Simulating the Crack Propagation Mechanism of Pre-Cracked Concrete Specimens under Shear Loading Conditions
Исследуется механизм роста трещин в бетонных образцах с исходными трещинами при нагружении сдвигом. Выполнены лабораторные испытания бетонных образцов с двумя предварительно нанесенными краевыми трещинами. Для численного моделирования механизма роста трещин в хрупких телах при нагружении чистым сдви...
Збережено в:
| Опубліковано в: : | Проблемы прочности |
|---|---|
| Дата: | 2015 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут проблем міцності ім. Г.С. Писаренко НАН України
2015
|
| Теми: | |
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/173371 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Simulating the Crack Propagation Mechanism of Pre-Cracked Concrete Specimens under Shear Loading Conditions / H. Haeri // Проблемы прочности. — 2015. — № 4. — С. 130-146. — Бібліогр.: 53 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860153589556576256 |
|---|---|
| author | Haeri, H. |
| author_facet | Haeri, H. |
| citation_txt | Simulating the Crack Propagation Mechanism of Pre-Cracked Concrete Specimens under Shear Loading Conditions / H. Haeri // Проблемы прочности. — 2015. — № 4. — С. 130-146. — Бібліогр.: 53 назв. — англ. |
| collection | DSpace DC |
| container_title | Проблемы прочности |
| description | Исследуется механизм роста трещин в бетонных образцах с исходными трещинами при нагружении сдвигом. Выполнены лабораторные испытания бетонных образцов с двумя предварительно нанесенными краевыми трещинами. Для численного моделирования механизма роста трещин в хрупких телах при нагружении чистым сдвигом используются концепция разрывности перемещений высокого порядка и специальные конечные элементы для вершины трещины для учета взаимовлияния берегов трещин. Предложен специальный метод моделирования эффекта перекрытия берегов трещины на процесс разрушения лигаментной зоны между двумя параллельными трещинами.
Досліджується механізм росту тріщин у бетонних зразках із початковими тріщинами при навантаженні зсувом. Виконано лабораторні випробування бетонних зразків із двома попередньо нанесеними крайовими тріщинами. Для чисельного моделювання механізму росту тріщин у крихких тілах при навантаженні чистим зсувом використовуються концепція розривності переміщень високого порядку і спеціальні скінченні елементи для вершини тріщини для урахування взаємовпливу берегів тріщин. Запропоновано спеціальний метод моделювання ефекту перекриття берегів тріщини на процес руйнування лігаментної зони між двома паралельними тріщинами.
|
| first_indexed | 2025-12-07T17:52:38Z |
| format | Article |
| fulltext |
UDC 539.4
Simulating the Crack Propagation Mechanism of Pre-Cracked Concrete
Specimens under Shear Loading Conditions
H. Haeri
Department of Civil Engineering, Sharif University of Technology, Tehran, Iran
haerihadi@gmail.com, hadihaeri@ymail.com
ÓÄÊ 539.4
Ìîäåëèðîâàíèå ìåõàíèçìà ðîñòà òðåùèí â îáðàçöàõ èç áåòîíà ñ èñõîäíûìè
òðåùèíàìè ïðè íàãðóæåíèè ñäâèãîì
Õ. Õàýðè
Ôàêóëüòåò ãðàæäàíñêîãî ñòðîèòåëüñòâà, Òåõíîëîãè÷åñêèé óíèâåðñèòåò Øàðèô, Òåãåðàí, Èðàí
Èññëåäóåòñÿ ìåõàíèçì ðîñòà òðåùèí â áåòîííûõ îáðàçöàõ ñ èñõîäíûìè òðåùèíàìè ïðè
íàãðóæåíèè ñäâèãîì. Âûïîëíåíû ëàáîðàòîðíûå èñïûòàíèÿ áåòîííûõ îáðàçöîâ ñ äâóìÿ ïðåä-
âàðèòåëüíî íàíåñåííûìè êðàåâûìè òðåùèíàìè. Äëÿ ÷èñëåííîãî ìîäåëèðîâàíèÿ ìåõàíèçìà
ðîñòà òðåùèí â õðóïêèõ òåëàõ ïðè íàãðóæåíèè ÷èñòûì ñäâèãîì èñïîëüçóþòñÿ êîíöåïöèÿ
ðàçðûâíîñòè ïåðåìåùåíèé âûñîêîãî ïîðÿäêà è ñïåöèàëüíûå êîíå÷íûå ýëåìåíòû äëÿ âåðøèíû
òðåùèíû äëÿ ó÷åòà âçàèìîâëèÿíèÿ áåðåãîâ òðåùèí. Ïðåäëîæåí ñïåöèàëüíûé ìåòîä ìîäåëè-
ðîâàíèÿ ýôôåêòà ïåðåêðûòèÿ áåðåãîâ òðåùèíû íà ïðîöåññ ðàçðóøåíèÿ ëèãàìåíòíîé çîíû
ìåæäó äâóìÿ ïàðàëëåëüíûìè òðåùèíàìè. Íà ïåðâîé ñòàäèè íàãðóæåíèÿ íàáëþäàåòñÿ ïîÿâëå-
íèå êðûëîâèäíûõ òðåùèí, äàëüíåéøèé ðîñò êîòîðûõ ïðîèñõîäèò â íàïðàâëåíèè íàãðóæåíèÿ
ñäâèãîì. Íàïðàâëåíèå ðîñòà äâóõ êðàåâûõ òðåùèí â ëèãàìåíòíîé çîíå â îñíîâíîì çàâèñèò îò
óãëà íàêëîíà è äëèíû òðåùèíû, â òî âðåìÿ êàê ïðî÷íîñòü ïðè ñäâèãå – îò òèïà ðàçðóøåíèÿ.
Àíàëèç ìåõàíèçìà ñëèÿíèÿ òðåùèí ïîêàçûâàåò, ÷òî ðàçðóøåíèå îáðàçöîâ èç áåòîíà ñ èñõîä-
íûìè òðåùèíàìè ïðîèñõîäèò ïî ñìåøàííîé ìîäå â ñëó÷àå íåïåðåêðûâàþùèõñÿ òðåùèí è ïî
ìîäå I äëÿ ïåðåêðûâàþùèõñÿ òðåùèí. Ñðàâíåíèå ðàñ÷åòíûõ ðåçóëüòàòîâ ñ ýêñïåðèìåíòàëü-
íûìè ïîäòâåðæäàåò âûñîêóþ òî÷íîñòü è ýôôåêòèâíîñòü ïðåäëîæåííîãî ÷èñëåííîãî ìåòî-
äà ìîäåëèðîâàíèÿ ðîñòà òðåùèí.
