Biomedical Engineering Reference
In-Depth Information
Fig. 25.1
Generalized
Maxwell model
model is
k
1
exp
−
k
i
t.
(25.1)
k
1
t
η
1
k
2
exp
−
k
2
t
η
2
k
i
−
1
exp
−
k
i
−
1
t
η
i
−
1
σ(ε,t)
=
+
+···+
+
To capture the nonlinear behavior with a reduced number of parameters, Fung's
quasi-linear viscoelasticity (QLV) theory has been widely used (Fung,
1972
). QLV
theory assumes that the relaxation or creep response can be separated into strain-
dependent and time-dependent components, as described in 1D as
E
t
(t
−
τ)
d
σ
d
ε
ε(t)
d
τ
d
τ,
σ(ε,t)
=
(25.2)
where
E
t
is the reduced relaxation function that depends on time and d
σ/
d
ε
is
the instantaneous elastic response. Both functions are obtained by curve-fitting the
experimental data. The QLV model is able to fit a single set of experimental data
(Weiss and Gardiner,
2001
) such as a stress relaxation experiment. However, this
model cannot fully describe or predict stress or strain profiles at different constraint
levels with the same set of parameters due to the fact that the reduced relaxation
function
E
t
depends on time only (Provenzano et al.,
2001
) whereas the actual tissue
response includes a strain-dependent relaxation function. The QLV model predicts
the same relaxation or creep rate regardless of strain or stress levels.
Schapery's single integral nonlinear theory, or modified superposition theory
(Provenzano et al.,
2002
; Duenwald et al.,
2010
), is similar to Schapery's single
integral nonlinear theory and therefore in this paper, we use Schapery's single inte-
gral theory to demonstrate this class of constitutive models. In 1D uniaxial loading,
Schapery's theory can be expressed as:
h
1
(ε)
E
ρ(t)
ρ
(τ)
d
h
2
(ε)ε
d
τ
σ(ε,t)
=
h
e
(ε)E
e
ε
+
−
d
τ,
(25.3)
where the reduced time
ρ
and the reduced time variable of integration
ρ
are func-
tions of strain and time and are defined as:
t
τ
d
t
d
t
a
e
[
ρ
=
ρ
=
,
.
(25.4)
ε(t
)
ε(t
)
a
e
[
]
]
0
0
