A constitutive equation for creeping snow
Bruno Salm
Abstract.
On the basis of the principle of least irreversible force (Ziegler, 1963) a constitutive
equation for creeping snow in a quasistationary state is established. As a consequence of this
principle, the constitutive equation depends only on the dissipation function of the system.
This function is developed in power series of the invariants of the stress tensor. Tests show that
because of the distinct non-newtonian behaviour of snow, terms up to the sixth degree of the
invariants have to be taken into account.
Résumé.
Une équation constitutive de la neige rampante en état quasistationnaire est donnée,
basée sur le principe de la force irréversible minimale (Ziegler, 1963). De ce principe suit, que
cette équation constitutive ne dépend que de la fonction de dissipation du système. Cette fonction
est développée en séries des puissances des invariants du tenseur des contraintes. Des expériences
montrent qu'il faut tenir compte des termes jusqu'au sixième degré des invariants, dû à l'état
non-newtonien distinct de la neige.
INTRODUCTION
The link between physics and phenomenological mechanics is represented by the
so-called constitutive equations. Such equations are based on the mechanical behaviour
of the microstructure of the material and are the foundation of fluid dynamics. They
describe the specific properties of an idealized material by means of a relation between
kinematic and dynamic variables. Examples of the former are strain, strain rate, etc.
and the latter stress, stress rate, etc.
Constitutive equations must be consistent with various requirements, for example
they must be independent of the system of coordinates and of the observer and his
state of motion. Obviously no discrepancy with the general principles of mechanics
and thermodynamics can be allowed.
The degree of idealization must be chosen in such a way that the model behaves
analogously to the real structure but remains manageable for fluid dynamic
calculations.
Two important idealizations are the assumption of isotropy and the neglect of
stress history. For snow under arbitrary conditions these cannot generally be made.
However, for the tested snow type and the chosen test procedure, isotropy can be
assumed and the influence of stress history can be simply represented by the total
deformation with acceptable accuracy. Therefore, the mechanical characteristics of
an initial state of snow just before the start of the tests can be calculated (Salm, 1971).
This initial state, a result of a long stress history and metamorphism, refers to the
instantaneous state of the natural snowpack at that time when the snow samples were
taken for the tests.
The most general constitutive equation for an initial state according to our
definition is that of a Reiner—Rivlin fluid (Reiner, 1958)
°H=f8ij+gVii+hVikVi!i
(1)
where ay is the stress, VVi the deformation rate and ôy the Kronecker delta. The
functions /, i and h depend on the three invariants of the deformation rate tensor as
well as on temperature, structure and density of snow. The summation convention
is used.
A constitutive equation for creeping snow
223
THE PRINCIPLE OF LEAST IRREVERSIBLE FORCE
In snow, reversible and irreversible work is performed analogously as in a Burger's
body (Fig. 1), which describes well the behaviour of snow under low stresses. The
reversible parts are the two Hooke elements with the mechanical characteristics Ei
and E 2 . Under a constant state of stress as soon as a quasistationary state is reached,
(E,
(E9)
FIGURE 1.
Burger's body.
the only element performing work is the Newton element N, which dissipates all
energy. It must be emphasized that the viscoelastic characteristics of snow given by
the Burger's body also exist in snow under the action of the isotropic part of the stress
tensor. Therefore, in contradiction to other materials, a quasistationary state under a
steady hydrostatic state of stress is also possible (Salm, 1971). If only the quasistationary states of deformation are considered only the irreversible part of work
must be taken into account. This assumption is made and therefore the rheological
model of snow is a simple (non-newtonian) dashpot.
In order to further restrict the number of free parameters, temperature and snow
type (that is, initial structure and density) are kept constant.
Ziegler (1963) demonstrated that the principle of least irreversible force (which
can also be stated as the principle of maximum rate of entropy production) narrows
the field of admissible constitutive equations. Applied to continua it yields
IhpD
Y1 dpD
oVi = pD {— Vklx
v3Fk
Wn
(2)
p is the density and D is the so-called dissipation function per unit mass of the
system. Instead of the three independent functions in equation (1), equation (2)
contains only one which has a simple physical meaning. It represents the rate of
dissipation work that is the irreversible power PW of the irreversible stress </')
D(V)
= P(i) = - o^vn
P
J
>0
(3)
The principle of least irreversible force is formulated in the velocity space. However,
Ziegler (1963) showed that it is possible to obtain a corollary in force space: the
principle of least velocity.
