Investigation of Fracturing Process of Rock-Like Brazilian Disks Containing Three Parallel Cracks under Compressive Line Loading
Выполнены испытания на одноосное сжатие образцов типа “бразильского диска” из скальной породы с тремя предварительно выращенными параллельными трещинами в средней части. Образцы из модельного материала (бетон на основе пуццоланового портландцемента, мелкого песка и воды) изготовляли в лаборатории...
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| Veröffentlicht in: | Проблемы прочности |
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| Datum: | 2014 |
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2014
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Investigation of Fracturing Process of Rock-Like Brazilian Disks Containing Three Parallel Cracks under Compressive Line Loading / H. Haeri, K. Shahriar, M.F. Marji, P. Moarefvand // Проблемы прочности. — 2014. — № 3. — С. 133-148. — Бібліогр.: 60 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859867077557354496 |
|---|---|
| author | Haeri, H. Shahriar, K. Marji, M.F. Moarefvand, P. |
| author_facet | Haeri, H. Shahriar, K. Marji, M.F. Moarefvand, P. |
| citation_txt | Investigation of Fracturing Process of Rock-Like Brazilian Disks Containing Three Parallel Cracks under Compressive Line Loading / H. Haeri, K. Shahriar, M.F. Marji, P. Moarefvand // Проблемы прочности. — 2014. — № 3. — С. 133-148. — Бібліогр.: 60 назв. — англ. |
| collection | DSpace DC |
| container_title | Проблемы прочности |
| description | Выполнены испытания на одноосное сжатие образцов типа “бразильского диска” из скальной породы с тремя предварительно выращенными параллельными трещинами в средней
части. Образцы из модельного материала (бетон на основе пуццоланового портландцемента,
мелкого песка и воды) изготовляли в лаборатории механики горных пород. Измерения критической нагрузки в таких дисках показали, что она незначительно зависит от ориентации
трещин и их взаимовлияния. Процесс разрушения образцов изучали при разных углах наклона
трех параллельных трещин относительно оси сжимающей нагрузки по обычной схеме испытаний бразильских дисков. Экспериментально установлено, что на первом этапе нагрузки
начинается бифуркация трещин и их распространение в направлении линии сжимающей
нагрузки. Проведено численное моделирование процесса разрушения образцов с помощью
косвенного метода граничных элементов высшего порядка, известного как “метод разрыва
перемещений”. Данные численных расчетов по инициированию и слиянию трещин сравнивали
с экспериментальными результатами. Получена тесная корреляция численных результатов с
экспериментальными.
Проведено випробування на одновісний стиск зразків типу “бразильського диска” зі
скальної породи, що мають три попередньо вирощені паралельні тріщини в середній
частині. Зразки з модельного матеріалу (бетон на основі пуцоланового портландцементу, дрібного піску та води) виготовляли в лабораторії механіки гірських порід.
Вимірювання критичного навантаження в таких дисках паказало, що воно несуттєво
залежить від орієнтації тріщин і їх взаємовпливу. Процес руйнування зразків вивчали за різних кутів нахилу трьох паралельних тріщин відносно осі стискаючого навантаження за звичайною схемою випробувань бразильських дисків. Експериментально
установлено, що на першому етапі навантаження починається біфуркація тріщин та
їх розповсюдження в напрямку лінії стискаючого навантаження. Проведено числове
моделювання процесу руйнування зразків за допомогою непрямого методу граничних елементів вищого порядку, що відомий як “метод розриву переміщень”. Дані
числових розрахунків по ініціюванню і злиттю тріщин порівнювали з експериментальними. Отримано тісну кореляцію числових результатів з експериментальними.
