is held as vibrational energy of the molecules about their mean position. The mean bond length increases becauses the amplitude of this vibration increases with temperature increase.

If you like mechanical strain energy is potential energy and temperature energy is kinetic energy, both of the molecules.

 

[Mapes]

Now thermal energy is stored in a different way in the lattice - it does not stretch or strain the bond lengths.
At any given temperature there is a mean bond length which corresponds to the solution of an energy equation. The actual energy is held as vibrational energy of the molecules about their mean position. The mean bond length increases becauses the amplitude of this vibration increases with temperature increase.

Thermal energy does stretch/strain bond lengths. If only the vibration amplitude increased, then the average of all the bond lengths would be unchanged and objects wouldn't expand. But the bond lengths do increase due to an asymmetry in the bond energy diagram. See http://books.google.com/books?id=Ze...y energy diagram "thermal expansion"&f=false", for example.

 

[Studiot]

It is difficult to put a complicated idea into simple terms, any simple explanation will blur the edges. You are correct that I should not have implied that all the thermal energy goes into increase vibration, some does appear as increased bond energy.

To me stretching or straining the bond lengths implies that some force has acted to increase the bond lengths from their equilibrium length and is therefore actively holding the molecules further apart than they would otherwise be and to which they would return if this 'force' let go.
This is indeed the 'force' the OP was enquiring about. It is also the case when the material suffers an external load.

In fact this extra force does not exist or need to exist. The increased bond lengths are an equilibrium solution to the energy equation at the new temperature, as I previously stated and as your diagram shows.

You are quite correct in that the diagram is asymmetric, and the resulting vibration is anharmonic. However amplitude does increase with temperature and the energy of the vibration increases with the amplitude.

Remember the question was effectively why do materials expand (thermally) without stress, not why do they expand (thermally) at all.
I am sorry if my clumsiness caused any confusion.

 

[Andy Resnick]

This is a question put to me by a colleague of mine which I don't quite know how to answer and has lead to a confusion ( or may be a misconception of physics)

<snip>

It can be difficult to extrapolate the concept of stress, which is a continuum concept, to that of individual molecules/bonds- that's a discretized concept.

Here's a possible model- a mass attached by a spring to a rigid wall:

http://blog.dotphys.net/wp-content/uploads/2008/10/screenshot-272.jpg

Consider a single atom (the mass) bound to the rest of the object (the wall). At rest, the mass requires no force to be held a distance 'x' from the wall. But, in order to change the distance to a distance x + d, a static force has to be applied to the mass. This is a model of thermal expansion- heat supplies energy to the atom.

If the wall is not free to move, the spring must develop a reaction force; otherwise, the force applied to the mass would cause the mass to accelerate into the wall.

Instead of 'reaction force', continuum mechanics uses 'compressive stress' (or tensile stress, depending on the sign of the strain 'd') to describe the situation.

For free expansion the wall is free to move; thus the bond length changes and there is no reaction force/stress that develops. When the wall is held in place; there is compressive stress developed within the molecular bond.

In terms of energy, the deformation of the bond from it's equilibrium configuration results in a potential energy term (like 1/2kd^2 for the most simple case).

 

[RohansK]

Thanks Mapes and Studiot for your replies.

I want to clarify on a few things. In the first place Studiot you have mentioned molecules for the example purpose, whereas I had a pure solid metal in mind so that the complication of having two different types of atoms bonding together to form a molecule and the molecules then vibrating, doesn't increases.

So we stick to atoms alone as in a pure metal. Secondly by Bond length you mean the average interatomic separation between the two adjacent atoms - right?

Even then if you take molecules for discussion then I think that as Mapes mentioned the thermal energy would partly be used to increase the amplitude of vibration of the molecules as a whole and partly to increase the interatomic separation between the constituting atoms of the molecuels ( bond length as you call it), and essentially the action happening between the atoms of the bond is exactly the same as what happens in increasing the vibrational amplitude of the molecules as whole.

And the total length increase in case of a molecule will be the total of : (Average increase in separation between the molecules due to increased amplitude of the molecules as a unit) + ( average increase in bond length between atoms constitutin each molecule)

And in case of a pure metal (where no molecules but only single atoms are present at each lattice point) the increase in length is purely a function of the increase in the amplitude of vibration - is this though of mine correct?


Secondly with respect to the Potential Energy v/s Atomic Seperation curve shown, What my conception was, is as follows:

1. The force of attraction is maximum at the lowest point on the curve where r = r0 and would go on decreasing for bond lengths ( or mean atomic seperations) greater than r0 ie for r1, r2, r3 etc, though remaining attractice in nature.

2. When any pair of atoms vibrates due to thermal energy they deviate from their mean position of separation and move away from each others > like simple harmonic motion - to and fro about a mean position > (amplitude increases and the attracticve forces start acting on the atoms, which reduce the velocity of the atoms as they move towards their maximum outer position, and in turn due to increasing separation the attractive force acting on the molecules during their progressive displacements outwards keeps decreasing > but in this course the velocity of the atoms keps on reducing till zero ( at outmost position) and they start to travel backwards under the influence of the weak attractive force acting at the maximum displacement

The atoms continue to move towards each others and come closer to each other > attraction force between them increases with reducing distance between them ( as the diagram suggests) > so does increase the velocity of the atoms > just like in S.H.M. > and when they cross the mean position then due to increased velocity (due to increasing attractive forces ) start moving into each others > due to the inertia they keep on moving closer to each other >

Now does the force of attraction decrease as shown in the diagram ? OR as soon as the atoms cross the mean positon the Repulsive Force starts acting which reduces the velocity of the atoms, brings it to zero and then owing to the exsisting repulsive force the atoms are repelled away from each others towards the mean positions ( here r<ro) and > the cycle repeats itself.

Is the concept I mentioned in point no. 2 correct.


Now the real question regarding the P.E. v/s Seperation curve:

But the diagram shows that the attractive force decreases with reducing distance beyond mean position ( in the region where r < ro) and that there is no repulsive nature in the force (and clearly seen in the diagram , as the nature of force remains attractive till Zero Potential energy mark, an only beyond this the Repulsive zone starts).

So if there is NO REPULSION but mere REDUCTION IN ATTRACTION then how does the inwards movements of the atoms stops ( for S.H.M. condition) as there is nothing significant to reduce the velocity to zero and start the reversal of motion towards the mean position.

Then shouldn't it be that as soon as the mean position is crossed ( atoms moving inwards) there should be Repulsive Force starting to act with increasing magnitude as 'r' decreases ?

Please explain / clarify on this. Then we shall move forward with the discussion.


And the main question in my ORIGINAL POST is still UNANSWERED :

Why does Thermal expansion is FREE EXPANSION ie WITHOUT STRESS INDUCED despite the interatomic distance INCREASED just like in a metal rod under Tensile Stress ( where stress is said to be induced due to increase in atomic seperation)

 

[Studiot]

Why does Thermal expansion is FREE EXPANSION ie WITHOUT STRESS INDUCED despite the interatomic distance INCREASED just like in a metal rod under Tensile Stress ( where stress is said to be induced due to increase in atomic seperation)

Mapes, Andy and I are all trying to offer simple versions of the same theory, and yes your question has been answered, I am sorry if not very clearly.

So please reread what has already been said, rather than trying to define the nature of the answer.

A metal rod cannot be subject to tensile stress unless it is subject to an external load.

It is this external load that is the difference between the two situations.

It is this external load that causes the tensile stress.

Contrary to what you have written above, it is this stress that causes the increase in interatomic separation, not the other way round.

If you like we can get more mathematical.

I am using molecules in the chemists sense (it is their word) to mean the smallest unit of a pure substance. This may be a single atom or it may be a chemical compound. And yes bond length equates to interatomic distance.

 

[koolraj09]

Hi,
I would like to reopen this post for the question in case of unrestrained expansion of a metal rod while it is heated. Now when heated the length of rod increases.i.e. we have strain induced in the rod equal to {change in length/original length}. So now that if we have strain induced in it then by hook's law _$$\sigma$$=E_{$$\epsilon$$.}
Now since the R.H.S of this equation is not zero,(since $$\epsilon$$ is not equal to zero and E is also not zero) then there is a stress induced in the rod, even if it is unrestrained.
How come this doesn't happen in practice??
Sorry for not using latex properly.

 

[Mapes]

That version of Hooke's law assumes constant temperature. A more general version is $\epsilon=\sigma/E + \alpha\Delta T$, and the stress is indeed zero for unconstrained thermal expansion.