Stress intensity solutions for cracked plates by the dual boundary method
This paper presents the application o f the dual boundary element method for the determination o f stress intensity factors in plate bending problems. The loadings considered include internal pressure, and also combined bending and tension. Mixed mode stress intensity factors are evaluated by a crac...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
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| Cite this: | Stress intensity solutions for cracked plates by the dual boundary method / A. Sahli, D. Boutchicha, A. Belarbi, O. Rahmani // Проблемы прочности. — 2007. — № 5. — С. 81-93. — Бібліогр.: 15 назв. — англ. |
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Sahli, A. Boutchicha, D. Belarbi, A. Rahmani, O. 2013-08-15T08:27:51Z 2013-08-15T08:27:51Z 2007 Stress intensity solutions for cracked plates by the dual boundary method / A. Sahli, D. Boutchicha, A. Belarbi, O. Rahmani // Проблемы прочности. — 2007. — № 5. — С. 81-93. — Бібліогр.: 15 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/48100 539.4 This paper presents the application o f the dual boundary element method for the determination o f stress intensity factors in plate bending problems. The loadings considered include internal pressure, and also combined bending and tension. Mixed mode stress intensity factors are evaluated by a crack surface displacement extrapolation technique and the J-integral technique. The boundary element results for the case studies considered in the paper have been compared with either analytical or finite element results and in all cases good agreement has been achieved. Описано застосування методу подвійних граничних елементів для визначення коефіцієнтів інтенсивності напружень у задачах, що пов’язані зі згином пластин. Умови навантаження включають внутрішній тиск та комбінацію згину з розтягом. Коефіцієнти інтенсивності напружень для змішаних мод оцінюються за допомогою методу екстраполяції переміщення поверхні тріщини та методу J-інтеграла. Результати розрахунків методом граничних елементів для досліджуваних випадків навантаження порівнюються із даними аналітичних чи скінченноелементних розрахунків. Отримано їх хорошу збіжність. Описано применение метода двойных граничных элементов для определения коэффициентов интенсивности напряжений в задачах, связанных с изгибом пластин. Исследуемые условия нагружения включают внутреннее давление и комбинацию изгиба с растяжением. Коэффициенты интенсивности напряжений для смешанных мод оцениваются с помощью метода экстраполяции перемещения поверхности трещины и метода J-интеграла. Результаты расчетов методом граничных элементов, полученные для рассматриваемых случаев нагружения, сравниваются с данными аналитических или конечноэлементных расчетов. Получено их хорошее соответствие. en Інститут проблем міцності ім. Г.С. Писаренко НАН України Проблемы прочности Научно-технический раздел Stress intensity solutions for cracked plates by the dual boundary method Расчет напряженно-деформированного состояния пластин с трещинами методом двойных граничных элементов Article published earlier |
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Stress intensity solutions for cracked plates by the dual boundary method |
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Stress intensity solutions for cracked plates by the dual boundary method Sahli, A. Boutchicha, D. Belarbi, A. Rahmani, O. Научно-технический раздел |
| title_short |
Stress intensity solutions for cracked plates by the dual boundary method |
| title_full |
Stress intensity solutions for cracked plates by the dual boundary method |
| title_fullStr |
Stress intensity solutions for cracked plates by the dual boundary method |
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Stress intensity solutions for cracked plates by the dual boundary method |
| title_sort |
stress intensity solutions for cracked plates by the dual boundary method |
| author |
Sahli, A. Boutchicha, D. Belarbi, A. Rahmani, O. |
| author_facet |
Sahli, A. Boutchicha, D. Belarbi, A. Rahmani, O. |
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Научно-технический раздел |
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Научно-технический раздел |
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2007 |
| language |
English |
| container_title |
Проблемы прочности |
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Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| format |
Article |
| title_alt |
Расчет напряженно-деформированного состояния пластин с трещинами методом двойных граничных элементов |
| description |
This paper presents the application o f the dual boundary element method for the determination o f stress intensity factors in plate bending problems. The loadings considered include internal pressure, and also combined bending and tension. Mixed mode stress intensity factors are evaluated by a crack surface displacement extrapolation technique and the J-integral technique. The boundary element results for the case studies considered in the paper have been compared with either analytical or finite element results and in all cases good agreement has been achieved.
Описано застосування методу подвійних граничних елементів для визначення коефіцієнтів інтенсивності напружень у задачах, що пов’язані зі згином пластин. Умови навантаження включають внутрішній тиск та комбінацію згину з розтягом. Коефіцієнти інтенсивності напружень для змішаних мод оцінюються за допомогою методу екстраполяції переміщення поверхні тріщини та методу J-інтеграла. Результати розрахунків методом граничних елементів для досліджуваних випадків навантаження порівнюються із даними аналітичних чи скінченноелементних розрахунків. Отримано їх хорошу збіжність.
Описано применение метода двойных граничных элементов для определения коэффициентов интенсивности напряжений в задачах, связанных с изгибом пластин. Исследуемые условия нагружения включают внутреннее давление и комбинацию изгиба с растяжением. Коэффициенты интенсивности напряжений для смешанных мод оцениваются с помощью метода экстраполяции перемещения поверхности трещины и метода J-интеграла. Результаты расчетов методом граничных элементов, полученные для рассматриваемых случаев нагружения, сравниваются с данными аналитических или конечноэлементных расчетов. Получено их хорошее соответствие.
|
| issn |
0556-171X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/48100 |
| citation_txt |
Stress intensity solutions for cracked plates by the dual boundary method / A. Sahli, D. Boutchicha, A. Belarbi, O. Rahmani // Проблемы прочности. — 2007. — № 5. — С. 81-93. — Бібліогр.: 15 назв. — англ. |
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| fulltext |
UDC 539.4
Stress Intensity Solutions for Cracked Plates by the Dual Boundary
Method
A. Sahli, D. Boutchicha, A. Belarbi, and O. Rahmani
Département de Génie Mécanique (USTO), Oran, Algérie
УДК 539.4
Расчет напряженно-деформированного состояния пластин с
трещинами методом двойных граничных элементов
A. Сахли, Д. Бучича, А. Беларби, О. Рахмани
Отделение технической механики, Оран, Алжир
Описано применение метода двойных граничных элементов для определения коэффициентов
интенсивности напряжений в задачах, связанных с изгибом пластин. Исследуемые условия
нагружения включают внутреннее давление и комбинацию изгиба с растяжением. Коэф
фициенты интенсивности напряжений для смешанных мод оцениваются с помощью мето
да экстраполяции перемещения поверхности трещины и метода J-интеграла. Результаты
расчетов методом граничных элементов, полученные для рассматриваемых случаев нагру
жения, сравниваются с данными аналитических или конечноэлементных расчетов. Получено
их хорошее соответствие.
К л ю ч е в ы е с л о в а : коэффициент интенсивности напряжений, метод гранич
ных элементов, изгиб пластин, J -интеграл.
In troduc tion . Stress intensity factor solutions for several simple plate
geometries loaded by tension and bending are available in a parametric form [1,
2]. However, for more general cases involving more complex geometry, numerical
methods such as the boundary element m ethod (BEM) or the finite element
method (FEM) m ust be used for evaluation o f fracture mechanics parameters.
Therefore, there have been considerable developments in the use o f the BEM for
fracture analysis and there are m any examples o f its use for linear elastic fracture
mechanics (LEFM) [3, 4]. A n alternative approach is to use the dual boundary
element method (DBEM) with a single region being used in the analysis and one
crack face being m odeled using the displacement boundary integral equation and
the other crack face being m odeled using the traction boundary integral equations.
The DBEM therefore requires no subregioning and eliminates the singular
algebraic equations.
The purpose o f the present paper is to illustrate the effectiveness o f the
DBEM for the LEFM analysis o f plate bending problems. The DBEM is used to
analyze several crack configurations in plates, including center crack, edge crack
and cracks emanating from a hole in a finite w idth plate loaded by either bending
and tension or uniform pressure, and the K l , K n , and K ш stress intensity
factors are presented. The results for the case studies discussed in this paper have
© A. SAHLI, D. BOUTCHICHA, A. BELARBI, O. RAHMANI, 2007
ISSN 0556-171X. Проблемы прочности, 2007, N 5 81
A. Sahli, D. Boutchicha, A. Belarbi, and O. Rahmani
been compared with either analytical or finite element results and in all cases the
boundary element results are in good agreement with the results from alternative
sources.
