SCIENCE CHINA Technological Sciences • RESEARCH PAPER •

February 2012 Vol.55 No.2: 555–567 doi: 10.1007/s11431-011-4654-z

A new constitutive law for the nonlinear normal deformation of rock joints under normal load RONG Guan1,2, HUANG Kai3, ZHOU ChuangBing1*, WANG XiaoJiang1 & PENG Jun1 1

State Key Laboratory of Water Resources and Hydropower Engineering Science (Wuhan University), Wuhan 430072, China; 2 Key Laboratory of Rock Mechanics in Hydraulic Structural Engineering (Wuhan University), Wuhan 430072, China; 3 Changjiang Institute of Survey, Planning, Design and Research, Wuhan 430010, China Received June 21, 2011; accepted October 31, 2011; published online December 28, 2011

In view of the deviation of the fitting results of the classical exponential model and the hyperbolic model (the BB model) from several experiment data during intermediate stress period, a new constitutive model for the nonlinear normal deformation of rock joints under normal monotonous load is established with flexibility-deformation method. First of all, basic laws of the deformation of joints under normal monotonous load are discussed, based on which three basic conditions which the complete constitutive equation for rock joints under normal load should meet are put forward. The analysis of the modified normal constitutive model on stress-deformation curve shows that the general exponential model and the improved hyperbolic model are not complete in math theory. Flexibility-deformation monotone decreasing curve lying between flexibility-deformation curve of the classical exponential model and the BB model is chosen, which meets basic conditions of normal deformation mentioned before, then a new normal deformation constitutive model of rock joints containing three parameters is established. Two main forms of flexibility-deformation curve are analyzed and specific math formulas of the two forms are deduced. Then the range of the parameters in the g model and the g model and the correlative influence factor in geology are preliminarily discussed. Referring to different experiment data, the validating analysis of the g model and the g model shows that the g model can be applied to both the mated joints and unmated joints. Besides, experiment data can be better fit with the g model with respect to the BB model, the classical exponential model and the logarithm model. rock joints, normal deformation, constitutive model, flexibility Citation:

1

Rong G, Huang K, Zhou C B, et al. A new constitutive law for the nonlinear normal deformation of rock joints under normal load. Sci China Tech Sci, 2012, 55: 555567, doi: 10.1007/s11431-011-4654-z

Introduction

Normal deformation constitutive model of rock joints is the basic theory of jointed rock mass mechanics and rock mass hydraulics. Nonlinear deformation of rock joints under normal load directly affects contact states, opening states and connectivity of rock joints, and furthermore influences permeability, deformation and strength characteristics of rock mass. It is of great significance to know deformation *Corresponding author (email: [email protected]) © Science China Press and Springer-Verlag Berlin Heidelberg 2011

laws of rock joints for the research in high-steep slope stability evaluation, underground surrounding rock mass deformation analysis and fractured rock mass seepage. A lot of researches have been carried out by domestic and foreign scholars and the research methods can be mainly divided into four categories. (1) Macroscopic phenomenological method. Functions according to the laws and trends of experiment data are adopted to fit joint deformation curves after systematic analysis of rocks of different lithology. For example, Shehate chose empirical logarithm formula to present the relatech.scichina.com

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tionship between normal stress and closed deformation of joints [1]. Goodman brought forward the BB model [2, 3]. Bandis and Barton suggested that the relationship between normal stress and normal deformation of rock joints was nonlinear on the basis of numerous indoor experiment data of rocks of different lithology including limestone, siltstone, slate, sandstone etc. [4, 5]. For the mated joint, the hyperbolic constitutive model, that is, the famous BB joint model, was recommended, which represented normal deformation by use of initial stiffness in addition with the maximum deformation of joints and size effect was taken into consideration. With regard to the unmated joints, Bandis found that the logarithm formula fit best. Saeb and Amadei established an incremental constitutive model of discontinuities, which could simulate the effect of compression and shear [6]. This model was accepted as the generalized hyperbolic model. Malama and Kulatilake put forward the generalized exponential model on the basis of the classical exponential model [7]. Zhou, Yu and others proposed the improved elastic nonlinear model of joints, which was based on the exponential model [8, 9]. Supposing that unload and reload curves have asymptotic line of maximum compressive deformation, Yin expanded the normal deformation hyperbolic constitutive model, and established normal constitutive functions with new parameters which can be applicable under cyclic load [10]. (2) Elastoplastic mechanics method. Theoretical relation is set up between stress increment and deformation increment for discontinuities with elastoplastic mechanics theory, such as Plesha’s model and Jing’s model [11, 12]. The basic view of Plesha’s model is that rock joints are taken as parallel walls on macro scale and microstructure consisting of interlocking asperity surfaces. This model can be used to describe dilatancy, softening and cycling load and unload of rock joints. (3) Hertzian contact method. According to tribology, assuming that deformation of asperities of discontinuities under compressive load is linear-elastic, nonlinear characteristics of discontinuities under normal load can be reflected by the modification of the quantity and contact area of asperities. This kind of research was carried out by experts at home and abroad, including Swan, Sun Matsuki, Xia et al. [13–16]. The model brought forward by Swan was effective only in smaller stress range, but unsuitable in overall range. (4) Damage mechanics method. With the introduction of damage factor, the actual state of relevant physical quantities can be expressed by the combination of the intact state and completely disturbance state. One of the representative examples is DSC model put forward by Desai [17]. In the discontinuous deformation analysis methods such as Discrete Element Method, it is obligatory to use appropriate normal constitutive model of joints to reflect closure and opening deformation of discontinuities in fractured rock under complex load. At present, Coulomb sliding model, continuous yield model and the BB model are in common

