1. INTRODUCTION

1.1. Rock dilation The mechanical behavior of rocks and rock masses has been extensively investigated in the fields of civil and mining engineering. Experimental and field observations of rock failure show that the failure process is closely associated with rock dilation, which is a phenomenon associated with micro-crack initiation and propagation, and increase in void space when the rock is loaded beyond a certain threshold. It is important to understand nonlinear characteristics of rocks before and after the peak stress and subsequently test the behavior under various loading conditions in numerical modeling with a constitutive model for a better understanding of rock failure process. However, it is a very challenging and difficult task to develop a constitutive model which represents the complete stress-strain behavior of rocks adequately, especially for the nonlinear response such as dilation.

Cook [1] proved that dilation during compression to failure was a pervasive volumetric property of rocks and not a superficial phenomenon. Dilation represents the true volumetric behavior of rocks, and it is closely related to the process of rock failure. Based on studies by many researchers [2-7], the failure process of brittle rocks can be divided into the following stages: (1) crack closure; (2) linear elastic deformation; (3) crack

initiation; (4) stable crack growth; (5) crack coalescence and damage; (6) unstable crack growth; (7) failure; (8) post peak behavior. A detailed illustration of the dilation process of rocks can be found in [8].

In continuum mechanics, the parameter most widely used to measure dilation is the dilation angle )(ψ , which can be obtained from triaxial compression tests by calculating plastic axial and volumetric strain increments [9]. For a joint, the dilation angle is determined, from direct shear tests, as the ratio of normal to tangential displacements along a joint [10]. The physical meaning of ψ can be understood by considering a frictional sliding, either along a rough joint or along particles as shown in Fig. 1.

However, in rock engineering, when the dilation angle is taken into consideration, especially for numerical modeling studies, the approach by most researchers is

ARMA 10-459 A confinement and deformation dependent dilation angle model for rocks Cai, M. School of Engineering, Laurentian University, Sudbury, Ontario, Canada Zhao, X.G. Beijing Research Institute of Uranium Geology, Beijing, China

Copyright 2010 ARMA, American Rock Mechanics Association This paper was prepared for presentation at the 44th US Rock Mechanics Symposium and 5th U.S.-Canada Rock Mechanics Symposium, held in Salt Lake City, UT June 27–30, 2010. This paper was selected for presentation at the symposium by an ARMA Technical Program Committee based on a technical and critical review of the paper by a minimum of two technical reviewers. The material, as presented, does not necessarily reflect any position of ARMA, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of ARMA is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgement of where and by whom the paper was presented.

ABSTRACT: A mobilized dilation angle model considering the influence of both confining stress and plastic shear strain is proposed in this paper. The model is used to predict the volumetric-axial strain relationships of a few rock samples and the results are found to be in good agreement with experimental results. Realistic post-failure dilation behavior of rocks can be captured using the proposed model in combination with Mohr-Coulomb strain-softening models. The model is then used to study the excavation-induced displacement around tunnels located in different rock masses. It is illustrated from a few examples that displacement distributions obtained from the dilation angle model are more reasonable, when compared with the general trend measured underground.

Fig. 1. Dilation associated with sliding along a rough joint and particles. Modified from [9].

ψ Velocity

ψ Velocity

often simplistic; it is generally assumed as either one of the two constants − zero in a non-associated flow rule and the same as the friction angle in an associated flow rule. In most popular failure criteria, such as linear Mohr-Coulomb failure criterion and non-linear Hoek-Brown failure criterion, the rock dilation is assumed to remain as a constant when the rock mass is deformed.

In many numerical analysis tools [11-14], the default value for dilation angle is often zero for all the nonlinear constitutive models. For the strain-hardening/softening Mohr-Coulomb model with non-associated shear flow rules in FLAC and FLAC3D, the user is able to define the dilation angle as a piecewise-linear decay function of plastic strain. However, only limited suggestions are given in the user’s manuals on how to determine plastic strain dependent dilation angle. It is logical to have a plastic strain dependent dilation angle, but the question to be asked is whether plastic strain is the only influencing factor on dilation.

In fact, a constant dilation angle is an approximation that is clearly not physically correct. This assumption of constant dilation is made largely because little is known about how the dilation of a rock changes past peak load. Some researchers [15-16] illustrate that it may be unrealistic and misleading to use a constant dilation angle. They also point out that dilation angle should be a function of plastic parameters and confining stress. Many experimental results of triaxial compression tests on rock samples showed that rock dilation gradually decreased with increasing confining pressure [17-19]. At low confining pressures, brittle behavior accompanied by volumetric dilation predominates with significant stress-drop [20]. With the increase of confining pressures, brittle failure characterized by axial splitting failure gradually transfers into shear failure and is characterized with the formation of localized shear bands, and the deformation process is associated with a systematic reduction in stress drop following strain localization and reduced dilation [21].

It is observed in the field that rock failure, deformation and associated radial dilation near the tunnel boundary are highly dependent on confinement. Under low confinement condition, spalling is the dominant failure mode around underground excavations in brittle hard rocks subjected to high stresses. Once the crack damage stress level is exceeded [7], volumetric deformation of the rock increases drastically. As noted by Kaiser et al. [22], the volume increase of stress-fractured rocks near an excavation results from three sources: (1) dilation due to new fracture growth, (2) shear along existing fractures or joints, and (3) dilation due to geometric incompatibilities when blocks of broken rocks move relative to each other as they are forced into the excavation. It should therefore be anticipated that the strength and dilation behavior of rocks near excavation

boundary should differ from those encountered at some distances away from the excavation boundary. As shown in Fig. 2, on the boundary of an unsupported tunnel,

03 =σ (the minimum principal stress) and the highest tangential stress (the maximum principal stress, 1σ ) exist. Under such a condition, maximum rock dilation may take place if the rock fails. With the increase of σ3 away from the excavation boundary, the dilation of surrounding rocks will decease significantly. This means that, from the boundary of excavation to deeper grounds, dilation decreases gradually and finally vanishes at high confining stresses. These behaviors of rocks have been observed especially in hard rock mines and deep civil tunnels [23-24].

1.2. Influence of confinement on dilation Based on the techniques of measuring volumetric strain, developed by Crouch [26] and later modified and used by Wawersik [27], Cipullo [28], Singh [29], and Medhurst [30] and others, a number of experimental studies on the behavior of rocks in triaxial compression have shown that dilation is strongly influenced by the magnitude of confining stress [30-32]. Fig. 3 shows typical volumetric responses under various confining stresses for medium-grained sandstones, and some important characteristics can be observed from the figure. It is observed that: (a) maximum dilation rate occurs at the post-peak deformation stage regardless of the magnitude of the confining stresses; (b) the onset of dilation is delayed with increasing confining stresses; (c) the gradient of dilation decreases with increasing confining stresses; (d) the rate of dilation, which can be defined as the tangent slope of a point in the volumetric-axial strain curve, deceases when rock undergoes a gradual transition from strength weakening to residual strength; (e) the maximum rate of dilation takes place at

Fig. 2. Stress and rock fracturing condition near the tunnel boundary. σx0, σy0 and σz0 are the in situ stress components [25].

In situ stress

σy0

σx

σz

σ1

σ3=0

σ2 σ1

σ2

σ3

Tunnel

Induced stress and rock fracturing state

the strain softening stage and higher confining stress results in small rate of dilation at this stage; (f) dilation gradually reaches to a constant value, i.e. at the end of the deformation stage there would be no additional volumetric strain changes; (g) dilation should be mainly governed by both confining stress and post-peak plastic strain of the rock.

