The authors deal with a simplified model to construct the unique solution for necking in a hyperelastic rod; in this study, the stored energy function depends not only on the local strain field, but also on the gradient of displacements. The authors obtain the solution through phase planes, by imposing non-deforming boundary conditions and by converting the force-controlled problem into a displacement-controlled problem. From the unique solution obtained, the authors describe the whole deformation process due to increase in displacements. The main focus of the paper is the construction of solution before and after bifurcations. The stress-strain curve plotted from the solution shows two main characteristics: i) after the stress peak is reached, a sudden stress drop occurs; ii) afterwards occurs a stress plateau. Mathematical explanations for these two phenomena are provided.