On the Rammsonde hardness equation

H. U. Gubler

Abstract. A new Rammsonde equation is developed which better describes the real energy transfer from the sonde to the snow. The energy transfer has been measured with the help of a built-in force transducer and a penetration length measurement device. The well-known simple Rammsonde equation can easily be improved to account for the energy losses. The correlation between ram hardness and tensile strength is measured and discussed.

Resume. Une nouvelle équation pour la sonde de battage est développée, qui décrit plus exactement le transfert réel d'énergie de la sonde à la couche de neige. Le transfert d'énergie a été mesuré à l'aide d'un capteur de force et d'un engin mesurant la longueur de pénétration. L'équation simple et bien connue pour la sonde de battage peut facilement être modifiée afin que les pertes d'énergie soient prises en considération. Une corrélation entre résistance au battage et résistance à la traction est mesurée et discutée.

INTRODUCTION

The Swiss Rammsonde has been universally used for many years to measure the displacement resistance to penetration of the layers of a snowpack. The Rammsonde was developed by Haefeli in 1936—1937 (Haefeli, 1936). It is a simple, sturdy, portable instrument which can be used after only a little instruction. This instrument is suitable for snow hardness measurements from low to very high hardness, but it cannot be adapted for very low hardness snow. The hardness profile is measured from the highest layer down. The material displacing probe has the shape of a cone with a point angle of 60° (Fig. 1).

DEFINITION OF THE DISPLACEMENT-HARDNESS

Usually the displacement hardness, which has the dimension of a force, is given by the penetration of a harder material than the tested one whereby the shape of the probe and the total work performed during penetration have to be defined. In contrast to a tensile or a compression strength test the stress distribution caused by a hardness test is very difficult to describe. The definition of an instrument-independent unit for the hardness is impossible. Furthermore, if the material strength is a function of the deformation rate (e.g. in a partially viscous material), the penetra­tion speed has also to be fixed.

PROBLEMS OF A HARDNESS MEASUREMENT IN SNOW

The experiments show that the deformation mechanism responsible for the hardness depends strongly on the rate of deformation (Fig. 2). The displacement—hardness changes one to two orders of magnitude as a function of penetration speed.

APPARATUS

The Swiss Rammsonde consists of at least three parts: the sonde shaft with the probe, the hammer guide and the hammer (Fig. 1). If the thickness of the snow deposition

On the Rammsonde hardness equation 111

Hammer Guide

-Hammer

-Shaft-extension

-Shaft with Scale

tf-Probe FIGURE 1. The Rammsonde.

Pene­

tration

Speed

[m/s]

.058•1CT

.2.6-10"3

1.9-10-3

.65

norma!]

Raiiiteftt

stsrtj r.f.;

sj'-sed

.7 u/&

Mean

Hard­

ness

[N]

S 900

600

400

120

100

Highest

measured

Hardness

[N]

900

800

140

Variatio

Lowest

measured

Hardness

[N]

400

100

n 15 N

Timedejpendence

of the

Hardness R

R

tx, R£

R

R

Ï- — i - ^ t

I 1—_ t

-vr-V-'n

R

Total

Penetra­

tion

[lû~5m]

17

15

20

186

10 mra

FIGURE 2. Hardness and deformation mechanism as a function of penetration speed.

is larger than about 1 m, the shaft can be lengthened in 1 m steps. Shaft, hammerguide and hammer are not fixed together in their longitudinal axis but there is considerable friction between shaft and guide. Probe with shaft including guide, extension and the hammer each have a mass of 1 kg. Guide and hammer are made of stainless steel, shaft and extensions consist of aluminium tubes. The connections of the extensions and the shaft are not rigid. The impact of the hammer on the guide excites damped oscillations in the loosely connected parts of the sonde system. These damped oscillations transfer portions of the original available mechanical energy into heat. The experiments show that this accounts for a considerable overall loss of mechanical energy.

Investigations with a slow-motion camera show the following time sequence of the motion of the different parts for the first impact of the hammer on the combina­tion of shaft and hammerguide: (1) Hammerguide and hammer rebound from the shaft simultaneously and with the same speed. (2) The hammerguide slows down faster than the hammer (friction between guide and shaft) and falls back on the shaft where it rebounds again. (3) In the meantime the hammer also falls back and hits the guide again. The ensuing motion is a function of the initial speed of the hammer and the motion of the shaft after the first impact. The force and penetration measurements described later demonstrate that hammer and guide are reflected from the shaft

112 H.U.Gubler

several times and that the relative motions of the different parts never die out before the probe penetration finally stops.

To measure the effective force of the probe on the snow a quartz force transducer was built in between probe and shaft. A simple arrangement of a lab-fixed reticle and a sonde shaft-fixed light and photocell allowed the measurement of the probe penetration as a function of time (Fig. 3). The real energy transfer from probe to snow was easily calculated as the force integrated over the penetration path. The two curves, force as a function of time and penetration as a function of time, were displayed on a two-channel storage oscilloscope and could be evaluated directly or photographed with a polaroid camera. The fundamental frequency of the system excited by the impact of the hammer was about 2 kHz. This oscillation was registered by the force transducer and masked the interesting force—time curve. An active low pass filter (Fig. 3) reduced this oscillatory part of the signal. The resulting risetime for a step force measurement amounted to 1.5 x 10~3s. Figure 4 shows examples of combined penetration and force measurements for low and high hardness. The resolution of the penetration measurement is 1mm for normal ram tests and 3 mm for constant speed tests (0.1 —1.0m/s). A weighted cylinder was fixed to the shaft for the constant penetration speed experiments in this range. For the constant speed tests in the range of 10~3m/s the sonde was fixed to the piston of a constant feed press.

Amplifier

Amplifier Lowpass Filter

c t or a g e Oszilloscope

<3CH 1 •oCH 2

FIGURE 3. Experimental arrangement.

$s«{K>* ^-'ï&^j!.^''.

' ^ l& ' - f -V- . ' J

' * :•" "' r - i t^V,"?. • <"«"lîîV".* '

f.'iSrV-'ïl"-" 1 ' **-i* -v*/. ' • : r - ^ : - ^ J.

fl " Ï*" '

:*#3rv <&;«*; :<&&::

r v.

~.i . i

,« ! - -.: * #"* '-r - . •

' j , '"»

.' * * • •

; • * :

, \ " vô' f -C

FIGURE 4. Combined penetration and force measurement. Height of hammer fall, 0.2 m ; time per unit, 5 X 10~3s; force per unit, 50 N; penetration distance between two peaks (lower curve), 10"3m.

On the Rammsonde hardness equation 113

PREPARATION OF THE SNOW SAMPLES

The snow samples were prepared from new snow and fine-grained snow. Before the snow was sieved into wooden sample boxes (0.4 x 0.25 m, 0.28 m high) it was kept at —10°C for at least 24 h. The sieve (grid spacing 2 mm) was moved by a vibrator with an amplitude of about 2 mm at a frequency of 50 Hz. The sample box was moved slowly forwards and backwards below the sieve to get a regular snow accumulation. At regular intervals coloured powder was dusted over the snow to indicate horizontal layers in the samples. After sieving the samples were sintered for 2—5 days at — 10°C. Some samples were compacted slowly (strain rate: 0.2 x 10~3s~') immediately after sieving.

TEST PROCEDURES

For each experiment at least two samples were prepared under the same conditions, one of them with coloured horizontal layers at vertical distances of about 0.02 m. The hardness tests were performed in the coloured samples to investigate the resulting deformation of the snow structure. Generally three test holes were made per sample, the first one was the result of an ordinary ram test with different dropping heights of the hammer, the second and the third one originated from the hardness tests with high and low constant speed penetration. For the tensile strength tests cylindrical samples (diameter 0.058 m, length 0.188 m) were cut from the uncoloured sample. After the hardness tests vertical sections were cut through the sample holes. Some vertical section surfaces were sprayed with blue ink and afterwards heated with a blower. The remaining colour intensity distribution shows the snow density changes along the holes (Fig. 5).

RESULTS OF THE MEASUREMENTS

The penetration per blow (consisting of a series of impacts between hammer and guide) varied from 3 x 10~3m (hard snow) to about 2 x 10~2m (soft snow) and the penetration time varied between 5 x 10_3s and l C ' s . For the calculation of the mean ram hardness of a sample the tests between 0.10 and 0.20 m measured from tlie sample surface were taken into account. The estimated uncertainty of the hardness for a single measurement (reading error) is 10 per cent. The penetration is measured to 0.5 mm. Additionally there is a possible systematic error of about 7 per cent given by the calibration of the force transducer, the friction produced by the penetration measurement device and the Teflon guides for the shaft (Fig. 3) and the uncertainty of the zero line for the force. This leads to a mean error for about 10 ram tests per sample of 10 per cent. Calibration, zero line and friction were examined before a new sample was measured.