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use in rock engineering research [18]. Coulomb model is a linear model with the fixed dilation angle. Exponential formula is used by continuous yield model and hyperbolic form by the formula of the BB model. The linear model is hard to illustrate deformation laws of joints under normal load. Now the main drawbacks of the present normal deformation constitutive model are that in the intermediate stress range, normal deformation simulated by the BB model is smaller compared to the experiment data, but normal deformation simulated by the classical exponential model is greater than the experiment data. In this paper, basic mathematical conditions which normal constitutive model should meet are analyzed, by taking normal deformation of rock joints as the research object in the macro phenomenological method. It is pointed out that the generalized exponential model slows down the speed of normal deformation in intermediate stress range to fit experiment data better, with the introduction of a new parameter based on the classical exponential model. In similar way, the improved hyperbolic model can be got by introducing a new parameter to accelerate the speed of normal deformation on the basis of the BB model. However, both the generalized exponential model and improved hyperbolic model are the improved models based on the classical exponential model or the BB model in the normal deformation-stress curve, in which fitting degree of experimental data needs to be raised and initial stiffness of stress-deformation curve irrationally trends to be infinite. Normal deformation relation in flexibility-deformation curve is studied and applicable function forms are chosen. A new normal deformation constitutive model which completely meets mathematical conditions is obtained. It is demonstrated that the new model can not only be applicable to joints of which normal stress-deformation curve accords with the hyperbolic model and the classical exponential model, but also can apply to joints of which normal stress-deformation relation complies with the logarithm model. That is to say, the new model can apply to both the mated joints and unmated joints. Besides, this proposed model fits experiment data better than the hyperbolic model, the classical exponential model, the logarithm model, and is valuable in rock deformation and hydro-mechanical coupling analysis.

2 Basic conditions for normal deformation model of joints In this paper, normal stress , normal deformation u and flexibility Cn are positive. Under normal compressive condition, the total deformation of jointed rock is the sum of deformation of the joints and deformation of the intact rock, meanwhile the flexibility of joints plus the flexibility of the intact rock is equal to the flexibility of jointed rock. As shown in Figure 1 [7], the flexibility of jointed rock and the

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flexibility of the intact rock are both equal to Ci when the normal stress increases to some extent. During this normal loading progress, the flexibility of joints gradually decreases from the initial flexibility Cni to zero, and normal accumulative deformation of joints is the maximal closure deformation Vm. When the joints under load begin to deform, the development of normal deformation is hindered by contact of the asperities of joints. The initial stiffness of joints is Kni, and the general progress of joints is shown in Figure 2. As normal stress gradually increases, normal contact area and stiffness of joints gradually increases, and closure deformation trends to be maximal normal deformation Vm little by little. Normal stress-deformation relation of joints is obviously nonlinear. Upon the analysis of the existing experiment progress and results, three basic conditions which constitutive model should meet are concluded. ① Normal closure deformation u gradually increases and closure deformation approaches to Vm, as normal stress increases. ② The flexibility Cn of joints gradually decreases from the initial flexibility as normal stress  increases, and the flexibility approaches to zero as closure deformation u approaches to Vm. ③ Normal closure

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deformation u is equal to zero while stress is equal to zero. All the constitutive models should meet conditions ① and ②, but condition ③ is not applicable for normal constitutive models considering stress history. The basic conditions can be expressed by five equations below: Cn |u  0  Cni  0,

(1)

Cn |u Vm  0,

(2)

Cn  0, u

(3)

u |σ   Vm ,

(4)

u |σ  0  0.

(5)

3 Discussion on the improved normal constitutive model of rock joints based on the stress-deformation method 3.1

Hyperbolic model and classical exponential model

Through the analysis of a large number of normal deformation experiment data of joints, Bandis and Barton et al. put forward the hyperbolic model of joints (known as the BB model) as shown below [4, 5]. u

σVm . σ  K niVm

(6)

The corresponding flexibility function is Cn  Cni (1  u / Vm ) 2 .

Figure 1 Normal deformation of joints and intact rock under compression.

(7)

The classical exponential model and the corresponding flexibility function proposed by Malama and Kulatilake are respectively shown as [7]   σ   u  Vm 1  exp   ,  K niVm   

(8)

Cn  Cni (1  u / Vm ).

(9)

The initial normal stiffness of joints Kni and the maximal closure deformation Vm can be calculated according to empirical equations by knowing the joint roughness coefficient JRC, the joint compressive strength JCS and the average width of the aperture aj. For one-dimensional problem, the initial normal flexibility of joints is the inverse of the initial normal stiffness, that is Cni=1/Kni. 3.2 Figure 2

Normal stress-deformation relationship of rock joints.

General exponential model

According to the experiment under normal monotonous load with rectangular joint specimens of Arizona diorite and

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granodiorite, Malama and Kulatilake found that the hyperbolic model and the classical model distinctly deviated from experiment data during intermediate stress period [7]. To be specific, the normal deformation of the hyperbolic model speeded up too slowly, while the classical exponential model too fast. At the same time, they put forward the concept of the half-closure stress (marked by 1/2). The half-closure stress is the value of normal stress when u reaches to Vm/2. For the BB model:  1/ 2  K niVm . As for the classical exponential model:  1/ 2  K niVm ln 2 . As long as Kni and Vm of the BB model and the classical exponential model are determined, the half-closure stress 1/2 and normal development speed will be determined. So the classical exponential model can be written as   σ  u  Vm 1  exp   ln 2   . σ1/ 2    

(10)

On the basis of the classical exponential model, Malama and Kulatilake put forward the general exponential model below by taking the power function form of stress [7]. n    σ    u  Vm 1  exp   ln 2    .   σ1/ 2    

(11)

The parameter n is the physical quantity reflecting surface morphological characteristics of joints. The range of the parameter n is between 0 and 1, not including 0. Compared with the classical exponential model, 1/2 is kept unchanged and the speed of deformation is slowed down by the general exponential model. From Figure 3 (n1
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the general exponential model will speed up more slowly than the BB model. As shown in Figure 3, the curve of the general exponential model with the parameter n1 does not lie between the BB model and the classical exponential model, and the fitting data deviate from the experiment data. According to eq. (11), the flexibility formula of the general exponential model is Cn 

ln 2  nVm  σ    σ1/ 2  σ1/ 2 

n   σ   exp   ln 2   .   σ1/ 2  

(12)

When normal stress is equal to zero, normal flexibility will tend to be infinite and the physical significance of the model is not explicit. 3.3

Improvement of the BB model

As for the general exponential model proposed by Malama and Kulatilake, stress is represented as power function form in the BB model  1/ 2  K niVm . By introducing a third parameter, the following improved hyperbolic model can be proposed: u

Vm σ m . σ1/ 2 m  σ m

(13)

The parameter m is in the range of 1 and +∞, and it also can reflect surface morphology characteristics of joints. Relative to the BB model, the improved hyperbolic model can remain 1/2 unchanged in the stage of middle stress, improving the BB model by increasing the growth rate of normal deformation along with normal stress. Figure 4 expresses the stress deformation relationship of the improved hyperbolic model and the BB model. For the improved hyperbolic model, m1
Figure 3 Normal deformation of joints in the BB model, the classical exponential model and the general exponential model.

n 1

mVm σ1/m 2 σ m 1 . (σ1/m 2  σ m ) 2

(14)

It can be obtained from the above formula that when  is equal to 0, Cn is close to infinity, which does not meet eq. (1). Whether it is the general exponential model or the improved hyperbolic model proposed by the authors, the essence is to base on a traditional model in u (stress –deformation) curve, by introducing a third parameter to change the speed of normal deformation of the traditional model, so that in the stage of middle stress new model curve

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Cnu relationship, the flexibilities of curves C1, C2, C3 are Cn1, Cn2, Cn3 respectively, and the flexibility is explicitly represented only by normal deformation u, as shown in eqs. (7) and (9). In u curve, when the normal deformation is v, the corresponding normal stress of the classical exponential model is 1, and the normal stress of the BB model is 2, and also the normal stress of the new proposed model is 3. Cn 2  Cn3  Cn1 ,  1 1  1 C  C  C , n3 n1  n2

 Figure 4 Normal deformation of joints in the BB model, the classical exponential model and the improved hyperbolic model.

which is between the classical exponential model and the BB model can fit the test data better. In the u curve, there is a large difference between the classical exponential model equation and the BB model equation, so when the third parameter changes in the domain, it cannot guarantee that the new model curve is always between the classical exponential model and the BB model. The two improved models have shortcomings that the physical meaning of initial flexibility is not clear when normal stress is zero. Then we try to take deformation as the independent variable, flexibility as the dependent variable in the relationship of normal deformation of joints, study the variation between Cn and u (flexibility deformation) in the normal deformation process, and propose a constitutive model in which normal stress deformation relationship fits experimental data of different types of joints.

4 Normal constitutive model of joints based on flexibility deformation method 4.1

v 0

v 1 v 1 1 u   u   u , 0 C 0 C Cn 2 n3 n1

(15)

(16)

1  u σ  Cn , C u  σ , n  σ  2 σ  σ3 σ  σ1 σ , 0 0   0

(17)

σ 2  σ 3  σ1 .

(18)

In Cnu relationship (shown in Figure 5), the curve will go through (0, Cni) and (Vm, 0), and it can meet eqs. (1) and (2) of basic conditions of normal deformation model of joints. In u relationship, there is an asymptote u=Vm in the BB model and the classical exponential model. The new proposed model is between the two models mentioned above, so the new model can meet basic condition of the asymptote u=Vm. As C3 is monotonically decreasing, normal deformation becomes greater and flexibility gets smaller. The condition meets eq. (3). As for eq. (5) of the basic conditions, integration path  = 0 can be set when calculating the integral, then the condition u = 0 is met. So a new constitutive model of normal deformation of joints can be received by integrating monotonically decreasing curve C3 which is between the two traditional models and goes through (0, Cni) and (Vm, 0).

Derivation of normal constitutive model of joints

According to eqs. (7) and (9), the Cnu curve which represents the relation between normal deformation of joints and flexibility is shown in Figure 5. The classical exponential model is the straight line C1 in which (0, Cni) is the starting point, and C1 and the u-axis intersect at (Vm, 0). The BB model is parabola C2 in which (0, Cni) is the starting point, and C2 and the u-axis are tangent to the (Vm, 0). It can be proved that if monotonically decreasing curve C3 of new model is between the BB model and the classical exponential model in Cnu relationship, the curve of the new model is between the BB model and the classical exponential model in u relationship. Then the concrete proving process is shown below. In the

Figure 5 Flexibility-deformation relationship in constitutive model of normal deformation of joints.

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Flexibility equation of curve C3 can be written as Cn  Cni (1  u / Vm ) g (u ).

(19)

In order to make sure that the curve C3 is always between C1 and C2 and goes through (0, Cni) and (Vm, 0), g(u) should meet the following conditions:  g (u ) |u  0  1,  1  u / Vm  g (u )  1.

(20)

According to the forms of flexible curve in the BB model and the classical exponential model, there are two simple integration types of g(u) which meet the above formula conditions. A: g (u )  1  δ  u / Vm ,   (0,1).

(21)

B: g (u )  (1  u / Vm ) λ ,   (0,1).

(22)

Next the constitutive models of normal deformation are deduced which are corresponding to g(u) of types A and B respectively. Subsequently, according to the different test data of normal deformation of joints, the two models are analyzed and compared. Finally the three-parameter constitutive model of normal deformation which has better applicability is recommended. The constitutive model of normal deformation which is corresponding to function g(u) of class A is called the g model. Specific flexibility equation is u  Cni (1  u / Vm )(1  δ  u / Vm ). σ

(23)

When  is equal to 0, C3 and C1 coincide; when  equals 1, C3 and C2 coincide; when  is between 0 and 1, the smaller  is, the more close C3 is to C1, and the greater  is, the more close C3 is to C2. According to eqs. (23) and (5), the following relationship can be received by integral method:



σ 0

σ    

σ

u 0 u 0

u Cni (1  u / Vm )(1  δu / Vm ) Vm u δCni (u  Vm )(u  Vm / δ )

Vm K ni δ 1



0

 1  1   u,  u  Vm u  Vm / δ 

Vm K ni [(ln | u  Vm |)u0  (ln | u  Vm / δ |)u0 ]. δ 1 σ

Vm K ni u  Vm ln . δ  1 δu  Vm

(24)

Through eq. (24), the following relation can be received:   σ (δ  1)   Vm  exp    1  Vm K ni    . u  σ (δ  1)  δ exp   1  Vm K ni 

Parameters Vm, Kni in eq. (25) are the maximal closure deformation and the initial normal stiffness of joints respectively. Parameter  of eq. (25) can be considered to be in association with joints weathering, roughness, fluctuation degree, the matching of joint surface and strength of the wall of rock joints. As  changes between 0 and 1, the g model can approach the BB model and the classical exponential model respectively. The constitutive model of normal deformation which is corresponding to g(u) of class B is called the g model. Specific flexibility equation is u  Cni (1  u / Vm ) λ 1 . σ

(26)

When  equals 0, C3 and C1 coincide; when  equals 1, C3 and C2 coincide; when  is between 0 and 1, the smaller  is, the more close C3 is to C1, and the greater  is, the more close C3 is to C2. Same as eq. (26), the following relationship can be received by integral method: K V σ  ni m λ

