7.1 Introduction: Why Contact Stress Governs Machine Failures

Gears, rolling bearings, cam-follower pairs, railroad wheels, and bolted joints fail not from bulk bending or tension but from localized contact stresses at the interface between two bodies. Hertzian contact theory (1882) predicts:

• Contact area much smaller than body dimensions.
• Peak pressures reaching hundreds of MPa to several GPa.
• Subsurface stress maxima that drive fatigue crack initiation below the surface.

Understanding contact mechanics is essential for:

1. Bearing and gear design — subsurface fatigue (spalling, pitting).
2. Fretting and wear (Chapter 8) — surface damage at contact interfaces.
3. Failure analysis — interpreting spall morphology and crack origin depth.
4. Tribology — bridging to film thickness and friction (EHL).

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7.2 Hertzian Contact: Fundamental Assumptions

Hertz developed his theory for non-conformal (counter-formal) contact between two elastic bodies with:

1. Linear elastic material behavior (Hooke's law, small strains).
2. Frictionless contact (no tangential tractions at the interface in the classical formulation).
3. Smooth surfaces (no roughness — real contact addressed in Section 7.8).
4. Small contact zone relative to body curvatures (half-space approximation).
5. Non-conforming geometry — bodies touch at a point (sphere-plane) or line (cylinder-plane) before loading.

When these assumptions are violated (conformal contact, plasticity, roughness, sliding friction), Hertz theory must be modified or supplemented.

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7.3 Reduced Modulus $E^*$

When two elastic bodies with moduli $E_1$, $E_2$ and Poisson's ratios $\nu_1$, $\nu_2$ come into contact, the combined stiffness is captured by the reduced modulus:

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

7.3.1 Identical materials

For two bodies of the same material ($E_1 = E_2 = E$, $\nu_1 = \nu_2 = \nu$):

$$E^* = \frac{E}{2(1-\nu^2)}$$

For steel ($E = 210$ GPa, $\nu = 0.3$): $E^* = 210/(2 \times 0.91) = 115.4$ GPa.

7.3.2 Rigid body limit

If body 2 is rigid ($E_2 \to \infty$): $E^* = E_1/(1-\nu_1^2)$.

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7.4 Sphere on Flat (Point Contact)

A sphere of radius $R$ pressed against a flat plane with normal force $P$ creates a circular contact patch of radius $a$:

7.4.1 Contact radius

$$\boxed{a = \left(\frac{3PR}{2E^*}\right)^{1/3}}$$

For sphere-on-sphere contact, use the equivalent radius $R = R_1 R_2 / (R_1 + R_2)$.

7.4.2 Peak (Hertz) pressure

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

The pressure distribution is semi-ellipsoidal:

$$p(r) = p_0 \sqrt{1 - \left(\frac{r}{a}\right)^2}, \qquad 0 \le r \le a$$

7.4.3 Approach (displacement)

The relative approach of the bodies:

$$\delta = \frac{a^2}{R} = \frac{3P}{2E^* a}$$

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7.5 Line Contact: Cylinder on Flat (or Two Cylinders)

A cylinder of radius $R$ and length $L$ pressed against a flat (or another cylinder) creates a rectangular contact strip of half-width $a$:

7.5.1 Contact half-width

$$\boxed{a = \sqrt{\frac{4PR}{\pi E^* L}}}$$

7.5.2 Peak pressure

$$\boxed{p_0 = \frac{2P}{\pi a L}}$$

Pressure distribution (plane strain):

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

7.5.3 Gear tooth contact

Spur gear teeth are approximated as cylinders in line contact with equivalent radius:

$$R_{\mathrm{eq}} = \frac{R_1 R_2}{R_1 + R_2}$$

where $R_1$, $R_2$ are the radii of curvature at the pitch point. AGMA and ISO gear standards use Hertzian contact stress for surface durability rating.

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7.6 Subsurface Stress Field and Maximum Shear

The Hertzian pressure at the surface creates a subsurface stress field. For point contact (sphere on flat), along the axis of symmetry ($x = y = 0$):

7.6.1 Principal stresses vs depth $z$

$$\sigma_z = -p_0 \left[1 - \frac{z^3}{(a^2 + z^2)^{3/2}}\right]$$

$$\sigma_r = \sigma_\theta = -\frac{p_0}{2}\left[1 - \frac{z}{\sqrt{a^2 + z^2}} - \frac{z^3}{(a^2 + z^2)^{3/2}}\right]$$

7.6.2 Maximum orthogonal shear stress

$$\boxed{\tau_{\max} = 0.31 \, p_0 \quad \text{at} \quad z \approx 0.48 \, a}$$

This is the classic result: fatigue cracks in bearings and gears initiate below the surface at a depth of roughly half the contact radius, not at the surface.

7.6.3 von Mises stress

Maximum von Mises stress occurs at $z \approx 0.48a$ for $\nu = 0.3$, with value $\approx 0.32 p_0$.

7.6.4 Line contact (cylinder)

$\tau_{\max} \approx 0.30 p_0$ at $z \approx 0.78a$ — slightly deeper than point contact.

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7.7 Assumptions, Limitations, and Extensions

7.7.1 When Hertz theory fails

7.7.2 Effect of friction on subsurface stress

With friction coefficient $\mu$, the maximum shear stress shifts toward the surface and can exceed the frictionless value. This is why sliding contacts (gears under scuffing, cam followers) show surface-initiated failures rather than subsurface spalling.

7.7.3 Elastic-plastic contact

When $p_0$ exceeds $\approx 2.8 S_y$ (von Mises) or $\approx 3.2 S_y$ (Tresca), plastic flow begins. The contact area enlarges and peak pressure is capped at $\approx 2.8 S_y$ (hardness limit). Meyer hardness and indentation theory extend Hertz to the elastic-plastic regime.

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7.8 Surface Roughness and the Plasticity Index

Real surfaces are not smooth. Greenwood and Williamson (1966) modeled contact between rough surfaces as asperity contact:

7.8.1 Composite roughness

$$\sigma = \sqrt{\sigma_1^2 + \sigma_2^2}$$

where $\sigma_1$, $\sigma_2$ are RMS roughness heights of the two surfaces.

7.8.2 Plasticity index

$$\boxed{\psi = \frac{E^* \sigma}{H}}$$

where $H$ is the hardness of the softer material.

• $\psi < 0.6$: elastic asperity contact (Hertz valid at asperity level).
• $\psi > 1.0$: plastic asperity contact (most asperities yield on first contact).
• $0.6 < \psi < 1.0$: mixed elastic-plastic.

7.8.3 Real contact area

The nominal contact area (Hertz prediction) overestimates the real area of asperity contact:

$$A_{\mathrm{real}} = \frac{P}{p_{\mathrm{asperity}}}$$

where $p_{\mathrm{asperity}}$ can approach hardness $H$ for plastic asperities. Real area is typically 1–10% of nominal area for machined surfaces.

7.8.4 Implications for design

• High $\psi$ (soft + rough): high wear, plastic deformation, poor fatigue life.
• Low $\psi$ (hard + smooth): elastic contact, Hertz predictions reliable, good fatigue resistance.
• Shot peening and superfinishing reduce $\sigma$, lowering $\psi$ and improving contact fatigue life.

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7.9 Contact Fatigue: Spalling and Pitting

7.9.1 Spalling mechanism

1. Cyclic Hertzian stress creates subsurface shear at $z \approx 0.48a$.
2. Micro-cracks initiate at material defects (inclusions, carbides) at the subsurface shear maximum.
3. Cracks propagate at $\approx 45°$ to the surface (Mode II shear).
4. Crack reaches the surface → spall (flake of material detaches).

7.9.2 Pitting

Surface-initiated fatigue (from friction, asperity interaction, or lubricant contamination) creates pits — small craters at the surface. More common in gears with boundary lubrication.

7.9.3 Design against contact fatigue

• Limit $p_0$ below the material's contact fatigue strength (from bench tests or standards).
• Ensure adequate lubricant film (EHL — Chapter 8) to separate surfaces.
• Use clean steel (low inclusion content, vacuum remelt) for bearing applications.
• Case hardening (carburizing, nitriding) creates a hard surface with compressive residual stress and a tough core.

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7.10 Design Procedure: Hertzian Contact Analysis

1. Identify contact geometry: sphere-plane, cylinder-plane, or general ellipsoid.
2. Compute equivalent radius $R_{\mathrm{eq}}$ and reduced modulus $E^*$.
3. Calculate contact dimensions: $a$ (radius or half-width) from Hertz formulas.
4. Compute peak pressure $p_0$.
5. Check yield: $p_0 < 2.8 S_y$ (elastic contact) or $p_0 < $ allowable contact fatigue stress.
6. Locate subsurface $\tau_{\max}$: $z \approx 0.48a$ (point) or $0.78a$ (line).
7. Assess roughness: compute $\psi$. If $\psi > 1$, Hertz overestimates performance.
8. Check lubrication: compute minimum film thickness (EHL, Ch.8). If $\lambda < 1$, boundary lubrication — expect surface-initiated failure.

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7.11 Summary

Design takeaway: Never size gears, bearings, or cams on nominal compressive stress ($P/A_{\mathrm{nominal}}$). Use Hertzian $p_0$, check subsurface shear, verify lubrication regime, and assess roughness via the plasticity index. Contact failures initiate subsurface — inspect for spalling at the predicted depth.