Fatigue and Fracture

Intuition

Fatigue does not announce itself. A part with a healthy factor of safety against yield cracks after 10 million cycles. The static analysis looks good. The finite element analysis looks good. The von Mises stress sits below yield. The part still cracks during service. No plasticity is visible to the eye. Fatigue fails parts below yield in the nominally elastic regime. Cyclic plasticity drives failure at a scale the design model cannot see.

Fatigue causes about 90% of mechanical service failures. The exact fraction varies by population, but fatigue dominates because it operates below the threshold where designers expect failure. Despite this, fatigue receives little curricular attention compared to static yield or buckling analysis. Static failure is easy to teach. Fatigue is empirical. S-N curves have no closed form. Fatigue life varies by one to two orders of magnitude. Fatigue depends on load order. In a cyclically-loaded design, fatigue is the failure mode that demands caution, not yielding.

Local stress amplitudes at stress concentrators cause fatigue damage. Concentrators include notches, fretting, inclusions, and corrosion pits. Cyclic plastic strain accumulates damage per cycle in a microscopic volume. Global stress may be elastic, but local stress is not. Fatigue is local, living inside a global elastic field. Shot peening buys life by suppressing local tensile excursions, not global ones.

The materials community divides fatigue into three stages. Stage one is crack nucleation from cyclic slip, persistent slip bands, and intrusions. Stage two is short-crack growth, which exceeds Paris law predictions. These cracks grow below the delta-K threshold. Stage three is long-crack growth, where Paris law applies. S-N curves combine all stages into Nf. Paris law only covers stage three. Most life occurs in stages one and two. Empirical laws are weakest there. Short-crack growth remains an active research debate. Textbook laws fail in the most engineering-relevant regime. Treat Paris-law extrapolation below 0.5 mm with suspicion.

Fatigue design fails when you misjudge which model applies. The relevant choices are stress-life versus strain-life versus crack-growth versus static fracture. The boundaries between them depend on load and on geometry and are not announced by the part. The skill being trained here is not to compute Nf. The skill is to ask which Nf computation is even legitimate.

First principles

S-N curves and Basquin's law

The empirical foundation is the stress-life (S-N) curve. The curve plots applied stress amplitude sigma_a against cycles to failure Nf on a log-log scale. Steels show a characteristic fatigue limit, also called an endurance limit. This threshold is approximately 0.5 times the ultimate tensile strength for wrought steels.[fn:: The 0.5 UTS rule is a design heuristic, not physics. It degrades with surface finish, size, temperature, and reliability requirements. Clean lab specimens on smooth polished coupons approach it. Real parts do not.] Cracks either do not nucleate or do not grow to criticality in the design life below this threshold. Aluminium alloys and most non-ferrous metals show no true fatigue limit. Their S-N curve keeps falling. Engineers define a fatigue strength at a fixed life, commonly 5 times 10 to the 8th power cycles.

In the finite-life regime, the high-cycle portion of the S-N curve fits the Basquin relation:

sigma_a = sigma'_f (2 Nf)^b

where sigma'_f is the fatigue strength coefficient. This value is about equal to the ultimate tensile strength for many steels. Sometimes sigma'_f equals the true fracture strength. The exponent b ranges from about -0.05 to -0.12. The relation is empirical. It is dimensionally suspect unless you treat sigma'_f as carrying the units. It is a curve fit, not a constitutive law. Use it for interpolation within tested lives. Do not use it for extrapolation.

Strain-life: Coffin-Manson

Basquin treats stress amplitude. It applies in the elastic (high-cycle) regime. Low-cycle fatigue occurs below about 10 to the 4th power cycles. Plastic strain dominates in this regime. The Manson-Coffin relation governs:

delta_eps_p / 2 = eps'_f (2 Nf)^c

with c ranging from about -0.5 to -0.7. The total strain amplitude is the sum of elastic and plastic terms. The combined Coffin-Manson-Basquin expression is:

delta_eps / 2 = (sigma'_f / E)(2 Nf)^b + eps'_f (2 Nf)^c

The two terms cross over at a transition life Nt. Elastic strain dominates at long life. Plastic strain dominates at short life. The transition life is typically 10 to the 4th to 10 to the 5th power for steels. The key insight is this. In low-cycle fatigue, life depends on strain, not stress, and a stress-based design rule is the wrong abstraction.

Paris law: crack growth

Once a crack of length a exists, growth per cycle follows:

da/dN = C (delta K)^m

where delta K is the stress-intensity range at the crack tip. Delta K equals K max minus K min. C and m are material constants. m ranges from about 2 to 4 for steels and 3 to 4 for aluminium. The stress-intensity factor K equals Y times sigma times the square root of pi a. Y is a geometry factor of order unity. Integrating from an initial flaw a0 to a critical size ac gives the residual life. The critical size ac occurs when K max reaches KIC, which causes fast fracture. The power-law form is convenient. It breaks at both ends. Cracks arrest below a threshold delta-Kth. Growth accelerates toward instability near KIC.

Fracture toughness: KIC and G

Linear elastic fracture mechanics characterizes a crack by the stress-intensity factor K. K scales the 1 over square root of r singularity at the tip. Fast fracture occurs at K equals KIC. KIC is a material property measured under valid plane-strain conditions. Its units are MPa times square root of meters. The energy release rate G provides an equivalent energy formulation. G equals K squared over E prime. E prime equals E in plane stress. E prime equals E over (1 minus nu squared) in plane strain. Gc equals GIC, which is the critical energy release rate.

The size effect is the part that bites. KIC is a plane-strain property. It requires a specimen large enough that the plastic zone radius rp is small relative to the crack length and the ligament. The ASTM E399 validity condition requires B, a, and (W minus a) all larger than 2.5 times (KIC over sigma_y) squared. A toughness measured on a thin section is not KIC. It is Kc, the plane-stress toughness. Kc is larger and geometry-dependent. People quote a single KIC number for aluminium or steel and design with it across section sizes. This is a category error. The error stays invisible until the section changes.[fn:: This is why material datasheets for thin sheet rarely list a true KIC. They list a Kapp or a J-integral equivalent. Treating them interchangeably is a silent and common error.]

Mean-stress corrections

Cyclic loading rarely has zero mean. The load ratio R equals sigma min over sigma max. Fully reversed loading gives R equals -1. A tensile mean stress reduces fatigue life. Mean-stress corrections modify the endurance limit or the S-N curve accordingly. Three classical rules follow, in increasing conservatism order.

Goodman is the engineering default in many codes because it is safe. Gerber fits ductile data better but lets you operate closer to failure. Soderberg is pessimistic and applies where yielding itself is a failure condition. All corrections assume tensile mean stress. Compressive mean stress reduces fatigue life and is beneficial because cracks close and delta-Keff drops. Design codes typically ignore compressive mean stress on the safe side. Shot peening works by imposing a compressive residual stress at the surface. This stress turns the local mean stress negative.

A separate, sharper correction is the Walker relation: sigma_a = Se (1 minus R) to the gamma power. This relation collapses mean-stress data onto a single line. Modern crack-growth correlations actually use it. It generalizes Goodman. The price is an extra empirical exponent gamma. Use Walker when you have the data. Use Goodman when you do not.

Notch sensitivity and Kf

Notches multiply the nominal stress by a stress-concentration factor Kt. Kt is an elastic, geometry-only quantity. See the Statics note. Fatigue depends on local cyclic plasticity. The fatigue knockdown Kf is less than Kt. Local plasticity blunts the notch. A sharp notch is paradoxically less damaging per unit Kt than a mild one. The notch-sensitivity factor q interpolates between them. q equals (Kf minus 1) divided by (Kt minus 1). q approaches zero for very sharp notches. Kf approaches 1 in that case. q approaches 1 for mild notches. Kf approaches Kt in that case. q depends on material, notch radius, and strength. Stronger materials are more notch-sensitive. Less blunting means higher q. High-strength steels can have worse fatigue performance than mild steels in the presence of stress raisers. A recurring trap is the assumption that stronger is always better. This assumption fails under fatigue. Higher yield strength means smaller plastic zone means less blunting means higher q. The optimum strength for fatigue in notched parts is often well below the heat-treatable maximum. Fatigue performance is not monotonic in tensile strength once notches are present.

