Experimental Determination of Viscosity of Rocks Paper deals with pertinent analytical considerations concerning solid viscosity and describes the procedures followed in the determination of parameters for structural and true viscosity of a Queenston limestone
by Mahmood H. Rana
ABSTRAcT--An i m p o r t a n t p a r a m e t e r i n v o l v e d in t h e viscoelastic d e f o r m a t i o n o f s t r u c t u r a l m a t e r i a l s is t h e coefficient of " s o l i d " viscosity. D e t e r m i n a t i o n of t h i s p a r a m e t e r is necessary, if it is t o b e used in s t r u c t u r a l design. T h i s p a p e r deals w i t h p e r t i n e n t a n a l y t i c a l c o n s i d e r a t i o n s c o n c e r n i n g solid viscosity a n d describes t h e procedures followed in t h e d e t e r m i n a t i o n of p a r a m e t e r s for s t r u c t u r a l a n d t r u e viscosity of a Q u e e n s t o n limestone, T h e following t h r e e t e c h n i q u e s were used: (1) R e l a x a t i o n t e c h n i q u e (2) U n i a x i a l c o m p r e s s i v e loading (3) C a n t i l e v e r - b e a m loading T h e results o b t a i n e d are in close m u t u a l a g r e e m e n t e x c e p t for (2) above, w h e r e e x p e r i m e n t a l c o n d i t i o n s were d i f f e r e n t f r o m t h o s e in (1) a n d (3). A quasi-periodic b e h a v i o r o f s t r a i n is indicated. I t has b e e n s h o w n t h a t t h e solid viscous p a r a m e t e r is a t r a n s i e n t p r o p e r t y a n d m a y d e p e n d on s u c h f a c t o r s as a p p l i e d load, t i m e , g r a i n size, g r a i n - p a c k i n g in a m a t e r i a l , a n d t h e d i r e c t i o n of t e s t i n g . I t h a s b e e n c o n c l u d e d t h a t coefficients of t r u e a n d s t r u c t u r a l solid viscosity of m a t e r i a l s c a n be d e t e r m i n e d for a g i v e n set of c o n d i t i o n s .
Nomenclature A = d i r e c t i o n parallel t o t h e b e d d i n g dolomite = coefficient o f viscosity ~R = coefficient o f r e l a x a t i o n viscosity ~RA ~ coefficient o f r e l a x a t i o n viscosity tion A ~R8 = coefficient o f r e l a x a t i o n viscosity tion B ~RC ~ coefficient of s t r u c t u r a l viscosity tion C ~s --- coefficient o f s t r u c t u r a l viscosity
plane of
in direcin direcin direc-
Mahmood H . Rana is Assistant Professor, Department of Mining Engineering, Queen's University, Kingston, Ontario, Canada. Paper was presented at the 1966 S E S A A nnual Meeting held in Pittsburgh, Pa. on November 6-9.
538
I December 1969
~r = coefficient of t r u e viscosity ~h = coefficient o f viscosity in s h e a r B = d i r e c t i o n n o r m a l to t h e b e d d i n g p l a n e of dolomite C = d i r e c t i o n n o r m a l t o b o t h directions A a n d B in d o l o m i t e d = effective t h i c k n e s s of c e m e n t i n g m a t e r i a l bet w e e n grains, also, t h e differential s y m b o l d P = s t r e s s differential d x = d i s t a n c e slipped b y grain along t h e g r a i n c e m e n t i n t e r f a c e in d o l o m i t e = a verage grain diameter Dg E = Y o u n g ' s m o d u l u s of elasticity e = 2.7183 (base of N a p i e r i a n logarithms) = unit = m a j o r principal s t r a i n ~1 ~2 = m i n o r principal s t r a i n (~1 - ~2) = s h e a r s t r a i n = shear-strain gradient 2/ = s h e a r s t r a i n = initial s h e a r s t r a i n = shear-strain gradient G = m o d u l u s of rigidity G~ = u n r e l a x e d m o d u l u s of r i g i d i t y I = m o m e n t of inertia l = l e n g t h o f a sample, of a m i n e s t o p e lo = initial l e n g t h of a sample, o f a m i n e s t o p e P = a p p l i e d force, c o m p r e s s i v e load = u n i t stress CrI = m a j o r principal s t r e s s if2 = m i n o r principal stress ( ~ , - ~) = s h e a r i n g stress t = time T = time of relaxation = s h e a r stress O = r a t e o f r e l a t i v e d i s p l a c e m e n t of grains on t w o sides of d W = w i d t h , also d i a m e t e r Y = deflection a t t h e e n d o f a c a n t i l e v e r b e a m = v e l o c i t y o f sagging o f t h e free e n d o f a c a n t i l e v e r b e a m , m a d e o f a viscous m a t e r i a l r = coefficient of viscosity in tension.
Introduction .~,.~
T h e concept of solid v i s c o s i t y is n o t new. Kelvin ~ defined t h e p a r a m e t e r as e a r l y as 1875. Since then, Jaffreys ~ and, more recently, R e i n e r ~ h a v e r e p o r t e d significant work in t h i s area. D u r i n g t h e d e v e l o p m e n t of t h e field of " r o c k m e c h a n i c s " in t h e l a s t 10 to 15 years, researchers h a v e shown enormous i n t e r e s t in t h e p h y s i c a l p r o p e r t i e s of rocks a n d rock-like m a t e r i a l s . I n t h e process of d e v e l o p m e n t of rock m e c h a n i c s a n d t h e t e c h n i q u e s of s t r a t a control in t h e mines, it h a s become a p p a r e n t t h a t t h e design c r i t e r i a b a s e d on t h e classical t h e o r y of e l a s t i c i t y a r e n o t e n t i r e l y v a l i d for rocks. As a result of c o n s i d e r a b l e work in rock mechanics, it is now well known t h a t rocks d e f o r m in a viscoelastic manner. Therefore, t h e c o n c e p t of solid v i s c o s i t y is of obvious i m p o r t a n c e . However, it h a s never been utilized for t h e solution of problems in s t r u c t u r a l design, i n v o l v i n g viscoelastic materials, since no values of t h e coefficient of v i s c o s i t y are a v a i l a b l e for rocks except a few rep o r t e d b y Griggs,~ T e r r y a n d M o r g a n s ~ a n d P o m e roy. ~ A t t e m p t s are now being m a d e to bring t h e concept of solid v i s c o s i t y into p r a c t i c a l design. T h i s p a p e r c o n s t i t u t e s t h e first s t e p in t h i s direction, namely, it describes suitable p r o c e d u r e s to o b t a i n v a l u e s of coefficients* of v i s c o s i t y a n d to distinguish a n d d e t e r m i n e t h e viscous d e f o r m a t i o n of rocks. T h e work was done on Q u e e n s t o n limestone.
Definitions Some definitions, p e r t i n e n t to t h i s s t u d y are given as fo]lows: Coefficient of relaxation viscosity, ~ ~, is defined as resistance (per u n i t a r e a per u n i t time) to r e l a x a t i o n of e i t h e r i n h e r e n t or i n d u c e d , s t r a i n in a solid granular material. Coefficient o f true viscosity, ~r, is t h e r e s i s t a n c e (per u n i t a r e a per u n i t time) to p e r m a n e n t (viscous) d e f o r m a t i o n of a solid g r a n u l a r m a t e r i a l , such t h a t , u n d e r t h e d e f o r m a t i o n - p r o d u c i n g forces, loss of cohesion or visible failure do n o t occur, b u t t h e m a t e r i a l is under c o n d i t i o n s of solid flow. Coefficient o f structural viscosity, 7,, m a y be defined as resistance (per u n i t a r e a per u n i t time) to overall s t r u c t u r a l (elastico-viscous) d e f o r m a t i o n of a solid g r a n u l a r m a t e r i a l .
Theoretical Considerations Some r e l a x a t i o n techniques, uniaxial c o m p r e s s i v e loading a n d c a n t i l e v e r - b e a m loading h a v e been used for t h e d e t e r m i n a t i o n of v a r i o u s v i s c o s i t y p a r a m eters. Here, some t h e o r e t i c a l a s p e c t s of these procedures a r e given.
* The use of the plural should become clear from the definitions that follow.
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~ . . ~ , . ; - ......~. ~ ..~'~S ~..:.i,~.."tr :. ~ Fig. 1--Grains
and
intergranular
cement
V i s c o s i t y by R e l a x a t i o n T e c h n i q u e s T h e b e h a v i o r of a r o c k specimen d e p e n d s on its loading h i s t o r y in-situ, s p r e a d over geological t i m e scales. W h e n a specimen is r e m o v e d f r o m i t s env i r o n m e n t , t h e s t r a i n s i n d u c e d in it relax. B y using t h e reflective p h o t o e l a s t i c technique, t-~~ it is possible to m e a s u r e t h e r e l a x a t i o n of shear s t r a i n s in a plane a n d to c a l c u l a t e t h e shear stresses a s s o c i a t e d w i t h it. Coefficient of r e l a x a t i o n v i s c o s i t y can be calcul a t e d from p h o t o e l a s t i c m e a s u r e m e n t s , using e i t h e r one of t h e following t w o a p p r o a c h e s : (1) T h e K e l v i n Solid M o d e l is a s s u m e d to describe t h e d e l a y e d - e l a s t i c b e h a v i o r of rocks. T h e rheological e q u a t i o n is w r i t t e n as: T = G7 -F ~
(1)
O m i t t i n g t h e s u b s c r i p t s for s i m p l i c i t y , t h e solut i o n o f e q (1) is: 3'
=
e -a/"'t
( if: Vo
-~- -
E q u a t i o n (1) c a n also be w r i t t e n (al -
a2) = E(el -
T ee/"td
0
(2)
as:
e~) -t- ~r (el -- e2)
(3)
E q u a t i o n (3) is different f r o m eq (1) i n t h a t E h a s r e p l a c e d G. I n case of Q u e e n s t o n limestone, t h e difference b e t w e e n E a n d G is n o t significant a n d e i t h e r can be used. E q u a t i o n (3) c a n now be used to d e t e r m i n e ~R, if ( ~ - ~2), t i m e t a n d E (or G) a r e known. (a) Zener u-x3 a n d K e ~4, 15 e v o l v e d a n a p p r o a c h t o s t u d y t h e i m p e r f e c t i o n s in t h e elastic b e h a v i o r of metals. A modified version of t h e i r a p p r o a c h c a n be used for d e t e r m i n a t i o n of y R. I n Fig. 1, if d is t h e effective t h i c k n e s s of t h e
Experimental Mechanics
I 539
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ISEOD~NG PLA.E ~
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(B.P.)
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.
OI.ECTmON
.
Fig. 2--Relaxation samples: preparation, terminology and nomenclature
intergranular matrix material in a rock t h a t consists of equiaxed grains, then the coefficient of viscosity at a shear stress of (al - ~2) is defined by:
G,.d.T
~R -
D0
(10)
Viscosity by Compressive L o a d i n g ~
-
(4)
(V/d)
where v is the rate of relative displacement of grains on two sides of the intergranular cement. I f dx is the distance slipped along the graincement interface due to shearing stress (~1 - a2) during time T, then: V ~
(5)
dx/T
where T is the time of relaxation. given by: ~
dx/D~
T h e strain is (6)
where D~ is the length of one side of the grain, and is equivalent to the average diameter of a spherical grain. I f a region is considered in an immediate vicinity of one of the grain-cement interfaces, then:
(~ -
-
~2)
G.
(7)
where G~ is the unrelaxed modulus of rigidity. T h e strains in eqs (6) and (7) m u s t be equal. Therefore: dx
D--~ =
(~
-
~)
G,
(S)
and therefore, v ~
(al -
~2)'Dg/G~'T
whence ~/n can be determined from:
540 I December 1969
(9)
E m e r y 16 considered the overall m o v e m e n t in the mine stopes due to redistribution of stresses as a consequence of mining, and the flow permitted b y the resistance t o deformation of a Kelvin solid. H e theoretically examined an anticipated state o f transition in the roof of a stope, dipping off the horizontal in a longitudinal direction, and derived the following equation for ~,: d P . W 2. t '7~ = :2(l~ _ Zo~)
(11)
E q u a t i o n (11) does not account for the mass involved and needs modification, before it is applied to determine y~ from measurements in mine stopes. However, the equation is valid for small specimens, tested in the laboratory. Viscosity by Cantilever-beam L o a d i n g
N a d a i 1~ maintains t h a t as long as Poisson's ratio does not enter into the problem, the solutions for the bending or the torsion of prismatic bars are the same for a compressible elastic material and for a viscous material. Although rocks have variable Poisson's ratios, the above s t a t e m e n t is valid if it is used towards estimating a coefficient of solid viscosity. T h e deflection of an elastic cantilever beam, loaded at its end b y a force P is given by: pl ~ Y - 3EI
(12)
The distance by which the loaded end of a cantilever beam of a viscous material descends per unit time, or, the velocity of sagging of its loaded end, is given by: s -
pl ~
3r
(13)
Also, ~b = 3~8h
(14)
E q u a t i o n (14) has been given both b y N a d a i 17 and Reiner. ~ E q u a t i o n s (13) and (14) can be used to calculate ~,h.
Preparation and Description of Test Samples Queenston limestone used for this investigation was available in the form of 3 X 6 X 12-in. blocks, each having an approximate bedding plane, running normal to the least dimension and passing t h r o u g h 3 X 12 and 3 X 6 faces of a block.
Samples for Relaxation Tests T w o types of samples were required and for convenience in identity, t h e y m a y be termed " h a n d samples" and "core samples." T h e samples were prepared so t h a t t h e y could be used for relaxation tests pertaining to both the Kelvin solid model and K e ' s formulation. Four limestone blocks were selected at random. Prisms 3 X 3 X 6 in. were sawn from the ends of each of the blocks with a rock saw. E a c h of these prisms, in turn, was cut into two "halves," each measuring 3 X 3 X 3 (in.) T h e remaining 3 X 6 X 9 portions of the limestone blocks were sawn for the preparation of core samples. Three mutually orthogonal fresh faces were cut on each of the hand samples. These faces were numbered, 1, 2 and 3 as shown in Fig. 2. While cutting these faces, samples each of thickness between 0.25 and 0.50 in. were removed from them for each of the hand samples. These were polished for subsequent microscopic measurements and studies. Three thin sections were prepared for faces 1, 2 and 3 of one hand sample to s t u d y geological features of limestone. Annealed photoelastic plastic disks of 2-in. size were bonded to faces 1, 2 and 3 of each of the h a n d samples. T h e bonding was obtained with reflective cement developed for use with the photoelastic material. T h e 3 X 6 X 9 portions of the limestone blocks, saved while preparing hand samples, were used to obtain core samples for determination of the unrelaxed moduli of rigidity Gu. The drilling was done with a standard floor-model drill press. T h e samples were drilled in directions A, B and C.
Fig. 3ITypical samples for compression tests
them. Anchor pins made of 2 X 2-in. steel plates, each having a well-reamed 2-in.-long steel pin of 5/16-in. size fixed in the middle, were bonded to t h e samples with a bonding cement which has a shearing strength of a b o u t 3000 psi when "set." T h e length of these samples corresponded to "direction A." Typical samples are shown in Fig. 3.
Sample for Cantilever Test Only one sample, a prism of limestone of 11/2 X ll/2-in, cross section for 10 in. of its 12-in. length and 41/2 X 11/2 in. for the rest of 2 in. was prepared. Plastic patches were bonded on two sides of the sample, corresponding to faces I a n d 3. T h e plastic covered the faces entirely, leaving out small areas near the 11/2 X 11/2 end. A mirror was bonded on face 1 on the area left without the plastic patch. Photoelastic observation points were m a r k e d on each of the two patches. T h e details are shown in Fig. 4. T h e sample was allowed to relax for a b o u t seven weeks before loading.
Experimental Procedure T h e procedure can be divided into three categories, used in the description of preparation of samples.
Samples for Compressive-loading Tests
Relaxation Tests
Five limestone blocks were selected at r a n d o m and prisms, each 2 X 2 X 12 in., were sawn from
T h e procedure was designed to measure variables involved in eqs (3) and (10) to calculate ~R. T h e
Experimental Mechanics I 541
following m e a s u r e m e n t s made:
and
observations were
m a d e with a microscope and the average values of grain size and effective thickness of intergranular m a t r i x were calculated f r o m these measurements. T h e thin sections corresponding to faces 1, 2 and 3 were studied using petrofabric techniques.
(a) shear strains and corresponding shear stresses in relaxation (b) t i m e a n d r a t e of relaxation (c) principal strain and stress m a g n i t u d e s (d) grain size and effective thickness of intergranular cement. (e) geological features of Queenston limestone (f) determination of d y n a m i c moduli of dolomite
Compressive Loading T h e procedure was designed to measure the variables involved in eq (11) to calculate ~, and ~r. T h e variables measured were: applied load, load increments, stress differential and change in length of a sample. J u s t before loading a sample of the t y p e shown in Fig. 3 in compression, a H o r s t m a n extensometer, shown in Fig. 5, was m o u n t e d on the anchor pins bonded to the sample. T h e i n s t r u m e n t consists of two anchoring blocks A and B, the m a i n b o d y C, a differential m i c r o m e t e r D a n d a photoelastic tensioning disk E. T h e disk E is strained b y means of a m i c r o m e t e r screw to give a predetermined n u m b e r of fringes. E a c h t i m e a reading is taken, an even pressure against the anchor pins is maintained b y reproducing the same color p a t t e r n on the plastic disk. T h e readings are t a k e n f r o m the differential m i c r o m e t e r and their difference gives the change in length (increment or decrement). T h e samples were tested using the a r r a n g e m e n t shown in Fig. 6. T h e e q u i p m e n t consists of a loading f r a m e with a 50,000-1b load cell a t t a c h m e n t , a h a n d - o p e r a t e d jack for application of load and a Baldwin strain-gage indicator. T h e values of the applied load a n d the changes in load with t i m e were given b y the strain indicator. W h e n first loading a sample, the value of the applied load was i m m e d i a t e l y read after the application of load. Thereafter, changes in lengths of
Points of highest shear strains (e~ - e~) were m a r k e d on the various faces of h a n d samples with the aid of a h a n d polariscope. These points were t h e n read for (el - e=) on a normal-incidence (reflective) polariscope ~ in degrees of compensation. As the points of highest shear strains moved, t h e y were " r e - m a r k e d " and read. T h e readings were t a k e n over a period of 27 days. T h e variation of shear strain with t i m e thus determined is the r a t e of relaxation of inherent strain in Queenston limestone. T h e strain m a g n i t u d e s in the principal strain directions were measured with an oblique-incidence polariscope, ~ in conjunction with the normalincidence polariscope. Core samples were used for the determination of unrelaxed modulus of rigidity G, a n d modulus of elasticity E, using a beat-frequency oscillator. =~ F o r the m e a s u r e m e n t s of grain size and effective thickness of intergranular cement in limestone, six spots were selected at r a n d o m on each of the polished samples. At each one of these spots, grain size a n d thickness of intergranular cement were m e a s u r e d for faces 1, 2 a n d 3 along a n u m b e r of r a n d o m l y selected lines. T h e m e a s u r e m e n t s were
i [
THE PLASTIC PATCHXr~ .
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FACE NO. 5
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:-.THE DEFLECTION-MEASURING MIRROR FACE NO. I Fig, 4--Cantileverbeam: pointsfor photoelasticmeasurementson faces 1 and 3 END VIEW.
542 [ December 1969
- -
. . . .
.... ..g_
A, B---ANCHORING C ---MAIN
BLOCKS.
BODY.
I, 2 , 3 - - - A D J U S T I N G
D---DIFFERENTIAL E---STRESS- X
SCREWS.
4---KEY AND
MICROMETER.
TENSIONING
FOR
SCREWS
FOR
ROTATING
DISC.
I, 2
AND
THE
MICROMETER. Fig. 5--The Horstman extensometer
1 2 -
HORSTMAN EXTENSOMETER LOADING FRAME LOAD CELL LOADING JACK LIGHT SOURCE STRAIN INDICATOR
the sample were recorded with the extensometer, corresponding to the changes in applied load. W h e n necessary, the load was increased b y operating the loading jack. A n u m b e r of samples were tested for various periods of t i m e ranging between 0.815 a n d 502 hr. After unloading, each sample was allowed to re-
A B C D
Fig. 6--Setup for compression tests
-
SCALE DOLOMITE CANTILEVER LIGHT SOURCE WEIGHT BUCKET
BEAM
Fig. 7--Setup for the "cantilever" test
Experimental Mechanics I 543
lax for at least as much time as it was under load. T h e permanent change of length for cumulative values of time and stress differential was considered as viscous deformation 9 I n using eq (11) for calculations, allowance was m a d e for the square cross section of the samples.
of 50 lb was placed when loading the sample. The small mirror, bonded at the end of the beam, was used to measure, optically, the deflection at the loaded end of the beam. The light from the source C reflected from the mirror B onto scale A. A n y deflection of t h e loaded end of the beam changed the reading on scale A. The load was applied normal to face 1. The observation points on the plastic patches were read for (Et - ~). These readings and those of end deflections were t a k e n before loading, immediately after loading, twice during the load test, immediately on unloading and after relaxation.
Cantilever Loading T h e general a r r a n g e m e n t is shown in Fig. 7. Although the sample was not loaded in a typical cantilever-beam fashion, the deviations from a s t a n d a r d test were considered small enough for eqs (13) and (14) to be applicable. A string passed over a pully, of which one end was attached to the free end of the cantilever " b e a m " and the other end had a bucket, in which a weight
Observations
and Results
The observations and results from the three tech-
140 130" 120"
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Fig. 8 - - V a r i a t i o n of (el - - e2) with t i m e , s a m p l e no. RI (A)
I~)0
2(30
300= TIME - (HOURS)
5OOI
99
""
9. '9": , -"
~
50-
-"
"JD.
s
".fl~ ; %e
,"
I
."
9
^
667
,
O.'''~176176 ~176
-
:
%
~.~'o.~"
~ /At,
s
o
"--'
"~
t"~
I
I
.-
"'.o.'"
9
/
/
:
~A k
%. \
\
- .X \
Fig. 9 - - V a r i a t i o n of - - ~) with t i m e , s a m p l e
&
"/
: ss
80
:
s 9
s
9"" m"
60
--:"
o
I
r
9 9
o~
9
9 ""
~-iio
70
6(~0
oO'b
120-
Q~ I-if) 0
[]
460
D 9 9 ~ ~ ~ "r~' 9176 []
130'
90
A
[] ......
3
0
140-
z
A
FACE
I / []
20
IO0
O----O
" .'~
i'/
I
FACE I FACE 2
I .'" ""
30]
(et
no, R2 (B)
LX
/ :
-
~
.
~
'
~
j
.-
40
.---o
.-
] December1969
F~E ~wo
o ...... o
i
i
i
i
i
~00
200
300
400
500
T I M E - (Hours)
544
&
~ ~
80-
70" rq
t~
,,,o
..
9176176
90"
/
u
%
CETHREE
s
(~
o
667
ing variation of shear strains with time are given in Figs. 8 and 9. The values of shear stresses were calculated from (~1 ~)'s. Relaxed values of (~t ~) and shear stresses were obtained by subtracting initial readings from the final readings. (el -~-~2)'s were determined from variations of (et - e~) with time. A study of the limestone from thin sections showed that the material consists of rod-like fragments of organic origin forming a matrix and fine granular cement. The constituents were present in 1 : 1 ratio. Figure 10 is a photomicrograph of a polished section of limestone. Coefficient of viscosity in relaxation was determined in directions A, B and C. The results of Tables 1 and 2 are based on eq (3) while those given in Table 3 were calculated using eq (10). Table 3 also contains the values of average grain size and the thickness of intergranular cement in limestone, determined from the measurements on the polished sections. From Tables 1 and 2, it should be noted that -
Fig. IO--A photomicrograph (limestone)
niques used are presented as follows.
Relaxation Viscosity Relaxation of shear strains was determined from photoelastic measurements. Typical curves show-
-
-
-
TABLE 1--COEFFICIENT OF RELAXATION VISCOSITY IN DIRECTION A, ~RA FACES I. SAMPLE NO
NORMAL TO C DIRECTION ( o"I - cr z ) ZERO COMPENSATION P.S.I
E = 1 . 2 6 x 109 L B S / F T ( 0 " I - O"z ) A T
RELAXED
RELAXED
PEAK COMPENSATION
( o " i - o- z )
(E~ .
LBS/FTZ
xlO'SFT/FT
P.S.I
~z)
2
RATE OF
~RA
RELAXATION
(POISES)
(•l
-
~a
)
x I 0 i7
x I0 "HFT/FT/SEC Ri (A)
212
668
65,664
69.1
R I (B)
293
701
58,752
8.2
6.94
15.8
1.48 :5.07
R 2 (A)
0
558
77,472
6.7
5.87
56.50
R 2 (8)
277
587
44,640
6.2
:5.22
4.98
R 3 (A)
326
656
44,640
6.3
2.80
5.94
R 3 (B)
212
815
87, 8 5 2
12.3
6.76
4,79
R 4 (A)
244
538
42,:556
5.9
5.28
2.90
R4 (B)
277
754
66,240
9.4
4.48
5.55
TABLE 2--COEFFICIENTS OF RELAXATION VISCOSITY IN DIRECTIONS B AND C, ~RB AND ~RC
E = 1.26 x 109 L B S / F T / a DIRECTION B-FACES 2 NORMAL TO DIRECTION A SAMPLE
RELAXED
NO
(~,-O-z ) LBS/FT z
R,(A) Rt(B)
Rz(A) R2(B) R3(A) R3(B) R4(A) R4(B)
21,168 82,224 58,7529 32,976 75,168 5,400 79,920 56,448
RELAXED ~z
(e-~-
)
x t0 "5 FT/FT
5.0 11.9 8.5 4.8 107 800 11.2 82
RATE OF RELAXATION ( ~ , " ~z ) x I0 "11 FT/FT/SEC
1.58 6.54 4.40 278 4.95 4.94 5.52 5.25
DIRECTION C-FACES "r]RB
RELAXED
(POISES)
( e-, - O-2)
x I017
LBS/FT z
5, NORMAL TO DIRECTION B
RELAXED ( e~ ez) x I0 "5 FT/FT "
RATE OF RELAXATION (e-~ - E z )
"r/RC (POISES) x I017
x 1041 FT/FT/SEC
5 05 4.95 4.98 4.74 5.75 4.54 551 428
44,496 103,248 I I , 808 5t,696 70,416 107,856 49,248 59,888
65 14.4 1.6 7.5 I0.0 15.1 8.0 5.4
5.66 7.78 0.95 4.14 6.54 7.76 12.00 8.19
4.89 4.80 4.20 4.66 4.07 5.07 2.01 1.65
Experimental Mechanics [ 545
Rhas different values in different directions. T h e results of Table 3, coupled with the observations of Figs. 8 and 9, indicate that ~R m a y vary with grain size, the level of inherent strain and the packing pattern of grains of a material. The results from the two relaxation techniques agree closely.
o_
i io
o
/
/
~
~
V1I S C O S7I T Y
STRESS
Structural and True Viscosities from Compression Tests
/
150q
~5
x - -
J 14004- 1400
All the samples were loaded parallel to the bedding plane. The following were recorded from these tests:
DIFFERENTIAL .
.
.
.
.
.
.
:
I 9
!
9
I
"1
iooo
~ooo~i
900
2
.I
i
"'...
900FIO I
/~
9
/
I
T h e values of ~ were calculated by using structural deformation while those of ~r were calculated by considering viscous deformation. Viscous deformation was differentiated from structural deformation by allowing the samples to relax after unloading. Figures 11, 12 and 13 give some of the typical results obtained. The time during which samples were under load ranged between 0.815-502.154 hr. The initial loads for the samples ranged between 228 and 872 psi. The total applied load (cumulative) for various samples was in the range 1078 to 2496 psi, whereas the final cumulative values of dP ranged between 676 and 1612 psi. The values of ~8 were calculated at each increment point. T h e y are valid for direction A only. The values of~r are given in Table 4.
700
700
6
!!
' FINAL LOAD
t iI
:
~176176176176 / ~oo- ~ooF
x
q-
_x x~
400.
400--10
~
~
o ................
-
ioo o
I IOO
J 200 TIME
/ 5oo
I 400
(HOURS)
laxation of strain for 15 days, the total end deflection was 0.00328 ft. On the basis of these readings, the values of 7, and ~r obtained were 1.35 X 1018 and 8.10 X 1016 poises, respectively. T h e values are in close agreement with the values of ~ R. The point to point true viscosity was determined from photoelastic measurements. The difference between the (El -- e2) readings on each point just before unloading and those taken after relaxation
TABLE 3--COEFFICIENTS OF RELAXATION VISCOSITY ~RA, ~RB AND 'TRC IN DIRECTIONS A, B AND C RESPECTIVELY, FROM KE'S EQUATION A
DIRECTION
INTERGRANULAR MATRIX SAMPLE
T
RELAXED
NO
(HRS)
( E l " E2) x IO-6 F T / F T
B
GRAIN SIZE SIZE = 0 4 4 5 m m
Gu xlO e P.S.I
DIRECTION = O,145mm
INTERGRANULAR MATRIX SIZE = 0 6 4 5 m m T
RELAXED
Gu
rlR8
T
(HRS)
(~E2 ) x 10eFT/FT
x I0 e
(POISES)
(HRS)
P S I
x l O Iz
R, {A)
160
91
3.67
448
528
82
51 7
67
565 560
9 I0 8 69
Rz(B)
554
62
356
14 50
525
= O[45mm
INTERGRANULAR MATRIX
?~RA
R2(A)
C
GRAIN SIZE
(POISES) x I0 '7
Ri (B)
RELAXED ( E ~ - E2 ) x106 F T / F T
SIZE = 0 2 4 5 m m Gu
~RC
xlO 6 PSI
(POISES) x l O 17
30 119 85
2.96
1715
493
65
J45
2 97
2.96 2.59
16. 50 1497
815
144
157
295
467
16
I I0
2 15
480
48
2.59
1570
490
75
[ 14
254
505 823.5
R3(A)
625
E5
2.60
12.40
600
107
198
1512
425
I O0
1.05
1.87
R3(B)
505
123
2.64
i020
450
80
275
15.65
R4(A} R4(B~
51~ 5828
89 94
4.79 4.70
30
585
112
2 64
1705
540 185
I8 i 80
I 09 I 20
2.47 093
20.80
454
82
2 60
12.45
I 85
54
I 26
0.90
546 I December 1969
II
I 500
Fig, ll--Variation of structural viscosity, stress differential and final loads with time, sample no, 1
Equations (13) and (14) were used to determine the coefficients of structural and true viscosity on the basis of the deflection at loaded end of the beam. The coefficients y, and ~,h are equivalent and eq (14) can be used. The total deflection (viscoelastic) at the end of 29 days was 0.0.9685 ft. After unloading and re-
GRAIN SiZE = O I 4 5 m m
o
x-x
.... :!I 200
Structural and True Viscosities from Cantilever Test
DIRECTION
x
I /
.... ( .... ~ ,T -
(a) Cumulative values of stress differential dP. (b) T i m e period during which d P was effective. (c) Decrements in original lengths of the samples corresponding to dP and time.
6
o . O / o ~
31-
,~oo= .
,6oo: moo- ~a
,400. ~
.~ F 2k 9 [0 II _
<~
s
~
.~
o .~
>
~
6
1200
.. 9 FINAL
+
a
400+
6
9
4
300,-
3 2
" 9176
r I0
90s
-
-~
,x
8OC - 4 o c
I
:
I STRESS I DIFFERENTIAL
"I :I
u. u-st. o 4F
x~ ....
~xl
.........
o
t S i
:VISCOSITY : o 9
60O ' 3 0 0 - ~
J ! !
ol
Zl-
'~li
2
50C
-IO
-
o ~
:'-....
99 40( -200I I I
:
6
h-
5
300
"
7 6
I00
200
100-
5 4
0
i
2
200
I
I00
-
-
4
~-
3
F-
I r" I10 -
-
O- - - I 0
0
I0
3
o -oh,o,
o~
x . . . . , - . 4x
8 h7h~ -
ZOO
2004-
I00
LOAD
0''0~
"I :1
Bff
- uJ 6 h
70(
II. F I N A L
:~
zT~
!
i
9 8
400
~
-191"-
I
500
sSTRESS 2; :v 9 DIFFERENTIAL
4~-
= Jf2
x | !
o" :
'.
- 600-
IOOO ~ 5 0 ( 1 - ~ 0 II -
.'
50O4- I 0
Jsa 9
I100
LOAD
,-
S
5
600
61-
o
:
700
-
5P
9
800
8P 7h
I100~
90C
-
~_ I0-'
1200. 8 0 0 ~
I000
70C
xsX
d
9F
o < J
z ~ ,=- ~: I~00
VISCOSITY
o2
20
30
40
50
TIME(HOURS)
Fig. 13--Variation of structural viscosity, stress
o.q o.'2 o.'3 0:4 o.% o!6 0/7 o!8 o19 TIME
(H
differential and final leads with time, sample no, 7
OURS)
Fig. 12--Variation of structural viscosity, stress differential and final loads with time, sample no. 3
much deformation in-situ. The differences mentioned above m a y be due to the following: (1) Variation in the thickness of the organic intergranular matrix (since the calcite grains are equiaxed). (2) Packing pattern of the calcite grains and the intergranular matrix. (3) Percentages of each one of the two types of grains in limestone. (4) Inherent shear-stress level of a location in a sample (although a relation between the stress
gave the relaxed (el - e=) values. The relaxed (r - r values were calculated. The results are given in Tables 5 and 6.
Discussion The results show that resistance to deformation is different in directions A, B and C of Queenston limestone. This limestone has not undergone
TABLE 4--COEFFICIENTS OF TRUE VISCOSITY FROM COMPRESSION TESTS SAMPLE
LENGTH
CUMULATIVE
NO
(INCHES)
d P PSI
TOTAL DEFORMATION ( F T ) x I04
VISCOUS DEFORMATION (FT)
x IO~
CUMULATIVE TIME (SECS)
VISCOUS
DE--
FORMATION AS
"~T (POISES)
REMARKS
% OF T O T A L DEFORMATION
20.67
I
7.92
1328
2
72
I000
3
3.9
1612
1695
142
2000
1807740
9675
2.51x1014
071
105720
5000 8849
2 98x1014
15.00
2930
I 1 5 x l O Iz
4
4
904
7.88
7.2[
141480
91.53
6 3 8 x I0 t3
5
I0
1542
0375
180960
14.72
6
IO
628
2.55 6.54
0.84
171090
I273
1 , 0 7 x I015 I. 8 5 x I014
7
I0
736
10.52
6.67
165660
6334
2.63x
8
t0
676
3 O0
0.46
153740
1528
3 2 6 x I0 t4
9 I0
I0 I0
684 1046
1342
767
207960
5713
122
089
212368
7328
2.67 263
SAMPLE NEAR FAILURE AT THIS POINT
SAMPLE JUST SHORT OF FAILURE
I013 x ~013 x l O '3
Experimental Mechanics I 547
TABLE 5--COEFFICIENTS OF TRUE VISCOSITY--FACE I, CANTILEVER TEST PERIOD ALLOWED
PERIOD OF LOADING = 696 HOURS
POINTS
NET PERMANENT
NO
(E~ - ~ ) F T / F T x IO.6
RATE
FOR
ELASTIC
GRAIN TO GRAIN "r/T
NET PERMANENT
OF
( G, - G2 )
VISCOUS DEFORMATION
RECOVERY = 5 6 0 HOURS E = 1 . 2 6 x 109 L B S / F T 2 % DEFORMATION DUE TO VISCOUS
(POISES) x 1017
LBS/FT z
(El - E 2 )
FLOW
F T / F T / S E C x IO'" I
59
156
28080
6.48
2
59
2.35
42192
654
2565
5
49
1.95
35136
4
55
1.32
25472
6.52 6.58
2258 I 5.94
5
9
0.36
7056
5.71
489
6 7
65 7
260 028
46944
6.46
28.65
4752
69-7
, 4.51
8
16
064
tl664
6.37
11,55
9
3
2304
I0
20
0.12 0.80
5.90 665
2.23 16.49
II 12
43 40
14112
1.72
6.64
30384 28080
160
20.4l
61.45
6.70
4820
TABLE 6--COFFEICIENTS OF TRUE VISCOSITY--FACE 3, CANTILEVER TEST PERIOD OF LOADING = 696 HOURS
POINTS
NET PERMANENT
NO
( E ~ - E2) F T / F T xlO -6
PERIOD
RATE
OF
ALLOWED
FOR
ELASTIC
NET PERMANENT
VISCOUS
(o-,-G2
)
LBS/FT 2
DEFOR MATION
(Er-E2)
RECOVERY = 560 HOURS E = 1 . 2 6 x I09 L B S / F T z
GRAIN TO
% DEFORMATION
GRAIN ~ T
DUE TO VISCOUS FLOW
(POISES) x I0 ~z
FT/FT/SECx I0 Hi
I
79
3,15
56504
6 57
64,75
2
98
591
70416
650
3590
5 4 5
42 56 62
I 67
30528 59888
6.40 658 651
52,81 4242 41.90
6
I0
7
16
8
6
9 10
35 62
II
50
44496 6912
064 024
I 1664 4608 25472
6.82
1515
6.38 5.90
1250 7674 539I
1.32 247
44496
6.58 651
1.20
21168
665
12
5
012
13 14
0 7
0 028
4752
697
26 92
15
15
0152
9560
648
3611
level and ~ has not been established). Relaxation Viscosity
A significant observation is that the relaxation (shear strain vs. time) curves exhibit quasiperiodicity. Some are typical damping curves, while a few exhibited an increasing amplitude with time. The latter indicate the local release of locked-up energy. These curves indicate that the physical properties of rocks depend on such factors as applied load, time, grain size, grain packing, percent composition of various types of grains in the material, etc.
548 I December 1969
2.3 247 040
2504
5.90
0
6.98 ALL RECOVERED
Structural and True Viscosities from Compression Tests
The results for ~ indicate that there are, broadly speaking, two phases involved in the rheological behavior of limestone. There is an initial, shortlived, low-viscosity phase, followed by a longer phase, marked with the quasi-periodic behavior of ~,. The latter stage is the Kelvin solid phase, represented by eq (1). The former phase can be represented by the Maxwell liquid equation: ~2)/~
(15)
where (~--~-- z2) is the shear-stress gradient.
This
(~--
~)
= (~, -
~)/V
+ (~, -
Fig. 14--Photomicrograph showing "slightly deformed," "moderately deformed" and "highly deformed" portions of a thin section of sand crystals (after Friedman, 1963)
behavior of limestone stands to reason when its packing properties are considered. During the initial stage, the m o r e c o m p e t e n t calcite grains t e n d to come closer to each other b y squeezing out the less-competent intergranular matrix. T h e process continues until the calcite grains come in contact, when the Kelvin phase sets in. Figure 14 shows a p h o t o g r a p h a f t e r F r i e d m a n 21 who deformed cylinders of sand crystals composed of single crystals of calcite t h a t enclosed detrital grains and calcite-cemented sandstones. T h e experimental conditions were different t h a n those of this investigation, b u t "slightly deformed," " m o d erately deformed" and "highly d e f o r m e d " portions of the p h o t o g r a p h s u b s t a n t i a t e w h a t h a s been said. I t would a p p e a r t h a t "slightly d e f o r m e d " and " m o d e r a t e l y d e f o r m e d " phases in Fig. 14 are analogous to the two phases of the b e h a v i o r of Queenston limestone. T h e coefficients of true viscosity given in T a b l e 4 are hard to relate to a n y of the variables in eq (11). T h e v a r i a t i o n s in experimental conditions concerning the samples m a y be p a r t l y responsible.
Structural and True Viscosities from the Cantilever Test T h e values of 70 a n d ~v determined f r o m deflection m e a s u r e m e n t s are in the ratio 1:8 approximately. T h e v a r i a t i o n s in point-to-point true viscosities on faces 1 and 3 of the cantilever are not surprising in view of the heterogeneous n a t u r e of limestone.
Conclusions T h e conclusions of this investigation are as follows: (1) T h e coefficients of viscosity of a rock can be e s t i m a t e d in a n y given direction b y various m e t h ods under a given set of conditions. (2) T h e coefficients of viscosity for limestone are directional.
(3) T h e viscosity p a r a m e t e r in the A direction of limestone exhibits Maxwell liquid b e h a v i o r during the initial phase and Kelvin solid b e h a v i o r in t h e second phase. T h e r e m a y be a c o m m o n a r e a between these two phases. (4) Average grain size, effective thickness of t h e less-durable organic material between relatively c o m p e t e n t calcite grains of limestone, percentages of the t y p e s of grains constituting limestone and the packing p a t t e r n s of these grains seem to be imp o r t a n t controls for the solid viscous p a r a m e t e r . (5) T h e v a r i a t i o n s of shear strain with t i m e exhibit quasi-periodicity. (6) T h e coefficient of structural viscosity decreases with increments in load a n d increases with decrements of (or c o n s u m p t i o n of) load. (7) A comparison between results f r o m the three techniques shows t h a t within the conditions of the experiments, a close a g r e e m e n t exists except for compression technique where ~ a n d ~r are considerably different t h a n those f r o m t h e other techniques. T h e difference m a y be a t t r i b u t e d to high loads, n a t u r e of t h e material a n d the t i m e element.
Acknowledgments Grateful acknowledgments are m a d e to E. J. Heidecker for assistance in the s t u d y of thin sections a n d to J. V. K e n n e d y a n d R. K. Williams for assistance in m e a s u r e m e n t s of grain sizes.
References 1. Kelvin (Sir W. Thomson), "'Elasticity," Encyclopaedia Britannica, 9th Edn. (1875). 2. Jeffreys, H., The Earth, 128, 146, Cambridge (1929). 3. Reiner, M., Deformation, Strain and Flow, 2nd Ed., H . K . Lewis and Co. Ltd., London (1960). 4. Griggs, D., "'Creep of Rocks," Jnl. Geol., 47 (3), 225-251 (1939). 5. Terry, N . B., and Morgans, W . T. A., "Studies of the Rheologicat Behaviour of Coal," Mechanical Properties of Non-Metallic Brittle Materials, Lon~,~n Conference, Butterworths Sci. Pub's., 239-255, London
(1958).
6. Pomeroy, C. D., Nature, London, 178-279 (1956). 7. Emery, C. L., " T h e Strain in Rocks in Relation to M i n e Openings," The Mining Engineer, Paper No. 3834, 54-59, (October 1960). 8. Corlett, A . V . , a n d Emery, C . L . , "'Pre-stressandStressRedistribution in Rocks Around M i n e Openings," Can. Mining Met. Bull., 52 (566) (1959). 9. Emery, C. L., "Photoelastic CoatinGs on Granular Materials," Ronead Paper, Presented to the Institute of Physics, England, (1960). 10. Van Duyse, H., "'Study of Influence of a Moving Face on a Cross-Cut Driven in the Rock Below I t , " Inter. Strata Control Conf., Paris, Paper D7, 313-320, (May, 1960). 11. Zener, C., "Phyeical Review," 60, 906 (1941). 12. Zener, C., "'Metals Technology," (August1946). 13. Zener, C., "'Elasticity and Anelasticity of Metals," The University of Chicago Press, Imp. 3, 69 (1948). 14. K e , T. S., "Experimental Evidence of the Viscous Behaviour of Grain Boundaries in Metal ,'" Phys. Rev., 71 (8), 553-546 (1947). 15. K e , T. S., "'Stress Relaxation Across Grain Boundaries in Metals," Phys. Rev., 72 (1), 41-46 (1947). 16. Emery, C. L., " A Study of Instrumentation Applied to the Determination of Physical Properties of Rocks in Mines," P h D Thesis, The University of Sheffield, England, 79-81 (1961). 17. Nadai, A., Theory of Flow and Fracture of Solids, McGraw-HiU Book Co., Inc., N e w York, 396-398 (1950). 18. Emery, C. L., "The Pre-stressed Conditions of the Rocks Around Mine Openings," M S c Thesis, Queen's University, Kingston, Ontario, Canada, Appendix A , ii (1958). 19. Smith, J. D., "'The Condition of Stress Surrounding a Simulated Mine Opening," M S e Thesis, Queen's University, Kingston, Ontario, Canada (1962). 20. Obert, L., Windes, S. L., and Duvall, W . I., "'Standard Tests for Determining the Physical Properties of M i n e Rock," United States Bureau of Mines Report of Investigation, No. 3891 (1946). 21. Friedman, N., "Petrofabric Analysis of Experimentally Deformed Calcite-Cemented Sandstone," Jnl. Geol. 71 (1), Plate 1 opp. page 20 (1963).
Experimental Mechanics I 549
