Tools › Structures › Contact › Contact Stress › Example
Hertz and Johnson: steel sphere on a flat, contact size, pressure and subsurface shear maximum
| Tool | Contact Stress |
| Source | Johnson, K. L. (1985). Contact Mechanics. Cambridge University Press, Cambridge. Sections 3.4 (Hertz pressure on a circular region) and 2.7 (Hertz line contact); surface stresses after Huber, M. T. (1904). |
| Level | Graduate |
Problem
A steel sphere of 10 mm radius (E = 207 GPa, ν = 0.3) is pressed against a flat of the same steel by 1,000 N. Find the contact radius, the peak pressure and the approach, the depth and value of the maximum shear stress, and the surface radial stresses at the centre and at the edge of the contact.
Inputs to Enter
- Case: sphere on a flat; R_1 = 10 mm (flat R_2 blank); P = 1,000 N; μ = 0
- Both bodies E = 207 GPa, ν = 0.3 (SI units)
Answers
- a = (3 P R / 4 E*)^(1/3) = 0.404 mm; p_0 = 3 P / (2 π a²) = 2,925 MPa; δ = 16.3 μm
- τ_max = 0.31 p_0 at z = 0.48 a (Johnson §3.4: 0.31 p_0 at 0.48 a; tool within 1 %)
- Surface radial stress at the centre −(1 + 2ν) p_0 / 2 = −0.80 p_0, at the contact edge the tension (1 − 2ν) p_0 / 3 = 0.133 p_0 (Huber; within 0.2 %)
- Line contact (cylinder on flat): τ_max = 0.300 p_0 at z = 0.786 b and the orthogonal shear 0.25 p_0 at z = 0.50 b, x = 0.87 b (Johnson §2.7)
Notes
The tool opens on a ball-on-raceway case; choose the sphere-on-flat case and SI units. The numbers are the library checks of test_hertz_contact.php (the printed a, p_0, δ) against Johnson's and Huber's closed forms.