Study of Characteristic Equation of the Elastic Stress Field near Bimaterial Notche
Fracture occurs at interface corners due to stress singularity which generates as a result of material discontinuity and geometrical configuration. In elastic stress field near a bimaterial notch tip, eigenvalues extracted from Airy’s stress function approach determine the order of singularity. In t...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2013
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| Назва видання: | Проблемы прочности |
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| Цитувати: | Study of Characteristic Equation of the Elastic Stress Field near Bimaterial Notches / H. Arabi, M.M. Mirsayar, A.T. Samaei, M. Darandeh // Проблемы прочности. — 2013. — № 5. — С. 119-129. — Бібліогр.: 20 назв. — англ. |
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irk-123456789-1120352020-12-17T15:15:50Z Study of Characteristic Equation of the Elastic Stress Field near Bimaterial Notche Arabi, H. Mirsayar, M.M. Samaei, A.T. Darandeh, M. Научно-технический раздел Fracture occurs at interface corners due to stress singularity which generates as a result of material discontinuity and geometrical configuration. In elastic stress field near a bimaterial notch tip, eigenvalues extracted from Airy’s stress function approach determine the order of singularity. In this paper, the characteristic equation of elastic stress field near bimaterial notches is investigated. The study is done on singular eigenvalues as well as the first non-singular eigenvalue which has not been well studied before. First, different combination of materials and geometrical configurations for two of the most applicable paths in the Bogy diagram (β = 0, β = α/4) were studied and the results were comprehensively discussed. It was shown that the geometrical and materials configurations near a bimaterial notch tip can significantly affect on the stress singularity near these corners. Finally, the areas between two lines β = 0 and β = α/4 in the Bogy diagram with high stress singularities were determined and discussed for both the first and second singular eigenvalue. Исследуется характеристическое уравнение для упругого поля напряжений у вершины надреза на стыке двух материалов. Установлено, что разрушение происходит в угловых точках их стыка из-за возникновения сингулярных напряжений вследствие разрыва сплошности материала и особенностей геометрической конфигурации. В поле упругих напряжений у вершины надреза на стыке двух материалов порядок такой сингулярности определяют собственные значения функции напряжений Эри. Выполнен анализ сингулярных собственных значений и малоизученного первого несингулярного собственного значения. Рассмотрены различные комбинации материалов и геометрических конфигураций для двух наиболее используемых траекторий на диаграмме Боги (β = 0, β = α/4) и детально проанализированы полученные результаты. Показано, что геометрические конфигурации и комбинации материалов у вершины надреза существенно влияют на сингулярность напряжений вблизи угловых точек надреза. Области с высокой сингулярностью напряжений были выделены между линиями β = 0 и β = α/4 на диаграмме Боги и проанализированы как для первого, так и второго сингулярного собственного значения. Досліджується характеристичне рівняння для пружного поля напружень у вістрі надрізу