COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Finite Element Analyses of Multi-Material Wedges and Junctions
with Singular Antiplane Stress Fields
X. C. Ping 1*, M. C. Chen 1, J. L. Xie 2
1
2
School of Mechatronics Engineering, East China Jiaotong University, Nanchang Jiangxi, 330013 China
School of Mechanical and Electronic control Engineering, Beijing Jiaotong University, Beijing, 100044
China
Email: [email protected]
Abstract In this paper a kind of wedge-tip hybrid element based on the eigenanalyisis solutions is
established and incorperated with the assumed hybrid-stress finite element model to slove the singular
antiplane stress fields in tips of multi-material wedges and junctions. The new model is verified by dealing
with the crack problem in an infinite body. As application, some typical multi-material wedge and junction
configurations are investiged. All the caculations show that present method gives converged solution with
fewer elements, and has a strong applicability.
Key words: Singular stress fields, antiplane, multi-material, eigenanalysis, hybrid element
INTRODUCTION
Junctions of multiple materials are often used to improve the material performance or realize special
function. However, stress singularities can exist in elastic solids when discontinuities are present in the
geometry and/or the material properties of the materials, such as multi-material wedge, bonded and
disbonded muti-material junctions. These materials are often exposed to not only inplane loadings
corresponding to mode I and II fracture, but also antiplane shear loadings corresponding to mode III
fracture.
In this paper, antiplane problems are mainly paid attention to. Singular stresses are often assumed to have
the asymptotic form: τ (r ,θ ) = kr λ τ% (θ ) ，in which (r ,θ ) are the polar coordinates originated at the
singular point, λ is the eigenvalues, τ% (θ ) gives the angular variation and k is stress intensity
coefficients depending on the given loadings and global geometries of the structure．Some analytical
methods can be used to solve singular antiplane stress fields of typical geometrcal discontinuity [1-7].
However, owing to the mathemetical complexity, certain practical problems can only be solved in terms of
numerical methods. Conventional finite element models with refined meshes near the singular point can be
used to predict singualr stresses, but it has been recognized that the results are not accurate. For this reason,
special elements have been developed over the years for evaluating the singular stresses in the vicinity of
the singular point. Quarter-point elements that are easily implemented by shifting the mid-side nodes were
proposed to include the square-root displacement field at the crack front [8, 9] . A triangular element
incorporating the correct order of singularity originating at one node of the element was created by Tracey
and Cook [10]. A variable-order singular finite element to simulate the variable orders of singularity by
adjusting the location of the mid-side nodes was proposed by Lim and Kim [11]. However, the accuarcy of
the obtained singular stresses depend sensitively on the asymptotic behaviour realised in the FEM solutions,
⎯ 590 ⎯
and solving this problem is not possible by means of finite element only. Moreover, the above metioned
techniques fail to some degree more general stress singularity problems. It is recommendable that a kind of
wedge-tip element based on the finite element eigenanalysis method has been developed to determine the
eigen-solutions (i.e., the eigenvalues and angular variaitions of field components) of muti-material wedges
and junctions by Pageau et al [12], which opens applications to many beyond those few cases for which
analytical fields are available and overcome the shortcommings of above mentioned singular elements.
Recently, a kind of ad hoc finite element eigenanalysis method has been derivated by the authors to solve
the eigen-solusions [13], which needs fewer elements and yields more accurate results than the original
eigenanalysis method [12]. To finially determine the singular stresses, the stress intensity coefficents k’s
must be decided. In this paper a kind of wedge-tip element based on the eigenanalyisis solutions [13] is
established and incorperated with the assumed hybrid-stress finite element model to solve k’s in the
multi-material wedge and junction configurations. The new model will be validated by the well known
crack problems in infinite body. As applications, some typical multi-material wedge and junction
configurations are also investiged.
GENERALIZED HELLINGER-REISSNER VARIATIONAL PRINCIPLE
y
∂Ωn
Ω2
α
Ω1
α2
nm
r
θ
x
o
α1
R
Figure 1: The wedge-tip domain of a bi-material wedge
The wedge-tip domain shown in Fig. 1 is considered. In terms of linear elastic theory, the straindisplacement relations are
γ (x, y )=Dm w( x, y )
(1)
in which (x, y ) denotes the electro-elastic field components in Cartesian coordinate systems.
