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Zonal disintegration mechanism of deep crack-weakened rock masses under dynamic unloading
Acta Mechanica Solida Sinica, Vol. 22, No. 3, June, 2009 Published by AMSS Press, Wuhan, China
ISSN 0894-9166
ZONAL DISINTEGRATION MECHANISM OF DEEP CRACK-WEAKENED ROCK MASSES UNDER DYNAMIC UNLOADING Xiaoping Zhou1
Qihu Qian2
Bohu Zhang1,3
1
( School of Civil Engineering, Chongqing University, Chongqing 400045, China) (2 PLA University of Science and Technology, Nanjing 210007, China) 3 ( School of Civil Engineering and Architecture, Southwest Petroleum University, Chengdu 610500, China)
Received 20 November 2008, revision received 15 May 2009
ABSTRACT Size and quantity of fractured zone and non-fractured zone are controlled by cracks contained in deep rock masses. Zonal disintegration mechanism is strongly dependent on the interaction among cracks. The strong interaction among cracks is investigated using stress superposition principle and the Chebyshev polynomials expansion of the pseudo-traction. It is found from numerical results that crack nucleation, growth and coalescence lead to failure of deep crackweakened rock masses. The stress redistribution around the surrounding rock mass induced by unloading excavation is studied. The effect of the excavation time on nucleation, growth, interaction and coalescence of cracks was analyzed. Moreover, the influence of the excavation time on the size and quantity of fractured zone and non-fractured zone was given. When the excavation time is short, zonal disintegration phenomenon may occur in deep rock masses. It is shown from numerical results that the size and quantity of fractured zone increase with decreasing excavation time, and the size and quantity of fractured zone increase with the increasing value of in-situ geostress.
KEY WORDS deep crack-weakened rock masses, interaction among cracks, stress superposition principle, zonal disintegration mechanism, dynamic unloading
I. INTRODUCTION During the excavation of a tunnel in the deep rock masses, fractured zone and non-fractured zone occur alternately around deep tunnels, which has been referred as the zonal disintegration phenomenon and has never been observed in the shallow rock engineering before[1–9] . Moreover, the zonal disintegration phenomenon cannot be clearly explained by conventional elastoplasticity theory. The zonal structure of fracturing has been discovered in the processes of drilling and blasting of the opening and in the experiment. And it is observed that size and quantity of fractured zone and non-fractured zone are dependent on the excavation time, the strength of rock masses, the size of tunnel, etc. In order to interpret the mechanism of the zonal disintegration phenomenon, to analyze the law of zonal disintegration phenomenon and to numerically simulate zonal disintegration phenomenon, a new branch of rock mechanics-nonlinear deep rock mechanics is established[1] . Zonal disintegration phenomenon around the tunnels at the 2000-3000 meters deep in Witwatersrand golden mines in South Africa is initially observed with a periscope[2] . Subsequently, zonal disintegration phenomenon is observed widely in
Corresponding author. Tel: +86-23-65120720, E-mail: [email protected] Project supported by the National Natural Science Foundation of China (Nos.50490275 and 50778184).
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the surrounding rock mass in deep golden mines in South Africa[3] . Similarly, zonal disintegration phenomenon is also discovered in the Taimyrskii and Mayak mines in Russia[4] . The phenomenon of zonal disintegration of rocks introduces a radical change into the existing notion about the mechanism of interaction of the rock mass with engineering structures[5] . Several hypotheses on the shape and size of the working influence zone, which include the arch hypothesis, the friable medium hypothesis have been maintained for a long time in rock mechanics. Some of the publications have tried to discover a non-monotonic variation of stresses distant from the mine surface. This nonmonotonic variation was associated either with the stress-filled attenuation along the working, or with the periodic variation of stress in time[6–9] . Some theories , which link the cracks parallel to the periphery of the working with the effect of seismic waves from outside sources[10], have been suggested. However, some of them did not consider the effect of the joint and the fracture on the failure of the deep rock mass. Moreover, the others can not take into account the zonal disintegration phenomenon in the deep rock mass in the processes of drill and blasting. Rock masses are characterized by the existence of distributed joints and fractures, and the mechanical properties of jointed rock masses are thus strongly dependent on the properties and geometry of these weak planes[11] . In tunnels, fracturing is initiated as a result of the removal of the confining stress and the loading by tangential stress concentration[12]. The excavation methods and the speed of excavation significantly affect the quantity and size of fractured zone and non-fractured zone in deep rock masses. Thus, the dynamic unloading speed significantly affects the mechanism of zonal disintegration phenomenon around the surrounding opening. Shemyakin who first studied the zonal disintegration phenomenon of deep rock mass in 1986, obtained good results by means of in-situ observation, experimental test and theoretical analysis, but the exact mechanism of zonal disintegration phenomenon is not revealed clearly[8] . In order to better understand the fracture and failure of deep rock masses, mechanism of zonal disintegration phenomenon around deep tunnels must be studied. In Ref.[9], it is assumed that deep rock masses are elastoplastic, and the effect of the excavation time on the mechanism of zonal disintegration phenomenon was investigated using the elastoplastic theory. In Ref.[13], the effect of high in-situ geostress on the mechanism of zonal disintegration phenomenon was studied using the strength criterion of deep rock masses. To the authors’ knowledge, mechanism of zonal disintegration phenomenon in deep weakened-crack rock masses subjected to dynamic unloading is not investigated previously. For a better understanding of the zonal disintegration mechanism in deep rock mass engineering it is helpful to reveal the failure mechanism of the surrounding rock under dynamic unloading. In this paper, an approach to reveal the mechanism of zonal fractured phenomenon around surrounding rock masses under dynamic unloading is proposed. Then, the size and quantity of fractured zone and non-fractured zone can be determined using the present method.
II. THEORETICAL MODEL The strength criterion of deep rock masses can be expressed by[14] 1 I1 (aσ2 + σ3 ) + W (σ1 + σ3 + bσ2 ) F1 = σ1 − 1+a σc n bσ1 d 2 −σc + (aσ2 + σ3 ) + b =0 when F1 ≥ F2 σc (1 + a)σc F2 =
1 I1 (σ1 + aσ2 ) − σ3 + W (σ1 + σ3 + bσ2 ) 1+a σc n dσ3 bσ1 + aσ2 2 + −σc +b =0 when F1 < F2 σc (1 + a) σc
(1)
(2)
where a is a parameter of the strength criterion associated with the intermediate stress; σc is the uniaxial compressive strength of an intact rock material; W is the negative dilatancy coefficient; b, d and n are rock material parameters, whose magnitude depend on the geological strength index (GSI) or the rock mass rating (RMR) which characterizes the quality of rock masses. To simplify the treatment of zonal disintegration phenomenon in deep rock masses, n = 0.5 is used.
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Under plane strain condition, Eq.(1) is applied to analyze the zonal disintegration mechanism in deep crack-weakened rock masses. Introducing a parameter υ, the intermediate principal stress σ2 can be expressed as σ2 = υ(σ1 + σ2 ). The expression of the strength criterion of deep rock masses involving the Lambe parameters, p = 12 (σ1 + σ3 ) and q = 12 (σ1 − σ3 ), permits a simplified and normalized treatment of the rock masses failure phenomenon. The important concept of ‘instantaneous friction angle’ is defined by[13] sin β =
dq dp
(3)
where β is instantaneous friction angle (see Fig.1).
Fig. 1. The envelope of Mohr’s failure circles for the strength criterion of deep rock masses.
