4.5 Generalized-Maxwell Model
175
10
−3 10
−2 10
−1 10
0 10
1 10
2 10
3
10
0
10
0.1
10
0.2
10
0.3
ω
E
E ∞
τ = 10
2
, 10
1
, 10
0
, 10
−1
, 10
−2
1 + e 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
[e − 1] ωτ
1 + ω 2 τ 2
10
−3 10
−2 10
−1 10
0 10
1 10
2 10
3
10
−0.3
10
−0.2
10
−0.1
10
0
ω
C
C ∞
τ = 10
2
, 10
1
, 10
0
, 10
−1
, 10
−2
1 + e ω
2
τ
2
1 + e 2 ω 2 τ 2
10
−3 10
−2 10
−1 10
0 10
1 10
2 10
3
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
ω
C
C ∞
τ = 10
2
, 10
1
, 10
0
, 10
−1
, 10
−2
[e − 1] ωτ
1 + e 2 ω 2 τ 2
Fig. 4.51 Standard-Linear-Solid 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 m and E ∞ = E m
E
:= E ∞
1 + e τ
2
m ω
2
1 + τ 2
m ω 2 and E
:= E ∞
[e − 1] τ m ω
1 + τ 2
m ω 2 ,
(4.215)
whereas C
and C
denote the so-called storage and loss compliance moduli, respectively, that are defined as
C
:= C ∞
1 + e τ
2
m ω
2
1 + e 2 τ 2
m ω 2 and C
:= C ∞
[e − 1] τ m ω
1 + e 2 τ 2
m ω 2 .
(4.216)
The storage and loss stiffness and compliance moduli are plotted against the
angular frequency ω for various relaxation times in Fig. 4.51.
.