1. Plate Bending Equations. Reissner’s [5, 6] plate bending equations are
used to obtain the fundamental solutions for the boundary integral equations:
*
A iju j + bi _ 0 (1)
where A j is the Navier operator and can be expressed as
D(1 — v)
A*
Aa P _ 2
2 .2 c (1+ V) d 2
(V 2 — 22)<5aP + (------ ) ----------
( ) ap (1 — v ) adxp (2a)
* * D(1 V) 2 ^
A“3 = —A- 3 _ — — ■ (2b)
2
where V is the Laplace operator and bt represents the loading on the plate,
vq ,a
b a _ l 2(1 — V ) , (2d)
b3 _ q. (2e)
In Eqs. (1), (2a)-(2e), v is Poisson’s ratio, q is the distributed load on the plate
surface, d ap is the Dirac delta function, and D and 2 are given by the
following equations:
E h 3
D _ 1 2 ^ ' (3a)
(3b)
h
in which E is Young’s module and h is the plate thickness.
2. The Dual Boundary Integral Equations. The dual equations, on which
the dual boundary element method is based, are the displacement and the traction
boundary integral equations. The displacement boundary integral equations for
Reissner plate model were reported in [7]. The traction integral equations for
Reissner plates have been reported independently in [8] and [9]. In this paper,
these boundary integral equations are used. The boundary integral representation
82 ISSN 0556-171X. npo6n.eMH npounocmu, 2007, N 5
Stress Intensity Solutions for Cracked Plates
of the displacement components wt can be written for an internal source point
X ' as
wt (X ') + J Pjj (X ', x )w j(x )dT(x) = J W j (X ', x )p j (x)dT(x) +
+ J(W i*(X ', X ) - \ W *,a (X ', X ))q3(X )dQ (X ), (4)
Q (1-V M
where wa are rotations of x- and y-axes, respectively, w3 is out of plane
displacement, p j are bending moment and shear tractions, respectively, and q3
is internal pressure. Roman indices vary from 1 to 3 and Greek indices vary from
* *
1 to 2. The values P j (X ', x ) and W j (X ', x ) represent the Reissner plate
fundamental solutions at the boundary point x .
The above integrals are regular provided r ^ 0. If the point X ' is placed at
the boundary, that is X ' ^ x' G T , the distance r tends to zero and, in the limit,
the fundamental solutions exhibit singularities. Assuming that the displacements
Wj satisfy the Holder continuity, Eq. (4) for the source points on the boundary
can be written as follows:
Cj (x ')Wj (x ') + J P j (x', x )Wj (x )dT(x ) = J W j (x', x ) p j (x )dT(x ) +
+J(W *3(x', X ) - ------ - 2 < >a(x', X ))q3(X )d Q (X ), (5)
Q (1-V M
where the integral in the left hand side denotes a Cauchy principal value integral,
x ', x e r are source and field points, respectively, and €„(x') are the jump
1
terms. The term C j(x') is equal to —o „ when x' is located on a smoothj 2 j
boundary.
By applying the divergence theorem, the last domain integral in (5) can be
transferred to boundary integral, in the case of a uniform load (q3 = const) to
give:
/ (Wf*3(x', X ) - \ W*,a (x', X ))q3(X )dQ (X ) =
q ( 1 - ^ ) A
= q 3 J (Vi,a (x' , x ) ----- 772 Wia (x ' , x ))n a (x )dF(x ), (6)
r ( 1 - v )A
where V*a are the particular solutions of the equation V*dd = W *3.
The stress resultant components are obtained by differentiation of equation
(4) with respect to the coordinate of the source point X ' and then substituting
them into the stress displacement relations for Reissner plate theory to give:
ISSN 0556-171X. npoôëeMbi npounocmu, 2007, N9 5 83
A. Sahli, D. Boutchicha, A. Belarbi, and O. Rahmani
M a,0(X ') = J W«0k(X ' , x )Pk(x)dr(x) - J P«0k(X ' , x )w k (x )d r(x ) +
r r
vv
+ q J Q«0( X ' ,x )d r (x ) + - — Tryqà a 0
r ( 1 - v ) A
(7)
Q0(X 0 = J W3*0*(X ' , x )P k (x )dr(x) - J P30k(X ' , x)wk(x)dr(x) +
r
+ q J Q«9( X ' , x )d r(x X
where M a 0 and Q 0 are bending moment and shear force stress resultants,
respectively.
* *The kernels W jk , Pjk are obtained from linear combination of derivatives
H* H* ,
of W j and P j , respectively, and Q ^ - from linear combination of derivatives
* *
of Wi3 and Wi0 e .
By moving the source point X ' to the boundary and multiplying all the
terms by the unit outward normal ng to the boundary at the source point x', the
traction integral equations are obtained as follows:
- p a (x ) + n g ( x ' ) J P«0y ( x ', x )wy ( x )d r( x ) + n g ( x' ) J ( x x )w 3 ( x )d r( x ) =
2 r r
= n 0 (x ')J Wa0y (x' , x )P / (x )dr(x ) + n 0 (x ' )J Wa0y (x' , x )P 3(x )dr(x ) +
r r
* 1 qv
+ qn0 (x ' )J Q«0( x \ x)d r(x ) + - ~ — TT2 na (x'X (8)
r 2 (1- v ) A v '
- P 3 (x) + n0 (x' ) J P3*0y (x ' , x)wy (x )dr(x) + n0(x' ) J P3*03(x ' , x)w3(x)dr(x) =
2 r r
= n 0 (x' ) J W3*0y ( x ' , x ) Py (x )dr(x ) + n 0 (x' ) J W3*03 ( x ' , x ) P 3(x )d r( x ) +
+ qn 0 (x' ) J Q 3*0 (x' , x )d r( x ), (9)
where p a = M «9 and p 3 = Q 9«9 . The second integral in the left hand side
stands for the Hadamard principal value integral.
Equations (8)-(9) represent three traction integral equations, and together
with the three displacement integral equations in Eq. (5) form the dual boundary
integral formulation.
84 ISSN 0556-171X. npoôëeMbi npounocmu, 2007, N 5
Stress Intensity Solutions for Cracked Plates
3. Numerical Implementation. In order to implement the DBEM
numerically, the boundary including crack surfaces are discretized into elements.
The general modelling strategies used in this work are similar to those used in
[10] and can be summarized as follows:
(i) crack boundaries are modelled with discontinuous quadratic elements, as
shown in Fig. 1 to satisfy continuity conditions of displacements and its derivatives
in all nodes for the existence of principal value integrals; in such a way that each
node of one of the crack surfaces is coincident with the node on the opposite
surface;
(ii) the traction equations (8)-(9) are applied for collocation on one of the
crack surfaces;
(iii) the displacement equation (5) are applied for collocation on the opposite
crack surface and the other non-crack boundaries;
(iv) continuous quadratic elements are applied along the remaining boundary
of the body, except at the intersection between a crack and an edge, where
discontinuous or semidiscontinuous ones are required on the edge, in order to
avoid a common node at intersection, and also at boundary corner, where
semidiscontinuous elements are preferred, in order to avoid a common node at the
corner.
Fig. 1. Dual boundary element model [(D) displacement equation; (T) traction equation; (•) element
node].
This simple strategy is robust and allows the DBEM to effectively model
general edge or embedded crack problems; on the other hand, crack tips, crack-
edge corners and crack kinks require no special treatment, since they are not
located at nodal points where the collocation is carried out.
4. Stress Intensity Factors Evaluation. The stress resultant intensity factors
K i, K 2 , and K 3 are usually used instead of the stress intensity factors K I, K II,
and K III. The relationship between the stress resultant intensity factors and the
stress intensity factors are [11]:
The stress resultant intensity factors can be evaluated in several ways. In this
work, the /-integral technique is employed to calculate the stress resultant
intensity factors. The path independent /-integral is defined for plate bending as
corner
[12]:
ISSN 0556-171X. npoôneMbi nponnocmu, 2007, № 5 85
A. Sahli, D. Boutchicha, A. Belarbi, and O. Rahmani
J db = f (Wb - qw3 )ndd r - f P i w i,dd? + f q,dw3dQ. ( i d
r r Q
The strain energy density for plate bending, Wb, is defined as
1
Wb = ~
I
M afi
wa ,fi + w fia
+ Q a (wa + w3,a ) (12)
where wt j is the strain tensor, and na are the components of the unit outward
normal to the contour path.
