COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Finite Element Analysis of Singular Inplane Stress Fields around an
Inclusion Corner Tip
M. C. Chen 1*, X. C. Ping 2
1
2
School of Civil Engineering, East China Jiaotong University, Nanchang Jiangxi 330013, China
School of Mechatronics Engineering, East China Jiaotong University, Nanchang Jiangxi 330013, China
Email: [email protected]
Abstract A kind of hybrid stress corner-tip element model based on a kind of ad hoc finite element numerical
solutions is established to study singular stress fields around an inclusion corner tip. Incorporated with stabdard
finite element method, the model can be used to analyze any elastic problems of complex structures containing stress
singular points. To illustrate the efficacy of the suggested procedure, Numerical examples for infinite elasticity
cortaining rectangular inclusion are given, and the stress intensity factors around the inclusion corner tips are
computed and discussed. Comparisons show present method yields satisfactory solutions with fewer element.
Key words: singular stress fields, FEM, inclusion, corner-tip, plane stress
INTRODUCTION
Many engineering applications involve a junction of multiple materials to enhance the mechanical performance of
engineering materials. However, due to the mismatches of material properties, the stresses at the bonded interfacial
edge are often unbonded and present high stress singularities which provide a source of crack initiation. In order to
study the fracture starting at the inclusion corner tips with the method of fracture mechanics, the singular stress fields
have to be taken into account.
It is well known that singular stresses are often assumed to have the asymptotic form: σ ij = kr λ f ij (θ ) , in which
(r ,θ ) are the polar coordinates originated at the singular stress point, λ is eigenvalues, fij (θ ) gives angular
variations and k is stress intensity coefficient．There are a lot of studies on the eigenvalues and the singular stress
fields. As for the inclusion problems, the work of Chen [2] based on complex function method is representative.
However, it should be emphasized that the analytical technique can be only applied to some simple geometrical
discontinuities and particular combinations of materials, therefore, numerical methods, e.g., FEM [3] and BEM[4], are
powerful alternative. Among several finite element methods dealing with complicated stress singularities, the hybrid
element method appears to be more powerful [5]. Based on the principle of a novel hybrid finite element technique
introduced in [1], in this paper a kind of corner-tip element (see Fig. 1) is established to study singular stress fields
around an inclusion corner tip. As applications, rectangular inclusions in substrate materials is considered, the
influences of the inclusion sizes, the distance between two inclusions and the material mismatches on the stress
intensity factors are investigated.
HELLINGER-REISSNER VARIATIONAL PRINCIPLE
Fig. 1 shows a fully bonded bimaterial junction. In terms of linear elastic theory, the strain-displacement relations is
ε (x, y ) = Dm u( x, y )
(1)
in which (x, y ) denotes the displacement and stress field components in the Cartesian coordinate system;
⎯ 566 ⎯
ε(x, y ) = {ε x , ε y , γ xy }T is the strain vectors, u( x, y ) = {u x , u y }T is the displacement vectors, Dm is a matrix
composed of differential operators, i.e.,
o
Figure 1: A corner-tip domain of a bi-material junction
0 ⎤
⎡ ∂ / ∂x
⎢
Dm = ⎢ 0
∂ / ∂y ⎥⎥
⎢⎣∂ / ∂y ∂ / ∂x ⎥⎦
(2)
The linear constitutive equations is
σ (x, y ) = Cε(x, y )
(3)
where σ (x, y ) = {σ x , σ y , τ xy }T is the stress vector, and C is the elastic constant matrix for the plane problem.
If there are no body forces, the stress equilibrium equation reads as
DmT σ ( x, y ) = 0
(4)
The traction boundary condition on the outward boundary is defined as
n m σ ( x, y ) = T
(5)
where
⎡ nx
nm = ⎢
⎣0
ny ⎤
,
nx ⎥⎦
0
ny
(nx , n y ) constitute the outward normal of the domain boundary. On the other hand, the displacement boundary
conditions on the outward boundary is
u ( x, y ) = u ( x, y )
(6)
herein all the components with upper bar are ones prescribed on the boundary.
The displacement and stress continuity conditions across the interfaces are defined as
u1 ( x, y ) θ =α = u 2 ( x, y ) θ =α , u1 ( x, y ) θ =α = u 2 ( x, y ) θ =α
1
1
2
(7a)
2
n mσ1 ( x, y ) θ =α = n mσ 2 ( x, y ) θ =α , n mσ1 ( x, y ) θ =α = n mσ 2 ( x, y ) θ =α
1
1
2
(7b)
2
The exact solution for a boundary value problem of the bimaterial junction has to satisfy all Eqs. (1-7). By taking the
constitutive equations (3) and the displacement and stress continuity conditions (7) as subsidiary conditions,
Hellinger-Reissner variational principle for plane elasticity can be obtained as (Washizu [6])
Π = ∫ [σ ( x, y )T Sσ ( x, y ) + σ ( x, y )T Dm u( x, y )]dV − ∫
Vn
∂Vn
⎡⎣{n m σ ( x, y )}T {u( x, y ) − u( x, y )}⎤⎦dS
⎯ 567 ⎯
(8)
where Vn denotes the domain of the plane elasticity, ∂Vn denotes the outward boundary of the domain.
ASYMPTOTIC SINGULAR DISPLACEMENT AND STRESS FIELDS
Based on the ad hoc finite element eigenanalysis method [1], a standard characteristic equation is obtained as
∑ ⎡⎣(δq )
e T
e
(λ 2 P e + λ Q e + R e )q e ⎤⎦ = 0
(9)
where ‘e’ denotes the element components, q is the eigenvector, Definitions of all other symbols see paper [1]. By
solving the characteristic equation, the eigen-solutions, i.e., the eigenvalues λ and eigenvectors q , can be obtained.
Accordingly, the angular variation of displacement u% (θ ) and stress σ% (θ ) are obtained from these eigen-solutions.
Generally speaking, the asymptotic singular displacement and stress fields near the corner-tip can be written as below:
N
u(r ,θ ) = ∑ kn r λn +1u% n (θ )
(10)
n =1
N
σ (r ,θ ) = ∑ kn r λn σ% n (θ )
(11)
n =1
where N indicates the number of the eigenvalues truncated. In Eqs. (2), it has been arranged as λn ≤ λn +1 (herein
equal means multiple roots). In the scope of fracture mechanics, Re(λ ) < −1 should be excluded in the series.
In order to examine influences of external loadings and wedge geometry on the displacement and stress singular fields,
coefficients k ’s must be solved. In the following section, a corner-tip element will be established to determine
coefficients k s and subsequently the singular displacement and stress fields.
ESTABLISHMENT OF THE SUPER CORNER-TIP ELEMENT
Sound variational basis and high coarse mesh accuracy of crack-tip and wedge-tip hybrid elements for conventional
materials have been discussed [1, 5]. Similarly, to formulate finite element calculations for the singular displacement
and stress fields in fully bonded bimaterial junction, a finite element formulation for the 8-node super corner-tip
element, as shown in Fig. 2, will be developed based on the computed numerical eigen-solutions from Eq. (9).
Figure 2: Definition of an eight-node super corner-tip element
Through integration by parts and the divergence theorem, the finite element form of Eq. (8) can be expressed in the
following boundary integration form as:
Πe = −
T
1
e
e
T e
[
(
,
)]
(
,
)
[
(
,
)]
n
σ
x
y
u
x
y
ds
+
n
σ
x
y
u e ( x, y )ds
m
m
∫
∫
∂
∂
V
V
n
2 n
(12)
where the element components u e ( x, y ) and σ e ( x, y ) can be given with Eqs. (10) and (11), i.e.,
u e ( x, y ) = Tdu(r , θ ) = U e 2× N {k }
(13)
σ e ( x, y ) = Tsσ (r , θ ) = S e3× N {k }
(14)
⎯ 568 ⎯
where stress intensity coefficient matrix
{k} = {k1 ,L , k N }T ,
U e and S e are self-defined matrix during derivation.
Td and Ts are transformation matrices between polar coordinates and Cartesian coordinates, i.e.,
⎡ cos(θ ) − sin(θ ) ⎤
⎥,
⎣ sin(θ ) cos(θ ) ⎦
[Td ] = ⎢
⎡ cos 2 (θ )
sin 2 (θ )
− cos(θ ) sin(θ ) ⎤
⎢
⎥
cos 2 (θ )
cos(θ ) sin(θ ) ⎥ .