|
| first_indexed | 2025-12-07T15:48:54Z |
| format | Article |
| fulltext |
UDC 539.4
Investigation of Fracturing Process of Rock-Like Brazilian Disks Containing
Three Parallel Cracks under Compressive Line Loading
H. Haeri,
a,1
K. Shahriar,
b
M. F. Marji,
c
and P. Moarefvand
b
a Department of Mining Engineering, Science and Research Branch, Islamic Azad University,
Tehran, Iran
b Department of Mining and Metallurgical Engineering, Amirkabir University of Technology,
Tehran, Iran
c Faculty of Mining and Metallurgy, Institution of Engineering, Yazd University, Yazd, Iran
1 hadihaeri@ymail.com
ÓÄÊ 539.4
Èññëåäîâàíèå ïðîöåññà ðàçðóøåíèÿ îáðàçöîâ òèïà “áðàçèëüñêîãî äèñêà”
èç ñêàëüíîé ïîðîäû ñ òðåìÿ ïàðàëëåëüíûìè òðåùèíàìè ïðè îäíîîñíîì
ñæàòèè
Õ. Õàýðè
à,1
, Ê. Øàõðèàð
á
, Ì. Ô. Ìàðæè
â
, Ï. Ìîàðåôâàíä
á
à Ôàêóëüòåò ãîðíîãî äåëà, Îòäåëåíèå íàó÷íûõ èññëåäîâàíèé, Èñëàìñêèé óíèâåðñèòåò Àçàä,
Òåãåðàí, Èðàí
á Ôàêóëüòåò ãîðíîãî äåëà è ìåòàëëóðãèè, Òåõíîëîãè÷åñêèé óíèâåðñèòåò èì. Àìèðà Êàáèðà,
Òåãåðàí, Èðàí
â Ôàêóëüòåò ãîðíîãî äåëà è ìåòàëëóðãèè, Îòäåëåíèå ìàøèíîñòðîåíèÿ, Óíèâåðñèòåò ã. Éåçä,
Èðàí
Âûïîëíåíû èñïûòàíèÿ íà îäíîîñíîå ñæàòèå îáðàçöîâ òèïà “áðàçèëüñêîãî äèñêà” èç ñêàëü-
íîé ïîðîäû ñ òðåìÿ ïðåäâàðèòåëüíî âûðàùåííûìè ïàðàëëåëüíûìè òðåùèíàìè â ñðåäíåé
÷àñòè. Îáðàçöû èç ìîäåëüíîãî ìàòåðèàëà (áåòîí íà îñíîâå ïóööîëàíîâîãî ïîðòëàíäöåìåíòà,
ìåëêîãî ïåñêà è âîäû) èçãîòîâëÿëè â ëàáîðàòîðèè ìåõàíèêè ãîðíûõ ïîðîä. Èçìåðåíèÿ êðèòè-
÷åñêîé íàãðóçêè â òàêèõ äèñêàõ ïîêàçàëè, ÷òî îíà íåçíà÷èòåëüíî çàâèñèò îò îðèåíòàöèè
òðåùèí è èõ âçàèìîâëèÿíèÿ. Ïðîöåññ ðàçðóøåíèÿ îáðàçöîâ èçó÷àëè ïðè ðàçíûõ óãëàõ íàêëîíà
òðåõ ïàðàëëåëüíûõ òðåùèí îòíîñèòåëüíî îñè ñæèìàþùåé íàãðóçêè ïî îáû÷íîé ñõåìå èñïû-
òàíèé áðàçèëüñêèõ äèñêîâ. Ýêñïåðèìåíòàëüíî óñòàíîâëåíî, ÷òî íà ïåðâîì ýòàïå íàãðóçêè
íà÷èíàåòñÿ áèôóðêàöèÿ òðåùèí è èõ ðàñïðîñòðàíåíèå â íàïðàâëåíèè ëèíèè ñæèìàþùåé
íàãðóçêè. Ïðîâåäåíî ÷èñëåííîå ìîäåëèðîâàíèå ïðîöåññà ðàçðóøåíèÿ îáðàçöîâ ñ ïîìîùüþ
êîñâåííîãî ìåòîäà ãðàíè÷íûõ ýëåìåíòîâ âûñøåãî ïîðÿäêà, èçâåñòíîãî êàê “ìåòîä ðàçðûâà
ïåðåìåùåíèé”. Äàííûå ÷èñëåííûõ ðàñ÷åòîâ ïî èíèöèèðîâàíèþ è ñëèÿíèþ òðåùèí ñðàâíèâàëè
ñ ýêñïåðèìåíòàëüíûìè ðåçóëüòàòàìè. Ïîëó÷åíà òåñíàÿ êîððåëÿöèÿ ÷èñëåííûõ ðåçóëüòàòîâ ñ
ýêñïåðèìåíòàëüíûìè.
Êëþ÷åâûå ñëîâà: èñõîäíûå òðåùèíû, äèñêîâûå îáðàçöû, ìàòåðèàëû èç ñêàëüíûõ
ïîðîä, ñëèÿíèå òðåùèí, ýêñïåðèìåíòàëüíûå ðàáîòû, ÷èñëåííîå ìîäåëèðîâàíèå.
Introduction. The presence of pre-existing cracks may reduce the fracture toughness
of brittle materials [1]. The mechanical behavior of brittle materials may be affected by the
micromechanical behavior of cracks. Nevertheless, the extension of cracks depends on the
properties of cracks such as size, location, orientation, and loading condition. The initiation
© H. HAERI, K. SHAHRIAR, M. F. MARJI, P. MOAREFVAND, 2014
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 3 133
and propagation of cracks play a vital role in predicting the cyclic fracture process of rock
specimens [2].
In the crack propagation process of brittle materials, such as precracked rock
specimens, usually two types of cracks are observed originating from tips of pre-existing
cracks (i.e., wing cracks and secondary cracks). Wing cracks are usually produced due to
tension, while secondary cracks may initiate due to shear. Therefore, initiation of wing
cracks in rocks is favored relative to secondary cracks because of the lower toughness of
these materials in tension than in shear [3]. The pre-existing cracks in rocks are normally
under compressive loading rather than under tension, shear or mixed mode loading [4]. It is
mainly expected that the crack initiation will follow in the direction (approximately)
parallel to the maximum compressive load [5].
Many experimental works have been devoted to study of the crack initiation,
propagation path, and eventual coalescence of the pre-existing cracks in specimens made of
various substances, including natural rocks or rock-like materials under compressive
loading [6–24]. Brazilian disk test is one of the most suitable tests in evaluating the static
and dynamic fracture toughness of rocks and rock-like specimens containing central
pre-existing crack or cracks. These tests may also be used to study the crack initiation,
propagation path and cracks coalescence of brittle substances such as rocks under
compressive line loadings [25–33]. This testing procedures used extensively to measure the
tensile strength, fracture toughness and mixed mode breakage process in the uncracked and
precracked disk specimens of various brittle substances under compressive line loading
[34–43]. It should be noted that in Brazilian disk specimens, the crack initiation and
fracture process in specimens often happen very soon under compressive line loading due
to the low tensile strength of rocks and rock-like materials. For example, Al-Shayea [43]
experimentally studied the crack propagation paths in the central straight through crack in
Brazilian disk (CSCBD) specimens of brittle limestone with different crack inclination angles
under mixed mode I/II loading and investigated the influence of confining pressure and
temperature on the crack initiation and propagation of the rock specimens. The
experimental results were compared with theoretical predictions of crack initiation angles.
Ghazvinian et al. [33] have performed analytical, experimental, and numerical studies for a
better understanding of crack propagation process in the CSCBD specimens under
compressive line loading. The existing experimental and numerical analyses also confirmed
the effect of crack inclination angle and crack length on fracturing processes of brittle
materials under various loading conditions.
Various numerical methods have been developed for simulation of crack propagation
in brittle substances, e.g., finite element method (FEM), boundary element method (BEM),
discrete element method (DEM), etc. Three important fracture initiation criteria were
proposed to study the crack propagation mechanism of brittle materials, i.e., (I) the
maximum tangential stress (� � -criterion) [44], (ii) the maximum energy release rate
(G-criterion) [45], and (iii) the minimum energy density criterion (S -criterion) [46]. Some
modified form of the above criteria, e.g., F-criterion which is a modified form of energy
release rate criterion proposed by Shen and Stephansson [47] may also be used to study the
breakage behavior of brittle substances [48–50]. Several computer codes were used to
model the breakage mechanism of brittle materials such as rocks, for example, FROCK
code [17], Rock Failure Process Analysis (RFPA2D) code [51], 2D Particle Flow Code
(PFC2D) [23, 33, 52].
However, in most of the published investigations usually specimens with a single
center crack have been studied. In the present study, the Brazilian disks of rock-like
materials containing three parallel cracks in the central part of the specimen are analyzed
both experimentally and numerically. The precracked disk specimens [prepared from
portland pozzolana cement (PPC), fine sands, and water] are tested under compressive line
loading. The stress and displacement fields, the crack propagation and crack coalescence
H. Haeri, K. Shahriar, M. F. Marji, and P. Moarefvand
134 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 3
through the specimens and in the bridge area (the area between the three cracks in the
specimens containing three cracks) of precracked rock-like disk specimens have been
studied both experimentally and numerically. Some of the experimental works are simulated
numerically by a modified higher-order displacement discontinuity method, while crack
propagation and crack coalescence in the bridge area are studied based on Mode I and
Mode II stress intensity factors (SIFs). The experimental data are compared with the
numerical results, which are in good agreement with each other and illustrate the accuracy
and validity of the present work.
1. Specimen Preparation and Testing. Initiation, propagation and coalescence of the
pre-existing cracks in rock-like specimens have been experimentally investigated by many
researchers [15–17]. One of the most difficult tasks in experimental investigations is the
preparation of specimens containing cracks. The rock-like disk specimens are specially
prepared from a mixture of PCC, fine sands, and water. The diameter and thickness of the
precracked rock-like disk specimens prepared for experimental tests are 100 and 30 mm,
respectively (Fig. 1).
Table 1 gives the mechanical properties of the prepared rock-like specimens tested in
the rock mechanics laboratory before inserting the cracks.
Note that the tensile strength (� t ) for uncracked rock-like disk specimens is
�
�t
F
BR
�
2
,
where F is the applied compressive load in kN, B is thickness of the disk specimen, and
R is radius of the disk specimen.
Various Brazilian disk tests were conducted on rock-like disk specimens containing
three parallel cracks. These cracks are created by inserting three thin metal shims with 10 mm
width and 1 mm thickness into the specimens during their casting in a mold.
The disk specimens with three cracks are prepared in such a manner that the cracks
are parallel to each other and oriented at different angles to the load line, i.e., at the angles
�� 0, 15, 30, 45, 70, and 90� (in a counterclockwise direction) as schematically shown in
Investigating the Fracturing Process of Rock-Like Brazilian Disks …
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 3 135
T a b l e 1
Some Mechanical Properties of the Uncracked Rock-Like Disk Specimens
Characteristic Parameter Value
Compressive strength �c , MPa 28
Tensile strength �t , MPa 3.81
Young modulus E, GPa 15
Poisson’s ratio � 0.21
Fig. 1. A typical rock-like disk specimen.
Fig. 2. The compressive line loading F was diametrically applied, and the loading rate of
0.5 MPa/s was maintained during the tests.
Figure 2 demonstrates a schematic view of the geometry of three parallel cracks (i.e.,
crack 1, crack 2, and crack 3, respectively) with equal lengths 2 10b� mm and the crack
length-to-diameter ratio b D� 0.1.
