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Stresses in Elastic Materials in Non-uniform Cooling



I am having confusions about the concepts of thermal stresses of an un-constrained elastic body when heated/cooled.

Let's say if I have an elastic body of an arbitrary geometry. If I subject it to a uniform temperature change without any constraints, then there will no thermal stresses at the end of the analysis. I guess I am right till here.

Now, let's say if I have an elastic body at an initial temperature say 500C at t=0 secs. I cool the entire body to room temperature say 25 C at t=100secs. Here the cooling is transient and has been brought by a non-uniform temperature change within the body, let's say by convection or conduction. Hence the temperature change is not uniform as in the first case, but let's assume that through the non-uniform temperature change, the elastic body has been brought to 25C from 500C.

Now, will there be any thermal stresses in the body? 






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Jayadeep U. B.'s picture

Dear Shrimad,

Thermal stresses develop when there are constraints on free thermal expansion or contraction.  Assuming isotropic elastic materials, there are three common situations (or their combinations), when thermal stresses develop:

  1. When there are external constraints on the body, which undergoes temperature change.
  2. When the body undergoing temperature change is made of materials with different coefficient of thermal expansion (internal constraint).
  3. Non-uniform temperature distribution in a homogenous body (internal constraint).

However, even non-uniform temperature distribution need not cause thermal stresses always.  For example, a linear temperature variation across the thickness of a slab, with uniform temperature distribution in other directions, will not cause any thermal stresses.  So the bottom-line is: thermal stresses are developed due to the constraints on free thermal expansion/contraction, and I think this general rule is applicable even for anisotropic or inelastic bodies.

To answer your question, in my opinion, the body you considered will not have any thermal stresses in the initial and final conditions, however, there could be thermal stresses developed during the temperature change, depending on the kind of non-uniform temperature changes present.

Hope this comment is helpful.  Regards,


I think there is no thermal stresses in the body whether it is heated uniformly or non-uniformly, unless you apply some constraints.

Imagine you devide this body into two half, since there is no constraints on the original surface of this body, there should not be any stress on the new surface of the body either, according to the static equilibrium. So you can see there is no stress within this body.


Thanks a lot for the replies. I think I get the idea, but I still wanted to clarify.

I was thinking that there will be no stresses if I cool the body (which is initially at a uniform temperature and stress free) through non-uniform temperature distribution, and then bring it back to the original temperature through the same temperature distribution through heating. This simply follows because the body is elastic.

Now, when I cool the body to lets say room temperature from an initial uniform temperature through the non-uniform temperature distribution, each point inside the body will expand or contract differently depending on the temperature at that point, hence generating transient thermal stresses inside the body. But my doubt is, will these stresses vanish as the entire body reaches room temperature or would these transient stresses affect the final stress state/shape of the body?








Jayadeep U. B.'s picture

Hi Shriram,

You are hinting at the possibility of residual
stresses (even though there is a change in temperature).  For the
conditions mentioned in your problem, there is no scope for residual
stresses.  Also, please remember that the elastic response is
independent of the loading history, and should depend only on the
initial (undeformed) and final configurations.  I believe that the
thermal stresses are no exception.  Therefore, if the stresses during
the process remains within the elastic limit, there will not be any
thermal stresses in the final configuration.  However, if yielding
occurred at any point during the process, there could be locked in
stresses in the final configuration




 I am working on a similar topic on what you guys are discussing. To my understanding, when we heat/cool a body, it is mostly by convection or radiation. In any of these cases, the outer surface of the body is heated/cooled first and the heat is transferred to the inside/outside by conduction (depending on the conductivity of the material).  During this process of conduction from outside to inside or vice-versa, there are transient thermal stresses arising. And in many cases these transient thermal streses may result in buckling of the part. These transient thermal stresses depend upon the temperature you are heating the sample (nearing to bonding temperature) or the rate at which you are heating the sample. These transient thermal stresses also depend on the co-efficient of thermal expansion. Since the properties of a material change with temperature (co-efficient of thermal expansion), expansions of the sample at different locations vary.

Hope I solved some of your basic questions. Please correct me if I am wrong.



- Gopi

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