на стику двох матеріалів. Установлено, що руйнування відбувається в кутових точках їх стику через виникнення сингулярних напружень внаслідок розриву суцільності матеріалу й особливостей геометричної конфігурації. У полі пружних напружень у вістрі надрізу на стику двох матеріалів порядок такої сингулярності визначають власні значення функції напружень Ері. Проаналізовано сингулярні власні значення і маловивчене перше несингулярне власне значення. Розглянуто різні комбінації матеріалів і геометричних конфігурацій для двох найбільш використовуваних траєкторій на діаграмі Богі (β = 0, β = α/4) та детально проаналізовано отримані результати. Показано, що геометричні конфігурації і комбінації матеріалів у вістрі надрізу суттєво впливають на сингулярність напружень поблизу кутових точок надрізу. Області з високою сингулярністю напружень виділено між лініями β = 0 і β = α/4 на діаграмі Богі і проаналізовано як для першого, так і другого сингулярного власного значення. 2013 Article Study of Characteristic Equation of the Elastic Stress Field near Bimaterial Notches / H. Arabi, M.M. Mirsayar, A.T. Samaei, M. Darandeh // Проблемы прочности. — 2013. — № 5. — С. 119-129. — Бібліогр.: 20 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/112035 539.4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Научно-технический раздел Научно-технический раздел |
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Научно-технический раздел Научно-технический раздел Arabi, H. Mirsayar, M.M. Samaei, A.T. Darandeh, M. Study of Characteristic Equation of the Elastic Stress Field near Bimaterial Notche Проблемы прочности |
| description |
Fracture occurs at interface corners due to stress singularity which generates as a result of material discontinuity and geometrical configuration. In elastic stress field near a bimaterial notch tip, eigenvalues extracted from Airy’s stress function approach determine the order of singularity. In this paper, the characteristic equation of elastic stress field near bimaterial notches is investigated. The study is done on singular eigenvalues as well as the first non-singular eigenvalue which has not been well studied before. First, different combination of materials and geometrical configurations for two of the most applicable paths in the Bogy diagram (β = 0, β = α/4) were studied and the results were comprehensively discussed. It was shown that the geometrical and materials configurations near a bimaterial notch tip can significantly affect on the stress singularity near these corners. Finally, the areas between two lines β = 0 and β = α/4 in the Bogy diagram with high stress singularities were determined and discussed for both the first and second singular eigenvalue. |
| format |
Article |
| author |
Arabi, H. Mirsayar, M.M. Samaei, A.T. Darandeh, M. |
| author_facet |
Arabi, H. Mirsayar, M.M. Samaei, A.T. Darandeh, M. |
| author_sort |
Arabi, H. |
| title |
Study of Characteristic Equation of the Elastic Stress Field near Bimaterial Notche |
| title_short |
Study of Characteristic Equation of the Elastic Stress Field near Bimaterial Notche |
| title_full |
Study of Characteristic Equation of the Elastic Stress Field near Bimaterial Notche |
| title_fullStr |
Study of Characteristic Equation of the Elastic Stress Field near Bimaterial Notche |
| title_full_unstemmed |
