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ГЕОМЕХАНИКА
УДК 622: 510.67
L.S. Ksendzenko, V.V. Makarov, N.A. Opanasiuk
LUDMILA S. KSENDZENKO, VLADIMIR V. MAKAROV, NICOLAI A. OPANASIUK, Department of Mining Engineering, School of Engineering, Far-Eastern Federal University, Vladivostok, Russia.
Email: vlmvv@mail.ru
Zonal Character of Periodical Failures Near Openings
in the Conditions of High-Stress Rock Mass
Rock mass failures at a high depth near underground openings often have a zonal character. The mechanism of the phenomenon is conditioned by periodical character of stresses in surrounding rock mass and the development of tensile macrocracks in the places (zones) of maximum tangential stresses. A mathematical model of the highly stressed rock mass has been developed on the basis of defect medium mechanics and non-equilibrium thermodynamics principles. A method of determining model parameters has also been developed and a satisfactory correspondence between the experimental research on faulted zonal structures near high depth openings and mathematical model calculation has been achieved. The main relationships between the width of cracking zones and the rock mass strength property have been established.
Key words: mathematical rock mass model, failure, zonal character.
Периодический характер зонального разрушения сильно сжатых горных пород вокруг выработок. Ксендзенко Л.С., к.ф.-м.н., доцент кафедры алгебры, геометрии и анализа Школы Естественных наук; Макаров В.В., д.т.н., профессор кафедры горного дела и комплексного освоения георесурсов Инженерной школы; Опанасюк Н.А., старший преподаватель кафедры горного дела и комплексного освоения георесурсов Инженерной школы (Дальневосточный федеральный университет, Владивосток).
Разрушение сильно сжатого массива вокруг подземных выработок зачастую носит зональный характер. Механизм этого явления заключается в периодическом характере напряжений в массиве горной породы, окружающей выработку, и развитии макротрещин отрыва в местах (зонах) максимальных касательных напряжений. Математическая модель сильно сжатого массива горной массы разработана на основе механики дефектных сред и принципов неравновесной термодинамики. Разработан метод определения параметров модели и получено удовлетворительное соответствие экспериментальных исследований структур зонального разрушения сильно сжатого массива вокруг горной выработки расчетам, полученным на основе модели. Установлена связь между протяженностью трещиноватых зон и свойствами сильно сжатой горной породы.
Ключевые слова: математическая модель горной породы, разрушение, зональный характер
© Makarov V.V., Ksendzenko L.S., Opanasiuk N.A., 2013
Introduction
Failure conditions can take place in the boundary parts of the openings at high depths of excavation and wall drilling. In some cases the failure has a zonal character, where tensile macrocrack zones alternate with relatively monolithic rock mass [Shemjakin E. I.,1986; Adams G.R., Jager A.J., 1980]. Many attempts have been made to describe the zonal character of rock mass failure near openings based on classical mechanics [Shemjakin E. I. et al. 1987; Reva V. N., Tropp E. A 1995; Odintsev V. N., 1996; Alexeys A. D. et al. 1996]. But no one has been able to explain all the properties of zonal failure structures without the introduction of new assumptions in every new case.
Recently, a new gauge theory has been applied to solids to describe the whirl fields of plasticity in high energy conditions [Kadich A., Edelen D. 1983; Panin V. E. et al. 1990; Panin V. E, 1990]. The main principle of gauge theory is incompatibility of the deformation in solid. But it does not apply to the zonal failure phenomenon of rock masses near openings. In this paper we demonstrate an example of a description of the phenomenon by employing the gauge mathematical model.
The description can be done from the position of consideration of the rock mass hierarchical block systems [Neng Xiong Xu 2009]. A rock sample in this conception is shown as the first level of a hierarchical system and the rock mass on the opening scale corresponds with the second one.
Mathematical model
Rock at a high depth is modelled by a faulted structure which is far from the state of thermodynamical equilibrium; it is in constant all-around compression. The boundary-value problem is assigned as a task concerning the stressed state of a weightless plane which is a solid damage structure with given on an infinity stresses, that models a gravity field, and loosened by a round hole that models an unlined underground opening (Fig. 1). Due to the polar symmetry of the task the equilibrium equations are as follows:
1
dr
= 0,
r0 < r < œ,
(1)
where Grr is the normal radial stress, G(№ stress.
is the normal tangential stress, and crrv is the shear
Figure 1. Design scheme of an unlined opening task
At the opening boundary (r = r0) and at infinity, the following applied forces exist: <rr = 0 at r = r0; <Jrr, <w at r
where cr^ = y • H, y is the rock density (N/ m3), and H is the opening depth (m).
The rock mass afa great depth is modelled by the structure, where commonly the conditions of deformation compatibility £iJ- are not met:
R = ^£ii - 2 | ^g22 ^ 0 . (3) 3c2 2 3ci3c2 ¿kx 2
The damage parameter R is expressed by the equation [Guzev M. A, Paroshin A. A. 2000]: A2 R -y2 R = 0, (4)
where A is the Laplace operator and y is the model parameter.
As the task is plane- and axi-symmetrical, Equation (4) in polar coordinates will become:
R = y2R. (5)
r _2 1 a 12D 2
V_
_r r _r
The solution of Equation (5) decreasing at r — œ is:
R(r) = aJ0^Jfr) + bN0(Jfr) + cK0(Jfr), (6)
where J0, N0, K0 are the Bessel, Neumann, and MacDonald functions of zero order, respectively. Non-classical boundary conditions and task solution
At the opening boundary the rock mass undergoes considerable destruction; therefore the damage parameter R should not equal zero. Assuming that all zones of rock destruction are equivalent and are of the same origin, we introduce the extremum of the function condition in the boundary R(r) and the following zones of destruction. Therefore boundary conditions for the function R( r ) are:
R'(r )| r=r = 0, R'(r )| r _r* = 0, (7)
where r * is determined experimentally.
The equation for the first invariant of stresses 7 = 7zz + <7rr + 7pp is:
E
À7 =-R, 7 —^2(1 + v)< , r — œ, (8)
2(1 -y) œ
with the determined function R, where E is the modulus of elasticity and y is the Poisson ratio. Solving the task (8) gives the formulas for the stress components:
f E 1
7rr 7œ
7pxp 7œ
,3/2 r
aJi [4rr ) + bNi (fir ) + cKi (fir )] ;
2 (1 -y2 )y3
aJo (fi- r )+bNo (fi- r )-cKo (fi- r )] +
\+v21__e_
r1 ) 2 (1 -y2 ) y
E 1
+-E—3J2 --[aJ1(4r- r) + bN1(4y- r) + cK^fi - r)], (9)
2(1 -y2)y r
where r is the distance from the centre of the opening to a selected point in the rock mass. Stress calculation and criteria of failure
Stresses surrounding the opening are of oscillating character (Fig. 2).
1 - R
2 _
3-
V 3,.
■M 2 1 \ 3 / \ 4 / \
/ \1
Figure 2. Oscillating character of the stresses and R - functions in the rock mass surrounding the underground opening
The zones of failure appear in the areas where the conditions of cracking under compression
are met:
K, =(*• l)"2-ai-r,^) a Kk.
(10)
where l is the half-length of fracture faults of the rock mass and is assumed to be equal to the minimum half-length of a tensile macrocrack which is unstable in stress conditions (m); a®, a0 are the maximum and minimum of major stresses, respectively (MPa); r, r are empirical factors; K1 is the coefficient of stress intensity (MPa *m1/2); and K is the fracture toughness of
1/2
rock material (MPa *m ).
As a criterion function the following dependence is taken: K(r) = Kj / KIc.
(11)
At Kj / KIc < 1 there is no failure around the opening; at K7 / KIc >1 the fracture starts to appear. The criterion function, as well as the stress functions and R, has an oscillating character (Fig. 3).
