Hertz Subsurface Stress Simulator — σz, σr and τmax versus Depth
Visualize the centerline depth profile of σ_z, σ_r and τ_max under a Hertzian sphere contact in real time. The peak shear stress sits at z≈0.48a — the origin of rolling-contact fatigue (pitting).
Parameters
Max contact pressure p_max
MPa
Contact radius a
mm
Poisson's ratio ν
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Evaluation depth z/a
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Elastic solution on the contact axis (r=0) for sphere contact (Johnson, Contact Mechanics). At the surface z=0, σz = -p_max.
Results
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σ_z (axial stress)
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σ_r (radial stress)
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τ_max (max shear stress)
—
τ_max peak depth z*/a (ν=0.3)
Centerline stress profile σ_z, σ_r, τ_max versus depth
X = z/a / Y = stress / p_max (red = σ_z, blue = σ_r, green = τ_max, orange dash = τ_max peak at 0.48, yellow dash = current evaluation depth)
Theory & Key Formulas
Stresses on the contact axis (r=0) of a Hertzian sphere contact. Here a is the contact radius, p_max is the maximum contact pressure, ν is Poisson's ratio and z is depth:
Maximum shear stress on the axis (half of the principal-stress difference):
$$\tau_{\max} = \frac{|\sigma_z - \sigma_r|}{2}$$
For ν=0.3, τ_max/p_max ≈ 0.31 occurs at z*/a ≈ 0.48, which is the origin of rolling-contact fatigue (pitting and spalling).
What is the Hertz Subsurface Stress Simulator
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When a ball rolls through a bearing, the surface gets pressed the hardest, right? So shouldn't failures start from the surface?
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It feels that way, but it is actually the opposite. Yes, the compressive stress σz reaches its maximum −p_max at the surface (z=0). But the shear stress that actually tears metal apart is largest a bit below the surface. With the default values p_max=1500 MPa, a=0.5 mm, ν=0.3 and z/a=0.5, the simulator should show σ_z = −1200 MPa, σ_r = −270 MPa and τ_max = 465 MPa. The peak of the green curve (τ_max) sits around z/a≈0.48 — that is the subsurface shear peak.
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Why exactly 0.48a? Does it fall out of the math?
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Yes, straight out of the elastic solution in Johnson's "Contact Mechanics". On the centerline, σz and σ_r=σ_θ have different depth profiles. At the surface the two are close together so τ_max is small. Going down, σ_z drops slowly while σ_r tends to zero much faster, so the difference τ_max=|σ_z−σ_r|/2 peaks at an intermediate depth. Slide the ν parameter and you can see σ_r — and the peak depth — shift. ν=0.5 moves it near 0.55a, ν=0 down to about 0.38a.
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How does this connect to bearing or gear life?
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It is the origin of rolling-contact fatigue (RCF). Every time a ball rolls over a point, a shear-stress pulse rises near 0.48a beneath the surface. Repeat that 10^6 to 10^9 times and tiny cracks nucleate at that depth, propagate to the surface, and finally pop a flake of metal off — that is bearing "flaking" or gear "pitting". In practice we keep p_max below 1.5–3 GPa so τ_max stays well under half the yield stress. And because a non-metallic inclusion (MnS, for example) at that depth accelerates crack initiation, clean bearing steels (VIM-VAR and similar) are used in long-life applications.
Frequently Asked Questions
Yes. For sphere contact, τmax/p_max ≈ 0.31 occurs at z/a ≈ 0.48, whereas for 2-D cylindrical line contact τmax/p_max ≈ 0.30 occurs at z/b ≈ 0.78 (where b is the contact half-width). Line contact therefore has a deeper shear peak, and roller-bearing damage initiates deeper than in ball bearings. This tool covers sphere contact only — see the cylinder mode of hertz-contact.html for line contact.
In pure rolling (zero friction), the ideal Hertz peak sits at z/a≈0.48. Adding sliding or tangential traction shifts the maximum shear stress toward the surface. At friction coefficients of about μ≈0.3, τmax reaches the surface and causes "surface-origin" pitting or micro-pitting. In gear design the lubricant film thickness ratio Λ is used to judge whether subsurface or surface origin dominates.
The classical RCF life theories (Lundberg-Palmgren) used the maximum shear stress τmax or the orthogonal shear stress τ0. Modern theories (Ioannides-Harris, the ISO 281 life modification factors) use the von Mises stress or stress-volume integrals. As a first-pass design rule, keeping τmax/p_max ≈ 0.31 (sphere contact, ν=0.3) below the design allowable also keeps the von Mises stress below about 0.62·p_max, well within the yield-safe envelope. This tool displays the most basic quantity, τmax.
Rolling bearing life design: In ball and roller bearings, the subsurface shear stress τmax generated by the contact between rolling elements and raceways is the origin of fatigue cracks. The Lundberg-Palmgren theory and ISO 281 predict the L10 life from the depth distribution of τmax (or τ0) and the number of rotations. Sweeping p_max in the simulator and watching τmax gives an intuitive feel for how the design contact pressure drives life.
Gear tooth pitting assessment: A gear mesh exposes the tooth flanks to repeated near-line Hertz stresses. The shear-stress peak around depth 0.5b is the typical origin of pitting (pit-shaped surface flaking). AGMA / JIS gear-strength calculations compare the surface (contact) stress with the material's fatigue strength to compute a safety factor, but the subsurface stress theory shown by this tool sits behind that calculation.
Cam-follower wear analysis: Combustion-engine cams against rocker arms, mechanical-press cam mechanisms and other machine elements with repeated near-line contact mix sliding and rolling, so both surface-origin and subsurface-origin damage modes appear. Subsurface stress analysis is the starting point, on top of which the effects of lubricant film thickness and surface roughness are layered to predict damage.
Wheel-rail contact in railways: The contact between a wheel tread and the rail head is a classic Hertzian contact, with p_max reaching 1-2 GPa. τmax appears a few millimetres below the rail head and is the origin of internal rail fatigue defects (detail fractures, transverse defects). Stress analyses essentially identical to this tool are used routinely in rail life prediction and maintenance planning.
Common Misconceptions and Cautions
The most common misconception is the short-circuit "maximum stress = maximum failure risk". The peak compressive stress σz = −p_max does occur at the surface (z=0), but pure compression contributes very little to metal failure. Ductile metals fail in shear, and τmax peaks slightly below the surface at z≈0.48a. Watch the green curve in the simulator carefully: τ_max is small at the surface, peaks at depth 0.48a and then decays slowly. The origin of fatigue failure is "where the shearing is strongest", not "where the pressing is strongest".
Next, do not forget that this tool only shows stresses on the contact axis (r=0). The full Hertzian stress field is three-dimensional. At the edge of the contact circle (r=a, z=0) a tensile stress arises — its maximum is at the surface, stretched radially — and this drives cone cracks (the failure mode of ceramics and glass) and surface-origin fatigue. Sliding or rolling shifts τmax toward the surface as well. What this tool shows is the ideal "pure-rolling, zero-friction, on-axis" case; real machines layer friction, sliding, residual stress and material inclusions on top.
Finally, mind the limits of the Hertzian elastic solution. Hertz theory assumes a perfectly elastic body, smooth surfaces and a small contact (a≪R). Once p_max exceeds about 1.6 times the yield stress (the plasticity onset for sphere contact, p_max ≈ 1.60·σ_y, Tabor), the material yields and the stress field redistributes. For steel, p_max > 4-5 GPa already enters the plastic regime. The simulator allows p_max up to 5000 MPa, but in real structural steel that value is well into yielding — read the results as a "what-if perfectly elastic" reference, not a literal stress.