Hertzian contact

What it is for

The Contact module rates one contact between two elastic bodies on its own: a ball in a groove, a roller on a raceway, crossed cylinders, a sphere on a plate, or any two bodies described by their principal radii. It gives the size of the contact, the peak and mean pressure, how far the bodies approach, the contact stiffness and the stresses below the surface, where rolling contact fatigue starts.

Use it to understand a contact, to try materials and radii quickly, or to check a single contact that is not part of a bearing. The Bearings module solves every rolling element of a bearing; for a ball in its groove both give the same contact.

The module sits in the Connections group of the navigation.

Inputs

Choose + Ball on raceway, + Roller on raceway or + Crossed cylinders at the top of the list as a starting point; every input can be changed afterwards.

Body 1 and Body 2

Each body has a Shape:

Sphere (ball): one radius.
Cylinder: radius and length; a cylinder against a body that is straight along its axis gives a line contact.
Plane: flat.
Groove (raceway): a concave groove given by its radius and its Conformity f = r/D of the ball it carries (0.52 is a typical inner groove).
Torus (crowned wheel), Crowned roller (length and a circular crown by its drop, or a logarithmic profile) and General radii Rx, Ry: principal radii in two planes, convex positive and concave negative.

Each body has a Material (bearing steel 100Cr6, ceramic Si₃N₄, stainless steel, tungsten carbide, grey cast iron, tin bronze, aluminium alloy) or Custom E and ν with your own Young's modulus E and Poisson's ratio ν.

Load and arrangement

Normal load F presses the bodies together.
Angle between the principal planes turns body 2 against body 1 (crossed cylinders at 90°).
For line contacts: Laminae (ISO/TS 16281), Tilt of the axes and Edge stress concentration (ISO/TS 16281, Eq. 60).
Traction coefficient μ (sliding) adds a surface shear μ·p; it moves the highest stress towards the surface. Traction direction sets where it acts: along x, the rolling direction across the minor axis b (across the line of a line contact), or along y, along the major axis a (along the line, as a roller's skew or a spin gives it). Turning a problem by 90° swaps the x and y results.
Curves up to the load sets how far the load curves run.

Requirements

Permissible pressure defaults to 4200 MPa for a point contact and 4000 MPa for a line contact, the values ISO 76 rates bearings with.
An optional Permissible τmax below the surface checks the largest shear stress below the surface.

Results

Peak pressure p0 and the mean pressure, against the permissible pressure.
Contact ellipse a × b, or Half-width b × length for a line contact.
Approach δ: how far the two bodies move together, and the stiffness dF/dδ.
Max shear τmax and von Mises stress below the surface, each with its depth.
Suggested case depth: three times the depth of τmax, a rule of thumb for case-hardened parts.

Warnings under the checks banner say what the contact leaves out and you should look at; they are not failing checks:

Edge loading of a point contact: the contact ellipse is longer than a finite cylinder or roller it lies on, so the cylinder's ends cut it off and raise the pressure at its edges. The point contact does not include that.
Edge loading of a line contact: the line is loaded up to the ends of the shorter cylinder, whose end faces raise the pressure there. Turn on Edge stress concentration (ISO/TS 16281, Eq. 60) under Line contact to raise the end laminae for it, or crown the roller. With it on, the warning becomes a note.

The Notes in the details list the model's other limits, such as a contact large against its radii of curvature, laminae a crown lifts, or a traction on unlike materials (see below).

The Details list shows the principal relative curvatures, the reduced modulus E*, the ellipticity and the exact elliptic integrals, the contact area and mean pressure, for line contacts the laminae, and with a traction its direction and Dundurs' β of the pair.

The three views

3D model: both bodies built from their radii, touching at the contact. Exaggeration magnifies the deformation; Pressure shows the pressure as a coloured dome on the contact patch; Stress slice cuts both bodies along xz or yz and colours the chosen stress below the surface; the rest of each cut face is hatched like the section of a technical drawing, the two bodies' hatching at opposite angles, so the cut reads clearly. Drag to orbit, scroll to zoom.
Stress maps: the stress below the surface in the xz and yz planes, as von Mises, τmax, the orthogonal shear or a single component, with the depths of the peaks marked. Hover to read values.
Curves: the stresses against depth, and the peak pressure, the ellipse and the approach against the load, with the current load marked.

The sliders above the views change the load and the two main radii while you drag; letting go applies the value as one step you can undo.

Example

The Ball and roller on raceways template's first contact presses a 12 mm ball (radius 6 mm) into an inner groove of radius 20 mm with a conformity of 0.52, both of bearing steel, with 2500 N. It gives a peak pressure of 2715 MPa (64.7 % of 4200 MPa), an ellipse of 2.05 × 0.215 mm, an approach of 18.43 µm, τmax 854.9 MPa at 0.165 mm depth, von Mises 1556 MPa at 0.154 mm and a suggested case depth of 0.496 mm.

Standards and scope

Point contacts follow Hertz's theory with exact elliptic integrals; line contacts are sliced like an ISO/TS 16281 roller, so crowned and tilted rollers carry an uneven line load. The stresses below the surface are exact for frictionless contact and include the surface traction when a traction coefficient is given: below an ellipse from Cerruti's solution in either direction, below a line from the Smith–Liu solution across it and the anti-plane shear along it.

Traction on unlike materials. The traction is laid on the Hertz pressure without changing it. That is exact when both bodies have the same elastic constants. For unlike ones a tangential traction also shifts the pressure and the contact (the coupling Dundurs' second parameter β measures, from 0 for alike bodies to about 0.29 for a rigid body on steel), which this model leaves out. When |β| is above 0.1 with a traction, a note says the traction's stresses are approximate: steel on aluminium (β about −0.12), on grey cast iron (about −0.12) or on tungsten carbide (about 0.11), for example. Steel on ceramic Si₃N₄ (about 0.04), bronze (about −0.07) or stainless steel (about −0.02) stays below it.

Not covered: the coupling of traction and pressure on unlike materials (only noted), plastic deformation, residual stresses from hardening, and the edge loading of a point contact longer than a finite cylinder (a warning).