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A MICROMECHANICAL VIEW OF THE FRACTURE TOUGHNESS OF ICE
W. Nixon, E. Schulson
To cite this version:
W. Nixon, E. Schulson. A MICROMECHANICAL VIEW OF THE FRACTURE TOUGHNESS OF ICE. Journal de Physique Colloques, 1987, 48 (C1), pp.C1-313-C1-319. �10.1051/jphyscol:1987144�.
�jpa-00226290�
JOURNAL DE PHYSIQUE
C o l l o q u e
C1, suppl6ment aun o
3 , Tome 48,mars
1987A MICROMECHANICAL VIEW OF THE FRACTURE TOUGHNESS OF ICE
W.A. NIXON and
E.M.
SCHULSONThayer School
of
Engineering, Dartmouth College, Hanover, NH 03755, U.S.A.R6sum6 - Des e s s a i s de tenacith ont 6t6 r 6 a l i s 6 s avec de l a glace d'eau douce en fonction de l a t a i l l e de grains, de l a temperature e t de l a v i t e s s e de chargement.
Les r e s u l t a t s montrent que l a t6nacit6 augmente avec l a temperature e t d6croit.avec l a t a i l l e des grains. La t6naciot6 e s t constante avec l a v i t e s s e de chargement ( 5 ) au dessus d'une v i t e s s e c r i t i ue
Ktqui depend de l a temp6rature. En dessous de
Ktl a t6nacit6 augmente lorsque (R decroit. Les r6sultat.s sont i n t e r p r s t 6 s 1 l ' a i d e dvun modsle microm6canique.
ABSTRACT
Fracture toughness tests have been performed on freshwater ice as a function of grain size, temperature, and loading rate. Results indicate that toughness increases as temperature and grain size decrease. Toughness is constavt with loading rate (K) aboye a critical loading rate, kt, which is temperature dependent. Below Kt the toughness increases as K decreases. The results are explained in terms of a micromechanical model.
INTRODUCTION
Under certain conditions of strain rate, temperature and grain size, both the tensile and compressive strength of ice may be controlled by crack propagation [I-51. From this it may be inferred that for a complete understanding of the strength of ice, its resistance to crack propagation must be understood. In particular, a micromechanistic description of the effects of grain size, temperature and loading rate is needed.
Many workers [6-161 have measured the fracture toughness of ice. However, only empirical descriptions have been given and the results are marked by scatter. Thus it has not hitherto been possible to distinguish trends unambiguously.
This paper presents fracture toughness, KI,, values for freshwater ice obtained over a range of grain size, temperature and loading rate and develops a theory, based on micromechanical behavior,.
PROCEDURE
The specimens were equiaxed, randomly oriented aggregates of freshwater ice Ih made in a mold of 91 mm diameter. The specimen length was 231 mm. Samples were frozen radially inwards while the system remained open to the flow of water, according to the method described by Lee at al. [17]. Because the earlier work of Nixon and Schulson [IS] had shown a grain size effect on toughness, samples were made with similar grain sizes. In practise this meant that seed grains from three adjacent sieve windows were used. These gave grain sizes in the ice (as measured by the linear intercept method) of 2.1 + 0.2 mm, 2.5 rt0.2 mm, and 2.8f 0.2mm respectively. While the use of different grain sizes contributed to scatter in the result this approach expedited specimen manufacture. Speci ens had an average density at -S°C of 916
3 'g
kg
m-and the melt water conductivity at 20°C was 8.8 x 10' mholcm.
The test specimen was a circumferentially notched cylinder. All samples were notched at -12°C on a lathe and then allowed to equilibrate to the test temperature for 16 hours. Notching was done with a special tool (of half angle 22") to a depth of 9.15 mm, then a lathe mounted razor blade was used to sharpen the notch a further 0.25 mm. A fresh blade was used for each specimen. This method of preparation appears to give a repeatably sharp notch [15].
Tests were performed at five temperatures (-2°C to -50°C) and at up to five different loading rates at each temperature (0.01 to lo5 k ~ a d m s-I). A total of 20 conditions were investigated, of which 11 were reproduced.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987144
J O U R N A L D E PHYSIQUE
Tests were performed on a 0.5 MN MTS (servohydraulic) testing machine situated in a old room.
The room temperature is controlled to + 0.2OC. For the high loading rate tests
(>10kPadm s- f
)a special high flow rate valve (90 gallonslminute) and accumulator were used to power the machine and data were recorded on a 20 MHz storage oscilloscope. At lower loading rates a conventional valve powered the machine and ar, x-y plotter was used.