Study of Characteristic Equation of the Elastic Stress Field near Bimaterial Notche |
| title_sort |
study of characteristic equation of the elastic stress field near bimaterial notche |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| publishDate |
2013 |
| topic_facet |
Научно-технический раздел |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/112035 |
| citation_txt |
Study of Characteristic Equation of the Elastic Stress Field near Bimaterial Notches / H. Arabi, M.M. Mirsayar, A.T. Samaei, M. Darandeh // Проблемы прочности. — 2013. — № 5. — С. 119-129. — Бібліогр.: 20 назв. — англ. |
| series |
Проблемы прочности |
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AT arabih studyofcharacteristicequationoftheelasticstressfieldnearbimaterialnotche AT mirsayarmm studyofcharacteristicequationoftheelasticstressfieldnearbimaterialnotche AT samaeiat studyofcharacteristicequationoftheelasticstressfieldnearbimaterialnotche AT darandehm studyofcharacteristicequationoftheelasticstressfieldnearbimaterialnotche |
| first_indexed |
2025-07-08T03:13:05Z |
| last_indexed |
2025-07-08T03:13:05Z |
| _version_ |
1837046837055324160 |
| fulltext |
UDC 539.4
Study of Characteristic Equation of the Elastic Stress Field near
Bimaterial Notches
H. Arabi,
a,1
M. M. Mirsayar,
b
A. T. Samaei,
b
and M. Darandeh
a
a Azad University of Garmsar, Garmsar, Iran
b Iran University of Science and Technology, Narmak, Tehran, Iran
1 habibarabi1@yahoo.com
ÓÄÊ 539.4
Àíàëèç õàðàêòåðèñòè÷åñêîãî óðàâíåíèÿ óïðóãîãî ïîëÿ íàïðÿæåíèé
âáëèçè íàäðåçîâ íà ñòûêå äâóõ ìàòåðèàëîâ
Õ. Àðàáè
à
, Ì. Ì. Ìèðñàÿð
á
, A. T. Ñàìàýé
á
, Ì. Äàðàíäåõ
à
à Óíèâåðñèòåò Àçàä Ãåðìñàð, Ãåðìñàð, Èðàí
á Èðàíñêèé óíèâåðñèòåò íàóêè è òåõíîëîãèè, Òåãåðàí, Èðàí
Èññëåäóåòñÿ õàðàêòåðèñòè÷åñêîå óðàâíåíèå äëÿ óïðóãîãî ïîëÿ íàïðÿæåíèé ó âåðøèíû íàäðå-
çà íà ñòûêå äâóõ ìàòåðèàëîâ. Óñòàíîâëåíî, ÷òî ðàçðóøåíèå ïðîèñõîäèò â óãëîâûõ òî÷êàõ èõ
ñòûêà èç-çà âîçíèêíîâåíèÿ ñèíãóëÿðíûõ íàïðÿæåíèé âñëåäñòâèå ðàçðûâà ñïëîøíîñòè ìàòå-
ðèàëà è îñîáåííîñòåé ãåîìåòðè÷åñêîé êîíôèãóðàöèè.  ïîëå óïðóãèõ íàïðÿæåíèé ó âåðøèíû
íàäðåçà íà ñòûêå äâóõ ìàòåðèàëîâ ïîðÿäîê òàêîé ñèíãóëÿðíîñòè îïðåäåëÿþò ñîáñòâåííûå
çíà÷åíèÿ ôóíêöèè íàïðÿæåíèé Ýðè. Âûïîëíåí àíàëèç ñèíãóëÿðíûõ ñîáñòâåííûõ çíà÷åíèé è
ìàëîèçó÷åííîãî ïåðâîãî íåñèíãóëÿðíîãî ñîáñòâåííîãî çíà÷åíèÿ. Ðàññìîòðåíû ðàçëè÷íûå êîì-
áèíàöèè ìàòåðèàëîâ è ãåîìåòðè÷åñêèõ êîíôèãóðàöèé äëÿ äâóõ íàèáîëåå èñïîëüçóåìûõ òðàåê-
òîðèé íà äèàãðàììå Áîãè (� � 0, � �� /4) è äåòàëüíî ïðîàíàëèçèðîâàíû ïîëó÷åííûå ðåçóëü-
òàòû. Ïîêàçàíî, ÷òî ãåîìåòðè÷åñêèå êîíôèãóðàöèè è êîìáèíàöèè ìàòåðèàëîâ ó âåðøèíû
íàäðåçà ñóùåñòâåííî âëèÿþò íà ñèíãóëÿðíîñòü íàïðÿæåíèé âáëèçè óãëîâûõ òî÷åê íàäðåçà.
Îáëàñòè ñ âûñîêîé ñèíãóëÿðíîñòüþ íàïðÿæåíèé áûëè âûäåëåíû ìåæäó ëèíèÿìè � � 0 è
� �� /4 íà äèàãðàììå Áîãè è ïðîàíàëèçèðîâàíû êàê äëÿ ïåðâîãî, òàê è âòîðîãî ñèíãóëÿðíîãî
ñîáñòâåííîãî çíà÷åíèÿ.
Êëþ÷åâûå ñëîâà: íàäðåç íà ñòûêå äâóõ ìàòåðèàëîâ, ñèíãóëÿðíûå ñîáñòâåí-
íûå çíà÷åíèÿ, ïåðâîå íåñèíãóëÿðíîå ñîáñòâåííîå çíà÷åíèå, õàðàêòåðèñòè-
÷åñêîå óðàâíåíèå, ïîëå óïðóãèõ íàïðÿæåíèé.
Introduction. Today, bonded joints are used in manufacturing of many
modern structures such as micromechanical devices, adhesively bonded joints and
composite materials [1–6]. In structures containing these bonds, combination of
materials and geometrical configuration are a critical source of stress singularity,
and failure usually occurs at these corners.
