DIN 2093 Relaxation Ratio: Load Decay Over Time Under Constant Displacement

1. Definition of Stress Relaxation

Under constant displacement (constant compression), disc springs experience microplastic deformation within the material (dislocation slip, creep), causing part of the elastic strain to gradually transform into plastic strain. This results in a progressive decrease of spring force over time. This phenomenon is called stress relaxation.

The degree of relaxation is expressed by the relaxation ratio or load retention rate:

• $F_0$ — Initial load (force at $t=0$, typically the preload after assembly)
• $F(t)$ — Residual load after time $t$

The closer this ratio is to 1, the smaller the relaxation and the better the material's resistance to relaxation.

2. Empirical Relaxation Model

For disc springs under constant displacement, the load decay curve can be described using an exponential power law or stretched exponential function. A commonly used generalized model in engineering is:

Where: - $R_{\infty}$ — Infinite-time relaxation rate, i.e., the asymptotic value of $F/F_0$ as $t \to \infty$ (e.g., 0.85 means 85% of the load is retained ultimately) - $\tau$ — Relaxation time constant (hours or days), characterizing the time scale of the relaxation process; smaller $\tau$ indicates faster relaxation - $\beta$ — Shape factor ($0 < \beta \le 1$), controlling the curve shape: - $\beta = 1$: Classical exponential decay, constant relaxation rate - $\beta < 1$: Fast initial relaxation, gradually slowing down, which matches the actual behavior of most metallic materials

3. Physical Meaning of Shape Factor $\beta$

indicates the presence of dispersion in the time spectrum during relaxation—the microscopic relaxation mechanisms of the material (e.g., dislocation motion with different activation energies) have different time constants, causing the macroscopic relaxation curve to drop rapidly initially and then slow down.

For quenched and tempered spring steel: - At room temperature: $\beta \approx 0.3 \sim 0.5$ - At elevated temperatures: $\beta$ may increase to 0.6~0.8, making the curve shape closer to simple exponential decay

In design calculations, if test data is unavailable, a conservative value of $\beta = 0.5$ can be used to predict the higher initial relaxation.

4. Determination of Time Constant $\tau$ and Infinite Relaxation Rate $R_{\infty}$

These two parameters are highly dependent on material, operating temperature, and initial stress level. Methods for obtaining them:

1. Experimental measurement: Conduct constant-displacement relaxation tests on actual disc springs at the target temperature, record the $F(t)/F_0$ curve, and fit the above model to obtain $\tau$ and $R_{\infty}$.
2. Manufacturer data: Some disc spring manufacturers provide relaxation curves for specific specifications at room temperature, 150°C, and 200°C.
3. Empirical estimation: Without data, for spring steel at room temperature, $\tau$ is on the order of $10^3 \sim 10^4$ hours, and $R_{\infty}$ is approximately 0.92~0.98; at 150°C, $\tau$ can shorten to hundreds of hours, and $R_{\infty}$ drops to 0.85~0.92.

5. Temperature Acceleration Effect

Relaxation accelerates significantly at high temperatures. The temperature dependence of the relaxation process can be described by the Arrhenius relationship (similar to the temperature derating factor). For spring steel, empirical evidence shows:

• $Q$ — Relaxation activation energy, approximately 150~220 kJ/mol for spring steel

Engineering reference: The relaxation rate at 200°C is approximately 10 times that at room temperature, meaning the load drop over the same time period is much larger. For example, if the load retention is 95% after 1000 hours at room temperature, it may be only 70%~80% at 200°C.

Therefore, disc springs for high-temperature applications must use materials with high relaxation resistance (e.g., special alloys, high-temperature spring steel) or compensate through periodic retightening.

6. Application in Bolted Joint Design

6.1 Preload Loss

The force reduction due to relaxation $\Delta F_{relax} = F_0 - F(t)$ must be included in the total preload loss $F_Z$ according to VDI 2230, to determine the minimum assembly preload:

Where $F_Z$ includes embedding settlement loss, thermal effect loss, and relaxation loss $\Delta F_{relax}$.

6.2 Long-Term Reliability Assessment

• For permanent connections (never disassembled during service life), the functional check should use the value of $F(t)$ at the target service life (e.g., 10^5 hours) as the residual preload.
• For maintainable connections, a periodic retightening interval can be specified to restore the preload to its initial value.

7. Calculation Example

Given: A disc spring at room temperature with initial preload $F_0 = 8\,000\ \text{N}$. Based on tests, the relaxation parameters are: - $R_{\infty} = 0.90$ (ultimately retains 90% load) - $\tau = 2\,000\ \text{h}$ - $\beta = 0.5$

Find: Residual load after 5 000 hours of operation.

Calculation:

Residual load $F(5000\ \text{h}) \approx 8\,000 \times 0.9206 \approx 7\,365\ \text{N}$.

Conclusion: After 5 000 hours, the load drops by approximately 8%. If this residual force still meets requirements for loosening prevention, slip resistance, etc., the design is acceptable.

8. Design Recommendations

• Room temperature applications: For critical connections, a long-term relaxation loss of 5%~10% of the initial force can be taken as a design margin.
• High-temperature applications: Actual parameters must be determined through high-temperature relaxation tests, and larger losses (up to 20%~40%) should be considered.
• Material selection: For high-temperature disc springs, chromium-silicon steel, stainless steel, or nickel-based alloys can be used to improve relaxation resistance.
• Life management: For disc springs in long-term service, it is recommended to specify inspection or replacement intervals in the equipment maintenance manual.

Summary: The relaxation ratio $F(t)/F_0$ model describes the load decay law of disc springs under constant displacement using three parameters: $R_{\infty}$, $\tau$, and $\beta$. High temperatures significantly accelerate relaxation. During design, relaxation loss must be incorporated into preload calculations based on test data to ensure the connection remains reliable throughout its service life.