Extended Barton-Bandis model for rock joints under cyclic loading: Formulation and implicit algorithm

Abstract

In this paper, the Barton-Bandis model for rock joints is extended to cyclic loading conditions, without any new material parameter. Also developed herein is an algorithm for implicit numerical solution of the extended Barton-Bandis model, which can also be used for the original Barton-Bandis model for which an implicit algorithm has been unavailable. To this end, we first cast the Barton-Bandis model into an incremental elasto-plastic framework, deriving an expression for the elastic shear stiffness being consistent with the original model formulation. We then extend the model formulation to cyclic loading conditions, incorporating the dependence of shear stress and dilation on the joint position and the shearing direction. The extension is achieved by introducing a few state-dependent variables which can be calculated with the existing material parameters. For robust and accurate utilization of the model, we also develop an implicit algorithm based on return mapping, which is unconditionally stable and guarantees satisfaction of the strength criterion. We verify that the proposed model formulation and algorithm produces the same results as the original Barton-Bandis model under monotonic shearing conditions. We then validate the extended Barton-Bandis model against experimental data on natural rock joints under cycling loading conditions. The present work thus enables the Barton-Bandis model, which has been exceptionally popular in research and practice, to be applicable to a wider range of problems in rock mechanics and rock engineering.

Full text

Extended Barton–Bandis model for rock joints under cyclic
loading: Formulation and implicit algorithm
Fan Feia, Jinhyun Choob,∗
aAtmospheric, Earth, and Energy Division, Lawrence Livermore National Laboratory, United States
bDepartment of Civil and Environmental Engineering, KAIST, South Korea
Abstract
In this paper, the Barton–Bandis model for rock joints is extended to cyclic loading conditions,
without any new material parameter. Also developed herein is an algorithm for implicit numerical
solution of the extended Barton–Bandis model, which can also be used for the original Barton–
Bandis model for which an implicit algorithm has been unavailable. To this end, we first cast the
Barton–Bandis model into an incremental elasto-plastic framework, deriving an expression for the
elastic shear stiffness being consistent with the original model formulation. We then extend the
model formulation to cyclic loading conditions, incorporating the dependence of shear stress and
dilation on the joint position and the shearing direction. The extension is achieved by introducing
a few state-dependent variables which can be calculated with the existing material parameters.
For robust and accurate utilization of the model, we also develop an implicit algorithm based
on return mapping, which is unconditionally stable and guarantees satisfaction of the strength
criterion. We verify that the proposed model formulation and algorithm produces the same results
as the original Barton–Bandis model under monotonic shearing conditions. We then validate the
extended Barton–Bandis model against experimental data on natural rock joints under cycling
loading conditions. The present work thus enables the Barton–Bandis model, which has been
exceptionally popular in research and practice, to be applicable to a wider range of problems in
rock mechanics and rock engineering.
Keywords: Rock joints, Constitutive model, Cyclic loading, Numerical algorithm, Implicit
method
1. Introduction
The Barton–Bandis model [1–3] is a physically-motivated empirical constitutive model for
rock joints, which has been extensively used for a wide range of rock mechanics problems in re-
search and practice. The unparalleled popularity of the model can be attributed to the combination
∗Corresponding Author
Email address: jinhyun.choo@kaist.ac.kr (Jinhyun Choo)
arXiv:2207.06026v1 [physics.geo-ph] 13 Jul 2022

of the following two features: (i) it can well reproduce the shearing and dilation responses of rock
joints and their dependence on normal stress and joint roughness, and (ii) its material parameters,
such as the residual friction angle, the joint roughness coefficient (JRC), and the joint compression
strength (JCS), are all physically meaningful and commonly characterized in practice.
When it comes to rock joints under cyclic loading, however, the Barton–Bandis model faces
some limitations. Since the model was originally formulated for monotonic loading, it does not in-
corporate an array of behavior characteristics that depend on the joint position and/or the shearing
direction (see e.g. [4–8] for experimental evidence). For this reason, when Barton [9] developed
the concept of mobilized joint roughness for describing pre- and post-peak monotonic shearing be-
havior and its size dependence, he also presented an attempt to extend the original model to cyclic
loading based on the mobilized joint roughness coefficient. However, this attempt was not con-
cerned with contractive volume change behavior during cyclic loading and was validated against a
single dataset only. Later, Asadollahi and Tonon [10] proposed a modified Barton–Bandis model
based on the results of eighteen cyclic shear box tests. While the modified model was thoroughly
validated, it is rather complex and not the same as the Barton–Bandis model under monotonic
shearing conditions. Thus it would require significant effort to implement and use the modified
model for practical problems.
From a numerical perspective, the lack of an implicit algorithm for the Barton–Bandis model
has been a critical hurdle for applying the model to every kind of problem that it can benefit.
Due to its empirical nature, the Barton–Bandis model is not formulated in an incremental (rate)
form. Therefore, although the Barton–Bandis model has been implemented in explicit numerical
methods (e.g. [11–13]), it has not been compatible with implicit methods such as finite elements for
discontinuities (e.g. [14–17]). This limitation has hampered the proper use of the Barton–Bandis
model for many important problems that involve a relatively long period of time and/or complex
coupling with environmental processes such as fluid flow and heat transfer. It also restricts the
applicability of the Barton–Bandis model under general cyclic loading conditions, because cyclic
shearing is often triggered by changes in fluid pressure and/or temperature over a long period
of time (e.g. [18–20]). It is also noted that the use of an implicit algorithm can significantly
improve the robustness and accuracy of numerical simulation of rock masses; see, e.g. Hashimoto
et al. [21] in the context of discontinuous deformation analysis. Therefore, an implicit algorithm
for the Barton–Bandis model, similar to those developed for a variety of other constitutive models
(e.g. [22–26]), is highly desired.
Motivated by the above-described limitations of the Barton–Bandis model, the work has two
objectives: (i) to extend the model formulation to cyclic loading conditions with minimal ingredi-
ents, and (ii) to develop an implicit algorithm that can be used for both the original and extended
Barton–Bandis model. To achieve the first objective, we introduce a few state-dependent variables
– but not a new material parameter – capturing the position- and direction-dependence of shear
2

stress and dilation in rough rock joints. To achieve the second, we cast the Barton–Bandis model
into an incremental elasto-plastic framework and develop an implicit return mapping algorithm
applicable to every type of shearing stage. The proposed formulation and algorithm is verified to
produce virtually the same results as the original Barton–Bandis model under monotonic shearing
conditions. The extended Barton–Bandis model is then validated to be in good agreement with ex-
perimental data on the shear stress and dilation behavior of natural rock joints under cyclic loading
conditions.
2. Incremental elasto-plastic formulation of the Barton–Bandis model
In this section, we cast the Barton–Bandis model into an incremental elasto-plastic framework
for rock joints. The purpose of this elasto-plastic formulation is to make the Barton–Bandis model
compatible with a robust implicit integration algorithm, which will be developed later in this paper.
2.1. General constitutive law and kinematics
The general incremental form of a joint constitutive law can be written as
˙
t=C·˙
u,(1)
where tis the traction vector, uis the displacement vector, and Cis the tangent stiffness of the
joint.
As standard, we first decompose the joint displacement vector into its normal and shear direc-
tions as
u=uNn+δm(2)
where uNand δare the magnitudes, and nand mare the unit vectors, of the joint normal and
shear displacements, respectively. Applying the same directional decomposition to the traction
vector gives
t=σNn+τm,(3)
where σNand τare the joint normal and shear stresses, respectively. In this work, we consider
compressive stress positive following the standard sign convention in rock mechanics.
Elasto-plastic modeling of rock joints begins by postulating that the joint displacement can be
additively decomposed into elastic and inelastic (plastic) parts as
u=ue+up.(4)
3