From Eq.(1), dq/dp can be obtained as below 1 dq = {−16pW σc(1 + a)(2 + a)(1 + υ)(1 + bυ) + 2aσc2 (2 + a)(−1 + 2υ) dp 2(2 + a)2 σc2 1 − (1 + a)σc2 [−32pW (2 + a)(b + ab − d)(1 + υ)(1 + bυ) + 8σc (2 + a)(b + d)(1 + aυ)] 2 /[−16p2 W (2 + a)(b + ab − d)(1 + υ)(1 + bυ) + 8pσc (2 + a)(b + d)(1 + aυ) 1
+17σc2 b2 + 18aσc2 b2 + 5a2 σc2 b2 − 2bdσc2 (1 + a) + d2 σc2 ] 2 }
(4)
The influence of W on the zonal fracturing phenomenon was discussed in Ref.[13]. For simplification, it is assumed that W = 0. From Eqs.(3) and (4), when W = 0, following equations can be obtained
where
σ = M − N sin β
(5)
τ = N cos β
(6)
σc −17b2 − 18ab2 − 5a2 b2 + 2bd + 2abd − d2 + H M= 8(2 + a)(b + d)(1 + aυ) σc [J + K cos(2β) + L sin β] 4(1 + a)2 (b + d)2 (1 + aυ)2 N= , H = 2 16(b + d)(1 + aυ)(a − 2aυ + 2 sin β + a sin β)2 [a − 2aυ + (2 + a) sin β] J = 4b2 + 27ab2 + 26a2 b2 + 15a3 b2 + 6abd + 6a2 bd − 4d2 − ad2 − 34ab2 υ −36a2 b2 υ − 58a3 b2 υ − 4abdυ + 8a2 bdυ + 12a3 bdυ − 10ad2 υ + 4a2 d2 υ + 96a3 b2 υ 2 −16a2 bdυ 2 − 16a3 bdυ 2 − 16a2 d2 υ 2 + 8a3 d2 υ 2 − 64a3 b2 υ 3 − 16a3 d2 υ 3 K = −4b2 − 25ab2 − 22a2 b2 − 5a3 b2 − 2abd − 2a2 bd + 4d2 + 3ad2 +30ab2 υ + 28a2 b2 υ + 6a3 b2 υ − 4abdυ − 8a2 bdυ − 4a3 bdυ + 6ad2 υ + 4a2 d2 υ L = 8b2 + 20ab2 + 48a2 b2 + 20a3 b2 + 16bd + 24abd + 8a2 bd + 8d2 + 4ad2 −128a2 b2 υ − 64a3 b2 υ + 32abdυ + 48a2 bdυ + 16a3 bdυ + 32ad2 υ + 16a2 d2 υ +128a2 b2 υ 2 + 64a3 b2 υ 2 + 32a2 d2 υ 2 + 16a3 d2 υ 2
Instantaneous friction angle β can be obtained as a function of σ, a, b and d from Eq.(5). (1) The stress field induced by excavation
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From works by Zhou and Qian[9] , the stress field induced by excavation can be obtained by 4 λ 1 −a (A1 + A2 + A3 ) + 2a4 R(A4 + A5 + A6 ) σr1 (r, t) = R (aR)3/2 (λ + 6μ)3 λ + 2μ 3a4 (A1 + A2 + A3 ) − 2a4 (A4 + A5 + A6 ) + 2a4 R(A7 + A8 + A9 ) + (7) R 2R 4 λ + 2μ 1 −a (A1 + A2 + A3 ) + 2a4 R(A4 + A5 + A6 ) σθ1 (r, t) = R (aR)3/2 (λ + 6μ)3 4 λ 3a (A1 + A2 + A3 ) − 2a4 (A4 + A5 + A6 ) + 2a4 R(A7 + A8 + A9 ) + (8) R 2R where A1 , A2 , A3 , A4 , A5 , A6 , A7 , A8 , A9 can be determined from Ref.[9]. (2) The elastic stress distribution induced by in-situ geostress For the geometry of the opening shown in Fig.2, the elastic stress distribution under plane strain condition can be written as
R2 R2 p1 − p2 p1 + p2 R4 σr2 = 1− 2 + 1 − 4 2 + 3 4 cos(2θ) (9) 2 r 2 r r
R2 R4 p1 − p2 p1 + p2 1+ 2 − 1 + 3 4 cos(2θ) (10) σθ2 = 2 r 2 r
R2 p1 − p2 R4 e 1 + 2 2 − 3 4 sin(2θ) (11) τrθ = − 2 r r where R is the radius of the opening, θ and r are the tangential and radial coordinate, respectively; σr and σθ are the radial and tangential stress, respectively; τrθ is the shear stress, p1 is the horizontal component of in-situ geostress (p1 = λγH); p2 is the vertical component of in-situ geostress (p2 = γH); H is the overburden depth, γ is the unit weight of the rock, and λ = p1 /p2 . From Eqs.(7)-(11), the total elastic stress fields can be determined by σθ = σθ1 + σθ2 ,
σr = σr1 + σr2 ,
e τrθ = τrθ
(12)
In this paper, it is assumed that rock masses containing a set of parallel cracks is planar, isotropic, and linear elastic. A global coordinate frame X-Y is attached to the body. Local coordinate frames xm -ym are attached to the center of main cracks, m = 1, 2 such that xm is along the crack line depicted in Fig.3. The half length of the crack is c, and the angle of local coordinate xm against the global coordinate X is θm .
Fig. 2. Geometry problem of application example.
Fig. 3. The parallel crack arrays around the opening.
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If crack interactions are neglected, the normal stress σ and shear stress τ acting at the any point (r, θ) on a plane dipping at an angle θm with respect to the X-axis can be written as σθ − σr σθ + σr + cos(2ξ0 ) − τrθ sin(2ξ0 ) 2 2 σθ − σr sin(2ξ0 ) + τrθ cos(2ξ0 ) τ= 2
σ=
(13)
where ξ0 = θm − θ, the angle θm and θ are defined in Fig.2. From Eqs.(5) and (13), instantaneous friction angle β can be obtained. Under high geostress condition, the stress redistribution induced by excavation leads to high compressive stress concentrations around the deep tunnel. Under high compression, it has been experimentally observed that some cracks may propagate into the matrix material in a nonself-similar fashion. These wing cracks typically grow gradually in the direction of tangential compressive stress with increasing tangential compression in a stable manner until certain lengths are attained, at which unstable growth begins and results in the ultimate failure of the rock material and the formation of fracturing zones. Therefore, each main crack m may have two kinks or branches pm = 2, as shown in Fig.4. The orientation of each main crack with respect to the X-direction is denoted by θm , while the kink orientation with respect to the X-direction is θm + θpm , where counterclockwise angles are positive. Furthermore, polar coordinates are defined with the origin at either main crack tip from the main crack plane, e.g. rpm and θpm denote positions on the kinks pm . The lengths of the main cracks and kinks are denoted by 2cm and lpm , respectively. To consider the influence of crack interactions on zonal fracturing phenomenon, the crack interaction can be taken into account by using stress superposition principle[15–18] . In order to study the problem of crack interactions, the original problem is decomposed into five subproblems, as shown in Fig.4. The first subproblem is a homogeneous problem of the uncracked solids under dynamic unloading. In the second and third subproblem, the main crack m is subjected to the internal loading by dislocations of its own kink. The main crack boundary conditions must be satisfied exactly in these subproblems for the proposed superposition method to yield accurate results. Exact satisfaction of these boundary conditions is easily achieved through the formulation of a Hilbert problem. In the fourth and fifth subproblems, initially unknown shear tractions qm act on the main crack m. For crack 1, the boundary condition of original problem under unloading excavation is τ + τx⊥11y1 + τx⊥22y2 + q1 + τxq12 y1 = f (σ + σy⊥11y1 + σy⊥12y1 + σyq21 y1 )
(14)
where f = tan β is the instantaneous friction coefficient. τx⊥11y1 and σy⊥11y1 are the shear stress and normal stress acting on crack 1 induced by the dislocation distribution of the kink of crack 1, respectively. τx⊥12y1 and σy⊥12y1 are the shear stress and normal stress acting on the crack 1 induced by the dislocation distribution of the kink of crack 2, respectively. q1 is the shear traction acting on crack 1. τxq12 y1 is the
Fig. 4. The problem of two ineracting kinked cracks.