DERIVATION OF A RAMMSONDE HARDNESS EQUATION

For simplicity the usual equation used to determine the ram hardness neglects energy losses:

- E 1 R = - = - [MH.g.h + (MH+Ms+MG).g. s] (1)

' S [H P] R = mean ram hardness s = penetration g = gravitational acceleration

114 H

.U.G

ub

ler

I

On the Rammsonde hardness equation 115

h = height of hammer fall MH = mass of hammer MG = mass of hammerguide Ms = mass of shaft including probe E = total energy transferred to the probe

The first term [1] of the sum represents the kinetic energy of the hammer immediately before the first impact occurs, the second term equals the potential energy loss of the total system during penetration.

If one considers the system of guide and shaft as rigid, an expression including the effect of partial restitution of energy can easily be developed (Waterhouse, 1966). The resulting equation agrees with the law of conservation of momentum. The assumptions are: (1) The probe is able to move. (2) The probe remains loose in the hole. (3) The impact results in a reduction of the speeds of the two masses in the centre of mass system by a factor e (coefficient of restitution e « 1, equal _1 for elastic reflection). If one considers the energy losses in the first impact only, R is expressed as:

MH.g.h .MH(Ms+MG)(l + e)2

- E 1 R = - = -

MH + MG +MS

[1]

MH.g.h *, », „ ( M H - e M s) + (MH + MG +MS). g. s MH + MG + Ms

[2] [3]

(2)

Term [1 ] : Kinetic energy of shaft including guide after the first impact. Term [2] : Remaining energy of the hammer after the first impact. Term [3] : Potential energy loss of the whole system during penetration.

If shaft and hammer are both already moving before the impact occurs (e.g. before the second impact), the velocity of the hammer FoH after the impact, as a function of the velocities of the hammer VH and the shaft Vg before the impact, is:

F ° H = T, ZTT,77 K ^ H - e(Ms +MG)} VH +MS(1 + e) Vs] (3) AfH +MG +MS

With the help of the law of conservation of momentum the velocity of the sonde after the impact Vos can easily be calculated. Knowing the velocities of the shaft before and after the impact the energy transfer of the hammer to the shaft can be computed. If one develops a computer program that calculates penetration and penetration velocity of the two masses immediately before the impact takes place, one is able to calculate the total effective energy transfer per blow at least for the two-body problem.

The starting velocity of the shaft after the first impact is

V* = V(2*/0 M " , ' (4) MH+MS +MG

and it can be determined experimentally. More precise measurements result when the snow is replaced by a soft spring. These measurements lead to a mean value for the coefficient of restitution e for the Rammsonde of 0.2 (for a sonde system with fixed hammerguide according to equation (2) e was measured to be 0.52). The calculation of the total energy transfer according to equation (2) with e = 0.2 leads to far too high values. The reasons are: (1) The shaft and the hammerguide do not have the same start velocity. (2) The remaining energies of the hammer and the guide after the first impact are not transferred to the probe without losses.

116 H.U.Gubler

Using equation (2) the calculated energy transfer through the first impact may already be larger than the whole experimentally measured energy transfer. (The kinetic energy of the sonde after the first impact calculated in equation (2) is too high because the effective mass is only Ms and not (Ms + MG).)

The attempt to develop an equation which better describes the real physical process of the first impact leads to the following equation:

Vo* (MH+MG)(l+e) MH

MH + MG +MS

[3]

Vv MH + MG

[2] [1]

(5)

Term [1] : Velocity of the hammer immediately before the first impact. Term [2] : Inelastic impact between hammer and guide system. Term [3] : Partially elastic impact between the combination hammerguide and shaft.

The remaining velocity of hammer and guide after this first impact is:

Vo» = (MH + MG) - eMs Mh

MH+MG+MS MH+MG

Vu (6)

The total energy transfer to the probe per blow is:

E=R.s = tiMsVol +Ms.g.s+8[1ÂMliVoll+y1/2MGVo2H + (MH +MG).g.s]

[1] [2] [3] [4] [5] ( y )

Term [1] : Kinetic energy of the shaft after the first impact. Term [2] : Potential energy loss of the shaft during penetration. Term [3] : Kinetic energy of the hammer after the first impact. Term [4] : Kinetic energy of the guide after the first impact. Term [5] : Potential energy loss of hammer and guide during penetration. 5 = Efficiency of further impacts, y = Friction loss factor for the friction between guide and shaft (estimated value 0.25).