Due to access limitation to true triaxial experimental facilities, not enough systematic study results from literature are available to develop a comprehensive dilation model with the consideration of the intermediate principal stress. Hence, the influence of the intermediate principal stress on dilation model is not considered in this work. Besides, some soft rocks with high-porosity such as Ohya tuff, Yokohama siltstone [33] present strong contractive behavior, even when a low confining stress (1 MPa) is applied. Hence, rocks like these will also be excluded in our analysis.

1.3. Previous study on dilation modeling There have been a number of early attempts to develop constitutive relations for rock dilation during

deformation. The objective has usually been to predict the form of the stress-volumetric strain relation, or the pressure dependency of this relation. These studies mainly focus on micromechanical models, such as crack sliding model [2] and physical model of crack growth [34]. Although such idealized models have been developed to give insight into a fundamental response of rock, it is generally believed that the complex behavior displayed by natural rock is beyond the reach of these approaches [35]. Besides, some constitutive laws are applicable to a single or a few studied rocks, and it is difficult to represent general dilation behavior of rocks using these models.

From laboratory data, Vemeer and de Borst [9] concluded that for soil, rock, and concrete, the dilation angle is at least 20° less than the friction angle. They also deduced a formula to estimate the dilation angle, as follows:

pv

p

pv

εεε

ψ&&

&

+−=

12sin (1)

where pvε& and p

1ε& are volumetric and axial plastic strain increments, respectively. For ψ > 0, an irreversible increase of volume occurs, while for ψ < 0, a decrease is predicted (plastic contraction). ψ = 0 is the special case of plastically volume-preserving flow.

Hoek and Brown [36], based on wide engineering experience, suggest the use of constant dilation angle values that are dependent on rock mass quality. For very good rock, they recommended that the dilation angle is about 1/4 of the friction angle; for the average quality rock, the value suggested is 1/8, and poor rock seems to have a negligible dilation angle. Ord [37], using numerical modeling, concluded that for most geological materials, particularly brittle ones, it is possible that the dilation angle may be greater than the friction angle, a situation not normally considered in soil or rock mechanics.

In order to characterize the changes in dilation that occur with varying confining stress, Yuan and Harrison [35] proposed an empirical dilation index, which was defined as the ratio of apparent dilation angle a rock possessed at any particular confining stress level to that under uniaxial compression, to describe dilation behavior of rocks. The dilation index is given by

01

1

0 )/arctan()/arctan(

pvp

ppvppdI

εεεε

θθ

∆∆

∆∆== (2)

where vpε∆ and p1ε∆ are the incremental plastic volumetric strain and the incremental plastic axial strain, respectively. Subscript 0 indicates quantities under uniaxial compression.

(a)

(b) Fig. 3. Volumetric-axial strain curves (b) associated with stress-strain curves (a), and peak stress corresponding to volumetric strain at the same axial strain level for a medium-grained sandstone (Bursnip’s Road) under various confining stresses. Modified from [32].

0 5 10 15 20 25 30 35 40 45 5024

12

0

-12

-24

-36

-48

-60

σ3=13.8MPa

σ3=27.6MPa

σ3=3.45MPa

σ3=1.72MPa

Vol

umet

ric s

train

(mill

istra

in)

Axial strain(millistrain)

σ3=0.345MPa

Axial strain (millistrain)

0 5 10 15 20 25 30 35 40 45 500

20

40

60

80

100

120

140

160

180

200

σ3=13.8MPa

σ3=27.6MPa

σ3=3.45MPa

σ3=1.72MPa

Com

pres

sive

stre

ss (M

Pa)

Axial strain(millistrain)

σ3=0.345MPa

Axial strain (millistrain)

However, this idealized dilation index is based on a linear dilation behavior, i.e., the rate of dilation is constant at a certain confining stress level and the volumetric dilation of rock is assumed to increase infinitely as the rock deforms. In fact, the rate of dilation is not constant, and it gradually decreases and eventually reaches zero when rock undergoes a large deformation, i.e., the volumetric dilation will vanish following large plastic straining because rocks cannot dilate infinitely. The phenomenon will be illustrated in the next section.

Detournay [16] also argued that the assumption of a constant dilation angle was unrealistic. Using an example of tunnel closure prediction employing a variable dilation angle model, the author proposed a tangent dilatancy factor ∗

pK that decays from an initial value pK (a function of the friction angle) according to an exponential function of the plastic shear strain γ as

∗−∗ ⋅−+= γγ /)1(1 eKK pp (3)

where the parameter ∗γ can most usefully be related to the maximum inelastic volume increase ∗∆ as

2/)1ln( +

∆=

∗∗

pKγ (4)

Based on plasticity dependency in the flow rule of Eq. (3) and a detailed analysis of dilation angle calculated from published test data including five rock types, Alejano and Alonso [15] proposed a peak dilation angle decay model in which the confining stress dependency was implemented following a comparison with the trend of the deformation behavior observed by Barton and his colleagues [38-39] for joints, as follows:

∗−⋅−+=

,/, )1(1

ppeKK peak

γγψψ (5)

where ∗,pγ is a plastic constant which should be calculated for each rock type, and peakψ can be expressed as

1.0

loglog1 3

1010 ++

=σ

σσ

φψ ci

cipeak (6)

where φ is peak friction angle, ciσ is the intact unconfined compressive strength of rock, and 3σ is confining stress.

This model incorporates dependence of dilation angle on confining stress and the plastic parameter, and it can be implemented in numerical modeling easily. However, a hypothesis was made in this model, i.e., once the plastic deformation starts, the dilation angle begins to drop from a peak value.

We have reviewed and summarized some empirical methods and theoretical models of determining dilation, and subsequently discussed the effect of confining stress and plastic shear strain on rock dilation. Subsequently, an empirical dilation angle model, which considers the influence of both confining stress and plastic shear strain, is developed, and the model is implemented in FLAC to predict the volumetric-axial strain relationships using experimental data. Finally, numerical simulations are performed to calculate excavation-induced displacement distribution around tunnels in different rock types using the proposed dilation angle model. The aim of the study is to evaluate the influence of the mobilized dilation angles on deformation behavior around tunnels. For comparison, simulations using constant dilation angles are also conducted.

2. A DILATION ANGLE MODEL FOR ROCKS

2.1. A mobilized dilation angle model for intact rocks

It is recognized that rocks and rock masses commonly exhibit post-peak strain-softening or strength-weakening behavior, which is described by the gradual loss of load-bearing capacity of a rock from a peak load condition to a residual one. A plastic parameter or softening parameter η can be introduced to account for the process and mode of strength transition, as shown in Fig. 4. The plastic parameter is null in the elastic region, and if 0>η , strain softening appears until the residual strength is reached. The rate of strength drop or the slope of the softening stage may be expressed by (– stan ), where s is the angle of the slope as shown in Fig. 4. If the rate of strength drop approaches to infinity, the perfectly brittle behavior takes place; if the rate is zero, the perfectly plastic behavior results. It is clear that perfectly brittle and perfectly plastic behavior models are particular cases of the strain-softening model.

The plastic parameter η can be defined in a few ways, but so far there is no generally agreed method as pointed

Fig. 4. Strain-softening behavior of rock and its particular cases [40].