In this study, six specimens were prepared. Three parallel cracks are located at the
centerline of each specimen with the spacing of 20 mm (S � 20 mm). Spacing is taken as
the vertical distance between the centers of two cracks expressed in mm.
2. Experimental Procedures and Results. The specially prepared (rock-like)
specimens are tested experimentally, and the results are used to analyze the breaking loads
and the crack propagation process of the precracked disk specimens.
2.1. Breaking Stress Analysis of the Precracked Disk Specimens. It is obvious that
the precracked rock-like disk specimens have a lower strength as compared to uncracked
specimens.
The breaking stress analysis of the precracked disk specimens containing three
parallel cracks with different orientations is of paramount importance to study the behavior
of brittle materials. The stresses causing new crack initiation and the crack coalescence
were also observed. Figure 3 describes the normalized breaking stress variation for six
136 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 3
Fig. 2 Geometry of three parallel cracks in a rock-like disk specimen under compression.
Fig. 3. Normalized breaking stress versus � � 0, 15, 30, 45, 70, and 90�.
H. Haeri, K. Shahriar, M. F. Marji, and P. Moarefvand
cases: �� 0, 15, 30, 45, 70, and 90�. The final breaking stress of the precracked disk
specimens is normalized by the average breaking stress of the uncracked specimens. The
average breaking stress of uncracked specimens is about 3.81 MPa. The normalized
breaking stress for the six cases �� 0, 15, 30, 45, 70, and 90� is usually less than one
because the pre-existing crack may reduce the final strength of specimen (Fig. 3). As
shown in Fig. 3, normalized breaking stress in �� 0 to 70� and �� �90 is larger than
that at different stages of crack propagation process with other inclination angles.
Thus, fracture toughness mode is more critical than tensile breakage mode in the
precracked disk specimens. It should be noted that in �� �30 , pure shear mode loading
was obtained and then the value of breakage load increased.
2.2. Crack Propagation of Precracked Disk Specimens. Crack coalescence
phenomenon may occur when the three pre-existing cracks combine due to propagation of
wings and/or secondary cracks (originating from the tips of pre-existing cracks) in brittle
substances under various compressive loadings. As shown in Fig. 4, crack coalescence in
the bridge area may also occur during the crack propagation process.
In the current experimental study, wing cracks are instantaneously initiated
quasi-statically (Fig. 4). The development and coalescence of wing cracks in the bridge
area (i.e., the area in-between the three pre-existing cracks) may be the main cause of
fracture paths in rock-like disk specimens.
The bridge areas may be considered as the areas starting from the tips of pre-existing
cracks for the cases shown in Fig. 4a–d (�� 0, 15, 30, and 45�). It should be noted that for
the case shown in Fig. 4e–g (�� 70 and 90�) the cracks may not be initiated at the tips of
pre-existing cracks. For �� �90 , the specimen might fail due to two possible crack
propagation processes starting from the right/left tips of pre-existing cracks in a tensile
splitting mode.
Figure 4a–g shows the observed wing cracks propagating toward each other and
causing crack coalescence in the bridge areas.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 3 137
� � 0 � � �15 � � �30 � � �45
a b c d
� � �70 � � �90 � � �90
e f g
Fig. 4. Experimental results illustrating the coalescence path of rock-like disk specimens containing
three pre-existing cracks.
Investigating the Fracturing Process of Rock-Like Brazilian Disks …
Figure 4a–g illustrates six cases of coalescence paths due to the propagation of the
wing cracks in the bridge areas that are experimentally observed.
3. Numerical Simulation of the Precracked Specimens by Indirect Boundary
Element Method.
3.1. Numerical Method. A displacement-based version of the indirect boundary
element method known as displacement discontinuity method (DDM), which was
originally proposed by Crouch [53] for the solution of elastostatic problems in solid
mechanics, is used in this study [54–58].
A higher accuracy of the displacement discontinuities along the boundary is obtained
by using quadratic displacement discontinuity (DD) elements. A quadratic DD element is
subdivided into three equal subelements, such that each subelement contains a central node
for which the nodal DD is evaluated numerically [48, 49].
Figure 5 shows the displacement distribution at the quadratic collocation point ‘m’,
which can be calculated as
D A Dj m j
m( ) ( ) , �
j x y� , , m� 1 2 3, , , (1)
where D j
1 , D j
2 , and D j
3 are the quadratic nodal displacement discontinuities in x and y
directions. Considering a quadratic element of length, 2c, with equal subelements (c1 �
� �c c2 3 ) and a quadratic shape function, Am ( ) for � � �c c , the following shape
functions of the quadratic collocation point ‘m’ can be defined [48],
A c c1 1 1
22 8( ) ( ) / ,
� � A c c2
2
1
2
1
24 4( ) ( ) / , �� � A c c3 1 1
22 8( ) ( ) / . � � (2)
The stresses and displacements for a straight crack in an infinite specimen along the
x-axis, in terms of single harmonic functions f x y( , ) and g x y( , ), are given by crouch and
star field [45] as
� � �
� �
xx xy xyy yy yyy
yy xy
f y f g y g
y f
� � � �
� �
2 2 2
2
[ , , ] [ , , ],
[ , y yy yyy
xy yy yyy xyy
g y g
f y f y g
] [ , , ],
[ , , ] [ ,
� �
� � � �
2
2 2 2
�
� � � ],
(3)
and the displacements are
u f y f g y g
u
x y xx x xy
y
� � � � � � �
� �
[ ( ) , , ] [ ( ) , , ],
[( )
2 1 1 2
1 2
� �
� f y f g y gx xy y yy, , ] [ ( ) , , ],� � � �2 1 �
(4)
138 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 3
Fig. 5. Quadratic collocations for the higher order displacement discontinuity variation.
