Influence of Akin singular element for intensity of stress singularity near interface edge of three
dimensional dissimilar material joints
Yutaro Watanabe Takahiko Kurahashi
Electrical Mechanical Advanced Course, Nagaoka University of Technology,
National Institute of Technology, Nagaoka College, 1603-1 Kamitomiokamachi, Nagaoka, Niigata 940-2188, JPAPN 888 Nishikatakai, Nagaoka Niigata 940-8532, JAPAN [email protected]
[email protected]
Toshimi Kondo Hideo Koguchi
National Institute of Technology, Nagaoka College, Nagaoka University of Technology,
888 Nishikatakai, Nagaoka, Niigata 940-8532, JAPAN 1603-1 Kamitomiokamachi, Nagaoka, Niigata 940-2188, JPAPN [email protected] [email protected]
Abstract— In this study, stress analysis is carried out by FEM with Akin singular element. Two computational models are employed in this study. One is dissimilar material joint model that consists of Mild steel–Aluminium, and the other is dissimilar material joint model that consists of Silicon-Resin. As external force 10 MPa is applied to the upper surface of Mild steel and Silicon respectively. As a result, it is found that stress components σzz obtained by using Akin singular element is higher than that obtained by normal element in both cases. Moreover, it is also found order of singularity with Akin singular element obtained by least square method is closer to order of singularity obtained by the finite element eigen analysis in both cases.
Keywords—FEM; Stress singular field; Akin singular element;
Order of singularity; Intensity of stress singularity
I. INTRODUCTION
In the case of analyzing dissimilar material joints applied external force, stress singular analysis is carried out based on element free Galerkin method [1], finite element method [2]
and boundary element method [3]. In the case that stress intensity factor around crack tip or vertex on interface of dissimilar material joints is obtained by methods based on numerical analysis, a lot of nodes and elements should be prepared around crack tip or near vertex on interface of dissimilar material joints. Therefore, a lot of researchers investigate type of singular element. Guzina et. al. investigated difference of stress intensity factor by order of interpolation and number of nodes in 3D boundary element analysis [4]. In addition, Ong et. al. proposed several types of singular element on singular line and at vertex in potential problems based on 3D boundary element analysis [5]. In both reference, better solution is obtained by using singular element in comparison with using conventional element. In this paper, we carry out stress analysis with singular element proposed by Akin [6]. This singular element was proposed by Akin for obtaining high accurate stress distribution near vertex on interface. However, this singular element is applied to only
two dimensional models. Therefore, we extended this singular element to three dimensions, and the purpose of this research is to investigate the influence of extended Akin singular element.
II. STRESS ANALYSIS BASED ON FEM USING 3D AKIN SINULAR ELMENT
A) Shape function for conventional and Akin singular element
In the case that finite element analysis is carried out, shape function in linear tetrahedron element is defined as
, , , . Here, , and indicatevolume coordinate. In shape function, there is a character that the sum of shape function is one. In 1976, Akin proposed special singular element considering stress distribution in singularity field under characteristic of shape function. Shape function in Akin singular element is shown as Eq.(1).
) 4 , 3 , 2 , 1 ) ( (
) , , 1 (
) (
) , ,
1 1(
1
N i SN SN N
vertex i i
vertex
(1) Calculating derivative of shape function
SN1,
SN2,
SN3and
SN4with respect to x, y and z, Eq.(2) is obtained.
i i i
i i i
SN SN SN
z y x
z y x
z y x
z SN y SN x
SN 1
(2)
Final form of right hand side vector is expressed as Eq.(3)- Eq.(5). Eq.(2) is applied to elements including singular point and derivative of shape function for conventional linear tetrahedron element is applied to the other elements.
Comparison of distribution of shape function between linear
tetrahedron and Akin singular elements in case of λ
vertex=0.50
is shown in Fig.1.
Fig.1 Comparison of distribution of shape function with linear tetrahedron (N
1-N
4) and Akin singular element (SN
1-SN
4)
) 1 ( 1 4 1 1
4
) 1 ( 1 3 1 1
3
) 1 ( 1 2 1 1
2
1 1 1
4 3 2 1
) 1 ( )
1 (
) 1 ( )
1 (
) 1 ( )
1 (
) 1 ( ) 1 (
vertex vertex
vertex
vertex vertex
vertex
vertex vertex
vertex vertex vertex
N N N N N
N N N N N
N N N N N
N N
SN SN SN SN
(3)
) 1 ( 1 4 1 1
4
) 1 ( 1 3 1 1
3
) 1 ( 1 2 1 1
2
1 1 1
4 3 2 1
) 1 ( )
1 (
) 1 ( )
1 (
) 1 ( )
1 (
) 1 ( ) 1 (
vertex vertex
vertex
vertex vertex
vertex
vertex vertex
vertex vertex vertex
N N N N N
N N N N N
N N N N N
N N
SN SN SN SN
(4)
) 1 ( 1 4 1 1
4
) 1 ( 1 3 1 1
3
) 1 ( 1 2 1 1
2
1 1 1
4 3 2 1
) 1 ( )
1 (
) 1 ( )
1 (
) 1 ( )
1 (
) 1 ( ) 1 (
vertex vertex
vertex
vertex vertex
vertex
vertex vertex
vertex vertex vertex
N N N N N
N N N N N
N N N N N
N N
SN SN SN SN
(5)
B)
Derivation of finite element equationMultiplying weighting function {u
*}(x)
T(={u
*(x)
v*(x)
w*(x)}) for Governing equations for elastic deformation analysis in three dimensional model, integrating over the element domain Ω
eand Applying Green’s theorem to Governing equations, Eq.(6) is consequently obtained. Here, [C] consists of differential operators.
