Mechanical Behavior · Fatigue
Fatigue and S-N curves: why parts break far below their strength
Start with the first picture: lower the stress amplitude until the dot on the steel S-N curve runs off to the right, then switch the material to aluminium and do it again.
What is an S-N curve?
A load that goes up and down again and again can crack a part at a stress well below its tensile strength. An S-N curve plots the stress amplitude S against the number of cycles N to failure. Most steels have a floor, the endurance limit, below which they last forever in lab tests in air. Aluminium alloys have no floor.
Try it: with SAE 1045 normalized and zero mean, slide the amplitude from 300 MPa down to 220 MPa. The life grows from about 105 cycles to infinite once you pass the 228 MPa endurance limit. Switch to 2024-T351 aluminium: the line keeps falling and the dot never runs off.
How does mean stress change fatigue life?
A tensile mean stress holds cracks open and shortens life at the same amplitude. A Haigh diagram plots amplitude against mean stress. Each line joins the fatigue strength at zero mean to a static strength at zero amplitude. Points under a line are predicted to survive the chosen life.
| Line | σar MPa | Safety | Life Nf |
|---|
Try it: keep SAE 1045 normalized at σa = 150 MPa and σm = 200 MPa. The dot sits under the Goodman and Gerber lines but above Soderberg, so Soderberg alone says it fails. Drag σm to 0 and all three agree again.
What is the strain-life (Coffin-Manson) curve?
At high loads a metal yields a little in every cycle, and strain is easier to control than stress. The strain-life curve adds an elastic line (Basquin) and a plastic line (Coffin-Manson). Where they cross is the transition life: plastic strain rules before it, elastic strain after it.
Try it: at εa = 1% compare the two SAE 1045 conditions. The soft normalized steel lasts longer (about 1,700 cycles against about 340), because it can take plastic strain. At εa = 0.2% the hard quenched and tempered steel wins by far (about 2.4 × 108 cycles against 1.9 × 105): it stays elastic.
How fast does a fatigue crack grow?
Once a crack exists, each cycle opens it and moves its tip forward a tiny step. The Paris law ties that step to the range of the stress intensity factor, ΔK = Y Δσ √(πa), which grows with the crack length a. So the crack speeds up, and the part breaks when Kmax reaches the fracture toughness.
Try it: with the ferrite-pearlite steel at Δσ = 150 MPa, move a0 from 1 mm to 0.1 mm. The life goes up more than three times, from about 3.1 × 105 to 1.1 × 106 cycles, because the slow early growth takes longest. Doubling Δσ cuts the life by 2m = 8 for m = 3.
Miner's rule: adding up damage from two load levels
What to take away
Cycles, not peak load
A stress far below the tensile strength can break a part if it repeats enough times. The S-N curve says how many.
Steel floor, no aluminium floor
Many steels show an endurance limit in lab tests in air. Aluminium alloys do not, so their design needs a finite life.
Tensile mean stress hurts
At the same amplitude, a tensile mean shortens life. Goodman is the usual design line; Soderberg is more cautious.
Short cracks take the longest
Crack growth speeds up as the crack grows. Finding a small flaw early buys most of the remaining life.
More detail: the equations and the limits of these models
Basquin and the endurance limit
Basquin's law is σa = σf' (2Nf)b, with 2Nf reversals (two per cycle). Solving for life gives Nf = ½ (σa/σf')1/b. The constants here are fits to strain-controlled tests on polished axial specimens (SAE J1099 for the steels, Dowling Table 14.1 for 2024-T351). For the steels the page puts the knee at 106 cycles and uses the Basquin value there as the endurance limit. That gives 0.32 to 0.38 of the tensile strength, below the common rule of thumb of 0.5 for polished rotating-bending bars. Real parts need further factors for surface finish, size, notches and corrosion. Even steels lose the floor in seawater or at very high cycle counts (108 and more).
Mean stress
Goodman: σa/Se + σm/σu = 1. Gerber: σa/Se + (σm/σu)2 = 1. Soderberg: σa/Se + σm/σy = 1. Se is the fully reversed strength at the design life. The equivalent fully reversed amplitude is σa divided by the second term taken from 1, for example σar = σa/(1 − σm/σu). For a compressive mean the page gives no credit (σar = σa), which is on the safe side. Test data for ductile steels usually fall between Goodman and Gerber.
Strain-life
εa = (σf'/E)(2Nf)b + εf'(2Nf)c. The transition life is where the two terms are equal: 2Nt = (εf' E/σf')1/(b−c). The loop uses the cyclic stress-strain curve εa = σa/E + (σa/K')1/n' with n' = b/c and K' = σf'/εf'n', so the loop and the life curve come from the same four constants. Branches are drawn with the Masing rule (twice the cyclic curve). Mean strain and mean stress are not included here.
