Can an Elastica curve have a discontinuity in the curvature?

Question

This question is related to this one: What are the conditions for the union of two Elastica curves to be an Elastica curve as well?

An Elastica curve is defined as one that minimises the bending energy, i.e. the net curvature squared. Since this integral definition is indifferent to derivatives of the curvature, it would seem that one could have Elastica solutions that have discontinuities in the curvature. But is this viable from the point of view of physical equilibrium, i.e. balancing forces and torques?

Some simple examples of what I am speaking: Consider a straight line (zero curvature) joined to a circle (constant non-zero curvature), or two circles of different curvature joined together, so that the slope is continuous at the joint. Are these valid Elastica curves?

Answer 1

No, a single elastic curve has continuous curvature.

There are two ways of understanding this. The most straightforward is that the solutions to the elastica are known (and have been since Euler). The solution is a fairly straightforward differential equation and has a closed-form solution in terms of elliptic functions. Those functions are continuous.

But it's also possible to understand it from physics principles. Take your example of a line joined to an arc. That certainly has a bending energy, and in fact it's straightforward to cause an elastic strip to take that form, by applying a constraint at the endpoint and another constraint at the join point. But the constraint is applying force (or torque) to the elastic strip - take the constraint away, and as the curve seeks its minimum energy, the discontinuity in curvature will go away. I'm sure it's possible to work out the relationship between the force and the discontinuity, but I don't remember it off the top of my head.

Answer 2

I figured out the physics principle underlying why a strip of material, like an Elastica, should have continuous curvature everywhere.

For this, we need something called the moment-curvature relation present in mechanics. This states the bending moment (or torque) $m$ in a rod -- or any other geometry, for that matter -- is proportional to the local curvature $\kappa$: $$ m \propto \kappa. $$ Discontinuous curvature at any point thus means that the torque applied by one side of the material does not equal that applied by the other side, i.e. there is a net torque on the material at that point. For the rod in equilibrium, this can only happen if there is an external torque being applied at the point.

Conclusion:

Thus, a free rod cannot have discontinuous curvature. There can only be discontinuities in the curvature if there are external torques/forces acting at those points.