4.5 Generalized-Maxwell Model
171
Thereby the convolution of the creep function with the stress rate history is abbreviated symbolically as C(t) ) ˙
σ(t).
Laplace Transformation Representation
Upon Laplace transformation, the convolution integral of the relaxation function E(t)
with a prescribed strain (rate) history, a causal signal (t) = H(t) (t) with (0) = 0,
results in
L{σ(t)} = L{E(t) ) ˙
(t)} = s L{E(t)} L{(t)}
(4.202)
= E ∞
1 + e τ m s
1 + τ m s
L{(t)}.
Note that the nominator E ∞ [1 + e τ m s] in the above expands as E ∞ [1 + τ m s] +
E m τ m s thus highlighting again the parallel arrangement of a Hooke and a Maxwell
element
L{σ(t)} =
E ∞ + E m
τ m s
1 + τ m s
L{(t)}.
(4.203)
Choosing, as a particular example, a causal harmonic strain history with (t) =
H(t) a sin(ω t) and thus L{(t)} = a ω/[ω
2
+ s
2
] renders, after inverse Laplace
transformation,
13 a causal phase shifted harmonic signal for the resulting stress history 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 m
a .
(4.204)
13 The inverse Laplace transformation for the stress history follows from the following step by step
computation:
L{σ(t)}
a
= E ∞
[1 + e τ m s]
[1 + τ m s]
ω
[ω 2 + s 2 ]
= E ∞
[1 + τ m s] ω − [1 − e] τ m ω s
[1 + τ m s] [ω 2 + s 2 ]
= E ∞
ω
ω 2 + s 2 + E
[1 + τ 2
m ω 2 ] s
[1 + τ m s] [ω 2 + s 2 ]
= E ∞
ω
ω 2 + s 2 + E
τ m ω
ω
ω 2 + s 2 + E
s − τ m ω 2
[1 + τ m s] [ω 2 + s 2 ]
= E
ω
ω 2 + s 2 + E
[1 + τ m s] s
[1 + τ m s] [ω 2 + s 2 ]
− E
τ m [ω 2 + s 2 ]
[1 + τ m s] [ω 2 + s 2 ]
= E
ω
ω 2 + s 2 + E
s
ω 2 + s 2 − E
τ m
1 + τ m s
= E
L{H(t) sin(ω t)} + E
L{H(t) cos(ω t)} − E
L{H(t) e
−t/τm }
Alternatively a more direct derivation that highlights the parallel arrangement of a Hooke and a
Maxwell element reads: