Therefore, a lot of researhes investigate type of singular element

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July20-25,2014,Barelona,Spain

STUDY ON EFFECT OF THREE DIMENSIONAL AKIN

SINGULAR ELEMENT FOR STRESS ANALYSIS OF

DISSIMILAR MATERIAL JOINTS

T.Kurahashi 1

, Y.Watanabe 2

, T.Kondo 3

and H.Koguhi 4

1 4

Departmentof Mehanial Engineering,Nagaoka UniversityofTehnology,Kamitomioka

1603-1, Nagaokam Niigata 940-2188,JAPAN,kurahashimeh.nagaokaut.a.jp

2

Eletrial andMehanial SystemsEngineeringAdvanedCourse,AdvanedCourse of

Nagaoka NationalCollege ofTehnology,Nishikatakai888,Nagaoka,Niigata 940-8532,

JAPAN,a25819xst.nagaoka-t .a .jp

3

Department of MehanialEngineering,Nagaoka National College ofTehnology,

Nishikatakai888,Nagaoka,Niigata 940-8532, JAPAN, tkondonagaoka-t.a.jp

Key words: Three-dimensional Akin Singular Element, Stress Analysis, Stress Singu-

larity, Dissimilar Material Joints, Order of Singularity, Finite Element Eigen Analysis

1 Introdution

In our researh group, stress singular analysis is arried out based on element free

Galerkin method [1℄, nite element method [2℄ and boundary element method [3℄. In

ase that stress intensity fator around rak tip or near vertex on interfae of dissim-

ilar material joints is obtained based on methods by numerial analysis, a lot of nodes

should be prepared around rak tip or near vertex on interfae of dissimilar material

joints. Therefore, a lot of researhes investigate type of singular element. Guzina et. al.

investigate dierene of stress intensity fator 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 boundaryelementanalysis [5℄. Inboth referene, better solutionisobtained by using

singularelementin omparison with ase using onventional element.

Inase ofnite elementanalysis, Meshiiet. al. and Akinhaveproposedsingularelement

around rak tip[6℄,[7℄. Moreover, Georgiou et. al. have investigated singularelement in

niteelementowanalysis[8℄. Inallofpapers,targetmodelis2Dmodel,anditisdiÆult

to nd tehnial papers for appliation of singular element for 3D model. In this study,

we fous on singular element proposed by Akin, and extend the element to 3D model,

examine appliability of the element to 3D dissimilarmaterial joint model. In addition,

(2)

model[9℄, itis diÆultto analitiallyobtainorder of singularityin 3D model. Therefore,

it is neessary to numerially obtain order of singularity in ase of 3D model. In this

study, numerial proedure shown in referenes[10℄, [11℄ is applied to obtain 3D order of

singularity inthis study. Moreover, welarifyvalidityof Akin singularelementextended

to3D modelby mathematialformulation.

2 Shape funtion for onventional and Akin singular elements in FEM

In ase that nite element analysis is arried out, shape funtion in linear tetrahedron

element iswritten as Eq.(1).

8

>

<

>

: N

1

=1

N

2

=

N

3

=

N

4

=

(1)

Here, N

1 , N

2 , N

3

and N

4

denote shapefuntion, and , and indiatevolume oordi-

nate. There is harateristi inshape funtionsuh that summation of shape funtion is

equal to one. In 1976, Akin proposed speial singular element onsidering stress distri-

butionin singularity eldunder harateristi ofshape funtion. Shapefuntion inAkin

singularelementis denoted as Eq.(2).

8

>

<

>

: SN

1

=1 1 N

1 (;;)

R(;;)

SN

2

=

N2(;;)

R(;;)

SN

3

= N

3 (;;)

R(;;)

SN

4

= N

4 (;;)

R(;;)

(2)

Here, Eq.(2) isdened suh that origin of - oordinate system is orrespondingto sin-

gularitypoint,and funtionR iswrittenas Eq.(3). InEq.(3), parameter

vertex

indiates

order of singularity atvertex.

R (;; )=(1 N

1 )

vertex

=(++ ) vertex

(3)

Here, alulating derivative of shape funtion SN

1 , SN

2 , SN

3

and SN

4

with respet to

x, y and z, Eq.(4) is obtained.

8

<

: SNi

x

SNi

y

SN

i

z 9

=

;

= 2

6

4 x

y

z

x

y

z

x

y

z

3

7

5 1

8

<

: SN

i

SN

i

SN

i

9

=

;

(4)

Final form of right hand side vetor is expressed as Eqs. Eq.(5) - Eq.(7). Eq.(4) is

applied to elements inluding singular point and derivative of shape funtion for on-

ventional linear tetrahedron element is applied to the other elements. Comparison of

(3)

distribution of shape funtion between linear tetrahedron and Akin singularelements in

ase of

vertex

=0.50 is shown inFig.1.

8

>

<

>

: SN1

SN2

SN3

SN4

9

>

=

>

;

= 8

>

<

>

:

(1

vertex )

N1

(1 N

1 )

vertex

N2

(1 N

1 )

vertex

+

vertex N1

N

2

(1 N

1 )

(1+

vertex )

N3

(1 N

1 )

vertex

+

vertex N1

N

3

(1 N

1 )

(1+

vertex )

N4

(1 N

1 )

vertex

+

vertex N1

N

4

(1 N

1 )

(1+vertex) 9

>

=

>

;

(5)

8

>

<

>

: SN

1

SN2

SN

3

SN

4

9

>

=

>

;

= 8

>

<

>

:

(1

vertex )

N

1

(1 N

1 )

vertex

N2

(1 N

1 )

vertex

+

vertex N1

N

2

(1 N

1 )

(1+vertex)

N

3

(1 N

1 )

vertex

+

vertex N

1

N

3

(1 N

1 )

(1+vertex)

N

4

(1 N

1 )

vertex

+

vertex N

1

N

4

(1 N

1 )

(1+vertex) 9

>

=

>

;

(6)

8

>

<

>

: SN1

SN2

SN3

SN4

9

>

=

>

;

= 8

>

<

>

:

(1

vertex )

N1

(1 N

1 )

vertex

N2

(1 N

1 )

vertex

+

vertex N1

N

2

(1 N

1 )

(1+vertex)

N3

(1 N

1 )

vertex

+

vertex N1

N

3

(1 N

1 )

(1+

vertex )

N4

(1 N

1 )

vertex

+

vertex N1

N

4

(1 N

1 )

(1+

vertex )

9

>

=

>

;

(7)

3 Mathematial Proof for Akin Singular Element Extended to 3D Model

UsingshapefuntionofAkinsingularelement,Unknownphysialvariableuisexpressed

as Eq.(8).

u(;; ) = SN

1 u

1 +SN

2 u

2 +SN

3 u

3 +SN

4 u

4

= 1

1 N

1

(;; )

R (;; ) u

1 +

N

2

(;; )

R (;; ) u

2 +

N

3

(;; )

R (;; ) u

3 +

N

4

(;; )

R (;; ) u

4

= u

1 +(u

2 u

1 )

(++ ) vertex

+ (u

3 u

1 )

(++ )

vertex +(u

4 u

1 )

(++ )

vertex

(8)

Representing ;; by spherial oordinatesystem, Eq.(8) is written asEq.(9)(Fig.2).

u(;; ) = u

1 +

(u

2 u

1 )

sin()os()

s(;)

vertex

+ (u

3 u

1 )

sin()sin()

s(;) vertex

+(u

4 u

1 )

os()

s(;) vertex

r 1

vertex

= u

1 +f

1 (;)r

1 vertex

(9)

where s(;) indiates (sinos +sinsin+os).

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X Y Z

0.95 0.85 0.75 0.65 0.55 0.45 0.35 0.25 0.15 0.05

(a)N

1

X Y Z

0.95 0.85 0.75 0.65 0.55 0.45 0.35 0.25 0.15 0.05

(b) SN

1 (

vertex

=0.50)

X Y Z

0.95 0.85 0.75 0.65 0.55 0.45 0.35 0.25 0.15 0.05

(a)N

2

X Y Z

0.95 0.85 0.75 0.65 0.55 0.45 0.35 0.25 0.15 0.05

(b) SN

2 (

vertex

=0.50)

X Y Z

0.95 0.85 0.75 0.65 0.55 0.45 0.35 0.25 0.15 0.05

(a)N

3

X Y Z

0.95 0.85 0.75 0.65 0.55 0.45 0.35 0.25 0.15 0.05

(b) SN

3 (

vertex

=0.50)

X Y Z

0.95 0.85 0.75 0.65 0.55 0.45 0.35 0.25 0.15 0.05

(a)N

4

X Y Z

0.95 0.85 0.75 0.65 0.55 0.45 0.35 0.25 0.15 0.05

(b) SN

4 (

vertex

=0.50)

Figure 1: Comparison of distributionof shape funtion between lineartetrahedron (N

1 -N

4

) and Akin

singularelements(SN

1 -SN

4 )

(5)

Figure2: Spherialoordinatesystem

In addition, Representing oordinate x by spherial oordinate system, Eq.(10) is ob-

tained.

x(;; ) = N

1 x

1 +N

2 x

2 +N

3 x

3 +N

4 x

4

= x

1 +(x

2 x

1

)+(x

3 x

1

)+(x

4 x

1 )

= x

1 +

(x

2 x

1

)sinos +(x

3 x

1

)sinsin+(x

4 x

1 )os

r

= x

1 +f

2

(;)r (10)

wheref

2

(;) denotes ((x

2 x

1

)sinos +(x

3 x

1

)sinsin+(x

4 x

1

)os ). Aording

toEq.(10), Eq.(11), isobtained.

f

2

(;) = 1

r

(x(;; ) x

1

); r = 1

f

2 (;)

(x(;; ) x

1

) (11)

Consequently, strain

zz

isdenoted asEq.(12).

zz

=

u(r;;)

z

= u

r r

z +

u

z +

u

z

= (1

vertex )f

1 (;)r

vertex 1

f

2 (;)

+ f

1 (;)

r

1 vertex 1

f

2 (;)

r

+ f

1 (;)

r

1 vertex 1

f

2 (;)

r

=

(1

vertex )f

1 (;)

1

f

2 (;)

+ f

1 (;)

1

f2(;)

+

f

1 (;)

1

f2(;)

r

vertex

= C(;)r

vertex

(12)

Therefore, it is found that relationship,

ij

;

ij /r

vertex

, isobtained.

(6)

4 Numerial Experiments

Analysisofstresssingularityeldforaluminiumandmildsteelbondedjointmodelshown

in Fig.3 is arried out. Material properties are shown in Tab.1 In this study, hanging

elementsizenear singularitypoint,relationshipbetween minimumelementsizeandorder

of singularity

vertex

or intensity of stress singularity K

1zz

is investigated. Total num-

ber of nodes and elements for eah ase is shown in Tab.2. In Tab.2, hmin indiates

harateristi minimum mesh size. hmin is alutated by hmin = (Vmin) (1=3)

=

((1=6)xminyminzmin) (1=3)

, ,and Vmin and x

i

min(i=1,2,3) represent min-

imum mesh volume and minimum mesh size for eah diretion. In addition, order of

singularity

vertex

near singularity point is obtained as shown in Tab.3 based on nite

elementeigen analysisfororder ofsingularity in3Dmodel[10℄,[11℄. Detailofthis proe-

dure is shown inAppendix. Stress analysis is arried out for eah minimum mesh size

Figure 3: Finiteelementmodelandboundaryondition

Table1: Materialproperties

Material Young'smodulus(GPa) Poisson's ratio

Mild steel (material1) 216.00 0.30

Aluminium(material2) 69.09 0.33

near singularity pointshown in Tab.2, and relationshipbetween minimummesh size and

order of singularity

vertex

and intensity of stress singularity K

1zz

is investigated based

on alulation results by least square method using tting equation

zz

= K

1zz r

vertex

.

Distributionof

zz

nearsingularitypointforeahminimummeshsizeareshowninFigs.5-

7, and plotted value is stress value for radius r diretion from singular point O on line,

=90deg. and =45deg. square point indiates result by using Akin singular element,

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Figure 4: Setofmeshsize ofAkinsingularelement

Table 2: Numberofnodesandelementsforeahase

Case xmin=ymin(mm) zmin(mm) hmin(mm) Nodes Elements

1 0.0120 0.03125 0.01145 51,303 234,000

2 0.0166 0.03125 0.01430 27,881 126,144

3 0.0300 0.12500 0.03347 6,773 29,760

Table3: Orderofsingularity

Charateristi rootp

vertex

Orderof singularity

vertex (

vertex

=1-p

vertex )

0.879 0.121

(8)

and irle point indiates result by normal element. In all results, it is seen that stress

value near singularpointobtained by Akin singularelementishigherthan that obtained

bynormalelement. Inaddition,itisfound thattheresultinase1islosetothatinase

2,butsmallstress valueisobtainedevenif Akinsingularelementisusedinaseofourse

meshes suh as ase 3. Moreover, lines shown in Figs.5-7 indiate tting urve by equa-

tion

zz

=K

1zz r

vertex

, and relationship between eah minimum mesh size and order of

singularity

vertex

orintensityofstresssingularityK

1zz

isshown inFig.8andTab.4. From

Fig.8, it is found that order of stress singularity

vertex

in ase of Akin singular element

islose tosolutionobtainedbynite elementeigen analysisfor orderof singularityin3D

model, omparing to that in ase of normal element. Moreover, from Tab.4, it is seen

that intensity of stress singularity K

1zz

obtained by normal element is higher than that

using Akin singular element. This results denotes that if singularelement is not applied

to stress analysis, the intensity of stress singularity is exessively evaluated. Therefore,

it an be said that the singular element should be appliedto evaluate intensity of stress

singularity appropriately.

10 12 14 16 18 20 22 24

0.001 0.01 0.1 1

Stress "Sigma_zz",MPa

Distance from point O "r",mm Akin singular elements without Akin singuler elements Akin (kij=12.7931595 lamda=0.0923196003) without Akin (kij=13.4285822 lamda=0.0770154893)

Figure 5: Comparisonofdistributionof stress

zz

from singularpointO (=90deg.,=45deg.) in ase

ofhmin=0.01145mm

Table 4: Intensityof stresssingularityK

1zz

foreah ase

Case without Akinsingular element with singularelement

1 (hmin=0.01145mm) 13.43 MPamm 0:077

12.79MPamm 0:092

2 (hmin=0.01430mm) 12.41 MPamm 0:096

11.89MPamm 0:111

3 (hmin=0.03347mm) 11.71 MPamm 0:045

11.21MPamm 0:064

(9)

10 12 14 16 18 20 22 24

0.001 0.01 0.1 1

zz

ofhmin=0.01430mm

10 12 14 16 18 20 22 24

0.001 0.01 0.1 1

zz

ofhmin=0.03347mm

(10)

0 0.05 0.1 0.15 0.2 0.25 0.3

0.01 0.1

Order of singularity

Characteristic minimum mesh size, mm Akin element normal element Numerical solution by Finite Element Eigen Analysis

Figure8: RelationshipbetweenminimummeshsizeandorderofsingularityinaseofAkinandnormal

elements

5 Conlusions

In this paper, a singularity element proposed by Akin was extended to 3D model using

orderofsingularity

vertex

obtainedbynite elementeigenanalysis,and thiselementwas

appliedtoobtainintensity ofstress singularity nearvertex oninterfae edgeof dissimilar

materialjoints. Asthe omputationalmodel,aluminium-mildsteel bondedstruturewas

employed, and eet of the singular element was investigated by hanging the minimum

mesh size near vertex on interfae. Conlusions inthis study are shown as follows.

Stress value near singular point obtained by Akin singular element is higher than

that obtained by normalelement.

Order of stress singularity

vertex

in ase of Akin singular element is lose to solu-

tion obtained by nite element eigen analysis for order of singularity in 3D model,

omparing tothat in ase of normalelement.

Ifsingularelementisnotappliedtostressanalysis,theintensityofstresssingularity

is exessively evaluated.

Aknowledgement

This work was supported by Grant-in-Aid for Young Sientists (B) (No. 25820015).

We wish to thank you sta of researh institute for information tehnology at Kyushu

university for use of super omputer system, FUJITSU PRIMERGY CX400.

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