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9
Rock Failure Mechanism / Механизмы разрушения горных
УДК 622.831
Ksendzenko L.
LUDMILA S. KSENDZENKO, Assistant Professor, Department of Algebra, Geometry and Analysis, School of Natural Sciences, Far-Eastern Federal University, Vladivostok, Russia. 8 Sukhanova St., Vladivostok, Russia, 690950, e-mail: ksendzenko@mail.ru
Modelling the focal stage of the strain field distribution in highly stressed rock
The article deals with the non-Euclidian continuum model applied to describe the stress-strain state of highly stressed rock at the pre-failure stage. The oscillating character of the distribution of stresses both along the perimetre of the rock and along its height has been demonstrated in it. A satisfactory convergence of the results of the theoretical researches with the experimental data has been obtained.
Key words: sample of rock; uniaxial compression simulation; non-classical field of stress.
Introducion
Determination medium-term and short-term precursors of geodynamic phenomena requires research the regularities of deformation of rocks in the pre-failure state.
Therefore, it is important to simulate the deformation in strongly compressed samples of rocks around the perimeter of the sample and its height.
Mathematical modeling of deformations in rock samples in the pre-failure state on the basis of models of mechanics of defective mediums is performed [1, 2], where the rock is in the pre-fracture stage is regarded as a defective medium in conditions far from thermodynamic equilibrium, when the conditions of compatibility of elastic deformations in general are not performed
Experimental studies the deformation of rock samples
Experimental study of the nature of the deformation of rock samples in pre-fracture stage loading were realized on the samples of dacite and rhyolite cylindrical shape with fixing of strain gages on the basis of 10 mm and the use of recording equipment tensometric station UIU-2002 (Fig. 1). The source of destruction is pointed by red color on the figure 1. The symbol 5 marked the acoustic sensors. Below all the calculations and graphs were made for a sample of rhyolite N 23.
Fig. 1. Scheme loading of the sample. Location of pairs strain gauges and sensors AE
© Ksendzenko L., 2015
[100] vestnikis.dvfu.ru/eng
fefu: school of engineering bulletin. 2015. n 3/24 / вестник инженерной школы двфу. 2015. № 3 (24)
Researching patterns of the deformation in rock samples in the pre-failure state when P > P (a threshold of dilatancy can be accepted as Critical load P* ) showed that the deformation of the sample have often reversible character, consisting in changing the sign of the increment of longitudinal and lateral strains, which were measured in local areas, the relevant scheme stickers sensors.
Research [3] revealed that the deformations of the sample along its height in the pre-failure stage have different signs consists in changing the sign of the increment of longitudinal and lateral strains in the area of formation cracks of tearing off and simultaneous strain of the same sign significantly (abnormally) exceeding the typical level for this breed.
Apart from of the increments of strains of different signs along height of the sample, the same character of strain observed along the perimeter.
Furthermore, it was found the geometrical position of source of destruction, and reversible (perilesional) areas I and II type.
Theoretical study of the deformation of rock samples
Theoretical studies of the patterns of deformation of samples at strong compression are performed using continuum model with the self-balanced stresses ([1]). In [1] describes the non-classical stationary distribution of the stress field in the cylindrical sample of rock using a non-Euclidean continuum model.
The non-classical normal uniaxial and tangential stresses Tz and T in the cylindrical system coordinate are determined by equalities:
Tzm'n =-kJ (V)cos(nP + Po)cosqmz ,
PP
i J (-/ ) n
dr
—2 Jn (kr) + К Jn (ksr) cos(np + Po) cos qmz,
z t • m
where ks = —, s = 1,2,... ; zs — are roots of equation J 1(z) = 0, and =-, m = 1,2
R h
R — is radius of the sample, 2h — is height of the sample, m is its height, m. Stress components represented as the sum of three terms:
T22 (r, p, z) = B^T^ + b1,1,1Tzz1'1'1 + B^Tz^,
T^ (r, p, z) = A^rpf + A™tp + ^rp,
where r^10 = —kJ0 (kxr) cos(p0 ) cos qz, T11'1 = —kJ (kxr) cos(p + p0) cos qz,
? • • • • 5
(1) (2)
Tl¡1 2 = -ki2J2 (kir) cos(2p + Po)cosqiz ; Tp'p'0 (r, P,z) =
Ji ( v )- ki2 Jo ( v )
r
cos (Po ) cos ( qiZ ),
С ( r,P, z ) =
4 - -i2 j Ji ( -r )--Jo ( -r )
cos (p + p0) cos ( qz ).
TP ( r,P, z ) =
£ - -i2 j J 2 ( -r )-- Ji ( -r )
cos ( 2p + p0) cos (qz ) .
Laboratory studies have shown that the source area of the sample is an ellipsoid.
The coordinates center of source area and the coordinates center area of the reversible deformations are S ^R ,0, -0,4 • h j and B ^ R ,0,0 j respectively.
In order to determine the six unknown coefficients of the expansion in the components of the stress need six equations. For the first of them use as fracture criterion:
k = m (( + tz). (3)
1/2
Here Kj - is Odintsev's stress intensity factor (MPa• m ), at the top of the mezo-crack in the
source of the sample; P = —, —i = 0,25; —3 = 0,2; M = • y1.
Y
1/2
The fracture toughness (MPa• m ) of the rock sample are finding by formula
K*c = M-a*, MPa•VM, with the proviso that the load exceeds a certain critical P* ( P > P*),
l - half length mezo-crack, m.
The normal uniaxial and tangential stresses T u Tz are given by (1) u (2). Where a* =ac- p*,
P* - is a critical load, above which in the sample begins to form mezo-cracking structure.
