An extension to the Saeb and Amadei constitutive model for rock joints to include cyclic loading paths

Int. J. Rock Mech. Min. Sei. & Geomech. Abstr. Vol. 32, No. 2, pp. 101-109, 1995

Pergamon

0148-9062(94)00039-5

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

An Extension to the Saeb and Amadei Constitutive Model for Rock Joints to Include Cyclic Loading Paths M. S O U L E Y t F. H O M A N D t B. AMADEI~

Joints in a given rock mass are subjected to a wide variety of boundary conditions and also to various cycles of loading and unloading in both normal and shear directions. This paper presents an extension of the Saeb and Amadei model to take into account joint loading and unloading in both normal and shear directions. In the normal direction, the cyclic behavior is hyperbolic and the irrecoverable normal closure depends on the joint loading history. Concerning the shear direction change, two assumptions are supposed: the pre-peak behavior is elastic; and, during the residual behavior, the joint is smooth (all the asperities are crushed) and the shear band developed by these fragments is not taken into account. Finally the applicability of the proposed model to predict the behavior of a joint under loading-unloading paths is verified using existing experimental results.

INTRODUCTION In a recent paper, Saeb and Amadei [1] proposed a non-linear model of rock joint behavior taking into account the stiffness of the surrounding rock mass. This model was given in both graphical and mathematical forms. In the mathematical form, the normal and shear stress increments are related to the normal and shear displacement increments through a (2 x 2) non-symmetric stiffness matrix whose terms depend on intrinsic parameters describing the joint behavior. These parameters can be obtained from the normal stress vs normal displacement curve of the joint under consideration and its shear stress vs shear displacement and dilatancy response curves under various levels of constant normal stress. The Saeb and Amadei model [1] can be seen as a generalization of the models of Goodman [2] and Bandis et al. [3]. Some of its general features are as follows:

(2)

(3)

(4)

(1) The model accounts for the coupling between the normal behavior of a dilatant joint and its shear and dilatant behavior. This allows one to predict the increase in normal deformability of an initially mated joint as it traverses a range of unmated

(5)

tLaboratoire de Geomecanique, Institut National Polytechnique de Lorraine, E.N.S.G., Rue du Doyen Marcel Roubault, B.P. 40, 54501 Vandoeuvre-les-Nancy Cedex, France. :~Department of Civil Engineering, University of Colorado, Boulder,

conditions during shearing. If the initial joint aperture is known, the model can be used to determine the variation in joint aperture during shearing. The model can also be used for joints that are initially unmated. The model accounts for the effects of rock mass stiffness on joint shear strength. In particular, it clearly shows that joint shear strength under constant (non-zero) or variable normal stiffness is higher than joint shear strength under constant normal stress. This is in good agreement with previous theoretical and experimental observations on rock joints published in the literature [4-15]. The input parameters to the model can all be determined from conventional normal compression tests and direct shear tests under constant normal stress [15, 16]. The model is incremental and therefore can be implemented into discrete (distinct) numerical codes such as UDEC [16]. The model can easily be modified to accomodate other constitutive models of joint normal or shear behavior [16].

Joints, in a given rock mass are not only subjected to various boundary conditions but also to different loading-unloading paths in directions normal or parallel tO the joints. For instance, the loading direction of a

CO 80309-0428, U.S.A. 101

102

SOULEY et al.: MODEL FOR CYCLIC LOADING OF ROCK JOINTS

block can change because of the movement of neighbouring blocks [17]. The main drawback of the Saeb and Amadei model [1] is that it is limited to monotonic loading and cannot predict joint unloading. This paper presents an extension of the Saeb and Amadei model to take into account the effect of cyclic loading on joint normal and shear behavior.

other model considers the peak and residual shear displacements constant (constant displacement model). The Ladanyi and Archambault [18] and Goodman and St. John [19] models are used to describe joint dilatancy occurring during shear [Fig. l(c)]. In this model, the peak rate of dilatancy, tan i, is related to the applied normal stress, an, as follows: I

0U n

tan i = ~

THE SAEB AND AMADEI MODEL

The main hypotheses and characteristics of the existing model are summarized below. The joint is assumed to be initially mated. Also, joint opening and compressive stresses are assumed to be positive.

