Mechanics of materials / Sheet 05
Fatigue
Formula reference for mechanical engineering interviews. Fatigue happens below yield, starts at the surface, and is governed by geometry, surface condition and mean stress rather than by material strength alone.
Scope: Metals, high and low cycleSections: 12Units: SI, MPaRevision: A
47 formulas
Cycle parameters
Alternating stress
σa = (σmax - σmin) / 2
The amplitude; this is what drives crack growth
Mean stress
σm = (σmax + σmin) / 2
Tensile mean is harmful, compressive mean helps
Stress range
Δσ = σmax - σmin = 2 σa
The quantity used in fracture mechanics
Stress ratio
R = σmin / σmax
R = -1 fully reversed, R = 0 zero-based, R = 1 static
Amplitude ratio
A = σa / σm
The slope of the load line on the Goodman diagram
Endurance limit
Se = ka kb kc kd ke · S′e
Marin equation: correct the lab value down to the real part.
Lab value, steel
S′e = 0.5 Sut
Cap at 700 MPa once Sut exceeds 1400 MPa
Aluminium
Sf ≈ 0.4 Sut at 5×10⁸ cycles
No true endurance limit; design for finite life
Surface factor
ka = a · Sutb
Ground 1.58/-0.085, machined 4.51/-0.265, forged 57.7/-0.718
Size factor
kb = 1.24 d-0.107
For 2.79 to 51 mm; kb = 1 under axial load
Load factor
kc = 1 / 0.85 / 0.59
Bending / axial / torsion
Temperature and reliability
kd, ke
ke = 0.897 at 90%, 0.814 at 99%, 0.753 at 99.9%
ka is usually the harshest factor, because cracks start at the surface. An as-forged finish can cost more than half the endurance limit on its own.
S-N and strain-life
Finite life range
Sf = a Nb
Between 10³ and 10⁶ cycles
Curve constants
a = (f Sut)² / Se, b = -⅓ log(f Sut / Se)
f ≈ 0.9 at 10³ cycles
Cycles to failure
N = (σa / a)1/b
Invert the S-N relation
Basquin, stress-life
σa = σ′f (2N)b
High cycle, elastic regime
Coffin-Manson
Δεp/2 = ε′f (2N)c
Low cycle, plasticity dominated
Total strain-life
Δε/2 = (σ′f/E)(2N)b + ε′f(2N)c
The two terms cross near 10³ cycles, the transition life
Below about 10³ cycles the part is yielding every cycle, so strain-life governs. Above 10⁶ you are in the infinite-life regime for steel only.
Mean stress criteria
Goodman
σa/Se + σm/Sut = 1/n
Linear, the standard design choice
Soderberg
σa/Se + σm/Sy = 1/n
Most conservative, also covers yielding
Gerber
n σa/Se + (n σm/Sut)² = 1
Parabolic, fits ductile steel test data best
ASME elliptic
(n σa/Se)² + (n σm/Sy)² = 1
Common for rotating shafts
Langer yield line
σa + σm = Sy / n
Always run alongside, for first-cycle yield
Combined loading
σ′a = √(σa² + 3τa²)
Build σ′m the same way, then enter Goodman
Order of conservatism, from safest to least: Soderberg, Goodman, ASME elliptic, Gerber. Report Goodman unless you have a reason, and say which one you used.
Notches
Fatigue notch factor
Kf = 1 + q (Kt - 1)
Apply to σa, and to σm unless local yielding relieves it
Notch sensitivity
q = (Kf - 1) / (Kt - 1)
Between 0 and 1; from Neuber or Peterson charts
Neuber form
q = 1 / (1 + √a / √r)
r is the notch root radius, √a a material constant
Typical Kt
2 to 3 hole, 1.5 to 3 fillet, 2 to 4 keyway
Always on the net section
q falls as the root radius shrinks, so Kf is always less than Kt. That does not license sharp corners: Kt rises faster than q falls, so the net effect still gets worse. Generous fillets remain the rule.
