Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-27T09:51:13.111Z Has data issue: false hasContentIssue false
  • English
  • Français

Degenerate Scale Problem for a Hypocycloid Hole in an Infinite Plate in Plane Elasticity

Published online by Cambridge University Press:  06 June 2016

Y.-Z. Chen
Y.-Z. Chen*
Affiliation:
Division of Engineering Mechanics Jiangsu University Jiangsu, China
*
*Corresponding author (chens@ujs.edu.cn)
Get access

Abstract

Based on the conformal mapping, this paper provides a closed form solution for the degenerate scale of the hypocycloid hole in plane elasticity. In the derivation, we assume the vanishing displacements along the boundary in the degenerate scale problem. Some functions in the boundary condition are decomposed into three parts with particular behavior. Even the displacements are vanishing along the boundary of an exterior region, the displacements and stresses are not equal to zero in the exterior region. This is a particular feature in the degenerate scale problem.

Keywords

Degenerate scale problemConformal mapping techniqueClosed form solution
Type
Technical Note
Information
Journal of Mechanics , Volume 32 , Issue 4 , August 2016 , pp. N7 - N10
Copyright
Copyright © The Society of Theoretical and Applied Mechanics 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Cheng, A. H. D. and Cheng, D. S., “Heritage and Early History of the Boundary Element Method,” Engineering Analysis with Boundary Elements, 29, pp. 286–302 (2005).CrossRefGoogle Scholar
2. Chen, J. T. and Chen, Y. W., “Dual Boundary Element Analysis Using Complex Variables for Potential Problems with or without a Degenerate Boundary,” Engineering Analysis with Boundary Elements, 24, pp. 671–684 (2000).Google Scholar
3. Chen, J. T., Kuo, S. R. and Lin, J. H., “Analytical Study and Numerical Experiments for Degenerate Scale Problems in the Boundary Element Method of Two-dimensional Elasticity,” International Journal for Numerical Methods in Engineering, 54, pp. 1669–1681 Google Scholar
4. Chen, J. T. and Wu, A. C., “Null-Field Approach for the Multi-Inclusion Problem Under Antiplane Shears,” Journal of Applied Mechanics, 74, pp. 469–487 (2007).Google Scholar
5. Chen, J. T. and Li, Y. T., “Torsional Rigidity of a Circular Bar with Multiple Circular Cnclusions Using the Null-field Integral Approach,” Computational. Mechanics, 44, pp. 221–232 (2009).CrossRefGoogle Scholar
6. Dong, C. Y. and Lee, K. Y., “A New Integral Equation Formulation of Two-dimensional Inclusion–crack Problems,” International Journal of Solids and Structures, 42, pp. 5010–5020 (2005).Google Scholar
7. He, W. J., Ding, H. J. and Hu, H. C., “Degenerate Scales and Boundary Element Analysis of Two Dimensional Potential and Elasticity Problems,” Computers and Structures, 60, pp. 155–158 (1996)Google Scholar
8. Vodicka, R. and Mantic, V., “On Invertibility of Elastic Single-layer Potential Operator,” Journal of Elasticity, 74, pp. 147–173 (2004).Google Scholar
9. Vodicka, R. and Mantic, V., “On Solvability of a Boundary Integral Equation of the First Kind for Dirichlet Boundary Value Problems in Plane Elasticity,” Computational. Mechanics, 41, pp. 817–826 (2008).Google Scholar
10. Chen, J. T., Lin, J. H., Kuo, S. and Chiu, Y. P., “Analytical Study and Numerical Experiments for Degenerate Scale Problems in Boundary Element Method Using Degenerate Kernels and Circulants,” Engineering Analysis with Boundary Elements, 25, pp. 819–828 (2001).Google Scholar
11. Chen, J. T., Kuo, S. R., Kao, S. K. and Jian, J., “Revisit of a Degenerate Scale: A Semi-Circular Disk,” Journal of Computational and Applied Mathematics, 283, pp. 182–200 (2015).Google Scholar
12. Chen, J. T., Lee, C. F., Chen, I. L. and Lin, J. H., “An Alternative Method for Degenerate Scale Problems in Boundary Element Methods for the Two-dimensional Laplace Equation,” Engineering Analysis with Boundary Elements, 26, pp. 559–569 (2002).CrossRefGoogle Scholar
13. Chen, J. T., Lin, S. R. and Chen, K. H., “Degenerate Scale Problem When Solving Laplace's Equation by BEM and Its Treatment,” International Journal for Numerical Methods in Engineering, 62, pp. 233–261 (2005).Google Scholar
14. Chen, Y. Z., Lin, X. Y. and Wang, Z. X., “Numerical Solution for Degenerate Scale Problem for Exterior Multiply Connected Region,” Engineering Analysis with Boundary Elements, 33, pp. 1316–1321 (2009).Google Scholar
15. Chen, J. T., Lee, Y. T., Ku, S. R. and Chen, Y. W., “Analytical Derivation and Numerical Experiments of Degenerate Scale for an Ellipse in BEM,” Engineering Analysis with Boundary Elements, 36, pp. 1397–1405 (2012).Google Scholar
16. Kuo, S. R., Chen, J. T. and Kao, S. K., “Linkage Between the Unit Logarithmic Capacity in the Theory of Complex Variables and the Degenerate Scale in the BEM/BIEMS,” Applied Mathematics Letters, 26, pp. 929–938 (2013).Google Scholar
17. Kuo, S. R., Chen, J. T., Lee, J. W. and Chen, Y. W., “Analytical Derivation and Numerical Experiments of Degenerate Scales for Regular N-Gon Domains in Two-dimensional Laplace Problems,” Applied Mathematics and Computation, 219, pp. 5668–5683 (2013).Google Scholar
18. Muskhelishvili, N. I., Some Basic Problems of the Mathematical Theory of Elasticity, Noordhoff, Groningen (1963).Google Scholar