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A photoelastic study of ice pressure in rock cracks
Cold Regions Science and Technology, 11 (1985) 141-153 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
141
A P H O T O E L A S T I C S T U D Y OF ICE PRESSURE IN ROCK C R A C K S G.P. David$on and J.F. Nye H.H. Wills Physics Laboratory, UniversiW of Bristol, Bristol BS8 ITL (England)
(ReceivedDecember20, 1984;acceptedin revisedform January30, 1985)
ABSTRACT Water was frozen in a slot made in a transparent material, and the resulting stresses in the material produced by the expansion o f the ice were measured by use o f the photoelastic effect. Because the stresses changed with time, a photoelastic technique was developed which used photometric recording and digital processing in place o f the more usual combination o f photography and optical compensators. The maximum water pressure was 11 bar. A theory is presented which accounts, in outline, for the observed stresses, and we discuss its implications for freezing in real rocks. The physics o f freezing water confined in a rock crack is different according to whether the ice does or does not begin to extrude. In our experiment there was some extrusion, and in this case the pressure can be calculated simply from the known yieM strength o f ice, the elasticity and strength o f the rock not being directly involved.
1. INTRODUCTION It is generally accepted that when water freezes in a crack in a rock the expansion of the ice sets up stresses that tend to propagate the crack. But the physical details are not well understood and, in spite of an extensive literature on the general subject (see, for example, Embleton and King, 1975, McGreevy and Whalley, 1982, and the references they cite), there seems to be no quantitative theoretical analysis of the basic process. The idea of the present study was to make an artificial crack, a slot, in a transparent material, to fill the slot with water and then to freeze it; one would then use the photoelastic effect to 0165-232x/85/$03.30
© 1985 Elsevier Science Publishers B.V.
measure the resulting stresses. It was not intended that the crack should propagate. Since it is not at all obvious how to calculate the stresses set up, even without crack propagation, our aim was to use the photoelastic measurements to help develop a quantiative understanding of the physics involved. In this way we hoped to throw light on at least one important element in the frost cracking process. The photoelastic method we used is outlined in §2. The other experimental methods used are described in §3 and the main results are reported in §4. In § 5 we consider the physics of the freezing process and outline a quantitative theory to show how the measured stresses and other observed effects arise; then in § 6 we discuss the implications of this theory for the stresses in real rocks.
2. THE PHOTOELASTIC MODEL AND METHOD Figure 1 shows the photoelastic model used. Ice frozen in the slot produces surface tractions on the walls of the slot. The surface tractions in turn produce a distribution of stress within the material, and in particular, at the bottom of the slot. Our main interest was to measure the distribution of surface tractions on the slot walls, because this is the primary quantity that theory has to predict; once it is known, it is a standard problem in elasticity to infer the stress throughout the rest of the material and especially at the root of the crack. The photoelastic method is ideal for measuring the total distribution of surface tractions on the slot walls rather than just at a single point (and it also incidentally gives the distribution of stress within the material). Moreover, there are no errors caused by interaction between a separate stress
142 transducer and the material in which the slot is made, because the material itself acts as the stress transducer. We chose perspex (lucite) as a convenient material to use.
.-
-'"
I
I Fig. 1. The perspex photoelastic model. More detail is shown in Figure 5. Conventional photoelastic techniques (e.g., Kuske and Robertson, 1974) for measuring the stresses proved inadequate because they were too slow to follow the changes in the stresses as the water froze. It thus became necessary to develop a technique which measured the photoelastic effect photometrically (rather than by using a compensator), and which processed the resulting signals digitally. Our method has features used in previous attempts to automate the photoelastic method (see, for example, Miiller and Saackel, 1979, and references given in that paper) but was developed independently. 2.1 Choice of polariscop~
The simplest form of polariscope uses diffuse light, but although it is intrinscially simple and economical, needing no lenses, it has severe disadvantages in the present context. First, for reasons to be described later, a thick model is required, and then the combined effects of perspective and uncollimated light mean that the definition at the borders of the model
is poor. Second, it is difficult to use fractional fringe techniques, which are needed in insensitive materials at low stress levels, because the image brightness is too low and the light is uncollimated. A lens polariscope, using collimated light from a point source, solves these problems, and we used a modification of it in our experiments. It needs large aperture lenses and a bright point-source of light. An image of the model can be simply projected on to another plane, which greatly facilitates the improved techniques, to be described later, for fractional fringe measurement. It is especially useful where physical access to the model is restricted, as in this case where the model was in a freezer. 2.2 Photoelastic stress measurements
Conventional photoelastic stress measurements record the lines of constant principal stress difference (the isochromats) and lines of constant principal stress direction (the isoclinics) on photographs in cases where the stresses are sufficiently large to produce high-order fringes. If the loads are small or the model material is insensitive, the fringes are usually broad and indistinct. Compensation methods must then be used (Dally and Riley, 1978); although precise, these are very slow and therefore inappropriate to a dynamic experiment. We now describe the technique we adopted; it is both fast and precise. High precision is needed, not for the final result, but because the method for the separation of the principal stresses ( § 2.3) uses numerical differentiation. Circularly-polarized light is incident upon the stressed photoelastic model parallel to the dimension marked h in Fig. 1. At any point the stress modifies the light so that, in general, it becomes elliptically polarized. From the standard theory of photoelasticity the orientation of the polarization ellipse is the same as that of the principal stress directions at the point in question. Moreover, the shape of the ellipse is related to the relative retardation A, which is proportional to the difference al " a2 of the principal stresses ( a l ) az): 27rhC n -
-
-
X
01
-
09
where h is the model thickness, X is the wavelength of the light, and C is the relative stress-optical coefficient.
143
tude of the modulation being related to A, and its phase to O. A, the relative retardation and 0, which gives the orientation of the principal stresses, are the two quantities we wish to measure. Because of the form of eqn. (1) (note that the modulation amplitude Io sin A can be positive or negative) there are certain ambiguities in deducing A and 0 (in addition to the obvious ambiguity of an integral multiple of 27r in A). However, in our measurements we were concerned with changes in stress with time at a single point, or in space as the model was scanned, so that in practice the ambiguities could be resolved by continuity.
Thus the task is to measure the shape and orientation of the polarization ellipse of the light emerging from each point of the model To do this we used a rotating linear analyser and measured the intensity of the transmitted light, for it varies at twice the rotation frequency, with amplitude and phase related to the shape and orientation of the polarization ellipse. To see this in more detail, consider Fig. 2, where OX and OY are parallel to a~ and o2, respectively, and make an angle 0 with the fixed axes Ox, Oy. The sense of OX is f'txed by stipulating that 0 ~< 0 < lr. If the X-component of the incident vibration is written as a sin wt, the Y-component is a cos wt (for right-handed circularly-polarized light). The birefringence in the model introduces a relative phase retardation A between the components. If the analyser instantaneously makes an angle ~k with X, the vibration it transmits is
2.3 Separation of principal stresses
which has frequency 6~ and amplitude Ao (~b) given by
The photoelastic measurements give values of al - 02, the principal stress difference, and O, the principal stress direction, at a chosen point in the model. However, we wish to know Ox, Oy and rxy, the components of stress on an element, and this requires a knowledge of el and o2 separately. To achieve this we use the equilibrium equations
[Ao(~)] 2 = a2(1 + sinA sin2~k).
aOx
A(~k) = a sin ( ~ t + A) cos ~ + a cos wt sin ~k
Ox
Orxy + .... Oy
0
Polariser
I L,aht~'~
~] Iqode/ X~x
,¢~_~1
Otly+ 07"xy =0 ~y ' ~X
Rotafint?
×
:natyser
X
I
~
andintegratetogive
Orxy
o~ = (O~)o xo
y Oy = (Oy)o -- / d
Yo
Fig. 2. Illustrating the theory of the polariscope.
dx
(y = constant)
(2)
dy
(x = constant)
(3)
~Y
Orxy 0x
In terms of the angle ¢ between the analyser and the fixed x-axis (~ = ¢ + O) this gives
The shear stress at a point, rxy, may be determined directly from the photoelastic measurements through the equation
I(~) =Io[1 + sinA sin 2(~--8)]
rxy = ½(ol -- o~) sin 20
( I o ) 0)
where we have replaced a 2 by Io and [A0(~k)]: by I(~). If the rotation frequency of the analyser is ~2, so that ¢ = ~2t, we have
I(t) =Io[1 + sinA sin 2(I2t--0)l
(1)
We see that the incident intensity I0 is modulated sinusoidally at an angular frequency 212, the ampli-
By measuring rxy along two neighbouring lines parallel to the x-axis the values of O'rxy/Oy may be calculated. Then, by using eqn. (2), we can start at a point with a known value of trx (for example, a vertical free boundary where Ox = 0) and integrate ~Zxy/Oy step by step along y = constant to give Ox at each point along the line. The other normal stress try may then
144 be determined from ay = Ox + [(al " a 2 ) 2 -- 4 7"~y]½
where the sign depends upon the sign of rxy and the value of O. Alternatively, and as a check, we can measure along lines parallel to the y-axis and use eqn. (3) to find Oy.
