Fracture and Fatigue Analysis for a Cracked Carabiner Using 3D Finite Element Simulations
Выполнено численное моделирование процесса усталостного разрушения соединительного карабина из сплава Al-7075 с малой краевой трещиной для типичных условий эксплуатационного нагружения. С использованием метода конечных элементов в трехмерной постановке определены характеристики разрушения материала...
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| Опубліковано в: : | Проблемы прочности |
|---|---|
| Дата: | 2015 |
| Автори: | , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут проблем міцності ім. Г.С. Писаренко НАН України
2015
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| Теми: | |
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/173401 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Fracture and Fatigue Analysis for a Cracked Carabiner Using 3D Finite Element Simulations / M.R.M. Aliha, A. Bahmani, S. Akhondi // Проблемы прочности. — 2015. — № 6. — С. 129-144. — Бібліогр.: 41 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859636200837480448 |
|---|---|
| author | Aliha, M.R.M. Bahmani, A. Akhondi, S. |
| author_facet | Aliha, M.R.M. Bahmani, A. Akhondi, S. |
| citation_txt | Fracture and Fatigue Analysis for a Cracked Carabiner Using 3D Finite Element Simulations / M.R.M. Aliha, A. Bahmani, S. Akhondi // Проблемы прочности. — 2015. — № 6. — С. 129-144. — Бібліогр.: 41 назв. — англ. |
| collection | DSpace DC |
| container_title | Проблемы прочности |
| description | Выполнено численное моделирование процесса усталостного разрушения соединительного карабина из сплава Al-7075 с малой краевой трещиной для типичных условий эксплуатационного нагружения. С использованием метода конечных элементов в трехмерной постановке определены характеристики разрушения материала по смешанной моде для различных условий нагружения и конфигураций трещины (включая ее длину, глубину и угол наклона). Согласно полученным результатам, деформацией исследуемого карабина по моде III ввиду ее малости можно пренебречь, вследствие чего разрушение происходит по смешанному механизму отрывасдвига (смешанная мода I/II). Проведен расчет коэффициентов интенсивности напряжений (KI и KII ) и сингулярных Т-напряжений. Показано, что знак и значение этих параметров разрушения определяются геометрией и ориентацией трещины. Рассчитаны критическая длина трещины и живучесть карабина с трещиной при повторно-переменном нагружении на основании известных критериев разрушения и моделей роста усталостной трещины.
Виконано числове моделювання процесу втомного руйнування з’єднувального карабіна зі сплаву Al-7075 із малою крайовою тріщиною для типових умов експлуатаційного навантаження. Із використанням методу скінченних елементів у тривимірній постановці визначено характеристики руйнування матеріалу за змішаною модою для різних умов навантаження і конфігурацій тріщини (включаючи її довжину, глибину і кут нахилу). Згідно з отриманими результатами деформацією досліджуваного карабіна по моді III через її малість можна знехтувати, внаслідок чого руйнування відбувається за змішаним механізмом відриву-зсуву (змішана мода I/II). Проведено розрахунок коефіцієнтів інтенсивності напружень (KI і KII ) і сингулярних Т-напружень. Показано, що знак та значення цих параметрів руйнування визначаються геометрією та орієнтацією тріщини. Розраховано критичну довжину тріщини та живучість карабіна з тріщиною при повторно-змінному навантаженні на основі відомих критеріїв руйнування і моделей росту втомної тріщини.
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| first_indexed | 2025-12-07T13:15:55Z |
| format | Article |
| fulltext |
UDC 539.4
Fracture and Fatigue Analysis for a Cracked Carabiner Using 3D Finite
Element Simulations
M. R. M. Aliha,
a,1
A. Bahmani,
a,b
and S. Akhondi
b
a Welding and Joining Research Center, School of Industrial Engineering, Iran University of Science
and Technology (IUST), Narmak, Tehran, Iran
b Department of Sport Engineering, Science and Research Branch, Islamic Azad University, Tehran,
Iran
1 mrm_aliha@iust.ac.ir
ÓÄÊ 539.4
Àíàëèç óñòàëîñòíîãî ðàçðóøåíèÿ ñîåäèíèòåëüíîãî êàðàáèíà ñ òðåùèíîé
ìåòîäîì òðåõìåðíûõ êîíå÷íûõ ýëåìåíòîâ
Ì. Ð. Ì. Àëèõà
à
, À. Áàõìàíè
à,á
, Ñ. Àõîíäè
á
à Èðàíñêèé óíèâåðñèòåò íàóêè è òåõíîëîãèè, Íàðìàê, Òåãåðàí, Èðàí
á Èñëàìñêèé óíèâåðñèòåò Àçàä, Òåãåðàí, Èðàí
Âûïîëíåíî ÷èñëåííîå ìîäåëèðîâàíèå ïðîöåññà óñòàëîñòíîãî ðàçðóøåíèÿ ñîåäèíèòåëüíîãî
êàðàáèíà èç ñïëàâà Al-7075 ñ ìàëîé êðàåâîé òðåùèíîé äëÿ òèïè÷íûõ óñëîâèé ýêñïëóàòàöèîí-
íîãî íàãðóæåíèÿ. Ñ èñïîëüçîâàíèåì ìåòîäà êîíå÷íûõ ýëåìåíòîâ â òðåõìåðíîé ïîñòàíîâêå
îïðåäåëåíû õàðàêòåðèñòèêè ðàçðóøåíèÿ ìàòåðèàëà ïî ñìåøàííîé ìîäå äëÿ ðàçëè÷íûõ óñëî-
âèé íàãðóæåíèÿ è êîíôèãóðàöèé òðåùèíû (âêëþ÷àÿ åå äëèíó, ãëóáèíó è óãîë íàêëîíà). Ñîãëàñíî
ïîëó÷åííûì ðåçóëüòàòàì, äåôîðìàöèåé èññëåäóåìîãî êàðàáèíà ïî ìîäå III ââèäó åå ìàëîñòè
ìîæíî ïðåíåáðå÷ü, âñëåäñòâèå ÷åãî ðàçðóøåíèå ïðîèñõîäèò ïî ñìåøàííîìó ìåõàíèçìó îòðûâà-
ñäâèãà (ñìåøàííàÿ ìîäà I/II). Ïðîâåäåí ðàñ÷åò êîýôôèöèåíòîâ èíòåíñèâíîñòè íàïðÿæåíèé
( KI è KII ) è ñèíãóëÿðíûõ Ò-íàïðÿæåíèé. Ïîêàçàíî, ÷òî çíàê è çíà÷åíèå ýòèõ ïàðàìåòðîâ
ðàçðóøåíèÿ îïðåäåëÿþòñÿ ãåîìåòðèåé è îðèåíòàöèåé òðåùèíû. Ðàññ÷èòàíû êðèòè÷åñêàÿ
äëèíà òðåùèíû è æèâó÷åñòü êàðàáèíà ñ òðåùèíîé ïðè ïîâòîðíî-ïåðåìåííîì íàãðóæåíèè íà
îñíîâàíèè èçâåñòíûõ êðèòåðèåâ ðàçðóøåíèÿ è ìîäåëåé ðîñòà óñòàëîñòíîé òðåùèíû.
Êëþ÷åâûå ñëîâà: ñîåäèíèòåëüíûé êàðàáèí, ïàðàìåòðû ðàçðóøåíèÿ, òðåõìåðíûé
êîíå÷íîýëåìåíòíûé ðàñ÷åò, ñìåøàííàÿ ìîäà ðàçðóøåíèÿ, óñòàëîñòíàÿ äîëãîâå÷íîñòü.
N o t a t i o n
A – crack width
L – a constant and characteristic length in the carabiner
r, � – crack tip coordinate
C m B m, , ,* *
– coefficients of fatigue crack growth
B – biaxiality ratio
E – elastic modulus
K K KI II III, , – mode I, mode II, and mode III stress intensity factors
K cI – mode I fracture toughness
© M. R. M. ALIHA, A BAHMANI, S. AKHONDI, 2015
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 6 129
K eff – effective stress intensity factor
N – fatigue life
P – applied load
T – T-stress
W – crack depth
Wc – critical crack depth
� – crack inclination angle relative to the applied load
� � �xx xy yy, , – stress components in Cartesian system
� �� – tangential stress component
� – Poisson’s ratio
�0 – fracture direction
Introduction. A carabiner is a metal or composite loop having a spring gate or screw
locking gate, which is frequently used by mountain- or rock-climbers to protect them from
sudden falling. One end of the carabiner is clipped around a rope that is attached to the
climber and the other end is clipped around a piece of webbing that is attached to the
mountainside or rocks. This equipment is also employed in other activities such as
industrial (window cleaning, construction work, fire service departments and rope rescue
operations), sports (sailing, climbing, hang gliding, canyoning, caving and slack lining),
camping equipments, marine equipments etc. Figure 1 shows a typical type of carabiner
and different applications of carabiners. Since these equipments are employed in dangerous
activities, their design and safe service conditions are of great importance. Hence the
strength, load bearing capacity and their failure resistance against sudden fracturing and
fatigue crack growth are key issues that should be known for the safe and reliable using of
carabiners. In the past decades some international groups and researchers have studied the
mechanical and strength properties of carabiners from different aspects. For example, the
ASTM has proposed a standard method for testing the strength of carabiners using a tensile
test procedure [1]. Accordingly, carabiners are designed to repeatedly withstand the loads
of climbing falls, which are typically between 2 and 20 kN, but in extreme cases they can
be as high as 20 kN. Walk [2] experimentally examined the high cycle fatigue behavior of
carabiners and showed that after almost 500,000 loading cycles with amplitude of 2 kN, a
plastic deformation of 1mm is induced in the tested carabiner. Custer and Dave [3]
experimentally proved that the mentioned range for load bearing capacity of carabiners
subjected to moderately high amplitudes of cyclic loads does not provide conservative
estimations for typical types of available carabiners. Blair et al. [4] studied the deformation
and crack growth behavior of D type aluminum carabiners under cyclic loading. Scott [5]
investigated the use of composite materials for manufacturing carabiners to improve
strength, weight and impact properties of carabiners. They showed that a composite
carabiner made of short carbon fibers can improve noticeably the strength and mechanical
properties. Meanwhile the weight of composite carabiner was also up to 40% less than the
similar aluminum type carabiners.
