Elasticity_ Barber

460 30 The boundary-value problem
Using this representation for ϕ and substituting into equations (30.1, 30.2),
we find that
−
∫ ∞
0
∫ ∞
0
ξA(ξ)J 0 (ξr)dξ = − µ
(1 − ν) u 0(r) ; 0 ≤ r < a (30.5)
ξ 2 A(ξ)J(ξr)dξ = 0 ; r > a . (30.6)
These constitute a pair of dual integral equations for the function A(ξ).
Sneddon used the method of Titchmarsh 3 and Busbridge 4 to reduce equations
of this type to a single equation, but a more recent solution by Sneddon 5 and
formalized by Gladwell 6 effects this reduction more efficiently for the classes
of equation considered here.
30.2 Collins’ Method
A related method, which has the advantage of yielding a single integral equation
in elementary functions directly, was introduced by Green and Zerna 7
and applied to a wide range of axisymmetric boundary-value problems by
Collins 8 .
30.2.1 Indentation by a flat punch
To introduce Collins’ method, we first examine the simpler problem in which
the punch is flat and hence u 0 (r) is a constant. This was first solved by
Boussinesq in the 1880s.
A particularly elegant solution was developed by Love 9 , using a series
of complex harmonic potential functions generated from the real Legendre
polynomial solutions of Chapter 24 by substituting (z + ıa) for z. This is
tantamount to putting the origin at the ‘imaginary’ point (0, −ıa). The real
and imaginary parts of the resulting functions are separately harmonic and
have discontinuities at r =a on the plane z =0.
3 E.C.Titchmarsh, Introduction to the Theory of Fourier Integrals, Clarendon Press,
Oxford, 1937.
4 I.W.Busbridge, Dual integral equations, Proceedings of the London Mathematical
Society, Ser.2 Vol. 44 (1938), pp.115–129
5 I.N.Sneddon, The elementary solution of dual integral equations, Proceedings of
the Glasgow Mathematical Association, Vol. 4 (1960), pp.108–110.
6 G.M.L.Gladwell, loc. cit., Chapters 5,10.
7 A.E.Green and W.Zerna, loc. cit..
8 W.D.Collins, On the solution of some axisymmetric boundary-value problems
by means of integral equations, II: Further problems for a circular disc and a
spherical cap, Mathematika, Vol. 6 (1959), pp.120–133.
9 A.E.H.Love, Boussinesq’s problem for a rigid cone, Quarterly Journal of Mathematics,
Vol. 10 (1939), pp.161–175.