For the cylindrically notched specimen the relationship between KIC and P, the peak load, and specimen geometry is given by [18]
K,
=Y.PD
where (1)
Y
=1.72 - D -1.27 (la) D,
and where D, Dn are the original and notch root diameter respectively.
RESULTS
Figures la-e show the results. There is some scatter, and this may be due in part to the variation of grain size between different s m l e s . The maximum variation from 1.9 to 5.0 mrn grain size would be expected to produce a variation in toughness of approximately 10% based on 1151. With one exception (discussed below) the results are reproducible to within 10%. Figures la-c (for -2, -5, -lO°C) show that the toughness is constant at high rates but increases as the rate drops below some (temperature dependent) critical value. While there are insufficient data to make such a statement for the low temperature results (figures Id-e), th behavior does not contradict the trends observed a t t h e higher temperatures.
Figure 2 shows the variation of toughness with temperature. Only rate independent values are shown. Although unusual, the toughness increases with decreasing temperature. Similar behavior has been noted before [6,10]. The increase i s slight, from a mean of 7 7 k P a d m at -2OC to a mean of 112 k ~ a d m at -50°C. The high value of
152kPadm at -50°C and the low value o 8 0 k ~ a d m at -2OC and 0.01 k ~ a d m s- f
are strange. These results are not thought to be true toughness measurements for reasons discussed below.
LOAOlNG RATE, k . kPo Jm-1
Figure la: The variation of toughness with loading rate at -2OC
LOAOlNG RATE, K, ~PO&;S-I
Figure lb: The variation of toughness with loading rate at -5OC
DISCUSSION
In attempting to understand the fracture
t o u g n e s s of ice i t i s instructive to ! 't
consider the work on rocks and ceramics (Table
1).The tensile strength of ice is
closer to rocks than ceramics, while the
4 0fracture energy of ice is 1 to 2 orders of
2 0magnitude lower than that for both rocks
and ceramics. We may surmise that if the
0.01
Figure lc: The variation of toughness with loading rate at -lO°C LOAOlNG RATE. k . kPo K m s - '
Table 1: Comparison between Mechanical Properties of Ice, Rocks and Ceramics
Elastic Modulus Tensile Strength Fracture v g y Surface nergy
(Gpa) ( m a ) J/m Jlm ??
I422
Freshwater 11.8[19] 0.5 -1.4 [3] 0.5 0.1 [26]
&& Limestones 11
-33[331 1.8
-5[33] 15 - 851331 0.5[34]
Sandstones 2
-3 [33] 0.3
-8[33] 20 - 275[33] 0.5[34]
Granite 63 [25] 100 [25]
280
-400[35] 150
-350[31] 30 - 110[31] 11271 125
-175[29] 100
-275[29] 8
-12 [29]
350[281 100
-250[28] 30 [28] 2[281
"fracture energy sinks" in ice are similar t o those in rocks and ceramics they will be much less in evidence than in the other two classes of materials.
Grain size Effects ""
Nixon and Schulson
[15]found a grain size (dl dependence of fracture toughness which they ex- pressed as (note that
inreference
[15]this equation is given incorrectly, with the constants transoosed)
twhere KIc is in k ~ a d m and d is in mm.
They chose the -0.5 exponent for d not because it gave the best fit (the grain size range was too small for any significant difference to be evident in r2 values for exponents between -1.0 and -0.3) but because a similar dependence had been found in other materials [20,21].
Such a dependence may arise in a number of ways. First, machining damage may occur at the notch tip.
Inceramics such damage is corrected for by increasing the effective crack length. If we assume that microcracks, proportional to the grain size [3], form as a result of machining damage, then our crack len th would be increased by an amount $.
We may calculate j3 by minimising the standard deviation of the toughnes results found at -lO°C and 10 kPa dm s- ' I?
(i.e. by assuming that there is no grain size effect and by choosing j3 so as to minimise scatter in the earlier [15]
results). If this is done a value of j3
=0.64 is found. Thus machining cracks in the largest grain sized samples (d
=8mm) should be
=5mm long. Three such large grained samples were notched in the standard way to see if such cracks were present. The samples were not tested, but cut into thick sections
(=15mm thick).
Each contained a notch which was viewed carefully under many different lighting conditions. No cracks were seen. This suggests that the grain size effect is a real materials effect and not an artefact.
LOADING RATE. I?, kpo&s~;ls-l
Figure Id: The variation of toughness with loading rate at -20°C
001 o I I o 10 102 10' 104 105
LOAOING RATE, K. hPa &s-1
Figure le: The variation of toughness with loading rate at -50°C
C1-316 JOURNAL
DE
PHYSIQUEA second possibility is that the ice experiences a specimen size effect. If this were so then we would expect the toughness to increase with grain size. Again the observed trend is contrary to this expectation.