e e
d t x u d
C x
u ( )}
T[ ]
T{ } { ( )}
T{ }
{
*
*(6)
Here, {t} indicates traction force. Substituting Hooke’s law {σ}= [D][ε] to Eq.(6), Eq.(7) is obtained.
e e
d t x u d x u C D C x
u( )}T[ ]T[ ][ ]{ ( )} { ( )}T{}
{* *
(7)
Here, applying shape function shown in previous section, interpolation functions of weighting function {u
*(x)} and displacement components {u(x)} for conventional and Akin singular elements are respectively defined as Eq.(8) and (9).
4 4 3 3 2 2 1 1
4 4 3 3 2 2 1 1
4 4 3 3 2 2 1 1
* 4 4
* 3 3
* 2 2
* 1 1
* 4 4
* 3 3
* 2 2
* 1 1
* 4 4
* 3 3
* 2 2
* 1 1
*
*
*
*
) ( ) ( ) ( ) (
) ( ) ( ) ( ) (
) ( ) ( ) ( ) (
) (
) (
) ( )}
( {
) ( ) ( ) ( ) (
) ( ) ( ) ( ) (
) ( ) ( ) ( ) (
) (
) (
) ( )}
( {
w x N w x N w x N w x N
v x N v x N v x N v x N
u x N u x N u x N u x N
x w
x v
x u x u
w x N w x N w x N w x N
v x N v x N v x N v x N
u x N u x N u x N u x N
x w
x v
x u x u
(8)
4 4 3 3 2 2 1 1
4 4 3 3 2 2 1 1
4 4 3 3 2 2 1 1
* 4 4
* 3 3
* 2 2
* 1 1
* 4 4
* 3 3
* 2 2
* 1 1
* 4 4
* 3 3
* 2 2
* 1 1
*
*
*
*
) ( ) ( ) ( ) (
) ( ) ( ) ( ) (
) ( ) ( ) ( ) (
) (
) (
) ( )}
( {
) ( ) ( ) ( ) (
) ( ) ( ) ( ) (
) ( ) ( ) ( ) (
) (
) (
) ( )}
( {
*
*
*
*
w x SN w x SN w x SN w x SN
v x SN v x SN v x SN v x SN
u x SN u x SN u x SN u x SN
x w
x v
x u x u
w x SN w x SN w x SN w x SN
v x SN v x SN v x SN v x SN
u x SN u x SN u x SN u x SN
x w
x v
x u x u
Akin Akin Akin Akin
Akin Akin Akin
Akin
(9)
Substituting Eq. (8) to Eq.(7) for conventional element or Eq.
(9) to Eq.(7) for Akin singular element, Eq.(10) is obtained.
e e
d t u u d N C D C N
u}T [ ][ ]T[ ][ ][ ]T { } { }T {}
{ * *
(10)
Eq.(10) is represented as Eq.(11).
e e
d t u u d k
u}T [ ] {} { }T {}
{ * *
(11) In Eq.(11), finite element equation is finally obtained as Eq.(12) because the weighting function is arbitrary.
e e
d t u d
k] {} {}
[
(12) As numerical integration for Eq.(12), Gauss-Legendre integration is employed, and three integration points are used in computation of this study.
III. EVALUATION OF INTENSITY OF STRESS SINGULARITY
Stress singular analysis for aluminum-mild steel dissimilar
joint model (case A) and Silicon-resin dissimilar joint model
(case B) shown in Fig.2 is carried out. Moreover, Fig.3 shows
finite element model around singular point. Material
properties are shown in Table.1. In this study, by changing
element size near singularity point, relationship between
minimum element size and order of singularity
λvertexis
investigated. Moreover, by changing material properties, the
influence of Akin singular element is investigated. In Fig.4, 5,
6, 7, 8 and 9, Δh
minindicates characteristic minimum mesh
size. Δh
min is calculated by Δhmin = ((1/6)×ΔxminΔy
minΔz
min)
(1/3).
