Copressibility as a function of porosity for a glass
containing spherically shaped pores.
by
Anthony Wayne England
Submitted in Partial Fulfillment
of the Requirements for the
Degree of Bachelor of Science
in
Geology and Geophysics
pS
INST TFEC
ITHDP,%
and the
Degree of Master of Science
MIti rRIES
in
Geology and Geophysics
at the
Massachusetts Institute of
Technology
January, 1965
Signature
of Author *.
seol.- -r*.Jer..-s-,D-----r
.
..
..
*.
Department ef Geology and Geophyvics, December 31, 1964
Certified by
............
A4/
Aoc
cpted by .
.......-
,
K.
,-w-riurr#r
CG
Thesis Supervisor
ti.......,......i..i....
Departmental Coittuee
on Graduate Students
-2-.
Contents
Page
Abstract
Introdtction
Procedure
Experimental ResAlts
Conclusion
Appendices
Appendix A.
Derivation of a limiting
compressiblity for a very
porous material
Appendix B.
Proof that a slight
elliptioity of a nearly
round hole has negligible
effect on compressibility.
Appendix C.
Proof that measured
compreseibilities ltU
always be greater than
Mackenrlse' s prediction.
Bibliography
Figures
Figure I.
Figure I,
Characteristic
compressiblity trials
eciproecal compressibility
1 porosity
25
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-C40
Abstract
The linear compressbilities of a suite of porous samples of a
glass were measured to 1 kilobar pressure.
Porosity in the form of
spherical voids, varied from sample to sample over the range of sero
to 70 per cent.
The relation between compressibility and porosity
ase compared with the theoretical predictions of Mackenzie (1950),
Walsh (1965), and an expression developed here for the compressibility
of a material as its porosity approaches 100 per cent.
Measured compressibility
as in good agreement at low porosities
with the Mackenie relation,
ofef
where
0 eff
Oo
2 T- 1
0
is the effective compressibility of a material with a
centrally located, spherically shaped pore space, 00 and v are,
respectively, the compressibility and Poisson's ratio of the material
surrounding the pore, and T is the porosity.
Within !5so the
(1)
n5-
experimental compressibility uniformly ncreased with porosity over
that of ?ackensit's model to a difference of 20 per cent at 70 per
cent porosity.
To at least this 70 per cent porosity, however, the
compressibilities do rt
sff
exceed the value given by
06 ( l+i *
(2)
v
which was derived for exceedingly porous material, i.e.e
walls separating pores may be considered thin and planar.
where the
Since
equations (1) and (2) behave identically as the porosity approaches
unity, equation (2) provides a useful upper limit to compressibilities
at high porosity.
Ineluded also are a proof that a slight ellipticity of a spherical
hole has a negligible effect on the functional dependence between
compressibility and porosity, and a proof that measured compressibility
will always be greater than ?ackensie's prediction.
-6Introduction
Often, in both geologic and engineering materials, knowledge of
intrinsic
(characteristic of the largest
truly homogeneous unit)
elastic properties alone is not enough to explain overall elastic
behavior.
One such situation is that of a material containing cracks
or scattered, variously shaped pores.
Adams and Williamson (1923),
Sisman (1933), Birch (1960), Brace (1965), and others have emphasised
the importance of these voids to an understanding of elastic behavior.
the effect on compressibility
This study is limited to that case of th
of spherically shaped pores.
Mackensie (1950), utilizing a method
proposed by Frhlish and Sach (1946), derived an expression for
effective compressibility of such a material but, because of its
complexity, ignored the elastic interaction between pores.
One would
then expect the *~ckensieprediction to be good at low porosities,
but increasingly poorer at higher porosities.
The Mackenzie relation
is
neff
aD 0(
+ 2
j 40v
b~
(1)
(1
-7-
where
effPand B are the effective and intrinsic compressibilities.
respectively, v is Poisson's ratio, and
T
is perosity.
the compressibility of the glass surrounding the pores.
Here, po is
Walsh (1965),
using a different method, rederived the KMakerme equation and, as
wells treated the eases of elliptic and peMnyWshaped cracks.
At porosities near unity# the assuption of unXferA
strain
throughout the material tnder hydrostatic pressure weuld see to be
valid.
It is then possible, in this extreme case of elastic Inter-
action between pores, to derive, as in Appendix A, an expression for
the effective compressbility, i.e.,
(2)
(in the previous notation).
Nearly spherical pores can be readily produced in glass.
Hasselman
and FiMrath (1964) produced glasses containing homogeeously dispersed,
spherically shaped pores.
The elastic properties of their material
-8.
were found to be in good agreement with the Mackensie relation.
porosity in their saples ranged from 0.5 to 2.5 per cent.
However,
At this
low porosity the interesting effect of elastie interation between
pores could not yet be observed,
Data by Coble and Kingery (1956) on slntered alumina indicate
a greater increase in copressibility with inoreasing perosity than
the Maekenzie relation predicte.
the alanam
Unfortunately, the pore spates in
were irregularly shaped, many not even approxizately
spherical, making difficult a comparison with theory.
It i
interesting
to note that the difference between the theoretical compressibility
for spherically shaped pores and the compressibility of their samples
is in the direction predicted by Walsh for the effect of pores of
increasing
lipt city.
Similarly, Duckrth,
ta
(1950), investigated the effect of
porosity on mechanical properties of cerwecls.
Flin
feldspar, and
clay were sintered together iith polystyrene pellet inclusions to
form a porous porcelain,
During the sinter process, the polystyrene
-9-
buaned out leaving spherically shapes pores.
obtained between zero and 28 per cent.
Five porosities were
At 25 per cent porosity, the
compressibility deduced from their measured valuee of Yoeng's modulus
and Poisson's ratio vas greater by about 10 per cent than the
theoreticl value.
It
aokezie
ill be seen that this 10 per cent deviation
is consistent with the results of this work.
The preset study extends the range of porosity to about 70
per cent.
Glass was used because of its homogeneity, and to avoid
any local anisotropy introduced by orystallinity, as in the porca'Iin
employed in the Duekworth study.
cally shaped pore,
0
The spread of porosities for spherti
2.5 per cent by Haselsan and Pulrath (1964),
and this extension to 70 per cent, should cover the spectrum of such
perosities of geologi
and engineering interest.
-10-
Procedure
The experimental procedure eansisted, first of fabriating
eample~
of varying porosity,~ and then of determining compressibility.
Sintering of a fine glass pewder seemed to hold greatest proise
for ease in developing the porous material.
The glass powder, or
friti was mau~saotured by the Pieao division of Gliden and had the
trade designatlon of P. 3l,
can be found in Table 1.
The chemical coaposition of the fit
To obtain the frit, Pemo
ate~r
uenOhed
a glass melt produoing
i
/8 inch glass shards.
pass a 200 mesh seive.
Although the shards are not generally
marketeds, temo
The glass
These were groad to
was kind enough to supply them for study.
rt
casombined
it h 3 per cent by vieght of the
binder, polyethylene glyo!, in hot water.
The mixture was then
dried, passed through & 14 mesh selve, and dywpressed into 518 inch
and I inch dia etr right circular oylinders about 1 inch long.
These pressed pellets were then ready to be Asntereds
found unnecessary to contain the sauple in avy way for snter
It was
taeperatures 50' or more belov the W860C malting point of the glass.
These samples, thch were placed on a fire brick, maintained their
overall form throughout the ensmuing contraction and expanson of
sintering.
i
each ale
For inter temperatures approaching the melting points
as contained in a crucible with inside disensiaos near
those desired for the finished pellet.
their low theral expanrsion
Clay crucibles, because of
were found unsatisfactory.
The pellets,
after welding to the cracible walls dring sintering, would crack
in tenslt
n upon cooling.
Aluina orucibles worked quite well,
Since
the theral expansion of the almina was greater than that of the
glass, the glass was actually under
ompression after cling.
During the heating of a pressed frit, the binder burns out at
about 20 0 4, and at approximately 580oC the sample begins t shrink;
this i
the
evidene of sitering.
st evidence
great as 10 per cent,
gether
Tobtal shrinkage may be as
As the individual grains become welded to-
the grain interstices become rounded and
solated peores form.
It was thought that soe sort of equilibrium exists between
-12-
pore pressures glas su rfae tension at the sinter teperature, and
the gas pressue
external to the tesple.
