Adaptive Quasi-Linear Viscoelastic Modeling - Computational Modeling in Tissue Engineering - page 56

Fig. 3 Schematics describing four common test protocols for calibration of QLV models. Tissue
strain is plotted in gray and tissue stress is plotted in black. a Relaxation tests with multiple
amplitudes. b Incremental relaxation tests. c Single large amplitude ramp-and-hold test.

d Incremental ramp-and-hold tests

The following notation is used to define the fitting procedure for such a series of
relaxation tests. R n (t) is the time history of stress recorded in the nth relaxation
test, whose strain amplitude is De n (n = 1, 2,…, N). The strain history for the nth
relaxation test, e n (t), involves a step function u(t), in which each test is treated as
being performed at time t = 0 on a specimen that has never before been stretched:

e n ð t Þ¼ De n u ð t Þ: ð 23 Þ

3.1.1 Adaptive QLV Model

The first step in fitting the Adaptive QLV model is to calculate the time course of
each viscoelastic strain in the nth relaxation test using Eq. ( 9 ):

ðÞ¼ Z

t

ds ¼ Z

t

Þ De n du ðÞ
ds

V ð e Þ
ni

g i t s ð g i t s ð Þ De n d ðÞ ds ¼ De n g i ðÞ; ð 24 Þ

1 1

where d(t) is Dirac's delta function. Substituting into the first line of Eq. ( 9 ):

r n ðÞ¼ r o De ðÞþ De n X

M

k i De ð g i ðÞ; ð 25 Þ

i ¼ 1

The first term in Eq. ( 25 ), r o (e), can be fit from the steady state value of R n (t)by
noting that all of the shape functions approach zero over time: