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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

50.2σc

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,tr3×1

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)

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