The numerical implementation of invariant-based viscoelastic formulations at finite strains. An anisotropic model for the passive myocardium

The present study developed a conceptual framework for finite strain viscoelasticity thought to be suitable to capture the salient features of a class of passive soft biological tissues like the myocardium. A superposition of a Maxwell Body and an Elastic Body defines the viscoelastic continuum, and its deformation is related to two independent reference configurations. The reference configuration of the Maxwell Body moves in space as it is described (apart from rigid body rotation) by a rate equation in strain space, and stores the history of the deformation. At thermodynamic equilibrium the reference configuration of the Maxwell Body coincides with the current configuration of the continuum. The Helmholtz free energy is expressed as a function of two independent strain variables and entirely renders the constitution of the viscoelastic body. Although this view is to some extent different from reported viscoelastic concepts for finite strains, its linearization around the thermodynamic equilibrium coincides with earlier suggested viscoelastic models. The linearized viscoelastic model has been implemented for a particular anisotropic constitutive model for the passive myocardium. Non-negative dissipation of the model is guaranteed. Material parameters were estimated from in vitro testing of porcine myocardium and the response due to pushing a rigid punch into the myocardium was studied. Results between anisotropic and isotropic descriptions of the myocardium differed significantly, which justified the implementation of an anisotropic model for the myocardium.