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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 experi- mental results, a two-dimensional constitutive
model was developed for rock joints undergoing monotonic or cyclic
loading sequences. The joint model was formu- lated in the
framework of non-associated plasticity, coupled with empirical
rela- tions 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 deforma- bility
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 asperi- ties
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 deforma- bility 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).
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216 L. Jing et al.
A number of constitutive models for rock joints have been devel-
oped 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 rough- ness 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
sam- ples 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 constitu- tive 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 rela-
tions will be briefly introduced, followed by the formulation,
implemen- tation 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 posi- tive.
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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 mate- rial. b is a material constant representing
the wearability of the joint mate- rial. 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)
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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 com- plex 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 devel- oped 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
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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 (clo- sure) of joints. Let k, be the normal
stiffness and u, be the normal displace- ment, 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
deg- radation, 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 nor- mal stress changes. This
model of roughness damage is used for the consti- tutive 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-
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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-lin-
ear with many small scale stick-slip oscillations, indicating
gradual failure of the smaller higher-order asperities. For the
second cycle, the normal dis- placement 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 rep- licas 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,
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Study of Rock Joints Under Cyclic Loading Conditions 221
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 "
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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 dis- placement 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
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Study of Rock Joints Under Cyclic Loading Conditions 223
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 gen- eral 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
spe- cial empirical relations suitable for rock joints to produce a
final constitu- tive 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, respec- tively. 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
deforma- tion, i. e.
d oi = kij du;, (13)
where kij is the stiffness tensor withk, n = k,, = 0, ktt = kt
and k,, = kn.
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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 consti- tutive 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
thermo- dynamic temperature and (~ be the heat, then the second law
of thermo-
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Study of Rock Joints Under Cyclic Loading Conditions 225
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 writ- ten
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 consti- tutive 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 empiri- cal relations about
some special properties of rock joints, besides the selec- tion 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 cur- rent 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 con- tact 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).
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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 follow- ing 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 rela- tion between increments in
the macroscopic stresses and relative displace- ments 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)
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Study of Rock Joints Under Cyclic Loading Conditions 227
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 fol- lowing 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 displace- ment 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,
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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
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Study of Rock Joints Under Cyclic Loading Conditions 229
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 con- stant 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
-
Study of Rock Joints Under Cyclic Loading Conditions 231
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 dila- tancy 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
num- ber of material parameters involved. Most of them, however,
can be easily determined by conventional shear tests, preferably
under cyclic shear con- ditions. 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.
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Authors' address: Dr. Lanru Jing, Division of Engineering
Geology, Royal Institute of Technology, S-10044 Stockholm,
Sweden.