Fatigue and fracture testing

Why it matters

Most mechanical failures in service are not single overloads. They are fatigueFailure in which a crack starts at a stress concentration and grows a little with every load cycle until the remaining section breaks.: a crack starts at a stress concentration and grows a little with every load cycle until the remaining section breaks. Axles, aircraft skins, welds, springs, medical implants and bicycle frames are all designed around fatigue. Fracture toughness tells you how big a crack a part can tolerate before it breaks suddenly. Together they connect lab testing to safe-life and damage-tolerant design.

Stress cycles

A cyclic load has a maximum sigma_max and a minimum sigma_min:

stress amplitude  sigma_a = (sigma_max - sigma_min) / 2
mean stress       sigma_m = (sigma_max + sigma_min) / 2
stress ratio      R       = sigma_min / sigma_max

A "fully reversed" test (R = -1) has zero mean stress. Rotating-beam and axial tests at R = -1 are the classic source of S-N data.

The S-N curve

Specimens are tested at several stress amplitudes until they fail, and amplitude is plotted against cycles to failure on a log scale. In the high-cycle region the curve is close to a straight line in log-log coordinates, described by Basquin's equationThe power law sigma_a = sigma_f' * (2N)^b that describes the high-cycle S-N curve as a straight line in log-log coordinates.:

sigma_a = sigma_f' * (2N)^b

2N is reversals (two per cycle), sigma_f' is the fatigue strength coefficient, and b is the fatigue strength exponent, typically between about -0.05 and -0.12 for metals. Because the line is straight in log-log space, you fit it with ordinary linear regression on log10(sigma_a) vs log10(2N).

Many steels show a fatigue (endurance) limitThe stress amplitude below which many steels appear not to fail, even after very many cycles; aluminium alloys show no clear limit.: below a certain amplitude, roughly 10^6 to 10^7 cycles and beyond, they appear not to fail. Aluminium alloys do not show a clear limit, so designers quote a fatigue strength at a stated life, for example 5 × 10^8 cycles.

Fatigue data scatter. At one amplitude, lives can vary by a factor of 10 or more. Real design curves are statistical (for example, a 95% survival curve), not a line through the averages. Surface finish, size, notches, environment and residual stress all shift the curve, which is why test reports describe the specimen and surface in detail.

Worked example

A steel tested at R = -1 gives these results:

A regression of log10(sigma_a) on log10(2N) gives b of about -0.089 and sigma_f' of about 1,280 MPa. The predicted life at 420 MPa is N = 0.5 * (420 / 1280)^(1 / -0.089), about 1.3 × 10^5 cycles. Treat it as a median estimate, not a guarantee.

Mean stress

A tensile mean stress shortens life. The simple GoodmanA mean-stress correction that converts a cycle to an equivalent fully reversed amplitude, sigma_a / (1 - sigma_m / S_u). correction converts a cycle with amplitude sigma_a and mean sigma_m into an equivalent fully reversed amplitude:

sigma_ar = sigma_a / (1 - sigma_m / S_u)

With sigma_a = 200 MPa, sigma_m = 150 MPa and S_u = 600 MPa, sigma_ar = 200 / 0.75 = 267 MPa. Goodman is conservative for many ductile metals; others (Gerber, Morrow, SWT) exist and your employer's design method decides which one you use.

Fracture toughness

Linear-elastic fracture mechanics describes the stress field at a crack tip with the stress intensity factorK = Y * sigma * sqrt(pi * a), which describes the stress field at a crack tip; fracture occurs when it reaches K_IC.:

K = Y * sigma * sqrt(pi * a)

a is the crack length (for an edge crack) or half-length (for a centre crack), and Y is a geometry factor (about 1.12 for a small edge crack in a wide plate). Fracture happens when K reaches the material's plane-strain fracture toughness K_IC, measured on pre-cracked specimens by a standard method such as ASTM E399. Rearranged, the critical crack size is

a_c = (1 / pi) * (K_IC / (Y * sigma))^2

Illustrative example. A high-strength aluminium alloy with K_IC = 25 MPa·sqrt(m) carries 300 MPa with a small edge crack (Y = 1.12): a_c = (1/pi) * (25 / 336)^2 = 0.00176 m, about 1.8 mm. Inspections must reliably find cracks well below that size. A steel with twice the toughness tolerates a crack four times longer at the same stress, because a_c scales with K_IC^2.

Crack growth between inspection intervals is modelled with the Paris lawThe relation da/dN = C (delta_K)^m that models crack growth per cycle, used in damage-tolerant design., da/dN = C (delta_K)^m, using constants measured in crack-growth tests. That is the basis of damage-tolerant design in aerospace.

Common mistakes

• Fitting cycles instead of reversals and then plugging the constants into the other form.
• Ignoring scatter and designing to the mean curve.
• Using smooth-specimen data for a notched or welded part: notches and welds need their own data or methods.
• Using a non-valid K_IC: thin specimens can give apparent toughness that is too high; the standard's validity checks matter.

Real fatigue and fracture-critical designs follow your industry's codes and are reviewed by qualified, often licensed, engineers.

Recap

• S-N data follow Basquin's power law; fit it in log-log space and respect scatter.
• Mean stress matters: Goodman converts to an equivalent fully reversed amplitude.
• K = Y sigma sqrt(pi a); the critical crack size scales with (K_IC / sigma)^2.

Try it

Open the Do tab to fit Basquin constants to S-N data, predict a life and compute a critical crack size.