Thermal Expansion of Two Holes

[Hlud]

I have a problem understanding the concept of thermal expansions of solids with a hole in it. I do not want to accept face value the traditional explanation i am given that hole will increase in size because i am told to imagine if the hole was never cut. I fail to see the congruency in these two examples.

My biggest issue can be posed as this question. Imagine a metal with two holes drilled in it, with a small amount of the metal (distance x) separating the two holes. When the metal is heated, the holes presumably will expand as well. Does this mean the holes expand into the metal, shortening this distance x?

 

[Student100]

I have a problem understanding the concept of thermal expansions of solids with a hole in it. I do not want to accept face value the traditional explanation i am given that hole will increase in size because i am told to imagine if the hole was never cut. I fail to see the congruency in these two examples.

My biggest issue can be posed as this question. Imagine a metal with two holes drilled in it, with a small amount of the metal (distance x) separating the two holes. When the metal is heated, the holes presumably will expand as well. Does this mean the holes expand into the metal, shortening this distance x?

No, the holes won't expand into the metal, the entire system expands. So what this means is the distance X is also expanding.

Think about what would happen were no holes cut into the metal.

 

[Doc Al]

I do not want to accept face value the traditional explanation i am given that hole will increase in size because i am told to imagine if the hole was never cut.

Imagine that instead of a hole there was just an outline of a hole. What would happen to that outline as the solid expands?

 

[Hlud]

The reason i do not accept it face value is because i do not see the congruency of those two situations. I understand the situation. Imagine if there was no hole, but actual metal. But why are they congruent?

 

[Student100]

The reason i do not accept it face value is because i do not see the congruency of those two situations. I understand the situation. Imagine if there was no hole, but actual metal. But why are they congruent?

Here is a good post from AlephZero:

There are materials (e.g. many crystals, and some composite matierals) where the coefficient of thermal expansion is different in different directions. In those materials, in general the hole would not expand uniformly, and would change shape.

But apart from that:
If the solid plate is at a constant temperature and the edges are free to expand, the stress in the plate will be zero everywhere, independent of the temperature.

So, imagine that you draw the circle, then change the temperature of the plate, then cut out the hole around the expanded or contracted shape of the line. Since the stress in the plate was zero everywhere before cutting, it will stay zero after cutting, and cutting the hole wll not change the shape of the plate.

Or more mathematically:
If there is no elastic stress in the plate, the strain field is $\epsilon_{xx} = \epsilon_{yy} = \epsilon_{zz} = \alpha \Delta T$ and the shear strains are all zero, where $\alpha$ is the corefficient of expansion and $\Delta T$ the temperature change.

Because the strain field is symmetrical in x y and z, this means that ANY two points that were originally a distance $d$ apart become a distance $d(1 + \alpha \Delta T)$ apart, indepedent of the shape of the object, and whether or not it contains holes.

 

[Hlud]

Ahh, that helps me understand it conceptually. Thanks for that!

 

[Hlud]

So, i am having a lot of trouble understanding conceptually that thermal expansion of a metal will also cause a hole in that metal to expand. So i put forth this example. Imagine two slabs of concrete, that are long and wide. If i put them next to each other, i would like to have an expansion joint in between them, right?

Now, imagine i put a small concrete connecter on each end of these slabs, so technically we now have one concrete slab with a really big hole in it. Previously, the separation between these two slabs decreased. With our new connecting pieces, the separation (in our case, the hole) increases.

Does this work around the need for an expansion joint? I mean, can we just have really tiny width expansion joints, since the hole will increase in size? Or am i not understanding how the expansion of holes really works? Thanks for any help.

 

[Nidum]

What actually happens with your two slabs and the links depends on details of the whole assembly .

If slabs are free to slide along then the whole assembly just gets a bit longer .

If slabs are anchored at the outer ends then inboard ends will try to move together but are prevented from doing so . In this case compressive stresses will be induced in the links and slabs .

In the real world constrained thermal expansion can generate very high and often dangerous stress levels .

The old type railway tracks had expansion joints everywhere to prevent thermally induced stresses from buckling the rails .

Expansion joints are also commonly seen on road bridges . In part these are for the same purpose as in rail tracks .

(Purely for interest their other purpose is to allow the bridge to change shape slightly under different traffic loads without inducing end loads in the structure .)

 

[A.T.]

So, i am having a lot of trouble understanding conceptually that thermal expansion of a metal will also cause a hole in that metal to expand.

Here is a good way to visualize this:
http://physics.stackexchange.com/a/22436

 

[russ_watters]

With our new connecting pieces, the separation (in our case, the hole) increases.