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shear stress acting on the crack 1 induced by the shear traction on main crack 2. σyq21 y1 is the normal stress acting on the crack 1 induced by the shear traction on the main crack 2. σ and τ are the normal stress and shear stress acting on the crack 1, which can be determined by Eq.(13), respectively. The form of Eq.(14) in this paper is different from that of Eq.(1) in Ref.[16]. For the crack 2, the boundary condition of the original problem under unloading excavation is τ + τx⊥22y2 + τx⊥21y2 + q2 + τxq21 y2 = f (σ + σy⊥22y2 + σy⊥21y2 + σyq12 y2 )
(15)
where same notations are used for the shear and normal stresses acting on the main crack 2. The form of Eq.(15) in this paper is different from that of Eq.(2) in Ref.[16]. By the superposition of subproblems, the boundary conditions of both kinks of the crack 1 are given as τr1 θ1 + τr⊥11θ1 + τr⊥12θ1 + τrq11θ1 + τrq12θ1 = 0
(16)
σθ1 θ1 + σθ⊥11θ1 + σθ⊥12θ1 + σθq11 θ1 + σθq12 θ1 = 0
(17)
σθ − σr sin(2ξ1 ) + τrθ cos(2ξ1 ), ξ1 = θm + θpm − θ. τr⊥11θ1 , τr⊥12θ1 , τrq11θ1 , τrq12θ1 are the shear where τr1 θ1 = 2 stress induced by the dislocation distribution of the kink of crack 1, the shear stress induced by the dislocation distribution of the kink of crack 2 , the shear stress induced by the shear traction q1 acting on the crack 1, and the shear stress induced by the shear traction q2 acting on the crack 2, respectively. σθ − σr σθ + σr + cos(2ξ1 ) − τrθ sin(2ξ1 ). And σθ1 θ1 = 2 2 A single crack is considered and the xm -ym frame is attached to the crack center with x along the crack plane. According to Muskhelishvili[19] , we have σyy + σxx = 2[ϕ (z) + ϕ (z)]
(18)
σyy − σxx + i2τxy = 2[zϕ (z) − ψ (z)]
(19)
2G0 (ux + iuy ) = k0 ϕ(z) − zϕ (z) − ψ(z)
(20)
where k0 = 3 − 4ν0 for plane strain case, ν0 is the Poisson’s ratio, G0 is the shear modulus, ux and uy are displacement components, and z = x + iy. Equation (14), which enforces the boundary conditions on the main crack, can be rewritten as the sum of the five terms (τ − f σ) +
pm
⊥ (τxmpm ym
−
⊥ f σympm ym )
+
pm =1
pm
⊥p
⊥p
qm qm m (τxm m ym − f σym ym ) + qm + (τxm ym − f σym ym ) = 0 (21)
pm =1
where the symbol ⊥ denotes the dislocation on kink p of the crack m . The first term on the left side can be determined by Eq.(13), the second term on the left side vanishes because the Hilbert problem is solved exactly for each isolated crack subjected to the loading by dislocations of the own kink. The third term can be obtained from[15–18]
lp ⊥pm ⊥pm m σym ym + iτxm ym = αp (zop )[U (zmm , zop , θmm ) + U (zmm , zop , θmm )] m m m m 0 +αp (zop ) U (zmm , zop , θmm ) + U (zmm , zop , θmm ) drop (22) m
m
m
m
m
where U (zmm , zopm , θmm ), V (zmm , zopm , θmm ),V (zmm , zopm , θmm ) , U(zmm , zopm , θmm ) were given by Zhou[16] . It is found that Chebyshev polynomials Zn , which have been used to analyze the interaction between straight cracks, can also be used here. Consequently, with respect to the local x-coordinate attached to the center of the main crack, the fourth term in Eq.(21) is given by qm =
N n=0
bm n [Zn (ξ)]
(23)
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where bm n denotes the (N + 1) unknown coefficients to be determined for each crack, ξ = x/c (−1 ≤ ξ ≤ 1) is the normalized local crack coordinate, and the recurrence formula for Zn (ξ) is n+1 n+1 ξ − ξ2 − 1 − ξ + ξ2 − 1 (24) Zn (ξ) = 2 ξ2 − 1
The Muskhelishvili potentials ϕn (z), ψ n (z) for Zn (ξ) are obtained by suitable superpositions of elementary potentials[16] . Using Eqs.(11), (22), (23) and (24), Eq.(21) can be rewritten as
lp pm N M m (τ − f σ) + bmn [−Ln (ξ)] + {Im{ αpm (zopm )[U (zmm , zopm , θmm ) m =1 p
n=0
m
0
=1
+U (zmm , zopm , θmm )]dropm } − f Re{
lp
m
0
αpm (zopm )[U (zmm , zopm , θmm )
+U (zmm , zopm , θmm )]dropm }}
lp pm M m + {Im{ αpm (zopm )[V (zmm , zopm , θmm ) + V (zmm , zopm , θmm )]dropm } m =1 p
m
−f Re{ +
M
lp
m
0
{Im[
m =1
0
=1
αpm (zopm )[V (zmm , zopm , θmm ) + V (zmm , zopm , θmm )]dropm }}
N
n bm n gmm (zmm )] − f Re[
n=0
N
n bm n gmm (zmm )]} = 0
(25)
n=0
Equation (25) becomes a system of M simultaneous real Fredholm integral equations of the first kind. Similarly, according to the boundary condition in Eq.(16), the complex equation for the kink pm becomes
lpm α (z pm l M p pm opm ) m + [αpm (zopm )U (zmm , zopm , θpmm ) 2 zpmm − zopm 0 0 m =1 p
m
+α
pm
(z
opm
)V (zmm , z
opm
=1
, θpmm )]dr
opm
+
pm M m =1 p
m
=1
0
lp
m
[αpm (zopm )U (zmm , zopm , θpmm )
+αpm (zopm )V (zmm , zopm , θpmm )]dropm +σ + iτ +
M N
n bm n gmm )=0 (zp mm
(26)
m =1 n=0
where zpmm = zcm − zcm + [(c + rpm eiθpm )eiθm ]e−iθm , θpmm = θpm + θm − θm . Equation (26) leads to a system of 2pm real singular equations of the Cauchy type. According to works by Gerasoulis[20],the mode I and mode II stress intensity factors KI and KII are easily determined by taking the limit of the product of the singular stress field. This yields KI + iKII = π 3/2 (2lpm )1/2 eiθpm Apm (1)
(27)
Under dynamic loads, crack growth velocities have a great influence on the dynamic SIF. In most cases, the mode I dynamic SIF can be expressed as[21, 22] KID = k1 (v)KI
(28)
where KID is the mode I dynamic SIF, KI is the mode I static SIF, k1 (v) is a function of crack growth velocity v. The function k1 (v) = 1.0, when v = 0, and k1 (v) = 0 when v reaches the critical velocity of the crack growth, which is normally regarded as the velocity of a Rayleigh wave in the rock material.
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The mode I dynamic stress intensity factor, for two-dimension in-plane crack growth, has been established[21] . The dynamic stress intensity factor can be estimated by multiplying the equivalent static stress intensity factor by an approximate formulation of k1 (v) for brittle rock material, i.e. vr − v k1 (v) = (29) vr − 0.75v when the tensile crack is subjected to a pair of splitting forces at its center and vr − v (30) k1 (v) = vr − 0.5v when the tensile crack is subjected only to far field uniform stress, where vr is the velocity of Rayleigh wave, v = dl/dt . For the mode II crack, the dynamic crack tip SIF KIID and the static crack tip SIF KII are correlated by KIID = k2 (v)KII (31) An approximate formulation of k2 (v) is given as[21, 22] vr − v (32) k2 (v) = vr − 0.9v when the crack is loaded only by a pair of concentrated forces at its center. vr − v k2 (v) = (33) vr − 0.65v when the crack is loaded only by far field uniform stresses, where vr is the velocity of the Rayleigh wave. The crack growth criterion can be determined by
2
2 KID KIID + =1 (34) d d KIC KIIC d d is the Mode I critical dynamic stress intensity factor, KIIC is the Mode II critical dynamic where KIC stress intensity factor. Cracks satisfying the criterion (34) will nucleate, propagate and coalesce. If cracks coalesce, the fractured zone will be observed.
III. NUMERICAL RESULTS AND DISCUSSION 3.1. Effect of Excavation Time t on Zonal Disintegration Phenomenon in Deep Rock Masses In the process of numerical simulation, it is assumed that the deep tunnel is constructed in granite. The excavation unloading rate is calculated by the failure strain divided by the unloading time t. Experimental studies on granite under dynamic unloading showed that failure strain of granite under strain rate 10−2 s−1 is almost 1000 micro-strain. Therefore, corresponding to strain rate of 10−2 s−1 , unloading time t is 1 × 102 ms. Under dynamic unloading, when Eq.(34) is satisfied, the crack starts to grow. The initial time of crack growth is t1 . After t1 , the crack grows continuously and results in the failure of the rock material. The crack growth time is then t − t1 . Under dynamic unloading, t1 is about 0.05 s. If the crack spacing is 2e = 4 m, the crack growth velocity v = 2e/(t − t1 ) is approximated to be 4/(1 × 102 × 10−3 − 0.05) = 80 (m/s). The velocity of the Rayleigh wave for the granite is about 2000 m/s, the function k1 (v) can then be determined from Eqs.(29) and (30), the function k2 (v) can be obtained from Eqs.(32) and (33). The crack growth length can be obtained from Eq.(34). In the numerical simulation, it is assumed that √ d c = 0.5 m, e = 2 m, h = 0.5 m, a = 0.5, mi = 10, RMR = 60, KIC = 2 MPa/ m, υ = 0.5 W = 0, R = 3 m, θm = −65◦, γ = 27.2 kN/m3 , E = 20 GPa,√ ν0 = 0.25, vr = 2000 m/s d p1 = 50 MPa, p2 = 70 MPa, σc = 30 MPa, KIIC = 0.9 MPa/ m The numerical results are shown in Figs.5-7. In Figs.5-7, the shadow parts are fractured zones, the blank parts are non-fractured zones. It is found from Figs.5-7 that when excavation time t is short, zonal fracturing may occur in deep rock masses. For certain excavation time t, the quantity of the fractured zone is defined, the size of the fractured zone decreases with increasing distance from the surface of opening; on the other hand, the quantity of the fractured zone increases with decreasing excavation time t.
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Fig. 5. Fractured zones around deep circular tunnel when t = 0.1 s.
Fig. 6. Fractured zones around deep circular tunnel when t = 1 s.
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Fig. 7. Fractured zones around deep circular tunnel when t = 10 s.
3.2. Influence of p2 /σc on Zonal Disintegration Phenomenon in Deep Rock Masses In the numerical simulation, it is assumed that √ d = 2 MPa/ m, υ = 0.5 c = 0.5 m, e = 2 m, h = 0.5 m, a = 0.5, mi = 10, RMR = 60, KIC 3 W = 0, R = 3 m, θm = −65◦ , γ = 27.2 kN/m√ , E = 20 GPa, ν0 = 0.25, vr = 2000 m/s d p1 = 50 MPa, p2 = 70 MPa, KIIC = 0.9 MPa/ m
Fig. 8. Fractured zones around deep circular tunnel when t = 0.1 s, p2 /σc = 2.33.
Fig. 9. Fractured zones around deep circular tunnel when t = 0.1 s, p2 /σc = 1.56.
Fig. 10. Fractured zones around deep circular tunnel when t = 0.1 s, p2 /σc = 1.17.
The numerical results are shown in Figs.8-16. In Figs.8-16, the shadow parts are fractured zones, the blank parts are non-fractured zones.
Fig. 11. Fractured zones around deep circular tunnel when t = 1 s, p2 /σc = 2.33.
Fig. 12. Fractured zones around deep circular tunnel when t = 1 s, p2 /σc = 1.56.
Fig. 13. Fractured zones around deep circular tunnel when t = 1 s, p2 /σc = 1.17.
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Fig. 14. Fractured zones around deep circular tunnel when t = 10 s, p2 /σc = 2.33.
Fig. 15. Fractured zones around deep circular tunnel when t = 10 s, p2 /σc = 1.56.
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Fig. 16. Fractured zones around deep circular tunnel when t = 10 s, p2 /σc = 1.17.
It is found from Figs.8-16 that: (1) Cracks satisfying the crack growth criterion in Eq.(34) will nucleate, propagate and coalesce, then the fractured zone shows up, rock masses fail; (2) The size and quantity of the fractured zone increase with the increasing in-situ stress; (3) The numerical results are in good agreement with the experimental results as by Shemyakin[6] . In order to determine the condition of the onset of zonal disintegration phenomenon, we also studied the zonal disintegration phenomenon when the value of the in-situ stress is less than the uniaxial compressive strength of rock masses. It is found from numerical results that the zonal disintegration phenomenon does not occur once the value of the in-situ stress is less than the uniaxial compressive strength of rock masses, that is, the zonal disintegration phenomenon occurs only when the value of in-situ geostress is higher than the uniaxial compressive strength of rock masses.
IV. CONCLUSIONS (1) The crack growth criterion in Eq.(34) is applied to analyze the zonal disintegration phenomenon of deep rock masses. Cracks satisfying the crack growth criterion will nucleate, propagate and coalesce, then, the fractured zone will occur, and finally rock masses fail. (2) Excavation time significantly affects the size and quantity of the fractured zone around the surrounding rock mass. When the excavation time is short, the zonal disintegration phenomenon may occur in deep rock masses. For certain excavation time, the quantity of fractured zone is defined. Moreover, the size of the fracture zone decreases with increasing distance from the opening surface. (3) It is found from numerical results that the quantity of the fractured zone increases with decreasing excavation time. The size and the quantity of fractured zone increase with the increase of in-situ geostress. (4) Zonal fracturing phenomenon occurs only when the value of the in-situ geostress is higher than the uniaxial compressive strength of rock masses.
References [1] Qian,Q.H., The current development of nonlinear rock mechanicsthe mechanical problems of deep rock mass. In: Chinese Society of Rock Mechanics and Engineering, Proceedings of the 8th Rock Mechanics and Engineering Conference. Beijing: Science Press, 2004, 10-17 (in Chinese). [2] Cloete,D.R. and Jager,A.J., The nature of the fracture zone in gold mines as revealed by diamond core drilling. Association of Mine Managers, 1972-1973. [3] Adms,G.R. and Jager,A.J., Petroscopic observations of rock fracturing ahead of stope faces in deep-level gold mine. Journal of the South African Institute of Mining and Metallurgy, 1980, 80(6): 204-209. [4] Shemyakin,E.I., Fisenko,G.L., Kurlenya,M.V. and Oparin,V.N., Zone disintegration of rocks around underground workings—Part I: Data of in-site observations. Journal of Mining Science, 1986, 22(3): 157-168. [5] Shemyakin,E.I., Fisenko,G.L., Kurlenya,M.V. and Oparin,V.N., Zone disintegration of rocks around underground workings-Part II: Disintegration of rocks on models of equivalent materials. Journal of Mining Science, 1986, 22(4): 223-232. [6] Shemyakin,E.I., Fisenko,G.L., Kurlenya,M.V. and Oparin,V.N., Zone disintegration of rocks around underground workings—Part III: Theroetical notions. Jounal of Mining Science, 1987, 23(1): 1-6.
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[7] Shemyakin,E.I., Fisenko,G.L., Kurlenya,M.V. and Oparin,V.N., Zone disintegration of rocks around underground workings—Part IV: Practical applications. Jounal of Mining Science, 1988, 24(1): 238-241. [8] Shemyakin,E.I., Kyrlenya,M.V. and Reva,V.N., Effect of zonal disintegration of rocks around underground workings. USSR, 1986, 289(5): 1088-1094. [9] Zhou,X.P. and Qian,Q.H., Zonal fracturing mechanism in deep tunnel. Chinese Journal of Rock Mechanics and Engineering, 2007, 26(5): 877-885 (in Chinese). [10] Mirzaev,G.G., Protosenya,A.G., Ogorodnikow,Y.N. and Vikhorev, V.I., Support Systemsof Deep Mines. Moscow: Bedra, 1976 (in Russian). [11] Cai,M. and Horii,H., A constitutive model and FEM analysis of jointed rock masses. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 1993, 30(4): 351-359. [12] Cai,M., Kaiser,P.K., Tasaka,Y., Maejima,T., Morioka,H. and Minami,M., Generalized crack initiation and crack damage stress thresholds of brittle rock masses near underground excavations. International Journal of Rock Mechanics and Mining Sciences, 2004, 41(5): 833-847. [13] Zhou,X.P., Wang,F.H., Qian,Q.H. and Zhang,B.H., Zonal fracturing mechanism in deep crack-weakened rock masses. Theoretical and Applied Fracture Mechanics, 2008, 50(1): 57-65. [14] Zhou,X.P., Qian,Q.H. and Yang,H.Q., Strength criteria of deep rock mass. Chinese Journal of Rock Mechanics and Engineering, 2008, 27(1): 117-123 (in Chinese). [15] Lo,K.K., Analysis of branched cracks. ASME Journal of Applied Mechanics, 1978, 45: 797-802. [16] Zhou,X.P., Qian,Q.H. and Zhang,Y.X., The constitutive relation of crack-weakened rock masses under axial-dimensional unloading. Acta Mechanica Solida Sinica, 2008, 21(3): 221-231. [17] Horii,H. and Nemat-Nasser,S., Brittle failure in compression: splitting, faulting and brittle-ductile transition. Philosophical Transactions of the Royal Society of London, 1986, A319: 337-374. [18] Niu,J. and Wu,M.S., Analysis of asymmetric kinked cracks of arbitrary size, location and orientation—Part I. Remote Compression. International Journal of Fracture, 1998, 89(1): 19-57. [19] Muskhelishvili,N.L., Some Basic Problems in the Mathematical Theory of Elasticity. Netherlands: Noordhoff, Groningen, 1953. [20] Gerasoulis,A., The use of quadratic polynomials for the solution of singular integral equations of cauchy type. Computer & Mathmatics with Applications, 1982, 8(1): 15-22. [21] Freund,L.B., Dynamic Fracture Mechanics. Cambridge University Press, 1990. [22] Cherepanov,G.P., Mechanics of Brittle Fracture. New York: Translated by R de Wit and W C Ccoley, McGraw-Hill, 1979.
