1993 Study of rock joints under cyclic loading conditions.pdf

Rock Mech. Rock Engng. (1993) 26 (3), 215--232 Rock Mechanics and Rock Engineering 9 Springer-Verlag 1993 Printed in Austria

Study of Rock Joints Under Cyclic Loading Conditions By

L. Jing, O. Stephansson, and E. Nordlund

Lule~ University of Technology, Lulegt, Sweden

Summary

A conceptual model for the behaviour of rock joints during cyclic shear and under constant normal stresses was proposed according to results from shear tests with 50 concrete replicas of rock joints. The shear strength and deformability of joint sam- pies were found to be both anisotropic and stress dependent. Based on these experimental results, a two-dimensional constitutive model was developed for rock joints undergoing monotonic or cyclic loading sequences. The joint model was formulated in the framework of non-associated plasticity, coupled with empirical relations representing the surface roughness degradation, appearance of peak and residual shear stresses, different rates of dilatancy and contraction, variable normal stiffness with normal deformation, and dependence of shear strength and deformability on the normal stress. The second law of thermodynamics was represented by an inequality and used to restrict the values of some of the material parameters in the joint model. The new joint model was implemented into a two-dimensional Dis- tinct Element Method Code, UDEC, and its predictions agreed well with some well-known test results.

I. Introduction

Samples of rock joints have been tested with both monotonic and cyclic shear sequences for years (Bandis, 1983; Kutter and Weissbach, 1980). Under a constant normal stress, an apparent peak shear stress, often called shear strength, can be observed for some joints, but may not appear for others. The joints will dilate in the normal direction while being sheared forward and contract when the shear direction is reversed. This is generally attributed to the roughness of the rock joint due to the presence of asperities on the joint surface. This roughness of the joint surface decreases with the accumulation of shear displacements due to the failure of asperities, and this is in turn called surface roughness degradation. For the deformability of rock joints, it has also been observed that the shear stiffness of rock joints increases with the increase of normal stress magnitude, and the normal stiffness will increase with the increase of the normal closure (Ban- dis, 1983; Barton et al., 1985).

216 L. Jing et al.

A number of constitutive models for rock joints have been developed over the years, for example, Goodman's empirical model (Good- man, 1976), Barton-Bandis' empirical model (Barton et al., 1985), Plesha's theoretical model (Plesha, 1987) and recently Amadei-Saeb's theoretical model (Amadei and Saeb, 1990). However, most of them were only developed for monotonic shear sequences without consider- ing the surface roughness degradation, except for Plesha's model. Plesha (1987) developed his model in the frame of theory of plasticity, with consideration for both cyclic shear sequences and surface roughness degradation. However, the peak -- residual behaviour of the shear stress was not considered in his model. The effects of normal stress on the shear strength and shear deformability of joints, and the variable normal stiffness with normal deformation, were also absent, perhaps due to lack of experimental evidence.

In the light of these shortcomings, a systematic experimental study was carried out to investigate joint behaviour during cyclic shear tests with special attention paid to the anisotropy and stress dependence of the shear strength and shear deformability of joints, and the general stress-displacement behaviour of joints during cyclic shear. In order that the test could be repeated with the identical initial surface condition of joint samples, totally 50 concrete replicas from two granite joint samples were made and tested under different normal stresses and in different shear directions in the plane of joint surface. The two natural joint samples have different initial surface conditions. One sample has no filling, with perfectly mated joint surfaces and no previous shear deformation. Another also has no filling, but shows a moderate degree of weathering and slightly mismated joint surfaces due to previous shear deformation. The 50 joint replicas were made of high strength concrete and tested on a servo-hydraulic shear frame (Jing, 1990). Based on the test results, a conceptual model for the general behaviour of rock joints during cyclic shear sequences was postulated, which exhibits the basic features of peak/residual shear stresses and dilatancy/contraction of the normal deformation during cyclic shear. Based on this conceptual model and using Plesha's model as a starting point, a new two-dimensional constitutive model for rock joints was developed with special consideration given to the stress-dependence of the friction angle and shear stiffness, surface roughness degradation, and peak/residual feature of shear stress. This new joint model was then implemented into a Distinct Element Code, UDEC, with good agreements between the model predictions and some well known test results conducted by different researchers, for example, Bandis et al. (1983) and Kutter and Weissbach (1980). In the following sections of this paper, the basic test results and empirical relations will be briefly introduced, followed by the formulation, implementation and validation of the new joint model. Details concerning the test equipment and test techniques can be found in Jing (1990) and will not be repeated here. The tensile stress and joint opening are taken as positive.

