Strain rate dependent mechanical properties of new albany reference shale

Int. J. Rock .l,tech. Min. Sci. & Geomech. Abxtr. Vol. 27. No. 3, pp. 199-205, 1990

0148-9062 90 $3.00 + 0.00 Copyright C 1990 Pergamon Press plc

Printed in Great Britain. All rights reserved

Strain Rate Dependent Mechanical Properties of New Albany Reference Shale K. P. CHONGt'~. A. P. BORESIt

Drilling, mining and blasting of rock-like materials, in situ fracturing, rock fragmentation size and permeability extension are strongly affected by strain rates of loading. For low strain rates, fragment si:es are large, and cracks propagate further. For high strain rates, fragment sizes are small and cracks are localized. The purpose of the present investigation is to determine the effects of strain rate on the initial (linear) value of Young's modulus and on Poisson's ratio for different grades of New Albany oil shale, a reference oil shale designated by the U.S. Department of Energy. Material anisotropy is considered. Cylindrical specimens cored in a particular orientation of the oil shale formation are tested in compression, and the rate effects are investigated. An efficient data acquisition/reduction system is dereloped, interfacing with an Instron closed-loop hydraulic testing machine. Strain gauge instrumentation is used to measure lateral and longitudinal strains. Strain rates of about lO-~-IO°/sec are studied.

INTRODUCTION Strain rate effects have been studied by numerous investigators [1-7,9]. Various properties of materials have been of primary concern. Particularly, the effects of strain rate on the modulus of elasticity, Poisson's ratio, yield strength, ultimate strength, toughness and fracture have been of interest, in as much as many of these properties affect material response [2-4,7]. The proper relation between these effects and the material response of solids subjected to rapid or impulsive loads has posed a difficult problem for the analyst, since a dynamic analysis is often required in such cases. To complicate the analysis further, often under conditions of rapid/impulsive loads, deviations from classical elastic-plastic behaviour occur. It has been observed [2,7] that high rates of loading (high rates of strain or high strain rates) have produced apparent increases in many properties of solids, such as yield strength, modulus of elasticity, ultimate strength, etc. On the other hand, certain properties which are ratios of strain components (e.g. Poisson's ratio) are essentially independent of strain rate. In a study of the effect of axial straining rate on properties of concrete [7], it was found that the strain energy absorbed increased proportional to the dynamic "l'Department of Civil Enginering. University of Wyoming. Laramie, WY 82071. U.S.A. **Presently at: The National Science Foundation, Washington, DC 20550, U.S.A.

modulus of elasticity and proportional to the square of the dynamic axial strain [7]. No significant differences in the failure modes of concrete cylinders in dynamic and static tests were observed. Dynamically and statically tested concrete cylinders (7.62 x 15.24 cm or 3 x 6in) both failed in a typically brittle manner, by developing cones at the end which cause the cylinder to split. Similar effects have been observed in metals. For example, increasing the axial strain rate from 10-: to 10"-/see results in an increase in yield stress of the titanium alloy Ti 6AI-4V by more than 20%. In the tests by Lindhoim et al. [6] and by Watstein [7], as is common in such tests, the average strain rate is considered to be the strain at failure divided by the duration of the test. The material properties related to this average strain rate are then used to model "failure conditions" for the material. In a perfectly elastic material [8,9], the elastic strain occurs by the development of small displacements between lattice points of the material. This deformation occurs instantaneously, with all lattice points being displaced almost simultaneously. Upon release of the load (strain), the strain is recoverable and the distance between lattice points is returned to its original value. These elastic deformations are resisted by the atomic bonding of the material. That is, they are not dependent upon the rate of deformation. Only the mass properties of the atoms in the lattice prevent the elastic strains from occurring instantaneously. However, this mass 199

200

C H O N G and BORESI:

STRAIN RATE PROPERTIES IN NEW A L B A N Y R E F E R E N C E SHALE

effect limits the propagation of elastic strains to the elastic wave speed of the material [10]. Nevertheless, for most applications, the elastic strains can be considered to occur instantaneously. It has been noted [11] that the total incremental strain can be divided into an elastic (recoverable) component and a plastic (nonrecoverable or permanent strain) component. This division is an actual model of the physical phenomenon. Its importance increases with increasing load. A similar division of strain has been employed in the study of fracture properties of rocks [12]. For large deformations, the separation of the total strain requires a careful definition of reference frames. However, for small (infinitesimal) strains, this requirement is unnecessary [8]. As noted above, the elastic (recoverable) strain occurs essentially immediately. However, the plastic (permanent) strain is related to incremental changes in the structural configuration of material. These incremental changes are caused by sequential microscopic phenomena such as movement of dislocations on glide planes [I 1]. For a given state, these effects (e.g. movement of dislocations) occur with a definite velocity determined by the average time to overcome a barrier in the glide plane and to undergo displacement. This sequential movement produces a localized plastic strain in the lattice [13, 14] and requires time to develop fully. As a consequence on the microscopic level, the total strain (elastic plus plastic) requires time to be developed and to be observed. The strength and elastic properties of most rocks are known to depend strongly on strain-rate, generally increasing with increased rate of loading [15]. Little was kno~n of the significance of this effect on oil shale until recently. Lankford [16] investigated the strain rate dependence of strength and ductility for oil shale from the U.S. Bureau of Mines test mine at Anvil Points. Colorado. He tested a total of 98 specimens at various confining pressures (0, 5, I0 and 20ksi: or 0. 35. 69 and 138MN,/m-') and various strain rates (from 1.73 × 10 ~ to 1.87× 103/see). Typical lean, medium and rich samples were tested. The slow intermediate strain rates were performed with a servocontrolled hydraulic ram supplying the axial load, while the high strain rate was done with a split Hopkinson pressure bar apparatus. A failure criterion [16] was developed in accordance with general first-order stresses for comparison with the data. The experimental results agreed reasonably well with the failure criterion over the range of strain rates and confining pressures. Of the three parameters (static confining pressure, strain rate and organic content) varied in Landford's experiment. strain rate was the dominate influence on fracture strength. From 18 unconfined tests, Lankford concluded that failure strength increased with strain rate, while the inelastic strain to failure was essentially invariant with respect to strain rate. The actual strength of the shale nearly tripled, going from about 62.1 to 165.6 MN m-' (9-24 ksi) as the strain rate varied from

1.7 × 10 a to 2 × 103 sec. Strength and ductility increase with static confining pressure, but since static confining pressures in place (overburden pressures) are not likely to exceed 13.8-20.7 M N m z (2-3 ksi), confining pressure is not included in this investigation. Chong et al. [2, 3, 17] presented a strain rate dependency mechanism modelling after the nonlinear viscoelastic solid. The effects of strain rate on the fracture strength and the elastic modulus of concrete has been investigated by Watstein [7]. The concrete was tested at strain rates ranging from [ 0 -6 t o 10/see. He found that the ratio of dynamic to static modulus at the highest strain rate was about 1.4. Rate effects on the elastic properties during subcritical loading (strain rate of about I/see) was not addressed by most investigators. Effects of strain rate on ultimate compressive strength, ultimate strain and for limited data on the initial value of Young's modulus in compression for Wyoming oil shale have been invested by Chong et aL [18]. They concluded that the initial value of Young's modulus in the direction perpendicular to the bedding planes of lean oil shale, containing an organic volume of 20%, increased by approx. 40% during a strain rate increment of five orders of magnitude. In a recent paper [17], it was observed that Young's moduli and, to a much smaller degree, Poisson's ratios, increase with strain rates. Improved understanding of the elastic and fracture properties, dynamic failure mechanisms and post-failure behaviour of rocks will help in designing any modification of rocks. Elastic properties of the medium are required for most of the failure models that have been proposed, see for example, K i p p e t al. [19] and Hommert [20]. During explosive detonation, strain rates in excess of 104/see are achieved near the borehole and the response changes from rubblization through inelastic deformation with macroscopic failure to transient elastic response. In general, elastic properties such as Young's modulus E depend on the strain rate, composition of rock (in this case, oil yield), in situ stress state, temperature and the anisotropy of the material. However, strain rate, oil yield and anisotropy are the dominate factors which influence the mechanical response during explosive fragmentation. Since, as noted previously, static confining pressure in place (overburden pressure) of oil shale formations is low and not likely to exceed 13.8-20.7 MN/m-" (2-3 ksi), it is not a significant factor. Also, in the case of explosive fragmentation, the temperature of the formation is not expected to rise very much, except very near to the borehole. Thus, mainly strain rate effects are studied here. EXPERIMENTATION

An Instron 1332 servo-hydraulic loading system with feedback control was used to load the specimens. A sophisticated and fully automated data acquisition/ reduction system, involving a Keithley System 501 and Zenith Z-248 PC (AT compatible) has been perfected, interfacing with the Instron loading system. More than

CHONG and BORESI:

STRAIN RATE PROPERTIES IN NEW ALBANY REFERENCE SHALE

1000 lines of Basic code have been written to interface the Keithley data acquisition system with the Instron hydraulic testing machine, extensometer and strain gauge instrumentation and the Zenith AT compatible computer. After extensive calibration, large amounts of data have been produced. About 1700 data points have been generated. Two sets of tests have been run for samples cored perpendicular to bedding planes, covering wet (saturated) and dry conditions. Tests are done on samples cored perpendicular and parallel to bedding planes. All samples are 2.54cm (l in) in diameter and 5.08cm (2in) long. For samples perpendicular to bedding planes, maximum loads of 10.9-11.6 kN (2450-2600 Ib) are used. For samples parallel to bedding planes, which have a tendency to split, maximum loads of 6.2-7.1 kN (1400-1600 lb) are applied. Strain gauges are used for their reliability (as compared to extensometers) especially for high rates of loading. Test periods run from 120 sec (corresponding to strain rate of about 10-~/sec) to 50msec (strain rate of about 10-~/sec). The 50 msec period is limited by the hydraulic testing machine. The time interval for the 120sec period is 600 msec, yielding 100 data points; whereas, the time interval for the 50 msec period is 2.1 msec. providing I I data points. The Zenith (AT), Keithley and Instron computer data acquisition/reduction and testing systems developed and interfaced by the investigators, are state-of-the-art types. Besides being accurate and automated, work that had previously taken approx. 15 hr of tedious manual labour now takes I hr using this system. The flowchart of the data acquisition routine is listed below: Program initiation Calibration of load cell offset Physical properties input routine Instrument and test data input routine Creation of titles for output and data files Calibration of bridge excitation and offset Data acquisition routine Data verification routine Data conversion routine Data analysis routine Output routine Retest routine Statistic routine One set of specimens cored perpendicular to the bedding planes was subjected to compressive loading at different strain rates in order to determine the compressive modulus E..: and the Poisson's ratio v:~ (- is the axis of symmetry of the material). Another set of specimens cored parallel to the bedding planes was subjected to compressive Ioadings at different strain rates to determine the modulus E~, and Poisson's ratio v~,. Both extensometers and strain gauges have been tested. Strain gauges were selected for their reliability, reproducibility and excellent responses to high strain rates. Axial and lateral strains in the test specimens were measured using the strain gauges attached to them driven by a full bridge circuit. Two strain gauges were

201

cemented to opposite sides of the specimen in each of the axial and lateral directions and the bridge circuit was completed to yield an average strain reading from each pair of gauges. Dummy gauges were also used for temperature compensation. A spherical seat, which is lightly lubricated with mineral oil so that it locks after the dead weight of the cross head is picked up, was used at the lower end of the specimen for compression tests. This setup helped to reduce any bending or torsional stress resulting from misalignment in the load transfer train. End constraints and friction at the ends of a test specimen affect the stress and strain distributions and ultimate strengths [21,22]. Two end platens made of aluminum were used to minimize end friction. The ratios of Young's modulus to Poisson's ratio are approximately equal for specimen and platen (E~/v~= Ep/%) to ensure the specimen and platen expand laterally about the same amount, minimizing the end friction. In addition, paraffin wax is applied between the specimen and the platens. Since the stress-strain relation for oil shale is almost linear, the initial value of Young's modulus (for the linear portion) and the Poisson's ratio are key parameters in constitutive modelling. A load cell of 22.3 kn (50001b) capacity was used to measure the load in specimens of all sizes. One pair of strain gauges to measure the axial strain and another pair to measure the lateral strain were attached to each specimen at its midheight. Specimens were then subjected to several loading cycles up to a maximum of about 25% of the estimated failure load in order to minimize any initial settlement of the stress-strain curve. Subsequently, each specimen was tested up to the same load level while measuring the load, axial strain and lateral strain at constant strain rates ranging from about 10 -5 to about 10°/sec.

MULTI-PARAMETER STATISTICAL ANALYSIS The initial linear value of Young's modulus and Poisson's ratio were determined for all the strain rate tests in each of the specimens. Moisture content is an important factor, since the oil shale is clay-rich. Water saturated samples and samples dried for 3 and 7 days [23] were tested. Multiple regression analysis of the initial Young's modulus (in ksi) and the Poisson's ratio with strain rate i and oil yields M (in G/T), as the independent variables were performed using the BMDP statistical computer program [24]. The following regression equations were derived: (a) saturated samples: E. = 2731.57 - 55.4751 M + 45.8639 log ~,

(I)

v:, = 0.105 ! + 0.006698 M + 0.004513 log ~,

(2)

E~ = 7412.98 - 204.4050 M + 116.209 log d,

(3)

v~, = 0.06256 + 0.008544 M + 0.002761 log i ;

(4)

CHONG and BORESI: STRAIN RATE PROPERTIES tN NEW ALBANY REFERENCE SHALE

_0_

Table 1. Statistical characteristics of equations (I-12) Dependent ',ariable

Equation No.

Coefficient of determination (%)

E.

I 5 9 2 6 10 3 7 I1 4 8 12

81.08 80.82 84.27 69.70 61.48 79,10 62.68 66.21 68.84 64,86 65.68 78,76

v:, E, v,

(b)

No. of data

Standard error of estimate

,'~t (G T)

229 230 230 137 93 69 115 115 92 63 61 61

42.74 (ksi) 65.77(ksi) 57.14 (ksi) 0.0032 0.0064 0.0056 183.76 (ksi) 138.05(ksi) 206.72(ksi) 0.00616 0.00420 0.00348

15.35 15.51 15.21 15.33 14.26 15.29 13.90 13.75 13.77 13.49 14.30 13.29

s a m p l e s d r i e d f o r 3 days:

E. = 3429.46 - 88.1809 M + 70.0809 log ~,

(5)

= 0.014968 + 0.010998 M + 0.00017607 log ~, (6) E, = 7242.32 - 163.781 M + 95.348 log g,

Mean values Moisture content

E (ksi)

-2.864 1748.48 -2.883 1859.53 -2.880 1837.22 - 2.875 -2.961 - 2.879 -3.461 4169.00 - 3.520 4654.20 - 3.493 4433.50 - 3.541 - 3.652 - 3.582

Saturated 3 day dried 7 day dried Saturated 3 day dried 7 day dried Saturated 3 day dried 7 day dried Saturated 3 day dried 7 day dded

0.2065 0.1713 0.1512

0.1682 0.1704 0.1528

Statistical c h a r a c t e r i s t i c s o f e q u a t i o n s (1-12) a r e s h o w n in T a b l e 1. T y p i c a l g r a p h s are p l o t t e d in F i g s 1-9 s h o w i n g p r e d i c t i o n e q u a t i o n s (in solid lines) vs raw data. R e a s o n a b l e fits are o b s e r v e d .

(7)

= 0.086888 + 0.0063782 M + 0.0020954 log f ; (8) (c) s a m p l e s d r i e d for 7 days:

E. = 3 4 1 2 . 4 7 -

Log ~

92.5614 M + 58.1418 l o g ? ,

(9)

v:, = 0.0780051 + 0.0047899 M - 0.19664 x 1 0 - ~ l o g ( ",

(10) (11)

E, = 8274.68 - 250.953 M + 110.334 log ~', v,= = 0.221655 - 0.00460297 M

(12)

+ 0.0021558 log ~'.

R E S U L T S AND D I S C U S S I O N F r o m the results o f the statistical analysis a n d experi m e n t a l o b s e r v a t i o n s , Y o u n g ' s m o d u l i are s t r o n g l y d e p e n d e n t o n strain rates, i n c r e a s i n g r a p i d l y with increases in strain rates. It d e c r e a s e s w i t h increases in oil yields ( M ) . T h e oil yield M increases as specific g r a v i t y (p) d e c r e a s e s [29]. P o i s s o n ' s ratios, o n the o t h e r h a n d , are a l m o s t i n d e p e n d e n t o f strain rates. H o w e v e r , they g e n e r a l l y i n c r e a s e w i t h increases in oil yields. T h e s e results are c o n s i s t e n t with e a r l i e r i n v e s t i g a t i o n s by 0.30

2000 -y: (/)

o 0.25

1800

2.1 6 4 ~x

2 -~ 0.20 o

1600

e-

xv

~

--

~x

xx~X~xx

i

. ¢ 4wwe

.

--

xxx eee

x •

II-e

2.246

a,. 0.15

I 1400 10-5

I i,

"11 10-4

I

I IJlldl! 10-3

Stroin

rote

d I r ppL==I I0-2

= p le~!HI 10-1

0.7~

t

i

/ Illll]

30 - 5

P

I ] ~lllll

10-4 Stroin

( per sec )

Fi~. I. Ne~ AIb.',ny oil shale, Young's modulus (E:) for medium and rich s;,turated samples with p =2.245 (13.9G/T) and 2.164 (17.1GT). Note: I k s i = 6 . 9 M P a : IG, T = 4 . 1 7 1 t .

I

I I tlllll

10-5 rote

I

I I IIllll

10-2

(per

10-I

see}

Fig. 3. New Albany oil shale, Poisson's ratio (v:,) for saturated samples for two grades of oil shale: p = 2.164 (I 7 G~T) and 2.245 (13.9 G~T). Note: I G, T =4.171/t. 0.30-

2200

2000

o

0.25

o 8 E

1800

oe

oe

o

•

eegoo

0. Ol t

1600

1400 10-s

2.206

-~ 0.20-

I

i

~,!II~

I

I lililll

10-4

J

~ lllllll

10-3 Stroin

rote

(per

I

10-Z

t

llerl]l

10-I

sec)

Fig. 2. New Albany oil shale. Moisture effects on Young's modulus (E:) for medium shales (p =_._4) or 13.9G T). Note: lksi=6.9MPa: IG T=4.171/t

O.10 ~ 10-5

o

ee

oeBe

2249 I

; llllllf 10 . 4

I

%

I ~]zj~ll 10-3

Stroin

rote

eoe e

r I IIIIH] 10"2

(per

]

I J llllll 10-~

sec)

Fig. 4. New Albany oil shale. Poisson's ratio (v:,) for samples dried for 3 days for two grades ofoil shale: p = 2.206 (15.5 G/T) and 2.249 (13.8 G/T). Note: I G/T = 4.17 l,'t.

CHONG and BORESI: STRAIN RATE PROPERTIES IN NEW ALBANY REFERENCE SHALE 0.3O -

203

0.30

0.25

0.25 .o

.2

°C

-¢ 0.20

2.164 I

0.1-~

XA

!

"! A~

21287

. . . .

•

:

-:-

.

.o

L-,.e 0.15

A

2.240 I

0.10 10-5

I IIII

I

Ill

I

•

vx

¥

I

I

IIIIIII

t

10-3

I

10-Z

IILIIII

10- I

Stroin rate (per sec)

0.1C 10-6

i iIiilll

i

•

oo

• x

i i tlllll

10-s

Strain

Fig. 5. New Albany oil shale, Poisson's ratio (v:,) for samples dried for 7 days for two grades ofoil shale: p = 2.164 (17G/T) and 2.240 (14.2 G/T). Note: I G/T = 4.17 Lt.

l •

~ x

o.~

--. ~.

"

2208

x

llllll]

10-4

•

i

i ]llJlll

10 - 4

rote

(per

10-3

i

i tllHll 10-z

see)

Fig. 9. New Albany oil shale. Poisson's ratio (v,,) for samples dried for 7 days for two grades •foil shale: p = 2.287 (12.1 G/T) and 2.208 (15.4G, T). Note: I G/T=4.171;t. Chong et al. [2], Lankford [16], Green and Perkins [151.

6~G

There was some hysteresis observed, especially for rich -~- 5 5 0 0 oq ~E 5OO(3

";2;;"

- - ' "

~

45~

.~

4OO<3

3000

I

I I ~11111

10" 5

I

I I111111

10-5

1

"'"

I III1HI

10-4

Strain

rate

"

I

I IIIIHI

lO-Z

10-3

(per sec)

Fig. 6. New Albany oil shale, Young's modulus (E~) for samples dried for 3 days for two grades •foil shale: p = 2.287 (12.1 G/T) and 2.208 (15.4G/T). Note: 1 ksi = 6.9MPa; I G/T =4.171/t.

0.25

0.20

2.243

no

0.10

x, •

x~ e a •

0.15

x~ x ~ =.ee •

x--

the stress rate applied, with the elastic response to a

~

change in stress rate occurring almost instantaneously.

-e e e

2.287

I

I IIIIII!

lO-G

I

t IPI~Itl

10-S

!

I ttlHl[

10 - 4

Strain

rote

~ t IlllllJ

10 - 3

(per

10-Z

see )

Fig. 7. New Albany oil shale, Poisson's ratio (v,,) for saturated samples for two grades •foil shale: p = 2.243 (14G/T) and 2.287 (12.1 G/T). Note: I G/T =4.171/t.

0,25

2 0.20

2.208 It X * r v X x

YX .

w '5

O.

X

component. In a static tension (or compression) test, the strain rate is extremely small, so that the plastic strain is allowed time to develop more or less fully before the next load increment is added. The resulting plot o f stress vs total

plastic strain has not fully developed and the total dynamic strain is smaller than the total static strains.

w:~ X ' ~ ; ~

2.254

0.15

0-10 P 10-6

However, the "time dependency" o f the total strain is due principally to the presence o f the plastic strain

strain is referred to as the static stress-strain curve, if the loading rate is sufficiently high (dynamic), the plastic strain component may not have time to develop fully before the next load increment is applied. Then, it appears as if the material has stiffened since the

0.30

° ~c

STRAIN RATE DEPENDENT FRACTURE MECHANISMS

As noted above, the rate dependency of material properties is related to the rate dependency of the inelastic strain components. To model this rate dependency, various spring-dashpot models have been employed (Chong et al. [2]; Boresi and Sidebottom [I !]). At the same time, the elastic strain rate depends on

0.30

-0'

oil shale in loading/unloading to higher loads beyond about 50% o f the ultimate load. However, the loading was limited to the 50% level. N o effort was made to determine stress-strain behaviour up to failure since the stress-strain curves are approximately linear up to failure. Moisture content also affects elastic coefficients. Dried samples have higher Young's moduli than saturated ones and generally lower Poisson's ratios. Possible mechanisms are explained in the following section.

I llltlll

I

l I llllll

10-S

l

t

11111

10-4

Stroin

rote

( per

10-3

t

I i i iiiii 10-Z

sec)

Fig. 8. New Albany oil shale. Poisson's ratio (v,) for samples dried for 3 days for two grades of oil shale: p = 2.208 ( 15.4 G/T) and 2.254 (13.5G/T).

As a result, the static stress-strain curve always lies below real dynamic stress-strain curves (curves generated with higher strain rates). Unfortunately, some plastic strain (e.g. dislocation movement) occurs even for very small stress levels [4, 25]. As the stress increases, a level is reached (depending on the material) at which a rather abrupt measurable increase o f plastic strain occurs. This abrupt increase in

204

C H O N G and BORESI:

STRAIN RATE PROPERTIES IN NEW ALBANY REFERENCE SHALE

plastic deformation has led to the concept of yield or flow stress [26]. The term "'plastic strain" for permanent changes of material structure is widely used in textbooks in material science. As used in material science, the term includes creep and relaxation effects (e.g. viscoelastic effects). Such effects have been observed to occur at stress levels, above the yield stress, in metals near room temperature [4, 26]. Creep which occurs in this manner is referred to as transient creep [I 1]. This type of creep is not steady state creep (which usually occurs in metals at room temperature greater than approximately one-third of the absolute melting temperature), but rather of the dislocation glide type. In developing a damage model for the dynamic fracture of brittle rock, Taylor and Flanagan [10] have employed a damage mechanism related to microcracking in the rock, under the assumption that the rock is permeated by an array of randomly distributed microcracks which grow and interact with one another under tensile stresses. These microcracks effectively degrade the material properties (e.g. Poisson's ratio, bulk modulus, etc.) of the fractured medium (rock). The computation for the degraded properties is related to a crack density parameter Cj which represents the volume fraction of the material made of flaws. Unfortunately, the parameter Cj is given only to within a proportionality factor [/ such that: C d = f l N A 3,

=k~it,

(14)

Since the factor ~ is strain rate sensitive, then the factor ? (a pseudo-Poisson's ratio) is also strain rate sensitive. In turn, the bulk modulus of the microcracked material is strain rate sensitive. However, it is noted that v, the undegraded Poisson ratio, is not strain rate sensitive. For a given state (organic content, etc.), the present data indicate that there is no degrading of Poisson's ratio for eastern oil shale for the strain rates and load levels employed. According to the Taylor-Flanagan model, this implies that the microcrack density Ca approaches 0 for eastern oil shale, under low load levels. Most geomaterials are weaker in tension than compression. Under uniaxial compression, or even under global compressive stresses, certain locations of the rock material may be subjected to tension, which give rise to the nucleation and growth of microcracks. These microcracks exist within the rock because of its heterogeneous nature. Therefore, elastic and fracture properties of rock

(15)

where: k~ = t = a constant, time of load application. Considering inertial forces, we have: AP - R m

(13)

where N is the number of flaws per unit volume active at a given pressure level p (positive in tension) and A is the nominal fragment size. The determination of // depends upon accurate laboratory data of fracture stress vs strain rate. The theory developed by Taylor and Flanagan relates the degraded properties to the undegraded properties and to C d. Poisson's ratio denoted by ~ and the crack density Cd are related by the formula (approximation): ~: = v(I - ~Cd).

under tension are important in modelling the material behaviour during dynamic events (17]. Due to end constraints, the stress system applied in a compressive test is complex and only the central portion away from the platens is expected to be free from shear stresses due to end friction (which has been minimized). Fundamental micromechanical processes need to be considered for formulation of constitutive models for material behaviour. Measurements of micromechanical parameters may become necessary. Material generally contains numerous existing microcracks and flaws [27, 28]. Up to about 50% of the critical load, there is generally no growth and no coalescence which would occur during failure and post failure regions only. Instead, shear and opening of these microcracks occur under low loads. Also, in the uncracked crystalline material, individual grains and masses of atoms are forced to move against intergranular and interatomic forces. Focussing attention on a single penny-shaped microcrack, the strain (mainly shear) during impulsive loading is governed by strain rate ~ and the time of load application, that is:

t'

(16)

where Ap = force required to overcome inertia (related to applied stress), R = frictional resistance, m = mass of material associated with each microcrack. Equation (16) shows that the applied stress (which depends on Ap) is related to strain rate assuming other variables are kept constant. Similarly, inertia forces associated with grain movement and crystal deformation have to be accounted for. (Probably, the last two items are temperature dependent, too, as intermolecular and intergranular forces are temperature dependent.) When applied stress increases due to strain rate effects, more microcracks are activated. That is, the number of microcracks activated (N) is directly proportional to the applied stress (a). At low strain rates only a few relatively large microflaws are activated (leading to a few large broken pieces of a specimen). Elastic properties, which depend on the applied stress and the resulting strain, are also affected by the strain rate. However, some investigators feel that the inertial forces are not significant at strain rates below 10~/sec. Alternatively, to explain the strain rate dependency, oil shale which is slightly viscoelastic (like most solids) can be characterized as a standard nonlinear solid [2]. Referring to Fig. 10, the viscosity t/ is furnished primarily by the intermolecular forces before nucleation of

CHONG and BORESI:

STRAIN RATE PROPERTIES IN NEW ALBANY REFERENCE SHALE

vvwvvw,

F

Fig. 10. Standard nonlinear solid model.

microcracks and by frictional forces during the crackpropagation stage. Under low strain rates, the dashpot will relax gradually, leaving the spring E,. to carry the total load F and to store strain energy which ultimately causes the maximum shear stress to be reached and failure to occur. For high strain rates, the dashpot stiffens and springs EKand E,. act in unison, giving higher E (Young's modulus) at higher strain rates. Dependence of Poisson's ratio on strain rates, on the other hand, is negligible. Acknowledgements--This paper is the result of part of a research contract on "'Mechanical Characterization of Eastern Reference Oil Shale", performed under Contract Number DE-FC21-83FE60177 for the U.S. Department of Energy IT. C. Bartke, COTR; C. E. Roosmagi, Oil Shale Projects Manager), by the Western Research Institute (V. E. Smith, Manager of Oil Shale Research; F. P. Miknis, Task Manager) through a subcontract with the University of Wyoming. Financial and technical support are gratefully acknowledged. The authors would like to thank J. S. Harkins, T. E. Gillum and P. E. Crouse for performing the experiments. They also wish to acknowledge the expert manuscript preparation by Lora Becker.

Accepted/or p,thlh'ation 17October 1989.

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