Crack growth
da/dN = C (ΔK)m with ΔK = Y Δσ √(πa). With Y fixed, the integral from a0 to the critical size ac = (1/π)(Kc/(Y σmax))2 is closed form for m ≠ 2. The page takes R = 0, so σmax = Δσ. It leaves out the threshold ΔKth (a few MPa√m in steels) below which long cracks do not grow, the fast growth near Kc, overload retardation, and short-crack behaviour. Striations appear in ductile metals, mostly in the middle (Paris) range of growth rates. One striation per cycle holds best near 0.1 to 1 µm per cycle; outside that range the count and the cycles often differ.
Miner's rule
Damage adds linearly and the order of the blocks does not matter. In tests, high then low loading often fails at D below 1 and low then high above 1. Design codes often use D = 0.3 to 1 for this reason.
On this site: Reading a tensile curve (where σy and σu come from) · Fracture toughness (Kc and the critical crack) · Dislocations and Burgers vectors · Strengthening mechanisms · Creep and deformation maps · Wear mechanisms (choose mode 3, rolling contact fatigue) · Scratch testing: multi-pass failure · EBSD KAM maps (plastic strain near a crack) · TEM two-beam defect contrast (dislocation structures in fatigued metals)
Questions people ask
What is metal fatigue?
Damage from a load that repeats. Each cycle moves dislocations back and forth, a crack starts at the surface, grows a little every cycle, and the part breaks when the crack is big enough. The peak stress can be well below the yield strength.
What is the endurance limit?
The stress amplitude below which a material survives an unlimited number of cycles in lab tests. Many steels and titanium alloys show one, often near 106 to 107 cycles. Aluminium and copper alloys do not, so a fatigue strength at a stated life (often 5 × 108 cycles) is quoted instead.
What is the difference between Goodman, Gerber and Soderberg?
All three shrink the allowed amplitude as the mean stress rises. Goodman ends at the tensile strength on a straight line. Gerber ends there on a parabola and allows more. Soderberg ends at the yield strength and allows the least.
What is the difference between low-cycle and high-cycle fatigue?
Low-cycle fatigue has visible plastic strain in each cycle and lives up to about 104 to 105 cycles; it is described by strain-life. High-cycle fatigue is nearly elastic and lasts longer; it is described by stress-life. The transition life in the third picture marks the change for each material.
What are fatigue striations?
Fine parallel lines on a fatigue fracture surface, seen in the SEM. In ductile metals each line usually marks one load cycle, so the spacing gives the local crack growth rate. The larger beach marks visible by eye mark changes in loading, not single cycles.
What is the Paris law?
da/dN = C (ΔK)m: the crack growth per cycle rises as a power of the stress intensity range. For steels m is about 2 to 3.5, for aluminium alloys about 3 to 4. It holds in the middle range of growth rates.
References
Show the 12 references
- N. E. Dowling, Mechanical Behavior of Materials, 4th ed., Pearson (2013): chapter 9 (stress-life and mean stress), chapter 11 (fatigue crack growth), chapter 14 (strain-life); Table 14.1 for the 2024-T351 constants.
- SAE J1099, Technical Report on Low Cycle Fatigue Properties: Ferrous and Non-Ferrous Metals, SAE International (2002): Tables 2A and 2B for SAE 1045 (normalized, 192 HB; quenched and tempered, 500 HB) and SAE 4340 (409 HB).
- O. H. Basquin, The exponential law of endurance tests, Proceedings of the ASTM 10, 625 to 630 (1910).
- L. F. Coffin, A study of the effects of cyclic thermal stresses on a ductile metal, Transactions of the ASME 76, 931 to 950 (1954).
- S. S. Manson, Behavior of materials under conditions of thermal stress, NACA Technical Note 2933 (1953).
- P. Paris and F. Erdogan, A critical analysis of crack propagation laws, Journal of Basic Engineering 85, 528 to 533 (1963). doi:10.1115/1.3656900
- J. M. Barsom and S. T. Rolfe, Fracture and Fatigue Control in Structures, 3rd ed., ASTM / Butterworth-Heinemann (1999): the crack growth equations for ferrite-pearlite, martensitic and austenitic steels.
- A. N. Savkin, A. V. Andronik and R. Koraddi, Test procedure to determine the Paris equation coefficients for crack growth rates exemplified by aluminum alloy 2024-T3, Materials Testing 57, 912 to 919 (2015). doi:10.3139/120.110787
- M. A. Miner, Cumulative damage in fatigue, Journal of Applied Mechanics 12, A159 to A164 (1945).
- C. R. Soderberg, Factor of safety and working stress, Transactions of the ASME 52, 13 to 28 (1930).
- S. Suresh, Fatigue of Materials, 2nd ed., Cambridge University Press (1998): chapters on crack initiation, striations and crack growth.
- J. Schijve, Fatigue of Structures and Materials, 2nd ed., Springer (2009): fractography, beach marks and striations, and Miner's rule in practice.
BibTeX
@misc{tripathy2026fatiguesn,
author = {Tripathy, Manisha},
title = {Fatigue and S-N Curves Lab},
year = {2026},
howpublished = {\url{https://untetheredatom.com/mechanical-behavior/fatigue-and-sn-curves}},
note = {Interactive web tool}
}