However, catastrophic failure due to the crack growth and fracture is one of the
possible damage modes for carabiners. Cracks or sharp notches can be initiated inside or in
the inner surfaces of the carabiner during its service life (for example due to extra and
sudden loads, cyclic fatigue loads, impact or contact between carabiner and webbing metal,
etc.) and especially at cold climates and under low temperatures. These cracks may then
grow due to repeated loading-unloading cycles induced by the weight of climbers. Hence it
is necessary to study the effect of such cracks in the reliability and load bearing capacity of
M. R. M. Aliha, A Bahmani, and S. Akhondi
130 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 6
carabiners. However, crack growth analysis of such equipments have been rarely investigated
in the past and therefore, in this paper the crack growth behavior of a carabiner containing a
very small crack initiated in the inner surface of carabiner is investigated numerically using
the finite element method. first the fracture parameters of a typical cracked carabiner are
computed from the finite element simulations for various loading conditions and crack
dimensions. The critical crack depth and the fatigue life of the investigated carabiner are
then computed numerically for the investigated carabiner.
Fracture Parameters for a Cracked Component. Crack faces in any desired
component subjected to an arbitrary loading can experience three basic modes of
deformations, namely mode I (crack opening or tensile), mode II (in-plane sliding or
shear) and mode III (out of plane sliding or tear) deformations. Fracture mechanics is a
suitable discipline for dealing with the problems containing cracked components subjected
to mechanical, thermal or environmental loads. In this framework, the state of stresses at
the vicinity of crack front is considered for evaluating the fracture behavior and load
bearing capacity of the cracked components. Due to the complexity of geometry and
applied loads, real cracked components often experience different combinations of three
basic fracture modes and in vast majority of practical cases, the fracture of most structures
occurs usually under mixed mode I/II (combined tensile–shear) loading conditions. The
crack tip stress field in a cracked component subjected to mixed mode I/II can be written as
an infinite series expansion outlined by Williams [6] as follows:
�
�
�
�
�
�
�
��
xx
yy
r
�
�
�
�
�
�
�
�
�
�
�
�
�
�
� �
1
2
2
1
2
3
2
2cos sin sin sin sin cos
cos sin sin
�
�
�
�
�
�
2
1
2
3
2
2
1
2
3
2
1
2
�
�
�
�
�
�
�
�
�
�
�
�
�
� sin cos
sin cos cos sin sin
�
�
�
� �
�
3
2
1
2
3
2 2
1
2
3
�
�
�
�
�
�
�
�
�
�
�
� �
�
2
�
�
�
�
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�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
K
K I
I
I
�
�
�
�
�
�
�
�
�
�
�
�
��
�
�
�
�
��
�
�
�
T O r
O r
O r
0
0
1 2
1 2
1 2
( )
( )
( )
/
/
/
, (1)
Fracture and Fatigue Analysis for a Cracked Carabiner ...
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 6 131
Fig. 1. Carabiner and its different applications.
where � xx , � xy , and � yy are the stress components in a Cartesian system, r and � are
the crack tip polar coordinates with origin located at the tip of crack (see Fig. 2). As seen
from Eq. (1), the first term is singular and the second term is a constant nonsingular term
independent of distance from the crack tip. It is now widely accepted that the mode I and
mode II stress intensity factors (K I and K II ) which are related to the singular terms and
the constant non singular stress term (called the T-stress) are three fundamental fracture
parameters for describing the crack tip stresses in a cracked component subjected to mixed
mode I/II (i.e., combined tension–shear) loading [7–13]. While the stress intensity factors
describe the severity of stress singularity around the crack tip, the T-stress can influence
significantly the initiation of fracture and the trajectory of fracture growth. Other terms
represented by O r( )/1 2 in Eq. (1) are usually negligible in the vicinity of the crack tip.
Accordingly, in order to study the fracture behavior of cracked components subjected to
mixed mode loading, it is necessary that the three fracture parameters (K I , K II , and
T-stress) are known for the given applied loading condition. Hence in the upcoming
sections of this paper, the fracture parameters of a carabiner containing a crack and
subjected to a typical loading (which it often experiences at service conditions) is obtained
numerically for different crack geometries and crack dimensions. By modeling and
analyzing a three dimensional carabiner in the ABAQUS finite element software, variations
of K I , K II , and T are computed for different crack geometries and mixed mode
conditions.
