CONTINUUM MECHANICS for ENGINEERS

and by defining the absolute modulus, as the magnitude of G * ˜G ω
(iω)
according to
(9.6-8)
together with the loss angle, δ between and G′(ω) as given by its
tangent
˜ G( ω)
the stress σ 12 (Eq 9.6-7), may now be written
From this equation we see that the peak value of the stress is
(9.6-9)
(9.6-10)
(9.6-11)
and that the strain lags behind the stress by the loss angle δ. Figure 9.10
provides a graphical interpretation of this phenomenon. In Figure 9.10a, the
two constant magnitude stress and strain vectors, separated by the constant
angle δ, rotate about a fixed origin with a constant angular velocity ω. The
vertical projections of these vectors, representing the physical values of the
stress and strain, are plotted against time in Figure 9.10b. From Figure 9.10a,
the portion of the stress in phase with the strain is σ o cos δ and by Eq 9.6-11
together with Eq 9.6-9, the storage modulus may be expressed as
Similarly, the loss modulus is written as
(9.6-12a)
(9.6-12b)
Consistent with the duality present in all of viscoelastic theory we reverse
the roles of stress and strain in the preceding portion of this section to define
the complex compliance, J * (iω) along with its associated real and imaginary
parts. Briefly, we assume an applied stress
( )
[ ] + [ ′′( ) ]
˜ G ω G ω G ω
( )= ′( )
ω
tan δ =
ω
′′( ) G
G
2 2
′( )
σ ˜ iδ
iωt ( ω δ)
G ω e γ e G˜ i t
ω γ e
12 = ( ) = ( ) +
o
o
σ G˜
ω γ
G′
=
G′′
=
= ( )
o o
σocosδ γ
o
σosinδ γ
o