Degenerate Scale Problem in Antiplane Elasticity or Laplace Equation for Contour Shapes of Triangles or Quadrilaterals

CHEN Yi-zhou

doi:10.3879/j.issn.1000-0887.2012.04.010

Volume 33 Issue 4

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CHEN Yi-zhou. Degenerate Scale Problem in Antiplane Elasticity or Laplace Equation for Contour Shapes of Triangles or Quadrilaterals[J]. Applied Mathematics and Mechanics, 2012, 33(4): 500-512. doi: 10.3879/j.issn.1000-0887.2012.04.010

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Degenerate Scale Problem in Antiplane Elasticity or Laplace Equation for Contour Shapes of Triangles or Quadrilaterals

doi: 10.3879/j.issn.1000-0887.2012.04.010

CHEN Yi-zhou

Division of Engineering Mechanics, Jiangsu University, Zhenjiang, Jiangsu 212013, P.R.China

Received Date: 2011-11-24
Rev Recd Date: 2012-01-20
Publish Date: 2012-04-15

Abstract

Abstract

Several solutions of the degenerate scale for shapes of triangles or quadrilaterals in an exterior boundary value problem of antiplane elasticity or Laplace equation were provided. The Schwarz-Christoffel mapping was used thoroughly. It is found that a complex potential with simple form in the mapping plane satisfies the vanishing displacement condition (or w=0) along the boundary of the unit circle when a dimension “R” reaches its critical value 1. This means the degenerate size in the physical plane is also achieved. Finally, the degenerate scales can be evaluated from some particular integrals that depend on some parameters in the mapping function. A lot of numerical results of degenerate sizes for shapes of triangles or quadrilaterals are provided.
- degenerate scale,
- conformal mapping technique,
- shapes of triangles or quadrilaterals,
- antiplane elasticity

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References(17)

References

[1]	Chen J T, Lin J H, Kuo S R, Chiu Y P. Analytical study and numerical experiments for degenerate scale problems in boundary element method using degenerate kernels and circulants[J]. Engineering Analysis With Boundary Elements, 2001, 25(9): 819-828.
[2]	Chen J T, Lee C F, Chen I L, Lin J H. An alternative method for degenerate scale problems in boundary element methods for the two-dimensional Laplace equation[J]. Engineering Analysis With Boundary Elements, 2002, 26(7): 559-569.
[3]	Petrovsky I G. Lectures on Partial Differential Equation[M]. New York: Interscience, 1971.
[4]	Jaswon M A, Symm G T. Integral Equation Methods in Potential Theory and Elastostatics[M]. New York: Academic Press, 1977.
[5]	Christiansen S. On two methods for elimination of non-unique solutions of an integral equation with logarithmic kernel[J]. Applicable Analysis, 1982, 13(1): 1-18.
[6]	Chen J T, Lin S R, Chen K H. Degenerate scale problem when solving Laplace’s equation by BEM and its treatment[J]. International Journal for Numerical Methods in Engineering, 2005, 62(2): 233-261.
[7]	Chen J T, Shen W C. Degenerate scale for multiply connected Laplace problems[J]. Mechanics Research Communication, 2007, 34(1): 69-77.
[8]	Chen Y Z, Lin X Y, Wang Z X. Degenerate scale problem for plane elasticity in a multiply connected region with outer elliptic boundary[J]. Archive Applied Mechanics, 2010, 80(9): 1055-1067.
[9]	He W J, Ding H J, Hu H C. Non-equivalence of the conventional boundary integral formulation and its elimination for plane elasticity problems[J]. Computers and Structures, 1996, 59(6): 1059-1062.
[10]	He W J, Ding H J, Hu H C. Degenerate scales and boundary element analysis of two dimensional potential and elasticity problems[J]. Computers and Structures, 1996, 60(1/3): 155-158.
[11]	Chen J T, Kuo S R, Lin J H. Analytical study and numerical experiments for degenerate scale problems in the boundary element method of two-dimensional elasticity[J]. International Journal for Numerical Methods in Engineering, 2002, 54(12): 1669-1681.
[12]	Vodicka R, Mantic V. On invertibility of elastic single-layer potential operator[J]. Journal of Elasticity, 2004, 74(2): 147-173.
[13]	Vodicka R, Mantic V. On solvability of a boundary integral equation of the first kind for Dirichlet boundary value problems in plane elasticity[J]. Computational Mechanics, 2008, 41(6): 817-826.
[14]	Chen Y Z, Lin X Y, Wang Z X. Evaluation of the degenerate scale for BIE in plane elasticity and antiplane elasticity by using conformal mapping[J]. Engineering Analysis With Boundary Elements, 2009, 33(2): 147-158.
[15]	Chen J T, Wu C S, Chen K H, Lee Y T. Degenerate scale for the analysis of circular thin plate using the boundary integral equation method and boundary element methods[J]. Computational Mechanics, 2006, 36(1): 33-49.
[16]	Driscoll T A, Trefethen L N.Schwarz-Christoffel Mapping[M]. London, New York: Cambridge University Press, 2002.
[17]	Chen Y Z, Hasebe N, Lee K Y. Multiple Crack Problems in Elasticity[M]. Southampton: WIT Press, 2003.