Adaptive Quasi-Linear Viscoelastic Modeling - Computational Modeling in Tissue Engineering - page 63

where, analogous to the previous section, c i (t) is the integral of g i (t):
c i ð t Þ¼ Z

t

g i ðÞ ds :

ð 44 Þ

0

The stress can be written:

8
<

r o ð n 1 Þ De þ D T t

þ D T P i

c i ð t Þ¼ r Pn ð t Þ;

k i ð n 1 Þ De þ D T t 0\t\T

r n ðÞ¼ ð 45 Þ

: r o nDe ð Þ

þ D T P i

k i nDe ð Þ c i ð t Þ c i ð t T Þ ð Þ ¼ r Hn ð t Þ; t [ T

where r Pn (t) and r Hn (t) are the predictions of the time variation of stress over the
nth ramp-loading and nth hold-relaxation phases, respectively.
As before, r o (e) can be calibrated at each strain level nDe through:

r o nDe ð Þ¼ H n t !1 ð Þ: ð 46 Þ

For known shape functions, the values of k i (nDe) may be calibrated using the
hold-relaxation data of the nth ramp-and-hold test so that the integrals I n are each
minimized:

I n ¼ Z

þ1

Þ 2 dt : ð H n ðÞ r Hn ðÞ ð 47 Þ

T

For parametric shape functions, the parameters of the shape functions and the
values of k i (nDe) for all n may be calibrated so that the following integral is
minimized:

Z 2

þ1

I ¼ X
n

H n ðÞ r Hn ðÞ
H n ð T Þ

dt : ð 48 Þ

T

As before, the values of the functions k i at intermediate stains may be estimated
by interpolation.
An advantage of this protocol amongst the four considered in this chapter is that
it affords an additional set of data with which to check or refine the predictive
capability of a QLV model: once the k i functions have been determined for all
strains using a fitting of the hold-relaxation data, the fitted model may be used to
predict the ramp-loading stress histories r Pn . These predictions can be compared
with the measured data to assess the predictive capability of the fitted constitutive
model.
This fitting procedure is far simpler than that needed for fitting the Fung QLV
and Generalized Fung QLV models. However, it is more complicated than that for