Studying a crack initiation in terms of elastic oscillations in stress strain rock mass
Purpose.Deriving the criterion of a crack (joint) initiating under simultaneous effect of the rock stress state and elastic oscillations generated by an external source is the research purpose. Determining the quantitative relations to estimate the contribution of oscillations to crack initiation an...
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УкрНДМІ НАН України, Інститут геотехнічної механіки НАН України
2016
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irk-123456789-1047382016-07-15T03:02:20Z Studying a crack initiation in terms of elastic oscillations in stress strain rock mass Sdvyzhkova, О. Golovko, Yu. Dubytska, M. Klymenko, D. Purpose.Deriving the criterion of a crack (joint) initiating under simultaneous effect of the rock stress state and elastic oscillations generated by an external source is the research purpose. Determining the quantitative relations to estimate the contribution of oscillations to crack initiation and creating a theoretical basis for the improvement of rock burst forecasting technique is a goal as well. Methods. The brittle failure theory and a time-space approach are applied to determine a critical length of initiating cracks depending on stress level and amplitude-frequency characteristics of acoustic oscillations. Analysis of experimental data and comparison with the numerical results are carried out. Findings. Quantitative ratios between the critical length of the crack, the stress intensity factor, oscillation amplitude and frequency are determined. It is shown that there are such values of the oscillation frequencies at which the critical crack length is especially sensitive to the amplitude alteration. The increase in the oscillation amplitude initiates starting the crack with small length. Numerical estimation is made for close-grained sandstone using such characteristics as crack resistance factor and Rayleigh’ wave velocity and tensile strength. Increasing the amplitude twice at the frequency of 1145 Hz causes the triple reduction of the starting crack length. Numerical results correlate with in situ data related to acoustic predicting the dynamic phenomena in the rock mass. Цель. Определить критерий страгивания трещины в породной среде при одновременном воздействии внутренних напряжений в породном массиве и акустических колебаний, генерируемых внешним источником, а также получить количественные соотношения для оценки влияния колебаний на процесс страгивания трещины и создать теоретическую основу для совершенствования метода акустического прогноза. Методика. Применение теории хрупкого разрушения в пространственно-временной постановке для получения количественных соотношений между критической длиной страгиваемой трещины и амплитудночастотными характеристиками колебаний. Сравнение численных результатов с данными экспериментального прогноза динамических явлений. Результаты. Даны количественные соотношения между критической длиной трещины, квазистатическим напряжением, амплитудой, и частотой упругих колебаний в породном массиве. Показано, что существуют частоты колебаний, на которых критическая длина трещин особо чувствительна к изменению амплитуды. Так, например, для песчаника увеличение амплитуды колебаний в 2 раза на частоте 1145 Гц приводит к уменьшению длины страгиваемых трещин в 3 раза. Численные результаты коррелируют с экспериментальными данными акустического прогноза динамических явлений в породном массиве. Мета. Визначити критерій зрушування тріщини у породному середовищі при одночасному впливі внутрішніх напружень у породному масиві й акустичних коливань, що генеруються зовнішнім джерелом, а також отримати кількісні співвідношення для оцінки впливу коливань на процес страгування тріщини та створити теоретичну основу для вдосконалення методу акустичного прогнозу. Методика. Застосування теорії крихкого руйнування у просторово-часовій постановці для отримання кількісних співвідношень між критичною довжиною тріщини, що стартує, та амплітудно-частотними характеристиками коливань. Порівняння чисельних результатів з даними експериментального прогнозу динамічних явищ. Результати. Надані кількісні співвідношення між критичною довжиною тріщини, квазістатичним напруженням, амплітудою і частотою пружних коливань у породному масиві. Показано, що існують частоти коливань, на яких критична довжина тріщин особливо чутлива до зміни амплітуди. Так, наприклад, для пісковику збільшення амплітуди коливань у 2 рази на частоті 1145 Гц призводить до зменшення довжини тріщин, що стартують, у 3 рази. Чисельні результати корелюють з експериментальними даними акустичного прогнозу динамічних явищ у породному масиві. 2016 Article Studying a crack initiation in terms of elastic oscillations in stress strain rock mass / О. Sdvyzhkova, Yu. Golovko, M. Dubytska, D. Klymenko // Розробка родовищ: Зб. наук. пр. — 2016. — Т. 10, вип. 2. — С. 72-77. — Бібліогр.: 8 назв. — англ. 2415-3435 DOI: http://dx.doi.org/10.15407/mining10.02.072 http://dspace.nbuv.gov.ua/handle/123456789/104738 622.831.3:531.36 en Розробка родовищ УкрНДМІ НАН України, Інститут геотехнічної механіки НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
Purpose.Deriving the criterion of a crack (joint) initiating under simultaneous effect of the rock stress state and elastic oscillations generated by an external source is the research purpose. Determining the quantitative relations to estimate the contribution of oscillations to crack initiation and creating a theoretical basis for the improvement of rock burst forecasting technique is a goal as well.
Methods. The brittle failure theory and a time-space approach are applied to determine a critical length of initiating cracks depending on stress level and amplitude-frequency characteristics of acoustic oscillations. Analysis of experimental data and comparison with the numerical results are carried out.
Findings. Quantitative ratios between the critical length of the crack, the stress intensity factor, oscillation amplitude and frequency are determined. It is shown that there are such values of the oscillation frequencies at which the critical crack length is especially sensitive to the amplitude alteration. The increase in the oscillation amplitude initiates starting the crack with small length. Numerical estimation is made for close-grained sandstone using such characteristics as crack resistance factor and Rayleigh’ wave velocity and tensile strength. Increasing the amplitude twice at the frequency of 1145 Hz causes the triple reduction of the starting crack length. Numerical results correlate with in situ data related to acoustic predicting the dynamic phenomena in the rock mass. |
| format |
Article |
| author |
Sdvyzhkova, О. Golovko, Yu. Dubytska, M. Klymenko, D. |
| spellingShingle |
Sdvyzhkova, О. Golovko, Yu. Dubytska, M. Klymenko, D. Studying a crack initiation in terms of elastic oscillations in stress strain rock mass Розробка родовищ |
| author_facet |
Sdvyzhkova, О. Golovko, Yu. Dubytska, M. Klymenko, D. |
| author_sort |
Sdvyzhkova, О. |
| title |
Studying a crack initiation in terms of elastic oscillations in stress strain rock mass |
| title_short |
Studying a crack initiation in terms of elastic oscillations in stress strain rock mass |
| title_full |
Studying a crack initiation in terms of elastic oscillations in stress strain rock mass |
| title_fullStr |
Studying a crack initiation in terms of elastic oscillations in stress strain rock mass |
| title_full_unstemmed |
Studying a crack initiation in terms of elastic oscillations in stress strain rock mass |
| title_sort |
studying a crack initiation in terms of elastic oscillations in stress strain rock mass |
| publisher |
УкрНДМІ НАН України, Інститут геотехнічної механіки НАН України |
| publishDate |
2016 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/104738 |
| citation_txt |
Studying a crack initiation in terms of elastic oscillations in stress strain rock mass / О. Sdvyzhkova, Yu. Golovko, M. Dubytska, D. Klymenko // Розробка родовищ: Зб. наук. пр. — 2016. — Т. 10, вип. 2. — С. 72-77. — Бібліогр.: 8 назв. — англ. |
| series |
Розробка родовищ |
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| first_indexed |
2025-07-07T15:46:02Z |
| last_indexed |
2025-07-07T15:46:02Z |
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1837003611464269824 |
| fulltext |
Founded in
1900
National Mining
University
Mining of Mineral Deposits
ISSN 2415-3443 (Online) | ISSN 2415-3435 (Print)
Journal homepage http://mining.in.ua
Volume 10 (2016), Issue 2, pp. 72-77
72
UDC 622.831.3:531.36 http://dx.doi.org/10.15407/mining10.02.072
STUDYING A CRACK INITIATION IN TERMS OF ELASTIC OSCILLATIONS
IN STRESS STRAIN ROCK MASS
О. Sdvyzhkova1*, Yu. Golovko1, M. Dubytska1, D. Klymenko1
1Higher Mathematics Department, National Mining University, Dnipropetrovsk, Ukraine
*Corresponding author: e-mail sdvyzhkova_e@nmu.org.ua, tel. +380676301048
ВИЗНАЧЕННЯ УМОВИ СТАРТУ ТРІЩИН, ЩО ІНІЦІЙОВАНИЙ КОЛИВАННЯМИ
В ПОРОДНОМУ НАПРУЖЕНО-ДЕФОРМОВАНОМУ СЕРЕДОВИЩІ
О. Сдвижкова1*, Ю. Головко1, М. Дубицька1, Д. Клименко1
1Кафедра вищої математики, Національний гірничий університет, Дніпропетровськ, Україна
*Відповідальний автор: e-mail sdvyzhkova_e@nmu.org.ua, тел. +380676301048
ABSTRACT
Purpose. Deriving the criterion of a crack (joint) initiating under simultaneous effect of the rock stress state and
elastic oscillations generated by an external source is the research purpose. Determining the quantitative relations to
estimate the contribution of oscillations to crack initiation and creating a theoretical basis for the improvement of
rock burst forecasting technique is a goal as well.
