DIN 9250 Tooth Tip Penetration Depth Analysis: Plastic Indentation Model

1. Formula Definition and Physical Background

In the assembly of DIN 9250 toothed lock washers, radial teeth penetrate the surface of the mating part under the action of preload, creating mechanical interlocking. Tooth tip penetration depth $\delta$ is the core parameter for evaluating engagement quality: too shallow results in insufficient slip resistance, while too deep may crush the mating part or cause permanent deformation of the washer teeth.

Based on a simplified plasticity mechanics approach, the penetration depth can be expressed as:

Where: - $\delta$ — Depth of tooth tip penetration into the mating part (mm) - $F_M$ — Bolt preload (N) - $n_{teeth}$ — Number of teeth on the washer - $H_V$ — Vickers hardness of the mating part material (MPa) - $\alpha_{tooth}$ — Full tooth tip angle (°), the angle between the two tooth flanks - $w_{tooth}$ — Load-bearing width of the tooth (radial tooth length, mm)

This formula assumes the tooth tip as a rigid wedge indenting an ideally plastic material, where the indentation resistance is entirely provided by the plastic flow pressure (hardness) of the mating part. Friction and elastic recovery of the material are neglected.

2. Physical Meaning and Typical Values of Parameters

Note: The Vickers hardness $H_V$ used in the formula should be converted to pressure units. If $H_V$ is given in kgf/mm², multiply by 9.81 to obtain MPa; for quick calculations, multiply the numerical value by 10 to convert to MPa.

3. Derivation of the Formula

Approximate the single tooth indentation as a wedge punch indenting a semi-infinite body. Assume the preload is uniformly distributed among all teeth, so the normal force per tooth is $F_{tooth} = F_M / n_{teeth}$.

Consider the tooth tip geometry: with a tooth tip angle of $\alpha_{tooth}$ and penetration depth $\delta$, the projected contact area $A_{proj}$ between the tooth and the mating part, perpendicular to the indentation direction, can be approximated as:

(Two triangular faces on each side, each area ≈ depth × width × tan(half-angle), with tooth width direction $w_{tooth}$)

When the tooth is fully indented, the mating part material undergoes complete plastic flow, and the average pressure on the contact surface reaches the indentation hardness $H$ of the material (approximately equal to Vickers hardness $H_V$). The force equilibrium equation is:

Substituting $A_{proj}$:

Rearranging gives:

Compared to the formula provided by the user, the difference in the denominator constant is the presence or absence of the factor 2. The user's formula may use an alternative definition of equivalent area (e.g., considering only one-sided projected area, or incorporating a comprehensive correction factor from experiments). In practical engineering, the coefficient is often combined into an empirical constant $C$, and the hardness value is adjusted. For safety, it is recommended to use the formula provided by the user as the basis; its essence remains the ratio relationship between force, hardness, and geometry, and the order of magnitude is correct.

4. Calculation Example

Given: - M10 bolt, assembly preload $F_M = 18\,000$ N - Number of washer teeth $n_{teeth} = 12$ - Full tooth tip angle $\alpha_{tooth} = 60°$, $\tan(30°) \approx 0.577$ - Tooth width $w_{tooth} = 1.5$ mm - Mating part: S235 structural steel, hardness $H_V = 140$ (approx. 1400 MPa)

Calculation:

This value is clearly too high, indicating that the parameter combination may be unreasonable. The actual issues are: - For steel mating parts, Vickers hardness 140 corresponds to 1400 MPa, but the tooth tip contact area formula is overly simplified; the actual contact width increases sharply with depth, limiting further indentation. - The formula does not account for elastic constraints around the plastic zone, leading to overestimation of penetration depth. Typically, the actual penetration depth for steel‑steel connections is in the range of 0.05 – 0.25 mm. Therefore, the formula is more suitable for soft materials or for trend estimation, and the calculation constant requires adjustment. - If used for aluminum alloy ($H_V=90$ → 900 MPa), $\delta = 18000/(12×900×0.577×1.5) ≈ 1.92$ mm, also too large.

Therefore, in practical application of this formula, a constraint factor $C$ (typically 2–4) is often added to the denominator, or the hardness unit conversion is made more precise. The user's formula can be considered as an uncorrected basic form, used to compare the relative penetration trends of different design options rather than absolute values.

5. Engagement Effectiveness Evaluation Criteria

Based on penetration depth, assess whether the engagement is effective:

• Minimum penetration depth $\delta_{min}$: Must be greater than the surface roughness $R_z$ of the mating part (approx. 0.01–0.03 mm) to ensure breakthrough of the oxide layer.
• Maximum penetration depth $\delta_{max}$: Limited by the thickness of the mating part, surface hardened layer, or acceptable plastic deformation; generally ≤ 0.3 mm (for steel).

If the calculated $\delta$ falls outside the range, adjustments should be made: - Reduce preload $F_M$ - Increase the number of teeth $n_{teeth}$ (reducing force per tooth) - Increase the tooth tip angle $\alpha_{tooth}$ (a blunter tooth penetrates less) - Select a mating part material with higher hardness

6. Application in DIN 9250 Washer Design

• Preliminary Estimation: Use this formula to quickly estimate the order of magnitude of penetration depth and determine if excessive indentation is likely.
• Material Matching: When the mating part is soft (e.g., aluminum), increase the number of teeth or reduce preload to avoid excessive $\delta$.
• Tooth Profile Optimization: Combine the penetration depth formula with contact pressure checks to select an appropriate tooth tip angle.
• Experimental Calibration: Ultimately, use metallographic cross-sections or profilometer measurements to determine actual penetration depth, back-calculate the implicit coefficients in the formula, and establish a more accurate product-specific model.

Summary:
$\delta = F_M / (n_{teeth} \cdot H_V \cdot \tan(\alpha_{tooth}/2) \cdot w_{tooth})$ provides an analytical estimation method for tooth tip penetration depth based on plastic hardness. Although the absolute values may have systematic deviations from experimental results, it correctly reflects the qualitative relationships between preload, number of teeth, hardness, and tooth geometry, making it an effective engineering tool for toothed washer engagement design.