26th US Symposium on Rock Mechanics / Rapid City, SD / 26-28 June 1985

A constitutive law for shear behavior of rock joints

R.JANARDHANAM

University of North Carolina, Charlotte, USA

WILLIAM F.KANE

University of Alabarna, Huntsville, USA

1 INTRODUCTION

Knowledge of the behavior of rock joints is essential in the design and construction of large structures like tunnels, underground power houses, dams and foundations. When a numerical solution technique, such as fi- nite element method is employed to analyze an underground structure, it is required that the realistic behavior of rock joints be known for a meaningful analysis. Many investigators have worked on the refinement of the constitutive relationship for a jointed rock mass. In this in- vestigation, an attempt was made to examine the direct shear behavior of rock joints and to formulate a representative Constitutive Law.

2 BACKGROUND

The failure theory for rock joints in its simplest form can be expressed as

F =•N

Where F and N are shear and normal forces and • is the coefficient of friction. Failure criteria most often used is Mohr - Couloumb's law

r = S O + #o

Where r and o are shear and normal stresses and S o is the inherent shear strength of contact surface. However, normal and shear stress relations for peak and residual strengths of rock joints can be represented gen- erally by Figure 1. Here the residual strength can be viewed as the strength of an uncemented discontinuity while the peak strength is the strength of the intact cemented joint.

Friction is the controlling factor between the surface of joints and fracture planes. Maximum values for the coefficient of friction may be high as 75 to 80 degrees. However, residual values rarely deviate be- yond the range of 25 to 30 degrees. For smooth surfaces, the coeffi- cient of friction will range, generally between 0.4 and 0.8 but mostly be about 0.5 and 0.6 Relatively smooth surfaces fail by stick-slip. Rough surfaces slide by stick-slip at low normal stresses.

The shear strength of a rock joint is sensitive to degree of surface roughness, compressive strength of the rock, degree of weathering,

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mineralogy, and the presence or absence of water. Irregular and poorly matching joint surfaces can cause stress distribution to be complex. The roughness of a joint is classified into first and second order roughness. First order roughness correspond to major undulations on the bedding plane. The small bumps and protrusions on the primary undulations are re- ferred as second order roughness. At low normal stresses, the second order projections control behavior and account for what is often called a slight cohesion. This will become less important as the normal stress increases and small asperities are sheared. The the primary or first order projections will come into play.

The behavior of a joint where the discontinuity surface is not exactly parallel to the direction of the shear stress, Figure 2, can be ex- pressed as

r.= rcos2i +•(sin i) (cos i)

and

•i =•cos2i + r(sin i) (cos i) Where i = the slope of the inclined plane. In case of a rough rock joint, Figure 3, due to dilation, any horizontal (shear) displacement, must be accompanied by a displacement in the vertical direction (normal). Con- traction is also possible and is the closing of the joint due to large normal stresses. This joint compression is not elastic and is, there- fore irreversible.

Studies have shown that experiments on similar surfaces will not nec- essarily yield the same values of the coefficient of friction but then, in general, coefficient determined on relatively small surfaces can be applied to larger one. Also the direct shear test has been called "a natural way to test properties of discontinuities, especially at low normal pressures." Therefore, in this investigation, the behavioral response of rock joint was studied by conducting direct shear test on rock samples. The details of test program, analysis of test results and the development of a constitutive law are presented in the follow- ing sections.

3 DESCRIPTION OF THE ROCK

The rock tested was obtained from the Metropolitan Atlanta Rapid Transit Authority which was in the process of driving subway tunnels beneath the city of Atlanta, Georgia. The rock was tested and was determined to be amphibole (hornblende) gneiss, dark gray-green in color and consisting of fine to coarse grained crystals. Enough biotite mica was also found in some samples to warrant their description as biotite gneiss; a small amount of quartz was also found. Foliation in the gneiss was faint to strong. The density of the rock was approximately 2.5g/cm 3.

4 TESTING PROGRAM

A number of tests were conducted in order to determine the behavior of the rock in shear. These tests fell into several catagories described below. All samples consisted of a top block and bottom block, Figure 4. The top one measured approximately 10 x 10 x 2 cm and the bottom one 16 x 13.5 x 2.5 cm. The use of a larger lower block eliminated the need

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v

Peak Shear

•/Strength -Re•tidr•na•t•hear Horizontal Displacement (u)

Figure 1. Normal and shear stress relations for peak and residual strengths.

Figure 2, An inclined joint surface.

v=u tan i

Figure 3.. Dialation in a rough rock joint.

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Figure 4. Direct shear test on a rock joint interface.

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5OO

400

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ol

1470KPA

0.476

1470KPA

0.37

dry surface

wet surface

• = 490KPA

• = 0.51

e = 490KPA

/ • = 0.49

I I I I

.1 .• .3 .4 .5

HORIZONTAL DISPLACEMENT IN CM.

.7

Figure 5. Effect of moisture presence on shear behavior of rock joints.

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for an area correction during shear testing. The first set of tests examined shear behavior of saw-cut rock sur-

faces. Surfaces were cut on a Highland Park Diamond Saw and tested with the samples as such. Tests were run on dry samples and samples soaked in water for a week preceding testing.