γ (x, y )={γ xz , γ yz }T is a column vectors composed of shear strain components γ xz and γ yz . w( x, y ) is
the out-of plane displacement. Dm is a matrix composed of differential operators, i.e.,
⎡ ∂ / ∂x ⎤
Dm = ⎢
⎥
⎣∂ / ∂y ⎦
(2)
The linear elastic constitutive equations can be collected as follows:
τ (x, y ) = C γ (x, y )
(3)
where τ (x, y )={τ xz , τ yz }T is a column vectors composed of shear stress components τ xz and τ yz . C is
the elastic matrix containing the elasticity constants.
It is assumed that inside the elastic body there are no body forces, thus the stress equilibrium equation
read:
DTm τ = 0
(4)
⎯ 591 ⎯
One of the boundary conditions at the outward boundary of the wedge-tip domain is
nm τ = 0
(5)
in which n m = [nx
n y ] is the unit normal vector of the outward domain boundary. nx and n y are the
direction cosines of n m with respect to x and y axes respectively. Another boundary condition at the
outward boundary is
w( x, y ) = w( x, y )
(6)
herein the component with upper bar is one prescribed on the boundary.
With Eqs. (1-6), a generalized Hellinger-Reissner variational principle for antiplane elastic body can be
obtained as
Π = ∫ [ τ ( x, y ) T Sτ ( x, y ) + τ ( x, y )T Dm w( x, y )]dV − ∫
Ωn
∂Ωn
⎡⎣{n m τ ( x, y )}T {w( x, y ) − w( x, y )}⎤⎦dS
(7)
in which Ω n denotes the domain of the two-dimensional body, and ∂Ω n denotes the boundary of the
domain.
ASYMPTOTIC SINGULAR DISPLACEMENT AND STRESS FIELDS
The asymptotic singular electro-elastic fields near the singular point can be written as below:
N +M
∑ k rλ
w(r ,θ ) =
n =1
τ (r ,θ ) =
n +1
n
w% n (θ )
(8)
(θ )
(9)
N +M
∑ k r λ τ%
n
n =1
n
n
in which w% (θ ) is the angular variation of out-of-plane displacement. N and M indicate the number of the
imaginary and real eigenvalues truncated. In Eqs. (8, 9), 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.
Based on the ad hoc finite element eigenanalysis method [13], a standard characteristic equation is obained
as
∑ ⎡⎣(δq )
e T
e
(λ 2 P e + λ Q e + R e )q e ⎤⎦ = 0
(10)
in which ‘e’ denotes the element components, q is the eigenvector, meanings of all other symbols are
defined in paper [13]. Though solving the characteristic equation, all the eigenvalues λ and eigenvectors
q can be obtained. Accordingly, the angular variation of displacement w% (θ ) and stress τ% (θ ) are
obtained from these eigensolutions.
In order to examine influences of external loadings and wedge geometry on the singular fields, stress
intensity coefficients k ’s must be solved. In the following, a kind of super wedge-tip element will be
established for the sake of determining the singular fields finally.
ESTABLISHMENT OF THE SUPER WEDGE-TIP ELEMENT MODEL
Sound variational basis and high coarse mesh accuracy of crack-tip and wedge-tip hybrid elements for
inplane problems have been discussed in [14, 15]. Similarly, to evaluate the antiplane singular fields in the
vicinity of the bi-material wedges shown in Fig. 1, a kind of super wedge-tip element, as shown in Fig. 2,
is established based on generalized Hellinger-Reissner variational principle and the computed numerical
eigensolutions from Eq. (10).
⎯ 592 ⎯
7
8
y
9
r
θ
α
o
1
2
6
p
3
(a )
x
7
8
y
r
θ
1
5
x
2
4
6
p
3
(b)
5
4
Figure 2: Definition of singular element: (a) a nine-node super wedge-tip element;
(b) a eight-node super wedge-tip element for bi-material junction;
Through integration by parts and the divergence theorem, The element form of Eq. (7) can be expressed as
the following boundary integration form:
Πe = −
1
n m τ e ( x, y )we ( x, y )ds + ∫ n m τ ( x, y )e we ( x, y )ds
∂Ω n
2 ∫∂Ωn
(11)
in which the element components we ( x, y ) and τ e ( x, y ) can be given according to Eqs. (8, 9) as follows
we ( x, y ) = w(r , θ ) = W e1×(2 N + M ) {k }
(12)
τ e ( x, y ) = [ Ts ] τ (r , θ ) = S e 2×(2 N + M ) {k}
(13)
in which coefficient matrix
{k} = [ k1R , k1I ,L kNR , kNI , k1 ,L , kM ]
T
.