where superscripts (·)eand (·)pdenote the elastic and inelastic parts, respectively. Accordingly, the
joint normal and shear displacements are decomposed into their elastic and inelastic parts as
uN=ue
N+up
N(5)
δ=δe+δp(6)
The elastic part corresponds to the recoverable joint displacement associated with the interlocking
behavior of rough joints. The inelastic part represents the unrecoverable joint displacement which
involves the mobilization of joint roughness.
In what follows, we introduce constitutive relationships to the elastic and inelastic joint defor-
mations based on the Barton–Bandis model. We first consider the inelastic deformation as it can
be fully described by Barton’s strength criterion.
2.2. Inelastic deformation
Let us first recapitulate a general plasticity framework for rock joints. The shear strength of a
joint is described by a yield function F, given by
F(t)=τ−σNtan φ≤0,(7)
where φis the friction angle, which may vary with the normal stress and the shear displacement.
As a non-associative flow rule is supported by experimental evidence, a potential function G,F
is introduced, of which the general form is given by [27]
G(t)=τ−Ztan ψdσN,(8)
where ψis the dilation angle. The flow rule gives the rate of the inelastic displacement as
˙
up=˙
λ∂G
∂t=˙
λ(m−ntan ψ),(9)
where λdenotes the plastic multiplier.
We now specialize the framework to the Barton–Bandis model. Recall that the Barton–Bandis
strength criterion is given by
τ=σNtan "φr+JRCmlog JCS
σN!#,(10)
where φris the residual friction angle, JRCmis the mobilized joint roughness coefficient, and JCS
is the joint wall compressive strength [1]. The residual friction angle can be empirically related to
the basic friction angle, φb, which can be obtained from tilt tests in the laboratory. For example,
4

Barton and Choubey [1] proposed the following empirical relationship between φrand φb
φr=(φb−20) +20(r1/r2) (in degrees), (11)
where r1is the Schmidt rebound number on wet and weathered joint surfaces, and r2is the Schmidt
rebound number on dry and unweathered surfaces. Notably, φris often assumed to be equal to φb
if there is no weathering. To make the yield function (7) equivalent to the Barton–Bandis strength
criterion (10), we set the total friction angle as
φ=φr+JRCmlog JCS
σN!(in degrees). (12)
The dilation angle of the Barton–Bandis model is given by [28],
ψ=1
MJRCmlog JCS
σN!(in degrees),(13)
where Mis the damage coefficient accounting for asperity degradation due to shearing. The dam-
age coefficient can be estimated by an empirical equation proposed by Barton and Choubey [1],
which is written as
M=0.7+JRCp
12 log [JCS/σN],(14)
where JRCpis the peak joint roughness coefficient [3,9]. Note that the damage coefficient in-
creases with the magnitude of the normal stress. This stress dependence of Mallows one to capture
that a joint experiences more asperity damage under a larger confining pressure, see e.g. [7,8,29]
for experimental evidence. We also note that both JRCpand JCS are subject to size effects [30–32],
which can be estimated by scaling laws proposed by Barton and Bandis [33]
JRCp=JRCp0 lj
l0!−0.02JRCp0
,(15)
JCS =JCS0 lj
l0!−0.03JRCp0
,(16)
where JRCp0and JCS0refer to the values of JRCpand JCS, respectively, measured from a joint of
length l0, and ljis the length of the joint of interest.
Calculation of the friction angle (12) and the dilation angle (13) requires one to evaluate the
mobilized joint roughness coefficient, JRCm, during the course of shearing. The original model
of Barton and coworkers [3,9] provides values of the normalized JRCmas a function of the nor-
malized shear displacement, as presented in Table 1. In this table, δpis the shear displacement at
5

which JRCmreaches its peak value, JRCp, which can be estimated as [3,33]
δp=lj
500 JRCp
lj!0.33
.(17)
Also, parameter iis defined as
i=JRCplog JCS
σN!,(18)
such that Eq. (10) gives zero shear stress when there is no shear displacement.
δ/δp0.0 0.3 0.6 1.0 2.0 4.0 10.0 25.0 100.0
JRCm/JRCp−φr/i0.0 0.75 1.0 0.85 0.70 0.50 0.40 0.0
Table 1: Normalized JRCmvalues in the Barton–Bandis model [3,9].
In the literature, JRCmhas usually been calculated by interpolating the values in Table 1in
a piecewise linear manner. However, this approach is not optimal from a numerical perspective,
because the derivative of JRCmis discontinuous at the shear displacements designated in the table.
Such discontinuities in the derivatives make it tricky to evaluate the flow rule (9) and may be
detrimental to the convergence behavior in an implicit numerical method.
Therefore, here we adopt a smoothed relation between JRCmand δ, which has recently been
proposed by Prassetyo et al. [34]. The smoothed relation can be written as
JRCm=























"7(1 +φr/i)δ
3δp−(3 −7φr/i)δ−1#φr
iJRCpif 0 ≤δ<δp,
"−0.217 ln δ
δp!+1#JRCpif δp≤δ.
(19)
Here, JRCmis a hyperbolic function of δwhen 0 ≤δ < δp, and it is a logarithmic function when
δp≤δ.
2.3. Elastic deformation
In the Barton–Bandis model, the elastic response in the joint normal direction is described by
the hyperbolic equation proposed by Bandis et al. [2], given by
σN=κue
N
1−ue
N/ue
max
,(20)
6

where κis the initial normal stiffness, and ue
max is the maximum joint closure. Both κand ue
max can
be evaluated using the following empirical equations [2]
κ=−7.15 +1.75JRCp+0.02 JCS
aj
,(21)
ue
max =0.296 +0.0056JRCp+2.241 JCS
aj!−0.245
.(22)
Here, ajis the initial (unstressed) joint aperture, which can be approximated as [9]
aj=JRCp
50.2σc
JCS −0.1,(23)
with σcdenoting the uniaxial compressive strength of the rock. If there is no weathering, σcis
assumed to be equal to JCS [1], and Eq. (23) simplifies to
aj=JRCp
50 .(24)
Unlike the joint elasticity in the normal direction, the elastic joint response in the shear direc-
tion is not explicitly described in the Barton–Bandis model. However, an explicit description of
the elastic shear stiffness is required in an elasto-plastic framework. Therefore, in the following,
we derive an expression for the elastic shear stiffness being consistent with the Barton–Bandis
model, without introducing any new parameter. The feasibility of the proposed shear stiffness will
be validated later through a comparison with experimental data on cyclic shear loading.
The joint elasticity in the shear direction is given by
τ=µδe,(25)
where µis the elastic shear stiffness. To derive an expression for µconsistent with the Barton–
Bandis model, we should first delineate the elastic shear regime in the Barton–Bandis model. As
can be seen from Fig. 1, JRCmis negative when δ < 0.3δpand JRCm=−φr/iat δ=0. These
negative JRCmvalues are intended to produce zero-dilation shearing, which takes place when the
joint surfaces are being interlocked, until δreaches 0.3δp. Therefore, it is consistent with the
Barton–Bandis model to postulate that joint shear deformation is elastic until δreaches 0.3δp. We
then derive the elastic shear stiffness such that the shear stresses calculated by Eqs. (10) and (25)
are identical when inelastic joint deformation emerges at δ=0.3δp. As JRCm=0 at this point, we
get
τ|δ=0.3δp=σNtan φr.(26)
7