Methods. The brittle failure theory and a time-space approach are applied to determine a critical length of initiating
cracks depending on stress level and amplitude-frequency characteristics of acoustic oscillations. Analysis of exper-
imental data and comparison with the numerical results are carried out.
Findings. Quantitative ratios between the critical length of the crack, the stress intensity factor, oscillation amplitude
and frequency are determined. It is shown that there are such values of the oscillation frequencies at which the criti-
cal crack length is especially sensitive to the amplitude alteration. The increase in the oscillation amplitude initiates
starting the crack with small length. Numerical estimation is made for close-grained sandstone using such character-
istics as crack resistance factor and Rayleigh’ wave velocity and tensile strength. Increasing the amplitude twice at
the frequency of 1145 Hz causes the triple reduction of the starting crack length. Numerical results correlate with in
situ data related to acoustic predicting the dynamic phenomena in the rock mass.
Originality. The crack initiation criterion has been identified.
Practical implications. Quantitative relations between stress components and amplitude-frequency characteristics
should be used to improve the outburst forecasting technique and increase the reliability of dynamic effect prediction.
Keywords: crack initiating, dynamic phenomena, stress, oscillations, failure
1. INTRODUCTION
Predicting the dynamic phenomena (rock burst, out-
burst) in coal mining is a great challenge when ensuring
the safety of workers. Some forecast techniques are
based on the registration and analysis of oscillations in
rock mass (Standart SOU…, 2005). However the criteria
of dangerous states emergence are not justified enough.
The forecast of dynamic effects can be defined as
prediction of the rock failure. That is why studying the
oscillations influence on crack propagation in previously
stress-strained rock mass is appropriate. It should be
noted that sources of oscillations can be different and
often not properly identified. Acoustic impulses are gen-
erated in a solid body during such violations of its integ-
rity as dislocations and cracks. A large number of scien-
tific works is devoted to generation of elastic impulses at
crack initiation in a solid body.
It is supposed that cracks are the centers of the acous-
tic activity in rocks because the rock mass can be consid-
ered a kind of a solid body as well. Internal contacts in
the rock structure are broken by cracks and the energy of
the rock stress strain state in the crack vicinity is a source
of the crack initiating. The crack propagates if the stress-
es exceed a certain level and releases the excess of elastic
energy into the coal seam. At the same time the crack is a
stress concentrator itself. The abrupt stress change
around the crack generates an impulse affecting other
cracks (Prykhodchenko, Sdvyzhkova, Khomenko &
Kovrov, 2008).
O. Sdvyzhkova, Yu. Golovko, M. Dubytska, D. Klymenko. (2016). Mining of Mineral Deposits, 10(2), 72-77
73
Elastic waves can stop or speed up the process of
crack propagation under loading resulting in the material
fracturing (Morozov & Petrov, 1997). They can contrib-
ute to appearance of new surfaces of weakening. Various
rock-destroying mechanisms are the sources of oscilla-
tions as well. In this case the generated oscillations are
“enriched” with oscillations caused by the destruction
processes in the working tools zone.
Joints (cracks) initially present in the rock mass are in-
fluenced both by slowly-changing stress strain state and
by much faster-changing oscillations. These are generated
by an external source which is individual for each crack
and can provoke increase in the crack dimensions.
Determining the conditions under which such crack
growth becomes possible (i.e. the conditions of the crack
initiation) is an important theoretical constituent in de-
veloping reliable methods of forecasting the rock bursts
and outbursts and other dynamic effects on the basis of
acoustic signals registered in the rock mass.
2. THE MAIN PART OF THE ARTICLE
2.1. Experimental data analysis
Results of studying the potentially dangerous areas of
the rock mass in situ are presented by the authors. The
coal seam has been sounded with acoustic signal gener-
ated by mining mechanisms. Registration of the acoustic
signal that is extending into the rock mass was imple-
mented using the acoustic equipment AK-1 (Shashenko,
Zhuravlev, Sdvizhkova & Dubytska, 2015).
The operation principle of the outburst monitoring
system AK-1M is based on the registration and analysis
of acoustic oscillations. Oscillations are generated both
by the rock mass itself at the stress state changing around
the excavations and by the mechanisms used for coal
mining. The acoustic signal produced by drifting tools is
picked up by an underground block of the equipment
with a build-in sensor (geophone 1 in Figure 1).
Figure 1. Functional scheme of equipment AK-1M: 1 – geo-
phone; 2 – link; 3 – surface part of the equipment
with intrinsically safe output circuit; 4 – attenuator;
5 – analysis complex
The sensor activity is based on using a piezoelectric
effect. It has high dynamic characteristics and ability to
perceive oscillations within a wide range of frequency
(from several Hz to dozens of MHz). An electric charge
appears under the influence of acoustic and seismic
waves pressure on external and internal sides of piezoe-
lectric couple plates. A total electromotive force between
an output wire and a case changes proportionally to the
pressure. Then the signal is transmitted to the surface
part of the equipment 3 (Fig. 1) over the intrinsically safe
communication line 2. The surface part of the equipment
is located in a seismic service department of the mine.
The communication line can be nearly 10 km length.
Then the signal is redirected to an attenuator 4. It is in-
tended for the smooth, stepped or fixed drop in voltage,
current strength, power of electric and electromagnetic
oscillations. After that the signal comes to an analytical
complex 5. The analysis, recording and preservation of
the sound signals are actualized in this complex.
The rock mass area investigated by the acoustic sig-
nals has to be located between a source of oscillations
and the sensor. Actually this condition is difficult to
implement with existing technologies of coal mining.
Reception of the acoustic signal in conditions of coal
deposit is possible due to the layered structure of the rock
mass and divergence in physical and mechanical proper-
ties of coal seam and rocks in the roof and floor.
In particular, sandstone layers have high acoustic
conductivity and other rocks in the roof are characterized
by significantly bigger acoustic hardness in comparison
with the coal seam. It gives a possibility to register the
signals far from the studied area.
The most important technical requirement is absence
of additional sound intervention when acoustic observa-
tions take place. For example, borehole drilling or drift
heading should be absent to register oscillations generat-
ed by shear equipment. That is why the sensor should be
placed out of working zone (Fig. 2).
Figure 2. Arrangement of sensors
Studying acoustic signals during a longwall retreat-
ing is carried out by setting geophones into the walls of
two parallel roadways. One of them is installed in the
main gate and the other is placed in the tailgate. The
geophones are located at a distance of 10 to 40 m in front
of the coal face if a panel method of coal field prepara-
tion is adopted (Fig. 2). The sensor should be tightly
installed in the borehole to ensure sufficient contact with
the rock mass. There is a risk of moisture seeping into
the electric circuit of the sensor due to the coal seam
inclination. Therefore the sensor should be placed into
the borehole which is drilled under some positive angle
regarding to the horizon.
surface part links
geophone
longwall
O. Sdvyzhkova, Yu. Golovko, M. Dubytska, D. Klymenko. (2016). Mining of Mineral Deposits, 10(2), 72-77
74
When the longwall advances at a distance of 10 – 40 m,
the space between the sound source and geophone is
reduced. Then the next borehole is prepared and other
geophone is installed. The surface unit switches over to
the next geophone respectively. The sensor installed at
the previous station is extracted and used to prepare the
further station.
Processing the acoustic signal results in an ampli-
tude-frequency spectrum. Analysis of the amplitude-
frequency components is conducted within a range of
0 – 300 Hz and 1250 – 4000 Hz. The statistical data
processing allowed to establish a correlation between
the amplitude-frequency characteristics (AFC) of elastic
oscillations and probability of the rock outburst. Nu-
merous experiments have been carried out in situ at
various coal mines of Central and Western Donbass for
this purpose. A real zone of potential danger in a coal
seam is characterized by the variability of AFC includ-
ing migration of the main frequency in a wide range.
