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Time to formation of first cracks in ice
Gold, L. W.
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BY
L.
W. GOLD
RETARD D E FORMATION DES P R E M I E R E S FISSURES DANS
SOMMAIRE
L ' a u t e u r a rCalisC d e s o b s e r v a t i o n s , 'a deux t e r n p g r a t u r e s diffCrentes, du r e t a r d d e f o r m a t i o n d e s p r e r n i ' e r e s f i s s u r e s i n t e r n e s d a n s l a g l a c e B s t r u c t u r e colonnaire sournise 'a une c o n t r a i n t e de c o m p r e s s i o n con- s t a n t e dirigCe n o r m a l e m e n t a u g r a n d a x e d e s c r i s t a u x . I1 a not6 que l a f o r m a t i o n d e s f i s s u r e s e s t u n phCnom'ene accClCrC e n fonction de 1'616- v a t i o n d e l a t e m p g r a t u r e . L e r e t a r d d e l a f o r m a t i o n de l a p r e m i ' e r e f i s s u r e i m p o r t a n t e ( p l u s d e 2 m m d e l a r g e u r e t plus d e 20 m m de longu- e u r ) obCit B 1tCquation gknCrale suivante:
( U 0 - ffp) t = t e x p
0 k T
L a v a l e u r d e Uo, Cnergie de c r C a t i o n d e l a c u n e p o u r une c o n t r a i n t e appliquCe p = 0 a CtC trouvCe Cquivalente 'a 25 k c a l / m o l e , to B e n v i r o n 4 . 6 x 10-17 s e c . e t
>
e n v i r o n 1 . 4 x s e c . L t a u t e u r 'a observC q u e l a dCformation a s s o c i C e>
l a f o r m a t i o n d e l a p r e m i ' e r e f i s s u r e c r o i s s a i t e n fonction i n v e r s e d e l a c o n t r a i n t e , m a i s q u ' e l l e Ctait indC- pendante d e l a t e m p C r a t u r e d a n s l e s l i m i t e s d e l a p r C c i s i o n d e s o b s e r - v a t i o n s . L e s r C s u l t a t s s t a c c o r d e n t a v e c l e modkle d 1 C t o i l e m e n t d e s f i s s u r e s p a r e m p i l e m e n t d e s d i s l o c a t i o n s CtudiC p a r S t r o h (1954) e t Bullough (1964). I1 a o b s e r v C que l e s f i s s u r e s de d i s l o c a t i o n pouvaient e n t r a r n e r l a f o r m a t i o n de f i s s u r e s de G r i f f i t h s i l a c o n t r a i n t e d e t e n s i o n d C p a s s a i t e n v i r o n 7 k g / c m 2 . Aucune f i s s u r e n l a CtC o b s e r v C e quand l a c o n t r a i n t e de t e n s i o n Ctait i n f C r i e u r e 'a e n v i r o n 5 kg/cm2.FROM
Physics
of Snow and
Ice
Proceedings of the International Conference on Low Temperature Science, 1966
VOLUME I
Part
1
Edited
by
Hirobumi ~ U R A
THE INSTITUTE OF LOW TEMPERATURE SCIENCE
HOKKAIDO UNIVERSITY
Time to Formation of First Cracks in Ice
Abstract
Observations have been matle at two temperatures on the time t o formation of the first internal craclis in columnar-grain ice subjected t o a constant compressive loacl applied l~erpentlicular to the long direction of the grains. Crack formation was obser\.ed to be a thermally activatecl process, with the time to formation of the first large craclc (2 o r m o r e nun wide ancl 20 o r m o r e m m long) following the general law
(U,(I./)
t = to e x p -
k7' '
'l'he value of Uo, the activation energy f o r zero appliecl s t r e s s ), was founcl t o be about 25 lccal/mole, to about 4.6X 10 l7 sec and tr about 1.4X 10kZO cn13/molecule. T h e a m o u n t of strain associated with t h e formation of the first cracli was observed t o increase with decreasing stress, but to be inclependent of temperature t o within the accuracy of the experiments. T h e results w e r e fount1 to be consistent \\:it11 the models for crack nucleation by pile-up of dislocations discussed by Stroll (1954) ant1 Bullough jlY(i4). It was observed that dislocation cracks coulcl initiate Grifith craclis if t h e stress exceeclecl about 7 kglcm', ant1 n o c r a c k s were observecl w h e n the applied srress was less than about 5 kg/cmZ.
I. Introduction
I11 earlier papers (Gold, 1960, 1962)
information
was presented on the formation of internal craclis during the creep of columnar-grain ice under constant compressive load. A series of esperiments has now been undertalien to determine the dependence of this cracliing activity on stress. time, and temperature. It is the purpose of the present paper to report results of observations on the dependence of the time to formation of the first large craclc on stress and temperature.11. Experimental Procedure
T h e ice used in the investigation was prepared according to the method described by Gold (1960). This produces a columnar-grain ice with grain size between 1 and 6 mm. A bias in crystallographic orientation develops during freezing such that within 2 c m of the ice-air interface there is a marlced preference for the basal plane to be parallel to the long direction of the columnar grains, but to have a random orientation in the plane perpendicular to that direction. Constant compressive loads, ranging in value between 4 and 20 lig/cm" were applied perpendicular to the long direction of the columns. T h e specimens were about 5 x 1 0 cnl in section and 25 cnl in length, with the long direction of the columns perpendicular to the 10 ~ 2 5 cm face.
Loads were applied t o the 5 x 1 0 c m face of the specimen with a simple lever system. T h e loading plates in contact with the ice were made of steel, with surfaces g r o u n d t o a flat "mirror" finish. W e i g h t s used f o r applying the load w e r e applied o r removed within about
2
sec.Creep strain was measured over a 15.25 cm gauge length with a n extensometer of variable sensitivity, t h e maximum being about 3 x 1 0 "
%.
Observations have been completed at two temperatures : -9.5k0.5"C a n d -31+0.5"C.Cracks t h a t formed in t h e columnar-grain ice d u r i n g creep under constant compres- sive load applied perpendicular t o the long direction of t h e columns were long and nar- row and usually involved only o n e o r two grains. T h e long direction of the crack was in t h e long direction of the grain, and the plane of the crack tended t o be parallel to the direction of t h e applied stress. Observations showed that the cracks tended t o be ~ a r a l l e l o r perpendicular t o the basal plane of grains in which they formed. T h e formation of a crack could be observed visually without difficulty by illuminating t h e specimen with a suitable light.
It was observed that there was a very considerable scatter in the time required for t h e first crack t o f o r m f o r a given applied load. If the load was g r e a t enough, very small cracks w e r e observed to f o r m d u r i n g a n d immediately following t h e application of the load. T h i s activity was followed by a period during which n o cracks formed, o r
Fig. 1. 1)epenclence o n stress of the time to formation of the first large crack, temperature: -9.5 i 0.5"C
CRACICS I N I C E 361 only a few that were very small. After a period of time larger cracks would form a t a rate that gradually increased with time to a maximum. T h e formation of a large crack (greater than
2
mm wide and2
cm long) appeared to be a reasonably well defined event. T h e times reported in this paper are those associated with the formation of the first of such large cracks.111. Observations
It has been found for several materials that for the condition of constant tempera- ture the stress dependence of the time to failure in tension is given by an empirical expression of the following form :
log tf =
r l + H p ,
(1) wheretf is time to failure,
p
is applied stress, and R are constants.T h e observations were analysed to see whether a similar dependence existed for the first large crack formed in ice by a compressive load
In Fig. 1 the logarithm of the time to formation of the first large cracks observed a t -9.5OC is plotted against the stress. T h e same information for the temperature equal to -31°C is given in Fig. 2. In both figures the line drawn is the least squares f i t t o the observations. For the tests at -9.5'C the equation of the line i s :
4 6 8 10 1 2 1 4 1 6 18 20
STRESS. p, k g l c r n 2
Fig. 2. Ilependence on stress of the ti~nc t o formation of the firs1 large crack, temperature= -31 ?0.5"C
log 1,
-
2.93 -O.l(ip, standard deviation -0.32,where t , is the time t o the formation of t h e first large crack ; and a t -31°C, log 1,
-
4.80-0.19p,standard deviation =0./12
.
Both liries are drawn in Fig. 2 for comparison.In calculating the least squares fit a t -9.5'C, the observations a t 16 and 18 lrg/cm? were omitted as the average of t h e logarithms of the times asscciated o;it!l these stresses appeared to be significantly greater than w o ~ ~ l c l have been expected lrom t h e c-~lxervations made a t lower stresses. T h e specinlens used for these tests \\-e!.c cut fro111 only t a r o blocks of ice, both of whic11 had an average grain size of less than 3 m m . If these observations a r e included, t h e equation for the least scluares fit becomes
log t , = 2.60-0.13)
,
.
