Hamilton’s principle as inequality for inelastic bodies

This paper is concerned with Hamilton’s principle for inelastic bodies with conservative external forces. Inelasticity is described by internal variable theory by Rice (J Mech Phys Solids 19:433–455, 1971), and the influence of strain change on the temperature field is ignored. Unlike Hamilton’s principle for elastic bodies which has an explicit Lagrangian, Hamilton’s principle for inelastic bodies generally has no an explicit Lagrangian. Based on the entropy inequality, a quasi Hamilton’s principle for inelastic bodies is established in the form of inequality and with an explicit Lagrangian, which is just the Lagrangian for elastic bodies by replacing the strain energy with free energy. The quasi Hamilton’s principle for inelastic bodies states that the actual motion is distinguished by making the action an maximum. The evolution equations of internal variables can not be recovered from the quasi Hamilton’s principle.