Biomedical Engineering Reference
In-Depth Information
E(t)
in which, with
=
E(t)
−
E(
∞
)
,
E
e
ω
∞
0
E
t
sin
ωt
dt
E
S
(ω)
=
(6.47)
is called the
storage modulus
,
ω
∞
0
E
t
cos
ωt
dt
E
D
(ω)
=
(6.48)
is called the
loss modulus
, and the
loss tangent
(dimensionless) is given by
E
D
(ω)
E
S
(ω)
tan
δ(ω)
=
(6.49)
The above equations give the dynamic, frequency-dependent mechanical properties
in terms of the relaxation modulus.
E
S
is the component of the stress-strain ratio in
phase with the applied strain, while
E
D
is the component 90 degrees out of phase.
These relations may be inverted in order to obtain
∞
2
π
(E
S
−
E
e
)
E(t)
=
E
e
+
sin
ωt dω
(6.50)
ω
0
∞
2
π
E
D
ω
E(t)
=
E
e
+
cos
ωt dω
(6.51)
0
Physically, the quantity
δ
represents the phase angle between the stress and strain
sinusoids. The dynamic stress-strain relation can be expressed as
=
E
∗
(ω)
ε
0
e
j(ωt
+
δ)
σ(t)
(6.52)
with
E
∗
=
jE
D
.
One can also consider the dynamic behavior in the compliance formulation. In
the modulus formulation,
σ(t)
E
S
+
E
∗
(ω)ε(t)
is an algebraic equation for sinusoidal
=
loading, hence the strain is
1
E
∗
(ω)
σ(t)
ε(t)
=
(6.53)
However, the complex compliance
J
∗
is defined by the equation
J
∗
(ω)σ (t),
ε(t)
=
(6.54)
with
J
∗
=
J
S
−
jJ
D
. Hence, the relationship between the dynamic compliance and
the dynamic modulus is
1
E
∗
(ω)
J
∗
(ω)
=
(6.55)
This is considerably simpler than the corresponding relation for the transient creep
and relaxation properties. Notice that tan
δ(ω)
=
J
D
(ω)/J
S
(ω)
.
Plots of dynamic viscoelastic functions may assume a variety of forms. For in-
stance, one may plot the dynamic properties versus frequency, with the frequency
scale given logarithmically. Alternatively, one may plot the imaginary part versus
the real part:
E
D
versus
E
S
or
J
D
versus
J
S
. Such a plot is often used in dielectric
