Tooth Tip Contact Pressure: Hertz Line Contact Theory Calculation (NFE 25-511)

1. Problem Background

The radial teeth of the NFE 25-511 single-sided toothed conical spring washer press into the mating part and the bolt head/nut surface under preload. A narrow band contact forms between the tooth tip and the pressed surface, resulting in extremely high contact pressure. If the pressure exceeds the material's yield limit, local plastic deformation will occur, affecting the locking reliability and reusability of the washer. Therefore, the tooth tip contact pressure must be calculated based on Hertz line contact theory, and the surface strength must be verified accordingly.

2. Simplification of Tooth Tip Contact Geometry

The tooth tip cross-section can be regarded as a cylindrical surface with curvature radius $R$ in contact with a plane along the radial direction (length $L$), forming a line contact.

• $R$ — Equivalent curvature radius at the tooth tip (mm), determined by the tooth geometry (typical values: 0.05–0.2 mm)
• $L$ — Length of the contact line for a single tooth (mm), i.e., the radial width of the tooth
• $z$ — Number of teeth

Normal force on a single tooth:

Force per unit contact line length (line load):

3. Hertz Line Contact Formula

Hertz solution for line contact between two elastic bodies (cylindrical surface–plane):

3.1 Contact Half-Width

3.2 Maximum Contact Pressure

3.3 Pressure Distribution within the Contact Zone

where $E^*$ is the equivalent elastic modulus:

• $E_1, \nu_1$ — Washer material (typically spring steel, $E \approx 206\,000\ \text{MPa}, \nu \approx 0.3$)
• $E_2, \nu_2$ — Mating part material (similar for steel; for aluminum alloy $E \approx 70\,000\ \text{MPa}, \nu \approx 0.33$)

4. Plasticity Verification Criterion

When the maximum contact pressure exceeds the material's indentation hardness (approximately $2.8 \sim 3.0 \, \sigma_y$, or directly taken as Vickers hardness $H_V$), local plastic yielding occurs at the tooth tip, the contact area expands, and the pressure redistributes.

The allowable contact pressure (elastic range) can be conservatively taken as:

For steel-on-steel contact, if $p_0 > 1800\ \text{MPa}$ (corresponding to a yield strength of approximately 600 MPa), micro-plasticity begins. Typically, tooth tip contact pressure is much higher than this, so local plasticity is inevitable — this is precisely how the tooth tip "embeds" into the surface.

The design should control the plastic zone depth to prevent complete tooth collapse. This is achieved by limiting $p_0$ to no more than $3\sigma_y$ of the material, thereby maintaining macroscopic elasticity.

5. Calculation Example

Given Conditions:

• M10 bolt, preload $F_M = 18\,000\ \text{N}$
• Washer teeth number $z = 12$, tooth tip curvature radius $R = 0.15\ \text{mm}$, tooth width $L = 1.5\ \text{mm}$
• Washer material: spring steel, $E_1 = 206\,000\ \text{MPa}, \nu_1 = 0.3$
• Mating part: medium carbon steel, $E_2 = 206\,000\ \text{MPa}, \nu_2 = 0.3$

Equivalent Elastic Modulus:

Line Load:

Contact Half-Width and Maximum Pressure:

Evaluation:

This pressure far exceeds the yield strength of steel (≈ 1000–1500 MPa), indicating that local plastic deformation at the tooth tip is inevitable. The actual contact half-width will increase due to plasticity, reducing the pressure. Based on experience, the stabilized contact pressure after plastic correction is approximately $p_{ult} \approx 3\sigma_y \approx 3\,000\ \text{MPa}$ (for hard steel), and the tooth tip indentation depth is about 2–3 times the Hertz theoretical estimate.

If the mating part is aluminum alloy ($E_2=70\,000\ \text{MPa}, \sigma_y\approx 300\ \text{MPa}$), then $E^*$ is lower and the allowable pressure is smaller, leading to deeper tooth indentation. Special attention must be paid to the risk of collapse.

6. Simplified Evaluation of Plastic Tooth Tip Contact

When the Hertz pressure exceeds the material hardness, the average contact pressure can be approximated by the material's constrained hardness:

where $C \approx 2.8 \sim 3.0$ (fully plastic constraint). The actual contact half-width $b_{eff}$ can then be determined from force equilibrium:

For the above example (steel–steel, $\sigma_y=1000\ \text{MPa}$):

This is about 4 times larger than the Hertz elastic half-width, and the pressure drops to approximately 3000 MPa, matching the material hardness.

7. Engineering Judgment Criteria

For NFE 25-511 washers, it is generally desirable to enter the micro-plastic state to ensure reliable locking, but macroscopic collapse of the mating part surface must be avoided. Therefore, the preload or number of teeth should be adjusted based on the strength of the mating part material to keep $p_0$ within a reasonable range.

8. Relationship with Surface Pressure Verification (VDI 2230 R10)

Hertz tooth tip contact pressure is a local micro-contact, while VDI 2230 R10's $p_{Bmax} = F_{Mmax}/A_p$ is the nominal bearing surface pressure. They are fundamentally different: - Tooth tip pressure determines the indentation depth and micro-plasticity; - Nominal pressure prevents macroscopic collapse of the entire bearing surface.

Both must be satisfied simultaneously. If the nominal pressure is already close to $p_G$, the additional ultra-high pressure at the tooth tip may cause severe damage to the mating part surface. In such cases, the washer outer diameter should be increased, or a washer with a larger tooth tip radius should be selected.

Summary:
Using the Hertz line contact formula $p_0 = \sqrt{F' E^* / (\pi R)}$, the theoretical elastic contact pressure at the tooth tip can be calculated, allowing assessment of the initial locking state. Since the pressure is extremely high, the tooth tip typically enters micro-plasticity, and the actual pressure is limited by the material hardness. The design should keep the plastic contact pressure below $3\sigma_y$ and coordinate with the overall surface pressure verification to ensure both reliable locking and prevention of mating part collapse.