How is the theoretical load deflection $P$ calculated for a multi-turn Crest-to-Crest wave spring, and what role does the material's modulus of elasticity play in the linear range?

The theoretical load $P$ for a Crest-to-Crest wave spring is derived from the beam deflection formula adjusted for the sinusoidal geometry. The standard equation is $P = \frac{E b t^3 N f K}{D_m^3 n}$ where $E$ is the Modulus of Elasticity, $b$ is the radial wall, $t$ is the material thickness, $N$ is the number of waves per turn, $f$ is the deflection, $D_m$ is the mean diameter, and $n$ is the number of turns. The Modulus of Elasticity $E$ is critical as it defines the stiffness; for instance, using 17-7PH CH900 ($E \approx 200$ GPa) versus Inconel X-750 ($E \approx 213$ GPa) significantly alters the spring rate $k = P/f$. The linear range is typically maintained between $20\%$ and $80\%$ of the available deflection before bottoming out or entering the non-linear high-stress zone where $f > 0.8(h-t)$.