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Fracture mechanics and fatigue

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?

  1. AIn real materials, the bonding force between the atoms themselves is only a few percent of the value assumed in the theoretical estimate
  2. BStress concentrates at the tips of tiny cracks inside the material or on its surface, locally reaching the theoretical strength
  3. CThe cross-sectional area shrinks under load, so the true stress becomes tens of times the nominal stress and reaches the theoretical strength
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Answer: B. Stress concentrates at the tips of tiny cracks inside the material or on its surface, locally reaching the theoretical strength

The theoretical strength is the stress needed to pull planes of atoms apart all at once, on the order of E/10 (about 7 GPa for glass). Griffith proposed that real materials contain tiny cracks, and because stress concentrates at their tips and locally reaches the theoretical strength, fracture can start even when the average stress is far lower. The fact that freshly drawn glass fibers with flawless surfaces reach strengths of several GPa supports this idea. The bonding force itself is already built into the estimate of the theoretical strength, so it is not smaller than assumed. The difference between true and nominal stress concerns ductile materials that neck, and does not explain the low strength of brittle materials.

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More Fracture mechanics and fatigue questions

  1. 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?
  2. 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))?
  3. Among the modes of crack deformation, which statement correctly describes Mode I?
  4. 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?
  5. 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?