H. Haeri, K. Shahriar, M. F. Marji, and P. Moarefvand
where � is shear modulus and f x, , g x, , f y, , g y, , etc. are the partial derivatives of the
single harmonic functions f x y( , ) and g x y( , ) with respect to x and y. These potential
functions (for the quadratic variation of DD along the element) can be written as
f x y D I I I
g x y
x
m
m
m
( , )
( )
( , , ),
( , )
(
�
�
�
�
�
�
�
1
4 1
1
4 1
0 1 2
1
3
� �
�
�
�
�
�
� )
( , , ).D I I Iy
m
m
m
� 0 1 2
1
3 (5)
The common function �m, can be defined as
�m mI I I A x y d( , , ) ( ) ln [( ) ] ,/
0 1 2
2 1 2� � �� m� 1 to 3. (6)
The integrals I 0 , I1 , and I 2 in Eq. (6) can be obtained as
I x y x y d
c
c
0
2 2 1 2( , ) ln [( ) ] /� � � �
�
�
� � � � � � �y x c x c c( ) ( ) ln( ) ( ) ln( ) ,� �1 2 1 2 2� � (7a)
I x y x y d
c
c
1
2 2 1 2( , ) ln [( ) ] /� � �
�
�
� � � � � �xy y x c cx( ) . ( ) ln( ) ,� �1 2
2 2 2
1 205 � � (7b)
I x y x y d
y
x y
c
c
2
2 2 2 1 2 2 2
1 23
3( , ) ln [( ) ] ( )( )/� � � � � � �
�
� ��
� � � � � � � �
1
3
3
1
3
3
2
3
2 3 3
1
2 3 3
2
2( ) ln( ) ( ) ln( )xy x c xy x c
c
x y� � 2
2
3
�
�
�
�
�
�
�
�
�
c
, (7c)
where �1 , �2 , �1 , and �2 can be derived as
�1 �
�
�
�
�
�
�
�arctan ,
y
x c
�2 �
�
�
�
�
�
�
�arctan ,
y
x c
�1
2 2 1 2� � �[( ) ] ,/x c y
�2
2 2 1 2� � �[( ) ] ./x c y
(8)
In order to eliminate the singularity of the displacements, provide stress calculation
near the crack ends, and increase the accuracy of order-higher displacement discontinuity
method around the original crack tip, a special treatment of the crack at the tip is necessary
[48, 49]. In previous works, usually, one or two elements for the specific crack tip
was/were used, but in the present study three special crack tip elements at the crack ends
are used in the general higher-order displacement discontinuity method. As shown in Fig.
6, the DD variation for three nodes can be formulated using a special crack tip element
containing three nodes (or having three special crack tip subelements) [48],
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 3 139
Investigating the Fracturing Process of Rock-Like Brazilian Disks …
D A D cj Tm j
m( ) ( ) ( ), � � j x y� , , m� 1 2 3, , . (9)
Considering a crack tip element with the three equal subelements (c c c1 2 3� � ), the
shape functions AT 1 ( ), AT 2 ( ), and AT 3 ( ) can be obtained as equations:
A
c c c
T 1
1 2
1
1 2
3 2
1
3 2
5 2
1
5 2
15
8 8
( ) ,
/
/
/
/
/
/
� � � A
c c c
T 2
1 2
1
1 2
3 2
1
3 2
5 2
1
5 2
5
4 3
3
2 3 4 3
( ) ,
/
/
/
/
/
/
�
�
� �
A
c c c
T 3
1 2
1
1 2
3 2
1
3 2
5 2
1
5 2
3
8 5 2 5 8 5
( ) .
/
/
/
/
/
/
� � �
(10)
The common function �T
m
T T TI I I( , , )1 2 3 is defined as
�T
m
T
m
Tm
c
c
I A x y d( ) ( ) ln [( ) ] ,/� � �
�
� 2 2 1 2 m� 1 2 3, , . (11)
The integrals IT
1 , IT
2 , and IT
3 can be expressed as
I x y x y d
I x y
T
c
c
T
1 1 2 2 2 1 2
2 3 2
( , ) ln [( ) ] ,
( , ) l
/ /
/
� � �
�
�
�
n [( ) ] ,
( , ) ln [( ) ]
/
/ /
x y d
I x y x y
c
c
T
� �
� � �
�
�
2 2 1 2
3 5 2 2 2 1 2 d
c
c
.