224
Bruno Salm
Applied to continua it yields
V1 dpD'
—
,/dpD'
Vii=pD'(-—oJ
\dakl
/
(4)
day
where
/j'( ff ) =/»0) = I a « F y > 0
(5)
P J
£> and£>' depend on the same quantities as the functions/, g and h in equation (1).
Assuming isotropy, the expressions on the right side of equation (4) are given by
(Ziegler, 1963)
dpD'
~—
dpD'
Ok\ = ~
9a kl
9p£>'
dpD'
0(i)
9a (1)
9p£>' 9a(i)
=
9a u
+
2
~
dpD'
°(2) + 3
9a (2)
dpD' 9a(2)
+
9 a a ) 9ay
a(3)
(6)
9a (3)
9pZ)' 9ff(3)
+
9a (2) 9ay
9a (3) 9a y
(7)
where
9a (1) _
9ay
(8)
«ij
9^(2) _
90y
9^(3) _
9 ay
°ii
0(i)«ij
(9)
Oik
Oik^kj - C T ( l ) f f i j ~ 0 ( 2 ) ^ i j
and a([), ff(2) a n ^CT(3)a r e
tne
(10)
invariants of the stress tensor.
THE DISSIPATION FUNCTION
General comments
Once the principle of the least irreversible force is accepted, the main problem is the
determination of the dissipation function. The following determination of this
function is restricted to snow, but with appropriate modifications it may be
applicable to ice.
In snow mechanics a hyperbolic sine function is very often used as the stress—strain
rate relation in the uniaxial state of compressive stress. This law is supported by
theoretical considerations in the microscopic scale (Kauzmann, 1941). Because snow
responds unsymmetrically to changes of the sign of stress, if we want to include the
uniaxial state of tensile stress one has to formulate this law in more general way, as
Vu = C[exp(aa n )-exp(-Z?a n )]
(11)
where C, a and b are constants of the material. The right-hand side of equation (11)
can be developed in a power series which converges for all an. The dissipated power
is obtained as
3
anVn = C (a+b)a^
+ (a2~b2)^+
2!
4
(a3 + b3) — +
3!
(12)
The nature of the relation between dissipated power and stress is ultimately based on
the mechanical behaviour of the bonds between ice grains and to some extent also on
A constitutive equation for creeping snow
225
plastic deformation of ice grains. There must exist a hidden regularity governing this
relation in all conceivable states of stress. It is now postulated that the dissipation
function can be generally represented in a form like equation (12). When the stresses
are replaced by the invariants of the stress tensor it is therefore possible to develop
the dissipation function in power series of these invariants. Furthermore, it is
assumed that this series is sufficiently rapidly convergent that only the first few terms
are required.*
Because of the distinct non-newtonian behaviour of snow terms up to the sixth
degree in the stress tensor invariants are tentatively taken into account. It is to be
noticed of the invariants that the first is linear, the second is quadratic and the third
is cubic in the magnitude of the stresses. The coefficients of the invariants are denoted
by a with subscripts 1, 2 and 3 which signify the invariants belonging to the term in
question. The sum of the subscripts equals the degree of the term. The terms up to
the sixth degree are given in Table 1.
TABLE 1.
Dissipation function pD'
Degree (/)
(0(1))'
2
3
4
5
6
^(2))i/2
(ff(l)) ( i ^ 2 ) °-(2)
<*n
a
m
<°(l)>0-4)<&)
au
«mi
"inn
J
a
-*ini2
1112
nm
(ff(3))' /3
«7(1))°
3)
ff(3)
ff
(2)CT(3)
tf(l)<7(2)°(3)
As the dissipation function is non-negative, the term ct^o^ cannot occur. Hence,
the first approximation, that of a newtonian liquid, is given by degree 2
pD' = ttuffp) +a20(2)
(13)
In the tests principal deformation rates exclusively were measured and as it is assumed
that the principal directions of the state of deformation and state of stress are
identical, the expressions of stresses are henceforth confined to these directions.
