Hamilton’s principle for Green-inelastic bodies

In this paper, Hamilton’s principle for Green-elastic bodies with conservative external forces (see Fung and Tong, 2001. Classical and Computational Solid Mechanics. World Scientific, Singapore, Chapter 11.1) is extended to Green-inelastic bodies. Inelastic constitutive laws are addressed within the thermodynamic framework with internal variables by Rice (1971. Inelastic constitutive relations for solids: an internal variable theory and its application to metal plasticity. J. Mech. Phys. Solids 19, 433–455). Generalized thermodynamic forces conjugate to internal variables are termed internal forces in this paper. The materials whose free energy functions are point functions of internal variables, are termed here Green-inelastic materials which embody Green-elastic materials as special cases and ensures that the internal forces are potential forces. It is shown that Hamilton’s principle for Green-elastic bodies can be extended to Green-inelastic bodies just by replacing the strain energy of a moving body with the corresponding free energy plus the potential energy of the internal forces.