Fatigue Life from Basquin's Law (S-N Curve)
Fatigue Life from Basquin's Law (S-N Curve)
Basquin's law is the standard way to describe the high-cycle fatigue behavior of metals: on a log-log plot, stress amplitude versus cycles to failure forms a straight line, which is exactly what a power-law relationship produces. This lets an entire S-N curve be summarized with just two material constants, the fatigue strength coefficient and exponent. Because fatigue life depends on stress amplitude raised to a large negative power (1/b, with b often around -0.1), even a modest increase in operating stress can cut fatigue life by an order of magnitude or more — this steep sensitivity is why fatigue-critical designs keep stress amplitudes conservatively low relative to yield.
Basquin's law gives cycles to failure as N = 0.5*(sigma_a/Sf)^(1/b). where S_f is the fatigue strength coefficient, sigma_a2 is the applied stress amplitude, and b_bas is the fatigue strength exponent (a negative material constant).
Inputs
S_f := 900 MPa=900 MPa
sigma_a2 := 150 MPa=150 MPa
b_bas := -0.12=-0.12
The stress ratio sigma_a/Sf compares the applied amplitude to the material's reference fatigue strength, and raising it to the 1/b power (a large negative exponent) is what makes life fall off so steeply as stress rises.
N_life := 0.5·(sigma_a2/S_f)^(1/b_bas)=1.53e+06
Results
At this stress amplitude the predicted life comes out in the high-cycle fatigue regime, on the order of hundreds of thousands to millions of cycles, which is a normal result well below the material's ultimate strength. Because the exponent is so sensitive, a design that seems to have generous fatigue margin at nominal load can lose most of that margin if actual service stresses run even 10-15% higher than assumed, which is why fatigue design typically carries its own dedicated safety factor.
Assumptions & limitations
- High-cycle fatigue regime — Basquin's law is not valid for low-cycle (plastic-dominated) fatigue.
- Constant amplitude, fully-reversed (or corrected mean-stress) loading assumed.
- Material constants Sf and b are specific to the alloy, surface finish, and test conditions used.
- No safety factor applied — this is the mean/nominal predicted life.