In order to study the fracture mechanism of these bonds, a sufficient knowledge
on the variation of eigenvalues extracted from characteristic equation of the elastic
stress field is definitely one of the most important steps. These eigenvalues
determine the order of singularity near the bimaterial notch tip. The characteristic
© H. ARABI, M. M. MIRSAYAR, A. T. SAMAEI, M. DARANDEH, 2013
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 5 119
equation for a bimaterial notch has extensively been studied by many researchers
in different applicable geometrical and material configurations [7–14]. In this field,
Bogy [8, 9] and Hein and Erdogan [7] have formulated the stress and displacement
fields of bimaterial corners and determined the order of stress singularity for
different combinations of materials and corner angles. They indicated that both
geometrical description and the elastic properties of the materials could strongly
influence the state of singularity at the interface corners.
Despite the interest in research on the characteristic equation of bimaterial
notch problem, the emphasis was put only on singular eigenvalues, while higher
order eigenvalues have not been thoroughly studied. On the other hand, the recent
investigations indicated that the higher order terms can significantly affect distribution
of stress field adjacent a homogeneous and bimaterial notch tip [15–20]. Among
these higher order terms, the first non-singular stress terms which are called
V-stress in homogeneous notches and I-stress in bimaterial notches are the most
effective terms, whereas disregard of these terms may result in a significant error in
estimating the stress distribution near the corner [16, 20].
In this paper, a comprehensive study is done on characteristic equation of
elastic stress field of bimaterial notches. In addition to singular eigenvalues, the
eigenvalue associated with I-stress term is studied as well. The study is done on
different combinations of materials and geometrical configuration. The combinations
of materials were selected for two of the most applicable lines of the Body
diagram, � �� 4 and ��0, for different values of �. The results indicate that
both geometrical configuration and material combination significantly affect the
order of singularity and the value of first non-singular eigenvalue. Finally, the area
with high stress singularity between these two lines was determined and discussed.
1. Characteristic Equation of Elastic Stress Field near a Bimaterial Notch
Tip. The asymptotic free-edge stress and displacement fields near a bimaterial
notch tip can be obtained using Airy’s stress function approach. Using this method,
it can be indicated that the stresses and displacements near a bimaterial notch tip,
shown in Fig. 1, subjected to a remote mechanical load can be represented as
� �
�
ij
m
k ijk
m
k
N
i
m
k ik
m
k
N
H r f
u H r g
k
k
�
�
�
�
�
�
�
1
1
1
,
,
(1)
where ( , ) ( , )i j r
are the polar coordinates located at the bimaterial notch tip,
m�1 2, represents the material number, and �k corresponds to the kth eigenvalue.
Also in Eq. (1), f ijk and g ik are known functions of elastic properties of the
materials, eigenvalues �k , the local geometry characterized by the angles
1 and
2 , and the polar coordinate
. The eigenvalues of a bimaterial problem are
obtained by solving numerically the characteristic equation and they depend not
only on the corner geometry characterized by
1 and
2 but also on the elastic
properties of each material.
H. Arabi, M. M. Mirsayar, A. T. Samaei, and M. Darandeh
120 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 5
Due to the presence of stress term containing expression r k� �1
, it is clear that
the real eigenvalues in the range of 0 1� ��k result in a stress singularity. The
term | |�k �1 which represents the intensity of the singularity near the interface
corner is referred to as the order of singularity and it increases when �k approaches
zero. There are one or two eigenvalues in the range of 0 1� ��k , depending on the
combination of materials and geometrical configuration. There are also different
numbers of higher order eigenvalues �k �1 representing non-singular terms of
stress field near the interface corner.
The eigenvalues �k can be extracted by solving the characteristic equation
for the values of �:
F e b c d� � � � �2 2 2 2 0, (2)
where
e� � � � � �( ){cos( ) cos( ) [cos( ) cos(� �
�
�
�
2 2 2 2 21 1 2
2
1
1 22� �)
� � � � � � � �1 2 1 1 2 1 1 22 1 2cos( )]} ( )[ cos( )] ( )[ cos( )
� �
�
], (3a)
b� � � � � �( ){sin( ) sin( ) [sin( ) sin(� �
�
�
�
2 2 2 2 21 1 2
2
1
1 22� �)
� � � � �sin( )]} ( )sin( ) ( )sin( ),2 1 2 1 22 1 2
� �
�
(3b)
c� � � � � �� � �
�
�
�{( )[cos( ) cos( ) cos( ) cos(2 2 2 2 21 1 2 2
2 12� �)
� � � � � � � �1 2 1 1 2 1 1 21 1 2cos( )] ( )[ cos( )] ( )[ cos( )]},
�
(3c)
d � � � � � � �� �
�
{( )[sin( ) sin( ) sin( ) sin(2 2 2 2 21 2 1 1 1 2 2
)�
� � � � �sin( )] ( )sin( ) ( )sin( )}.2 1 2 1 22 1 2�
�
(3d)
Study of Characteristic Equation of the Elastic Stress Field ...
