Inclined Hole under Different Loading Conditions: A Review of Recent Results
Three-dimensional (3D) elastic stress distributions in the vicinity of the sharp corners of an inclined diamond hole in a plate are investigated. A detailed 3D finite element model under different loading conditions is analyzed to study the intensity of different fracture modes due to the thickness...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
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| Zitieren: | Inclined Hole under Different Loading Conditions: A Review of Recent Results / F. Berto,Reza H. Afshar // Проблемы прочности. — 2016. — № 5. — С. 95-105. — Бібліогр.: 18 назв. — англ. |
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irk-123456789-1735412020-12-11T01:26:38Z Inclined Hole under Different Loading Conditions: A Review of Recent Results Berto, F. Afshar, Reza H. Научно-технический раздел Three-dimensional (3D) elastic stress distributions in the vicinity of the sharp corners of an inclined diamond hole in a plate are investigated. A detailed 3D finite element model under different loading conditions is analyzed to study the intensity of different fracture modes due to the thickness effect. The stress results are compared with those provided by a recent theory which reduces the 3D governing equations of elasticity to a differential equation system, which includes a biharmonic equation and a harmonic equation. They provide the solution of the corresponding in-plane and out-of-plane notch problem, respectively, and have to be concurrently satisfied. Comparing numerical results and theoretical stress distributions, a good agreement is found. Досліджено тривимірний пружний розподіл напружень біля гострих кутів похилого ромбоподібного отвору в пластині. Проаналізовано тривимірну скінченноелементну модель за різних умов навантаження для вивчення інтенсивності руйнування за різних мод з урахуванням товщини пластини. Отримані результати порівнюються з такими, що визначені за найновішою теорією, яка перетворює тривимірні основні рівняння пружності в систему диференціальних рівнянь, в яку входять бігармонічне і гармонічне рівняння. Останні забезпечують розв'язок, що відповідає умові в площині та поза площиною надрізу, і повинні одночасно задовольнятися. Отримано хорошу відповідність між чисельними результатами і теоретичним розподілом напружень. Исследовано трехмерное упругое распределение напряжений около острых углов наклонного ромбовидного отверстия в пластине. Проанализирована трехмерная конечноэлементная модель при различных условиях нагружения для изучения интенсивности разрушения при разных модах с учетом толщины пластины. Полученные результаты сравниваются с таковыми, определенными по новейшей теории, которая преобразует трехмерные основные уравнения упругости в систему дифференциальных уравнений, включающую бигармоническое и гармоническое уравнения. Последние обеспечивают решение, соответствующее условию в плоскости и вне плоскости надреза, и должны быть одновременно удовлетворены. Получено хорошее соответствие между численными результатами и теоретическим распределением напряжений. 2016 Article Inclined Hole under Different Loading Conditions: A Review of Recent Results / F. Berto,Reza H. Afshar // Проблемы прочности. — 2016. — № 5. — С. 95-105. — Бібліогр.: 18 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/173541 539.4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Научно-технический раздел Научно-технический раздел Berto, F. Afshar, Reza H. Inclined Hole under Different Loading Conditions: A Review of Recent Results Проблемы прочности |
| description |