[ Ts ]
is the transformation matrix
between polar coordinates and Cartesian coordinates, i.e.,
⎡cos(θ ) − sin(θ ) ⎤
⎥.
⎣ sin(θ ) cos(θ ) ⎦
[Ts ] = ⎢
W e = ⎣⎡ r λ1R +1w% 1R (θ ) r λ1I +1w% 1I (θ ) L r λNR +1w% NR (θ ) r λ1I +1w% 1I (θ ) r λ1 +1w% 1 (θ ) L r λM +1w% M (θ ) ⎤⎦ ,
% 1R (θ ) r λ1I +1T
% 1I (θ ) L r λNR +1T
% NR (θ ) r λ1I +1T
% 1I (θ ) r λ1 +1T
% 1 (θ ) L r λM +1T
% M (θ ) ⎤ ,.
S e = ⎣⎡ r λ1R +1T
⎦
% is defined during
in which “R” and “I” denote the real parts and imaginary parts respectively. T
derivation
The boundary components we ( x, y ) in Eq. (11) can be expressed in terms of the wedge-tip element nodal
vector W e ( x, y ) as
we ( x, y ) = [ L]W e ( x, y )
(14)
where the matrix [ L] is set up by one-dimensional Lagrangian linear 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 / l )
s / l],
in which s is the distance measured from point p, and l is the length between the two nodes.
Substituting Eqs. (12-14) into Eq. (11) yields
⎯ 593 ⎯
1
e
e
T
T
{k } [ H ] { k } + { k } [ G ] W e ( x , y )
2
in which
1
e
[ H ] = ∫∂Ωn {[nmSe ( x, y)]T W e ( x, y) + [ W e ( x, y)]T nmSe ( x, y)}ds ,
2
Πe = −
[G ]
e
=∫
∂Ωn
(15)
[nm S e ( x, y )]T [ L ] ds .
The stationary value of the functional Π e of Eq. (15) reads
{k} = ([ H ] )−1 [G ]
e
e
W e ( x, y )
(16)
Inserting Eq. (16) into Eq. (15) leads to
1
e
(17)
Π e = W e ( x, y ) T [ K s ] W e ( x, y )
2
e
e
e
e
where [ K s ] = ([G ] )T ([ H ] ) −1 [G ] denotes the element matrix of the super wedge-tip hybrid element for
the elastic wedge.
ESTABLISHMENT OF GLOBAL FINITE ELEMENT EQUATION
A kind of assumed inplane stress hybrid element was derived in Paper [16]. In the same way, the assumed
antiplane stress hybrid element is established as the global elements, and the element displacement and
assumed element stresses are obtained as
⎡1 0 a1η
w ( x, y ) = [ N g ] [qg ] , τ = ⎢
⎣0 1 b1η
e
e
⎧β ⎫
a3ξ ⎤ ⎪ 1 ⎪
⎨ M ⎬ = [ P ]{β }
b3ξ ⎥⎦ ⎪ ⎪
⎩β4 ⎭
(18)
in which [ N g ] is the interpolation function matrix, and [qg ] is the nodal displacement vector. In order to
assemble with the super wedge-tip hybrid element, 4-node quadrilateral elements are used.
After invoking Eq. (18) into Eq. (7) the elementwise functional becomes
1
T
T
Π e = − {β } [H g ]{β } + {β } [Gg ][qg ] − Load terms
2
in which
Hg = ∫
V
[ P ] S [ P ] dV , G g = ∫V [ P ] Dm [ N g ]dV ,
T
e
(19)
T
e
in which V e denotes the domain of two-dimensional body. As the stationary condition of Π e , we have
1
[ H g ]{β } = [Gg ][qg ] and thus Π e = [qg ]T ⎡⎣ K g ⎤⎦ [qg ]
2
T
−1
in which ⎡⎣ K g ⎤⎦ = [Gg ] [ H g ] [Gg ] the generalized element matrix.