3D Modeling of a Carabiner. A three dimensional model of a typical D type
carabiner (manufactured by PETZL company in France) with real geometry and dimensions
were created by laser scanning of the carabiner. Then the data points obtained from the
laser scanning, were imported into a GEOMAGIC software to create a real three dimensional
(3D) solid model of carabiner (see Fig. 3 for the created model). The use of finite element
technique is a power full and sometimes the only available method for stress analysis of
such complex 3D structures. The use of this technique has been widely increased in recent
years for analyzing and simulating the mechanical behavior of many types of sport devices
and equipments (such as sport balls containing rigid foams, golf ball and club, racing car
chassis, tennis ball, baseball bat and footwear) [14–18]. Hence, in order to analyze a
carabiner, the created 3D solid model was imported into the ABAQUS finite element
132 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 6
M. R. M. Aliha, A Bahmani, and S. Akhondi
Fig. 2. Crack tip stress component under mixed mode I/II loading.
software and a finite element model of carabiner was created using a total number of
153,274 tetragonal and hexagonal solid elements. Also, the critical region A (shown in
Fig. 3) was chosen for considering a crack at the inner surface of the investigated carabiner.
Finite element stress analysis of the un-cracked carabiner under a typical service loading,
showed that this location experiences high stress concentrations and due to the type of
applied loads in this area which are a combinations of wear, cyclic loading-unloading
forces, contact and dynamic shocks, the location A is more vulnerable area for nucleation of
cracks. Hence a small 3D straight thorough thickness edge crack with width a and depth
W was also modeled in this region. Figure 4 shows the geometry of crack initiated at the
inner surface of carabiner with a typical direction of load applied to the carabiners in real
situations. Crack plane can make an angle (�) relative to this loading direction which is also
shown in Fig. 4. In order to produce a square root singularity of the stress field, singular
(quarter point) elements are used in the first ring of finite elements surrounding the 3D
crack front. Figure 5 shows the mesh pattern generated for the modeled carabiner.
The load and boundary conditions that is applied in practice to the carabiners was also
considered in the finite element model. Accordingly, a reference load of 1200 N (i.e.,
typical weight of a person with his/her equipments) was applied to the top inner surface of
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 6 133
Fracture and Fatigue Analysis for a Cracked Carabiner ...
a b
Fig. 3. 3D model of a D type carbineer created by laser scanning technique (a) and the location of
critical point obtained from stress analysis of carabiner for introducing a crack (b).
Fig. 4. Geometry, location and orientation of 3D surface crack initiated at the inner surface of
carabiner.
the carabiner and the opposite bottom inner surface along the line of loading was fixed.
Material properties of a 7075 aluminum alloy used for manufacturing the most of metal
carabiners were also used in the models as E� 71 GPa and �� 0.33. For the modeled
carabiner the fracture parameters are functions of the crack geometry (i.e., the width a and
depth W), applied load (P) and the crack angle (�), and can be written as
K f W l a l PI � ( , , , ),� (2)
K f W l a l PII � ( , , , ),� (3)
T f W l a l P� ( , , , ),� (4)
where l is a constant length in the modeled carabiner which has been defined in Fig. 4. By
performing several 3D finite element analyses, the fracture parameters were obtained
directly from J-integral method built in ABAQUS code [19] for different geometry and
loading conditions of the cracked carabiner. In this method by performing a static analysis
under monotonic loading for the cracked component, the strain energy release rate per unite
fracture surface in front of the crack tip is obtained using a path independent contour
integral (called J-integral). Using the value of this energy, K I , K II , and T can be
determined from the available relations [19]. Figure 6 shows the contours of von Mises
stress for the whole analyzed carabiner and a zoomed view of von Mises crack tip stresses
that revels the severity of stresses in this region as a critical location of the total model. In
the next section, the obtained results are presented and discussed.
Results and Discussion. The variations of mode I and mode II stress intensity factors
(K I and K II ) for different crack inclination angles and crack width and depth ratios (W l
and a l) have been presented in Figs. 7 and 8. The finite element results showed that the
mode III stress intensity factor (K III ) was nearly zero and negligible in the whole analyzed
models and in general a combined mixed mode I/II (tension–shear) fracture mechanism
would control the failure of the investigated cracked carabiner in this research. As seen
from Fig. 7, K I increases by increasing the depth and width of crack, but it becomes
smaller for greater crack inclination angles (�). However, according to Fig. 8, the variations
of mode II stress intensity factor with crack geometry is ascending for small cracks and
descending for cracks with greater depth. Figure 9 also compares the K KI II ratio that
134 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 6
M. R. M. Aliha, A Bahmani, and S. Akhondi
Fig. 5. Mesh pattern created for finite element modeling of carabiner.
reveals the mode I component is very much noticeable than the mode II in the analyzed
carabiner especially for the larger cracks. For mixed mode loading conditions an effective
stress intensity factor is usually used for estimating the equivalent value of fracture
toughness. A review of literature shows that several theoretical and empirical based models
have been proposed for calculation of effective mixed mode stress intensity factor. For
example, Richard [20] proposed an empirical estimation for calculation of the effective
mixed mode I/II stress intensity factor. Based on the maximum tangential stress and the
maximum energy release rate criteria, Yan et al. [21], Rhee and Salama [22], and Forth et
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 6 135
Fracture and Fatigue Analysis for a Cracked Carabiner ...
