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Pressures developed in a porous granular system as a result of ice
segregation
Penner, E.
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Ser
TH1
N21r2
no.
81
c.
2
NATIONAL
RESEARCH
COUNCIL
C A N A D A
BLDG
DIVISION OF BUILDING RESEARCH
Pressures Developed in a Porous
Granular System a s a Result of
Ice Segregation
Edward Penner, Soil Mechanics Section
Division of Building Research
National Research Council of Canada
RESEARCH PAPER N O . 81
O F T H E
DIVISION OF BUILDING RESEARCH
Reprinted from Special Report 40
Highway Research Board, Washington, D.C.
---
OTTAWA
PRICE 25 CENTS
JULY
1959
Pressures Developed in a Porous Granular
System as a Result
of
Ice Segregation
EDWARD PENNER, Soil Mechanics Section, Division of Building Research National Research Council of Canada, Ottawa
Introductory Remarks by the Chairman
The g r e a t French mathematician and philosopher, ~ o i n c a r 6 , has pointed out on a number of occasions that even in mathematics true pioneering advance is usually made by the geometric and intuitive mind. This advance is subsequently r e n d e r e d "secure" f o r science by the analyst o r logician. Unfortunately, a s Poincare a l s o points out, t h i s "securingtt by the logician may often be likened to building a complicated scaffold around the simple intuitive creation, obscur- ing i t s beautiful lines and diminishing also i t s i n s p i r a t o r y power f o r further intuitive creation. In the p r e s e n t age, scientific and engineer- ing education is becoming m o r e and m o r e analytical with the deplor- able r e s u l t that p r e s e n t college c u r r i c u l a often deprive engineering
students of an opportunity to develop geometric and intuitive creative thinking. It is hoped that the fine example of geometrical thinking shown in the paper by Edward Penner and i n other important contri- butions to t h i s symposium will help to r e s t o r e the needed balance between geometrical and analytical approach not only in this field, but a l s o in other a r e a s of engineering science.
@ FROST ACTION in soils has been studied for a number of y e a r s by the Division of Building Research with a view to developing m o r e adequate f r o s t action c r i t e r i a for s o i l s engineering in regions of seasonal f r o s t . The normal pattern of events that leads to the destruction of r o a d s and a i r p o r t s i s heaving of the s o i l when it f r e e z e s a n d loss of supporting strength when i t thaws. The displacement of shallow foundations a n d the damage to cold storage warehouses due to f r o s t action normally involves only the frost heaving aspect, the l o s s of supporting strength being of little consequence.
The expansion of water when i t f r e e z e s accounts for a s m a l l fraction of the total heave. Although this in itself may be particularly destructive in many building mate- r i a l s , a l a r g e percentage of the displacement in s o i l s i s rightfully attributed to the formation of i c e l e n s e s a t the freezing plane. The flow of m o i s t u r e is normally a s - sumed to be largely in the liquid phase under the influence of a so-called "suction" gradient. As a consequence, the development of t h i s suction a t t h e ice/water interface
is the most essential element i n the mechanism of f r o s t heaving since i t m u s t precede and continue during the growth of i c e lenses. In a closed s y s t e m the suction increases to a maximum when ice lensing stops.
Recent experiments by the author (1) showed that the magnitude of the suction (soil moisture tension) a t the ice/water interface, when the growth of the ice l e n s s t o p s , de- pends upon the dimensions of the pore system in the soil. Such m e a s u r e m e n t s w e r e made in the absence of any appreciable external p r e s s u r e on the i c e phase which, how- ever, exists in nature and i t s effect i s therefore of significance in the formation of ice l e n s e s . This paper gives the experimentally determined relationships between over- burden p r e s s u r e s and the magnitude of the s o i l moisture tension when the i c e l e n s growth h a s d e c r e a s e d to zero, that i s , under equilibrium conditions. Some t h e r m o - dynamic a s p e c t s of the freezing of moist s o i l s have been d e s c r i b e d by Edlefsen and Anderson (2) and the i c e lensing s y s t e m has been described by Winterkorn (3), Powers
(4),
and Gold (5). Based on somewhat s i m i l a r treatment the mechanism of i c e lensing is discussed in-the light of the experimental r e s u l t s .EXPERIMENTAL too Material and Specimen Preparation
The granular system used consisted
of powdered quartz known in the c e r a m i c
,
80 industry a s p o t t e r ' s flint. The grain- Is i z e distribution based on the hydrometer analysis is given in Figure 1. Selection
of this m a t e r i a l was on the b a s i s of chem- 60
i c a l purity and fineness. m
The specimens w e r e p r e p a r e d a t vari- n ous densities in a Harvard Miniature Com-
2
paction Apparatus by controlling the mois- 40 t u r e content and compactive effort. The2
dimensions of the specimens w e r e 15/16 in. w 0i n diameter and approxin~ately 3 in. long.
