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Theory Manual Version 3.6
 Subsection 5.6.3: Damage Criterion Up Section 5.6: Reactive Damage Mechanics Subsection 5.6.5: Constitutive Models for Damage and Yield Criteria 

5.6.4 Reaction Kinetics and Thermodynamics

The axiom of mass balance in a reactive constrained mixture reduces to where is the material time derivative of and is a function of state representing the referential mass supply density to constituent due to reactions with all other constituents. In the damage framework, the above relations show that Substituting these expressions into eq.(5.6.4-1) shows that the referential mass density supplies are given by where In these expressions, the damage is advancing when increases over two consecutive time points. In this expression for we need to evaluate where we defined to represent the tensorial normal to the damage hypersurface, which needs to be evaluated at .
In this isothermal damage framework it can be shown from the energy balance that a heat supply density must radiate the bond-breaking energy out of the continuum to maintain isothermal conditions, where Since is a monotonically increasing function of , its material time derivative is always positive when the damage is increasing, and zero otherwise as per eq.(5.6.4-5). Since the specific strain energy is always positive, it follows that the specific heat supply in eq.(5.6.4-7) is negative or zero, consistent with the expectation that heat needs to leave the continuum to maintain isothermal conditions.
 Subsection 5.6.3: Damage Criterion Up Section 5.6: Reactive Damage Mechanics Subsection 5.6.5: Constitutive Models for Damage and Yield Criteria