Êëþ÷åâûå ñëîâà: ïàðíûå êðàåâûå òðåùèíû, áåòîííûå îáðàçöû, ðîñò òðåùèíû, èñïû-
òàíèÿ íà ÷èñòûé ñäâèã, ïåðåêðûâàþùèåñÿ òðåùèíû.
Introduction. Presence and arrangement of discontinuities and fractures (or joints) in
a rock masses or concrete structures may drastically reduce the strength of such structures.
A crack may act as a nucleus of initiation and extension of new discontinuities in a brittle
solid which may propagate and coalesce with other flaws. These phenomena may further
reduce the strength and the stiffness of the rock materials. Recognizing, the geometrical and
mechanical properties of the discontinuities and joints (or fractures) in a solid may improve
the understanding of cracking mechanism and cracks coalescence in the failure process of
big rock structures such as surface and underground mines, tunnels, rock slopes, etc. [1].
However, the mechanical characteristics of rock masses and concrete structures may
be affected by the mechanical behaviors of the discontinuities and cracks (or joints)
subjected to various loading conditions. The extension and propagation of cracks mainly
depends on the size, persistence, location, spacing, opening, roughness and loading
conditions [2, 3].
© H. HAERI, 2015
130 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 4
The wing (tensile) and secondary (shear) cracks may be observed in the crack
propagation process of brittle solids such as rocks, concretes and rock-like specimens.
Wing cracks are usually the first cracks that are produced due to tension while secondary
cracks initiate due to shear. In brittle solids, the initiation of wing cracks is favored relative
to secondary cracks because of the lower toughness of these materials in tension than in
shear [4].
Wing cracks are tensile cracks that mainly initiate at or near the tips of pre-existing
cracks and propagate in a curved path (with increasing load) while the secondary cracks
may grow after the propagation of the wing cracks (they may also initiate from the original
tips of the cracks). Direct shear and compressive (indirect shear) loading conditions may
both produce the secondary cracks in brittle solids. The breaking stages of the pre-cracked
concrete-like specimens under shear loading have attracted the attentions of some
researchers. Also many numerical studies have been carried out in this field [5, 6].
Savilahti et al. [7] carried out direct shear tests on plaster specimens containing
non-overlapping and overlapping joints. Wong et al. [8] conducted direct shear tests on
specimens made from plaster materials and natural rocks containing open non-persistent
joints and investigated the shear strength and breaking process of the pre-cracked brittle
solids. According to their findings, the breakage patterns of these specimens are mainly
occurred due to joint separation. In addition, Gehle and Kutter [9] also studied the breakage
and shear behaviors of intermittent rock joints under direct shear loading conditions. It can
be concluded that the shear resistance of pre-cracked rock specimens may be affected by
the crack inclination. Ghazvinian et al. [10, 11] and Sarfarazi et al. [12] used particle flow
code (PFC2D) to study the effect of non-overlapping and overlapping joints on the breakage
behavior of a rock bridge under direct shear tests. They compared the numerical simulations
and experimental tests and confirmed that the simulated breaking paths within the
specimens are similar to those obtained by experiments.
Some experimental works have been devoted to study the crack initiation, propagation,
interaction and eventual coalescence of the pre-existing cracks inspecimens prepared from
various brittle materials, including natural rocks and concrete-like materials under
compressive loading conditions [13–26].
Several numerical methods have also been used to simulate the crack propagation
process in brittle solids under various loading conditions, e.g., finite element method
(FEM), boundary element method (BEM), discrete element method (DEM), etc. [27–30].
Based on the numerical and analytical methods, some computer codes were proposed to
model the breaking mechanism of brittle materials such as geo-materials, e.g., FROCK
code [19], Rock Failure Process Analysis (RFPA2D) code [31], 2D Particle Flow Code
(PFC2D) [10–12].
Mainly, three classical fracture initiation criteria were used to study the crack
propagation mechanism of brittle materials based on the linear elastic fracture mechanics
(LEFM) concepts: (i) maximum tangential stress criterion (�-criterion) [32], (ii) maximum
energy release rate criterion (G-criterion) [33], and (iii) minimum energy density criterion
(S-criterion) [34]. F-criterion which is a modification of the maximum energy release rate
criterion has been proposed by Shen and Stephansson [35]. It may also be used to study the
fracturing behavior of brittle solids [31–42].
In this study, single and double edge cracked concrete specimens were specially
prepared from a proper mixture of portland pozzolana cement (PPC), sand and water in a
concrete laboratory and tested under shear loading. The crack propagation mechanism of
these pre-cracked concrete specimens is experimentally investigated by carrying out some
indirect shear tests. The same specimens are numerically simulated by a modified higher
order displacement discontinuity method. A computer code is prepared by using a cubic
variation of displacement discontinuities with four equal subelements and treating the crack
tips by using some special crack tip elements near the crack ends.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 4 131
Simulating the Crack Propagation Mechanism ...
The proposed computer code estimates the mode I and mode II stress intensity factors
(based on the LEFM concepts) and the �-criterion (which is implemented in this code) can
predict the possible crack propagation and the crack initiation direction. This numerical
scheme may be regarded as a mesh reducing dual boundary element method [43, 44] for the
crack analysis (in two dimensions) where the cracks are discretized as two overlapping
straight lines (not as two separate lines as considered in the conventional dual boundary
element method).
In this study, the cracks coalescence mechanism for the double edge cracked
specimens indicated that the jointed specimen break in the mixed mode in the case of
non-overlapping joints configuration and in the tensile mode for the overlapping joints
form. The effects of non-overlapping and overlapping cracks configurations on the final
fracturing path in the bridge areas (in between the two edge cracks) are also studied.
Finally, a comparison is accomplished between the results of the present numerical analysis
with those cited in the literature. These comparisons showed that the numerical results are
well fitted with experimental and numerical results and approve the accuracy and efficiency
of the proposed numerical method in the study of the breaking process under direct shear
loading conditions.
