Don thanks for the kind words. Well deserved, you are doing a very good job of keeping this discussion on debate points.
A spring's spring rate goes up with the displacement.
If so, please explain how a simple linear "milk" (spring) scale works.
At 2" displacement the spring rate might 100lb/in. at 3" displacement the spring rate might be 500 lbs/in.
Only if the spring is designed to be non-linear. Such springs can be made, but they require collapse of some element(s) of the spring.
So my contention that the receiver gets stiffer (spring rate goes up) as the receiver is flexed more is correct.
Only if the receiver acts as a non-linear spring. Since there would be no element to collapse, I seriously doubt that the receiver would be non-linear in bending.
In the beam example the displacement (stretching) at the bottom of the beam is twice as much at the surface as it is half-way between the surface and the line that is neither in compression or tension.
And since the surface has had the yield point raised by case hardening the surface can stretch more before it yields and goes into plastic deformation. And since the point half way between the surface and the zero tension line is only stretching half as much as the surface it doesn't stretch enough to reach it's yield point even though it's yield point is lower.
I understand you to be saying that the first point to yield would be the area immediately below the case layer; the first point where there is no heat hardening.
If so, I agree. However, this point/layer will yield before the case layer has sufficient deflection to add significant extra force. The layer/case of heat hardened steel (carbon penetration required) would have to be quite thick to add increased force resistance.
DDA
And since the maximum deformation is at the surface and since the spring rate of the metal in the surface goes up as it deforms (like a spring has a higher spring rate at 3" displacement than it does at 1") the bulk of the load is carried by the exterior of the frame.
Don I repeat, it is not a simple matter of calculating the percentage of the cross section that has been case hardened and then pro-rating that increased yield strength into the strength of the frame. The exterior of the frame carries more because it has %deformed more and so is in higher tension than the interior. The interior point at the zero tension line is not providing any resistance at all because it is, by definition, at zero tension.
So case hardening a frame increases it's ability to resist plastic deformation.