[Td ] = ⎢ sin 2 (θ )
⎢ 2 cos(θ ) sin(θ ) −2 cos(θ ) sin(θ ) cos2 (θ ) − sin 2 (θ ) ⎥
⎣
⎦
The boundary value u e ( x, y ) of the corner-tip element in Eq. (12) can be expressed with its nodal vector U e ( x, y ) as
u e ( x, y ) = [ L]U e ( x, y )
(15)
where the matrix [ L] is set up by one-dimensional Lagrangian interpolation method, in which the inter-element
displacement compatibility is satisfied automatically. The interpolation shape function [ L] between two adjacent
nodes can be expressed as
[ L] = ⎡⎢(1 −
⎣
s
)I 2
l
s ⎤
I2 ,
l ⎥⎦
where s is the distance measured from point p, and l is the length between the two nodes; I 2 is the second order
identity matrix.
Substituting Eqs. (13-15) into Eq. (12) yields
Πe = −
1
e
e
T
T
{k } [ H ] {k } + {k } [ G ] U e ( x, y )
2
(16)
in which
1
{[n m S e ]T U e ( x, y ) + U e ( x, y )T [n m S e ]}ds ,
∫
∂
V
n
2
[H ]
e
=
[G ]
e
= ∫ [n m S e ]T [ L ] ds .
∂Vn
The stationary value of the functional Π e of Eq. (16) reads
{k} = ([ H ] )−1 [G ]
e
e
U e ( x, y )
(17)
Inserting Eq. (17) into Eq. (16) leads to
Πe =
1 e
e
U ( x, y ) T [ K c ] U e ( x, y )
2
(18)
where [ K INC ] = ([G ] )T ([ H ] ) −1 [G ] denotes the element matrix of the super corner-tip hybrid element around
e
e
e
e
the singular point.
Finally, the 8-node corner-tip element is incorporated with standard four-node quadrilateral hybrid elements [7], and
the final finite element algebraic equation for the whole elasticity is expressed as
[ K ] U( x, y ) = {F}
(19)
where [K] is the global stiffness matrix assembled from the 8-node super corner-tip element matrix [ K INC ] and
e
standard four-node quadrilateral hybrid element matrix; U ( x, y ) denotes the global displacement vector and {F}
⎯ 569 ⎯
denotes the global external mechanical loading vector. By solving Eq. (19), the global displacement U ( x, y ) is
determined. Consequently, the coefficient k ’s can be retrieved with Eq. (17). To satisfy the LBB conditions (see Tong
et al [5]), the number of eigenvalues truncated should be chosen greater than or equal to the total number of DOF
minus the rigid body modes of the hybrid element. In the plane case, the total rigid body modes are three.
THE EXPRESSIONS OF STRESS INTENSITY FACTORS
According to the eigen-solutions of Eq. (9), the singular stress fields around the corner-tip for isotropic bimaterial
junction can be expressed as a sum of two terms: one written in a form of r λ1 is corresponding to the mode I
deformation and the other written in a form of r λ2 is corresponding to the mode II deformation. λ1 and λ2 are two
orders of singularities obtained from Eq. (9). For the sake of comparing with the analytical solutions of Chen [2], the
dimensionless stress intensity factors FI,λ1 and FII,λ2 are expressed in the following form:
FI,λ1 = lim ⎡⎣σ θ (r , 45o ) ⋅ r − λ1 ⎤⎦ / ⎡⎣ S ∞ π b − λ1 fθI,1 (0o ) ⎤⎦
(20)
FII,λ2 = lim ⎣⎡σ rθ (r , 45o ) ⋅ r − λ2 ⎦⎤ / ⎡⎣ S ∞ π b − λ2 f rθII,1 (0o ) ⎤⎦
(21)
r →0
r →0
∞
. The
where σ θ (r , θ ) , σ rθ (r , θ ) are obtained from Eq. (11). S ∞ denotes the external loading σ x∞ , σ y∞ or τ xy
expressions of fθI,1 (θ ) and f rθII,1 (θ ) are given in Chen [2].
