Basquin S-N Relation

Plot alternating stress against life on log-log paper and the high-cycle part of an S-N curve is a straight line. Basquin noticed that in 1910 and wrote it as $\sigma_a = \sigma_f'(2N_f)^b$, where $2N_f$ counts reversals rather than cycles and $b$ is a small negative number, typically −0.05 to −0.12 for metals. With $\sigma_f' = 1000$ MPa and $b = -0.1$, a life of 500,000 cycles (a million reversals) gives $\sigma_a = 1000 \times 10^{-0.6} = 251$ MPa.

The exponent is where the intuition lives, and it is brutal. At $b = -0.1$, dropping the stress by a factor of two multiplies life by $2^{10} = 1024$. Run the same numbers the other way and a 10% stress overrun costs you about 63% of the life. This is why fatigue predictions are quoted in orders of magnitude rather than percentages, and why a load spectrum measured slightly wrong produces an answer that is not slightly wrong.

The limit to remember is the far right of the curve. Ask this formula for the stress at $10^9$ cycles and it will answer, but for steels and titanium the real curve flattens onto an endurance limit somewhere near $10^6$–$10^7$ cycles, below which the part effectively lives forever — so the extrapolation is optimistic in a way the algebra cannot see. Aluminium and copper alloys have no such plateau and keep sloping down, which is why aircraft structures get finite-life inspection intervals rather than infinite-life ratings. Note also that $2N_f$ is reversals: entering cycles where the formula wants reversals is a factor-of-two error, small on the stress axis and a factor of two on the life axis.