Theoretical and Numerical Modeling of Nonlinear Electromechanics with applications to Biological Active Media

Abstract

We present a general theoretical framework for the formulation of the nonlinear electromechanics of polymeric and biological active media. The approach developed here is based on the additive decomposition of the Helmholtz free energy in elastic and inelastic parts and on the multiplicative decomposition of the deformation gradient in passive and active parts. We describe a thermodynamically sound scenario that accounts for geometric and material nonlinearities. In view of numerical applications, we specialize the general approach to a particular material model accounting for the behavior of fiber reinforced tissues. Specifically, we use the model to solve via finite elements a uniaxial electromechanical problem dynamically activated by an electrophysiological stimulus. Implications for nonlinear solid mechanics and computational electrophysiology are finally discussed. A. Gizzi et al. / Commun. Comput. Phys., 17 (2015), pp. 93-126 configuration of electrostatic or electrodynamic fields. The variation of the assigned configuration of the electric field lines triggered by the electromechanical coupling is called mechanic-electric feedback (MEF). Typically, in EA systems deformations may induce a change of the eventual initial isotropy of a body.Historically, the most well known example of electro-elastic systems has been the piezoelectric crystal. In linearized kinematics, it can be proved that an isotropic dielectric immersed in an electric field develops polarization charges, inducing internal stresses proportional to the square of the electric field [37]. A similar dynamics characterizes piezoelectrics. The origin of the electromechanical coupling in piezoelectric materials stems from a phase transition that breaks the symmetry, and that leads also to a spontaneous polarization. By this spontaneous polarization there is a linear coupling between deformation and electric field [36,44], so that MEF effects are enhanced. Piezoelectrics manifest also the reverse feedback: imposed deformations induce an internal electric field proportional to the magnitude of the deformations. A second important class of materials where MEF is of relevance are electro-active polymers (EAP), that typically exhibit changes in size or in shape when stimulated by an electric field [53]. Among the recent literature addressing this class of materials, it is worth to mention contributions concerning electro-visco elastic polymers [4,5,74], and proposing thermodynamic formulations for electro-active synthetic materials [46].The MEF effect is observed also in materials that are in focus in the present study, i.e., biological media with contractile properties, such as the heart, the intestines, and several types of muscles. It is evident that biological systems undergo rather large deformations, therefore the underlying biophysical dynamics cannot be described accurately by means of the infinitesimal theory of elasticity. In particular, according to single cell and tissue specimen measurements, during a normal heart beat myocytes change their...

Temperature, geometry, and bifurcations in the numerical modeling of the cardiac mechano-electric feedback

“…Interesting, but rather complex descriptions based on active strains and thermodynamics have been proposed for instance in [26][27][28][29]. Here we will instead consider the active stress approach, which amounts to simply assume that the stress tensor can be expressed as a linear superposition of an elastic contribution S M N elastic , describing the passive properties of cardiac tissue, and an active part S M N active , resulting from the contractile behavior of the muscle:…”

A 3d Electromechanical Fem-Based Model for Cardiac Tissue

“…Both approaches are considered to lead to comparable solutions [29]. Since the experimental parametrization was based on the level of stresses, the latter approach was used.…”

Integrated Heart—Coupling multiscale and multiphysics models for the simulation of the cardiac function

“…3.2 for additional details). Although other choices for the definition of W P and W A are possible, see for example [99,103], we prefer the simplest case W P = 0. In fact W P represents the fraction of the tissue that cannot be mechanically activated.…”

“…The coupled multiphysics system (5.17)-(5.23) consists of the two-equation electrophysiology model of AlievPanfilov [4] equipped with the mechano-electric coupling terms from [182], the crossbridge force generation mechanism based on the distribution moment model from [27], an active stress model similar to those that were derived on thermodynamical principles in [215,99], and an isotropic Neo-Hookean passive material simplified to one-dimensional strains and stresses. For details and derivation of the sub-systems we refer to those works.…”