Study of Rock Joints Under Cyclic Loading Conditions

2. Basic Experimental Results

217

2.1 Effect of Normal Stress on the Shear Strength

Let the total friction angle represent the shear strength of the joints. From the shear test under constant normal stresses, the mobilized total friction angle of these concrete replicas of joints decreases with the increase of the magnitude of normal stress (see Fig. 1 a, 0 denotes shear directions in the plane of joint surface). In this figure, the friction angles corresponding to o-, --- 0 are obtained by tilting tests of the joint samples, in which the only normal force is the self-weight of the upper block of the joint sample and can be regarded as the initial friction angle without normal loading. To represent this effect of normal stress on the shear strength of joints under constant normal stresses, it is assumed that the total friction angle, r can be taken as the sum of a residual friction angle, gS,, and an asperity angle, c~, in any particular direction on the joint surface (Jing, 1990)

r = r + G (1)

and the effect of the normal stress on the shear strength of joints under constant normal stresses can then be approximately represented by an empirical relation

= d - (2)

where a'0 is the initial asperity angle, o-, is the magnitude of normal stress and o-~ is the magnitude of the uniaxial compressive strength of the material. b is a material constant representing the wearability of the joint material. The variation of the asperity angle a~ with normal stress accord-

50 X]

111 .d O z L5.

Z O

~- 40. (D

rY LL

35 0

~ o ,:, = 9o o

9 = 120 ~ = 150 ~ + =180 ~ o =210 ~

i i

NORMAL STRESS {MPa)

1'0

d n ,b o~=% 1-57 j (%=1o ~ ) 12-

~ 10 c-,.~: oL ID -" "O~

"-..,. ",,...'~.~'o, Ld "'*. ~'.~':'~ , 6- \ ,,+, ~,,,~. (D z "% ,+ "~'~; < 4- , OI >- +,'I

2- Ld

tZl

< 0 . . . . ~ - - 0.0 012 0.4 06 0.8 1.0 112

,, b= 1 9 = 2

b, = :3

9 = 4

x = 5

+ = 0.2

,,, = 0.3 = 0.4

o = 05

NORMAL STRESS/COMPRESSIVE STRENGTH, Un/d c

a b

Fig. 1. Variation of the total mobilized friction angle with respect to the shear direction and the magnitudes of normal stresses, a Measured results from shear tests on concrete replicas;

b behaviour of the empirical model by Eq. (2)

218 L. Jing et al.

ing to Eq. (2), with different values of b, is shown in Fig. lb. This model can serve as a conceptual model for joints under constant normal stresses. However, it cannot predict correct shear stress behaviour under more complex normal stress conditions because Eq. (2) may cause reversibility of the roughness by decreasing the normal stress magnitude. A more proper method is to use accumulated dissipated energy along the shear path as an index for the accumulated damage on the joint surface, as first proposed by Plesha (1987) and described in Section 2.3.

2.2 Effects of Normal Stress on Shear Deformability

The shear deformability of joints can be represented by the shear stiffness. It was observed during tests that the shear stiffness, k,, increases with the increase of magnitude of normal stress (see Fig. 2a, 0 denotes the shear direction in the plane of joint surface). An empirical relation is then developed to represent this property, written as (Jing, 1990)

(2_ -) k, 0

(0 >__ a. >_ ~c)

> o) (3)

where kT' is the maximum shear stiffnes and is obtained when the normal stress reaches the magnitude of ac. This relation becomes invalid when the magnitude of normal stress is beyond that of o-c because in this case the rock material fails. The variation of the shear stiffness k, with respect to the

A

5 r8 \

~6

03

~2

i , i , , i , i J

2 4 6 8

o O = 330 ~ = 300 ~ = 270 ~

o = 240 ~ + = 210 ~ , = 180 ~ 9 = 150 ~ 9 =120 ~

9 = 90 ~ 9 = 60 ~

-r K t/(Jc I--

W w )--

~3.

ILl

W

It.

I'--

13 Km/Oc =0 2(

=0J , I

=0.61

=081

=1.01

=] .b l

=2.0 I

=2.5 I

=3.0[

=3.51

tY

Study of Rock Joints Under Cyclic Loading Conditions 219

normal stress o-, is shown in Fig. 2 b with different values of k; ~ from this empirical relation. All quantities in this figure are normalized by o-~.

2.3 The Normal Stiffness and Roughness Degradation

Bandis et al. (1983) and Goodman (1976) reported hyperbolic relations between normal stiffness of rock joints and magnitude of normal stress, based on experimental results. However, it can also be reformulated as a relation between the normal stiffness and the normal displacement (closure) of joints. Let k, be the normal stiffness and u, be the normal displacement, the following empirical form can be directly derived from Bandis' hyperbolic relation (Jing, 1990),

kO kn = (1 - Un/U ) 2' (4 )

where k ~ is the initial normal stiffness and u m is the maximum closure of the joints.

Plesha (1987) proposed an exponential law for surface roughness degradation, based on his experimental results

cr = or0 exp [ - D WP], (5)

where WP is the dissipated energy during shear and D is an experimentally determined parameter. Hutson (1987) later verified this model by experi- ments and proposed that D = -0.114 JRC(N/~yc) or - 1.07 fl(N/~c) where fl is a material constant, JRC is the Joint Roughness Coefficient as defined by Barton (1976), N is the normal force during shear test, and o-c as defined above. A modified form of D in (5), using normalized normal stress instead of normal force, is adopted here (Jing, 1990)

D =Dm [crn/ac], (6)

where O m > 0 is a material constant dependent more on the geometry of asperities and is determined by shear tests. Because quantity on/o-c is always positive (or zero) and the minus sign in the exponential function in Eq. (5), the damage model represented by Eq. (5) is not reversible with normal stress changes. This model of roughness damage is used for the constitutive model developed later in this paper.

2.4 Stress-Displacement Behauiour During Cyclic Shear

The shear stress -- shear displacement behaviour of concrete samples of joints, under constant normal stress during cyclic shear, is very similar to the stress-displacement behaviour of a plastic body during a cyclic loading-

220 L. Jing et al.

unloading sequence (see Fig. 3). The peak/residual shear stress behaviour occurs only for the mated joint without previous shear deformation during the first shear cycle (Fig. 3 a), and no peak shear stress occurs for second shear cycle. Corresponding to the shear stress variation, for the first cycle, the normal displacement (dilatancy or contraction) curve is highly non-linear with many small scale stick-slip oscillations, indicating gradual failure of the smaller higher-order asperities. For the second cycle, the normal displacement curve is much smoother and linear, indicating smoother joint surfaces with much reduced higher-order asperities. We may conclude from these results that the peak shear stress is related to the higher-order asperities and the rate of dilatancy/contraction is basically controlled by the primary asperity angle.

(MPa) U~(mm) 3-

- ii ~ Cycle 1 ~ e 2 -12j ] -6 -31 3 6/] 12 -12 -g 6 9)2

Cycle 1 -3 J 1.0

a b

Fig. 3. Experimental joint behaviour of mated concrete replicas without previous shear. a Shear stress versus shear displacement; b normal displacement versus shear displacement

The above test results were observed during tests of concrete joint replicas under constant normal stresses and empirical relations (1--3) were developed to represent the behaviour of these concrete joint samples. Since the surface topography of these concrete samples is identical to those of the rock joint samples from which these concrete samples were cast, we think that these empirical models can also be extended to represent the behaviour of natural rock joints under similar loading conditions. The values of material constants b, Dm and k~ will of course be different. Simi- lar experimental shear stress -- shear displacement curves during cyclic shear, under constant normal stresses, are also reported by Kutter and Weissbach (1980). These experimental observations can be used to estab- lish a conceptual model for rock joints during cyclic shear, as given in the next section.

3. A Conceptual Model of Joint Deformation Under Cyclic Shear

Let T and U~ be the shear stress and shear displacement, respectively. The complete curve for the first shear cycle consists of segments OA, AB, BC,


v ! "C O .

K

I I U~ J r - utP-o Ut Ut

-~b i H ~--7

G

!I' !j,

!'-,L \ " ,

I -t"-.-,",\ i i "

222 L. Jing et al.

tinuing the forward shear, the shear stress passes peak point B (U~ p, rp+), the residual point C(UT, rr +) and reaches point C(U~ D, r~ +) where the shear direction reverses. The peak shear stress (i. e. strength of the joints) and residual shear stress are defined by

"t-p = o-n tan [@r "~- CO'], (7)

rr = ~rn tan [r (8)

where c~ is given by Eq. (5) under general loading conditions. The increase and decrease of shear stress about peak point B (U p, rp + ) with shear displacement is called the strengthening and weakening effects of joints with shear displacement, similar to the hardening and softening of a plastic body. The shear stress after the residual point will remain at the residual level of rr, as shown in segment CD. Corresponding to stress path OABCD in Fig. 4a, the joint dilates along the path O' A' B' C' D' in Fig. 4b with the gradually decreasing rate of dilatancy.

The shear direction reverses at point D (U, ~, r~ + ) and the shear stress decreases linearly along the straight segment DE of the same slope kt as that of segment O-A until it reaches point E (U e, %). The segment DE is called the unloading stage because it represents the proportional decrease of shear stress over shear displacement immediately after the reverse of shear direction, rb is defined by

vb = o-, tan [r (9)

where Cb < ~r is called the basic friction angle of the joint. Corresponding to this unloading stage of shear stress (segment DE in Fig. 4a), joint will contract with a constant slope, as shown by segment D' E ' in Fig. 4b. With continuing shear in the reversed direction, the shear stress will main- tain a constant magnitude of vb (as shown by segment EF in Fig. 4 a) until it reaches the original point 0 of shear displacement. The contraction will also continue, but at a different rate, represented by the segment E' F' of a steeper slope, see Fig. 4b. These two different rates of contraction, rl and r2, can be approximated by

rl = A u , /A u, = - tan [cr (10)

r2 = A u JA ut = - tan [c~ ~ + c~)/2], (11)

where oc) is the current asperity angle at point 0' and becomes smaller for subsequent shear cycles due to the damage incurred on the asperities.

When shear continues in the negative direction after the original point 0, the shear stress and normal deformation will have similar features as in the positive part of shear, see Figs. 4a and 4b.

For subsequent shear cycles, the peak shear stress will not occur and the stress path will follow curved segments KC and FH (dashed lines in


Fig. 4 a) and the normal deformation will follow the dashed curves in Fig. 4b. The dilatancy curves and contraction curves will not intersect each other, unlike that in the case of first cycle when peak shear stress occurs. For joints with previous shear histories, no peak shear stress will occur even for the first cycle, and the dilatancy curves will be much less non- linear and not intersect the contraction curves, see Fig. 4 c.

The shear stress -- shear displacement behaviour of the joints, as idealized in Fig. 4a by the conceptual model, may represent the joint behaviour under constant normal stresses only. For more complex and general loading conditions in both normal and shear directions, a more com- prehensive constitutive model is needed. By observing the similarity between the shear stress -- shear displacement behaviour of a joint and stress-strain behaviour of a plastic solid with hardening and softening effects during cyclic loading, it might be possible to formulate a general constitutive model for rock joints in the mathematical frame of plasticity theory. This general model may then be enhanced by combining with special empirical relations suitable for rock joints to produce a final constitutive model representing a more realistic behaviour of rock joints. This is described in the following sections, based on Plesha's original work (Plesha, 1987).

4. Mathematical Formulation of the General Joint Model

4.1 A General Model for Solid Interfaces

The basic assumptions are: the problem is a two-dimensional plane strain problem; the thermal effects, fluid flow, and time and rate-effects are not considered. The convention of the summation over repeated subindex are assumed unless stated otherwise. The tensile stress and joint opening are taken as positive.

Let -01 and -622 be two rock blocks in contact along a joint as shown in Fig. 5a. Let ui (i = t, n) be the relative shear and normal displacements between -01 and -02, and o-i (i = t, n) be the shear and normal stress compo- nents in the directions tangential and normal to the joint surface, respectively. It is assumed that the total displacement increments, dui, can be decomposed into a reversible part, duT, and an irreversible part, du p, written

dui = du7 + duf. (12)

An elastic response is assumed for the changes in the reversible deformation, i. e.

d oi = kij du;, (13)

where kij is the stiffness tensor withk, n = k,, = 0, ktt = kt and k,, = kn.

224 L. Jing et al.

(~1) ] n / I / / / I / / I I / A I I / / I / / / / / I I /

/ / i / i / i / / / /} l / / l l l l l /z l l

(Q2)

l~1) c

t ~ IL ~--

(Q2) 5

Fig. 5. Idealized rock joint: a macroscopic scale; 5 microscopic scale

The irreversible part of the incremental displacement is described by a sliding rule given in the form {0

duP= 2 F> O ' (14)

where 2 is a non-negative scalar. F and Q are called the slip function and sliding potential, respectively. Slip between two opposite surfaces of an interface will occur whenever the condition F > 0 is satisfied. The sliding rule is called a non-associated sliding rule if Q differs from F. The energy dissipated during the shearing process can be written.

dWp = cr i du( = )1 cr i 3Q/3 ~. (15)

By using the consistency coa~dition dF= 0 and standard mathematical derivations from the theory of plasticity, the scalar )1 and a general constitutive model for solid interface can be obtained in the forms

3F - - k u duj

>l= A + mB (16) and

with

( .t du,, d~ = k u A + roB/ (17)

gF k~s 3Q 3Q 3F 3Q A - 3 cr~ c9 as ' kp = k,r 3at --cv~s ksj, B = ak 3ak ' (18)

where m is a scalar representing strengthening (hardening) or weakening (softening) effects.

4.2 Thermodynamic Restr ict ion

Let S be the entropy of the sliding system, ~- be the absolute thermodynamic temperature and (~ be the heat, then the second law of thermo-


dynamics can be written as (Martin, 1975)

d5 = (d 9 + d WP)/Y > O. (19)

Since Y > 0 and dQ = 0 for isothermal problems, as assumed in our case, the second law of thermodynamics for our joints system can be simply written

d W p : ~b o" i (8 Q/3 cri) > O, (20)

i. e. the dissipative work must be non-negative. In reality, there is always heat generation during frictional sliding in contact-friction problems of material interfaces, so that in general, d 9 > 0. However, the surfaces of rock joints are usually rough and the contact area is relatively very small, the heat generated during shear is usually so little that it can be ignored without introducing serious errors. The inequality (20) must be met by the constitutive model.

The constitutive model expressed by Eq. (17) and inequality (20) is a general model for solid interfaces, which provide the foundation for constitutive models for interfaces of different materials if proper functional forms of F and Q can be supplied. A special constitutive model for rock joints can be obtained if the general model is further coupled with empirical relations about some special properties of rock joints, besides the selection of proper forms of slip function F and sliding potential Q. The result- ant joint model could then both satisfy the second law of thermodynamics and represent some special features of rock joints. This spezialization of the general model for rock joints is described in the following section.

