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Fracture mechanics and fatigue

Which statement correctly describes the curve of fatigue crack growth rate da/dN plotted against the stress intensity factor range ΔK on log-log axes?

  1. AThe middle region is a straight line given by the Paris law da/dN = C(ΔK)ᵐ, and below ΔK_th the crack hardly grows
  2. Bda/dN is constant regardless of ΔK, set only by the crack length and the material's yield stress
  3. CThe slope of the line decreases as ΔK grows, and crack growth stops when the maximum stress intensity factor approaches K_IC
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Answer: A. The middle region is a straight line given by the Paris law da/dN = C(ΔK)ᵐ, and below ΔK_th the crack hardly grows

The curve is usually divided into three regions. In the low region where ΔK approaches the threshold ΔK_th, the growth rate drops sharply, and below ΔK_th the crack effectively does not grow. The intermediate region is a straight line on log-log axes and can be described by the Paris law da/dN = C(ΔK)ᵐ, with an exponent m of roughly 2–4 for metals. In the high region where the maximum stress intensity factor K_max approaches the fracture toughness K_IC, the rate rises sharply and leads to unstable fracture, so the claim that growth stops is backwards. ΔK = YΔσ√(πa) depends on both the stress range and the crack length, so the rate is not determined by crack length alone.

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More Fracture mechanics and fatigue questions

  1. 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.
  2. The theoretical strength of a perfect crystal is estimated at roughly one tenth of the Young's modulus, yet the actual fracture strength of brittle materials such as glass is often only a few tens to a few hundreds of times smaller than that. What is the main reason?
  3. An elliptical hole in a wide plate (radius along the major axis a = 1.0 mm, radius of curvature at the end of the major axis ρ = 0.040 mm) carries a tensile stress of 100 MPa perpendicular to the major axis. Estimating the maximum stress at the end of the hole with σmax = σ₀(1 + 2√(a/ρ)), which value is closest?
  4. A glass (Young's modulus E = 70 GPa, surface energy γs = 1.0 J/m²) has a surface crack of length a = 1.0 µm. Which value is closest to the fracture stress estimated with the Griffith equation σf = √(2Eγs/(πa))?
  5. Among the modes of crack deformation, which statement correctly describes Mode I?