Contact of Cylinders and Rollers

The reduced elastic modulus of two isotropic contacting bodies is:

$$\frac{1}{E^*}=\frac{1-\nu_1^2}{E_1}+\frac{1-\nu_2^2}{E_2}.$$

• $E^*$ — reduced elastic modulus;
• $E_1$, $E_2$ — Young's moduli of the contacting bodies;
• $\nu_1$, $\nu_2$ — Poisson's ratios of the contacting bodies.

Equivalently:

$$E^*=\left(\frac{1-\nu_1^2}{E_1}+\frac{1-\nu_2^2}{E_2}\right)^{-1}.$$

For two convex spherical surfaces along a corresponding principal direction, the reduced radius can be written as:

$$\frac{1}{R^*}=\frac{1}{R_1}+\frac{1}{R_2}.$$

• $R^*$ — reduced radius of curvature;
• $R_1$, $R_2$ — radii of curvature of the contacting surfaces.

Equivalently, for two convex surfaces:

$$R^*=\frac{R_1R_2}{R_1+R_2}.$$

For other combinations of curvature, the sign of an individual radius follows the adopted geometric convention. General three-dimensional contact requires the principal curvatures of both surfaces.

The calculator below corresponds to two convex surfaces with positive radii.

For two long parallel cylindrical surfaces in the plane Hertz contact model, use the normal load per unit length:

$$F'=\frac{F}{L}.$$

The half-width of the contact strip is:

$$b=\sqrt{\frac{4F'R^*}{\pi E^*}}.$$

The maximum contact pressure is:

$$p_0=\frac{2F'}{\pi b}.$$

The pressure distribution across the strip is:

$$p(x)=p_0\sqrt{1-\frac{x^2}{b^2}},\qquad |x|\le b.$$

• $F$ — total normal force;
• $L$ — effective contact length;
• $F'$ — normal load per unit length;
• $b$ — contact-strip half-width;
• $R^*$ — reduced radius in the contact plane;
• $E^*$ — reduced elastic modulus;
• $p_0$ — maximum contact pressure;
• $p(x)$ — pressure at transverse coordinate $x$.

End effects along the roller length are not represented by this idealized two-dimensional model.