ISSN 0894-9166
ZONAL DISINTEGRATION MECHANISM OF DEEP CRACK-WEAKENED ROCK MASSES UNDER DYNAMIC UNLOADING Xiaoping Zhou1
Qihu Qian2
Bohu Zhang1,3
1
( School of Civil Engineering, Chongqing University, Chongqing 400045, China) (2 PLA University of Science and Technology, Nanjing 210007, China) 3 ( School of Civil Engineering and Architecture, Southwest Petroleum University, Chengdu 610500, China)
Received 20 November 2008, revision received 15 May 2009
ABSTRACT Size and quantity of fractured zone and non-fractured zone are controlled by cracks contained in deep rock masses. Zonal disintegration mechanism is strongly dependent on the interaction among cracks. The strong interaction among cracks is investigated using stress superposition principle and the Chebyshev polynomials expansion of the pseudo-traction. It is found from numerical results that crack nucleation, growth and coalescence lead to failure of deep crackweakened rock masses. The stress redistribution around the surrounding rock mass induced by unloading excavation is studied. The effect of the excavation time on nucleation, growth, interaction and coalescence of cracks was analyzed. Moreover, the influence of the excavation time on the size and quantity of fractured zone and non-fractured zone was given. When the excavation time is short, zonal disintegration phenomenon may occur in deep rock masses. It is shown from numerical results that the size and quantity of fractured zone increase with decreasing excavation time, and the size and quantity of fractured zone increase with the increasing value of in-situ geostress.
KEY WORDS deep crack-weakened rock masses, interaction among cracks, stress superposition principle, zonal disintegration mechanism, dynamic unloading
I. INTRODUCTION During the excavation of a tunnel in the deep rock masses, fractured zone and non-fractured zone occur alternately around deep tunnels, which has been referred as the zonal disintegration phenomenon and has never been observed in the shallow rock engineering before[1–9] . Moreover, the zonal disintegration phenomenon cannot be clearly explained by conventional elastoplasticity theory. The zonal structure of fracturing has been discovered in the processes of drilling and blasting of the opening and in the experiment. And it is observed that size and quantity of fractured zone and non-fractured zone are dependent on the excavation time, the strength of rock masses, the size of tunnel, etc. In order to interpret the mechanism of the zonal disintegration phenomenon, to analyze the law of zonal disintegration phenomenon and to numerically simulate zonal disintegration phenomenon, a new branch of rock mechanics-nonlinear deep rock mechanics is established[1] . Zonal disintegration phenomenon around the tunnels at the 2000-3000 meters deep in Witwatersrand golden mines in South Africa is initially observed with a periscope[2] . Subsequently, zonal disintegration phenomenon is observed widely in
Corresponding author. Tel: +86-23-65120720, E-mail: [email protected] Project supported by the National Natural Science Foundation of China (Nos.50490275 and 50778184).
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the surrounding rock mass in deep golden mines in South Africa[3] . Similarly, zonal disintegration phenomenon is also discovered in the Taimyrskii and Mayak mines in Russia[4] . The phenomenon of zonal disintegration of rocks introduces a radical change into the existing notion about the mechanism of interaction of the rock mass with engineering structures[5] . Several hypotheses on the shape and size of the working influence zone, which include the arch hypothesis, the friable medium hypothesis have been maintained for a long time in rock mechanics. Some of the publications have tried to discover a non-monotonic variation of stresses distant from the mine surface. This nonmonotonic variation was associated either with the stress-filled attenuation along the working, or with the periodic variation of stress in time[6–9] . Some theories , which link the cracks parallel to the periphery of the working with the effect of seismic waves from outside sources[10], have been suggested. However, some of them did not consider the effect of the joint and the fracture on the failure of the deep rock mass. Moreover, the others can not take into account the zonal disintegration phenomenon in the deep rock mass in the processes of drill and blasting. Rock masses are characterized by the existence of distributed joints and fractures, and the mechanical properties of jointed rock masses are thus strongly dependent on the properties and geometry of these weak planes[11] . In tunnels, fracturing is initiated as a result of the removal of the confining stress and the loading by tangential stress concentration[12]. The excavation methods and the speed of excavation significantly affect the quantity and size of fractured zone and non-fractured zone in deep rock masses. Thus, the dynamic unloading speed significantly affects the mechanism of zonal disintegration phenomenon around the surrounding opening. Shemyakin who first studied the zonal disintegration phenomenon of deep rock mass in 1986, obtained good results by means of in-situ observation, experimental test and theoretical analysis, but the exact mechanism of zonal disintegration phenomenon is not revealed clearly[8] . In order to better understand the fracture and failure of deep rock masses, mechanism of zonal disintegration phenomenon around deep tunnels must be studied. In Ref.[9], it is assumed that deep rock masses are elastoplastic, and the effect of the excavation time on the mechanism of zonal disintegration phenomenon was investigated using the elastoplastic theory. In Ref.[13], the effect of high in-situ geostress on the mechanism of zonal disintegration phenomenon was studied using the strength criterion of deep rock masses. To the authors’ knowledge, mechanism of zonal disintegration phenomenon in deep weakened-crack rock masses subjected to dynamic unloading is not investigated previously. For a better understanding of the zonal disintegration mechanism in deep rock mass engineering it is helpful to reveal the failure mechanism of the surrounding rock under dynamic unloading. In this paper, an approach to reveal the mechanism of zonal fractured phenomenon around surrounding rock masses under dynamic unloading is proposed. Then, the size and quantity of fractured zone and non-fractured zone can be determined using the present method.
II. THEORETICAL MODEL The strength criterion of deep rock masses can be expressed by[14] 1 I1 (aσ2 + σ3 ) + W (σ1 + σ3 + bσ2 ) F1 = σ1 − 1+a σc n bσ1 d 2 −σc + (aσ2 + σ3 ) + b =0 when F1 ≥ F2 σc (1 + a)σc F2 =
1 I1 (σ1 + aσ2 ) − σ3 + W (σ1 + σ3 + bσ2 ) 1+a σc n dσ3 bσ1 + aσ2 2 + −σc +b =0 when F1 < F2 σc (1 + a) σc
(1)
(2)
where a is a parameter of the strength criterion associated with the intermediate stress; σc is the uniaxial compressive strength of an intact rock material; W is the negative dilatancy coefficient; b, d and n are rock material parameters, whose magnitude depend on the geological strength index (GSI) or the rock mass rating (RMR) which characterizes the quality of rock masses. To simplify the treatment of zonal disintegration phenomenon in deep rock masses, n = 0.5 is used.
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Under plane strain condition, Eq.(1) is applied to analyze the zonal disintegration mechanism in deep crack-weakened rock masses. Introducing a parameter υ, the intermediate principal stress σ2 can be expressed as σ2 = υ(σ1 + σ2 ). The expression of the strength criterion of deep rock masses involving the Lambe parameters, p = 12 (σ1 + σ3 ) and q = 12 (σ1 − σ3 ), permits a simplified and normalized treatment of the rock masses failure phenomenon. The important concept of ‘instantaneous friction angle’ is defined by[13] sin β =
dq dp
(3)
where β is instantaneous friction angle (see Fig.1).
Fig. 1. The envelope of Mohr’s failure circles for the strength criterion of deep rock masses.