Inserting this shear stress and δe=0.3δpinto Eq. (25) gives
µ=σNtan φr
0.3δp
.(27)
Note that the above expression accounts for pressure-dependent shear stiffness, which has been
observed by many experimental investigations [3,6,8,9].
0 0.3 1 2 3 4 5
−ϕr/i
−1
0
1
inelastic deformation initiates
elastic deformation
δ/δp
JRCm/JRCp
Discrete (Barton, 1982)
Smoothed (Prassetyo et al., 2017)
Figure 1: Elastic and inelastic regimes interpreted for the elasto-plastic formulation of the Barton–Bandis model,
along with the smoothed JRCmcurve proposed by Prassetyo et al. [34], Eq. (19), and the discrete JRCmvalues in
Barton [9], Table 1.
3. Extension of the Barton–Bandis model to cyclic loading
In this section, we extend the Barton–Bandis model to rough rock joints under cyclic loading.
No additional material parameter is introduced, so as to retain the practical merits of the original
Barton–Bandis model.
3.1. Shear stress and dilation behavior of a joint under cyclic loading
Let us first review the shear stress and dilation behavior of a joint under cyclic loading. As
an illustrative example, in Fig. 2we show the experimental results of a granite joint obtained by
Lee et al. [7]. Note that this joint had an irregular configuration like a typical joint in natural rock
masses, unlike regular-shaped joints studied in some other laboratory experiments [4,5].
Depending on the current joint position and the shearing direction, the load cycle in Fig. 2can
be divided into four stages. The behavior characteristics in these four stages can be summarized
as follows. (The stage names are adopted from Lee et al. [7].)
8

(a) Shear stress
−18 −12 −6 0 6 12 18
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
a
b
c
d
e
f
g
hi
1st cycle
2nd cycle
1st cycle
2nd cycle
Shear displacement (mm)
Shear stress (MPa)
(b) Dilation
−18 −12 −6 0 6 12 18
−0.5
0
0.5
1
1.5
2
2.5
3
ab
c
d
e
f
gh
i
1st cycle
2nd cycle
1st cycle
2nd cycle
Shear displacement (mm)
Dilation (mm)
Figure 2: Experimental results of a granite joint under cyclic loading. From Lee et al. [7].
•Forward advance (a-b-c). In this initial loading stage, the shear stress first increases until it
reaches the peak strength (a-b), and then it decreases and gradually approaches the residual
shear strength (b-c). Meanwhile, the joint keeps dilating after roughness is mobilized.
•Forward return (c-d-e). When the shearing direction is reversed, the joint first undergoes
a small amount of elastic unloading (c-d) and then a reversed shearing (d-e). In this stage,
the joint contracts throughout and shows almost no residual volume change when it returns
to the initial mated position. It is noted that the shear stress in the reversed shearing phase
(d-e) is almost constant and lower than the residual shear stress at the end of the forward ad-
vance stage. This is because dilation in the forward advance stage provides shear resistance,
whereas contraction takes place in the return stage.
•Backward advance (e-f-g). After the joint passes its mated position, the shear stress in-
creases to a new peak strength in the backward stage (e-f) followed by a softening phase
(f-g). This new peak strength is lower than that in the forward stage. Also, the joint shows
a dilation behavior similar to that in the forward advance stage, but the amount of dilation
is slightly lower. These strength and dilation responses suggest that the joint has a lower
level of roughness in the backward stage, which in turn indicates that asperity damage is
dependent on the direction of shear displacement.
•Backward return (g-h-i). As the shear displacement reverses in the backward stage, the
joint experiences a small amount of elastic unloading (g-h) followed by a constant shear
stress (h-i), similar to its response in the forward return stage. Also, the joint shows contrac-
tion throughout and nearly zero volume change at its mated position.
9

In addition, the shearing behavior also shows some hysteresis. Due to asperity damages during
the first cycle, the amounts of both dilation and contraction are reduced in the second cycle. This
change in turn makes the shear strength lower in advance stages where dilation takes place, and
make it slightly greater in return stages where contraction takes place. The shear strength differ-
ence is particularly pronounced in the forward advance stage, because the very first peak strength
in the first cycle emanated from undamaged asperities.
3.2. State variables for the joint position and the shearing direction
Recall that the original Barton–Bandis model is concerned with the forward advance stage
only. Therefore, to extend the Barton–Bandis model to the other three stages, we first need a
quantitative approach to accounting for the current joint position (forward vs. backward) and the
shearing direction (advance vs. return).
For this purpose, we define two types of variables (but not new material parameters). The first
is a variable that has different signs in advance and return stages. Inspired by the formulation of
White [24], we define the variable as
α=˙
δ
|˙
δ|
δ
|δ|(28)
By definition, α= +1 in an advance stage and α=−1 in a return stage. As forward and backward
stages can be distinguished by the sign of slip displacement δ, the current stage can be mathemat-
ically identified as
stage =























forward advance if δ≥0 and α= +1,
forward return if δ≥0 and α=−1,
backward advance if δ < 0 and α= +1,
backward return if δ < 0 and α=−1.
(29)
Second, we introduce two variables representing accumulated slip displacements in the forward
and backward positions separately. Specifically, we define the accumulated slip displacement in
the forward stage, Λf, and that in the backward stage, Λb, as
Λf=0.3δp+Zinelastic
H(δ)H(α)|˙
δ|dt,(30)
Λb=0.3δp+Zinelastic
H(−δ)H(α)|˙
δ|dt,(31)
10

respectively, where H(·) denotes the Heaviside function, defined as
H(x)=








1 if x≥0,
0 if x<0,
(32)
and Rinelastic means that the integration is performed during an inelastic deformation (F=0). The
first Heaviside function in Eqs. (30) and (31) is to ensure that Λfand Λbare updated only in forward
and backward stages, respectively. The second Heaviside function is to let the accumulated slip
variables remain unchanged in their corresponding return stages. Also, because the joint roughness
evolves in the inelastic deformation regime (δ≥0.3δp), an initial value of 0.3δphas been added
to the accumulated slip variables. For notational convenience, we also define a unified variable
representing the accumulated slip displacement as
Λ = 








Λfif δ≥0,
Λbif δ < 0.
(33)
3.3. Extension of the shear strength to cyclic loading
We now extend the shear strength of the Barton–Bandis model, Eq. (10), to cyclic loading.
Recall that the physical mechanism why the shear strength is different during cyclic loading is
that the sense (dilative vs. contractive) and degree of roughness mobilization depend on the joint
position and shearing direction. Also, the effect of roughness mobilization on the shear strength in
the Barton–Bandis model is encapsulated in JRCm.
Therefore, we modify the sign and degree of JRCmas follows. First, we multiply αdefined in
Eq. (28) to JRCmsuch that the value of JRCmis positive in an advance stage and negative in an
return stage. In this way, φ > φrin an advance stage where dilation gives rise to additional shear
resistance, and φ<φrin an return stage where contraction makes the shear strength lower than the
residual value. It is noted that the same relation between φand φris common in other physically
motivated models for rock joints under cyclic loading (e.g. [24,35–37]). Second, to account for
that the peak strength in the backward stage is lower than that in the forward stage, we reduce the
value of JRCpin the backward stage. Particularly, adopting an empirical equation in Asadollahi
and Tonon [10], we define (JRCp)τ– the peak value of JRC associated with the shear strength – as
(JRCp)τ=








