Fatigue Life (cycles)⚠ unverified

Understanding the Parameters

• Alternating stress $S_a$ — the fully-reversed amplitude. Life is extraordinarily sensitive to it: the exponent $1/b$ is large and negative (typically $\sim -6$ to $-12$), so a $10\%$ stress increase can cut life by more than half. This steepness is the central fact of finite-life design.

• Fatigue-strength fraction $f$ — sets the high-stress anchor of the S-N line: the fatigue strength at $10^3$ cycles is $f\,S_{ut}$ ($f \approx 0.9$ for steel, lower for stronger steels). It fixes where the line starts.

• Endurance limit $S_e$ — the low-stress anchor: the S-N line runs to $S_e$ at $10^6$ cycles (the knee). Use the Marin-corrected value (Marin Endurance Limit).

• Constants $a$, $b$ — $a$ (a stress) and $b$ (a small negative slope) are just the two-point line fit between $(10^3, f S_{ut})$ and $(10^6, S_e)$; they are derived, not physical.

• Regime limits — below $S_e$ the model gives no finite life (infinite); above $\sim f S_{ut}$ (low-cycle, $<10^3$) plastic strain dominates and a strain-life (Coffin–Manson) model is needed instead.

Derivation (Approaching a Proof)

Basquin observed that in the finite-life region the S-N data plot as a straight line on log–log axes:

$$S_a = a\,N^{\,b} \;\Longleftrightarrow\; \log S_a = \log a + b\log N.$$

Fit the line through its two known endpoints. At $N = 10^3$ cycles the strength is $S_f = f\,S_{ut}$; at $N = 10^6$ cycles it is the endurance limit $S_e$. Substituting both points and solving for the constants:

$$b = -\frac{1}{3}\log_{10}\!\left(\frac{f\,S_{ut}}{S_e}\right), \qquad a = \frac{(f\,S_{ut})^2}{S_e}.$$

(The $\tfrac13$ comes from the three decades between $10^3$ and $10^6$.) Inverting Basquin's law for the life at a given stress:

$$S_a = a\,N^{\,b} \;\Longrightarrow\; N = \left(\frac{S_a}{a}\right)^{1/b}.$$

Dimensional check. $a$ carries stress units ($(f S_{ut})^2/S_e = \text{Pa}^2/\text{Pa} = \text{Pa}$), so $S_a/a$ is dimensionless; raised to the pure-number power $1/b$, $N$ is dimensionless — a cycle count, as required.

History and Development

O.H. Basquin introduced the power-law S-N form in 1910, giving analytic shape to August Wöhler's foundational fatigue curves. The two-point fit between the $10^3$-cycle strength $f S_{ut}$ and the $10^6$-cycle endurance limit is the Shigley construction for the finite-life line. For lives below $\sim10^3$ cycles the elastic Basquin law breaks down and the strain-life (Coffin–Manson) approach takes over — together they span the low- and high-cycle fatigue regimes.

Related Concepts: S-N Curve, S-N Fatigue Life Prediction, Endurance Limit steel, Marin Endurance Limit, Modified Goodman Factor, Fatigue Strength Coefficient

Notes: Constants derived internally: $a=(f S_{ut})^2/S_e$, $b=-\tfrac13\log_{10}(f S_{ut}/S_e)$. For fully-reversed stress — convert a mean-stress case to an equivalent $S_a$ (Goodman) first. Valid $S_e < S_a < f S_{ut}$; below $S_e$ infinite life, below $10^3$ cycles use strain-life.