model with linear elements, leads to (Fig. A1)
in this manner implies the assumption that they
are independent of each other so that, for ex-
1
E t
1
t
ample, changes in the magnitude of the perma-
ε (t) = σ0 +
+
1 - exp - 2 .
(A1)
E1 η1 E2
η2
nent strain component with time do not affect the
parameter that determines the instantaneous
elastic response. To our knowledge, this effect
A plot of this equation approximates a creep curve,
has never been studied experimentally, but it
so that, as described in Yosida et al. (1956), Bader
seems likely that at some strain, the assumption
(1962a), Nadai (1963) and Mellor (1964), and
will no longer be satisfied. Another point of inter-
shown in Figure A1, the parameters in eq A1 can
est is that, since the three strains are independent
be determined from a single experimentally de-
of each other, it is easy to omit one or two of them
rived creep curve (although, obviously, more
curves would be required for accuracy). This was
depending upon the application. Thus, for ex-
done by Yosida et al. (1956) and Shinojima (1967)
ample, for a problem in which a small stress is
from experimental data for snow deformed to
applied for a long period of time, the permanent
small strains. A similar procedure could also be
strain would become much larger than the com-
used (although more experiments would be re-
bined instantaneous and time-dependent elastic
strains. As a result, the elements that contribute
quired) to determine the parameters if some or all
these strain components can be ignored. Con-
of the model elements followed nonlinear stress-
versely, if the loads are applied for only a short
strain rate relationships.
Next, we show the relationship between the
time, then the permanent strain can be neglected
four-parameter viscoelastic fluid model and the
with only the elastic components being retained.
Note also that the "two types of stress" re-
more general constitutive relationships for non-
ferred to by Nadai (1963) are simply the stresses
linear viscoelastic materials. The intent is to show
across the arms of the Voigt model (Fig. A1). Equi-
the assumptions required to make the transition
librium requires that they sum to the magnitude
between the different constitutive relationships.
of the stress, σ, applied across the model.
The constitutive relationships for the four-
parameter model with linear elements can be de-
The summation and integration of the strains
for a constant stress σo applied at time t=0 to the
rived from conditions of equilibrium and the sum-
ε3
ε2
ε1
σ1
σ1
η2
σ
σ
E1
η1
E2
σ2
σ2
Voigt Model
Maxwell Model
σ0
dε
=
η1
dt
Strain
σ0
ε1 + ε 2
E2
Figure A1. The four-parameter spring-dashpot model of
σ0
ε1
a viscoelastic fluid showing nomenclature, stress-strain
E1
laws and relationships for determining values of the
parameters from a creep curve.
Time
22
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