I h e logarithm of the creep strain r in percent associatecl with the formation of t h e first crack is plotted against the logarithm of the stress for -9 5 C in F i g . 3 , a n d l o r -31'C in Pig. 4. Again, the line drawn is the least squares fit t o the observations. A t -9.S'C:
Fig. 3. I)el~enclence on stresi of the amount of crcep strain to the formation of the first large crack, Lemperature; - 9 . X
Fig. 4. Dependence on stress ofzthe-amount of creep strain to t h e formation of the first large crack, temperature
-
-31°Clog e = 0.74-1.77 log j
,
standard deviation - 0.23 If the points for p = 1 6 and 18 lcg/cm2 are included,log e = 0.90-1.91 log
/I.
log e - 1.6-2.5 log
P ,
standard deviation = 0.33112 Fig. 3 the best fit lines for -9.5 and -31°C are presented for comparison.
In Fig. 5 is presented on normal probability paper the frequency distribution of (log t,-log t,) for the observations at --9.5OC, where t , is the observed time to formation
of the first large craclc for a given experiment, and
i,
is the time for the same stress determined from eq. (1). A similar plot is given in Fig. 6 for (log e,-loge,) for -9.S0C, where e, is the observed creep strain a t the time of formation of a crack, ande,
is the value calculated from eq. (4).T h e straight line drawn in each figure was calculated assuming a normal distribution f o r the observatioils and standard deviations given with eqs. (2) and
(4,
respectively. These figures indicate that the observations had a random distribution about the calcu- lated best fit line. A plot of (log t,-logt,)
and (log e,-log 2,) against grain size indicatedP R O B A B I L I T Y O F F O R M A T I O N O F 1 S T L A R G E C R A C K
P R O B A B I L I T Y O F F O R h l A T l O N O F 1 S T L A R G E C R A C K
Q Q Q
-
w w a - a w m Q Q o o . Q . Q ? - w w o o o o o o o o o yl m a y l m aCRACIiS IN ICE 365 that these quantities did not depend significantly on grain size lor the r a n g e of sizes associated w i t h the tests.
IV.
DiscussionZhurlcov and his co-worlcers (1958, 1960) ancl Sanfirovo et trl. (1963) have shown f o r a number of metals and alloys that failure is a thermally activated process described by t h e general law
t , = t o exp
(Uo-(YP,
k?' ' ((5)w h e r e
t~
-
time to failure f o r a n applied tensilc stressP,
,,I
- temperature in OK,lJo, t o ancl n = constants that define the s t r e n g t h of the material, k = Boltzmann's constant.
Uo
and to have been observed to be independent of temperature and amount of creep strain ; Sanfirova et tz1. (1963) and Zhurleov et trl. (1963) have obtained evidence that ndepends on the s t a t e of t h e structure.
Figures 1 and 2 indicate that f o r t h e range of stress and temperature of t h e experi- ments t h e time to formation of the first large craclc in ice appears to satisfy eq. (5). If it is assumed that this is so, then from eqs. (1) and (6)
uo
'7.30A =1111~+,,
L
,
IValues f o r Uo, (Y and to calculated f r o m eqs. (2) and (3) a r e given in ?'able 1 (Uo-rrp) Table 1. Calculatecl values for n , Uo ant1 /a in I, =Io e s p -
k 7 ,
'Temperature cr
cro
"C cmVmolecule ergimolecule
I0 sec
T h e value f o r Uo, t h e activation energy for ) = O , was found to be equal t o 25.4 lccal/mole.
If
t h e observations f o r 1 6 and 18 1cg/cm2 a t -9.5OC are included, it is equal to about 30 lrcal/mole. T h e value obtained f o r a was about 1 . 4 x 1 0 'O sec. I t must be recognized, however, that these results a r e of a preliminary n a t u r e only because of t h e scatter in thc observations and t h e fact that measurements w e r e made a t only two temperatures.It is of interest t h a t according to t h e result shown in Figs. 3 and 4 t h e a m o u n t of creep strain to the formation of the first large craclr f o r a given stress is independent of temperature t o within t h e accuracy of the experiments. T h i s indicates that formation of a craclc nucleus of critical size depends primarily o n t h e amount of creep strain, that
is, on geometrical factors and not on factors controlling the rate at which the deforma- tion occurs.
