COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Special Hybrid Multilayer Finite Elements for 3-D Stress Analyses
around Quasi-Elliptic Hole in Laminated Composites
Z. S. Tian 1*, Q. P. Yang 1, Z. R. Tian 2
1
2
Department of Mechanics, Graduate School, Chinese Academy of Sciences, PO Box 2706, Beijing, 100080 China
Department of Mathematics, Northwestern Polytechnical University, Xi’an, Shaanxi, 710072 China
Email: zxq@mail.etp.ac.cn
Abstract The combination of two kinds of 3-dimensional special assumed stress hybrid elements — one contains a
traction-free cylindrical surface and another contains a traction-free planar surface — has been used for efficient
analysis of stress distributions around quasi-elliptic hole in thin to thick laminated composites.
Key words: laminated composites, special hybrid finite element, quasi-elliptic hole, 3-D stresses analyses
INTRODUCTION
For laminated composites it is necessary to pay extra attention since the interlayer delamination failure is the special
and common failure mode for these composites. When there are many cutouts, it may not only cause severe
cross-sectional warping but also cause high local stress and greatly enhance the possibility of delamination failure.
Therefore studying the stress distributions around the cutouts, analyzing the failure mechanism and ensuring the
structural components safety have important practical sense.
Since there is no analytical solution in closed-form available to predict the interlayer stresses, various finite element
methods have been suggested to analyze the problem. Numerical results show that, in general, the convergence rate for
the finite element method is dominated by the nature of the solution in the vicinity of steep stress gradient. The regular
high-accuracy element obtained by using polynomials of high order as interpolating functions cannot improve the rate
of convergence [1-5]. Unless using extremely fine meshes, it will be very difficult to obtain the reasonable stresses in
the narrow rim zone of the hole.
One objective of present study is to develop a special three-dimensional element which contains a traction-free
cylindrical surface. Another objective is to study the stress distribution of laminated composites with quasi-elliptic
hole.
THE ELEMENT STIFFNESS MATRIX
The present element is based on a modified complementary energy principle [6]. The energy functional for multiplayer
element can be expressed as
⎧1
Π mc (σ , u% ) = ∑∑ ⎨ ∫
n
i ⎩2
vn i
σ iT Sσ i dV − ∫
∂ vn i
T iT u% i dS + ∫
sσ n i
⎫
TT u% i dS⎬
⎭
(1)
where σ i stand for stresses of layer i; S : elasticity material property matrix; Vni : volume of layer i for element n; T i:
surface tractions of layer i; u% i : surface displacements of layer i; ∂Vni : boundary of layer i for element n; T : prescribed
tractions; Sσ ni : portion of ∂Vni over which tractions are prescribed.
The stresses in the layer i are assumed in terms of finite number of stress parameters β in the form
σ i = Pi βi
(2)
⎯ 722 ⎯
where P i satisfies the homogeneous equilibrium conditions, the tractions on the interlayer surface and the boundary
conditions on the traction-free surface. The corresponding boundary tractions T i for the layer i are determined by
T i = ν T σ i = ν T P i β i = Ri β i
(3)
where ν is matrix of direction cosine; R i = ν T P i.
The displacements u% i in the layer i are interpolated in terms of the nodal displacements qi.
u% i = Li q i
(4)
In this way the displacement continuity on the interlayer surface is maintained.
Substituting Eq. (2) to (4) into Eq. (1) from the variation of Πmc the element stiffness matrix is obtained.
K = G T H −1G
(5)
where
⎡H 1
⎤
⎢
⎥
⎢ H2
⎥
⎢
⎥
H =⎢
⎥
O
⎢
⎥
⎢
⎥
⎢ (Diag)
⎥
⎢⎣
H k ⎥⎦
Gi =
∫
∂
Hi =
∫P
iT
⎧ G1 ⎫
⎪ 2⎪
⎪G ⎪
G = ⎨ ⎬ ( i = 1,…, k)
⎪M ⎪
⎪G k ⎪
⎩ ⎭
Si P i d V
Vn i
(6)
R i T Li d S
Vn i
k is the total layer number of the element.
ζ
z
z
F
B
y
η
A G
Layer2
Layer1
C
D
o
1
y
x
o
(a) Geometry of the element
6
5
4
H
ξ
3
8
E
7
z
2
2hi
a
θ2
r
θ1
θ
x
(b) One layer of the element
Figure 1: Geometry of the special element with a traction-free cylindrical surface
FINITE ELEMENT FORMALATION
The laminated element with a traction-free cylindrical surface is shown in Fig. 1. The stress assumption of the typical
layer i is chosen as:
⎛
σ ri = ⎜1 −
⎝
⎛ a4 ⎞ i
a 2 ⎞ i ⎛ 4a 2 3a 4 ⎞ i
i
i
⎡
⎤
+
−
+
+
+
1
cos
2
sin
2
r
β
β
θ
β
θ
⎟ 1 ⎜
⎜ 1 − 4 ⎟ ⎡⎣ β 4 cosθ + β 5 sinθ ⎤⎦
3
2
4 ⎟⎣ 2
⎦
r2 ⎠
r
r
r
⎝
⎠
⎝
⎠
⎧⎪⎛ a 2 ⎞
⎛ 5a 4 4 a 6 ⎞
+ r ⎜1 − 4 + 6 ⎟⎟ ⎡⎣ β 6i cos3θ + β 7i sin3θ ⎤⎦ + ( di + hiζ ) ⎨⎜1 − 2 ⎟ β8i
r
r ⎠
⎝
⎩⎪⎝ r ⎠
2
4
4
⎛ 4a
⎛ a ⎞
3a ⎞
+ ⎜ 1 − 2 + 4 ⎟ ⎡⎣ β 9i cos 2θ + β10i sin 2θ ⎤⎦ + r ⎜ 1 − 4 ⎟ ⎡⎣ β11i cosθ + β12i sinθ ⎤⎦
r
r ⎠
⎝
⎝ r ⎠
⎯ 723 ⎯
⎫⎪
⎛ 5a 4 4 a 6 ⎞
+ r ⎜ 1 − 4 + 6 ⎟ ⎣⎡ β13i cos3θ + β14i sin3θ ⎦⎤ ⎬
r
r ⎠
⎝
⎭⎪
2
4
⎛ a ⎞
⎛ 3a ⎞
⎛
a4
σ θi = ⎜ 1 + 2 ⎟ β1i − ⎜ 1 + 4 ⎟ ⎡⎣ β 2i cos 2θ + β 3i sin 2θ ⎤⎦ + r ⎜ 3 + 4
r ⎠
r
⎝ r ⎠
⎝
⎝
⎞ i
i
⎟ ⎡⎣ β 4 cosθ + β 5 sinθ ⎤⎦
⎠
⎧⎪⎛ a 2 ⎞
⎛ a 4 4a 6 ⎞
− r ⎜ 1 − 4 + 6 ⎟ ⎡⎣ β 6i cos3θ + β 7i sin3θ ⎤⎦ + ( d i + hiζ ) ⎨⎜1 + 2 ⎟ β8i
r ⎠
r ⎠
⎝ r
⎩⎪⎝
⎛ 3a 4 ⎞
⎛
⎛ a 4 4a 6 ⎞
a4 ⎞
⎪⎫
− ⎜1 + 4 ⎟ ⎡⎣ β 9i cos 2θ + β10i sin 2θ ⎤⎦ + r ⎜ 3 + 4 ⎟ ⎡⎣ β11i cosθ + β12i sinθ ⎤⎦ − r ⎜ 1 − 4 + 6 ⎟ ⎡⎣ β13i cos3θ + β14i sin3θ ⎤⎦ ⎬
r ⎠
r ⎠
r ⎠
⎪⎭
⎝
⎝
⎝ r
⎛
(7)
⎞
i
i
⎟ ⎡⎣ − β 4 sinθ + β 5 cosθ ⎤⎦
⎝
⎠
4
6
⎧⎪⎛ 2a 2 3a 4 ⎞
⎛ 3a
4a ⎞
+ r ⎜1 + 4 − 6 ⎟ ⎡⎣ − β 6i sin3θ + β 7i cos3θ ⎤⎦ + ( d i + hiζ ) ⎨⎜ 1 + 2 − 4 ⎟
r
r ⎠
r
r ⎠
⎝
⎩⎪⎝
4
⎛ a ⎞
⎡⎣ − β 9i sin 2θ + β10i cos 2θ ⎤⎦ − r ⎜ 1 − 4 ⎟ ⎡⎣ − β11i sinθ + β12i cosθ ⎤⎦
⎝ r ⎠
τ ri θ = ⎜1 +
2a 2 3a 4
− 4
r2
r
⎞
⎛ a4
i
i
⎡
⎤
β
sin
θ
β
cos
θ
r
−
+
−
2
2
⎟⎣ 2
3
⎦ ⎜1 − r 4
⎠
⎝
⎫⎪ ⎛ a 2 ⎞
⎛ 3a 4 4a 6 ⎞
+ r ⎜ 1 + 4 − 6 ⎟ ⎡⎣ − β13i sin3θ + β14i cos3θ ⎤⎦ ⎬ − r ⎜ 1 − 2 ⎟ β15i
r
r ⎠
⎝
⎭⎪ ⎝ r ⎠
⎛ a4 ⎞
⎛ a4 ⎞
⎛ a6 ⎞
− r 2 ⎜ 1 − 4 ⎟ ⎣⎡ β16i cosθ + β17i sinθ ⎦⎤ − r ⎜ 1 − 4 ⎟ ⎣⎡ β18i cos 2θ + β19i sin 2θ ⎦⎤ − r 2 ⎜ 1 − 6 ⎟ ⎡⎣ β 20i cos3θ + β 21i sin3θ ⎤⎦
⎝ r ⎠
⎝ r ⎠
⎝ r ⎠
⎧⎛
⎞
4 ⎛ a4 ⎞
⎟⎟ − β 16i sinθ + β 17i cosθ + ⎜⎜1 + 4 ⎟⎟ − β 18i sin 2θ + β 19i cos 2θ
r⎝
r ⎠
⎠
⎩⎝
⎫⎪
⎛ a6 ⎞
h