Furthermore, as in the tests an is never vanishing and the magnitude of two principal
stresses is always the same, the stress invariants can be written as
a(1) =
ffii=a„(l+2X)
(14)
2
a (2) = ^(ffijffij - ajjOjj) = -a?i(2X + X )
(15)
2
a(3) = VepCTjj 0 jk aki - 3ffu ojjCTkk+ anffjja k k ) = a\x X
(16)
* A further example can be found in assuming a power law (Glen's law) instead of equation (11 ).
This law holds for ice and snow and can be developed in convergent power series.
22$
Bruno Sa\m
where
^22
=
ff
33
'
(17)
and
a-ri
\ =—
Ou
(18)
The task is now to determine the 21 free constants of Table 1 by means of the
test results. As many more measurements than the number of unknowns are available,
the method of least squares is used for this purpose. Because stresses are measured
without appreciable error compared with the measurement of the deformation rates,
they are treated as the independent variables. It is for this reason that equation (4)
instead of equation (2) is used. The principle that any manipulations with error
equations are forbidden creates some problems in the formulation of the normal
equations. Stresses and deformation rates were both measured, and therefore the error
equations must be formulated between these same quantities and the dissipation
function cannot be obtained directly.
Uniaxial state of stress
Vu, the quotient V22/Vn = V33/Vn and an, the only nonvanishing stress component
were measured. In this state of stress the second and third invariants of the stress
tensor are zero, and equation (4) then yields for i=j = \,
pD'
Vu =
.
s
= a 11 a 11 + a 1 „(7 11 + . . . a,inii<7 u
(19)
On
and with pD' = anVu, for i=j = 2
V22
(dpD'V1
dpD'
i -auoJLJl=lu
i Ji_
(20)
Vn
do g,) \9a ( i)
From equations (19) and (20) two sets of normal equations corresponding to the
measured deformation rates are obtained. The solution of the normal equations
originating from equation (19) gives the coefficients in the first column of Table 1,
and the solution of those originating from equation (20) gives a2 of the second
column and the coefficients of the third column. Associated with each degree / a
mean error
«,_!=/-!-!(21)
V n—m
is obtained to be used as a criterion for the quality of the approximation /. [v2] is the
sum of the square of the residua, n the number of measurements and m the number
of unknown coefficients. If our postulate of the general comments section is true,
that is that equation (19) has the nature of equation (12), mi_1 should decrease with
increasing degree i.
Uniaxial state of deformation
Vu and the quotient a22lau = a33lau
(4) yields for / = / = 1
IdpD'
\_1 doD'
\oa k j
/
Vn = pD' I — oJ
-—-
0CTn
= X and the stress au were measured. Equation
(22)
A constitutive equation for creeping snow
227
and for i=j = 2
lapD'
V1 dpD'
V22 = pD' — - a kl
—-=0
\9a k l
/ do22
(23)
As the first two factors on the right-hand side of equation (23) are always nonzero,
this leads to the condition
dpD'
r— = 0
(24)
aa22
From equations (22) and (24) the remaining free coefficients of the dissipation
function - i.e. those which are not already fixed by the tests under uniaxial state of
stress - can be determined. For this purpose the components of the gradient of the
dissipation function have to be adjusted to the appropriate measured (or given)
deformation rates.
Vn is developed in a power series
Vn = a.'llon+anio]1
+ ...
(25)
Equation (24) inserted into equation (7) for / = / = 2, yields
dpD'
9^(2)
1
tfu(l
dpD'
+ A) L9ff(i)
+ Xa2n
dpD'
9<?(3)~
(26)
and equation (26) inserted into equation (7) for / =/ = 1, yields
dpD'
1 - X dpD'
9ff„
1+ X
9tf(i)
• dpD'
Y a
"
(27)
9CT(3)J
On the other hand, equation (5) gives pD' = anVn which when differentiated becomes
dpD'
dVu
— = VU + on — (28)
dan
dau
The practical calculation of the remaining free a-values is made in three steps.
Firstly, the a'-values are calculated by adjusting equation (25) to the measured
values of deformation rate and stress. Secondly, using equations (27) and (28), the
most probable a-values of the fourth to the eighth column of Table 1 are obtained.
