3.1 Hooke Model
69
0
2
4
6
8
10
0
2
4
6
8
10
−10
−5
0
5
10
t
σ(t)/E
−10
−5
0
5
10
t
(t) E
Fig. 3.4 Specific Hooke model with E = 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 E
resulting from sinusoidal stress history with σ a = 5 and ω = 2 π/4 (right)
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{ = L{C(t) ) ˙
σ(t)} = s L{C(t)} L{σ(t)} = C L{σ(t)}.
(3.16)
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,
3 also a causal in-phase harmonic signal for the resulting strain history,
whereby the initial condition = C σ(0) is captured
= H(t) C
sin(ω t) σ a .
(3.17)
Thereby C
:= C = 1/E is here formally introduced as abbreviation, in terminological accordance to E
it is denoted the storage compliance modulus. The strain
history resulting from a sinusoidal stress history is shown in Fig. 3.4 (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.
(3.18)
3 The inverse Laplace transformation for the strain history follows from the following step by step
computation:
L{
σ a
= C
ω
ω 2 + s 2 = C
L{H(t) sin(ω t)}.