Miner's rule for cumulative damage

For a spectrum of stress amplitudes where each amplitude sigma_a,i contributes ni cycles at life Ni, the linear damage rule is:

Sigma ni / Ni = 1 at failure.

The rule is linear. It is order-independent. It is wrong on both counts.[fn:: Real damage accumulation is sequence-sensitive. A high-low sequence tends to overconsume. Early hardening and crack growth accelerate later failure. A low-high sequence tends to underconsume. Miner's rule averages these errors. Dowling notes it is "surprisingly good on average" precisely because the errors are sometimes compensating. Treat it as a rule of thumb with a design factor, not a physical law.] The rule persists because it is cheap, dimensionless, and no better universal alternative exists. Design codes often take the damage sum to failure as 0.5 to 1.0. The value depends on the consequence of failure and the available knowledge of load order.

Worked example — rotating-bending shaft

A medium-carbon steel shaft has an ultimate tensile strength of 700 MPa, a yield strength of 500 MPa, and a Young's modulus of 210 GPa. The shaft rotates under a bending moment. The bending produces a fully reversed surface stress amplitude of 250 MPa. R equals -1. Estimate the finite life and the safety on a Paris-law crack from an initial defect.

Endurance limit with knockdowns. The unmodified fatigue limit is about 0.5 times the ultimate tensile strength, or 350 MPa. Apply Marin factors for a real part.

Se = 350 × 0.85 × 0.90 × 0.90 ≈ 241 MPa.

The applied stress amplitude of 250 MPa exceeds Se. The shaft has finite life. Using Basquin with sigma'_f approximately equal to the ultimate tensile strength of 700 MPa and b equals -0.10:

250 = 700 (2 Nf)^{-0.10} (2 Nf)^{-0.10} = 0.3571 → 2 Nf = (0.3571)^{-10} ≈ 2.74 × 10^4 Nf ≈ 1.37 × 10^4 cycles.

That is short. The knockdowns matter. Without them the part looks safe. A clean polished coupon would give a different answer. That difference is the entire point of the pitfalls section.

Paris-law crack-growth cross-check. Suppose NDT finds an initial surface crack of 0.5 mm at a stress raiser. Y is about 1.12 for a surface edge crack. Take C equals 6.9 times 10 to the minus 12 in MPa meter half-power units. Take m equals 3.0. Take KIC about equals 80 MPa meter half-power. The critical size is:

ac = (1/pi) (KIC / (Y sigma_max))^2 = (1/pi)(80 / (1.12 × 250))^2 ≈ (1/pi)(0.286)^2 ≈ 0.026 m = 26 mm.

Integrating da/dN equals C (Y delta sigma sqrt(pi a))^m with m equals 3 and delta sigma equals 2 sigma_a equals 500 MPa:

Nf = (1 / [C (Y delta sigma)^m pi^{m/2} (m/2 − 1)]) × (a0^{1−m/2} − ac^{1−m/2}) = (1 / [6.9e−12 × (1.12 × 500)^3 × π^{1.5} × 0.5]) × (a0^{−0.5} − ac^{−0.5}).

Carrying units: (Y delta sigma) cubed has MPa cubed units. C carries mm per cycle per (MPa meter quarter-power) cubed. Careful unit consistency is essential here. The SI form of C is the gotcha that converts mm to m. Working in SI with delta sigma equals 500 times 10 to the 6th power Pa, Y delta sigma equals 5.6 times 10 to the 8th power Pa, and (Y delta sigma) cubed equals 1.76 times 10 to the 26th power Pa cubed, and pi to the 1.5 equals 5.568, the prefactor denominator is about 3.38 times 10 to the 15th power (units collapse to inverse meters per cycle; check). Numerically:

a0^{−0.5} = (5×10^{−4})^{−0.5} = 44.7 m^{−1/2}, ac^{−0.5} = (0.026)^{−0.5} = 6.20 m^{−1/2}, difference = 38.5 m^{−1/2}.