From Eq.(1), dq/dp can be obtained as below 1 dq = {−16pW σc(1 + a)(2 + a)(1 + υ)(1 + bυ) + 2aσc2 (2 + a)(−1 + 2υ) dp 2(2 + a)2 σc2 1 − (1 + a)σc2 [−32pW (2 + a)(b + ab − d)(1 + υ)(1 + bυ) + 8σc (2 + a)(b + d)(1 + aυ)] 2 /[−16p2 W (2 + a)(b + ab − d)(1 + υ)(1 + bυ) + 8pσc (2 + a)(b + d)(1 + aυ) 1
+17σc2 b2 + 18aσc2 b2 + 5a2 σc2 b2 − 2bdσc2 (1 + a) + d2 σc2 ] 2 }
(4)
The influence of W on the zonal fracturing phenomenon was discussed in Ref.[13]. For simplification, it is assumed that W = 0. From Eqs.(3) and (4), when W = 0, following equations can be obtained
where
σ = M − N sin β
(5)
τ = N cos β
(6)
σc −17b2 − 18ab2 − 5a2 b2 + 2bd + 2abd − d2 + H M= 8(2 + a)(b + d)(1 + aυ) σc [J + K cos(2β) + L sin β] 4(1 + a)2 (b + d)2 (1 + aυ)2 N= , H = 2 16(b + d)(1 + aυ)(a − 2aυ + 2 sin β + a sin β)2 [a − 2aυ + (2 + a) sin β] J = 4b2 + 27ab2 + 26a2 b2 + 15a3 b2 + 6abd + 6a2 bd − 4d2 − ad2 − 34ab2 υ −36a2 b2 υ − 58a3 b2 υ − 4abdυ + 8a2 bdυ + 12a3 bdυ − 10ad2 υ + 4a2 d2 υ + 96a3 b2 υ 2 −16a2 bdυ 2 − 16a3 bdυ 2 − 16a2 d2 υ 2 + 8a3 d2 υ 2 − 64a3 b2 υ 3 − 16a3 d2 υ 3 K = −4b2 − 25ab2 − 22a2 b2 − 5a3 b2 − 2abd − 2a2 bd + 4d2 + 3ad2 +30ab2 υ + 28a2 b2 υ + 6a3 b2 υ − 4abdυ − 8a2 bdυ − 4a3 bdυ + 6ad2 υ + 4a2 d2 υ L = 8b2 + 20ab2 + 48a2 b2 + 20a3 b2 + 16bd + 24abd + 8a2 bd + 8d2 + 4ad2 −128a2 b2 υ − 64a3 b2 υ + 32abdυ + 48a2 bdυ + 16a3 bdυ + 32ad2 υ + 16a2 d2 υ +128a2 b2 υ 2 + 64a3 b2 υ 2 + 32a2 d2 υ 2 + 16a3 d2 υ 2
Instantaneous friction angle β can be obtained as a function of σ, a, b and d from Eq.(5). (1) The stress field induced by excavation
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From works by Zhou and Qian[9] , the stress field induced by excavation can be obtained by 4 λ 1 −a (A1 + A2 + A3 ) + 2a4 R(A4 + A5 + A6 ) σr1 (r, t) = R (aR)3/2 (λ + 6μ)3 λ + 2μ 3a4 (A1 + A2 + A3 ) − 2a4 (A4 + A5 + A6 ) + 2a4 R(A7 + A8 + A9 ) + (7) R 2R 4 λ + 2μ 1 −a (A1 + A2 + A3 ) + 2a4 R(A4 + A5 + A6 ) σθ1 (r, t) = R (aR)3/2 (λ + 6μ)3 4 λ 3a (A1 + A2 + A3 ) − 2a4 (A4 + A5 + A6 ) + 2a4 R(A7 + A8 + A9 ) + (8) R 2R where A1 , A2 , A3 , A4 , A5 , A6 , A7 , A8 , A9 can be determined from Ref.[9]. (2) The elastic stress distribution induced by in-situ geostress For the geometry of the opening shown in Fig.2, the elastic stress distribution under plane strain condition can be written as
R2 R2 p1 − p2 p1 + p2 R4 σr2 = 1− 2 + 1 − 4 2 + 3 4 cos(2θ) (9) 2 r 2 r r
R2 R4 p1 − p2 p1 + p2 1+ 2 − 1 + 3 4 cos(2θ) (10) σθ2 = 2 r 2 r
R2 p1 − p2 R4 e 1 + 2 2 − 3 4 sin(2θ) (11) τrθ = − 2 r r where R is the radius of the opening, θ and r are the tangential and radial coordinate, respectively; σr and σθ are the radial and tangential stress, respectively; τrθ is the shear stress, p1 is the horizontal component of in-situ geostress (p1 = λγH); p2 is the vertical component of in-situ geostress (p2 = γH); H is the overburden depth, γ is the unit weight of the rock, and λ = p1 /p2 . From Eqs.(7)-(11), the total elastic stress fields can be determined by σθ = σθ1 + σθ2 ,
σr = σr1 + σr2 ,
e τrθ = τrθ
(12)
In this paper, it is assumed that rock masses containing a set of parallel cracks is planar, isotropic, and linear elastic. A global coordinate frame X-Y is attached to the body. Local coordinate frames xm -ym are attached to the center of main cracks, m = 1, 2 such that xm is along the crack line depicted in Fig.3. The half length of the crack is c, and the angle of local coordinate xm against the global coordinate X is θm .
Fig. 2. Geometry problem of application example.
Fig. 3. The parallel crack arrays around the opening.
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If crack interactions are neglected, the normal stress σ and shear stress τ acting at the any point (r, θ) on a plane dipping at an angle θm with respect to the X-axis can be written as σθ − σr σθ + σr + cos(2ξ0 ) − τrθ sin(2ξ0 ) 2 2 σθ − σr sin(2ξ0 ) + τrθ cos(2ξ0 ) τ= 2
σ=
(13)
where ξ0 = θm − θ, the angle θm and θ are defined in Fig.2. From Eqs.(5) and (13), instantaneous friction angle β can be obtained. Under high geostress condition, the stress redistribution induced by excavation leads to high compressive stress concentrations around the deep tunnel. Under high compression, it has been experimentally observed that some cracks may propagate into the matrix material in a nonself-similar fashion. These wing cracks typically grow gradually in the direction of tangential compressive stress with increasing tangential compression in a stable manner until certain lengths are attained, at which unstable growth begins and results in the ultimate failure of the rock material and the formation of fracturing zones. Therefore, each main crack m may have two kinks or branches pm = 2, as shown in Fig.4. The orientation of each main crack with respect to the X-direction is denoted by θm , while the kink orientation with respect to the X-direction is θm + θpm , where counterclockwise angles are positive. Furthermore, polar coordinates are defined with the origin at either main crack tip from the main crack plane, e.g. rpm and θpm denote positions on the kinks pm . The lengths of the main cracks and kinks are denoted by 2cm and lpm , respectively. To consider the influence of crack interactions on zonal fracturing phenomenon, the crack interaction can be taken into account by using stress superposition principle[15–18] . In order to study the problem of crack interactions, the original problem is decomposed into five subproblems, as shown in Fig.4. The first subproblem is a homogeneous problem of the uncracked solids under dynamic unloading. In the second and third subproblem, the main crack m is subjected to the internal loading by dislocations of its own kink. The main crack boundary conditions must be satisfied exactly in these subproblems for the proposed superposition method to yield accurate results. Exact satisfaction of these boundary conditions is easily achieved through the formulation of a Hilbert problem. In the fourth and fifth subproblems, initially unknown shear tractions qm act on the main crack m. For crack 1, the boundary condition of original problem under unloading excavation is τ + τx⊥11y1 + τx⊥22y2 + q1 + τxq12 y1 = f (σ + σy⊥11y1 + σy⊥12y1 + σyq21 y1 )
(14)
where f = tan β is the instantaneous friction coefficient. τx⊥11y1 and σy⊥11y1 are the shear stress and normal stress acting on crack 1 induced by the dislocation distribution of the kink of crack 1, respectively. τx⊥12y1 and σy⊥12y1 are the shear stress and normal stress acting on the crack 1 induced by the dislocation distribution of the kink of crack 2, respectively. q1 is the shear traction acting on crack 1. τxq12 y1 is the
Fig. 4. The problem of two ineracting kinked cracks.
