Biomedical Engineering Reference
In-Depth Information
ε 0 t n , with ε 0 as the strain amplitude. Substi-
The creep strain is ε(t)
=
J(t)σ c =
tution leads to
t
f 1 (t
+···
τ) n
τ) n (t
τ) 2 n
f 3 ε 0 (t
τ) 2 n (t
τ) 3 n
1
=
+
f 2 ε 0 (t
+
0
× g 1 + g 2 σ + g 3 σ 2
+··· n 1
(6.38)
Factorization of the stress-dependent and the time-dependent parts delivers:
f 3 ε 0 +··· t
0
= g 1 +
+··· f 1 +
g 3 σ 2
τ) n τ n 1
(6.39)
1
g 2 σ
+
f 2 ε 0 +
n(t
1
sin ) identical to the linear case by Laplace
transformation of the integral and an identity involving the gamma function. This
is in fact quite similar to the definitions of fractional derivative as from fractional
calculus (see the Appendix for detailed information). Again, we stumble on the ob-
vious origin of genesis of lumped fractional-order models characterizing viscoelas-
tic properties in the lungs. However, we need to make an explicit link between the
constitutive equations of viscoelastic behavior and the fractional-order terms in FO
models.
Before proceeding further with our theoretical development, it is necessary to
discuss the implications of stress-strain response in relation to history dependence.
Consider a stress history that is triangular in time (i.e. quite close to the actual
breathing patterns in some patients):
σ(t)
The integral part gives results (
=
0 or t< 0 ,
σ(t)
=
0 /t 1 )t,
for 0 <t<t 1 ,
(6.40)
σ(t)
=
2 σ 0
0 /t 1 )t,
for t 1 <t< 2 t 1 ,
and
σ(t)
=
0 or2 t 1 <t<
Use the Boltzmann integral and consider that slopes are piecewise constant:
ε(t)
=
0 or t< 0 ,
t
σ 0
t 1
ε(t)
=
J(t
τ)dτ,
for 0 <t<t 1
0
t 1
t
τ)dτ ,
σ 0
t 1
(6.41)
ε(t)
=
J(t
τ)dτ
J(t
for t 1 <t< 2 t 1 ,
and
0
t 1
t 1
2 t 1
τ)dτ ,
σ 0
t 1
ε(t)
=
J(t
τ)dτ
J(t
for 2 t 1 <t<
0
t 1
Substitute the given creep function and decompose the exponential as follows:
b
1
exp
τ c e
τ c
t
τ
t b
τ c
t a
e
=
b
a
(6.42)
τ c
a
A more meaningful approach to theoretical basis of viscoelasticity is to look at
the response of viscoelastic materials to sinusoidal load, referred to as dynamic be-
havior . Notice, however, that the term dynamic in this context has no connection
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