Library / Mechanical / MEC-94
Mechanical
Hertz maximum contact pressure for a sphere on a flat surface
Peak pressure at the centre of the circular patch where an elastic sphere presses on an elastic flat.
- Ref
- MEC-94
- Rev
- v1
- Field
- Mechanical
- Result
- p0 [MPa]
For a sphere pressed on an elastic flat, the contact pressure is greatest at the centre of the circular patch: p₀ = 3F/(2πa²), which is 1.5 times the mean pressure F/(πa²). Substituting the contact radius a gives p₀ = [6FEeff²/(π³R²)]1/3, with the combined modulus Eeff = [(1 − ν₁²)/E₁ + (1 − ν₂²)/E₂]−1 (often written E*). Inputs are in N, mm and GPa, and the answer is in MPa. The theory is for smooth, non-adhesive, frictionless, linear elastic contact; once the pressure approaches the yield stress it no longer applies.
Peak pressure at the centre of the circular patch where an elastic sphere presses on an elastic flat. It is one and a half times the mean pressure over the patch. The contact is smooth, non-adhesive and purely elastic.
Calculator
- Contact radius, a0.3191 mm
- Mean pressure over the patch, 2p₀/31,563 MPa
- Yield stress at which first yield would begin, about p₀/1.61,465 MPa
- Load that would double the peak pressure, 8F4,000 N
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 peak pressure of 2,344 MPa, 1,563 MPa on average over the patch. First yield begins below the surface when p₀ reaches about 1.6 times the yield stress, so only a very hard material (yield stress above about 1,465 MPa) stays elastic here; a mild steel or aluminium flat would yield.
- The peak pressure rises only as the cube root of the load: eight times the force doubles it (to 4,688 MPa), and making the sphere eight times larger in radius halves it. Increasing the radius is the most effective way to cut contact stress.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (MPa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| R Sphere radius | 0.5 to 200 mm | 318.2 | 17,270 | 16,950 | +10% | −6.16 % |
| F Normal force | 1 to 50,000 N | 295.3 | 10,880 | 10,590 | +10% | +3.23 % |
| E1 Young’s modulus of the sphere | 1 to 500 GPa | 105 | 2,945 | 2,840 | +10% | +3.15 % |
| E2 Young’s modulus of the flat | 1 to 500 GPa | 105 | 2,945 | 2,840 | +10% | +3.15 % |
| ν1 Poisson’s ratio of the sphere | 0 to 0.49 | 2,270 | 2,483 | 212.7 | +10% | +0.698 % |
| ν2 Poisson’s ratio of the flat | 0 to 0.49 | 2,270 | 2,483 | 212.7 | +10% | +0.698 % |
Sweep of Normal force
| Normal force (N) | Maximum contact pressure (MPa) |
|---|---|
| 1 | 295.3 |
| 3.867 | 463.6 |
| 14.95 | 727.6 |
| 57.82 | 1,142 |
| 223.6 | 1,793 |
| 864.7 | 2,814 |
| 3,344 | 4,416 |
| 12,930 | 6,932 |
| 50,000 | 10,880 |
Sweep one variable
Variables and how each one contributes
Maximum contact pressure
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 peak pressure with the cube root of the force: a large increase in load gives only a modest rise in pressure. Raising it by 50 N (to 550 N) moves the result by +3.23 %, from 2,344 to 2,420 MPa.
Sphere radius
The radius of the sphere (not its diameter), in mm. Lowers the peak pressure with R−2/3: a larger sphere spreads the load over a larger patch. Raising it by 1 mm (to 11 mm) moves the result by −6.16 %, from 2,344 to 2,200 MPa.
Young’s modulus of the sphere
Young’s modulus of the sphere material, in GPa (about 210 for steel). Raises the peak pressure: a stiffer sphere deforms less, so the patch is smaller and the pressure higher. Raising it by 10 GPa (to 220 GPa) moves the result by +1.54 %, from 2,344 to 2,380 MPa.
Poisson’s ratio of the sphere
Poisson’s ratio of the sphere material. Raises the peak pressure slightly: a larger ν stiffens the combined modulus. Raising it by 0.02 (to 0.32) moves the result by +0.457 %, from 2,344 to 2,355 MPa.
Young’s modulus of the flat
Young’s modulus of the flat body, in GPa. Raises the peak pressure: a stiffer flat deforms less, so the patch is smaller and the pressure higher. Raising it by 10 GPa (to 220 GPa) moves the result by +1.54 %, from 2,344 to 2,380 MPa.
Poisson’s ratio of the flat
Poisson’s ratio of the flat body. Raises the peak pressure slightly, for the same reason as ν₁. Raising it by 0.02 (to 0.32) moves the result by +0.457 %, from 2,344 to 2,355 MPa.
Worked example with the values above
- Combined modulus: Eeff = 115.4 GPa.
- Contact radius: a = (3FR/4Eeff)1/3 = 0.3191 mm = 3.191×10−4 m.
- p₀ = 3F/(2πa²) = 3 × 500 / (2π × 1.018×10−7) = 2,344 MPa.
Assumptions and limits
- Smooth, frictionless, non-adhesive contact between homogeneous, isotropic, linear elastic bodies.
- Small deformations, with the contact 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.
- Comparing p₀ with the yield stress as if it were a uniform stress: it is a peak compressive pressure, and the maximum shear stress lies below the surface.
- Using the formula beyond first yield, or for conforming contacts such as a ball in a close-fitting race.
Statistics and uncertainty
Because p₀ varies as F1/3R−2/3Eeff2/3, a 3 % error in the radius gives a 2 % error in pressure, and a 3 % error in the load gives 1 %. The moduli of real steels vary by a few per cent, so the pressure is known to about 3 to 5 %. Real contacts are rougher than the theory assumes, which raises local peaks.
Where it is used
- Contact stress in ball bearings, ball screws, rolling contacts and spherical supports.
- Choosing a ball radius or material so that the contact pressure stays below the allowable limit.
Maths you need
- Contact radius: radius of the circular patch of contact
- Combined modulus: a single stiffness for the pair of bodies in contact