This step adjusts the gradient of the dissipation function to the measured deformation
rates Vn according to equation (25). Thirdly, the coefficients a22 and a222 are
calculated using equation (26). This step adjusts the gradient of the dissipation
function to the condition of vanishing lateral movement.
At each step a mean error calculated by, equation (21) is associated with degree i
as a criterion for the quality of the approximation i.
Finally, the following remarks must be made:
The first step is necessary because the derivative of Vn with respect to anis
required. A direct numerical differentiation of the measured Vn would be too
inaccurate.
Only the first step represents an adjustment between measured quantities. The
normal equations corresponding to the second and third step are no more based on
real error equations.
In equation (27) it is also feasible to express dpD'/don in the derivatives of pD'
with respect to the first and second or the second and the third invariant. There may
be differences in the results depending on the quality of the approximation.
228
Bruno Salm
TEST PROCEDURE AND CHARACTERISTICS OF TESTED SNOW
Tests were carried out under uniaxial compressive and tensile state of stress and under
uniaxial state of deformation with constant principal stress in ^-direction. For the
measurement of the deformation rate Vn, mechanical dial gauges with a reliability of
±lCT 4 cm were used. In the case of uniaxial state of stress the volume change of the
snow sample was measured by means of the inflow or outflow of the air, with a
reliability of about ±l(T 2 cm 3 . In the uniaxial state of deformation the lateral stress
<?22 = ^33 was measured in addition to Vu. In this apparatus the tubular side wall
containing the snow sample, is split into two halves. One of these halves is horizontally
freely movable (x 2 -direction). Under an acting stress an a certain force is necessary
to hold the two sides together without movement. This force is measured with strain
gauges. Friction forces on the side walls are controlled and compensated (Salm, 1967).
The initial length of the snow samples was 10 cm and the diameter 5.8 cm for the
uniaxial state of stress and 8.2 cm for the uniaxial state of deformation. A preliminary
weighting (about 2 per cent of the maximum stress, in all cases a compressive stress)
caused a complete adaption of the snow samples' surface to the parts of the apparatus
which transmit forces.
The samples were taken from a homogeneous snow layer of the natural snow pack at
Weissfluhjoch and afterwards stored in the cold room for about 2 months. The
following characteristics were measured at the initial state:
Density: 432 k g m - 3
Hardness: 0.488 kpcm~ 2 = 4.787 x 10 4 Nm~ 2 (temperature: —5.3°C, instrument:
hardness gauge H 45 SLF) (kp = kilopond)
Air permeability : 51.6 m 3 s kg"1 (temperature: —5.3°C, apparatus: de Quervain—
Wolgroth)
Grain diameter: 0.123-0.155 mm
Void diameter: 0.334-0.481 mm (apparatus: tomograph (Good, 1974))
Tensile strength: 2.055 kpcmf 2 = 20.16 x l 0 4 N i r f 2 (temperature: -5.3°C)
Isotropy: Maximum difference of Vn between snow samples tested under uniaxial
compressive state of stress w i t h x r a x i s parallel and perpendicular to the natural snow
layer: 6.5 per cent
Temperature during the tests: —5.2 ± 0.3°C
The aim was to keep the change of bonding conditions (number of bonds per grain
and area of bonds) and the influence of metamorphosis of snow grains as small as
possible during the tests, as to go not too far away from the initial state. In spite of
test periods of only some hours and total deformations and density changes of less
than 10 per cent, the change of mechanical characteristics was surprisingly high, one
order of magnitude larger than for instance the change of density. Obviously
metamorphism is of no importance in this phenomenon. In accordance with results
of Kojima (1958), Yosida (1963) and Salm (1971) the following empirical relations
were used for the calculation of the initial state.