To decouple the related stress intensity factors of a mixed mode problem
from /-integral, an extension of a simple procedure which has been developed in
[13] based on the decomposition of the elastic fields into symmetric and
anti-symmetric mode component is used (Fig. 2). Using this procedure, only one
component of /-integral is required.
Fig. 2. Symmetric and anti-symmetric moment and shear force stress resultants at two points
located symmetrically relative to the crack axis.
The relationship between the component of /-integral in x 1 direction and
the stress resultant intensity factors for plate bending is given as
/ 1b —
12rt
1b E h3
h 2
K 12 + K 22 + 10(1 + V )K 32 (13)
The decomposition of the elastic fields are as follows:
M 11
M 12
22M
Q 1
Q 2s
m 11 + m 11
M 12 _ M 12
M 22 + M 2 2
Q 1 + Q1
Q 2 - Q2
M AS' M 11 - M 11
M1 1
’ — 2 *
M 12 + M 12
m AS M 22 - M 22
Q1AS Q1- Q1
Q AS Q 2 + Q2
(14)
86 ISSN 0556-171X. npoôëeMbi npounocmu, 2007, N9 5
Stress Intensity Solutions for Cracked Plates
Applying the decomposition procedure, the integral J j can be represented
by the sum of two integrals as follows:
(15)
where the superscript indicates the pertinent deformation mode.
Finally, the stress intensity factors are obtained from the following
relationship:
rS _ r I _ 12^ K 1
J 1 b _ J 1 b _
2
Eh 3
TAS _ rII , rIII _
J 1 b _ J 1 b + J 1 b _
12j i
Eh3 K 22 +
10 . (16)
To split mode II and mode III components from J j^ f , the displacement ratio
as proposed in [14] are used
48 r—
Aw1 _ w1(+180°) _ w1(-180°) _ Eh W 2r K 2 ,
3 _ w3(+180°) w3(-180°)
24(1 + v )
5Eh
Awi 10 K 2
Aw3 (j+ v )h 2 K 3
Substituting Eq. (19) into Eq. (16) the following relations hold:
(17)
(18)
(19)
AS _ 12^ (1+ v ) K 2
J 1 b _ -- K *
or
J
10
AS _ 12^ (1+ v ) K 2
h 2(1 + v ) (Aw1 x2
10Eh ^Aw3
+1
1b Eh3 1+
10 A w -i
h 2(1 + v )^ Awj
(20)
(21)
To implement this procedure into the boundary element analysis, a circular
contour path around the crack tip is defined as a set of internal points located at
symmetrical positions relative to the crack plane, as shown in Fig. 3. The numerical
integration along the contour path is accomplished using the trapezoidal rule.
5. Case Studies.
5.1. Square Plate with a Central Crack and Edge Twisting Moments (Fig. 4).
An infinite plate subjected to twisting moments, as shown in Fig. 4, is considered
in this case study. Because of the symmetrical nature of the loading and geometry,
only half of the plate was analyzed, with the line of symmetry being perpendicular
to the crack. The boundary conditions on the line of symmetry were assumed to
be x j = 0, 0 2 = M 11 = w = 0.
2
ISSN 0556-171X. npoôëeMbi npounocmu, 2007, N9 5 87
A. Sahli, D. Boutchicha, A. Belarbi, and O. Rahmani
P'
Fig. 3. Local crack tip coordinate reference system and contour path for the /-integral.
2 a
Fig. 4. Plate with edge twisting moments.
h/aVlO
Fig. 5. Normalized intensity factors for plate with edge twisting [lines correspond [11]; (A ) quarter
point; (□ ) /-integral].
