On a holistic variational formulation for material modeling including dissipative evolution

Based on Hamilton’s principle of stationary action, we present a holistic variational formulation for material modeling including dissipative evolution. To this end, we recall the definition of the action as path integral of the momentum vector. Reformulation of the action and inserting the 1$\mathrm{st}$ and 2$\mathrm{nd}$ Law of Thermodynamics yield an extended Hamilton functional. We show that the stationarity conditions yield well-known expressions as well as new conditions in an extended nested time domain. Introducing an asymptotic two-scale approach transforms the expressions in the nested time domain back to the physical time. Hereby, we receive usual differential equations, e.g., heat conductivity equation, diffusion equation, and Biot equation, and the constitutive laws for, e.g., temperature, entropy, and chemical potential, all from one holistic stationarity principle. Moreover, the formulation in the nested time domain produces additional, virtual conditions that naturally lead to the concept of dissipation distances. Due to its variational origin, our approach yields in a consistent manner a coupled space–time formulation.