Arrhenius Temperature Dependence
Arrhenius Temperature Dependence
Formula Expression
Parameters
| Symbol | Name | Unit |
|---|---|---|
| temp_C | temp_C | °C |
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DIN 2093 Arrhenius Temperature Dependence: Variation of Relaxation Time Constant with Temperature
1. Application of Arrhenius Relationship in Relaxation
The stress relaxation of disc springs under constant displacement is a thermally activated process, whose rate is extremely sensitive to temperature. The relationship between the relaxation time constant $\tau$ (characterizing the speed of relaxation) and absolute temperature $T$ can be described by the Arrhenius equation:
or equivalently, the relaxation rate $r \propto 1/\tau$ satisfies:
where: - $\tau(T)$ — relaxation time constant at temperature $T$ (hours or seconds) - $\tau_0$ — pre-exponential factor (constant related to the microscopic state of the material) - $Q$ — activation energy (J/mol), representing the energy required for the relaxation mechanism to overcome the energy barrier; for stress relaxation of spring steel, $Q$ is typically in the range of 50 ~ 100 kJ/mol - $R = 8.314\ \text{J/(mol·K)}$ — universal gas constant - $T$ — absolute temperature (K), $T = \theta + 273.15$, where $\theta$ is the temperature in Celsius
2. Amplification Effect of Temperature Increase on Relaxation Rate
Using the Arrhenius relationship above, the acceleration of the relaxation rate due to temperature increase can be quantified. Let the reference temperature be $T_1$ and the operating temperature be $T_2$; the ratio of relaxation rates is:
Engineering estimation: Taking room temperature 20°C (293 K) as the baseline and an activation energy $Q \approx 75\ \text{kJ/mol}$ (typical for spring steel), the amplification factor of the relaxation rate for every ≈ 50°C increase in temperature is calculated:
| Temperature | $T$ (K) | $r(T)/r(20°C)$ |
|---|---|---|
| 20°C | 293 | 1 |
| 70°C | 343 | ≈ 12 |
| 120°C | 393 | ≈ 85 |
| 170°C | 443 | ≈ 370 |
| 220°C | 493 | ≈ 1300 |
It can be seen that for every ≈ 50°C increase in temperature, the relaxation rate increases by approximately one order of magnitude (about 10 times). This is consistent with the engineering rule of thumb that "the relaxation rate at 200°C is about 10 times that at room temperature" (in fact, from 20°C to 220°C, there are approximately four 50°C steps, and the cumulative amplification factor exceeds 10⁴ times, but the previous statement is a rough estimate). In the range of 150–200°C, relaxation is already very significant and must be considered in design.
3. Temperature Correction in Relaxation Models
In the previously mentioned empirical relaxation ratio model:
Temperature primarily affects the time constant $\tau$, while the shape factor $\beta$ and the infinite relaxation ratio $R_{\infty}$ are relatively less affected by temperature (although $R_{\infty}$ may also decrease when approaching the tempering temperature). Therefore, the time scale of the relaxation curve is significantly compressed at high temperatures.
In practical calculations, given the known $\tau_1$ at one temperature, $\tau_2$ at another temperature can be derived using the Arrhenius formula:
4. Determination of Activation Energy $Q$
$Q$depends on the material, heat treatment condition, and relaxation mechanism:
| Material/Mechanism | $Q$ (kJ/mol) |
|---|---|
| Stress relaxation of spring steel (dislocation climb) | 60 – 100 |
| Creep of spring steel (diffusion-controlled) | 150 – 220 |
| Relaxation of stainless steel springs | 80 – 120 |
| High-temperature alloys (Inconel, etc.) | 200 – 300 |
Design recommendations: - For conventional spring steel applications below 150°C, a conservative estimate of $Q \approx 70 \sim 80\ \text{kJ/mol}$ can be used. - More precise values must be obtained through relaxation tests at at least three temperatures, using an Arrhenius plot (linear regression of $\ln\tau$ vs. $1/T$).
5. Application in Disc Spring Design
- Preload loss estimation: For disc springs operating at high temperatures, calculate $\tau(T)$ based on the operating temperature $T$, then substitute into the relaxation model to determine the remaining preload within the target service life.
- Accelerated testing: Using the Arrhenius relationship, short-term relaxation tests can be conducted at higher temperatures to extrapolate long-term relaxation behavior at the operating temperature (time-temperature equivalence).
- Material selection: If the operating temperature exceeds 200°C, materials with higher activation energy (e.g., high-temperature alloys) should be selected to reduce the relaxation rate.
6. Calculation Example
Given: A disc spring undergoes a 500 h relaxation test at 150°C (423 K), with a time constant $\tau(150°C) \approx 800\ \text{h}$. Activation energy $Q = 80\ \text{kJ/mol}$.
Find: The time constant and relaxation acceleration factor at 220°C (493 K).
Solution:
The relaxation rate increase factor $\approx 1/0.0395 \approx 25.3$ times. That is, relaxation at 220°C is about 25 times faster than at 150°C, requiring more frequent retightening or the use of higher-performance materials.
Summary: The Arrhenius model provides a theoretical basis for the time-temperature transformation of disc spring relaxation. Through the activation energy $Q$ and temperature $T$, the relaxation time constant can be quantitatively estimated, thereby predicting load retention capability at high temperatures. For disc springs operating at temperatures exceeding 150°C, relaxation assessment based on the Arrhenius relationship is essential, and sufficient preload margin must be reserved in the design.