102
4 Visco-Elasticity
0
2
4
6
8
10
−10
−5
0
5
10
t
σ(t)/E
0
2
4
6
8
10
−5
0
5
t
(t)/C
˜
C sin(ωt) − ˜
C cos(ωt)
˜
C e
−t/τ
Fig. 4.14 Specific Kelvin model with τ = 1.0: Normalized stress history σ(t)/E resulting from
sinusoidal strain history with a = 5 and ω = 2 π/4 (left) and normalized strain history
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)} = C
1
1 + τ s
L{σ(t)}.
(4.56)
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,
5 a phase shifted causal harmonic signal for the resulting strain history
superposed by an exponentially decaying signal that is needed to enforce the initial
condition = 0
= H(t)
C
sin(ω t) − C
cos(ω t) + C
e
−t/τ
σ a .
(4.57)
Thereby C
:= C/[1 + τ
2
ω
2
] and C
:= C τ ω/[1 + τ
2
ω
2
] are here formally
introduced as abbreviations, in terminological accordance to E
and E
they are
5 The inverse Laplace transformation for the strain history follows from the following step by step
computation:
L{
σ a
= C
1
[1 + τ s]
ω
[ω 2 + s 2 ]
= C
1
[1 + τ 2 ω 2 ]
ω [1 + τ 2 ω 2 ]
[1 + τ s] [ω 2 + s 2 ]
= C
1
[1 + τ 2 ω 2 ]
[ω − τ ω s] [1 + τ s] + τ ω τ [ω 2 + s 2 ]
[1 + τ s] [ω 2 + s 2 ]
= C
1
[1 + τ 2 ω 2 ]
ω − τ ω s
ω 2 + s 2 +
τ ω τ
1 + τ s
= C
ω
ω 2 + s 2 − C
s
ω 2 + s 2 + C
τ
1 + τ s
= C
L{H(t) sin(ω t)} − C
L{H(t) cos(ω t)} + C
L{H(t) e
−t/τ }
.