Hall-Petch relation, worked example
In this video I show you how to get the parameters of the Hall-Petch relation (σ0 and k) from the flow curves of the same material with different grain sizes. This is a typical exam question, so it is good to know how to solve it.
Steps:
Get Rp0.2 from the flow curves. Fit a linear on the elastic part of the curve, then shift it to 0.02 true strain. Where the line intersects the flow curve, you get Rp0.2.
Plot Rp0.2 (σy) on the Y-axis and 1/sqrt(d) on the X-axis.
Fit a linear on the points. The slope and y-intercept parameters of the equation will be the k and sigma_0 parameters.
Once you have sigma_0 and k, you can answer the following questions:
What was the grain size, if the yield strength (Rp0.2, σy) was ... MPa?
What is the yield strength (Rp0.2, σy) if the grain size is ... nm? You just have to substitute the correct numbers to the Hall-Petch equation and rearrange it for the unknown variable.
The image was used from the following article:
Khan, Akhtar S., Babak Farrokh, and Laszlo Takacs. "Compressive properties of Cu with different grain sizes: sub-micron to nanometer realm." Journal of materials science 43.9 (2008): 3305-3313.