Appearance of high-amplitude and high-frequency har-
monics is the main feature noted in all registered cases
of rock outburst. Therefore doubling or tripling of the
amplitude of the registered oscillation at frequencies of
1000 – 1300 Hz is considered an empirical warning of
the possible dynamic effect of rock pressure. However
this criterion has not been proved theoretically.
It can be assumed hypothetically that the change in
amplitude and frequency of acoustic oscillations at the
moment of outburst is connected with formation of new
free surfaces (cracks). We consider the crack initiation in
terms of elastic oscillation impact.
2.2. Crack initiation criterion
Let us consider a single crack at infinite space under
the action of time-dependent stress field. We determine
the loading conditions under which the increase of the
crack dimensions is probable, i.e. the crack initiates.
The following assumptions are adopted. The rocks
are represented as an elastic body and the crack is a disk
one. The rock destruction is considered as a brittle failure
and the stress is supposed to be normal to the crack
plane. The effects of fatigue failure are not considered.
Rock stresses are represented as a sum of the harmonic
and the quasi-static components.
It is known that any crack is a discontinuity in a con-
tinuum body. It creates stress concentration in the sur-
rounding area, so the stress value at the top of the crack
can exceed the yield strength. In terms of fracture me-
chanics, this means exceeding the limit value of the
stress intensity factor, which causes the crack growth and
creates the condition under which the new cracks appear
and a certain amount of the rock fails.
There are different approaches to define a criterion of
rock failure under the impact of stress changing in time
and space. Morozov & Petrov (1997) proposed a general
kind of the criterion:
( ) c
x
dx
t
t
dtdxtx
d 11
0
0
0
0
,
1 σσ
τ τ
≥ ⋅ − −
, (1)
where:
( )tx,1σ – principal stress;
c1σ – tensile strength; τ is a time parameter charac-
terizing the response delay of the failed material on the
considered structural level;
d – length parameter; х is an axial coordinate (axis X
is perpendicular to the principal stress);
t – current time value;
0x – X-coordinate of a point in the rock mass;
0t – time of failure.
The stress field is determined by the stress intensity
factor ( )tK1 defined by the formula:
( ) ( )
x
tK
tx
⋅
=
π
σ
2
, 1
1 . (2)
Morozov & Petrov (1997) obtained the condition of
the destruction as
2
1
12
c
cK
d
πσ
= :
( ) c
t
t
KtK 11
0
0
1 ≥
−ττ
, (3)
where cK1 is the critical value of 1K which is the crack
resistance factor determining the ability of a solid to
resist loading without crack initiation. The validity crite-
rion is illustrated by experimental results (Atroshenko,
Krivosheev & Petrov, 2002).
2.3. Criterion of crack initiation under harmonic
stress component
Let us express the stress intensity factor in terms of
regular principal stress 1σ for a disk crack. Let the radi-
us of the crack be l. Using (3) we obtain a condition of
the crack initiation at the given stress level:
( )
l
K
dtt c
t
t
1
1 2
1 0
0
πσ
τ τ
=
−
. (4)
Not only quasi-static stresses are acting in rocks in
the general case. The sign-variable and quick-changing
stresses occur in elastic waves due to the action of impul-
sive load in the rock mass. The elastic oscillations are
generated by an internal source during the rock brittle
destruction.
Under the simultaneous action of quasi-static stress
and elastic vibration, a stress tensor ( )tTσ can be repre-
sented as a sum of a quasi-static component ( )tkсT and an
oscillation one ( ) ( ) ( ) ( )tвtkсttв TTTT +=σ: .
The stress acting normally to the plane of the crack
should be considered in the vicinity of the initiation time t.
We can represent this stress in the form:
( ) ( ) ( )[ ]00001 2cos ϕπνσσ +−⋅⋅+−⋅+= ttattkt , (5)
where:
( ) ( )00 ttkt −⋅+= σσ – a component not related to
oscillations;
0σ – actual tensile stress;
O. Sdvyzhkova, Yu. Golovko, M. Dubytska, D. Klymenko. (2016). Mining of Mineral Deposits, 10(2), 72-77
75
0ttdt
d
k
=
= σ
– a component associated with the
stress change in time during the development of mining
operations;
0,, ϕνa – amplitude, frequency and phase of elastic
oscillations respectively.
