Post on 05-Jan-2016
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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
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Outlines
Motivation and literature review Formulation of the problem Method of solution Adaptive observer system Linear algebraic system Numerical examples Conclusions
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Outlines
Motivation and literature review Formulation of the problem Method of solution Adaptive observer system Linear algebraic system Numerical examples Conclusions
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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.
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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.
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Outlines
Motivation and literature review Formulation of the problem Method of solution Adaptive observer system Linear algebraic system Numerical examples Conclusions
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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
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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 :
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Formulation of the problem
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Outlines
Motivation and literature review Formulation of the problem Method of solution Adaptive observer system Linear algebraic system Numerical examples Conclusions
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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
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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
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Outlines
Motivation and literature review Formulation of the problem Method of solution Adaptive observer system Linear algebraic system Numerical examples Conclusions
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Adaptive observer system
1 1( , )r f
collocation collocation pointpoint
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Outlines
Motivation and literature review Formulation of the problem Method of solution Adaptive observer system Linear algebraic system Numerical examples Conclusions
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Linear algebraic system
By collocating the null-field point, we have
, where
and N is the number of circular holes.
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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
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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
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Outlines
Motivation and literature review Formulation of the problem Method of solution Adaptive observer system Linear algebraic system Numerical examples Conclusions
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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
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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
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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
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Numerical examples
Case4: Two circular holes touching to each other
The stress around the hole of radius Honein’s data
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Numerical examples
Case5: Three holes whose centers located at the axis
around the hole of radius using the present formulation
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Numerical examples
Case6: Three holes whose centers located at the axis
around the hole of radius using the present formulation
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Numerical examples
Case7: Three holes whose centers located at the line making 45 degree
around the hole of radius using the present formulation
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Outlines
Motivation and literature review Formulation of the problem Method of solution Adaptive observer system Linear algebraic system Numerical examples Conclusions
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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.
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The end
Thanks for your kind attentions.