- Email: [email protected]
Stress–strain relationship in elastic stage of fractured rock mass
Journal Pre-proof Stress–strain relationship in elastic stage of fractured rock mass
Faquan Wu, Yi Deng, Jie Wu, Bo Li, Peng Sha, Shenggong Guan, Kai Zhang, Keqiang He, Handong Liu, Shuhao Qiu PII:
S0013-7952(19)31191-3
DOI:
https://doi.org/10.1016/j.enggeo.2020.105498
Reference:
ENGEO 105498
To appear in:
Engineering Geology
Received date:
20 June 2019
Revised date:
14 January 2020
Accepted date:
16 January 2020
Please cite this article as: F. Wu, Y. Deng, J. Wu, et al., Stress–strain relationship in elastic stage of fractured rock mass, Engineering Geology (2020), https://doi.org/10.1016/ j.enggeo.2020.105498
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2020 Published by Elsevier.
Journal Pre-proof
Stress–Strain Relationship in Elastic Stage of Fractured Rock Mass Faquan Wua, Yi Denga,* [email protected], Jie Wub, Bo Lia, Peng Shaa, Shenggong Guana, Kai Zhanga, Keqiang Hec, Handong Liud, Shuhao Qiua
Qingdao University of Technology, Shandong, China, 273400
d
-p
c
Rock Innovation Technology Co. Ltd, Zhejiang, China, 312000
ro
b
of
Shaoxing University, Shaoxing, Zhejiang, China, 312000
North China University of Water resources and electric Power, Henan, China 450046
re
a
*
na
Abstract:
lP
Corresponding author.
Jo ur
Discontinuity is the main factor affecting the stress–strain relationship of rock mass. Based on the statistical parameters of the fracture network of rock mass and combining them with the strain energy theory from continuum and fracture mechanics, the elastic stress–strain relationship of fractured rock mass is proposed and its elastic modulus is derived. The stress–strain relationship can truly reflect the weakening and anisotropy of the mechanical properties of rock mass. On this basis, experimental and calculated stress–strain curves are compared, both of which exhibited the same slope. Additionally, monitoring data from a tunnel excavation is compared with the numerical simulation results, indicating that the displacement of both is basically consistent. Then, we compare three constitutive models with the statistical mechanics of rock mass (SMRM) model to highlight the advantages of the latter in terms of 1
Journal Pre-proof
reliability and adaptability. Finally, based on the results of a bearing plate test of a rock mass at Jindong Bridge, the anisotropy of the elastic modulus and its influencing factors are studied. The results show that the weakening and anisotropy of the elastic modulus of the rock mass are notable, and the modulus changes according to the inter-angle between the stress axis and the normal of the discontinuity, reaching the minimum at an angle of 57.5°. The modulus is positively related to the friction angle and the cohesion
of
of the plane and is inversely correlated to the average radius and density of the discontinuities. When the average radius of the discontinuities exceeds 3 m, the weakening of the elastic modulus of the rock mass
-p re
increases with the normal stress at the planes.
ro
tends to be stable. The stress environment also has an effect on the elastic modulus, which rapidly
na
1 Introduction
lP
Keywords: fractured rock mass; stress–strain relationship; elastic modulus
It is well known that approximately half of all infrastructure construction projects, e.g. slopes, railway
Jo ur
tunnels, highways, hydro-power facilities, mines, and deep engineering and civil engineering projects in mountain areas, are based on rock engineering(Liu and Tang, 1999). The prediction of the deformation and strength behaviour of rock masses and the artificial structures attached to them is a basic task in the study of rock mechanics, and the stress–strain relationship is a fundamental tool for describing the deformation of rock masses (Gerrard, 1982).
The most widely used models today for the calculation of rock deformation have been introduced from the field of continuum mechanics, such as elastics, elastoplasticity, etc., which are based on the hypothesis of homogeneity, continuity, and isotropy of the medium (Saeidi et al., 2014). However, real 2
Journal Pre-proof
rock masses vastly different from the ideal medium because they contain a large number of discontinuities that are randomly distributed in terms of location, size, and orientation. The mechanical effects of discontinuities are not accurately reflected in the aforementioned models(Bao et al., 2019).
Since the occurrence of certain disastrous events, such as the Vaiont reservoir landslide in Italy in the 1960s, which caused the deaths of more than 2000 people, engineering geologists and rock dynamicists
of
began to investigate the negative mechanical effects of the discontinuities in rock mass (Gu, 1979). A
ro
large number of tests have been carried out in laboratories and construction sites to determine the
-p
mechanical features of the rock planes. Near the end of the last century, scientists began to establish
re
models and study the global behaviour of rock mass (Fan, 1978; Main et al., 1993; Hoek et al., 2002; Bai
lP
and Wierzbicki, 2010; Han et al., 2011; Alejano and Bobet, 2012; Bidgoli et al., 2013)
Two methods have been employed for studying the mechanical behaviours of rock mass. One consists
na
of empirically weakening the mechanical properties and performing rock quality classification by
Jo ur
considering the degree of fragmentation, which only yields weak results (Wu, 1976; Hoek and Brown, 1980; Wei and Hudson, 1986; Zhou et al., 2002). However, this classification approach hardly reflects the structural and mechanical anisotropy of the rock mass, which leads to significant directional variability in its behaviour and frequently causes the unexpected collapse of engineering structures and huge economic losses.
The other method consists of trying to establish a perfect model based on objective data of geological phenomena and rigorous mechanical theory. Many people have tried to develop models by combining the mechanical responses of rocks and infinite discontinuities (Fossum, 1985; Yashinaka and Yamabe, 1988;
3
Journal Pre-proof
Hoek et al., 1992; Gehle and Kutter, 2003; Chen et al., 2012; Xu et al., 2013, Bao et al., 2020). Oda (1984 and 1986) made great effort to set up a fabric tensor to describe the geometric structure of rock mass and then establish a stress–strain model by employing fracture mechanics for the behaviours of joints. Kawamato et al. (1988) introduced the concept of the damage tensor to develop a 3D damage mechanical model for jointed rock mass. Zhou (2010) extended this model and made some modifications
of
to fit the compressive stress condition of the joints. Wu et al. (1994 and 2001) and Hu et al. (2011) proposed some stress–strain models for jointed rock mass based on the geometric probability of joint
-p
ro
systems, which was first proposed by Hudson and Priest (1983), and the strain energy of cracks.
re
All the above mentioned efforts have definitely promoted the advance of rock mechanics, but very few models have been widely accepted; either because the geometric model does not reflect the actual
lP
geological structure of rock mass or the hypothesis does not meet the actual mechanical principles, or
na
even because of the complexity of expression of available models.
Jo ur
The objective of this study is to propose a model based on the statistical mechanics of rock mass (SMRM) model, which was first proposed by the authors, that combines the theories of fracture and continuum mechanics, and the energy additivity principle.
2 Proposed Model 2.1 Hypothesis The proposed model for a unit of fractured rock mass will be established based on two basic hypotheses:
4
Journal Pre-proof
1. The rock mass is composed of an isotropic rock block that is cut by a discontinuity network, where the distribution of discontinuities obeys geological rules and geometric-probability constraints. The global strain energy of the rock-mass unit under a certain stress tensor corresponds to the sum of the two parts.
2. The deformation of any discontinuity obeys the rules of fracture mechanics, where the shear displacement is driven by the residual shear stress on the crack plane, i.e. the shear strength is
of
subtracted from the shear stress, and the shear strength is determined through parameters such as
ro
cohesion and roughness.
-p
2.2 Stresses on a discontinuity
re
A hexahedron unit of rock mass in Cartesian coordinate systems ( x1 , x2 , x3 ) , such as the one shown in
lP
Fig. 1, is analysed. It has three pairs of side faces parallel to three coordinate planes. A stress tensor ij acts on the unit as shown in the figure. A buried circular crack is present in the unit, and the normal of the
Jo ur
na
crack surface is described by the directional cosine n (n1 , n2 , n3 ) .
Fig. 1 Unit and stress conditions
5
Journal Pre-proof The normal stress and the shear stress t acting on the crack and their coordinate elements can be written as kj nk n j， i kj nk n j ni ,
(1)
t ti ti ， ti ij n j kj nk n j ni .
For a water pressure p in the crack, the stress changes to effective stress as e p .
of
Based on Coulomb strength theory, the shear strength of the crack surface is
s c tan ,
ro
(2)
-p
where c and tan are the cohesion and frictional coefficients of the crack, respectively. 0 ),
the shear strength
re
Considering that the crack will close when the normal stress is compressive (
open when the stress is tensional (
lP
s takes effect, and the fracture effect in the normal direction disappears. On the other hand, the crack will 0 );
thus, the shear strength will be absent and the fracture effects
na
both in the normal and shear directions will take place.
Jo ur
Here, the stress state function is defined as
1, 0 k k ( ) 0, 0
(3)
The residual shear stress on the plane after the shear strength s is deducted is 0， k 0, t s t s， k 0, t s t， k 1, s 0
We define the residual shear stress ratio as
6
(4)
Journal Pre-proof h
t
.
(5)
Because the shear deformation is driven by the residual shear stress , we will use h more often in our discussion. It can be seen that the value of h is affected by the stress. When the shear stress is less than the shear strength, = 0, and therefore h = 0. When the shear stress on the crack surface is greater than the
of
shear strength, h = /t (t s) / t 1 s / t 1 . When the normal stress on the crack surface is equal to the
-p
Therefore, the range of possible values of h is [0,1].
ro
tensile stress, then s = 0 and h = 1.
re
We can get
t (c tan ) , t
lP
h
(6)
na
where t is the shear stress on the crack surface and c are the shear strength parameters of the crack
Jo ur
surface, namely its cohesion and frictional angle, respectively.
2.3 Strain energy density
2.3.1 Strain energy due to a single crack
As derived from fracture mechanics, for a type-I disk-shaped crack with radius a (Fig. 2), the work
7
Journal Pre-proof
(a)
(b)
ro
of
Fig. 2. Crack and displacement. (a) Plane of the crack; (b) cross-section of the crack.
2E
k a 2 r 2 (0 ≤ r ≤a),
(7)
re
lP
vI
-p
done at the area element ds rd dr by a normal force dp k ds on the displacement is
2 where 8(1 ) and is Poisson’s ratio of the medium around the crack.
na
Jo ur
When the crack is displaced from I and becomes closed, the stress on the crack surface increases from zero to . Because the process is elastic, the work done to close the crack on the area element ds rd dr is
1 2 2 2 2 dWI vI dp k a r rd dr . 2 4E
(8)
By integrating over the entire crack area and considering the work done by the displacements of both the upper and lower walls, the strain energy stored in the surrounding medium due to the crack opening (or closing) can be obtained as U I 2 dWI
2E
k 2 2
a
0
8
2
0
3E
a 2 r 2 rdr d
k 2 2 a .
(9)
Journal Pre-proof
In the same way, we can obtain the strain energy caused by the shear stress and displacement by integrating in both the x and y directions. Thus, dTx cos rd dr, dTy sin rd dr , vx
2E
cos a 2 r 2， v y
2E
sin a 2 r 2 ,
(11)
(12)
of
1 1 2 2 2 dU II III vx dTx v y dTy a r d dr , 2 2 4E
(10)
ro
where x and y are the local coordinate axes of the crack and δ is the inter-angle between the shear stress
-p
and the x axis.
re
The work done by the shear stress is
2
and subscripts II and III indicate type-II and type-III cracks in terms of fracture
Jo ur
mechanics.
2
na
where
(13)
lP
1 1 2 3 , U II III 2 ( vx dTx vy dTy ) a 2 2 3E
Finally, the total strain energy of a crack due to normal and shear stresses is U c U I U II III
2 2 (k 2 )a3 . 3E
(14)
2.3.2 Strain energy density due to a crack network In the field of geology, the discontinuities in rock mass can be divided into several sets, each of which has a certain dominant dip direction and dip angle. Let there be m sets of discontinuities in a volume V of a rock-mass unit and Np be the number of the planes in the p-th set. Considering that the stresses ( and ) on the same set of planes are not significantly variable, the 9
Journal Pre-proof total strain energy caused by the crack network can be written as m Np
p1 q1
3E
U c U cpq
m Np
p1 q1
3E
2 2 2 3 (k )a
m
Np
p1
q1
2 2 2 3 (k ) a .
(15)
We have verified, based on the work of Gu (1979), that if the radius of the cracks follow a Np
3
p 1
negative-exponential distribution, then a 3 a V , where and a are the normal density and
of
average radius of a set of cracks, respectively.
-p
Uc m a (k 2 2 2 ) , V E p1
(16)
re
uc
ro
Thus, the strain energy due to the crack network can be written in the form of an energy density:
obtain
lP
where , are as defined above. Using (1) and through simple derivation via tensor algebra, we can
na
2 ij ni n j ns nt st
(17)
Jo ur
2 ij h2 ( it n j ns ni n j ns nt ) st
2.4 Stress–strain relationship of a rock-mass unit The strain energy density of a continuous rock mass can be taken from the theory of the mechanics of elasticity as 1 u0 ij C0ijst st . 2
(18)
Let the global strain energy density of a rock-mass unit be 1 u ij Cijst st . 2 10
(19)
Journal Pre-proof According to the principle of energy additivity, this global energy u equals the sum of u0 and uc , i.e.
u u0 uc .
(20)
By substituting (16), (17), and (18) into (19) and subtracting ij from both sides of the equation, we get the elastic stress–strain relationship of the fractured rock mass:
st
are the strain and stress tensors, respectively, and Cijst is the compliance tensor of
ro
where eij and
(21)
of
eij Cijst st (Coijst Ccijst ) st ,
through elastic theory as
1 v v ( is jt it js ) ij st , 2E E
lP
C0ijst
re
-p
the rock mass. C0ijst is the part of the tensor corresponding to a continuous rock, which can be obtained
(22)
na
where E and are the elastic modulus and Poisson’s ratio of the rock, respectively, and i j is the
Jo ur
Kronecker symbol ( i j = 1 for i = j and i j = 0 for i ≠ j ). Ccijst is the part of the tensor caused by the discontinuity network, which can be expressed as Ccijst
E
m
2 2 2 a[(k h )ni n j ns nt h it n j ns ] ,
(23)
p 1
where m is the number of crack sets, and a are the normal density and average radius of the crack sets, respectively, k is the normal stress state factor of the cracks, and h is the residual shear stress ratio of the structural surface.
2.5 Elastic modulus of a rock mass and modulus ratio The elastic modulus is an index derived from elasticity theory, which is based on the hypothesis of 11
Journal Pre-proof having a homogeneous, continuous, and isotropic medium. Although this is does not represent real cases, the ideas and methodology of elasticity theory can be used for defining the deformation properties of fractured rock mass, even if it is not an ideal elastic medium. Moreover, the concept of elastic modulus has been commonly used in engineering practice because it provides simplifications and clarifications in a mechanical sense. Here, we still define the elastic modulus for a rock mass in the same way as in elasticity theory.
of
However, this index will have much deeper mechanical and geological meanings.
ro
Through basic understanding of elasticity theory and the stress–strain model expressed in (21), we can
11 1 . e11 C01111 Cc1111
(24)
lP
re
Em
-p
define the elastic modulus Em for fractured rock mass in the 11 direction as
Substituting C01111 and Cc1111 from (22) and (23) into (24), the elastic modulus of the rock mass
na
becomes
Jo ur
Em
E m
,
(25)
1 a k h n h n p 1 2
2
4 1
2
2 1
where all the parameters in (25) have been previously defined. In the case of 2 3 , which is commonly used in engineering practice and scientific research, let
n1 cos and 1 n12 sin 2 ; considering that for a crack in a compressive state k 0 and when the crack is opened due to tensional normal stress k h 1 , then (25) can be transformed into the following form:
12
Journal Pre-proof 1 ， 0 m 2 2 2 1 ah sin cos Em p 1 1 E ， 0 m 4 2 2 1 p1 a cos sin cos
(26)
Here, Em / E is defined as the elastic modulus ratio, i.e. the ratio of the elastic modulus of the rock mass to that of an intact rock.
ro
of
3. Verification
-p
A series of laboratory tests, comparisons with field monitoring data, and numerical simulations were conducted to examine the stress–stain relationships described in Section 2 and the elastic modulus derived
re
from the proposed theoretical model.
na
3.1.1 Laboratory experiments
lP
3.1 Stress–strain relationship verification
Jo ur
Experimental data gathered by Li et al. (2013) were selected to verify the stress–strain relationship derived in (21). They performed a series of uniaxial compression tests on a model material with a standard size of 50 mm in diameter and 100 mm in height. One joint was prefabricated in each of the specimens with angles of 15°, 30°, 40°, and 50° with the loading plane. Because only one discontinuity was prefabricated in the standard samples, the density of the plane can be taken as 1/0.1 × cos (m) and the radius of the plane can be taken as 0.025/ cos (m). The physical and mechanical parameters of the model material are shown in Table 1, and the geometric parameters of the joints are shown in Table 2.
13
Journal Pre-proof Table 1 Physical and mechanical parameters of the model material
Material density (kg/m3)
Uniaxial compressive strength (MPa)
Compressive elastic modulus
Uniaxial tensile strength (MPa)
Poisson’s ratio
(GPa) 2030
36.9
2.95
9.2
0.205
of
Table 2
Normal density
Average radius (m)
Friction angle (°)
Cohesion (MPa)
0.0259/0.0289/0.0326/0.0389
28
0.01
-p
Dip angle (o)
ro
Geometric parameters of joints
−1
(m )
re
9.66/8.66/7.66/6.43
lP
15/30/40/50
na
The stress–strain curves obtained from Li et al.’s uniaxial compression tests were full stress–strain
Jo ur
curves, whereas the stress–strain curves derived in this paper are only valid in the elastic stages. Therefore, the elastic deformation region of the full stress–strain curves obtained from the aforementioned experiments were extracted to validate the proposed model. The elastic regions of the test curves extracted are shown in Fig. 3(a) for comparison with the results calculated using (21), which are shown in Fig. 3(b). It can be seen that both the trends and values of the curves in both groups are very similar. The very similar variations can be observed for each of the inter-angle δ values; i.e. the slopes, or the elastic moduli, of the curves are very close.
14
Journal Pre-proof
(b)
of
(a)
lP
re
3.1.2 In situ monitoring and numerical simulation
-p
ro
Fig. 3. Comparison of stress–strain curves. (a) Experimental data and (b) calculated data.
Monitoring data from a tunnel section of the Lan-Yu Railway (Lanzhou–Chongqing) were used to
na
validate the proposed model. This tunnel is 10 m high and 10 m wide and crosses the western Qinling mountains at a depth of 346 m through lamellar carbonaceous slate. A set of dominant discontinuities is
Jo ur
well-developed at an attitude of SW240°∠73° with a spacing of 2–5 cm. A primary support system with steel arches, shotcrete and bolts, and permanent lining made of 25 cm thick C25 steel concrete is installed. A series of pressure cells are placed at the interface of the primary support and the terminal lining. A numerical simulation using FLAC3D by Itasca Co. Ltd was carried out for comparison with the in situ monitoring data. The shape and size of the tunnel and the two support systems used in the simulation replicated the structure of the monitored tunnel section. The parameters of the rock and discontinuities are shown in Table 3. Table 3 Parameters for numerical simulations 15
Journal Pre-proof Density Materials Rock
3
（kg/m ) 2200
Elastic Modulus (MPa)
Poisson's ratio
Cohesion
1000
0.4
(MPa)
Friction angle (°)
Tensile strength (MPa)
1.356
32.36
0.061
0
22
0
Discontinuity
A comparison chart between the monitoring data and the numerical simulation results is shown in Figs. 4 and 5. This figure shows the displacement distributions of the tunnel wall, which exhibited great
na
lP
re
-p
ro
of
consistency between the field monitoring data and the numerical simulation results.
