DIN 9250 Temperature Derating Factor: Arrhenius Thermal Activation Model

1. Definition and Physical Background

In high-temperature environments, the mechanical properties of tooth lock washers (such as locking torque $M_{lock}$, preload retention capacity, and material yield strength) gradually degrade over time. The degradation rate is controlled by thermally activated processes (creep, stress relaxation, diffusion oxidation) and can be described using the Arrhenius relationship.

The temperature derating factor $f_T$ represents the retention ratio of material properties at temperature $T$ relative to the room temperature baseline:

• $f_T$ — Temperature derating factor (dimensionless, $0 < f_T \le 1$)
• $Q$ — Activation energy (J/mol), characterizing the energy barrier of the material's thermal degradation mechanism, related to material composition, heat treatment state, and failure mode
• $R = 8.314\ \text{J/(mol·K)}$ — Universal gas constant
• $T$ — Absolute temperature (K), $T = \theta + 273.15$, where $\theta$ is temperature in Celsius

Physical meaning: The higher the temperature, the greater the probability that atoms acquire sufficient energy to overcome the energy barrier, accelerating degradation processes such as creep, relaxation, and oxidation, causing performance indicators to decrease exponentially.

2. Application in Force Value Derating

Using $f_T$ as the performance retention factor over high-temperature service time $t$, the ratio of a given mechanical property (e.g., locking torque $M_{lock}$, allowable load $F_{zul}$, yield strength $R_{p0.2}$) at temperature $T$ relative to its room temperature value can be expressed as:

Note: This formula describes the trend of the stabilized value after long-term service, not the instantaneous elastic modulus derating (which follows a linear relationship, see DIN 6796 temperature derating section). The essential difference between the two:

• Elastic modulus derating: Instantaneous effect, resulting from increased atomic spacing and weakened bonding, decreasing linearly with temperature.
• Long-term force value (strength, locking torque) derating: Time-dependent, resulting from creep and relaxation, requiring the Arrhenius model.

3. Activation Energy $Q$ Values

is closely related to the material's microscopic mechanisms. For spring steels commonly used in DIN 9250 washers (e.g., 50CrV4, C75S), typical reference values for activation energy are:

Engineering design recommendations: - In the absence of specific test data, for spring steel washers in long-term service at 200–300°C, a conservative value of $Q \approx 180$ kJ/mol can be used. - More precise values require high-temperature endurance tests (testing locking torque retention at at least three temperature levels) and linear regression.

4. Comprehensive Temperature Correction Including Force Value Derating

In practical design, the Arrhenius derating factor can be combined with the linear elastic modulus derating to obtain a comprehensive temperature correction coefficient $\eta_T$:

where $E(T)/E_{20} = 1 - \beta (T-20)$, $\beta \approx 2.0\times10^{-4}$ K⁻¹.

The allowable load or locking torque at high temperature is then:

Example: - Spring steel washer, operating temperature 300°C (573 K), using $Q = 180$ kJ/mol - Elastic modulus derating factor: $E/E_{20} = 1 - 2\times10^{-4} \times 280 \approx 0.944$ - Arrhenius factor: $f_T = \exp(-180000 / (8.314 \times 573)) = \exp(-37.78) \approx 3.9\times10^{-17}$

This is an extremely small value, indicating nearly complete loss of mechanical properties during long-term service at 300°C. In reality, if the activation energy is based on diffusion creep, this result is reasonable—ordinary spring steel should not be used for long-term load-bearing above 300°C. If only short-term temperature resistance is considered, or if the activation energy is taken as 80 kJ/mol (oxidation mechanism), then:

Still extremely low, indicating that ordinary washers are indeed unsuitable for long-term application at 300°C.

If the temperature is 150°C (423 K), with $Q=180$ kJ/mol:

This is also unreasonable, as spring steel relaxation at 150°C is not significant. The contradiction arises from the absence of a time term. A more accurate model would be:

Or using the Larson-Miller parameter to combine temperature and time. In engineering practice, for temperatures below 150°C, relaxation is negligible, and $f_T \approx 1$ can be assumed.

Therefore, the formula $f_T = \exp(-Q/RT)$ is typically used together with a reference time or rate factor; used alone, it can only indicate a trend. In design, it is preferable to consult the material supplier's high-temperature operating limit tables rather than directly applying this formula to calculate absolute values.

5. Engineering Application Methods

1. Obtain material high-temperature performance data: Prioritize obtaining the allowable load ratio table or $M_{lock}$ retention curve from the washer manufacturer at the target temperature.
2. If only activation energy data is available: Follow the method above, incorporating a safety factor, and conservatively set the derating.
3. Consider multiple temperature ranges: If the joint experiences different temperatures, use the $f_T$ at the highest temperature as the design basis.
4. Experimental verification: For critical high-temperature joints, conduct locking torque decay tests under simulated operating conditions to directly calibrate the derating factor.

Summary:
The temperature derating factor $f_T = \exp(-Q/RT)$, based on the Arrhenius thermal activation theory, describes the exponential decay of material properties over time at high temperatures. Combining this factor with the linear elastic modulus derating allows a comprehensive evaluation of the force retention capability of DIN 9250 washers in high-temperature service. Accurate selection of the activation energy $Q$ and consideration of time effects are key to applying this model, and design should be supplemented with experimental data.