Learning topic

Stress State Beneath the Contact Surface

Subsurface contact stresses: normal, principal, and shear components, depth variation, friction effects, and identifying critical regions for strength.

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This topic explains the three-dimensional stress state beneath a contact surface. It covers the localization and depth variation of normal, principal, shear, and equivalent stresses, the role of friction, and why subsurface stresses are important for rolling-contact fatigue and contact-strength assessment.

Beneath a contact surface, a complex three-dimensional stress state develops. Surface contact pressure $p(x,y)$ is only a boundary condition; several normal and shear stress components arise inside the material.

Localization of the field

The highest contact-related stresses are concentrated in a volume whose characteristic dimensions are of the same order as the contact-patch radius $a$ or contact-strip half-width $b$. Stresses decrease rapidly with distance from the contact.

Normal stresses

Within the contact area, compressive normal pressure $p(x,y)$ acts on the surface. Beneath it, three-dimensional elastic interaction also creates normal stresses in other directions, so the state cannot be represented by a single value $-p$.

Shear and principal stresses

Even in frictionless normal contact, differences between principal normal stresses produce nonzero maximum shear stress:

$$\tau_{max}=\frac{\sigma_1-\sigma_3}{2},$$

$\tau_{max}$ — maximum shear stress at the considered point;
$\sigma_1$ — algebraically largest principal stress;
$\sigma_3$ — algebraically smallest principal stress.

In classical Hertz problems, $\tau_{max}$ often occurs at a finite depth rather than directly at the surface. Its exact magnitude and position depend on contact type and Poisson's ratio $\nu$, so a universal numerical value should not be used without specifying the problem.

Why subsurface stress matters

During repeated rolling, the subsurface region experiences a changing multiaxial stress state many times. This can promote fatigue-crack initiation below the surface and subsequent pitting or spalling.

Effect of friction

With tangential force or sliding, surface shear tractions are added to the normal Hertz problem. They change the principal-stress field and may move the critical region closer to the surface. A frictionless model should therefore not be applied automatically to contacts with substantial traction or sliding.

Assessment criteria

Depending on material and damage mechanism, engineers may examine maximum contact pressure $p_0$, principal stresses, $\tau_{max}$, equivalent stress, or specialized contact-fatigue criteria. For ductile isotropic materials, local equivalent stress may be assessed with Tresca or von Mises criteria, but rolling-contact life requires a separate fatigue model.

For ductile isotropic materials under a multiaxial stress state, the Tresca and von Mises criteria are widely used.

Tresca:

$$\sigma_{\mathrm{eq,T}}=\max\left(|\sigma_1-\sigma_2|,|\sigma_2-\sigma_3|,|\sigma_3-\sigma_1|\right).$$

von Mises:

$$\sigma_{\mathrm{eq,VM}}=\sqrt{\frac{(\sigma_1-\sigma_2)^2+(\sigma_2-\sigma_3)^2+(\sigma_3-\sigma_1)^2}{2}}.$$

$\sigma_1$, $\sigma_2$, $\sigma_3$ — principal stresses;
$\sigma_{\mathrm{eq,T}}$ — Tresca equivalent stress;
$\sigma_{\mathrm{eq,VM}}$ — von Mises equivalent stress.

In allowable-stress design, the corresponding equivalent stress is compared with an allowable value consistent with the material properties and the adopted design method.

Practical procedure

Determine the contact-pressure distribution $p(x,y)$.
Use the contact solution to obtain stress components in the subsurface region.
Calculate principal, shear, or equivalent stresses.
Locate the critical point.
Relate the result to the expected damage mechanism and the appropriate strength or life criterion.