How do you calculate the theoretical spring rate (k) for a Crest-to-Crest wave spring with shim ends, and what are the primary variables affecting linearity?

The spring rate $k$ for a multi-turn Crest-to-Crest wave spring is derived from the formula $k = \frac{E \cdot b \cdot t^3 \cdot N}{D_m^3 \cdot Z^4} \cdot \frac{4 \cdot Z}{N}$, where $E$ is the Modulus of Elasticity, $b$ is the radial wall, $t$ is the material thickness, $N$ is the number of turns, $D_m$ is the mean diameter, and $Z$ is the number of waves per turn. Linearity in the load-deflection curve is maintained between 20% and 80% of the available deflection. Beyond 80%, the 'bottoming out' effect occurs where the waves begin to touch, causes an exponential increase in $k$. Conversely, at low deflections (<20%), the rate may be lower due to the initial settling of the wave peaks against the mating surfaces.