Substituting (5) into (4) and carrying out the mathe-
matical transformation, we obtain the inequality:
( ) cK
l
k
ca 10
22
sin
⋅
>+⋅−⋅ πστπντ . (6)
This expression defines the moment of the crack initia-
tion considering the impact of elastic oscillations in the
rock mass. Inequality (6) can be used at high and low
frequencies of oscillations (including the value ν = 0).
Incubation time τ is not a uniquely specific parameter
and its value can be chosen and interpreted differently
(Parton & Borisov, 1988). In any case, τ should be con-
sidered (as it was mentioned above) as a parameter char-
acterizing the response delay of failed material to the
considered structural level at unsteady loading. If a fail-
ure criterion is expressed through the full stress, the pa-
rameter τ is determined as the transfer of energy between
neighboring elementary structures of failure with the
characteristic size d. In case of using only a regular stress
component (as it has been done in (4)), the response time
can be estimated in terms of a consecutive expansive
wave falling upon the crack of the finite length. The
numerical solution of this problem and its analysis
showed that the stress intensity factor at the crack tip
increases monotonically (Alekseev & Nedodaev, 1982).
The stress intensity factor reaches maximum at the time
of a Rayleigh wave coming from the opposite top. On
this basis we get
Rc
l=τ , where Rc is a rate of the Ray-
leigh wave.
Hence, we transform the inequality (6) to the form:
( ) 1
2
sin −⋅+≥⋅ l
l
K
lca c απ ν , (7)
where:
R
c
c
R c
K
K
ka
a
c
l
l
πν
σνσ
α
σ
ν
ν ⋅=
⋅
===
0
1
00 2
,,, .
These values are dimensionless by assumption.
Equation (7) determines the criterion of the crack ini-
tiation. If values νσ ,,,, 0 kcK Rc are constants, then the
parameters νcK and α are invariants as well. Then only
parameters l and a are variables in the criterion (7), i.e.
the initiation of the crack of length l is determined only
by the amplitude of oscillation a . The functional link
between these values can be obtained as a solution of the
transcendental equation (7).
It should be noted that if we consider the crack initia-
tion caused by elastic oscillations, then both sides of (7)
must be positive. If the right-hand side is negative then
the crack initiation is caused only by quasi-stationary
stress. If lKc <ν , the failure is provided by the con-
stant stress component.
2.4. Numerical analysis of the criterion
Let the crack resistance factor cK , the velocity of
Rayleigh’ waves spreading Rc and parameters of stress
state k,0σ be constant. Then the equation (7) gives a
relationship between the relative amplitude of the elastic
oscillation a and the relative crack length l (Fig. 3). The
crack initiation occurs for the given level of amplitude.
The results of the numerical solution of the equation
(7) are presented in Table 1. The calculations are carried
out for close-grained sandstone. The crack resistance
factor is assumed to be 47.11 =cK MPa m , the Ray-
leigh’ waves spreading velocity 2400=Rc m/s and the
tensile strength 9.00 =σ MPa. We also assume that the
stress component does not change in time. This means
the parameter α is equal to zero (α = 0).
Table 1. The dimensionless and real length of the initiating
crack at different values of the oscillation amplitude
and frequency
Frequency,
ν
Crack
resistance
factor,
νcK
Amplitude,
a
Dimen-
sionless
length,
l2
Real
length,
l2
733 0.8
0.0 1.2 4.2
0.2 1.0 3.4
0.4 0.8 2.4
0.6 0.6 1.8
0.8 0.4 1.4
1.0 0.4 1.2
1.5 0.2 0.6
2.0 0.14 0.4
928 0.9
0.0 1.6 4.2
0.2 1.4 3.6
0.4 1.0 2.8
0.6 0.8 2.0
0.8 0.6 1.4
1.0 0.4 1.2
1.5 0.2 0.6
2.0 0.18 0.4
1145 1.0
0.0 1.996 4.2
0.2 1.994 4.18
0.4 1.980 4.16
0.6 1.12 2.4
0.8 0.76 1.6
1.0 0.6 1.2
1.5 0.4 0.8
2.0 0.2 0.4
1386 1.1
0.0 2.4 4.2
0.2 2.4 4.2
0.4 2.4 4.2
0.6 2.4 4.2
0.8 2.4 4.2
1.0 0.8 1.4
1.5 0.4 0.7
2.0 0.28 0.48
The changes of an initiating crack length depending
on the amplitude of the oscillations are presented in the
graphs in Figure 3.