5, A New Constitutive Model for Rock Joints

5.1 Formulation of Rock Joint Model for Distinct Element Method

It is assumed that the rock joints are nominally planar surfaces consisting of many small asperities with regular shapes, of the same size, and are uniformly distributed over the joint surface (see Figs. 5 a and 5 b). Because of an explicit time-marching scheme used in Distinct Element Method using time steps, the stiffness parameters, the asperity angle ~ and the current peak shear stress are kept constant during a time step, but should be updated according to relations (3), (4), (5) and (7) at the end of the current time step for following calculations. Based on these assumptions, the contact between two opposite surfaces of joints will, during a shearing process, occur only on one slope of the asperities with the angle o~. This angle is called the "active" asperity angle c~ and can be one of two slope angles of the asperity, c~L or a~R (Fig. 6 a), determined by the shear direction. Let o5 and o-~ be macroscopic shear and normal stresses uniformly distributed at the base of asperity and o-f and o -c be the contact shear and normal stresses uniformly distributed on the active asperity slope with angle ~ (Fig. 6 b).

226 L. Jing et al.

,R

4

L o d . , 0 t a b

Fig. 6. a Definition of asperity angles; b stress transformation at the microscopic scale

Then equil ibrium of the stresses acting on the asperity leads to the following relations (Plesha, 1987):

{ a, ~ = 7/(at cos c~ + an sin o~) a~ = 7 / ( - at sin c~ + an cos ~) '

(21)

where 7/is a scalar independent of the stresses. By adopting the Coulomb friction law on the active asperity surface, the slip function F and sliding potential Q may be written as (see also Plesha, 1987)

F= la;I +~a~- C = [Gcos ~+ Gs in c~l +#( - atsin c~+ a, cos c~) - C

Q= la~l = la , cos c~+ ansin ~1,

(22)

(23)

where # = tan (G) for the forward shear stage and # = tan (~b) for the reversal shear stage, respectively. C is the cohesion of rock joints. Substitu- t ion of Eqs. (22) and (23) into Eq. (17) leads to an explicit constitutive relation between increments in the macroscopic stresses and relative displacements of the joint, given in the form

aa k~ da~= (k t - a lk t+a2k .+mQ)dUt -

a4k~kn ( dan=- al kt + a2 kn + mQ du, + k,, -

with

a3 kt k, dun al kt + a2 kn + m Q

a2 k~ ) al kt + az kn + mQ dun

(24)

al = cos 2 c~ - # sgn (0-, c) sin c~ cos c~ a2 = sin z c~ - # sgn (0-~) sin c~ cos cr

a3 = sin cc cos a~ + # sgn (a~) cos z c~ a4 = sin cr cos o~ +/.t sgn (G c) sin z

(25)


where sgn (oi) is the sign of contact shear stress o- i, k,, kn, and er are current values of the shear stiffness, normal stiffness and the asperity angle of rock joints.

Let the scalar m = h for the case of work-strengthening and m = s for the case of work-weakening, empirical relations for h and s have the following empirical form (Jing, 1990)

(u~- u~) k, (26) h- (u; u ~

(u;- uO (u,- S = -- s c sin 2 cr kSQ. (27)

(u ; - 2

When - z /4 k,, r < z /4 and 0 _< Sc -< 4, the inequality (20) will not be violated. For unloading and reverse shear stages, m = 0 is maintained.

5.2 General Behaviour of the New Joint Model

For the case in which the normal displacement maintains constant (du, = 0), from Eq. (24), the increment of normal stress is given by

don = -[a4 k,k,/(al kt + a2 k, + mQ)] du, < O. (28)

The increment do-, equals zero only when cc = 0 (a4 = 0 when cr = 0). This means that for a rough surface (a~ not equal to zero), normal stress will increase with the increase of the shear displacement while normal displacement is maintained constant.

For the case in which the normal stress is kept constant (do-,, = 0), by Eq. (24), the dilatancy of the joint is given as

dun = [a4 kt/(al k, + m Q)] du~. (29)

If, in addition, m = 0, then

du,, = (a4/aO du, = (tan c~) du,. (30)

It can be proved that du,, > 0 for forward shear stages (dilatancy), in either the positive or the negative direction, and du,

228 L. Jing et al.

~ 0.0

A -o.5 CO CO w -1.0 mr, l - co -1.5 --I

~" -2.0 r r O z -2.5

0

Din= O.OOq - - - - - Dm= 0.01 [

- - - Din= 0.0/,5

' 1'o ' 2'0 ' 3'o SHEAR DISPLACEMENT (ram)

E >.- C.3 Z


0.12 t -

0.08- F-- tO 127

0.04 SIS tO

O I O D _ -

9 , , o , Experimental data - - Prediction by 2-D model

0.00 . . . .