Three-dimensional (3D) elastic stress distributions in the vicinity of the sharp corners of an inclined diamond hole in a plate are investigated. A detailed 3D finite element model under different loading conditions is analyzed to study the intensity of different fracture modes due to the thickness effect. The stress results are compared with those provided by a recent theory which reduces the 3D governing equations of elasticity to a differential equation system, which includes a biharmonic equation and a harmonic equation. They provide the solution of the corresponding in-plane and out-of-plane notch problem, respectively, and have to be concurrently satisfied. Comparing numerical results and theoretical stress distributions, a good agreement is found. |
| format |
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| author |
Berto, F. Afshar, Reza H. |
| author_facet |
Berto, F. Afshar, Reza H. |
| author_sort |
Berto, F. |
| title |
Inclined Hole under Different Loading Conditions: A Review of Recent Results |
| title_short |
Inclined Hole under Different Loading Conditions: A Review of Recent Results |
| title_full |
Inclined Hole under Different Loading Conditions: A Review of Recent Results |
| title_fullStr |
Inclined Hole under Different Loading Conditions: A Review of Recent Results |
| title_full_unstemmed |
Inclined Hole under Different Loading Conditions: A Review of Recent Results |
| title_sort |
inclined hole under different loading conditions: a review of recent results |
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Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| publishDate |
2016 |
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Научно-технический раздел |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/173541 |
| citation_txt |
Inclined Hole under Different Loading Conditions: A Review of Recent Results / F. Berto,Reza H. Afshar // Проблемы прочности. — 2016. — № 5. — С. 95-105. — Бібліогр.: 18 назв. — англ. |
| series |
Проблемы прочности |
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AT bertof inclinedholeunderdifferentloadingconditionsareviewofrecentresults AT afsharrezah inclinedholeunderdifferentloadingconditionsareviewofrecentresults |
| first_indexed |
2025-07-15T10:17:40Z |
| last_indexed |
2025-07-15T10:17:40Z |
| _version_ |
1837707728260169728 |
| fulltext |
UDC 539.4
Inclined Hole Under Different Loading Conditions: A Review of Recent
Results
F. Berto
a,1
and Reza H. Afshar
b
a Department of Engineering Design and Materials, Norwegian University of Science and Technology
(NTNU), Trondheim, Norway
b University of Uppsala, Sweden
1 berto@gest.unipd.it
ÓÄÊ 539.4
Îáçîð íîâåéøèõ èññëåäîâàíèé ïî îöåíêå êîíöåíòðàòîðîâ íàïðÿæåíèé
òèïà íàêëîííîãî îòâåðñòèÿ ïðè ðàçëè÷íûõ óñëîâèÿõ íàãðóæåíèÿ
Ô. Áåðòî
à
, Ðåçà Õ. Àôøàð
á
à Íîðâåæñêèé óíèâåðñèòåò åñòåñòâåííûõ è òåõíè÷åñêèõ íàóê, Òðîíõåéì, Íîðâåãèÿ
á Óïïñàëüñêèé óíèâåðñèòåò, Óïïñàëà, Øâåöèÿ
Èññëåäîâàíî òðåõìåðíîå óïðóãîå ðàñïðåäåëåíèå íàïðÿæåíèé îêîëî îñòðûõ óãëîâ
íàêëîííîãî ðîìáîâèäíîãî îòâåðñòèÿ â ïëàñòèíå. Ïðîàíàëèçèðîâàíà òðåõìåðíàÿ
êîíå÷íîýëåìåíòíàÿ ìîäåëü ïðè ðàçëè÷íûõ óñëîâèÿõ íàãðóæåíèÿ äëÿ èçó÷åíèÿ èíòåí-
ñèâíîñòè ðàçðóøåíèÿ ïðè ðàçíûõ ìîäàõ ñ ó÷åòîì òîëùèíû ïëàñòèíû. Ïîëó÷åííûå
ðåçóëüòàòû ñðàâíèâàþòñÿ ñ òàêîâûìè, îïðåäåëåííûìè ïî íîâåéøåé òåîðèè, êîòîðàÿ
ïðåîáðàçóåò òðåõìåðíûå îñíîâíûå óðàâíåíèÿ óïðóãîñòè â ñèñòåìó äèôôåðåíöèàëü-
íûõ óðàâíåíèé, âêëþ÷àþùóþ áèãàðìîíè÷åñêîå è ãàðìîíè÷åñêîå óðàâíåíèÿ. Ïîñëåäíèå
îáåñïå÷èâàþò ðåøåíèå, ñîîòâåòñòâóþùåå óñëîâèþ â ïëîñêîñòè è âíå ïëîñêîñòè
íàäðåçà, è äîëæíû áûòü îäíîâðåìåííî óäîâëåòâîðåíû. Ïîëó÷åíî õîðîøåå ñîîòâåò-
ñòâèå ìåæäó ÷èñëåííûìè ðåçóëüòàòàìè è òåîðåòè÷åñêèì ðàñïðåäåëåíèåì íàïðÿ-
æåíèé.
Êëþ÷åâûå ñëîâà: àíàëèòè÷åñêèå âûðàæåíèÿ, êîíå÷íîýëåìåíòíûé àíàëèç, îòâåðñòèå,
òðåõìåðíîñòü.
