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5.5 Reactive Viscoelasticity
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 . Strong bonds produce the equilibrium elastic response, whereas weak bonds produce the transient viscous response. Strong bonds are in a stress-free state when in their reference configuration . Their deformation gradient is defined as usual, . When weak bonds break in response to loading at some time , they reform instantaneously in a stress-free configuration that coincides with the current configuration at time , thus, . Therefore, a reaction transforms intact loaded bonds into reformed unloaded bonds. Weak bonds that reform at time may be called generation bonds. The deformation gradient of generation weak bonds relative to their reference configuration is denoted by , which may be evaluated from the chain rule, The strain energy density in a reactive viscoelastic material 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. In this expression, is the mass fraction of generation weak bonds, which evolves over time as described below. The summation is taken over all generations that were created prior to the current time . The Cauchy stress in a reactive viscoelastic material is similarly given by where is the stress in the strong bonds and is the stress in the weak bonds. These stresses are related to the respective strain energy densities of strong and weak bonds according to The mass fractions are obtained by solving the equation of mass balance for reactive constrained mixtures, where the mass fraction supply must be specified as a constitutive function of the deformation gradient and the mass fractions from all generations. Since mass must be conserved over all generations, it follows that Any number of valid solutions may exist for , based on constitutive assumptions for . For example, 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 of (5.5-7), 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.
The finite element implementation of reactive viscoelasticity stores the value of every time that a bond-breaking deformation is detected; this value can be used to evaluate using (5.5-1). It also stores , where is the time step immediately following , which is then used for evaluating for subsequent time steps when using Type I bond kinetics, using the expressions of (5.5-7) and (5.5-8). These measures are then used to evaluate the stress from (5.5-3), with the summation taken over the time steps that correspond to bond-reforming generations.