Select contact geometry (sphere, cylinder, or flat), enter material and load parameters, and instantly compute contact radius, peak pressure, deflection, and subsurface stress distribution.
What exactly is Hertz contact? I see it mentioned for bearings and gears, but what's the core idea?
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Basically, it's the theory that predicts how two elastic, curved surfaces deform when pressed together. In practice, it tells us the size of the contact patch and the pressure distribution. For instance, when a steel ball bearing presses into its raceway, Hertz theory calculates the tiny, flattened area where they actually touch. Try selecting "Sphere on Sphere" in the simulator above and see how the contact radius changes when you increase the Normal Force.
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Wait, really? So the pressure isn't uniform? And what's this "Reduced Modulus" in the formulas?
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Exactly! The pressure is highest at the center and tapers off to zero at the edge—a "Hertzian" elliptical distribution. The Reduced Modulus $E^*$ is a clever trick. It combines the stiffness (Young's Modulus, $E$) and "squishiness" (Poisson's ratio, $\nu$) of both materials into one effective value. In the simulator, if you set one material's $E$ to a very high value (like 1000 GPa), you're simulating a rigid surface, and you'll see $E^*$ approach the modulus of the other material.
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Okay, that makes sense. But why is there a "Yield Strength" parameter? What happens if the pressure is too high?
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Great question! The maximum contact pressure $p_0$ is a critical design limit. If it exceeds the material's yield strength, permanent deformation (like a dent or brinelling) occurs. The tricky part? For spheres, yielding actually starts below the surface, at about half the contact radius deep, where shear stress is highest. A common case is an overloaded ball bearing. In the simulator, watch the "Max. Pressure" result. If it turns red and approaches your input Yield Strength, you know the contact is at risk of failure.
Physical Model & Key Equations
The foundation is the concept of an Effective Stiffness, or Reduced Modulus. It accounts for the elastic properties of both contacting bodies, allowing us to treat the problem as one equivalent elastic surface pressing against a rigid flat.
Here, $E_1$, $E_2$ are Young's Moduli and $\nu_1$, $\nu_2$ are Poisson's ratios. $E^*$ is the effective modulus used in all subsequent contact calculations.
For the classic case of Two Spheres in Contact, Hertz derived the contact radius and peak pressure. The geometry is combined into an effective radius $R^* = \left( \frac{1}{R_1}+ \frac{1}{R_2}\right)^{-1}$.
$$
a = \left( \frac{3 F R^*}{4 E^*}\right)^{1/3}, \quad p_0 = \frac{3F}{2 \pi a^2}
$$
$a$ is the radius of the circular contact area, $F$ is the applied normal force, and $p_0$ is the maximum pressure at the center of the contact. The average pressure is $F/(\pi a^2)$, so $p_0$ is 1.5 times the average.
Frequently Asked Questions
Select "sphere-sphere" and set the radius of one sphere to a very large value (e.g., 1e10 mm). This will make the surface of that sphere approximately flat, allowing you to calculate it as sphere-plane contact.
It shows the depth at which the maximum subsurface shear stress occurs. If this position exceeds the material's yield point, cracks may initiate from below the surface. Use it to evaluate the fatigue life of bearings and gears.
The unit for load is Newtons (N). This simulator is designed only for static normal loads and does not consider dynamic loads or tangential forces with friction. Please use it for pure pressing contact analysis.
For the material you want to treat as rigid, input a very large value for Young's modulus (e.g., 1e15 MPa) and 0.3 for Poisson's ratio. This will make the term for that material effectively zero in the equivalent elastic modulus calculation, correctly computing it as rigid contact.
Real-World Applications
Rolling Element Bearings: This is the quintessential application. Hertz theory is used to size ball and roller bearings, ensuring the contact pressure between the rolling elements and the raceways stays below the fatigue limit to prevent spalling and ensure long service life.
Gear Tooth Design: The contact between meshing gear teeth is modeled as cylinders in contact. Engineers use Hertz formulas to calculate contact stress (often called "Hertzian stress") to prevent surface pitting, a common gear failure mode due to repeated high contact loads.
Automotive Brake Pads & Discs: The initial contact between a new brake pad and a rotor is often analyzed using Hertz contact models to understand wear-in behavior and pressure distribution, which affects braking performance and noise.
Probe Indentation Testing: In materials science, nanoindentation tests use a small diamond tip (modeled as a sphere or pyramid) pressed into a sample. Hertz contact mechanics provides the framework to extract the sample's elastic modulus from the load-displacement data.
Common Misconceptions and Points to Note
First, are you under the impression that "the contact radius is proportional to the load"? In reality, for spherical contact, the relationship is $a \propto F^{1/3}$. Doubling the load only increases the contact radius by about 1.26 times. Conversely, the maximum contact pressure $p_0$ increases proportionally to $F^{1/3}$. The key point is that increasing the load expands the contact area only slightly while causing the pressure to rise sharply. For example, increasing the load eightfold doubles the contact radius, but also doubles the maximum contact pressure.
Next, input errors for material constants. While Poisson's ratio $\nu$ is dimensionless, entering 0.3 as 0.03 will significantly alter the equivalent elastic modulus $E^*$ and greatly skew the results. Also, while the Young's modulus for steel is approximately 206 GPa, always check the tool's specifications to see if you should input "206000" with the unit set to "MPa" or "206" with the unit set to "GPa". Inconsistent unit systems are one of the most common sources of error.
Finally, the applicability limits of Hertzian theory. This theory is based on ideal conditions: "perfectly elastic bodies", "smooth contact surfaces", and "no friction". In practice, factors like surface roughness, lubricant films, and material plastic deformation will cause calculated results to deviate from reality. For instance, even if the calculated maximum contact pressure $p_0$ exceeds the material's yield stress $\sigma_y$, stress may be relieved by localized plastic deformation. Treat the simulator's results as a "first approximation"; the golden rule for critical designs is to verify them with more detailed CAE analysis.