An explanation based upon crack tip creep is unlikely. We can represent the fracture energy, Gc, as
GC =
2 ~ s
+%a (3)
where ys is the solidivapour surface energy of ice and ydam is the energy dissipated by other micromecXanisms. For ice (see table 1) yd , and 2 y,, are of the same order of magnitude. If ydam arises from creep deformation around the crack tip then the toughness would be expected to increase with increasing temperature. The opposite trend is found.
This leaves the possibility of an athermal damage process . For rocks [22] and ceramics 123,241 microcracking may cause toughening. For rocks [22] the toughening may arise simply from the extra surface area. For ceramics, two other mechanisms are suggested[23], crack tip compliance and microcrack dilatation. Both compliance and dilatation toughnening require a high microcrack density to become effective.
The question ~ s e s : If we assume microcracking is the toughening process in ice, which of the three mechanisms outlined above is most applicable? It is useful at this point to compare for ice, rocks and ceramics the ratio of ydam/ 2 ysv . For ice this has a value of -2, for rocks -25 and for ceramics -35. Thus we would expect fewer microcracks in ice than in rocks or ceramics, and thus should not consider, at least initially, processes relying on high microcrack density such as the compliance and dilatation mechanisms. Thus the model which we favor assumes that the fracture energy sink arises essentially from the energy required to form new microcrack surfaces.
Microcracks in ice are proportional in size
tothe grain size,
d[3]. Thus the
energy arising frommicrocracks formed per unit area of crack advance is:
where N is the number of microcracks per grain, fi is the constant of proportionality between crack size and grain size, and 1 is the ditance from the crack flank at which microcracking will occur. Equation 4 is similar to that given% c e et
a1[24], which shows the same lid d e p e f p of energy. If yd a lid then since KI, a G, Yi%e would expect K p to be proportional to d- and thus, as observe#5]:
If we consider the ratio of yd 12y at a grain size of lmm we fin from equation 2 a value of 2.8. 4
Assuming that P is of order 1 we%us klve a value of (N x 1
)of
=10- m. From [I] a suitable value of shear stress would be 0.25 MPa. For an applied K1 of 100 d a d m the calculated value of $J would ne
=10mm. This in turn suggests a value of N of
;-0.1; i.e. 10% of the grains within Id would contain a microcrack. These values do not seem unreasonable.
A more exact estimate of K can be made. If we hypothesj75 an imaginary polycrystalline specimen (that is, not a single crystal) wit% infinitely large grains, the d
-term in equation 5 would disappear and there would be no microcracking. KO is thus the toughness arising from the macrocrack surface energy.
Hence
/ \
112
(6)
1 - v2
Taking E
=11.8 GPa [26] and v
=0.3 we find KO
=50.9 kPa dm. This is very close to the value [IS] of 58.2 k ~ a d m .
Temperature Effects
The variation of toughness with temperature rules out the possibility of creep deformation as an important fracture energy sink. We can achieve a partial explanation of the variation by considering how E and y vary with temperature. In general it may be assumed that ys, a E and that E varies linearly from Eo at bk to E0/2 at 273K. Using these values we find an expected increase in toughness of - 18%
between -2°C and -50°C, whereas we found an increase of - 40%, twice what was predicted. However,
other factors may also be influenced by temperature, such as the stress for microcrack initiation. Also
important may be the presence [26] of a semi liquid surface layer which decreases in thickness as
temperature decreases, finally disappearing at about -40°C. This may explain the curvature in figure 3 at
the higher temperatures. Finally, scatter may amplify the trend artificially. Nonetheless, the variation of
toughness with temperature is of the same order of magnitude as predicted by the model derived above.