SHEAR DISPLACEMENT (mm)

Fig. 8. Measured shear stress from Bandis et al. (1983) and predicted shear stress by the new joint model

6.2 Validation Against Kutter's Cyclic Shear Test

The first cycle of Kutter and Weissbach's (1980) cyclic shear tests subjected to a constant normal stress (or n = -2.5 MPa), was simulated by the new joint model with the results shown in Fig. 9. The model parameters are:

4- I

zr 2-

f : ! ! i ..I- . ~ ~ 9 " /i'"--:'" m- 2. . . __ "-r-

to I I

, I J ,

-4 -4'o -2'o o 2'0 SHEAR DISPLACEMENT (mm}

Z0

I ooo Experimental data ] t Oo . . ~:~o, I - - Predlctuon by 2-D model

4 oo L _ _ _ _

~ go ~

~ ~ .

0 0 --'--- , ,

-40 -20 0 20 40 SHEAR DISPLACEMENT {mm) b

Fig. 9. Measured results by Kutter and Weissbach (1980) and predictions by the new joint model for cyclic shear tests, a Shear stress versus shear displacement; b normal displacement

versus shear displacement

230 L. Jing et al.

k~ = 320 MPa/m, k~ = 1 GPa/m, C = 0, ~ = 39 o, cr = 9 ~ in the positive shear direction, c~ = 7 ~ in the negative shear direction, o-c = -75 MPa, u p = 11.3 ram, u~ = 29 ram, D~ = 0.003, and s~ = 1.05. The parameters ~r, c~, D,~ and s~ were back-calculated f rom the publ ished data. The basic features of the shear stress variation, di latancy during forward shear and contrac- t ion during unloading and reversal shear stages are reproduced by the joint model with reasonably good agreement.

6.3 Validation Against the Author's Cyclic Shear Test

The cyclic shear tests per formed on one concrete joint repl ica under constant normal stress (crn = -2 MPa) was simulated by using the new joint model (Jing, 1990). The model parameters are: k~" = 18.6 GPa/m, k ~ = 13,3 GPa/m, cr c =-52 MPa, b= 2.0, u, " =-0 .15 mm, u p = 2 ram, u~ = 6.4 ram, qS~ = 45 o, ~b = 33 o, Dm= 0.002, sc = 1.5 and cc = 5 ~ and 3.4 ~ in posit ive and negative shear directions, respectively. Figures 10 and 11 show

First shear cycle c(MPo) 3

- i2/ -6 -'_3 ' - - '~" Ut(mml

-3

-i2

Second shear cycle z(MPa) 3-

~ 3~. 'g 1~2 ,,,,.~.-""~" Ut(mm) -3"

. . . . Experimental data - -P red ic t ion by 2-D model

a b

Fig. 1O. Measured and predicted shear stresses versus shear displacement for a concrete replica of rock joints, a The first cycle and b the second cycle

U. (mmJ Un(mm) 1.0- , 1.0

%

. . . . . . . . . 1' Ut(mm) I U t (mm)

i t I 1to -0.5J -05- - -Exper imena da a - .

a . . . . Prediction by 2-D mode[ b

Fig. 11. Measured and predicted dilatancy and contraction versus shear displacement for a concrete replica of mated joint, a The first cycle and b the second cycle


the measured and predicted shear stresses and normal deformations for the two consecutive shear cycles. The shear stress - shear displacement curves for two consecutive cycles are shown separately in Fig. 10. The measured and predicted joint dilatancy and contraction are presented, also separately for the two consecutive cycles, in Fig. 11. The model simulation agrees well with the measured results.