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 5 121
Fig. 1. General configuration of a bimaterial notch.
The parameters � and �, called Dundurs’ elastic mismatch parameters, are
defined as
�
� � � �
� � � �
�
�
1 2 2 1
1 2 2 1
1 1
1 1
( ) ( )
( ) ( )
, �
� �� � �� � �
� ��
�
� �
� � �
� � �
�
�
1 1
1 1
1
1
( )
( )
. (4)
In this equation, Em , � m , and � �m m mE�
[ ( )]2 1 are the Young modulus,
Poisson’s ratio and shear modulus associated with the material m (m�1, 2),
respectively. The parameter �m is equal to 3 4� � m for the plane strain and
( ) ( )3 1�
� �m m for the plane stress conditions. It can be seen when � �1 2 ��
or � �1 2 0� , the parameter � approaches 1 or �1, respectively. On the other
hand, by considering the restriction of 0 1 2� �� m and assuming the plane strain
conditions, the parameter � varies only between ( )�
1 4 and ( )��1 4 in terms
of parameter �. For the plane strain condition, all the values of mismatch
parameters � and � are contained within a parallelogram enclosed between
���1 and � �� ��4 1 in the � �� plane. The point � �� �0 represents the
condition associated with a homogeneous notch with identical materials. It is also
noteworthy that the sign of � depends on the relative stiffness of materials 1 and 2.
The mismatch parameter � is positive when material 2 is more compliant
than material 1, and negative when material 2 is stiffer than material 1. In this
paper, the material 1 is considered to be stiffer than material 2, and thus the
combinations of material can be selected from the first and fourth quadrants of
� �� plane. On the other hand, Fig. 2 illustrates the values of mismatch parameters
for some mostly-used combinations of material [16]. It can be seen that for a wide
range of the bimaterial joints, the mismatch parameters are concentrated along and
between lines � �� 4 and ��0 in the � �� space, with more data points near
� �� 4. Therefore, in the current paper the lines � �� 4 and ��0 are considered
for study of the characteristic equation.
122 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 5
Fig. 2. Values of elastic mismatch parameters � and � in plane strain condition for selected
combination of materials [16].
H. Arabi, M. M. Mirsayar, A. T. Samaei, and M. Darandeh
It should be noted that the angles
1 and
2 are assumed to be equal in this
paper giving
� �1 2 2� � � ( ). Therefore, by considering different values of �
in the range of 0 1� �� and
� �1 2 2� � � ( ), the eigenvalues corresponding
to both singular and first non-singular terms can be calculated by solving Eq. (2)
numerically.
It is also noteworthy that the first singular eigenvalue represents the mode I
(opening mode) and the second singular one is associated with the mode II (sliding
mode) of the edge motion after applying the external load [3, 15].
2. Results and Discussions.
2.1. Singular Eigenvalues. In this section, singular eigenvalues are studied as
a function of opening angle � and mismatch parameters � and �.
2.1.1. First Singular Eigenvalue. Figures 3 and 4 indicate the variation of the
first singular eigenvalue corresponding to the mode I versus opening angle �,
across two lines ��0 and � �� 4. It can be seen that, in case of ��0, the first
singular eigenvalue increases with increasing opening angle � and decreases with
increasing mismatch parameter � for a constant value of �. It also can be seen that
for all opening angles and mismatch parameter �, the first singular eigenvalue is a
real value.
