CC BY
6
УДК 539.3
DOI: 10.18303/2618-981X-2018-5-285-291
ВЛИЯНИЕ ДЕФЕКТА В БЛОЧНОЙ СТРУКТУРЕ НА РАСПРОСТРАНЕНИЕ ВОЛН
КайсинВан
Ляонинский технический университет, 123000, Китай, г. Фусинь, кандидат технических наук, преподаватель факультета механики и инженерии, e-mail: kaixing_wang@163.com
Ишан Пан
Ляонинский технический университет, 123000, Китай, г. Фусинь, доктор физико-математических наук, профессор, факультет механики и инженерии, e-mail: panyish_cn@sina.com
Виктор Николаевич Опарин
Институт горного дела им. Н. А. Чинакала СО РАН, 630091, Россия, г. Новосибирск, Красный пр., 54, доктор физико-математических наук, профессор, член-корреспондент РАН, зав. отделом экспериментальной геомеханики, тел. (383)-205-30-30, e-mail: оparin@misd.m
Надежда Ивановна Александрова
Институт горного дела им. Н. А. Чинакала СО РАН, 630091, Россия, г. Новосибирск, Красный пр., 54, доктор физико-математических наук, главный научный сотрудник, тел. (383)205-30-30, e-mail: nialex@misd.ru
Экспериментально исследуется поведение блочной среды при динамическом воздействии, когда один из блоков имеет дефект, например, трещина перпендикулярная или параллельная к направлению приложенной нагрузки. Блочная среда моделируется вертикально расположенной стопкой гранитных блоков, в которой в одном из блоков имеет место указанный дефект. Блоки разделены резиновыми прослойками. По данной системе блоков производится динамический удар. С помощью датчиков акселерометров записываются осциллограммы ускорений двух блоков в стойке. Исследуется характер изменения осциллограмм ускорения и их спектральных свойств при распространении волны по физической модели блочной среды с дефектом. Мы анализируем скорость волны маятникового типа и структуру сигнала, проходящего через блочную среду, в следующих трех случаях: (а) дефекта вообще нет, (б) дефект реализуется как трещина, перпендикулярная к направлению приложенной нагрузки, и (в) дефект реализуется как трещина, параллельная направлению наложенной нагрузки. Показано, что при появлении дефекта скорость маятниковой волны в блочной среде уменьшается.
Ключевые слова: блочная среда, ударная нагрузка, дефект блока, маятниковая волна, скорость волны, спектр ускорения.
EFFECT OF A DEFECT IN A BLOCK MEDIUM ON PROPAGATION OF WAVES
Kaixing Wang
Liaoning Technical University, Fuxin, 123000, China, Ph. D., Lecturer, Department of Mechanics and Engineering, e-mail: kaixing_wang@163.com
Yishan Pan
Liaoning Technical University, Fuxin, 123000, China, D. Sc., Professor, Department of Mechanics and Engineering, e-mail: panyish_cn@sina.com
Victor N. Oparin
Chinakal Institute of Mining SB RAS, 54, Krasny Prospect St., Novosibirsk, 630091, Russia, D. Sc., Professor, Corresponding Member of the RAS, Head of Department of Experimental Geomechanics, e-mail: oparin@misd.nsc.ru
Nadezhda I. Aleksandrova
Chinakal Institute of Mining SB RAS, 54, Krasny Prospect St., Novosibirsk, 630091, Russia, D. Sc., Chief Researcher, e-mail: nialex@misd.ru
The behavior of a block medium under a dynamic load is investigated experimentally in the case when one of the blocks has a defect, for example, a crack which is perpendicular or parallel to the direction of the load applied. The block medium is modeled by a vertically disposed stack of granite blocks in which the above described defect occurs in one of the blocks. The blocks are separated by rubber interlayers. This system of blocks is subject for a dynamic impact. Oscillograms of the accelerations of two blocks in the stack are recorded with the help of the sensors of accelerome-ters. We study the character of changes of the oscillograms of accelerations and their spectral properties of the propagation of a wave in the physical model of the block medium with a defect. We analyze the velocity of the pendulum-type wave and the structure of the signal, passing through the block medium, in the following three cases: (a) there is no defect at all, (b) the defect is realized as a crack perpendicular to the direction of the load applied, and (c) the defect is realized as a crack parallel to the direction of the load applied. It is shown that when a defect appears, the velocity of the pendulum-type wave in the block medium decreases.