As far as the isotropic part of the deformation rate tensor is concerned the relation
can be written as
—
= - ^
expl-c
—
Index u means the initial state, c" is a constant, Ap is the change of
the initial density. The tests, however, clearly demonstrate that the
only governed by a density variation or by the isotropic part of the
tensor, but also by the deviatoric part of this tensor. Therefore, the
(29)
density and p u
phenomenon is not
deformation rate
total deformation
A constitutive equation for creeping snow
229
in ^-direction A/, which contains the deviatoric part, will be introduced as further
parameter in a second relation
_
=
_ e x p ( - c - )
(30)
c is a constant and lu is the initial length of the snow sample. An example for the
calculation of the initial state according to equation (30) is given in Fig. 2 and shows
a good agreement with the assumed relation. With equation (29) the quotients
[V22IV11V of the uniaxial state of stress and Xu of the uniaxial state of deformation
were calculated. Finally, equations (29) and (30) can be used also in the case of
general states of stresses. For a state of deformation satisfying Ap = 0 the two
equations are consistent, the same is true for the state of hydrostatic pressure as long
as the total deformations remain small.
The influence of stress history was investigated by four types of different sequences
of an which are schematically given in Fig. 3. The total duration of the tests varied
considerably according to Table 2.
FIGURE 2.
Calculation of the initial state of snow structure.
230
Bruno Salm
TABLE 2.
Total duration of the experiments
68.012
1675
Experiment
Duration (min)
68.013
701
68.014
1878
68.015
4690
68.016
3308
68.017
3384
RESULTS
All the calculations were made with a computer PDP-11.
Ex p é r i m e n t
B
Stress history
stress
M
mm j
68.012.1/2
Û
68.013.1/2
W777A
O
68.014.1/2
ezÊSsa
m
68.015.1/2
r7f7%foM»r,™
O
68.016.1/2
A
68.017.1/2
V///////A
1 2
3 4 5 6
degree (i-1) of the polynomial
FIGURE 3.
Adjustment of Vn in the uniaxial state of stress.
A constitutive equation for creeping snow
231
Uniaxial state of stress
The adjustment of equation (19) to the test results was made for each degree i of the
dissipation function up to the sixth. Figure 3 shows an optimum for degree i — \ = 2
or i — 1 = 3 , because the mean error remains approximately constant for higher degrees.
The fourth degree of the dissipation will be chosen because in the lower degree Vu
decreases for about an > 0.5 kp crrf2.
The mean error was furthermore separately calculated for the tests 68.012—68.014
and 68.015—68.017. The smaller error for each group compared with the error of all
tests gives the impression of two slightly different snow types.
Experiment
Stress history
\
y_22
V11
0
68.012.1/2
A 68 013.1/2
o
68 014.1/2
•
68 015.1/2
o
016.1/2
,W^%%%W,
rJ7>r^??^
A
i degree i-1 = 2
mean error rrtj_i
[1] I
0.2
0.15
68.012.1/2*68.017.1/2
1 2
3 4 5
degree (i-1) of the polynomial
FIGURE 4.
Adjustment of V2JVn in the uniaxial state of stress.
232
Bruno Salm
The adjustment of equation (l 1) to the results confirms the statement of the
general comments section.
In Fig. 4 the adjustment of equation (20) to the results is given. Here again the
fourth degree i seems to be the best.
The obtained coefficients for the uniaxial state of stress are given in Table 3.
Uniaxial state of deformation
Figure 5 represents the mean errors occurring with the calculations of the second
step and shows an effect of some importance. As by the uniaxial state of stress tests
the coefficients of lower degrees are already fixed, only terms of higher order are
remaining free. If the degree of the dissipation function is assumed with 4, then in
equation (27) a, and aXi and in equation (26) only a22 are the remaining free
coefficients. It follows that under more complex states of stress higher degrees of the
dissipation function have to be taken into account. Figure 6 demonstrates that at
least in the adjustment of equation (26) degree 6 has to be considered.
As a final check Vu and pD' were calculated with the resultant a-values, assuming
a degree i of 6 for the uniaxial state of deformation tests. The mean error referring
to the test results is about the same as obtained from the adjustment of equation (25).
m e a n error
m
|_fj_]
mmj
Adjustment
and
of E q u a t i o n s
(27)
(28)
10
degree
FIGURE 5.
Adjustment of the axial and lateral deformation rate under uniaxial state of
deformation. Mean errors as a function of degree i.
A constitutive equation for creeping snow
233
,1 PD' [ k p cm mm
Oil
kp]
cm2J
( l k p cm 2 =9.81»10 4 N m 2 )
FIGURE 6.
