Adaptive Quasi-Linear Viscoelastic Modeling - Computational Modeling in Tissue Engineering - page 64

the Adaptive QLV model in a single, large amplitude ramp-and-hold test. In the
incremental ramp-and-hold test, the variation of the relaxation function with tissue
length can be modeled. This precludes the use the exceptionally simple fitting
procedure from the single, large amplitude ramp-and-hold protocol (Eq. ( 40 )) to
determine the k i functions at intermediate stains ((n-1)De \ e \ nDe).

3.4.2 Generalized Fung Model

Substituting the strain function from Eq. ( 41 ) into Eq. ( 9 ), stress history for the nth
relaxation test is:

8
< r o ð n 1 Þ De þ D T

t

R ds ¼ r Pn ð t Þ;

þ D T P i

t

Þ A i ð n 1 Þ De þ D T s g i t s ð 0\t\T

0

r n ðÞ¼

: r o nDe ð Þ

þ D T P i

R ds ¼ r Hn ð t Þ;

T

Þ A i ð n 1 Þ De þ D T s

g i t s ð t [ T

0

ð 49 Þ

The nonlinear unknown functions A i (e) appear inside the convolution integral in
both the ramp-loading stress history r Pn (t) and the hold-relaxation stress history
r Hn (t). Therefore, both r Pn (t) and r Pn (t) depend on the functions A i (e) for all the
strains between (n-1) De and nDe. This complicates the optimization procedure as
in the case of the Fung model for the single large amplitude ramp-and-hold pro-
tocol. Similarly, the optimization protocol is based on several trial-and-error steps.
However, because in the incremental ramp-and-hold protocol the incremental
strain De is small, we may approximate the A i (e) functions as piecewise linear. For
the case of a stress-free, fully relaxed specimen at zero strain, we may assume all
A i (0) to be zero. In the nth ramp-and-hold test, A i may be written as a linear
function in terms of a single parameter, a slope m ni , as:

¼ A i ð n 1 Þ De

ð n 1 Þ De þ De
T

Þ þ m ni De
T

A i t ð t ; ð 50 Þ

where A i ((n-1) De) was determined in the (n-1)th ramp-and-hold test. For known
shape functions, the parameters m ni may be determined from data in the nth ramp-
and-hold test by minimizing the integral I n given in Eq. ( 47 ). For parametric shape
functions, the parameters of the shape functions and all the m ni parameters may be
calibrated so that the integral I given by Eq. ( 48 ) is minimized.