Fig. 5. Field stress monitoring results
Jo ur
Fig. 4. Displacement monitoring results
However, it is reasonable that there were a big differences in rock pressure between the field monitoring data (Fig. 5) and the simulation results, which were 11.5, 1.2, 12.2, 6.1, 10.3, 3.1, and 14.9 MPa, respectively, from the left toe of the wall to the right one, mainly because the primary support system overcomes most of the pressure from the surrounding rock mass.
3.2 Elastic modulus verification The elastic modulus is an important index for characterising rock–mass deformation. A verification of the elastic modulus can indirectly reflect whether the obtained stress–strain relationship is correct. 16
Journal Pre-proof Field test data from the anchor tunnel of Jindong suspension bridge in Yunnan, southwestern China were used to validate the calculation method for the elastic modulus of rock mass proposed in this paper. The rock mass in this area is composed of slightly metamorphic rock with a phyllitic structure. There are two sets of dominant structural planes, both of which are rough and undulate. An in situ loading test using a bearing plate was carried out to obtain the pressure–settlement relationship and the elastic modulus of the rock mass. The parameters of the rock block were obtained via laboratory tests, and an in situ direct shear test was conducted to measure the shear strength of the structural planes. The parameters are shown
of
in Table 4. The in site direct shear test uses the flat push method, and the shear area is about 900cm2. The
ro
normal load is perpendicular to the pre-shear plane, the direction of the shear force is parallel to the
-p
pre-shear plane and points to the sliding direction of the sliding mass. Take out the samples(30cm ×
re
30cm × 30 ~ 40cm) from the rock mass with structural planes for testing. The installation image of
Jo ur
na
lP
direct shear test is shown in Fig. 6.
Fig. 6. Installation image of direct shear test of rock mass with structural plane
Table 4 Parameters of the rock and joints
17
Journal Pre-proof Group number
Parameters for the rock
Parameters for structural plane
Elastic modulus (GPa)
Poisson's ratio
Normal density (m−1)
Average radius (m)
Inclination
1
12.7
0.23
20
2
12.7
0.23
12
(°)
Friction angle (°)
Cohesion (MPa)
0.28
28
22.8
0.045
0.30
50
19.8
0.172
of
The stress–deformation diagram of the rock mass obtained after the field test is shown in Fig. 6(a). A part of the pressure curve was selected and converted into a stress–strain curve, shown in Fig. 6(b), via
pD(1 m2 ) W
,
(27)
re
Em
-p
ro
Boussek’s formula (Yin, 1990):
lP
where p is the pressure on the bearing plate in MPa, D is the diameter or side length of the bearing plate in mm, W is the settlement of the rock mass related to p , is a coefficient related to the shape and
na
stiffness of the bearing plate (for a circular plate, = 0.785; for a square plate, = 0.886), and m is
Jo ur
the Poisson’s ratio of the rock mass.
By substituting the parameters of the rock and the structural plane in Table 4 into (25), we can obtain the elastic modulus of the rock mass. As shown in Fig. 7(b), the results of the in situ tests and the calculations were in good agreement. A possible explanation for the slightly lower values observed in the test results than in the calculations is that a few irregular joints were not considered in the calculations.
18
Journal Pre-proof
(b)
of
(a)
Fig. 7. (a) Deformation curves made using field test data. (b) Comparison between stress–strain curves obtained using
re
-p
ro
test data and calculated data.
lP
4. Discussion
elastic modulus are discussed.
na
In this section, the SMRM model is compared with previous models, and the features of the proposed
Jo ur
4.1 Comparison between the SMRM model and previous models Developing a constitutive model of rock mass is the core subject of the study of rock mechanics. In the past, there were three main approaches employed to this end. The first was to regard the discontinuities in the rock mass as an infinite plane and superimpose their mechanical effects on the continuous rock to derive the constitutive relationships. The second was Oda’s constitutive model, which is based on the fabric tensor of rock mass. The derivation process of this model is relatively complicated, which makes it difficult for real applications. The third was the damage mechanics model. In this model, the damage variables, stresses, and strains 19
Journal Pre-proof are all expressed in tensors. It is difficult to reasonably define these tensors and make their physical meaning clear. Compared with the three aforementioned models, the SMRM model reasonably combines fracture mechanics, continuum mechanics, and the energy addition principle to make the derivation process simpler and easier to understand. Moreover, the derived parameters are comprehensive and applicable.
of
4.2 Discussions on the elastic modulus
ro
The SMRM model properly reflects the parameters of the rock mass, such as its elastic modulus. In this
-p
subsection, we will discuss the characteristics of the elastic modulus and its influencing factors.
re
4.2.1 Fully directional elastic modulus of rock mass
lP
Test data from another test site in Jindong bridge were used to further analyse the characteristics of the elastic modulus of the rock mass, namely weathered altered diabase with a set of well-developed
Table 5
Jo ur
na
structural planes. The parameters related to the rock block and the structural planes are listed in Table 5.
Parameters of altered diabase and joints Parameter for the rock Elastic modulus (GPa)
Poisson’s ratio
Parameter for the structural plane Normal density (m−1)
Average radius (m)
Inclination
Friction angle (°)
Cohesion (MPa)
20.3
0.087
(°) 14.4
0.23
10
0.46
45
A stereogram of the elastic modulus Em obtained using the ‘SMRM Calculation’ software through (25) 20
Journal Pre-proof is shown in Fig. 8, where any point in the chart area represents a spatial direction and the colour is related
-p
ro
of
to the value of the rock mass in that direction.
lP
re
Fig. 8. Stereogram of the elastic modulus of a rock mass.
The results show that the elastic modulus of the rock mass exhibits significant anisotropy; its maximal
na
value reached 14.0648 GPa, but its minimum value was 1.5244 GPa. The central point of the chart, i.e.
Jo ur
the vertical direction, shows a value of Em = 2.47 GPa in the direction of the loading. Comparing this with the value of 2.04 GPa measured in situ, the relative error was 17.4%. 4.2.2 Influencing factors of Em
It can be seen from Fig. 8 that the elastic modulus of the rock mass is significantly weaker compared with the rock block and also exhibits strong anisotropy (Horii, 1983). Taking the same model presented in Section 4.2.1 as an example, we can investigate the influencing factors and variation characteristics of the elastic modulus of the rock mass derived using (25). (1) Anisotropy caused by the orientation of discontinuities
21
Journal Pre-proof Fig. 9(a) shows the significant influence of the dip angle of structural planes on the elastic modulus of the rock mass in the following situations: Em E , Em E , Em E ,
j j
(28)
2
2
of
where j and are the frictional angle of the plane and the inter-angle between the normal of the plane
ro
and the loading direction, respectively. Apparently, Em becomes minimum at
4
j 2
, which is
re
-p
entirely reasonable.
lP
(2) Effect of the radius and normal density of discontinuities Fig. 9(b) shows the inverse correlation between the average radius of a set of discontinuities a and
na
the elastic modulus of the rock mass. The influence of the radius is significant for relatively smaller
Jo ur
cracks. However, when the size of the plane become large, e.g. 4 m in this case, the elastic modulus will tend toward lower values. It can be seen in (25) that the influence of the normal density of a set of planes λ is on the same scale and under the same conditions as a . (3) Effect of the strength of discontinuities The strength of discontinuities, i.e. the friction angle and cohesion, exhibits a positive correlation with the elastic modulus ratio of the rock mass (Figs. 9(c) and (d)). The elastic modulus of the rock mass increases with the friction angle and cohesion of the planes increase, reaching the modulus of the rock block while the discontinuities are locked by shear strength. (4) Effect of the stress state of discontinuities 22
Journal Pre-proof The stress state of discontinuities is also an important factor affecting the elastic modulus of rock mass. First, the influences of the compressive and tensional states at the discontinuities are absolutely different, as indicated in (26); i.e. the modulus of the rock mass under a tensional stress state is much less than that under a compressive state. Secondly, under a compressive stress state, the elastic modulus of the rock mass rapidly increases as the normal stress at the planes increases. Below a certain normal stress, the modulus of the rock mass will tend to be stable because the discontinuities are gradually locked (Fig.9(e)).
lP
re
-p
ro
of
This can be defined as the ‘stress lock’ of the discontinuities.
(b)
(c)
Jo ur
na
(a)
(d)
(e)
Fig. 9. Effects of various joint parameters on the elastic modulus of a rock mass: (a) inter angle, (b) average radius, (c) friction angle, (d) cohesion, and (e) stress state.
23
Journal Pre-proof
5. Conclusion The stress–strain relationship of fractured rock mass was derived based on fracture mechanics, rock-mass structure theory, and strain energy theory. This stress–strain relationship explains the influence of various geological factors on the deformation behaviour of rock mass. The elastic modulus of rock mass calculated through the proposed constitutive relationship fully accounts for the influence of
of
discontinuities in all directions and can reflect the weakening and anisotropic characteristics of the
ro
rock-mass elastic modulus. Compared with other constitutive models, the advantages of the SMRM
-p
model lie in its reliability, simplicity, and applicability. Additionally, our new model can better reflect the
re
geo-mechanical effects of discontinuous networks and can be widely applied as a theoretical fundamental model for mechanical research on rocks and practical engineering geological issues. However, the SMRM
na
deformation stage needs further research.
lP
model can only reflect the elastic deformation stage of fractured rock mass, the content of the plastic
Jo ur
Discontinuities are the main factor that weakens the elastic modulus of rock mass and produces anisotropy. The derived elastic modulus of rock mass is distributed as a ‘U’ curve according to the changes in axial stress and the normal angle of the discontinuities; it is lowest at an angle of /4+ /2 .
