DIN 9250 Mating Surface Indentation Assessment: Contact Stress and Hardness Limits

1. Indentation Mechanism and Significance of Verification

Under preload, the teeth tips of DIN 9250 serrated lock washers press into the mating part surface with extremely high local pressure, forming permanent plastic indentations. Moderate indentation is a prerequisite for mechanical interlocking anti-loosening, but excessive indentation can lead to:

• Excessive damage to the mating part surface (especially for parts with high surface quality requirements)
• Reduction of effective clamping thickness and additional preload loss
• Decreased positioning accuracy during repeated assembly/disassembly
• Penetration of thin-walled parts or surface-hardened layers, causing corrosion or stress corrosion cracking

Therefore, a quantitative assessment of indentation depth $h_{indent}$ is essential, and it must be controlled within permissible limits.

2. Functional Relationship of Indentation Depth

Indentation depth is a function of the maximum contact pressure at the tooth tip $p_{max}$ and the hardness of the mating part material $HV_{mating}$:

The physical relationship is: the higher $p_{max}$ and the lower the material hardness, the deeper the indentation. The quantitative relationship must distinguish between elastic contact and fully plastic contact.

2.1 Indentation Depth under Elastic Contact (Hertz Theory)

When $p_{max} < 1.6\,\sigma_y$ (approximately $1.6\sigma_y \approx HV/3$), the tooth tip and mating part are in elastic contact. For line contact, the maximum elastic indentation depth (approach of the two contacting bodies' centers) is:

Where $F'$ is the load per unit length ($F_M/(n \cdot w_{tooth})$), $R = r_{tip}$ is the tooth tip radius of curvature, and $b$ is the contact half-width:

Since the $p_{max}$ of DIN 9250 tooth tips typically far exceeds the elastic limit, a purely elastic state is rare; this equation is mainly used for reference.

2.2 Indentation Depth under Fully Plastic Contact (Recommended Method)

When $p_{max}$ exceeds the material hardness, the tooth tip enters plastic flow, and the actual contact pressure is limited to approximately the material's constraint hardness $H_{con}$ (about $3\sigma_y$, or directly converted from Vickers hardness $HV$). The plastic indentation depth is then determined by force equilibrium:

The projected area of the tooth tip $A_{proj}$ is related to the indentation depth $h_{indent}$ and tooth tip geometry (half-angle $\theta = \alpha_{tooth}/2$, tooth width $w_{tooth}$). For a wedge-shaped tooth:

Substituting into the force equilibrium $F_{tooth} = F_M / n$ and solving yields:

Where: - $H_{con}$ — Constraint hardness of the mating part (MPa), can be conservatively taken as $H_{con} \approx H_V$ (Vickers hardness in MPa, 1 HV ≈ 9.81 MPa, often taken as 1 HV = 10 MPa in engineering) - Other parameters are the same as in the tooth tip penetration depth analytical formula (see "Tooth Tip Penetration Depth Analysis" section)

This formula is identical to the penetration depth analytical formula from the previous section, meaning the tooth tip indentation depth is the permanent plastic indentation depth $h_{indent}$.

3. Permissible Indentation Depth

Based on engineering experience and surface quality requirements, the following is recommended for ordinary structural steel connections:

This value considers: - Surface roughness (general machined surfaces $R_z \approx 0.02 \sim 0.05$ mm) - Oxide layer or coating thickness - Avoidance of visually apparent indentations - No impact on positioning accuracy after multiple assembly/disassembly cycles

Special Requirements: - Parts with high surface quality (e.g., sealing surfaces, precision guides): Permissible depth should be stricter (≤ 0.02 mm) - Non-critical external surfaces or single-use connections: Can be relaxed appropriately (≤ 0.10 mm) - Soft materials like aluminum alloys, plastics: Although the permissible value can be slightly larger (e.g., 0.10 mm) due to ease of crushing, controlling surface pressure is more critical to avoid overall collapse.

4. Calculation Example

Given: - M10 bolt, preload $F_M = 18\,000\ \text{N}$ - Number of teeth $n = 12$, tooth width $w_{tooth} = 1.5\ \text{mm}$, tooth tip half-angle $\theta = 30°$ ($\tan 30° = 0.577$) - Mating part: S235 steel, hardness $H_V = 140$, converted $H_{con} \approx 1400\ \text{MPa}$

Calculation:

Reorganizing: Denominator = $2 \times 12 \times 1400 \times 0.577 \times 1.5 = 24 \times 1400 \times 0.8655 \approx 24 \times 1211.7 \approx 29\,080$

This value far exceeds 0.05 mm, indicating excessive indentation under 18 kN preload. Improvement directions: - Reduce preload to 5 000 N → $h \approx 0.17\ \text{mm}$, still exceeds the limit. - Increase the number of teeth to 20 or increase tooth width to reduce force per tooth. - Use a mating part with higher hardness (e.g., quenched and tempered steel 300 HV, then $h \approx 0.29\ \text{mm}$, still large). - In practice, once the DIN 9250 washer tooth tip penetrates to a certain depth, the tooth width increases sharply due to plastic expansion, making the actual indentation much smaller than this simplified calculation. This formula does not account for plastic zone expansion and the self-limiting effect of the tooth, thus overestimating the absolute value. In engineering, it is often used to calculate trends, or a correction factor (typically between 0.3 and 0.5) is determined through testing.

Corrected (factor 0.4): $h_{indent,real} \approx 0.4 \times 0.619 \approx 0.25\ \text{mm}$, still greater than 0.05 mm. This indicates that 18 kN combined with mild steel may indeed be overloaded, requiring parameter optimization.

5. Measures to Control Indentation Depth

1. Reduce tooth tip contact pressure: Increase the number of teeth $n$, increase the tooth tip radius of curvature $r_{tip}$ (blunting), increase tooth width $w_{tooth}$.
2. Increase mating part hardness: Use high-strength steel, surface hardening, carburizing.
3. Limit preload: Minimize $F_{Mmax}$ while maintaining connection functionality.
4. Use an intermediate flat washer: Place a harder flat washer between the serrated washer and the mating part to transfer tooth indentation to a replaceable washer, protecting the main structure.
5. Coating management: Thin coatings (≤ 10 μm) are typically penetrated by the teeth and do not affect indentation depth; thick coatings may hinder penetration but also reduce the anti-loosening effect.

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

Mating surface indentation assessment is control of local plastic deformation, while VDI 2230 R10's $p_{Bmax} \le p_G$ is prevention of overall crushing due to macroscopic surface pressure. They are not interchangeable: - Local indentation may be acceptable (depth ≤ 0.05 mm), but if the overall surface pressure exceeds the limit, the mating part may still undergo overall plastic sinking. - Conversely, macroscopic pressure may be satisfied, but excessive local indentation at the tooth tip can still damage the surface.

Therefore, both criteria must be satisfied simultaneously during design.

Summary:
The indentation depth $h_{indent}$ on the mating surface of a DIN 9250 washer is calculated based on plastic contact force equilibrium and is a function of preload, tooth geometry, and material hardness. The permissible depth is typically limited to 0.05 mm. By reasonably adjusting the aforementioned parameters, the indentation can be controlled within an acceptable range, balancing the anti-loosening effect with surface integrity.