DIN 9250 Tooth Surface Contact Stiffness: Asperity Contact Model

1. Definition and Role of Contact Stiffness

In bolted joints, contact stiffness $k_{contact}$ describes the ability of the interface between the tooth tips of a toothed washer and the mating part (or bolt head/nut) to resist normal deformation. It is one link in the total stiffness chain of the joint system, influencing:

• Axial deformation distribution during preload establishment;
• Dynamics of fretting wear under vibration;
• Slope characteristics of the torque-angle curve.

After the tooth tips indent the mating surface, the actual contact area is much smaller than the nominal area. The high pressure in the contact zone causes local elastic/plastic deformation. Contact stiffness determines the change in interface compression under preload fluctuations.

2. Basic Formula and Derivation

For a single asperity contact, if the contact region can be approximated as a circle of radius $a$, according to the flat-bottomed cylindrical indenter solution in Hertzian contact theory (i.e., a rigid flat punch indenting an elastic half-space), the normal contact stiffness is given by:

where: - $E^*$ — Equivalent elastic modulus (MPa), see below; - $a$ — Contact radius (mm).

The contact area $A_{contact} = \pi a^2$, hence $a = \sqrt{A_{contact} / \pi}$. Substituting gives:

This formula directly relates contact stiffness to the actual load-bearing area and is applicable when elastic contact dominates; if significant plastic deformation occurs, the area is determined by hardness, and this formula can still serve as an approximation for the elastic-plastic state.

3. Equivalent Elastic Modulus $E^*$

The tooth tip and mating surface are two materials (e.g., steel washer and steel/aluminum joint member). The equivalent modulus is calculated based on springs in series:

• $E_1, \nu_1$ — Washer material (spring steel: $E\approx206\,000$ MPa, $\nu=0.3$)
• $E_2, \nu_2$ — Joint member material

For steel-on-steel contact:

For steel-on-aluminum alloy contact ($E_2=70\,000$ MPa, $\nu_2=0.33$):

4. Determination of Actual Contact Area $A_{contact}$

Under preload, the tooth tip first undergoes elastic deformation. Due to extremely high stress (> material hardness), the tip enters plastic yielding, and the actual contact area is determined by the load capacity and material hardness. For fully plastic contact:

• $F_{tooth}$ — Normal force per tooth (N), $F_{tooth} = F_M / z$, where $z$ is the number of teeth
• $H$ — Indentation hardness of the softer material (MPa), approximately $H \approx 3 \sigma_y$ ($\sigma_y$ is the yield strength)

If the contact is in the elastic-plastic transition, the real area lies between the elastic and plastic values. In engineering practice, the plastic area can be taken as a conservative estimate (overestimating the area, leading to a slightly higher calculated stiffness, but the effect is not significant).

Example: Washer $z=12$, preload $F_M=18\,000$ N → $F_{tooth}=1500$ N; steel joint member, hardness $H \approx 3000$ MPa ($\sigma_y\approx1000$ MPa), then:

5. Total Contact Stiffness for Multiple Teeth

The entire washer has $z$ teeth. The contact stiffnesses of individual teeth are in parallel. The total tooth surface contact stiffness is:

This assumes uniform load distribution and equal contact area for each tooth. Substituting the single-tooth formula yields:

It can be seen that the total contact stiffness increases with the square root of the preload $F_M$ and increases with the number of teeth $z$.

Note: This stiffness only reflects the normal elasticity of the tooth tip interface and does not include the bending stiffness of the washer's disc body. In system analysis, this contact stiffness should be combined in series with the washer structural stiffness, bolt stiffness, etc.

6. Application in Bolted Joint Systems

Tooth surface contact stiffness is usually much greater than the elastic stiffness of the disc spring washer itself (because although the contact area is small, the modulus is extremely high). In the compliance chain of VDI 2230, the contact compliance $\delta_{contact} = 1/K_{contact,total}$ is often neglected, unless the joint member is very soft (plastic, magnesium alloy) and the number of teeth is very small.

If precise calculation is required, the total system compliance is:

Typically $\delta_{contact} \ll \delta_W$, and its effect on the result is within 5%.

7. Calculation Example

Conditions: - Steel washer ($E_1=206$ GPa, $\nu_1=0.3$) + steel joint member ($E_2=206$ GPa, $\nu_2=0.3$) → $E^* = 113\,200$ MPa - Preload $F_M=20\,000$ N, number of teeth $z=12$ → force per tooth $1\,667$ N - Joint member hardness $H = 3\sigma_y = 3 \times 800 = 2400$ MPa - Single-tooth plastic area $A_{contact} = 1667 / 2400 \approx 0.694\ \text{mm}^2$ - Single-tooth stiffness $k_{contact} = 2 \times 113\,200 \times \sqrt{0.694 / \pi} = 226\,400 \times \sqrt{0.221} \approx 226\,400 \times 0.470 = 106\,400$ N/mm

Total contact stiffness $K_{total} = 12 \times 106\,400 \approx 1.28 \times 10^6$ N/mm.

Compared to the washer disc body stiffness (on the order of $10^4$ N/mm), the contact stiffness is two orders of magnitude higher, so contact deformation can be neglected.

8. Notes and Limitations

1. Plasticity Correction: The above calculation is based on the fully plastic contact area. In reality, the tooth tip undergoes work hardening after repeated loading, and the area tends to stabilize, but this estimation can still be used.
2. Surface Roughness: The formula does not account for the reduction in real contact area due to roughness, which may overestimate stiffness. The error is small for smooth surfaces.
3. Coating Effects: Soft coatings (e.g., Dacromet) introduce additional compliance, resulting in lower actual contact stiffness than bare metal.
4. Dynamic Effects: Fretting wear under vibration changes the contact geometry, and stiffness will vary.
5. Experimental Determination: For precise analysis, contact stiffness can be obtained by directly measuring the force-indentation depth curve using ultrasonic or indentation methods.

Summary:
The contact stiffness of DIN 9250 toothed washers can be estimated using $k_{contact} = 2E^*\sqrt{A_{contact}/\pi}$. Combined with the plastic contact area and multiple teeth in parallel, the total interface stiffness is obtained. This stiffness is typically much greater than the washer body stiffness and can often be neglected in design, but it should be included in the compliance chain for very soft joint members or in precision analyses.