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shear stress acting on the crack 1 induced by the shear traction on main crack 2. σyq21 y1 is the normal stress acting on the crack 1 induced by the shear traction on the main crack 2. σ and τ are the normal stress and shear stress acting on the crack 1, which can be determined by Eq.(13), respectively. The form of Eq.(14) in this paper is different from that of Eq.(1) in Ref.[16]. For the crack 2, the boundary condition of the original problem under unloading excavation is τ + τx⊥22y2 + τx⊥21y2 + q2 + τxq21 y2 = f (σ + σy⊥22y2 + σy⊥21y2 + σyq12 y2 )
(15)
where same notations are used for the shear and normal stresses acting on the main crack 2. The form of Eq.(15) in this paper is different from that of Eq.(2) in Ref.[16]. By the superposition of subproblems, the boundary conditions of both kinks of the crack 1 are given as τr1 θ1 + τr⊥11θ1 + τr⊥12θ1 + τrq11θ1 + τrq12θ1 = 0
(16)
σθ1 θ1 + σθ⊥11θ1 + σθ⊥12θ1 + σθq11 θ1 + σθq12 θ1 = 0
(17)
σθ − σr sin(2ξ1 ) + τrθ cos(2ξ1 ), ξ1 = θm + θpm − θ. τr⊥11θ1 , τr⊥12θ1 , τrq11θ1 , τrq12θ1 are the shear where τr1 θ1 = 2 stress induced by the dislocation distribution of the kink of crack 1, the shear stress induced by the dislocation distribution of the kink of crack 2 , the shear stress induced by the shear traction q1 acting on the crack 1, and the shear stress induced by the shear traction q2 acting on the crack 2, respectively. σθ − σr σθ + σr + cos(2ξ1 ) − τrθ sin(2ξ1 ). And σθ1 θ1 = 2 2 A single crack is considered and the xm -ym frame is attached to the crack center with x along the crack plane. According to Muskhelishvili[19] , we have σyy + σxx = 2[ϕ (z) + ϕ (z)]
(18)
σyy − σxx + i2τxy = 2[zϕ (z) − ψ (z)]
(19)
2G0 (ux + iuy ) = k0 ϕ(z) − zϕ (z) − ψ(z)
(20)
where k0 = 3 − 4ν0 for plane strain case, ν0 is the Poisson’s ratio, G0 is the shear modulus, ux and uy are displacement components, and z = x + iy. Equation (14), which enforces the boundary conditions on the main crack, can be rewritten as the sum of the five terms (τ − f σ) +
pm
⊥ (τxmpm ym
−
⊥ f σympm ym )
+
pm =1
pm
⊥p
⊥p
qm qm m (τxm m ym − f σym ym ) + qm + (τxm ym − f σym ym ) = 0 (21)
pm =1
where the symbol ⊥ denotes the dislocation on kink p of the crack m . The first term on the left side can be determined by Eq.(13), the second term on the left side vanishes because the Hilbert problem is solved exactly for each isolated crack subjected to the loading by dislocations of the own kink. The third term can be obtained from[15–18]
lp ⊥pm ⊥pm m σym ym + iτxm ym = αp (zop )[U (zmm , zop , θmm ) + U (zmm , zop , θmm )] m m m m 0 +αp (zop ) U (zmm , zop , θmm ) + U (zmm , zop , θmm ) drop (22) m
m
m
m
m
where U (zmm , zopm , θmm ), V (zmm , zopm , θmm ),V (zmm , zopm , θmm ) , U(zmm , zopm , θmm ) were given by Zhou[16] . It is found that Chebyshev polynomials Zn , which have been used to analyze the interaction between straight cracks, can also be used here. Consequently, with respect to the local x-coordinate attached to the center of the main crack, the fourth term in Eq.(21) is given by qm =
N n=0
bm n [Zn (ξ)]
(23)
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where bm n denotes the (N + 1) unknown coefficients to be determined for each crack, ξ = x/c (−1 ≤ ξ ≤ 1) is the normalized local crack coordinate, and the recurrence formula for Zn (ξ) is n+1 n+1 ξ − ξ2 − 1 − ξ + ξ2 − 1 (24) Zn (ξ) = 2 ξ2 − 1
The Muskhelishvili potentials ϕn (z), ψ n (z) for Zn (ξ) are obtained by suitable superpositions of elementary potentials[16] . Using Eqs.(11), (22), (23) and (24), Eq.(21) can be rewritten as
lp pm N M m (τ − f σ) + bmn [−Ln (ξ)] + {Im{ αpm (zopm )[U (zmm , zopm , θmm ) m =1 p
n=0
m
0
=1
+U (zmm , zopm , θmm )]dropm } − f Re{
lp
m
0
αpm (zopm )[U (zmm , zopm , θmm )
+U (zmm , zopm , θmm )]dropm }}
lp pm M m + {Im{ αpm (zopm )[V (zmm , zopm , θmm ) + V (zmm , zopm , θmm )]dropm } m =1 p
m
−f Re{ +
M
lp
m
0
{Im[
m =1
0
=1
αpm (zopm )[V (zmm , zopm , θmm ) + V (zmm , zopm , θmm )]dropm }}
N
n bm n gmm (zmm )] − f Re[
n=0
N
n bm n gmm (zmm )]} = 0
(25)
n=0
Equation (25) becomes a system of M simultaneous real Fredholm integral equations of the first kind. Similarly, according to the boundary condition in Eq.(16), the complex equation for the kink pm becomes
lpm α (z pm l M p pm opm ) m + [αpm (zopm )U (zmm , zopm , θpmm ) 2 zpmm − zopm 0 0 m =1 p
m
+α
pm
(z
opm
)V (zmm , z
opm
=1
, θpmm )]dr
opm
+
pm M m =1 p
m
=1
0
lp
m
[αpm (zopm )U (zmm , zopm , θpmm )
+αpm (zopm )V (zmm , zopm , θpmm )]dropm +σ + iτ +
M N
n bm n gmm )=0 (zp mm
(26)
m =1 n=0
where zpmm = zcm − zcm + [(c + rpm eiθpm )eiθm ]e−iθm , θpmm = θpm + θm − θm . Equation (26) leads to a system of 2pm real singular equations of the Cauchy type. According to works by Gerasoulis[20],the mode I and mode II stress intensity factors KI and KII are easily determined by taking the limit of the product of the singular stress field. This yields KI + iKII = π 3/2 (2lpm )1/2 eiθpm Apm (1)
(27)
Under dynamic loads, crack growth velocities have a great influence on the dynamic SIF. In most cases, the mode I dynamic SIF can be expressed as[21, 22] KID = k1 (v)KI
(28)
where KID is the mode I dynamic SIF, KI is the mode I static SIF, k1 (v) is a function of crack growth velocity v. The function k1 (v) = 1.0, when v = 0, and k1 (v) = 0 when v reaches the critical velocity of the crack growth, which is normally regarded as the velocity of a Rayleigh wave in the rock material.
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The mode I dynamic stress intensity factor, for two-dimension in-plane crack growth, has been established[21] . The dynamic stress intensity factor can be estimated by multiplying the equivalent static stress intensity factor by an approximate formulation of k1 (v) for brittle rock material, i.e. vr − v k1 (v) = (29) vr − 0.75v when the tensile crack is subjected to a pair of splitting forces at its center and vr − v (30) k1 (v) = vr − 0.5v when the tensile crack is subjected only to far field uniform stress, where vr is the velocity of Rayleigh wave, v = dl/dt . For the mode II crack, the dynamic crack tip SIF KIID and the static crack tip SIF KII are correlated by KIID = k2 (v)KII (31) An approximate formulation of k2 (v) is given as[21, 22] vr − v (32) k2 (v) = vr − 0.9v when the crack is loaded only by a pair of concentrated forces at its center. vr − v k2 (v) = (33) vr − 0.65v when the crack is loaded only by far field uniform stresses, where vr is the velocity of the Rayleigh wave. The crack growth criterion can be determined by
2
2 KID KIID + =1 (34) d d KIC KIIC d d is the Mode I critical dynamic stress intensity factor, KIIC is the Mode II critical dynamic where KIC stress intensity factor. Cracks satisfying the criterion (34) will nucleate, propagate and coalesce. If cracks coalesce, the fractured zone will be observed.
III. NUMERICAL RESULTS AND DISCUSSION 3.1. Effect of Excavation Time t on Zonal Disintegration Phenomenon in Deep Rock Masses In the process of numerical simulation, it is assumed that the deep tunnel is constructed in granite. The excavation unloading rate is calculated by the failure strain divided by the unloading time t. Experimental studies on granite under dynamic unloading showed that failure strain of granite under strain rate 10−2 s−1 is almost 1000 micro-strain. Therefore, corresponding to strain rate of 10−2 s−1 , unloading time t is 1 × 102 ms. Under dynamic unloading, when Eq.(34) is satisfied, the crack starts to grow. The initial time of crack growth is t1 . After t1 , the crack grows continuously and results in the failure of the rock material. The crack growth time is then t − t1 . Under dynamic unloading, t1 is about 0.05 s. If the crack spacing is 2e = 4 m, the crack growth velocity v = 2e/(t − t1 ) is approximated to be 4/(1 × 102 × 10−3 − 0.05) = 80 (m/s). The velocity of the Rayleigh wave for the granite is about 2000 m/s, the function k1 (v) can then be determined from Eqs.(29) and (30), the function k2 (v) can be obtained from Eqs.(32) and (33). The crack growth length can be obtained from Eq.(34). In the numerical simulation, it is assumed that √ d c = 0.5 m, e = 2 m, h = 0.5 m, a = 0.5, mi = 10, RMR = 60, KIC = 2 MPa/ m, υ = 0.5 W = 0, R = 3 m, θm = −65◦, γ = 27.2 kN/m3 , E = 20 GPa,√ ν0 = 0.25, vr = 2000 m/s d p1 = 50 MPa, p2 = 70 MPa, σc = 30 MPa, KIIC = 0.9 MPa/ m The numerical results are shown in Figs.5-7. In Figs.5-7, the shadow parts are fractured zones, the blank parts are non-fractured zones. It is found from Figs.5-7 that when excavation time t is short, zonal fracturing may occur in deep rock masses. For certain excavation time t, the quantity of the fractured zone is defined, the size of the fractured zone decreases with increasing distance from the surface of opening; on the other hand, the quantity of the fractured zone increases with decreasing excavation time t.