Earlier observations (Gold, 1960) had shown that for the first 2 to 3
O/o
creep strain of columnar-grain ice with the basal plane of grains tending to be parallel to the long direction of the columns, the deformation was almost wholly two dimensional; the creep strain in the long direction of the grains was much less than perpendicular to that direction. T h e average creep rate in the direction of the applied stress, therefore, should be equal but of opposite sign to that in the perpendicular direction. Since it is much more difficult to cause slip to occur on non-basal planes of ice than on basal planes, and since the dependence of the creep rate of ice on stress is the same for both tension and compression (Steinemann, 1954), it would be expected that those grains with their basal planes parallel or perpendicular to the applied stress would be subject to a transverse tensile stress whose maximum value would be about equal to the applied stress. It is, in fact, in grains with this general orientation that cracks are usually observed to form. T h e orientation of the basal planes with respect to the grain boundaries and thel n d u c e d T r a n s v e r s e S t r e s s p
Fig. 7 a. Nucleation oi crack in grain A hy pile-up of tlislocations at grain boundary in grain B
D ~ r e c t i o n o f M o t i o n
1 :
o f D ~ s l o c a t i o n si
-
A p p l i e dI
l n d u c e d T r a ' n s v e r s e S t r e s s papplied stress in the present experiments w u l d appear to hc most :;~ritahle for the nu- cleation of c r ~ c l ~ s by t h e pile-up of pure edge dislocations after tile manner discussed by Stroll (195.1, 1957) and Rullough 11961). Consider the situation shon,n in Fig. 7 a, where the craclc in grain
A
has been nucleatecl due to the stress i~nposecl I i l r the pile-up of d i s l o c a t i o ~ ~ s a t t h e boundary inB.
A possible example of such a craclr is shown inFig. S a. I'ossihle esamplc of mechanism illuslratcd in I:ig. 7 a
Fig. 8 b. I'ossil>lc cxarnl>le of mcchrmisrn illustrated in Vig. 7 b. Note crack l>erpcndic~~lar to slip I)lancs in grain on left-hand side
of the specimen, but rather to the relaxation of the transverse stress,
p.
At a grain with a relatively large resistance to deformation the stressp
would act on an area of width about equal to the grain diameter. T h e formation of a Griffith crack would involve, therefore, only one or two grains, particularly during the initial stages of de- formation. It was observed that at -9.5OC the formation of such large craclrs occurred only for stresses greater than about 7 kg/cm2 = 6.86x
lo6
dyn/cm2. For stresses less than this amount the craclrs were small, with widths that appeared to be a millimeter or less. For stress less than about 5 lrg/cm2=4.90 dyn/cm2 no craclts were observed to form. In both cases the creep strain was significantly greater than that required for the for- mation of a large crack according to Fig. 3.If it is assumed that
r
= 105 erg/cm2 (Iiesstvedt, 1964), G/l -a = 4x
10'O dyn/cm2 (Gold, 1958),p
= 6.86x
10' dyn/cm2 (equal but opposite in sign to the applied compressive load),then substituting these values into eq. (9) gives
for the case shown in Fig. 7 a .
Substituting these values into eq. (13) gives ( ~ , , i ~ = 1 1 . 4 ~ cm for the case shown in Fig. 7 b. These values are in reasonable agreement with the observations. T h e corresponding values for 11b are 3.1
x
10 j cm. If it is assumed that the length of theBurgers vector for ice, b , is 4.5 X 10 cm then the corresponding values for 7 1 are 680
and 1360.
T h e fact that the observations are in reasonable agreement with theory (when the energy associated with the formation of a unit area of crack is assumed equal to the surface energy for the ice-vapour interface) indicates that there is little or no plastic flow associated with the formation of a crack. A similar observation was made for cracks formed in ice by the application of a thermal shoclr (Gold, 1963). This indicates that for the stress situations studied it is easier for non-uniform internal stresses to nucleate and propagate craclrs parallel and perpendicular to the basal plane than to cause significant slip on non-basal planes.
V.
Conclusions
This study has indicated that the formation of craclrs in ice is a thermally activated process. T h e amount of creep strain that occurs to the formation of the first large crack for a given applied compressive stress appears to be largely independent of tem- perature. This suggests that the process or processes controlling creep behavior are associated also with the nucleation of cracks. T h e characteristics of the initial part of the crack-forming process are consistent with dislocation models of craclr nucleation discussed by Stroll (1957) and Bullough (1964). Dislocation craclts can nucleate Griffith craclts when the maximum applied shear stress (assumed equal to one-half of the applied compressive load) is about 3.43X 10' dyn/cm2. Internal craclts do not form when the
m a s i n ~ u m a p p l i e d s h e a r s t r e s s is less thari a b o u t
2.5
x 1 0 V y n / c m ' .Acknowledgments
7 .
I h e t ~ u t h o r ~ v i s h e s to e x p r e s s h i s a p p r e c i a t i o n to W. Ubbinl<. L. Gooclrich a ~ l c l
F.
F y f e f o r t h e i r a s s i s t a n c e in c a r r y i n g o u t t h e o l , s e r v a t i o ~ ~ s . 'This paper is a c o n t r i b u t i o n f r o m the Division of Building Liesearch, N a t i o n a l I i e s e a r c h C o u n c i l , C a n a d a , a n d is pub- lished w i t h t h e a p p r o v a l of t h e D i r e c t o r of t h e T)ivision.References
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