Hi Derek.
Yeah, it is an interesting process to consider.
A possible analogy that I have thought of is the bending of flexible pipe. Some use a tubing bender with formed members that match the desired radius. Some use the tubular spring that goes around the outside. I have used sand in the tube for many years and it works very well.
The sand (kern) aids the tube (mantle) retain its shape throughout the bending process. The core won't compress much with the cover becoming more inflexible as the forces come to bear against each other.
SS
Scott, I like your analogy of considering a sand filled pipe. However, I think the kernmantle taking a non zero diameter turn is an almost opposite model.
Consider the soft copper pipe to be filled with a bunch of high tensile steel wires. The pipe is as you say, the braided (mantle) while the wires are the (kern). The cores of some ropes are braided, giving the rope bend flexibility, but my climbing rope is full of a bunch of aligned straight monofibres - like a copper pipe filled with steel wires.
Now we bend them.
In the case of the sand filled pipe, the pipe is ductile, so it responds to the bending forces by thickening on the inside of the bend and thinning on the outside of the bend. The sand is essentially incompressible, but it can flow, so as the inner curve volume shrinks, the outer curve volume expands and the sand flows / shifts but maintains an approximate round pipe section.
If on the other hand we try to bend the pipe filled with steel wires, the tube stretches easily, but the wires inside it refuse to stretch. Because the pipe is full of wires, the tube keeps the wires in their places - the inner wires cannot move or 'give' their excess length to the outer wires that require extra length to accommodate the larger outer radius. We are struck immediately with the fact that the pipe full of thin wires is now behaving as if it were a bar of solid iron.
In some climbing ropes the core fibres are given a light braiding, in others like my kernmantle, the core fibres run straight and parallel from one end of the rope to the other, and although the individual fibres are much more elastic that the iron wires in the coper pipe, the effect is the same as we try to bend the rope, the outer braid constrains the fibres to remain 'in their place', so the outer fibres remain running all the way around the outer radius of the curve.
These fibres are more elastic than the iron wires, so they stretch under the tension imposed by the larger outer radius. But the amount of stretch required to expand the outer radius to accommodate the turn quickly escalates to the working load of the fibres. If I were to use a mechanical system to continue to tighten the curve, eventually I would overload the tensile limit of the outer fibres and they would fail.
A rope however, with a laid or braided core is able to allow the outer tension to be 'fed' from the inner fibres that are under compression, the flow of the braid from one side of the cable to the other is responsible for giving us the amazing flexibility of cordage so critical in its usefulness not only to pass round pulleys, spars etc, but also to pass around the cordage itself and engage in the aspect we are all here for - the creation of knots.
Derek