Biomedical Engineering Reference
In-Depth Information
Fig. 6.14
Example of
history-dependent strain as a
function of constant stress
divided into Heaviside step
functions
Based on the definition of the relaxation function each step strain in the summation
gives rise to a relaxing component of stress. Assuming there is no effect from the
interaction between the step components, we have
N
σ
c
=
ε(
0
)E(t, ε)
+
ε
i
E(t
−
t
i
,ε)
(6.33)
i
=
0
Divide by
σ
c
, and use the definition of the creep compliance from (
6.4
),
N
J
i
E
t
−
t
i
,ε(t
i
)
1
=
J(
0
)E(t, ε)
+
(6.34)
i
=
0
If we consider in the limit that we have infinitely many fine step components (i.e.
similar rationale to the decomposition of the recurrent ladder network elements), we
obtain a Stieltjes integral, with
τ
as a time variable of integration:
t
E
t
τ,ε(τ)
dJ(τ,σ
c
)
dτ
1
=
J(
0
)E(t, ε)
+
−
dτ
(6.35)
0
This relationship is implicit and is analogous to the one for the linear case. In order to
develop an explicit form, one needs to assume a specific form for the creep behavior.
First, assume the creep function to be separable into a stress-dependent portion and
a power law in time. Such a form, called quasi-linear viscoelasticity has been used
widely in biomechanics [
49
]:
=
g
1
+
+···
t
n
g(σ)t
n
g
3
σ
2
J(t,σ)
=
g
2
σ
+
(6.36)
where
g
1
,g
2
,g
3
are constants. Consider the following relaxation function, with
f
1
,f
2
,f
3
constants, as a trial solution. A separable form does not give rise to a
solution:
E(t,σ)
=
f
1
t
−
n
+
f
2
ε(t)t
−
2
n
+
f
3
ε(t)
2
t
−
3
n
+···
(6.37)
