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Fracture mechanics and fatigue

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))?

  1. AAbout 149 MPa
  2. BAbout 374 MPa
  3. CAbout 211 MPa
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Answer: C. About 211 MPa

2Eγs/(πa) = 2×70×10⁹×1.0/(3.14×1.0×10⁻⁶) ≈ 4.46×10¹⁶ Pa², and its square root is about 2.11×10⁸ Pa, or about 211 MPa. The Griffith equation comes from the condition that a crack grows unstably when the elastic strain energy released as it extends exceeds the energy 2γs of the two new surfaces it creates. About 149 MPa drops the factor of 2 that comes from forming two surfaces, and about 374 MPa forgets to divide by π. Because σf is inversely proportional to √a, making the crack 4 times longer halves the fracture stress.

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More Fracture mechanics and fatigue questions

  1. Among the modes of crack deformation, which statement correctly describes Mode I?
  2. 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?
  3. 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?
  4. Which pairing of a feature seen when observing a fracture surface in a scanning electron microscope with the type of fracture is correct?
  5. 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?