+9 ⎜ 1 + 6 ⎟ ⎡⎣ − β 20i sin3θ + β 21i cos3θ ⎤⎦ ⎬ + σ z(i −1) ζ =+1 − i (1 + ζ ) τ rz(i −1) ζ =+1
r ⎠
r
⎝
⎭⎪
σ zi = − hi2 (1 + ζ )2 ⎨⎜⎜ 3 +
a4
r4
[
]
[
(
− hi (1 + ζ )
(
∂ τ rz(i −1)
∂r
⎪⎧⎛
τ θi z = hi (1 + ζ ) ⎨⎜ 1 −
⎩⎪⎝
ζ =+1
) − 1 (1 + ζ ) ∂ (τ
r
(i −1)
ζ =+1
θz
]
)
)
∂θ
⎛
a2 ⎞ i
a4 ⎞ i
i
i
⎡
⎤
r
cos
sin
+
+
+
+
β
4
β
θ
β
θ
3
[β18cos 2θ + β19i sin2θ ]
⎟ 15
17
4 ⎟
⎣ 16
⎦ ⎜
r2 ⎠
r
⎝
⎠
⎞ i
⎪⎫ (i −1)
i
⎟ ⎡⎣ β 20 cos3θ + β 21sin3θ ⎤⎦ ⎬ + τ θz ζ =+1
⎠
⎭⎪
4
⎛ a4 ⎞
⎪⎧ ⎛ a ⎞
τ ri z = hi (1 + ζ ) ⎨r ⎜1 − 4 ⎟ ⎣⎡ − β16i sinθ + β17i cosθ ⎦⎤ + 2 ⎜ 1 − 4 ⎟ ⎣⎡ − β18i cos 2θ + β19i sin 2θ ⎦⎤
⎝ r ⎠
⎩⎪ ⎝ r ⎠
⎛ a6 ⎞
⎪⎫
+3r ⎜ 1 − 6 ⎟ ⎣⎡ − β 20i sin3θ + β 21i cos3θ ⎦⎤ ⎬ + τ rz(i −1) ζ =+1
⎝ r ⎠
⎭⎪
where
⎛
a
+2r ⎜ 2 + 6
r
⎝
6
di =
1
( zi + zi −1 )
2
hi =
1
( zi − zi −1 )
2
(8)
di is the co-ordinate z of midsurface of layer i; hi is the half-thickness of layer i (Fig. 1).
The stress assumption satisfies the following conditions exactly:
(1) The homogeneous equilibrium equations in each layer.
(2) The continuity of transverse stresses σ z , τ rz and τ θ z in the z direction within the entire laminate (while the in-plane
stresses σ r , σ θ and τ rθ are allowed to be discontinuous in the z direction at the interlayer surface, as it is usually
expected in an anisotropic laminate).
⎯ 724 ⎯
(3) The traction-free boundary condition over the cylindrical surface.
(4) The stresses σ z , τ rz and τ θ z assumed to be zero at the bottom surface of the element (The point differs from above
three points, since this point is neither the necessary condition of the variational principle nor the necessary condition
of the element.).
NUMERICAL EXAMPLES
Figure 2: Top view of rectangular laminate [+45/-45]s with a quasi-elliptic hole (θ = tg-1y/x)
Figure 3: Present mesh pattern for 1/8 of the laminate
The example is a 4-layer [+45/-45]s cross ply square laminate including a quasi-elliptic hole under uniform tension σ 0 .
The radius of half-circular = a, side length of the plate = 12a, thickness of each layer = 0.4a. The top view is shown in
Fig. 2. The material properties are given by EL/ET=17.5, GLT/ET=0.69, GZT/ET=0.38, vLT=0.28, vZT=0.33.
