Adaptive Quasi-Linear Viscoelastic Modeling - Computational Modeling in Tissue Engineering - page 66

increment involved a 2.0 mm stretch over T = 20 s; each hold lasted 2000 s,
sufficient for specimens to relax to an almost constant level of isometric force.
Force was recorded at 10 Hz. Before the first test, the loading bars were adjusted
so that the stress was zero at the initial (zero strain) length of the specimen. After
the final ramp-and-hold increment, the loading bars were returned to their pre-test
separation, and the specimen was allowed to relax for 2000 s.
The predictions of the models were thereafter checked against data from a
single, large amplitude ramp-and-hold protocol. The same specimens were stret-
ched 8.0 mm at a constant rate over T = 10 s [ 55 ].

4.2 Calibration

Here we present detailed examples of how to calibrate the Adaptive QLV model
and Generalized QLV model to the incremental ramp-and-hold test data. Strain
and stress were calculated from experimentally measured displacement and force
using Eqs. ( 21 ) and ( 22 ). Three exponential shape functions were chosen:

g 1 ð t Þ¼ e t = s 1 ; g 2 ð t Þ¼ e t = s 2 ; g 3 ð t Þ¼ e t = s 3 ; ð 51 Þ

where s 1 , s 2 and s 3 are the three parameters of these shape functions that should be
calibrated. The incremental strain in each test was:

De ¼ Dl
l o

¼ 2

30 ¼ 0 : 0667 ;

ð 52 Þ

and the strain function in the nth test (n = 1, 2, 3, or 4) was:

e n ð t Þ¼ 0 : 0667 n 1

ð Þ 0 : 0667

20

t ;

0\t\20

; ð 53 Þ

0 : 0667n ; t [ 20

so that the four final strains following the four incremental tests were 0.0667,
0.1333, 0.2000 and 0.2667.
In the plot of the experimentally recorded stress data for the four incremental
tests (Fig. 5 )[ 55 ], the first 20 s (ramp-loading phase) is expanded 5 times in the
time domain for clarity, and the initial stress (stress at the beginning of each ramp
loading), the maximum stress (stress at the end of each ramp loading, and at the
beginning of the subsequent hold relaxation), and the final stress (at the end of
each hold relaxation) are labeled in each test. Following the fitting procedures
described above, we calibrated the QLV models using the hold relaxation time
courses of stress, H n , recorded during the four incremental tests.

4.2.1 Adaptive QLV Model

According to Eq. ( 46 ) and using the experimental recordings, the elastic part of
stress, r o , was 1, 6, 19 and 38 Pascal at 4 strains of 0.0667, 0.1333, 0.2000 and