1

Stress Field Around Hole Under Antiplane Shear Using Null-field Integral Equation

Reporter: An-Chien WuDate: 2005/05/19Place: HR2 ROOM 306

Jeng-Tzong Chen, Wen-Cheng Shen and An-Chien Wu

2

Outlines

Motivation and literature review Formulation of the problem Method of solution Adaptive observer system Linear algebraic system Numerical examples Conclusions

3

Outlines

Motivation and literature review Formulation of the problem Method of solution Adaptive observer system Linear algebraic system Numerical examples Conclusions

4

Motivation and literature review

In this paper, we derive the null-field integral equation for a medium containing circular cavities with arbitrary radii and positions under uniform remote shear.

The solution is formulated in a manner of a semi-analytical form since error purely attributes to the truncation of Fourier series.

To search a systematic method for multiple circular holes is not trivial.

5

Motivation and literature review

Honein et al.﹝1992﹞- Mobius transformations involving the complex potential.

Bird and Steele﹝1992﹞- Using a Fourier series procedure to solve the antiplane elacticity problems in Honein’s paper.

Chou﹝1997﹞- The complex variable boundary element method.

Ang and Kang﹝2000﹞- The complex variable boundary element method.

6

Outlines

Motivation and literature review Formulation of the problem Method of solution Adaptive observer system Linear algebraic system Numerical examples Conclusions

7

Formulation of the problem

The antiplane deformation is defined

For a linear elastic body, the stress components

The equilibrium equation can be simplified

then, we have

Consider an infinite medium subject to N traction-free circular holes

The medium is under antiplane shear at infinityor equivalently under the displacement

Bounded by contour

as the displacement field:

are

8

Formulation of the problem

Let the total stress field in the medium be decomposed into and the total displacement

can be given as

The problem converts into the solution of the Laplacesubject to the following

Neumann boundary condition

where the unit outward normal vector on the hole is

problem for :

9

Formulation of the problem

10

Outlines

Motivation and literature review Formulation of the problem Method of solution Adaptive observer system Linear algebraic system Numerical examples Conclusions

11

Boundary integral equation and null-field integral equation

2 ( ) ( , ) ( ) ( ) ( , ) ( ) ( ),B B

u x T s x u s dB s U s x t s dB s x Dp = - Îò ò

0 ( , ) ( ) ( ) ( , ) ( ) ( ), e

B BT s x u s dB s U s x t s dB s x D= - Îò ò

x

Dx

x

D

x

where

( , ) ln lnU s x x s r= - =

( , )( , )

s

U s xT s x

n

¶=

¶

( )( )

s

u st s

n

¶=

¶

eD

eD

12

Boundary integral equation and null-field integral equation

-10 -8 -6 -4 -2 0 2 4 6 8 10-10

-8

-6

-4

-2

0

2

4

6

8

10

-10 -8 -6 -4 -2 0 2 4 6 8 10-10

-8

-6

-4

-2

0

2

4

6

8

10

o

( , )s R q=

R

( , )x r f=

iU

eU

r

13

Outlines

Motivation and literature review Formulation of the problem Method of solution Adaptive observer system Linear algebraic system Numerical examples Conclusions

14

Adaptive observer system

1 1( , )r f

collocation collocation pointpoint

15

Outlines

Motivation and literature review Formulation of the problem Method of solution Adaptive observer system Linear algebraic system Numerical examples Conclusions

16

Linear algebraic system

By collocating the null-field point, we have

, where

and N is the number of circular holes.

17

Flowchart of present method

0 [ ( , ) ( ) ( , ) ( )] ( )BT s x u s U s x t s dB s= -ò

Degenerate Degenerate kernelkernel

Fourier Fourier seriesseries

Null-field Null-field equationequation

Algebraic systemAlgebraic system Fourier Fourier CoefficientsCoefficients

PotentiPotentialal

AnalyticAnalyticalal

NumericNumericalal

18

Collocation points

01

01

( ) ( cos sin )

( ) ( cos sin )

M

n nn

M

n nn

u s a a n b n

t s p p n q n

q q

q q

=

=

= + +

= + +

å

å

2M+12M+1 unknown Fourier unknown Fourier coefficientscoefficients

collocation collocation pointpoint

19

Outlines

Motivation and literature review Formulation of the problem Method of solution Adaptive observer system Linear algebraic system Numerical examples Conclusions

20

Numerical examples

Case1: Two circular holes whose centers located at the axis

The stress around the hole of radius with various values of

Honein’s data

21

Numerical examples

Case2: Two circular holes whose centers located at the axis

The stress around the hole of radius with various values of

Honein’s data

22

Numerical examples

Case3: Two holes lie on the line making 45 degree joining the two centers making 45 degree

The stress around the hole of radius with various values of

Honein’s data

23

Numerical examples

Case4: Two circular holes touching to each other

The stress around the hole of radius Honein’s data

24

Numerical examples

Case5: Three holes whose centers located at the axis

around the hole of radius using the present formulation

25

Numerical examples

Case6: Three holes whose centers located at the axis

around the hole of radius using the present formulation

26

Numerical examples

Case7: Three holes whose centers located at the line making 45 degree

around the hole of radius using the present formulation

27

Outlines

Motivation and literature review Formulation of the problem Method of solution Adaptive observer system Linear algebraic system Numerical examples Conclusions

28

Conclusions

A semi-analytical formulation for multiple arbitrary circular holes using degenerate kernels and Fourier series in an adaptive observer system was developed.

Regardless of the number of circles, the proposed method has great accuracy and generality.

Through the solution for three circular holes, we claimed that our method was successfully applied to multiple circular cavities.

Our method presented here can be applied to problems which satisfy the Laplace equation.

The proposed formulation has been generalized to multiple cavities in a straightforward way without any difficulty.

29

The end

Thanks for your kind attentions.