Because of symmetry, the analysis is carried out only one-eighth of the plate. The present mesh pattern is shown in Fig.
3. The computed stresses are obtained by combination of present special elements around the circular, the special
elements each of them contains a traction-free planar surfaces [4] along the straight side of the hole and the ordinary
assumed stress multiplayer elements [5] in the remaining regions.
Present elements
Referernce solution
3
Present elements
Reference solution
5
4
σθ/σ0
σθ/σ0
2
1
3
2
1
0
0
o
15
o
30
o
45
o
60 o
75 o
θ
90
o
(z = 2h)
0
0o
15 o
45 o
30 o
(z = h)
Figure 4: σ θ distributions along the rim of the hole
⎯ 725 ⎯
60 o
75 o
θ
90
o
The computed distributions of the normalized stresses σ θ , σ z , τ θ z along the rim of the hole at the interface of layer are
shown in Fig. 4 and Fig. 5 respectively. The distributions of normalized stresses σ r and τ rθ , τ rz and τ θ z along radial
direction (θ = 67.5˚) at the interface of layer are shown in Fig. 6 and Fig. 7. Apparently, there are no analytical
solutions for the problem. The values obtained by using ordinary 3-dimensional 8-node multilayer assumed
displacement anisotropic element [7] are considered as the references which are also shown in the figures for
comparison.
Present elements (σz)
Present elements (σz)
0.3
Present elements (τθz)
Reference solution (τθz)
Present elements (τθz)
τθz
σz
σz/σ0, τθz/σ0
σz/σ0, τθz/σ0
0.2
Reference solution (σz)
0.9
Reference solution (σz)
0.1
0.0
Reference solution (τθz)
0.6
0.3
σz
-0.1
τθz
-0.2
0o
15o
30o
45o
0.0
60 o
75o
θo
0o
90
15 o
(z = 2h)
30
o
45 o
60 o
θ
75 o
90
o
(z = h)
Figure 5: σ z and τ θ z distributions along the rim of the hole
0.8
Present element (σr)
Reference solution (σr)
Reference solution (τrθ)
σr
σr/σ0, τrθ/σ0
σr/σ0, τrθ/σ0
0.4
0.0
τrθ
0.0
-0.3
Present elements (σr)
-0.6
-0.4
2.60
3.35
4.10
τrθ
Reference solution (σr)
Present elements (τrθ)
Reference solution (τrθ)
r/a
1.85
σr
0.3
Present element (τrθ)
-0.9
1.85
4.85
2.60
(z = 2h)
r/a
3.35
4.10
4.85
(z = h)
Figure 6: σ r and τ rθ distributions along the radial direction (θ = 67.5º)
0.4
Present elements (τrz)
0.2
Reference solution (τrz)
Reference solution (τrz)
Present elements (τθz)
Present elements (τθz)
Reference solution (τθz)
Reference solution (τθz)
0.1
τrz
0.0
-0.1
1.85
Present elements (τrz)
τθz
0.2
τrz/σ0, τθz/σ0
τrz/σ0, τθz/σ0
0.3
τθz
r/a
2.60
3.35
4.10
4.85
(z = 2h)
0.1
τrz
0.0
-0.1
r/a
2.25
3.00
3.75
(z = h)
Figure 7: τ rz and τ θ z distributions along the radial direction (θ = 67.5º)
⎯ 726 ⎯
4.50
Figure 8: Sub-structuring technique of the laminate
It can be seen that the present results are very close to the references. The computed results also show that the
maximum stress σ θmax / σ 0 = 4.73 (at θ = 70.38˚) which is quite close to the reference value σ θmax / σ 0 = 4.95 (at
θ=70.38˚). However, these references are obtained by the use of sub-structuring technique with three steps as shown in
Figure 8 when the mesh is refined [7]. The degree-of-freedom (dof) used for each step is 779, 1924 and 5128
respectively and the total dof is 7831. The problem is analyzed by using only one mesh pattern for present method
(Figure 3). The total dof is 439, which is 1/18 of the multilayer assumed displacement anisotropic solid element. It
shows that the use of the present special elements to solve the stress concentration problems in a laminate is extremely
efficient.
CONCLUSIONS
The combination of the special element with a traction-free cylindrical surface and the special element with a
traction-free planar surface is applicable to efficient analyses of 3-dimensional stresses around some cutouts in
laminated composites — such as U-shaped notches, semi-circular holes etc.
Acknowledgements
This work was supported by the Knowledge Innovation Program of Chinese Academy of Sciences.
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⎯ 727 ⎯