λ   u  1    1 .  Vm  

(27)

Regulating eq. (27), the following relation can be received: 1    λ u  Vm 1   λσ  1  . K V    ni m  

(28)

Similarly, parameter  can be considered to be in association with joints weathering, roughness, fluctuation degree, the matching of joint surface and strength of the wall of rock joints. As  changes between 0 and 1, the g model can approach the BB model and the classical exponential model respectively. 4.2 Discussion on the range of parameters  and  in the two constitutive models

2

u

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(25)

In the above section it has been proved that when  and  are between 0 and 1 in the g model and the g model based on flexibility deformation method, flexibility deformation curve is in the area between C1 and C2, and stress deformation curve is between the BB model and the classical exponential model. A brief discussion about the possibility of parameter  and  in other range and the change of corresponding flexibility curve is shown below. Based on the basic mathematical expression of the g model, the g model and the physical meaning of  and , these two parameters are considered to be greater than 0. When  is bigger than 1, the two intersections of the curve in the g model and abscissa are (Vm, 0) and (Vm/, 0), and the curve is shown as C5 in Figure 5. When the curve is in the area of u  (0, Vm ) , the discipline that flexi-

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bility monotonically decreases as deformation increases is not met (eq. (3)), so parameter  in the g model is in the range of 0 and 1. When  is equal to 1, the g model is the BB model. When  is bigger than 1, flexibility deformation curve of the g model is C4 in Figure 5. C4 meets the discipline that flexibility monotonically decreases as deformation increases. In addition, Cn 

1 u  σ K ni

 λσ   K V  1  ni m 

 (11/ λ )

.

(29)

When  and  are large, the relationship of flexibility Cn and 1/ is approximately linear. Bandis simulated the test data of the uncoupled joints of slate, limestone, sandstone, and siltstone by logarithmic model, and the relationship between normal stress and deformation was log σ  p  q  u.

(30)

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stage by Newton iteration method. δ



Vm K ni  u  Vm ln  σ  δu  Vm K niVm



   1. 

(32)

[(1  u / Vm )    1].

(33)

The results of parameters  and  are shown in Table 1 and the simulated curve is shown in Figure 6. The g model and the g model are consistent with experimental data with respect to the BB model and the classical exponential model. The mean residual (the average distance between fitting points and test points) of the BB model and test points is 0.0372 and the mean residual of the classical exponential model and test point is 0.0104. The mean residual of the g model and test points is 0.0081. For the test data in ref. [7], the g model fits best.

The relationship between flexibility Cn and 1/ is linear, and the flexibility equation is Cn 

log10 e 0.4343  . qσ qσ

(31)

Those p and q are the parameters of the logarithmic model. So it can be speculated that when  is greater than 1, the g model can be used to simulate normal deformation discipline of rock joints of which normal stress deformation law meets the logarithmic relationship. The verification is shown in Sections 4.3 and 4.4. Thus parameter  of the g model is in the range of 0 and +∞.

5 Validation of the constitutive model of normal deformation of joints 5.1 Validation and analysis of the model based on Malama test According to ref. [7], sample size of Arizona granodiorite Grd-1 is 93 mm×106 mm×93 mm, and the initial stiffness is 4.142 MPa mm1. The maximal closed normal deformation Vm is 0.505 mm. First, the experimental results are fit by the BB model and the classical exponential model. The results are shown in Figure 6. Under the middle stress level, the normal deformation of the BB model is less than test data, and the normal deformation of the classical exponential model is greater than it. Under the high stress level, the approach speed of normal deformation in the BB model to Vm is lower than that of the test data. The g model and the g model of normal deformation of joints are used to analyze the test results. According to the results, parameters  and  can be got by eqs. (32) and (33) respectively after calculating the corresponding value in intermediate stress

Figure 6 Experiment of joints in Arizona granodiorite [7] under compression and simulation.

Table 1

Experiment results and parameters of the model

 (MPa)

u (mm)





3.508

0.3774

0.380

0.300

4.007

0.3901

0.450

0.290

4.520

0.3995

0.520

0.400

5.089

0.4139

0.530

0.400

5.671

0.4239

0.560

0.400

6.170

0.4330

0.560

0.400

6.891

0.4330

0.660

0.500

7.543

0.4447

0.640

0.450

8.403

0.4560

0.630

0.440

9.193

0.4599

0.660

0.460

9.845

0.4646

0.660

0.450

10.718

0.4715

0.650

0.430

0.580

0.410

Average of , 

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5.2 Validation and analysis of the model in granodiorite test of the Swedish Nuclear Fuel and Waste Management Co. (SKB) According to ref. [19], SKB and The Swedish National Testing and Research Institute (SP) got core samples from the 230 m, 390 m, 700 m deep underground respectively in Fushi Mark KFM01A drilling on April 24, 2003. After making the sample in the SP laboratory, it was sent to Norwegian Geotechnical Institute (NGI) to do normal and tangential experiment of joints. After that the normal load test data of number F01-117-6 was simulated. The sample was got from about 230 m deep underground, which was fine-

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grained granodiorite, and the angle of the joint plane and hole axial was 0°–30°. When the normal load stress was 0.5 MPa, the normal strain sensor was adjusted to 0, and deformation curve was recorded when the normal stress varied from 0.5 MPa to 10.0 MPa. On the hypothesis that the deformation x happens under the load of 0.5 MPa, this normal deformation should be considered when normal stress deformation curve is simulated. For the experimental data after being treated, by use of Software Origin 8.5, the normal stress-deformation relationship is fit with the BB model, the classical exponential model, the g model and the g model respectively. Results are shown in Figure 7 and the parameters are in Table 2.

Figure 7 Fitting curves of the sample F01-117-6 [19] under compressive load using (a) fitting curve of the BB model; (b) fitting curve of classical exponential model; (c) fitting curve of the g model; (d) fitting curve of the g model. Table 2 Fitting parameters of joints in ref. [19] BB model Kni (MPa mm1) Vm (mm) x (mm) Average remaining squared residual Square of the coefficient of correlation

Classical exponential model 12.235 0.193 0.044 4.4×10-6 0.995

Kni (MPa mm1) Vm (mm) x (mm) average remaining squared residual square of the coefficient of correlation

12.370 0.192 0.044 0.987 4.5×106 0.995

Kni (MPa mm1) Vm (mm) x (mm)

g model Kni (MPa mm1) Vm (mm) x (mm)

 Average remaining squared residual Square of the coefficient of correlation

22.538 0.138 0.022 5.8×106 0.993

g model

 average remaining squared residual square of the coefficient of correlation

11.216 0.203 0.047 1.180 4.5×106 0.995

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The smaller the average remaining squared residual is, the greater the square of correlation coefficient will be, and the results are better. From Table 2, the simulated results of SKB granodiorite by the BB model and the classical exponential model are compared, and the BB model fits better. Kni and Vm in the g model are very close to the results of the BB model. The parameters  and  are close to 1, and the curves of the g model and the g model are close to the curve of the BB model too. It can be indicated that the test data in line with the BB model can also be analyzed by the g model and the g model. Also it can be inferred that when the parameters  and  are close to 0, the data in line with the classical exponential model will also meet the law of the g model and the g model. However, this law needs further verification by the appropriate test data or normal compression test of joints.

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the classical exponential model, the g model and the g model, normal stress deformation relationship of joints is simulated by Software Origin 8.5. Results are shown in Figure 9 and parameters are shown in Table 3. For test data in line with logarithmic relationship, the results of the BB model, the classical exponential model and the g model are not good. According to the stress deformation curve of IG25 joints in ref. [20], conclusion can be reached that Kni is about 4.36 MPa mm1 by the tangent slope of origin. When normal stress  is 25 MPa, normal

5.3 Validation and analysis of the model by Matsuki test According to ref. [20], when the normal load of joints is 57.6 KPa, wait until the test system is stable, and then load to 25 MPa at the speed of 0.13 MPa s1. Test curve IG25 of normal stress deformation is met with logarithmic relationship u  0.0163ln   0.0603 (Figure 8). For the BB model,

Figure 8 Test curve of normal stress deformation and the logarithmic curve [20].

Figure 9 Fitting curves of the sample IG25 [20] under compressive load using (a) fitting curve of the BB model; (b) fitting curve of the classical exponential model; (c) fitting curve of the g model; (d) fitting curve of the g model.

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Table 3 Fitting parameters of joints in ref. [20] BB model

Classical exponential model

Kni (MPa mm1)

10.160

Vm (mm)

0.112

Average remaining squared residual

2.0×10

Square of the coefficient of correlation

0.968

5

Kni (MPa mm1)

17.929

Vm (mm)

0.103

average remaining squared residual square of the coefficient of correlation

g model 1

7.0×105 0.882

g model 1

Kni (MPa mm )

10.103

Kni (MPa mm )

2.140

Vm (mm)

0.112

Vm (mm)

0.490



0.990

Average remaining squared residual

2.0×10

Square of the coefficient of correlation

0.968

 5

24.490

average remaining squared residual square of the coefficient of correlation

deformation u is 0.112 mm, and Vm should be greater than 0.112 mm by analysis. It is shown in Table 3 that Vm which is fitted by the g model is 0.490 mm and it is less than 0.112 mm when fitted by the other three models. So relative to the other three models the g model can meet experimental data, and also parameters have their actual physical meaning. From the analysis in this section, for the normal deformation of joints test data which are in line with logarithmic relationship can be fit better by the g model rather than the BB model, the classical exponential model and the g model. 5.4 Validation and analysis of the model based on the test data of Bandis [4] There are 6 groups of uncoupled joint test data which are fit