σ

ε

Elastic Softening Residual

s

0=η0>η

σε

Perfectly brittle behavior

Perfectly plastic behavior

∞→− stan

0tan →− s

ε

σ

out by Alejano and Alonso [15]. Generally, there are two ways of determining the parameter; one is defining η as a function of internal variables, and the other is based on the incremental plastic strain. In the former case, the most widely accepted parameter is plastic shear strain, which is introduced into our proposed dilation angle model and can be obtained as the difference between the major and minor principal plastic strains:

ppp 31 εεγη −== (7)

For the latter, the incremental softening parameter η& depends on plastic strain increments, and the most commonly used expression is [9]:

)(32

332211pppppp εεεεεε

τηη &&&&&&& ++=

∆∆

= (8)

In FLAC or FLAC3D, strain-softening model based on Mohr-Coulomb model provides the incremental plastic parameter psε∆ to measure plastic shear strain and control the evolution of the strength parameters in the

softening regime. In order to implement the dilation angle model in FLAC, the relationship between between the plastic shear strain in the proposed dilation angle model and characteristic plastic parameter in strain-softening model needs to be known. According to the principle of non-associated flow rule in the strain-softening models in FLAC, the relationship of the two plastic parameters has been given by [40].

The modified triaxial compression apparatus developed by Crouch [26] provides an effective means of investigating the dilation behavior of rocks. Under this important precondition, the irrecoverable strain directly associated with the plastic strain can be obtained using the technique developed by Elliott and Brown [41] and the technique was later used by Medhurst to study yield characteristics of coal samples [42]. The technique uses loading-unloading cycles to differentiate between the recoverable and irrecoverable components of the observed rock behavior. The method assumes that the rock sample has a behavior “memory,” i.e., if a sample is unloaded to zero deviator stress at different stages of the complete stress-strain curve and then reloaded, the reloading path will follow the previous unloading path until it comes to the initial loading point. For each test, an irrecoverable strain locus can be created by linking all points at intersection of the value of axial strain at zero deviator stress and the value of volumetric strain at zero deviatoric stress for different unloading cycles. An illustration of constructing the irrecoverable strain locus is shown in Fig. 5.

The irrecoverable strain locus provides a complete relationship between plastic volumetric strain p

vε and plastic axial strain p

1ε , and indirectly implies the plastic lateral strain, as follows:

2/)( 13pp

vp εεε −= (9)

The dilation angles can be calculated in the course of deformation by a method of average strain provided by Alejano and Alonso [15]. For example, consider points a, b, and c in Fig. 5 and the plastic strain components for the two intervals ( ab , bc ) are given by:

2/)( _,_,_,p

bvip

avip

abvi εεε +=

2/)( _,_,_,p

cvip

bvip

bcvi εεε += 3,1=i (10) The corresponding increment is obtained from:

pabvi

pbcvi

pacvi _,_,_, εεε −=& 3,1=i (11)

The dilation angle can be calculated according to Eq. (1):

)2

arcsin(__1

_p

acvp

ac

pacv

εε

εψ

&&

&

+−= (12)

Fig. 5. Irrecoverable strain locus for constructing dilation angle model under triaxial loading-unloading cycle test. Modified from [30].

σ 1-σ

3

ε1

ε v

Irrecoverable strain locus

ε1

ε1p - εv

p

Crack damage threshold

a

b

c

γp

ψ

(a)

(c)

(b)

Shift of the dilation angle with respect to plastic strain

Crack damage threshold

Crack damage threshold

where pacv

pacv

pacv __3_,1_ 2εεε &&& += . The corresponding

plastic shear strain )( pγ can also be calculated using the same method.

As can be seen from Fig. 5 (a-b), when the stress is smaller than the crack damage stress, p

3ε is negative and p

1ε is very small, and a negative dilation angle can result according to Eq. (1). When the stress is higher than the crack damage threshold, the dilation angle first increases at a high rate until the peak strength is reached, and then decreases gradually. Hence, the definition of dilation in our present study makes sense only from the crack damage stage to the post-failure regime. As shown in Fig. 5 (c), the calculated dilation angles can be negative. For simplicity, these negative dilation angles may be ignored, and plastic shear strain pγ can be considered to start from null and correspond to zero dilation; meanwhile, a complete shift for all dilation angle data is made as can be seen in Fig. 5 (c). In this manner, a set of estimated dilation angles and corresponding pγ for Moura coal specimens [44] under different confining stresses were obtained (see Fig. 6).

Triaxial compression tests performed by other authors on different rocks did not follow Medhurst’s cycle loading-unloading method. Hence, the plastic strains have to be calculated by subtracting the elastic strain from the total strain according to plasticity theory:

E

ve 311

2 σσε

−= and ep

111 εεε −= (13)

E

vev

)2)(21( 31 σσε

+−= and e

vvpv εεε −= (14)

where 1ε is total axial strain, e1ε is axial elastic strain,

vε is total volumetric strain and evε is elastic volumetric

strain; E and ν are Young’s modulus and Poisson’s ratio, respectively.

Using the approach described above, the relation between dilation angle and plastic shear strain (γp) under different confining stresses can be captured for other six rock types studied in the work. For example, the dilation angle-plastic parameter relationship of weak medium-grained sandstone (Bursnip’s Road) under different confining stresses and their best fits are shown in Fig. 7. Obviously, the irrecoverable strain locus proposed by Medhurst is preferred to obtain the dilation angle due to its reasonable test procedure. Nevertheless, other data, such as ones shown in Fig. 7, also provide very valuable information for studying rock dilation and establishing our empirical dilation angle model.

Fig. 6 and Fig. 7 show the variation of the dilation angle with both plastic shear strain and confining stress for the seven rock types investigated. It can be seen that with increasing pγ , the dilation angle starts to increase rapidly from a small value to its peak value, after which it gradually deceases. A low confining stress results in a high peak dilation angle and with increasing confining stress, the peak dilation angle drops and the rate of decrease becomes less when the confining stresses are sufficiently high.

A fitting equation of plastic shear strain-dependent dilation angle for all seven rocks under different confining pressures can be expressed as

)/()]exp()[exp( bccbab pp −−−−= γγψ (15) where a, b, and c are fit coefficients; and γp is the plastic shear strain in %.

Among the coefficients, a and b show a trend of gradually decreasing from their peak values under unconfined or very low confining stress conditions to

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00

5

10

15

20

25

30

35

40

45

50 0.2MPa(300mm) 0.2MPa(146mm) 0.4MPa(300mm) 0.8MPa(300mm) 1MPa(146mm) 3MPa(146mm) 4MPa(146mm_a) 4MPa(146mm_b)

Dila

tion

angl

e (o )

Plastic shear strain (%)

Fig. 6. Mobilized dilation angle-plastic shear strain curves of Moura coal under different confining stresses in cycle loading-unloading experiments by Medhurst [42], and their best fits (dashed lines).

0 1 2 3 4 5 6 7 8 9 1005

1015202530354045505560

0.345MPa0.69MPa1.72MPa3.45MPa6.9MPa

0.345MPa 0.69MPa 1.72MPa 3.45MPa 6.9MPa

Dila

tion

angl

e (o )

Plastic shear strain (%)

(Individual fit)

Fig. 7. Mobilized dilation angle-plastic shear strain curves of weak medium-grained sandstone (Bursnip’s Road) [32] under different confining stresses and their best fits (dashed lines).

low or residual values at high confinements, whereas c shows a trend of gradually increasing its value from very low to high confinement conditions until an asymptotic value is reached. Here, we apply the empirical approach again to determine the relationships between coefficient a , b , c and confining stress σ3. These coefficients vary with σ3 according to the following equations:

)/exp( 3321 aaaa σ−+= (16)

)/exp( 3321 bbbb σ−+= (17)

3321

cccc σ+= (18)

where ia , ib and ic ( 3,2,1=i ) are fit coefficients (Table 1) and σ3 is the confining pressure in MPa.