O. Sdvyzhkova, Yu. Golovko, M. Dubytska, D. Klymenko. (2016). Mining of Mineral Deposits, 10(2), 72-77
76
a
l
0 1 2 3
0.4
0.8
1.2
1.6
K = cv 1.1
K = cv 1.0
K = cv 0.9
K = cv 0.8
Figure 3. The critical crack length as a function of the
relative oscillation amplitudes (α = 0)
We consider the curves with a sharp drop in the criti-
cal crack length l ( 0.1=νcK and 1.1=νcK ).This fact
is observed after sections that are parallel to the X-axis.
Note that the horizontal portions of the curves corre-
spond to the criterion:
01
2
≤−⋅+ l
l
Kc αν . (8)
This means that crack initiation occurs only because
of the quasi-stationary stress component. There is no
connection with the oscillations. At small values of com-
plex parameter νcK the critical crack length decreases
smoothly enough depending on the increase in the oscil-
lation amplitude. But at large values of parameter νcK
the horizontal sections increase and significant reduction
in the critical crack length reaches unrealizable values of
1>a . Therefore, we consider the value 0.1≈νcK as a
critical one when a small change in amplitude causes a
sharp decrease in the length of those cracks which are
ready to initiate. We can see on the plot that at
1.1=νcK the doubling the dimensionless amplitude
leads to approximately triple reduction of the initiated
crack length.
In particular, this effect can be seen clearly in case of
sandstone. When the oscillation frequency is 1145 Hz,
the value of complex parameter is approximately
0.1≈νcK , and we can observe from the table that insig-
nificant increase in oscillation amplitude (from 0.4 to 0.8)
provokes initiation of the short cracks. The length of
initiated cracks decreases very sharply (from 4 to 1.6 m).
This fact can be interpreted as a significant risk of cata-
strophic rock failure.
The results of theoretical research coincide with the
empirical data described above. Doubling and tripling
the amplitude of registered oscillation at frequencies of
1000 – 1300 Hz lead to the “short” crack initiation in the
rock mass. As a result, new free surfaces are created and
possibility of sudden release of potential energy occurs.
This means a high probability of dynamic effects in the
rock mass (rock bursts and outbursts).
Quantitative assessment shows that the amplitude
alteration within the high frequency range is extremely
important for predicting dynamic effects in hard rocks.
Therefore, forecasting technique should be improved by
analyzing the high-frequency part of the acoustic signal
spectrum. Several prognostic indicators (maximum am-
plitude in the high-frequency part of the spectrum, the
spectrum area, the ratio of amplitude and spectrum area)
can be identified to improve the accuracy of dynamic
effect prediction (Maslennikov, 1999).
3. CONCLUSIONS
The criterion of a crack initiation under unsteady
loading has been developed on the basis of the general
space-time approach to the description of a solid fracture.
The stress component acting normally to the plane of the
crack is supposed to be a sum of quasi-stationary and
harmonic components. The quantitative relation has been
obtained to determine the critical value of the crack
length at which the crack initiates depending on the qua-
si-static stress and oscillation amplitude and frequency.
In particular case of a sandstone layer the increase of
elastic oscillation amplitude in 2 times at frequency
1145 Hz reduces the critical length of initiating crack in
2 – 3 times.
Numerical results correlate with experimental data
regarding the acoustic prediction of dynamic effect in the
rock mass. In situ data indicates that increasing the am-
plitude of the recorded oscillations in the rock mass in
2 – 3 times at frequencies 1000 – 1300 Hz is a sign of
possible dynamic effects (rock bursts and outbursts).
The analysis of oscillation amplitude alteration is ex-
tremely important within the high frequency part of the
acoustic signal spectrum. Several prognostic indicators
(maximum amplitude in the high-frequency part of the
spectrum, the spectrum area, the ratio of amplitude and
spectrum area) can be identified to improve technique of
acoustic sounding and increase the accuracy of dynamic
effect prediction.
ACKNOWLEDGEMENT
We are very grateful to the staff of the mine “Kras-
nolymanska” PSSC for providing statistical data.