The elastic modulus of rock mass is positively related to the friction angle and cohesion of the discontinuities and is inversely correlated with the average radius and density of the discontinuities. Similarly, when the normal stress at the planes increase, the elastic modulus of the rock mass rapidly increases. However, the influence of the structure on the elastic modulus of the rock mass has a scaling effect. When the average radius of the discontinuities and the normal stress exceed a certain range, the
24
Journal Pre-proof
elastic modulus of the rock mass remains basically unchanged.
Acknowledgements We acknowledge the financial support of the National Natural Science Foundation of China (41831290) and Dr. Yihu Zhang from Changjiang River Scientific Research Institute for his help with the site
ro
re
-p
reviewers for their insightful and valuable comments.
of
monitoring data of Jindong Bridge test data. The authors also thank the editor Juang and two anonymous
References
Jo ur
na
lP
Conflicts of Interest: The authors declare no conflict of interest.
Alejano, L.R., Bobet, A., 2012. Drucker-Prager criterion. The ISRM Suggested Methods for Rock Characterization,
Testing
and
Monitoring:
2007-2014,
pp
247-252.
https://doi.org/10.1007/978-3-319-07713-0_22.
Bai, Y., Wierzbicki, T., 2010. Application of extended Mohr-Coulomb criterion to ductile fracture. Int. J. Fracture. 161(1), 1–20. https://doi.org/10.1007/s10704-009-9422-8.
Bao, H., Zhai, Y., Lan, H.X., Zhang, K.K., Qi, Q., Yan, C.G., 2019. Distribution characteristics and 25
Journal Pre-proof
controlling factors of vertical joint spacing in sand-mud interbedded strata. J Struct Geol. 128, 103886. https://doi.org/10.1016/j.jsg.2019.103886.
Bao, H., Zhang, G.B., Lan, H.X., Yan, C.G., Xu, J.B., Xu, W., 2020. Geometrical heterogeneity of the joint roughness coefficient revealed by 3D laser scanning. Eng Geol. 265, 105415. https://doi.org/10.1016/j.enggeo.2019.105415.
of
Bidgoli, M.N., Zhao, Z., Jing, L., 2013. Numerical evaluation of strength and deformability of fractured
ro
rocks. J. Rock Mech. Geotech. Eng. 5(6), 419–430. https://doi.org/10.1016/j.jrmge.2013.09.002.
-p
Chen, X., Liao, Z, Peng, X., 2012. Deformability characteristics of jointed rock masses under uniaxial
re
compression. Int. J. Min. Sci. Tech. 22(2), 213–221. https://doi.org/10.1016/j.ijmst.2011.08.012.
na
pp. 56–86.
lP
Fan, T.Y., 1978. Fracture Mechanics Foundation. Chinese Jiangsu Science and Technology Press, Nanjing,
Jo ur
Fossum, A.F., 1985. Effective elastic properties for a random jointed rock mass. Int. J. Rock Min. Sci. 22(6), 467–70. https://doi.org/10.1016/0148-9062(85)90011-7.
Gehle, C., Kutter, H.K., 2003. Breakage and shear behaviour of intermittent rock joints. Int. J. Rock Mech. Min. Sci. 40(5), 687–700. https://doi.org/10.1016/S1365-1609(03)00060-1.
Gerrard, C.M., 1982. Equivalent elastic moduli of a rock mass consisting of orthorhombic layers. Int. J. Rock Mech. Min. Sci. 19(1), 9–14. https://doi.org/10.1016/0148-9062(82)90705-7.
Gu, D.Z., 1979. Foundation of Rock and Soil. Engineering Geology. Chinese Beijing: Science Press.
Han, J.X., Li, S.C., Li, S.C., Tong, X.H., Li, W.T., 2011. Model study of strength and failure modes of 26
Journal Pre-proof
rock
mass
with
persistent
cracks.
Chin.
J.
Rock
Soil.
Mech.
32(2),
178–184.
https://doi.org/10.16285/j.rsm.2011.s2.019.
Hoek, E., Brown, E.T., 1980. Empirical strength criterion for rock masses. J. Geotech. Geo-environ. Eng. 106, ASCE 15715. https://doi.org/10.1016/0148-9062(81)90766-x.
Hoek E, Carranza-Torres C, Corkum B, 2002. Hoek-Brown failure criterion-2002 edition. Proceedings of
of
NARMS-Tac. Jul 7;1(1):267-73.
ro
Hoek E, Wood D, Shah S, 1992. A modified Hoek–Brown failure criterion for jointed rock masses. Rock
re
-p
Characterization: ISRM Symposium, Eurock'92, Chester, UK, 14–17 September.
Horii, H., Nemat-Nasser, S., 1983. Overall moduli of solids with microcracks: Load-induced anisotropy. J.
lP
Mech. Phys. Solids. 31(2), 155–171. https://doi.org/10.1016/0022-5096(83)90048-0.
na
Hu, X., Wu, F., Sun, Q., 2011. Elastic modulus of rock mass based om the two parameter
Jo ur
negative-exponential (TPNE) distribution of discontinuity spacing and trace length. Bull. Eng. Geol. Environ. 70(2), 255–263. https://doi.org/10.1007/s10064-010-0321-z.
Hudson, J.A., Priest, S.D., 1983. Discontinuity frequency in rock masses. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 20(2), 73–89. https://doi.org/10.1016/0148-9062(83)90329-7.
Kawamoto, T., Ichkawa, Y., Kyoya, T., 1988. Deformation and fracturing behavior of discontinuous rock mass and damage mechanics theory. Int. J. Num. Methods Geomech. 12(1), 1–30. https://doi.org/10.1002/nag.1610120102.
Li, S.C., Wang, L., Li, S.C., Han, J.X., 2013. Experimental study on post-peak deformation and failure of 27
Journal Pre-proof
rock specimens with different dip angles through joints. Chin. J. Rock Mech. Eng. 32(2), 3391– 3395.
Liu Y R, Tang H M, 1999. Rock mechanics. China University of Geosciences Press.
Main, I.G., Sammonds, P.R., Meredith, P.G., 1993. Application of a modified Griffith criterion to the evolution of fractal damage during compressional rock failure. Geophys. J. Int. 115(2), 367–380.
of
https://doi.org/10.1111/j.1365-246X.1993.tb01192.x.
ro
Oda, M., 1984. Similarity rule of crack geometry in statistically homogeneous rock masses. Mech. Mater.
re
-p
3(2), 119–29. https://doi.org/10.1016/0167-6636(84)90003-6.
Oda, M., 1986. An equivalent continuum model for coupled stress and fluid flow analysis in jointed rock
lP
masses. Water Resour. Res. 22(13), 1845–56. https://doi.org/10.1029/WR022i013p01845.
na
Saeidi, O., Rasouli, V., Vaneghi, R.G., Gholami, R., Torabi, S.R., 2014. A modified failure criterion for
Jo ur
transversely isotropic rocks. Geosci. Front. 5(2), 215–225. https://doi.org/10.1016/j.gsf.2013.05.005.
Wei, Z.Q., Hudson, J.A., 1988. The influence of joints on rock modulus. Proceedings of the International Symposium
on
Engineering
in
Complex
Rock
Formations.
Jan
1
(pp.
54-62).
https://doi.org/10.1016/B978-0-08-035894-9.50010-X.
Wu, F.Q., 1988. A 3D model of a jointed rock mass and its deformation properties. Int. J. Min. Geol. Eng. July, Volume 6, Issue 2, pp 169–176. https://doi.org/10.1007/BF00880806.
Wu, F.Q., Wang, S.J., Song, S.W., Lu, J.T., 1994. Statistical principles in mechanics of rock masses. Chin. Sci. Bull. 6, 493–503. 28
Journal Pre-proof
Wu, F.Q., Wang, S.J., 2001. A stress–strain relation for jointed rock masses. Int. J. Rock Mech. Min. Sci. 38(4), 591–598. https://doi.org/10.1016/S1365-1609(01)00027-2.
Xu, T., Ranjith, P.G., Wasantha, P.L.P., Zhao, J., Tang, C.A., Zhu, W.C., 2013. Influence of the geometry of partially-spanning joints on mechanical properties of rock in uniaxial compression. Eng. Geol. 167, 134–147. https://doi.org/10.1016/j.enggeo.2013.10.011.
of
Yashinaka, R, Yamabe, T., 1986. Joint stiffness and the deformation behavior of discontinuous rock. Int J
ro
Rock Mech Min Sci Geomech Abstr, 23(1), 19–28. doi:10.1016/0148-9062(86)91663-3
re
-p
Yin, S.Y., 1990. Elastoplastic mechanics. China University of Geosciences Press.
1729–1753.
lP
Weiyuan, Z.H., 2010. Structure stability of rock mass engineering. Chin. J. Rock Mech. Eng. 29(09),
na
Zhou, W., Zhao, J., Liu, Y., Yang, Q., 2002. Simulation of localization failure with
Jo ur
strain-gradient-enhanced damage mechanics. Int. J. Numer. Anal. Methods Geomech. 26(8), 793– 813. https://doi.org/10.1002/nag.225.
29
Journal Pre-proof
CRediT author statement
Faquan Wu: Conceptualization, Methodology, Writing - Review & Editing, Funding acquisition. Yi Deng: Writing- Original draft preparation, Software, Validation, Investigation. Jie Wu: Software, Project administration.
of
Bo Li: Supervision.
ro
Peng Sha: Formal analysis.
Kai Zhang: Writing - Review & Editing Preparation.
lP na
Shuhao Qiu: Validation.
Jo ur
Handong Liu: Resources.
re
Keqiang He: Investigation.
-p
Shenggong Guan: Visualization.
30
Journal Pre-proof Declaration of interests
The authors declare that they have no known competing financial interests or personal relationships that could have
Jo ur
na
lP
re
-p
ro
of
appeared to influence the work reported in this paper.
31
Journal Pre-proof
Highlights
A model for the elastic stress–strain relation of fractured rock mass is proposed.
The weakening and anisotropy of the elastic modulus of rock mass are notable.
The modulus is positively related to the normal stress at the planes.
It is positively related to the friction angle and cohesion of the plane.
It is inversely correlated to the average radius and density of discontinuities.