�
�
(12)
The Mode I and Mode II stress intensity factors K I and K II can be estimated based
on the linear elastic fracture mechanics (LEFM) theory as the opening and sliding
displacements [48]:
K
c
D cyI �
�
�
�
�
�
�
�
�
�
�
4 1
2 1 2
( )
( ),
/
K
c
D cxII �
�
�
�
�
�
�
�
�
�
�
4 1
2 1 2
( )
( ).
/
(13)
140 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 3
Fig. 6. A special crack tip element with three equal subelements.
H. Haeri, K. Shahriar, M. F. Marji, and P. Moarefvand
3.2. Numerical Simulation of the Experimental Works. A modified higher-order
displacement discontinuity method based on the versatile boundary element method [48] is
used for the numerical simulation of the proposed experimental works (i.e., to study the
crack coalescence in the bridge area and crack propagation process of brittle substances
under compressive line loading). Six different specimens (Fig. 4a–g) are numerically
simulated by the proposed numerical method, and the results are plotted in Fig. 7a–g. The
LEFM approach based on the concept of SIFs proposed by Irwin [59] is implemented in the
boundary element code and the maximum tangential stress criterion given by Erdogan and
Sih [44] are used in a stepwise procedure to estimate the propagation path of wing cracks.
An iterative method explained by Marji [60] has been used to investigate the crack
propagation directions and paths after each crack extension step, �b b� 0 1. , successively.
The propagation path of each crack has been estimated by this iterative method, and finally
the coalescence of the cracks has been observed (after propagation of wing cracks). As it
follows from the results shown in Fig. 7, the numerically simulated propagation paths are in
good agreement with the corresponding experimentally observed paths (Fig. 4). It should
be noted that the numerical results are based on the crack propagation process originating
from the cracks tips but as shown in Fig. 4, while some experimental specimens do not
include any wing cracks starting from the tips of horizontal cracks.
Table 2 compares the numerical and experimental results considering the crack
initiation and coalescence stresses. The wing crack initiation stresses for various specimens
vary from 1.12–2.32 MPa for the numerical approach and from 0.92–1.86 MPa for the
experimental one. The crack coalescence stress varies from 2.25–2.76 MPa for the
numerical analysis and from 2.16–2.61 MPa for the experimental one.
4. Discussion. The crack propagation process in rock specimens has been studied by
several researchers using the CSCBD problem as shown in Fig. 8 [32].
Recently Ghazvinianet et al. [33] have experimentally and numerically investigated
the crack propagation paths for different crack inclination angles, �� 0, 15, 30, 45, and
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 3 141
� � 0 � � �15 � � �30 � � �45
a b c d
� � �70 � � �90 � � �90
e f g
Fig. 7. Numerical simulation of crack coalescence path for disk specimens containing three parallel
cracks with equal crack lengths 2 10b � mm.
Investigating the Fracturing Process of Rock-Like Brazilian Disks …
60� for CSCBD rock-like specimens. They have used PFC2D code (a discrete element
approach based on the finite difference method) to conduct a number of numerical
simulations to reproduce their experimental works on CSCBD specimens. Table 3 shows
the mechanical properties of rock-like specimens.
Figures 9 and 10 illustrate the experimental data of Ghazvinian et al. [33] and PFC2D
simulations of the crack propagation paths in (CSCBD) specimens with different crack
inclination angles �� 0, 15, 30, 45, and 60�, respectively.
142 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 3
T a b l e 2
Comparison of Wing Crack Initiation and Crack Coalescence Stresses
(Using the Proposed Numerical Method and the Experimental Data)
Crack geometry Wing crack initiation stress (MPa) Cracks coalescence stress (MPa)
Numerical Experiment Numerical Experiment
� � 0 Crack 1
Crack 2
Crack 3
2.13
2.24
2
1.74
–
1.68
2.7 2.52
� � �15 Crack 1
Crack 2
Crack 3
1.91
2
1.83
1.54
–
1.68
2.43 2.28
� � �30 Crack 1
Crack 2
Crack 3
1.95
1.97
1.93
1.77
–
1.67
2.52 2.31
� � �45 Crack 1
Crack 2
Crack 3
1.13
1.25
1.12
0.92
–
0.95
2.53 2.44
� � �70 Crack 1
Crack 2
Crack 3
1.47
1.54
1.43
1.14
–
1.28
2.62 2.42
� � �90 Crack 1
Crack 2
Crack 3
2.29
2.32
2.28
1.86
–
1.79
2.76 2.61
Fig. 8. Schematic view of rock-like CSCBD specimen.
H. Haeri, K. Shahriar, M. F. Marji, and P. Moarefvand
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 3 143
T a b l e 3
Mechanical Properties of Rock-Like Specimens [33]
Characteristic Parameter Value
Crack length 2b, mm 30
Compressive strength �c , MPa 6.6
Tensile strength �t , MPa 1
Density � , kg/m3 1200
� � 0 � � �30� � �15
� � �45 � � �60
Fig. 9. The crack propagation path in CSCBD specimens with different crack inclination angles
� � 0, 15, 30, 45, and 60� [33].
� � 0 � � �15 � � �30
� � �45 � � �60
Fig. 10. PFC2D simulation of the propagating paths in (CSCBD) specimens with different crack
inclination angles � � 0, 15, 30, 45, and 60� [33].
Investigating the Fracturing Process of Rock-Like Brazilian Disks …
The crack propagation process in rock-like CSCBD specimens has also been
numerically studied by using the higher-order boundary element method proposed in this
study. The numerical results obtained by the boundary element simulation of CSCBD
specimens are shown in Fig. 11. The numerically simulated crack propagation paths shown
in Fig. 11 are in good agreement with the experimental results given by Ghazvinian et al
[33] in Fig. 9. Comparative analysis of the results plotted in Figs. 10 and 11 and the
experimental data in Fig. 9 clearly demonstrates the accuracy, validity and superiority of the
boundary element method results, as compared to those obtained by PFC2D code (Fig. 10).
The boundary element code is much faster and user-friendly, because the boundary element
method essentially reduces one dimension of the problem and alternatively reduces the
mesh size sharply and makes the discretization of the problem simpler and quicker [33].
Conclusions. The crack propagation mechanism of brittle substances has been studied
by comprehensive experimental and numerical works in the recent years. This is a
complicated process, which requires further efforts to investigate the crack propagation,
crack coalescenc in the bridge area, and final fracture paths of the rocks and rock-like
materials under compressive line loading. In this study, a detailed analysis of the fracturing
process of the precracked rock-like disk specimens was accomplished both by experimental
tests and numerical simulation. Effects of fracturing on the breaking load of the precracked
rock-like materials are discussed. It is shown that the crack propagation mechanism in the
brittle substances due to the crack coalescence phenomenon in the bridge area occurs
mainly by propagation of wing cracks emanating from the pre-existing crack tips. The
secondary cracks may also be produced after propagation of the wing cracks in the
specimens under compressive line loading, but it is experimentally shown that the wing
cracks are mainly responsible for the crack coalescence and the final cracks propagating
paths. The tested specimens are also numerically simulated by the indirect boundary
element method, and the corresponding numerical results show a good fit with the
experimental results. The experimental and numerical models properly illustrate initiation
of the wing cracks, as well as crack propagation paths produced by the coalescence
phenomenon of the three parallel pre-existing cracks in the bridge area.
144 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 3
� � 0 � � �15 � � �30
� � �45 � � �60
Fig. 11. Boundary element simulation of the crack propagation process in CSCBD specimens.
H. Haeri, K. Shahriar, M. F. Marji, and P. Moarefvand
Ð å ç þ ì å
Ïðîâåäåíî âèïðîáóâàííÿ íà îäíîâ³ñíèé ñòèñê çðàçê³â òèïó “áðàçèëüñüêîãî äèñêà” ç³
ñêàëüíî¿ ïîðîäè, ùî ìàþòü òðè ïîïåðåäíüî âèðîùåí³ ïàðàëåëüí³ òð³ùèíè â ñåðåäí³é
÷àñòèí³. Çðàçêè ç ìîäåëüíîãî ìàòåð³àëó (áåòîí íà îñíîâ³ ïóöîëàíîâîãî ïîðòëàíä-
öåìåíòó, äð³áíîãî ï³ñêó òà âîäè) âèãîòîâëÿëè â ëàáîðàòî𳿠ìåõàí³êè ã³ðñüêèõ ïîð³ä.
Âèì³ðþâàííÿ êðèòè÷íîãî íàâàíòàæåííÿ â òàêèõ äèñêàõ ïàêàçàëî, ùî âîíî íåñóòòºâî
çàëåæèòü â³ä îð³ºíòàö³¿ òð³ùèí ³ ¿õ âçàºìîâïëèâó. Ïðîöåñ ðóéíóâàííÿ çðàçê³â âèâ÷à-
ëè çà ð³çíèõ êóò³â íàõèëó òðüîõ ïàðàëåëüíèõ òð³ùèí â³äíîñíî îñ³ ñòèñêàþ÷îãî íàâàí-
òàæåííÿ çà çâè÷àéíîþ ñõåìîþ âèïðîáóâàíü áðàçèëüñüêèõ äèñê³â. Åêñïåðèìåíòàëüíî
óñòàíîâëåíî, ùî íà ïåðøîìó åòàï³ íàâàíòàæåííÿ ïî÷èíàºòüñÿ á³ôóðêàö³ÿ òð³ùèí òà
¿õ ðîçïîâñþäæåííÿ â íàïðÿìêó ë³í³¿ ñòèñêàþ÷îãî íàâàíòàæåííÿ. Ïðîâåäåíî ÷èñëîâå
ìîäåëþâàííÿ ïðîöåñó ðóéíóâàííÿ çðàçê³â çà äîïîìîãîþ íåïðÿìîãî ìåòîäó ãðàíè÷-
íèõ åëåìåíò³â âèùîãî ïîðÿäêó, ùî â³äîìèé ÿê “ìåòîä ðîçðèâó ïåðåì³ùåíü”. Äàí³
÷èñëîâèõ ðîçðàõóíê³â ïî ³í³ö³þâàííþ ³ çëèòòþ òð³ùèí ïîð³âíþâàëè ç åêñïåðèìåí-
òàëüíèìè. Îòðèìàíî ò³ñíó êîðåëÿö³þ ÷èñëîâèõ ðåçóëüòàò³â ç åêñïåðèìåíòàëüíèìè.