1. Shear Fracturing Process. Consider a body of box shape with an arbitrary size of
a crack, subjected to arbitrary shear loading as shown in Fig. 1. When a crack is subjected
to pure shear loading, the normalized mode I stress intensity factor K aI ( ) ( )� � � is
reached to its maximum value at a propagation angle � (�� 75.50�), and at propagation
angles 90 40�� � �� its value may be larger than the normalized mode II stress intensity
factor K aII ( ) ( ).� � �
The compressive stress applied on the original crack plan may not be sufficient to
reduce or eliminate the high tension stress near the crack tip (i.e., at �
�| |40 ) resulting in a
mode I failure.
Since the singularities of the tensile (wing) stresses near the crack ends may reduce
their accuracy, low normal compressive stress were applied to the surface of original crack.
Therefore, a shear fracture model as shown in Fig. 1 is proposed here.
Figure 2 illustrates the loading mode and crack tip stress for shear box test. The forces
in Cartesian coordinates are obtained as [45]:
p px � sin ,� p py � cos .� (1)
These forces produce the shear and compressive stresses (� xy and � y ), respectively.
The � xy and � y on original crack plane can express as [46]:
� �y
p
A
� cos , � �xy
p
A
� sin . (2)
Fig. 1. Loading condition for shear box test.
132 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 4
H. Haeri
Based on the LEFM principles, the mode I and mode II stress intensity factors �K I ( )�
and �K II ( )� (expressed in MPa m
1 2/ ) can be written in terms of the normal and shear
stresses [46, 47] obtained for the last special crack tip element as
� � K
p
A
aI ( ) cos ,� � � � �K
p
A
aII ( ) sin .� � � (3)
The shear �� � and tangent stresses � �r in polar coordinates at the tip of a crack, or a
measure of the crack tip elastic stress field are obtained as
� �
�
�
�
�
�
�
�
��
�
�
� �
�
� �
�
K
r
K
r
I II( )
cos
( )
sin cos ,
2 2 2
3
2 2
3 2
(4)
�
�
�
� � �
�
� �
�r
K
r
K
r
�
�
�
�
�
I II( )
sin cos
( )
cos sin
2 2 2 2 2
1 3
2
2 2�
�
�
�. (5)
The final results for a mixed mode I and II are given here for completeness
K K K KI I II II( ) ( )cos ( ) sin cos� �
�
�
� �
� � �
�
�
�
�
�
�� ��3 2
2
3
2 2
( )cos ,
( ) ( )sin cos ( )cos s
�
�
� �
� �
�
�
2
2
2
2 2 2
1 3K K KII I II� � � in .2
2
��
�
�
�
�
�
(6)
2. Specimen Simulations and Testing. Various numerical direct shear tests were
conducted on concrete-like specimens containing double edge cracks. These specimens
with different ligament angles (the ligament angle is the counterclockwise angle between
the ligament length and shear axis and crack lengths are prepared.
Two non-overlapping/overlapping cracks with different locations are shown in Figs. 3
and 4. The crack geometries are defined by two parameters namely � and b showing the
ligament angle and length of the two cracks, respectively. For different ligament angles, the
lengths of edge cracks are different.
Fig. 2. Loading and stress mode at the tip of crack.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 4 133
Simulating the Crack Propagation Mechanism ...
Ligament angle (�): 0, 20, 25, 45, 90, 105, 115, 125, 135, 145, and 155�.
The mechanical properties of the prepared concrete-like shear specimens are:
compressive strength � c � 28 MPa, the Young modulus E�15 GPa, Brazilian tensile
strength � t � 3.81 MPa, and Poisson’s ratio �� 0.21.
134 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 4
H. Haeri
a b c d
Fig. 3. Non-overlapped crack geometries with different ligament angles �: (a) � � 0; (b) � � �20 ;
(c) � � �25 ; (d) � � �45 .
a b c d
e f g
Fig. 4. Overlapped crack geometries with different ligament angles �: (a) � � �90 ; (b) � � �105 ;
(c) � � �115 ; (d) � � �125 ; (e) � � �135 ; (f) � � �145 ; (g) � � �155 .
3. Numerical Simulation. A displacement based version of the indirect boundary
element method known as displacement discontinuity method (DDM) originally proposed
by Crouch [48] for the solution of elastostatic problems in solid mechanics is used in this
study [39, 49–51].
3.1. Displacement Discontinuity Method Using Cubic DD Elements. In this research,
a higher accuracy of the displacement discontinuities along the boundary of the problem is
obtained by using cubic displacement discontinuity (DD) elements is used. A cubic DD
element, Dk ( )� is divided into four equal subelements that each subelement contains a
central node for which the nodal DD van be calculate two fundamental variables of each
element (the opening displacement discontinuity Dy and sliding displacement discontinuity
Dx ) [52],
D Dk i k
i
i
( ) ( ) ,� ��
�
��
1
4
k x y� , , (7)
where Dk
1 (i.e., Dx
1 and Dy
1 ), Dk
2 (i.e., Dx
2 and Dy
2), Dk
3 (i.e., Dx
3 and Dy
3), and Dk
4
(i.e., Dx
4 and Dy
4), are the cubic nodal displacement discontinuities and,
�
�
1 1
3
1
2
1
2 3
1
3
2 1
3
1
3 3 48
9 9
( ) ( ) ( ),
( ) (
� � � �
�
� �
�
a a a a
a a2
1
2 3
1
3
3 1
3
1
2
1
2 3
1
16
9 9 16
� � �
� � � �
� �
a a
a a a a
) ( ),
( ) ( ) (� 3
4 1
3
1
2
1
2 3
1
33 3 48
),
( ) ( ) ( ),� � � � �� � a a a a
(8)
are the cubic collocation shape functions using a a a a1 2 3 4� � � . As shown in Fig. 5, a
cubic DD element is divided into three equal subelements (each subelement contains a
central node for which the nodal displacement discontinuities are evaluated numerically).
The potential functions f x y( , ) and g x y( , ) for the cubic case can be find from
f x y D F I I I
g x y
x
i
i
i
( , )
( )
( , , ),
( , )
(
�
�
�
�1
4 1
1
4 1
0 1 2
1
4
� �
�
�
�
� )
( , , ),D F I I Iy
i
i
i
0 1 2
1
4
(9)
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 4 135
Simulating the Crack Propagation Mechanism ...
Fig. 5. Cubic shape function showing the variation of higher order displacement discontinuities along
an ordinary boundary element.
in which the common function Fi is defined as
F I I I I N x y di i( , , , ) ( ) ln[( ) ] ,
/
0 1 2 3
2 2 1 2
� �� � � � i�1 4, ... , , (10)
where the integrals I 0, I1, I 2, and I 3 are expressed as follows:
I x y x y d
I x y x
a
a
0
2 2 1 2
1
2
( , ) ln[( ) ] ,
( , ) ln[( )
/� �
� �
� � �
� � y d
I x y x y d
I x
a
a
a
a
2 1 2
2
2 2 2 1 2
3
] ,
( , ) ln[( ) ] ,
(
/
/
�
� � �
�
�� �
, ) ln[( ) ] ./y x y d
a
a
� �
� � � �3 2 2 1 2
(11)
The singularities of the stresses and displacements near the crack ends may reduce
their accuracies, special crack tip elements can be effectively used to increase the accuracy
of the DDs near the crack tips [36]. As shown in Fig. 6, the DD variations for three nodes
can be formulated using a special crack tip element containing three nodes (or having three
special crack tip subelements),
D D a D a D ak C k C k C k( ) [ ( )] ( ) [ ( )] ( ) [ ( )] ( )� � � �� � �� � �1
1
2
2
3
3 , k x y� , , (12)
136 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 4
H. Haeri
Fig. 6. Special crack tip element with three equal subelements.
where each crack tip element has a length a a a a1 2 3 4� � � . Considering a crack tip
element with the three equal subelements ( a a a1 2 3� � ), the shape functions �C1 ( )� ,
�C 2 ( ),� and �C 3 ( )� can be obtained as
�
�
C
C
a a a
1
1 2
1
1 2
3 2
1
3 2
5 2
1
5 2
2
15
8 8
5
( ) ,
( )
/
/
/
/
/
/
�
� � �
�
� �
�
� � �
�
�
1 2
1
1 2
3 2
1
3 2
5 2
1
5 2
3
1 2
8
3
2 3 4 3
3
8 5
/
/
/
/
/
/
/
,
( )
a a a
C
�
��
a a a1
1 2
3 2
1
3 2
5 2
1
5 22 5 8 5/
/
/
/
/
, �
� �
(13)
F x y D x y dC k
a
a
( , )
( )
( ) ln[( ) ] ,/�
�
�1
4 1
2 2 1 2
� �
� � � k x y� , . (14)
Inserting the common displacement discontinuity function Dk ( )� [Eq. (12)] in Eq. (14)
gives
F x y x y dC C
a
a
( , )
( )
( ) ln[( ) ] /�
�
�
�
�
�
�1
4 1
1
2 2 1 2
� �
� � �� !
"
#
$
%$
�Dk
1
� �
�
�
�
�
! �
��C
a
a
kx y d D2
2 2 1 2 2( ) ln[( ) ] /� � �
� �
�
�
�
�
!
&
'
$
($
��C
a
a
kx y d D3
2 2 1 2 3( ) ln[( ) ] ,/� � � k x y� , . (15)
Inserting the shape functions �C1 ( )� , �C 2 ( ),� and �C 3 ( )� in Eq. (15) after some
manipulations and rearrangements the following three special integrals are deduced:
I x y x y d
I x y
C
a
a
C
1
1 2 2 2 1 2
2
3 2
( , ) ln[( ) ] ,
( , ) l
/ /
/
� �
�
� � � �
� n[( ) ] ,
( , ) ln[( ) ]
/
/ /
x y d
I x y x y
a
a
C
�
� �
� � �
� �
2 2 1 2
3
5 2 2 2 1 2 d
a
a
�.
�
(16)
Based on the LEFM principles, the mode I and mode II stress intensity factors K I and
K II (expressed in MPa m
1 2/ ) can be written in terms of the normal and shear displacement
discontinuities [47] obtained for the last special crack tip element as
K
a
D ayI �
�
�
��
�
�
��
)
�
�
4 1
2
1
1 2
1
( )
( )
/
and K
a
D axII �
�
�
��
�
�
��
)
�
�
4 1
2
1
1 2
1
( )
( ),
/
(17)
where ) is the shear modulus and � is Poisson’s ratio of the brittle material.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 4 137
Simulating the Crack Propagation Mechanism ...
3.2. Numerical Simulation of the Edge Cracked Specimens. A modified higher order
displacement discontinuity method based on the versatile boundary element method is used
for the numerical simulation of the double edge cracked specimens proposed in this
research to study the cracks coalescence and final breakage path of a bridge area under
shear box loading conditions. The LEFM approach based on the concept of mode I and
mode II stress intensity factors proposed by Irwin [53] is implemented in the boundary
element code and the maximum tangential stress criterion given by Erdogan and Sih [32] is
used in a stepwise procedure to estimate the propagation path of the propagating wing
cracks.
Since the experimental analysis of crack propagation is somewhat time-consuming,
expensive, difficult and complex, some more numerical simulations of crack propagation
process are also accomplished by boundary element method in this study.
The numerical simulation is done for several cases with different ligament angles and
ligament lengths as schematically shown in Figs. 3 and 4.
Figures 7–9 present the results of numerical simulation considering different ligament
angles (i.e., at angles �� 0, 20, 25, 45, 105, 115, 125, 135, 145, and 155�). Comparing
Figs. 7–9, it can be concluded that the final breakage path of the pre-cracked specimens
may be affected by ligament angles and the lengths of edge crack.
3.3. Effect of Crack Overlapping on the Fracturing Behavior of Bridge Areas. The
cracks coalescence and breakage paths of double edge cracked specimens with non-
overlapping and overlapping cracks are described in this section.
3.3.1. Breakage Paths of Specimens Containing Non-Overlapped Cracks. In the current
simulations, the wing cracks are instantaneously initiated quasi-statically (Figs. 7–9). The
development and coalescence of wing cracks in the bridge area (the area in between the two
cracks in the specimens containing double edge cracks) may be the main cause of the
breakage paths in rock-like disc specimens.
Figures 7 and 8 illustrate the breakage paths of non-overlapping cracks for cases
�� 0, 20, 25, 45, and 90�. Three specimens with different crack lengths for �� 0 are
modeled as shown in Fig. 7.
138 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 4
H. Haeri
a b c
Fig. 7. The breakage paths of non-overlapped cracks for � � 0 with different lengths b: (a) b�
� 26 2. mm; (b) b� 48 7. mm; (c) b� 67 5. mm.
For �� 0, when the crack lengths are b� 26.2 and 48.7 mm (Fig. 7a, b), the wing
cracks initiated at the tips of the both cracks and then the cracks coalesced with each other
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 4 139
Simulating the Crack Propagation Mechanism ...
a b c
Fig. 8. Breakage paths of non-overlapped cracks for � � �20 (a) � � �25 (b), and � � �45 (c).
a b c d
e f g
Fig. 9. The breakage paths of overlapped cracks: (a) � �*+�; (b) � � +,1 �; (c) � � �115 ; (d) � �
� �125 ; (e) � � �135 ; (f) � � �145 ; (g) � � �155 .
at the propagating crack tips in the bridge area, but for the case shown in Fig. 7c (for �� 0)
with the cracks length b� 67.5 mm, the cracks may first start to initiate at the tips of
cracks and then the specimen may fail in the direction of the original cracks plane.
It should be noted that for the case shown in Fig. 8a (for �� �20 ) first the wing
cracks initiated at the tips of cracks and then the cracks coalesced with each other at the
propagating crack tips in the bridge area.
For cases shown in Fig. 8b, c (for �� 25 and 45�), the wing cracks initiated at the tip
of the cracks and propagated in a curved path until they coalesced with the tip of the other
crack, and also, no coalescence have occurred at the tips of the propagating cracks.
3.3.2. Breakage Paths of Specimens Containing Overlapped Cracks. Figure 9a–g
illustrate the fracturing paths of overlapped cracks for cases �� 90, 105, 115, 125, 135,
145, and 155�.
For case shown in Fig. 9a (for �� �90 ) the wing cracks initiated and continued their
growth in a direction (approximately) vertical to the direction of maximum shear force and
coalesced with each other at the propagating crack tips in the bridge area. In this breakage
mode, the specimen may fail with a single breakage surface.
For case shown in Fig. 9b, c, and f (for ��105, 115, and 145�), the wing cracks
initiate at the cracks and propagated to that of the other crack wall.
It should be noted that for the case shown in Fig. 9d, e, and g (for ��125, 135, and
155�), the cracks initiated at the tip of upper/lower crack first and then the specimen might
fail due to crack propagation process starting from the tip of upper/lower and also, no
coalescence might occur at the tips of cracks.
4. Discussion. The crack propagation process in concrete samples has been studied by
several researches using the edge cracked problem under shear loading.
Recently Sarfarazi et al. [12] have been experimentally and numerically investigated.
The effect of crack overlap on the final breakage behavior of a bridge area in the shear box
test for different specimens, the lengths of edge joints are different; for the same specimen,
the lengths of those two joints are the same. The crack lengths b are 52.5, 66.8, 75, or
78.8 mm which are associated with ligament angles �� 0, 25, 90, and 115� for double edge
cracked concrete-like samples (Fig. 10). They have used PFC2D code (a discrete element
approach based on finite difference method) to conduct a number of numerical simulations
to reproduce their experimental works on double edge cracked samples. Table 1 shows the
mechanical properties of concrete-like specimens.
Figures 11 and 12 illustrate the Sarfarazi et al. [12] experimental works and PFC2D
simulations of the crack propagating paths in concrete-like specimens with variable
ligament angles �� 0, 25, 90, and 115�, respectively [12].
140 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 4
H. Haeri
a b c d
Fig. 10. Non-overlapped (a, b) and overlapped (c, d) crack geometries with different ligament angles
�: (a) � � 0; (b) � � �25 ; (c) � � �90 ; (d) � � �115 .
The crack propagation process of concrete like 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 the propagating paths in
the edge cracked specimens are shown in Fig. 13. The numerically simulated crack
propagation paths shown in Fig. 13 were in good agreement with the experimental results
given by Sarfarazi et al. [12] in Fig. 11. Comparing the results graphically shown in Figs. 12
and 13 with the experimental works in Fig. 11 clearly demonstrate the accuracy, validity
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 4 141
Simulating the Crack Propagation Mechanism ...
T a b l e 1
Mechanical Properties of a Concrete-Like Specimen [12]
Description Parameter Value Unit
Crack length b – mm
Average uniaxial compressive strength �c 6.6 MPa
Density - 1,200 kg/m3
Average Brazilian tensile strength �t 1 MPa
Average Young’s modulus in compression E 5 GPa
Cohesion – 2.2 MPa
Internal angle of friction – 20.4 deg
a b c d
Fig. 11. The crack propagation paths in concrete-like specimens with different ligament angles � � 0
(a), 25 (b), 90 (c), and 115� (d) [12].
a b c d
Fig. 12. PFC2D simulation of the propagating paths in concrete-like specimens with variable ligament
angles � � 0 (a), 25 (b), 90 (c), and 115� (d) [12].
and superiority of the boundary element results compared to those obtained by PFC2D code
(given in Fig. 12). The boundary element code is much faster and it is quite easy to work
with it 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.
Conclusions. The mechanism of crack propagation in brittle solids under shearing
mode has been studied by comprehensive experimental and numerical studies in the recent
years. This mechanism is a complicated process and further research may be devoted to
investigate the crack propagation, cracks coalescence and final breakage paths of a bridge
area under direct shear loading condition (the condition of shear box test).
A modified higher order displacement discontinuity method, (which is a category of
the broad boundary element method) was especially developed to simulate the mechanism
of crack propagation and cracks coalescence in the specimens and in the bridge areas of the
edge cracks. The linear elastic fracture mechanics theory based on mode I and mode II
stress intensity factors is used in the numerical simulation.
Effects of ligament angle and crack lengths in the bridge area on the fracturing
process of edge cracked specimens containing non-overlapping and overlapping cracks
have been modeled. It has been shown that the breaking of edge cracked specimens occur
mainly by the propagation of wing cracks emanating from the tips of the two pre-existing
cracks.
The numerical models well illustrate the production of the wing cracks and the cracks
propagation paths produced by the coalescence phenomenon of the non-overlapping and
overlapping cracks in the bridge area. In this study, it has been shown that there is a good
agreement between the corresponding numerical and experimental results which enables
one to clearly understand the fracturing mechanism of concrete like specimens containing
edge cracks.
