4.4 Maxwell Model
151
Complex Harmonic Oscillation Representation
The differential equation relating the total stress and strain reads in complex representation as
σ(t) + τ ˙
σ(t) = η ˙
(t).
(4.164)
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
i τ ω
1 + i τ ω
∗
=: E
∗
∗
.
(4.165)
Thereby the quantity relating the complex amplitudes of the total strain and stress
is denoted the complex stiffness modulus
E
∗
(ω) = E
τ
2
ω
2
1 + τ 2 ω 2 + i
τ ω
1 + τ 2 ω 2
=: E
+ i E
,
(4.166)
its inverse is the complex compliance modulus (so that E
∗ C
∗
= 1)
C
∗
(ω) = C
1 − i
1
τ ω
=: C
− i C
.
(4.167)
Note that for the specific 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
E
:= E
τ
2
ω
2
1 + τ 2 ω 2 and E
:= E
τ ω
1 + τ 2 ω 2 ,
(4.168)
whereas C
and C
denote the so-called storage and loss compliance moduli, respectively, that are defined as
C
:= C and C
:= C
1
τ ω
.
(4.169)
The storage and loss stiffness and compliance moduli are plotted against the
angular frequency ω for various relaxation times in Fig. 4.39.
Finally, the (real) amplitude E a and the phase shift angle δ of the complex stiffness
modulus are defined as
E
∗
(ω) =:
[E ] 2 + [E ] 2 e
i tan
−1 (E
/E
)
=: E a e
i δ
,
(4.170)
151
Complex Harmonic Oscillation Representation
The differential equation relating the total stress and strain reads in complex representation as
σ(t) + τ ˙
σ(t) = η ˙
(t).
(4.164)
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
i τ ω
1 + i τ ω
∗
=: E
∗
∗
.
(4.165)
Thereby the quantity relating the complex amplitudes of the total strain and stress
is denoted the complex stiffness modulus
E
∗
(ω) = E
τ
2
ω
2
1 + τ 2 ω 2 + i
τ ω
1 + τ 2 ω 2
=: E
+ i E
,
(4.166)
its inverse is the complex compliance modulus (so that E
∗ C
∗
= 1)
C
∗
(ω) = C
1 − i
1
τ ω
=: C
− i C
.
(4.167)
Note that for the specific 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
E
:= E
τ
2
ω
2
1 + τ 2 ω 2 and E
:= E
τ ω
1 + τ 2 ω 2 ,
(4.168)
whereas C
and C
denote the so-called storage and loss compliance moduli, respectively, that are defined as
C
:= C and C
:= C
1
τ ω
.
(4.169)
The storage and loss stiffness and compliance moduli are plotted against the
angular frequency ω for various relaxation times in Fig. 4.39.
Finally, the (real) amplitude E a and the phase shift angle δ of the complex stiffness
modulus are defined as
E
∗
(ω) =:
[E ] 2 + [E ] 2 e
i tan
−1 (E
/E
)
=: E a e
i δ
,
(4.170)