Adaptive Quasi-Linear Viscoelastic Modeling - Computational Modeling in Tissue Engineering - page 58

the strain at time 0 - (the initial strain). Assuming the initial strain to be the strain
at time zero forces these models to behave linearly for all relaxation tests starting
from the same initial strain. Note that this problem does not exist for the Adaptive
QLV model because the Adaptive QLV model incorporates nonlinear functions in
terms of the instantaneous strain, which is the final strain level of a relaxation test.
Using the predictions of Eq. ( 29 ), r o can be calibrated using Eq. ( 26 ). Subse-
quently, the A i functions can be calibrated at each De n so that the integral I n given
by Eqs. ( 27 ) or the integral I given by Eq. ( 28 ) is minimized for known or
parametric shape functions, respectively. As with the Adaptive QLV model,
interpolation may be used for intermediate straining for the case of the Generalized
Fung QLV model. Note that this procedure would apply as well for the fitting of
the Fung QLV model to any of the N individual tests, but only a single parameter
set would result. If these parameters are dependent upon strain, then the Fung QLV
model is not valid and the Generalized Fung QLV model must be used.

3.2 Incremental Relaxation Tests

Incremental relaxation testing is a faster protocol to assess the nonlinear behavior
of tissues and calibrate QLV models. The time savings arises from the reduced
number of force relaxation intervals required: in the previous protocol the speci-
men must be shortened back to its initial length following each test, and a waiting
period is required before performing any subsequent relaxation tests. This wait
should be long enough to ensure that any viscous effect associated with the
shortening back to the initial specimen length have settled, and should be of a
duration equal to that of the relaxation test. To avoid these wait periods the
relaxation tests may be performed in an incremental manner with a series of small
(and usually equal amplitude) stretches that are performed with no unloading
between the stretch increments (Fig. 3 b). In this protocol the initial length of each
relaxation test is the final length from the previous test. For equal strain increments
of amplitude De on a specimen that is strain free before the application of the first
increment, the initial strain in the nth relaxation test is (n-1)De. The strain
function can therefore be written:

ð n 1 Þ De ; t\0

e n ð t Þ¼ : ð 30 Þ

nDe ; t [ 0

3.2.1 Adaptive QLV Model

Substituting the strain function from Eq. ( 30 ) into Eq. ( 9 ) the viscoelastic strains in
the nth relaxation test are: