The detection and evaluation of closed cracks are of prime interest in industry. Whereas conventional ultrasonic methods fail to detect these defects, nonlinear methods based on activation of the nonlinear behavior of closed cracks constitute an interesting alternative. The aim of this article is to give a better understanding of interactions between cracks and a longitudinal elastic wave for a quantitative investigation into nonlinear signatures. Using a 1D approach based on the literature, the nonlinear signature of the contact interface is analyzed in two cases. In the first, the interface is initially open and in the second, it is initially closed before interaction with an elastic wave. These signatures were qualitatively observed experimentally in real cracks. Next, in order to investigate the influence of the coexistence of open and closed zones within the interface, a numerical 2D-study is proposed. Two configurations are considered involving two steel blocks in contact, with different contact interface morphologies. The first configuration is a perfectly plane contact interface, while the second one involves an interface between a concave surface and a plane surface. A non-plane wave is also considered. This study attempts to establish a link between local second harmonic generation and interface parameters (pre-stress, gap) that can be exploited for the nondestructive quantitative evaluation of interfaces or cracks.