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

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

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

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.

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.

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

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

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

Formulation of the problem

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

Boundary integral equation and null-field integral equation where

Boundary integral equation and null-field integral equation

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

Adaptive observer system collocation point

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

Linear algebraic system By collocating the null-field point, we have , where and N is the number of circular holes.

Flowchart of present method Analytical Degenerate kernel Fourier series Potential Null-field equation Numerical Algebraic system Fourier Coefficients

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

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

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

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

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

Numerical examples Case4: Two circular holes touching to each other The stress around the hole of radius Honein’s data

Numerical examples Case5: Three holes whose centers located at the axis around the hole of radius using the present formulation

Numerical examples Case6: Three holes whose centers located at the axis around the hole of radius using the present formulation

Numerical examples Case7: Three holes whose centers located at the line making 45 degree around the hole of radius using the present formulation

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

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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