6
20 In preparation f o r compaction, uniform amoisture distribution was attained by ap- plying water with a fine s p r a y and subse-
quently storing the m a t e r i a l in a sealed o
polythene bag. Uniformity of moisture 0.001 0.0 1
content w a s ascertained by sampling and G R A I N S I Z E ( M M
determining the moisture content a t vari- Figure 1. The grain-size distribution of
ous locations in the moist material. the potter's flint used in these experi-
After molding in the compaction appa- ments
.
r a t u s in %-in. l a y e r s the specimens were
oven-dried a t 105 C. Upon placement in the sample holder of the f r o s t c e l l (Fig. 2) almost perfect saturation w a s achieved by permitting d e - a i r e d water to e n t e r the spec- imen a t the base and to s p r e a d unidirectionally. When the saturation of the specimen was complete the external s o u r c e of water w a s cut off (closed system). Using this method of preparation the specimens showed no tendency to swell when saturated o r shrink when desaturated. The specimens a l s o had sufficient strength to r e s i s t defor- mation a t the overburden p r e s s u r e s used i n the freezing experiments. It may be noted h e r e that the sample holder of the f r o s t c e l l w a s designed to accommodate specimens p r e p a r e d i n the Harvard compaction apparatus.
Method of Freezing
The specimens w e r e frozen unidirectionally, s i m i l a r to the freezing of s o i l s in na- ture. Antifreeze mixtures w e r e circulated in the outer compartments of the frost c e l l to control the t e m p e r a t u r e of the specimen. In the bottom two compartments the c i r - culating fluid was maintained a t +1
'12
C. The temperature of the fluid circulating through the top compartment w a s - 3 C. The variation in temperature was approxi- mately 2 0.05 C in the conditioning fluid but w a s a s m a l l fraction of t h i s within the specimen. Temperature measurements w e r e made with a specially built, high preci- sion millivolt r e c o r d e r , using copper-constantan thermocouples. To avoid any dis- turbance in the material a f t e r preparation thermocouples w e r e placed s o that they would not coincide with the location of the equilibrium ice/water interface.The method of freezing w a s f i r s t to t e m p e r a t u r e condition the lower end of the s a t - urated specimen to +1% C and then s t a r t the circulation of the - 3 C fluid. This would f r e e z e the specimen unidirectionally f r o m the top, leaving a t the completion of the ex- periment approximately 2 in. of unfrozen m a t e r i a l beneath the ice lens. Ice crystalli- zation w a s mechanically induced a t about 2 C of supercooling. Greased lucite rings w e r e provided around the specimen to reduce the side friction caused by heaving. It must a l s o be emphasized that the freezing of the specimens did not c o m p r i s e of a g r a d - ual lowering of temperature of the conditioning fluid. No f u r t h e r changes i n the tem- p e r a t u r e w e r e imposed on the specimen a f t e r the - 3 C conditioning fluid w a s introduced.
T H E R M O C O U P L E T H E R M O C O U P L C O M P A R T M POROUS P L A T E I
YaTER
INLET I I I I I I I I I I I - -0 1 2 S C A L E IN INCHES F i g u r e 2. S e c t i o n t h r o u g h f r o s t c e l l .Soil Moisture Tension Measurements and the Application of Positive P r e s s u r e s Heaving normally continued for several days depending largely on the overburden p r e s s u r e . At the completion of heaving the unfrozen portion of the specimen was quickly removed from the f r o s t cell and sliced in
'/z
in. lengths for moisture content determinations.Two methods were used to determine the soil moisture tension induced in the un- frozen portion of the soil due to i c e lens growth. From separately determined mois- t u r e content/soil moisture tension curves the induced tension could be evaluated from the moisture contents of the unfrozen portion. More details of this method may be found in a n e a r l i e r publication (1).
In the second method a mercury manometer with a water connection to the base of the unfrozen portion of the specimen was used to give continuous moisture tension readings throughout the experiment. This second method was considered superior to the f i r s t since i t did not involve the separate determination of the moisture tension curve. It i s also considerably more accurate in the region where the moisture content change i s small with increasing tension. One drawback of the second method i s that i t i s limited by a maximum of about one atmosphere of tension.