2.7×107 0.999

by logarithmic formula in ref. [4]. Two groups of slate, two groups of limestone, one group of sandstone, and one group of siltstone are included. Bandis found that in the low stress range there was a great deviation between test data and fitting results based on logarithmic relationship. Then 6 groups of test data were simulated by the g model and results are shown in Figure 10 and Table 4. According to the results, in the whole range of stress, the g model is in good agreement with test data. The g model applies to simulation of the normal deformation of uncoupled joint. Because plastic deformation of uncoupled joint under normal load is greater than plastic deformation of coupled joint, accordingly the parameter  is large. So it can be inferred that parameter  can be expressed by plastic work or plastic deformation of rock joints, but the specific relation needs further research. According to the analysis of test data in this section and

Table 4 Fitting parameters of uncoupled joints in ref. [4] Model and parameters

No.1 slate joints

No.2 slate joints

p

0.626

0.571

q

23.349

16.329

Kni (MPa mm1)

18.793

17.381

Logarithmic model g model

Vm (mm)

0.440

0.316



17.475

6.777

No.10 limestone joints

No.9 limestone joints

p

0.606

0.606

q

11.158

8.872

Kni (MPa mm-1)

11.942

6.630

Vm (mm)

0.379

1.000



5.036

15.000

Model and the Parameters Logarithmic model g model Model and parameters Logarithmic model

No.2 sandstone joints

No.2 mudstone joints

p

0.758

0.731

q

6.987

7.081

Kni (MPa mm )

6.281

4.014

Vm (mm)

0.482

0.789



4.190

8.572

-1

g model

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Figure 10 Fitting curves of uncoupled joints in the g model ((a) No.1 slate joints; (b) No.2 slate joints; (c) No.10 limestone joints; (d) No.9 limestone joints; (e) No.2 sandstone joints; (f) No.2 mudstone joints under compressive load in ref. [4].

the above section, it can be preliminarily verified that the g model can be more useful for normal deformation of joints of which stress deformation relation meets logarithmic variation. Through the validation and analysis for normal deformation by four different test data, the normal constitutive g model of joints not only suits for normal deformation of coupled joint, but also suits for normal deformation of uncoupled joint. At the same time, the g model can better fit test result than the BB model, the classical exponential model and the logarithmic model. So the g model based on flexibility-deformation method and complete math conditions is a better new nonlinear normal deformation constitutive model of rock

joints and specific stress deformation relationship is shown in eq. (28).

6 Preliminary discussion on the basic relation of parameters in g model According to the derivation of the model and the verification based on the test results of normal deformation of rock joints in the above section, it can be inferred that the g model is applicable for the normal deformation of the mated joints and unmated joints. There are three parameters in this model: Vm, Kni, . The first two parameters are the maximal closure deformation and the initial normal stiffness respec-

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tively. These two parameters have specific physical significance and can be easily obtained from test. In general, parameter  can be considered to be in association with joints weathering, roughness, fluctuation degree, the matching of joint surface and strength of the wall of rock joints. Parameter  is large in the analysis of several uncoupled joints in Section 5.4, so it can be inferred that parameter  can be expressed by plastic deformation of rock joints. In order to reflect the relationship between Vm, Kni,  in the g model and JCS, JRC, E (E represents the jointed rock mass modulus of elasticity), statistic analysis of basic parameters of different joints in limestone, siltstone, slate, sandstone in ref. [4] has been implemented. Vm, Kni, JCS, JRC and E can be obtained from ref. [4] and parameter  from Table 4. The results are shown in Table 5. It can be inferred from the table that Kni, JCS of joints are positively correlated with the modulus of rock, Vm is negatively correlated with Kni, JCS and E, and parameter  is positively correlated with Kni, JCS and E. For the results of ref. [4], the relation between JRC of joints and Vm, Kni, JCS,  is not explicit. In order to reflect the influence of parameter  on the basic law of the normal deformation of joints, supposing Vm, Kni in the g model are 0.505 mm, 4.142 MPa respectively, when parameter  is 0.2, 0.4, 0.6 and 1.0 respectively, the variation of normal deformation of joints in the g model can be seen in Figure 11. From Figure 11, it can be indiTable 5

Statistic of parameters of four different rock joints in ref. [4]

Kni Type of rock joints (MPa mm1) Sandstone 8.400

Vm (mm) 0.293

JCS (MPa) 53.0

7.9

E (GPa) 24.0

Siltstone

13.40

0.320

80.0

5.9

28.5

8.570

Limestone

19.10

0.193

103.0

9.8

49.0

10.000

Slate

20.30

0.159

125.0

5.4

66.0

12.150

JRC

 4.190

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cated that the bigger the  is, the slower the speed of normal deformation approaching to Vm is. It can also be indicated that  is positively correlated with Kni, JCS.