Finally, element matrix of the wedge-tip element ⎡⎣ K g ⎤⎦ is incorporated with element matrix ⎡⎣ K g ⎤⎦ of the
global elements, and the final finite element algebraic equations is expressed as
[ K ] W ( x, y ) = { F }
(20)
where [K] is the assembled element matrix. W ( x, y ) denotes the global displacement vector including all
the nodes components and { F } denotes the global external mechanical loading vector. By solving Eq.
(20), the global displacement W ( x, y ) is determined. Consequently, the coefficients k ’s can be retrieved
by Eq. (16). To satisfy the LBB conditions (see Tong et al. [16]), the number of stress and electric
displacement parameters should be chosen greater than or equal to the total number of DOF minus the
⎯ 594 ⎯
rigid body modes of the hybrid element. In the antiplane problem, there is only one rigid body mode.
NUMERICAL EXAMPLES
In this section, several numerical examples are conducted to demonstrate the validity and versatility of our
proposed super wedge-tip element.
1. Bimaterial interfacial crack As shown in Fig. 3, an interfacial crack exists between two bonded
dissimilar elastic layers. Longitudinal shear stresses are applied at the layer surfaces. Because of symmetry,
only the right half of the infinite plate is analyzed.
y
τ0
h2
Mat . 2
−2 a
x
h1
Mat .1
τ0
Figure 3: Geometry for modeling the interfacial crack in two bonded material layers
For the crack problems (notch angle α → 0 ), the mode III stress intensity factor (SIF) is expressed as
K III = lim 2r τ θ z (r , θ )
r →0
(21)
θ =0
in which τ θ z (r , θ ) can be calculated from Eq. (9).
To evaluate the accuracy of present model, a central crack in an infinite plate ( h1 =h 2 → ∞ ) is firstly
analyzed. Three kind of meshes with one crack-tip element (wedge angle α = 0o ) and different number of
4-node quadrilateral elements are investigated respectively. With the increase of 4-node elements, the size
of the crack-tip element is also reduced correspondingly. The computed results are listed in Table 1. It can
be seen the refinement mesh with 404 4-node elements gives the most accurate solutions. On the other
hand, the dimension of coefficient {k} ’s influence the convergence effects remarkably. Dim ({k} )=6 leads
to higher errors than Dim ( {k} )=8 and Dim ( {k} )=10, which can be explained by the LBB condition, i.e.,
Dim ({k} ) should be equal to or larger than 8. Dim ({k} )=8 seems to be the best choice in present case,
and it is needless to increase the number of Dim ({k} ) to 10. It can be seen present results are very close to
the exact solution ( K III /(τ 0 a ) =1.0) with only 404 elements.
Table 1 K III /(τ 0 a ) for the central cracks in infinite panels, h2/h1=1, h=20a
4-node Elem.
Dim ( {k} )=6
Dim ( {k} )=8
Dim ( {k} )=10
80
1.01042
1.00518
1.00589
108
1.00722
1.00132
1.00348
404
1.00504
1.00000
1.00005
Exact
1.00000
The SIFs are inevitably influenced by layer thicknesses and material constants. With h1/a=1, the influences
of h2/h1 to SIFs is shown in Fig. 4(a), it can be seen that the SIFs decrease with the increase of h2/h1. The
⎯ 595 ⎯
ratio of material properties G2/G1 also influences the SIFs. As h2/h1<1, the lower G2/G1 leads to the
increase of SIF, while it is opposite as h2/h1>1. With h1/a=2, the influences of h2/h1 to SIFs are shown in
Fig. 4(b), compared with Fig. 4(a), it can be seen that the increase of h1 leads to the decrease of SIFs.
