4.3 Generalized-Kelvin Model
125
0
2
4
6
8
10
−4
−2
0
2
4
t
σ(t)/E
˜
E sin (ω t) +
˜
E
co s( ωt )
− ˜
E
e
−t/[cτ ]
0
2
4
6
8
10
−10
−5
0
5
10
t
(t)/C
˜
C sin(ωt) − ˜
C cos(ωt)
˜
C
e
−
t / τ
Fig. 4.26 Standard-Linear-Solid Kelvin model with τ k = 1.0 and E 0 = E k : Normalized stress
history σ(t)/E k resulting from sinusoidal strain history with a = 5 and ω = 2 π/4 (left) and normalized strain history k resulting from sinusoidal stress history with σ a = 5 and ω = 2 π/4
(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.
(4.103)
Observe, furthermore, that direct application of the Laplace transformation to the
differential equation relating the total stress and strain
L{σ(t)} + c τ k s L{σ(t)} = E ∞ L{ + E ∞ τ k s L{
(4.104)
renders immediately the relation
L{σ(t)} = E ∞
1 + τ k s
1 + c τ k s
L{
(4.105)
with inverse
L{ = C ∞
1 + c τ k s
1 + τ k s
L{σ(t)} =
C k
1
1 + τ k s
+ C 0
L{σ(t)},
(4.106)
that are entirely conforming with the convolution integral representations.
Complex Harmonic Oscillation Representation
The differential equation relating the total stress and strain reads in complex representation as
σ(t) + c τ k ˙
σ(t) = E ∞ + E ∞ τ k ˙
(4.107)