Adaptive Quasi-Linear Viscoelastic Modeling - Computational Modeling in Tissue Engineering - page 48

2 Nonlinear and Quasi-Linear Viscoelasticity

The most general framework for nonlinear viscoelasticity considers the stress (or
force) as a function of the entire history of the straining (or elongation) of a
material [ 57 ]. Such a model is too general for any practical application. Imposing
some symmetry constraints and the principle of fading memory, Coleman and Noll
derived a general nonlinear viscoelastic model that can be approximated by a
Taylor-like series about a zero strain history [ 45 , 46 ]. In this approximation, stress
is calculated in terms of first and higher order integrals of the strain history. Up to
the first order integral, the constitutive law is linear; it becomes nonlinear when
second and higher order integrals are included in the approximation. Alternative
integral series were derived by Rivlin and Spencer [ 58 , 59 ] and by Pipkin and
Rogers [ 60 , 61 ]. In the models of Pipkin, the stress is calculated in terms of the
time derivative of the strain history rather than the strain history itself.
The single-integral approximation of the Pipkin-Rogers model can be rewritten
as the Fung QLV for one-dimensional viscoelastic modeling of biological tissues.
The central feature of the Fung model is that the dependence of the force response
on the stretch history can be obtained from a linear convolution integral, which
allows the nonlinear model to retain all of the benefits of linearity. Fung incor-
porated nonlinearity into a linear viscoelastic constitutive law by replacing strain
with a nonlinear function of strain. The resultant viscoelastic model is linear with
respect to a pure function of strain instead of strain itself, and hence ''quasi-
linear.''
The term ''quasi-linear viscoelastic'' has since become more general, and
encompasses any nonlinear viscoelastic constitutive law in which loading history
dependence can be modeled by a linear convolution integral or a summation of
linear convolution integrals [ 55 ]. A feature of such QLV models is that the stress
and strain (or force and displacement) are related by an intermediate variable that
separates the nonlinearity from the viscoelasticity. An example of this is the
''elastic stress'' in the Fung QLV model, which serves as an intermediate variable
that allows use of a linear convolution integral to derive the stress. This is
described in detail in the next section.

2.1 The Fung QLV Model

In a linear viscoelastic model, stress is calculated in terms of strain history using a
linear convolution integral as:

r ðÞ¼ Z

t

de ðÞ
ds

Gt s

ð Þ ds ð 1 Þ

1