The yield strength sigmay of many metals depends on the size

The yield strength sigma_y, 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: sigma_y = sigma_0 + kd^-1/2 The following results are from measurements of average grain diameter and yield stress: Using curve fitting, determine the constants sigma_0 and k in the Hall-Petch equation for this material. Using the constants determine with the equation the yield stress of material with grain size 0.05 mm. Make a plot that shows the data points with circle markers and the Hall-Petch equation with a solid line. Use linear interpolation to determine the yield stress of material with grain size of 0.05 mm. Make a plot that shows the data points with circle markers and linear interpolation with a solid line. Repeat part (b) using cubic spline interpolation.

Solution

d Actual value 0.005 205 0.009 150 0.016 135 0.025 97 A_) 0.04 89 At x= 0.05 0.062 80 y= -1021 (x) +156.5 = 100.45 0.085 70 The yield stress of the material with grain size 0.05 is 100.45 Mpa 0.11 67 B_) When x=0.05 , put this value in cubic spline equation : -45264 x-cube + 98227- x-squared - 6695 x + 219.3 y= 124..4 Mpa