Adaptive Quasi-Linear Viscoelastic Modeling - Computational Modeling in Tissue Engineering - page 70

Fig. 7 Elastic part of the stress (a) and three nonlinear functions k 1 (b), k 2 (c) and k 3 (d) for the
fitted Adaptive QLV model [ 55 ]. Gray circles are calibrated values based on experimental data
and black curves are cubic spline interpolations

e t = s i

X

3

Þ þ 0 : 0667
20

e T = s i 1 r Hn ð t Þ¼ r o 0 : 0667n ð A i 0 : 0667 ð n 1 Þ ð Þ s i

i ¼ 1

2 e t = s i :

X

3

þ 0 : 0667
20

1 e T = s i þ T m ni s i

s i e T = s i

ð 60 Þ

i ¼ 1

Because stress is zero at zero strain, we may assume A i (0) = 0. Thus, using Eq.
( 50 ), we may write:

Þ¼ De X

n

A i nDe ð m ni : ð 61 Þ

m ¼ 1

This allowed Eq. ( 60 ) to be rewritten in terms of s i and m ni .
These parameters were calibrated so that the integral I given by Eq. ( 48 ) was
minimized. The fitted curves for the Generalized QLV model with pricewise linear
approximations of the A i functions (Fig. 8 ) again produced an excellent fit to the
hold stress in all four tests (Fig. 9 ); optimum time constants and slopes are listed in
Table 3 . The optimum time constants were very close to those found for the
Adaptive QLV model. Indeed, one may use a different fitting procedure (that