(For a diseUasson of the
meehanies of sintering see ackesmie and Shuttlelartih
1949).
Onee
the pores were elosed, a change in external pressure should alter
that equilibrium and force an adjustment in pore size.
Thus, it
ould seem that differing porosities could be obtained ftam initially
identical pellets by varying this pressure.
900C ws built into a pressure vessel.
A furnace capable of
Pressure combinations from
a vacum to 1500ps were tried without success.
With the limited
internal dimensions of the furnace, it was necessary to use a metal
tube auffle to contain the saxple.
su*acessively tried.
Tantalum and stainless were
Even in a nitrogen atmosphere there remained
snough contaminant to immediately corrode either muffle at the
firing temerature.
Staultan0ously, some material in the glass
would be reduced, turning the sample black and freeing a gas which
rendered the sample a pumice if left at temperature for a sufficient
time
These attempts always produced extreely inhomogeneous pore
distributions as well as changing the chemistry of the glass.
-10.
During these early trials, it was found that the volatiles
(probably bonded
0, C02, and Wa2 0) naturally exhausted from the
heated frit could be trapped in pores to varying amunts simply by
altering the thermal histories of the samples.
It was this effect
which finally was employed to obtain suitable material, that is,
firing time and temperature were varied to produce materials of
different porosity.
Sample size affected slightly the final porosity.
This suggested that the pore distribution might not be entirely
homogeneous.
However, only a central core of each pellet vas used
and thin sections of these portions reveal little, if any, inhomogeneity,
Plates I and II show the actual shape and distribution
of pores for two of the samples.
In the preparation of the samples, a clam-type Nevi Duty
Electronic Co. (K.2012) furnace in conjunction with a Tagliabue
controller was used.
furnace.
A 1200 watt rheostat controlled power in the
Temperature was measured at the sample loceation in the
center of the muffle tube to an accuracy of better than 50C by a
Leeds and Northrop portable potentiometer.
Three solid glass samples were to be produced for determination
of intrinnsic compressibility.
To check the effect of different thermal
histories on the intrinsic elastic properties of the glass, one
sample, denoted (G),
was obtained by melting together the 1/8 inch
shards supplied by Peace, the second, (F), was derived from the
made use of discarded sinter products.
frltt and the third, (S),
These three samples were melted at 12000C for one hour in alumina
crucibles and allowed to weld to the orucibles during cooling.
Actually, sample (F) was found to have a few inhomogeneously
distributed bubble inclusions.
Although this sample behaved
anomalously, data for it
are included to illustrate the effects of
a alight inhomogeneity.
These bubbles were not discovered until a
close examination was prompted by its high comressibility.
Such
bubbles cannot be seen in samples (G) and (S).
It was next necessary to determine the densities and, thus,
porosities of the samples.
Tofreeth
free the
from the
rucibles and to
0015-
eliminate surface inh ogeneities of those samples not contained
in
crucibles, all of the samples were cored with diamond-core drills
producing final diameters of either 10mns
25rm.
or 16m and lengths of about
Since the pores were non.oontiguous,
an immersion method of
density determination was used with carbon tetrachloride (chosen for
its
low gas solubility) as the buoying medium.
Weight was measured
with an accuracy of better than -ti mg and a reproducibility of ±0,3 ag.
The density of the carbon tetrachdoride was obtained from the Inter-
national Critical Tables.
The densities are accurate to
and the perosities to ±+.009.
M.003
gfcc
These measurements are included in
Table III.
The final step in
compressibilities,
the study was to determine te
the linear
Eectric resistance strain gages were attached
directly to the samples with CHI1
araldite resin, type 502.
The
gages were HLH Co. constantan foil with epoxy backing (type Fa-Z3-2Z 33)
with a resistance of 12030
± 2 ohms and a gage factor of 2.05
1 %.
Brace (1964a) determined the pressressure effect on a representative of
-16-
this type gage to be + 0.5440"
bar "1 . and this correction was added
to all indicated strains to give true strain.
For details of such
compressibility measurweents see arce (1965).
A standard piston and cylinder type pressure vessel was used
for the comprssibility determination.
The pressure was measured by
changes in resistance of a 200 ohm manganin coil, and both pressure
and strain were recorded on a Model 136 Moseley X
recorder.
The
procedure involved pressure seasoning the sample to properly seat
the attached strain gage.
trials AI and $.
This step is represented in Figure 1 by
The actual measurement was taken as the average
of two runs to I kilobar, except in the case of the 70 per cent
porosity easople which began to ooellapse at about 500 bars.
compressibility,
Linear
L P * vas determined graphically from the record
on the X-Y recorder.
conditions prevailed.
Pressure changes were slow so that isothermal
Copressibility,
times linear copressibility.
be three
was assed to to
Overall accuracy of the
T recorder
is 0.2 per cent and the 200 oshe mganin coil was calibrated by
E-17-
Brace (1964b) to an absolute aouracy of 0,3 per cent,
the compressibilty detert ination
Accuracy of
as limited by the graphical
measurement of the XTY recorded slope.
These recorded slopes,
representatives of which may be seen in Figure 1, were remarkably
regular and linear.
As in Figure 1, the first complete ru on each
sample to 1 kilobar and back to room pressure was aecompanied by
noticeable hysteresis.
This hysteresis decreased with repeated runs,
thus, probably representing some aooomdation between the strain
gage and the sample surface not aecomplished in the initial pressure
seasoning.
Since such aeoomodation is,
the pressure-increasing traces of A,
for the most part, irreversible,
and
are considerably more
boved than pressare.decreasing halves of the same trials.
Therefore,
the graphical determination of the linear compressibility was always
taken from the pressureedecreasing half of each pressure cycle.
Since even this pressure-decreasing curve exhibited about a
15 minute decrease in slope
ostly at the low pressure extreme of
the pressure run, the actual measurement was made upon the high
.18.
pressure half of each decreasing-pressure trace.
Thues
for pressures
between 1/3 and I kbt the +5 minute inaccuracy in graphically
determining the slope allowed an inaccuracy in the compressibility
determination for the curves measured of less than ± 1 per cent.
Experimental Results
The corrected elastic compressabilities of the porous samples
are listed in
Tale
IV.
To properly relate this to the values
predicted by the Mackenzie relation (equation (1)) the Poisson's
ratio, v, must be knom.
W. F. Brace (1964b), using a solid glass
sample, similar to sample (G),
perfoed a uniaxial compression test
recording the axial and circumferential strains with separate strain
gages.
Thus, Young*s modulus and Poisson's ratio could be determined.
HiS value for Poisson' a ratio was 0.23 with an uncertainty of 5 per
cent.
Also, the compressibility determined using the Brace values for
E (0 75x*1 6 bars) and v in the relation
$
E
-19-
was 2.20 x 10
6
bars "l which is# within the experiment error, equivalent
to the values obtained here (2.16, 2.17) for the solid samples (G) and
(s).
Using this value of Poisson's ratio, Mackenzie's theoretical
prediction, plotted as BI0
is shown in Figure 11 wth the
experimentally determined omipressibilities.
Agreement is good at
low porosities with the percentage difference of measured over
theoretical compressibility increasing with parosity to a %axis=
of 20 per cent at ?0 per cent porosity.
The scatter is certainly related in part to a partial
crystallization of samples 25, 26, and 27 that were maintained
at high temperature for considerable longer periods than were the
other samples (Table 11).
This crystallization is evident in the
photomicrograph of sample 21 J.1 Plate I .
to approximate the tre
If a seooth ourve were
bulk modulus-porosity behavior in Figure II,
these samples, which were fired for extended periods, alone, t0wld
aliwys lie about 5 per cent beloiw the taekensie eurve.
This indicates
-20.
an increase of compressibility with either crystallization, or with
soeme unknown change of chemistry occuring during the extended period
at temperature.
The effect of exsolution of volatiles upon intrinsic elastic
properties cannot be adequately treated here.
A general impression
from Morey (1938) is that the small amount of Na20 in this glass
available for exaolution could lower the compressibility by two or
three per cent, which is counter to that apparent in Figure UI.
Coble and Cooper (1964) have estimated this particular eff0t to
be negligible.
Another possible source of deviation could be nonsphericity
of the pores.
Plates I and II,
which are photomaicrographs of thin
sections of samples 720 and 25 which have porosities of 50 1 and
35.6 per cent respectively, show the voids to be very nearly spherical.
(Care must be taken in examining these plates.
The large holes
which pass completely through the thin section have undoubtedly
undergone some shipping during manufacture of the thin section.
-21#111
wz" Qpr
The shape of the dark shadows east by those pores eontained entirely
within the thin section are more to be trusted).
Since slight
deviation from sphericity does not alter the compressibility for
samples of equivalent porosity
as show
be discounted as a source of error.
in Appendix
, this must
The increasing elastia interaction
between pores as the porosity increases is,
thus, the dominant factor
is explaining the difference between the measured and the Mackensie
theoretical compressibility.
It is shown in Appendix C that the
effective compreseibility of a material with spherical pores will
always lie on or above Mackenste's value because of this ignorance
of elastic intermtion,
As previously memtoned, visual examination of the solid sample
(F) revealed inhmogeneously distributed, small bubble inclusions
which undoubtedly accounts for its anorolously high compressibility.
(A few bubbles near the strain gage wruld greatly affect measurements
of compressibility
ithout noticeably altering the sample density).
-220
Conclusions
A series of fourteen glass samples with porosities varying from
0 to 70 per cent were fabricated.
The pores were homogeneous in
distribution and approximately spherically shaped.
The porosities
having been determined by the Archiedes imersion method electrical
resistance strain gages were attached and the linear compressibilities
found for confining pressures to 1 kb.
%'iese results were then
compared with ththeoretical prediction originally due to Mackensie
(1950), (equation (1)).
within a few per cents
For the intrinsie
Poisson-s ratio of 0.23,
the experimental results followed Mackentie' s
relation,
eff
(i +
to porosities up to 13 per cent.
f
V
At 24 per cent porosity, the
compressibility was greater than the theoretical value by 10 per
cent (about 5 per cent of which is felt to be due to crystallisation
(1)
4-23-
incurred in this particular sample).
At the mud
porosity of
70 per cents there was a 20 per cent increase of compressibility
over Mackenziets value.
These results can be seen in Figure II,
The expression
off ,*
(12+
il4j - rS
(2)
for compressibility derived in Appendix A for materials vith very
high porosities, unlike Mackenzie's relation, includes effects of
pore interaction.
This relation is plotted as
I
in Figure II.
It is important to note that the limit of this relation as the
porosity approaches unity is identical with this same limit in
Mackenzie's theory.
Therefore,
since all compressibilities at the
hig r porosities fall between b Vl ad
-1 (Figure I),
Mackenzies
equation and equation (2) can be considered as useful lower and
upper bounds, respectively, for the compressibilities at higher
porosities.
-.24.
Acknowledgments
The author is indebted to Professors W. F. Brace and Ro. L
and to Dr. J. B. Walsh for their invaluable counsel.
Coble,
The skills
provided by teohancians F. Gripper and D. Stanton are also sincerely
appreciated.
Credit is also due Dr. Walsh (1964), for the theory and method
in Appendix Bo
-25-
'
if
SI
i
"-
I
--
--
_.j at,,,-. 'rr
-4J.
I
r
I
b28-
Table I:
Coaposition of Paeo P-311 frit
Empirical formula
K20
.02
Percentage weight
.7
Table II
Thersal
Ca0
Ma0
.69
.29
6.5
Eistory of Samples.
A1203
.27
14.1
"203
.57
10.0
3102
2.049
7olwt.3
245.34
54.4
Before firing, all samples
(except G, F, and 3) in form of right circular cylinders
about 1 inch in length.
Sample
Sample Diameter
Chronological temp-tine history
Contained in 1 1/2"
1200C - 2 bro.
alumina crucible
680C
680
750
20 min*
15 min.
730
26
30
750 c - 7 hrsa
1 1/2" (unpacked frit in
23
27
720
1
- 2 hra.
- 7300ooled
780 C - 4 bre.
crucible)
25
29
2 hre.
810
1 1/2' (unpaoked frit in
crucible)
1*
1"
'1"
.- 7 hrs.
850 C - 45 min.
5 min. cooled - 800c - 4 hrs.
8100C - 7 trs.
67210 - 2 hrs.
61aOc - 1 1/2 Bhr.
1/2 hr.
- 750oQC
820 C - 25 min.
-29-.
Porosity is determined by timersion
Table III: Porosity of samples.
in CC14.
Density of CC14 at 20ec is 1.5941 gm/cc.
Measurements are acurate to
1
Wt. CC0014
g
Sample
Wt, Dry, g
ago reproducible to
Density, g/o
0.3 ag.
Porosity
(based upon
= 2.511)
4.3390
11.8848
G
7.1093
F
12.1510
S
680
750
730
26
30
25
29
11.6766
9.9792
10.8791
4. 3026
5.3784
5.5559
23
27?
720
1
Table IVs
6.0488
7.1177
-1*4mi~
26
30
25
29
23
27
720
.048
.111
*247
.33',
.389
.460
-1.0661
-1.9917
3.1861
.008
.130
.2221
.0320
.2174
- .2807
.04
.698
Tabulation of porosity and compressibility
i
Balk Modulus K in bars).
Sample Diameter
Strain Gage
Mounting
16 am
Axial
Cirouferential
Axial
Axial
Axial
Axial
16 ma
a
2*511
2.509
2,512
.7005
2.0612
(0 in bars
Sample
2.5930
4.4410
3.8920
2.8637
7.9562
16
16
16
10
mm
anm
am
m
16 ma
10 am
16 mm
10 am
16ma
16 am
16 mm
Circumferential
Cirouaerential
Circusferoential
ariouneerental
Cirwd erental
Axial
Axial
Axial
Porosity
&* 009
.008
.048
.111
.130
.334
.356
.389
442
.460
.504
.698
9.10
1%
2.157
2,394
2o714
2*718
4.116
4.635
5. 331
5.505
6.465
7.278
8.171
14.849
-6
K1 0
S1%
.426
.461
.0418
36$
.243
.216
*188
.182
.155
.137
.122
.067
Appendix A
Derivation of a limiting compressibility for a
very porouas material
We here consider a very porous or frothy material.
that the nll
It is assumed
thickness, Zt, between pores is relatively uniferm and
that if b is a dimension of the sallest pore, then 2t << b.
A
necessary oritical assumption is that under hydrostatic compres.ion,
p. the strain, e be unifore
throughout the material and that the shape
of each porereraina fixed,
This
s not an unreasonable stipulation.
If the walls between pores are uniformly thick and are planar, the
boundary displacemets of each wall section would generally be uniforta
and, thus, the strains and stresses would be constant except at wall
interstions whi h are here volametrically negligible.
With these assumptions, and writing the
olume a pore as
v"/~td
where x# y, and z are an arbitmary set of cartesian coordinates, the
volume of that pore under hydrostatic compression is
v
J (1+.)dx(1+e)dy(1+e)dz
V
where V implies the same limits of integration.
If e << l
V' is
approximately
V1 = (1+3e)v
since e is a constant.
(A-1)
The magnitude of the biazial stress, pt in
the valls can be written
(A2)
where p is defined positive in compression, and E and v are,
respectively, Toung's audulus and Poisson's ratio.
The compressibility,
0, of the wall material can always be substituted for
Thus, the change in the pore volume per total volume the pore is,
from equations A-l and A-Z,
-32-
A.
(A-3)
C1.V) ....
ON
If per f denotes the effective co
ressibility of the porous
material as a whole, a necessary energy balance is written
2p (3.)
(A-4)
p (2e)()
where p is the external pressure and 11 is the porosity, or (1- )
is the fraction of solid material undergoing biaxial strain eo
the volume of the associated wall material is
total volume of pore V,
A.
Thusl
Since
small compared to the
Vis effectively the change in porosity,
from equations (A-3) and (A-4)
(A-5)
Walsh (1965) shows that for a porous material
ef f
B - p
Th refore. in this situations
off
a
+
t
).)
Appendix B
Proof that a slight ellipticity of a nearly round hole
has negligible effect on compressibility
Rather than attempting to solve the more oomplicated problem
of the displacement field around an ellipsoidal hole in a material
subjected to hydrostatic tension, we treat here the oonceptually
stilar, tv-diensional, plane-strain problem of an elliptical hole
in a plate under biaxial tension, R.