Table 1. Fit coefficients for confining stress dependent a, b, c of seven rocks

a b c a1 a2 a3 b1 b2 b3 c1 c2 (%) c3

1 63.17 11.92 2.80 5.83 36.25 6.77 0.14 1.14 1.232 14.63 34.90 3.40 4.06 15.56 5.54 0.08 0.40 0.583 10.34 34.76 4.90 10.14 17.77 16.26 0.07 1.13 0.554 20.93 35.28 2.34 0.99 44.39 0.73 0.37 3.54 0.475 20.03 35.64 0.89 10.47 26.58 1.31 0.15 17.5 0.826 17.19 32.40 3.37 0.09 2.23 23.6 0.03 8.75 0.257 12.57 27.23 2.09 1.49 4.02 6.62 0.07 2.90 1.60Note: 1-Witwatersrand quartzite [26]; 2-Sandstone (strong) [31]; 3- Silty sandstone [31]; 4-Sandstone (weak) [32]; 5-Moura coal [42]; 6-mudstone [31]; 7- Seatearth [32].

According to Eqs. (15) to (18), an overall fit for different rocks is made to illustrate the dilation angle model considering both plastic shear strain and confining stress simultaneously. Fig. 8 (a-b) indicates that the results of overall fit using the dilation angle model agree well with the results of individual fit based on experimental data of two rock types. When confining stress increases, a general trend that the peak dilation angles decrease and the locations of peak dilation angle gradually shift towards right with more plastic shear straining, can be observed from all models. Due to the variation of dilation behavior with rock types, the shape of dilation angle – plastic shear strain relations are different for different rocks, i.e., the fit coefficients in the model differ for different rocks. Judging from the form of curves combined with the grain description, uniaxial compressive strength and post-peak stress-strain curve for every rock, an empirical classification of fit coefficients can be made to represent general dilation angle models for four special rock types (see Fig. 9 ):

(a) Coarse-grained hard rock (e.g., quartzite). There exists a very rapid increase of dilation angle with plastic shear strain, and then a rapid decrease of dilation angle

post peak. This coincides with its brittle failure mode (axial splitting failure) which is accompanied by large dilation.

(b) Medium-grained hard rock (e.g., strong sandstone and silty sandstone). The model behaves in a similar way as that of coarse-grained hard rock before the peak value dilation angle is reached. However, compared to quartzite, the peak values are relatively small and rate of the dilation angle decrease with plastic parameter is small. The experimental results [31] show that there are predominantly vertical fractures occurred at peak stress associated with large dilation at low confinements. (c) Fine-medium-grained soft rock (e.g., coal and weak sandstone). At low confinements, the pre-peak behavior of the dilation angle model is similar to the two models shown above, i.e., the peak dilation angles are reached at a plastic shear strain level of 2-3 millistrain. In addition, the magnitude of the peak dilation angle is similar to that of the medium-grained hard rock, and the post-peak behavior of the model is similar to that of the coarse- grained hard rock. In this rock group, all specimens

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00

5

10

15

20

25

30

35

40

45

50

1MPa

0.8MPa

(model)1MPa

(model)

Individual fit Overall fit (model)

4MPa

3MPa

0.8MPa0.4MPa(model)0.4MPa

0.2MPa(model)0.2MPa(146mm)

Dila

tion

angl

e (o )

Plastic shear strain (%)

0.2MPa(300mm)

(a) Moura coal

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

40

45

50

55

0.345 MPa0.69 MPa1.72 MPa3.45 MPa6.9 MPa

Individual fit Overall fit (model)

Dila

tion

angl

e (o )

Plastic shear strain (%) (b) Weak medium-grained sandstone (Bursnip’s Road)

Fig. 8. Comparison of overall fit using the dilation angle and individual fit of seven rocks at different confining pressures.

exhibited strain-softening behavior [32, 42]. Especially for coal, a transition from axial splitting to shearing

failure can be observed when the confinement is high.

(d) Fine-grained soft rock (e.g., mudstone and seatearth). This model is characterized by a progressive increase of the dilation angle, and it takes more plastic shear strains for the peak dilation angle to be reached. After the peak value is reached, dilation angle drops gradually in a manner similar to that of the medium-grained hard rock. According to the test results [31-32], the stress-strain curves of this rock type show a transition from strain-softening to strain-hardening when the confining stresses are very high.

It should be pointed out that the four types of dilation angle behavior are summarized based on limited test data and the classification factors include uniaxial compressive strength (UCS) and grain size only. When more test data are available, the model parameters can be fine-tuned to suit the dilation behavior of a particular rock. In engineering application when test data are not available, the model with suggested parameters in the inserted table in Fig. 9 can be used in combination with Eqs. (15) and (16)-(18) to approximate the plastic shear strain and confinement dependent dilation behavior of rocks.

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

70

80

90 coal sandstonefarmer sandstoneroad mudstone Quartzite seatearth Siltysanstone

c=c1+c2σc3

3

b=b1+b2exp(-σ3/b3)

a=a1+a2exp(-σ3/a3)

Quartzite(a) Coarse-grained hard rock

ψ=ab[exp(-bγp)-exp(-cγp)]/(c-b)(b)Medium-grained hard rock

(d) Fine-grained soft rock

(c) Fine-medium-grained soft rock

CoalSeatearth

Mudstone

Sandstone (strong)Silty sandstone

Dila

tion

angl

e (o )

Plastic shear strain (%)

Coarse-grained hard rock

Medium-grained hard rock

Fine-medium-grained soft rock

Fine-grained soft rock

a1=63.17 a1=10.34~14.63 a1=20.03~20.93 a1=12.57~17.19a2=11.92 a2=34.76~34.90 a2=35.28~35.64 a2=27.23~32.4a3=2.8 a3=3.4~4.9 a3=0.89~2.34 a3=2.09~3.37b1=5.83 b1=4.06~10.14 b1=0.99~10.47 b1=0.089~1.49b2=36.25 b2=15.56~17.77 b2=26.58~44.39 b2=2.23~4.02b3=6.77 b3=5.54~16.26 b3=0.73~1.31 b3=6.62~23.6c1=0.14 c1=0.07~0.08 c1=0.15~0.37 c1=0.03~0.07c2=1.14% c2=0.4%~1.13% c2=3.54%~17.5% c2=2.9%~8.75%c3=1.23 c3=0.55~0.58 c3=0.47~0.82 c3=0.25~1.60

Sandstone(weak)

Rock type Grain UCS (MPa)

Quartzite Coarse 205Sandstone(strong) Medium 97Silty sandstone Medium 61Sandstone(weak) Medium 44.18Coal Fine 32.7Mudstone Fine 55Seatearth Fine 20.17

Overall fit(model)

Fig. 9. The variation of mobilized dilation angle for different rock types at a confinement of 1 MPa. For other confining pressures, the curves can be obtained using the parameters shown in the inserted table.

