The tensile stress generated in clay through drying

J. agric. Engng Res. (1987) 37, 279-289

The Tensile Stress Generated in Clay through Drying G. D. TOWNER* A tensile stress is generated in a drying bar of clay that is prevented from shrinking in one direction. Theory predicts that after an initial period of drying the stress is equal to a constant fraction of the equivalent soil-water suction P (as inferred from the water content). The latter stage is independent of the initial water content of the bar. Calculations of tensile stress using critical state soil mechanics on published data for seven remoulded clays gave values in the range 0-15P to 0.22P. The clay cracks at a unique water content, which is independent of the initial water content, and can be determined if the relationship between tensile strength and water content is known. Published experimental results on kaolinite confirmed the uniqueness of the water content at cracking but provided poor agreement with the predicted value. Some inadequacies of the experiments are discussed.

1. Introduction Towner 1 described investigations on the mechanics of cracking of drying clay. Bars of clay were allowed to dry from different initial water contents but prevented from shrinking in the longitudinal direction. Their water contents were measured when they cracked and were found to be essentially the same independent of the initial water content. Parallel experiments were performed to determine the tensile strength of the clay as a function of water content, and the soil-water suction as a function of water content. The tensile stress generated by the shrinkage can be shown to be equal to the total change in soil-water suction when the shrinkage is isotropic. The experiments showed that the change in suction was considerably greater than the measured tensile strength, except at the start of drying, so that the tensile stress was less than the change in suction. The aim of the present paper is therefore to examine in more depth the mechanics of the stress system leading to cracking. Though the principles of the method involved can be extended to cracking in bodies of clay with different boundary conditions, the present paper is restricted to those pertaining to the clay bars described above. Other conditions, such as semi-infinite bodies that occur in the field, will form the subject of a further paper. 2. Theory Consider the shrinkage of a drying bar of clay with ends in the longitudinal direction fixed as indicated in Fig. 1 (left). The shrinkage takes place through a decrease in the width and depth of the bar until the increasing tensile stress induced in the longitudinal direction exceeds the increasing tensile strength, at which stage the bar cracks across the longitudinal direction: The bar is subjected to the system of stresses shown in Fig. 1 (middle) together with a negative pressure, -luol, in the pore water. Though the induced tensile stress, aT, is a direct consequence of the changing soil-water content and pressure, Fig. 1 (middle) can also be regarded as representing a bar of clay subjected to the combined actions of a negative pressure in the pore water and a tensile stress applied to each end in the longitudinal direction of such a magnitude that the ends remain fixed in position. * Rothamsted Experimental Station, Harpenden, Herts AL5 2JQ, UK Received 30 June 1986; accepted in revised form 18 October 1986

279 0021-8634/87/080279+ II $03.00/0

9 1987 The British Society for Research in Agricultural Engineering

280

TENSILE STRESS IN DRYING CLAY

Notation a2, b2

a3, b3

k P P2, P3 q u Uo

F(P) go

P 7

6e1, ger, cSes, 6~v

coefficients in Eqn (A2) coefficients in Eqn (A1) ratio of radial stress to longitudinal stress spherical pressure ( = (0-~+ 2at)/3) spherical pressure for anisotropic and isotropic stresses respectively deviator stress ( = a~-at) pore-water pressure pore-water pressure during drying, a negative quantity (I/~ol = -Uo, a positive quantity specific volume of saturated clay a mapping function that relates the anisotropic spherical pressure that is in equilibrium with a specified water content to the isotropic spherical pressure, P, that is in equilibrium with the same water content (Fig. 2) earth pressure at rest isotropic pressure applied in a normal consolidation test specific gravity of the clay mineral solids longitudinal, radial, shear and volumetric strains respectively

t/ stress ratio q/p I12 stress ratio r/for anisotropic compression

gravimetric water content gravimetric water contents defined in Fig. 5 K slope of the swelling curve in the plot of specific volume against the natural logarithm of the pressure slope of the normal consolidation curve in the plot of specific volume against the natural logarithm of the pressure 0-D fir, 0-T longitudinal, radial and tensile stresses respectively 0-1, 0"3 major and minor stresses respectively F , M critical state soil constants A ( 2 - x)l k 0 0o, 01

It then follows from Terzaghi's effective stress principle saturated the water content of the clay under the system of shown in Fig. 1 (middle) is the same as under that shown water pressure. The longitudinal stress, luol-aT, is positive, End

anchored

I

/

.