K
K1C 2
1
-1
1 2 3 4
rO
Figure 3. Character of criterion function
Parameters of model
The research on rock mass zonal failure was carried out for an unlined opening. In order to run the calculations, algorithms and programs were developed that included formulas for calculating the damage function R(r), stresses, and criterion function Kr(r). On the basis of the developed programs the experiment was carried out with three types of parameters being analysed.
The first type included parameters of model y, c, which are determined from the experiment taking into account the all-around compression at a high depth. The second type includes parameters that characterize the mechanical properties of rock mass: E, the modulus of elasticity (MPa); v, the Poisson ratio; oc, the uniaxial compression strength (MPa); and also the value of gravitational stresses in the rock mass, o^ (MPa). The third type includes parameters in Formula (10) which characterize the cracking structure of the rock mass: the half-length of rock fracture faults l and the fracture toughness of rock material KIc, and also the coefficients yx, y3(below he dependence y3/y1 is used).
Parameter y can be determined using the procedure of statistical analysis of the natural research results of the zonal failure process in deposits in Russia (the Far Eastern part, Siberia, Donbass) and China. A linear character of dependence between the relative distance from the opening contour to the middle point of the first failure zone and uniaxial strength of the rock was determined:
r */r0 = 0.0083oc + 0.748, (12)
where r* is the distance from the opening contour to the middle point of the first failure zone which has been found from experimental data (m).
The relationship between parameters y and r* / r0 is linear too but avoid the long description the determination of it number can be achieved according with Table. 1
Table 1
Meanings of the parameter y of the model
Middle part of the I-st zone, r*=1-r/r0 r0=1.75 r0=2.0 r0=2.5 r0=3.0
0.7 26.486 20.279 12.978 9.012
0.8 20.313 15.552 9.953 6.912
0.9 16.080 12.311 7.879 5.471
1.0 13.050 9.991 6.394 4.440
According to recent research the rock mass can be shown as a hierarchical block medium [3-7]. When the physical character of failure on the neighboring hierarchical levels is the same, the macrodefect size of the lower level can be determined as a mesodefect of the corresponding higher level [3, 4]. So this low is reflected in the conservation shear-tensile character of the rock failure on the neighboring levels of samples and mass in conditions of high stress [8].
The algorithm for the determination of the mathematical model's parameters consists in the next steps:
1. After the rock sample strength reaches the limit of strength oc, the limit of residual strength ores, Young modulus E, and Poisson ratio v are determined. Then by using
* /
Formula (12) the emplacement of the first failure zone's middle point r / rn can be
found. Then after the substitution of these data in the Table 1, the first correction parameter of the mathematical model y can be determined.
2. The maximum diameter r of the rock sample minerals dmax and maximum mesocrack disclosure h « dmax are determined and after that the minimum half-length of the
tensile mesocrack lmezo ~(2,5 — 5)dmax , is calculated.
3. The critical half-length of the tensile macrocrack in the sample is determined by the
h 'E
formula l*=—f-^-, and the stress intensity factor of the rock mass is
4[1—v2 yvac
calculated by the formula K™' = y • crrCec -JkI™ .
4. Then, the stress intensity factors of both the rock sample and the rock mass are written in the equation of equality and the parameter of the model "C" is determined as a result of the calculation. For this purpose the equation of equality is applied:
5. Kmass = y '^Cec 'TO = Ki = (Til)1/2 — y3arr), (13) where the "C" parameter is used in the right-hand part of Formula (13) (see (9)).
6. Programs for the determination of the last destruction zone (for lined and unlined openings: B1 and B2 respectively) and programs for calculation of the radial length of fracture zones (for lined and unlined openings: Cl and C2 respectively) were developed. Program charts with a brief description were also developed. The patterns of the changes in the zonal structure of rock mass failure depending on various factors were obtained.
7. The main parameters of the zone structure were identified: the number of zones of failure, the location of the furthest fracture zone from the opening boundary (the last zone of failure); the creation of relative critical stresses of failure zones; and also the value of the radial length of failure zones.