Key words: block medium, impact load, defect of a block, pendulum-type wave, velocity of wave, spectrum of acceleration.
Introduction
The contemporary geomechanical and geophysical sciences describe a rock mass as a block structure of complex hierarchy. As per this concept, a rock mass consists of blocks of various scale that are nested in one another and mutually linked by intermediate layers composed of weaker and fractured rocks [1, 2]. In such a block medium, a specific phenomenon of dynamic response, called pendulum-type waves, was discovered in [3]. Pendulum-type waves occur due to deformation of intermediate layers while the blocks move as rigid bodies. They are closely related to geomechanical structure of the rock mass. Pendulum-type waves in a block-structure medium were studied theoretically and experimentally in [3-9]. But dynamic response of a block medium will be changed if any block is broken. In this article, we experimentally study the effect of a block failure in the horizontal direction (or in the vertical direction) on the dynamic response of the block medium.
Description of the laboratory experiment
In this article, a stack of granite blocks separated by rubber interlayers is used as a physical model of a one-dimensional block medium. A photograph of the installation on which the experiment was conducted is shown in Fig. 1 (a). The dimensions of an individual granite block are equal to (100 x 100 x 100) mm. Its weight M is
equal to 2.8 kg. The longitudinal velocity of sound Vp in granite is equal to 5400 m/s.
The Young's modulus of the rubber interlayers is determined experimentally at a press and is equal to 2.2 MPa. The process of the propagation of the elastic waves in this block medium is recorded by accelerometers with sensitivity 1 mV/g, measuring range ± 500 g, and maximum frequency 100 kHz. The sensors of the accelerometers are mounted at the block 3 (measuring point 1) and the block 9 (measuring point 2), see Fig. 1(a). As a wave exciter, a ball of mass m = 0.3 kg is used, which falls on the block 1 from the height h = 200 mm. In theory, pendulum-type waves arise if the energy of impact W = mgh satisfies the conditions 4 -10"11 < k < 4 -10"9, where k = W(MVp) [3]. With the above parameters of the physical model, k = 1.84x 10"9 and therefore the conditions for the energy of impact are satisfied.
a) b) c)
Fig. 1. Physical model of a one-dimensional block medium
We experimentally study the following three cases:
(a) block 2 is not destroyed, see Fig. 1(a);
(b) block 2 is destroyed by a horizontal crack as shown in Fig. 1(b); and
(c) block 2 is destroyed by a vertical crack as shown in Fig. 1(c).
Experimental data and their analysis
Fig. 2 and 3 show typical oscillograms of accelerations of the block 3 (measuring point 1) and accelerations of the block 9 (measuring point 2) for the above mentioned cases (a)-(c). These oscillograms demonstrate the passage of a pendulum-type wave through the physical model of the block medium.
The main parameters that determine the behavior of the pendulum-type wave as it passes through the block medium are the wave propagation velocity and the attenuation of the maximum and minimum amplitudes of the accelerations. The wave propagation velocity is defined as the ratio of the distance between the sensors to the difference in the initial moments of the elastic wave arrival at the sensors.
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!-h 0/
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£ -500
03
10 15 20
9 18 27
2000
-200
0 9 18 27
t (m s) a)
t (m s) b)
t (ms)
c)
Fig. 2. Accelerations of the measuring point 1
0
0
100
-100
5 10 15 70 0
t (ms) a)
100 r 0
-1001
9 18 27 t (m s)
0 9
18
27
t (m s)
b) c)
Fig. 3. Accelerations of the measuring point 2
The wave propagation velocity takes the following values:
(a) 272 m/s, if the block 2 is not destroyed;
(b) 241 m/s, if the block 2 is destroyed by a horizontal crack; and
(c) 224 m/s, if the block 2 is destroyed by a vertical crack.
Observe that, the velocity of propagation of the pendulum-type wave in the case (c) is less than in the case (b) and is less than in case (a).
Denote time by t; the displacement of a block at the time moment t by x(t); the acceleration of a block at the time moment t by x(t). For the whole period of time of the wave propagation, the energy E of the acceleration signal may be calculated by the formula
Et = \\x(t)f dt.
(1)
The time width at of the acceleration signal may be calculated by the formula
]1/2
1 ?
—\{t-<t >) |*(i)| dt
Ef
0
0
1 2 where <t>=—dt.
