Learning topic
Contact Stresses
Learn contact stress and Hertz theory for spheres, cylinders, and rollers: pressure distributions, contact geometry, subsurface stresses, and contact strength.
Subtopics
Contact stresses arise in a localized region through which two bodies transmit a normal force $F$. Because the contact area can be small, local stresses may be much higher than nominal stresses elsewhere in the components.
Contact geometry
Before loading, smooth curved surfaces may touch at a point or along a line. Elastic deformation under $F$ creates a finite contact patch or strip whose shape depends on the local surface curvatures $R_1$ and $R_2$.
Hertz theory
Classical Hertz theory describes local elastic contact of smooth bodies for small deformation and a contact region that is small compared with the characteristic body dimensions. The material properties of the two bodies are combined through the reduced modulus $E^*$, while geometry is represented through the reduced radius $R^*$.
The reduced elastic modulus of two isotropic contacting bodies is:
$$\frac{1}{E^*}=\frac{1-\nu_1^2}{E_1}+\frac{1-\nu_2^2}{E_2}.$$
- $E^*$ — reduced elastic modulus;
- $E_1$, $E_2$ — Young's moduli of the contacting bodies;
- $\nu_1$, $\nu_2$ — Poisson's ratios of the contacting bodies.
Equivalently:
$$E^*=\left(\frac{1-\nu_1^2}{E_1}+\frac{1-\nu_2^2}{E_2}\right)^{-1}.$$
For two convex spherical surfaces along a corresponding principal direction, the reduced radius can be written as:
$$\frac{1}{R^*}=\frac{1}{R_1}+\frac{1}{R_2}.$$
- $R^*$ — reduced radius of curvature;
- $R_1$, $R_2$ — radii of curvature of the contacting surfaces.
Equivalently, for two convex surfaces:
$$R^*=\frac{R_1R_2}{R_1+R_2}.$$
For other combinations of curvature, the sign of an individual radius follows the adopted geometric convention. General three-dimensional contact requires the principal curvatures of both surfaces.
The calculator below corresponds to two convex surfaces with positive radii.
Contact pressure
Pressure $p$ in a Hertz contact region is nonuniform: it reaches the maximum value $p_0$ near the center and decreases to zero at the idealized contact boundary.
For a circular Hertz contact area of radius $a$, the pressure varies as:
$$p(r)=p_0\sqrt{1-\frac{r^2}{a^2}},\qquad 0\le r\le a.$$
The maximum pressure at the center is:
$$p_0=\frac{3F}{2\pi a^2}.$$
- $p(r)$ — contact pressure at distance $r$ from the center;
- $p_0$ — maximum contact pressure;
- $a$ — contact radius;
- $r$ — radial coordinate within the contact patch;
- $F$ — normal force.
At the contact boundary $p(a)=0$, while the maximum $p_0$ occurs at $r=0$.
Point and line contact
Axisymmetric spherical contact produces a circular patch of radius $a$, while long parallel cylinders produce a narrow strip of half-width $b$. Contact dimensions increase as load and material compliance increase.
Subsurface stresses
The contact pressure produces a three-dimensional stress state beneath the surface. Maximum shear or equivalent stresses may occur below the surface rather than at it, which is important for rolling-contact fatigue and pitting.
Contact strength
Machine-component design requires more than the maximum Hertz pressure $p_0$ alone. Repeated loading, material hardness and microstructure, roughness, lubrication, misalignment, residual stresses, and other service factors can strongly affect damage.
Calculation procedure
- Identify the initial contact type and local radii of curvature $R_1$ and $R_2$ of the surfaces.
- Check the assumptions of elastic Hertz theory: small contact area, small deformation, smooth surfaces, and no significant plasticity.
- Calculate the reduced elastic modulus $E^*$.
- Determine the reduced curvature or reduced radius $R^*$ for the relevant geometry.
- Use the normal force $F$, or load per unit length $F'=F/L$ for line contact, to determine the dimensions of the contact area or strip.
- Calculate the maximum contact pressure $p_0$ and, where needed, the pressure distribution $p(r)$ or $p(x)$.
- Assess the subsurface stress state and the contact-strength criterion relevant to the component.
- Check cyclic loading, friction, lubrication, roughness, misalignment, and edge effects when they are significant.
The child topics develop contact geometry, Hertz theory, sphere and cylinder contact, subsurface stress state, and contact strength in greater detail.
About this topic
This section introduces contact stresses in engineering components. It covers contact geometry, reduced elastic and curvature parameters, Hertz pressure distributions, point and line contact, subsurface stress states, and the factors governing contact strength and fatigue damage.