JRCpif δ≥0,
0.87JRCpif δ < 0.
(34)
Note that (JRCp)τwill only replace JRCpin calculating JRCm. It will not be used for evaluating
the damage coefficient (14) and the peak shear displacement (17).
11

In addition to the above modification, we replace δin Eq. (19) with Λ, to account for the
direction-dependence of asperity damage. This gives
JRCm=























α"7(1 +φr/iτ)Λ
3δp−(3 −7φr/iτ)Λ−1#φr
iτ
(JRCp)τif 0 ≤Λ< δp,
α"−0.217 ln Λ
δp!+1#(JRCp)τif δp≤Λ,
(35)
where
iτ=(JRCp)τlog JCS
σN!.(36)
Finally, inserting Eq. (35) into Eq. (10), the shear strength is extended to cyclic loading.
It is noted that here we have used the same function form of JRCm(but with different signs
determined by α) for both forward and backward stages, whereas Asadollahi and Tonon [10]
proposed a new function for the evolution of JRCmin the backward stage.
3.4. Extension of the dilation angle to cyclic loading
We further extend the dilation angle of the Barton–Bandis model, which was originally pro-
posed for the forward advance stage, to the other three stages. Firstly, we recall that the dilation
behavior in the backward advance stage is qualitatively similar to that in the forward stage, while
the amount of dilation is lower. Therefore, the dilation angle in both the forward and backward ad-
vance stages can be expressed as the original equation (13), using JRCmin Eq. (35) which accounts
for the reduced peak mobilization in backward stages via (JRCp)τ(34). Next, for the dilation an-
gles in the forward and backward return stages, we assume that it is negative (contractive) and its
magnitude decreases linearly to zero after elastic unloading such that there is no residual volume
change in the initial mated position. A specific expression for this dilation angle in return stages
can be obtained by combining the flow rule (9) and the directional decomposition (2). Eventually,
we get
ψ=























1
MJRCmlog JCS
σN!if α= +1,
−arctan up
N
|δ|!180
πif α=−1,
(in degrees). (37)
Note that 180/π is multiplied to the second equation to express the dilation angle in degrees, as in
the original Barton–Bandis model.
12

3.5. Comparison with Barton’s suggestions
In his 1982 paper [9], Barton presented an attempt to extend the original Barton–Bandis model
to cy‘clic loading conditions. Barton’s attempt differs from the model extended here in the follow-
ing four aspects.
•To incorporate position- and direction-dependence, Barton extended the JRCmevolution
function (Table 1) to the forward return, backward advance, and backward return stages.
Instead, our model has introduced the state-dependent variables (α,Λf,Λb) for this purpose,
keeping the same form of JRCmfunction originally proposed for the forward advance stage.
•In Barton’s attempt, the (constant) shear stress in a return stage is larger than the residual
shear stress. Conversely, in our model, the shear stress in a return stage is lower than the
residual shear stress, as JRCmin Eq. (35) is negative in a return stage (α=−1).
•Barton neither considered contractive volume change in a return stage nor imposed a con-
straint on the amount of remaining dilation at the mated position. However, as in Eq. (37),
the dilation in our model is contractive in a return stage and designed to make the remaining
dilation zero at the mated position.
•The magnitudes of (JRCp)τin a backward stage, the case of δ < 0 in Eq. (34), is different.
Barton suggested that (JRCp)τ=0.75JRCp, whereas we have adopted (JRCp)τ=0.87JRCp
from Asadollahi and Tonon [10].
Comparing the two versions of extension – one following Barton’s suggestion [9] and the
other described in the present work – with several sets of experimental data in the literature, we
have found that the model extended in this work consistently shows better agreement with the
experimental data. This paper has thus proposed a new version of extension to cyclic loading.
4. Implicit algorithm
In this section, we introduce an algorithm for an implicit update of the extended Barton–Bandis
model. The goal of the algorithm is as follows: given values at tn(the current time instance) and the
displacement increment between tnand tn+1(the next time instance) ∆u, find the traction vector
tand the consistent tangent operator Cat tn+1. Hereafter, we shall denote quantities at tnwith
subscript (·)n, and write quantities at tn+1without any subscript for brevity.
Leveraging the incremental elasto-plastic formulation of the Barton–Bandis model, we develop
a return mapping procedure as in Algorithm 1. Similar to return mapping algorithms for other
inelasticity models (e.g. [24,38]), the algorithm uses a predictor–corrector approach that proceeds
as follows:
1. Calculate a trial state assuming that the displacement increment is fully elastic.
13

2. If the trial state is elastic (F<0), it is accepted as the final state.
3. Otherwise, the trial state is corrected such that the final state satisfies constraints imposed
by inelasticity.
As is well known, this kind of return mapping algorithm has two main advantages over explicit al-
gorithms: (i) it is unconditionally stable, and (ii) it guarantees satisfaction of the strength criterion
(F≤0), without any overestimation of the strength. In the following, we elaborate several points
that are specific in the proposed return mapping algorithm for the Barton–Bandis model.
Type of shearing stage. To utilize the proposed model for a general cyclic loading problem, one
must identify the type of the current shearing stage as Eq. (29). Note that the total slip displacement
δis used for this purpose. In a discrete setting, αin Eq. (28) can be calculated as (∆δ/|∆δ|)(δ/|δ|),
where ∆δ:=δ−δn.
Stress-dependent shear stiffness. One challenge in implicit integration of the proposed elasto-
plastic formulation is that the elastic shear stiffness µis a function of the normal stress, see Eq. (27).
While the stress dependence of the shear stiffness can be incorporated in the algorithm, it gives rise
to non-symmetry in the tangent operator, which may be undesirable for numerical performance.
Therefore, similar to how some return mapping algorithms have handled pressure-dependent elas-
tic moduli in constitutive models for soils (e.g. [22]), here we evaluate the shear modulus using the
normal stress at the current time instance, tn. In this way, the elastic tangent operator is calculated
as
Ce=Kn⊗n+µnm⊗m.(38)
where
K=κ(ue
max)2
(ue
max −ue
N)2, µn=(σN)ntan φr
0.3δp
.(39)
Accummulated slip displacements. The proposed model uses two variables for accumulated slip
displacements, Λfin Eq. (30) and Λbin (31), to distinguish between asperity damages in for-
ward and backward stages, respectively. To produce the initial shear strengths, both variables are
initialized to be 0.3δp, namely, Λf,0=0.3δpand Λb,0=0.3δp, where subscript (·)0denotes the
initial value of a time-dependent variable. When inelastic deformation occurs, the two variables
are updated as
Λf=(Λf)n+H(δ)H(α)|∆δ|,(40)
Λb=(Λb)n+H(−δ)H(α)|∆δ|.(41)
14

