Technology / Technical guide
Hertz Contact Mechanics: Area, Pressure & Elastic Conditions
A curved contact that appears to touch at a point in rigid geometry develops a finite loaded area when the bodies deform. Hertz theory describes a particular elastic contact model. Its local pressure is different from a force divided by an arbitrary whole-component area.
Let contact area change with the elastic load.
Geometry, elastic properties and force are linked in the ideal model. This explains why contact pressure need not scale linearly with load when the area changes, and why the geometry and assumptions must remain visible.
01 / Finite contact
Loaded elastic geometry differs from rigid touching.
A rigid sphere touching a plane has a geometrical point of contact.
A rigid sphere touching a plane has a geometrical point of contact. Elastic deformation creates a finite circular area in the ideal normal-contact case. Parallel cylinders instead have a line-contact representation with a finite width and a different load-per-length formulation.
Abaqus’s Hertz verification example explicitly separates the cylinder geometry, pressure distribution and near-contact stresses. Its numerical example does not identify an actual generator bearing.
02 / Assumptions
A Hertz model has a restricted physical scope.
The elementary model uses smooth, nonconforming surfaces, small contact dimensions relative to the relevant body geometry, linear elastic material behaviour and the stated normal-load condition.
The elementary model uses smooth, nonconforming surfaces, small contact dimensions relative to the relevant body geometry, linear elastic material behaviour and the stated normal-load condition. Frictionless normal contact does not include every possible tangential traction or sliding effect.
Plastic deformation, roughness, adhesion, coatings, thin layers or strongly conforming contact can require other descriptions. A general bearing label does not guarantee all these assumptions. Naming the Hertz model identifies an approximation, rather than a product certification or lifetime method.
03 / Elastic quantities
Both bodies contribute to the contact compliance.
For the isotropic sphere–plane model, an effective modulus satisfies 1/E* = (1 − ν1²)/E1 + (1 − ν2²)/E2.
For the isotropic sphere–plane model, an effective modulus satisfies 1/E* = (1 − ν1²)/E1 + (1 − ν2²)/E2. Each E and Poisson ratio belongs to its own body. The Inria-hosted CompuTiX theory presents this relation with explicit geometry.
A rigid-plane limit removes that plane’s elastic contribution, but does not make the sphere rigid. Conversely, inserting one material’s E without the associated Poisson factor would generally represent a different compliance. Direction-dependent or layered materials need a model appropriate to those conditions.
04 / Radius scaling
Load and area grow together in the ideal sphere case.
For sphere radius R and compressive force magnitude F, contact radius is a = [3FR/(4E*)]^(1/3) in the stated ideal model.
For sphere radius R and compressive force magnitude F, contact radius is a = [3FR/(4E*)]^(1/3) in the stated ideal model. Holding R and E* fixed, increasing F eightfold doubles a and increases the circular area fourfold.
That comparison is hypothetical scaling, not a permissible load change. The contact indentation also changes, and the small-contact assumptions still have to remain applicable. A cylinder case uses different geometry and cannot inherit this sphere’s cube-root relationship merely because both are called Hertz contacts.
05 / Pressure shape
Peak pressure differs from the area average.
In this ideal circular contact, mean pressure is F/(πa²), while centre peak pressure is p0 = 3F/(2πa²).
In this ideal circular contact, mean pressure is F/(πa²), while centre peak pressure is p0 = 3F/(2πa²). The radial distribution decreases from that peak toward zero at the contact edge. The peak is therefore 1.5 times the mean for this stated shape.
With eightfold force and fourfold area in the preceding comparison, mean and peak pressures double. They do not increase eightfold. This is an original consequence of the ideal relationships, not a measured bearing load, material allowable or predicted damage threshold.
06 / Scope
A local contact is only one part of the assembly.
An actual bearing distributes forces across several contacts and its surrounding structure.
An actual bearing distributes forces across several contacts and its surrounding structure. Internal clearance and preload can affect that distribution, while lubrication changes additional contact behaviour. The preload guide and film guide retain those separate mechanisms.
No MARSTON bearing geometry, E*, contact force, contact pressure or elastic limit is assigned here. A local elastic contact solution alone cannot determine complete bearing life or machine structural safety.
Related technical guides: Blade Root Stress Concentration: Nominal & Local Quantities · Bearing Internal Clearance, Preload & Elastic Contact · Bearing Lubrication Films: Full Film, Mixed & Boundary States.
General background: Abaqus: ideal local elastic cylinder-contact scope · Inria-hosted CompuTiX: sphere–plane Hertz relationships. These references explain industry concepts; they are separate from the applicable MARSTON product and project documents.
Key concepts and conditions.
Read each quantity with its reference, units and stated conditions. A general relation does not establish a model-specific value.
| Concept / quantity | Meaning / relation | Conditions / limits |
|---|---|---|
| Rigid touching | Point or line geometry | Loaded elastic contact has finite area. |
| Effective modulus | E* in Pa | Both bodies and Poisson ratios contribute. |
| Sphere contact radius | [3FR/(4E*)]^(1/3) | Restricted elastic sphere–plane model. |
| Mean pressure | F/(πa²) in Pa | Uses the loaded contact area. |
| Centre peak | 3F/(2πa²) in Pa | 1.5 times the mean for the stated shape. |
Engineering questions.
Does geometric point contact mean infinite loaded pressure in the Hertz model?
No. Elastic deformation produces a finite contact area under its assumptions.
Can sphere and cylinder load relations be interchanged?
No. Their geometry and load representations differ.
Does a rigid plane make both contacting bodies rigid?
No. The other body’s elastic contribution remains.
Does eightfold force produce eightfold pressure in the ideal sphere case?
No. Contact area also changes; fixed R and E* give doubled peak pressure.
Is the Hertz peak identical to force divided by a whole bearing area?
No. Its reference is the local loaded contact area and pressure shape.
Does one contact calculation establish bearing life or product approval?
No. Complete assembly conditions and appropriate failure evidence remain separate.
Technical enquiry
Discuss the technical question.
A curved contact that appears to touch at a point in rigid geometry develops a finite loaded area when the bodies deform. Share the model and applicable operating or design information for an engineering discussion.