Jo ur
na
lP
re
-p
ro
of
32
Faquan Wu, Yi Deng, Jie Wu, Bo Li, Peng Sha, Shenggong Guan, Kai Zhang, Keqiang He, Handong Liu, Shuhao Qiu PII:
S0013-7952(19)31191-3
DOI:
https://doi.org/10.1016/j.enggeo.2020.105498
Reference:
ENGEO 105498
To appear in:
Engineering Geology
Received date:
20 June 2019
Revised date:
14 January 2020
Accepted date:
16 January 2020
Please cite this article as: F. Wu, Y. Deng, J. Wu, et al., Stress–strain relationship in elastic stage of fractured rock mass, Engineering Geology (2020), https://doi.org/10.1016/ j.enggeo.2020.105498
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2020 Published by Elsevier.
Journal Pre-proof
Stress–Strain Relationship in Elastic Stage of Fractured Rock Mass Faquan Wua, Yi Denga,* [email protected], Jie Wub, Bo Lia, Peng Shaa, Shenggong Guana, Kai Zhanga, Keqiang Hec, Handong Liud, Shuhao Qiua
Qingdao University of Technology, Shandong, China, 273400
d
-p
c
Rock Innovation Technology Co. Ltd, Zhejiang, China, 312000
ro
b
of
Shaoxing University, Shaoxing, Zhejiang, China, 312000
North China University of Water resources and electric Power, Henan, China 450046
re
a
*
na
Abstract:
lP
Corresponding author.
Jo ur
Discontinuity is the main factor affecting the stress–strain relationship of rock mass. Based on the statistical parameters of the fracture network of rock mass and combining them with the strain energy theory from continuum and fracture mechanics, the elastic stress–strain relationship of fractured rock mass is proposed and its elastic modulus is derived. The stress–strain relationship can truly reflect the weakening and anisotropy of the mechanical properties of rock mass. On this basis, experimental and calculated stress–strain curves are compared, both of which exhibited the same slope. Additionally, monitoring data from a tunnel excavation is compared with the numerical simulation results, indicating that the displacement of both is basically consistent. Then, we compare three constitutive models with the statistical mechanics of rock mass (SMRM) model to highlight the advantages of the latter in terms of 1
Journal Pre-proof
reliability and adaptability. Finally, based on the results of a bearing plate test of a rock mass at Jindong Bridge, the anisotropy of the elastic modulus and its influencing factors are studied. The results show that the weakening and anisotropy of the elastic modulus of the rock mass are notable, and the modulus changes according to the inter-angle between the stress axis and the normal of the discontinuity, reaching the minimum at an angle of 57.5°. The modulus is positively related to the friction angle and the cohesion
of
of the plane and is inversely correlated to the average radius and density of the discontinuities. When the average radius of the discontinuities exceeds 3 m, the weakening of the elastic modulus of the rock mass
-p re
increases with the normal stress at the planes.
ro
tends to be stable. The stress environment also has an effect on the elastic modulus, which rapidly
na
1 Introduction
lP
Keywords: fractured rock mass; stress–strain relationship; elastic modulus
It is well known that approximately half of all infrastructure construction projects, e.g. slopes, railway
Jo ur
tunnels, highways, hydro-power facilities, mines, and deep engineering and civil engineering projects in mountain areas, are based on rock engineering(Liu and Tang, 1999). The prediction of the deformation and strength behaviour of rock masses and the artificial structures attached to them is a basic task in the study of rock mechanics, and the stress–strain relationship is a fundamental tool for describing the deformation of rock masses (Gerrard, 1982).
The most widely used models today for the calculation of rock deformation have been introduced from the field of continuum mechanics, such as elastics, elastoplasticity, etc., which are based on the hypothesis of homogeneity, continuity, and isotropy of the medium (Saeidi et al., 2014). However, real 2
Journal Pre-proof
rock masses vastly different from the ideal medium because they contain a large number of discontinuities that are randomly distributed in terms of location, size, and orientation. The mechanical effects of discontinuities are not accurately reflected in the aforementioned models(Bao et al., 2019).
Since the occurrence of certain disastrous events, such as the Vaiont reservoir landslide in Italy in the 1960s, which caused the deaths of more than 2000 people, engineering geologists and rock dynamicists
of
began to investigate the negative mechanical effects of the discontinuities in rock mass (Gu, 1979). A
ro
large number of tests have been carried out in laboratories and construction sites to determine the
-p
mechanical features of the rock planes. Near the end of the last century, scientists began to establish
re
models and study the global behaviour of rock mass (Fan, 1978; Main et al., 1993; Hoek et al., 2002; Bai
lP
and Wierzbicki, 2010; Han et al., 2011; Alejano and Bobet, 2012; Bidgoli et al., 2013)
Two methods have been employed for studying the mechanical behaviours of rock mass. One consists
na
of empirically weakening the mechanical properties and performing rock quality classification by
Jo ur
considering the degree of fragmentation, which only yields weak results (Wu, 1976; Hoek and Brown, 1980; Wei and Hudson, 1986; Zhou et al., 2002). However, this classification approach hardly reflects the structural and mechanical anisotropy of the rock mass, which leads to significant directional variability in its behaviour and frequently causes the unexpected collapse of engineering structures and huge economic losses.
The other method consists of trying to establish a perfect model based on objective data of geological phenomena and rigorous mechanical theory. Many people have tried to develop models by combining the mechanical responses of rocks and infinite discontinuities (Fossum, 1985; Yashinaka and Yamabe, 1988;
3
Journal Pre-proof
Hoek et al., 1992; Gehle and Kutter, 2003; Chen et al., 2012; Xu et al., 2013, Bao et al., 2020). Oda (1984 and 1986) made great effort to set up a fabric tensor to describe the geometric structure of rock mass and then establish a stress–strain model by employing fracture mechanics for the behaviours of joints. Kawamato et al. (1988) introduced the concept of the damage tensor to develop a 3D damage mechanical model for jointed rock mass. Zhou (2010) extended this model and made some modifications
of
to fit the compressive stress condition of the joints. Wu et al. (1994 and 2001) and Hu et al. (2011) proposed some stress–strain models for jointed rock mass based on the geometric probability of joint
-p
ro
systems, which was first proposed by Hudson and Priest (1983), and the strain energy of cracks.
re
All the above mentioned efforts have definitely promoted the advance of rock mechanics, but very few models have been widely accepted; either because the geometric model does not reflect the actual
lP
geological structure of rock mass or the hypothesis does not meet the actual mechanical principles, or
na
even because of the complexity of expression of available models.
Jo ur
The objective of this study is to propose a model based on the statistical mechanics of rock mass (SMRM) model, which was first proposed by the authors, that combines the theories of fracture and continuum mechanics, and the energy additivity principle.
2 Proposed Model 2.1 Hypothesis The proposed model for a unit of fractured rock mass will be established based on two basic hypotheses:
4
Journal Pre-proof
1. The rock mass is composed of an isotropic rock block that is cut by a discontinuity network, where the distribution of discontinuities obeys geological rules and geometric-probability constraints. The global strain energy of the rock-mass unit under a certain stress tensor corresponds to the sum of the two parts.
2. The deformation of any discontinuity obeys the rules of fracture mechanics, where the shear displacement is driven by the residual shear stress on the crack plane, i.e. the shear strength is
of
subtracted from the shear stress, and the shear strength is determined through parameters such as
ro
cohesion and roughness.
-p
2.2 Stresses on a discontinuity
re
A hexahedron unit of rock mass in Cartesian coordinate systems ( x1 , x2 , x3 ) , such as the one shown in
lP
Fig. 1, is analysed. It has three pairs of side faces parallel to three coordinate planes. A stress tensor ij acts on the unit as shown in the figure. A buried circular crack is present in the unit, and the normal of the
Jo ur
na
crack surface is described by the directional cosine n (n1 , n2 , n3 ) .
Fig. 1 Unit and stress conditions
5
Journal Pre-proof The normal stress and the shear stress t acting on the crack and their coordinate elements can be written as kj nk n j， i kj nk n j ni ,
(1)
t ti ti ， ti ij n j kj nk n j ni .
For a water pressure p in the crack, the stress changes to effective stress as e p .
of
Based on Coulomb strength theory, the shear strength of the crack surface is
s c tan ,
ro
(2)
-p
where c and tan are the cohesion and frictional coefficients of the crack, respectively. 0 ),
the shear strength
re
Considering that the crack will close when the normal stress is compressive (
open when the stress is tensional (
lP
s takes effect, and the fracture effect in the normal direction disappears. On the other hand, the crack will 0 );
thus, the shear strength will be absent and the fracture effects
na
both in the normal and shear directions will take place.
Jo ur
Here, the stress state function is defined as
1, 0 k k ( ) 0, 0
(3)
The residual shear stress on the plane after the shear strength s is deducted is 0， k 0, t s t s， k 0, t s t， k 1, s 0
We define the residual shear stress ratio as
6
(4)
Journal Pre-proof h
t
.
(5)
Because the shear deformation is driven by the residual shear stress , we will use h more often in our discussion. It can be seen that the value of h is affected by the stress. When the shear stress is less than the shear strength, = 0, and therefore h = 0. When the shear stress on the crack surface is greater than the
of
shear strength, h = /t (t s) / t 1 s / t 1 . When the normal stress on the crack surface is equal to the
-p
Therefore, the range of possible values of h is [0,1].
ro
tensile stress, then s = 0 and h = 1.
re
We can get
t (c tan ) , t
lP
h
(6)
na
where t is the shear stress on the crack surface and c are the shear strength parameters of the crack
Jo ur
surface, namely its cohesion and frictional angle, respectively.
2.3 Strain energy density
2.3.1 Strain energy due to a single crack
As derived from fracture mechanics, for a type-I disk-shaped crack with radius a (Fig. 2), the work
7
Journal Pre-proof
(a)
(b)
ro
of
Fig. 2. Crack and displacement. (a) Plane of the crack; (b) cross-section of the crack.
2E
k a 2 r 2 (0 ≤ r ≤a),
(7)
re
lP
vI
-p
done at the area element ds rd dr by a normal force dp k ds on the displacement is
2 where 8(1 ) and is Poisson’s ratio of the medium around the crack.
na
Jo ur
When the crack is displaced from I and becomes closed, the stress on the crack surface increases from zero to . Because the process is elastic, the work done to close the crack on the area element ds rd dr is
1 2 2 2 2 dWI vI dp k a r rd dr . 2 4E
(8)
By integrating over the entire crack area and considering the work done by the displacements of both the upper and lower walls, the strain energy stored in the surrounding medium due to the crack opening (or closing) can be obtained as U I 2 dWI
2E
k 2 2
a
0
8
2
0
3E
a 2 r 2 rdr d
k 2 2 a .
(9)
Journal Pre-proof
In the same way, we can obtain the strain energy caused by the shear stress and displacement by integrating in both the x and y directions. Thus, dTx cos rd dr, dTy sin rd dr , vx
2E
cos a 2 r 2， v y
2E
sin a 2 r 2 ,
(11)
(12)
of
1 1 2 2 2 dU II III vx dTx v y dTy a r d dr , 2 2 4E
(10)
ro
where x and y are the local coordinate axes of the crack and δ is the inter-angle between the shear stress
-p
and the x axis.
re
The work done by the shear stress is
2
and subscripts II and III indicate type-II and type-III cracks in terms of fracture
Jo ur
mechanics.
2
na
where
(13)
lP
1 1 2 3 , U II III 2 ( vx dTx vy dTy ) a 2 2 3E
Finally, the total strain energy of a crack due to normal and shear stresses is U c U I U II III
2 2 (k 2 )a3 . 3E
(14)
2.3.2 Strain energy density due to a crack network In the field of geology, the discontinuities in rock mass can be divided into several sets, each of which has a certain dominant dip direction and dip angle. Let there be m sets of discontinuities in a volume V of a rock-mass unit and Np be the number of the planes in the p-th set. Considering that the stresses ( and ) on the same set of planes are not significantly variable, the 9
Journal Pre-proof total strain energy caused by the crack network can be written as m Np
p1 q1
3E
U c U cpq
m Np
p1 q1
3E
2 2 2 3 (k )a
m
Np
p1
q1
2 2 2 3 (k ) a .
(15)
We have verified, based on the work of Gu (1979), that if the radius of the cracks follow a Np
3
p 1
negative-exponential distribution, then a 3 a V , where and a are the normal density and
of
average radius of a set of cracks, respectively.
-p
Uc m a (k 2 2 2 ) , V E p1
(16)
re
uc
ro
Thus, the strain energy due to the crack network can be written in the form of an energy density:
obtain
lP
where , are as defined above. Using (1) and through simple derivation via tensor algebra, we can
na
2 ij ni n j ns nt st
(17)
Jo ur
2 ij h2 ( it n j ns ni n j ns nt ) st
2.4 Stress–strain relationship of a rock-mass unit The strain energy density of a continuous rock mass can be taken from the theory of the mechanics of elasticity as 1 u0 ij C0ijst st . 2
(18)
Let the global strain energy density of a rock-mass unit be 1 u ij Cijst st . 2 10
(19)
Journal Pre-proof According to the principle of energy additivity, this global energy u equals the sum of u0 and uc , i.e.
u u0 uc .
(20)
By substituting (16), (17), and (18) into (19) and subtracting ij from both sides of the equation, we get the elastic stress–strain relationship of the fractured rock mass:
st
are the strain and stress tensors, respectively, and Cijst is the compliance tensor of
ro
where eij and
(21)
of
eij Cijst st (Coijst Ccijst ) st ,
through elastic theory as
1 v v ( is jt it js ) ij st , 2E E
lP
C0ijst
re
-p
the rock mass. C0ijst is the part of the tensor corresponding to a continuous rock, which can be obtained
(22)
na
where E and are the elastic modulus and Poisson’s ratio of the rock, respectively, and i j is the
Jo ur
Kronecker symbol ( i j = 1 for i = j and i j = 0 for i ≠ j ). Ccijst is the part of the tensor caused by the discontinuity network, which can be expressed as Ccijst
E
m
2 2 2 a[(k h )ni n j ns nt h it n j ns ] ,
(23)
p 1
where m is the number of crack sets, and a are the normal density and average radius of the crack sets, respectively, k is the normal stress state factor of the cracks, and h is the residual shear stress ratio of the structural surface.
2.5 Elastic modulus of a rock mass and modulus ratio The elastic modulus is an index derived from elasticity theory, which is based on the hypothesis of 11
Journal Pre-proof having a homogeneous, continuous, and isotropic medium. Although this is does not represent real cases, the ideas and methodology of elasticity theory can be used for defining the deformation properties of fractured rock mass, even if it is not an ideal elastic medium. Moreover, the concept of elastic modulus has been commonly used in engineering practice because it provides simplifications and clarifications in a mechanical sense. Here, we still define the elastic modulus for a rock mass in the same way as in elasticity theory.
of
However, this index will have much deeper mechanical and geological meanings.
ro
Through basic understanding of elasticity theory and the stress–strain model expressed in (21), we can
11 1 . e11 C01111 Cc1111
(24)
lP
re
Em
-p
define the elastic modulus Em for fractured rock mass in the 11 direction as
Substituting C01111 and Cc1111 from (22) and (23) into (24), the elastic modulus of the rock mass
na
becomes
Jo ur
Em
E m
,
(25)
1 a k h n h n p 1 2
2
4 1
2
2 1
where all the parameters in (25) have been previously defined. In the case of 2 3 , which is commonly used in engineering practice and scientific research, let
n1 cos and 1 n12 sin 2 ; considering that for a crack in a compressive state k 0 and when the crack is opened due to tensional normal stress k h 1 , then (25) can be transformed into the following form:
12
Journal Pre-proof 1 ， 0 m 2 2 2 1 ah sin cos Em p 1 1 E ， 0 m 4 2 2 1 p1 a cos sin cos
(26)
Here, Em / E is defined as the elastic modulus ratio, i.e. the ratio of the elastic modulus of the rock mass to that of an intact rock.
ro
of
3. Verification
-p
A series of laboratory tests, comparisons with field monitoring data, and numerical simulations were conducted to examine the stress–stain relationships described in Section 2 and the elastic modulus derived
re
from the proposed theoretical model.
na
3.1.1 Laboratory experiments
lP
3.1 Stress–strain relationship verification
Jo ur
Experimental data gathered by Li et al. (2013) were selected to verify the stress–strain relationship derived in (21). They performed a series of uniaxial compression tests on a model material with a standard size of 50 mm in diameter and 100 mm in height. One joint was prefabricated in each of the specimens with angles of 15°, 30°, 40°, and 50° with the loading plane. Because only one discontinuity was prefabricated in the standard samples, the density of the plane can be taken as 1/0.1 × cos (m) and the radius of the plane can be taken as 0.025/ cos (m). The physical and mechanical parameters of the model material are shown in Table 1, and the geometric parameters of the joints are shown in Table 2.
13
Journal Pre-proof Table 1 Physical and mechanical parameters of the model material
Material density (kg/m3)
Uniaxial compressive strength (MPa)
Compressive elastic modulus
Uniaxial tensile strength (MPa)
Poisson’s ratio
(GPa) 2030
36.9
2.95
9.2
0.205
of
Table 2
Normal density
Average radius (m)
Friction angle (°)
Cohesion (MPa)
0.0259/0.0289/0.0326/0.0389
28
0.01
-p
Dip angle (o)
ro
Geometric parameters of joints
−1
(m )
re
9.66/8.66/7.66/6.43
lP
15/30/40/50
na
The stress–strain curves obtained from Li et al.’s uniaxial compression tests were full stress–strain
Jo ur
curves, whereas the stress–strain curves derived in this paper are only valid in the elastic stages. Therefore, the elastic deformation region of the full stress–strain curves obtained from the aforementioned experiments were extracted to validate the proposed model. The elastic regions of the test curves extracted are shown in Fig. 3(a) for comparison with the results calculated using (21), which are shown in Fig. 3(b). It can be seen that both the trends and values of the curves in both groups are very similar. The very similar variations can be observed for each of the inter-angle δ values; i.e. the slopes, or the elastic moduli, of the curves are very close.
14
Journal Pre-proof
(b)
of
(a)
lP
re
3.1.2 In situ monitoring and numerical simulation
-p
ro
Fig. 3. Comparison of stress–strain curves. (a) Experimental data and (b) calculated data.
Monitoring data from a tunnel section of the Lan-Yu Railway (Lanzhou–Chongqing) were used to
na
validate the proposed model. This tunnel is 10 m high and 10 m wide and crosses the western Qinling mountains at a depth of 346 m through lamellar carbonaceous slate. A set of dominant discontinuities is
Jo ur
well-developed at an attitude of SW240°∠73° with a spacing of 2–5 cm. A primary support system with steel arches, shotcrete and bolts, and permanent lining made of 25 cm thick C25 steel concrete is installed. A series of pressure cells are placed at the interface of the primary support and the terminal lining. A numerical simulation using FLAC3D by Itasca Co. Ltd was carried out for comparison with the in situ monitoring data. The shape and size of the tunnel and the two support systems used in the simulation replicated the structure of the monitored tunnel section. The parameters of the rock and discontinuities are shown in Table 3. Table 3 Parameters for numerical simulations 15
Journal Pre-proof Density Materials Rock
3
（kg/m ) 2200
Elastic Modulus (MPa)
Poisson's ratio
Cohesion
1000
0.4
(MPa)
Friction angle (°)
Tensile strength (MPa)
1.356
32.36
0.061
0
22
0
Discontinuity
A comparison chart between the monitoring data and the numerical simulation results is shown in Figs. 4 and 5. This figure shows the displacement distributions of the tunnel wall, which exhibited great
na
lP
re
-p
ro
of
consistency between the field monitoring data and the numerical simulation results.
