Stress Life Curve

Stress-Life (S-N) Fatigue Curve (structures/fatigue/stress-life-curve)

Use when fatigue test data must be turned into a usable curve: fit the Basquin equation to S-N test points, determine the endurance limit from the runout stress level, and predict the fatigue life at a constant stress amplitude. The leaf covers curve construction from test data, the log-log fit, the endurance limit, and life prediction; mean-stress corrections, cumulative damage, and notch effects are separate leaves.

Domain quick reference

• Basquin equation: S = A * N^b, with stress amplitude S, cycles to failure N, fatigue strength coefficient A (the stress amplitude at N = 1 cycle), and fatigue strength exponent b (negative for metals, typically -0.05 to -0.15).
• Worked curve, A = 1000 MPa, b = -0.1 (verified by running scripts/stress_life_curve_logic.py): S(1e4) = 398.1 MPa, S(1e5) = 316.2 MPa, S(1e6) = 251.2 MPa, S(1e7) = 199.5 MPa. Each decade of life costs roughly 20 to 25 percent of the amplitude.
• Life prediction: N = (S / A)^(1 / b). At S = 300 MPa on the worked curve, N = (0.3)^-10 = 1.69e5 cycles.
• Log-log fit: log S = log A + b * log N; least squares over the (log N, log S) pairs recovers A and b. The three exact points (1e3, 501.2), (1e4, 398.1), (1e5, 316.2) refit to A = 1000 MPa, b = -0.1.
• Endurance limit: Se = A * N_runout^b at the runout threshold; for runout at 1e7 cycles the worked curve gives Se = 1000 * 1e7^-0.1 = 199.5 MPa. Equivalently, the endurance limit is the highest tested stress level whose test survived the runout life, read directly off the data.
• Typical metallic values: A near 0.9 * Sut and b between -0.05 and -0.15; the endurance limit of steels sits near 0.5 * Sut.
• Equivalent life-form parameterization N = C * S^-m appears in some references, with m = -1 / b and C = A^(-1 / b); convert before mixing parameters.

Workflow

1. Gather the S-N test data as (cycles to failure N, stress amplitude S) pairs in one stress unit; keep runout tests (test stopped without failure at the runout life) separate from failures.
2. Fit the Basquin curve: run the log-log least squares regression over the failed specimens to get (A, b).
3. Determine the endurance limit: evaluate the fitted curve at the runout threshold, or take the highest runout stress level from the data when the curve is not needed.
4. Predict the life at the applied stress amplitude with N = (S / A)^(1 / b), or the allowable amplitude for a required life with S = A * N^b.
5. Report A, b, the endurance limit, the runout threshold, and the predicted life with the stress unit stated.

Pitfalls

• Confusing the S-N curve with the Goodman mean-stress correction: the Basquin curve assumes fully reversed loading (zero mean stress); a fluctuating cycle on a mean stress must be corrected with the goodman-diagram leaf before the amplitude is read against the curve.
• Confusing curve construction with cumulative damage: the S-N curve gives the life of a constant-amplitude cycle; a varying-amplitude spectrum needs the miner-damage leaf to sum the damage fractions per cycle block.
• Confusing the smooth-specimen curve with notched behavior: a stress concentration shortens the life at the same nominal amplitude; reduce the amplitude with the fatigue notch factor from the notch-sensitivity leaf first.
• Mixing parameterizations: S = A * N^b and N = C * S^-m carry the exponent with opposite signs; using b as m (or the reverse) silently corrupts every prediction.
• Mixing stress units between A and the applied amplitude, or between the test data and the design case.
• Extrapolating far beyond the tested life range: fatigue lives are log-normally scattered, so predictions outside the data band are weak regardless of the fit quality.
• Claiming an endurance limit where none exists: aluminum and other non-ferrous alloys have no true fatigue limit, only a finite-life curve; report the runout-level stress, not an infinite-life guarantee.
• Fitting in linear space: the regression must run on (log N, log S) so every decade of life weighs equally.

Behavior contract (gate 3)

The S-N analysis logic is exercised by the gate 3 contract test: scripts/test_stress_life_curve.py against scripts/stress_life_curve_logic.py (stdlib unittest, offline). Run:

python3 scripts/test_stress_life_curve.py

The 21 test methods cover the worked anchors above, the trend property that life grows as the stress amplitude drops, and the ValueError cases for non-positive inputs, a zero exponent, too few fit points, identical lives, and missing runouts.

Compliance

• FAR-25, CS-25, and MMPDS are cited as reference-only certification context (compliance: STANDARDS-REF, gated: false); no text is quoted from any of them. The S-N methodology itself is standard mechanical engineering practice.