Dissipation function in the stress space for o12 = a33 = 0.
The obtained coefficients from the uniaxial state of deformation are given in
Table 3,
Final remarks
In Figs. 6 and 7, pD' is given as a function of the stresses an and a21 = a33. The
reliability is strictly confined to the tested range of stresses. An example where pD' is
beyond this range is given in Fig. 7. Here for au = - 0 . 3 kp cirf2 and a22 = "0.11 kp cm-2
the dissipation function becomes negative, which is of course impossible.
In the following Table 3 the utilized coefficients of the dissipation function are
given.
The establishment of a constitutive equation for snow, based on the principle of
least irreversible force, is possible by developing the dissipation function in power
TABLE 3.
Coefficients of the dissipation function (expressed in units
of kp, cm and min)
Uniaxial state of stress
an = 0 . 3 1 3 2 X 1 0 anl = -0.4040 X 1 0 a m l = 0.4586 X 1 0 -
a2 = 0.8023 X 1 0 a,2 = - 0 . 1 3 4 2 X 1 0 '
3
aln, = 0.2054 X10"
Uniaxial state of deformation
a3 = - 0 . 8 2 1 2 X 1 0 "
aJ3 = 0 . 5 8 6 7 X 1 0 a,„ =0.1427
«,„, = 0.1136
a
n3 = - 0 . 3 6 2 6
a,„ = 0.1789 XlO 2
^ 23 = -0.1150 XlO'
i n , 2 = - 0 . 8 8 3 3 X10" 2
y~ =0.4403 XlO 2
234
Bruno Salm
PD' [kp crrrmirVJ (IkpcrrV muï1 = 5.884x10° N m ' s 1 )
+0.05
( 1 kp c r n 2 * 9 . 8 h lO^Nm 2 )
FIGURE 7.
Dissipation function in the stress space.
series. Because of the distinct non-newtonian behaviour terms up to a high degree
need to be taken into account.
Acknowledgement.
The author is indebted to Mr M. Heimgartner for his worthwhile help
for the numerical evaluation of the test results (computer programs).
A constitutive e q u a t i o n for creeping s n o w
235
REFERENCES
Good, W. (1974) Numerical parameters to identify snow structure (Proceedings of the Grindelwald
Symposium, 1974), pp. 9 1 - 1 0 2 : this publication.
Kauzmann, W. (1941) Flow of solid metals from the standpoint of the chemical-rate theory.
Trans. Amer. Inst. Min. Metal. Engrs 143, 57—83.
Kojima, K. (1958) Sekisetsusô no nensei asshuku, IV. (Viscous compression of natural snow
layers. IV). Low Temp. Sci., Series A, 17, 5 3 - 6 4 .
Reiner, M. (1958) Rheology. Encyclopedia of Physics (edited by S. Fliigge), vol. VI, pp. 4 3 4 - 5 5 0 :
Springer-Verlag, Berlin-Gottingcn-Heidelberg.
Salm, B. (1967) An attempt to clarify triaxial creep mechanics of snow. Physics of Snow and Ice,
Sapporo, Japan, Institute of Low Temperature Science, pp. 8 5 7 - 8 7 4 .
Salm, B. (1971) On the rheological behavior of snow under high stresses. Contribution 1129.
Low Temp. Sci., Series A, 2 3 , 4 3 .
Yosida, Z. (1963) Physical properties of snow. Ice and Snow (edited by W. D. Kingery),
pp. 4 8 5 - 5 2 7 : MIT Press, Cambridge, Mass.
Ziegler, H. (1963) Some extremuum principles in irreversible thermodynamics with application
to continuum mechanics. Progress in Solid Mechanics (edited by I. N. Sneddon and
R. Hill), vol. IV, p. 9 1 : North-Holland Publishing Company, Amsterdam.
DISCUSSION
Mikio Shoda:
Why is there a difference of the viscosity in compression and in tension?
Bruno Salm:
In compression there is pressure melting at the bonds due to high stresses. Furthermore
the structure of snow, i.e. the 'lines' of grains carrying the main part of the stress do
certainly not behave in a same manner under compression and tension. Under the
latter state of stress these 'lines' are somehow stiffer.