4.5 Generalized-Maxwell Model
173
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,
14 a causal phase shifted 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
−c t/τ m
σ a .
(4.206)
Thereby C
:= C ∞ [1 + e τ
2
ω
2
]/[1 + e
2
τ
2
ω
2
] and C
:= C ∞ [e − 1] τ ω/[1 +
e
2
τ
2
ω
2
] are here formally introduced as abbreviations, in terminological accordance
to E
and E
they are denoted the storage and the loss compliance moduli (note that
C
/C
= [e − 1] τ ω/[1 + e τ
2
ω
2
]). The strain history resulting from a sinusoidal
stress history is shown in Fig. 4.50 (right).
It is interesting to note that the relaxation function and the creep function are
related via their Laplace transformations as
s
2
L{E(t)} L{C(t)} = 1.
(4.207)
Observe, furthermore, that direct application of the Laplace transformation to the
differential equation relating the total stress and strain
L{σ(t)} + τ m s L{σ(t)} = E ∞ L{(t)} + E 0 τ m s L{(t)}
(4.208)
renders immediately the relation
14 The inverse Laplace transformation for the strain history follows from the following step by step
computation:
L{(t)}
σ a
= C ∞
[1 + τ m s]
[1 + e τ m s]
ω
[ω 2 + s 2 ]
= C ∞
[1 + e τ m s] ω − [e − 1] τ m ω s
[1 + e τ m s] [ω 2 + s 2 ]
= C ∞
ω
ω 2 + s 2 − C
[1 + e 2 τ 2
m ω 2 ] s
[1 + e τ m s] [ω 2 + s 2 ]
= C ∞
ω
ω 2 + s 2 − C
e τ m ω
ω
ω 2 + s 2 − C
s − e τ m ω 2
[1 + e τ m s] [ω 2 + s 2 ]
= C
ω
ω 2 + s 2 − C
[1 + e τ m s] s
[1 + e τ m s] [ω 2 + s 2 ]
+ C
e τ m [ω 2 + s 2 ]
[1 + e τ m s] [ω 2 + s 2 ]
= C
ω
ω 2 + s 2 − C
s
ω 2 + s 2 + C
e τ m
1 + e τ m s
= C
L{H(t) sin(ω t)} − C
L{H(t) cos(ω t)} + C
L{H(t) e
−t/[e τm] }
.