The present work represents a progress towards the understanding of the mechanical behavior of the fiber/matrix interface during push-in tests of fiber-reinforced polymer-matrix composites. Finite element simulations incorporating a cohesive zone model are used for this purpose. Different values of interface strength, interface fracture toughness, fiber diameter and friction coefficient are considered to study how they affect the load-displacement curves. A critical value of the displacement exists, being independent of the fiber diameter for given values of interface strength and fracture toughness, marking the separation between two regimes: (i) a cohesive-dominated zone interaction and (ii) a frictional contact between debonded fiber and matrix. Maps showing the different regimes are constructed, proving their helpfulness to tune the mechanical properties of the interface in order to favor a certain mechanical response. Finally, we study the debonding velocity and how this is affected by the mechanical properties of the interface providing an empirical relation.