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Subsubsection 4.5.1.7: Power Distortional Up Chapter 4: Materials Subsection 4.6.1: General Specification of Damage Materials

## 4.6 Reactive Damage Mechanics

A material may accumulate damage over a single cycle or multiple cycles of loading which alters its properties. In the classical framework of damage mechanics this attenuation of the material properties is described by a single scalar damage variable when the material is isotropic (. For anisotropic materials however, classical frameworks require that we introduce a function of the fourth-order damage tensor to account for anisotropic damage. In FEBio, we use a reactive damage mechanics framework where the elastic response is proportional to the total number of intact bonds in the material and where, at any given time in the loading history, represents the mass fraction of bonds that have broken. In this reactive framework, it is possible to also model damage in anisotropic materials by assuming that multiple bond types exist in the material, each of which may get damaged under different circumstances. Each bond type may be described by a distinct solid constituent within a solid mixture (see Sections 4.1.2.14↑ and 4.1.3.22↑), each having its own scalar damage variable .
For a given bond type, the strain energy density of a damaged material is given by where is the strain energy density when all bonds of that type are intact. Here, represents the mass fraction of bonds that remains intact. Similarly, the Cauchy stress of the damaged material is given by where is the stress in the intact material, at a given strain, as derived from . The intact material may be based on any of the elastic materials described in Sections 4.1.2↑ and 4.1.3↑.
The evolution of the damage variable is determined by a user-selected scalar damage criterion measure ( is the capital form of . For example, may represent the strain energy density, or von Mises stress, or maximum principal normal strain, etc. If exceeds a given threshold at some state of deformation , then damage may initiate or progress further. If all bonds fail at a single threshold value , the material undergoes fracture. More commonly, bonds may fail with increasing probability as increases over a given range. Consequently, the evolution of damage may be based on a user-selected cumulative distribution function (c.d.f.) , such that where is the maximum value of achieved over the loading history up until the current time .

Subsubsection 4.5.1.7: Power Distortional Up Chapter 4: Materials Subsection 4.6.1: General Specification of Damage Materials