SIMULATION OF THE RAM TEST

The following rules for the calculation of the energy transfer through an impact gave the best fit to the experimental data: (1) Partially elastic impact between hammer and guide. (2) Inelastic impact between guide and shaft and (3) partially elastic impact of the hammer with the combination of guide and shaft. The parameters to describe the hardness of the snow as a function of the penetration velocity are defined in Fig. 6. For the snow type chosen in the described experiments e (defined in Fig. 6) is about zero. But it has to be mentioned that, for example, in wet snow e is not zero

R(v ) = y i

R(v) = a v v= v [ R L i R h . v ; VK ]

III R(v) = R „ • £ -v

FIGURE 6. Definition of the parameters to describe the snow hardness.

On the Rammsonde hardness equation 117

and this results in a dependence of the measured mean ram hardness on the dropping height of the hammer.

DERIVATION OF A SIMPLE RAMMSONDE HARDNESS EQUATION

Figure 7 shows that the efficiency factor 5 is only a function of h/s and remains constant for h/s > 40. Figure 8 compares the ram hardness calculated from

1,0

0,5

1

A

Jr""~—r

A n r * p

o h = 0,1 m x h =0,2 m A h = 0,3 m o h =0,4 m v h = 0,5 m

^ 1

h 5

40 80 120 160 200 240 260

FIGURE 7. Efficiency factor 6 as a function olh/s (equation (7)) for Afs = 1.18 kg MG = 0.313 kg, MH = 0.982 kg.

500

400

20 40 60 100 120 140

FIGURE 8. Ram hardness calculated from equation (1) , and equation (7) -compared with measured data.

118 H.U.Gubler

equations (1) and (7). The coefficient of restitution has been determined experi­mentally in. different ways by measuring the speed either of the shaft or the hammer immediately after the first impact or by determining the time until either the second impact or the first halt of the probe. The simplest way to correct the old ram hardness equation is to adjoin a coefficient c depending only on h/s to the first term in equation (1). c turns out to be constant within 5 per cent for h/s > 40 (Fig. 9).

0,3

0,2

i C

R

N

= C M H

<1M= M s

g | 3 + ih

= 1kg

•4H * M.J g

h s

10 20 30 40 50 60 70

FIGURE 9. Correction coefficient for equation (1) as a function of h/s.

HYPOTHESIS FOR A CORRELATION OF THE RAM HARDNESS AND THE TENSILE STRENGTH

For the tensile strength tests the well-known centrifugal technique was used. The angular acceleration was 0.3 turns.s-2 for all samples. The mean error of at least three tensile tests per sample is about 15 per cent. Figure 10 shows the ram hardness as a function of tensile strength. In Fig. 11 the ram hardness for compacted and uncom-pacted samples is plotted as a function of density. The tensile strength shows a similar dependence on density and compaction. The mean diameter of the grains (which may consist of several crystals) of all samples should be about the same because the same sieve size was used for all samples. But the shape of the grains is different depending on which kind of snow was sieved, low density samples consist of new snow, higher density samples of rounded grains. For low density snow the mean number of contact points per grain does not change very much and is not a function of density if the snow is not compacted. In contrast, the investigation of the thin sections prepared from all samples shows that the densification of the samples yields an increase in the number of active contact points per grain. For a given mean grain and bonding diameter the tensile strength is a function of the intergranular structure, which is a function of the mean active (stressed) bonding surface, and approximately linearly dependent on the density (for constant temperature and stress rate):

ot = cJ(SB).(8-c) (8)

S = density cx = parameter f(SB) = function of the active (stressed) intergranular bonding surface c = parameter, function of the mean grain diameter

The change of the time dependence of the hardness as a function of penetration speed (Fig. 2) leads to the following hypothesis for the mechanism of the displacement hardness. Step 1 : Breaking of the bondings between the grains in a region in front of the probe. Step 2: Densification of the loose grains against the friction between them (plastic wave front propagating from the sonde probe into the snow). The total work

On the Rammsonde hardness equation 119

1200

1000

200

FIGURE 10. Ram haidness R as a function of tensile strength oB .

A R [N]

18 20 22 0B [10' N/m2]

200

100

R-1281 A R-695A

1

§ S

V

P[l03kg/nf

0.1 0.2 0.3

FIGURE 11. Ram hardness .R as a function of density p.

consists of two terms, the first one describing the breaking of the bonds, the second one equals the frictional work during densification. Again the first term is proportional to the function/(^g) and approximately linearly dependent on the density, the friction term depends on the mean density during compaction.

