On the Use of Reactive Solids to Model Finite Deformation Elastoplasticity and Elastoplastic Damage Mechanics: A Constrained Reactive Mixture Theory and Finite Element Formulation

The familiar hyperelastic framework is generally unable to account for inelastic responses without modification involving internal state variables [1]. The purpose of this study is to model elastoplasticity and elastoplastic damage by treating materials as reactive solids whose composition may evolve in response to applied loading. This approach does not appeal to hidden variables and produces a purely hyperelastic model of plasticity and damage, where energy dissipation is due to changes in material composition.  
An elastic solid may be defined as a material with bonds that store energy without dissipating it. In a reactive framework, any dissipation of stored energy occurs as a result of bonds breaking and reforming, producing inelastic behavior. The state at which bonds reform determines alternative frameworks of reactive mechanics. Based on our prior successes describing growth and remodeling [2], damage mechanics [3], and viscoelasticity [4], this study adopts the theory of reactive constrained solid mixtures [5, 6] to describe concurrent mechanisms by which bonds may alternatively plastically deform or sustain damage. The evolution equations controlling the material’s bond compositions are governed by the axiom of mass balance, with constitutively prescribed mass supply terms relating loading measures to internal reactions that modify the material’s composition.
Plastic deformation is accounted for by allowing loaded bonds (e.g. chemical bonds) within the material to break and reform in a stressed state with a new reference configuration, a concept is adopted from the literature on plasticity in biological tissues and fibrous networks [7]. Bonds which break and reform represent a new generation with a new reference configuration, thus introducing energy dissipation. Importantly, this new reference configuration is time-invariant and provided by constitutive assumption, a significant departure from the classical Kroner-Lee approach. There is great freedom in postulating the new reference configuration, provided the Clausius-Duhem inequality is not violated. Under the dual assumptions of normality and an associated flow rule, the present mixture formulation exactly reduces to classical Prandtl-Reuss plasticity in the limit of infinitesimal strains and rotations.
A coupled theory of elastoplastic damage is developed by synthesizing our previous work on reactive damage mechanics with reactive elastoplasticity. In reactive damage mechanics, damage emerges due to bonds which break permanently and do not reform, representing another source of energy dissipation. The addition of damage involves no further state variables beyond allowing for the mass content of bonds to decrease as damage occurs; further, the evolution equation for the damage variable is not postulated but rather emerges by satisfying the axiom of mass balance.
The theoretical models developed in this study were implemented into the finite element software FEBio [8]. Verification was completed against fundamental benchmarks with analytical and commercial finite element solutions. Validation was accomplished by comparing computational predictions to experimental results from a series of finite deformation forming experiments, demonstrating both fitting and predictive capabilities.
In our reactive approach, the material response depends only on observable state variables. By adding scalar composition measures to the list of state variables, all functions of state are uniquely determined by the loads applied to the material and its composition. The benefit of structuring a theoretical model in this manner is that all evolution equations are directly obtained from fundamental balance axioms, e.g. the axiom of mass balance, and the modeler need not provide any evolution equations for internal variables. A further benefit lies in the ability of the present model to recover elastoplastic and elastoplastic damage behaviors with only an additional set of scalar state variables, thus obviating the need for tensorial back stresses or plastic/damage potentials.
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