174
4 Visco-Elasticity
L{σ(t)} = E ∞
1 + e τ m s
1 + τ m s
L{(t)} =
E ∞ + E m
τ m s
1 + τ m s
L{(t)},
(4.209)
with inverse
L{(t)} = C ∞
1 + τ m s
1 + e τ m s
L{σ(t)},
(4.210)
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) + τ m ˙
σ(t) = E ∞ (t) + E 0 τ m ˙
(t).
(4.211)
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 e τ m ω
1 + i τ m ω
∗
=: E
∗
∗
.
(4.212)
Thereby the quantity relating the complex amplitudes of the total strain and stress
is denoted the complex stiffness modulus
15
E
∗
(ω) = E ∞
1 + i e τ m ω
1 + i τ m ω
=: E
+ i E
,
(4.213)
its inverse is the complex compliance modulus (so that E
∗ C
∗
= 1)
C
∗
(ω) = C ∞
1 + i τ m ω
1 + i e τ m ω
=: C
− i C
.
(4.214)
Note that for the Standard-Linear-Solid Maxwell model the complex moduli E
∗
and C
∗ have indeed real and imaginary parts. Here, E
and E
denote the so-called
storage and loss stiffness moduli, respectively, that are defined as
16
15 Observe that the complex stiffness modulus may alternatively be expressed as
E
∗ (ω) = E ∞ + E m
i τ m ω
1 + i τ m ω
.
.
16 Observe that the storage and loss stiffness moduli may alternatively be expressed as
E
:= E ∞ + E m
τ 2
m ω 2
1 + τ 2
m ω 2 and E
:= E m
τ m ω
1 + τ 2
m ω 2 .