In case of � �� 4, the first singular eigenvalue has the imaginary part, as
well as the real part when the opening angle is small. With increasing opening
angle the imaginary part disappears, while the real part decreases with increasing
opening angle. It should be noted that only real part is plotted in Fig. 4 as a
function of opening angle. It is seen that the real part attains its maximum value at
the point, where the imaginary part disappears. It also can be seen that the first
singular eigenvalue increases with increasing � for a constant opening angle.
It should be mentioned that only real parts of the eigenvalues (singular and
nonsingular) are studied in this paper. In fact, only real parts of the eigenvalues
directly affect stress distribution near homogeneous and bimaterial notches. This is
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 5 123
Fig. 3. Variation of first singular eigenvalue versus opening angle � for � � 0.
Study of Characteristic Equation of the Elastic Stress Field ...
because the complex eigenvalues of the notch problem (homogeneous or bimaterial)
are conjugate and their effects counteract each other finally [15, 19].
2.1.2. Second Singular Eigenvalue. In Figs. 5 and 6, the variation of the
second singular eigenvalue (associated with sliding mode) is illustrated as a
function of opening angle � , for ��0 and � �� 4. It can be seen that for ��0,
the second singular eigenvalue is always a real value and increases with increasing
� having a higher slope than the first singular eigenvalue (see Fig. 3). Also,
second singular eigenvalue increases by increasing � for a constant value of �.
For � �� 4, the second singular eigenvalue is a complex value which changes into
a real value with increasing opening angle �. In fact, for � �� 4 the second
singular eigenvalue is the conjugate value of the first singular eigenvalue in small
values of �. It also can be seen that for both ��0 and � �� 4, the real part of the
second eigenvalue is increased with increasing opening angle �.
124 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 5
Fig. 4. Variation of real part of the first singular eigenvalue versus opening angle � for � �� 4.
Fig. 5. Variation of second singular eigenvalue versus opening angle � for � � 0.
H. Arabi, M. M. Mirsayar, A. T. Samaei, and M. Darandeh
2.2. First Non-Singular Eigenvalues. As it was mentioned in the introduction,
recent studies on the homogeneous and bimaterial notched specimens indicate that
the first non-singular stress term can significantly influence on the stress distribution
near the notch tip .Therefore, study of variation of the first non-singular eigenvalue
as a function of opening angle and mismatch parameters is necessary. This section
focuses on variation of this eigenvalue versus geometry and mismatch parameters
in a wide range of opening angles and applicable combinations of materials.
Figures 7 and 8 show variation of the first non-singular eigenvalue as a
function of � for ��0, 0 1� �� and � �� 4, 0 1� �� . It can be seen that for
both cases, the eigenvalue is a real value at first and changes into a complex value
by increasing the opening angle �. In fact, by increasing the value of the opening
angle, the first nonsingular eigenvalue chaotically changes from a real value to a
complex value but the first transition does not occur at angles less than 44.9�.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 5 125
Fig. 6. Variation of real part of the second singular eigenvalue versus opening angle � for � �� 4.
Fig. 7. Variation of real part of the first non-singular eigenvalue versus opening angle � for � � 0.
Study of Characteristic Equation of the Elastic Stress Field ...
The opening angle corresponding to the first transition from real to complex
value which is called critical opening angle, � cr , is around 44.9� for homogeneous
notches and varies in bimaterial notches for different values of �. In Fig. 9, the
critical opening angle � cr is plotted as a function of mismatch parameter �. It is
seen that at ��0 (associated with the homogeneous notches), the critical opening
angle � cr is 44.9� and this value increases with increasing �. It also can be seen
that for � �� 4, value of � cr increases with a higher slope than that for ��0.
2.3. Areas with Higher Order of Singularity. As it was mentioned in Section 1,
the term | |�k �1 is referred to as order of singularity, and it increases when �k
approaches zero. In this section, the areas with higher order of singularity in the
126 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 5
Fig. 8. Variation of real part of the first non-singular eigenvalue versus opening angle � for � �� 4.
Fig. 9. Variation of critical opening angle for the first non-singular eigenvalue.
H. Arabi, M. M. Mirsayar, A. T. Samaei, and M. Darandeh
Bogy diagram are determined for singular eigenvalues. The results plotted in Figs.