2.2. Model verification The axisymmetric model in FLAC is used to simulate triaxial compression tests. The axisymmetry provides an analysis of rock deformation closely resembles the test conditions [8]. The size of the model is chosen as half of the intact rock sample, i.e., 73 × 292 mm, and the length of every square mesh is 14.6 mm, as shown in Fig. 10. The y-displacement at the bottom boundary is restricted and when the model runs, and the x-displacement on the symmetric axis is fixed automatically. Confinement stresses parallel to the x-direction are applied to the right boundary. A constant y-velocity is imposed on the upper boundary, and the displacement control loading will result in a compression stress field throughout the specimen. The material density is 2500 kg/ m3 and gravity is not applied.

Fig. 10. Model mesh and boundary conditions.

Vy

Sym

met

ric a

xis

σ3

y

x 0

The dilation behaviors based on the Mohr-Coulomb model are presented in Fig. 11. The elastic parameters are given as E =1.7 GPa and v = 0.28, and the strength parameters are c = 3.56 MPa and φ = 46.9°. Clearly, under the same confinement condition for σ3 = 1 MPa, the initial slope of volumetric-axial strain relationship up to the onset of failure for all curves are the same, but the post-failure responses depend on the assigned dilation angle values. Once plastic yield happens, the volumetric strain increases linearly with the increasing dilation angle, and the rate of volumetric strain is higher for larger dilation angles. When a zero dilation angle is given, the volumetric strain remains constant after yielding, which means that there is no volumetric dilation beyond yielding. In Fig. 11, the volumetric strain-axial strain relations are also shown for different confining stresses with a dilation angle of ψ = 20°. A higher confining stress causes the onset of dilation to be delayed due to occurring of more elastic contraction, and the volumetric strain decreases while the rate of dilation remain constant with increasing confinement.

A strain-softening model, with peak and residual c and φ estimated from reference [42] and defined in Fig. 12, is used in the simulation. For simplicity, the plastic shear strain for both c and φ to reach their residual values is chosen as psε = 0.028. This selection of material parameter can present a typical strain-softening behavior, and constant dilation angles are assumed in all cases.

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

0.83 (Residual)

3.56 (Peak)

c (M

Pa)

εps (%)

35.44o (Residual)

46.88o (Peak)

φ (o )

εps (%)

Fig. 12. Illustration of the c and φ as a bilinear decay function of plastic shear strain.

Fig. 13 illustrates the dilation behaviors for the strain-softening models. In Fig. 13, the pre-peak linear elastic compression is the same as that by perfectly plastic model because the same deformation parameters are used. However, due to strain softening from peak to residual strengths, bilinear dilation behaviors are exhibited. The most obvious characteristic is that the rate of dilation at the strain softening stage is higher than that at the residual deformation stage, even though constant dilation angles are assumed. Hence, strain-softening of the material can contribute to more volumetric dilation than perfectly plastic models when the same axial compressive deformation is experienced. The increase of constant dilation angle can enhance the rate of dilation, which is similar to the response of the perfectly plastic model. With the increase of confining stress, the onset of dilation is postponed, i.e., more compressive deformation has to be experienced before yielding; meanwhile, the magnitude of dilation is reduced. One thing to be noted is that the confining stress has no effect on the rate of dilation at both the softening and the residual deformation stages.

0 10 20 30 40 50 6020

10

0

-10

-20

-30

-40

-50

-60 σ3=1 MPa

ψ = 20o

5 MPa4 MPa3 MPa

2 MPa

1 MPa

0 MPa

10o

15o

20o

25o

0oVol

umet

ric st

rain

(mill

istra

in)

Axial strain (millistrain)

5o

Fig. 13. Volumetric-axial strain curves of 146 mm diameter rock samples with different constant dilation angle under constant confining stress (a), subjected to different confining stresses with constant dilation angle (b).

0 10 20 30 40 50 6020

15

10

5

0

-5

-10

-15

-20

-25

-30

-35

-40 σ3=1 MPa

ψ = 20o

5 MPa

4 MPa

3 MPa

2 MPa

1 MPa

0 MPa

10o

15o

20o

25o

0oVol

umet

ric st

rain

(mill

istra

in)

Axial strain (millistrain)

5o

Fig. 11. Volumetric-axial strain curves of 146 mm rock samples with different constant dilation angles under 1 MPa confining stress, and subjected to different confining stresses with a dilation angle of 20°.

The mobilized dilation angle model for coal is utilized here to illustrate the nonlinear volumetric response and the dependence of dilation on confining stress and plastic shear strain in the strain-softening model, in which cohesion and friction angle follow bilinear softening laws. As shown in Fig. 14 (a), there is a remarkable difference in terms of rate of dilation change with increasing confining stress and plastic shear strain when compared with the results for the constant dilation angle model shown in Fig. 13. At low confining pressure such as 13 ≤σ MPa, the material demonstrates relatively higher volumetric dilation. With the increase of confining stress, the dilation decreases drastically. The magnitude of dilation strongly depends on confining stress, but the rate of dilation is more sensitive to the characteristic plastic shear strain ( psε ) than to the confining stress (see Fig. 14 (b)). The gradual decrease of dilation rate with increased deformation, which is seen in the laboratory test data, can only be captured by the proposed dilation angle model. Constant dilation angle values based on perfectly elasto-plastic model and strain-softening model can only predict linear increase of volumetric strain which is physically incorrect.

In order to further verify the proposed dilation angle model, a series of numerical experiments are performed on Moura coal and Witwatersrand quartzite samples. According to Hoek-Brown peak and residual strength parameters estimated from triaxial compression tests conducted by Medhurst [42] and Crouch [26], the equivalent Mohr-Coulomb peak and residual strength parameters (cohesion and friction angle) under different confining stresses were calculated. Young’s modulus and Poisson’s ratio were obtained based on test data and all the input parameters are listed in Table 2. The size and boundary conditions of the numerical model are the same as shown in Fig. 10. For simplicity, cohesion and friction angle bilinear functions using same characteristic plastic shear strains were adopted to fit the test results. The simulation results together with the original test results are presented in Fig. 15.

Table 2. Parameters in the Mohr-Coulomb strain-softening model based on test data of coal and quartzite

Rock type σ3 (MPa)

E (GPa) v cp

(MPa) φp

(o) cr

(MPa) φr

(o) εps

Coal 0.2 1.30 0.32 2.02 61.72 0.12 58.13 0.030

1 1.52 0.28 2.14 57.69 0.33 46.59 0.0323 1.91 0.36 2.60 52.09 0.72 37.73 0.0334 2.06 0.36 2.83 50.24 0.87 35.37 0.033

Quartzite 0.345 65.93 0.17 17.27 71.78 0.39 71.04 0.011

10.34 67.14 0.24 20.25 66.00 3.95 51.19 0.01713.79 65.39 0.24 22.35 64.61 4.85 48.86 0.02134.48 68.09 0.20 29.58 60.03 9.39 41.70 0.029

0 10 20 30 40 5020

15

10

5

0

-5

-10

-15

-20

-25

-30

-35

-40

εps=0.028

σ3=5MPa

σ3=4MPa

σ3=3MPa

σ3=2MPa

σ3=1MPa

σ3=0MPa

Vol

umet

ric st

rain

(mill

istra

in)

Axial strain (millistrain) (a)

0 10 20 30 40 50 6010

8

6

4

2

0

-2

-4

-6

-8

-10

-12

-14

-16

σ3=1MPa

εps=0.021εps=0.028εps=0.056εps=0.112

Vol

umet

ric st

rain

(mill

istra

in)

Axial strain (millistrain) (b)

Fig. 14. Volumetric-axial strain curves of rock samples with mobilized dilation angle model under different confining stress (a), and subjected to variable plastic parameters under constant confining stress σ3=1 MPa based on Mohr-Coulomb strain-softening model in FLAC (b).