It has also been shown that the specimens containing overlapping cracks with high
ligament angles may fail at a single fracturing surface. This failure pattern is highly
dependent on joint overlapping. In non-overlapping cracks with lower ligament angles, the
cracks are initiated from both two cracks tips and the elliptical breaking surface may occur
in the models.
142 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 4
H. Haeri
a b c d
Fig. 13. Boundary element simulation of the crack propagation process in double edge cracked
specimens with variable ligament angles � � 0 (a), 25 (b), 90 (c), and 115� (d).
Ð å ç þ ì å
Äîñë³äæóºòüñÿ ìåõàí³çì ðîñòó òð³ùèí ó áåòîííèõ çðàçêàõ ³ç ïî÷àòêîâèìè òð³ùèíàìè
ïðè íàâàíòàæåíí³ çñóâîì. Âèêîíàíî ëàáîðàòîðí³ âèïðîáóâàííÿ áåòîííèõ çðàçê³â ³ç
äâîìà ïîïåðåäíüî íàíåñåíèìè êðàéîâèìè òð³ùèíàìè. Äëÿ ÷èñåëüíîãî ìîäåëþâàííÿ
ìåõàí³çìó ðîñòó òð³ùèí ó êðèõêèõ ò³ëàõ ïðè íàâàíòàæåíí³ ÷èñòèì çñóâîì âèêîðèñ-
òîâóþòüñÿ êîíöåïö³ÿ ðîçðèâíîñò³ ïåðåì³ùåíü âèñîêîãî ïîðÿäêó ³ ñïåö³àëüí³ ñê³í÷åí-
í³ åëåìåíòè äëÿ âåðøèíè òð³ùèíè äëÿ óðàõóâàííÿ âçàºìîâïëèâó áåðåã³â òð³ùèí.
Çàïðîïîíîâàíî ñïåö³àëüíèé ìåòîä ìîäåëþâàííÿ åôåêòó ïåðåêðèòòÿ áåðåã³â òð³ùèíè
íà ïðîöåñ ðóéíóâàííÿ ë³ãàìåíòíî¿ çîíè ì³æ äâîìà ïàðàëåëüíèìè òð³ùèíàìè. Íà
ïåðø³é ñòà䳿 íàâàíòàæåííÿ âèíèêàþòü êðèëîâèäí³ òð³ùèíè, ÿê³ â ïîäàëüøîìó ðîñ-
òóòü ó íàïðÿìêó íàâàíòàæåííÿ çñóâîì. Íàïðÿìîê ðîñòó äâîõ êðàéîâèõ òð³ùèí ó
ë³ãàìåíòí³é çîí³ â îñíîâíîìó çàëåæèòü â³ä êóòà íàõèëó ³ äîâæèíè òð³ùèíè, â òîé ÷àñ
ÿê ì³öí³ñòü ïðè çñóâ³ – â³ä òèïó ðóéíóâàííÿ. Àíàë³ç ìåõàí³çìó çëèòòÿ òð³ùèí ïîêàçóº,
ùî ðóéíóâàííÿ çðàçê³â ³ç áåòîíó ç ïî÷àòêîâèìè òð³ùèíàìè â³äáóâàºòüñÿ ïî çì³øàí³é
ìîä³ ó âèïàäêó íåïåðåêðèâíèõ òð³ùèí ³ ïî ìîä³ I äëÿ ïåðåêðèâíèõ òð³ùèí. Ïîð³âíÿí-
íÿ ðîçðàõóíêîâèõ äàíèõ ç åêñïåðèìåíòàëüíèìè ï³äòâåðäæóº âèñîêó òî÷í³ñòü ³ åôåê-
òèâí³ñòü çàïðîïîíîâàíîãî ÷èñåëüíîãî ìåòîäó ìîäåëþâàííÿ ðîñòó òð³ùèí.
1. H. H. Einstein, D. Veneziano, G. B. Baecher, and K. J. O’Reilly, “The effect of
discontinuity persistence on rock slope stability,” Int. J. Rock Mech. Min. Sci.
Geomech. Abstr., 20, No. 5, 227–236 (1983).
2. N. R. Barton, “Review of a new shear strength criterion for rock joints,” Eng. Geol.,
7, 287–332 (1973).
3. B. Ladany and G. Archambault, “Simulation of shear behavior of a jointed rock
mass,” in: Proc. of the 11th Symp. Rock Mechanics (ATME) (1970), pp. 105–125.
4. Z. T. Bieniawski, “Mechanism of brittle fracture of rock. Part II. Experimental
studies,” Int. J. Rock Mech. Min. Sci. Geomech. Abstr., 4, No. 4, 407–423 (1967).
5. A. Bobet and H. H. Einstein, “Fracture coalescence in rock-type materials under
uniaxial and biaxial compression,” Int. J. Rock Mech. Min. Sci., 35, 863–888
(1998a).
6. A. Bobet and H. H. Einstein, “Numerical modeling of fracture coalescence in a model
rock material,” Int. J. Fract., 92, 221–252 (1998b).
7. T. Savilahti, E. Nordlund, and O. Stephansson, “Shear box testing and modeling of
joint bridge,” in: N. Barton and O. Stephansson (Eds.), Rock Joints (Proc. of Int.
Symp. on Rock Joints, June 4–6, 1990, Loen, Norway), Balkema, Rotterdam (1990),
pp. 295–300.
8. R. H. C. Wong, W. L. Leung, and S. W. Wang, “Shear strength study on rock-like
models containing arrayed open joints,” in: D. Elsworth, J. P. Tinucci, and K. A.
Heasley (Eds.), Rock Mechanics in the National Interest, Swets & Zeitlinger Lisse,
Leiden (2001), pp. 843–849.
9. C. Gehle and H. K. Kutter, “Breakage and shear behavior of intermittent rock joints,”
Int. J. Rock Mech. Min. Sci., 40, 687–700 (2003).
10. A. Ghazvinian, M. R. Nikudel, and V. Sarfarazi, “Effect of rock bridge continuity and
area on shear behavior of joints,” in: Proc. of the 11th ISRM Congress (July 9–13,
2007, Lisbon, Portugal), Lisbon, Portugal (2007).
11. A. Ghazvinian, V. Sarfarazi, W. Schubert, and M. Blumel, “A study of the failure
mechanism of planar non-persistent open joints using PFC2D,” Rock Mech. Rock
Eng., 45, 677–693 (2011).
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 4 143
Simulating the Crack Propagation Mechanism ...
12. V. Sarfarazi, A. Ghazvinian, W. Schubert, et al., “Numerical simulation of the process
of fracture of echelon rock joints,” Rock Mech. Rock Eng., 47, No. 4, 1355–1371
(2013), doi: 10.1007/s00603-013-0450-3.
13. K. Ravi-Chandar and W. G. Knauss, “An experimental investigation into dynamic
fracture: III. On steady-state crack propagation and crack branching,” Int. J. Fract.,
26, 141–154 (1984).
14. H. Horii and S. Nemat-Nasser, “Compression-induced microcrack growth in brittle
solids: axial splitting and shear failure,” J. Geophys. Res., 90, 3105–3125 (1985).
15. J. F. Huang, G. L. Chen, Y. H. Zhao, and R. Wang, “An experimental study of the
strain field development prior to failure of a marble plate under compression,”
Tectonophysics, 175, 269–284 (1990).
16. P. S. Theocaris and M. Sakellariou, “Crack propagation in brittle materials under
compressive stresses studied by caustics,” J. Mater. Sci., 26, No. 6, 1640–1646
(1991).
17. R. H. C. Wong, K. T. Chau, C. A. Tang, and P. Lin, “Analysis of crack coalescence in
rock-like materials containing three flaws – Part I: experimental approach,” Int. J.
Rock Mech. Min. Sci., 38, 909–924 (2001).
18. Y. P. Li, L. Z. Chen, and Y. H. Wang, “Experimental research on pre-cracked marble
under compression,” Int. J. Solids Struct., 42, 2505–2516 (2005).
19. C. H. Park, Coalescence of Frictional Fractures in Rock Materials, Ph.D. Dissertation,
Purdue University, West Lafayette, Indiana (2008).
20. S. Q. Yang, Y. H. Dai, L. J. Han, and Z. Q. Jin, “Experimental study on mechanical
behavior of brittle marble samples containing different flaws under uniaxial
compression,” Eng. Fract. Mech., 76, 1833–1845 (2009).
21. C. H. Park and A. Bobet, “Crack coalescence in specimens with open and closed
flaws: a comparison,” Int. J. Rock Mech. Min. Sci., 46, 819–829 (2009).
22. C. H. Park and A. Bobet, “Crack initiation, propagation and coalescence from
frictional flaws in uniaxial compression,” Eng. Fract. Mech., 77, 2727–2748 (2010).
23. R. P. Janeiro and H. H. Einstein, “Experimental study of the cracking behavior of
specimens containing inclusions (under uniaxial compression),” Int. J. Fract., 164,
83–102 (2010).
24. S. Q. Yang, “Crack coalescence behavior of brittle sandstone samples containing two
coplanar fissures in the process of deformation failure,” Eng. Fract. Mech., 78,
3059–3081 (2011).
25. P. Cheng-zhi and C. Ping, “Breakage characteristics and its influencing factors of
rock-like material with multi-fissures under uniaxial compression,” Trans. Nonferrous
Met. Soc. China, 22, 185–191 (2012).
26. K. Wallin, “A simple fracture mechanical interpretation of size effects in concrete
fracture toughness tests,” Eng. Fract. Mech., 99, 18–29 (2013).
27. A. R. Ingraffea and F. E. Heuze, “Finite element models for rock fracture mechanics,”
Int. J. Num. Anal. Meth. Geomech., 4, 25–43 (1980).
28. M. Fatehi Marji and I. Dehghani, “Kinked crack analysis by a hybridized boundary
element/boundary collocation method,” Int. J. Solids Struct., 47, 922–933 (2010).
29. M. Fatehi Marji, H. Hosseini-Nasab, and A. H. Kohsary, “A new cubic element
formulation of the displacement discontinuity method using three special crack tip
elements for crack analysis,” JP J. Solids Struct., 1, 61–91 (2007).
144 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 4
H. Haeri
30. C. A. Tang, P. Lin, R. H. C. Wong, and K. T. Chau, “Analysis of crack coalescence in
rock-like materials containing three flaws – Part II: Numerical approach,” Int. J. Rock
Mech. Min. Sci., 38, 925–939 (2001).
31. R. H. C. Wong, C. A Tang, K. T. Chau, and P. Lin, “Splitting failure in brittle rocks
containing pre-existing flaws under uniaxial compression,” Eng. Fract. Mech., 69,
1853–1871 (2002).
32. F. Erdogan and G. C. Sih, “On the crack extension in plates under plane loading and
transverse shear,” J. Basic Eng., 85, 519–527 (1963).
33. M. A. Hussian, E. L. Pu, and J. H. Underwood, “Strain energy release rate for a crack
under combined mode I and mode II,” in: Fracture Analysis, ASTM STP 560 (1974),
pp. 2–28.
34. G. C. Sih, “Strain-energy-density factor applied to mixed mode crack problems,” Int.
J. Fract., 10, 305–321 (1974).
35. B. Shen and O. Stephansson, “Modification of the G-criterion for crack propagation
subjected to compression,” Eng. Fract. Mech., 47, 177–189 (1994).
36. M. F. Marji, H. Hosseinin–Nasab, and A. H. Kohsary, “On the uses of special crack
tip elements in numerical rock fracture mechanics,” Int. J. Solids Struct., 43, 1669–
1692 (2006).
37. M. F. Marji, “On the use of power series solution method in the crack analysis of
brittle materials by indirect boundary element method,” Eng. Fract. Mech., 98, 365–
382 (2013).
38. H. Haeri, K. Shahriar, M. F. Marji, and P. Moarefvand, “A coupled numerical-
experimental study of the breakage process of brittle substances,” Arabian J. Geosci.,
8, No. 2, 809–825 (2013), doi: 10.1007/s12517-013-1165-1.
39. H. Haeri, K. Shahriar, M. F. Marji, and P. Moarefvand, “On the strength and crack
propagation process of the pre-cracked rock-like specimens under uniaxial compression,”
Strength Mater., 46, No. 1, 140–152 (2014).
40. H. Haeri, M. F. Marji, K. Shahriar, and P. Moarefvand, “On the HDD analysis of
micro cracks initiation, propagation and coalescence in brittle substances,” Arabian J.
