Understanding Fluctuating Stresses
In a typical fatigue scenario (like a rotating axle), the stress goes through maximum ($\sigma_{max}$) and minimum ($\sigma_{min}$) peaks. We characterize this cycle using two key metrics:
Alternating Stress ($S_a$)
$$S_a = \frac{\sigma_{max} - \sigma_{min}}{2}$$
The amplitude of the fluctuation. This is the primary driver of fatigue crack growth.
Mean Stress ($S_m$)
$$S_m = \frac{\sigma_{max} + \sigma_{min}}{2}$$
The average background stress. Tensile mean stress accelerates failure, while compressive retards it.
The Wöhler (S-N) Curve
Developed by August Wöhler, the S-N curve plots alternating stress ($S_a$) against the number of cycles to failure ($N_f$) on a logarithmic scale.
- ➤ Low Cycle Fatigue ($N < 10^3$): Governed by plastic deformation. High stress, few cycles.
- ➤ High Cycle Fatigue ($10^3 < N < 10^6$): Elastic deformation dominates. Classic fatigue regime.
- ➤ The Endurance Limit ($S_e$): Notice the "knee" in the graph for ferrous metals (like steel) around $10^6$ cycles. If the stress is kept below $S_e$, the material has an theoretically infinite life! Non-ferrous metals (like Aluminum) do not have this horizontal knee.
Mean Stress Failure Envelopes
The S-N curve assumes Mean Stress ($S_m = 0$)—a fully reversed cycle. But in reality, components often operate with a non-zero mean stress. To predict failure under combined $S_a$ and $S_m$, we use the Haigh Diagram.
Soderberg (Most Conservative)
$$\frac{S_a}{S_e} + \frac{S_m}{S_{yt}} = \frac{1}{N_f}$$
Guards against both fatigue and static yielding.
Modified Goodman (Standard for Design)
$$\frac{S_a}{S_e} + \frac{S_m}{S_{ut}} = \frac{1}{N_f}$$
Connects endurance limit to ultimate strength. widely used for brittle materials and conservative design.
Gerber (Parabola)
$$\frac{S_a}{S_e} + \left(\frac{S_m}{S_{ut}}\right)^2 = \frac{1}{N_f}$$
Fits experimental failure data best for ductile metals.
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