4.1 Newton Model
83
0
2
4
6
8
10
0
2
4
6
8
10
−10
−5
0
5
10
t
σ(t)/η
−10
−5
0
5
10
t
(t) η
− ˜
C cos(ωt)
˜
C
Fig. 4.4 Specific Newton model with η = 1.0: Normalized stress history σ(t)/η resulting from
sinusoidal strain history with a = 5 and ω = 2 π/4 (left) and normalized strain history η
resulting from sinusoidal stress history with σ a = 5 and ω = 2 π/4 (right)
L{ = L{C(t) ) ˙
σ(t)} = s L{C(t)} L{σ(t)} =
1
η s
L{σ(t)}.
(4.19)
Choosing, as a particular example, a causal harmonic stress history with σ(t) =
H(t) σ a sin(ω t) and thus L{σ(t)} = σ a ω/[ω
2
+ s
2
] renders, after inverse Laplace
transformation,
2 a causal harmonic signal (shifted by −π/2) for the resulting strain
history superposed by a constant signal that is needed to enforce the initial condition
= 0
= H(t)
−C
cos(ω t) + C
σ a .
(4.20)
Thereby C
:= 1/[η ω] is here formally introduced as abbreviation, in terminological accordance to E
it is denoted the loss compliance modulus. The strain history
resulting from a sinusoidal stress history is shown in Fig. 4.4 (right).
2 The inverse Laplace transformation for the strain history follows from the following step by step
computation:
L{
σ a
= −
1
η s
−ω
[ω 2 + s 2 ]
= −
1
η ω
−ω 2
s [ω 2 + s 2 ]
= −
1
η ω
s 2 − [ω 2 + s 2 ]
s [ω 2 + s 2 ]
= −
1
η ω
s
ω 2 + s 2 −
1
s
= −C
s
ω 2 + s 2 + C
1
s
= −C
L{H(t) cos(ω t)} + C
L{H(t)}.