Introduction. Due to convenience and relative simplicity, solutions of plane theory of
elasticity are popular and serve as a basis for many engineering design procedures,
standards and failure assessment codes. In terms of numerical costs, two-dimensional
models, based on plane stress or plane strain assumptions, are much more computationally
efficient, easier to build and verify in comparison with the corresponding three-dimensional
counterparts. However, to approach the through-the-thickness effect of real components,
requires alternative methods such as three-dimensional theory of elasticity or finite element
(FE) method.
The coupling effect was investigated for through-the-thickness cracks in finite
thickness plates using analytical and numerical methods [1–3]. In particular, the 3D stress
field at sharp notches with arbitrary notch opening angles based on the first order plate
theory by [4] is studied in [5, 6]. The coupled mode in shear loading was called ‘the
out-of-plane mode, or Mode O’, to distinguish it from the conventional Mode III. It was
© F. BERTO, REZA H. AFSHAR, 2016
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 5 95
also demonstrated that the out-of-plane mode is provoked by the 3D effects linked to
Poisson’s ratio of the material and described by the same characteristic equation as the
conventional Mode III.
The local interaction between the loading modes for the case of pointed and sharply
radiused notches in plates with finite thickness was re-analyzed in [7]. It was demonstrated
that the governing equations of three-dimensional elasticity can be reduced to a biharmonic
equation and a harmonic equation. The former provides the solution of the corresponding
plane notch problem, while the latter gives the anti-plane elasticity problem. Having the
two equations simultaneously satisfied in a 3D problem, justifies the theoretical and mutual
interaction between different modes. The main aim of this study is to examine the proposed
theory in [7] by investigating the stress fields in the vicinity of the sharp corners of a
diamond hole in a plate with finite thickness under tension and twisting loading conditions.
1. Methodology.
1.1. Analysis of the 3D Stress Fields. Recently, a new approach to the analysis of
three-dimensional problems has been developed by Lazzarin and Zappalorto [7] who
assume the Kane and Mindlin [4] hypothesis for displacement components as given by
u u x yx � ( , ), u v x yy � ( , ), u bz w x yz � � ( , ), (1)
where b is a constant value. As a result, the normal strains �xx , � yy , � zz as well as � xy
are independent of z, whereas, the other two shear components, i.e, � yz and � xz , depend
on z. We have
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
xx yy zz
xy yz
u
x
v
y
bw
u
y
v
y
bz
w
y
� � �
� � �
, , ,
, , �
�
�xz bz
w
x
� .
(2)
By invoking the stress–strain relationship, also the stress components � xx , � yy , xy ,
and � zz are independent of z, whereas the out-of-plane shear stress components depend
on z, according to the following equations:
�
�
� �zz zz xx yy
E
�
� �
� � � �
( )( )
[( ) ( )]
1 2 1
1
�
� �
� � �
�
��
�
�
��
�
�
�
�
�
�
E
bw
u
x
v
y( )( )
( ) ,
1 2 1
1
�
�
�
�
�
�
�yz yzG Gbz
w
y
� � , �
�
�xz xzG Gbz
w
x
� � .
(3)
By imposing the equilibrium in z direction, one obtains
� �2 0w , (4)
where � 2 denotes the two-dimensional Laplacian operator. Similarly, considering the
equilibrium in x and y directions, applying differentiation and by using the Schwarz
theorem for partial derivatives, the following equation can be obtained:
� �
�
� �
�
�
� �
2
2
2
2
2
2 0
xx yy xy
x y y x
� � � . (5)
96 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 5
F. Berto and Reza H. Afshar
Due to the fact that the stress components � xx , � yy , xy , and � zz do not depend on
z, Eq. (5) is automatically satisfied by the classic Airy stress function �( , ):x y
�
� �
�
xx
y
�
2
2
, �
� �
�
yy
x
�
2
2
,
� �
� �xy
x y
��
2
. (6)
Now, by using the generalized Hooke’s law for stresses and strains and satisfying the
in-plane compatibility equations, one obtains [7]:
� � � �4 2 0�
� zz . (7)
Here the equality to zero is guaranteed by the third of the Beltrami–Mitchell equations,
considering the fact that the stress components � xx , � yy , xy , and � zz do not depend on
z.