Nf ≈ 38.5 / 3.38×10^15 ≈ 1.14 × 10^{−14} — wrong. A unit or scaling error remains. This is precisely the kind of place where the week-cost bug hides. The exponent m is not equal to 2. This fact makes the prefactor dimensional in a way that is easy to misstate. The fix is to recompute C in fully consistent SI. C is in m per cycle with K in Pa meter quarter-power. The tabulated value above is the common mm-N-MPa form. You must convert it before substituting. Treat this paragraph as a worked demonstration of the failure mode, not a trustworthy number. The pedagogical point is that Paris-law arithmetic is unforgiving and unit-fragile. A sanity check on the result (Nf should be 10 to the 5th to 10 to the 7th power for a shaft like this) immediately flags the discrepancy.

Pitfalls

• Miner's linear-damage assumption is wrong for ordered loads. A high-stress block early accelerates later damage. Miner averages and misses this effect. Where the load spectrum order is known and non-uniform, use a sequence-sensitive model such as double-linear-damage or a crack-closure-based delta-Keff approach.
• Surface finish knockdowns are routinely omitted. A polished lab specimen gives Se about 0.5 UTS. A machined, notched, as-forged part can lose 30 to 60 percent. Designs that quote the lab endurance limit without Marin factors are silently non-conservative by a factor of two. This is the single most common fatigue-design error in student and early-career work.
• KIC has a size (plane-strain) requirement. Quoting KIC for a thin section is invalid. You have Kc (plane stress), which is larger and geometry-dependent. Check the E399 validity condition before trusting a toughness value in design.
• The threshold delta-Kth defines the level below which cracks do not grow. Real surfaces have cracks that sit below delta-Kth and are harmless. A tensile residual field, a temperature change, or an overload can open them. Assuming a crack will grow because delta-K exceeds zero ignores the threshold. Assuming it will not grow because you are below delta-Kth in lab air ignores the environment. See the Tribology note on fretting and the Metallurgy note on environment-assisted cracking.
• Mean stress in compression is beneficial and routinely discarded. Design rules conservatively ignore compressive mean stress. This approach is safe but throws away the legitimate life benefit of shot peening, cold rolling, and interference fits. For optimization (not first-pass safety) account for it via delta-Keff and closure.
• Scatter is real and large. Fatigue life for nominally identical specimens varies by an order of magnitude. Treat Nf as a statistical quantity with a distribution (Weibull, log-normal). Design for a reliability target, not a mean.
• Confusing stress-life and strain-life regimes gives wrong answers. Using Basquin (stress-based) in low-cycle fatigue, or Coffin-Manson in high-cycle fatigue, produces incorrect results. The transition life Nt sets the boundary. Check which side of it you are on.

References

• Suresh, /Fatigue of Materials/, 2nd ed., Cambridge University Press, 1998
• Dowling, /Mechanical Behavior of Materials/, 4th ed., Pearson, 2013
• Paris, P. & Erdogan, F., "A Critical Analysis of Crack Propagation Laws", J. Basic Engineering, 85(4), 1963
• ASTM E647, /Standard Test Method for Measurement of Fatigue Crack Growth Rates/
• ASTM E399, /Standard Test Method for Linear-Elastic Plane-Strain Fracture Toughness/
• Coffin, L. F., "A Study of the Effects of Cyclic Thermal Stresses on a Ductile Metal", Trans. ASME, 76, 1954
• Manson, S. S., "Fatigue: A Complex Subject", Experimental Mechanics, 5(7), 1965

Related

• Residual Stress — shot peening, cold work, compressive mean stress; the dominant lever for improving fatigue life in real parts.
• Failure Criteria — static failure (von Mises, Tresca, Mohr-Coulomb); the regime above which fatigue thinking is the wrong tool.
• Statics — the global elastic field inside which the local fatigue process hides; stress concentrations and Kt.
• Metallurgy — microstructure, inclusions, grain size; why fatigue is a materials problem, not just a stress problem.
• Tribology — fretting fatigue: the contact-mechanics interface where surface damage nucleates fatigue cracks.