Algorithm 1 Implicit update algorithm for the extended Barton–Bandis model.
Input: The displacement increment ∆uat tn+1.
1: Calculate the total displacement, u=un+ ∆u, and decompose it into uNand δas Eq. (2).
2: Calculate the trial elastic displacement, ue,tr =ue
n+ ∆u, and decompose it into ue,tr
Nand δe,tr as
Eq. (2).
3: Calculate the trial normal stress, σtr
N, as Eq. (20) with ue,tr
N→ue
N.
4: Calculate the trial shear stresses, τtr =µnδe,tr. Here, µnis evaluated as in Eq. (27) with (σN)n.
5: Calculate the damage coefficient, M, as Eq. (14) with σtr
N→σN.
6: Calculate the peak joint roughness coefficient for shear strength, (JRCp)τ, as Eq. (34).
7: Calculate parameter iτas Eq. (36), with σtr
N→σN.
8: Calculate the mobilized joint roughness coefficient JRCmas Eq. (35). If δ≥0, (Λf)n→Λ;
otherwise, (Λb)n→Λ.
9: Calculate the friction angle φas Eq. (12), using JRCmcomputed above.
10: Evaluate the yield function F(σtr
N, τtr, φ).
11: if F<0then
12: Elastic joint displacement.
13: Update t=σtr
Nn+τtr m,Λf=(Λf)n, and Λb=(Λb)n.
14: Calculate the elastic tangent operator Ceas Eq. (38), and let C=Ce.
15: else
16: Inelastic joint displacement.
17: Update Λfand Λb, as Eqs. (40) and (41).
18: Update JRCmas Eq. (35). If δ≥0, Λf→Λ; otherwise, Λb→Λ.
19: Calculate the dilation angle ψas Eq. (42), with α=(∆δ/|∆δ|)(δ/|δ|).
20: Solve for ueand λusing Newton’s method, as Eqs. (43)–(46).
21: Calculate σNand τusing uecomputed above, and update t=σNn+τm.
22: Calculate the elasto-plastic tangent operator Cep as Eq. (56), and let C=Cep.
23: end if
Output: tand Cat tn+1.
Once Λfand Λbare updated, Λis determined according to the current joint position (see Eq. (33))
and then used to evaluate the mobilized joint roughness coefficient, JRCm.
Dilation angle. As in Eq. (37), the proposed model uses two expressions for the dilation angle to
produce dilation in an advance stage (as in the original Barton–Bandis model) and contraction in
an return stage. The dilation angle in the return stage is calculated such that it gives zero volume
15

change when the joint goes back to the initial position. Therefore, the dilation angle at the next
time instance should be calculated with the remaining amount of dilation and slip displacement at
the current time instance. So Eq. (37) is evaluated as follows:
ψ=























1
MJRCmlog JCS
σN!if α= +1,
−arctan (up
N)n
|δn|!180
πif α=−1,
(in degrees). (42)
Note that the first equation is calculated with quantities at tn+1and the second one with those at tn.
Newton’s method for inelastic correction. In the corrector step, Newton’s method is used to solve
for the elastic displacement vector, ue, and the discrete plastic multiplier, ∆λ. The unknown vector
can be written as
x=









(ue)3×1
∆λ








4×1
.(43)
The residual vector is composed of four equations that need to be satisfied, i.e.
r(x)=











ue−ue,tr + ∆λ∂G
∂t!3×1
F











4×1
→0.(44)
At each Newton iteration, the unknown vector is updated by solving
J·∆x=−r(x),(45)
where the Jacobian matrix J, given by
J=




















1+ ∆λ∂2G
∂t⊗∂t·Ce!3×3 ∂G
∂t!3×1
∂F
∂t·Ce!|
1×3
0



















4×4
,(46)
with 1denoting the second-order identity tensor. Specific expressions for the derivatives of the
yield function Fand the potential function Gare provided in Appendix A. It is noted that we have
used the fact that the friction and dilation angles of the Barton–Bandis model are independent of
16

the plastic multiplier. Note also that at the end of each iteration, all variables related to the elastic
displacement – the traction vector and other stress-dependent variables – should be updated.
Consistent tangent operator. To use the model in a stress-controlled problem as well as in an
implicit numerical method, the consistent tangent operator should be calculated. Depending on
whether the deformation is fully elastic or not, the consistent tangent operator takes different forms
as
C=∂t
∂ue,tr =








Ceif F<0,
Cep if F=0.
(47)
The elastic tangent Cecan be calculated as Eq. (38), and the elasto-plastic tangent Cep can be
derived as follows. First, we linearize Eq. (44) with respect to ue,tr and obtain














δue+ ∆λ∂2G
∂t⊗∂t·Ce·δue+∂G
∂tδ∆λ!3×1
∂F
∂t·Ce·δue













4×1
=








δue,tr3×1
0








4×1
.(48)
where δ(·) denotes the linearization operator. Inserting δue=(Ce)−1·δtinto the above equation,
we obtain
"(Ce)−1+ ∆λ∂2G
∂t⊗∂t#·δt+∂G
∂tδ∆λ=δue,tr,(49)
∂F
∂t·δt=0.(50)
Then we rearrange Eq. (49), which gives
δt=P· δue,tr −∂G
∂tδ∆λ!,(51)
where Pis defined as
P= 1+ ∆λ∂2G
∂t⊗∂t·Ce!−1
·Ce.(52)
By differentiating the two sides of Eq. (51) with respect to ue,tr, we get Cep as
Cep =∂t
∂ue,tr =P· 1−∂G
∂t
∂∆λ
∂ue,tr !.(53)
17

The only unknown in Eq. (53) is ∂∆λ/∂ue,tr. To evaluate this, we insert Eq. (51) into Eq. (50) and
obtain
∂F
∂t·P· δue,tr −∂G
∂tδ∆λ!=0.(54)
Rearranging the above equation and differentiating it with respect to ue,tr gives
∂∆λ
∂ue,tr =
∂F
∂t·P
∂F
∂t·P·∂G
∂t
.(55)
Finally, inserting the above equation into Eq. (53), we get
Cep =P− P·∂G
∂t!⊗ ∂F
∂t·P!
∂F
∂t·P·∂G
∂t
.(56)
Remark 1.The implicit algorithm developed in this work (Algorithm 1) can also be used for the
original Barton–Bandis model, by specializing it to a forward advance stage (δ≥0 and α=1).
5. Verification and validation
This section has two objectives: (i) to verify the elasto-plastic formulation of the Barton–
Bandis model and the implicit integration algorithm, and (ii) to validate the extension of the for-
mulation to cyclic loading. For the first purpose, we simulate the monotonic shear tests modeled
by Barton et al. [3] and compare the results of the elasto-plastic formulation and the original one.
For the second purpose, we apply the extended Barton–Bandis model to simulate the cyclic shear
tests of Lee et al. [7] conducted on four rock joint samples.
5.1. Verification against the original Barton–Bandis model under monotonic loading
For verification, we use the proposed elasto-plastic formulation to simulate two sets of mono-
tonic shear box tests modeled by Barton et al. [3]. The first set demonstrates how the model cap-
tures the stress dependence of rock joint behavior, by shearing a 0.3-m long rock joint under three
different normal stresses, namely 3 MPa, 10 MPa and 30 MPa. The second set is concerned with
size (scale) effects on rock joint behavior, simulating three rock joints of different sizes, namely
0.1 m, 1 m, and 2 m, sheared under the same normal stress of 2 MPa. Table 2presents the material
parameters used in the two sets, which are adopted from Barton et al. [3]. To keep the normal
18

