4.2 Kelvin Model
103
denote the storage and the loss compliance moduli (note that C
/C
= τ ω). The
strain history resulting from a sinusoidal stress history is shown in Fig. 4.14 (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.
(4.58)
Observe, furthermore, that direct application of the Laplace transformation to the
differential equation relating the total stress and strain
L{σ(t)} = η s L{(t)} + E L{(t)}
(4.59)
renders immediately the relation
L{σ(t)} = E [1 + τ s] L{(t)},
(4.60)
that is entirely conforming with the convolution integral representation.
Complex Harmonic Oscillation Representation
The differential equation relating the total stress and strain reads in complex representation as
σ(t) = E (t) + η ˙
(t).
(4.61)
Then, for a stationary harmonic oscillation of the total stress and strain with
σ(t) = σ
∗ e
i ω t and (t) =
∗ e
i ω t the relation between the corresponding complex
amplitudes
∗
= a e
i δ and σ
∗
= σ a e
i δ σ (where δ = −π/2 for sinusoidal strain
control and δ σ = −π/2 for sinusoidal stress control) follows as
σ
∗
= E [1 + i τ ω]
∗
=: E
∗
∗
.
(4.62)
Thereby the quantity relating the complex amplitudes of the total strain and stress
is denoted the complex stiffness modulus
E
∗
(ω) = E [1 + i τ ω] =: E
+ i E
,
(4.63)
its inverse is the complex compliance modulus (so that E
∗ C
∗
= 1)
C
∗
(ω) = C
1
1 + τ 2 ω 2 − i
τ ω
1 + τ 2 ω 2
=: C
− i C
.
(4.64)