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Fracture mechanics and fatigue

Check Griffith theory, stress concentration and the plastic zone at a crack tip, fracture modes, reading fracture surfaces, fatigue crack initiation and growth, the Paris law, mean stress corrections and life estimates with Miner's rule through three-choice questions.

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Fracture mechanics and fatigue: key points

The actual strength of materials is far below the theoretical strength because stress concentrates at the tips of tiny cracks. Griffith proposed that fracture proceeds when the elastic energy released as a crack extends exceeds the energy needed to create new surfaces. In metals, the work consumed by plastic deformation at the crack tip is far larger, and the tendency to fracture is organized in terms of the stress intensity factor K and its critical value.

Under repeated loading, cracks arise from slip at the surface even at stresses below the yield stress, and they advance a little in each cycle. The growth rate is set by the stress intensity factor range ΔK, and the intermediate region can be described by the Paris law. In design, the effect of mean stress is corrected with the Goodman diagram, and damage from varying loads is added up with Miner's rule to estimate life.

All Fracture mechanics and fatigue questions (15)
  1. 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?
  2. 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?
  3. 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))?
  4. Among the modes of crack deformation, which statement correctly describes Mode I?
  5. In a ductile metal such as steel, the energy needed for a crack to advance (the critical energy release rate G_c) is orders of magnitude larger than 2γs (a few J/m²) estimated from the surface energy. What is the main reason?
  6. In a steel with a yield stress of 500 MPa, the stress intensity factor at a crack tip is K = 30 MPa·√m. Estimating the size of the plane-stress plastic zone (its distance from the crack tip) with r_p = (1/2π)(K/σy)², which value is closest?
  7. Which pairing of a feature seen when observing a fracture surface in a scanning electron microscope with the type of fracture is correct?
  8. Which statement correctly describes the initiation of fatigue cracks when a stress amplitude below the yield stress is repeatedly applied to a metal with a smooth surface?
  9. The fracture surface of a part that failed by fatigue shows shell-like patterns visible to the naked eye (beach marks) and fine lines visible in an electron microscope (striations). Which statement about them is correct?
  10. A cyclic stress with a mean stress σm = 200 MPa is applied to a steel with a fully reversed fatigue limit σw = 300 MPa and a tensile strength σB = 800 MPa. Using the modified Goodman relation σa = σw(1 − σm/σB), which value is closest to the stress amplitude corresponding to the fatigue limit?
  11. Which surface treatment can be expected to improve the fatigue strength of a steel part?
  12. A steel part fails by fatigue after 1.0×10⁵ cycles at stress amplitude σ₁ and after 1.0×10⁶ cycles at σ₂. After it is used for 4.0×10⁴ cycles at σ₁, it continues to be used at σ₂. Which value is closest to the remaining life estimated with Miner's rule (failure when Σn/N = 1)?
  13. Which statement about low-cycle fatigue (roughly 10⁴ cycles or fewer to failure) is correct?
  14. 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?
  15. 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.

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