stress constant during shearing, we make use of a global Newton iteration similar to how to solve
a mixed boundary-value problem. Being consistent with Barton et al. [3], the damage coefficient
is set to be M=2 for all the cases.
Parameter Unit Value
Stress effects (Fig. 3) Size effects (Fig. 4)
φrdegrees 30.0 30.0
JRCp0- 10.0 15.0
JCS0MPa 100.0 150.0
l0m 0.1 0.1
Table 2: Material parameters for the two sets of verification examples, adopted from Barton et al. [3].
Figure 3compares the simulation results for the first set (σN=3 MPa, 10 MPa, and 30 MPa)
obtained by the elasto-plastic formulation and the original version [3]. Under all the three normal
stresses, the shear stress and dilation responses simulated by the proposed model are virtually the
same as those by the original model. The results also demonstrate that the Barton–Bandis model
can capture the effects of normal stress on the shear strength, stiffness, and the amount of dilation.
Remarkably, all the initial slopes of the shear stress–displacement curves agree well with each
other. This agreement indicates that Eq. (27) incorporates the stress dependence of elastic shear
stiffness as implied in the Barton–Bandis model.
(a) Shear stress
0 5 10 15 20 25
0
5
10
15
20
25
3 MPa
10 MPa
30 MPa
Shear displacement (mm)
Shear stress (MPa)
Proposed model
Modeled by Barton et al. (1985)
(b) Dilation
0 5 10 15 20 25
0
0.5
1
1.5
2
3 MPa
10 MPa
30 MPa
Shear displacement (mm)
Dilation (mm)
Proposed model
Modeled by Barton et al. (1985)
Figure 3: Verification of the proposed model under different normal stresses: (a) shear stress and (b) dilation compared
with the results of the original Barton–Bandis model in Barton et al. [3].
19

Next, in Fig. 4we compare the two models’ simulation results for the second set (lj=0.1 m,
1 m, and 2 m). It can be seen that the proposed model can also capture size (scale) effects in the
same way as in the original Barton–Bandis model. The initial slopes of the two models again show
excellent agreement, indicating that Eq. (27) correctly incorporates size effects on the elastic shear
stiffness (through the size dependence of δp). It further suggests that the slight difference between
the two results are due to the algorithm, rather than the proposed formulation for the elastic shear
stiffness. In other words, because the implicit algorithm exactly satisfies the strength criterion in
each step – unlike an explicit algorithm – the slight difference is natural. Taken together, it has
been verified that the proposed elasto-plastic formulation inherits the capabilities of the original
Barton–Bandis model.
(a) Shear stress
0 4 8 12 16
0
0.5
1
1.5
2
2.5
3
3.5
0.1 m
1 m
2 m
Shear displacement (mm)
Shear stress (MPa)
Proposed model
Modeled by Barton et al. (1985)
(b) Dilation
0 4 8 12 16
0
0.5
1
1.5
2
2.5
0.1 m
1 m
2 m
Shear displacement (mm)
Dilation (mm)
Proposed model
Modeled by Barton et al. (1985)
Figure 4: Verification of the proposed model for joints with different joint sizes: (a) shear stress and (b) dilation
compared with the results of the original Barton–Bandis model in Barton et al. [3].
Lastly, in Fig. 5we present typical Newton convergence profiles in local return mapping and
global stress control for the foregoing simulations. As shown in Fig. 5a, the Newton iterations
during local return mapping displays asymptotically quadratic convergence, which affirms the
correctness of the Jacobian matrix (46). The global convergence behavior shown in Fig. 5b also
exhibits nearly quadratic rates, which verifies the elasto-plastic tangent operator (56). These re-
sults indicate that the proposed algorithm allows one to use the Barton–Bandis model with high
robustness and efficiency.
5.2. Validation against experimental data on rock joints under cyclic loading
Having verified our formulation and algorithm, we validate the extended Barton–Bandis model
against the responses of real rock joints measured in cyclic shear box tests. Particularly, we use the
20

(a) Local convergence
0 1 2 3
100
10−4
10−8
10−12
Iteration
Relative residual norm
(b) Global convergence
0 1 2 3
100
10−4
10−8
10−12
Iteration
Relative residual norm
Figure 5: Verification of the proposed model: Newton convergence profiles during (a) local return mapping and (b)
global stress control.
experimental data on rock joint samples from Hwangdeung granite and Yeosan marble in Lee et
al. [7]. We choose four joint samples named GH18, GH27, GH45, and MH34 therein. The first
three joints are from Hwangdeung granite joints and the last one from Yeosan marble. The normal
stress was 1 MPa for the GH18 and GH27 samples and 3 MPa for the GH45 and MH34 samples.
Table 3presents the material parameters of the four joint samples. All the values are adopted from
Lee et al. [7], except for JRCp0whose values are calibrated to match the experimental data.
Parameter Unit Value
Granite
GH18
Granite
GH27
Granite
GH45
Marble
MH34
φrdegrees 34.6 34.6 34.6 38.3
JRCp0- 9.0 7.8 9.0 13.0
JCS0MPa 151.0 151.0 151.0 72.0
l0m 0.12 0.12 0.12 0.12
ljm 0.12 0.12 0.12 0.12
Table 3: Material parameters for the validation examples. All the values except JRCp0are adopted from Lee et al. [7].
Figures 6–9compare the simulation results and experimental data of the four joint samples. It
can be seen that the simulation results show excellent qualitative agreement with the experimental
data. The quantitative agreement is also satisfactory, considering that no additional parameter is
21

introduced for cyclic loading. It is noted that, while not presented for brevity, we have confirmed
that the extension proposed in this work match the experimental data better than the attempt made
in Barton [9]. Detailed discussions on the model behavior in each shearing stage are provided
below.
•Forward advance. The model behavior in the first load cycle is the same as the original
Barton–Bandis model, as demonstrated earlier in Figs. 3and 4. However, after asperities
have been damaged in the first cycle, the model shows less dilation and thus a lower peak
stress. This hysteresis is consistent with the experimental observations.
•Forward return. During the elastic unloading phase, the slope of the simulated shear stress
curve matches well with the experimental data. This indicates that the elastic shear mod-
ulus derived as Eq. (27) is not only consistent with the original Barton–Bandis model but
also physically realistic. In the following phase of reversed shearing, the model produces a
constant shear stress and a contractive normal displacement such that no volume change re-
mains when the joint returns to the initial mated position. The shear stress in the second load
cycle is slightly lower than that in the first cycle. All these model behaviors are consistent
with the experimental observations.
•Backward advance. When the joint position is backward, the model gives a lower peak
strength and less dilation than those in the forward position. The peak strength and dilation
become further reduced in the second load cycle. So it can be seen that the model captures
not only asperity damage during cyclic loading but also its dependence on the direction of
shear displacement.
•Backward return. Similar to the forward return stage, the model produces a constant shear
stress following a short period of elastic unloading. The shear strength in the backward
position is lower than that in the forward position. Also, the amount of contraction is lower
in the second cycle, because less dilation has taken place in the backward advance stage of
the second cycle. As a result of this less contraction, the shear stress slightly increases in the
second cycle. Note that the same difference also exists in the experimental data.
To conclude, the extended Barton–Bandis model can reproduce all the salient features of rock
joint behavior under cyclic loading, without any free parameter introduced to the original model.
Therefore, the proposed model is believed to be one of the most capable and practical means to
simulate and predict the behavior of rock joints under cyclic loading.
6. Closure
In this paper, we have extended the Barton–Bandis model to rock joints under cyclic loading
conditions, developing an algorithm for the robust and accurate use of the model in numerical
22