Figure 2: The variation of toughness with temperature .0011
0 -10 -20 -30 4 0 -50
TEMPERATURE, S:
Figure 3: The variation of
kt
with temperatureRate Effects
The authors have proposed elsewhere [16] that the rate effect may arise from changes in size of the creep zone at the crack tip. If this zone becomes "too
large" then plane strain conditions no longerdominate and we would expect the measured toughness to increase. Nixon and Sclplson [16] proposed a threshold rate, Kt, below which the creep zone violates plane strain conditions. Kt i s a function of the applied K1, the test temperature, and the material creep parameters. The variation of Kt with temperature is shown in figure 3, calculations having been made using data from Barnes et al. [30] talang note of the change in activation energy at -8OC. The +pparent discontinuity between -5°C and -10°C occurs because of this change. The appropriate values of Kt are shown on figures la-e and it can be seen that at -2", -5' and -10" the values appear to be reaspnable. The situation at -20' and -50" is less clear because of the lack of high-rate data, but the values of Kt are at least reasonable. We may conclude that the variation of $e fracture toughness of freshwater ice with loading rate occurs only below a transtional loading rate, Kt, which can be predicted by consideration of the creep zone size. Two points should be noted here. tirst, the critical value of rc, and thus the value of k t , is specific to a certain specimen size and geometry. Kt is
Q&
0.01 kPadm s- a material arameter. Second, at high temperatures and low loading rates (e.g., -10°C and above, P
)it seems unlikely that we have measured the fracture toughness at all. The creep zone size under these conditions extends across the sample. Given this much creep fracture probably occurs as the immediate result of initiating a new crack from a now blunted notch. This latter point is considered in more detail elsewhere [32] but may explain the two anomalous results alluded to above. At -50°C (see figure le) one value at the lowest rate was considerably higher than the other (152 kPadm as against 110 kPadm). In this case the lower result may be true crack propagation, as at higher rates, while the higher value may arise because a new crack had to initiate at the root of a blunted notch (the notch having blunt
I' 7'
during the test). At -2OC (see figure la), the toughness ex ibits a peak at a loading rate of 0.3 kPadm s' .
It may be that the failures at both 0.3 and 0.01 kPadm s- were due to crack initiation rather than crack propagation, in which case, from previous work [3], we would expect the failure load to drop as the rate decreases.
CONCLUSIONS
1) The toughness increases of ice by
=40% as the temperature drops from -2 to -50°C for those rates at which toughness is rate independent.
2) At each temperature, the toughnesswas constant with loading rate until that rate decreased below a threshold value, K,. The value of K, is determined by consideration of the creep zone size at the crack tip.
3) A model has been presented, bearing similarities to toughness models for both rocks and ceramics, which explains the observed variation of toughness with grain size, temperature , and loading rate.
ACKNOWLEDGEMENTS
The work was supported by the Ice Research Laboratory
(IRL)of the Thayer School of Engineering,
Dartrnouth College, Hanover, N.H., U.S.A. The IRL is a cooperative UniversitylIndustrylGovemment
Cl-318 JOURNAL
DE PHYSIQUE
laboratory supported by the following organisations: Amoco Production Company, Exxon Production Research Company, Mobil Research and Development Corporation, Shell Development Company, Sohio Petroleum Company, the U.S. Army Research Office, the National Science Foundation, the Maritime Administration, the Minerals Management Service, and The U.S. Coast Guard. This support
i sgratefully acknowledged.
REFERENCES
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2 Schulson, E.M., "The Fracture of Ice", in press, J. de Physique (1986).
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Fifth Intl OMAE Conference, 4 (1986): 298-302.
4 Cannon, N.P., "The Influence of Grain Size on the Compressive Strength of Polycrystalline Ice", MS Thesis (1985), Thayer School of Engineering, Dartmouth College.
5 Laughlin, J.L., "The Effect of a Bimodal Grain Size Distribution on the Compressive Strength of Polycrystalline Ice", ME Thesis (1986), Thayer School of Engineering, Dartmouth College.
6 Liu,H.W. and Loop,S.W., "Fracture Toughness of Freshwater Ice", Draft Report, CRREL, Hanover, NH (1974).
7 Liu,H.W. and Miller,K.J., "Fracture Toughness of Fresh-Water Ice", Joumal of Glaciology, 22 (1977): 135-143.
8 Goodman,D.J. and Tabor,D., "Fracture Toughness of Ice: a Preliminary Account of Some New Experiments", Joumal of Glaciology, 21(1978): 651-660.
9 Goodman, D.J., "Critical Stress Intensity Factor
(K1,)
Measurements at High Loading Rates for Polycrystalline Ice", Proceedings of IUTAM Symposium on the Physics and Mechanics of Ice, Copenhagen (1979), 129-146.10 Hamza,H. and Muggeridge,D.B., "Plane Strain Fracture Toughness (K1,) of Freshwater Ice", Proceedings of POAC (1979), 697-707.
11 Goodman,D.J., "The Fracture Toughness of Ice", Data Report, Hokkaido University, Dept. of Applied Physics (1980).
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13 Timc0,G.W. and Frederking,R.M.W., "Comparative Strengths of Freshwater Ice",Cold Regions Science and
~ e c h n o l o ~ ~ , 6 (1982): 21-27.-
14 Nixon, W.A., "Some Aspects of the Enginnering Properties of Ice", Ph.D. Thesis (1984). Cambridge University.
15 Nixon, W.A., and Schulson, E.M., "The Fracture Toughness of Ice over a Range of Grain Sizes", Proc. Fifth Intl OMAE Conference, 4 (1986): 349-353.