(

=

k2

1 - a..~,~ tan i0 when ut ~
ltan i = 0

when

u t > Utr

or a,/> OT

(Sb)

Joint normal behavior

The hyperbolic model of Bandis et al. [3] is used to described joint behavior under increasing normal stress [Fig. l(a)]. The normal stress, an, and joint normal displacement, u,, are related as follows: unKni 1/m On = - -

(a)

Normal stress On

(1)

Vm+U.

or

u. =

V m o"n

VmKni -

(2)

an

where Vm is the maximum joint closure and K.i is the initial normal stiffness of the joint which is negative for the sign convention adopted here. At any other normal stress level, the joint tangent normal stiffness, which is the tangent to the (an vs un) curve, is equal to: an

un

-vm

Closure opening Normaldisplacement

2

(b) Also, the joint is assumed to have zero tensile strength. Zp

Joint shear and dilatant behavior

The shear response of a joint to shear loading under constant normal stress is idealized as shown in Fig. l(b) [2]. The shear stress vs shear displacement curve consists of a pre-peak, post-peak and residual regions. In the model, the joint shear stress, ~, and shear displacement, ut are related as follows:

Ut "Jr-( yrutp -

~" = \U,p

T

Zr

-

U,r/

\ Utp-- U,r /

(if

U,p

I I Utp utr Shear displacement

Ut

~< u t ~. Utr )

(if u~ > ut~)

(4)

(c) Un

where % and Utp are the peak shear strength and displacement, respectively. Likewise, zr and utr are the residual shear strength and displacement, respectively. Finally, Ks is the shear stiffness. In general, Zp, Utp, ~, Utr and Ks depend on the normal stress. G o o d m a n [2] recommended two models to represent the shear behavior of a joint under various constant normal stresses. One model assumes that the shear stiffness is independent of the normal stress (constant stiffness model); whereas, the

model

Utp

Utr

ut

Shear displacement

Fig. 1.' (a) Normal stress vs normal displacementcurve for a joint; (b) idealized shear stress vs shear displacement curve; and (c) idealized dilatancy curve.

SOULEY et al.:

MODEL FOR CYCLIC L O A D I N G OF ROCK JOINTS

where o-x is a transitional stress beyond which no further dilatancy takes place, tan i0 is the peak rate of dilatancy at zero normal stress and k2 is an empirical constant. Ladanyi and Archambault [18] suggested k2 = 4 whereas Wibowo et al. [15] found experimental values of k2 ranging between 0.25 and 3.55. The variation of the peak shear strength, Zp, with the normal stress, o.n, is described by the failure criterion of Ladanyi and Archambault [18] modified by Saeb [20] which takes the following form: % = (1 -- a~)o., tan(q~. + i) + a~s~.

(6)

In equation (6), as is the proportion of total joint area sheared through the asperities, (1 - as) is the proportion on which sliding takes place, tk, is the angle of friction for sliding along the asperities and the angle, i, is defined in equations (5a)-(5b). Finally, Sr represents the shear strength of the intact rock in the asperities which can be approximated by the Mohr-Coulomb criterion or Fairhurst's parabolic criterion. According to Ladanyi and Archambault [18], as is normal stress dependent with: a~=l-(1-

o.n'/"o.T/

(7)

where kj is another empirical constant. Ladanyi and Archambault [18] suggested k~ = 1.5, whereas Wibowo et al. [15] found experimental values of kl ranging between 0.05 and 0.3. The residual shear strength, rr, varies with the normal stress, following the Goodman model [2]:

Tr :

Zp

Bo +

f (

0.. o-T

zr = zp

)

if 0.~ < o'T if 0.~ i> o'T"

(8)

In equation (8), B0(0 ~
(9)

Since equation (9) must also represent the normal joint behavior of a smooth joint (i.e. tan i0 = 0), it follows that f(o.,) in equation (9) must be equal to the right-hand side of equation (2). Then, equation (9) reduces to: u. = ut 1 -- °'"_ k2 tan io + O'T,]