Does this work around the need for an expansion joint? I mean, can we just have really tiny width expansion joints, since the hole will increase in size? Or am i not understanding how the expansion of holes really works? Thanks for any help.

Only if you think you can move the entire slab of concrete without breaking it!

 

[Hlud]

What actually happens with your two slabs and the links depends on details of the whole assembly .

If slabs are free to slide along then the whole assembly just gets a bit longer .

So, does this mean, if the links were not there, and the slabs were still free to slide along, then the separation will get bigger as well?

 

[russ_watters]

So, does this mean, if the links were not there, and the slabs were still free to slide along, then the separation will get bigger as well?

No, it gets smaller.

The reason the hole gets bigger when there is a link is that the link expands. So in order for the link to expand without buckling, it would need to be really, really strong.

 

[Doc Al]

So, i am having a lot of trouble understanding conceptually that thermal expansion of a metal will also cause a hole in that metal to expand.

Try this. Imagine a square sheet of metal. Now divide that square into 9 smaller squares, in a 3 x 3 arrangement. So the original square is now 9 smaller squares. Do you agree that when heated each small square will expand? And thus the original square expands?

Now remove the center square and redo the thought experiment. Do you agree that when heated each small square (8 of them now) will expand? And thus the hole in the middle (the missing square) must have expanded as well?

 

[Khashishi]

I think what might be confusing you Hlud is that the answer depends on what kind of stresses are being applied. Since you didn't mention any sort of external barriers, we assume that the metal is simply free to expand in all directions. In this case, every feature in the metal will simply expand together. The whole thing just becomes bigger.

Now if you put the metal in some kind of box that doesn't expand, then you try to heat the metal to expand it, then there will be all sorts of compressive forces keeping the metal from expanding. Then, if the box doesn't break, maybe the holes will get smaller as the material is forced inward because there is nowhere else for it to go.

When you are talking about expansion joints, those are needed because there are external constraints. Different parts of the ground expand at different rates, since the ground isn't one material and it isn't at one temperature.

 

[Nidum]

Just consider a ring of metal . When heat is applied metal expands and increases circumference of hole .Hole gets bigger .

Even easier consider a square ring each side expands so square gets bigger .

 

[Hlud]

Here is a good way to visualize this:
http://physics.stackexchange.com/a/22436

Let me explain why i have trouble with this analogy, using one of mine own. Imagine nine squares, each containing a baby. In our square, the baby is given enough space for it to live in. As the baby grows into an adult, the nine squares are expanded to accommodate the bigger humans living inside. Because there is a human inside the center square, the designers of these nine squares decide to build out.

Now going back to your link. Assuming there is no hole, we see similarity to my analogy above. However, what happens if we first remove the center square, thus creating a hole. Let's map the original eight squares on top of the expanded eight squares. Clearly we can see that all of the eight squares did not just expand, but expanded outward only as if the center square was still there!

No, it gets smaller.

The reason the hole gets bigger when there is a link is that the link expands. So in order for the link to expand without buckling, it would need to be really, really strong.

So, in other words, the four squares in contact with the missing center square can be thought of as the 'links'. Ok, that is making a lot more sense. Does this mean that the four squares in contact with the missing center square have more stress on them?

 

[russ_watters]

So, in other words, the four squares in contact with the missing center square can be thought of as the 'links'. Ok, that is making a lot more sense. Does this mean that the four squares in contact with the missing center square have more stress on them?

There is more stress on them than if the center square was there, yes -- because stress is a function of area and there is less area when there is a hole. Note that in the expanding sheet of metal examples, there are no stressess because the entire sheet is allowed to expand unconstrained -- a concrete sidewalk, however, is not so free to expand.

 

[Nidum]

A common practice in engineering is to shrink fit components together .

Typically a shaft into a flange or pulley .

Several ways of actually doing it but simplest is to heat flange or pulley so that the hole in it increases in diameter enough to allow shaft to be easily assembled into place .

As flange or pulley cools the hole in it closes down again in diameter and a very strong interference fit on shaft is generated .

 

[Hlud]

There is more stress on them than if the center square was there, yes -- because stress is a function of area and there is less area when there is a hole. Note that in the expanding sheet of metal examples, there are no stressess because the entire sheet is allowed to expand unconstrained -- a concrete sidewalk, however, is not so free to expand.

Ok, then that makes a lot of sense. Now it ties in what AlephZero said. Thanks for the help!

 

[A.T.]

Clearly we can see that all of the eight squares did not just expand, but expanded outward only as if the center square was still there!

That is the only way they can expand, while preserving their square shape and interconnection with the other squares. You should read the explanations in the link, especially the part about expanding rigid body vs. expanding fluid/dough/clay.