The yield strength, cry, of many metals depends on the size of the grains. For these metals, the relationship between the yield stress and the average grain diameter d can be modeled by the Hall-Petch equation:
σy = σ0 + kd(€“1/2)
The following are results from measurements of average grain diameter and yield stress.

(a) Using curve fitting, determine the constants σ0 and k in the Hall-Petch equation for this material. Using the constants determine with the equation the yield stress of material with a grain size of 0.05 mm. Make a plot that shows the data points with circle markers and the curve derived from the Hall-Petch equation with a solid line.
(b) Use linear interpolation to determine the yield stress of material with a grain size of 0.05 mm. Make a plot that shows the data points with circle markers and the linear interpolation with a solid line.
(c) Use cubic interpolation to determine the yield stress of material with a grain size of 0.05 mm. Make a plot that shows the data points with circle markers and cubic interpolation with a solid line.

Transcribed Image Text:

d (mm)0.005 0.009 0.016 00250.040 0.062 0.0850.110 , MPa) | 205 | 150 | 135 | 97 | 89 | 80 | 70 | 67