16 Nixon, W.A., and Schulson, E.M., "Fracture Toughness of Freshwater Ice as a Function of Loading Rate", Proc. First INTI,. Conference on Ice Technology (1986): 287-296.
17 Lee,R.W., Currier,J.H., Lim,P.N. and Schulson,E.M., "A Procedure for Testing Polycrystalline Ice in Uniaxial Tension", Journal of Glaciology, 30 (1984): 246-247.
18 Br0wn.W.F. and Srawlev,J.E., "Plane Strain Crack Toughness Testing of High Strength Metallic Materials", ASTM
STP 410 (1966). -
19 Gamm0n.P.H.. Kiefte.H., Clouter,M.J., and Denner,W.W., "Elastic Constants of Artificial and Natural Ice Samples by Brillouin Spectroscopy", Journal of Glaciology, 29 (1983): 433-460.
20 Curry, D.A. a d Knott, J.F., "The Relationship between Fracture Toughness and Microstructure in the Cleavage Fracture of M i d Steel", Metal Science, 10 (1976): 1-6.
21 Stonesifer, F.R. and Armstrong, R.W., "Effect of Prior Austenite Grain Size on the Fracture Toughness Properties of A355 B Steel", Fracture 1977, Vol. 2, ICF4, Waterloo, Canada.
22 Hoagland, R.G., Hahn, G.T., and Rosenfield, A.R., "Influence of Microstructure on Fracture Propagation in Rock, Rock Mechanics, 5 (1973): 77-106.
23 Evans, A.G. and Fu, Y., "Some Effects of Microcracks on the Mechanical Properties of Brittle Solids
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11. Microcrack Toughnening", Acta Met., 33,8 (1985): 1525-1531.24 Rice, R.W., Frieman, S.W., Pohanka, R.C., Mecholsky, J.J., and Wu, C.Cm., "Microstructural Dependence of Fracture Mechanics Parameters in Ceramics", in Fracture Mechanics of Ceramics (1978), ed. by R.C. Bradt, D.P.H.
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Rolla, August 2-6 (1982).27 Davidge, R.W. and Tappin, G., "The Effects of Temperature and Environment on the Strength of Two Polycrystalline Aluminas", Proc. Brit. Ceram. Soc., 15 (1970): 47.
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COMMENTS
Remark o f S.KIRBY
One way o f d e t e c t i n g i n d i r e c t l y t h e p r e s ~ n c e o f a m i c r o c r a c k p r o c e s s z o n e i s t o m e a s u r e t h e c h a n g e i n s p e c i m e n c o m p l i a n c e w i t h i n c r e a s i n g t e n s i l e l o a d b y s u p e r i m p o s i n g a s m a l l - a m p l i t u d e f o r c e o s c i l l a t i o n a n d c o m p a r i n g t h e c h a n g e i n c o m p l i a n c e w i t h t h a t e x p e c t e d f r o m y o u r s p e c i m e n g e o m e t r y w i t h o u t m i c r o c r a c k i n g . Answer :
I f e a r t h i s method may h a v e l i m i t e d a p p l i c a t i o n f o r i c e , s i n c e t h e number o f m i c r o c r a c k s is v e r y
small
a n d it may t h u s b e d i f f i c u l t t o d e t e c t t h e c h a n g e i n c o m p l i a n c e a r i s i n g f r o m t h e i r p r e s e n c e .J . W . GLEN
Your e v i d e n c e f o r c l a s s i c a l f a t i g u e b e h a v i o u r s e e m s t o b e much more c o n d u s i v e f o r b u b b l e s t h a n c l e a r i c e . Do y o u f e e l t h e c l e a r i c e i s s h o w i n g t h e b e h a v i o u r ?
Answer :
Y e s , I f e e l t h a t t h e c l e a r i c e d o e s show c l a s s i c a l f a t i g u e b e h a v i o r . A l t h o u g h t h e r e are f e w e r r e s u l t s f o r t h e c l e a r i c e , t h e t r e n d i s s i m i l a r ; t h e i n c r e a s e i n l i f e t i m e f r o m 2 7 0 , 0 0 0 c y c l e s a t 3 9 . 8 N e n d l o a d t o N, 8 8 0 , 0 0 0 c y c l e s a t
