What is Hertzian contact stress theory, and what factors does it consider when calculating pressure between two touching surfaces?

Hertzian contact stress theory, developed by Heinrich Hertz, allows us to determine the stress and deformation when two curved surfaces come into contact. The theory uses a set of equations that consider material properties (like Young's modulus and Poisson's ratio), geometry (radii of curvature), and applied forces to calculate contact pressure and the size of the contact area. It assumes homogenous, isotropic materials obeying Hooke's Law, with small deformations and only normal stresses at the contact surface. However, it doesn't account for tangential forces or friction, which could be significant in some real-world scenarios.

How do you apply Hertzian contact stress formulas to determine stress when a sphere presses against a flat plane, and where does the largest shear stress occur in this scenario?

When dealing with a sphere on a flat plane, we can still utilize the sphere-on-sphere formulas from Hertzian contact stress theory. The radius of the flat surface is considered to approach infinity, which simplifies the equations. Using this simplification, you can calculate the maximum contact stress and the radius of the contact area. The largest shear stress doesn't occur at the surface, but at a depth z = 0.47a, where 'a' is the contact area radius, and its magnitude is max τ = 0.31·σο, where σο is the maximum contact stress. Keep in mind this only accounts for normal forces and does not consider any friction.

In the context of Hertzian contact stress, how is the contact area and maximum contact stress determined when two cylinders are pressed together?

In the scenario of a cylinder on a cylinder under Hertzian contact stress theory, the contact area forms a rectangle with width 2a and length l. The contact stress is distributed semi-circularly across this width. To calculate the maximum contact stress (σο), you'd use the formula σο = √(F*E / (2πrl(1-v^2))), where F is the applied force, E is the effective Young's modulus, r is the effective radius, l is the length of the cylinder, and v is Poisson's ratio. The half-width of the contact area (a) is calculated as a = √(8Fr(1-v^2) / (πEl)). If the cylinder has a limited length, 'q = F/l' represents a uniformly distributed linear load, which factors into the stress calculations. The theory simplifies the stress calculation, assuming the cylinders are perfectly aligned and no other forces are in play.

How does Hertzian contact stress theory describe the pressure distribution and maximum contact stress when a curved surface is pressed against a flat plane?

For curved surfaces on a flat plane, Hertzian contact stress theory predicts that the pressure distribution forms an ellipse, following an ellipsoid pattern. The maximum contact stress (σο) is calculated as σο = 1.5|F / (πab)|, where F is the applied force, and a and b are the semi-major and semi-minor axes of the elliptical contact area, respectively. Determining 'a' and 'b' often requires solving more complex equations involving the geometry of the curved surface and the material properties, which is not explicitly covered.

How does the calculation of contact stress change using Hertzian contact stress theory when the materials of the two contacting surfaces are different?

When materials are different, Hertzian contact stress calculations require an effective Young's modulus (E) is calculated using the individual moduli (E1 and E2) as follows: E = 2E1E2 / (E1 + E2). Poisson's ratio (v) is often taken as 0.3 for many materials. The effective Young’s modulus is an important part of the calculations. This approach assumes that both materials deform elastically and the contact area remains small compared to the radii of curvature.

Contact stress between two spheres

Crush Contact Stress: How to Calculate Pressure Between Two Touching Surfaces

Nico Varela in Tech & Innovation September 2025 • 5 min read.

"Understanding Hertzian Contact Stress: A Guide to Calculating Pressure and Deformation"

Have you ever wondered what happens when two surfaces touch? It's more than just a simple meeting. At the point of contact, forces are exchanged, leading to stress and deformation. This phenomenon is crucial in engineering, design, and material science, and understanding it helps us predict how materials behave under pressure.

The study of contact stress dates back to Heinrich Hertz, who developed foundational theories to calculate the stresses and deformations arising from the contact of curved surfaces. His work, known as Hertzian contact stress theory, provides a set of equations that relate material properties, geometry, and applied forces to the contact pressure and the size of the contact area.

This article simplifies Hertz's formulas, focusing on practical applications and insights. We'll explore the key concepts, equations, and assumptions, making it easier for engineers, designers, and curious minds to understand and apply this essential knowledge. Whether you're designing machine components or analyzing material behavior, this guide provides the tools to master contact stress calculations.

Understanding Hertzian Contact Stress: Formulas and Applications

When two objects meet at a single point or along a line, applying force causes them to deform, creating stress within the materials. Developed by Heinrich Hertz, this theory relies on the Boussinesq equations and assumes that the materials are homogenous and isotropic, obey Hooke's Law, and experience only normal stresses on the contact surface. The deformation, or approach, between the two bodies must be small compared to their overall dimensions. When dealing with different materials, an effective Young's modulus, E, is calculated using the individual moduli (E1 and E2) as follows: E = 2E1E2 / (E1 + E2). Poisson's ratio is typically set to 0.3.

Let's examine some common scenarios:

Sphere on Sphere: Use the formula 1/r = 1/r1 + 1/r2, where r1 and r2 are the radii of the two spheres. The maximum contact stress (max στ) is given by max στ = σο = (1.5|F|E^2) / (π|r^2|(1-v^2)^2), where F is the applied force, E is the effective Young's modulus, r is the effective radius, and v is Poisson's ratio. The contact area radius a = 1.5|√((1 – v^2)|F|r / E). If the larger sphere encloses the smaller one (a hollow sphere), r2 is negative.
Sphere on Flat Plane: When a sphere presses against a flat surface, the radius of the flat surface (r2) approaches infinity. In this case, the formulas for sphere-on-sphere contact can still be used, simplifying the calculation. The largest shear stress occurs at z = 0.47a, with a magnitude of max τ = 0.31·σο.
Cylinder on Cylinder: The contact area is a rectangle with width 2a and length l. The contact stress is distributed semi-circularly across the width. Using 1/r = 1/r1 + 1/r2, the maximum contact stress is max στ = σο = √(FE / (2πrl(1-v^2))), and the half-width of the contact area is a = √(8Fr(1-v^2) / (πEl)). If the cylinder has a limited length, q = F/l represents a uniformly distributed linear load.
Cylinder on Flat Plane: Similar to the sphere on a flat plane, the radius of the flat plane (r2) approaches infinity. The corresponding equations can be derived from the cylinder-on-cylinder scenario.
Curved Surface on Flat Plane: When a curved surface contacts a flat plane, the pressure distribution forms an ellipse. The pressure distribution follows an ellipsoid pattern. The maximum contact stress is max στ = σο = 1.5|F / (πab). Here, a and b are the semi-major and semi-minor axes of the elliptical contact area, respectively.

These formulas provide a foundation for calculating contact stresses in various scenarios. They help engineers ensure that components can withstand the forces applied during operation, preventing failures and extending the lifespan of mechanical systems.

Mastering Contact Stress Analysis

Hertzian contact stress theory offers engineers and designers the necessary tools to assess and predict the mechanical behavior of contacting bodies. With this knowledge, one can design robust components, prevent failures, and optimize the performance of mechanical systems. Continue to explore these formulas, test their applications, and refine your understanding to excel in this fascinating field.