Technical information

3-11
the stress amplitude (constant) and
ω
as the angular frequency (units of radians per
second), the resulting steady state strain response is also sinusoidal. The quotient of
stress and strain in the frequency domain is may be represented by a complex number:
21
)sin(cos** EiEiEE ⋅+=⋅+⋅= φφ .......................................................... (3.3.3)
in which
*
E
is the material’s complex modulus, *E is the dynamic modulus and
φ
denotes the phase lag by which the strain lags behind the applied stress. These
quantities, although not shown explicitly, are functions of both
ω
and
0
T .
The components of the complex modulus,
1
E and
2
E in equation 3.3.3, can be
expressed using one fundamental viscoelastic function known as the relaxation
spectrum
h
and an additional material constant known as the equilibrium modulus
∞
E :
)(ln
1
),(),(
0
22
22
001
τ
τω
τω
τω
dThETE ⋅
⋅+
⋅
⋅+=
∫
∞
∞
............................................. (3.3.4)
)(ln
1
),(),(
0
22
002
τ
τω
τω
τω dThTE ⋅
⋅+
⋅
⋅=
∫
∞
..................................................... (3.3.5)
As can be seen, the relaxation spectrum has units of stress and is a function of time
τ
and temperature
0
T , i.e., ),(
0
Thh
τ
=
. The equilibrium modulus
∞
E is temperature
independent, defined as: )(lim)(lim
10
tEEE
t ∞→→∞
=
=
ω
ω
in which )(tE is the
viscoelastic relaxation modulus (units of stress).
Equations 3.3.4 and 3.3.5 are appropriate for a given constant test or reference
temperature,
0
T . The assumption of thermo-rheological simplicity (Schwarzl and
Staverman, 1952) states that these relations can remain applicable for a different
(constant) temperature,
T
, simply by replacing physical time,
τ
, with reduced (or
pseudo) time,
r
τ
, defined as:
Tr
a/
τ
τ
=
, where ),(
0
TTaa
TT
=
is the so-called time-
temperature shift factor which is a unitless function of temperature only. Due to the
reciprocal nature of time and frequency, the reduced angular frequency,
r
ω
, is simply
obtained by
Tr
a⋅=
ω
ω
with
ω
as the applied angular frequency. For a certain class of