DIN 2093 Fatigue Life Estimation: Fatigue Groups and S‑N Curves

1. Basis for Fatigue Grouping

The fatigue performance of disc springs depends not only on material and surface treatment but also strongly on section thickness $t$. Due to higher internal defect probability and greater stress gradients, thick-section springs exhibit significantly lower fatigue strength than thin-section springs. Based on this, DIN 2093 divides disc springs into three fatigue groups:

Note: The fatigue limit stress amplitude is the allowable value under pulsating compression* ($R=0$) at the OM point, for quenched and tempered spring steel (e.g., 50CrV4) with shot peening. Specific values should be taken from DIN 2093 standard charts and are influenced by mean stress.

2. Mathematical Model of S‑N Curves

The DIN 2093 S‑N curve approximates two straight segments on a log-log scale: - Finite life segment: from approximately $N = 10^4$ to the knee point $N_D$, where stress amplitude and life follow a power-law relationship; - Infinite life segment: when $N \ge N_D$, the stress amplitude becomes constant, representing the fatigue limit $\sigma_{A}$.

Formula for the finite life segment:

Where: - $\sigma_a(N)$ — allowable stress amplitude (MPa) corresponding to life $N$ (OM point, absolute value) - $\sigma_{A}$ — fatigue limit stress amplitude (MPa) at the knee point $N_D$, i.e., the allowable stress amplitude for infinite life in that group - $N_D$ — knee point cycles (Group 1: $10^7$, Group 2: $10^6$, Group 3: $2\times10^5$) - $k$ — fatigue strength exponent (absolute value of the inverse log-log slope), typical for disc springs $k \approx 5 \sim 10$; a conservative design may use $k = 5$ (steeper slope, more conservative)

If $\sigma_a \le \sigma_{A}$, then life $N \ge N_D$, considered infinite life, requiring no further calculation.

3. Determination of Stress Amplitude $\sigma_a$

Before fatigue evaluation, the stress amplitude at the OM point must be calculated from the minimum deflection $s_{min}$ and maximum deflection $s_{max}$ of the working cycle:

The OM point stress is given by the Almen‑Laszlo formula (see previous sections). Note that when the mean compressive stress is high, $\sigma_{A}$ must be corrected using criteria such as Goodman (DIN 2093 provides corresponding fatigue limit diagrams). In the life formulas below, $\sigma_{A}$ should be understood as the allowable stress amplitude after mean stress correction.

4. Life Estimation Procedure

1. Determine disc spring thickness $t$ → Identify fatigue group, obtain $N_D$ and the group's base fatigue limit $\sigma_{A0}$ (pulsating $R=0$).
2. Calculate working stress amplitude $\sigma_a$ and mean stress $\sigma_m$ (based on OM point).
3. Mean stress correction: Using the DIN 2093 Haigh diagram or formula, correct $\sigma_{A0}$ to the allowable stress amplitude $\sigma_{A}$ under the actual mean stress.
4. Comparison:
5. If $\sigma_a \le \sigma_{A}$ → Infinite life ($N \ge N_D$).
6. If $\sigma_a > \sigma_{A}$ → Enter finite life region, estimate $N$ using $N = N_D \cdot (\sigma_{A} / \sigma_a)^k$.
7. Safety factor: Require $N \ge N_{req} \cdot S_D$, or equivalently $\sigma_a \le \sigma_{A} / S_D$, with $S_D \ge 1.2$.

5. Calculation Example

Given: - Disc spring specification: $t = 2.0\ \text{mm}$, belongs to Group 2, $N_D = 10^6$, base fatigue limit $\sigma_{A0} = 500\ \text{MPa}$ (shot peened). - Working cycle: $s_{min}=0.3\ \text{mm}$, $s_{max}=0.9\ \text{mm}$, calculated $\sigma_{OM,min} = -1200\ \text{MPa}$, $\sigma_{OM,max} = -3200\ \text{MPa}$. - Stress amplitude $\sigma_a = (3200-1200)/2 = 1000\ \text{MPa}$, mean stress $\sigma_m = (3200+1200)/2 = 2200\ \text{MPa}$.

Mean stress correction (Goodman):

This indicates that the disc spring cannot withstand fatigue under this mean stress; the maximum deflection must be reduced.

Adjust working cycle to $s_{min}=0.3\ \text{mm}$, $s_{max}=0.6\ \text{mm}$: Calculated $\sigma_{OM,max} \approx -2100\ \text{MPa}$, $\sigma_{OM,min} = -1200\ \text{MPa}$

After correction, $\sigma_{A} = 500 \times (1 - 1650/1600) \approx 500 \times (-0.03125) \rightarrow$ still negative, but close to zero. This shows that Group 2 disc springs are unsuitable for such high mean stress; a thinner disc spring (Group 1) or reduced load should be considered.

If using Group 1 ($t=1.2\ \text{mm}$) with a similar outer diameter, $\sigma_{A0}=700\ \text{MPa}$, the OM stress may differ under the same deflection ratio (due to thickness change). This is only a method demonstration. Assuming after adjustment, actual $\sigma_a = 300\ \text{MPa}$, $\sigma_m=1400\ \text{MPa}$, $R_m=1600\ \text{MPa}$, then $\sigma_A = 700 \times (1-1400/1600) = 700 \times 0.125 = 87.5\ \text{MPa}$, still insufficient.

Thus, disc spring fatigue design often requires iterative optimization of dimensions and preload, prioritizing mean stress reduction.

6. Engineering Application Recommendations

• Thin disc springs (Group 1) have the highest fatigue limit and should be prioritized for high-cycle applications; if force is insufficient, use parallel stacking.
• Shot peening can increase $\sigma_{A}$ by approximately 20%~30%, making it the most effective method for improving fatigue performance.
• Avoid flattening: Maximum deflection $s_{max} \le 0.75 h_0$ not only protects strength but also significantly reduces mean stress.
• Multi-point verification: In addition to the OM point, fatigue evaluation should also be performed at tensile stress points (Point I), especially for thick-section disc springs.

Summary: DIN 2093 fatigue life estimation is based on thickness-grouped S‑N curves, with stress amplitude $\sigma_a$ and mean-stress-corrected allowable value $\sigma_{A}$ as core parameters. Correct grouping, accurate stress calculation, and Goodman correction enable effective prediction of disc spring fatigue life or determination of infinite life.