(a) Shear stress
−18 −12 −6 0 6 12 18
−1.6
−0.8
0
0.8
1.6
2.4
1st cycle
2nd cycle
1st cycle
2nd cycle
Shear displacement (mm)
Shear stress (MPa)
Proposed model
Experiment
(b) Dilation
−18 −12 −6 0 6 12 18
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
1st cycle
2nd cycle
1st cycle
2nd cycle
Shear displacement (mm)
Dilation (mm)
Proposed model
Experiment
Figure 6: Validation of the proposed model: (a) shear stress and (b) dilation compared with the experimental results
on the GH18 joint in Lee et al. [7].
(a) Shear stress
−18 −12 −6 0 6 12 18
−1.5
0
1.5
3
1st cycle
2nd cycle
1st cycle
2nd cycle
Shear displacement (mm)
Shear stress (MPa)
Proposed model
Experiment
(b) Dilation
−18 −12 −6 0 6 12 18
−0.5
0
0.5
1
1.5
2
2.5
3
1st cycle
2nd cycle
1st cycle
2nd cycle
Shear displacement (mm)
Dilation (mm)
Proposed model
Experiment
Figure 7: Validation of the proposed model: (a) shear stress and (b) dilation compared with the experimental results
on the GH27 joint in Lee et al. [7].
simulation. The extended model is free of any new material parameter and equivalent to the
original model under monotonic shearing conditions. As such, the main features and practical
merits of the Barton–Bandis model have been carried over to cyclic loading conditions. Also,
the implicit algorithm developed herein enables one to use the Barton–Bandis model, in both
original and extended forms, to be compatible with state-of-the-art numerical methods for fracture
propagation and/or coupled multiphysical problems (e.g. [39–45]). The contributions of this work
23

(a) Shear stress
−18 −12 −6 0 6 12 18
−4
−2
0
2
4
6
1st cycle
2nd cycle
1st cycle
2nd cycle
Shear displacement (mm)
Shear stress (MPa)
Proposed model
Experiment
(b) Dilation
−18 −12 −6 0 6 12 18
−0.5
0
0.5
1
1.5
2
2.5
1st cycle
2nd cycle
1st cycle
2nd cycle
Shear displacement (mm)
Dilation (mm)
Proposed model
Experiment
Figure 8: Validation of the proposed model: (a) shear stress and (b) dilation compared with the experimental results
on the GH45 joint in Lee et al. [7].
(a) Shear stress
−18 −12 −6 0 6 12 18
−5
−2.5
0
2.5
5
7.5
1st cycle
2nd cycle
1st cycle
2nd cycle
Shear displacement (mm)
Shear stress (MPa)
Proposed model
Experiment
(b) Dilation
−18 −12 −6 0 6 12 18
−0.5
0
0.5
1
1.5
2
2.5
1st cycle
2nd cycle
1st cycle
2nd cycle
Shear displacement (mm)
Dilation (mm)
Proposed model
Experiment
Figure 9: Validation of the proposed model: (a) shear stress and (b) dilation compared with the experimental results
on the MH34 joint in Lee et al. [7].
will thus help address a large number of rock joint problems in research and practice.
Acknowledgments
The authors wish to express their deep gratitude to Dr. Nick Barton for his careful review of
the manuscript and constructive suggestions. This work was supported by the National Research
Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No. 2022R1F1A1065418).
24

Portions of this work were performed under the auspices of the U.S. Department of Energy by
Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.
Appendix A. Derivatives in the implicit algorithm
This appendix provides specific expressions for the derivatives in the implicit solution algo-
rithm. First, the derivatives of the yield function (7) and the potential function (8) are given by
∂F
∂t=m+ntan φ+σN
∂tan φ
∂σN
n,(A.1)
∂G
∂t=m+ntan ψ(A.2)
∂2G
∂t⊗∂t=∂tan ψ
∂σN
n⊗n,(A.3)
The derivatives of φand ψwith respect to σN, respectively, are given by
∂tan φ
∂σN
=(1 +tan2φ)"−JRCm
σNln 10 +log JCS
σN!∂JRCm
∂σN#,(A.4)
and
∂tan ψ
∂σN
=





















1+tan2ψ
M2"−JRCmM
σNln 10 +Mlog JCS
σN!∂JRCm
∂σN
−JRCmlog JCS
σN!∂M
∂σN#if α= +1,
0 if α=−1.
(A.5)
Here, the derivative of JRCmwith respect to σNcan be calculated as
∂JRCm
∂σN
=














αφr
i2
τ
(JRCp)τ
σNln 10
∂JRCm
∂(φr/iτ)if 0 ≤Λ< δp,
0 if δp≤Λ,
(A.6)
where
∂JRCm
∂(φr/iτ)=3(10Λ−3δp)(δp−3Λ)
h3δp−(3 −7φr/iτ)Λi2.(A.7)
25

Lastly, the derivative of Mwith respect to σN, which appears in Eq. (A.5), can be calculated as
∂M
∂σN
=JRCp
12 log (JCS/σN)2
1
σNln 10.(A.8)
References
[1] N. Barton, V. Choubey, The shear strength of rock joints in theory and practice, Rock Mechanics 10 (1-2) (1977)
1–54.
[2] S. Bandis, A. Lumsden, N. Barton, Fundamentals of rock joint deformation, in: International Journal of Rock
Mechanics and Mining Sciences & Geomechanics Abstracts, Vol. 20, Pergamon, 1983, pp. 249–268.
[3] N. Barton, S. Bandis, K. Bakhtar, Strength, deformation and conductivity coupling of rock joints, in: Interna-
tional Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, Vol. 22, Elsevier, 1985, pp.
121–140.
[4] R. Hutson, C. Dowding, Joint asperity degradation during cyclic shear, in: International Journal of Rock Me-
chanics and Mining Sciences & Geomechanics Abstracts, Vol. 27, Elsevier, 1990, pp. 109–119.
[5] X. Huang, B. Haimson, M. Plesha, X. Qiu, An investigation of the mechanics of rock joints—Part I. Laboratory
investigation, in: International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts,
Vol. 30, Elsevier, 1993, pp. 257–269.
[6] F. Homand, T. Belem, M. Souley, Friction and degradation of rock joint surfaces under shear loads, International
Journal for Numerical and Analytical Methods in Geomechanics 25 (10) (2001) 973–999.
[7] H. S. Lee, Y. J. Park, T. F. Cho, K. H. You, Influence of asperity degradation on the mechanical behavior of
rough rock joints under cyclic shear loading, International Journal of Rock Mechanics and Mining Sciences
38 (7) (2001) 967–980.
[8] F. Meng, H. Zhou, Z. Wang, C. Zhang, S. Li, L. Zhang, L. Kong, Characteristics of asperity damage and its
influence on the shear behavior of granite joints, Rock Mechanics and Rock Engineering 51 (2) (2018) 429–
449.
[9] N. Barton, Modelling rock joint behavior from in situ block tests: implications for nuclear waste repository
design, Vol. 308, Office of Nuclear Waste Isolation, Battelle Project Management Division, 1982.
[10] P. Asadollahi, F. Tonon, Degradation of rock fracture asperities in unloading, reloading, and reversal, Interna-
tional Journal for Numerical and Analytical Methods in Geomechanics 35 (12) (2011) 1334–1346.
[11] S. O. Choi, S.-K. Chung, Stability analysis of jointed rock slopes with the Barton–Bandis constitutive model in
UDEC, International Journal of Rock Mechanics and Mining Sciences 41 (2004) 581–586.
[12] Q. Lei, J.-P. Latham, J. Xiang, Implementation of an empirical joint constitutive model into finite-discrete el-
ement analysis of the geomechanical behaviour of fractured rocks, Rock Mechanics and Rock Engineering
49 (12) (2016) 4799–4816.
[13] S. Ma, Z. Zhao, W. Nie, J. Nemcik, Z. Zhang, X. Zhu, Implementation of displacement-dependent Barton–
Bandis rock joint model into discontinuous deformation analysis, Computers and Geotechnics 86 (2017) 1–8.
[14] F. Liu, R. I. Borja, A contact algorithm for frictional crack propagation with the extended finite element method,
International Journal for Numerical Methods in Engineering 76 (10) (2008) 1489–1512.
[15] F. Fei, J. Choo, A phase-field method for modeling cracks with frictional contact, International Journal for
Numerical Methods in Engineering 121 (4) (2020) 740–762.
[16] F. Fei, J. Choo, C. Liu, J. A. White, Phase-field modeling of rock fractures with roughness, arXiv preprint
arXiv:2105.14663 (2021).
26