Fig. 5. Field stress monitoring results
Jo ur
Fig. 4. Displacement monitoring results
However, it is reasonable that there were a big differences in rock pressure between the field monitoring data (Fig. 5) and the simulation results, which were 11.5, 1.2, 12.2, 6.1, 10.3, 3.1, and 14.9 MPa, respectively, from the left toe of the wall to the right one, mainly because the primary support system overcomes most of the pressure from the surrounding rock mass.
3.2 Elastic modulus verification The elastic modulus is an important index for characterising rock–mass deformation. A verification of the elastic modulus can indirectly reflect whether the obtained stress–strain relationship is correct. 16
Journal Pre-proof Field test data from the anchor tunnel of Jindong suspension bridge in Yunnan, southwestern China were used to validate the calculation method for the elastic modulus of rock mass proposed in this paper. The rock mass in this area is composed of slightly metamorphic rock with a phyllitic structure. There are two sets of dominant structural planes, both of which are rough and undulate. An in situ loading test using a bearing plate was carried out to obtain the pressure–settlement relationship and the elastic modulus of the rock mass. The parameters of the rock block were obtained via laboratory tests, and an in situ direct shear test was conducted to measure the shear strength of the structural planes. The parameters are shown
of
in Table 4. The in site direct shear test uses the flat push method, and the shear area is about 900cm2. The
ro
normal load is perpendicular to the pre-shear plane, the direction of the shear force is parallel to the
-p
pre-shear plane and points to the sliding direction of the sliding mass. Take out the samples(30cm ×
re
30cm × 30 ~ 40cm) from the rock mass with structural planes for testing. The installation image of
Jo ur
na
lP
direct shear test is shown in Fig. 6.
Fig. 6. Installation image of direct shear test of rock mass with structural plane
Table 4 Parameters of the rock and joints
17
Journal Pre-proof Group number
Parameters for the rock
Parameters for structural plane
Elastic modulus (GPa)
Poisson's ratio
Normal density (m−1)
Average radius (m)
Inclination
1
12.7
0.23
20
2
12.7
0.23
12
(°)
Friction angle (°)
Cohesion (MPa)
0.28
28
22.8
0.045
0.30
50
19.8
0.172
of
The stress–deformation diagram of the rock mass obtained after the field test is shown in Fig. 6(a). A part of the pressure curve was selected and converted into a stress–strain curve, shown in Fig. 6(b), via
pD(1 m2 ) W
,
(27)
re
Em
-p
ro
Boussek’s formula (Yin, 1990):
lP
where p is the pressure on the bearing plate in MPa, D is the diameter or side length of the bearing plate in mm, W is the settlement of the rock mass related to p , is a coefficient related to the shape and
na
stiffness of the bearing plate (for a circular plate, = 0.785; for a square plate, = 0.886), and m is
Jo ur
the Poisson’s ratio of the rock mass.
By substituting the parameters of the rock and the structural plane in Table 4 into (25), we can obtain the elastic modulus of the rock mass. As shown in Fig. 7(b), the results of the in situ tests and the calculations were in good agreement. A possible explanation for the slightly lower values observed in the test results than in the calculations is that a few irregular joints were not considered in the calculations.
18
Journal Pre-proof
(b)
of
(a)
Fig. 7. (a) Deformation curves made using field test data. (b) Comparison between stress–strain curves obtained using
re
-p
ro
test data and calculated data.
lP
4. Discussion
elastic modulus are discussed.
na
In this section, the SMRM model is compared with previous models, and the features of the proposed
Jo ur
4.1 Comparison between the SMRM model and previous models Developing a constitutive model of rock mass is the core subject of the study of rock mechanics. In the past, there were three main approaches employed to this end. The first was to regard the discontinuities in the rock mass as an infinite plane and superimpose their mechanical effects on the continuous rock to derive the constitutive relationships. The second was Oda’s constitutive model, which is based on the fabric tensor of rock mass. The derivation process of this model is relatively complicated, which makes it difficult for real applications. The third was the damage mechanics model. In this model, the damage variables, stresses, and strains 19
Journal Pre-proof are all expressed in tensors. It is difficult to reasonably define these tensors and make their physical meaning clear. Compared with the three aforementioned models, the SMRM model reasonably combines fracture mechanics, continuum mechanics, and the energy addition principle to make the derivation process simpler and easier to understand. Moreover, the derived parameters are comprehensive and applicable.
of
4.2 Discussions on the elastic modulus
ro
The SMRM model properly reflects the parameters of the rock mass, such as its elastic modulus. In this
-p
subsection, we will discuss the characteristics of the elastic modulus and its influencing factors.
re
4.2.1 Fully directional elastic modulus of rock mass
lP
Test data from another test site in Jindong bridge were used to further analyse the characteristics of the elastic modulus of the rock mass, namely weathered altered diabase with a set of well-developed
Table 5
Jo ur
na
structural planes. The parameters related to the rock block and the structural planes are listed in Table 5.
Parameters of altered diabase and joints Parameter for the rock Elastic modulus (GPa)
Poisson’s ratio
Parameter for the structural plane Normal density (m−1)
Average radius (m)
Inclination
Friction angle (°)
Cohesion (MPa)
20.3
0.087
(°) 14.4
0.23
10
0.46
45
A stereogram of the elastic modulus Em obtained using the ‘SMRM Calculation’ software through (25) 20
Journal Pre-proof is shown in Fig. 8, where any point in the chart area represents a spatial direction and the colour is related
-p
ro
of
to the value of the rock mass in that direction.
lP
re
Fig. 8. Stereogram of the elastic modulus of a rock mass.
The results show that the elastic modulus of the rock mass exhibits significant anisotropy; its maximal
na
value reached 14.0648 GPa, but its minimum value was 1.5244 GPa. The central point of the chart, i.e.
Jo ur
the vertical direction, shows a value of Em = 2.47 GPa in the direction of the loading. Comparing this with the value of 2.04 GPa measured in situ, the relative error was 17.4%. 4.2.2 Influencing factors of Em
It can be seen from Fig. 8 that the elastic modulus of the rock mass is significantly weaker compared with the rock block and also exhibits strong anisotropy (Horii, 1983). Taking the same model presented in Section 4.2.1 as an example, we can investigate the influencing factors and variation characteristics of the elastic modulus of the rock mass derived using (25). (1) Anisotropy caused by the orientation of discontinuities
21
Journal Pre-proof Fig. 9(a) shows the significant influence of the dip angle of structural planes on the elastic modulus of the rock mass in the following situations: Em E , Em E , Em E ,
j j
(28)
2
2
of
where j and are the frictional angle of the plane and the inter-angle between the normal of the plane
ro
and the loading direction, respectively. Apparently, Em becomes minimum at
4
j 2
, which is
re
-p
entirely reasonable.
lP
(2) Effect of the radius and normal density of discontinuities Fig. 9(b) shows the inverse correlation between the average radius of a set of discontinuities a and
na
the elastic modulus of the rock mass. The influence of the radius is significant for relatively smaller
Jo ur
cracks. However, when the size of the plane become large, e.g. 4 m in this case, the elastic modulus will tend toward lower values. It can be seen in (25) that the influence of the normal density of a set of planes λ is on the same scale and under the same conditions as a . (3) Effect of the strength of discontinuities The strength of discontinuities, i.e. the friction angle and cohesion, exhibits a positive correlation with the elastic modulus ratio of the rock mass (Figs. 9(c) and (d)). The elastic modulus of the rock mass increases with the friction angle and cohesion of the planes increase, reaching the modulus of the rock block while the discontinuities are locked by shear strength. (4) Effect of the stress state of discontinuities 22
Journal Pre-proof The stress state of discontinuities is also an important factor affecting the elastic modulus of rock mass. First, the influences of the compressive and tensional states at the discontinuities are absolutely different, as indicated in (26); i.e. the modulus of the rock mass under a tensional stress state is much less than that under a compressive state. Secondly, under a compressive stress state, the elastic modulus of the rock mass rapidly increases as the normal stress at the planes increases. Below a certain normal stress, the modulus of the rock mass will tend to be stable because the discontinuities are gradually locked (Fig.9(e)).
lP
re
-p
ro
of
This can be defined as the ‘stress lock’ of the discontinuities.
(b)
(c)
Jo ur
na
(a)
(d)
(e)
Fig. 9. Effects of various joint parameters on the elastic modulus of a rock mass: (a) inter angle, (b) average radius, (c) friction angle, (d) cohesion, and (e) stress state.
23
Journal Pre-proof
5. Conclusion The stress–strain relationship of fractured rock mass was derived based on fracture mechanics, rock-mass structure theory, and strain energy theory. This stress–strain relationship explains the influence of various geological factors on the deformation behaviour of rock mass. The elastic modulus of rock mass calculated through the proposed constitutive relationship fully accounts for the influence of
of
discontinuities in all directions and can reflect the weakening and anisotropic characteristics of the
ro
rock-mass elastic modulus. Compared with other constitutive models, the advantages of the SMRM
-p
model lie in its reliability, simplicity, and applicability. Additionally, our new model can better reflect the
re
geo-mechanical effects of discontinuous networks and can be widely applied as a theoretical fundamental model for mechanical research on rocks and practical engineering geological issues. However, the SMRM
na
deformation stage needs further research.
lP
model can only reflect the elastic deformation stage of fractured rock mass, the content of the plastic
Jo ur
Discontinuities are the main factor that weakens the elastic modulus of rock mass and produces anisotropy. The derived elastic modulus of rock mass is distributed as a ‘U’ curve according to the changes in axial stress and the normal angle of the discontinuities; it is lowest at an angle of /4+ /2 .
The elastic modulus of rock mass is positively related to the friction angle and cohesion of the discontinuities and is inversely correlated with the average radius and density of the discontinuities. Similarly, when the normal stress at the planes increase, the elastic modulus of the rock mass rapidly increases. However, the influence of the structure on the elastic modulus of the rock mass has a scaling effect. When the average radius of the discontinuities and the normal stress exceed a certain range, the
24
Journal Pre-proof
elastic modulus of the rock mass remains basically unchanged.
