Demonstrate that Equation \(6.16\), the expression defining true strain, may also be represented by $$ \epsilon_{T}=\ln \left(\frac{A_{0}}{A_{i}}\right) $$ when specimen volume remains constant during deformation. Which of these two expressions is more valid during necking? Why?

We are given that the volume of the specimen remains constant during deformation, which means that \(V_0 = V_i\), where \(V_0\) is the initial volume, and \(V_i\) is the volume after deformation. For a cylindrical specimen, the volume is given by \(V = Ah\), where \(A\) is the cross-sectional area and \(h\) is the height of the cylinder (length). Thus, $$A_0 l_0 = A_i l_i$$.

We have derived the relationship between the initial and final volume as \(A_0 l_0 = A_i l_i\). We will now express the ratio \(\frac{A_{0}}{A_{i}}\) in terms of lengths \(l_0\) and \(l_i\). From the volume relationship, we can write $$\frac{A_{0}}{A_{i}} = \frac{l_i}{l_0}.$$

We now have the area ratio \(\frac{A_{0}}{A_{i}}\) in terms of \(l_0\) and \(l_i\). To demonstrate that the true strain equation can also be represented by the area ratio, we will apply a natural logarithm to the area ratio expression we derived in step 2. Using the natural logarithm, we have $$\epsilon_{T} = \ln\left(\frac{l_i}{l_0}\right) = \ln \left(\frac{A_{0}}{A_{i}}\right).$$ This shows that the true strain equation can indeed be represented by the area ratio when the volume of the specimen remains constant during deformation.

During the necking process, the specimen undergoes localized deformation, concentrating stress and strain in a smaller region. In this situation, the volume of the specimen may not remain constant. The expression $$\epsilon_{T} = \ln \left(\frac{A_{0}}{A_{i}}\right)$$ based on the area ratio relies on the assumption of constant volume during deformation. Therefore, this expression may not accurately represent the true strain during necking because of potential volume changes. On the other hand, the expression $$\epsilon_{T} = \ln\left(\frac{l_i}{l_0}\right)$$ does not rely on the assumption of constant volume and is a more direct measure of length changes involved in the deformation process. So during necking, the more accurate and valid expression for true strain is: $$\epsilon_{T} = \ln\left(\frac{l_i}{l_0}\right)$$ because it does not depend on the assumption of constant volume during deformation.