Compute rolling contact stress via Hertz contact theory. Real-time visualization of contact ellipse semi-axes a,b, peak pressure p₀, subsurface stress distribution σ,τ, and Dang Van fatigue assessment.
Contact Parameters
Contact Geometry
Load P [N]
N
Radius R1 [mm]
mm
Radius R2 [mm]
mm
Max value = flat (∞)
Elastic Modulus E1 [GPa]
GPa
Elastic Modulus E2 [GPa]
GPa
Poisson's Ratio ν1
Friction Coefficient μ
Yield Stress σy [MPa]
MPa
Results
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Contact radius a [μm]
—
Contact radius b [μm]
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Peak pressure p₀ [MPa]
—
Maximum τ [MPa]
—
τmax depth z [μm]
—
Dang Van status
Contact Ellipse Pressure Distribution (Ellipsoidal Model)
What exactly is "rolling contact stress"? I see it in bearing specs, but what's physically happening where the ball touches the raceway?
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Basically, when two curved surfaces like a ball and a raceway are pressed together by a load, they don't just touch at a point. They deform elastically, creating a tiny, flattened contact patch. The stress inside that patch, especially right at the surface and just below it, is Hertzian contact stress. In practice, this is the root cause of rolling contact fatigue (RCF) in bearings. Try increasing the Load (P) in the simulator above and watch how the contact ellipse grows and the peak pressure shoots up.
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Wait, really? So it's not just surface wear? And the simulator shows a red "subsurface" zone... is that more important?
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Exactly! For rolling contacts like bearings, the most dangerous stress is often below the surface. Hertz theory predicts that the maximum shear stress occurs at a depth of about 0.48a for pure rolling, where 'a' is the contact radius. That's where fatigue cracks usually start. A common case is a bearing failing from a subsurface spall that eventually pits the surface. In the simulator, note how the red high-stress zone is buried. Change the Friction Coefficient (μ) to see how surface traction pulls the maximum stress closer to the surface.
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So how do engineers use this? If I have a steel ball on a steel plate, can I just plug in the numbers from the simulator to see if it will fail?
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You've got it. The calculated peak pressure (p₀) is compared to the material's yield stress (σy). If p₀ is much higher, you'll get plastic deformation immediately. For fatigue life, engineers use p₀ in empirical formulas. For instance, a ball bearing's dynamic load rating 'C' is directly linked to the Hertz stress it can withstand for a million cycles. Try this: set both materials to steel (E=210 GPa), give them a small radius, and apply a load so p₀ exceeds the Yield Stress. The simulator will show a safety factor below 1, indicating potential failure.
Physical Model & Key Equations
The core of Hertz theory is reducing two contacting bodies to an equivalent single elastic sphere pressed against a rigid flat. This requires calculating an Equivalent Radius (R*) and an Equivalent Modulus (E*) that capture the combined geometry and stiffness.
E₁, E₂: Elastic moduli of the two bodies. ν₁, ν₂: Their Poisson's ratios. R₁, R₂: Radii of curvature (positive for convex surfaces). A larger E* means stiffer contact, leading to a smaller contact area.
With the equivalent properties defined, we can solve for the Contact Radius (a) and the Maximum Contact Pressure (p₀) at the center of the circular contact patch.
$$
a = \left(\frac{3PR^*}{4E^*}\right)^{1/3}, \quad p_0 = \frac{3P}{2\pi a^2}
$$
P: The applied normal load. Notice that the contact radius 'a' grows with the cube root of the load, while the peak pressure p₀ is proportional to P^(1/3). This is why doubling the load doesn't double the stress; the increased contact area helps share the load.
Real-World Applications
Ball & Roller Bearing Design: This is the classic application. The dynamic load capacity (C) of a bearing, which determines its L₁₀ life (the life at which 90% of bearings survive), is calculated directly from the permissible Hertz contact stress. Engineers use these calculations to select bearing size and preload for applications from electric motors to jet engines.
Gear Tooth Contact Analysis: The meshing of gear teeth is a rolling and sliding contact problem. Hertz stress at the pitch line is a key factor in pitting failure. Modern gear design software uses extended Hertz theory to account for the elliptical contact patch and sliding friction.
Railway Wheel-Rail Interface: The contact between a train wheel and the rail is a critical Hertzian problem. Managing the contact stress is essential to prevent rail head checking, squats, and other forms of rolling contact fatigue that can lead to catastrophic derailments.
Prosthetic Joints (Hip/Knee Implants): The contact between the metal/ceramic femoral head and the polymer/acetal cup is analyzed using Hertz theory to minimize contact pressure. This reduces wear particle generation, which is a primary cause of inflammation and implant loosening.
Common Misconceptions and Points to Note
When you start using this tool, there are a few points beginners often stumble on. First and foremost is the "sign of the radius of curvature". The rule is to input convex surfaces (like spheres or cylinders) as positive and concave surfaces (like grooves or sockets) as negative. Getting this wrong will result in a completely different equivalent radius R*. For example, for a bearing where a ball with a 10mm radius fits into a groove with a 100mm radius, you input R1=+10 and R2=-100. A "flat" plane has an infinite radius of curvature, so you treat it as 1/R2=0. The second point is "the realism of the calculated contact radius a". Increasing the load increases 'a', but for instance, with steel-on-steel, a 1kN load, and a 10mm ball diameter, the contact radius will be on the order of 0.1mm. It's helpful to visualize that the contact you thought was a "point" is actually a tiny "area". The third is "the reliability of material constants". The tool uses convenient default values, but in practice, using the actual Young's modulus and Poisson's ratio for your material is essential. For example, even within aluminum alloys, E can range from 60 to 75 GPa. Finally, don't forget the "premises of Hertzian theory". This is an idealized model assuming perfectly elastic bodies, no slip at the contact interface, and smooth surfaces. In reality, effects like plastic deformation and surface roughness exist, so treat the tool's results as a "first approximation". For critical designs, the golden rule is to verify with CAE or experiments.