Adaptive Quasi-Linear Viscoelastic Modeling - Computational Modeling in Tissue Engineering - page 51

in which the strain-dependent amplitudes A i ð e Þ of the relaxation modes correspond
to strain-dependent time constants s i ð e Þ . Note that the Fung QLV model is
recovered by restricting A i ð e Þ and s i ð e Þ to be independent of strain. Following the
Fung QLV model (Eq. ( 2 )), the stress-strain relation for the Generalized Fung
QLV can be written as the sum of the convolution integrals associated with the
different shape functions:

Z

t

r ðÞ¼ r o ð e ð t ÞÞ þ X

M

Þ A i ð e ð t ÞÞ de ðÞ
ds

g i t s ð ds ; ð 7 Þ

i ¼ 1

1

where r o ð t Þ is the steady state tissue stress for a given tissue strain, e(t), and is
taken out of the convolution integral to simplify model calibration. g i ð t Þ is the ith
shape function and is considered usually as an exponential term (i.e. exp ð t = s i Þ ).
A i ð e Þ is the nonlinear function of strain and is equivalent to the derivative of the

elastic stress in the Fung QLV model (i.e. dr ð e Þ ðÞ de). Since the steady state
tissue stress is taken out of the convolution integral, all shape functions should
approach zero as time approaches infinity.

2.3 The Adaptive QLV Model

The Adaptive QLV model differs from the Fung and Generalized Fung QLV
models in that it uses an alternative approach to incorporate nonlinearity into a
linear viscoelastic model. This alternative approach simplifies the calibration
procedure significantly, and is conceptually simple as well because it enables
representation of the resultant QLV model in terms of an infinite array of Maxwell
elements (springs and dashpots).
In the Adaptive QLV model, stress and strain are related through an interme-
diate variable termed the viscoelastic strain, V ð e Þ ð t Þ , and through a linear convo-
lution integral as:

r ð t Þ¼ k ð e ð t ÞÞ V ð e Þ ð t Þ
V ð e Þ ð t Þ¼ Z

t

ð 8 Þ

g ð t s Þ de ð s Þ
ds

ds

1

where k(e) is a pure nonlinear function of strain and g(t) is a reduced relaxation
function that can be expressed as a sum of exponentials with different time con-
stants. V ð e Þ ð t Þ represents the dependence of the stress on the history of straining.
Nonlinearity enters the model through k(e), which converts the strain history
(viscoelastic strain) to stress through a simple multiplication.