The solution for the displacement#
due to Inglis (1913), is
-((p-l)
cosh 2 a - (p+l) cos 20 + 2 cos o
*=0
(3B.l)
where: a and 0 are elliptical coordinates of a system centered on
the hole,
and
are the correspondong displacements,
h is the modulus of transfomantion,
_
c (cosh 2acoos2 )
a is
the half length of the focal line
Sis the modulus of rigidity of the material, equal to
where
E and v ares respectively, Y~ang's modulus and Poisson's ratio,
p is (3-v) for the oase of plane strain, and
%o is the coordinate of the
urface of the elliptical hole.
The change in the veolne per unit volme of the hole and per nwit
thickness of the plate is
A
1us tip
*c~S
"ab
(B4)
0
where a and b are the minor and major axes, respectively, of the
elliptical hole.
This is a fuction whtoh, if maltiplied by porosity,
gives the change in porosity with a given pore shape and hydr a vb
stress, RI
Integral (92) m4 be written in ten
a =%. p = 3-4,
adwth
sa
=9
of equatian (.14)
with
e36.
2r
J osha2ao - cd
f
of
2
Since aosh Zoo
ao
W% 4 , or,
ash2 no +
(3"3)
2a,#
)
+ (
)21 squton
(0-3) ±0
a+1Sf
If a g is defined suh that gA=
A
E
b
(b/a)3_s~
then
(S.w4)
g
g is the oea-alled aspect rato, or degree of elipticity.
The desired result is achloeed if the change in
porosity) with a ohange in elliptAety at unit inial
i seroi or, equivslently, if ---bg
(11 is
efliptiotty
= 0. This is satisfied since
..37a
2
g4l
As is showh by alsh (1965).
where Oeff is the effective oomprossibtilty of a porous aterial,
P0 is the cwpressibility of the solid material, and
is the ohange in porosity with pressure.
~Then(
ag'
Thuas at least in the tw*
onal case# a slight deviation from oircularity should not effoot
dimena
perf
l = 0 by equation (5-5).
(
).* A similar behavior is opeoted for three dimensional
situatIons.
(0-5)
-e3B
Appendix C
Proof that measured copressibilities will always be
greater than Mackensie's prediotion.
The as=p tion *ade by Macklene (1950) in his derivation of
the effective compressibility for a material containing spherical
pores is that there is a domain associated with each pore whose
boundary undergoes a unifrm displaoment when the material is
This is tantamount to
subjected to a hydrostatic tensions, t
assuing a displacemnt field for the sample as a whole; such an
approximate field, as will be shown, results in a low calculated
compressibility.
By the minimam potential energy principlo
TiuidS
~arere
+
fV
*
3tud
zed
+1
2 a
aij
S is the surface of the material,
V indicates the vola2e enclosed by the surface S#
state in V#
Ui are the displacements of the natural
are arny assumed set of displacements in V, in this
(ro1)
n51'94
instance, Mackenie'
S
a =
u1 5
+
*2
ljej)
+
ij= 2 'etg
+
Uji)
set,
are true strains,
are Mackenie's strais,
ajj = *ijj'o ter cijke
are the elastic constants of the
material,
t
subject to the boundary condition which is
l
cf j
-
n
surfac,
her nS is the vector normal to the
on
and
3 is hydrostatic tension on surface.
Relation (C-1), in conjunction tith the divergence theorem# yields
fS
2
,
fT,
ds
or, since T is a constant on S,
401is ,
If this Is mtiplied by 'a
V (AP)
is
he
re
rt. V ips the total volae and
(C-2)
Ap is the change in pressure (since Ap is defined in a sense
opposite to that of Ti the inequality mut be reversed), then (0o3)
can be written as
V Ap
(04)
V Ap
Since the definiton of copressibility is " Vedp
frterial is assumed H*akeen), etrtion (C04)
,
(and sinoe the
implies that
*
approxisate
r the uniqueness theorem4 the equality in equation (C.-5) holds if
only if the displaoUments such as those in Maekene's derivation
are identical to the natural displaceents; a requirement which
Maokenzie's theory does not onerally meet.
For a disaussion of those methods, the reader is referred to
Sokelaikoff (1956) *
(C-05)
true
t1d
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