Cumulative damage
Miner's rule
Σ (ni / Ni) = 1
ni applied cycles, Ni life at that stress level
Real scatter
Σ (ni / Ni) = 0.7 to 2.2
Quote this when you quote the rule
Known weakness
sequence independent
High-to-low ordering fails early, low-to-high fails late
Cycle counting
rainflow method
Converts a random load history into discrete cycles
Fracture mechanics
Stress intensity
KI = Y σ √(π a)
Y is the geometry factor, a the crack length
Range per cycle
ΔK = Y Δσ √(π a)
Only the tensile part of the cycle counts
Paris law
da / dN = C (ΔK)m
m is roughly 3 for steel, 3 to 4 for aluminium
Fast fracture
KI ≥ KIC
The end of stable growth
Critical crack size
ac = (1/π)(KIC / Y σ)²
Sets the inspection interval
Three growth regimes
threshold, Paris, unstable
Below ΔKth cracks do not propagate at all
Reading a fracture surface
Initiation site
at the surface, at a stress raiser
Ratchet marks point back to it
Propagation zone
smooth, beach marks
Striations under a microscope, one per cycle
Final fracture zone
rough, dull, overload
Ductile dimples or brittle cleavage
Zone ratio
small final zone = low load
The part simply ran out of cycles
What helps and what hurts
| Improves fatigue life | Mechanism | Reduces fatigue life |
|---|---|---|
| Shot peening | compressive residual stress | Tensile residual stress |
| Polishing, grinding | removes initiation sites | As-forged or corroded surface |
| Nitriding, carburizing | hard compressive case | Decarburization |
| Generous fillets | lowers Kt | Sharp corners, keyways, cross holes |
| Correct bolt preload | joint absorbs the load swing | Loose or underpreloaded bolts |
| Cold rolling threads | grain flow plus compression | Cut threads, chrome plating |
Worked chain: infinite life check
Step 1
σmax, σmin → σa, σm
From the load history at the critical location
Step 2
S′e = 0.5 Sut, then apply Marin
Surface and size usually dominate
Step 3
Kf from Kt and q, apply to σa
Do not skip this; the notch is where it breaks
Step 4
Goodman for n, plus Langer for yield
Report the lower of the two
Step 5
if finite life, get N from the S-N curve
Mandatory for aluminium, which has no limit
Numbers worth memorizing
| Quantity | Value | Comment |
|---|---|---|
| S′e for steel | 0.5 Sut | capped at 700 MPa |
| Endurance limit knee | 10⁶ cycles | steel and titanium only |
| Aluminium at 5×10⁸ | 0.4 Sut | curve keeps falling |
| Low to high cycle boundary | 10³ cycles | strain-life below this |
| Paris exponent m, steel | ≈ 3 | 3 to 4 for aluminium |
| Share of service failures | ≈ 90% | fatigue, not static overload |
Concepts that decide the interview
- Fatigue happens below yield, so every static check in the previous sheets can pass while the part still fails. If asked why something that passed FEA broke in service, start here.
- Cracks start at the surface, which is why surface finish is the harshest Marin factor and why every effective fix is a surface fix.
- Compressive residual stress is the cheapest improvement available. Shot peening shifts the operating point left on the Goodman diagram at zero weight cost.
- A correctly preloaded bolt survives because the joint carries most of the load fluctuation, leaving the bolt with a high mean stress and a small alternating one. Tightening a bolt more makes it last longer, which sounds wrong and is not.
- The endurance limit is a steel and titanium phenomenon. Designing an aluminium part for infinite life is a real error, and it surfaces every time lightweighting comes up.
- Weld fatigue strength is governed by toe geometry and residual stress, and is nearly independent of base metal strength. Specifying a stronger steel does not fix a cracking weld.
- Kf is less than Kt because notch sensitivity falls with radius, but Kt rises faster, so sharp corners still lose. Generous fillets remain the rule.
- Miner's rule is linear and sequence independent, and real data scatters from 0.7 to 2.2. Say that when you quote it.
- Corrosion, fretting and elevated temperature remove the endurance limit entirely, so infinite-life design is off the table in those environments regardless of material.
Sheet 05 of 07 / Rev AConstant amplitude unless noted, room temperature, no corrosion