Fig. 8. Variation of mode II stress intensity factor K II with a l and W l for different crack
inclination angles �.
a b
Fig. 6. Contours of von Mises stress in the analyzed cracked carabiner (a) and a zoomed view of the
crack tip stresses (b).
Fig. 7. Variation of mode I stress intensity factor K I with W l and a l for different crack
inclination angles �.
al. [23] proposed different equations for estimating the effective mixed mode I/II and mixed
mode I/II/III stress intensity factors. Tanaka [24] and Liu and Mahadevan [25] proposed
also some models for obtaining the effective stress intensity factors based on the fatigue
crack growth assumptions. Among the mentioned models, the effective mixed mode I/II
fracture toughness K eff derived based on the maximum energy release rate criterion and
defined as [22, 23]:
K K Keff � �I II
2 2
(5)
136 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 6
M. R. M. Aliha, A Bahmani, and S. Akhondi
Fig. 9. Variation of K KI II ratio in the analyzed carabineer with a l and W l for different crack
inclination angles �.
Fig. 10. Variation of effective stress intensity factor Keff with a l and W l for different crack
inclination angles �.
Fig. 11. Variation of T-stress with a l and W l for different crack inclination angles �.
is the simplest and the most convenient one which is a measure of cracked material against
mixed mode I/II fracture; since the length of chord passing from the origin of K I versus
K II failure envelopes shows the magnitude of mixed mode fracture toughness.
Figure 10 shows the variations of K eff in the analyzed carabiner for different
geometry and loading conditions. As seen from this figure, the value of effective fracture
toughness increases for greater cracks and smaller crack inclination angles.
Figure 11 presents the variations of T-stress in the analyzed cracked carabiner. As is
seen from this figure, the T-stress generally increases when the depth and length of crack
becomes higher. According to the previous research studies, mixed mode I/II fracture
toughness depends on the sign and magnitude of T-stress [26–30]. Hence, the negative
T-stresses that exist in the whole models of the analyzed carabiner can increase its load
bearing capacity [26–28]. Also these negative T-stresses decrease the angle of fracture
initiation and hence can influence the path of fracture trajectory [9, 26–28, 31]. Moreover,
as seen from Fig. 11, the influence of T-stress is more pronounced for smaller cracks and
lower crack inclination angles.
For investigating the influence of non singular stress term (T-stress) relative to the
singular terms (i.e., K I and K II ) a nondimensional parameter called the biaxiality ratio (B)
is commonly used [32]. Value of B is defined as
B
T W
K eff
�
�
. (6)
Hence, a greater value of B reveals that the T-stress has significant role in the process
of mixed mode fracture. The values of B for the investigated carabiner and for different
geometry and loading conditions have been determined from Eq. (6) and presented in
Fig. 12. In comparison with other cracked components and specimens [26–29], the
obtained B data for the analyzed carabiner are very noticeable, which shows the
significant influence of non singular stress term (T-stress) on fracture behavior of the
analyzed cracked carabiner and especially for small cracks.
Under mixed mode loading, crack growth direction is not along the plane of initial
crack and fracture may initiate along the noncoplanar direction (i.e., along a curvilinear
path). Different criteria have been proposed to estimate the direction of mixed mode I/II
fracture initiation, e.g., [33–36]. For example, the maximum tangential stress criterion
states that mixed mode fracture occurs along the direction where the tangential stress in
front of the crack tip is maximum. The tangential stress component can be written as
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Fracture and Fatigue Analysis for a Cracked Carabiner ...
Fig. 12. Variation of biaxiality ratio B with a l and W l for different crack inclination angles �.
�
�
� �
� ��� � �
�
��
�
��
� �
1
2 2 2
1 52 2 1
r
K K T O rcos cos . sin sin (I II
/ ).2
(7)
Hence the crack initiation direction can be obtained from
��
��
��
� �� �
0
0,
� �
��
��
� �
2
2 0
0� . (8)
By using the fracture parameters computed for the investigated carabiner, the value of
�0 were determined from Eq. (8) for different loading and geometrical conditions. Figure 13
shows the variations of �0 with a l, W l, and � in the analyzed carabiner.
Finite element analysis of the investigated carabiner containing a small edge crack
showed that the influence of both stress intensity factors (K I and K II ) and the T-stress on
the onset of fracture and load bearing capacity of the cracked carabiner can be noticeable.
Therefore, in order to increase the reliability of such devices and avoid catastrophic failures
due to crack growth from probably initiated notches or cracks in these equipments, it is
necessary that fracture mechanics based design of such devises to be considered in addition
to the conventional design disciplines. At a critical level of applied load or at a critical
crack length or depth, the value of fracture parameters reach their critical values such that
the required conditions for failure of body is satisfied. The fracture parameters obtained in
this research can be help full for estimating the onset of crack growth in the analyzed
carabiner using the available mixed mode fracture criteria. Accordingly, when the parameters
such as stress, strain or energy at the tip of crack, exceed the critical fracture toughness
( )K cI of the cracked material (which is a constant material property), fracture of
component would be expected to occur under mixed mode tension–shear conditions.
