Is there a formula to calculate the radius that a rod of a given radius can be bent around while staying in the elastic range? For example, if I had a rod that was 1/4" in diameter and made of HDPE (which I think has a modulus of elasticity of 0.8 GPa), how would I calculate the minimum radius the rod could be bent around while still staying in the elastic range so that it could be unwound without deformation?
I think what you need to answer your question is the proper application of Timoshenko beam theory. The beam equations we engineers are most familiar with (Euler-Bernoulli) neglect shear strain and rotation. If you try to apply Euler-Bernoulli theory to this problem, the error will be large. I'm going to do it anyway because my college text didn't cover Timoshenko. For a material with very low shear modulus, this might be OK. HDPE has a shear modulus about 1/600^{th} that of steel. The fact that the cross section is rotating significantly w.r.t. the neutral axis doesn’t stress the HDPE material much compared to steel.
I derived an expression that might be good enough to steer you in the right direction for testing. A 1/4" HDPE rod will bend pretty easily by hand, so you could just start trying it and seeing for yourself. Also, there's going to be some time-dependent behavior of that material, which I'm neglecting here. I assumed the beam (here, rod) would be bent into a circular arc with a sector angle of 90° or pi/2 radians. Then the distance from the midpoint of the chord to the midpoint of the arc is equal to R*(1-0.5*sqrt(2)) where R is the arc's radius. If you set that distance equal to the deflection of a simply supported beam with a uniformly distributed load, make some relational substitutions, solve for R, then you get:
R=0.7*E*d/Y
R is the radius of curvature
E is elastic modulus
d is the diameter of your rod
Y is the yield strength
Again, this is way outside of the realm of what Euler-Bernoulli beam theory models accurately; the deflection is large compared to the beam's dimensions.
Use this just to choose which pipe you grab to start testing with first and go from there.
If you use .8 GPa for E, 26 MPa for Y, d=1/4", then you get an R of 5.4 inches, which seems believable.
Just out of curiosity, suppose it's a 1/4" mild steel rod, with 53 ksi .2% offset yield strength, and 29,000 ksi Young's modulus. Then the expression above gives an R of 96 inches. That might seem like a reasonable number, but keep in mind one of the relationships I used in deriving my expression was that the deflection would be significant compared to the radius of curvature. If R is 96 inches, then the deflection at the middle of the rod is 28 inches, and the rod is 135 inches long to start with. Would an 11-foot ¼” steel rod deflect 2.3 feet in the middle without yielding? I don’t have that piece laying around the shop unfortunately, or I would test it.