Annals qf Glaciology 26 1998
© International Glaciologica l Society
Elastic properties of snow-ice fortnations in their whole
density range
ANATOLY D. FROLOV, rGOR v FEDYU KIN
United Scientific Council on Earth Cryology of the Russian Academy of Sciences, Fersman Street 11, lv[oscow 117312, Russia
ABSTRACT. The available experim enta l data on acoustic pa rameters a nd d ynamic
elastic moduli of dry coherent snow-ice form ations in th eir whole density range are a nalysed and generalized in this paper. A set of critical densities at which there a re suffici entl y
sharp changes in the laws of th ese property variations have been discovered a nd are di scussed. The empirical equations describing the elastic-wave propagation velocities (Vp a nd
Vs ), acoustic resistiviti es and elastic moduli as functions of the porosity factor of the medium are obtained for fi ve sub-ra nges in the density range from light snow to massive ice.
INTRODUCTION
The structure- texture transform ations which occ ur during
dry-snow densification and di age nesis cause cha nges in its
aco ustic, elastic, strength and other properti es. The direct
experimenta l study of these transform ations is difficult
because data acc umulation occurs rather slowly. However,
up to the present time, considerable experimenta l data have
been obtained a nd some th eo reti cal co ncepts abo ut the
cha racter of the snow-structure re-organization have been
developed thanks to the work of many scientists. For
example, it has been known for some time that there are
two critical densities: 550 a nd 820 kg m- 3 (Anderson and
Benson, 1963). The first one cha rac terizes th e state of dry
dense snow with completely proxim ate grains, i. e. some limit of the medium rando m densification and secondly the
transition to ice with closed pores. It is also known that fresh
light snow is characterized by a very friable chained structure with a coordination number varying from 2 to 3. H owever, how the structural changes occur during a medium
densification from a light to a dense state and at what density the new dominant space order a rises rema ins unclear.
Are snow-property changes monotonous during densification? Known critical densiti es (mentioned above ) testify
to none of these monotonous processes. It is ha rdl y possibl e
to imagine an identical and continuous mechanism of snow
densification from a light (p '" 30- 50 kg m ~) to a dense
(p '" 550 kg m 3) one. H owever, until recently, there were
practically no data about critical densiti es in these snow
states. Only in a few papers (for instance, Samoylyuk, 1992;
Voronkov and Frolov, 1992) were there indications of an opport unity for critical densities below 550 kg m 3. It is obvious that the problems of critical densities a re important
for the development of the concept of snow mechanical
p roperty-fo rm ation processes and laws for its changes. They
are closely related to the snow- tructure transformation regularities of importance in snow-cover strength and estimates of bearing ability, achi evements of explosions in
snow, registration of aco ustic signals from various moving
a nd immobile sou rces of elastic oscillations, etc. Moreover,
the acoustic a nd elas tic parameters of porous media a re
very sensitive to their structural organizations. This refers
to the elastic waves which propagate in the skeleton of the
porous med ia (the waves of first arriva l), with velociti es
determined by the compressibi lit y of both th e ice grains
a nd the pores (Bourbie and others, 1986). The waves of the
second kind (air waves) have not been considered.
Therefore, we have underta ken the pre ent investigation
to a nalyse in detail the avail able seismo-acoustic data in order to reveal th e m ain regul arities of the tra nsformati on features of the mechanical properties in the who le density
range of snow- ice form ations.
BACKGROUND
The experimental stud y of illler-relations between acoustic
properties, elastic moduli of snow- ice formations and th eir
densities have been the objective of many researchers
(Bentley, Brockamp, Chernigov, Durynin, Kohnen, Maeno,
Pounder, R obin, Smith, Yamada a nd others ). Reviews of th e
res ults obtained by these authors, given by M acheret (1977),
Melior (1977), Voitkovski y (1977), Bogorod ski y a nd Gavril o
(1980) a nd Paterson (1994), co ntain the ex perimenta l data
and empirical formul ae, which have been proposed to describe th e changes in the aco ustic parameters with in some
density intervals. H owever, a uthors have usua lly aimed to
present laws in a form of co ntinuous (linear or non-linear)
functions and spread them over a wide density range. It is
unlikely that by using this expedient general regulariti es of
aco ustic- and elastic-property changes can be found. In addition, most of studies have been conducted using snow of
densities of 350- 400 kg m- 3 a nd higher.
The properti es of friabl e snow have been poorly investigated. Thi s is due to the difficulties of their stud y by ultraso ni c
methods, because it is necessa ry to remove the sensor 's dimensions and to create a perfect contact between the sensors and
the snow. In our experiments, in order to ensure perfec t contact we used both special synthetic oils a nd ice bridges.
Th e m ai n diffic ult y in ge nera lizing from the differen t
data is th e difference between experime11la l proced ures
and that none of the snow- ice media studied are identical
or even at the same or similar densities and temperatures.
For example, in M elior's review, the ex perimental d ata of
55
Frolov and f<""'edyukin: Elastic properties rifsnow-iceforma tion
4000
...
~
i
3000
XI
I
~
~
U
0
"ii
"
I
x
2000
•
I
.
x. "
x
x
1000
ijj
*
* "
0
200
"
-:
Kohnen ,1972
x Benne! , 1972
" Our data
XI
~
u
I
.. '"
Xl
>
~
...
'Il"
0
0"
0
400
600
800
1000
Snow density (kg m- 3)
Fig. 1. Longitudinal wave velocity
different authors.
V;, vs snow density from
many authors have been presented in the form of intervals
(bands ) of values, the width a nd representativeness of whi ch
are difficult to estimate.
Therefore, we have selected as the basis for analysis the
seismic fi eld data for the elastic-wave velocities of Bennet
(1972), K ohn en (1977) a nd K ohnen and Bentley (1973) for
Antarctica a nd Greenl a nd (snow densit y 350 kg m 3 and
higher ) a nd our partly publish ed (Savel'yev a nd others,
1967; Frolov, 1976) data obtain ed in the Kolskiy Penin sul a
(snow density from 30- 50 to 500 kg m 3).
Our data for light snow were obtained by an ultrasonic
pulsing method (with the P- and S-wave sensors of a dominant frequ ency of 50- 100 kH z) in special pits dug in the
snow cover. The measurements were achieved by profiling
(with a specia l sensor-assembly plan) a nd transm itting
technology. The necessar y acoustic con tact between the sensors and the grains of snow were ensured by using silicon oil,
glycerine or by freezing. The measurements were repeated
30-50 times for those layers with a close density, age a nd
structure, a nd for each such series the m ean (the most probable) values of elastic-wave velocities were determined. We
did not ta ke into account the dependency of the acoustic
parameters on the temperature and humidity of the snow.
All the data presented below correspond to dry coherent
snow at temperatures near - 10° to - 15°C.
The first comparison of these data has shown a very good
convergence of the elastic-wave velocity values obtained in
the overlapping interval of snow density (Fig. I) a nd has
revealed a new critical density P cv 330 kg m - 3 (Voronkov
a nd Frolov, 1992). The dependency of the velocities Vp a nd
Vs on densit y appeared to be linear with a different slope in
the various intervals between the critical densities. Compa rison with other published data (Macheret, 1977; M elior, 1977;
Voitkovskiy, 1977; and others) has not revealed any important contradictions.
On the basis of the above-mentioned experimental d ata
on elastic velociti es a nd densities, we have calculated (using
as a first approximation the known formul ae from the th eory of elasticity of continuo us media) the values of the acoustic resistivities Zp and Zs and Young's elastic modulus E,
shear modulus C, bulk modulus K, Poisson's ratio v and
L ame's constant ..\ as param eters for snow- ice formation.
Statistical analysis of th ese data led us to conclude that
snow- ice formation acoustic- a nd elastic-parameter changes
56
in the whol e density range cannot be described satisfactorily
by a sp ecific smooth function for density (or porosity),
including a polynome of a reasonabl e degree, up to one-fifth .
The avail ability of 'ufficiently sharp cha nges in th e density dependencies of aco ustic para meters refl ects the fact that
during snow densification and metamorphism the structuretexture re-organization has different prevailing mechanisms
of transformation in the various density intervals. The transitions of a medium from one state to another (which can be
considered as analogues of structura l-phase transitions of
the second type ) should take place near some critical values
(narrow intervals) of density. At these densities, the degree of
structure- space order (structural entropy) and rigidity appear to reach some bounda ry values, which lead to replacement of the dominant mechani sms and regul arit y of
mechanical-property changes. H owever, representation of
experimental data as a function of snow density P or poros ity
P allows one to reveal the boundary states of structural transform ations only to a rough approximation. Therefore, we
have undertaken a search for a new parameter of compari son
which is more sensitive to snow-medium structural changes.
RESULTS AND DISCUSSION
We have established that the best parameter of comparison,
which enables the revelation of the boundary states of snowstructural transformations more precisely and approxim ates
the laws of acoustic- a nd elas tic-property changes, is the porosity factor K p of a m edium:
Kp
= Vpor/Vsol = P /( 1 - P ) = (Pi - p) / p
where Pi = 917 kg m - 3 is the density of massive ice, Vpor and
Vsol are the volume of pores and the solid part of a medium,
resp ectively. For snow- ice formations, the value of K p lies
within the limits of 0 to 30. According to the theory of g ranular media, the porosity factor K p correlates with the grainsize and the corresponding pore volume. So, this para meter
is linked more to structura l changes than to density or porosity. Th erefore, furth er a nalysis and a search for critical
densiti es have been made on the basis of the snow- ice aco ustic and elastic properti es as a functions of K p. This has been
achi eved for the m ean experimental d ata on elastic-wave
velocities 11;, and Vs, and its squared ratio A = (Vp/Vs)2,
and also for a ll dynamic moduli of elasticity calculated from
these parameters.
For example, as one can see from Figure 2, the Poisson's
.35
.30
~
0
.25
1::
.20
"0
.15
~
..
...
0
c..
.10
.05
850
550 350
150
0
.001
.01
.1
1.0
10
100
Porosity factor Kc and corresponding density (kg m- 3.)
Fig. 2 Poisson's ratio vs the logarithm qfporosityJactor K p.
Frolov and Fedyukin: E lastic properties qfsnow-iceJonnation
o
107
ZP ",--
.. ,..
~
.s
<{
~
'0
iD
CD
CD
~
CD
.,
..,c
-1
f!
u::
iii
c
'6
c
0
~
.<:::
10 5
c
c
'6
10 4
vp"
<:::-
on
850
550 350
150
-2 t------.----~._----L,~--~~----_r--
.001
.01
1.0
.1
10
100
Porosity factor Kp and corresponding density (kg m- 3)
~
C
0
<::>
:l:' 103
0;
iii
.
'0
..
on
;;;;
Fig. 3. FiTSt derivative ifparameter A vs porosityJacto r K p.
Cl
u
;;;
u
0
0;
.-/
ZS
~
.." ..
. .."
Cl
106
Cl
. ....
§- ..,
.
~
-:;;
la
..,~
'"'E
/
Vs
~
!!
u
1ii 102
0
"
~
>
ratio dependence as a function of the logarithm of K p has
severa l m a rkedly sharp cha nges, within intervals between
which the dependence is near to linear. A simil ar situation
has been obtained for the oth er acoustic a nd elastic parameters for snow- ice formati ons. But, for Poisson's ratio,
which d epends highly upon the structure of the medium
a nd is very sensitive to its tra nsform ation, the cha nges stand
o ut m ost clearly a nd show a di stinct possibility for pointing
out the critical densities.
For the purpose of refining the va lues of cri tical densities, we have a na lysed th e first a nd second deri vatives of
the aco ustic a nd elas tic pa ra meters as functi ons of K p. Figure 3 shows a n example of such a dependency.
Th e critical densities have been identifi ed on the basis of
the study of these derivative behavi oural d ata. We have identified th em as the na rrow density intervals corresponding to
cha nges in th e dominant mecha nisms of the snow- ice medium structural transformati ons which occur du r ing densification. The cri tical densities obta ined a re:
P crl
= 130 - 160kg m - 3 ,
Pcr2
= 330 - 360kg m - 3 ,
Pcr3
= 530 -570kg m - 3 ,
P cr4
= 820 -840kg m- 3 .
