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

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 *c*σ) 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 = *E*ε 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

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).

21