5. Strengthening Mechanisms in Metals You may find it useful to read Ch. 7 in Callister. a) List the four main ways to strengthen metals and describe the main principles behind each. b) Explain how grain boundaries affect the ability of a metal to plastically deform. c). The relationship between yield stress and grain size of a polycrystalline material is described by the Hall-Petch equation: σy = 0 + kyd-1/2 where Oy is the yield stress, d is the average grain diameter, and σ and ky are the Hall- Petch constants. i. Data for the mechanical response of steel as a function of the average grain size is shown below. Plot the data and determine σo and ky. Yield stress, oy (MPa) 117.23 Grain size, d'1/2 (μm-12) 0.077 0.090 123.53- 0.106 131.40 0.123 137.69 0.130 146.38 0.154 152.65 0.185 172.35 0.212 183.37 0.258 204.62 0.263 210.93 0.292 223.52 0.336 243.20 0.353 253.44 0.462 306.19 0.484 302.20 0.594 356.53 0.662 390.389 Why does the yield strength increase as the grain size decreases?