to the late 1960s was largely in response to needs
did measure Young's modulus of snow in static
arising from the expansion of U. S. military activi-
uniaxial compression tests. Later, Yosida et al.
ties in the polar regions. The sheer size of the
(1956) discussed the interpretation of the four-
Soviet Union, and its range of arctic, subarctic
parameter model and found the parameters for it
and alpine environments, made the study of snow
from creep tests on snow under uniaxial com-
mechanics important in that nation, although, un-
pressive stress. Bader (1962a) also suggested that
fortunately, only a small fraction of the resulting
the one-dimensional hyperbolic sine relationship:
literature is available in translation.
dε
= ε o sinh (Aσ)
The objective of most of the work through this
(1)
dt
period was to determine the parameters required
(where ε is the strain, σ is the stress and t is time
for application of linear elasticity, viscosity and
and εo and A are constants) might be used to
viscoelasticity to problems involving snow me-
chanics. The effort followed the recognition that
describe creep in snow; that is, it could replace
some patterns of deformational behavior in snow
the linear relationship for the dashpot of the Max-
samples in a laboratory or field setting could be
well element of the four-parameter model. Mellor
described by linear relationships. For example,
(1964) introduced an additional term into eq 1 by
Bader et al. (1939) discussed the creep of snow
dividing the coefficient of the hyperbolic sine by
a viscosity coefficient, η. He also discussed the
(they used the term "plasticity") in connection
with investigations of snow settlement. They did
use of exponential and power relationships to
experiments on samples in both uniaxial confined
represent compactive viscosity (i.e., the viscosity
and unconfined compression, but since they did
determined from the compaction of natural snow-
not attempt to formulate a constitutive relation-
packs, or from confined compression experiments
ship to describe the process there was no frame-
in the laboratory) in terms of the snow density as
work within which parameters could be defined.
derived from data sets collected by various inves-
Thus, they made no mention of any particular
tigators. Other determinations of the constants
mechanical property or constitutive relationship,
for the four-parameter model from creep test data
although the patterns of deformation certainly
have been done in Russia by Kuvaeva et al. (1967)
suggested a combination of linear elastic and vis-
and by Shinojima (1967). Parameters for these lin-
ear relationships, along with the available values,
cous behavior. In fact, Yosida et al. (1956) were
are summarized in Appendix B.
able to use data from Bader et al. (1939) to calcu-
Even as efforts continued to find parameters
late values for the coefficient of Newtonian vis-
for linear relationships, it was apparent that the
cosity of snow.
ranges were too limited to solve many problems
The most general constitutive relationship used
in snow mechanics. Bader (1962a) recognized the
for snow prior to about 1970 was the equation for
problem and suggested that the ranges of the lin-
a four-parameter viscoelastic fluid with linear ele-
ear relationships might be extended if they were
ments (App. A). According to Yosida et al. (1956),
applied incrementally, as the values of the pa-
it was first used in snow mechanics by de
rameters change with deformation. We have found
Quervain (1946) to interpret the results of torsion
no references in which attempts to use this ap-
experiments.* Bucher (1948) included a sketch of
proach were made, although Desrues et al. (1980)
a Maxwell model (a spring and dashpot in series
did devise a similar method involving simple non-
as shown in Fig. A1 in App. A) and used the
linear relationships. Mellor (1975) stated that there
constitutive relationship for a linear viscous fluid
were still no alternatives to linear relationships,
to find the coefficient of Newtonian viscosity for
and that 1) there were no constitutive relation-
compacted snow as a function of temperature,
ships for use in solving problems involving mul-
duration of loading and a variety of types of snow,
tiaxial stress states, and 2) the data to develop
grain sizes, and ages. Interestingly, although the
such relationships did not exist. He credited B.
Maxwell model includes a spring element, Bucher
Salm with initiating efforts to address the need
made no mention of the elastic properties (or lack
for such relationships. In fact, Salm (1967) did
of them) of snow, although Yosida et al. (1948)
consider the extension of the hyperbolic sine rela-
tionship to cases of the creep of snow in triaxial
*Kuvaeva et al. (1967) reported that the viscosity of snow was
stress states. Later Salm (1971) used the relation-
first determined by "the group of K. S. Zavriev in 1937."
ships in exponential form to develop a failure
Unfortunately, the reference they gave for this work appears
criterion based on energy storage and dissipa-
to be incorrect and we could not locate the paper.
3
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