Here, Δi
min(i=x,y,z) represents minimum mesh size for each
direction. In addition, order of singularity
λvertexnear
singularity point is obtained as shown in Table.2 based on
finite element eigen analysis for order of singularity in 3D
model[7].
Fig.2 Computational model
Fig.3 Finite element model around singular point Table.1 Material properties
Material Young’s modulus(GPa) Possion’s ratio
Mild steel(Fe) 216 0.30
Aluminum(Al) 69.0 0.33
Silicon(Si) 166 0.26
Resin(Re) 5.49 0.32
Stress singular analysis is carried out for each minimum mesh size Δh
minnear singularity point.
Table.2 Order of singularity
Characteristic root p Order of singularity λ(λ=1-p)
Pvertex(Fe-Al)=0.879
λvertex(Fe-Al)=0.121
Pvertex(Si-Re)=0.605
λvertex(Si-Re)=0.395 In cases A and B, relationship between minimum mesh size and order of singularity
λvertexand intensity of stress singularity K
1zzis investigated based on calculation results by least square method using fitting equation σ
zz= K
1zzr
-λvertex. In Fig.4 and 5, distribution of
σzznear singularity point for each minimum mesh size Δh
min=0.043mm in case A and distribution of
σzznear singularity point for each minimum mesh size Δh
min=0.008mm in case A are shown respectively.
Square point indicates stress component for
zdirection obtained by using Akin singular element, and circle point indicates stress component for z direction obtained by normal element. In these results of case A, it is found that stress value near singular point obtained by Akin singular element is
higher than by normal element. Line shown in Fig.4 and 5 indicate fitting curve by equation
σzz= K
1zzr
-λvertex. In case A, relationship between each minimum mesh size and order of singularity λ
vertexis shown in Fig.6. From Fig.6, it is found that order of stress singularity
λvertexin case of Akin singular element is close to solution obtained by finite element eigen analysis for order of singularity in 3D model, compared to that in case of normal element. In Fig.7 and 8, distribution of
σzznear singularity point for each minimum mesh size Δh
min=0.0054mm in case B and distribution of
σzznear singularity point for each minimum mesh size Δh
min=0.0050mm in case B are shown respectively. In these results of case B, it is found that stress value near singular point obtained by Akin singular element is higher than by normal element. In case B, relationship between each minimum mesh size and order of singularity λ
vertexis shown in Fig.9. From Fig.9, it is found that order of stress singularity
λvertexin case of Akin singular element is close to solution obtained by finite element eigen analysis for order of singularity in 3D model, compared to that in case of normal element. In addition, it appears that difference of stress values for
z direction between Akinsingular element and normal element become large because meshes are not enough fine in case B. Therefore, it is necessary to do the stress analysis by using much more fine mesh.
Fig.4 Distribution of σ
zzat line at φ =45 [deg]
(Δh
min=4.33×10
-2mm) (Mild steel and Aluminum)
Fig.5 Distribution of σ
zzat line at φ =45 [deg]
(Δh
min=8.04×10
-3mm) (Mild steel and Aluminum) a) Mild steel and
Aluminum(case A)
b) Silicon and Resin (case B)
Expansion
Fig.6 Variation of order of singularity λ
vertexfor each minimum mesh size Δh
min(Mild steel and Aluminum)
IV. CONCLUSION
In this paper, a singularity element proposed by Akin was extended to 3D model using order of singularity
λvertexobtained by finite element eigen analysis, and this element was employed near vertex on interface edge of dissimilar material joint model to obtain intensity of stress singularity.
As the computational model, aluminum-mild steel dissimilar material joint model and silicon-resin dissimilar material joint model were employed, and influence of the singular element was investigated by changing the minimum mesh size near vertex on interface and material properties. Conclusions in this study are shown as follows.
Stress value near singular point obtained by Akin singular element is higher than that obtained by normal element.
Order of stress singularity λ
vertexin case of Akin singular element is close to solution obtained by finite element eigen analysis for order of singularity in 3D model, compared to that in case of normal element.
ACKONWLEGMENT
This work was supported by Grant-in-Aid for Young Scientists (B) (No. 25820015). We wish to thank you staff of research institute for information technology at Kyushu university for use of super computer system, FUJITSU PRIMERGYCX400.
Fig.7 Distribution of σ
zzat line at φ =45 [deg]
(Δh
min=5.43×10
-3mm) (Silicon and Resin)
Fig.8 Distribution of σ
zzat line at φ =45 [deg]
(Δh
min=4.92×10
-3mm) (Silicon and Resin)
Fig.9 Variation of order of singularityλ
vertexfor each minimum mesh size Δh
min(Silicon and Resin)
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