Fatigue and Fracture Mechanics

Why it matters

Repeated loading can initiate and grow cracks even when nominal stress stays below monotonic yield strength. Fatigue-life estimates require an appropriate tested stress–life or strain–life model; an endurance-limit value alone does not determine a finite life.

Stress-cycle quantities

σ_m = (σ_max + σ_min)/2 and σ_a = (σ_max − σ_min)/2. The stress range is Δσ = 2σ_a. Fully reversed loading has σ_m = 0 and stress ratio R = −1 when σ_max is nonzero.

An S–N curve relates stress amplitude to cycles to failure for specified conditions. Some materials exhibit an approximate endurance limit; others require a fatigue strength quoted at a specified life. Surface finish, size, notches, environment, temperature, mean stress, and statistical scatter matter. An “infinite-life” idealization is not a guarantee for every component.

Worked example: specified S–N relation

For an illustrative material under fully reversed loading, a fitted relation valid from 10³ to 10⁶ cycles is σ_a = C N_f^b, with C = 1000 MPa and b = −0.1. Estimate the life at σ_a = 316.228 MPa.

N_f = (σ_a/C)^(1/b) = (0.316228)^(−10) ≈ 100000 cycles. This lies inside the stated fit range. C has stress units and b is dimensionless. A formulation using reversals 2N_f has different constants and must not be mixed with this convention.

Crack assessment

For linear-elastic mode-I loading, K_I = Yσ√(πa). Y depends on geometry, loading, and the definition of a. K has units MPa√m when stress is in MPa and crack length is in metres. It is not the dimensionless stress concentration factor K_t.

For a centre crack of total length 2a in an ideal infinite plate, Y = 1. If a = 0.005 m and σ = 100 MPa, K_I = 100√(π × 0.005) = 12.53 MPa√m. Compare with a toughness appropriate to the constraint, thickness, temperature, and loading conditions. K_IC denotes plane-strain fracture toughness when its validity requirements are satisfied.

A common empirical crack-growth relation is da/dN = C_p(ΔK)^m in its fitted Paris regime. It does not describe every growth regime; threshold effects, load ratio, and unstable fracture require separate treatment.

Common mistakes

• Treating an endurance limit as a complete S–N curve.
• Confusing stress amplitude with stress range or maximum stress.
• Applying an empirical relation outside its calibration range.
• Mixing cycles and reversals or crack half-length and full length.

Quick check

1. If σ_max = 200 MPa and σ_min = −100 MPa, what are σ_m and σ_a? 50 MPa and 150 MPa.
2. Can fatigue occur below yield stress? Yes.
3. Does every metal have a distinct endurance limit? No.