Geosci., 8, No. 5, 2841–2852 (2015), doi: 10.1007/s12517-014-1290-5.
41. H. Haeri, K. Shahriar, M. F. Marji, P. Moarefvand, “Investigation of fracturing
process of rock-like Brazilian disks containing three parallel cracks under compressive
line loading,” Strength Mater., 46, No. 3, 404–416 (2014).
42. H. Haeri, K. Shahriar, M. F. Marji, P. Moarefvand, “Simulating the effect of disc
erosion in TBM disc cutters by a semi-infinite DDM,” Arabian J. Geosci., 8, No. 6,
(2014), doi: 10.1007/s12517-014-1489-5.
43. J. T. Chen and H. K. Hong, “Review of dual boundary elements methods with
emphasis on hypersingular integrals and divergent series,” Appl. Mech. Rev., 52,
17–33 (1999).
44. M. H. Aliabadi, Fracture of Rocks, Computational Mechanics Publications,
Southampton, UK (1998).
45. Q. Rao, Z. Sun, O. Stephansson, et al., “Shear fracture (mode II) of brittle rock,” Int.
J. Rock Mech. Min. Sci., 40, 355–375 (2003).
46. S. Kahraman and R. Altindag, “A brittleness index to estimate fracture toughness,”
Int. J. Rock Mech. Min. Sci., 41, 343–348 (2004).
47. K. J. Shou and S. L. Crouch, “A higher order displacement discontinuity method for
analysis of crack problems,” Int. J. Rock Mech. Min. Sci. Geomech. Abstr., 32, 49–55
(1995).
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 4 145
Simulating the Crack Propagation Mechanism ...
48. S. L. Crouch, Analysis of Stresses and Displacements around Underground
Excavations: An Application of the Displacement Discontinuity Method, University of
Minnesota Geomechanics Report, Minneapolis, Minnesota (1967).
49. H. Guo, N. I. Aziz, and R. A. Schmidt, “Linear elastic crack tip modeling by
displacement discontinuity method,” Eng. Fract. Mech., 36, 933–943 (1990).
50. C. Scavia, “Fracture mechanics approach to stability analysis of crack slopes,” Eng.
Fract. Mech., 35, 889–910 (1990).
51. M. H. Aliabadi and D. P. Rooke, Numerical Fracture Mechanics, Computational
Mechanics Publications, Southampton, UK (1991).
52. M. F. Marji, H. Hosseini-Nasab, and A. Hosseinmorsgedy, “Numerical modeling of
the mechanism of crack propagation in rocks under TBM disc cutters,” J. Mech.
Mater. Struct., 2, 439–457 (2009).
53. G. R. Irwin, “Analysis of stress and strains near the end of a crack traversing a plate,”
J. Appl. Mech., 24, 361–364 (1957).
Received 20. 07. 2014
146 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 4
H. Haeri
|
| id | nasplib_isofts_kiev_ua-123456789-173371 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0556-171X |
| language | English |
| last_indexed | 2025-12-07T17:52:38Z |
| publishDate | 2015 |
| publisher | Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| record_format | dspace |
| spelling | Haeri, H. 2020-12-01T15:46:23Z 2020-12-01T15:46:23Z 2015 Simulating the Crack Propagation Mechanism of Pre-Cracked Concrete Specimens under Shear Loading Conditions / H. Haeri // Проблемы прочности. — 2015. — № 4. — С. 130-146. — Бібліогр.: 53 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/173371 539.4 Исследуется механизм роста трещин в бетонных образцах с исходными трещинами при нагружении сдвигом. Выполнены лабораторные испытания бетонных образцов с двумя предварительно нанесенными краевыми трещинами. Для численного моделирования механизма роста трещин в хрупких телах при нагружении чистым сдвигом используются концепция разрывности перемещений высокого порядка и специальные конечные элементы для вершины трещины для учета взаимовлияния берегов трещин. Предложен специальный метод моделирования эффекта перекрытия берегов трещины на процесс разрушения лигаментной зоны между двумя параллельными трещинами. Досліджується механізм росту тріщин у бетонних зразках із початковими тріщинами при навантаженні зсувом. Виконано лабораторні випробування бетонних зразків із двома попередньо нанесеними крайовими тріщинами. Для чисельного моделювання механізму росту тріщин у крихких тілах при навантаженні чистим зсувом використовуються концепція розривності переміщень високого порядку і спеціальні скінченні елементи для вершини тріщини для урахування взаємовпливу берегів тріщин. Запропоновано спеціальний метод моделювання ефекту перекриття берегів тріщини на процес руйнування лігаментної зони між двома паралельними тріщинами. en Інститут проблем міцності ім. Г.С. Писаренко НАН України Проблемы прочности Научно-технический раздел Simulating the Crack Propagation Mechanism of Pre-Cracked Concrete Specimens under Shear Loading Conditions Моделирование механизма роста трещин в образцах из бетона с исходными трещинами при нагружении сдвигом Article published earlier |
| spellingShingle | Simulating the Crack Propagation Mechanism of Pre-Cracked Concrete Specimens under Shear Loading Conditions Haeri, H. Научно-технический раздел |
| title | Simulating the Crack Propagation Mechanism of Pre-Cracked Concrete Specimens under Shear Loading Conditions |
| title_alt | Моделирование механизма роста трещин в образцах из бетона с исходными трещинами при нагружении сдвигом |
| title_full | Simulating the Crack Propagation Mechanism of Pre-Cracked Concrete Specimens under Shear Loading Conditions |
| title_fullStr | Simulating the Crack Propagation Mechanism of Pre-Cracked Concrete Specimens under Shear Loading Conditions |
| title_full_unstemmed | Simulating the Crack Propagation Mechanism of Pre-Cracked Concrete Specimens under Shear Loading Conditions |
| title_short | Simulating the Crack Propagation Mechanism of Pre-Cracked Concrete Specimens under Shear Loading Conditions |
| title_sort | simulating the crack propagation mechanism of pre-cracked concrete specimens under shear loading conditions |
| topic | Научно-технический раздел |
| topic_facet | Научно-технический раздел |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/173371 |
| work_keys_str_mv | AT haerih simulatingthecrackpropagationmechanismofprecrackedconcretespecimensundershearloadingconditions AT haerih modelirovaniemehanizmarostatreŝinvobrazcahizbetonasishodnymitreŝinamiprinagruženiisdvigom |