Biomedical Engineering Reference
In-Depth Information
Consequently,
ε
A
=
B
sin
(
−
δ)
at the intersection. However, the ellipse is symmet-
ric, so
A
B
sin
δ
.
Now, with the aim of obtaining a different relation between
δ
and the param-
eters of the ellipse, consider the point of maximum stress, which occurs when
sin
(ωt
+
δ)
=
=
π
1. Then
ωt
+
δ
=
2
, and referring to Fig.
6.15
,wehave
B
sin
π
δ
B
sin
π
2
δ)
cos
π
2
C
=
ε(σ
max
)
=
2
−
=
cos
(
−
δ)
+
sin
(
−
(6.60)
thus
B
cos
δ
(6.61)
Since
A
=
B
sin
δ
, then tan
δ
=
A/C
, it follows that the width of the elliptic Lis-
sajous figure is a measure of the loss angle
δ
of a linearly viscoelastic material.
To obtain a relation for the
storage modulus E
S
in connection with the elliptic
stress-strain curve, suppose
ε
ε(σ
max
)
=
=
ε
max
sin
ωt
. Then, the stress
σ
at the maximum
strain is
σ
E
S
ε
max
so the slope of the line from the origin to the point of maximum strain is
E
S
,as
shown in Fig.
6.15
.
To obtain a relation for the
storage compliance J
S
in connection with the ellip-
tic stress-strain curve, suppose
σ
=
E
S
ε
max
sin
ωt
+
E
D
ε
max
cos
ωt
.For
ε
max
,ωt
=
π/
2, then
σ
=
=
σ
max
sin
ωt
. Then, the strain at the maximum
J
∗
|
stress is, by the definition of
|
,
ε
=
(J
S
−
jJ
D
)σ
max
. Hence
ε
=
J
S
σ
max
sin
ωt
−
J
D
σ
max
cos
ωt
, because
j
=
√
−
1 implies a 90-degree phase shift. For
σ
max
,ωt
=
π/
2, then
ε
=
J
S
σ
max
or
σ
max
=
ε/J
S
. Therefore, the slope of the line from the
origin to the point of maximum stress is 1
/J
S
, as shown in Fig.
6.15
. Observe that
1
/J
S
≥
E
S
equality occurs if the material is elastic (i.e.
δ
=
0).
Consider now the time
t
=
0, and
ε
=
0. It follows that
σ(t)
=
D
sin
δ
, with
σ(t)
=
ε
max
[
E
S
sin
ωt
+
E
D
cos
ωt
]
.For
t
=
0,
σ
0
=
ε
max
E
D
, which is marked as
an intercept on the ordinate in Fig.
6.15
.
Also observe that
σ
max
=|
ε
max
, so, again referring to Fig.
6.15
, this modulus
corresponds to a line of intermediate slope between that for
E
S
and that for 1
/J
S
.
To conclude this line of thoughts, the loss angle
δ
, or the loss tangent tan
δ
,may
be considered as the fundamental measure of damping in a linear material.
However, the above derivation is not limited to linear viscoelastic properties.
Also nonlinear viscoelastic materials can be excited by sinusoidal loads. The sim-
plest dynamic example of the effect of nonlinearity is nonlinear elasticity in one
dimension. Suppose that
E
∗
|
σ
=
f(ε)
(6.62)
If we write
f(ε)
as a power series we obtain
a
1
ε
1
a
2
ε
2
a
3
ε
3
σ
=
+
+
+···
(6.63)
and if the strain is sinusoidal, say
ε
=
cos
ωt
and using trigonometric identities, for
example,
1
2
(
1
cos
2
ωt
=
+
cos 2
ωt)
(6.64)
