Learning topic
Contact of Two Bodies: Types and Geometry
Point, line, and surface contact: contact-zone geometry, surface curvature, reduced radius, and choosing an appropriate contact model.
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Contact geometry determines the shape of the contact region and strongly influences local pressure $p$. It is important to distinguish the initial geometric contact of unloaded bodies from the finite contact region created by elastic deformation.
Initial point contact
Two curved surfaces, such as two spheres or a sphere and a plane, may ideally touch at one point. Under a normal force $F$, that point expands into a small contact patch. It is circular for an axisymmetric case and generally elliptical when the principal curvatures differ in two directions.
Initial line contact
Two parallel cylinders or a cylinder and a plane ideally touch along a line. Deformation produces a narrow contact strip of finite half-width $b$.
Surface contact
If unloaded bodies already have a finite nominal contact area, the problem may not correspond to the classical Hertz model of initially point or line contact. Pressure then depends on geometry, compliance, restraint, and actual surface conformity.
Curvature
The local shape of a smooth surface is characterized by principal radii of curvature. In simple axisymmetric problems, two surfaces can be combined into one reduced radius $R^*$.
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 convex-concave contact, curvature signs depend on the adopted convention. Very conformal surfaces may produce a large $R^*$ and broad contact area, so the small-contact assumption of Hertz theory requires careful checking.
Material properties
The size of the elastic contact region depends not only on geometry but also on the combined compliance of both bodies, represented by $E^*$.
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:
$$E^*=\left(\frac{1-\nu_1^2}{E_1}+\frac{1-\nu_2^2}{E_2}\right)^{-1}.$$
Choosing a model
- Identify the initial contact type: point, line, or finite area.
- Determine local principal radii of curvature $R_1$ and $R_2$.
- Estimate whether the contact region is small compared with body dimensions and radii.
- For a small elastic patch or strip, consider Hertz theory.
- For conformal contact, sharp edges, large contact regions, or complex geometry, use a more detailed contact model.
Examples
A sphere on a plane is initially point contact; a long roller on a plane is initially line contact; a flat bearing plate on a foundation is surface contact. The same normal force $F$ can therefore produce fundamentally different pressure distributions.