In this case study Young’s modulus was assumed to be 2-105 MPa with
Poisson’s ratio being set to zero. Results are given in Fig. 5 for the quarter point
and the J-integral methods for a range of crack lengths and one plate thickness for
the normalized moment and shear force intensity factors F 2 and F 3 , which are
obtained from the following equations:
,7 K 2
F2 _ . , ^ (22)MoV a
88 ISSN 0556-171X. npoôneMû npoHuocmu, 2007, № 5
Stress Intensity Solutions for Cracked Plates
(1 + v )hK 3
M пл/löaF 3 Ayr .R e d • (23)
The DBEM results are also compared in Fig. 5 with Sih’s [2] analytical
results. The J-integral results for F 2 and F 3 are also presented in Table 1.
T a b l e 1
Normalized Moment and Shear Force Intensity Factors for a Cracked Plate
Subjected to Twisting Moment Calculated from /-Integral Paths
h F 2 F 3
a-J 1 0 Path 1 Path 2 Path 3 Path 1 Path 2 Path 3
0 . 1 0 0 0.2301 0.2207 0.2123 0.1242 0.1191 0.1146
0.125 0.2794 0.2727 0.2663 0.1253 0.1223 0.1194
0.250 0.4164 0.4141 0.4126 0.1056 0.1050 0.1046
0.500 0.5961 0.5933 0.5932 0.0684 0.0681 0.0681
0.750 0.7166 0.7132 0.7136 0.0457 0.0455 0.0455
1 . 0 0 0 0.7942 0.7905 0.7911 0.0305 0.0303 0.0304
1.250 0.8295 0.8254 0.8260 0.0198 0.0197 0.0197
1.500 0.8321 0.8279 0.8282 0.0126 0.0125 0.0125
5.2. Cantilever Rectangular Plate with Edge Moment, Central Crack, and
Central Hole (Fig. 6).
Results are given in Fig. 7 for this case study for a range of crack lengths and
two plate thicknesses. The results for the F 1 normalized intensity factors have
been obtained using the quarter point and the /-integral methods. The DBEM
results have been verified using a 500-element FEM model. The /-integral results
for this case study are also shown in Table 2.
Fig. 6 . Cantilever plate with internal crack and internal hole.
The normalized stress intensity factor, Fx, is related to the moment intensity
factor K 1 by the expression
j-, K i
F i 12 /— (24)qb V a
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A. Sahli, D. Boutchicha, A. Belarbi, and O. Rahmani
T a b l e 2
Normalized Moment Intensity factors for a Plate with a Central Crack and a Hole
Subjected to Edge Moment Calculated from /-Integral Paths
a h h
d
—j= = 1.0
aV 10
—j= = 0.1
aV 10
Path 1 Path 2 Path 3 Path 1 Path 2 Path 3
0.1 1.1841 1.1852 1.1853 1.3265 1.3338 1.3372
0.2 1.0965 1.0973 1.0974 1.3122 1.3146 1.3147
0.3 1.0671 1.0672 1.0672 1.3219 1.3229 1.3245
0.4 1.0658 1.0659 1.0658 1.3495 1.3493 1.3493
0.5 1.0845 1.0844 1.0927 1.3926 1.3928 1.4028
0.6 1.1230 1.1228 1.1313 1.4591 1.4591 1.4689
0.7 1.1890 1.1977 1.0846 1.5623 1.5754 1.4050
0.8 1.3088 1.3177 1.1981 1.7355 1.7482 1.5668
0.9 1.5883 1.4657 - 2.1172 1.9153 -
Fig. 7. Normalized intensity factors for a plate with internal crack and internal hole [(□ ) quarter
point; (A ) FEM; lines correspond J-integral: (1) h/dVlO = 1.0; (2) h /dV l0 = 0.1].
5.3. Rectangular T-Shaped Plate with Edge Moment and Edge Cracks (Fig. 8).
This case study comprises a cantilever T-shaped plate subjected to a
uniformly distributed bending moment with perpendicular edge cracks at the
junction of the T as shown in Fig. 8. DBEM quarter point, stress extrapolation
and J-integral results for the F normalized intensity factor for a range of crack
lengths and a single plate thickness are presented in Fig. 9. The J-integral results
for this case study are also given in Table 3.
