The true stress (in MPa) versus true strain relationship for a metal is given by $σ = 1020 ϵ^{0.4}$

The cross-sectional area at the start of a test (when the stress and strain values are equal to zero) is $100 m m^{2}$ . The cross-sectional area at the time of necking ( $i n m m^{2})$ is (correct to two decimal places).

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Solution

Given, $σ = 1020 ϵ^{0.4}$

at UTS, $ϵ_{T} = n = 0.4$

at UTS, neck formation starts,

$ϵ_{T} = 0.4 = I n (\frac{A_{0}}{A_{f}})$

0.4 = $I n (\frac{100}{A_{f}})$

$\frac{100}{A_{f}} = e^{0.4}$

$A_{f} = \frac{100}{e^{0.4}}$

Cross-section area at the time of necking,

$A_{f} = 67.0.32 m m^{2}$

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The true stress (in MPa) versus true strain relationship for a metal is given by σ=1020ϵ0.4 The cross-sectional area at the start of a test (when the stress and strain values are equal to zero) is 100mm2. The cross-sectional area at the time of necking (inmm2) is (correct to two decimal places).

Given, σ=1020ϵ0.4 at UTS, ϵT=n=0.4 at UTS, neck formation starts, ϵT=0.4=In(A0Af) 0.4 = In(100Af) 100Af=e0.4 Af=100e0.4 Cross-section area at the time of necking, Af=67.0.32mm2

Given, $σ = 1020 ϵ^{0.4}$

at UTS, $ϵ_{T} = n = 0.4$

at UTS, neck formation starts,

$ϵ_{T} = 0.4 = I n (\frac{A_{0}}{A_{f}})$

0.4 = $I n (\frac{100}{A_{f}})$

$\frac{100}{A_{f}} = e^{0.4}$

$A_{f} = \frac{100}{e^{0.4}}$

Cross-section area at the time of necking,

$A_{f} = 67.0.32 m m^{2}$