High-carbon steel with a fully pearlitic microstructure was used to form a high-strength bolt (H.-C. Lee et al., J. Mater. Proc. Tech. 211, p. 1044, 2011). It was found that the pearlite in the bolt head had an average interlamellar spacing of 257 nm whereas the average spacing in the body of the bolt was 134 nm.

 Assuming that dislocation pile-up is the primary mechanism responsible for the strength of this alloy, what ratio of strength (or hardness) might be expected between the head and body of the bolt? Hint: 

Use the Hall–Petch equation to compare the yield strength at various points in the bolt.

Hall–Petch Eq.: Sigma_yield strength = (k_y)*(d^(-0.5))

I am in a Mechanical Metallurgy class. The book being used is Deformation and Fracture Mechanics of Engineering Materials 5^th Edition by: Hertzberg, Vinci, & Hertzberg.

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ATTACHED IMAGE Both the balt head and body ... View the full answer