Results of research
As a result of the modelling experiment on the basis of the adopted mathematical model we determined that the parameters of the zonal structure depend slightly on the values of the elastic modulus of rocks E and the Poisson ratio v. This conclusion corresponds to the data obtained during laboratory studies (when the elasticity varies in 10 times, the critical stresses of zone creation change by 2-5% on average). The research on fracture zones was carried out for rather solid rock (ac = 150 MPa) and for weak rock (ac = 15 MPa). In order to run the calculations, algorithms and programs were developed that included formulas for calculating the expressions of defectiveness R(r), stress, and the criterion function K (r).
The results of the solid rock case study are demonstrated by the application of the developed method to the problem of zonal failure in the Nikolaevskij ore mine (Dalnegorsk, Russia). The forecasted depth of development of the cracking zone is shown in Table 2.
Table 2
Forecasted depth of development of zonal failure in the Nikolaievskij ore mine
Number of failure zones I II III IV
Relative critical stress of zone formation 1.3 2.3 2.9 3.3
Depth of zone appearance (m) 520 920 1160 1320
The amplitude parameter "C" is dependent on the modulus of deformation and the Poisson ratio of the rock mass (Fig. 4).
Figure 4. The relationships of the model amplitude parameter with the deformation modulus E and Poisson ratio V under conditions of different values of rock mass failure strength: 1 - KIC = 1.5 MPa ■ m12,
2 - K/c = 2.0 MPa ■ m12, 3 - K/c = 2.5 MPa ■ m12.
The precision of the correlation between theory and experiment has been estimated by comparing the results of in situ measurement of the radial displacements near the openings at high depth (Nikolaevskij ore mine) with the model calculation results (Fig. 5). It was determined that the difference between the forecasted and measured data was no more than 50% in the four radius field around the opening.
Figure 5. Comparison between theoretical (1) and experimental (2) data of radial deformations
The comparison of the results of the analytical and experimental studies of the weak rock also shows their good convergence (Table 3). The research on zonal failure of the rock mass was carried out for an unlined opening.
Table 3
Comparison of the theoretical and experimental [Neiman L.K., 1991] results
(unlined opening)
Parameter Method Elements of zonal failure structure
1stzone 2ndzone 3rdzone 4thzone
1. Location of the furthest zone boundary, r/r0 Experiment 1.03 2.23 3.40 4.54
Theory 1.28 2.17 3.09 3.97
Deviation, % 24.3 -2.7 -9.1 -12.6
2. Relative critical zone stresses, J / J™"' Experiment 1.1 2.2 2.7 -
Theory 0.95 2.1 3.1 -
Deviation, % -13.9 -4.5 14.8 -
It was determined that the basic factor that influences the parameters of zonal failure structure is the value of stresses that act within the rock mass (opening depth). With the increase of stresses, the number of failure zones increases and their radial length increases until it reaches neighboring zones. The closer the zone is located to the opening boundary, the faster the process. The boundary of the last zone of failure moves further into the rock mass (Fig. 6, left).
The parameters of the fracture structure of the rock mass also influence the character of zone failure. The radial length of the zones of failure decreases if the rock fracture toughness rises (Fig. 6, right). When the rock fracture toughness decreases, zones of failure appear at lower relative stresses and the distance of the last failure zone from the opening boundary increases.
Figure 6. Dependence of the position of the last failure zone on the relative stresses acting within the rock mass (left) and the radial length of the first zone of failure on the rock fracture toughness (right)
The activity of rock destruction also has a strong influence on the parameters of the zone failure structure. As the fracture faults length increases, the radial length of failure zones increases. This parameter decreases if the dependence y3 /y rises. Regularities determined for weak rock mentioned above are true for solid rock also.
Conclusion
Thus, the research conducted shows that as the depth of the opening rises the zonal character of rock failure becomes more expectable, which should be taken into consideration when designing a lining for such conditions.
A method for determining of the mathematical model parameters of zonal failure structure near to deep openings has been developed. A full quality and good quantitative correlation between theoretical forecasting and experimental research has been achieved.
Acknowledgements
The paper was supported by the grants FPP № 14.A18.21.1980, №7.8652.2013 and "Scientific Found" of FEFU.
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