However, based on the numerical results of this research, the magnitudes of static
loads applied due to the weight of a person with a climbing equipment (typically about
120 kg or 1200 N) are much less than the required value of critical stress intensity factor
for propagating the fracture from the very small cracks considered in the investigated
carabiner. While, for the given load and crack geometries the magnitudes of computed
stress intensity factors for the analyzed carabiner were typically less than 2 MPa m! 0 5. , the
critical fracture toughness of Al-7075 material varies in the range of 20 and 30 MPa m! 0 5.
[37]. This means that the probability of crack growth from such very small cracks in the
138 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 6
M. R. M. Aliha, A Bahmani, and S. Akhondi
Fig. 13. Variation of fracture initiation angle �0 with a l and W l for different crack inclination
angles �.
carabiners subjected to general tensile service loads (for example the weight of climbers) is
very low. However, for larger crack depths (which can be occurred due to the gradually
fatigue crack growth in the carabiner subjected to repeated loading-unloading static loads)
the risk of crack propagation becomes more in the investigated carabiner. The critical crack
depth (Wc ) for each crack orientation angle analyzed in this paper was thus determined
using several extra finite element analyses. In order to estimate Wc , crack depth was
increased in the finite element models by increment of 1 mm in each step and the
corresponding values of K I and K II were computed from the finite element analyses.
The obtained stress intensity factors were then compared with the fracture toughness of
Al-7075 material (i.e., equal with 24 MPa m! 0 5. [37]) by employing the available mixed
mode fracture criteria to predict whether the fracture has occurred or not? For example,
according to the maximum tangential stress criterion, fracture of a body subjected to mixed
mode loading condition takes place when the contribution of mode I and mode II stress
intensity factors presented in the right hand side of Eq. (9) reaches or exceeds the critical
fracture toughness (K cI ) of the material,
K K KcI I II� �
�
��
�
��
cos cos sin .
� �
�0 2 0
0
2 2
3
2
(9)
Table 1 presents the corresponding values of stress intensity factors for different rack
depths in the carabiner loaded by 1200 N computed. Corresponding values of K I and K II
computed for each crack depth can then be replaced into Eq. (9) to assess the onset of
fracture. Accordingly, Table 2 shows the critical crack depth (Wc ) for which the unstable
crack growth would be expected to occur based on Eq. (9). As seen from this table, Wc
varies between 7.6 to 8.2 mm depending on the crack inclination angle in the analyzed
carabiner. It was also found that by increasing the crack depth, the value of mode II stress
intensity factor (K II ) decreases drastically for all crack inclination angles and hence the
failure of carabiner at the onset of unstable crack growth would be controlled mainly by
dominantly mode I (tensile or opening) type mechanism.
Fracture and Fatigue Analysis for a Cracked Carabiner ...
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T a b l e 1
Corresponding Values of Mode I and Mode II Stress Intensity Factors
for Different Crack Depths in the Investigated Carabiner
�,
deg
K ,
MPa m! 0 5.
W , mm
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 8.5
0 K I 2.21 2.90 2.99 3.23 4.30 6.73 11.62 21.26 30.76
K II 0.34 0.60 0.046 0.017 1.01 0.78 1.09 2.01 2.24
15 K I 1.85 2.42 2.86 3.72 5.30 8.00 13.35 26.62
K II 0.18 0.07 0.10 0.24 0.23 0.43 0.53 0.29
30 K I 2.03 2.59 3.22 3.86 5.49 8.05 12.50 25.76
K II 0.08 0.54 0.27 0.15 0.12 0.13 0.05 0.03
45 K I 1.81 2.37 3.00 3.72 5.27 7.60 12.78 27.64
K II 0.30 0.58 0.51 0.44 0.35 0.50 0.74 1.61
It should be also noted that although the risk of sudden fracturing in the investigated
cracked carabiner containing small cracks and subjected to static service loads is very low
but repeated loads applied to these devices may facilitate the required conditions for
graduall fatigue crack growth. In practice, a carabiner experiences a large number of
loading and unloading cycles during its service life. Consequently, it is important and
necessary to investigate the remaining life of the cracked carabiners subjected to repeated
cyclic fatigue loads.
