152
4 Visco-Elasticity
10
−3 10
−2 10
−1 10
0 10
1 10
2 10
3
10
−11
10
−8
10
−5
10
−2
10
1
ω
E
E
τ = 10
2
, 10
1
, 10
0
, 10
−1
, 10
−2
ω
2
τ
2
1 + ω 2 τ 2
10
−3 10
−2 10
−1 10
0 10
1 10
2 10
3
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
ω
E
E
τ = 10
2
, 10
1
, 10
0
, 10
−1
, 10
−2
ωτ
1 + ω 2 τ 2
10
−3 10
−2 10
−1 10
0 10
1 10
2 10
3
10
−0.4
10
−0.2
10
0
10
0.2
10
0.4
ω
C
C
τ = 10
2
, 10
1
, 10
0
, 10
−1
, 10
−2
1
10
−3 10
−2 10
−1 10
0 10
1 10
2 10
3
10
−6
10
−3
10
0
10
3
10
6
ω
C
C
τ = 10
2
, 10
1
, 10
0
, 10
−1
, 10
−2
1
ωτ
Fig. 4.39 Specific Maxwell model: Normalized storage stiffness modulus E (ω)/E (top left) and
normalized loss stiffness modulus E (ω)/E (top right) together with normalized storage compliance
modulus C (ω)/C (bottom left) and normalized loss compliance modulus C (ω)/C (bottom right)
plotted against the angular frequency ω for five decades of relaxation times τ
likewise the (real) amplitude C a and the phase shift angle δ of the complex compliance
modulus are defined as
C
∗
(ω) =:
[C ] 2 + [C ] 2 e
−i tan
−1 (C
/C
)
=: C a e
−i δ
,
(4.171)
so that σ a = E a a (or a = C a σ a ) and δ σ = δ + δ. Specifically, the angular frequency dependent amplitude E a (ω) and phase shift angle δ(ω) follow from
E a :=
[E ] 2 + [E ] 2 = E
τ ω
√
1 + τ 2 ω 2
and tan δ :=
E
E =
1
τ ω
, (4.172)
correspondingly, the angular frequency dependent amplitude C a (ω) and phase shift
angle δ(ω) follow from