3. EXPERIMENTAL METHODS 3.1 Experimental control To freeze the water in the slot of the model (Fig. 1) in a predictable way we found it was best to maintain an ambient temperature of 0°C and at the same time to apply local freezing to the upper surface of the block. A domestic freezer was modified by fitting doubleglazed viewing ports, allowing optical access by the polariscope. The ambient temperature was controlled by a proportional temperature controller, which consisted of a pulsed heater (250 W electric bulb), the duty cycle of which was automatically adjusted to be proportional to the temperature difference between the freezer and a preset level. The freezer cooling was switched on continuously; thus the heat input was balanced against the cooling action of the freezer. A fan in the freezer helped to maintain a uniform temperature. Local cooling on the model was provided by a semiconductor thermoelectric module using the Peltier effect. At maximum current the module developed a cold-face temperature of--17°C in an ambient temperature of 0°C. The temperatures of the freezer and cooling module were monitored with a simple multiprobe electronic thermometer which used silicon diodes as economical and accurate probes. As the experiment progressed the growth of ice in the slot (Fig. 1) was monitored by viewing the ice perpendicular to the plane of the slot through the side of the block. For this purpose (as distinct from the photoelastic measurements made in the perpendicular direction) the block was between crossed polaroids and illuminated by white light. The bireffingence of the ice crystals made them stand out against the water. The ice was photographed in col-
our by use of a 200 mm telephoto lens mounted on a bellows to obtain the necessary magnification. 3.2 Acquisition of photoelastic data The polariscope is shown in Fig. 3. The light source was a 2.5 mW He-Ne laser, the beam from which was passed through a polaroid film and a mica ¼X plate, cut for a wavelength of 632,8 nm, and oriented with its axes at ¼rr to that of the polaroid. The resulting circularly-polarized light beam was expanded by a microscope objective; at the same time it was spatially filtered to remove non-uniformities generated by off-axis modes within the laser tube and also by dust on the preceding optical elements. The expanding beam was then collimated by a lens and passed into the freezer to be incident upon the model. After passing through the model and out of the freezer the now elliptieally-polarized beam was analysed by a polaroid film rotating at a frequency of 125 Hz. The light, now modulated at 250 Hz, passed through the field lens, which was set to form an image of the model in the plane P. This image was viewed by a microscope in which the eye piece had been replaced by a sensitive photodiode. The microscope was capable of movement in both x- and ydirections. Its objective lens was chosen so that the sensitive area of the photodiode was equivalent to 0.3 × 0.3 mm on the model. Figure 4 shows the path of the subsequent signal processing. The photodiode detector output was amplified and processed by several modules to defive dc voltages proportional to the modulation amplitude IolsinAI, the dc level Io and the phase of the modulated signal relative to a reference signal. This phase reference was generated from the rotating polaroid analyser by using a fight-emitting diode and photodiode arranged to detect a black-white pattern on the hub of the rotating f'tim. The x,y coordinates of the microscope were converted to dc voltages by linear potentiometers linked to the driving stage. All these dc values were passed via an analogueto-digital converter to a PET microcomputer where they were further processed and stored on tape. The program for the PET was written in BASIC, except for the critical data-input and averaging routines, which were written in machine-code.
145 polariser laser
I,
image I p[anePl
model
I
/
, II < ~ ,w~ I "
1/4 av spati~ plate fdter
"~ rotating motor analyser -
microscope & detector = I I
I
Fig. 3. The polariscope.
I
I
AmplitudeImax~--~-11[,L~Data Acquisition&Analysis
Po[ariscope~ I Detector
I i
I ,,- orl !
- ~level ~~ d.c. I
l iIl ~
Phase
,I
II'
coordinates J l
I
sensor
I ; I
I
Channel selector AtoD converter
PET microcomputer]
IT
I
] Analyser
"1 cycle trigger , Signal processing
I I
Fig. 4. Dataacquisitionand analysis. 3.3 The photoelastic models Constructing a suitable model (Fig. 1) for twodimensional photoelastic stress measurement proved quite difficult. The main constraints were (0 that the stress generated by the ice should be two.dimensional, and (ii) that as the pressure in the water increased, there should be no leaking of the water from the sides of the slot (that is, the front and back faces in Fig. 1), which would relax the stress. Constraint (i) was met by using a thick model where the thickness of the block was large compared with the width B of the slot, so that a condition of plane strain was approached. Constraint (ii) proved more difficult to satisfy. At first, glass plates were clamped on either side of the perspex models, separated by a layer of grease, which would both lubricate the block, allowing two-dimensional strain, and also
prevent escape of water from the slot in the third dimension. This was unsuccessful on two counts: (i) the water was forced into the lubricant layer, distorting the optical path and, of course, causing a relaxation of stress; (ii) the temporary birefringence from the thermal distortion of the glass clamping plates proved to be of the same order of magnitude as the birefringence in the model, so causing large errors. Figure 5 shows the final solution. The perspex block AA' was made in two halves cut from a single plate. The profile BCD and the outer faces were machined, a fly-cutter on a vertical mill being used to reduce machining stresses to a minimum. The two halves were joined together with cyanoacrylate adhesive, which was very effective and had the advantage of setting rapidly and producing a very thin glue line.
146
scan lines
elevation
A
adhes~ E
cyanoacrylot e I ptan
silicone rubber
Fig. 5. Construction of the perspex model. Two perspex plates E and E' were glued to the faces of block A' with cyanoacrylate adhesive. Note that they are f'Lxed only to A' and not to A, to allow free relative movement between A and A'. The crucial elements are the two very thin seals of silicone rubber arranged to prevent the water escaping from the slot. They constrain the water in the slot even under considerable pressure and deform so slightly that there is very little relaxation of the pressure. At the same time they provide almost no hindrance to the free relative movement between A and A'. This technique was vital for obtaining reliable results.
4. RESULTS 4.1 Preliminary experiments Before the main quantitative experiments on the freezing of water in a slot we did a number of simple qualitative experiments. In these we found that the temperature environment of the block affected the way in which the ice grew in the slot. Pre-cooling the block to --10°C followed by surface cooling resulted
in supercooled water, which then froze suddenly with much dendrite formation. Seeding the water with silver iodide crystals prevented the supercooling but encouraged a number of large dendrite-like structures. On the other hand, cooling the surface of the block in an ambient temperature of 0°C produced a smooth ice front which progressed evenly down the slot; these were the conditions we chose for the main experiment to be described in § 4.2. In these preliminary experiments we also measured how much the ice extruded during freezing. This is clearly an important parameter; no extrusion would imply that the change in volume of the water on freezing was entirely taken up by deformation of the perspex block, with resulting high stresses. We found that the water level in the slot changed during the initial cooling of the block, and before ice began to form; this had to be taken into account before deducing the extrusion. For cylindrical holes in large blocks the extrusion was close to the 10% expected from the volume change on freezing. This was also the case for parallelsided slots sealed well at their sides, but not by using the technique of Fig. 5, for the sealing not only prevented the water leaking out but also restricted the opening of the slot. Whenever the extrusion was zero there was leakage of the water. In the main experiment (§4.2), however, the measured extrusion was 3% and no water leaked; as already explained (§ 3.3) the sealing in this case (Fig. 5) allowed the slot to open freely while still keeping the water in. 4.2 The main experiment Water was frozen in a parallel-sided slot made in a block (70 X 70 × 32 mm) constructed as described in § 3.3. The slot had width 1 mm, length 32 mm and depth 25 mm. Before the experiment the block was carefully annealed to minimise residual stresses, although they were allowed for in the analysis as explained below. The freezer was set to an ambient temperature of 0°C and the cooling module made the minimum temperature on the upper surface of the block --17°C. Axes are taken with Ox (horizontal) perpendicular to the slot andOy (vertical) parallel to it. The photoelastic parameters in the perspex during the cooling were measured along lines parallel and perpendicular to the slot faces (Fig. 5). These were
147
subsequently analysed as described in § 2 to give the stress distribution along the slot as a function of time. The constant (Oy)o in the integration of eqn. (3) along the vertical scan lines was uncertain, because the corner B was in contact with the cooling module and had high residual stresses. Therefore we relied on the horizontal scan lines, which start on the vertical free surface, to provide the necessary constant of integration, (Ox)0 in this case. Residual stresses, which were of the same order of magnitude as the water stresses, were allowed for by subtracting them. The maximum non-residual stress measured, which was 10 bar, was a small fraction (0.19) of a fringe. Figure 6 shows the progression of the ice front down the slot. The measurements are taken from photographs of the ice during the experiment. Three lines are plotted: (I) the ice surface, showing a small amount of extrusion; (II) the position of the icewater interface; (III) the position of the boundary of
1.0
Top of slot
I -x.--
m~ x
--,x - -~x--~l¢-- x "°'x'x~--)<'x~<~'-)e'cx~<-x-xl<~<'~x
0.7
c'~\Cloudy Ice
(b) ,,
0-6
o
~
0.5
(0 "%
0.4
....
(e)>,,
Water
"o,
0-3
(e) " Q
,
o,
0.2
Ice
IT ,Q
/,o (~J "~%.%.~
0-1
o.
0
I
I
I
1
I
I
I
I
C(jJ ~¥
2 time [hrs}
Fig. 6. The motion of the ice front down the slot. l is the length of the slot and y is the distance measured from the bottom. The letters in parentheses refer to Figs. 7a-g. (I) top of ice plug; (II) ice-water interface; fill) boundary between clear and cloudy ice, interpreted as the upper limit of contact between the ice plug and the slot walls.
a further feature that we now describe. As the ice progressed it became cloudy and indistinct over its upper portion, a front dividing cloudy from clear ice being formed parallel to, but some way behind, the ice-water front. The stress data in fact suggest that the cloudy area is caused by the ice separating from the slot walls rather than pressing on them. We discuss the reason for this not entirely unexpected phenomenon in § 5, but here we remark that it provides a check on the accuracy of the stress measurements, because both the normal traction ax and the shear traction rxy should be zero along this part of the slot. By this criterion we estimate the random error in each individual measurement of Ox and rxy as +0.5 bar. Figures 7 a - g show the normal and shear stress components ax and rxy along a line near to the slot wall and parallel to it, as a function of time. The positions of the ice-water front and the cloudy front are also shown. Each of Figs. 7 a - g is indicated on Fig. 6 by the corresponding letter. The stress component ey parallel to the slot face is not shown on the diagrams because its value was strongly affected by the presence of residual stresses, which had a large gradient perpendicular to the slot face and close to it. After correction for the residual stresses the values of ey were consequently much less certain than the values of ax and rxy, which were much less severely affected by such effects. Three zones of interest are evident: (a) Below the ice-water front, in a region subject to hydrostatic pressure, we might expect Ox to be uniform and rxy to be zero. The measured value of ax is not quite uniform; it sometimes shows a peak near the ice-water front itself (see especially Figs. 7c, d). The shear traction rxy, on the other hand, shows an unexpected departure from zero at the lowest measured point. We attribute both these apparent non-uniformities to systematic errors of measurement which arise from the following sources: first, the patch contributing to each stress measurement is of finite size, 0.3 X 0.3 ram, and so the vertical scan lines necessarily cannot be exactly at the slot wall, and second, there are substantial spatial gradients in the stresses both near the ice-water interface and near the bottom of the slot. (b) Above the cloudy-clear front both ax and rxy are essentially zero, within the random error of the
148
measurements. The explanation is simply that the ice has separated from the slot wall. (c) Between the cloudy-clear front and the i c e water front is a transitional region where the surface traction changes from hydrostatic pressure in (a) to zero in (b). In this region the ice is in contact with the wall and exerts a frictional force as it tends to extrude. The measured shear stress is of the correct sign to be explained as the effect of friction. Figure 8 shows the hydrostatic pressure p in the water as a function of the fraction of water frozen.
The maximum pressure is 11 bar and the relation is essentially linear. 5. A theoretical model
The purpose of this section is to show how the quantitative observations of §4 can be understood, and then in § 6 we apply this understanding to ice in cracks in real rocks. Suppose the slot of length l (Fig. 9a) were initially completely filled with water (actually the water level was just below the top) and that
/x
? o ¢D
/:
-cloudy front
u~
i o
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--1
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ul
(/i x
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-2
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bars
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-2
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~ /x
x
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/
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,///~
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~j
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-6
(c)
-L.
-2
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i
~
,
(d)
~
~rs
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2
149
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q
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i
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,
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-2
~x i
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(f)
x
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=
40
°
,
,
. o~
J
l
-8 -6 bars
'-Io" -8 -~ bars
-~
l
l
l
-4
"
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~
j
~q I~ '~
-2
o ' "2
xj xj
(g)
P
0
i
Fig. 7. Distribution of shear traction "rxy and normal traction trx on the walls of the slot at different stages (a)-(g) of freezing (the width of the slot is exaggerated). The lines simplyjoin the experimental points.
freezing has progressed to PQ, a distance s from the open end. Now consider a small advance ds of the freezing front to P'Q'. Because ice expands by 10% on freezing, the pressure in the water will increase and will force the walls of the slot further apart. There is an initial difficulty in understanding how this is possible without breaking the water seal and releasing the pressure. If the ice were rigid rather than elastic, and if the slot walls remained plane, they would
be forced outwards as in Fig. 9b and the water seal would consist merely of line contacts at P and Q. In fact the line PQ in the ice is under lateral elastic compression, and as the slot walls are forced apart by the increased water pressure the compression of PQ is relaxed slightly but not destroyed, thus preserving the seal. RS in Fig. 9a marks the observed boundary between the clear ice below and the cloudy ice above. As mentioned in § 4.2 we interpret this observation
150 angle is smoothed out and the walls do not consist of plane segments, but nevertheless the model of Fig. 9c is a useful starting point. We assume that the sharp angle moves down with the ice front. Then, ignoring any extrusion of the ice plug at this stage, we can readily calculate the angle t~ by conservation of mass. To first order the mass of water in the length ds equals the mass of ice in the length ds, plus the mass of water in the new volume made by expansion of the slot below PQ. The latter is (for unit length perpendicular to the diagram)
1Z P (bars)
o
O
8
0 °
4
0
0.2
0.4
o
0.6
08 F
pwot(l -- s)ds
1.0
Thus
Fig. 8. The pressure p rises approximately linearly with the fraction F of water frozen.
PwBds = piBds + pwa(l -- s)ds where Pw and Pi are the densities of water and ice respectively, and where B is the length PQ. Hence
to mean that the forcing apart of the slot walls has caused the ice to lose contact with them above RS. This is confirmed by the fact that the measured traction on the slot wall is zero above RS. (The ice appears cloudy probably because of the condition of the separated surfaces.) Thus the actual contact areas are PR and QS. To explain the observations we must take account of the deformation of the slot walls from planes and Fig. 9c displays the essential feature. In this idealization the slot walls are supposed plane and parallel down to PQ where they bend sharply inwards by an angle a, but continuing plane. In practice the sharp
pi ) B
Putting B = 1.0 mm, l -- s = 9.3 mm (corresponding to Fig. 7e) and Pi/Pw = 0.91 gives ~ = 0.010 rad = 0.55 ° . Thus the 10% change of volume made by freezing can be accommodated if the slot wall bends through this angle at the moving freezing front. (As l - s decreases to zero the required ot obviously approaches infinity, which implies either cracking or plastic deformation at the bottom of the slot.)
',/I I// S ,
t I
Ip.t
,'/
' \ ).ga,
lee
,a f ds
'
(4)
I
\
ds .
.
.
.
s
]
P'
-- I -- --
t t
Winter
4
-s
(a)
t
(c)
(d)
Fig. 9. Not to scale. (a) shows notation; PQ is the ice-water interface; (b) simplified model with rigid slot walls; (c) simplified model with the slot wails bent through an angle ~; (d) model allowing for extrusion and non-parallel slot walls.
151 Let us now make the model more realistic by allowing the slot walls below PQ to be slightly curved and eliminating the sharp angle in the walls at P and Q. This is easily done by noticing that the above calculation is still correct provided ads is reinterpreted as the average opening of the slot in the water space below PQ. As a further step we should allow some extrusion of the ice plug, for it was observed (Fig. 6) that during freezing the ice was extruded from the slot (by 3%) but not by enough (10%) to account for the whole volume change. As the freezing front moves down by ds suppose the ice plug moves up by fds (Fig. 9a), where f i s a fraction lying between zero, for no extrusion, and 0.1 for maximum extrusion. Then expression (4) for a is modified to -
kB l--s
(5)
de -
pds'
(7)
Ec
where E is the appropriate elastic modulus of ice. The ratio of the first and third terms in eqn. (6) is, using eqn. (7)
Bp
Bde m-
where k = (#w -- Oi --fPi)/Ow, a measure of the fractional expansion after allowing for extrusion. If f = 0 (no extrusion), k = 0.09; while if f = 0.1 (maximum extrusion), k = 0. Let us now look at the extrusion process in a little more detail. When extrusion is allowed in the model it is important to take account of the fact that the slot walls above PQ will not be quite parallel. So let them (Fig. 9d) be inclined inwards, at an angle ~. As the freezing front advances from PQ to P'Q' the ice particles at P,Q move up by fds and outwards to P", Q", the slot walls meanwhile moving out by da. Thus the opening by 2da is the result of a wedging action of the tapering ice plug as it is extruded upwards. The ice fibre PQ is initially formed under compression along its length and as it moves up the slot the pressure on it is observed to fall. It will therefore experience an elastic relaxation strain de (an extension). Hence by geometry
Bde= 2da -- 2f/3ds
from RS to R'S'. By observation RR' and PP' are respectively 20 and 30 times greater than PP"; therefore, in estimating the lowering of pressure on the ice fibre PQ, we can ignore its movement. Since both the pressure in the water and the contact length RP increase approximately linearly with the fraction of water frozen, the gradient of pressure along the contact remains approximately the same. The reduction in pressure on PQ is then p(ds'/c), where ds' = R R ' and c = RP, and the corresponding strain is
2f/3ds
ds' •
2Ec
_
_
ds
With the observed values (corresponding to Fig. 7e) p = 6 bar, ds'/ds = 0.55,c = 6.4 m m , f = 0.03, w i t h e = 9 × 104bar, the ratio is 9.5 × 10-s//3. So the ratio is small if/3 >> 9.5 × 10-s, which corresponds to a taper of merely 2 /am over the length of the slot. The machining and polishing of the slot will certainly have produced variations much greater than this and so the third term in eqn. (6), which is due to taper, will certainly overwhelm the first one, which is due to the elasticity of the ice. This means that eqn. (6) becomes da =f~ds
(8)
which simply says that the wedging action by extruding ice takes place as if the ice were rigid. Now we can compare the opening of the slot just above PQ, namely 2da, with the average opening in the water below PQ, namely ads, a being given by eqn. (5). Thus
(6)
2da We now show that, in our experiment, the elastic term Bde is negligible compared with the other two terms in this equation. To do this we have to estimate the lowering of pressure on the ice fibre PQ. The normal pressure on the sides of the ice plug falls from p, the hydrostatic pressure in the water, at P to zero at R (Fig. 9a), and it is consistent with observation to take the variation as linear. As the ice-water boundary advances by ds the line of zero pressure advances
ads
-
2f/3(l-- s) kB
(9)
Putting in the same numerical values as before (f = 0.03 corresponds to k = 0.063) gives 8.9/3. Since/3 is certainly much less than 1/8.9 in this experiment the ratio is small. That is, the elastic widening of the slot takes place predominantly in the unfrozen water zone, rather than by any wedging action as the ice plug is extruded upwards.
152 Our main conclusion from this analysis is that the simple model of Fig. 9c (with some extrusion and associated wedging action included, and with the sharp bend in the wall of the slot smoothed out) does in fact capture the essence of the expansion process that occurred in the experiment. It is clear that the process we have described would not work if a slot with plane walls tapered downwards (~ negative). Even a slot with rough or wavy walls would allow the ice plug to extrude freely if it tapered downwards sufficiently. Our model for a build-up of pressure relies on there being hindrance to extrusion; this is represented in the model with plane walls by a positive angle ~, and more generally we assume it is provided by non-parallelism of the walls. We note, however, that in the experiment the walls did part company with the ice above RS (Fig. 9a); this is evidently because, above this level, the pressure distribution forces the wails apart by an amount that is not compensated by any wedge-like upward motion of the ice plug. It is now apparent that, since the slot in the experiment was not plane parallel to optical standards (the 2 t~m calculated above), and was not intended to be, and since some extrusion was observed to occur in spite of the waviness of the interfaces, plastic deformation took place in the ice, certainly near the two interfaces and possibly within its bulk too. In either case it is plausible to use a plastic friction law r = constant = ko, where ~" is the average shear traction at the wall. Considering the forces on the ice plug, the upward force from the water on the lower end balances the downward frictional force from the walls 2k0 pB = 2 koc or p = c (10) B
Figure 6 shows that c increases approximately proportionally to the fraction of water frozen and hence, from eqn. (10), so does p. Thus a plastic friction law accords very well with the linear increase of p observed in Fig. 8. The value of ko deduced is 0.5 bar, which is plausible. A plastic friction mechanism of this kind was first suggested by Professor K.E. Puttick (private communication).
6. IMPLICATIONS FOR REAL ROCKS
In the light of the experiments and the model which explains them, what can we infer about the stresses produced by ice in real rocks? Most real rocks contain water in pores of various sizes quite apart from macroscopic cracks. Mellor (1970, 1971, 1973), in particular, has drawn attention to the fact that pore water in typical rocks freezes progressively as the temperature is lowered and that some is virtually unfreezable at natural temperatures. As the pore water freezes in wet rock there are sharp discontinuities in the measured thermal strain. Such strains would be especially effective in producing cracks when they are inhomogeneous, and two sources of inhomogeneity are readily available, namely, gradients in temperature and in water content. Such considerations suggest that one should be cautious in relating frost damage simply to the expansion of bulk water in macroscopic cracks. Nevertheless, assuming that such freezing does occur, it is important to trace its consequences. A feature that emerges very clearly from our discussion is the importance of the contact area between ice and rock, in a given crack, in determining the possible water pressure. Fresh cracks in rocks do not have plane faces; they are commonly sinuous and their faces may be rough on a variety of scales. It is true that the expansion of the crack as freezing progresses tends to release the ice plug, but our analysis suggests that this is a small effect because most of the expansion occurs in the water-riffled region below the level of the ice front. In any case, the shape of the crack means that only a small additional extrusion of the plug can be enough to preserve the water seal. The roughness and sinuosity of real cracks thus increase the contact area between ice and rock so that it may well be effectively the whole area of the crack. One can now distinguish two possible regimes: (1) where the ice plug is either extruding or is on the point of extruding, and (2) where the ice plug is essentially fixed and grows in situ. In case (1) the irregularity of real cracks leads one to expect plastic deformation of the ice, and relation (10) would be applicable with c taken as the distance from the outer
153 surface to the freezing front. With 2k0 in the range 1 - 1 0 bar, and with the ratio of crack depth to crack width c/B in the range 100-1000, pressures of 102104 bar, quite sufficient to extend the crack, would be reached. Thus in case (1) the pressure can be calculated just from the yield strength of ice; we do not need to know the elasticity of the rock. On the other hand, in case (2) the pressure developed depends essentially on the elastic and strength properties of the rock, because these control the opening of the crack. What then decides which regime is applicable? As c/B increases, the pressure for ice extrusion eventually exceeds the pressure needed simply to expand the unfrozen water by 10% against the elasticity and strength of the rock. It is at this point that case (1) changes to case (2). Apart from the matter of the progressive freezing of pore water mentioned above, there are two main differences between our experiment with perspex and conditions in real rocks: first, the shape of the crack is different, plane and fairly smooth rather than sinuous and rough, and second, perspex is elastically softer than most rocks. Our experiment is dearly in regime (1), with extrusion. This suggests that it would be useful to do further photoelastic experiments with perspex in which the slot was made deliberately sinuous. One would also wish to vary the ratio c/B to test relation (10). No new experimental difficulties would be expected except perhaps with the artificial seal used to prevent the water leaking out sideways. This was a crucial element in the experiments; it is effective up to 10 bars, but its upper limit is unknown. As c/B was increased there would come a point where the ice ceased to extrude and a new theory for determining p, appropriate to regime (2) and explicitly involving the elasticity and strength of perspex, would have to be invoked. It is worth remarking that
an artificial seal was only necessary because, in order to apply simple photoelastic methods, we needed to make the experiment two-dimensional. In the real, three-dimensional case a wave of freezing penetrating from the exposed surface of the rock would make a natural seal. ACKNOWLEDG EME NTS
The experimental technique described was evolved after a number of preliminary photoelastic experiments had been carried out in this laboratory by J.R. Cave, Miss L.J. Hill and Mrs. K.M. Branson, and after helpful discussions on the physical mechanism of frost cracking with Professor K.E. Puttick. We acknowledge with gratitude their work, the advice of Dr. M.E.R. Walford, and correspondence with Dr. M. Mellor. REFERENCES
Dally, J.W. and Riley, W.F., 1978. Experimental Stress Analysis. McGraw-Hill,New York, NY. Embleton, C. and King, C.A.M., 1975. Periglacial Geomorphology. Arnold, London. See index heading freezethaw,
Kuske, A. and Robertson, G., 1974. Photoelastic Stress Analysis. Wiley, London. McGreevy, J.P. and WhaUey,W.B., 1982. Arct. Alp. Res., 14: 157-162. Mellor, M., 1970. U.S. Army Cold Regions Research and Engineering Laboratory, Research Report 292. Mellor, M., 1971. U.S. Army Cold Regions Research and Engineering Laboratory, Research Report 294. Mellor, M., 1973. In: Permafrost: the North American Contribution to the Second International Conference. ISBNO309-02115-4, National Academy of Sciences, Washington, DC, pp. 334-344. Mfiller, R.K. and Saackel, L.R., 1979. Proc. Soc. Exp. Stress Anal., Exp. Mech., 36: 245-251.
141
A P H O T O E L A S T I C S T U D Y OF ICE PRESSURE IN ROCK C R A C K S G.P. David$on and J.F. Nye H.H. Wills Physics Laboratory, UniversiW of Bristol, Bristol BS8 ITL (England)
(ReceivedDecember20, 1984;acceptedin revisedform January30, 1985)
ABSTRACT Water was frozen in a slot made in a transparent material, and the resulting stresses in the material produced by the expansion o f the ice were measured by use o f the photoelastic effect. Because the stresses changed with time, a photoelastic technique was developed which used photometric recording and digital processing in place o f the more usual combination o f photography and optical compensators. The maximum water pressure was 11 bar. A theory is presented which accounts, in outline, for the observed stresses, and we discuss its implications for freezing in real rocks. The physics o f freezing water confined in a rock crack is different according to whether the ice does or does not begin to extrude. In our experiment there was some extrusion, and in this case the pressure can be calculated simply from the known yieM strength o f ice, the elasticity and strength o f the rock not being directly involved.
1. INTRODUCTION It is generally accepted that when water freezes in a crack in a rock the expansion of the ice sets up stresses that tend to propagate the crack. But the physical details are not well understood and, in spite of an extensive literature on the general subject (see, for example, Embleton and King, 1975, McGreevy and Whalley, 1982, and the references they cite), there seems to be no quantitative theoretical analysis of the basic process. The idea of the present study was to make an artificial crack, a slot, in a transparent material, to fill the slot with water and then to freeze it; one would then use the photoelastic effect to 0165-232x/85/$03.30
© 1985 Elsevier Science Publishers B.V.
measure the resulting stresses. It was not intended that the crack should propagate. Since it is not at all obvious how to calculate the stresses set up, even without crack propagation, our aim was to use the photoelastic measurements to help develop a quantiative understanding of the physics involved. In this way we hoped to throw light on at least one important element in the frost cracking process. The photoelastic method we used is outlined in §2. The other experimental methods used are described in §3 and the main results are reported in §4. In § 5 we consider the physics of the freezing process and outline a quantitative theory to show how the measured stresses and other observed effects arise; then in § 6 we discuss the implications of this theory for the stresses in real rocks.
2. THE PHOTOELASTIC MODEL AND METHOD Figure 1 shows the photoelastic model used. Ice frozen in the slot produces surface tractions on the walls of the slot. The surface tractions in turn produce a distribution of stress within the material, and in particular, at the bottom of the slot. Our main interest was to measure the distribution of surface tractions on the slot walls, because this is the primary quantity that theory has to predict; once it is known, it is a standard problem in elasticity to infer the stress throughout the rest of the material and especially at the root of the crack. The photoelastic method is ideal for measuring the total distribution of surface tractions on the slot walls rather than just at a single point (and it also incidentally gives the distribution of stress within the material). Moreover, there are no errors caused by interaction between a separate stress
142 transducer and the material in which the slot is made, because the material itself acts as the stress transducer. We chose perspex (lucite) as a convenient material to use.
.-
-'"
I
I Fig. 1. The perspex photoelastic model. More detail is shown in Figure 5. Conventional photoelastic techniques (e.g., Kuske and Robertson, 1974) for measuring the stresses proved inadequate because they were too slow to follow the changes in the stresses as the water froze. It thus became necessary to develop a technique which measured the photoelastic effect photometrically (rather than by using a compensator), and which processed the resulting signals digitally. Our method has features used in previous attempts to automate the photoelastic method (see, for example, Miiller and Saackel, 1979, and references given in that paper) but was developed independently. 2.1 Choice of polariscop~
The simplest form of polariscope uses diffuse light, but although it is intrinscially simple and economical, needing no lenses, it has severe disadvantages in the present context. First, for reasons to be described later, a thick model is required, and then the combined effects of perspective and uncollimated light mean that the definition at the borders of the model
is poor. Second, it is difficult to use fractional fringe techniques, which are needed in insensitive materials at low stress levels, because the image brightness is too low and the light is uncollimated. A lens polariscope, using collimated light from a point source, solves these problems, and we used a modification of it in our experiments. It needs large aperture lenses and a bright point-source of light. An image of the model can be simply projected on to another plane, which greatly facilitates the improved techniques, to be described later, for fractional fringe measurement. It is especially useful where physical access to the model is restricted, as in this case where the model was in a freezer. 2.2 Photoelastic stress measurements
Conventional photoelastic stress measurements record the lines of constant principal stress difference (the isochromats) and lines of constant principal stress direction (the isoclinics) on photographs in cases where the stresses are sufficiently large to produce high-order fringes. If the loads are small or the model material is insensitive, the fringes are usually broad and indistinct. Compensation methods must then be used (Dally and Riley, 1978); although precise, these are very slow and therefore inappropriate to a dynamic experiment. We now describe the technique we adopted; it is both fast and precise. High precision is needed, not for the final result, but because the method for the separation of the principal stresses ( § 2.3) uses numerical differentiation. Circularly-polarized light is incident upon the stressed photoelastic model parallel to the dimension marked h in Fig. 1. At any point the stress modifies the light so that, in general, it becomes elliptically polarized. From the standard theory of photoelasticity the orientation of the polarization ellipse is the same as that of the principal stress directions at the point in question. Moreover, the shape of the ellipse is related to the relative retardation A, which is proportional to the difference al " a2 of the principal stresses ( a l ) az): 27rhC n -
-
-
X
01
-
09
where h is the model thickness, X is the wavelength of the light, and C is the relative stress-optical coefficient.
143
tude of the modulation being related to A, and its phase to O. A, the relative retardation and 0, which gives the orientation of the principal stresses, are the two quantities we wish to measure. Because of the form of eqn. (1) (note that the modulation amplitude Io sin A can be positive or negative) there are certain ambiguities in deducing A and 0 (in addition to the obvious ambiguity of an integral multiple of 27r in A). However, in our measurements we were concerned with changes in stress with time at a single point, or in space as the model was scanned, so that in practice the ambiguities could be resolved by continuity.
Thus the task is to measure the shape and orientation of the polarization ellipse of the light emerging from each point of the model To do this we used a rotating linear analyser and measured the intensity of the transmitted light, for it varies at twice the rotation frequency, with amplitude and phase related to the shape and orientation of the polarization ellipse. To see this in more detail, consider Fig. 2, where OX and OY are parallel to a~ and o2, respectively, and make an angle 0 with the fixed axes Ox, Oy. The sense of OX is f'txed by stipulating that 0 ~< 0 < lr. If the X-component of the incident vibration is written as a sin wt, the Y-component is a cos wt (for right-handed circularly-polarized light). The birefringence in the model introduces a relative phase retardation A between the components. If the analyser instantaneously makes an angle ~k with X, the vibration it transmits is
2.3 Separation of principal stresses
which has frequency 6~ and amplitude Ao (~b) given by
The photoelastic measurements give values of al - 02, the principal stress difference, and O, the principal stress direction, at a chosen point in the model. However, we wish to know Ox, Oy and rxy, the components of stress on an element, and this requires a knowledge of el and o2 separately. To achieve this we use the equilibrium equations
[Ao(~)] 2 = a2(1 + sinA sin2~k).
aOx
A(~k) = a sin ( ~ t + A) cos ~ + a cos wt sin ~k
Ox
Orxy + .... Oy
0
Polariser
I L,aht~'~
~] Iqode/ X~x
,¢~_~1
Otly+ 07"xy =0 ~y ' ~X
Rotafint?
×
:natyser
X
I
~
andintegratetogive
Orxy
o~ = (O~)o xo
y Oy = (Oy)o -- / d
Yo
Fig. 2. Illustrating the theory of the polariscope.
dx
(y = constant)
(2)
dy
(x = constant)
(3)
~Y
Orxy 0x
In terms of the angle ¢ between the analyser and the fixed x-axis (~ = ¢ + O) this gives
The shear stress at a point, rxy, may be determined directly from the photoelastic measurements through the equation
I(~) =Io[1 + sinA sin 2(~--8)]
rxy = ½(ol -- o~) sin 20
( I o ) 0)
where we have replaced a 2 by Io and [A0(~k)]: by I(~). If the rotation frequency of the analyser is ~2, so that ¢ = ~2t, we have
I(t) =Io[1 + sinA sin 2(I2t--0)l
(1)
We see that the incident intensity I0 is modulated sinusoidally at an angular frequency 212, the ampli-
By measuring rxy along two neighbouring lines parallel to the x-axis the values of O'rxy/Oy may be calculated. Then, by using eqn. (2), we can start at a point with a known value of trx (for example, a vertical free boundary where Ox = 0) and integrate ~Zxy/Oy step by step along y = constant to give Ox at each point along the line. The other normal stress try may then
144 be determined from ay = Ox + [(al " a 2 ) 2 -- 4 7"~y]½
where the sign depends upon the sign of rxy and the value of O. Alternatively, and as a check, we can measure along lines parallel to the y-axis and use eqn. (3) to find Oy.
3. EXPERIMENTAL METHODS 3.1 Experimental control To freeze the water in the slot of the model (Fig. 1) in a predictable way we found it was best to maintain an ambient temperature of 0°C and at the same time to apply local freezing to the upper surface of the block. A domestic freezer was modified by fitting doubleglazed viewing ports, allowing optical access by the polariscope. The ambient temperature was controlled by a proportional temperature controller, which consisted of a pulsed heater (250 W electric bulb), the duty cycle of which was automatically adjusted to be proportional to the temperature difference between the freezer and a preset level. The freezer cooling was switched on continuously; thus the heat input was balanced against the cooling action of the freezer. A fan in the freezer helped to maintain a uniform temperature. Local cooling on the model was provided by a semiconductor thermoelectric module using the Peltier effect. At maximum current the module developed a cold-face temperature of--17°C in an ambient temperature of 0°C. The temperatures of the freezer and cooling module were monitored with a simple multiprobe electronic thermometer which used silicon diodes as economical and accurate probes. As the experiment progressed the growth of ice in the slot (Fig. 1) was monitored by viewing the ice perpendicular to the plane of the slot through the side of the block. For this purpose (as distinct from the photoelastic measurements made in the perpendicular direction) the block was between crossed polaroids and illuminated by white light. The bireffingence of the ice crystals made them stand out against the water. The ice was photographed in col-
our by use of a 200 mm telephoto lens mounted on a bellows to obtain the necessary magnification. 3.2 Acquisition of photoelastic data The polariscope is shown in Fig. 3. The light source was a 2.5 mW He-Ne laser, the beam from which was passed through a polaroid film and a mica ¼X plate, cut for a wavelength of 632,8 nm, and oriented with its axes at ¼rr to that of the polaroid. The resulting circularly-polarized light beam was expanded by a microscope objective; at the same time it was spatially filtered to remove non-uniformities generated by off-axis modes within the laser tube and also by dust on the preceding optical elements. The expanding beam was then collimated by a lens and passed into the freezer to be incident upon the model. After passing through the model and out of the freezer the now elliptieally-polarized beam was analysed by a polaroid film rotating at a frequency of 125 Hz. The light, now modulated at 250 Hz, passed through the field lens, which was set to form an image of the model in the plane P. This image was viewed by a microscope in which the eye piece had been replaced by a sensitive photodiode. The microscope was capable of movement in both x- and ydirections. Its objective lens was chosen so that the sensitive area of the photodiode was equivalent to 0.3 × 0.3 mm on the model. Figure 4 shows the path of the subsequent signal processing. The photodiode detector output was amplified and processed by several modules to defive dc voltages proportional to the modulation amplitude IolsinAI, the dc level Io and the phase of the modulated signal relative to a reference signal. This phase reference was generated from the rotating polaroid analyser by using a fight-emitting diode and photodiode arranged to detect a black-white pattern on the hub of the rotating f'tim. The x,y coordinates of the microscope were converted to dc voltages by linear potentiometers linked to the driving stage. All these dc values were passed via an analogueto-digital converter to a PET microcomputer where they were further processed and stored on tape. The program for the PET was written in BASIC, except for the critical data-input and averaging routines, which were written in machine-code.
145 polariser laser
I,
image I p[anePl
model
I
/
, II < ~ ,w~ I "
1/4 av spati~ plate fdter
"~ rotating motor analyser -
microscope & detector = I I
I
Fig. 3. The polariscope.
I
I
AmplitudeImax~--~-11[,L~Data Acquisition&Analysis
Po[ariscope~ I Detector
I i
I ,,- orl !
- ~level ~~ d.c. I
l iIl ~
Phase
,I
II'
coordinates J l
I
sensor
I ; I
I
Channel selector AtoD converter
PET microcomputer]
IT
I
] Analyser
"1 cycle trigger , Signal processing
I I
Fig. 4. Dataacquisitionand analysis. 3.3 The photoelastic models Constructing a suitable model (Fig. 1) for twodimensional photoelastic stress measurement proved quite difficult. The main constraints were (0 that the stress generated by the ice should be two.dimensional, and (ii) that as the pressure in the water increased, there should be no leaking of the water from the sides of the slot (that is, the front and back faces in Fig. 1), which would relax the stress. Constraint (i) was met by using a thick model where the thickness of the block was large compared with the width B of the slot, so that a condition of plane strain was approached. Constraint (ii) proved more difficult to satisfy. At first, glass plates were clamped on either side of the perspex models, separated by a layer of grease, which would both lubricate the block, allowing two-dimensional strain, and also
prevent escape of water from the slot in the third dimension. This was unsuccessful on two counts: (i) the water was forced into the lubricant layer, distorting the optical path and, of course, causing a relaxation of stress; (ii) the temporary birefringence from the thermal distortion of the glass clamping plates proved to be of the same order of magnitude as the birefringence in the model, so causing large errors. Figure 5 shows the final solution. The perspex block AA' was made in two halves cut from a single plate. The profile BCD and the outer faces were machined, a fly-cutter on a vertical mill being used to reduce machining stresses to a minimum. The two halves were joined together with cyanoacrylate adhesive, which was very effective and had the advantage of setting rapidly and producing a very thin glue line.
146
scan lines
elevation
A
adhes~ E
cyanoacrylot e I ptan
silicone rubber
Fig. 5. Construction of the perspex model. Two perspex plates E and E' were glued to the faces of block A' with cyanoacrylate adhesive. Note that they are f'Lxed only to A' and not to A, to allow free relative movement between A and A'. The crucial elements are the two very thin seals of silicone rubber arranged to prevent the water escaping from the slot. They constrain the water in the slot even under considerable pressure and deform so slightly that there is very little relaxation of the pressure. At the same time they provide almost no hindrance to the free relative movement between A and A'. This technique was vital for obtaining reliable results.
4. RESULTS 4.1 Preliminary experiments Before the main quantitative experiments on the freezing of water in a slot we did a number of simple qualitative experiments. In these we found that the temperature environment of the block affected the way in which the ice grew in the slot. Pre-cooling the block to --10°C followed by surface cooling resulted
in supercooled water, which then froze suddenly with much dendrite formation. Seeding the water with silver iodide crystals prevented the supercooling but encouraged a number of large dendrite-like structures. On the other hand, cooling the surface of the block in an ambient temperature of 0°C produced a smooth ice front which progressed evenly down the slot; these were the conditions we chose for the main experiment to be described in § 4.2. In these preliminary experiments we also measured how much the ice extruded during freezing. This is clearly an important parameter; no extrusion would imply that the change in volume of the water on freezing was entirely taken up by deformation of the perspex block, with resulting high stresses. We found that the water level in the slot changed during the initial cooling of the block, and before ice began to form; this had to be taken into account before deducing the extrusion. For cylindrical holes in large blocks the extrusion was close to the 10% expected from the volume change on freezing. This was also the case for parallelsided slots sealed well at their sides, but not by using the technique of Fig. 5, for the sealing not only prevented the water leaking out but also restricted the opening of the slot. Whenever the extrusion was zero there was leakage of the water. In the main experiment (§4.2), however, the measured extrusion was 3% and no water leaked; as already explained (§ 3.3) the sealing in this case (Fig. 5) allowed the slot to open freely while still keeping the water in. 4.2 The main experiment Water was frozen in a parallel-sided slot made in a block (70 X 70 × 32 mm) constructed as described in § 3.3. The slot had width 1 mm, length 32 mm and depth 25 mm. Before the experiment the block was carefully annealed to minimise residual stresses, although they were allowed for in the analysis as explained below. The freezer was set to an ambient temperature of 0°C and the cooling module made the minimum temperature on the upper surface of the block --17°C. Axes are taken with Ox (horizontal) perpendicular to the slot andOy (vertical) parallel to it. The photoelastic parameters in the perspex during the cooling were measured along lines parallel and perpendicular to the slot faces (Fig. 5). These were
147
subsequently analysed as described in § 2 to give the stress distribution along the slot as a function of time. The constant (Oy)o in the integration of eqn. (3) along the vertical scan lines was uncertain, because the corner B was in contact with the cooling module and had high residual stresses. Therefore we relied on the horizontal scan lines, which start on the vertical free surface, to provide the necessary constant of integration, (Ox)0 in this case. Residual stresses, which were of the same order of magnitude as the water stresses, were allowed for by subtracting them. The maximum non-residual stress measured, which was 10 bar, was a small fraction (0.19) of a fringe. Figure 6 shows the progression of the ice front down the slot. The measurements are taken from photographs of the ice during the experiment. Three lines are plotted: (I) the ice surface, showing a small amount of extrusion; (II) the position of the icewater interface; (III) the position of the boundary of
1.0
Top of slot
I -x.--
m~ x
--,x - -~x--~l¢-- x "°'x'x~--)<'x~<~'-)e'cx~<-x-xl<~<'~x
0.7
c'~\Cloudy Ice
(b) ,,
0-6
o
~
0.5
(0 "%
0.4
....
(e)>,,
Water
"o,
0-3
(e) " Q
,
o,
0.2
Ice
IT ,Q
/,o (~J "~%.%.~
0-1
o.
0
I
I
I
1
I
I
I
I
C(jJ ~¥
2 time [hrs}
Fig. 6. The motion of the ice front down the slot. l is the length of the slot and y is the distance measured from the bottom. The letters in parentheses refer to Figs. 7a-g. (I) top of ice plug; (II) ice-water interface; fill) boundary between clear and cloudy ice, interpreted as the upper limit of contact between the ice plug and the slot walls.
a further feature that we now describe. As the ice progressed it became cloudy and indistinct over its upper portion, a front dividing cloudy from clear ice being formed parallel to, but some way behind, the ice-water front. The stress data in fact suggest that the cloudy area is caused by the ice separating from the slot walls rather than pressing on them. We discuss the reason for this not entirely unexpected phenomenon in § 5, but here we remark that it provides a check on the accuracy of the stress measurements, because both the normal traction ax and the shear traction rxy should be zero along this part of the slot. By this criterion we estimate the random error in each individual measurement of Ox and rxy as +0.5 bar. Figures 7 a - g show the normal and shear stress components ax and rxy along a line near to the slot wall and parallel to it, as a function of time. The positions of the ice-water front and the cloudy front are also shown. Each of Figs. 7 a - g is indicated on Fig. 6 by the corresponding letter. The stress component ey parallel to the slot face is not shown on the diagrams because its value was strongly affected by the presence of residual stresses, which had a large gradient perpendicular to the slot face and close to it. After correction for the residual stresses the values of ey were consequently much less certain than the values of ax and rxy, which were much less severely affected by such effects. Three zones of interest are evident: (a) Below the ice-water front, in a region subject to hydrostatic pressure, we might expect Ox to be uniform and rxy to be zero. The measured value of ax is not quite uniform; it sometimes shows a peak near the ice-water front itself (see especially Figs. 7c, d). The shear traction rxy, on the other hand, shows an unexpected departure from zero at the lowest measured point. We attribute both these apparent non-uniformities to systematic errors of measurement which arise from the following sources: first, the patch contributing to each stress measurement is of finite size, 0.3 X 0.3 ram, and so the vertical scan lines necessarily cannot be exactly at the slot wall, and second, there are substantial spatial gradients in the stresses both near the ice-water interface and near the bottom of the slot. (b) Above the cloudy-clear front both ax and rxy are essentially zero, within the random error of the
148
measurements. The explanation is simply that the ice has separated from the slot wall. (c) Between the cloudy-clear front and the i c e water front is a transitional region where the surface traction changes from hydrostatic pressure in (a) to zero in (b). In this region the ice is in contact with the wall and exerts a frictional force as it tends to extrude. The measured shear stress is of the correct sign to be explained as the effect of friction. Figure 8 shows the hydrostatic pressure p in the water as a function of the fraction of water frozen.
The maximum pressure is 11 bar and the relation is essentially linear. 5. A theoretical model
The purpose of this section is to show how the quantitative observations of §4 can be understood, and then in § 6 we apply this understanding to ice in cracks in real rocks. Suppose the slot of length l (Fig. 9a) were initially completely filled with water (actually the water level was just below the top) and that
/x
? o ¢D
/:
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Fig. 7. Distribution of shear traction "rxy and normal traction trx on the walls of the slot at different stages (a)-(g) of freezing (the width of the slot is exaggerated). The lines simplyjoin the experimental points.
freezing has progressed to PQ, a distance s from the open end. Now consider a small advance ds of the freezing front to P'Q'. Because ice expands by 10% on freezing, the pressure in the water will increase and will force the walls of the slot further apart. There is an initial difficulty in understanding how this is possible without breaking the water seal and releasing the pressure. If the ice were rigid rather than elastic, and if the slot walls remained plane, they would
be forced outwards as in Fig. 9b and the water seal would consist merely of line contacts at P and Q. In fact the line PQ in the ice is under lateral elastic compression, and as the slot walls are forced apart by the increased water pressure the compression of PQ is relaxed slightly but not destroyed, thus preserving the seal. RS in Fig. 9a marks the observed boundary between the clear ice below and the cloudy ice above. As mentioned in § 4.2 we interpret this observation
150 angle is smoothed out and the walls do not consist of plane segments, but nevertheless the model of Fig. 9c is a useful starting point. We assume that the sharp angle moves down with the ice front. Then, ignoring any extrusion of the ice plug at this stage, we can readily calculate the angle t~ by conservation of mass. To first order the mass of water in the length ds equals the mass of ice in the length ds, plus the mass of water in the new volume made by expansion of the slot below PQ. The latter is (for unit length perpendicular to the diagram)
1Z P (bars)
o
O
8
0 °
4
0
0.2
0.4
o
0.6
08 F
pwot(l -- s)ds
1.0
Thus
Fig. 8. The pressure p rises approximately linearly with the fraction F of water frozen.
PwBds = piBds + pwa(l -- s)ds where Pw and Pi are the densities of water and ice respectively, and where B is the length PQ. Hence
to mean that the forcing apart of the slot walls has caused the ice to lose contact with them above RS. This is confirmed by the fact that the measured traction on the slot wall is zero above RS. (The ice appears cloudy probably because of the condition of the separated surfaces.) Thus the actual contact areas are PR and QS. To explain the observations we must take account of the deformation of the slot walls from planes and Fig. 9c displays the essential feature. In this idealization the slot walls are supposed plane and parallel down to PQ where they bend sharply inwards by an angle a, but continuing plane. In practice the sharp
pi ) B
Putting B = 1.0 mm, l -- s = 9.3 mm (corresponding to Fig. 7e) and Pi/Pw = 0.91 gives ~ = 0.010 rad = 0.55 ° . Thus the 10% change of volume made by freezing can be accommodated if the slot wall bends through this angle at the moving freezing front. (As l - s decreases to zero the required ot obviously approaches infinity, which implies either cracking or plastic deformation at the bottom of the slot.)
',/I I// S ,
t I
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ds .
.
.
.
s
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t t
Winter
4
-s
(a)
t
(c)
(d)
Fig. 9. Not to scale. (a) shows notation; PQ is the ice-water interface; (b) simplified model with rigid slot walls; (c) simplified model with the slot wails bent through an angle ~; (d) model allowing for extrusion and non-parallel slot walls.
151 Let us now make the model more realistic by allowing the slot walls below PQ to be slightly curved and eliminating the sharp angle in the walls at P and Q. This is easily done by noticing that the above calculation is still correct provided ads is reinterpreted as the average opening of the slot in the water space below PQ. As a further step we should allow some extrusion of the ice plug, for it was observed (Fig. 6) that during freezing the ice was extruded from the slot (by 3%) but not by enough (10%) to account for the whole volume change. As the freezing front moves down by ds suppose the ice plug moves up by fds (Fig. 9a), where f i s a fraction lying between zero, for no extrusion, and 0.1 for maximum extrusion. Then expression (4) for a is modified to -
kB l--s
(5)
de -
pds'
(7)
Ec
where E is the appropriate elastic modulus of ice. The ratio of the first and third terms in eqn. (6) is, using eqn. (7)
Bp
Bde m-
where k = (#w -- Oi --fPi)/Ow, a measure of the fractional expansion after allowing for extrusion. If f = 0 (no extrusion), k = 0.09; while if f = 0.1 (maximum extrusion), k = 0. Let us now look at the extrusion process in a little more detail. When extrusion is allowed in the model it is important to take account of the fact that the slot walls above PQ will not be quite parallel. So let them (Fig. 9d) be inclined inwards, at an angle ~. As the freezing front advances from PQ to P'Q' the ice particles at P,Q move up by fds and outwards to P", Q", the slot walls meanwhile moving out by da. Thus the opening by 2da is the result of a wedging action of the tapering ice plug as it is extruded upwards. The ice fibre PQ is initially formed under compression along its length and as it moves up the slot the pressure on it is observed to fall. It will therefore experience an elastic relaxation strain de (an extension). Hence by geometry
Bde= 2da -- 2f/3ds
from RS to R'S'. By observation RR' and PP' are respectively 20 and 30 times greater than PP"; therefore, in estimating the lowering of pressure on the ice fibre PQ, we can ignore its movement. Since both the pressure in the water and the contact length RP increase approximately linearly with the fraction of water frozen, the gradient of pressure along the contact remains approximately the same. The reduction in pressure on PQ is then p(ds'/c), where ds' = R R ' and c = RP, and the corresponding strain is
2f/3ds
ds' •
2Ec
_
_
ds
With the observed values (corresponding to Fig. 7e) p = 6 bar, ds'/ds = 0.55,c = 6.4 m m , f = 0.03, w i t h e = 9 × 104bar, the ratio is 9.5 × 10-s//3. So the ratio is small if/3 >> 9.5 × 10-s, which corresponds to a taper of merely 2 /am over the length of the slot. The machining and polishing of the slot will certainly have produced variations much greater than this and so the third term in eqn. (6), which is due to taper, will certainly overwhelm the first one, which is due to the elasticity of the ice. This means that eqn. (6) becomes da =f~ds
(8)
which simply says that the wedging action by extruding ice takes place as if the ice were rigid. Now we can compare the opening of the slot just above PQ, namely 2da, with the average opening in the water below PQ, namely ads, a being given by eqn. (5). Thus
(6)
2da We now show that, in our experiment, the elastic term Bde is negligible compared with the other two terms in this equation. To do this we have to estimate the lowering of pressure on the ice fibre PQ. The normal pressure on the sides of the ice plug falls from p, the hydrostatic pressure in the water, at P to zero at R (Fig. 9a), and it is consistent with observation to take the variation as linear. As the ice-water boundary advances by ds the line of zero pressure advances
ads
-
2f/3(l-- s) kB
(9)
Putting in the same numerical values as before (f = 0.03 corresponds to k = 0.063) gives 8.9/3. Since/3 is certainly much less than 1/8.9 in this experiment the ratio is small. That is, the elastic widening of the slot takes place predominantly in the unfrozen water zone, rather than by any wedging action as the ice plug is extruded upwards.
152 Our main conclusion from this analysis is that the simple model of Fig. 9c (with some extrusion and associated wedging action included, and with the sharp bend in the wall of the slot smoothed out) does in fact capture the essence of the expansion process that occurred in the experiment. It is clear that the process we have described would not work if a slot with plane walls tapered downwards (~ negative). Even a slot with rough or wavy walls would allow the ice plug to extrude freely if it tapered downwards sufficiently. Our model for a build-up of pressure relies on there being hindrance to extrusion; this is represented in the model with plane walls by a positive angle ~, and more generally we assume it is provided by non-parallelism of the walls. We note, however, that in the experiment the walls did part company with the ice above RS (Fig. 9a); this is evidently because, above this level, the pressure distribution forces the wails apart by an amount that is not compensated by any wedge-like upward motion of the ice plug. It is now apparent that, since the slot in the experiment was not plane parallel to optical standards (the 2 t~m calculated above), and was not intended to be, and since some extrusion was observed to occur in spite of the waviness of the interfaces, plastic deformation took place in the ice, certainly near the two interfaces and possibly within its bulk too. In either case it is plausible to use a plastic friction law r = constant = ko, where ~" is the average shear traction at the wall. Considering the forces on the ice plug, the upward force from the water on the lower end balances the downward frictional force from the walls 2k0 pB = 2 koc or p = c (10) B
Figure 6 shows that c increases approximately proportionally to the fraction of water frozen and hence, from eqn. (10), so does p. Thus a plastic friction law accords very well with the linear increase of p observed in Fig. 8. The value of ko deduced is 0.5 bar, which is plausible. A plastic friction mechanism of this kind was first suggested by Professor K.E. Puttick (private communication).
6. IMPLICATIONS FOR REAL ROCKS
In the light of the experiments and the model which explains them, what can we infer about the stresses produced by ice in real rocks? Most real rocks contain water in pores of various sizes quite apart from macroscopic cracks. Mellor (1970, 1971, 1973), in particular, has drawn attention to the fact that pore water in typical rocks freezes progressively as the temperature is lowered and that some is virtually unfreezable at natural temperatures. As the pore water freezes in wet rock there are sharp discontinuities in the measured thermal strain. Such strains would be especially effective in producing cracks when they are inhomogeneous, and two sources of inhomogeneity are readily available, namely, gradients in temperature and in water content. Such considerations suggest that one should be cautious in relating frost damage simply to the expansion of bulk water in macroscopic cracks. Nevertheless, assuming that such freezing does occur, it is important to trace its consequences. A feature that emerges very clearly from our discussion is the importance of the contact area between ice and rock, in a given crack, in determining the possible water pressure. Fresh cracks in rocks do not have plane faces; they are commonly sinuous and their faces may be rough on a variety of scales. It is true that the expansion of the crack as freezing progresses tends to release the ice plug, but our analysis suggests that this is a small effect because most of the expansion occurs in the water-riffled region below the level of the ice front. In any case, the shape of the crack means that only a small additional extrusion of the plug can be enough to preserve the water seal. The roughness and sinuosity of real cracks thus increase the contact area between ice and rock so that it may well be effectively the whole area of the crack. One can now distinguish two possible regimes: (1) where the ice plug is either extruding or is on the point of extruding, and (2) where the ice plug is essentially fixed and grows in situ. In case (1) the irregularity of real cracks leads one to expect plastic deformation of the ice, and relation (10) would be applicable with c taken as the distance from the outer
153 surface to the freezing front. With 2k0 in the range 1 - 1 0 bar, and with the ratio of crack depth to crack width c/B in the range 100-1000, pressures of 102104 bar, quite sufficient to extend the crack, would be reached. Thus in case (1) the pressure can be calculated just from the yield strength of ice; we do not need to know the elasticity of the rock. On the other hand, in case (2) the pressure developed depends essentially on the elastic and strength properties of the rock, because these control the opening of the crack. What then decides which regime is applicable? As c/B increases, the pressure for ice extrusion eventually exceeds the pressure needed simply to expand the unfrozen water by 10% against the elasticity and strength of the rock. It is at this point that case (1) changes to case (2). Apart from the matter of the progressive freezing of pore water mentioned above, there are two main differences between our experiment with perspex and conditions in real rocks: first, the shape of the crack is different, plane and fairly smooth rather than sinuous and rough, and second, perspex is elastically softer than most rocks. Our experiment is dearly in regime (1), with extrusion. This suggests that it would be useful to do further photoelastic experiments with perspex in which the slot was made deliberately sinuous. One would also wish to vary the ratio c/B to test relation (10). No new experimental difficulties would be expected except perhaps with the artificial seal used to prevent the water leaking out sideways. This was a crucial element in the experiments; it is effective up to 10 bars, but its upper limit is unknown. As c/B was increased there would come a point where the ice ceased to extrude and a new theory for determining p, appropriate to regime (2) and explicitly involving the elasticity and strength of perspex, would have to be invoked. It is worth remarking that
an artificial seal was only necessary because, in order to apply simple photoelastic methods, we needed to make the experiment two-dimensional. In the real, three-dimensional case a wave of freezing penetrating from the exposed surface of the rock would make a natural seal. ACKNOWLEDG EME NTS
The experimental technique described was evolved after a number of preliminary photoelastic experiments had been carried out in this laboratory by J.R. Cave, Miss L.J. Hill and Mrs. K.M. Branson, and after helpful discussions on the physical mechanism of frost cracking with Professor K.E. Puttick. We acknowledge with gratitude their work, the advice of Dr. M.E.R. Walford, and correspondence with Dr. M. Mellor. REFERENCES
Dally, J.W. and Riley, W.F., 1978. Experimental Stress Analysis. McGraw-Hill,New York, NY. Embleton, C. and King, C.A.M., 1975. Periglacial Geomorphology. Arnold, London. See index heading freezethaw,
Kuske, A. and Robertson, G., 1974. Photoelastic Stress Analysis. Wiley, London. McGreevy, J.P. and WhaUey,W.B., 1982. Arct. Alp. Res., 14: 157-162. Mellor, M., 1970. U.S. Army Cold Regions Research and Engineering Laboratory, Research Report 292. Mellor, M., 1971. U.S. Army Cold Regions Research and Engineering Laboratory, Research Report 294. Mellor, M., 1973. In: Permafrost: the North American Contribution to the Second International Conference. ISBNO309-02115-4, National Academy of Sciences, Washington, DC, pp. 334-344. Mfiller, R.K. and Saackel, L.R., 1979. Proc. Soc. Exp. Stress Anal., Exp. Mech., 36: 245-251.