As shown in Fig. 15 (a-b), a good agreement between numerical and experimental results for coal and quartzite from low to high confining stress indicates that the mobilized dilation angle model can capture the nonlinear stress-strain and dilation behavior satisfactorily. The failure and dilation behavior for coal and quartzite are similar and show strongly confining stress dependent characteristics. Compared to coal, quartzite shows more dilation, especially at high confining stresses. The dilation increase rates of quartzite are also higher than that of coal. On the other hand, the behavior of coal is more sensitive to confinement.

0 5 10 15 20 25 30 35 40 4515

10

5

0

-5

-10

-15

-20

-25

-30

-35

-40

Model

Test

σ3=4MPa

σ3=3MPa

σ3=1MPa

Vol

umet

ric st

rain

(mill

istra

in)

Axial strain (millistrain)

σ3=0.2MPa

(a)

0 2 4 6 8 10 12 14 1610

5

0

-5

-10

-15

-20

-25

-30

-35

-40

-45

-50

-55

Model

Test

σ3=34.48MPa

σ3=13.79MPa

σ3=10.34MPa

Vol

umet

ric st

rain

(mill

istra

in)

Axial strain (millistrain)

σ3=0.345MPa

(b)

Fig. 15. Actual test and simulation results of volumetric-axial strain of Moura coal (a) and Witwatersrand quartzite samples (b) subjected to different stresses.

3. DILATION ANGLE FOR ROCK MASSES

3.1. Peak friction angle of rock masses Intact rock strength and deformation properties determined from laboratory tests are seldom applicable to field conditions due to the effect of joints contained in rock masses. Various strength criteria have been developed in the past to describe the behavior of rock masses under different stress state. Of these, the linear Mohr-Coulomb failure criterion and the generalized Hoek-Brown failure criterion are widely used in rock engineering. For a jointed rock, the nonlinear Hoek-Brown criterion would be more suitable than the Mohr-Coulomb criterion [43]. The Hoek-Brown strength parameters can be estimated based on the GSI system, which provides a method for rock mass peak strength

estimation. The latest version of the generalized Hoek-Brown criterion for jointed rock masses is defined by [44]:

31 3 ( )a

ci bci

m sσσ σ σσ

= + + (19)

where 3σ and ciσ are the minimum principal stress and the uniaxial compressive strength (UCS) of the intact rock, respectively. bm , s and a are rock mass strength parameters which depend upon the characteristics of the rock mass. They are determined using the GSI index and

im value as shown in [44]. Further quantification of the GSI system using block volume and joint surface condition factor for the estimation of the peak strength and residual strength parameters are developed by Cai et al. [45-46]. A simple method to estimate the im value from uniaxial compression test is proposed recently by Cai [47].

The equivalent Mohr-Coulomb parameters, rock mass cohesion ( c ) and friction angle (φ ), can be obtained based on the Hoek-Brown envelope and a chosen range of 3σ . In the 1σ - 3σ space, the Mohr-Coulomb failure criterion is expressed as

1 32 cos 1 sin1 sin 1 sin

c φ φσ σφ φ

+= +

− − (20)

where 2 cos1 sin

c φφ−

is the unconfined compressive strength

of the rock mass and 1 sin1 sin

φφ

+−

is the slope of the failure

envelope.

For deep tunnels, Hoek et al. [44] suggest the maximum confining level ( max3σ ) from the following equation

0.94

3max 0.47 cm

cm Hσ σσ γ

−⎛ ⎞

= ⎜ ⎟⎝ ⎠

(21)

where cmσ is the rock mass strength, γ is the unit of weight of the rock mass, and H is the depth of the tunnel below surface. When the horizontal stress is higher than the vertical stress, the horizontal stress value should be used in place of Hγ in Eq. (21) [44].

As an example, for a hard rock mass with UCS of 100 MPa, a set of curves for the equivalent φ following the variation of GSI and mi values are presented in Fig. 16.

3.2. Dilation angle for rock masses When the plastic potential surface is the same as the yield surface, the plastic flow rule is called the associated flow (or normality) rule, which is characterized by the convex of the yield surface and the normality of the plastic shear strain increment vector to the surface. The concepts of associated plastic flow were developed for perfectly plastic and strain-hardening materials such as metals. For materials that follow associated flow rule, we have ψφ = . However, for most geo-materials, the plastic flow (given by the plastic potential) does not comply with the normality rule. Early attempts at modeling the yield and flow of soils, by applying associated flow rule, revealed that calculated volumetric strains were much larger than those observed from experiments [48-49]. Furthermore, from a theoretical point of view, there is no energy dissipation for φψ ≥ when plastic deformation occurs in geo-materials. In reality, the dissipated energy should be non-negative for possible stress cycles of loading and reloading, otherwise the material would produce energy [26]. Hence, non-associated flow rules (i.e. the plastic potential and yield surface are not identical) with φψ < have been widely supported and adopted to describe cohesive frictional geo-materials in laboratory tests and engineering practices [50-54].

It is a very challenging and difficult task to investigate the volumetric variation characteristics of rock masses under different confining pressures in the field. However, based on the conclusion of φψ < and peakpeak φψ ≈ at the null confinement from theoretical analysis and

experimental observations, it is feasible to estimate the peak dilation angle from the peak friction angle of rock mass determined by the GSI system. In numerical tools such as FLAC and FLAC3D, even though they do not prevent the user from prescribing a dilation angle greater than the friction angle, a recommendation that the dilation angle should be less than the friction angle during modeling is put forward to make the analysis stable and avoid energy to be generated by the model [11]. Hence, in the dilation angle model for rock mass, an assumption of instantaneous peakpeak φψ = , experiencing a given plastic shear strain at null confinement, can be made, and this coincides with the suggestions given by Alejano and Alonso [15].

As discussed in Section 2, the proposed dilation angle model is for intact rocks. For jointed rock masses, the dilation behavior may exhibit a different behavior but in this study we assume that the dilation behavior of jointed rock masses follow similar trend as observed for intact rocks so that the empirical relations established for intact

10 20 30 40 50 60 70 80 9010

15

20

25

30

35

40

45

50

55

60

65

70

30

mi

2520

35

161310

7

φ (o )

GSI

5

Fig. 16. Estimation of friction angles for GSI and mi values. Note that the confinement range assumed in obtaining the figure is from 0 to 5 MPa. Modified from Cai et al. [45].

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

10

20

30

40

50

60

70

80

Dilation angles of silty sandstone

Dilation angles of sandstone (strong)Friction angles of silty sandstone

Peak

fric

tion

and

dila

tion

angl

e (o )

Confining stress (MPa)

Friction angles of sandstone (strong)

(a)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

10

20

30

40

50

60

70

80

Dilation angles of sandstone (weak)

Dilation angles of coal

Friction angles of sandstone (weak)

Peak

fric

tion

and

dila

tion

angl

e (o )

Confining stress (MPa)

Friction angles of coal

(b)

Fig. 17. The relationships between peak dilation and friction angles for two rock types under different confinements. Medium-grained hard rock (a), and fine to medium-grained soft rock (b).

rocks can be applied to jointed rock masses. All we have to do is to estimate the peak friction angle of the rock mass using the method shown in Section 2.1 and take peakpeak φψ ≈ . Variation of ψ with confinement and plastic shear strain is governed by functions shown Fig. 9 for different rock types.