A general fatigue crack growth model for a cracked body subjected to mixed mode
I/II loading can be written as [38]:
da dN B K eff
m� * ( ) ,
*
" (10)
where the parameters a and N are the crack length (or depth) and the number of cycles,
respectively. The coefficients B* and the exponents m* can be found by test for mixed
mode [38]. As previously mentioned, by increasing the crack depth the value of K II
becomes negligible in comparison with K I . Therefore, the mixed mode I/II fatigue crack
propagation models can be simplified to the pure mode I condition. Other papers (e.g.,
[39]), have also used such a simplification for fatigue life predictions of structural
components in practical applications. For estimating the fatigue life of the cracked
carabiner containing a growing defect under mode I loading conditions the well-known
Paris–Erdogan fatigue crack growth model presented by Paris and Erdogan [40]:
da dN C K m� ( )" (11)
can be utilized. The parameters C and m could be obtained for the material by conducting
standard fracture tests and from the available handbooks. The values of C and m for the
Al-7075 material were obtained equal to 2 7 10 11. ! � (m/cycle) and 3.7, respectively [41].
The cycles to failure for the investigated carabiner could be calculated as follows:
dW
dN
C K N
C
dW
K
m
m
W
W
i
c
� # �
�
$"
"
1
1 mm
. (12)
The parameters Wi and Wc are the initial and the critical crack depths, respectively.
Accordingly, the remaining service life cycles of the investigated carabiner subjected to
repeated 1200 N to zero loading-unloading cycles can be simply obtained from Eq. (12) by
knowing the dependency of K on crack depth W. By performing several finite element
analyses, variations of stress intensity factor with crack depth were obtained by curve
fitting of a polynomial to the K W� data. Table 2 presents the number of required loading
M. R. M. Aliha, A Bahmani, and S. Akhondi
140 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 6
T a b l e 2
Permissible Defect Depth and the Number of Loading-Unloading Fatigue Cycles
Required for Fracture of the Investigated Carabiner
�, deg 0 15 30 45
Wc , mm 8.15 7.64 7.66 7.62
N , cycles 281,950 283,740 267,350 295,230
cycles computed based on Eq. (12) for growing a crack of depth Wi �1mm to Wc in the
investigated carabiner for different crack inclination angles.
It should be finally noted that although the Al-7075 material exhibits elastic-plastic
behavior at the onset of fracture but the range of applied loads in the real and practical
situations to the carabineres are very less than the ultimate strength of the material
(typically about one tenth of the yield strength of Al-7075). This implies that the service
loads applied to the carabiners are limited to the elastic portion of stress–strain curve and
the use of linear elastic relations for obtaining the fracture parameters and also the fatigue
crack growth are valid and applicable.
C o n c l u s i o n s
1. A three-dimensional carabiner containing a 3D edge crack was modeled and
analyzed to obtain its fracture parameters.
2. While mode III deformation was negligible for the analyzed carabiner, the fracture
behavior of the carabiner was mainly controlled by a mixed mode I/II mechanism for small
cracks.
3. Values of stress intensity factors (K I and K II ) and T-stress were computed for
different crack depths, crack lengths and crack orientation relative to the loading
direction.
4. Mode I stress intensity factor increases generally for longer and deeper cracks with
smaller crack inclination angles. But variation of K II depends on the geometry and
loading conditions of the investigated crack. The T-stress had also noticeable nehative
values in the analyzed carabiner.
5. Permissible defect size was determined for the investigated carabiner using the
critical fracture toughness of Al-7075 alloy and a mixed mode fracture criterion.
6. Using a fatigue crack growth model, the remaining service life of the investigated
cracked carabiner subjected to repeated loading-unloading cycles was estimated for different
crack orientation angles.
Ð å ç þ ì å
Âèêîíàíî ÷èñëîâå ìîäåëþâàííÿ ïðîöåñó âòîìíîãî ðóéíóâàííÿ ç’ºäíóâàëüíîãî êàðà-
á³íà ç³ ñïëàâó Al-7075 ³ç ìàëîþ êðàéîâîþ òð³ùèíîþ äëÿ òèïîâèõ óìîâ åêñïëóàòà-
ö³éíîãî íàâàíòàæåííÿ. ²ç âèêîðèñòàííÿì ìåòîäó ñê³í÷åííèõ åëåìåíò³â ó òðèâèì³ðí³é
ïîñòàíîâö³ âèçíà÷åíî õàðàêòåðèñòèêè ðóéíóâàííÿ ìàòåð³àëó çà çì³øàíîþ ìîäîþ äëÿ
ð³çíèõ óìîâ íàâàíòàæåííÿ ³ êîíô³ãóðàö³é òð³ùèíè (âêëþ÷àþ÷è ¿¿ äîâæèíó, ãëèáèíó ³
êóò íàõèëó). Çã³äíî ç îòðèìàíèìè ðåçóëüòàòàìè äåôîðìàö³ºþ äîñë³äæóâàíîãî êàðà-
á³íà ïî ìîä³ III ÷åðåç ¿¿ ìàë³ñòü ìîæíà çíåõòóâàòè, âíàñë³äîê ÷îãî ðóéíóâàííÿ
â³äáóâàºòüñÿ çà çì³øàíèì ìåõàí³çìîì â³äðèâó-çñóâó (çì³øàíà ìîäà I/II). Ïðîâåäåíî
ðîçðàõóíîê êîåô³ö³ºíò³â ³íòåíñèâíîñò³ íàïðóæåíü (K I ³ K II ) ³ ñèíãóëÿðíèõ Ò-íàïðó-
æåíü. Ïîêàçàíî, ùî çíàê òà çíà÷åííÿ öèõ ïàðàìåòð³â ðóéíóâàííÿ âèçíà÷àþòüñÿ
ãåîìåòð³ºþ òà îð³ºíòàö³ºþ òð³ùèíè. Ðîçðàõîâàíî êðèòè÷íó äîâæèíó òð³ùèíè òà
æèâó÷³ñòü êàðàá³íà ç òð³ùèíîþ ïðè ïîâòîðíî-çì³ííîìó íàâàíòàæåíí³ íà îñíîâ³
â³äîìèõ êðèòåð³¿â ðóéíóâàííÿ ³ ìîäåëåé ðîñòó âòîìíî¿ òð³ùèíè.
