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Mechanical
Hertz contact radius of a sphere on a flat surface
Radius of the circular patch where an elastic sphere presses on an elastic flat surface under a normal force.
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- MEC-93
- Rev
- v1
- Field
- Mechanical
- Result
- a [mm]
When an elastic sphere of radius R is pressed on an elastic flat surface with a normal force F, the contact is a circle of radius a = (3FR/4Eeff)1/3, where the combined (reduced) modulus Eeff (often written E*) is Eeff = [(1 − ν₁²)/E₁ + (1 − ν₂²)/E₂]−1. The inputs here are in N, mm and GPa, and the contact radius is given in mm. The theory is for smooth, non-adhesive, frictionless contact between homogeneous, isotropic, linear elastic bodies with a contact much smaller than the bodies; it does not describe plastic indentation.
Radius of the circular patch where an elastic sphere presses on an elastic flat surface under a normal force. It depends on the combined elastic modulus Eeff (often written E*) of the two bodies. The contact is smooth, non-adhesive and purely elastic.
Calculator
- Combined modulus E*115.4 GPa
- Contact area, πa²0.3199 mm²
- Maximum contact pressure, 3F/(2πa²)2,344 MPa
- Approach of the two bodies, a²/R10.18 µm
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A 500 N load on a 20 mm diameter sphere gives a contact patch of radius 0.3191 mm (diameter 0.6383 mm). The load is spread over 0.3199 mm², so the peak pressure is about 2,344 MPa, much higher than the average pressure 1,563 MPa.
- The radius depends on the cube root of the load, so doubling the force raises the contact radius by 26 % but the peak pressure by the same 26 %, not 100 %: contact stress rises slowly with load, yet is still very high. Yielding begins below the surface when the peak pressure reaches roughly 1.6 times the yield stress, and this theory stops being valid beyond that.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (mm) | Local step | Change in result |
|---|---|---|---|---|---|---|
| F Normal force | 1 to 50,000 N | 0.04021 | 1.481 | 1.441 | +10% | +3.23 % |
| E1 Young’s modulus of the sphere | 1 to 500 GPa | 0.2847 | 1.508 | 1.223 | +10% | −1.54 % |
| E2 Young’s modulus of the flat | 1 to 500 GPa | 0.2847 | 1.508 | 1.223 | +10% | −1.54 % |
| R Sphere radius | 0.5 to 200 mm | 0.1176 | 0.8662 | 0.7487 | +10% | +3.23 % |
| ν1 Poisson’s ratio of the sphere | 0 to 0.49 | 0.3101 | 0.3243 | 0.0142 | +10% | −0.347 % |
| ν2 Poisson’s ratio of the flat | 0 to 0.49 | 0.3101 | 0.3243 | 0.0142 | +10% | −0.347 % |
Sweep of Normal force
| Normal force (N) | Contact radius (mm) |
|---|---|
| 1 | 0.04021 |
| 3.867 | 0.06311 |
| 14.95 | 0.09906 |
| 57.82 | 0.1555 |
| 223.6 | 0.244 |
| 864.7 | 0.3831 |
| 3,344 | 0.6012 |
| 12,930 | 0.9437 |
| 50,000 | 1.481 |
Sweep one variable
Variables and how each one contributes
Contact radius
What the equation gives you. Everything else feeds it.
Normal force
The normal compressive force pressing the sphere on the flat, in N. Raises the contact radius with the cube root of the force: doubling the load increases the radius by 26 %. Raising it by 50 N (to 550 N) moves the result by +3.23 %, from 0.3191 to 0.3294 mm.
Sphere radius
The radius of the sphere (not its diameter), in mm. Raises the contact radius with the cube root of R: a larger sphere spreads the load over a bigger patch. Raising it by 1 mm (to 11 mm) moves the result by +3.23 %, from 0.3191 to 0.3294 mm.
Young’s modulus of the sphere
Young’s modulus of the sphere material, in GPa (about 210 for steel, 70 for aluminium). Lowers the contact radius: a stiffer sphere deforms less, so the patch is smaller. Raising it by 10 GPa (to 220 GPa) moves the result by −0.763 %, from 0.3191 to 0.3167 mm.
Poisson’s ratio of the sphere
Poisson’s ratio of the sphere material (about 0.3 for steel). Lowers the contact radius slightly: a larger ν reduces the 1 − ν² term, which stiffens the combined modulus. Raising it by 0.02 (to 0.32) moves the result by −0.228 %, from 0.3191 to 0.3184 mm.
Young’s modulus of the flat
Young’s modulus of the flat body, in GPa. Lowers the contact radius: a stiffer flat deforms less, so the patch is smaller. Raising it by 10 GPa (to 220 GPa) moves the result by −0.763 %, from 0.3191 to 0.3167 mm.
Poisson’s ratio of the flat
Poisson’s ratio of the flat body. Lowers the contact radius slightly, for the same reason as ν₁. Raising it by 0.02 (to 0.32) moves the result by −0.228 %, from 0.3191 to 0.3184 mm.
Worked example with the values above
- Combined modulus: E* = 1 / [(1 − 0.3²)/2.1×1011 + (1 − 0.3²)/2.1×1011] = 115.4 GPa.
- Inside the root: 3FR/(4E*) = 3 × 500 × 0.01 / (4 × 1.154×1011) = 3.25×10−11 m³.
- a = (3.25×10−11)1/3 = 3.191×10−4 m = 0.3191 mm.
Assumptions and limits
- Smooth, frictionless, non-adhesive contact between homogeneous, isotropic, linear elastic bodies.
- Small deformations, with the contact radius much smaller than the sphere radius and the body dimensions; no plastic yielding.
Common mistakes
- Using the sphere diameter in place of the radius R.
- Mixing units, such as moduli in GPa with force in N and radius in m without converting.
- Using the formula after yield has begun, or for a conforming contact where the contact is not small.
Statistics and uncertainty
The radius varies as the cube root of F, R and 1/E*, so a 3 % error in any one of them gives only about a 1 % error in the radius, which makes the contact size a robust estimate. The stress is less forgiving: it depends on F1/3R−2/3E*2/3, and the moduli of real materials vary by several per cent between batches.
Where it is used
- Contact size and stress in ball bearings, rolling contacts and ball-on-plate fixtures.
- Estimating the indentation of a steel ball on a hardness-tester or the local stress under a spherical support.
Maths you need
- Young’s modulus: stress divided by strain in the elastic range
- Poisson’s ratio: sideways strain over axial strain