128
4 Visco-Elasticity
E a :=
[E ] 2 + [E ] 2 = E ∞
1 + τ
2
k ω 2
1 + c 2 τ
2
k ω 2 ,
(4.115)
correspondingly, the angular frequency dependent amplitude C a (ω) follows as
C a :=
[C ] 2 + [C ] 2 = C ∞
1 + c 2 τ
2
k ω 2
1 + τ
2
k ω 2 .
(4.116)
Finally the phase shift angle δ(ω) is expressed as
tan δ :=
E
E =
C
C =
[1 − c] τ k ω
1 + c τ
2
k ω 2 .
(4.117)
The amplitudes and (the tangent of) the phase shift angle of the complex stiffness
modulus and the complex compliance modulus are plotted against the angular frequency ω for various relaxation times in Fig. 4.28. The phase shift angle δ between the
harmonically oscillating total stress and strain and its tangent tan δ are also denoted
the loss angle and the loss factor, respectively. Obviously, the loss factor and thus the
loss angle tend to zero for large angular frequencies, since in this limit the viscous
dashpot is too inert to react.
Equivalence to Standard-Linear-Solid Maxwell Model
The Standard-Linear-Solid Kelvin model is characterized by differential equations
for (i) the global response relating total stress and total strain as well as for (ii) the
local response relating viscous strain and total strain as
σ + c τ k ˙
σ = E ∞ + E ∞ τ k ˙
and v + c τ k ˙
v = [1 − c] .
(4.118)
By re-parametrization of the relaxation time as τ m := c τ k the global response
of the Standard-Linear-Solid Kelvin model and the Standard-Linear-Solid Maxwell
model (as sketc.hed in Fig. 4.29) coincides, however the local response or rather the
viscous strain predicted by of the two models obviously still differs
σ + τ m ˙
σ = E ∞ + E 0 τ m ˙
and ˜
v + τ m ˙ ˜
v = .
(4.119)
Here the viscous strain ˜
v as predicted by the equivalent Standard-Linear-Solid
Maxwell model is related to the corresponding viscous strain v as predicted by the
underlying Standard-Linear-Solid Kelvin model via
˜
v =
v
1 − c
.
(4.120)
If furthermore the instantaneous elastic stiffness E 0 and the equilibrium elastic
stiffness E ∞ of the Standard-Linear-Solid Maxwell model and the Standard-Linear-