Fragmentation fractal of sandstone under acid corrosion and coupled static-dynamic loads
The analysis results show that the fragment of broken rock is a fractal distribution, and the smaller the impact pressure is, the less specimen fragments is,the lower degree of fragmentation degree is, and the lower fractal dimension is. Research shows that fractal dimension increases with the dynam...
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| Цитувати: | Fragmentation fractal of sandstone under acid corrosion and coupled static-dynamic loads / LIU Yong-sheng, LI Jin, ZOU Jia-yu, WU Yun, LU Shao-yong // Functional Materials. — 2018. — Т. 25, № 1. — С. 122-127. — Бібліогр.: 15 назв. — англ. |
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irk-123456789-1540522019-06-16T01:26:27Z Fragmentation fractal of sandstone under acid corrosion and coupled static-dynamic loads LIU Yong-sheng LI Jin ZOU Jia-yu WU Yun LU Shao-yong Characterization and properties The analysis results show that the fragment of broken rock is a fractal distribution, and the smaller the impact pressure is, the less specimen fragments is,the lower degree of fragmentation degree is, and the lower fractal dimension is. Research shows that fractal dimension increases with the dynamic compression strength of rock increasing, and the incident energy and the absorb energy increase linear with the fracture fractal dimension increasing. The fragmentation distribution of the specimens becomes more and more uniform with the increasing of the incident energy, and the characteristic scale of rupture decreased gradually. 2018 Article Fragmentation fractal of sandstone under acid corrosion and coupled static-dynamic loads / LIU Yong-sheng, LI Jin, ZOU Jia-yu, WU Yun, LU Shao-yong // Functional Materials. — 2018. — Т. 25, № 1. — С. 122-127. — Бібліогр.: 15 назв. — англ. 1027-5495 http://dspace.nbuv.gov.ua/handle/123456789/154052 DOI: https://doi.org/10.15407/fm25.01.122 en Functional Materials НТК «Інститут монокристалів» НАН України |
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Characterization and properties Characterization and properties |
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Characterization and properties Characterization and properties LIU Yong-sheng LI Jin ZOU Jia-yu WU Yun LU Shao-yong Fragmentation fractal of sandstone under acid corrosion and coupled static-dynamic loads Functional Materials |
| description |
The analysis results show that the fragment of broken rock is a fractal distribution, and the smaller the impact pressure is, the less specimen fragments is,the lower degree of fragmentation degree is, and the lower fractal dimension is. Research shows that fractal dimension increases with the dynamic compression strength of rock increasing, and the incident energy and the absorb energy increase linear with the fracture fractal dimension increasing. The fragmentation distribution of the specimens becomes more and more uniform with the increasing of the incident energy, and the characteristic scale of rupture decreased gradually. |
| format |
Article |
| author |
LIU Yong-sheng LI Jin ZOU Jia-yu WU Yun LU Shao-yong |
| author_facet |
LIU Yong-sheng LI Jin ZOU Jia-yu WU Yun LU Shao-yong |
| author_sort |
LIU Yong-sheng |
| title |
Fragmentation fractal of sandstone under acid corrosion and coupled static-dynamic loads |
| title_short |
Fragmentation fractal of sandstone under acid corrosion and coupled static-dynamic loads |
| title_full |
Fragmentation fractal of sandstone under acid corrosion and coupled static-dynamic loads |
| title_fullStr |
Fragmentation fractal of sandstone under acid corrosion and coupled static-dynamic loads |
| title_full_unstemmed |
Fragmentation fractal of sandstone under acid corrosion and coupled static-dynamic loads |
| title_sort |
fragmentation fractal of sandstone under acid corrosion and coupled static-dynamic loads |
| publisher |
НТК «Інститут монокристалів» НАН України |
| publishDate |
2018 |
| topic_facet |
Characterization and properties |
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http://dspace.nbuv.gov.ua/handle/123456789/154052 |
| citation_txt |
Fragmentation fractal of sandstone under acid corrosion and coupled static-dynamic loads / LIU Yong-sheng, LI Jin, ZOU Jia-yu, WU Yun, LU Shao-yong // Functional Materials. — 2018. — Т. 25, № 1. — С. 122-127. — Бібліогр.: 15 назв. — англ. |
| series |
Functional Materials |
| work_keys_str_mv |
AT liuyongsheng fragmentationfractalofsandstoneunderacidcorrosionandcoupledstaticdynamicloads AT lijin fragmentationfractalofsandstoneunderacidcorrosionandcoupledstaticdynamicloads AT zoujiayu fragmentationfractalofsandstoneunderacidcorrosionandcoupledstaticdynamicloads AT wuyun fragmentationfractalofsandstoneunderacidcorrosionandcoupledstaticdynamicloads AT lushaoyong fragmentationfractalofsandstoneunderacidcorrosionandcoupledstaticdynamicloads |
| first_indexed |
2025-07-14T05:33:49Z |
| last_indexed |
2025-07-14T05:33:49Z |
| _version_ |
1837599272422342656 |
| fulltext |
122 Functional materials, 25, 1, 2018
ISSN 1027-5495. Functional Materials, 25, No.1 (2018), p. 122-127
doi:https://doi.org/10.15407/fm25.01.122 © 2018 — STC “Institute for Single Crystals”
Fragmentation fractal of sandstone under acid
corrosion and coupled static-dynamic loads
LIU Yong-sheng, LI Jin, ZOU Jia-yu, WU Yun, LU Shao-yong
School of Civil Engineering and Architecture, East China Jiaotong
University, Nanchang 330013, P.R.China;
Received November 30, 2017
The analysis results show that the fragment of broken rock is a fractal distribution, and the
smaller the impact pressure is, the less specimen fragments is, the lower degree of fragmenta-
tion degree is, and the lower fractal dimension is. Research shows that fractal dimension in-
creases with the dynamic compression strength of rock increasing, and the incident energy and
the absorb energy increase linear with the fracture fractal dimension increasing. The fragmen-
tation distribution of the specimens becomes more and more uniform with the increasing of the
incident energy, and the characteristic scale of rupture decreased gradually.
Key words: rock dynamics; fragmentation fractal; acid corrosion; coupled static-dynamic
loads; strength; energy.
Обсуждается взаимосвязь между фрактальной размерностью и динамической
прочностью на сжатие, диссипацией энергии в породе под кислотной коррозией и
связанными статически-динамическими нагрузками. Результаты анализа показывают, что
фрагменты разрушенной породы имеют фрактальное распределение, и чем меньше ударное
давление, тем меньше фрагментов, меньше степень фрагментации и меньше фрактальная
размерность. Исследования показывают, что фрактальная размерность возрастает с ростом
динамической прочности породы на сжатие. Фрактальная размерность линейно возрастает
с вложенной энергией и поглощённой энергией. Распределение фрагментов образцов
становится все более однородным с ростом вложенной энергии, а характерный их масштаб
постепенно уменьшается.
Фрагментаційний фрактал пісковика під кислотною корозією та пов’язані
статично-динамічні навантаження. LIU Yong-sheng, LI Jin, ZOU Jia-yu, WU Yun,
LU Shao-yong.
Обговорюється взаємозв’язок між фрактальної розмірністю і динамічною міцністю
на стиск, дисипацією енергії в породі під кислотної корозією і пов’язаними статично-
динамічними навантаженнями. Результати аналізу показують, що фрагменти зруйнованої
породи мають фрактальний розподіл, і чим менше ударний тиск, тим менше фрагментів,
менше ступінь фрагментації і менше фрактальна розмірність. Дослідження показують,
що фрактальна розмірність зростає з ростом динамічної міцності породи на стиск.
Фрактальна розмірність лінійно зростає з вкладеною енергією і поглиненої енергією.
Розподіл фрагментів зразків стає все більш однорідним з ростом вкладеної енергії, а
характерний їх масштаб поступово зменшується.
Functional materials, 25, 1 2018 123
LIU Yong-sheng et al. / Fragmentation fractal of sandstone under ...
1. Introduction
Rock is a kind of heterogeneous brittle me-
dia. the internal micro cracks will be develop
and expand continuously under the influence
of the deep complex environment and the high
ground stress. The degree of difficulty, energy
consumption and fragmentation distribution of
rock fragmentation are important parameters
and measure index in rock drilling, blasting,
mining and mineral processing. With the de-
velopment of fractal theory in the 70s of last
century, the study of rock fracture has entered
a new stage. A large number of studies [1-3]
showed that the development of the rock frag-
ment from microscopic damage to the macro-
scopic crack has fractal characteristics, and the
fragment is fractal distribution.
In recent years, fractal theory has been wide-
ly used in the field of rock fragmentation and en-
ergy analysis, and a lot of research results have
been obtained. Xu Jin-yu et al. [7] analyzed the
fragmentation lumpiness distribution of marble
under impact loading test by fractal geometry.
Nagahama[8] studied the process of the damage
and energy dissipation of rock by fractal theory.
Wang Qi-sheng et al.[9] studied the fragmenta-
tion fractal characteristics of granite under the
static and dynamic coupled loads, and analyzed
the change of rock fragmentation fractal dimen-
sion with different loads.
In previous studies, they all did not take
into account the effects of deep underground
environment. In fact, the influence of deep un-
derground complex environment on the rock
properties can not be ignored in engineer-
ing practice, and most of the deep rock are in
acidic environment of underground water. In
addition, the deep rock is in a the coupled of
static and dynamic stress state. So there are
great practical significance to study the fractal
of deep rock considering comprehensively the
influence of deep underground environment. In
this paper, the fragmentation fractal character-
istics of red sandstone under the coupled static-
dynamic loads are comprehensively considered,
and relationship of the fractal dimension and
absorbed energy and dynamic strength were
analyzed.
2. Calculate of fractal dimension
In the early 70s of last century, French sci-
entist B. B. Mandelbrot put forward the concept
of fractal theory for the first time. The fractal
dimension can be used to describe the features
of a graph or object with fractal features[11].
The fragmentation process of rock is very com-
plex, which is the result of the interaction of
the external loading and the internal cracks in
the rock mass. Based on large numbers of theo-
retical and experimental studies, experts and
scholars [12-13]had put forward a statistical
model of rock fragmentation distribution, and
the most representative distribution function
are R-R and G-G-S .
The expression of R-R distribution:
y
r
r
a
= - -
æ
è
çççç
ö
ø
÷÷÷÷÷
é
ë
ê
ê
ê
ù
û
ú
ú
ú
1
0
exp (1)
Here a is the distribution parameter of rock
fragments, r0 is the characteristic size of the
rock fragments, which is the size of the block
when the cumulative amount of the sieve is
1 1 0-( )/ e %.
The expression of G-G-S distribution:
y
r
rm
b
=
æ
è
çççç
ö
ø
÷÷÷÷÷
(2)
Here b is the distribution parameter of rock
fragment, which is the linear slope in log-log co-
ordinate. rm is the distribution function. When r
= rm, the quantity under sieve is 100% , which
is the maximum size of the rock fragments.
Expanding the formula (1) in series and re-
moving of the higher order terms, then we can
find that the final results of the formula (1) and
formula (2) are the same. In the G-G-S distribu-
tion function, the M represents the total mass
of the fragments, the cumulative mass under
the sieve is m r( ) when the feature size is r .
And the formula (2) can be converted into:
m r
M
r
rm
b
( )
=
æ
è
çççç
ö
ø
÷÷÷÷÷
(3)
Derivative the formula (3):
dm r drbµ -1 (4)
Considering the relationship between the
increment of rock fragments and the mass
growth:
dm r dNµ 3 (5)
Based on the results of Turcotte et al.[14]
the relationship of the fractal dimension D and
the number of fragments N which is larger
than current size and the linear characteristic
size r can be show as:
N r Dµ - (6)
Combining formula (4) and (5), the calcula-
tion formula of fractal dimension D can be de-
duced:
W t
A C
E
t dtI
e e
e
I
t
( ) ( )= ò σ 2
0
(7)
124 Functional materials, 25, 1, 2018
LIU Yong-sheng et al. / Fragmentation fractal of sandstone under ...
Table 1 The loading scheme
Loading scheme Impact pressure (MPa) Axial static pressure
(MPa)
Combined static and
dynamic loading 0.45 0.50 0.55 0.60 0.65 8
Fig. 1 Coupled static-dynamic loading device
Fig. 2. The stress-strain curves of specimens
In the formula
W t
A C
E
t dtR
e e
e
R
t
( ) ( )= ò σ 2
0
is the total mass,
W t
A C
E
t dtT
e e
e
T
t
( ) ( )= ò σ 2
0
is the cumulative quality of the all fragment
which diameter is less than R.
According to the formula (7), the fractal di-
mension D can be calculated from the cumula-
tive mass under sieve of the different particle
size.
3. Experimental scheme and results
The experiment is carried out on the basis of
the coupled static-dynamic loading testing sys-
tem as shown in Figure 1. The loading scheme
is shown as Table 1.
Red sandstone was chosen as the research
object. The specimen size is cylinders with
50mm in diameter and 25mm high. In order
to simulate the acidic environment of deep un-
derground, the acid solution with pH = 4 were
prepared according to the characteristics of the
deep water, and the specimen are immersed in
the acid solution for 30 days.
After the experiment of coupled static-dy-
namic loads. All the broken fragments of rock
under different impact pressure were collected
by the self-made simple recovery device, and
the fragments of rock specimen were sieved
according to the relevant national standards.
The square hole sieves based on new standard
were used to sieve the rock fragments, which
hole diameter are 2.5mm, 5mm, 10mm, 16mm,
20mm, 25mm, 31.5mm, 40mm and 50mm. After
screening, the residue quantity under the sieve
is weighed by electronic balance each times.
The stress and strain curves of the speci-
mens are obtained as shown Figure 2.
According to [15], the incident energy and
absorbed energy of the specimens can be calcu-
lated by the formula.
W t
A C
E
t dtI
e e
e
I
t
( ) ( )= ò σ 2
0
(8)
W t
A C
E
t dtR
e e
e
R
t
( ) ( )= ò σ 2
0
(9)
W t
A C
E
t dtT
e e
e
T
t
( ) ( )= ò σ 2
0
(10)
where WI(t), WR(t) and W tT ( ) respectively rep-
resent the incident, reflection and transmis-
sion energy; A0 is the cross-sectional area of
elastic rod; C0 is the elastic rod wave velocity;
E0 is the elastic modulus of specimen; σ I t( ),
σR t( ) and σT t( ) are the incident, reflection and
transmission stresses.
The energy absorbed WS under coupled
static-dynamic loads can be showed as:
W W W W WS I O R T= + - +( ) (11)
Where WO is the energy of static loads pres-
sure,
W t d tO = ò σ ε
ε
( ) ( )
0
.
Functional materials, 25, 1 2018 125
LIU Yong-sheng et al. / Fragmentation fractal of sandstone under ...
the incident energy and absorbed energy of
the specimens can be calculated as showed
Table 2.
The fragmentation mode of rock under acid
corrosion and coupled static-dynamic loading
are shown in Figure 3, and they were screened
by the above experimental scheme, the sieving
results is shown in Table 3.
4. Analysis of fragmentation fractal
According to the results of the screening ex-
periment, the fractal dimension of the red sand-
stone under acid corrosion and coupled static-
dynamic loads can be calculated by the formula
(7), and the results are shown in Table 4.
From Table 4, we knew that the fractal
dimension of the specimen with 0.45 impact
pressure is smallest in Table 4, and the frac-
tal dimension increased with the impact pres-
sure increasing. This show the fragmentation
degree of the specimen with 0.45 impact pres-
sure is lowest, and it rise with impact pressure
increasing, which is in agreement with the ex-
perimental results form the Figure 2. Through
the comparison of the broken mode and frac-
tal dimension, it was found that the fewer the
Table 2 Energy at different impact pressure
Impact pressure (MPa) 0.45 0.50 0.55 0.60 0.65
Dynamic compressive strength(MPa) 59.14 67.72 68.21 68.91 73.42
Incident energy (J) 44.44 67.84 76.88 90.53 105.41
Absorbed energy (J) 20.11 36.32 34.35 45.88 54.70
Table 3 Results of screening experiment
Residual mass (g)
Hole diameter of sand sieve (mm)
2.5 5.0 10 16 20 25 31.5 40 50
Impact
pressure
(MPa)
0.45 0.6 1.4 4.3 2.5 0 0 0 0 99.2
0.50 7.2 3.6 14.1 15.3 11.9 0 0 54.9 0
0.55 3.8 2.9 11.7 14.6 4.2 0 0 71.8 0
0.60 17.6 8.9 23 16.4 0 0 40.1 0 0
0.65 22.7 12.9 29.2 41.2 0 0 0 0 0
Fig. 3. Fragmentation mode of the rock
126 Functional materials, 25, 1, 2018
LIU Yong-sheng et al. / Fragmentation fractal of sandstone under ...
