Hertzian contact explained

Hertzian contact stresses describe the localised stress distribution that occurs when two curved surfaces (e.g., spheres, cylinders, gears) are pressed against each other.

This theory, developed by Heinrich Hertz in the late 19th century, is used to predict the behaviour of materials under compressive loads where contact is made over a small area, like in bearings, gear teeth, or ball and roller systems.

Hertzian contact theory is primarily applied when dealing with elastic materials, meaning that the deformations caused by contact will disappear when the load is removed. However, at high loads or in certain cases, plastic deformation can occur.

Key Concepts

When two elastic bodies come into contact, the area of contact is not a point but a small region, due to the deformation of the surfaces. Hertzian stress analysis is used to determine the contact area, the pressure distribution within this area, and the maximum stresses in the bodies.

The key assumptions in Hertz’s theory are:

The materials are homogeneous and isotropic.
The contact area is small compared to the dimensions of the bodies.
The surfaces are frictionless, so only normal stresses are considered.

Types of Contact Scenarios

Sphere on Flat Surface (Point Contact): A spherical object contacting a flat surface results in a circular contact area.
Cylinder on Flat Surface (Line Contact): A cylindrical object contacting a flat surface results in a rectangular contact area, typically much longer than it is wide.
Two Spheres: When two spherical objects come into contact, the contact area is circular.
Two Cylinders (Line Contact): Two cylindrical objects in contact produce a line or rectangular contact area, with stresses distributed along the line of contact.

Hertzian Contact Stress Equations

1. Contact Between Two Spheres (Point Contact)

Consider two spheres of radii $R_1$ and $R_2$ , made of materials with Young’s moduli $E_1$ , $E_2$ and Poisson’s ratios $\nu_1$ , $\nu_2$ , under a normal load $F$ .

Contact Radius (a):
The radius of the circular contact area is given by:

$a = \left( \frac{3FR}{4E^*} \right)^{\frac{1}{3}}$

where $R$ is the reduced radius:

$\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}$

and $E^*$ is the reduced modulus of elasticity:

$\frac{1}{E^*} = \frac{1 - \nu_1^2}{E_1} + \frac{1 - \nu_2^2}{E_2}$

Maximum Contact Pressure (p₀):
The maximum contact pressure occurs at the centre of the contact area:

$p_0 = \frac{3F}{2\pi a^2}$

Stress Distribution:
The contact pressure at a point within the contact area is given by:

$p(r) = p_0 \left( 1 - \frac{r^2}{a^2} \right)^{\frac{1}{2}}$

where $r$ is the radial distance from the centre of the contact area.

2. Contact Between a Sphere and a Flat Surface

For the case of a sphere of radius $R$ contacting a flat surface (assumed to be an infinite plane), the equations are simplified versions of those for two spheres. The contact radius $a$ and maximum pressure $p_0$ are calculated similarly:

$a = \left( \frac{3FR}{4E^*} \right)^{\frac{1}{3}}$

$p_0 = \frac{3F}{2\pi a^2}$

The stress distribution also follows the same form as for two spheres.

3. Contact Between Two Cylinders (Line Contact)

In the case of two cylinders of radii $R_1$ and $R_2$ , the contact area forms a rectangle rather than a circle. The pressure distribution along the length of contact is similar to that of two spheres, but the expressions for the contact width $2b$ (since the contact is a line, the “contact radius” is replaced by a contact half-width $b$ ) and maximum pressure change.

Contact Half-Width (b):

$b = \sqrt{\frac{4FR}{\pi L E^*}}$

where $L$ is the length of the cylinders, $R$ is the reduced radius, and $E^*$ is the reduced modulus, as defined above.

Maximum Pressure (p₀):

$p_0 = \frac{2F}{\pi L b}$

Stresses Beneath the Contact Surface

In Hertzian contact theory, the contact stresses beneath the surface can be quite complex. There are three principal stresses:

Radial Stress ( $\sigma_1$ ): The stress directed radially outward from the centre of contact.
Tangential Stress ( $\sigma_2$ ): The stress directed tangentially along the contact surface.
Normal Stress ( $\sigma_3$ ): The stress in the direction normal to the surface of contact.

These stresses are highly concentrated near the surface and tend to decrease rapidly as depth increases. The maximum shear stress typically occurs below the surface, which can lead to subsurface fatigue and cracking under repeated loading.

Applications of Hertzian Stress Theory

Bearings: Rolling-element bearings, such as ball or roller bearings, rely on Hertzian stress calculations to predict load capacity and fatigue life.
Gear Teeth: The contact between gear teeth can be analysed using Hertzian theory to ensure that the teeth can withstand the applied loads without excessive wear or failure.
Railway Wheels: The contact between the wheel and the rail is another example where Hertzian stress theory is applied, particularly for wear prediction and crack formation analysis.

Limitations of Hertzian Contact Theory

Elastic Assumption: Hertzian theory assumes that the materials remain elastic under the applied load. For loads beyond the elastic limit, plastic deformation occurs, invalidating Hertz’s predictions.
Frictionless Contact: Real-world contact often involves friction, which can alter the stress distribution, especially under tangential loading.
Contact Geometry: Hertzian theory assumes that the bodies are initially non-conforming. In cases where the bodies are heavily loaded or conforming, the theory needs modifications or alternative methods.

Conclusion

Hertzian contact stresses provide essential insights into the stress distribution between curved surfaces under compressive loads, making it a vital tool for engineers and designers in fields like mechanical systems, material science, and tribology. By understanding the relationships between contact area, load, material properties, and stress, Hertz’s theory helps predict failure points, optimise design, and extend the life of components under repeated or high loads.

Product Development Engineers Ltd

Key Concepts

Types of Contact Scenarios

Hertzian Contact Stress Equations

1. Contact Between Two Spheres (Point Contact)

2. Contact Between a Sphere and a Flat Surface

3. Contact Between Two Cylinders (Line Contact)

Stresses Beneath the Contact Surface

Applications of Hertzian Stress Theory

Limitations of Hertzian Contact Theory

Conclusion

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Hertzian contact explained

Key Concepts

Types of Contact Scenarios

Hertzian Contact Stress Equations

1. Contact Between Two Spheres (Point Contact)

2. Contact Between a Sphere and a Flat Surface

3. Contact Between Two Cylinders (Line Contact)

Stresses Beneath the Contact Surface

Applications of Hertzian Stress Theory

Limitations of Hertzian Contact Theory

Conclusion

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