1. Reporter: An-Chien Wu Date: 2005/05/19 Place: HR2 ROOM 306
Stress Field Around Hole Under Antiplane Shear Using Null-field Integral Equation
Jeng-Tzong Chen, Wen-Cheng Shen and An-Chien Wu
Reporter: An-Chien Wu
Date: 2005/05/19
Place: HR2 ROOM 306

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
as the displacement field:
For a linear elastic body, the stress components
are
The equilibrium equation can be simplified
then, we have
Consider an infinite medium subject to N traction-free circular holes
Bounded by contour
The medium is under antiplane shear at infinity
or equivalently under the displacement

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 Laplace
problem for :
subject to the following
Neumann boundary condition
where the unit outward normal vector on the hole is

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
where

12. Boundary integral equation and null-field integral equation

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
collocation point

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
Analytical
Degenerate kernel
Fourier series
Potential
Null-field equation
Numerical
Algebraic system
Fourier Coefficients

18. 2M+1 unknown Fourier coefficients
Collocation points
2M+1 unknown Fourier coefficients
collocation point

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. Thanks for your kind attentions.
The end
Thanks for your kind attentions.