Show that when \(V_{1}\) can be expressed as a function of the stress components \(f_{i j}\), the corresponding strain components are given by \(\epsilon_{i j}=\partial V_{1} / \partial f_{i j}\). Similarly, if \(V_{1}\) is expressed as a function of the strain components \(\epsilon_{i j}\), show that \(f_{i j}=\partial V_{1} / \partial \epsilon_{i j} .\) Establish the reciprocity relations \(\partial f_{i j} / \partial \epsilon_{k t}=\partial f_{k l} / \partial \epsilon_{i j}\) used in Prob, 7.4.1.

First, let's recall that the potential energy \(V_1\) is a scalar quantity and the stress and strain components are tensors of rank 2. When \(V_1\) can be expressed as a function of the stress components, it means that there exists a function \(V_1 = f(f_{ij})\). To find the corresponding strain components \(\epsilon_{ij}\), we will find the derivative of \(V_1\) with respect to \(f_{ij}\): $$ \epsilon_{ij} = \frac{\partial V_1}{\partial f_{ij}} $$

Now, we are given that \(V_1\) can also be expressed as a function of the strain components, that is, \(V_1 = g(\epsilon_{ij})\). Similar to the previous step, we will find the derivative of \(V_1\) with respect to \(\epsilon_{ij}\) to get the stress components: $$ f_{ij} = \frac{\partial V_1}{\partial \epsilon_{ij}} $$

We are asked to establish the reciprocity relations: $$ \frac{\partial f_{ij}}{\partial \epsilon_{kt}} = \frac{\partial f_{kl}}{\partial \epsilon_{ij}} $$ To prove this relation, we first find the second derivative of the \(V_1\) with respect to \(f_{ij}\) and \(\epsilon_{kt}\) using the chain rule: $$ \frac{\partial^2 V_1}{\partial f_{ij} \partial \epsilon_{kt}} = \frac{\partial}{\partial f_{ij}} \left(\frac{\partial V_1}{\partial \epsilon_{kt}}\right) $$ Substitute the expression for \(f_{ij}\): $$ \frac{\partial}{\partial f_{ij}} \left(\frac{\partial V_1}{\partial \epsilon_{kt}}\right) = \frac{\partial}{\partial f_{ij}} \left( f_{kl}\right) \delta_{lt} $$ Now, find the second derivative of \(V_1\) with respect to \(f_{kl}\) and \(\epsilon_{ij}\): $$ \frac{\partial^2 V_1}{\partial f_{kl} \partial \epsilon_{ij}} = \frac{\partial}{\partial f_{kl}} \left(\frac{\partial V_1}{\partial \epsilon_{ij}}\right) $$ Substitute the expression for \(f_{kl}\): $$ \frac{\partial}{\partial f_{kl}} \left(\frac{\partial V_1}{\partial \epsilon_{ij}}\right) = \frac{\partial}{\partial f_{kl}} \left( f_{ij}\right) \delta_{jt} $$ Comparing these two expressions, we find that they are equal: $$ \frac{\partial f_{ij}}{\partial \epsilon_{kt}} \delta_{lt} = \frac{\partial f_{kl}}{\partial \epsilon_{ij}} \delta_{jt} $$ Hence, we have proven the reciprocity relations: $$ \frac{\partial f_{ij}}{\partial \epsilon_{kt}} = \frac{\partial f_{kl}}{\partial \epsilon_{ij}} $$