Vmo'n

(10)

Vm Kni - o.n

An incremental formulation can be obtained by differentiating equation (10). After rearrangement, the

103

normal stress increment can be expressed as a function of the increments of normal and shear displacement as follows: do"n = K.. du, + K,t dut (11) where K,, and K,~ are two normal stiffness coefficients whose expression can be found in Saeb and Amadei [1] [equations (22) and (23)]. These two stiffness coefficients depend on the joint normal stress and shear displacement. An equation similar to equation (11) can be expressed for the shear stress increment. Differentiating equation (4) and after rearrangement gives: dr

= Ktn

du~ +

gtt d u t

(12)

where Kt~ and gtt are two shear stiffness coefficients whose expressions can be found in Saeb and Amadei [1] [equations (25)-(34)] for the constant displacement and stiffness models. MODELING JOINT R E S P O N S E U N D E R N O R M A L LOADING AND UNLOADING Although it is now widely accepted that the response of a joint to normal loading can be described by an hyperbolic model, several models have been proposed in the literature to describe the response of a joint to normal unloading. For instance, Goodman [2] assumed that the unloading curve follows essentially the same path as for the intact rock. Heuz6 et al. [21], Jing [22] and Jing et al. [23] assumed that joint unloading is linear and that the unloading normal stiffness remains constant and equal to its values just before unloading. On the other hand, experimental results reported by Bandis et aL [3] and others [26-28] reveal (1) that joint normal unloading follows an hyperbolic response with a steep drop in normal stress at the beginning of the unloading phase, (2) that a non-negligible permanent deformation exists at the end of each loading-unloading cycle, (3) that the joint normal stiffness is higher upon unloading than in loading, and (4) that hardening takes place as the joint undergoes several cycles of loadingunloading-reloading [27]. These different characteristics are represented in Fig. 2. In general, assuming a linear unloading response as done by Heuz6 et al. [21] and Jing [22] would result in underestimating the joint normal stiffness (in fact, joints exhibit higher unloading stiffnesses than the solid rock [3]) and in overestimating the amount of permanent deformation (difference between hyperbolic and linear models). In the model proposed below, the unloading curve is assumed to be hyperbolic. Also, for instance, the maximum joint closure is assumed not to vary with the number of loading-unloading cycles. Furthermore, the irrecoverable normal displacement depends on the level of normal stress reached prior to unloading. Consider the geometry of Fig. 2 and the following variables: • 0..0 = the normal stress reached before unloading, • u.0 = the normal closure induced by 0..0,

104

SOULEY et al.: MODEL FOR CYCLIC LOADING OF ROCK JOINTS

-

°nO (~;\~o~lx \ ",. \ ~I\ k \\

1

a =

(17a)

Kni(unl) a

Normal stress

U.ir~+ ~ = -- Vm

(17b)

Un0 -- Unirr O-n0

Kni(unl) [ ~

,

~

x

unirr Normal displacement

Fig. 2. Normal stress vs normal displacement in the proposed model.

• un~, = the irrecoverable (plastic) normal closure, • Kn~(,n~= the initial normal stiffness for unloading (which is larger than the initial normal stiffness for loading Kni) • V~ = the maximum joint closure which is independent on the number of cycles, • x = the current joint normal displacement, • y = the current joint normal stress.

X -- Unirr a -- b ( x -- Unirr)

(Un0 -- Unirr)(Vm "1- Unirr)Kni(unl) V m --F Un0

O'nO=

(18)

After rearrangement, equation (18) becomes a second order polynomial in Uni~r:

Kn~(o.0u ~. + Kn~(on~)(Vm -- Un0)Uni. -

Kni(un0UnoVm -- O-n0(Vm+ Un0) = 0

Unirr ( 1 )

2 2 N/Kni(unl) ( Vm -1- Un0) -- 4Knit.n,)O-n0( Vm + Un0) 2Kni(unl) Vm -- Un0

2

Unirr (2) =

(13)

2gni(unl) Vm -- Uno

2 ay

(14)

where a and b are constants. The joint tangent normal stiffness, which is the tangent to the unloading hyperbola is equal to

(20)

2 2 N/Kni(unl)(Vm+un°)-4Kni(unl)o-n°(Vm+Un°)

or X =Unirr-¥-1 +by

(19)

with tWO roots Un~r(1) and Unir~(2) equal to:

As for loading, joint opening and compressive stresses are assumed to be positive, which means that un0, Un~r and gni(unl ) are negative. The unloading response curve [curve (C) in Fig. 2] is and assumed to be described by a hyperbola with equation:

Y

(17c)

.

Equations (17a) and (17b) can be solved for a and b in terms of Unirr, Vm and Kn~(un0. Substituting the resulting expressions of a and b into equation (17c) gives

\\

1-. N

-Vm unO

a -- b (t/n0 - Unirr)

(21)

Since in equations (20) and (21), Kni(~nl)< 0, O-not> 0 and ( V m --FUno ) > 0, then --4Kni(unl)o-no(V m + Un0) > 0. C o n s e quently:

u,i~(1) < IKni(unl)(Vm + Un0)l 2Kni(unl )

Vm -- Un--~0~--- -- Vm 2

(22)

and ay a Kn = ~xx = [a - b (x - Unirr)]2

(15) Unirr (2) >

The three unknowns, a, b and Unirrcan be determined by using three basic constraints: (1) curve (C) must pass by the point with coordinates (Un0, O-n0),(2) as the normal stress increases, x approaches - Vm and (3) at x = Un~,, the joint normal stiffness Kn~ =Kni<~.l). These three constraints can be expressed mathematically as follows:

Ignitunl)(Vm -4- u,0)l 2Kni(unl )

Vm -- Un0 = 2

-

It is clear that Uni~r(1)cannot be the solution. Moreover, if we add the additional constraint than the joint is stiffer during unloading than loading, it follows that: Unirr(2) < 0.

/ni(un,, ~< ~ 0 ( 1 -F- ~m0) x

=

-

Vm

(23)

(24)

Equation (24) leads to

"(Un0, O-n0)e (C) lim

Un0.

(25)

(16)

) ' - - > O0

lim x--

K.

=

Kni(.nD

> Unirr

This leads to a system of three non-linear equations with three unknowns a, b and umrr, e.g.:

with u,0 ~<0, a.0 1> 0 and Vm > 0. Let Kni(,n~)be equal to the secant to the normal loading curve at point (Uno,ano) in Fig. 2, i.e.: O-n0 Kni(unl) ----- - . Un0

(26)

SOULEY et al.: MODEL FOR CYCLIC LOADING OF ROCK JOINTS

Combining equations (21) and (26) gives: u,i, = ½x/(Vm + Un0)(Vm-- 3Un0)

Vm-- u.0 2

(27)

The unloading hyperbola has the following equation: (Vm -[- Unirr)O'n Un = ( V m "+" Unirr)Kni(unl) -- a n '

(28)

part with gni(unl ) replaced by a new value Kni(0. The same substitution must be made in equations (31)-(33). In general, Kni0)must be larger than or equal to the normal stiffness K,i associated with the loading part of the first cycle in order to capture the experimentally observed joint stiffening. Souley [16] proposed the following empirical relation to account for stiffening:

(29)

The two quantities Unir~and K.i(u,t) appearing in equations (28) and (29) are defined in equations (26) and (27). For unloading, equation (9) replaced by:

)

( V m --{-Unirr)O"n

u, = ut 1 -- a"_ k2tan i0 + aT//

(Vm -~- Unirr)gni(unl ) - a n

(30) Differentiating equation (30) yields 0U n

du. = ~

dU n

dut + ~

da. =

(

1-

O'n~k2

aT/

k2ut(1--O'n~ k2-1 tan io da~ O'T \

-t

aT//

gni(unl) ( Vm + Unirr)

2

. . . . . .

