---
title: "Contact stresses and Hertz assumptions"
subject: "Mechanics of Solids"
chapter: 11
key: "semester_5/mechanics_of_solids:hertz-contact-stress-notes"
source: https://www.bctengineer.com/q/semester_5/mechanics_of_solids/hertz-contact-stress-notes
site: BCT Engineering
---

# Contact stresses and Hertz assumptions

Mechanics of Solids · Chapter 11: Contact Stresses

- Asked in: year not given, 5 marks
- Video: [Hertz Contact Law Derivation | Why Normal Force Scales as δ^(3/2)](https://www.youtube.com/watch?v=dLjHUyoQZMo) — DEM & Granular Mechanics Lab
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/mechanics_of_solids/hertz-contact-stress-notes
- Whole chapter: https://www.bctengineer.com/md/semester_5/mechanics_of_solids/chapter-11

## Question

Write short notes on contact stresses. State the assumptions of Hertz theory and give the expressions for the contact width and the maximum pressure for (a) point contact between two spheres and (b) line contact between two parallel cylinders. Where do the principal stresses attain their maximum values?

## Short answer

Contact stresses develop when curved elastic bodies are pressed together, with small contact area and high subsurface shear. Assumptions: homogeneous, isotropic, linear elastic, frictionless, small deformations. For spheres: a = (3FR/4E*)^(1/3), p_max = 3F/(2πa^2). For cylinders: b = √(4FR/(πlE*)), p_max = 2F/(πbl). Maximum shear stress is subsurface, at depth ~0.48a (spheres) or ~0.78b (cylinders).

## Answer

#### Contact stresses

When two curved elastic bodies are pressed together, they touch over a very small area, and high compressive stresses (**contact or Hertz stresses**) develop at and below the surface. They govern the life of ball and roller bearings, gears, cams, wheels on rails, and rollers. Failure is by pitting, spalling or surface cracking, usually starting below the surface.

#### Assumptions (Hertz)

1. The materials are homogeneous, isotropic and linearly elastic.
2. The contact area is very small compared with the dimensions of the bodies and their radii of curvature.
3. Surfaces are smooth and frictionless, so only normal pressure acts.
4. Loads are static and act normal to the contact area.
5. Deformations are small. In general the contact area is elliptical; it is a circle for two spheres and a narrow strip for long parallel cylinders.

#### Point contact: two spheres

With radii $R_1, R_2$, load $F$, and $\dfrac{1}{E^*} = \dfrac{1-\nu_1^2}{E_1} + \dfrac{1-\nu_2^2}{E_2}$, $\dfrac{1}{R} = \dfrac1{R_1} + \dfrac1{R_2}$:

$$
a = \left(\frac{3FR}{4E^*}\right)^{1/3},\qquad p_{max} = \frac{3F}{2\pi a^2}
$$

The contact area is a circle of radius $a$, with a hemispherical pressure distribution.

#### Line contact: two parallel cylinders (length $l$)

$$
b = \sqrt{\frac{4FR}{\pi l\,E^*}},\qquad p_{max} = \frac{2F}{\pi b\,l}
$$

The contact is a strip of width $2b$ with a semi-elliptical pressure distribution.

For a cylinder on a plane, $R = R_1$.

#### Stresses

The principal stresses are compressive and are greatest on the load axis, at the surface. They fall off rapidly with depth.

- Line contact: $\sigma_z = \sigma_y = -p_{max}$ and, along the cylinder axis, $\sigma_x = -2\nu p_{max}$.
- Spheres: the stress normal to the surface is $-p_{max}$, and the in-plane radial and tangential stresses are $-\dfrac{(1+2\nu)}{2}\,p_{max}$ (about $-0.8\,p_{max}$ for $\nu = 0.3$).

The **maximum shear stress** is not on the surface but at a depth below it: $\tau_{max} \approx 0.30\,p_{max}$ at $z \approx 0.78b$ for line contact, and $\approx 0.31\,p_{max}$ at $z \approx 0.48a$ for spheres ($\nu = 0.3$). This subsurface shear explains why fatigue cracks start below the surface.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which set of conditions is assumed in Hertz theory?
   A. Rough surfaces with friction and plastic deformation
   B. Large contact area compared with the body sizes
   C. Smooth frictionless surfaces, linearly elastic isotropic bodies, contact area small compared with the radii
   D. Dynamic loads acting at an angle to the contact
   Answer: C. Hertz theory assumes elastic, isotropic bodies, frictionless surfaces, static normal loads and a small contact area. (Key point: Hertz assumptions)

2. What are the contact area and pressure distribution for two spheres?
   A. A strip of width 2b with semi-elliptical pressure
   B. A circle of radius a with hemispherical pressure
   C. A square with uniform pressure
   D. A point with infinite pressure
   Answer: B. Spheres give a circular contact with a hemispherical distribution and p_max = 3F/(2 pi a^2); cylinders give a strip. (Key point: Contact shape)

3. Where does the maximum shear stress occur in contact loading?
   A. At the surface at the centre of contact
   B. At the edge of the contact area
   C. At a depth below the surface, about 0.78b for line contact
   D. At the far end of the body
   Answer: C. tau_max (about 0.30 p_max) occurs below the surface, which is why fatigue cracks start subsurface. (Key point: Location of maximum shear)

4. For line contact, what are the surface principal stresses on the load axis?
   A. All tensile and equal to p_max
   B. About -p_max in two directions and -2 nu p_max along the cylinder axis, all compressive
   C. Zero in all directions
   D. +p_max and -p_max
   Answer: B. The stresses are compressive, with sigma_z = sigma_y = -p_max and sigma_x = -2 nu p_max. (Key point: Surface principal stresses)

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