Viscoelasticity in polymers: Phenomenological to molecular mathematical modeling

Abstract

We report on two recent advances in the modeling of viscoelastic polymers: (i) a new constitutive model that combines the virtual stick-slip continuum "molecular-based" ideas of Johnson and Stacer with the Rouse bead chain ideas; (ii) a two-dimensional version of a model that accounts for stenosis-driven shear wave propagation in biotissue.

A brief review of some approaches to hysteresis in viscoelastic polymers

“…We mention briefly two recent advances: (i) a new constitutive model [10] that has been developed which combines the virtual stick-slip continuum "molecular-based" ideas of Johnson and Stacer [43] with the Rouse bead chain (see Figure 2 below) ideas as described in Doi and Edwards [34]; (ii) a two dimensional version [14,15,50] of a model that accounts for stenosis driven shear wave propagation in biotissue.…”

“…In the biotissue efforts [14,15,50], we employ an internal variable formulation of Boltzmann type hysteresis laws to investigate the propagation of stenosis generated waves in biotissue where it has been demonstrated that a viscoelastic (as opposed to an elastic) formulation is important and that waves generated in a twodimensional cylindrical geometry with inner radius partial occlusions can be readily modelled and simulated. Specifically we have carried out efforts on two and three dimensional models that employ an internal variable approach to model wave propagation.…”

“…The goal is to model the propagation of the wave generated from the stenosis to the chest wall, and ultimately, to create an inverse problem methodology which can be utilized to determine the location of an arterial stenosis. In [14,15,50] we also discuss comparison of the viscoelastic model to an elastic one as well as present typical simulations for a biologically motivated example.…”

Inverse Problems, Control and Modeling in the Presence of Uncertainty

“…In [25,26] it is shown that similar models (with different nonlinearities in (4.7), (4.8)) are important if one replaces Fung kernels [43] with equations for distributed molecular mechanisms in describing shear response in biotissue. In another important application [15,16], the systems are the usual Maxwell's equations for the electromagnetic fields E and H in a heterogeneous dielectric and are given by…”

“…Let Ri(t) i = 1, ... , 4, denote the number of times that the ith transition occurs by time t. Then R, is a counting process with intensity Aj(X(t)), and the corresponding stochastic process can be defined by 26) where the Y are independent unit Poisson processes. That is, sample paths ri(t) of R(t) are given in terms of sample paths x(t) of X(t) by…”