End

~/lll~l

Lateral / shrinkage

I

i Verlicol shrinkage (a)

0

anchored ~

/

rllllll~.

o/

/ 1

I

0

U:Uo
that provided the clay remains stresses and pore-water pressure in Fig. 1 (right) with zero porebecause otherwise the bar would luol /

0

~l//llJ_~ lu I ~

#L-

rl//ll~

/

luol

/

luol luol-~r

I

I-ol

u: 0 (c)

Fig. 1. Shrinking clay bar: (left) conditions of shrinkage; (middle) prevailing stress system; (right) equivalent compressive stress system

G.

D.

281

TOWNER

extend in the longitudinal direction, which is contrary to the specified conditions. Thus, the original problem involving tensile stresses and a pore-water pressure which is negative, has been transformed into one involving compressive stresses and zero pore-water pressure for which there are potentially extensive theoretical methods available for its solution. Indeed, the system represented in Fig. 1 (right) corresponds to a drained extension test in the triaxiai compression apparatus in which the cell pressure is increased whilst the ram is locked in position against the top of the clay to prevent its extension. To facilitate comparisons with soil mechanics concepts, the following more familiar variables are introduced: at for the longitudinal stress; at corresponds to the axial stress, also often represented by a,, in the triaxial compression test: and o-, for the lateral stress; o-, corresponds to the radial stress, also often written as o3, in the triaxial test. Thus at---lU01--aT,

(1)

o-~ = In01.

(2)

Under certain stress conditions, of which that under analysis is one (q.o.), the stress ratio k = o-r/o-t is constant. The ratio k is analogous to the coefficient of earth pressure at rest (usually symbolized by Ko). Hence, from Eqns (1) and (2), crT = ( k - 1)lUol/k.

(3)

Since it is easier to measure water contents rather than soil-water pressure, and since normal consolidation (or the equivalent water release) measurements are more routine than those involving anisotropic compression, it is convenient to use the normal consolidation curve as a reference curve. The relationship between the water content, 0, and the spherical pressure p ( = (o-~+ 2o-,)/3) is typically of the form sketched in Fig. 2. There is a separate curve (dash line) for each stress ratio, k, for a given clay. An infinite number of such curves are possible in general, but they all lie within the region defined by the positive axes and the normal consolidation curve (full line). 2 (N.B. There is no need to distinguish between effective and total stresses because u = 0 so that they are equal.)

x,

JB

,D

Spherical pressure, p

Fig. 2. Curves o f water content versus spherical pressure. - - - , anisotropic stresses --, isotropic stresses (normal consolidation curve)

(hypothetical);

282

TENSILE

STRESS

IN

DRYING

CLAY

Two specimens of clay represented by points A and C respectively in Fig. 2 are at the same water content but under different stress systems. There is a one-to-one relationship at each water content between the corresponding pressures indicated by B and D respectively, say P2 =F(p3); where the subscripts 2 and 3 denote anisotropic and isotropic pressures respectively. Hence, if the water content of a clay sample under anisotropic stresses is measured, then the magnitude of the spherical pressure can be inferred from the normal consolidation curve in conjunction with the F(p3) function. Hence, from P2 =

trr = [3k/(1 +2k)qF(p3).

F(p3),

Furthermore,

(4)

F(pa) = F(P),

where P is the isotropic pressure applied in the normal consolidation apparatus. Hence, Eqns (4), (2) and (3) give tr~r = [ 3 ( k - 1)/(2k + 1)IF(P).

(5)

The values of k and F(P), and hence a T, can be calculated for a range of clays using critical state soil mechanics. The analysis, which is based on the modified Cam-Clay model, ~ is given in Appendix A, and only the essential results are given here. The value of a T depends on two critical state soil parameters, M and A (both dimensionless). Very roughly, M can be regarded as a measure of the shear strength expressed as a fraction of the compressive load, while ( 1 - A ) is the ratio of the slope of the swell-back curve to that of the normal consolidation curve in the semi-log form of plot. The parameter M can theoretically take on any value between zero and infinity, but in nature will only be very large for hard rock. All values measured for soils to date lie s within the range 0.7 to 1.2. The parameter A can theoretically lie between zero and unity, but it will probably lie well within these values because a zero value corresponds to a material that on rewetting would swell back up the consolidation curve, which is very unlikely; while a value of unity corresponds to a non-swelling material, which is contrary to our assumption. Indeed, for five clays listed by Schofield and Wroth, 4 0-45 < A < 0.81. The analysis predicts that, except close to the initial water content, 0.08P~
for

0.7~
and

0~
and for the more restricted but probably more realistic range of A values, 0.09P ~
for M (as above)

and

0.45 ~
Values of a T calculated for a range of clays reported in the literature are presented in Table I. It is noteworthy that a T ~<0.22P for all the clays, while a T = (0-20+0.02)P for six of them. This suggests that the soil constants M and A occur together in nature in such a way that the range of practical values for a T is narrower than given above. The values for the three kaolinite samples lie within the range (0.18_+ 0.03)P. The induced stress a T is zero at the beginning when the clay bar is moulded, and increases as the water content decreases until it is equal to the value given above. A consequence of this is that aT for a given clay is independent of the initial water content for water contents greater than a certain value. Hence, irrespective of the initial (stress-free) water content the given clay bar will crack at a unique water content.

3. Comparison with experiments The experimental results reported by Towner 1 provide a basis for testing the theory. In these, clay bars cracked at a water content of 0.32 kg/kg. Thus, the tensile stress ~rr generated by the shrinkage was less than the tensile strength at all water contents larger than

G. D. T O W N E R

283 Table 1 Critical state soil constants and calculated tensile stresses

Soil

Klein Belt Ton Wiener Tegel V London Clay Weald Clay Kaolin Kaolin Kaolin

M

A

Calculated aT/P

0"845* 1"01"

0.483* 0.788* 0-614" 0.628* 0.807* 0.85f 0.78t

0-22 0-20 0-20 0.22 0.20 0.15 0.18

0"888* 0"95* 1"02" 0"90? 0'90t

Sources: *Schofieldand Wroth4; 1Roscoe and Burlands 0'32 kg/kg. T h e calculated aT is plotted in Fig. 3 as a function of water content, 0, for aT = 0"I5P (dashed line) and aT = 0"20P (dotted line), using the regression line of In P and 0 given in Fig. 6 of Towner 1 (reproduced in Fig. 4). The corresponding tensile strength data points are also plotted in Fig. 3. Even allowing for the possibility that the water content change over which aT changes from zero to its fully developed value is not small, these curves are clearly wrong, intercepting the tensile strength curve at 0 << 0-32 kg/kg. In fact, the curves need to be steeper. This poses the question whether it might be possible to put a steeper curve through the data points in the plot of suction versus water content (Fig. 4) that would still be a valid representation of the data points and lead to agreement with theory (i.e. in effect perform a sensitivity test). The line regressing 0 on In P (not shown) is marginally steeper, but not sufficiently so. Hence, the dashed line has been drawn by eye as approximately the steepest feasible curve that represents at least some of the data. The stress calculated from it using aT = 0"20P is shown by the d a s h - d o t line in Fig. 3. The dotted line in Fig. 4 represents a parallel curve that leads to the same stress curve shown in Fig. 3 when aT = 0"15P. \

%

2.o'

I

I

i

9 ~ID'B .-

t.,,,~

,,%,.

.

I-0

z~,

9

\.

9

0.5 -~. 0.2

.~

"\

' , 9 ...

~ ~ ~.

0-1

\

'.....

I~

'~" -.....E . ~ x . ~ "'''''''.

"~O

o-o5

",i'- oI

0.30

0"35

I

I

0"40

0"45

Woter content, kg/kg

Fig. 3. Plot o f predicted tensile stress, crT, versus water content, O, and measured tensile strength versus water content respectively. Predicted tensile stress f r o m original soil-water suction data, assuming crT = 0.15P, . . . . , assuming a T = 0. 20 P ; f r o m revised soil-water suction data, - . . . . . Tensile strength f o r different initial water conents, O , 0.80 kg/kg; Z~, 0.64 kg/kg

284

TENSILE STRESS IN DRYING CLAY i

20-0

~1.

ko

i

~\ -"

~

~..

I0"0 -~ 5<) E 6 o 2"O

o

1.0

0"5 ~z~- o \ ~ ". \o \.. o\ \'. o

0.2

\

0.1

i

0.20

i

i

i

0.40 0'60 Water content, kq/kg

i

i

0"80

Fig. 4. Plot o f soil-water suction versus water content f o r different initial water contents, O , 0-80 k g / k g ; A , 0.64 kg/kg. - - . regression line through 0 data points," revised line ( - - - ) Jbr a T = 0.20P; ( . . . . ) f o r a T = 0.15P

The result is encouraging. Indeed, given that the dashed line in Fig. 4 represents fairly closely the triangles (which correspond to an initial water content of 0.64 kg/kg), and assuming that it is valid on extrapolation to smaller water contents, then the predicted stress in Fig. 3 is approximately correct for this case if trr = 0.20P. There is uncertainty on the value of 0 at which a r attains its fully developed value, but it is conceivable that aT remains smaller (perhaps only slightly) than the tensile strength until at least 0 = 0.45. In fact, the dashed line in Fig. 4 does not represent all the data points, but neither does the original regression line: The best line is one in between. However, that may not be the correct line. The data, though adequate for testing the earlier hypothesis, include too much scatter to use with the present one. Further discussion of it is therefore unprofitable. A prediction of the theory is that the clay bars crack at a unique water content independent of their initial (stress-free) water content. The results confirmed this very accurately. In fact, further experiments with clay drying under a different equivalent stress system and hence to a different unique water content at cracking have added further confirmation to this prediction. 4. Discussion

The principle of the analysis, by which a system of tensile stresses acting on clay with a negative pressure in the pore water is replaced by an equivalent stress system involving only compressive stresses and zero pore-water pressure, can be extended to other situations. For example, vertical cracking in the field during drying corresponds to uniaxial compression in the oedometer. Another example of practical interest is the drying, shrinkage and disintegration of soil crumbs, which, though more complicated to analyse, should be amenable to a similar approach. The present paper exploited critical state soil mechanics to calculate results for a range of clays to provide background data for examining some published experimental results.

G, D. T O W N E R

285

Fortunately, the theory predicted that the tensile stress did not depend very strongly on the type of clay, so that the calculated results could be regarded as applicable to the experimental measurements. But, the method of analysis does not depend on critical state soil mechanics. The required data can be obtained by direct measurement on the given clay. Such measurements are not necessarily routine. Indeed, those involving zero strain on some boundaries must include stress measurements on those boundaries, and these are difficult to make without serious error. The analysis assumed that the cracking occurs while the clay is still saturated. This is probably realistic, since the largest shrinkage takes place during this stage of the drying. However, if the potential cracking predicted on this assumption lies in the unsaturated state, then a modified analysis will have to be applied for water contents less than that at the initiation of desaturation. Such analysis will inevitably be very complicated, and the effort is not warranted at this stage, at least not until it is shown to be necessary. If the clay attains the shrinkage limit throughout its mass, then it will not crack at all. The uniqueness of the cracking water content, independent of the initial (stress-free) water content, suggests that it can be regarded as a soil property. However, its value depends also on the stress system applied to the soil. For example, it is found (and this is the subject of current research) that for a clay subjected to a stress system that is related to that causing vertical cracking in the field, aT = 0-34P instead of 0"20P. Thus, unlike metals, it is not possible to obtain cracking data for any general situation from measurements made on standard test pieces. On the other hand, calculations performed on the seven soils listed in Table 1 yielded a value of 1.67 (S.D. = 0.05) for the ratio of the two tensile stresses, so that if this proves to be a general result, then an estimate based on applying this factor to measurements made on such standard test pieces might be accurate enough for many purposes. The analysis for the beginning of drying requires further development in order to ascertain the water content at which the clay attains its fully developed anisotropic state. However, this transitional region may be an artifact of the experimental procedure (which requires an initial placement of the clay in a mould) and therefore of little practical significance, though it may represent recently deposited soils such as in spoil heaps. A tentative solution is to assume that, when the predicted tensile stress versus water content line (dash-dot line in Fig. 3) intersects the tensile strength curve in two positions as in Fig. 3, the relevant intersection is that at the smallest water content. In other words, to assume that the fully developed anisotropic state is not attained until the water content is less than that of the intersection at the largest water content. 5. Conclusions A tensile stress is generated in a drying bar of clay that is prevented from shrinking in one direction. During an initial drying stage the stress system changes from zero in a currently indeterminate way until the anisotropic stress ratio becomes constant. Thereafter, the tensile stress is a constant fraction of the equivalent soil-water suction P (inferred from the water content), and is independent of the initial water content at which the clay was stress free. In particular, the water content at cracking is independent of the initial water content. Quantitative agreement with published measurements is poor, but scatter in the results of the measurements of water content in equilibrium with suction made them of questionable value for testing the theory. However, the water contents measured at cracking were in good agreement with each other irrespective of the initial water content. Tensile stresses for seven remoulded clays calculated from published data lay in the range 0.15P to 0.22P, suggesting that there may not be very large variations in practice. Further analysis is required to determine the increase in tensile stress during the initial drying period. Further analysis should also include conditions when cracking occurs after

286

TENSILE

STRESS

IN

DRYING

CLAY

the clay has become unsaturated, but this will prove very difficult because the physics and mechanics of the unsaturated state is far from completely understood. The analysis can be extended in principle to other drying clay systems in which cracking occurs.

References 1 Towner, G. D. The mechanics of cracking of drying clay. Journal of Agricultural Engineering Research 1987, 36: 115-124 z Atkinson, J. H.; Bransby, P. L. The Mechanics of Soils. New York: McGraw-Hill, 1978 a Roscoe, K. H.; Burland, J. B. On the generalised stress-strain behaviour of 'wet' clay. In Engineering Plasticity. Cambridge University Press, 1968, pp. 535-609 4 Schofield, A. N.; Wroth, C. P. Critical State Soil Mechanics. New York: McGraw-Hill, 1968 s Corte, A.; Higashi, A. Experimental research on desiccation cracks in soil. Research Report 66, US Army Snow, Ice and Permafrost Research Establishment, Corps of Engineers, New Hampshire, USA, 1960, 48 pp. e Nieber, J. L. Simulation of fracturing of a desiccating soil:stress analysis. Paper No. 812-2512. American Society of Agricultural Engineers, Winter Meeting, Chicago, 1981

Appendix A The normal consolidation curve can be a p p r o x i m a t e d very closely by an expression of the form 0 = a 3 -b

3

In p,

(A1)

where 0 is the gravimetric water content; p the spherical pressure (u = 0);,a a and b a are constants, and the subscript 3 indicates that the stresses are the same in the three coordinate directions. Such a formulation is assumed in critical state soil mechanics. The corresponding relation for anisotropic compression in which the ratio of the stresses in the two anisotropic directions is constant is of the same form, viz. /9 = a 2 - - b 2

In p,

(A2)

where the subscript 2 indicates that the stresses are the same in two of the three coordinate directions. Furthermore, a2 < a3

and

b 2 = b 3.

Hence, the water content of an anisotropic stress system defined by the spherical pressure P2 (u = 0) is the same as that under an isotropic pressure P ( u - - 0 ) when a 2 --b 3

In

P2 = a 3 -

b3

In P,

so that P2 = P exp ((a 2 - a a ) / b 3 ) .

(A3)

F ( P ) = P exp ((a 2 - a3)/b3).

(A4)

Thus F(P) in Eqn (5) is given by The constants a2, a 3 and b a can be expressed in terms of fundamental critical state soil constants. In general, the water content is a function of p and q, where p is as already defined, and q - - t h - a r. In anisotropic stress systems in which the ratio 17--q/p is constant, it is convenient, and without any loss in generality, to use p and t/ as the variables. Hence, according to the modified C a m - C l a y model, a o = F + (2 - x) In 2 - (2 - x) In [(M 2 + t/2)/M 2] - 2 In p,

(AS)

G.

D.

TOWNER

287

where v is the specific volume of the clay; F and M are critical state soil constants, 2 and x are the slopes of the normal consolidation and swelling curves respectively in the plot of v against In p. The gravimetric water content, 0, is related to v by v = 1 +)'0, where )' is the specific gravity of the clay mineral solids. Isotropic compression corresponds to the condition q = 0, i.e. t/= 0, so that the normal consolidation curve is obtained by substituting this value in Eqn (A5) to give v = F+(2-x)

ln 2 - 2 In p.

Hence a 3 = (r' -

1 q- ( 2 - h:)

In 2)/),

and b 3 = 2/y.

Anisotropic compression is given in Eqn (A5) with r/= r/2, and hence a 2 = [ F - 1 + ( 2 - x ) In 2 - ( 2 - x )

In

{(M2+rl2)/M2)}]/y,

b 2 = 2/)'. Thus, Eqn (A4) may be expressed in terms of critical state soil constants, viz.

F(P) = {(M 2 + q~)/M2}AP, (A6) where A = ( 2 - K)/2. In order to progress further, it is necessary to obtain an expression for the stress ratio q2 for anisotropic compression with zero longitudinal strain. The general expressions for the volumetric strain, &v, and shear strain, &s, for the modified Cam-Clay model are given by fev

&s -

1[ (2-x)

= v

2-,r v

2

2r/f r/ ~1 ~M 2 + n + 2 ,

6_ffp]

"~F 2r/&l

\ M 2--

(A7)

+

(a8)

respectively) To simplify the next stage in the analysis, r/2 will be assumed to be constant (i.e. &12 = 0) in order to obtain an expression for ~2 to satisfy the condition that the longitudinal strain, let, is zero. The expression for r/2 will then be substituted into that for the radial strain, re,, to confirm that it leads to the expected result. By definition, fez = fes + fe,/3 and (A9) fie, = - 8 9 + &v/3. Hence, after substituting for fes and cSevfrom Eqns (A7) and (A8) respectively, setting r/= r/2 and fir/= 0, and rearranging, the following equations are obtained, 6el =

6r/2 A + (M 2 - r/2) 3v(M 2 _r/22) 2 Vfp,

(A10)

-3r/2A+(M2-r/2) @ fe r =

3v(M2_ r/22)

2 --.p

Therefore fe~ = 0 when 6r/2A + (M 2 -r/2) = 0, i.e. r/2 = 3All _+x/1 +(M/3A)2].

(A11)

288

T E N S I L E STRESS IN D R Y I N G

CLAY

An anisotropic stress system with zero longitudinal strain implies that tr~ < ~r,, so that q and hence r/2 is negative. Therefore '72 = 3 A l l - x / l

+(M/3A)2].

(AI2)

The substitution of r/2 into the expression for re,, leads after simplification to 2 6p re, - 2v p Eqn (A8) gives for fir/= 0, f~v -

2 fp o p

For strains occurring in only two directions, 6ev = 2fer,

so that

1 2 fp

Hence, the expression obtained for r/2 leads to the correct value of 6e, and satisfies the condition that l e t = 0 , and must therefore be the required solution. Thus our not unreasonable assumption that r/2 is constant is confirmed. By definition, r~ = q/p,

so that

(trr/th) = (3 - r/)/(3 + 2r/).

Hence, the tensile stress trx at a given water content corresponding to an equivalent isotropic pressure P is given by a T = [ 3 ( k - l)/(2k + 1)IF(P); where

(5, bis)

k = ( 3 - r/2)/(3 +2r/2);

and

F ( P ) = [(M 2 + r/2)/M2]p;

(A6, bis)

I

80 . . . . . .

8

ao a,

P

Fig. 5. Hypothetical curves of water content 0 versus spherical pressure, p, indicating the probable path, l ~ 2 ~ 3 , f o l l o w e d by the clay during the initial drying stage. 0 o is the initial water content when the clay is stress free. O~ is that when the anisotropic state becomes fully developed

G.

D.

TOWNER

289

and ~12 = 3A[ 1 - x/1 - (M/3A) 2 ],

(n 12,

bis)

in which M and A are critical state soil constants, and P is the magnitude to the soil-water pressure corresponding to the given water content. The analysis predicts that a T = 0 when P = 0, because the theory tacitly assumed that the clay was initially at zero soil-water pressure. In experiments, such as described by previous workers. 1'5"6 the clay was deposited at a finite water content, so that a T would have been zero at a non-zero value of P. This corresponds in the equivalent compressive system to first compressing the clay isotropically until it is at the initial water content, 0 o (stage 1 in Fig. 5); then increasing a t and ar so that a~/ar increases monotonically from unity to k (stage 2) to bring water content to 01 on the anisotropic curve (stage 3). Hence, in the original system, a T will increase from zero at 0 o (k = 0 in Eqn (5)) to the value given by the analysis at 01, and increase thereafter according to the theory. The determination of the form of the curve during stage 2 is very complicated. The change in water content, 0 o - 0 1 , is probably small, but this will be left as an open question for the present. We note that a lower limit for the corresponding change in p can be established by setting a~ equal to the isotropic pressure corresponding to Oo and k to its final constant value, whence 0 0 - 0 1 ,,~ 0.02 kg/kg.