[17] J. Choo, Y. Zhao, Y. Jiang, M. Li, C. Jiang, K. Soga, A barrier method for frictional contact on embedded
interfaces, arXiv preprint arXiv:2107.05814 (2021).
[18] P. Bettinelli, J.-P. Avouac, M. Flouzat, L. Bollinger, G. Ramillien, S. Rajaure, S. Sapkota, Seasonal variations
of seismicity and geodetic strain in the Himalaya induced by surface hydrology, Earth and Planetary Science
Letters 266 (3-4) (2008) 332–344.
[19] S. Zhou, C. Xia, Y. Hu, Y. Zhou, P. Zhang, Damage modeling of basaltic rock subjected to cyclic temperature
and uniaxial stress, International Journal of Rock Mechanics and Mining Sciences 77 (2015) 163–173.
[20] G. Preisig, E. Eberhardt, M. Smithyman, A. Preh, L. Bonzanigo, Hydromechanical rock mass fatigue in deep-
seated landslides accompanying seasonal variations in pore pressures, Rock Mechanics and Rock Engineering
49 (6) (2016) 2333–2351.
[21] R. Hashimoto, T. Sueoka, T. Koyama, M. Kikumoto, Improvement of discontinuous deformation analysis in-
corporating implicit updating scheme of friction and joint strength degradation, Rock Mechanics and Rock
Engineering (2021) 1–25.
[22] R. I. Borja, S. R. Lee, Cam-Clay plasticity, Part I: Implicit integration of elasto-plastic constitutive relations,
Computer Methods in Applied Mechanics and Engineering 78 (1) (1990) 49–72.
[23] R. I. Borja, K. M. Sama, P. F. Sanz, On the numerical integration of three-invariant elastoplastic constitutive
models, Computer Methods in Applied Mechanics and Engineering 192 (9-10) (2003) 1227–1258.
[24] J. A. White, Anisotropic damage of rock joints during cyclic loading: constitutive framework and numerical
integration, International Journal for Numerical and Analytical Methods in Geomechanics 38 (10) (2014) 1036–
1057.
[25] R. I. Borja, J. Choo, Cam-Clay plasticity, Part VIII: A constitutive framework for porous materials with evolving
internal structure, Computer Methods in Applied Mechanics and Engineering 309 (2016) 653–679.
[26] J. Choo, Mohr–Coulomb plasticity for sands incorporating density effects without parameter calibration, Inter-
national Journal for Numerical and Analytical Methods in Geomechanics 42 (18) (2018) 2193–2206.
[27] B.-K. Son, Y.-K. Lee, C.-I. Lee, Elasto-plastic simulation of a direct shear test on rough rock joints, International
Journal of Rock Mechanics and Mining Sciences 41 (2004) 354–359.
[28] R. Olsson, N. Barton, An improved model for hydromechanical coupling during shearing of rock joints, Inter-
national Journal of Rock Mechanics and Mining Sciences 38 (3) (2001) 317–329.
[29] N. Barton, A relationship between joint roughness and joint shear strength, in: Rock Fracture-Proc, Int. Symp.
on Rock Mechanics, Nancy, France, 1971, pp. paper1–8.
[30] N. Barton, S. Bandis, Some effects of scale on the shear strength of joints, International Journal of Rock Me-
chanics and Mining Sciences 17 (1980) 69–73.
[31] N. Barton, Some size dependent properties of joints and faults, Geophysical Research Letters 8 (7) (1981) 667–
670.
[32] S. Bandis, A. Lumsden, N. Barton, Experimental studies of scale effects on the shear behaviour of rock joints,
in: International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, Vol. 18, Elsevier,
1981, pp. 1–21.
[33] N. Barton, S. Bandis, Effects of block size on the shear behavior of jointed rock, in: The 23rd US symposium
on rock mechanics (USRMS), OnePetro, 1982.
[34] S. H. Prassetyo, M. Gutierrez, N. Barton, Nonlinear shear behavior of rock joints using a linearized imple-
mentation of the Barton–Bandis model, Journal of Rock Mechanics and Geotechnical Engineering 9 (4) (2017)
671–682.
[35] M. E. Plesha, Constitutive models for rock discontinuities with dilatancy and surface degradation, International
Journal for Numerical and Analytical Methods in Geomechanics 11 (4) (1987) 345–362.
[36] L. Jing, O. Stephansson, E. Nordlund, Study of rock joints under cyclic loading conditions, Rock Mechanics
27

and Rock Engineering 26 (3) (1993) 215–232.
[37] S. Stupkiewicz, Z. Mr ´
oz, Modeling of friction and dilatancy effects at brittle interfaces for monotonic and cyclic
loading, Journal of Theoretical and Applied Mechanics 39 (2001) 707–739.
[38] R. I. Borja, Plasticity: Modeling & Computation, Springer, 2013.
[39] R. R. Settgast, P. Fu, S. D. Walsh, J. A. White, C. Annavarapu, F. J. Ryerson, A fully coupled method for mas-
sively parallel simulation of hydraulically driven fractures in 3-dimensions, International Journal for Numerical
and Analytical Methods in Geomechanics 41 (5) (2017) 627–653.
[40] J. Choo, W. Sun, Cracking and damage from crystallization in pores: Coupled chemo-hydro-mechanics and
phase-field modeling, Computer Methods in Applied Mechanics and Engineering 335 (2018) 347–379.
[41] J. Choo, Stabilized mixed continuous/enriched Galerkin formulations for locally mass conservative porome-
chanics, Computer Methods in Applied Mechanics and Engineering 357 (2019) 112568.
[42] F. Fei, J. Choo, A phase-field model of frictional shear fracture in geologic materials, Computer Methods in
Applied Mechanics and Engineering 369 (2020) 113265.
[43] B. Lepillier, K. Yoshioka, F. Parisio, R. Bakker, D. Bruhn, Variational phase-field modeling of hydraulic frac-
ture interaction with natural fractures and application to enhanced geothermal systems, Journal of Geophysical
Research: Solid Earth 125 (7) (2020) e2020JB019856.
[44] F. Fei, J. Choo, Double-phase-field formulation for mixed-mode fracture in rocks, Computer Methods in Applied
Mechanics and Engineering 376 (2021) 113655.
[45] T. Kadeethum, S. Lee, F. Ballarin, J. Choo, H. M. Nick, A locally conservative mixed finite element framework
for coupled hydro-mechanical–chemical processes in heterogeneous porous media, Computers & Geosciences
152 (2021) 104774.
28