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nenko, V. (2016). Effect of time-transgressive faults upon
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Shashenko, A.N., Zhuravlev, V.N., Sdvizhkova, Ye.A., &
Dubytska, M.S. (2015). Forecast of disjunctives based on
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ABSTRACT (IN UKRAINIAN)
Мета. Визначити критерій зрушування тріщини у породному середовищі при одночасному впливі внутріш-
ніх напружень у породному масиві й акустичних коливань, що генеруються зовнішнім джерелом, а також
отримати кількісні співвідношення для оцінки впливу коливань на процес страгування тріщини та створити
теоретичну основу для вдосконалення методу акустичного прогнозу.
Методика. Застосування теорії крихкого руйнування у просторово-часовій постановці для отримання кількіс-
них співвідношень між критичною довжиною тріщини, що стартує, та амплітудно-частотними характеристиками
коливань. Порівняння чисельних результатів з даними експериментального прогнозу динамічних явищ.
Результати. Надані кількісні співвідношення між критичною довжиною тріщини, квазістатичним напру-
женням, амплітудою і частотою пружних коливань у породному масиві. Показано, що існують частоти коли-
вань, на яких критична довжина тріщин особливо чутлива до зміни амплітуди. Так, наприклад, для пісковику
збільшення амплітуди коливань у 2 рази на частоті 1145 Гц призводить до зменшення довжини тріщин, що
стартують, у 3 рази. Чисельні результати корелюють з експериментальними даними акустичного прогнозу ди-
намічних явищ у породному масиві.
Наукова новизна. Визначено умову старту тріщини.
Практична значимість. Кількісні співвідношення між компонентами напружень і амплітудно-частотними
характеристиками коливань є основою для вдосконалення методики прогнозу й підвищення його достовірності.
Ключові слова: тріщина, напруга, амплітуда, руйнування, коливання
ABSTRACT (IN RUSSIAN)
Цель. Определить критерий страгивания трещины в породной среде при одновременном воздействии внут-
ренних напряжений в породном массиве и акустических колебаний, генерируемых внешним источником, а
также получить количественные соотношения для оценки влияния колебаний на процесс страгивания трещины
и создать теоретическую основу для совершенствования метода акустического прогноза.
Методика. Применение теории хрупкого разрушения в пространственно-временной постановке для полу-
чения количественных соотношений между критической длиной страгиваемой трещины и амплитудно-
частотными характеристиками колебаний. Сравнение численных результатов с данными экспериментального
прогноза динамических явлений.
Результаты. Даны количественные соотношения между критической длиной трещины, квазистатическим
напряжением, амплитудой, и частотой упругих колебаний в породном массиве. Показано, что существуют ча-
стоты колебаний, на которых критическая длина трещин особо чувствительна к изменению амплитуды. Так,
например, для песчаника увеличение амплитуды колебаний в 2 раза на частоте 1145 Гц приводит к уменьше-
нию длины страгиваемых трещин в 3 раза. Численные результаты коррелируют с экспериментальными данны-
ми акустического прогноза динамических явлений в породном массиве.
Научная новизна. Определено условие страгивания трещины.
Практическая значимость. Количественные соотношения между компонентами напряжений и амплитуд-
но-частотными характеристиками колебаний являются основой для совершенствования методики прогноза и
повышения его достоверности.
Ключевые слова: трещина, напряжение, амплитуда, разрушение, колебания
ARTICLE INFO
Received: 5 April 2016
Accepted: 26 May 2016
Available online: 30 June 2016
ABOUT AUTHORS
Olena Sdvyzhkova, Doctor of Technical Sciences, Head of the Higher Mathematics Department, National Mining
University, 19 Yavornytskoho Ave., 5/33, 49005, Dnipropetrovsk, Ukraine. E-mail: sdvyzhkova_e@nmu.org.ua
Yurii Golovko, Candidate of Physics and Mathematics Sciences, Associate Professor of the Higher Mathematics
Department, National Mining University, 19 Yavornytskoho Ave., 5/26, 49005, Dnipropetrovsk, Ukraine. E-mail:
y_golovko@mail.ru
Mariia Dubytska, Candidate of Technical Sciences, Assistant Professor of the Higher Mathematics Department, National
Mining University, 19 Yavornytskoho Ave., 5/33, 49005, Dnipropetrovsk, Ukraine. E-mail: dubitskayam@gmail.com
Dina Klymenko, Senior Lecturer of the Higher Mathematics Department, National Mining University, 19 Yavornytsko-
ho Ave., 5/26, 49005, Dnipropetrovsk, Ukraine. E-mail: dinklim@mail.ru
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