Complete Question
The complete question is shown on the first uploaded image
Answer:
The value of the true strain at the onset of the necking is proved as, [tex]n = \epsilon_T[/tex]
Explanation:
From the question we see that necking begins when
[tex]\frac{d \sigma_T}{d \epsilon_T} = \sigma_T ---(1)[/tex]
Now we are told that
[tex]\sigma_T = K \epsilon ^n _T[/tex]
So substituting this into equation 1
[tex]\frac{d}{d \epsilon_T} (K \epsilon^n_T) = \sigma_T[/tex]
[tex]K n \epsilon^{n-1}_T = \sigma_T[/tex]
But we are told in the question that [tex]\sigma_T = K \epsilon ^n _T[/tex] So,
[tex]K n \epsilon^{n-1}_T = K \epsilon ^n _T[/tex]
Dividing both sides with [tex]K \epsilon ^n _T[/tex]
We have
[tex]\frac{K n \epsilon^{n-1}_T}{ K \epsilon ^n _T} =\frac{ K \epsilon ^n _T}{K \epsilon ^n _T}[/tex]
[tex]n \epsilon_T^{-1} =1[/tex]
[tex]n = \epsilon_T[/tex]