The behaviour of the derivatives between 550 a nd 800 kg m 3,
as well as the results obtained by other auth ors (Maeno a nd
oth ers, 1978), give some r eason to ass ume th at the density
Pcr5 = 700- 730 kg m - 3 is also critical; however, this fact
requires furth er ve rification.
Th e critical-density values, evaluated from the a nalysis
of th e behaviour of different parameters, differ somewha t
within the limits of the above-mentioned intervals. Thi s is
because cha nges in the domin ant mechani sm s of structuretexture transform ations and formation of a new space order
within the m edium, occur in somewhat (na rrow) interva ls
of density, a nd a furt her refining of the cri tical densities
becomes physically unreaso nable.
Our analysis has shown that, in the sub-ranges limited
by the critical densities, the dependence of the acoustic
p a rameters a nd alm ost all the elas tic moduli on K p can be
described in do uble loga rithmic scale by lin ear functions
with a good degree of accuracy (th e root-mean-squ are error
o f approxim ati on does not exceed the error of the experimenta l d ata ):
log Y =
Yl X
log K p + Y2
850
10 1
.001
.01
550 350
.1
Porosity factor
150
1.0
10
100
Kp and corresponding densHy (kg m- 3)
Fig. 4. Dependences ifprojJagation velocities VI' and Vs and
acoustic resistivities Zp and Zs qf elastic waves in snow-ice
media on the /JorosityJactor K p. Vertical dOlled lines mark the
critical densities.
a re: propagation velociti es of longitudina l V;) and shear Vs
elastic wavcs in m s- [ (Fi g. 4); aco ustic resisti vity longitud ina l
Zp a nd shear Zs, kg m 2 s- [ (Fi g. 4-); elas ti c mod uli : Young's E,
shear C, bulk K and Lame's '\ in M Pa (Fig. 5).
For Poisson's ratio v a nd pa rameter A, th e best approxim ation is obtained by using a half-logarit hmic scale (Fig. 2):
Y = Yl
X
log K p + Y2 .
(2)
Th e values of approx im ate factors (coeffi cients ofEqu ati ons (1) and (2)) for each of the fi ve density sub-ranges a re
given in Table I (see the units of parameters above in th e
tex t). It is necessa ry to note, th at near the critical densities
the processes of formati on of snow- ice medium mecha nica l
properties a re so complicated that their desc ription inj oints
oflin ear in ter vals is m ore co mplicated a nd the errors of ap105
E ___
104
G-;-
104
0..
0..
K -..
103
'A ---
..
-CD
E
11
-'
'0
c
:;
10 1
10 1
100
Cl
c
>-"
0
'0
0
E
..
u
'lii
'0
0
E
C
:;
'§
.!!
.
'0
lU
10 2
.><
10 2
103
.
:.:
..,
~
CD
~
"'\
..
'-'..
-:;;
~
10- 1 iii
100
iii
III
iii
10- 1
10-2
.001
.01
.1
1.0
10
100
Porosity factor Kp and corresponding density (kg m- J )
(1)
where Yl and Y2 are factors of approx imation.
The Y-averaged values for studying medium pa rameters
Fig. 5. Dependence qfelastic modul£ qf E , C , 1< and ,\ on the
pomsityJactor K p.
57
Frolou and Fedyukin: Elastic properties ofsnow---iceformation
Table 1. Values ofapproximatefactors ofEquations (1) and (2)
Density sub-ranges (kg m" )
Parameters
50- 150
V;,
VPI
v p~
V.
VSI
vs:!
