Adaptive Quasi-Linear Viscoelastic Modeling - Computational Modeling in Tissue Engineering - page 60

In a ramp-and-hold protocol the tissue length is increased at a constant rate to
its final value during the ramp-loading phase, and kept constant to allow the tissue
force to relax to its steady state value during the hold relaxation phase (Fig. 3 c).
The strain rate in the ramp-loading phase can be adjusted to prevent tissue damage
and mechanical waves during the loading [ 64 ]. Unlike the stepwise stretch tests
(relaxation tests) in which the tissue experiences only two levels of strain (the
initial strain and the final strain), the ramp loading tests expose the tissue to a
continuous range of strain over the loading ramp. Therefore, the ramp loading
stress reflects the tissue characteristics for all strains between the initial and final
strain. As will be described, this can be a benefit or a drawback. On the one hand,
this can complicate model fitting, but on the other, a single large amplitude ramp-
and-hold may be sufficient to calibrate a QLV model to the viscoelastic properties
of a tissue.
We again assume that initial configuration of the tissue is its unstretched, zero
strain, zero strain history configuration. In the ramp-loading phase of a single large
strain ramp-and-hold test, the tissue strain increases by De at a constant rate over
T seconds. In the hold-relaxation phase, then the tissue is maintained at its final
strain (De) until the stress relaxes to its steady state value. The strain function in
this protocol can be written:

8
<
:

0 ; t\0

De
T

e n ð t Þ¼ t ; 0\t\T : ð 35 Þ

De ; t [ T

We denote the experimentally recorded stress during the ramp-loading phase
(0 B t B T)asP(t), and that recorded during the hold-relaxation phase (t C T) as
H(t).

3.3.1 Adaptive QLV Model

A single ramp-and-hold test does not provide sufficient information to calibrate the
generalized form of the Adaptive QLV model presented in Eq. ( 9 ). However, this
protocol does provide sufficient information for calibration of the simple form of
the Adaptive QLV model presented in Eq. ( 8 ). Substituting the strain from Eq. ( 35 )
into Eq. ( 8 ), the viscoelastic strain may be calculated as:

8
<

R

t

Þ D T ds ¼ D T c ð t Þ;

gt s ð 0\t\T

V ð e Þ ðÞ¼

0

ð 36 Þ

R

T

:

Þ D T ds ¼ D T

gt s ð ð c ð t Þ c ð t T Þ Þ; t [ T

0

where c(t) is the integral of g(t) defined as: