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Subsubsection 4.15.2.2: in-situ stretch Up Chapter 4: Materials Subsection 4.16.1: Damage Fiber Power

## 4.16 Continuous Damage

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's Continuous Damage model, the elastic response of the fibers chosen for the material is mathematically related (which is material dependent) to the total number of intact bonds in the material. At any given time in the loading history, represents some mass fraction of bonds that have broken in the fibers. In this 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 (see Section 4.1.2.14↑ , e.g. to describe the matrix) in conjunction with a Continuous Damage solid constituent (See sub-sections , e.g. to describe the fibers) within a solid mixture (see Section 4.1.3.22↑) of a material, each having its own scalar damage variable .
The scalar damage variable is embedded in the strain energy function of the fibers, its relationship is dependent on the Continuous Damage solid constituent chosen. The scalar damage variable has continuous and discontinuous damage components defined by damage internal damage variables and .
The following material damage parameters must be defined:
 Parameter ( [M] Parameter ( [ ] Parameter ( [M] Parameter ( [t]
Note the parameter is the time in the current loading situation where the damage model is initiated.
The scalar damage variable is a function of , the continuous damage variable, and is given by
where is defined by
with ,
and is the value of the internal material variable that has reached a certain fraction () of the maximal damage value for a fixed maximum load level. At , the overall damage . For example, in Figure 1, and , meaning the overall damage in this cycle has reached 50% of the maximal damage value . Overall Damage variable as a function of Continuous Damage variable
To make sure damage evolution begins to accumulate only when damage is accrued, is the internal variable at an initial damage state and is the undamaged effective strain energy density of the fibrils.
is a function increasing the maximally reachable damage value for increased maximum load levels. It is assumed that the maximum damage is itself a function of the internal variable , describing the discontinuous damage variable and is given by
where is defined by
and is the value of the internal material variable that has reached a certain fraction of the maximal damage value for a fixed maximum load level. At , the overall damage (the same relationship can be adapted from Figure 1 with ). denotes the strain energy density at an initial damage state obtained at the limit of the physiological domain (e.g. in relation to arterial expansion). It is the initial damage strain energy threshold value for the fibers. The time associated with the loading history is denoted by and the time associated with the current loading situation is represented by .
Subsubsection 4.15.2.2: in-situ stretch Up Chapter 4: Materials Subsection 4.16.1: Damage Fiber Power