Acknowledgements We acknowledge the financial support of the National Natural Science Foundation of China (41831290) and Dr. Yihu Zhang from Changjiang River Scientific Research Institute for his help with the site
ro
re
-p
reviewers for their insightful and valuable comments.
of
monitoring data of Jindong Bridge test data. The authors also thank the editor Juang and two anonymous
References
Jo ur
na
lP
Conflicts of Interest: The authors declare no conflict of interest.
Alejano, L.R., Bobet, A., 2012. Drucker-Prager criterion. The ISRM Suggested Methods for Rock Characterization,
Testing
and
Monitoring:
2007-2014,
pp
247-252.
https://doi.org/10.1007/978-3-319-07713-0_22.
Bai, Y., Wierzbicki, T., 2010. Application of extended Mohr-Coulomb criterion to ductile fracture. Int. J. Fracture. 161(1), 1–20. https://doi.org/10.1007/s10704-009-9422-8.
Bao, H., Zhai, Y., Lan, H.X., Zhang, K.K., Qi, Q., Yan, C.G., 2019. Distribution characteristics and 25
Journal Pre-proof
controlling factors of vertical joint spacing in sand-mud interbedded strata. J Struct Geol. 128, 103886. https://doi.org/10.1016/j.jsg.2019.103886.
Bao, H., Zhang, G.B., Lan, H.X., Yan, C.G., Xu, J.B., Xu, W., 2020. Geometrical heterogeneity of the joint roughness coefficient revealed by 3D laser scanning. Eng Geol. 265, 105415. https://doi.org/10.1016/j.enggeo.2019.105415.
of
Bidgoli, M.N., Zhao, Z., Jing, L., 2013. Numerical evaluation of strength and deformability of fractured
ro
rocks. J. Rock Mech. Geotech. Eng. 5(6), 419–430. https://doi.org/10.1016/j.jrmge.2013.09.002.
-p
Chen, X., Liao, Z, Peng, X., 2012. Deformability characteristics of jointed rock masses under uniaxial
re
compression. Int. J. Min. Sci. Tech. 22(2), 213–221. https://doi.org/10.1016/j.ijmst.2011.08.012.
na
pp. 56–86.
lP
Fan, T.Y., 1978. Fracture Mechanics Foundation. Chinese Jiangsu Science and Technology Press, Nanjing,
Jo ur
Fossum, A.F., 1985. Effective elastic properties for a random jointed rock mass. Int. J. Rock Min. Sci. 22(6), 467–70. https://doi.org/10.1016/0148-9062(85)90011-7.
Gehle, C., Kutter, H.K., 2003. Breakage and shear behaviour of intermittent rock joints. Int. J. Rock Mech. Min. Sci. 40(5), 687–700. https://doi.org/10.1016/S1365-1609(03)00060-1.
Gerrard, C.M., 1982. Equivalent elastic moduli of a rock mass consisting of orthorhombic layers. Int. J. Rock Mech. Min. Sci. 19(1), 9–14. https://doi.org/10.1016/0148-9062(82)90705-7.
Gu, D.Z., 1979. Foundation of Rock and Soil. Engineering Geology. Chinese Beijing: Science Press.
Han, J.X., Li, S.C., Li, S.C., Tong, X.H., Li, W.T., 2011. Model study of strength and failure modes of 26
Journal Pre-proof
rock
mass
with
persistent
cracks.
Chin.
J.
Rock
Soil.
Mech.
32(2),
178–184.
https://doi.org/10.16285/j.rsm.2011.s2.019.
Hoek, E., Brown, E.T., 1980. Empirical strength criterion for rock masses. J. Geotech. Geo-environ. Eng. 106, ASCE 15715. https://doi.org/10.1016/0148-9062(81)90766-x.
Hoek E, Carranza-Torres C, Corkum B, 2002. Hoek-Brown failure criterion-2002 edition. Proceedings of
of
NARMS-Tac. Jul 7;1(1):267-73.
ro
Hoek E, Wood D, Shah S, 1992. A modified Hoek–Brown failure criterion for jointed rock masses. Rock
re
-p
Characterization: ISRM Symposium, Eurock'92, Chester, UK, 14–17 September.
Horii, H., Nemat-Nasser, S., 1983. Overall moduli of solids with microcracks: Load-induced anisotropy. J.
lP
Mech. Phys. Solids. 31(2), 155–171. https://doi.org/10.1016/0022-5096(83)90048-0.
na
Hu, X., Wu, F., Sun, Q., 2011. Elastic modulus of rock mass based om the two parameter
Jo ur
negative-exponential (TPNE) distribution of discontinuity spacing and trace length. Bull. Eng. Geol. Environ. 70(2), 255–263. https://doi.org/10.1007/s10064-010-0321-z.
Hudson, J.A., Priest, S.D., 1983. Discontinuity frequency in rock masses. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 20(2), 73–89. https://doi.org/10.1016/0148-9062(83)90329-7.
Kawamoto, T., Ichkawa, Y., Kyoya, T., 1988. Deformation and fracturing behavior of discontinuous rock mass and damage mechanics theory. Int. J. Num. Methods Geomech. 12(1), 1–30. https://doi.org/10.1002/nag.1610120102.
Li, S.C., Wang, L., Li, S.C., Han, J.X., 2013. Experimental study on post-peak deformation and failure of 27
Journal Pre-proof
rock specimens with different dip angles through joints. Chin. J. Rock Mech. Eng. 32(2), 3391– 3395.
Liu Y R, Tang H M, 1999. Rock mechanics. China University of Geosciences Press.
Main, I.G., Sammonds, P.R., Meredith, P.G., 1993. Application of a modified Griffith criterion to the evolution of fractal damage during compressional rock failure. Geophys. J. Int. 115(2), 367–380.
of
https://doi.org/10.1111/j.1365-246X.1993.tb01192.x.
ro
Oda, M., 1984. Similarity rule of crack geometry in statistically homogeneous rock masses. Mech. Mater.
re
-p
3(2), 119–29. https://doi.org/10.1016/0167-6636(84)90003-6.
Oda, M., 1986. An equivalent continuum model for coupled stress and fluid flow analysis in jointed rock
lP
masses. Water Resour. Res. 22(13), 1845–56. https://doi.org/10.1029/WR022i013p01845.
na
Saeidi, O., Rasouli, V., Vaneghi, R.G., Gholami, R., Torabi, S.R., 2014. A modified failure criterion for
Jo ur
transversely isotropic rocks. Geosci. Front. 5(2), 215–225. https://doi.org/10.1016/j.gsf.2013.05.005.
Wei, Z.Q., Hudson, J.A., 1988. The influence of joints on rock modulus. Proceedings of the International Symposium
on
Engineering
in
Complex
Rock
Formations.
Jan
1
(pp.
54-62).
https://doi.org/10.1016/B978-0-08-035894-9.50010-X.
Wu, F.Q., 1988. A 3D model of a jointed rock mass and its deformation properties. Int. J. Min. Geol. Eng. July, Volume 6, Issue 2, pp 169–176. https://doi.org/10.1007/BF00880806.
Wu, F.Q., Wang, S.J., Song, S.W., Lu, J.T., 1994. Statistical principles in mechanics of rock masses. Chin. Sci. Bull. 6, 493–503. 28
Journal Pre-proof
Wu, F.Q., Wang, S.J., 2001. A stress–strain relation for jointed rock masses. Int. J. Rock Mech. Min. Sci. 38(4), 591–598. https://doi.org/10.1016/S1365-1609(01)00027-2.
Xu, T., Ranjith, P.G., Wasantha, P.L.P., Zhao, J., Tang, C.A., Zhu, W.C., 2013. Influence of the geometry of partially-spanning joints on mechanical properties of rock in uniaxial compression. Eng. Geol. 167, 134–147. https://doi.org/10.1016/j.enggeo.2013.10.011.
of
Yashinaka, R, Yamabe, T., 1986. Joint stiffness and the deformation behavior of discontinuous rock. Int J
ro
Rock Mech Min Sci Geomech Abstr, 23(1), 19–28. doi:10.1016/0148-9062(86)91663-3
re
-p
Yin, S.Y., 1990. Elastoplastic mechanics. China University of Geosciences Press.
1729–1753.
lP
Weiyuan, Z.H., 2010. Structure stability of rock mass engineering. Chin. J. Rock Mech. Eng. 29(09),
na
Zhou, W., Zhao, J., Liu, Y., Yang, Q., 2002. Simulation of localization failure with
Jo ur
strain-gradient-enhanced damage mechanics. Int. J. Numer. Anal. Methods Geomech. 26(8), 793– 813. https://doi.org/10.1002/nag.225.
29
Journal Pre-proof
CRediT author statement
Faquan Wu: Conceptualization, Methodology, Writing - Review & Editing, Funding acquisition. Yi Deng: Writing- Original draft preparation, Software, Validation, Investigation. Jie Wu: Software, Project administration.
of
Bo Li: Supervision.
ro
Peng Sha: Formal analysis.
Kai Zhang: Writing - Review & Editing Preparation.
lP na
Shuhao Qiu: Validation.
Jo ur
Handong Liu: Resources.
re
Keqiang He: Investigation.
-p
Shenggong Guan: Visualization.
30
Journal Pre-proof Declaration of interests
The authors declare that they have no known competing financial interests or personal relationships that could have
Jo ur
na
lP
re
-p
ro
of
appeared to influence the work reported in this paper.
31
Journal Pre-proof
Highlights
A model for the elastic stress–strain relation of fractured rock mass is proposed.
The weakening and anisotropy of the elastic modulus of rock mass are notable.
The modulus is positively related to the normal stress at the planes.
It is positively related to the friction angle and cohesion of the plane.
It is inversely correlated to the average radius and density of discontinuities.
Jo ur
na
lP
re
-p
ro
of
32