RESULTS
The maximum soil moisture tensions induced by the growth of the i c e lens were de- termined f o r t h r e e different overburden p r e s s u r e s : 23, 229, and 535 ~ I I I per sq cm.
The lowest overburden p r e s s u r e used r e p r e s e n t s the weight of the plunger only and the highest p r e s s u r e was low enough s o that some ice segregation would be permitted. The freezing experiments were c a r r i e d out with specimens prepared a t various dry densities ranging from 1.592 to 1.750 gm p e r cu cm. When heaving ceased, the de- g r e e of saturation was calculated from the moisture contents of the unfrozen soil. The degree of saturation was plotted versus dry density for the three different overburden p r e s s u r e s a s shown in Figure 3. The quantities beside the plotted points in the figure give the amount of water lost p e r 100 gm of unfrozen soil. Knowing the relationship between percent saturation and moisture tension (Fig. 4) the moisture tensions induced a t various densities and overburden p r e s s u r e s could be determined. These data have been plotted in Figure 5, c u r v e s A, B, and C .
To determine in a m o r e direct way the relationship between moisture tension and overburden p r e s s u r e a further s e r i e s of
experiments was conducted using the s a m e loo type of material. These experiments w e r e similar to those already described with the exception that a mercury manometer
9 0
was used to measure the moisture tension in the unfrozen soil. This permitted di- z
r e c t readings of tensions a s the experi- O _ c
ment progressed. The final moisture
2
3 80 tensions a t various overburden p r e s s u r e s5
a r e plotted in Figure 5, c u r v e D. The cr e s u l t s were similar to those obtained by 5 70 0
the former method. E w
F r o m the r e s u l t s shown in Figure 5
the equilibrium conditions with respect to 60
overburden p r e s s u r e s and soil moisture tensions may be summarlzed a s follows
for the particular granular material used: 50
1.60 1 65 1.70 1.75 1.80
(a) the maximum moisture tension induced DRY DENSITY grn/~rn3
in the unfrozen portion due to i c e segre- Figure 3. The relationship between dry
gatio? d e c r e a s e s with increasing over- density, overburden pressure and percent burden p r e s s u r e s ; (b) the g r e a t e r the s a t u r a t i o n beneath the f r e e z i n g zone when
packing density of the granular material heaving i s reduced t o zero i n a closed
195
a t the s a m e overburden p r e s s u r e ; (c) f r o s t heaving due to i c e segregation can be limited either by the developnlent of a sufficiently high moisture tension, by ap- plying overburden p r e s s u r e s , o r by a combination of both; and (d) e x p r e s s e d in gm p e r s q c m the limiting m o i s t u r e ten- sion is about half the limiting overburden p r e s s u r e . (The overburden p r e s s u r e was I-
calculated in t e r m s of load p e r unit a r e a of the specimen.)
DISCUSSION O F THEORY To facilitate the discussion a diagram (Fig. 6) has been constructed showing the relative position of the ice/water i n t e r -
face and i c e lens with r e s p e c t to t h e s t r u c -
'
100 1000t u r e of the pore system. MOISTURE TENSION IN CM WATER
The s i z e of a stable s p h e r i c a l c r y s t a l Figure 4. The r e l a t i o n s h i p between mois-
of a solid in i t s own melt depends on tem- t u r e tension and percent s a t u r a t i o n .
p e r a t u r e and may be e x p r e s s e d in the f o r m used by Sill and Skapski (6). - Using t h e s u b s c r i p t s i f o r i c e and w f o r w a t e r :
where:
r = r a d i u s of the c r y s t a l , c m Pi = density of the ice, g m p e r c u c m uiw = interfacial energy, e r g s p e r s q c m
T = t e m p e r a t u r e of melting a t z e r o c u r v a t u r e of the solid/liquid interface, K
AT = freezing point depression below T
Assuming that this relationship holds for the freezing of water in small s p a c e s , it fol- lows that the temperature a t which i c e will propagate through a soil depends on the pore dimensions. By freezing a saturated soil specimen unidirectionally the advance of the i c e is impeded a s predicted by Eq. 1. T h i s mechanism is believed to bring about the freezing point depression of such a s y s t e m but the reaction of all p a r t s of the s y s - t e m to this is not the s a m e and brings about the growth of i c e phase which r e s u l t s in the phenomenon of i c e lensing.
The conditions in a s a t u r a t e d salt-free porous system immediately a f t e r crystalli- zation i s induced a r e a s follows: (a) the p r e s s u r e in the p o r e water is z e r o and the ad- sorbed l a y e r i s essentially fully developed; (b) the overburden p r e s s u r e is small, s i n c e the i c e l e n s has just begun to grow; (c) the temperature along the ice/water interface
F i g u r e 6. An e n l a r g e d schematic diagram showing a s e c t i o n of t h e i c e l e n s w i t h r e s p e c t t o t h e s o i l p a r t i c l e and s o i l p o r e .
is everywhere the s a m e and has been lowered a s predicted by Eq. 1.
In the p o r e the ice/water interface h a s a s s u m e d a c r i t i c a l radius and the r a t e of melting and freezing h e r e is equal. Above the soil particle, however, the ice/water interface is either flat o r h a s assumed a negative curvature. The freezing point of the water a t the ice/water interface above the soil particle i s c l o s e to that of f r e e water,
since the adsorbed phase is considered to be fully developed, but the f r e e z i n g point of the pore water is given by Eq. 1. Consequently, the ice phase will grow above the soil p a r t i c l e s a t the expense of the adsorbed phase. If such a s y s t e m is kept saturated, a n
external p r e s s u r e must be applied to stop the growth of i c e above the s o i l particles. To fix the thermodynamic equilibrium condition, two variables, in this c a s e tempera- t u r e and p r e s s u r e , a r e considered simultaneously. If the two phases a r e to be in equi- librium the f r e e energy of the i c e must be equal to the f r e e energy of the w a t e r , that is,
fi = fw. T h i s means that when any change o c c u r s in the s y s t e m the f r e e energy of each phase changes by an equal amount: dfi = dfw. It follows that:
vi d P i
-
si d T = ww dPw - sw dT where:vi and vw = specific volume of ice a n d water, respectively,
dPi and dPw = change in p r e s s u r e on the i c e and water, respectively, si and sw = specific entropy of the i c e and water, respectively,
Rearranging the t e r m s and dividing by dT:
Q f
Substituting - for sw
-
s i , where Qf is the latent heat of fusion and considering finiteT
changes, the expression becomes:
vw APw
-
vi APi - - - QfAT T
Since the i c e above the soil p a r t i c l e is postulated to be in d i r e c t contact with the ad- sorbed phase, any change in total p r e s s u r e is f e l t equally by both the ice and water phases. As a consequence, APw = APi and Eq. 2 in t e r m s of the total p r e s s u r e be- comes:
APiw is the p r e s s u r e required to bring about equilibrium e x p r e s s e d in t e r m s of the freezing point depression. Eq. 3 p r e d i c t s that for a freezing point depression, A P is
positive, since Qf is positive and the quantity vw
-
V i is negative. The equation holds equally when A P is negative but in this case AT is positive.Example 1. Ice Lensing Stopped with a Positive P r e s s u r e by Loading the Specimen A sample calculation will i l l u s t r a t e the maximum p r e s s u r e required to stop i c e lens growth when the s y s t e m r e m a i n s saturated, assuming a p o r e r a d i u s r with a freezing temperature of -0. 01 C compatible with Eq. 1. Substituting the following values into Eq. 3: Qf = (79.8) 4.184 x lo7 e r g s p e r gm T = 273K vw = 1.00 c u c m p e r gm vi = 1.09 cu c n ~ per gm AT = -0.01 C
gives a p r e s s u r e of 1.356 x l o 6 dynes p e r sq c m . Applying this p r e s s u r e to t h e sys- tem c a u s e s the temperature of freezing of the p o r e water a n d the ice/water interface above the soil particle to be equal.
Stating this in another way, before the p r e s s u r e was applied to the ice/water inter- face the freezing point a t that location was T l which, according to the moisture condi- tions imposed, was approximately 273 K. In the p o r e water the freezing point w a s Tz a s predicted by Eq. 1, s o that Tz < T I . Since the influence of the p r e s s u r e is not felt by the pore water T Z does not change but T1 d e c r e a s e s by an amount At with increasing P until T I - At = Tz. As a consequence there is no further tendency for the i c e l e n s to grow.
In the above example the c a s e h a s been considered where i c e lensing is stopped by applying a positive p r e s s u r e to the ice/water interface. This load is not transmitted to the pore water because the porous system is considered sufficiently rigid to r e s i s t deformation.
The growth of the lens can a l s o be stopped by applying a suction to the water i n the pore system. When suction is applied to the water in the s y s t e m some of the l a r g e r p o r e s may empty but the water in those that r e m a i n filled is under tension. At the s a m e time the f i l m s of water that surround the individual p a r t i c l e s decrease in thick- ness. Returning to Figure 6, applying a suction to the water has the effect of drawing the ice/water interface into the f o r c e field of the particle a n d can then be considered to be under compression. A s a r e s u l t of this p r e s s u r e the temperature of freezing a t the ice/water interface above the soil particle d e c r e a s e s a s the s o i l grain is approached.
Eq. 3 p r e d i c t s that the freezing point i n c r e a s e s when water and ice together a r e placed under tension. A s a consequence the water in a p o r e of s i z e r which f r e e z e s normally a s predicted by Eq. 1 now h a s a higher freezing temperature. The opposite
is t r u e a t a position above the soil p a r t i c l e because h e r e the tension tends to draw the ice/water interface into the adsorptive field and the resulting p r e s s u r e lowers the freezing point. The maximum tension is developed when the resulting freezing point of the p o r e water and the water a t the ice/water interface above the s o i l p a r t i c l e a r e equal. If the influence of the tension w e r e not considered to place the ice/water inter- f a c e above the soil particle in a s t a t e of compression and thus lower i t s freezing point, one would a r r i v e a t the f a l s e conclusion f r o m Eq. 3 that the tension n e c e s s a r y to stop i c e lens growth would be numerically equal to the p r e s s u r e given in Example 1. Re- stating this, in the c a s e where i c e lens growth is permitted to induce a tension in the p o r e water (closed system), the freezing point of the water a t the ice/water interface above the soil particle and of the pore water a r e again T I and Tz, respectively, where Tz > T I . T I is approximately 273 K and Tz is s e t by Eq. 1. A s the tension i n c r e a s e s in the p o r e water i t s freezing point,increases to some value equal to Tz
+
Atz. At the ice/water interface above the soil particle the freezing point d e c r e a s e s due to the r e a - sons given above to some value equal to T I-
At2. When T2+
At2 = TI-
A t l , no f u r t h e r i c e growth can take place a n d d e s c r i b e s the equilibrium condition. The c a s e where At2 = A t l , d e s c r i b e s the situation where the positive p r e s s u r e in the i c e / w a t e r inter- face above the soil particle is equal to the tension in the p o r e water. In fact whether this is a special c a s e o r if equilibrium conditions a r i s e where At2 and Atl, a r e not equal but have values such that T2 + At2 = T I-
A t l , is not known. In any c a s e when Atl = At2 the theoretical magnitude of the maximum tension is one half that given in Example 1 f o r a sample of s i m i l a r pore dimensions.Curve D in Figure 5 shows that in a s a t u r a t e d system the external p r e s s u r e r e - quired to stop i c e lens growth is approximately twice a s g r e a t a s the tension when no
external load is applied. Although this is supporting evidence i t is still too limited to be accepted a s complete confirmation of the theory.
T h e r e a r e two additional complicating f e a t u r e s of the s y s t e m which must a l s o be considered. Firstly, the a r e a of the ice/water interface is not known in the experi- mental determinations of p r e s s u r e . P r e s s u r e s a r e e x p r e s s e d in this paper a s loads applied p e r unit a r e a of sample. Secondly, a p r e s s u r e applied a t one point in the ice, f o r example, above the soil particle, is not fully transmitted to the section of the ice above the pore, in fact, i t tends to decrease. It is thought that adjustments in the i c e a r e accomplished by the i c e assuming changes in the c u r v a t u r e and, consequently, in the internal p r e s s u r e , both above the p a r t i c l e and above the pore. F u r t h e r , a granu- l a r system which is heterogeneous with respect to particle s i z e such a s was used in these experiments adds to the complexity of the system, in particular because of the emptying of the l a r g e r p o r e s a s the moisture tension in the p o r e system is increased.
In spite of these uncertainties t h e r e is general agreement between the theory and experimental r e s u l t s . It follows f r o m the theory that the maximum m o i s t u r e tension o r suction induced becomes l e s s if any external overburden p r e s s u r e is applied. This is also shown by the r e s u l t s i n F i g u r e 5.
If the freezing point depression in such s y s t e m s is related to the dimensions of the p o r e s compatible with Eq. 1 the maximum heaving p r e s s u r e s o r the maximum soil
moisture tensions induced should be g r e a t e r if the density of the granular material is increased. The curves relating overburden p r e s s u r e s and corresponding moisture tensions determined experimentally fall in t h e o r d e r predicted by the theory. This was shown in another way in previous experiments with r e s p e c t to soil texture and maximum moisture tension. In coarse-grained soil the m o i s t u r e tension developed was much l e s s than in fine-grained soils.
Experimentally, the limiting overburden p r e s s u r e e x p r e s s e d in g r a m s p e r unit a r e a of sample is approximately double the limiting tension in curve D, Figure 5. Theo- retically the magnitude of tension when i c e l e n s growth s t o p s can be shown to be l e s s than the external p r e s s u r e r e q u i r e d when the system is kept saturated. Whether the
2 to 1 relationship shown in Figure 5 holds f o r a l l soils is not known. F u r t h e r sub- stantiating proof would involve determining p r e s s u r e v e r s u s tension relationships a t equilibrium f o r s o i l s with different grain-size distributions. At present, no way is known of evaluating the a r e a of contact between the ice phase and the s o i l particle,
unit a r e a of sample. T h e r e a r e a l s o thought to be important f a c t o r s unaccounted f o r by using the relationship between APiw and AT. T h e s e involve t h e adjustments i n the c u r v a t u r e of the i c e resulting f r o m difference i n overburden p r e s s u r e a t a point above the p a r t i c l e to a point above t h e pore.
The theory p r e d i c t s that i c e lensing may be halted by overburden p r e s s u r e , by ten- sion i n the p o r e w a t e r , o r a combination of both. This h a s been shown to b e possible experimentally.
Finally, attention is drawn t o the different relationships t h a t m a y be d e r i v e d be- tween APi and AT, APw and AT, and APiw a n d AT f r o m Eq. 2. A detailed discussion may be found in the publication "Thermodynamics of Soil Moisture" by Edlefsen a n d Anderson (2). In a n e a r l i e r paper (1) the author tended to be in agreement with t h e theoretical-treatment by o t h e r s .
~ f i s
t r e a t m e n t , however, a p p e a r s valid only when the i c e and w a t e r p h a s e s a r e c o n s i d e r e d to b e isolated f r o m each o t h e r in t h e s y s t e m . Experience with the model d e s c r i b e d in this p a p e r , i n which'the i c e and w a t e r p h a s e s a r e in intimate contact, sbpported by the experimental r e s u l t s , suggests that t h e r e i s no way in which e i t h e r phase c a n b e independently subjected t o tension o r p r e s s u r e in a s y s t e m w h e r e i c e l e n s growth i s possible. Nevertheless, s i n c e only g e n e r a l a g r e e - ment between the theory and the experimental r e s u l t s can b e c l a i m e d , complete con- fidence in the i c e lensing mechanism discussed is a t p r e s e n t not possible.ACKNOWLEDGMENTS
The author gratefully acknowledges the help of h i s colleagues i n the Soil Mechanics Section and in p a r t i c u l a r w i s h e s t o thank D. R. E l d r e d and J. A. 0. Guibord 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 out the experimental work.
This paper i s a contribution f r o m the Division of Building R e s e a r c h , National Re- s e a r c h Council of Canada and is published with t h e approval of the Director of t h e Division.
REFERENCES
1. Penner, E . , "Soil Moisture Tension and I c e Segregation. Highway R e s e a r c h Board Bulletin 168 (1957).
2. Edlefsen, N. E . , and Anderson, A. B. C . , "Thermodynamics of Soil Moisture." Hilgardia, 15:31-298 (1943).
3. Winterkorn, H. F . , Discussion of "Suction F o r c e in Soils upon Freezing" by A. R. Jumikis, Proceedings, A m e r i c a n Society of Civil Engineers, Vol. 81, S e p a r a t e No. 656, pp. 6-9 (1955).
4. Powers, T. C.
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"Resistance of Concrete t o F r o s t a t E a r l y Stages."
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"A P o s s i b l e F o r c e Mechanism Associated with the F r e e z i n g of Water in P o r o u s Materials. Highway R e s e a r c h Board Bulletin 168 (1957).6. Sill, R. C . , and Skapski, A. S . , "Method f o r the Determination of Surface Ten- sion of Solids f r o m T h e i r Melting Points in Thin Wedges.
