A study of the behaviour of brittle rocks under plane strain and triaxial loading conditions

Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. Vol. 32, No. 7, pp. 725-733, 1995

Pergamon

0148-9062(95)00025-9

Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0148-9062/95 $9.50+ 0.00

Technical Note A Study of the Behaviour of Brittle Rocks Under Plane Strain and Triaxial Loading Conditions M. Y U M L U t M. U. OZBAYt:~

INTRODUCTION Many excavations in rock are treated as two-dimensional (2-D) and the state of stress around these excavations approximates that of plane strain. The linearly elastic analysis of these excavations is usually carried out using 2-D models which incorporate plane strain loading conditions. Non-linear modelling of 2-D excavations, however, is more complicated and the selection of the constitutive laws and parameters governing rock behaviour are not always straightforward. One of the most common constitutive laws used for non-linear rock behaviour is the strain softening model, proposed by Vermeer and de Borst [1] which is based on the Mohr-Coulomb failure criterion. Traditionally, input parameters for 2-D modelling with strain softening (typically using FLAC [2]) are obtained from triaxial tests and the validity of this approach is still to be verified. The main parameters required for strain softening modelling are cohesion, internal friction and dilation angle, as well as the elasticity parameters, namely elastic modulus, E, and Poisson's ratio, v. Stavropoulou [3] conducted unconfined plane strain tests up to the point of failure and found that plane strain loading gives rise to slightly increased strength and decreased elastic modulus and Poisson's ratio values compared to uniaxial loading. Little is known of the effect of plane strain loading on the other constitutive parameters such as the internal friction angle, dilation and residual strength. According to Vardoulakis [4], erroneous results may arise if data obtained from axisymmetric tests are directly applied to plane strain problems. The main objective of this Technical Note is to identify the differences in rock behaviour characteristics when tested under plane strain and triaxial loading conditions. A brief description of the test rig developed for plane strain testing is given. The results from the tests conducted on coal, sandstone, norite and quartzite

tDepartment of Mining Engineering, University of the Witwatersrand, Private Bag 3, WITS 2050, Johannesburg, Republic of South Africa. :~To whom all correspondence should be addressed. 725

specimens, both under plane strain and triaxial loading conditions are presented. The differences between the results from the plane strain and triaxial tests are interpreted in terms of Vermeer and de Borst [1] strain softening constitutive model as used in the computer program FLAC [2]. THE PLANE STRAIN TEST RIG A variety of plane strain and polyaxial test rigs have been used by researchers in the past to investigate the specific aspects of rock behaviour, such as the effect of intermediate stress on strength [3, 5, 6] and more recently, the bifurcation phenomenon [7-10]. It appears that two different approaches have been adopted by previous researchers for developing plane strain testing devices

[8]: (a) Hydraulic system is an active system whereby the displacements in the plane strain direction are restricted by means of appropriately applied confining stresses using hydraulics [3, 9]. (b) Stiff frame system is constructed to limit the displacements along the plane strain direction. The confining pressure is generated in the frame as the test specimen dilates during loading. This type of test rig can only approximate plane strain loading conditions since the frame is never stiff enough to inhibit specimen dilation completely [7,8]. The hydraulic system allows accurate plane strain conditions as long as the system provides the required pressure levels for limiting strains in the desired direction. Its design, however, is complicated and requires elaborate hydraulic systems for controlling the confining stresses during testing. A stiff frame system can be simple and inexpensive to construct, but the compromise is that the system provides at the best a "quasi-plane strain" condition as the resistance to displacements in the plane strain direction is determined by the relative stiffness of the frame and the rock specimen. A stiff frame test rig was developed and used in this study, mainly because of cost consideration. Figure 1 shows the schematic representations of the test rig. The

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rig comprises two steel U-frames, which are connected via two left/right threaded bolts to restrict displacements in the x2 direction (plane strain direction). The U-frames are made of 30 mm thick steel pieces to minimize bending at the specimen-steel contact and it is assumed that the U-frames remain "rigid" during testing. By turning the steel bolts of the U-frame simultaneously the specimen is pre-stressed along the x2 direction. The confinement along the x3-direction (comparable

to confinement in triaxial testing) is provided by means of two steel plates located on the sides of a 30 x 30 x 10 mm size prismatic specimen. The confinement to the specimen is provided by four bolts located at the corners of the side plates. The bolts are designed in such a way that, during loading, the changes to the confinement of the specimen, due to stretching of the bolts, remain within 5% of the initially set confinement value. Confining pressure, a3, up to 10 MPa could be applied to the

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TECHNICAL NOTE

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The tests were carried out using an MTS series 815 model closed-loop servo-controUed stiff rock testing system. The specimens were loaded in stroke (deformation) control mode using a strain rate of 10-5/sec. The confinement levels used were 0, 3, 5 and 8 MPa for both test types and the triaxial tests were carried to the standards as prescribed by ISRM.

Sc

RESULTS Material properties

Fig. 2. Simplified sketch of

specimen-frame interaction.

specimen by means of initial tightening of the four steel bolts. The confining pressure, tr3, could be monitored during testing by means of the strain gauges mounted on the steel tubes placed around the bolts. The test rig designed only approaches the plane strain loading condition. According to the diagram given in Fig. 2, the force F applied to the sample due to displacement Su - S¢ along x2 is F=k,(Su-S¢),

where kr is the stiffness of the specimen, Su and Sc are the displacements that would occur due to Poisson's effect, with and without confinement along x2, respectively. The force applied on the steel bolts during specimen dilation along x2 is

where k b is the stiffness of the steel bolt. For equilibrium, 2 k ~ & = k~(& - & ) ,

or, kr

(kr -t- 2k b)

(i) considerably higher strength values, (ii) the stress-strain curve remains largely linear, with much narrower yield zone, and (iii) higher residual strength values. Strength

F = 2k bS¢,

S c ~-~-

The typical stress-strain curves obtained from triaxial and plane strain testing of the four rock types, namely coal, sandstone, norite and quartzite, are given in Fig. 3 and the relevant properties of the four rock types are summarized in Table 2. Figure 3 and Table 2 show that the elasticity parameters E and v are about the same for both test types. The rate of strength increase with increasing confinement is also similar for both loading conditions, as could be noticed from the internal friction values listed in Table 2. The most marked differences resulting from plane strain tests are:

su.

The above formula implies that the magnitude of the displacements along x2 depends on the parameters E and v of the rock, and the stiffness of the bolts. In Table 1, the amount of strain likely to occur in the plane strain direction is given as proportions of the unconfined strain along x2 and the maximum principal strain along x~. As seen, in the case of coal, 99% on the unconfined strain along x2 direction is restricted, and this amounts to less than 0.2% of the major principal strain along x~ direction. For norite, however, the test rig could only restrict 66% of the unconfined strain along x2 direction, and this is about 4% of the major principal strain.

The higher strength values observed under plane strain loading condition are believed to be due to the strengthening effect of the intermediate principal stress, a2, as previously shown in polyaxial and plane strain testing [3,5,6]. The most significant differences occur between an unconfined axisymmetric (conventional uniaxial compressive strength) and unconfined plane strain (prismatic samples loaded using al > a2~0, a3=0) tests. Increasing tr3 does not cause any significant differences in strength between the two test types. This can be noticed from the failure envelopes given in Fig. 4, where almost a constant strength difference is noticeable between the strength envelopes obtained from the two different test types. The most changes in strength among the four rock types are related to coal and sandstone (39 and 42%, respectively), loading of which were closer to true plane strain than the other two rock types. Yield zone

Table 1. The effectiveness of the test rig (e¢ and eu are confined and unconfined strains along plane strain direction respectively, and e t maximum principal strain) Rock types

Sc/S~

Coal Sandstone Norite Quartzite

0.01 0.09 0.24 0.20

/~C/Sl ~ 0.00 0.02 0.07 0.04

One of the most significant effects of the plane strain loading on the stress-strain curve is the suppression of the non-linear region preceding the failure as compared to triaxial compression. As can be seen in Fig. 2, in triaxial loading, the deviation from the linear behaviour at the onset of "yielding" takes place at stress levels about 2/3 of the peak stress attained by the sample. The rate of strength increase with increasing strain continuously reduces, finally becoming zero at the peak stress

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Fig. 3. Stress-strain characteristics of the four different rock types as obtained from plane strain and triaxial tests.

value. Also, the strain range over which the yielding takes place is larger at increased confinement compared to plane strain tests. However, under plane strain loading, the yield zone is suppressed considerably. The stress-strain curve remains mostly linear in the pre-peak regime, and failure occurs at much smaller strain values than it does in triaxial testing. The reduction in failure

strain, due to increased intermediate principal stress, is also reported by Mogi [5].

Residual strength In the post peak regime, the residual strength values are greater under plane strain loading than they are in triaxial loading. The plane strain tests show "strain

YUMLU and OZBAY: TECHNICAL NOTE

729

Table 2. Material properties of the four rock types used as obtained from axisymmetric(AXI) and plane strain (PS) tests SD UCS (MPa) (%) E (GPa) v q~(°) No. of Rock type tests AXI PS AXI PS AXI PS AXI PS AXI PS Coal 10 38 53 4.9 5.1 3.9 4.2 0.19 0.19 30 30 Sandstone 10 86 122 1.7 1.9 28.0 29.3 0.25 0.26 52 52 Norite 10 226 270 1.2 1.4 85.0 85.4 0.29 0.30 53 55 Quartzite 10 264 310 6.8 8.0 66.7 68.2 0.18 0.19 65 68

hardening" in the post peak regime. This is probably due to the stiffer confinement applied by the steel plates and also because of the increasing confining stresses as the bolts react to the dilation of the specimen. Shear band localization

To further investigate the suppression of the yield zone during plane strain testing, a number of tests were stopped immediately before and after failure and sections were taken from the samples for visual inspections. In triaxial testing, the shear band formation was noted to start with the onset of yielding in the sample and fully

forming immediately after the peak stress. Similar observations were noted by Hallbauer et al. [11] and Wawersik et al. [9]. In contrast to these observations, in plane strain loading, the formation of shear plane was noted to be more sudden and coinciding with the peak stress. No noticeable evidence of faulting in sandstone, quartzite and norite specimens was observed prior to peak stress. These observations are consistent with other recent studies of bifurcation in rock materials tested under plane strain conditions [7, 9]. The results observed imply that the intermediate principal stress, a2, suppresses specimen fracturing in its

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YUMLU and OZBAY: TECHNICALNOTE

Fig. 5. Faultingin plane strain compression tests. direction, thereby resulting in dilatation to take place only in the minimum principal direction, x3. Similar observations were reported by previous studies on both soil [12] and rock materials [3,5].

The exception to the above observations was the onset of shear band formation well before peak stress in coal specimens loaded under plane strain. In these specimens, a fault zone initiation in form of a slanted S shape was observed at the centre of coal specimens well before peak stress at low confinements (see Fig. 3, Coal--3 MPa).

Mode of failure COAL

0 MPa

3 MPa

5 MPa

8 MPa

0 MPa

3 MPa

5 MPa

8 MPa

0 MPa

3 MPa

5 MPa

8 MPa

0 MPa

3 MPa

5 MPa

8 MPa

SANDSTONE

NORSE

QUARTZITE

Fig. 6. Mode of failure observed in plane strain compression tests.

Like in triaxial tests, the failure was along a shear plane in plane strain tests. Figure 5 shows typical faulting occurred during plane strain testing of the rock types used. The failure modes observed are summarized in Fig. 6. The prismatic specimens tested under uniaxial compression show similar failure modes to those observed in uniaxially tested cylindrical samples, that is, two cones at the top and bottom of the specimen form an "hour glass" shape. Under low confinement, the shear plane was, in some cases, less steep at the central portion (see diagrams labelled norite 3 MPa, and sandstone 0 and 3 MPa in Fig. 6). Steepening towards the corners in these low confinement tests is possibly due to platen interference. The inclination of the shear zone at the central portion of the specimen, therefore, is more likely to be the actual failure angle. With increasing confinement, the shear plane becomes steeper and more uniform (norite 8 MPa, sandstone 8 MPa) along a plane connecting the two diagonal edges, as also observed by other researchers [7]. However, the possibility of interference from platens in affecting the orientation of the shear plane cannot be ruled out. In coal specimens tested under 0-5 MPa confinements, the shear band tends to go through only one edge, emerging at the sides about 10-15 mm away from the loading platens. At 8 MPa confinement, however, the fault goes through both edges dividing the sample diagonally into two discrete pieces. This steepening effect is probably realistic as the loading conditions closely approximated those of plane strain in the case of coal.

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PLANE STRAIN LOADING AND STRAIN SOFTENING MODEL Strain softening as implemented in FLAC is a plasticity model based on the M o h r - C o u l o m b failure criterion, the main difference being that the strain softening model incorporates "load shedding" upon loading beyond peak strength. In this model, the complete stress-strain characteristic is defined by the two elastic constants E and v, and the three plasticity parameters cohesion C, friction qb, and dilation v as a function of plastic strain. The plasticity parameters for the rock types tested were obtained using the computer program SS [13], which allows approximate re-construction of the experimental data by means of defining C, ~b (and to a lesser

extent v) as a function of plastic strain. Figures 7 and 8 show the results of program SS on strain softening behaviour of the sandstone and coal samples tested under axisymmetric and plane strain loading conditions. The cohesion and internal friction angle are the most influential parameters in approximating the experimental results. The figures suggest that the friction angle remains constant during plane strain testing, while in triaxial testing the friction angle starts at a low value and increases (friction hardening) until it reaches the point at which the cohesion stops falling and becomes constant. The constant friction behaviour results in suppression of the non-linear region (yield zone) in the plane strain loading conditions. Also, friction hardening in the triaxial loading appears to be the reason for lower cohesion values (at peak strength) in this loading condition.

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CONCLUSIONS The stiff frame test rig developed provided allowed plane strain loading conditions for the prismatic coal specimens, while only quasi-plane strain loading conditions could be achieved for the sandstone, quartzite and norite specimens. The results from comparing a series of triaxial and plane strain tests conducted on these four different rock types showed that, under plane strain loading:

Otherwise, the elastic modulus, Poisson's ratio and dilation angle were observed to be less affected by the loading conditions of triaxial and plane strain testing. Further evaluation of the results for a commonly used strain softening model showed that the response of rock to plane strain loading could be simulated by keeping the internal friction angle constant with increasing plastic strain.

(i) the strength is about 30-42% greater,

(ii) the non-linear yield region preceding failure is suppressed, (iii) residual strength is higher, and (iv) the shear plane angle increases with increasing confinement.

Acknowledgements--This

r e s e a r c h is f u n d e d b y t h e S a f e t y in M i n e s Research Advisory Committee (SlMRAC) of the Ministry of Mineral and Energy Affairs of South Africa under project COL014. The ideas a n d g u i d a n c e p r o v i d e d b y D r J. A . R y d e r a r e g r e a t l y a p p r e c i a t e d .

Accepted for publication 26 January 1995.

YUMLU and OZBAY: REFERENCES 1. Vermeer P. A. and de Borst. Non-associated plasticity for soils, concrete and rock. Heron 29, 1-64 (1984). 2. Cundall P. A. FLAC Users' Manual. ITASCA Consulting Co. (1987). 3. Stavropoulou V. G. Behaviour of a brittle sandstone in planestrain loading conditions. Proceedings of the 23rd U.S. Symposium on Rock Mechanics, Univ. of California, pp. 351-358 (1982). 4. Vardoulakis I. Shear band inclination and shear modulus of sand in biaxial test. Int. J. Num. Anal. Meth. Geomech. 4, 103-119 (1980). 5. Mogi K. Effect of intermediate principal stress on rock failure. J. Geophys. Res. 72, 5117-5131 (1967). 6. Hojem J. M. P. and Cook N. G. W. The design and construction of a triaxial and polyaxial cell for testing rock specimens. S. Aft. Mech. Engng 1, 57~1 (1968). 7. Ord A., Vardoulakis I. and Kajewski R. Shear Band Formation in Gosford Sandstone. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 28, 397409 (1991).

TECHNICAL NOTE

733

8. Labuz J. F. and Papamichos E. Preliminary results of plane strain testing of soft rocks. Rock Mechanics as a Multi-Disciplinary Science, pp. 667~74. Balkema, Rotterdam (1991). 9. Wawersik W. R., Rudnicki J. W., Ollsson W. A., Holcomb D. J. and Chau K. T. Localization of deformation in brittle rock: theoretical and laboratory investigations. Proceedings of the International Conference on Micromechanics o f Failure o f Quasi-Brittle Materials, Albuquerque, N.M., pp. 115-124 (1990). 10. Haied A., Kondo D. and Henry J. P. Experimental detection of shear band in a sandstone. Assessment and Prevention o f Failure in Rock Engineering (Edited by Pasamehmetoglu et al.), pp. 149-154. Balkema, Rotterdam (1993). 11. Hallbauer D. K., Wagner H. and Cook N. G. W. Some observations concerning the microscopic and mechanical behaviour of quartzite specimen in stiff triaxial compression tests. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 10, 714-726 (1973). 12. Finn W. D. L., Wade N. H. and Lee K. L. Volume changes in triaxial and plane strain tests. Proc. ASCE, Soil Mech. Found. Div. SM6, 287-308 (1967). 13. Ryder J. A. Personal communication (1992).