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5.6.2 Free 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 [43, 64, 26, 51, 50, 71], 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 .