4.4 Maxwell Model
149
−5
0
5
t
σ(t)/E
˜
E sin(ωt) + ˜
E cos(ωt)
− ˜
E
e
−t/τ
0
2
4
6
8
10
0
2
4
6
8
10
−5
0
5
10
t
(t)/C
˜
C sin(ωt) − ˜
C cos(ωt)
˜
C
Fig. 4.38 Specific Maxwell 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)
tory superposed by an exponentially decaying signal that is needed to enforce the
initial condition σ(0) = 0
σ(t) = H(t)
E
sin(ω t) + E
cos(ω t) − E
e
−t/τ
a .
(4.158)
Thereby E
:= E τ
2
ω
2
/[1 + τ
2
ω
2
] and E
:= E τ ω/[1 + τ
2
ω
2
] are here formally introduced as abbreviations, however as will become transparent in the sequel,
they denote the so-called storage and loss stiffness moduli (note that E
/E
=
1/[τ ω]). The stress history resulting from a sinusoidal strain history is shown in
Fig. 4.38 (left).
Upon Laplace transformation, the convolution integral of the creep function C(t)
with a prescribed stress (rate) history, a causal signal σ(t) = H(t) σ(t) with σ(0) = 0,
results in
L{σ(t)}
a
= E
τ s
[1 + τ s]
ω
[ω 2 + s 2 ]
= E
τ ω
[1 + τ 2 ω 2 ]
s [1 + τ 2 ω 2 ]
[1 + τ s] [ω 2 + s 2 ]
= E
τ ω
[1 + τ 2 ω 2 ]
[τ ω 2 + s] [1 + τ s] − τ [ω 2 + s 2 ]
[1 + τ s] [ω 2 + s 2 ]
= E
τ ω
[1 + τ 2 ω 2 ]
τ ω 2 + s
ω 2 + s 2 −
τ
1 + τ s
= E
ω
ω 2 + s 2 + E
s
ω 2 + s 2 − E
τ
1 + τ s
= E
L{H(t) sin(ω t)} + E
L{H(t) cos(ω t)} − E
L{H(t) e
−t/τ }
.