4.3 Generalized-Kelvin Model
123
σ(t) = H(t)
E
sin(ω t) + E
cos(ω t) − E
e
−e t/τ k
a .
(4.99)
Thereby E
:= E ∞ [1 + c τ
2
k ω
2
]/[1 + c
2
τ
2
k ω
2
] and E
:= E ∞ [1 − c] τ k ω/[1 +
c
2
τ
2
k 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 − c] τ k ω/[1 + c τ
2
k ω
2
]). The stress history resulting from a
sinusoidal stress history is shown in Fig. 4.26 (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{(t)} = L{C(t) ) ˙
σ(t)} = s L{C(t)} L{σ(t)}
(4.100)
= C ∞
1 + c τ k s
1 + τ k s
L{σ(t)}.
Note that the nominator C ∞ [1 + c τ k s] in the above expands as C k + C 0 [1 +
τ k s] thus highlighting again the serial arrangement of a Kelvin and a Hooke element
in the Standard-Linear-Solid Kelvin model
L{(t)} =
C k
1
1 + τ k s
+ C 0
L{σ(t)}.
(4.101)
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
L{σ(t)}
a
= E ∞
[1 + τ k s]
[1 + c τ k s]
ω
[ω 2 + s 2 ]
= E ∞
[1 + c τ k s] ω − [c − 1] τ k ω s
[1 + c τ k s] [ω 2 + s 2 ]
= E ∞
ω
ω 2 + s 2 + E
[1 + c 2 τ 2
k ω 2 ] s
[1 + c τ k s] [ω 2 + s 2 ]
= E ∞
ω
ω 2 + s 2 + E
c τ k ω
ω
ω 2 + s 2 + E
s − c τ k ω 2
[1 + c τ k s] [ω 2 + s 2 ]
= E
ω
ω 2 + s 2 + E
[1 + c τ k s] s
[1 + c τ k s] [ω 2 + s 2 ]
− E
c τ k [ω 2 + s 2 ]
[1 + c τ k s] [ω 2 + s 2 ]
= E
ω
ω 2 + s 2 + E
s
ω 2 + s 2 − E
c τ k
1 + c τ k s
= E
L{H(t) sin(ω t)} + E
L{H(t) cos(ω t)} − E
L{H(t) e
−t/[c τk ] }
.