G 2 / G1 = 0.1
G 2 / G1 = 0.1
G 2 / G1 = 0.01
G 2 / G1 = 10
G 2 / G1 = 0.01
G 2 / G1 = 10
(b)
(a)
Figure 4: Normalized SIF, K III / τ 0 a , versus h2/h1 with (a) h1/a=1 and (b) h1/a=2
2. Crack terminating at bimaterial interface Fig. 5 shows a bimaterial junction with a crack
perpendicular with and terminating at the interface. Due to different material properties, the loadings are
not added uniformly across the layer surface. To fulfill the shear strain continuity, τ 0(1) / τ 0(2) = G1 / G 2 has
been satisfied artificially. A nine-node crack-tip element is used around the right hand size crack tip. The
results of SIFs are shown in Fig. 6. As G2/G1=1, the SIFs reach their lowest values regardless of the ratio of
h2/h1. As G2/G1>1, the SIFs present a stronger increase with the increase of G2/G1. On the other hand, the
decrease of h2/h1 leads to a slight increase of SIFs.
y
τ 01
τ 02
h2
Mat .1
h1
−2 a
x Mat . 2
τ 02
τ 01
Figure 5: Geometry for modeling the crack terminating at the bimaterial interface
h 2 /h1 =0.25
h 2 /h1 =0.5
h 2 /h1 =1
h 2 /h1 =2
Figure 6: Normalized SIF, K III / τ 0 a , versus G2/G1 with h1/a=1 for the crack terminating at the interface
⎯ 596 ⎯
3. Single-edge notch in a bimaterial layer As shown in Fig. 7, single-edge notch exists at the free-edge
of a bimaterial layer. To investigate the singular stress states near the notch tip, a nine-node wedge-tip
element is used here. For the wedge and junction problems, following Tan and Meguid [17], we use the
following expression of stress intensity parameter (SIP) to describe the singular stress fields:
τ θ zL =
0
1
L0
∫
L0
0
τ θ z (r , θ ) dr
(22)
τ0
y
−a
Mat . 2
x
Mat .1
h2
h1
τ0
Figure 7: Geometry for modeling the single-edge notch problem in a bimaterial layer
G 2 / G1 = 10
G 2 / G1 = 1
G 2 / G1 = 0.1
Figure 8: The SIP, τ zL 0 , versus angle α with h1 → ∞ and h 2 → ∞
in which L0 is a characteristic length which is determined experimentally for each bimaterial systerm.
The influence of north angle α (= 180o − α 2 ) and G2/G1 on SIP is plotted in Fig. 8. It can be seen that the
SIPs increase with the increase of north angle α . On the other hand, the increase of G2/G1 leads to the
increase of SIPs. As notch angle α = 180o , all SIPs converged to the same value regardless the ratio of
G2/G1. On the contrary, as notch angle α = 120o , the results of SIPs strongly depend on the ratio of G2/G1,
i.e., the increase of G2/G1 causes remarkably increase of SIPs.
4. Fully bonded bimaterial junction A kind of fully bonded bimaterial junction is shown in Fig. 9. As
discussed in example 3, τ 0(1) / τ 0(2) = G1 / G 2 has been satisfied to fulfill the shear strain continuity for the
interface OA. τ 0 across the upper layer surface is set to be equal to the smaller one of τ 0(1) and τ 0(2) . A
eight-node singular element as shown in Fig. 2(b) is used around point o, which can be considered as the
assembly of wedge-tip elements. The influences of layer thicknesses to the stress intensity coefficients k’s
are shown in Fig. 10. As the ratio of material properties G2/G1>1, coefficients k decrease with the increase
of h2/h1. On the other hand, as G2/G1<1, coefficient k’s increase with the increase of h2/h1. It is noted that
all k tend to be constant values with the increase of h2/h1.
⎯ 597 ⎯
τ0
y
h2
o
h1
Mat .1
τ 01
A
x
Mat . 2
τ 02
Figure 9: Geometry for modeling the fully bonded bimaterial junction
Figure 10: Stress intensity coefficients, k , versus h 2 /h1 with h1/a=1 for the bimaterial junction
CONCLUSIONS
A kind of wedge-tip element based on the eigenanalysis method is derived to analysis the singular
antiplane stress fields in tips of multi-material wedges and junctions. According to the numerical examples,
it is proved that present method converges to exact solution with less elements. Most importantly, it has a
strong applicability, thus can be used to deal with various kinds of muti-material wedges and junctions.
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.0350062 are gratefully acknowledged.
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