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Certain two-dimensional mixed boundary-value problems for wedge-shaped regions and dual integral equations

Published online by Cambridge University Press:  20 January 2009

R. P. Srivastav
R. P. Srivastav
Affiliation:
Department of Mathematics, Indian Institute of Technology, Kanpur, U.P., India
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Finding the distribution of stress in earth dams containing cracks is an outstanding problem of soil mechanics. Even the simplest mathematical model, viz., that of a wedge containing a plane crack which is symmetrically situated along the bisector plane of the angle of the wedge, with the plane strain assumption of the infinitesimal theory of elasticity, presents a difficult problem of solving the bi-harmonic equation subject to mixed boundary conditions. While elasticity problems related to wedge-shaped bodies have been investigated, it appears little attention has been paid to the mixed boundary-value problems.As a first step towards the solution of the mixed boundary value problem for the biharmonic equation, we discuss in this paper the solution of Laplace's equation

for wedge-shaped regions subject to mixed type of conditions on the boundary. If we assume that φ does not depend on z, the equation (1.1) is reduced to the equation

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 14 , Issue 4 , December 1965 , pp. 321 - 332
Copyright
Copyright © Edinburgh Mathematical Society 1965

References

REFERENCES

(1) Sneddon, I. N. and Srivastav, R. P.Dual series relations—I, Dual relations involving Fourier Bessel series, Proc. Roy. Soc. Edin. A 66 (1964), 150–160.Google Scholar
(2) Titchmarsh, E. C.Introduction to the Theory of Fourier Integrals (Oxford, 1948), p. 46.Google Scholar