7

(1) The basic conditions for stress-deformation relationship of joints under normal monotonous load are analyzed, and basic mathematical conditions which are indispensable are summarized. (2) In u curve, the improved hyperbolic model with the new parameter m fits experiment data of normal deformation better. But the general exponential model and the improved hyperbolic model based on the stress-deformation relationship are not mathematically complete. (3) Based on the flexibility-deformation method, the monotonously decreasing curves lying between the BB model and the classical exponential model is chosen through the point (0, Cni) and the point (Vm, 0) in Cnu relationship. (4) Through verification and analysis of the g model and the g model it can be indicated that the g model is applicable for the normal deformation of the mated joints and unmated joints. Simultaneously, the g model fits the experiment data more rationally than the BB model, the exponential model and the logarithmic model. (5) The g model is a better constitutive model of nonlinear normal deformation, but specific physical significance of the parameter  and the values of  for different types of rock demand further research. This work was supported by the National Natural Science Foundation of China (Grant Nos. 50879063 and 50979081) and the National Basic Research Program of China (“973” Program) (Grant No. 2011CB013501). 1

2

3 4

5

6

7

Figure 11

Variation of normal deformation of joints in the g model.

Conclusions

8

Shehata W M. In fundamental considerations on the hydraulic characteristics of joints in rock. In: Proceedings of the Symposium on Percolation Through Fissured Rock, paper no. T1-F, Stuttgart, 1972 Goodman R E. The mechanical properties of joints. In: Proceedings of the Third Congress on ISRM, Denver, vol. 1A, Washington, DC: National Academy of Sciences, 1974. 127–40 Goodman R E. Methods of Geological Engineering in Discontinuous rock. New York: West Publishing Co, 1976 Bandis S C, Lumsden A C, Barton N R. Fundamentals of rock fracture deformation. Int J Rock Mech Min Sci Geomech Abs, 1983, 20(6): 249–268 Barton N R, Bandis S C, Bakhtar K. Strength, deformation and conductivity coupling of rock joints. Int J Rock Mech Min Sci Geomech Abs, 1985, 22(3): 121–140 Saeb S, Amadei B. Modelling rock joints under shear and normal loading. Int J Rock Mech Min Sci Geomech Abs, 1992, 29(3): 267–278 Malama B, Kulatilake P H S W. Models for normal fracture deformation under compressive loading. Int J Rock Mech Min Sci, 2003, 40(6): 893–901 Zhou J M, Xu H F. Power function model of normal deformation for rock joints (in Chinese). J Rock Mech Eng, 2000, 19(supp): 853–855

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10

11

12 13

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Yu J, Zhao X B, Zhao W B, et al. Improved nonlinear elastic constitutive model for normal deformation of rock fractures (in Chinese). J Geotech Eng, 2008, 30(9): 1316–1321 Yin X J, Wang G L, Zhang C H. Study of constitutive model for rock interfaces and joints under cyclic shear loading (in Chinese). Eng Mech, 2005, 24(7): 1158–1163 Plesha M E. Constitutive models for rock discontinuities with dilantancy and surface degradation. Int J Num Anal Meth Geomech, 1987, (11): 345–362 Jing L, Stephansson O, Nordlund E. Study of rock joints under cyclic loading conditions. Rock Mech Rock Eng, 1993, 26(3): 215–232 Swan G. Determination of stiffness and other joint properties from roughness measurements. Rock Mech Rock Eng, 1983, (16): 19– 38 Sun Z, Gerrard C, Stephansson O. Rock joint compliance tests for compression and shear loads. Int J Rock Mech Min Sci Geomech Abs, 1985, 22(4): 197–213

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Matsuki K, Wang E Q, Sakaguchi K, et al. Time dependent closure of a fracture with rough surfaces under constant normal stress. Int J Rock Mech Min Sci, 2001, 38(5): 607–619 Xia C C, Yue Z Q, Tham L G, et al. Quantifying topography and closure deformation of rock joints. Int J Rock Mech Min Sci, 2003, 40(2): 197–220 Desai C S, Ma Y. Modeling of joints and interfaces using the disturbed-state concept. Int J Num Anal Meth Geomech, 1992, (16): 623–653 Itasca Consulting Group Inc. Universal Distinct Element Code User’s Guide. Minneapolis, USA: Itasca Consulting Group Inc., 2006 Jacobsson L. Forsmark Site Investigation-KFM01A–The normal stress and shear tests on joints. Svensk Kärnbränslehantering AB (SKB), 2004 Matsuki K, Wang E Q, Giwelli A A, et al. Estimation of closure of a fracture under normal stress based on aperture data. Int J Rock Mech Min Sci, 2008, 45(2): 194–209