NUMERICAL EXAMPLES
1. Single rectangular inclusion The plane problem of an infinite plate containing an inclusion, shown in Fig. 3, is
considered, the singular stress fields around the inclusion corner-tip o is analyzed. The elastic constants of the two
materials, namely the Poisson’s ratio are μ1 and μ2 respectively. To simulate the infinite effect, the width and the
height of material 1 are set to be w=20b and h=10l respectively, in which b and l are half height and width of the
inclusion. Due to symmetry, only one-quarter of the geometry is considered. To verify present method, The relation
between the dimensionless stress intensity factors, FI,λ1 and FII,λ2 , with l/b is investagated again, and it can be seen
through Fig. 4 and Fig. 5 that present solutions coincide with those of Chen [2].
Figure 3: A rectangular inclusion in an infinite substrate material
Under uniaxial tension loading σ y∞ , the dimensionless stress intensity factors for the square inclusion (l/b=1) are
listed in Table 1. It can be seen that the sizes of the corner-tip elements influence the precision of the numerical
solutions directly. For example, as μ2 / μ1 = 10 , the analytical solutions of Chen [2] are FI,λ1 = 0.2402 and
FII,λ2 = −0.4876 respectively, the errors of present solutions are -23.15% and -2.85% for the largest element and
-0.21% and –0.33% for the smallest element. the errors decrease with the refinement of the corner-tip element. Due to
the usage of corner-tip element, element numbers are reduced effectively, e.g., only 286 elements (one corner-tip
element and 285 four-node quadrilateral elements)are used even in the most refined mesh.
⎯ 570 ⎯
Figure 4: FI for the corner tip o under uniaxial tension loading σ y∞
Figure 5:
FII for the corner tip o under uniaxial tension loading σ y∞
∞
The numerical solutions and convergence effect under biaxial tension loadings σ x∞ = σ y∞ and shear loadings τ xy
are
also computed and listed in Table 2 ~ 3. Obviously, for the square inclusion (l/b=1), these symmetric loadings make
FI,λ1 reach its maximum value and make FII,λ2 tend to be zero. From this point of view, present solutions reflect the
objective fact.
Table 1 FI,λ1 and FII,λ2 for the square inclusion under uniaxial tension loadings σ y∞
Sizes (mm)
2.0b × 2.0b
1.0b × 1.0b
0.2b × 0.2b
SIFs
μ2 / μ1
FI,λ1
0.1
0.3875
2
0.2015
10
0.1846
100
0.2527
FII,λ2
2.4033
-2.3271
-0.4737
-0.3675
FI,λ1
0.3861
0.1988
0.2237
0.2530
FII,λ2
2.4549
-2.3293
-0.4795
-0.3687
FI,λ1
0.3837
0.1967
0.2397
0.2547
FII,λ2
2.4727
-2.3462
-0.4860
-0.3546
⎯ 571 ⎯
Table 2 FI,λ1 and FII,λ2 for the square inclusion under biaxial tension loadings σ x∞ = σ y∞
Sizes (mm)
SIFs
2.0b × 2.0b
1.0b × 1.0b
0.2b × 0.2b
μ2 / μ1
FI,λ1
0.1
7.4832
2
0.3891
10
0.4023
100
0.5134
FII,λ2
−0.0026
−0.0085
−0.0027
−0.0047
FI,λ1
7.5618
0.3934
0.4764
0.5118
FII,λ2
−0.0019
−0.0078
−0.0008
−0.0000
FI,λ1
7.5442
0.3988
0.4765
0.5113
FII,λ2
−0.0016
−0.0056
−0.0004
−0.0000
∞
Table 3 FI,λ1 and FII,λ2 for the square inclusion under shear loadings τ xy
Sizes (mm)
SIFs
2.0b × 2.0b
FI,λ1
1.0b × 1.0b
0.2b × 0.2b
μ2 / μ1
0.1
−11.7980
2
0.1516
10
0.6130
100
0.6673
FII,λ2
−0.0005
−0.0043
0.0043
0.0029
FI,λ1
−11.8577
0.1528
0.6346
0.6525
FII,λ2
−0.0016
−0.0019
−0.0004
0.0005
FI,λ1
−11.7653
0.1524
0.6341
0.6319
FII,λ2
−0.0021
−0.0018
−0.0010
−0.0002
2. Double squre inclusions The extent of mutual stress interaction between two square inclusions as shown in Fig. 5
depends on the distant d. the numerical solutions of FI,λ1 and FII,λ2 under three kinds of loadings types ( σ x∞ , σ y∞
∞
and τ xy
) are computed and listed in Table 4- 7. Under uniaxial tension loadings σ y∞ (see Table 4), the magnitude of
FI,λ1 and FII,λ2 decrease with the increase of distant d as μ2 / μ1 < 1 and increase with the increase of distant d as
μ2 / μ1 > 1 . Under uniaxial tension loadings σ x∞ (see Table 5) and shear loadings τ xy∞ (see Table 6), the
magnitude of FI,λ1 and FII,λ2 decrease with the increase of distant d regardless the ratio of μ2 / μ1 .