3–6 illustrated the variation of singular eigenvalues as a function of corner
geometry and combination of materials.
In order to determine the high order of singularity areas, the range
0.5� �� 0.55 for the first singular eigenvalue (eigenvalue of mode I) and
0.5� �� 0.7 for the second singular eigenvalue (eigenvalue of mode II) are
considered. Figure 10 indicates a limited area in the Bogy diagram between two
lines ��0 and � �� 4 representing the first singular eigenvalue in the range of
0 5 0 55. .� �� and 0 90� � �� . It is seen that for �� 0.6 the first singular
eigenvalue may fall outside the selected range.
Figure 11 shows the area in the Bogy diagram corresponding to the range of
0 5 0 7. .� �� for the second singular eigenvalue. It is seen that in this case, the
restriction is on the opening angle �. Therefore, the second singular eigenvalue is
in the range of 0 5 0 7. .� �� for all values of mismatch parameters by considering
the restriction in � as shown in the corners of the shown area. For higher opening
angle, the second singular eigenvalue falls outside the above-mentioned range.
It is clear from Figs. 10 and 11 that for first singular eigenvalue the order of
singularity is sensitive of mismatch parameter � , while the second singular
eigenvalue is sensitive to the opening angle �.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 5 127
Fig. 10. Area with higher order of singularity for the first singular eigenvalue in the range of
0 5 0 55. .� �� and 0 90� � �� limited by � � 0 and � �� 4.
Fig. 11. Area with high order of singularity for the second singular eigenvalue in the range of
0 5 0 7. .� �� limited by the maximum opening angles.
Study of Characteristic Equation of the Elastic Stress Field ...
Conclusions. The characteristic equation of elastic stress field near a bimaterial
notch tip was studied. The analysis was done on the most applicable combination
of materials and geometrical configurations in the Bogy diagram between lines
��0 and � �� 4. In addition to singular eigenvalues, the first nonsingular
eigenvalue was also studied. For different combination of materials and geometrical
configuration, singular and first nonsingular eigenvalues were comprehensively
studied. Finally, the areas between lines ��0 and � �� 4 with more singularity
were determined. It was shown that the first singular eigenvalue is sensitive of
mismatch parameter � , while the second singular eigenvalue is sensitive to the
opening angle �. It also was shown that, in the case of ��0, both the first and
second singular eigenvalues are increased with increasing opening angle for the
same combination of materials. However, the first singular eigenvalue decreases
and the second singular eigenvalue increases with increasing � at the constant
value of the opening angle. The results presented in this paper can be used for
estimating the order of singularity as a function of combination of materials and
corner geometry in the most applicable bonded joints.
Ð å ç þ ì å
Äîñë³äæóºòüñÿ õàðàêòåðèñòè÷íå ð³âíÿííÿ äëÿ ïðóæíîãî ïîëÿ íàïðóæåíü ó
â³ñòð³ íàäð³çó íà ñòèêó äâîõ ìàòåð³àë³â. Óñòàíîâëåíî, ùî ðóéíóâàííÿ â³äáó-
âàºòüñÿ â êóòîâèõ òî÷êàõ ¿õ ñòèêó ÷åðåç âèíèêíåííÿ ñèíãóëÿðíèõ íàïðóæåíü
âíàñë³äîê ðîçðèâó ñóö³ëüíîñò³ ìàòåð³àëó é îñîáëèâîñòåé ãåîìåòðè÷íî¿ êîí-
ô³ãóðàö³¿. Ó ïîë³ ïðóæíèõ íàïðóæåíü ó â³ñòð³ íàäð³çó íà ñòèêó äâîõ ìàòå-
ð³àë³â ïîðÿäîê òàêî¿ ñèíãóëÿðíîñò³ âèçíà÷àþòü âëàñí³ çíà÷åííÿ ôóíêö³¿ íà-
ïðóæåíü Åð³. Ïðîàíàë³çîâàíî ñèíãóëÿðí³ âëàñí³ çíà÷åííÿ ³ ìàëîâèâ÷åíå ïåðøå
