A finite element-based approach is proposed to compute the time response of infinite periodic structures with local nonlinearities. Under the assumption that the excitation sources and the nonlinear effects are localized, an infinite periodic structure can be modeled as a finite one with two left and right semi-infinite linear parts which are described via appropriate absorbing boundary conditions (BCs) formulated in the time domain using the wave finite element (WFE) method as recently proposed by the authors in [1]. The formulation of the absorbing BCs involves the usual vectors of displacements, velocities and accelerations, as well as vectors of supplementary variables. In this way, a classical second-order time differential equation for a nonlinear finite periodic structure with absorbing BCs can be formulated and integrated into a Newmark's algorithm for the temporal variable and a Newton Raphson's algorithm for the nonlinear equations. Numerical experiments are proposed which highlight the relevance of the approach for several types and levels of nonlinearities.