124
4 Visco-Elasticity
transformation
7, ,
8 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) = 0
(t) = H(t)
C
sin(ω t) − C
cos(ω t) + C
e
−t/τ k
σ a .
(4.102)
Thereby C
:= C ∞ [1 + c τ
2
k ω
2
]/[1 + τ
2
k ω
2
] = C k /[1 + τ
2
k ω
2
] + C 0 and C
:=
C ∞ [1 − c] τ k ω/[1 + τ
2
k ω
2
] = C k τ k ω/[1 + τ
2
k ω
2
] are here formally introduced as
abbreviations, in terminological accordance to E
and E
they are denote the storage
and the loss compliance moduli (note that C
/C
= [1 − c] τ k ω/[1 + c τ
2
k ω
2
]). The
strain history resulting from a sinusoidal stress history is shown in Fig. 4.26 (right).
7 The inverse Laplace transformation for the strain history follows from the following step by step
computation:
L{(t)}
σ a
= C ∞
[1 + c τ k s]
[1 + τ k s]
ω
[ω 2 + s 2 ]
= C ∞
[1 + τ k s] ω − [1 − c] τ k ω s
[1 + τ k s] [ω 2 + s 2 ]
= C ∞
ω
ω 2 + s 2 − C
[1 + τ 2
k ω 2 ] s
[1 + τ k s] [ω 2 + s 2 ]
= C ∞
ω
ω 2 + s 2 − C
τ k ω
ω
ω 2 + s 2 − C
s − τ k ω 2
[1 + τ k s] [ω 2 + s 2 ]
= C
ω
ω 2 + s 2 − C
[1 + τ k s] s
[1 + τ k s] [ω 2 + s 2 ]
+ C
τ k [ω 2 + s 2 ]
[1 + τ k s] [ω 2 + s 2 ]
= C
ω
ω 2 + s 2 − C
s
ω 2 + s 2 + C
τ k
1 + τ k s
= C
L{H(t) sin(ω t)} − C
L{H(t) cos(ω t)} + C
L{H(t) e
−t/τk }
.
8 Alternatively a more direct derivation that highlights the serial arrangement of a Kelvin and a
Hooke element reads:
L{(t)}
σ a
= C 0
ω
[ω 2 + s 2 ]
+ C k
1
[1 + τ k s]
ω
[ω 2 + s 2 ]
= C 0
ω
[ω 2 + s 2 ]
+ C k
1
[1 + τ 2
k ω 2 ]
ω [1 + τ 2
k ω 2 ]
[1 + τ k s] [ω 2 + s 2 ]
= C 0
ω
[ω 2 + s 2 ]
+ C k
1
[1 + τ 2
k ω 2 ]
[ω − τ k ω s] [1 + τ k s] + τ k ω τ k [ω 2 + s 2 ]
[1 + τ k s] [ω 2 + s 2 ]
= C 0
ω
[ω 2 + s 2 ]
+ C k
1
[1 + τ 2
k ω 2 ]
ω − τ k ω s
ω 2 + s 2 +
τ k ω τ k
1 + τ k s
= C
ω
ω 2 + s 2 − C
s
ω 2 + s 2 + C
τ k
1 + τ k s
= C
L{H(t) sin(ω t)} − C
L{H(t) cos(ω t)} + C
L{H(t) e
−t/τk }
.