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Strengthening mechanisms and dislocations

In a substitutional solid solution, assume that the increase in yield stress Δσ is proportional to the square root of the solute concentration c (the Fleischer model). If Δσ = 30 MPa at 1 at% solute, which is closest to Δσ at 4 at% solute? Ignore interactions between solute atoms and precipitation.

  1. AAbout 60 MPa
  2. BAbout 120 MPa
  3. CAbout 76 MPa
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Answer: A. About 60 MPa

Since Δσ ∝ c^(1/2), a fourfold increase in concentration doubles Δσ (√4 = 2), giving 30 × 2 = 60 MPa. About 120 MPa assumes that Δσ is proportional to concentration, and about 76 MPa uses the Labusch model, in which Δσ is proportional to concentration to the power 2/3 (30 × 4^(2/3) ≈ 76), which is not the premise of this question. The Fleischer model treats a dislocation as being caught one at a time on widely scattered solute atoms, and the resistance of each is set by the difference in atomic size between solute and matrix (the size effect) and the difference in shear modulus (the modulus effect). Because the gain in strength slows as the concentration increases, choosing elements with large size or modulus differences is what pays off when seeking high strength from solid-solution strengthening.

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More Strengthening mechanisms and dislocations questions

  1. When annealed low-carbon steel is pulled in tension, the stress drops after an upper yield point and a yield-point phenomenon appears in which Lüders bands travel along the specimen. Which statement about this phenomenon and its remedy is correct?
  2. The flow stress of a work-hardened metal is estimated with the Taylor equation σ = MαGb√ρ (ρ is the dislocation density). For cold-worked copper with M = 3.06, α = 0.3, G = 48 GPa, b = 0.256 nm and ρ = 1.0×10¹⁴ m⁻², which is closest to σ?
  3. When a metal is plastically deformed in tension, unloaded and then compressed in the reverse direction, the yield stress in compression is lower than the flow stress it would have had if tension had continued. Which is the correct explanation of this phenomenon (the Bauschinger effect)?
  4. As precipitates grow during aging, the strength rises to a maximum and then falls (overaging). Which is the correct explanation in terms of the interaction between precipitates and dislocations? Take the volume fraction of precipitates as constant.
  5. In an aluminum alloy, the increase in shear stress due to the Orowan mechanism, in which dislocations bow out and pass between precipitates, is estimated as Δτ ≈ Gb/L (L is the precipitate spacing). With G = 26 GPa, b = 0.286 nm and L = 100 nm, and taking the increase in the yield stress of the polycrystal as Δσ = MΔτ (Taylor factor M = 3.06), which is closest to Δσ?