(Kni(unl)(Vm + Unirr) -- O'n)

a

2aan

(31)

which after rearrangement can be expressed in the same form as equation (11). However, the two normal stiffness coefficients Kn° and K,, are now equal to: Knn = 1

k2 u t O'T

Q

I 1-

a,]2fiT.]

tan i0 -¢

gni(unl) ( V m "t- Unirr) 2

(Kni(unl)( Vm + Unirr )

-- O.n )2

(32)

and K.t= -(1-

ia~Y2 ° a x /tan

k2Ut ( l _ a. y2-t tan io.k O"T

O'T//

2 The unloading part of the new cycle is handled as discussed above. The proposed constitutive model for loading and unloading was implemented into the two-dimensional distinct element program UDEC [16]. Figure 3 shows a numerically simulated series of two cycles of loading and unloading on a joint with the following input parameters: K , ~ = - 5 M P a / m m , Vm= 5ram. In this example, equation (34) was used to model the increase in joint initial stiffness with the number of cycles. MODELING

tan io dut

(34)

Kni(i ) --- Kni(unl) + Kni

The joint tangent stiffness for unloading is then equal to: [Kni(unl)(Vm + Un0) -- O'n'~2 I n = Kni(un0 L ~ f f ~ m ' + ; n 3 ,]"

105

JOINT RESPONSE TO CHANGES SHEAR DIRECTION

IN

In the present model, it is assumed that in the pre-peak region, joint characteristics do not vary when the shear direction changes, since joint surface asperities are not damaged but only mobilized. Consider the shear stress vs shear displacement response curve of Fig. 4. A change in shear direction occurs in the post-peak region originating at point A. Let Uta, ZA, ZTA and tan iA be, respectively, the shear displacement, shear stress, residual shear strength and rate of dilatancy corresponding to point A. When the shear direction changes, a new joint is created with new properties that depend on the old ones and the position of point A. The properties that may change due to the change in shear direction are for instance: the strength of the joint walls, the initial dilatancy angle, the peak and residual shear stresses and displacements and B0. Since point A is in the post-peak region, joint surface asperities have undergone damage. Then the rate of dilatancy cannot be equal to its initial value. Moreover,

Kni(unl)( Vrn-l- Unirr)2 (Kni(unl)(Vm 21- Unirr) -- O'n) 2

Normal stress

(33)

%(MPa) Equation (32) provides an analytical expression for the joint tangent normal stiffness when the joint has been sheared by an amount equal to ut, and has a non-recoverable (plastic) normal deformation Umrr. Equation (32) reduces to equation (29) when u, = 0 and to equation (3) when Unirr = 0. Note that equations (30)-(33) are valid as long as ut ~ ut, or an/> aT, dilatancy vanishes, Km= 0 and K,n reduces to its value defined in equation (29). If the joint undergoes a new cycle of loading and unloading, equation (28) can be used to relate the normal stress to the normal displacement in the loading

60

30

un -5

-4

-3 -2 -1 Normal displacement (mm)

0

Fig. 3. Numerical example of two cycles of normal loading and unloading with K.i = - 5 MPa/mm and using equation (34).

106

SOULEY et al.: MODEL FOR CYCLIC LOADING OF ROCK JOINTS Using G o o d m a n ' s model [equation (8)], the new value of B 0 is equal to:

•~p . . . . . . . . . . . . . .

I

/

~

u tp (new)~ijAutr (newel

!7i u tp (old)

/

-i

_

u tA u tr(old) ut Shear displacement Fig. 4. Shear stress vs shear displacement with change of shear direction. the damage is not complete yet as long as UtA is less than utr. Thus the rate of dilatancy tan i, cannot be equal to zero, contrary to the empirical model of Cundall and H a r t [24] or the elastoplastic model of Heuz6 e t al. [21]. In the present model, the initial rate of dilatancy of the "new" joint, is assumed to be equal to the value of the rate of dilatancy preceding the change in shear direction, i.e.: tan i0 = tan iA

TrAO"T -- O"nTA

B0t,ew)- ZA(aT -- a , )

when

O"n <

Bo~.ew)= 1

when

Gn ~ a T

{

Autr~new)= Utr(oJd)-- UtA Utr(new) = Utp(new) + AUtr(new)

(37)

where Ksi is the initial pre-peak shear stiffness.

rb = a. tan (~b

(39)

where ~bb represents the basic friction angle. The shear stiffness coefficients connected to the change in shear direction change are the same as those given by Saeb and Amadei [1] [equations (25)-(30)] for the constant displacement model, with values of initial dilatancy, peak and residual displacements and B0 defined in equations (35)-(38).