We assume that in the center of the source area the stress intensity factor reaches a value equal the fracture toughness of the rock sample:
Kj|s = K*c. (4)
The second equation is obtained on the assumption that the first invariant of the stress tensor at the
point (R,0,0) reaches a**, when the load exceeds a critical value P*.
The remaining four equations derived from the conditions to achieve the extreme uniaxial and lateral stresses in the center of the source region and the center region of the reversing type I adjacent to the source region from the top along the axis of the sample.
Solving the system of equations, we find the coefficients of the equations (1) and (2):
Tgxp( r,9, z ) = 0,001bTf (r, z ) + 0,00012 (r,q>, z )-0,00015-T^2 (r,q>, z ), Tz ( r, p, z) = 0,0006 • TZ 10 ( r, z) - 0,0024 • T¿ ^ 1 (r, p, z ) + 0,00 1 2 • Tlz; 12 ( r, p, z ) .
On the Fig. 2.3 shows the distribution of linear stresses on the side surface of the sample. It is clearly seen that the stress distribution is oscillatory nature along the perimeter of the sample and its height.
Fig. 2. The oscillatory nature of the stresses on the lateral surface of the highly compressed sample on its
perimeter (the middle, z = 0)
МРа б
4 2 О -2 -4
Tzziv) / <p=0 i = R
/
h -0.03 -0.01 0.01 0.03 h
Fig. 3. The oscillatory nature of the uniaxial stresses on the lateral surface of the highly compressed
sample at its height
0
Fig. 4. The isolation of the source of failure region at the side surface of the sample
You can see that in the center of the fault zone longitudinal negative voltage that corresponds to the maximum compression of the material and the hoop stresses are positive, indicating that the tension in the circumferential direction of the rock in the fault zone. From experiments follows:
E (p) = 1\_rz (p)—2-v-Up)], (5)
where E = —66474 MPais the average modulus of total deformation; v = 0,4 — the coefficient of
transverse expansion. We assume that T^ (p) = T (p).
Comparison the non-classical experimental and theoretical uniaxial deformations on the side surface of the sample in the neighbourhood of the source region shows that discrepancy between theoretical and experimental strains reaches 37%.
Conclusion
The values of experimental and theoretical deformations correlate satisfactorily. The maximum discrepancy between non-classical theoretical and experimental strains reaches 37%.
Acknowledgement
The paper was supported by grants No. 13-06-0113m_a from "Scientific Fund" of Far Eastern Federal University and No. 5.2535.2014K from the Ministry of Education and Science of the Russian Federation.
REFERENCES
1. Guzev M.A. Description of the stress field in the sample rocks based on non-Euclidean model. Proceedings of 4th Russian-Chinese Scientific Forum Nonlinear Geomechanics and Geodynamics in Deep Level Mining. July, 27-31th, 2014, Vladivostok, FEFU, p. 65. (in Russ.).
2. Guzev M.A. & Makarov, V.V. Deforming and failure of the high stressed rocks around openings. Vladivostok, Dalnauka, 2007, 232 p. (in Russ.).
3. Ksendzenko L.S., Makarov V.V., Opanasyuk N.A., Golosov A.M. The patterns of deformation and failure of highly compressed rocks and Massifs: monograph [Electronic resource]. School of Engineering FEFU. Vladivostok, Far Eastern Federal University, 2014. [219 p.]; 1 CD. (Series "Geology and exploration of mineral resources"). (in Russ.).
4. Odintsev V.N. Separation of Hard Rock Mass. Institute of Earth's Interior Complex Development Problems RAS, Moscow, 1996, 68 p. (in Russ.).
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Механизмы разрушения горных / Rock Failure Mechanism
Л.С. Ксендзенко
КСЕНДЗЕНКО ЛЮДМИЛА СТЕПАНОВНА - кандидат физико-математических наук, доцент кафедры алгебры, геометрии и анализа Школы естественных наук (Дальневосточный федеральный университет, Владивосток). Суханова ул., 8, Владивосток, 690950. Е-mail: ksendzenko@mail.ru
Моделирование очаговой стадии распределения поля деформаций в сильно сжатом образце горной породы
В работе применяется неевклидова модель сплошной среды для описания напряженно-деформированного состояния сильно сжатого образца в стадии предразрушения. Показан периодический характер распределения напряжений как по периметру образца, так и по его высоте. Получена удовлетворительная сходимость теоретических и экспериментальных деформаций.
Ключевые слова: образец горной породы, одноосное сжатие, неклассическое поле напряжений.
СПИСОК ЛИТЕРАТУРЫ
1. Гузев М.А. Описание распределения поля напряжений в образце горных пород на основе неевклидовой модели // Нелинейные геомеханико-геодинамические процессы при отработке месторождений полезных ископаемых на больших глубинах: материалы 4-й Российско-китайской науч. конф., 27-31 июля, 2014, Владивосток / отв. ред. В.В. Макаров; Инженерная школа ДВФУ. Владивосток: ДВФУ, 2014. С. 65.
2. Гузев М.А., Макаров В.В. Деформирование и разрушение сильно сжатых горных пород вокруг выработок. Владивосток: Дальнаука, 2007. 232 с.
3. Ксендзенко Л.С., Макаров В.В., Опанасюк Н.А., Голосов А.М. Закономерности деформирования и разрушения сильно сжатых горных пород и массивов: монография [Электронный ресурс] / Инженерная школа ДВФУ. Владивосток: Дальневост. федерал. ун-т, 2014. [192 с.] - 1 CD. (Сер. «Геология, поиск и разведка полезных ископаемых»).
4. В.Н. Одинцев. Отрывное разрушение массива скальных горных пород М.: ИПКОН РАН, 1996. 166 с.
