Adaptive Quasi-Linear Viscoelastic Modeling - Computational Modeling in Tissue Engineering - page 50

Fig. 1 Four normalized
relaxation functions for a
reconstituted collagen tissue
specimen, at four different
initial strains. The relaxation
function of these tissues is
clearly a function of initial
strain, meaning that the Fung
QLV model is inadequate for
modeling this tissue. Each
curve represents a 3-term
exponential fit to
experimentally recorded data,
as reported in [ 55 ]

2.2 The Generalized Fung QLV Model

The Fung QLV model requires that isometric stress relaxation following an
instantaneous stretch be proportional to a single reduced relaxation function g(t),
regardless of the level to which a tissue has been strained prior to the application of
the instantaneous stretch, and regardless of the amplitude of the instantaneous
stretch. This behavior is inconsistent with that observed in many biological and
bio-artificial tissues. An example, shown in Fig. 1 , is of reconstituted collagen
stretched to four different (nearly) instantaneous levels of strain. The Fung QLV
model requires that g(t) be independent of strain level, and that the four curves
shown in Fig. 1 be a single curve; however, the four curves are significantly
different. This limits accuracy of the Fung QLV model in representing the
mechanical characteristics of these and other biological and bio-artificial tissues
[ 50 , 54 , 55 ].
To overcome this limitation the Fung QLV model has been generalized by
allowing the reduced relaxation function to vary with the tissue strain level:

r ðÞ¼ Z

t

de ðÞ
ds

g G ð t s; e Þ ds : ð 5 Þ

1

In this Generalized Fung QLV model, the reduced relaxation function g G ð t; e Þ is
considered as a weighted summation of multiple shape functions; and for each
shape function there is a distinct nonlinear function of strain that determines its
weight in the relaxation curve. For an exponential representation of the reduced
relaxation function analogous to Eq. ( 3 ), g G ð t; e Þ can be written to depend
explicitly on both time and strain:

t; ðÞ¼ A o ð e Þþ X

M

g G A i ð e Þ e t = s i ð e Þ ð 6 Þ

i ¼ 1