# Question regarding thermal expansion

If I have a hollow steel cylinder and I heat it up. The inner diameter and the outer diameter should increase.

If I put a shell around the cylinder to restrict the expansion,

Q1 Does the inner diameter reduce?

Q2 How to calculate the amount of thermal expansion force that act on the shell?

• I believe this depends quite a bit on the properties of the exterior shell. – JMac Aug 9 '17 at 18:19
• This really isn't a question about the physics of thermal expansion. This is more of an engineering question about the effects of thermal expansion. Suggest moving this question to the engineering stack exchange. – user93237 Aug 9 '17 at 19:08
• As chemomechnics' answer demonstrates, that's a question that forces us to look at the simple models' limits and consider generalizations. This modeling work is definitely physics. – stafusa Aug 9 '17 at 21:35

Q1: Whether the inner diameter decreases depends on the stiffness of the shell. If the shell is perfectly rigid, then yes, the inner diameter decreases, partly because the Poisson's ratio of steel is positive and you've effectively compressed the cylinder hoopwise (i.e., circumferentially) and partly because of the thermal expansion of the thickness of the wall.

(Here, I am assuming that no buckling occurs from this compression. If buckling occurs hoopwise, then the inner diameter would still effectively decrease, but the effect would be strongly location-dependent: think crumpling. I don't think this is where you were going with this question, though.)

If the stiffness of the shell is finite, though, then Q1 becomes more complicated. Let's move on first to Q2 on estimating the magnitudes of the forces to get quantitative and learn how to simplify and model this system.

Q2: If the cylinder wall thickness is very small relative to its diameter, then you can apply the thin-wall assumption and treat the cylinder wall essentially as a plate (whose ends happen to touch). I've written a little about generalized Hooke's Law, which is useful when calculating 3D stresses and strains in linear elastic materials. For normal stresses, generalized Hooke's Law gives $$\epsilon_{11}=\frac{1}{E}\sigma_{11}-\frac{\nu}{E}\sigma_{22}-\frac{\nu}{E}\sigma_{33}+\alpha\Delta T$$ where $E$ is the Young's modulus, $\nu$ is Poisson's ratio, $\alpha$ is the coefficient of linear thermal expansion, and you can rotate through the indices to calculate $\epsilon_{22}$ and $\epsilon_{33}$. To start, though, let's imagine the $1$ direction to be the circumferential direction in any small element and the other directions to be the longitudinal and thickness directions.

Note that this equation captures a key element to thinking about thermal expansion problems (assuming small deformations and linear elasticity): Both stresses and strains are additive; therefore, preventing an object from thermally expanding is identical under these conditions to allowing it to expand and then applying a compressive load to force it back to its original dimension.

When the cylinder is heated alone, under the thin-wall approximation, you have a stress-free plate that expands in all directions (and is then curved into a cylinder and the ends bonded). The diameter $\phi$ increases as $\phi=(1+\alpha\Delta T)\phi_0$, the circumference increases as $(1+\alpha\Delta T)\pi\phi_0$, and the wall thickness $t$ increases as $(1+\alpha\Delta T)t_0$.

If a rigid shell prevents the cylinder from expanding hoopwise, then generalized Hooke's Law tells us that a compressive hoopwise stress of at least $\alpha E\Delta T$ must appear in the thin-walled cylinder to correspond to the constraint of zero hoopwise strain. (Here, things get even more complex―and the at least appears―because the shell may also prevent the cylinder from expanding lengthwise. I'll leave this additional stress to you to calculate and incorporate.) This compressive stress causes Poisson expansion in the wall thickness in addition to thermal expansion of the wall.

OK, so this much allows you to estimate the components and magnitude of wall thickness decrease relevant to Q1 under certain assumptions. In addition, you can calculate that the force associated with this hoopwise stress on the cylinder (with length $L$) must be the stress multiplied by the relevant cross-sectional area, or $\alpha E\Delta T L t$. Perhaps a more convenient parameter, though, is the pressure $P$ exerted by the rigid shell to restrict any outward expansion of a long, thin-walled hollow cylinder. Because $\sigma_\rm{hoopwise}=P\phi/2t$ for these conditions, we have $P=2t\alpha E\Delta T/\phi$. If the long cylinder were sealed, for example, this would be the amount of gauge pressure required to maintain a constant outer diameter of the cylinder for a temperature increase of $\Delta T$.

Since this is getting long enough, I'll stop here and leave the implications of a non-rigid shell and a non-thin-walled cylinder to further discussion. In general, though, generalized Hooke's Law can take you a long way when calculating the relevant stresses and strains (assuming that you stay in the linear regime).