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Fig. 5. Fractured zones around deep circular tunnel when t = 0.1 s.
Fig. 6. Fractured zones around deep circular tunnel when t = 1 s.
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Fig. 7. Fractured zones around deep circular tunnel when t = 10 s.
3.2. Influence of p2 /σc on Zonal Disintegration Phenomenon in Deep Rock Masses In the numerical simulation, it is assumed that √ d = 2 MPa/ m, υ = 0.5 c = 0.5 m, e = 2 m, h = 0.5 m, a = 0.5, mi = 10, RMR = 60, KIC 3 W = 0, R = 3 m, θm = −65◦ , γ = 27.2 kN/m√ , E = 20 GPa, ν0 = 0.25, vr = 2000 m/s d p1 = 50 MPa, p2 = 70 MPa, KIIC = 0.9 MPa/ m
Fig. 8. Fractured zones around deep circular tunnel when t = 0.1 s, p2 /σc = 2.33.
Fig. 9. Fractured zones around deep circular tunnel when t = 0.1 s, p2 /σc = 1.56.
Fig. 10. Fractured zones around deep circular tunnel when t = 0.1 s, p2 /σc = 1.17.
The numerical results are shown in Figs.8-16. In Figs.8-16, the shadow parts are fractured zones, the blank parts are non-fractured zones.
Fig. 11. Fractured zones around deep circular tunnel when t = 1 s, p2 /σc = 2.33.
Fig. 12. Fractured zones around deep circular tunnel when t = 1 s, p2 /σc = 1.56.
Fig. 13. Fractured zones around deep circular tunnel when t = 1 s, p2 /σc = 1.17.
Vol. 22, No. 3 Xiaoping Zhou et al.: Deep Crack-weakened Rock Masses under Dynamic Unloading
Fig. 14. Fractured zones around deep circular tunnel when t = 10 s, p2 /σc = 2.33.
Fig. 15. Fractured zones around deep circular tunnel when t = 10 s, p2 /σc = 1.56.
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Fig. 16. Fractured zones around deep circular tunnel when t = 10 s, p2 /σc = 1.17.
It is found from Figs.8-16 that: (1) Cracks satisfying the crack growth criterion in Eq.(34) will nucleate, propagate and coalesce, then the fractured zone shows up, rock masses fail; (2) The size and quantity of the fractured zone increase with the increasing in-situ stress; (3) The numerical results are in good agreement with the experimental results as by Shemyakin[6] . In order to determine the condition of the onset of zonal disintegration phenomenon, we also studied the zonal disintegration phenomenon when the value of the in-situ stress is less than the uniaxial compressive strength of rock masses. It is found from numerical results that the zonal disintegration phenomenon does not occur once the value of the in-situ stress is less than the uniaxial compressive strength of rock masses, that is, the zonal disintegration phenomenon occurs only when the value of in-situ geostress is higher than the uniaxial compressive strength of rock masses.
IV. CONCLUSIONS (1) The crack growth criterion in Eq.(34) is applied to analyze the zonal disintegration phenomenon of deep rock masses. Cracks satisfying the crack growth criterion will nucleate, propagate and coalesce, then, the fractured zone will occur, and finally rock masses fail. (2) Excavation time significantly affects the size and quantity of the fractured zone around the surrounding rock mass. When the excavation time is short, the zonal disintegration phenomenon may occur in deep rock masses. For certain excavation time, the quantity of fractured zone is defined. Moreover, the size of the fracture zone decreases with increasing distance from the opening surface. (3) It is found from numerical results that the quantity of the fractured zone increases with decreasing excavation time. The size and the quantity of fractured zone increase with the increase of in-situ geostress. (4) Zonal fracturing phenomenon occurs only when the value of the in-situ geostress is higher than the uniaxial compressive strength of rock masses.
References [1] Qian,Q.H., The current development of nonlinear rock mechanicsthe mechanical problems of deep rock mass. In: Chinese Society of Rock Mechanics and Engineering, Proceedings of the 8th Rock Mechanics and Engineering Conference. Beijing: Science Press, 2004, 10-17 (in Chinese). [2] Cloete,D.R. and Jager,A.J., The nature of the fracture zone in gold mines as revealed by diamond core drilling. Association of Mine Managers, 1972-1973. [3] Adms,G.R. and Jager,A.J., Petroscopic observations of rock fracturing ahead of stope faces in deep-level gold mine. Journal of the South African Institute of Mining and Metallurgy, 1980, 80(6): 204-209. [4] Shemyakin,E.I., Fisenko,G.L., Kurlenya,M.V. and Oparin,V.N., Zone disintegration of rocks around underground workings—Part I: Data of in-site observations. Journal of Mining Science, 1986, 22(3): 157-168. [5] Shemyakin,E.I., Fisenko,G.L., Kurlenya,M.V. and Oparin,V.N., Zone disintegration of rocks around underground workings-Part II: Disintegration of rocks on models of equivalent materials. Journal of Mining Science, 1986, 22(4): 223-232. [6] Shemyakin,E.I., Fisenko,G.L., Kurlenya,M.V. and Oparin,V.N., Zone disintegration of rocks around underground workings—Part III: Theroetical notions. Jounal of Mining Science, 1987, 23(1): 1-6.
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[7] Shemyakin,E.I., Fisenko,G.L., Kurlenya,M.V. and Oparin,V.N., Zone disintegration of rocks around underground workings—Part IV: Practical applications. Jounal of Mining Science, 1988, 24(1): 238-241. [8] Shemyakin,E.I., Kyrlenya,M.V. and Reva,V.N., Effect of zonal disintegration of rocks around underground workings. USSR, 1986, 289(5): 1088-1094. [9] Zhou,X.P. and Qian,Q.H., Zonal fracturing mechanism in deep tunnel. Chinese Journal of Rock Mechanics and Engineering, 2007, 26(5): 877-885 (in Chinese). [10] Mirzaev,G.G., Protosenya,A.G., Ogorodnikow,Y.N. and Vikhorev, V.I., Support Systemsof Deep Mines. Moscow: Bedra, 1976 (in Russian). [11] Cai,M. and Horii,H., A constitutive model and FEM analysis of jointed rock masses. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 1993, 30(4): 351-359. [12] Cai,M., Kaiser,P.K., Tasaka,Y., Maejima,T., Morioka,H. and Minami,M., Generalized crack initiation and crack damage stress thresholds of brittle rock masses near underground excavations. International Journal of Rock Mechanics and Mining Sciences, 2004, 41(5): 833-847. [13] Zhou,X.P., Wang,F.H., Qian,Q.H. and Zhang,B.H., Zonal fracturing mechanism in deep crack-weakened rock masses. Theoretical and Applied Fracture Mechanics, 2008, 50(1): 57-65. [14] Zhou,X.P., Qian,Q.H. and Yang,H.Q., Strength criteria of deep rock mass. Chinese Journal of Rock Mechanics and Engineering, 2008, 27(1): 117-123 (in Chinese). [15] Lo,K.K., Analysis of branched cracks. ASME Journal of Applied Mechanics, 1978, 45: 797-802. [16] Zhou,X.P., Qian,Q.H. and Zhang,Y.X., The constitutive relation of crack-weakened rock masses under axial-dimensional unloading. Acta Mechanica Solida Sinica, 2008, 21(3): 221-231. [17] Horii,H. and Nemat-Nasser,S., Brittle failure in compression: splitting, faulting and brittle-ductile transition. Philosophical Transactions of the Royal Society of London, 1986, A319: 337-374. [18] Niu,J. and Wu,M.S., Analysis of asymmetric kinked cracks of arbitrary size, location and orientation—Part I. Remote Compression. International Journal of Fracture, 1998, 89(1): 19-57. [19] Muskhelishvili,N.L., Some Basic Problems in the Mathematical Theory of Elasticity. Netherlands: Noordhoff, Groningen, 1953. [20] Gerasoulis,A., The use of quadratic polynomials for the solution of singular integral equations of cauchy type. Computer & Mathmatics with Applications, 1982, 8(1): 15-22. [21] Freund,L.B., Dynamic Fracture Mechanics. Cambridge University Press, 1990. [22] Cherepanov,G.P., Mechanics of Brittle Fracture. New York: Translated by R de Wit and W C Ccoley, McGraw-Hill, 1979.