Question

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 necking.

Answer 1

Complete Question

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Answer:

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

Explanation:

From the question we see that necking begins when

                 $\frac{d \sigma_T}{d \epsilon_T} = \sigma_T ---(1)$

Now we are told that

               $\sigma_T = K \epsilon ^n _T$

So substituting this into equation 1

                 $\frac{d}{d \epsilon_T} (K \epsilon^n_T) = \sigma_T$

               $K n \epsilon^{n-1}_T = \sigma_T$

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

                $\frac{K n \epsilon^{n-1}_T}{ K \epsilon ^n _T} =\frac{ K \epsilon ^n _T}{K \epsilon ^n _T}$

                    $n \epsilon_T^{-1} =1$

                     $n = \epsilon_T$