The second set of tests were run of natural cleavage surfaces. Rock samples containing partings were cut so that the partings were parallel to the flat face of the cut. Once the samples had been cut, it was split along the parting to produce a top and bottom block. Samples were then cut to a refinished size and tested.

Artificially pitted surfaces were also tested to study the shear be- havior. A hammer and punch were used to fracture the amphibole along its crystal cleavages to obtain a surface very similar to the rough surfaces on the actual partings. Tests were run on dry and wet surfaces to study the effect of the presence of moisture in the joint.

5 TEST PROCEDURE

A direct shear machine was employed. All tests were strain controlled and a very slow rate (0.0183 cm/min) was chosen to eliminate any dynamic effects. The large sample was placed in the bottom of the shear box. The bottom box remained stationary. The top sample, placed in the top half of the shear box was rested on the large sample. A normal stress was ap- plied through a loading mechanism. A horizontal (shear) load was applied through a motor driven piston on the top sample. The horizontal (shear) displacement was recorded as the test progressed. The tests were termi- nated once the horizontal displacement exceeds 10% strain (with refer- ence to top sample).

6 ANALYSIS OF TEST RESULTS

The results of the tests on saw-cut surfaces generally showed an in- crease in shear stress linearly with horizontal displacement up to a maximum value where slip occurred. Further displacement resulted in stick-slip behavior shown in Figure 5. In stick-slip, the shear force increases up to a point at which a sudden slip with a resulting decline in shear force occurs. Then the surfaces lock together again and the shear force increases til another slip occurs. The stick slip can there- fore be regarded as relaxation oscillations due to irregular fluctua- tions in frictional force caused by shearing irregularities during sliding.

The presence of moisture at the joint is seen to increase the fre- quency of the oscillations of the stick-slip even though the amplitude is decreased, Figure 5. However, there is no substantial effect on the shear strength at failure. Water changes, rather reduces the coefficient of friction of the surface at the joints, as shown in Figure 6.

The shear stress-displacement responses of artificially pitted sur- faces for different normal stresses are shown in Figure 7. The shear stress is seen greater when the coefficient of friction is higher, for the same normal stress and horizontal displacement. The presence of water in such joints does not influence the magnitude of shear stress significantly. However moderate oscillations are noticed once the shear stress reaches the maximum. The natural cleavage surfaces are observed to behave in similar way. The results are not presented here.

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7 CONSTITUTIVE LAW

The observed test results (shown in Figure 7) are used to develop a con- stitutive law. The shape of the curve is assumed here to be hyperbola. The constitutive relation is expressed as

G *AH (1 + G * AH)

rmax

Where G = initial maximum tangent modulus rmax = maximum shear stress

•H = horizontal displacement

Introducing R i = •/Mi, where M i = •i• (i = 1,2,3,..n) and expressing G i interms of R i , •iand •i, a constitutive equation is determined by curve fitting as

13.2 a •AH 0.55 + 13.2

For verification, the developed equation is used to back predict the shear stress displacement responses for different normal stresses, shown in Figure 8. The prediction by the constitutive model compares well with the observations.

7OO

ß Dry, smooth surface p• 0.37

I ß Wet, smooth surface p• 0.15 500 ß

' 300

100

Figure 6.

500 1000 1500

NORMAL STRESS (KPA)

The Effect of Water on Coefficient of Friction - Saw Cut Smooth Surface.

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8 CONCLUSIONS

A comprehensive series of tests have been conducted to determine the shear behavior of rock joints. The results of tests on smooth surfaces generally showed an increase in shear stress linearly with displacement up to a maximum value where slip occurred. The presence of water was ob- served to decrease the magnitude of the stick-slip oscillations, and to increase the frequency of the stick-slip oscillations. In case of natural cleavage surfaces and artificially pitted surfaces, the shear stress was observed to increase with the shear displacement for greater normal stress. The constitutive equation developed can be used in incremental form for boundary value problems.

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l •=0.511 •=0

=1470KPA

=0.59)

•=0

a =980KPA

•=0.561

--•=490KPA

=0.225

0 .1 .2 .3 .4 .5 .6 .7

HORIZONTAL DISPLACEMENT IN CM.

Figure 7. Shear stress - horizontal displacement response of rock and artificially pitted surfaces.

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8OO

7OO

6OO

5OO

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ß ß

__ (7 =980KPA • =0. 597

ß observed

O predicted

o =490KPA =0.225

0 .1 .2 .3 .4 .5 .6 .7

HORIZONTAL DISPLACEMENT IN CM.

Figure 8. Comparisons of predictions with observed test results.

REFERENCES

Barton, N.R. 1972. A model study of rock-joint deformation. Internation- al journal of rock mechanics and mining sciences. 9:579-602.

Bock, H. 1979. Simple failure criterion for rough joints and compound shear surfaces. Engineering geology. 14, no. 4:241-254.

Coulson, J.H. The effects of surface roughness on the shear strength of joints in rock. Thesis Univ. Illinois, U.S. Dept. Army Corps. Eng., Miss. River Div., Omaha, Tech. Rept. MRD-2-70:283 pp.

Jaeger, J.C. 1970. The behaviour of closely jointed rock. Proc. 11th symp. on rock mech.:57-68.

Landanyi, B. & G. Archambault 1970. Simulation of shear behavior of a jointed rock mass. Proc. 11th symp. on rock mech.:105-125.

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