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Received 30. 06. 2013
148 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 3
H. Haeri, K. Shahriar, M. F. Marji, and P. Moarefvand
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| id | nasplib_isofts_kiev_ua-123456789-112729 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0556-171X |
| language | English |
| last_indexed | 2025-12-07T15:48:54Z |
| publishDate | 2014 |
| publisher | Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| record_format | dspace |
| spelling | Haeri, H. Shahriar, K. Marji, M.F. Moarefvand, P. 2017-01-26T20:41:05Z 2017-01-26T20:41:05Z 2014 Investigation of Fracturing Process of Rock-Like Brazilian Disks Containing Three Parallel Cracks under Compressive Line Loading / H. Haeri, K. Shahriar, M.F. Marji, P. Moarefvand // Проблемы прочности. — 2014. — № 3. — С. 133-148. — Бібліогр.: 60 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/112729 539.4 Выполнены испытания на одноосное сжатие образцов типа “бразильского диска” из скальной породы с тремя предварительно выращенными параллельными трещинами в средней части. Образцы из модельного материала (бетон на основе пуццоланового портландцемента, мелкого песка и воды) изготовляли в лаборатории механики горных пород. Измерения критической нагрузки в таких дисках показали, что она незначительно зависит от ориентации трещин и их взаимовлияния. Процесс разрушения образцов изучали при разных углах наклона трех параллельных трещин относительно оси сжимающей нагрузки по обычной схеме испытаний бразильских дисков. Экспериментально установлено, что на первом этапе нагрузки начинается бифуркация трещин и их распространение в направлении линии сжимающей нагрузки. Проведено численное моделирование процесса разрушения образцов с помощью косвенного метода граничных элементов высшего порядка, известного как “метод разрыва перемещений”. Данные численных расчетов по инициированию и слиянию трещин сравнивали с экспериментальными результатами. Получена тесная корреляция численных результатов с экспериментальными. Проведено випробування на одновісний стиск зразків типу “бразильського диска” зі скальної породи, що мають три попередньо вирощені паралельні тріщини в середній частині. Зразки з модельного матеріалу (бетон на основі пуцоланового портландцементу, дрібного піску та води) виготовляли в лабораторії механіки гірських порід. Вимірювання критичного навантаження в таких дисках паказало, що воно несуттєво залежить від орієнтації тріщин і їх взаємовпливу. Процес руйнування зразків вивчали за різних кутів нахилу трьох паралельних тріщин відносно осі стискаючого навантаження за звичайною схемою випробувань бразильських дисків. Експериментально установлено, що на першому етапі навантаження починається біфуркація тріщин та їх розповсюдження в напрямку лінії стискаючого навантаження. Проведено числове моделювання процесу руйнування зразків за допомогою непрямого методу граничних елементів вищого порядку, що відомий як “метод розриву переміщень”. Дані числових розрахунків по ініціюванню і злиттю тріщин порівнювали з експериментальними. Отримано тісну кореляцію числових результатів з експериментальними. en Інститут проблем міцності ім. Г.С. Писаренко НАН України Проблемы прочности Научно-технический раздел Investigation of Fracturing Process of Rock-Like Brazilian Disks Containing Three Parallel Cracks under Compressive Line Loading Исследование процесса разрушения образцов типа “бразильского диска” из скальной породы с тремя параллельными трещинами при одноосном сжатии Article published earlier |
| spellingShingle | Investigation of Fracturing Process of Rock-Like Brazilian Disks Containing Three Parallel Cracks under Compressive Line Loading Haeri, H. Shahriar, K. Marji, M.F. Moarefvand, P. Научно-технический раздел |
| title | Investigation of Fracturing Process of Rock-Like Brazilian Disks Containing Three Parallel Cracks under Compressive Line Loading |
| title_alt | Исследование процесса разрушения образцов типа “бразильского диска” из скальной породы с тремя параллельными трещинами при одноосном сжатии |
| title_full | Investigation of Fracturing Process of Rock-Like Brazilian Disks Containing Three Parallel Cracks under Compressive Line Loading |
| title_fullStr | Investigation of Fracturing Process of Rock-Like Brazilian Disks Containing Three Parallel Cracks under Compressive Line Loading |
| title_full_unstemmed | Investigation of Fracturing Process of Rock-Like Brazilian Disks Containing Three Parallel Cracks under Compressive Line Loading |
| title_short | Investigation of Fracturing Process of Rock-Like Brazilian Disks Containing Three Parallel Cracks under Compressive Line Loading |
| title_sort | investigation of fracturing process of rock-like brazilian disks containing three parallel cracks under compressive line loading |
| topic | Научно-технический раздел |
| topic_facet | Научно-технический раздел |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/112729 |
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