APPENDIX A: CONSTITUTIVE RELATIONSHIPS USED
TO DESCRIBE SNOW DEFORMATION
cous fluid is σ = η(dε/dt) where η is the coeffi-
The motive for making many of the measure-
cient of Newtonian viscosity. Unlike the case
ments of the mechanical properties of snow has
for elasticity, the parameters for nonlinear con-
been to find parameters for applying the constitu-
stitutive equations for viscous fluids have been
tive relationships for ideal elastic solids and vis-
determined under some conditions for both
cous fluids to problems involving snow deforma-
snow and ice. The relationships that have been
tion. The purpose of this appendix is to describe
(
)
used are the power law dε /dt = A σn , the ex-
these "ideal" materials and show how the rela-
(
)
ponential law dε /dt = Be σbn and the hyperbolic
tionships that describe their response to stress are
sine law (dε /dt = C sinh cσ) where A, B, b, C and c
related to more general constitutive relationships.
are constants that may depend on the temperature,
Note that we use the term "ideal" to describe
pressure and physical properties.
these materials and their constitutive relationships
The idea of using the law for what is now
because they are idealizations based on data from
called the four-parameter viscoelastic fluid (called
experiments on many types of materials. Thus,
the Burger's material or general linear substance
their origin is empirical, rather than analytical.
by some authors) to represent the deformation of
For our purposes we only need to consider
a material was apparently first proposed by Nadai
ideal elastic solids and viscous fluids under iso-
(1963, p. 166*) based on observations of experi-
thermal conditions. An ideal elastic solid is de-
mental creep curves. He defined a material that
fined by the property that the strain at any time
responds to loads by "three distinct types of strain
depends only on the instantaneous magnitude of
ε1, ε2, ε3 and two types of stress, σ1, σ2 ..." The
the stress and is independent of the stress history.
strains are 1) an ideal elastic strain that responds
Further, if the stress is removed from a deformed
instantly to changes in stress, 2) a component of
sample of an ideal elastic material, then the strain
permanent strain that changes as a function of
disappears, the sample returns to its original state
time and load, and 3) a semi-permanent, recover-
and all the strain energy stored during deforma-
able strain which represents a time-dependent
tion is recovered. Note that the definition does
elastic response to the applied load. The familiar
not require the stress-strain relationship to be lin-
spring-dashpot model for this material (shown in
ear. However, determining a value of Young's
Fig. A1 with linear springs and dashpots) illus-
modulus (E) for snow from a static test in uniaxial
trates how the total strain results from the sum-
loading includes the assumption that it is linear,
mation of the "three distinct types of strain..."
because the experimental data are fit to the one-
dimensional form of Hooke's law, σ = Eε where σ
listed above. The first two are from the Maxwell
is stress and ε is strain in the same direction.
model, while the third is the Voigt model, (called
the Kelvin, firmoviscous, or Bingham material by
Similar arguments apply to the other elastic con-
some authors). Summing the strain components
stants (the shear and bulk moduli, Poisson's ratio
and Lame's constant, λ).
Ideal linear viscous behavior is represented by
a constitutive equation in which the stress is pro-
*The discussion of the section titled "Composite, Viscoelastic
portional to the strain rate. As a result, the strain
Substance Disclosing Recovery Strains" in Nadai (1963, p. 166),
begins "This leads us to propose a third, ideal, composite, viscoelas-
at any time depends on the complete history of
tic, recovery-sensitive substance..." (italics his). Nadai then contin-
the stress, rather than its instantaneous magni-
ues to describe the model of the four-parameter viscoelastic
tude. Further, when the load is removed from a
fluid as given in the text above. Further, in a footnote on p. 170,
sample of a linear viscous fluid undergoing de-
Nadai noted the spring-dashpot models analyzed by Burgers
formation, the strain rate goes to zero and none of
and referred to one that "...demonstrates our composite sub-
stance having the three types of strain..." Note that Nadai (1963)
the strain is recovered. Thus, no strain energy is
is a version of a volume which was published originally in 1931,
stored during deformation, and all of the work
and in revised form in 1951. There is no similar discussion in
done by external stresses is nonrecoverable.
either of the earlier editions nor is there reference to earlier
The one-dimensional constitutive relation-
publication of the model, although it is was included without
ship for a homogeneous, isotropic, linear vis-
reference in Jaeger (1962).
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