Et
The dynamic response of the physical model of the block medium at the measuring points 1 and 2 is given in Tab. 1. The values of E and ot are obtained according to formulas (1) and (2) by numerical integration of the experimental data.
Table 1
Dynamic response of the physical model of the block medium
Measuring points Cases Energy Et (m2/ms3) Time width ct (ms) Maximum x(t) (g) Minimum x(t) (g)
Measuring point 1 (a) 175.0 1.7 488.8 -374.3
(b) 52.5 3.8 145.3 -171.8
(c) 37.0 4.6 86.3 -142.2
Measuring point 2 (a) 28.7 2.6 116.2 -100.7
(b) 18.0 3.1 76.5 -82.3
(c) 20.0 3.6 86.5 -83.3
From Tab. 1, we conclude that, in the cases (b) and ((c), the energy E at the measuring points 1 and 2 is less than in the case (a), while the time width at is bigger than in the case (a). Moreover, the changes in the values of E are more significant at the measuring point 1 rather than at the measuring point 2. We conclude also that, in the case (c), the time width at is bigger than in the case (b).
Let us analyze the spectral properties of the acceleration x(t). The spectrum of the acceleration i s computed by the formula
F(<o) = lx(t)e-i(atdt. (3)
The spectra F(co) of the accelerations x(t), represented in Fig. 2 and Fig. 3, are shown in Fig. 4 and Fig. 5.
50 r
^ 25 £
0
0 5 10 15 20
40 r 20 0
£
40
20
0
Q (Hz) a)
5 10 15 20 0 5 10 15 20
Q (Hz)) Q (Hz)
b) c)
Fig. 4. The spectrum of the acceleration at the measuring point 1
40
Si' 20
Ph
0
20
'is
S 10
Ph
0 5 10 15 20
co /Hz a)
0
a
Ph
20 10 0
0 5 10 151 20
00/Hz b)
0 5 10 15 20 O /Hz
<0
Fig. 5. The spectrum of the acceleration at the measuring point 2
The center of the spectrum of acceleration is calculated by the following formula
<O>
= ■ jœ| F (o)|2d O ,
Eo
(4)
2
where E© = J|F(©)| d©. The bandwidth of the spectrum of acceleration is calculated by the following formula
-|1/2
a,
O
-1 j (o- < o >)2 |F (o)|2 d o
Eo
(5)
The dynamic response in frequency domain at the measuring points 1 and 2 is given in Tab. 2. The values of the frequency center <©> and the bandwidth of the spectrum of acceleration ara are obtained according to the formulas (3)—(5) by numerical integration of the experimental data.
Table 2
The dynamic response in frequency domain at the measuring points 1 and 2
Measuring points Cases Frequency center <o> (Hz) Bandwidth of frequency a0 (Hz) Main frequency o (Hz)
Measuring point 1 (a) 4.8 2.3 3.0
(b) 2.6 1.1 2.2
(c) 2.1 1.0 2.2
Measuring point 2 (a) 2.4 1.0 3.0
(b) 1.9 1.0 1.5
(c) 1.8 1.0 1.5
Here, we say that o is the main frequency if F(o) takes the maximum value at o = O.
From Tab. 2, we conclude that, at the measuring point 1, in the cases (b) and (c), all the values of < o >, a0, and O are significantly less than the same values in the case (a). At the measuring point 2, in the cases (b) and (c), the values of < o > and O
are less than the same values in the case (a). This means that both of the measuring points are vibrating with a lower frequency in the cases (b) and (c).
Conclusions
1. If the block 2 is destroyed by a horizontal or vertical crack,
- the energy of the acceleration signal and the amplitude of acceleration of the blocks 3 and 9 decrease, while the time width of the acceleration signal increases;
- the blocks 3 and 9 are vibrating with a lower frequency.
2. The dynamic response of the physical model at the measuring point 1 changes more significantly for the case of a vertical crack rather than for the case of a horizontal crack.
3. The velocity of propagation of the pendulum-type wave in the case when the block 2 is destroyed by a vertical crack is less than in the case when the block 2 is destroyed by a horizontal crack and is less than in the case when the block 2 is not destroyed.
This work was supported by project no. AAAA-A17-117122090002-5.
REFERENCES
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© Kaixing Wang, Yishan Pan, V. N. Oparin, N. I. Aleksandrova, 2018
