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why thermal stress is self equilibrating

Thermal stress does not develop when the free displacement is not constrained. How is this related to the term self equilibrating?


Say, I have a plate with the upper and lower surfaces at two different temperatures. The plate is not having any other constraint to prevent themal expansion. Why thermal stresses will not be generated in this case ( because of temperature gradient)?



tlaverne's picture

Dear ranababu, 

May be I have missed something, but I think it is because your solver

doesn't handle properly the unconstrained displacement. If your displacement aren't constrained, 

then the pure mechanical part of the problem is singular (3 rigid translation and 3 rotations in 3D).  Most modern solvers are able to detect

those kind of singularity, but other don't. May be you should try to fixe at least two arbitrary points in 3D (one in 2D)

to see if in that case you get a  thermal field.



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If there's nothing that prevents the plate from expanding there shouldn't be any thermal stress either. Now I'm not sure what kind of plate you had in mind, as well as whether the plate just starts with different temperatures or if it's heated continually over time. Every little piece of information is golden and will change the outcome. Golden Retriever Information

Dear tlaverne,


Thanks a lit fro your response. My question was intended to find out the physics behind the "self equilibrating" nature of thermal stress i.e. how do we mathematically prove that thermal stresses are self equilibrating.


The way I look at it( say a beam free at both ends whose temp. has been increased uniformly) if we divide the body into infinitesimally small elements, each element is trying to expand under increase in temp. The surrounding material also is trying the same and this results in the same type of stresses at the ends ( compressive stresses at each end for thermal expansion)of these elements.Hence the sum total of stresses on each individual element is zero and hence it is self equilibrating. Is my explanation valid?Kindly advise.


Now coming to the second problem,  the upper and lower surfaces of a plate are at different temperature but the plate is not supported in any way.Hence it will take a curvature . However this curvature will generate self equilibrating stresses. My question is how the stresses that are generated in this case are "self equilibrating" i.e. how can it be explained using freebody diagram?


Kindly advise.



Dear Ranababu

Second question:

With through-tickness thermal gradient and no displacement constraints, Newton's 2nd law reads: sigma F = m * a (0 = 0). The resulting thermal deformation (curvature) is still stress-free (thermal strains will be there, though). Stresses will be generated only if permanent thermal deformation remains in the plate due to an imposed mechanical constraint (think of welding residual stresses or shrink fit problems). In this case, these stresses have to be in self equilibrium since all other mechanical forces are absent, otherwise RBM shall appear (which is essentially impossible now since a mechanical constraint is in place!!)

I hope this helps !


Jayadeep U. B.'s picture

Dear Ranababu,

1. The thermal stresses could develop even when there are no external forces on the system.  Hence they must be self-equilibrating, or else there would be accelerations... Even when there are external loads, we usually decompose the problem into two: one with only thermal stresses and other corresponding to the external loads, and then use the principle of superposition to combine them.  Therefore, we have the thermal stresses to be self-equilibrating in that case also.

2. Please refer to an earlier discussion on a related topic (   As far as I understand, if a deformed configuration is possible, where there are no restraints on thermal deformations, thermal stresses will not develop.  In the problem you have cited, the plate can take the shape of a circular arc, and there won't be any thermal stresses.  Such a configuration should be preferred based on an energy perspective also.  Extending this idea, I believe that we can find more complicated temperature distributions, which do not cause any thermal stresses!



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As others might have pointed out this, it is noteworthy to mention that a simply connected (no holes in the structure) free (unrestrained) structure won't undergo any stress due to temperature if the temperature change distribution is linear, in any of the three independent coordinates, i.e. when T=a+bx+cy+dz (a,b,c, and d are arbitrary coordinate-independent values). In case temperature change is anything but the above relation, even if the structure is free, there will be thermal stress within the body.



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