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

1. 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 subject to the following problem for : 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. Degenerate kernel Fourier series Potential Null-field equation Algebraic system Fourier Coefficients Analytical Numerical Flowchart of present method

18. Collocation points 2M+1unknown 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. The end Thanks for your kind attentions.