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arsen [322]
3 years ago
10

The yield strength for an alloy that has an average grain diameter of 4.5 x 10-2 mm is 109 MPa. At a grain diameter of 6.8 x 10-

3 mm, the yield strength increases to 263 MPa. At what grain diameter, in mm, will the yield strength be 224 MPa? d = Enter your answer in accordance to the question statement mm
Physics
1 answer:
AveGali [126]3 years ago
3 0

First we start withe the Hall-Pethch relation,

So,

\sigma_{y1} = \sigma_{0}+\frac{k_y}{\sqrt{d_1}}

Where,

\sigma_{y1} = Initial yield Strenght

d_1 = Initial grain size of the alloy

Substituting,

109MPa = \sigma_0 + \frac{k_y}{\sqrt{4.5*10^-2mm}}

We make the same relationship but now for the yield strength of 224MPa, so,

263MPa = \sigma_0 + \frac{k_y}{\sqrt{6.8*10^-3mm}}

Solving,

\sigma_0 = 11.0655 MPa

K_y= 20.7751 MPa.mm^{1/2}

Hall-Petch relation is for the constant

\sigma_y = 11.0655MPa +\frac{20.7751MPa.mm^{1/2}}{\sqrt{d}}

At end we make the sustitution for 224Mpa, so

\sigma_y=224MPa

224MPa = 11.0655MPa +\frac{20.7751MPa-mm^{1/2}}{\sqrt{d}}

d=\sqrt{0.0975657mm}

d= 0.9877mm

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