Two points read from a plot of measured copper yield stress against

PHYSICS · Multi-part · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Crystal structure, bonding and diffraction · Crystal structure, bonding and diffraction

Condensed Matter Physics, Ch Real crystals: vacancies, dislocations and the strength of metals

Problem

Two points read from a plot of measured copper yield stress

\sigma_{\rm y}

against

d_g^{-1/2}

, the empirical Hall-Petch form, are given. Data: for grain size

d_g=25\,\mu\mathrm{m}

d_g^{-1/2}=200\,\mathrm{m^{-1/2}}

and

\sigma_{\rm y}=53\,\mathrm{MPa}

; for

d_g=4\,\mu\mathrm{m}

d_g^{-1/2}=500\,\mathrm{m^{-1/2}}

and

\sigma_{\rm y}=95\,\mathrm{MPa}

. ((a)) Using the two points, find the slope

k_{\rm HP}

and the intercept

\sigma_{\rm y,0}

\sigma_{\rm y}=\sigma_{\rm y,0}+k_{\rm HP}\,d_g^{-1/2}

. ((b)) Predict the yield stress of copper with a grain size of

9\,\mu\mathrm{m}

, a point not on the table.

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