When interfaces between materials are numerically spread over a few mixed cells (a strategy often designated as "diffuse interfaces") the thermodynamic states and dissipation processes must be fully specified. Beyond the usual conservation laws which ensure accurate partial jump conditions at shocks, numerical endeavors should further include thermodynamic conditions in the mixture. In particular: - For applications with fast pressure relaxation processes, material pressures may be considered equal and partial volumes should evolve so as to ensure this condition. This assumption is expected to provide physically sounder results compared to the commonly used equal strain assumption although it requires specific numerical techniques. - Entropy in all material must increase over time in order to provide stability to the scheme and ensure that shocks arise from vanishing regularizations. Besides, relative entropy production in materials are a priori not constrained and different values may produce different solutions inside shocks. This characteristic feature of multimaterial flows is closely related to the presence of non-conservative products inside the equations which prevents uniqueness of jump conditions inside shocks. Benefiting from these considerations, a cell-centered Lagrangian scheme is here presented for a multi-material hydrodynamics model (with equal pressure assumption) which extends the single material EUCCLHYD scheme. It is conservative in mass, total momentum and total energy. Semi-discrete entropy production is positive for each material and is given as an arbitrary proportion of the global entropy production, hence mimicking different viscosity operators. The scheme is assessed on various 1 and 2-dimensional test cases where materials have highly contrasted equations of state. These test cases confirm the robustness of the scheme, emphasize the crucial role of entropy production terms, and show that pressures are kept equal up to the scheme order, or even strictly if an additional equilibration procedure is added.