Theory Manual Version 3.6
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Subsection 5.6.1: Bond-Breaking Reaction Up Section 5.6: Reactive Damage Mechanics Subsection 5.6.3: Damage Criterion

### 5.6.2 Strain Energy Density and Stress

Let the specific free energy stored in intact bonds be represented by ; that of broken bonds is zero. Therefore, the free energy density of the mixture is We may define the mass fraction of bond species as Now, the mixture mass balance in eq.(5.6.1-2) may be rewritten as , or more specifically, We may also rewrite the mixture free energy density in eq.(5.6.2-1) as where we have made use of eq.(5.6.2-3). The corresponding Cauchy stress may be evaluated using the standard hyperelasticity formula, These relation show that the free energy density and stress of a damaged material are scaled by the mass fraction of remaining intact bonds. Comparing these formulas to those of classical damage mechanics [59, 85, 33, 69, 68, 98], it becomes immediately apparent that the classical damage variable appearing in those theories is equivalent to the mass fraction of broken bonds, To further clarify this equivalence, we may let represent the free energy density of an intact elastic solid, such that eq.(5.6.2-4) may be rewritten as . Similarly, eq.(5.6.2-5) may be rewritten as , where is the stress in the intact elastic solid, derived from the hyperelasticity relation .
For nearly-incompressible hyperelastic materials (Section 2.4.3↑), the strain energy density of the intact material has the form of eq.(2.4.3-2), thus . In this case, we assume that the damage only affects the distortional part of the strain energy density , consistent with the general framework advocated in [52]. Thus, for uncoupled damage, we assume that . The resulting damage stress similarly takes the form , consistent with eq.(2.4.3-10), where is evaluated from as given in eq.(2.4.3-11) and is evaluated from as given in eq.(2.4.3-4).
When investigating the damage mechanics of tension-bearing fibrous materials, described in Section 2.4.5↑, it is important to use the unconstrained version of the fiber and damage mechanics models, even when the fibers are embedded in a ground matrix with a nearly-incompressible response (uncoupled formulation). This is a necessary requirement since uncoupled fiber formulations are now understood to be non-physical. Nevertheless, for historical reasons, FEBio allows users to use uncoupled fiber formulations in an uncoupled damage material.
Subsection 5.6.1: Bond-Breaking Reaction Up Section 5.6: Reactive Damage Mechanics Subsection 5.6.3: Damage Criterion