82
4 Visco-Elasticity
η ˙
(t) = H(t) σ 0 =⇒ (t) =: C(t) σ 0 .
(4.14)
Here C(t), i.e. the normalized strain history as response to an imposed constant
unit stress step σ(t) = H(t), has been introduced as the linearly increasing creep
function that is illustrated in Fig. 4.3 (right)
C(t) := H(t) t/η.
(4.15)
Based on the Boltzmann superposition process, the strain history (t) for t ≥ 0 as
response to an arbitrary stress history σ(t) ≡ H(t) σ(t) follows from the convolution
integral
(t) =
t
0
C(t − t
) ˙
σ(t
) dt
=: C(t) ) ˙
σ(t).
(4.16)
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)} = η s L{(t)}.
(4.17)
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,
1 a causal harmonic signal (shifted by π/2) for the resulting stress
history, whereby the initial condition σ(0) = η ˙
(0) is captured.
σ(t) = H(t) E
cos(ω t) a .
(4.18)
Thereby E
:= η ω is here formally introduced as abbreviation, however as will
become transparent in the sequel, it denotes the so-called loss stiffness modulus. The
stress history resulting from a sinusoidal strain history is shown in Fig. 4.4 (left).
Upon Laplace transformation, the convolution integral of the creep function C(t)
with a prescribed stress (rate) history, supposed a causal signal σ(t) = H(t) σ(t)
with σ(0) = 0, results in
1 The inverse Laplace transformation for the stress history follows from the following step by step
computation:
L{σ(t)}
a
= η s
ω
[ω 2 + s 2 ]
= η ω
s
[ω 2 + s 2 ]
= E
L{H(t) cos(ω t)}.