7. Conclusions

A two-dimensional constitutive model is developed for rock joints, based on the results from a series of systematic laboratory investigations and the theoretical formulation of Plesha (1987). The introduction of empirical relations for work-strengthening, work-weakening, variable stiffness parameters, surface roughness degradation, and different rates for dilatancy and contraction at different stages of shearing gives much greater flexibility and generality to this new constitutive model in simulating the mechanical behaviour of rock joints under complex loading conditions. The second law of thermodynamics is used to restrict the selection of some empirical parameters so that the model could be applied under physically meaningful conditions. Three validation tests have been performed, and the predictions from the new constitutive model agree well with the test results published by independent researchers and our own test results.

The major disadvantage of this joint model seems to be the large number of material parameters involved. Most of them, however, can be easily determined by conventional shear tests, preferably under cyclic shear conditions. The maximum shear stiffness k2 has to be determined by extrapo- lation from several sample tests. The parameters Dm and Sc, have to be determined through a trial-and-error process. Further research work is in progress to group these parameters in a proper fashion so that the number of parameters may be reduced. The model is formulated under the assump- tion that no fluid is present in the rock joints. The contact stresses, there- fore, are total stresses. However, the model could work with minimum change for problems in which fluids are present and effective stresses, rather than the total stresses, are used.

The model is valid for problems in which the shear displacement is not larger than the largest wave length of primary asperities. For problems with larger shear displacement, special care should be taken for joint dilatancy.

Acknowledgement

The financial support from the Swedish Natural Science Research Council (NFR) is gratefully acknowledged. The authors thank Drs. R. D. Hart, L. Lorig and Jose Lemos in the ITASCA Consulting Group, INC., MN, USA, for their encourage- ment and assistance during the development and implementation of the model. Thanks are also given to J. Forslund, K. Havnesk61d, M. Holmberg, U. Matila and

232 L. Jing et al.: Study of Rock Joints Under Cyclic Loading Conditions

Mrs. M. Lejon at the Division of Rock Mechanics, Luleft University of Technology, Sweden, for their assistance in the laboratory work and preparation of all the figures.

References

Amadei, B., Saeb, S. (1990): Constitutive models of rock joints. In: Rock joints, Proc. Int. Symp. on Rock Joints, Loen, Norway, 581--594.

Bandis, S. C., Lumsden, A. C., Barton, N. R. (1983): Fundamentals of rock joint deformation. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 20, 249--268.

Barton, N. (1976): Rock mechanics review: the shear strength of rock and rock joints. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 13, 255--279.

Barton, N., Bandis, S., Bakhtar, K. (1985): Strength, deformation and conductivity coupling of rock joints. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 22, 121--140.

Cundall, P. A. (1980): UDEC -- A generalized distinct element program for modelling jointed rock. Final technical report to European Research Office, U. S. Army, Contract DAJA37--79 -0548.

Goodman, R. E. (1976): Methods of geological engineering in discontinuous rocks. West Publishing Company, San Francisco.

Hutson, R. W. (1987): Preparation of duplicate rock joints and their changing dilatancy under cyclic shear. Ph.D. Thesis, Northwestern University, Evanston, Illinois.

Jing, L. (1990): Numerical modelling of jointed rock masses by Distinct Element Method for two and three-dimensional problems. Ph.D. Thesis, Lulefi Uni- versity of Technology, Lulefi, Sweden.

Kutter, H. K., Weissbach, G. (1980): Der EinfluB yon Verformungs- und Belas- tungsgeschichte auf den Scherwiderstand yon Gesteinsklifften unter beson- derer Berticksichtigung der Mylonitbildung. Final Report. DFG Research Pro- ject, Ku361/2/4.

Martin, J. B. (1975): Plasticity. MIT Press. Cambridge, Mass. Plesha, M. E. (1987): Constitutive models for rock discontinuities with dilatancy

and surface degradation. Int. J. Numer. Anal. Methods Geomech. 11, 345 --362.

Authors' address: Dr. Lanru Jing, Division of Engineering Geology, Royal Institute of Technology, S-10044 Stockholm, Sweden.

1993 Study of rock joints under cyclic loading conditions.pdf

Documents

Transcript of 1993 Study of rock joints under cyclic loading conditions.pdf