Any number of valid solutions may exist for $w^{u}$ , based on constitutive assumptions for the weak bond mass fraction supply $\hat{w}^{u}$ . In particular, for $u-$ generation bonds reforming in an unloaded state during the time interval $u\leqslant t<v$ , and subsequently breaking in response to loading at $t=v$ , Type I bond kinetics provides a solution of the form $$w^{u}\left(\mathbf{X},t\right)=\begin{cases} 0 & t<u\\ f^{u}\left(\mathbf{X},t\right) & u\leqslant t<v\\ f^{u}\left(\mathbf{X},v\right)g\left(\mathbf{F}^{u}\left(v\right);\mathbf{X},t-v\right) & v\leqslant t \end{cases}$$ where $$f^{u}\left(\mathbf{X},t\right)=1-\sum\limits _{\gamma<u}w^{\gamma}\left(\mathbf{X},t\right)$$ and $g\left(\mathbf{F}^{u}\left(v\right);\mathbf{X},t-v\right)$ is a reduced relaxation function which may assume any number of valid forms. (A reduced relaxation function $g\left(t\right)$ satisfies $g\left(0\right)=1$ and $g\left(t\to\infty\right)=0$ , and decreases monotonically with $t$ .) In particular, $g$ may depend on the strain at time $v$ relative to the reference configuration of the $u-$ generation. In the recursive expression above, the earliest generation $u=-\infty$ , which is initially at rest, produces $w^{u}\left(t\right)=1$ for $t<v$ and $w^{u}\left(t\right)=g\left(\mathbf{F}^{u}\left(v\right);\mathbf{X},t-v\right)$ for $t\geqslant v$ ; this latter expression seeds the recursion for subsequent generations. Therefore, providing a functional form for $g$ suffices to produce the solution for all bond generations $u$ .

For Type II bond kinetics, the solution for the mass fractions is given by $$w^{u}\left(t\right)=\begin{cases} 0 & t<u\\ 1-g\left(t-u\right) & u\leqslant t<v\\ g\left(t-v\right)-g\left(t-u\right) & v\leqslant t \end{cases}\,.$$ For this type of bond kinetics, the reduced relaxation function $g$ cannot depend on the magnitude of the strain, because strain-dependence might violate the constraint $0\leqslant w^{u}\leqslant1$ .

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 $$\Psi_{r}\left(\mathbf{F}\right)=U\left(J\right)+\tilde{\Psi}_{r}^{e}\left(\mathbf{\tilde{F}}\right)+\sum\limits _{u}w^{u}\tilde{\Psi}_{0}^{b}\left(\mathbf{\tilde{F}}^{u}\right)$$ where $\mathbf{\tilde{F}}=J^{-1/3}\mathbf{F}$ .

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:

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>