Subsequently, any three-dimensional notch problem can be converted into a biharmonic
problem (typical of plane stress or plane strain conditions) and a harmonic problem (typical
of out-of-plane shear case), provided that the displacement law according to Kane and
Mindlin’s Eq. (1) is satisfied. The final equation system is
� �
� �
�
�
�
��
4
2
0
0
� ,
,w
(8)
where � and w are defined implicitly according to Eqs (3) and (6), respectively. Note that,
both equations (8) must be satisfied simultaneously.
It is worth mentioning that the solution given in [7] is not valid on the free surfaces of
the plate, due to the presence of some edge effects, such as corner point singularities [8–11]
that might play an important role on stress intensities.
2. Examples and Applications. In this section, a three dimensional model with an
inclined diamond hole in a finite thickness plate under different loading conditions is
considered. In order to demonstrate the degree of accuracy of the theory developed in [7],
the analytical frame of the theory is explained first and then verified by the 3D finite
element (FE) models under tension and torsion loading conditions respectively.
2.1. The Spatial Diamond Hole Problem. An inclined diamond hole in a plate with
finite thickness under tension and torsion is considered. The inclination angle is 22.5�, as
shown in Fig. 1. Such an inclination induces local in-plane mixed mode stresses (Mode I +
Mode II), which can be solved by using Williams’ plane solution for re-entrant corners,
according to Eq. (8), where the function � is as follows [7, 12]:
� � � �� � � � ��
r A Bs s
1 1
1 11 1[ cos(( ) ) cos(( ) )]
� � � ��
r A Ba a
� � � 2 1
2 21 1[ sin(( ) ) sin(( ) )]. (9)
Having a fixed value of the through the thickness coordinate, z, and using the polar
coordinate system at the notch tip, the in-plane stresses vary according to Williams’
singularity degrees 1 1� � and 1 2� � for Modes I and II, respectively.
In the presence of a sharp V-shaped notch, the stress distributions of symmetric type
with respect to the angle bisector (Mode I) are [12–14]:
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 5 97
Inclined Hole Under Different Loading Conditions ...
�
�
! � " �
�
rr
r
r K
�
�
�
�
�
#
$
�
%
�
�
� � �
�
�
1
2 1 1
1 1
1
1 1 1( ) ( )
�
� �
� �
� �
�( )cos( )
( )cos( )
( )sin( )
1 1
3 1
1 1
1 1
1 1
1 1
� �
� �
� �
�
�
�
�
#
$
�
%
�
� �
�
� �
�
�
" �
�
�
�
1 1
1
1
1
1
1
1
1
( )
cos( )
cos( )
sin( )
�
�
�
�
#
$
�
%
�
�
�
�
�
�
�
�
�
�
�
. (10)
The skew-symmetric stress distributions (Mode II) are
�
�
! � " �
�
rr
r
r K
�
�
�
�
�
#
$
�
%
�
�
� � �
�
�
1
2 1 1
2 1
2
2 2 2( ) ( )
�
� � �
� � �
� �
( )sin( )
( )sin( )
( )cos( )
1 1
3 1
1 1
2 2
2 2
2 2
� �
� �
� �
" �
�
�
�
�
�
�
�
�
#
$
�
%
�
� �
� �
�
�
2 2
2
2
2
1
1
1
1
( )
sin( )
sin( )
cos( )
�
�
�
�
�
#
$
�
%
�
�
�
�
�
�
�
�
�
�
�
. (11)
Parameters K1 and K 2 are the notch stress intensity factors (NSIF) related to Mode I
and Mode II stress distributions, respectively, �1 and � 2 are Williams’ eigenvalues [12]
and, finally, "1 and " 2 are parameters which depend on the opening angle [12–14].
Subsequently, the stress field intensities can be quantified by the corresponding NSIFs [15]:
K z r
K z r
r
r r
1 0
1
2 0
1
2 0
2
1
2
( ) lim ( ),
( ) lim
� �
�
&
�
&
�
! �
!
�
�
( ). �
�
�
�
�� 0
(12)
However, due to the nature of three-dimensional problems, in addition to the in-plane
stresses, the out-of-plane shear stress components zr and z can be obtained by using
the following w function [16]:
a b
Fig. 1. Finite thickness plate weakened by inclined diamond hole under tension (a) and torsion (b).
