Thermodynamics of plastic hinges with damage

Elastic structural members are considered with plastic hinges at which damage may occur. Equations of balance of linear momentum, angular momentum and energy are obtained expressing the fact that energy may be absorbed by damage at a plastic hinge. It is shown that all the work done by the moments in bending of a plastic hinge that does not appear as heat must be absorbed by damage. The fraction of the bending work which is transformed into damage is thecoefficient of damage. In general, this coefficient is smaller than unity, which implies that the plastic hinge is a heat source. The heat source manifests itself by producing a jump discontinuity in axial heat flux and thus, in virtue of Fourier's law, a jump discontinuity in temperature gradient ensues. The general thermodynamics of the system is formulated: It is shown that agreement with the second law of thermodynamics prevails if the coefficient of damage does not exceed unity. Using the condition that a plastic hinge must break after absorbing a critical amount of damage energy, we show that a structure composed of such structural members with a finite number of plastic hinges must fail before it absorbs an infinite amount of energy through a cyclic loading and unloading program. Although the failure criterion for a single plastic hinge may be simple, the complex distribution of damage among the plastic hinges of an entire structure can make the failure criteria for a structure defy simple rules. It is also argued that the introduction of a distributed damage energy rate does not seem feasible in view of the second law of thermodynamics. This suggests that damage in a microstructure is localized rather than uniformly distributed.