90
Fig. 8. T-shaped cantilever plate.
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Stress Intensity Solutions for Cracked Plates
T a b l e 3
Normalized Moment Intensity for a Cantilever T-Shaped Plate
Calculated for /-Integral Paths
a Fi
b Path 1 Path 2 Path 3
0.1 0.2828 0.2862 0.2828
0.2 0.3112 0.3114 0.3142
0.3 0.3489 0.3481 0.3539
0.4 0.3758 0.3753 0.3788
0.5 0.3980 0.3978 0.4003
0.6 0.4171 0.4168 0.4195
0.7 0.4354 0.4349 0.4377
0.8 0.4565 0.4561 0.4590
0.9 0.4841 0.4837 0.4865
0.5393
0 0 2 0 4 0 6 0 8 I
a/b
Fig. 9. Normalized intensity factors for a T-shaped cantilever plate.
6. Discussion of Results. Results are presented graphically in Figs. 5, 7, and 9
for the various normalized intensity factors for the case studies shown in Figs. 4,
6, and 8. Plates with a range of crack lengths, loading conditions, plan forms and
boundary conditions were analyzed to investigate the effectiveness of the DDEM
for the evaluation of the stress intensity factors in cracked plates. The displacement
extrapolation, quarter point, stress extrapolation and /-integral methods have been
used to calculate the moment and shear force intensity factors and in all cases the
DBEM results are in close agreement with each other. Moreover, the DBEM
results for the first case study (Fig. 4) are in good agreement with results obtained
by Sih [2], who used the FEM in conjunction with the /-integral method to
evaluate the stress intensity factors at the crack tip. However, for the third case
study (Fig. 8), the FEM model with its 500 elements compares unfavourably with
the DBEM model, which required only 40 boundary elements. Quarter point
elements were used at the crack tip for the finite element study and the /-integral
approach was used to evaluate the stress intensity factors at the crack tip. For this
case study, the DBEM and the FEM results are in good agreement. Tables 1-3
give details of the normalized intensity factors for the various case studies
investigated. The results show a high degree of stability for each /-integral path
for all crack lengths, plate thicknesses, loading conditions and boundary conditions.
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A. Sahli, D. Boutchicha, A. Belarbi, and O. Rahmani
The values of the normalized crack tip intensity factors which are presented in
this paper can be converted to bending moment, twisting moment, and shear force
intensity factors, K 1, K 2, and K 3, from which Eq. (10) can be used to evaluate
the stress intensity factors, K j, K n , and K ш , at the crack tip.
Conclusions. The application of the DBEM for the LEFM analysis of plate
bending problems is discussed. The method has been extended in this paper to
enable the stress intensity factors in plate bending problems to be evaluated. In
particular, the displacement ratio approach for the evaluation of /-integrals for
three-dimensional fracture analysis has been developed further to deal with
/-integrals arising in the fracture analysis of plate bending problems. Results for
the normalized intensity factors for a number of case studies, including plates
with cracks at boundary discontinuities are presented. The displacement, bending
moments and twisting moments for the plates under consideration were evaluated
using the DBEM. These DBEM results were then used in conjunction with the
/-integral method to determine the normalized stress intensity factors at the crack
tips. The crack tip stress intensity factors can then be evaluated from the
normalized intensity factors. The BEM results were compared with either
analytical or finite element results and in all cases the DBEM results were in good
agreement with the results from alternative sources.
Р е з ю м е
Описано застосування методу подвійних граничних елементів для визначен
ня коефіцієнтів інтенсивності напружень у задачах, що пов’язані зі згином
пластин. Умови навантаження включають внутрішній тиск та комбінацію зги
ну з розтягом. Коефіцієнти інтенсивності напружень для змішаних мод оці
нюються за допомогою методу екстраполяції переміщення поверхні тріщини
та методу /-інтеграла. Результати розрахунків методом граничних елементів
для досліджуваних випадків навантаження порівнюються із даними аналітич
них чи скінченноелементних розрахунків. Отримано їх хорошу збіжність.
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Stress Intensity Solutions for Cracked Plates
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boundary element method in linear elastic crack problem in plate bending,”
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Received 10. 05. 2007
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