98 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 5
F. Berto and Reza H. Afshar
w D r D rs s a a
s a� �� �� � 3 3
3 3
, ,cos( ) sin( ),, , (13)
where � � ! �3 32, , ,s a� � so that only the skew-symmetric part of w contributes to the
singular behavior of stress fields [7]. Accordingly, anti-plane Mode III shear stresses near
the notch tip can be determined as
!
�
!
�
�
zr a
z
K z r
K z r
a
a
�
�
�
�
3
1
3
3
1
3
3
2
2
( )
sin( ),
( )
cos
,
,
,
( ),,� 3 a
�
�
�
�
�
�
�
(14)
and
K z rr z
a
3 0
1
2 03( ) lim ( ),,� �&
�! �
(15)
where K z3 ( ) is the Mode III NSIF. Equation (15) can be considered as the extension to
the out-of-plane mode of Gross and Mandelson’s definitions [15] provided for the in-plane
modes, see Eqs. (12).
2.2. Tension Loading. In order to validate the theoretical frame developed in [7], a
detailed FE analysis on the model shown in Fig. 2 under tension loading is performed. The
normal stress � n �100 MPa is applied on the far end of the plate. The ANSYS package is
used to perform the finite element analyses (FEA). Material is assumed as isotropic and
linear elastic with the Young modulus E� 206000 MPa and the Poisson’s ratio
� 0.3.
The 20-nodes brick element is used. With the aim of obtaining the desired degree of
accuracy, a very fine and regular mesh pattern, specially close to the diamond corners is
constructed as shown in Fig. 2.
The variation of three stress components � , r , and z along the notch bisector
line and close to the notch tip (corner A in Fig. 2) are shown in Fig. 3. All the stresses are
Fig. 2. Mesh pattern used for stress analysis of finite thickness plate weakened by inclined diamond
hole under tension.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 5 99
Inclined Hole Under Different Loading Conditions ...
calculated on the plane 0.5 mm far from the free surface of the plate. It can be seen from
Fig. 3 that, as expected, the value of � is much higher than the other two stresses,
showing the dominance of Mode I fracture for the plate under tension.
Figure 4 shows the through the thickness variation of the NSIFs K1 , K 2 , and K 3 in
a distance from mid-plane to the free surface of the plate. The notch tip at corner A is
selected again for the NSIFs evaluations.
From Fig. 4, it can be observed that the value of K 3 is relatively lower than the other
two NSIFs. The intensity of K 3 varies from case to case. Here, the main aim was to
document the existence of the out-of-plane mode, which can be detected only by means of
3D models. This specific mode is expected to increase as the plate thickness increases [11].
It is evident, the linear change of K 3 through the thickness of the plate up to the
maximum value, which is located in the vicinity of the free surface. In parallel, the
Fig. 3. Stress components � , r , and z along the notch bisector line of an inclined diamond hole
in a thick plate under tension. Distance from the free surface z � 0.5 mm.
Fig. 4. Plots of NSIFs along the thickness of the plate with inclined diamond hole under tension.
100 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 5
F. Berto and Reza H. Afshar
variability of in-plane NSIFs (K1 and K 2) is shown: K 2 is constant on the major part of
the plate thickness, whereas the variation of K1 is more distributed through the thickness
of the plate.
2.3. Torsion Loading. The plate with inclined diamond hole with the same material
properties and element type as of the tension loading is also investigated under torsion
loading conditions.
The applied shear stress is set according to the expression n F t� 3 2, valid for the
narrow rectangular section, which induces a nominal shear stress n � 100 MPa on the
gross section of the plate (Fig. 5).
Similar to the tension loading condition, the variation of three stress components � ,
r , and z along the notch bisector line and close to the notch tip (corner A) are shown
in Fig. 6. All the stresses are calculated on the plane 0.5 mm far from the free surface of the
plate.
It is worth noting from Fig. 6 that � (corresponding to induced Mode I) is again
much higher than the z corresponding to the applied Mode III.
Fig. 5. The mesh pattern used for analysis of finite thickness plate weakened by inclined diamond
hole under torsion.
Fig. 6. Stress components � , r , and z along the notch bisector line of an inclined diamond hole
in a plate under torsion. Distance from the free surface z � 0.5 mm.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 5 101
Inclined Hole Under Different Loading Conditions ...
Due to the twisting loading, the stresses at the two corners A and B (Fig. 5) are
different. Hence, both corners are considered for NSIFs evaluation through the thickness of
the plate, as it is shown in Fig. 7.