Table 4 Stress intensity factors for the corner-tip o shown in Fig. 5 under uniaxial tension loadings σ y∞
d/b
0.2
0.4
0.6
0.8
1.0
μ2 / μ1
SIFs
0.1
2
10
100
FI,λ1
0.4494
0.1888
0.2075
0.2090
FII,λ2
2.6723
-2.3365
-0.4886
-0.2971
FI,λ1
0.4309
0.1904
0.2104
0.2118
FII,λ2
2.6094
-2.3418
-0.4878
-0.2965
FI,λ1
0.4201
0.1919
0.2135
0.2150
FII,λ2
2.5711
-2.3448
-0.4868
-0.2959
FI,λ1
0.4131
0.1932
0.2165
0.2180
FII,λ2
2.5450
-2.3464
-0.4859
-0.2952
FI,λ1
0.4082
0.1942
0.2191
0.2206
FII,λ2
2.5261
-2.3474
-0.4850
-0.2947
⎯ 572 ⎯
o
Figure 5: Double square inclusions in an infinite substrate material
Table 5 Stress intensity factors for the corner-tip o shown in Fig. 5 under uniaxial tension loadings σ x∞
d/b
0.2
0.4
0.6
0.8
1.0
μ2 / μ1
SIFs
0.1
2
10
100
FI,λ1
0.3816
0.2167
0.2921
0.2912
FII,λ2
-2.4089
2.4286
0.5420
0.3295
FI,λ1
0.3815
0.2128
0.2811
0.2801
FII,λ2
-2.4041
2.4172
0.5297
0.3217
FI,λ1
0.3813
0.2099
0.2731
0.2720
FII,λ2
-2.4108
2.4072
0.5206
0.3160
FI,λ1
0.3811
0.2078
0.2671
0.2660
FII,λ2
-2.4003
2.3986
0.5137
0.3117
FI,λ1
0.3803
0.2061
0.2626
0.2613
FII,λ2
-2.4000
2.3913
0.5083
0.3084
∞
Table 6 Stress intensity factors for the corner-tip o shown in Fig. 5 under shear loadings τ xy
d/b
0.2
0.4
0.6
0.8
1.0
μ2 / μ1
SIFs
0.1
2
10
100
FI,λ1
-1.4130
0.5344
0.3076
0.3078
FII,λ2
0.4695
0.1453
0.1115
0.0677
FI,λ1
-1.3999
0.5322
0.3008
0.3009
FII,λ2
0.4032
0.1097
0.1064
0.0645
FI,λ1
-1.3781
0.5311
0.2962
0.2963
FII,λ2
0.3242
0.0835
0.1037
0.0627
FI,λ1
-1.3559
0.5306
0.2929
0.2929
FII,λ2
0.2540
0.0637
0.1024
0.0619
FI,λ1
-1.3359
0.5304
0.2904
0.2903
FII,λ2
0.1967
0.0485
0.1020
0.0616
CONCLUSIONS
A novel corner-tip element is developed to investigate the singular stress fields around the inclusion corner tips.
Compared with the conventional FEM, mesh refinement near the singular points are avoided successfully by the
⎯ 573 ⎯
application of the corner-tip elements. Though verification examples, it is shown that present model yields satisfactory
results with fewer elements. Due to the use of numerical but not analytical eigen-solutions, the present model can be
used to more complicated geometry configurations and material types.
Acknowledgements
The support of National Natural Science Foundation of China through Grant No. 10362002 and the Jiangxi Provincial
Natural Science Foundation of China through Grant No.512003 are gratefully acknowledged.
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⎯ 574 ⎯