sample fragments are , the larger the volume
is, the lower the fragmentation degree is, then
the lower the fractal dimension is. The fractal
dimension of the fragment size distribution can
be used to reflect the fragmentation degree of
material quantitatively. which shows that the
fragmentation distribution of rock specimen
has good self-similarity, that is a fractal distri-
bution.
The fractal is not only related to the mac-
roscopic damage of the material, but also to
its micro-structure and mechanical properties.
According to the results the dynamic test, The
changing law of the fractal dimensions and dy-
namic compression strength and energy of the
rock are shown in Figure 4 and Figure 5.
It can be seen from Figure 4 that the dy-
namic compression strength of rock increases
linear with the fractal dimension increasing.
The dynamic compression strength rise when
the impact pressure increasing, then the crack
will develop more fully, and the fragments be-
come more, so the fractal dimension increase.
Therefore, the fractal dimension can be used to
quantitatively describe the dynamic strength
of the specimens. From Figure 5 we can found
that the fractal dimension of the acid red sand-
stone shows a significant upward trend with
Table 4 Fractal dimension of different specimens
Impact pressure (MPa) Slope Fractal dimension Correlation coefficent
0.45 1.303 1.70 0.8402
0.50 0.781 2.22 0.9428
0.55 0.936 2.06 0.9332
0.60 0.542 2.46 0.9125
0.65 0.449 2.55 0.9787
Fig. 4. Relation between dynamic compression
strength and fractal dimensions
Fig. 5. Relation between energy andfractal
dimensions
the increasing of the incident energy under
the action of static and dynamic combination.
Those results are consistent with the other re-
search results on the fractal dimension of rock
subjected to impact dynamic load [11,13]. The
fragment size distribution of rock sample be-
come more and more uniform with the incident
energy increasing, and the characteristic scale
of fragment decreases accordingly. The de-
struction of rock is closely related to the devel-
opment, expansion and penetration of the in-
ternal micro-crack . It is also the process of the
development of the initial meso-damage in the
rock structure to the macroscopic fracture. The
more energy is absorbed, the more the crack ex-
pands, and the more the fragment is produced,
the higher the degree of fragmentation is, and
the fractal dimension is bigger.
5. Conclusions
The fractal characteristics of the rock frag-
mentation under coupling effect of acid corro-
sion and static-dynamic loading are analyzed
in this paper , the results show that:
Research showed that the fractal theory can
be applied to analyze the rock properties under
acidic environment and coupled loads. There
is a close relationship between the fractal di-
Functional materials, 25, 1 2018 127
LIU Yong-sheng et al. / Fragmentation fractal of sandstone under ...
mension, strength of rock, absorbed energy
and fragment shape. The fragment size distri-
bution of rock specimens under acid corrosion
and coupled static-dynamic loads have good
self-similarity, which is the characteristics of
fractal distribution. The study on the fractal of
rock under acid corrosion and coupled static-
dynamic loads can provide reference for the
safe mining of deep coal seam.
Experiment results showed that the lower
the impact pressure is, the fewer the sample
fragments are, the larger the volume of the
fragment are, the lower the fragmentation de-
gree is, and the lower the fractal dimension is.
The fractal dimension of the fragment can be
used to quantitatively described the fragmen-
tation degree of the material.
The fractal dimension increases with the
increasing of the incident energy and absorbed
energy of the specimens, and the dynamic com-
pression strength of rock samples increases
with the increasing of fractal dimension. The
fractal dimension can be used to analyze the
strength, energy and other internal perfor-
mance, to explore the inherent law of material
failure process.
Acknowledgments
This work is supported by the Natu-
ral Science Foundation of China (51664014,
51274101) and Science and technology project
of Jiangxi Provincial Department of Education
(GJJ160474).
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