¢-I

Moel I

.

.

.

.

.

.

~D

;

-0.25

o

-0.2

-0.15

-0.1

-0.05

Normal displacement(ram)

0

(38)

with TrA being the new residual shear strength corresponding to the new peak shear strength ~AIn fact, if the shear direction changes at point A, the created joint has the following characteristics (1) unmating, and (2) filling with some fragments due to the asperities damage. In other words, these fragments are first sheared, and in the second place a new joint whose properties were described previously is sheared. This subject has not addressed in the literature and requires further attention (experimental and analytical investigations). However in a recent paper, Jing e t al. [29] assume that the shear behavior of these fragments is similar to the shear behavior of a plate joint cut out in the rock (i.e. linear elastic-plastic behavior). In this model, the shear stiffness is equal to the initial shear stiffness of joint before shearing, and the shear stress is limited to the quantity

(35)

The new values of the peak and residual displacements are equal to: TA ut~.ew) = Ks---~ (36)

O"T

-0.25

-0.2

-0.15

-0.1

-0.05

0

Normal displacement(ram)

Fig. 5. Predicted response of cyclic normal behavior for experimental results of Benjelloun [26].

S O U L E Y et al.:

MODEL FOR CYCLIC LOADING OF ROCK JOINTS

CONFIRMATION OF THE PROPOSED METHOD

107

new loading. Also, in this example, the normal stiffness Kin,) at the beginning of each loading cycle is assumed to be constant and equal to the initial normal stiffness K,~ at the first cycle. Figure 5 shows an increase in the predictive irrecoverable closure compared to the experimental results. For instance, the predicted and observed values of the irrecoverable closure are equal to 0.137 and 0.1 mm, respectively after the first cycle. This overvaluation of irrecoverable closure results probably from the empirical relation of Kni(unl) [equation (26)] and also, explains the difference of the normal closure for thin level stress between prediction and experience. Finally, more realistic values of K.i(~) and Koi(..~)could be chosen, taking specific experimental results into consideration.

Numerical simulations of normal and shear laboratory tests were carried out using the distinct element program UDEC [24] in which the previous models were implemented [25]. Example I: normal direction This example tries to reproduce the experiments conducted by Benjelloun [26]. A mated granite joint is subjected to three cycles of loading and unloading. Observed and predicted normal stress vs normal displacement response curves are shown in Fig. 5. In this example, the maximum closure (Vm) and initial normal stiffness (K,i) introduced into the model were determined from experimental data (i.e. Kni = - 5 MPa/mm, Vm = 0,245 mm). To perform a new cycle the equations (30)-(33) are used with an irrecoverable displacement calculated at unloading in the previous cycle. A new initial normal stiffness Kniu) m u s t be introduced, for the

Example 2: shear direction As direct application of the formulation presented in the previous section, a cyclic shear test under constant normal

(a) 8

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Shear load (kip~

46 : ii '

z

i;iiii

. . . . . ~ ' - - 1 - -,,- - - - -

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0 _3 . .

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1l

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--".,",, ~ ,

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I I ~ . . . . . . . . . . I

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j

q--~'- " . . . . . r . . . . . . . . . .

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~5

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-8

I -0,4

I -0,2

, I I 0,2

-0,6

Shear displ

ement (in

I 0,4

I 0,6

(b)

.

Shear load ( K i ~ ) ........... J ................................ I I . . . . . . . . . . .

0

I . . . . . . . . . .

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I I

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Ill

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a . . I

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@ . . . . . . . . . . I I I L . . . . . . . . . . i

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Shear displacement (in) I I 0,4

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I I I I I i

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t 0,6

Fig. 6. Predicted response of cyclic shear behavior for experimental results of W i b o w o et al. [14]: shear load vs shear displacement (sample R4F3: normal load = 7.5 Kips) (a) experiment and (b) model.