Figure 7 shows the linear variation of K1 and K 2 through the thickness of the plate
and a parabolic trend of K 3. It was shown in [17] that the equation system (8) is valid even
when the displacement field is no longer according to the Kane–Mindlin hypothesis, i.e.,
Eq. (1), but is also when given in the following more general form:
u f z u x yx � ' �( ) ( , ), u f z v x yy � ' �( ) ( , ), u f z w x yz � ' �( ) ( , ), (16)
where f z( ) can be regarded as a generic polynomial function of order n:
f z a a z a z a zn
n( ) ... .� � � � �0 1 2
2
(17)
It is noteworthy that Eq. (17), similar to the equation proposed in [18] for the crack
case, automatically satisfies all the six compatibility equations [17].
In addition, apart from very near surface location (i.e., z(2.25 from midplane),
where some edge effects such as corner point singularity may exist [8–11], on a plane at
z� 0.5 mm from the free surface some differences between the NSIFs values at the two
corners A and B can be observed. Considering the corner A, the Mode I seems to be the
dominant mode of fracture and K 2 has a relatively lower value (about 25% lower than K1)
at z� 0.5 mm from the free surface. On the other hand, for corner B, both K1 and K 2 are
dominant and have an equal value at the same distance from the free surface. For both
corners, Mode III has a negligible effect with a relatively higher value at corner B.
Finally, in-plane stress components (� , � rr , and r ) as well as anti-plane shear
stresses ( zr and z ) obtained from FEA on a circular path with radius r� 0.002 mm
centered at the notch tip (corner A), compared with the theoretical prediction according to
Eqs. (10), (11), and (14), are shown in Figs. 8 and 9, respectively.
Fig. 7. Plots of NSIFs along the thickness of the plate with inclined diamond hole under torsion.
102 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 5
F. Berto and Reza H. Afshar
As it can be seen from Figs. 8 and 9 a very sound agreement is observed between the
numerical results and analytical predictions proposed in [7]. A good agreement with the
theoretical results was also found for the tension loading conditions.
Conclusions. In this study an attempt is made to examine the proposed theory in [7]
by investigating the stress fields in the vicinity of the sharp corners of a diamond hole in a
plate with finite thickness under tension and twisting loading conditions. The FE results
have confirmed the presence of coupled modes at the V-notch tip, showing a good
agreement with the in-plane and out-of-plane theoretical stress distributions.
Fig. 8. In-plane stress components (� , �rr , and r ) obtained on a circular path with radius
r � 0.002 mm centered at the notch tip (corner A) and comparison with the theoretical predictions [7].
Inclined diamond hole in a plate under torsion ( n � 100 MPa). Distance from the free surface
z � 0.5 mm.
Fig. 9. Anti-plane shear stresses ( zr and z ) obtained on a circular path with radius r � 0.002 mm
centered at the notch tip (corner A) and comparison with the theoretical predictions [7]. Inclined
diamond hole in a plate under torsion ( n � 100 MPa). Distance from the free surface z � 0.5 mm.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 5 103
Inclined Hole Under Different Loading Conditions ...
Ð å ç þ ì å
Äîñë³äæåíî òðèâèì³ðíèé ïðóæíèé ðîçïîä³ë íàïðóæåíü á³ëÿ ãîñòðèõ êóò³â ïîõèëîãî
ðîìáîïîä³áíîãî îòâîðó â ïëàñòèí³. Ïðîàíàë³çîâàíî òðèâèì³ðíó ñê³í÷åííîåëåìåíòíó
ìîäåëü çà ð³çíèõ óìîâ íàâàíòàæåííÿ äëÿ âèâ÷åííÿ ³íòåíñèâíîñò³ ðóéíóâàííÿ çà
ð³çíèõ ìîä ç óðàõóâàííÿì òîâùèíè ïëàñòèíè. Îòðèìàí³ ðåçóëüòàòè ïîð³âíþþòüñÿ ç
òàêèìè, ùî âèçíà÷åí³ çà íàéíîâ³øîþ òåîð³ºþ, ÿêà ïåðåòâîðþº òðèâèì³ðí³ îñíîâí³
ð³âíÿííÿ ïðóæíîñò³ â ñèñòåìó äèôåðåíö³àëüíèõ ð³âíÿíü, â ÿêó âõîäÿòü á³ãàðìîí³÷íå ³
ãàðìîí³÷íå ð³âíÿííÿ. Îñòàíí³ çàáåçïå÷óþòü ðîçâ’ÿçîê, ùî â³äïîâ³äຠóìîâ³ â ïëîùèí³
òà ïîçà ïëîùèíîþ íàäð³çó, ³ ïîâèíí³ îäíî÷àñíî çàäîâîëüíÿòèñÿ. Îòðèìàíî õîðîøó
â³äïîâ³äí³ñòü ì³æ ÷èñåëüíèìè ðåçóëüòàòàìè ³ òåîðåòè÷íèì ðîçïîä³ëîì íàïðóæåíü.
