Thermal expansion of a rectangular metal shape

Question

This question is inspired by this interesting answer which left me somewhat uncomfortable for the following reasons:

Suppose we have the following notched metal shape: Following the linked answer, we could decompose this picture into 4 rectangles (Rectangle 1 is the left side, Rectangle 2 is the right side, Rectangle 3 is the empty notch, and Rectangle 4 is the the rectangle directly above the notch) and predict that the gap will increase to length $d$ after some temperature increase.

We could repeat this experiment, modifying the shape slightly by making the notch taller: and imagine doing this until there was an infinitesimally thin "bridge" connecting the two sides. In each step the (horizontal) length of the gap increases after the temperature increase. In fact it can be proven that the resulting length is the same for all the notched configurations*.

However once we get to this picture, the result is that the gap will decrease in length upon temperature increase. This is known and even factored into the design of railroad tracks so that end-to-end tracks won't expand into each other and buckle.

My question How could an infinitesimally thin bridge be fully responsible for the expansion of the (possibly enormous) gulf between left and right sides? If we had many such shapes lying parallel, all with bridges of varying thickness, and a final one with no bridge, could it really happen that upon increasing the temperature, the gaps in all of them expand, except for the final one which contracts?

(Edit in response to a comment+answer; let’s ask what happens if the metal is on top of a frictionless surface and the metal is not fixed to anything. The frictionless setting might help to avoid issues of the bridge needing to overcome the inertia of the sides)

(* to see why the notched gap ends up the same length $d$ no matter how thin the bridge is, consider that $d = M\times \text{originalSideLength}(R_4)$ where $M$ is a constant depending only on the expansivity of the material and the beginning and ending temperatures).

Answer 1

My question How could an infinitesimally thin bridge be fully responsible for the expansion of the (possibly enormous) gulf between left and right sides? ...let’s ask what happens if the metal is on top of a frictionless surface and the metal is not fixed to anything.

When you propose an arbitrarily small force pushing on an object with absolutely zero constraints or resistance, you must expect movement, as predicted by Newton's Second Law. This is certainly counterintuitive because we rarely encounter solid systems with exactly zero equilibrium resistance to displacement. (A rare example is a person pushing a large boat weighing hundreds of tons veeerrrrryyy slowly away from the dock—in still air and water, naturally—with their own strength alone.) But it's the only reasonable expectation.

It's a binary situation: the two sides are either connected by a solid bridge, however minimal, or they aren't. And the corresponding result is also binary: the sides expand collectively around the center of the bridge or they expand individually around their own centers. It's equivalent to asking why an idealized ball sits on level ground but rolls on a slope, however slight. Here, any strain energy stored in a bridge metaphorically tilts the energy landscape to induce separation.

Answer 2

As the bridge is getting thinner, if the centers of mass of the two rectangles are held in place, the bridge will want to expand in length, but it will be constrained by the rectangles. So compressive stress will develop in the bridge. (If the bridge were free to expand, the rectangles would move apart.) For a very thin constrained bridge, the compressive stress will cause it to buckle out of plane. So the force it would then exert on the two rectangles would become even less than if it didn't buckle. And the buckling will allow the rectangles to remain closer together. In the limit of an infinitely thin bridge, the separating force will approach zero even if the bridge did not buckle, and, with the buckling, the rectangles won't even separate.