Fracture mechanics and fatigue
Fatigue crack growth in a steel plate is described by the Paris law da/dN = C(ΔK)ᵐ (C = 1.0×10⁻¹², with da/dN in m/cycle, ΔK in MPa·√m, and m = 4), and ΔK = Δσ√(πa) (geometry factor 1). Under cycling with a stress range Δσ = 200 MPa, which value is closest to the number of cycles for the crack to grow from 1.0 mm to 5.0 mm? Assume that ΔK over this range (about 11–25 MPa·√m) is well above the threshold and within the range where the Paris law holds.
- AAbout 2.5×10⁵ cycles
- BAbout 5.1×10⁴ cycles
- CAbout 6.3×10⁴ cycles
Show answer and explanation
Answer: B. About 5.1×10⁴ cycles
For m = 4, da/dN = C(Δσ√π)⁴a², so separating variables and integrating gives N = (1/a₀ − 1/a_f)/[C(Δσ√π)⁴]. C(Δσ√π)⁴ = 1.0×10⁻¹²×200⁴×π² ≈ 1.58×10⁻² m⁻¹ and 1/a₀ − 1/a_f = 1000 − 200 = 800 m⁻¹, so N ≈ 800/(1.58×10⁻²) ≈ 5.1×10⁴ cycles. About 2.5×10⁵ cycles assumes the rate at the initial 1.0 mm, 1.58×10⁻⁸ m/cycle, continues to the end, ignoring that ΔK rises and the crack accelerates as it grows. About 6.3×10⁴ cycles drops the 1/a_f term. Most of the life is spent while the crack is still short, early on.