Fig. 17 shows the relationships between peak dilation angles obtained from the developed dilation angle model [8] and the friction angles calculated from the Hoek-Brown failure criterion for two rock types under different confining stresses. At low confinement levels ( 03 ≈σ ), the dilation angles of rocks are close to their friction angles. However, with increasing confining stress, this assumption is not valid anymore as a more rapid drop of the dilation angles can be seen. It is observed that the dilation angle is more dependent on confining stress than on friction angle in the low confinement range (0~3 MPa).

Parameter a in Fig. 9 plays a critical role in determining the magnitude of peak dilation angle under different

confinement conditions, i.e., the peak dilation angle decreases as a decreases. In order to keep the curves’ shape of the dilation angle model for both intact rocks and jointed rock masses, one can adjust the coefficient a to ensure peakpeak φψ σ ==0_ 3

. For example, if max3σ is within the range of 0 to 5 MPa, the rock mass peak friction angle of medium-grained soft sandstone [32] is 37.5°, which is estimated from the generalized Hoek-Brown strength parameters ( ciσ = 44 MPa, GSI = 50, and

im =10.5). Accordingly, a transformation of the dilation angle model from rock to rock mass based on the inserted model parameters in Fig. 18 can be made and the result is presented in the same figure. For jointed rock masses, no test data are available to define complete curves of dilation angle as a function of plastic shear strain and confinement. Here, in order to make the problem solvable, an important assumption has been made – the dilation behavior of a rock mass resembles that of intact rocks.

4. INFLUENCE OF ROCK DILATION ON DISPLACEMENT AROUND UNDERGROUND EXCAVATION

4.1. Tunnel model description A tunnel excavation model with a radial grid is made up of 32,400 quadrilateral elements. The cross section of the arched tunnel is 4.9 m in width and 5.95 m in height. The outer boundaries have been modeled at a distance of around six tunnel widths in both directions to minimize the boundary effect on the analysis results. The tunnel is assumed to be constructed using the full-face excavation method and the rock support system is ignored. The objective of the simulation is to investigate the influence of plastic shear strain and confinement dependent dilation on displacement distribution around the excavation. A close-up view of the model, and displacement measurement lines in the right sidewall, roof, and left arch shoulder of the tunnel are presented in Fig. 19.

4.2. Rock mass properties Two rock types, including medium-grained hard rock, fine-medium-grained soft rock, are used to demonstrate the potential applications of the proposed plastic shear strain and confinement dependent dilation angle model in predicting deformation distributions around tunnels. This dilation angle model is implemented in FLAC for use with the cohesion weakening and frictional strengthening (CWFS) model [55] for medium-grained hard rocks, and Mohr-Coulomb strain-softening model for the fine-medium-grained soft rock.

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

0MPa0.5MPa1MPa2MPa3MPa4MPa5MPa

Dila

tion

angl

e (o )

Plastic shear strain (%)

a b ca1=20.93 b1=0.99 c1=0.37a2=35.28 b2=44.39 c2=3.54%a3=2.34 b3=0.73 c3=0.47

Coefficients of dilation angle model formedium-grained soft sandstone:

(a)Confinement:

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

Dila

tion

angl

e (o )

Plastic shear strain (%)

a b ca1=10 b1=0.99 c1=0.37a2=29 b2=44.39 c2=3.54%a3=2.34 b3=0.73 c3=0.47

Coefficients of dilation angle model formedium-grained soft sandstone:

(b)0MPa0.5MPa1MPa2MPa3MPa4MPa5MPa

Confinement:

Fig. 18. The transformation of dilation angle model from intact rock (a) to rock mass (b) for medium-grained soft sandstone.

In the CWFS model, the initial friction angle ( iφ ) is assumed to be zero, which indicates that the strength of hard rocks are entirely cohesive before the onset of plastic straining. The cohesion loss envelope is set to be slightly above the crack initiation stress envelope in the principal stress space, as suggested by Hajiabdolmajid [55] and Diederichs [24]. Hence, if the crack initiation stress level for hard rocks is assumed to be 1/3σci, the initial cohesions (ci) can be 20 MPa for medium-grained hard rocks in our analysis. Once spalling occurs, the strength of the broken rocks is mainly frictional and hence the residual cohesion cr is set to a relative small value for εp

c ≥ 0.2%, reflecting the small strain nature of the problem [44], and residual friction angle rφ is set to 52° with εp

f ≥ 0.5% (see Table 3). In the strain-softening model, σci, mi, Young’s modulus E and Poisson’s ratio v of the rocks were estimated using triaxial compression test data [32, 42]. Equivalent peak and residual Mohr-Coulomb strength parameters (cohesion and friction angle) of the corresponding rock masses are obtained using the GSI system. The residual strength parameters are obtained using a residual GSI value of 30 [46]. The equivalent Mohr-Coulomb strength parameters are listed in Table 4. Table 3. Input parameters in CWFS model for medium-grained hard rocks

E (GPa) v

Initial strength criterion

Residual strength criterion Plastic strain

cp (MPa) φp (o) cr (MPa) φr (o) εpc εp

f 21 0.22 20 0 1 52 0.002 0.005

Table 4. Equivalent parameters in Mohr-Coulomb strain-softening model for fine-medium-grained soft rock

GSI σci (MPa) mi

E (GPa) v

Initial strength criterion

Residual strength criterion

cp (MPa) φp (o) cr (MPa) φr (o)70 50 13 7 0.25 3.15 41.13 1.53 29.47

4.3. In situ stress The in situ stress magnitudes initialized in the elements of each model depend on two parameters, vertical stress and horizontal stress to vertical stress ratio (K0). The vertical stress is assumed to be induced by the weight of the rock mass overlying the excavation with a density of 2500 kg/m3 in all models. The intermediate principal stress parallel to the opening axis is assumed equal to the minimum principal stress in the plane strain analysis. In order to capture rock failure and dilation occurring on the sidewall of the tunnel, vertical stress as the maximum principal stress and horizontal stress as minimum principal stress are considered. On the contrary, if K0 > 1, the failure of rock around the excavation often occurs at the floor and roof.

4.4. Influence of rock mass dilation on the displacement near excavation boundary

4.4.1 Medium-grained hard rock mass For medium-grained hard rock, such as sandstone and silty sandstone [31], a dilation angle model is presented in Fig. 20 (a). The maximum and minimum in-situ stresses are 26 MPa (roughly 1000 m deep) and 13 MPa, respectively. The angle between the maximum principal stress direction and the vertical direction is 30°. This in-situ stress condition is used because we are focusing on rock failure and dilation on the arch shoulder and floor corner of the tunnel.

In Fig. 20 (b), the curve of displacement distribution from our variable dilation angle model, which is implemented into FLAC using the CWFS model approach, differs significantly from displacement distributions by the constant dilation angle model. It again illustrates that in the failure zone, when compared with the variable dilation angle model results, constant dilation angle models with small dilation angles ( 2/φψ < ) may underestimate the displacement close to the excavation boundary and overestimate the displacement away from the opening when a large dilation angle is used. When a large constant dilation angle is used, it overestimates the displacement in all depths significantly. People with numerical modeling experience know intuitively that dilation angle is a parameter that can be adjusted to get the desired displacement amount near the tunnel surface. The problem with a constant dilation angle model is that when the displacement at the tunnel surface is right, the displacement inside the rock masses may not be correct.

