For a tensile test, it can be demonstrated that necking begins when dσTdεT=σT. Using Equation σT=K(εT)n, determine an expression for the value of true strain at this onset of ne…

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asked Feb 13, 2021 11.9k views

1 Answer

Ken Wei · Answer 1 · 2021-02-20T08:02:43+0000

Complete Question

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Answer:

The value of the true strain at the onset of the necking is proved as,
$n = \epsilon_T$

Step-by-step explanation:

From the question we see that necking begins when

$(d \sigma_T)/(d \epsilon_T) = \sigma_T ---(1)$

Now we are told that

$\sigma_T = K \epsilon ^n _T$

So substituting this into equation 1

$(d)/(d \epsilon_T) (K \epsilon^n_T) = \sigma_T$

$K n \epsilon^(n-1)_T = \sigma_T$

But we are told in the question that
$\sigma_T = K \epsilon ^n _T$ So,

$K n \epsilon^(n-1)_T = K \epsilon ^n _T$

Dividing both sides with
$K \epsilon ^n _T$

We have

$(K n \epsilon^(n-1)_T)/( K \epsilon ^n _T) =( K \epsilon ^n _T)/(K \epsilon ^n _T)$

$n \epsilon_T^(-1) =1$

$n = \epsilon_T$