Zp
ZPI
ZP2
ZS
ZSI
ZS ~
E
el
e1
G
K
gl
g2
kl
k2
A
11
12
1/
nl
11 2
A
al
al
- 0.9646
3.0981
- 0.9235
2.8845
- 1.87 12
2.9209
- 1.8307
2.7079
- 2.8270
2.9940
- 2.7476
2.5891
- 3.0159
2.7683
- 3.6582
2.7717
- 0. 1596
0.2322
- 0.4199
2.6352
150- 350
- 0.8152
2.9975
-0.6752
2.7175
- 1.5485
2.7058
- 1.4078
2.4255
- 2.1955
2.5758
- 2.0883
2. 1454
- 2.6581
2.5420
- 3.2422
2.5106
- 0.2979
0.3417
- 1.7661
3.5160
350- 550
550- 800
850- 917
- 1.5891
3. 1616
- 1.5651
2.9176
- 2. 1376
2.8216
- 2.1148
2.5777
- 3.7397
2.8984
- 3.7355
2.4989
- 3.8573
2.7389
- 3.8864
2.5269
-0.0364
0.257 1
- 0.3542
3.0765
-0.2530
3.3373
0.2031
3.0990
- 0.5023
3.0451
- 0.4522
2.8068
- 0.6826
3.3024
- 0.6588
2.9039
-0.8320
3. 1221
- 0.9201
2.8020
0.0769
0.2486
- 0.7728
2.9802
-0.0326
3.5306
- 0.0228
3.2524
- 0.0630
3.4279
- 0.5318
3.1499
- 0.0797
3.8195
- 0.0760
3.4022
-0. 1083
3.7536
-0. 1201
3.6015
0.0105
0.3093
- 0.1752
3.5878
proximation inc r ease. Moreover, th e ch a nges of depende n ce
a t c riti cal densiti es are not uniform a nd equall y clear. For
example, as o n e can see from Figure 4, the ch a nges of elastic
moduli
E, G, K
n ear Pcrl
=
130- 160 kg m - 3 are slig ht but
nevertheless th ey exist (see Table I).
Apparently, there is a sufficiently diffe re nt influence of
snow st ructure-tex ture transformation on the diverse param ete rs of the m edium e las ticity, es p ecially in friabl e snow.
Thus, the c h anges in Poisson's r atio at a critical d e n si t y Pcrl
are more di stinc t (Fig. 2), than for oth er elastic moduli.
Model calc ul ations (Golubev and Frolov, 1998) h ave
shown that the a bove-me ntioned critical densities correspond to snow- ice m edium-g rain packings with increasing
domin a nt values of coo rdination numbers: 3, 4, 6 a nd 1012, an d 8 for Pcr5. These good correlatio n s co nfirm the objectivity of the c ritical d ensities. As a result, th e numb er of c ritical d ensiti es, determin ed or co nfirmed by o ur analysis,
div ides all vari e ty of dr y coh er ent sn ow- ice fo rmati o n s on
th e uccessio n o f the transition states with different dominant mechani sm s of stru c ture- texture transformations
during the m edium d en sification. The essence of these m ech an ism s is not yet clear and requires further studies.
The set of critical d en sity for dr y co h ere nt snow- ice form atio n s h as first a llowed us to d escribe without co ntrad iction the m ai n regul ariti es of th e ir aco ustic- and elasticprope rt y changes in th e whole density r a n ge. A sim ila r
a pproac h could b e ex tend ed to th e st re ngth and other physical properties of snow m edia.
CONCLUSIONS
Th e a n a lysis d ealing with p eculi ariti es of acoustic a nd elastic properties of the snow- ice form a tions leads to the following co nclusions:
1. The law of changes of acousti c p aram eters a nd e lastic
moduli of dry snow- ice media from light snow to massive ice cannot b e d escribed by the universal continu o us
function of d e nsity or porosity with an identical e rror in
their whol e range of d e nsities.
58
2. There is a set of c ritical d e n sities (",,150, ",,340, ",,550,
"" 720 and ",,830 kg m 3) limiting th e density intervals
in whi ch a m ed ium-structure re-organization is co nditioned by different domin ant mechanisms of d en sification of the snow- ice formalions.
3. The b es t co mparative p a r a m eter for the d escriptio n of
the laws is a porosity factor K p. The base regularities of
the aco ustic-para m e tel-s a nd e lastic-modu li c h anges can
be a pproximated by simil a r smooth functions of K p
within th e d ensity intervals, which a r e limited by th e
critical values.
4. The values of th e a pproxim ation coefficients for a ll acoustic and clastic paramete rs are d e term ined in each a llocated interval of d ensity that a llow prediction of these
prope rti es for dry snow- ice formation's known d ensity.
ACKNOWLEDGEMENTS
The studi es described in thi s paper w e re achi eved partly
b eca use of th e fin ancia l support of the International Scientific Foundation (ISF ) a nd Russian Governme nt Uoint
g ra nt NJ5AI00), to which the a uthors are grateful.
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