Question 1

Where does the maximum Hertz contact pressure p₀ occur?

Accepted Answer

The peak contact pressure p₀ occurs at the center of the contact area (x = 0). For sphere contact: p₀ = 3P/(2πa²) = (6PE*²/π³R*²)^(1/3). The pressure distribution follows a semi-elliptical profile p(x) = p₀√(1−(x/a)²). For two identical steel spheres (R = 20 mm, E = 200 GPa, ν = 0.3, P = 1000 N), this gives a ≈ 127 μm and p₀ ≈ 9.4 GPa.

Question 2

At what depth does the maximum subsurface shear stress τ_max occur?

Accepted Answer

For sphere contact, τ_max ≈ 0.310 × p₀ occurs at depth z* ≈ 0.480 × a below the surface. For cylinder contact, τ_max ≈ 0.300 × p₀ at z* ≈ 0.786 × a. This subsurface stress is the initiation site for rolling contact fatigue (RCF) cracks in bearings and gears. For p₀ = 3 GPa in sphere contact, τ_max ≈ 930 MPa at approximately 0.48a below the contact surface.

Question 3

What are the limitations of Hertz contact theory?

Accepted Answer

Hertz theory assumes: (1) linear elastic bodies, (2) contact area small relative to radii (a ≪ R), (3) frictionless surfaces, and (4) negligible surface roughness. Errors exceed 5% when a/R > 0.1. The theory is invalid when plastic deformation occurs (p₀ > ~1.1 × yield strength) or surface roughness Ra is comparable to a. Practical bearing design applies dynamic load factors and surface finish corrections to the Hertz stress.

Question 4

What is the difference between sphere–flat and sphere–sphere contact?

Accepted Answer

Sphere–flat contact is the special case of sphere–sphere with R₂ → ∞. The equivalent radius R* = 1/(1/R₁ + 1/R₂) reduces to R* = R₁ when R₂ = ∞. For R₁ = 20 mm and P = 1000 N on steel: sphere–flat gives R* = 20 mm, while sphere–sphere (R₂ = 20 mm) gives R* = 10 mm. The sphere–flat case therefore has a contact radius a ≈ 1.26× larger and peak pressure p₀ ≈ 0.63× (about 37% lower) than equal-radius sphere–sphere contact.

Hertz Contact Stress Calculator

Theoretical Background