R =-[c2f(SB).(8-c)+F(8)) s

(9)

120 H.U.Gubler

R 1 r . F(S) — = c2 + at s.cil f(SB)(8-c)

(10)

s = penetration c\ < Ci = parameters depending on grain structure, sintering time, temperature, stress rate -F(ô) = function describing the frictional work during densification

Figure 10 shows that R/at decreases with increasing densification. This leads to the conclusion that f(S%) is an increasing function ofS%.

SOME REMARKS ON THE RAM HARDNESS

Some vertical sections are shown in Fig. 5. It can be seen that the compacted volume per penetration length becomes smaller the larger the hardness. The fast penetrating probe performs inviscous elastic deformation, so that elastic energy is stored in the bondings until they simultaneously break in the stressed zone. This zone in front of the probe in which the breaking occurs throughout is smaller for harder snow (for larger bonding surface per unit volume). The descending probe compacts the resulting free grains. The sequence can be seen in Fig. 5.

CONCLUSIONS

Figure 9 shows that a simple ram hardness equation for the Swiss Ramsonde can be derived. Shaft, extensions and hammerguide of the Rammsonde should be fixed together. This would improve the energy transfer by 30—60 per cent and at the same time simplify the energy loss calculation. Also the reproducibility of the ram tests would be better.

For an improved understanding of the correlation between ram hardness and tensile strength, the dependence of the strength on the intergranular structure has to be investigated. A correlation without knowing further characteristics of the snow sample will not be possible.

REFERENCES

Haefeli, R. (1936) Beitrâge zur Géologie der Schweiz Geotechn. Serie-Hydrol. Lieferung 3. Waterhouse, R. (1966) Re-evaluation of the Rammsonde hardness equation. CRREL Special

Report 100.

DISCUSSION

L. Gold: Have you made measurements with a Rammsonde which has the hammerguide fixed to the shaft? If so, what were the results?

H. U. Gubler: I have made such measurements. For the system used (impact steel—steel) the coefficient of restitution amounts to 0.515 and the efficiency factor S is about 0.65 for a hammer mass of 1 kg and a sondeshaft mass of 1.2 kg. The overall losses of mechanical energy are significantly smaller than with hammerguide not fixed to the-shaft and the ram hardness depends linearly on h/s.

R. Kry: I would like to confirm a strain rate dependence of the Young's modulus of snow based on measurements I have made by a quasistatic method. If the surface effects

On the Rammsonde hardness equation 121

play a role as suggested by Dr Gubler in the hardness tests, this would indicate new grain contacts occurring during the course of a Young's modulus test. To evaluate this, especially for oscillation methods of measuring Young's modulus, the amplitude of oscillation must always be known.

H. U. Gubler: I think that measurements with monocrystalline and polycrystalline ice may also help to solve the problem for we do not know yet the temperature dependence of the mechanical properties of the interface layer between crystals of different c-axis directions.

R. A. Sommerfeld: The scatter of the tensile strengths was low. Is this typical of the artificial snow which you used?

H. U. Gubler: Of course snow samples have to be as homogeneous as possible. But I think a large part of the scatter arises because of a too high stress rate. The tensile strengths obtained with a high stress rate are very dependent on the stress rate itself; an asymptotic strength is reached only if the stress rate is less than about 150 Nra"2s"' for low density snow. I think most field experiments are performed with much higher stress rates.

M.Shodâ: I have made a measurement of the impact forces of snow blocks against a rigid plate comparing the values of the velocities before and after the impact. I found that the values of kinetic energy decreased by about one half.

Do you think that there is some correlation between the above-mentioned result and your result?

H. U. Gubler: I think not. This is a very different experiment. I measured the energy losses in the sonde system whereas you measured energy losses in the snow sample during the impact.

R. I. Perla: Dr Gubler's careful experiments have important implications for routine snowpack observations. The assumption of e = 1.0 which is used in the commonly accepted ram formula results in ram profiles that show more discrimination than may in fact exist in nature. Using a more realistic value, say e = 0.5, would therefore give more realistic ram profiles, at least from a qualitative viewpoint.

H. U. Gubler: For a Rammsonde system with fixed hammerguide a change of the coefficient of resistitution e only results in a scale factor, the uncertainty amounts in any case at least 10 per cent for a single measurement. I think the main difficulty with ram hardness measurements arise from the interpretation of the results and their corre­lation with other mechanical parameters of snow.