The rapid displacement increase captured by our mobilized dilation angle model reflects the influence of both plastic shear strain and confinement on rock dilation. This type of rapid displacement change is observed in some field monitoring data and it is seen that constant dilation angle models cannot capture this kind of behavior satisfactorily (see Fig. 21). The distribution of rock dilation angles around the excavation boundary

Fig. 19. A zoomed-in arched tunnel grid showing the displacement monitoring lines in the sidewall, roof, and left arch shoulder.

-5.000

-3.000

-1.000

1.000

3.000

5.000

-5.000 -3.000 -1.000 1.000 3.000 5.000

y

x

is shown in Fig. 20 (c). It is seen that the confinement and plastic deformation play a critical role in determining the magnitude and distribution of dilation angles around the excavation boundary.

4.4.2 Fine to medium-grained soft rock mass Fig. 22 (a) shows the plastic shear strain and confinement dependent dilation angle model for fine to medium-grained soft rocks. In this rock group, strain-softening behavior dominates [32, 42]. With increasing confining stresses, rock samples present gradual transition from splitting to shear failure, which leads to a right shift of peak dilation angles following the plastic shear strains, i.e., more plastic shear strains need to be accumulated to reach the maximum volumetric deformation under high confinement (see Fig. 22 (a)). In this case, the vertical stress is assumed to be 20 MPa (roughly 800 m deep) and K0 = 0.5.

Fig. 20. (a) The mobilized dilation angle model for the medium-grained hard rock (left top figure); (b) Influence of medium-grained hard rock dilation on displacement near the excavation (left lower figure); (c) The distribution of dilation angle (in degree) in the rock mass.

In the strain-softening model, a large characteristic plastic shear strain of 2.5% is assumed and it presents a strength softening process which is characterized by the rock mass approaching its residual strength. In Fig. 22 (b), different constitutive models including elastic model, perfectly elasto-plastic Mohr-Coulomb model, and the strain-softening model are used to investigate the displacement response in the tunnel sidewall. The stain-softening model results in the maximum displacement

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

40

45

50

55

10MPa

3MPa

2MPa

1MPa

5MPa

σ3=0MPa

6MPaDila

tion

angl

e (o )

Plastic shear strain (%)

4MPa

a b ca1=10 b1=4.06 c1=0.08a2=43.15 b2=15.56 c2=0.004a3=3.5 b3=5.54 c3=0.58

Coefficients of dilation angle model forhard medium-grained sandstone:

(a)

1.0 1.2 1.4 1.6 1.8 2.0 2.2-2

0

2

4

6

8

10

12

14

16

(b)

Constant dilation angles:

Mobilized dilation angle model

Dis

plac

emen

t (m

m)

r/r1

ψ=5φ/8ψ=φ/2ψ=φ/4ψ=φ/8ψ=0

Fig. 21. Comparison of measured roof subsidence and calculation results based on constant dilation angle model used in three underground powerhouses (Okuyoshino, Tanbara and Arimine in sequence) in Japan. Modified from [56].

0 5 100

10

20

30 Measurement Calculation

0 5 100

10

20

30

Relative subsidence of ceiling rock (mm)0 5 10 15

0

10

20

30

40

50

Mea

surin

g le

ngth

l (m

)

0.00E+00 8.00E+00 1.60E+01 2.40E+01 3.20E+01 4.00E+01

m_dil

30°

26 MPa

13 MPa

(c)

y

x

Displacement monitoring line

on the tunnel boundary, followed by the perfectly elasto-plastic model and the elastic model. Once plastic yield happens, the lateral displacement in the plastic region increases with increasing dilation angle (constant), and the gradient of displacement also shows a trend of increase for both perfectly elasto-plastic model and the strain-softening model. However, compared with the displacements obtained from the strain-softening model, the displacements obtained from the elastic and the perfectly elasto-plastic models are small in general. The predicted horizontal displacement along a line in the sidewall and the distribution of dilation angle around the excavation are depicted in Fig. 22 (b) and Fig. 22 (c), respectively. The displacement distribution obtained from the mobilized dilation angle model differs from the displacement distributions obtained from constant dilation angle models with ψ ranges from 0 to 3φ/8. Again, our model captures the rapid displacement increase near the excavation boundary.

Fig. 22. (a) The mobilized dilation angle model for the fine to medium-grained soft rock (left top figure); (b) Influence of rock dilation on displacement near the excavation (left lower figure); (c) The distribution of dilation angle around the excavation.

5. CONCLUSION By analyzing and summarizing the characteristics of rock dilation during deformation, a mobilized dilation angle model, which considers the influence of confining stress and plastic shear strain, is developed. According to numerical simulation using FLAC, a constant dilation angle using either perfectly plastic or strain-softening model produces unrealistic dilation behavior which cannot be supported by experimental data. On the other hand, using the proposed dilation angle model, and in combination with the Mohr-Coulomb strain-softening model, realistic post-failure dilation behavior of rocks can be captured. The simulation results for soft coal and hard quartzite samples are found to be in good agreement with experimental results, indicating that the model can represent the plastic shear strain and confinement dependent dilation behavior of rocks correctly.

Based on the proposed dilation angle model for intact rocks, the plastic shear strain and confinement dependent dilation angle model for different rock mass types is developed, and this model is used to illustrate the importance of considering variable dilation on simulating rock mass deformation around tunnels. It is seen from the simulation results that the mobilized dilation angle model can characterize rock dilation behaviors of two different rock mass types near the

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

40

45

0.5MPa

3MPa

2MPa

1MPa

5MPa

σ3=0MPa

Dila

tion

angl

e (o )

Plastic shear strain (%)

4MPa

a b ca1=5 b1=2 c1=0.25a2=37.55 b2=35 c2=0.1a3=1.5 b3=1 c3=0.65

Coefficients of dilation angle model forfine-medium-grained soft rock:

(a)

1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.62

0

-2

-4

-6

-8

-10

-12

-14

-16

-18

-20

-22

Strain-softening model

Elasticmodel

Perfectly elasto-plastic model

Model

X-d

ispl

acem

ent (

mm

)

r/r1

ψ=φ/2ψ=3φ/8ψ=φ/4ψ=φ/8ψ=0

(b)

0.00E+007.00E+001.40E+012.10E+012.80E+013.50E+01

m_dil

X-displacement monitoring line

y

x

(c)

excavation boundary reasonably, i.e., the generation of large deformation near the excavation boundary is attributed to the existence of low to zero confinements. The displacement decreases rapidly as confinement increases. The effective dilation angles in the model are not constant but are rather variables depending on the plastic shear strain experienced and confinement.

When applying the mobilized dilation angle model, one must be aware of the limitation that pre-peak nonlinear deformation behavior is not covered by the model because the stain-softening model in FLAC only presents linearly elastic behavior before peak load. However, compared with the post-peak deformation of rocks, the pre-peak volumetric variation is negligibly small. Hence, a simplified pre-peak linearly elastic behavior will not have a large influence on the overall dilation behavior of rocks, especially at the post-peak stage. Additional experimental data are needed to calibrate and fine tune the dilation angle model parameters for various rock types. In the next phase of our research, it is planned to use the developed dilation angle model and numerical tool to conduct a few case studies and analyze the influence of rock mass dilation on rock support during tunnel excavation.

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