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Theory Manual Version 3.6
 Subsection 5.7.3: Constitutive Modeling of Yield Response Up Chapter 5: Constitutive Models Subsection 5.8.1: Theoretical Formulation 

5.8 Reactive Elastoplastic Damage Mechanics

Plastic deformation (Section 5.7↑) is often coupled with damage, as the finite deformation and plastic flow of a loaded material typically induces some amount of failure. Within the constrained reactive mixture framework adopted in FEBio (Section 2.8↑), damage is produced by bonds breaking permanently (Section 5.6↑), which reduces the generation mass fractions [32]. In our treatment of elastoplastic damage we assume that both intact and yielded bonds may become damaged. Damage to intact bonds may represent some initial damage value for a material with defects, or damage due to intermolecular failure of bonds that never yielded; we refer to this as elastic damage. Damage to yielded bonds represents plastic damage. The mechanism of damage and the failure measure may be different for these two types of bonds, particularly since a stress- or energy-based failure measure may not be appropriate for plastic damage. Intact bonds belong to the generation which is present at . Once yielding occurs, all successive generations of that family are labeled as yielded bonds . This distinction is necessary so we can then distinguish between damage to intact bonds (elastic damage) and damage to yielded bonds (plastic damage), since intact bonds which get damaged never have the ability to yield. It is important to note that the nature of the plastic deformation described in Section 5.7↑ remains unchanged. Damage modifies the material behavior by reducing the fraction of bonds in various generations, which scales the response accordingly.
In a reactive constrained mixture framework the insertion of damage into the reactive plasticity formulation is straightforward. Since bonds break permanently in a damage reaction, there is no need to define a function of state to describe a (non-existing) reformed configuration. Furthermore, the specific free energy of broken bonds is zero. The scalar elastic damage criterion , which is taken to have the same functional form for all bond families , is the analog to the yield criterion for plasticity. As shown in Section 5.6↑, the main contrast with reactive plasticity is that not all bonds in the family break simultaneously at a single elastic damage threshold . Instead, the fraction of broken bonds varies as a function of , denoted by , such that . Here, is a function of state; it must be a monotonically increasing function of its argument to satisfy the Clausius-Duhem inequality [78]. We may view as a cumulative distribution function (CDF), whose corresponding probability distribution function (PDF) represents the probability of damage at a particular value of .
 Subsection 5.7.3: Constitutive Modeling of Yield Response Up Chapter 5: Constitutive Models Subsection 5.8.1: Theoretical Formulation 

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