The mechanical behavior of granular systems can be investigated using the discrete element method (DEM), in which the motion of each particle is governed by the Newton–Euler equations, and interactions between particles arise mainly from contact. The contact problem consists in determining the material points on the particle surfaces that interact with each other besides computing contact forces. Polyhedra [1] and non-uniform rational B-splines (NURBS) [2] are possibilities for modeling the particle geometry. To handle contact between such geometries, it is possible to use a master-to-master contact approach in which the contact interaction is considered pointwise [3]. To the authors’ knowledge, only implicit methods and preliminary ideas on explicit methods of integration of the equations of motion have been used in conjunction with the master-to-master technique. Although commonly employed in conjunction with finite element methods, implicit methods are not usually employed in DEM simulations due to their relatively high computational cost. The objective of the work is to handle explicit methods to integrate the equations of motion of particles in the context of a master-to-master contact approach. The focus is on the development of the normal and tangential contact contributions. Such methods also require a review of classical and fundamental concepts of finite rotations and incremental schemes, which are presented and discussed. Numerical applications using both implicit and explicit methods are presented to demonstrate their robustness and compatibility.