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Subsection 4.4.2: Unconstrained Viscoelastic Materials Up Chapter 4: Materials Subsection 4.5.1: Relaxation Functions

## 4.5 Reactive Viscoelastic Solid

Reactive viscoelasticity models a material as a mixture of strong bonds, which are permanent, and weak bonds, which break and reform in response to loading [13]. Strong bonds produce the equilibrium elastic response, whereas weak bonds produce the transient viscous response. For a compressive reactive viscoelastic solid, the strain energy density is given by where is the strain energy density of strong bonds and is the strain energy density of weak bonds, when they all belong to the same generation. is the deformation gradient of the strong bonds and the initial weak bond generation, wherease is the relative deformation gradient for the generation weak bonds, such that at time . In this expression, is the mass fraction of generation weak bonds, which evolves over time as described next. The summation is taken over all generations that were created prior to the current time .
Any number of valid solutions may exist for , based on constitutive assumptions for the weak bond mass fraction supply . In particular, for generation bonds reforming in an unloaded state during the time interval , and subsequently breaking in response to loading at , Type I bond kinetics provides a solution of the form where and is a reduced relaxation function which may assume any number of valid forms. (A reduced relaxation function satisfies and , and decreases monotonically with .) In particular, may depend on the strain at time relative to the reference configuration of the generation. In the recursive expression above, the earliest generation , which is initially at rest, produces for and for ; this latter expression seeds the recursion for subsequent generations. Therefore, providing a functional form for suffices to produce the solution for all bond generations .
For Type II bond kinetics, the solution for the mass fractions is given by For this type of bond kinetics, the reduced relaxation function cannot depend on the magnitude of the strain, because strain-dependence might violate the constraint .
For all bond kinetics, it is also possible to constrain the occurrence of the breaking-and-reforming reaction to specific forms of the strain. For example, the reaction may be allowed to proceed only in the case of dilatational strain, or only in the case of distortional strain.
For a material with an uncoupled formulation, the strain energy density has the form where .
The material type for a compressive reactive viscoelastic solid is “reactive viscoelastic”. For the uncoupled formulation the material type is “uncoupled reactive viscoelastic”. The following parameters need to be defined:
 Bond kinetics type [ ] Strain invariants that trigger weak bond breakage and reformation [ ] Elastic (strong bond) material Weak bond material Reduced relaxation function
The <kinetics> parameter should be set to 1 for Type I bond kinetics or 2 for Type II bond kinetics. The <trigger> parameter should be set 0 when weak bonds break and reform in response to any form of the strain; it should be set to 1 when the trigger is distortional strain; and it should be set to 2 when the trigger is dilatational strain. The <elastic> and <bond> materials may be selected from the list of unconstrained elastic materials given in Section 4.1.3↑ (for “reactive viscoelastic”) or from the list of uncoupled elastic materials in Section 4.1.2↑ (for “uncoupled reactive viscoelastic”). The <relaxation> material may be selected from the list provided in Section 4.5.1↓.
Example:
<material id="1" name="RV solid" type="reactive viscoelastic">
<kinetics>1</kinetics>
<trigger>0</trigger>
<elastic type="Holmes-Mow">
<density>1</density>
<E>1</E>
<v>0.3</v>
<beta>0.5</beta>
</elastic>
<bond type="Holmes-Mow">
<density>1</density>
<E>1</E>
<v>0.3</v>
<beta>0.5</beta>
</bond>
<relaxation type="relaxation-exponential">
<tau>4</tau>
</relaxation>
</material>

Subsection 4.4.2: Unconstrained Viscoelastic Materials Up Chapter 4: Materials Subsection 4.5.1: Relaxation Functions