Adaptive Quasi-Linear Viscoelastic Modeling - Computational Modeling in Tissue Engineering - page 46

between force and displacement or between stress and strain, are nonlinear
viscoelastic. The focus of this chapter is phenomenological constitutive models
that can be used for computational prediction of such nonlinear viscoelastic
behavior.
Before delving into these, a word is needed about an alternative approach that is
not the focus of this chapter: the micromechanical approach, in which continuum
constitutive laws are constructed from knowledge of the microstructures and
deformation mechanisms of a tissue. For a number of biological tissues, some
aspects of nonlinearity are understood in terms of the underlying deformation
mechanisms of the protein structure. An example is the mechanics of tendons and
ligaments, in which the uncrimping of type I collagen likely underlies non-linear
stiffening of the tissue at low strains [ 28 - 30 ]. Lanir developed the first approach to
developing tissue-level constitutive relations for such tissues by averaging fiber-
level models over an orientation distribution of fibers [ 31 - 35 ], and important
extensions to this approach have been made through studies on planar collagenous
tissues [ 36 ], and through studies of related systems such as the anterior cruciate
ligament [ 37 ], the tendon enthesis [ 38 , 39 ], and reconstituted collagen matrices
[ 22 - 24 , 40 - 43 ]. Such models offer clear advantages over phenomenological
models in a great number of situations, but require detailed advance knowledge of
tissue microstructure and deformation mechanisms [ 44 ].
1.1 Overview of Phenomenological Nonlinear Viscoelastic
Frameworks and Their Limitations
Phenomenological models involve frameworks that can be fit to the response of a
material with limited information about the material's make-up. The simplest class
of phenomenological models for viscoelastic behavior is linear viscoelasticity.
Linear viscoelasticity can be described completely in terms of a relaxation func-
tion and a constant modulus of elasticity.
The problem is more complicated for nonlinear viscoelastic behavior, with
numerous approaches available. Any nonlinear viscoelastic model can be con-
sidered a subset of the most general nonlinear viscoelastic form derived by
Coleman and Noll [ 45 , 46 ]. This formulation involves a summation of three dif-
ferent terms: elastic stress, linear pure viscoelastic stress, and nonlinear pure
viscoelastic stress. The constitutive law of a nonlinear material can also be
modeled mathematically using nonequilibrium thermodynamics through two
approaches: functional thermodynamics and state-variable thermodynamics [ 47 ].
In these approaches, scalar response quantities like free energy are modeled as
functionals of the strain (or stress) tensor, and stress (or strain) is calculated using
functional differentiation of those scalar quantities.
The most commonly used phenomenological model to describe the nonlinear
viscoelastic behavior of biological tissues is a quasi-linear viscoelastic (QLV)