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Received 20. 09. 2014
144 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2015, ¹ 6
M. R. M. Aliha, A Bahmani, and S. Akhondi
|
| id | nasplib_isofts_kiev_ua-123456789-173401 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0556-171X |
| language | English |
| last_indexed | 2025-12-07T13:15:55Z |
| publishDate | 2015 |
| publisher | Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| record_format | dspace |
| spelling | Aliha, M.R.M. Bahmani, A. Akhondi, S. 2020-12-02T16:55:01Z 2020-12-02T16:55:01Z 2015 Fracture and Fatigue Analysis for a Cracked Carabiner Using 3D Finite Element Simulations / M.R.M. Aliha, A. Bahmani, S. Akhondi // Проблемы прочности. — 2015. — № 6. — С. 129-144. — Бібліогр.: 41 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/173401 539.4 Выполнено численное моделирование процесса усталостного разрушения соединительного карабина из сплава Al-7075 с малой краевой трещиной для типичных условий эксплуатационного нагружения. С использованием метода конечных элементов в трехмерной постановке определены характеристики разрушения материала по смешанной моде для различных условий нагружения и конфигураций трещины (включая ее длину, глубину и угол наклона). Согласно полученным результатам, деформацией исследуемого карабина по моде III ввиду ее малости можно пренебречь, вследствие чего разрушение происходит по смешанному механизму отрывасдвига (смешанная мода I/II). Проведен расчет коэффициентов интенсивности напряжений (KI и KII ) и сингулярных Т-напряжений. Показано, что знак и значение этих параметров разрушения определяются геометрией и ориентацией трещины. Рассчитаны критическая длина трещины и живучесть карабина с трещиной при повторно-переменном нагружении на основании известных критериев разрушения и моделей роста усталостной трещины. Виконано числове моделювання процесу втомного руйнування з’єднувального карабіна зі сплаву Al-7075 із малою крайовою тріщиною для типових умов експлуатаційного навантаження. Із використанням методу скінченних елементів у тривимірній постановці визначено характеристики руйнування матеріалу за змішаною модою для різних умов навантаження і конфігурацій тріщини (включаючи її довжину, глибину і кут нахилу). Згідно з отриманими результатами деформацією досліджуваного карабіна по моді III через її малість можна знехтувати, внаслідок чого руйнування відбувається за змішаним механізмом відриву-зсуву (змішана мода I/II). Проведено розрахунок коефіцієнтів інтенсивності напружень (KI і KII ) і сингулярних Т-напружень. Показано, що знак та значення цих параметрів руйнування визначаються геометрією та орієнтацією тріщини. Розраховано критичну довжину тріщини та живучість карабіна з тріщиною при повторно-змінному навантаженні на основі відомих критеріїв руйнування і моделей росту втомної тріщини. en Інститут проблем міцності ім. Г.С. Писаренко НАН України Проблемы прочности Научно-технический раздел Fracture and Fatigue Analysis for a Cracked Carabiner Using 3D Finite Element Simulations Анализ усталостного разрушения соединительного карабина с трещиной методом трехмерных конечных элементов Article published earlier |
| spellingShingle | Fracture and Fatigue Analysis for a Cracked Carabiner Using 3D Finite Element Simulations Aliha, M.R.M. Bahmani, A. Akhondi, S. Научно-технический раздел |
| title | Fracture and Fatigue Analysis for a Cracked Carabiner Using 3D Finite Element Simulations |
| title_alt | Анализ усталостного разрушения соединительного карабина с трещиной методом трехмерных конечных элементов |
| title_full | Fracture and Fatigue Analysis for a Cracked Carabiner Using 3D Finite Element Simulations |
| title_fullStr | Fracture and Fatigue Analysis for a Cracked Carabiner Using 3D Finite Element Simulations |
| title_full_unstemmed | Fracture and Fatigue Analysis for a Cracked Carabiner Using 3D Finite Element Simulations |
| title_short | Fracture and Fatigue Analysis for a Cracked Carabiner Using 3D Finite Element Simulations |
| title_sort | fracture and fatigue analysis for a cracked carabiner using 3d finite element simulations |
| topic | Научно-технический раздел |
| topic_facet | Научно-технический раздел |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/173401 |
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