This study presents a computational framework for modeling and simulating plasticity within hyperelastic materials. It relies on a rigorous thermodynamical framework that incorporates both fluid and elastic behaviors. Plasticity is introduced into this framework through a multiplicative decomposition of the deformation gradient to represent plastic deformation. The internal energy is treated as a potential depending on the density, entropy, and elastic deformation, providing conditions to ensure entropy growth during the plastic process. The numerical method employs finite volume schemes optimized for updated-Lagrangian frameworks, provably preserving entropy growth at the semi-discrete level. The conservation of mass, momentum, and total energy, as well as the entropic behavior of the plastic process, are maintained at the fully discrete level. The method is extended to second-order accuracy in both space and time. The study discusses the choice of the plastic tensor and the temporal integration of plastic relaxation in the case of a perfectly plastic material governed by a von Mises constraint. Future work aims to refine the method through local time-stepping, reducing computational costs while preserving the scheme’s inherent properties.