íåñèíãóëÿðíå âëàñíå çíà÷åííÿ. Ðîçãëÿíóòî ð³çí³ êîìá³íàö³¿ ìàòåð³àë³â ³ ãåî-
ìåòðè÷íèõ êîíô³ãóðàö³é äëÿ äâîõ íàéá³ëüø âèêîðèñòîâóâàíèõ òðàºêòîð³é íà
ä³àãðàì³ Áîã³ (��0, � �� 4) òà äåòàëüíî ïðîàíàë³çîâàíî îòðèìàí³ ðåçóëü-
òàòè. Ïîêàçàíî, ùî ãåîìåòðè÷í³ êîíô³ãóðàö³¿ ³ êîìá³íàö³¿ ìàòåð³àë³â ó â³ñòð³
íàäð³çó ñóòòºâî âïëèâàþòü íà ñèíãóëÿðí³ñòü íàïðóæåíü ïîáëèçó êóòîâèõ
òî÷îê íàäð³çó. Îáëàñò³ ç âèñîêîþ ñèíãóëÿðí³ñòþ íàïðóæåíü âèä³ëåíî ì³æ
ë³í³ÿìè ��0 ³ � �� 4 íà ä³àãðàì³ Áîã³ ³ ïðîàíàë³çîâàíî ÿê äëÿ ïåðøîãî, òàê
³ äðóãîãî ñèíãóëÿðíîãî âëàñíîãî çíà÷åííÿ.
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128 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 5
H. Arabi, M. M. Mirsayar, A. T. Samaei, and M. Darandeh
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Received 06. 11. 2012
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 5 129
Study of Characteristic Equation of the Elastic Stress Field ...
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/NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken voor kwaliteitsafdrukken op desktopprinters en proofers. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
/NOR <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>
/PTB <FEFF005500740069006c0069007a006500200065007300730061007300200063006f006e00660069006700750072006100e700f50065007300200064006500200066006f0072006d00610020006100200063007200690061007200200064006f00630075006d0065006e0074006f0073002000410064006f0062006500200050004400460020007000610072006100200069006d0070007200650073007300f5006500730020006400650020007100750061006c0069006400610064006500200065006d00200069006d00700072006500730073006f0072006100730020006400650073006b0074006f00700020006500200064006900730070006f00730069007400690076006f0073002000640065002000700072006f00760061002e0020004f007300200064006f00630075006d0065006e0074006f00730020005000440046002000630072006900610064006f007300200070006f00640065006d0020007300650072002000610062006500720074006f007300200063006f006d0020006f0020004100630072006f006200610074002000650020006f002000410064006f00620065002000520065006100640065007200200035002e0030002000650020007600650072007300f50065007300200070006f00730074006500720069006f007200650073002e>
/SUO <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>
/SVE <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>
/ENU (Use these settings to create Adobe PDF documents for quality printing on desktop printers and proofers. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
>>
/Namespace [
(Adobe)
(Common)
(1.0)
]
/OtherNamespaces [
<<
/AsReaderSpreads false
/CropImagesToFrames true
/ErrorControl /WarnAndContinue
/FlattenerIgnoreSpreadOverrides false
/IncludeGuidesGrids false
/IncludeNonPrinting false
/IncludeSlug false
/Namespace [
(Adobe)
(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AddBleedMarks false
/AddColorBars false
/AddCropMarks false
/AddPageInfo false
/AddRegMarks false
/ConvertColors /NoConversion
/DestinationProfileName ()
/DestinationProfileSelector /NA
/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements false
/GenerateStructure true
/IncludeBookmarks false
/IncludeHyperlinks false
/IncludeInteractive false
/IncludeLayers false
/IncludeProfiles true
/MultimediaHandling /UseObjectSettings
/Namespace [
(Adobe)
(CreativeSuite)
(2.0)
]
/PDFXOutputIntentProfileSelector /NA
/PreserveEditing true
/UntaggedCMYKHandling /LeaveUntagged
/UntaggedRGBHandling /LeaveUntagged
/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice
|