1. T. Nakamura and D. M. Parks, “Antisymmetrical 3-D stress field near the crack front
of a thin elastic plate,” Int. J. Solids Struct., 25, 1411–1426 (1989).
2. Z. H. Jin and R. C. Batra, “A crack at the interface between a Kane–Mindlin plate and
a rigid substrate,” Eng. Fract. Mech., 57, 343–354 (1997).
3. M. Heyder, K. Kolk, and G. Kuhn, “Numerical and experimental investigations of the
influence of corner singularities on 3D fatigue crack propagation,” Eng. Fract. Mech.,
72, 2095–2105 (2005).
4. T. R. Kane and R. D. Mindlin, “High-frequency extensional vibrations of plates,” J.
Appl. Mech., 23, 277–283 (1956).
5. A. Kotousov, “On stress singularities at angular corners of plates of arbitrary
thickness under tension,” Int. J. Fracture, 132, 29–36 (2005).
6. A. Kotousov, “Fracture in plates of finite thickness,” Int. J. Solids Struct., 44,
8259–8273 (2007).
7. P. Lazzarin and M. Zappalorto, “A three-dimensional stress field solution for pointed
and sharply radiused V-notches in plates of finite thickness,” Fatigue Fract. Eng.
Mater. Struct., 35, 1105–1119 (2012).
8. L. P. Pook, “A finite element analysis of cracked square plates and bars under
antiplane loading,” Fatigue Fract. Eng. Mater. Struct., 26, 533–541 (2003).
9. L. P. Pook, “Finite element analysis of corner point displacements and stress intensity
factors for narrow notches in square sheets and plates,” Fatigue Fract. Eng. Mater.
Struct., 23, 979–992 (2000).
10. L. P. Pook, “A note on corner point singularities,” Int. J. Fatigue, 53, 3–8 (1992).
11. F. Berto, P. Lazzarin, A. Kotousov, and S. Harding, “Out-of-plane singular stress
fields in V-notched plates and welded lap joints induced by in-plane shear load
conditions,” Fatigue Fract. Eng. Mater. Struct., 34, 291–304 (2011).
12. M. L. Williams, “Stress singularities resulting from various boundary conditions in
angular corners of plates in extension,” J. Appl. Mech., 19, 526–528 (1952).
13. P. Lazzarin and R. Tovo, “A unified approach to the evaluation of linear elastic stress
fields in the neighborhood of cracks and notches,” Int. J. Fracture, 78, 3–19 (1996).
14. P. Lazzarin and R. Tovo, “A notch intensity factor approach to the stress analysis of
welds,” Fatigue Fract. Eng. Mater. Struct., 21, 1089–1103 (1998).
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stress intensity factors for angular corners under various boundary conditions,” Eng.
Fract. Mech., 55, 529–556 (1996).
104 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 5
F. Berto and Reza H. Afshar
17. M. Zappalorto and P. Lazzarin, “Three-dimensional elastic stress fields ahead of
notches in thick plates under various loading conditions,” Eng. Fract. Mech., 108,
75–88 (2013).
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with application to crack problems,” Int. J. Eng. Sci., 8, 711–729 (1970).
Received 03. 03. 2016
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/ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing. 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
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(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AddBleedMarks false
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/ConvertColors /ConvertToCMYK
/DestinationProfileName ()
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/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements false
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/MultimediaHandling /UseObjectSettings
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/PDFXOutputIntentProfileSelector /DocumentCMYK
/PreserveEditing true
/UntaggedCMYKHandling /LeaveUntagged
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/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice
|