Thermodynamics of plastic hinges with damage (Q1204043)

This nice study treats the embedding of the planar bending problem of an elastic bar with plastic hinges in equilibrium thermodynamics. At first, the balances of momentum, angular momentum and energy are stated, then the entropy balance is formulated and discussed after introducing a damage hypothesis. At regular points the structure behaves ideal elastic, that means only reversible processes are occurring. Dissipation is concentrated at the singular plastic hinges. Correspondingly the entropy is introduced as equilibrium quantity in the elastic region by means of Gibbs' equation. The internal energy of the system consists of the equilibrium portion assigned to the Gibbs' function by Legendre transformation and of the damage energy realized in the hinges. The only working variables are curvature or the bending angle difference, respectively. At regular points the usual linear elastic relation between curvature and bending moment is applied, at the hinges a special condition for the rate of damage is defined. On this base it is possible to calculate and to discuss the damage energy produced in any cyclic loading process. Finally, the (im)possibility of defining a spacially distributed damage rate is examined.