108

SOULEY et al.: MODEL FOR CYCLIC LOADING OF ROCK JOINTS

load performed by W i b o w o e t al. [13-15] has been predicted. N o t e that, the test [Figs 6(a) and 7(a)] provides some series o f cyclic direct shear experiments conducted on several replicas o f natural fractures o f Welded Tuff from Y u c c a montain. Test and predictions were made under a constant n o r m a l load o f 7.5 Kips (initial normal stress o f 416.67 psi). In order to use the present model, these o f experimental data performed and analyzed by W i b o w o e t a L [15] were selected and summarized as follows: - - j o i n t m a x i m u m closure - - j o i n t initial n o r m a l stiffness - - j o i n t initial tangent stiffness - - j o i n t basic friction angle - - t r a n s i t i o n a l normal stress - - r a t i o o f residual to peak shear strengths - - r e s i d u a l displacement

Vm = K.i = Ksi = ~b~ = aT =

- - k l = 0.25, k 2 = 0.81, --initial dilatancy rate - - i n t a c t rock cohesion - - i n t a c t rock friction angle

tan/0 = 0.18, So = 1110 psi, ~b0 = 31.5 °.

Figures 6(a), 6(b), 7(a) and 7(b) show, respectively, the observed and predicted shear load and the dilatancy vs shear displacement curves. Comparisons between the test response and the simulation show a g o o d agreement particularly for shear load vs shear displacement. The difference between observed and predicted dilatancy provides from:

0.029 in., - 50,000 psi/in., 18,400 psi/in., 30 °, 1860 psi,

• in simulation the input data k2 is constant c o n t r a r y to experience in which case depends on the sense o f shearing and the n u m b e r o f cycles. • the value o f the quantity (1 - G / a T ) k2 [equation (5a)] approaches 0.84, and does not affect significantly the new value o f the initial dilatancy [equation (35)] when the sense o f shearing was changed.

= 0.48, Utr = 0.55 in.

B0

Further, the Ladanyi and A r c h a m b a u l r s criterion used for peak shear strength and dilatancy [equations 5(a), 6], we have chosen the same values o f input data as the authors i.e.:

N o t e that the shear strength (peak stress) decreases with the cycles. This strength increases with the peak dilation angle, which decreases with the new initial

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SOULEY et al.: MODEL FOR CYCLIC LOADING OF ROCK JOINTS dilatancy as soon as the shear direction changes [equation (23)] in the post-peak region. The B0 coefficient, defined as the ratio of residual shear stress to peak stress, increases with the cycles up to unity (value when all asperities are crushed).

CONCLUSIONS The model presented in this paper is a generalization of the Saeb a n d A m a d e i model to loading with changes of n o r m a l a n d shear loading directions. As is observed in experiments, the cyclic n o r m a l behavior is hyperbolic a n d the irrecoverable n o r m a l closure depends o n the loading j o i n t history. The proposed model takes into account a j o i n t h a r d e n i n g behavior as a function of l o a d i n g - u n l o a d i n g cycles, as in the experiments. C o n c e r n i n g the shear direction change, two hypotheses have been made: the pre-peak behavior is elastic; and, d u r i n g the residual behavior, the j o i n t is s m o o t h (all the asperities are crushed) a n d the shear b a n d developed by these fragments is not taken into account. F o r instance, a linear elastic-plastic model is used to describe the behavior of these fragments. Also the shear behavior u n d e r cyclic loading (i.e. changes of shear direction) depends mainly o n the current damage grade of asperities a n d on its initial characteristics. The proposed m e t h o d points out a n increasing of j o i n t stiffness a n d a decrease of n o r m a l deformability with l o a d i n g - u n l o a d i n g cycles along the n o r m a l direction. The model takes into a c c o u n t a decreasing of peak stress a n d a n increase of asperities damage as a function of the n u m b e r of shear direction changes. The applicability of the proposed model to predict the behavior of rock j o i n t u n d e r cyclic conditions from the results of c o n v e n t i o n a l n o r m a l closure a n d direct shear tests was verified using existing test results. G o o d agreement was f o u n d between prediction a n d what was observed in the experiments. Accepted for publication 2 November 1994.

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109

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