Boussinesq's classical solution for the problem of the normal loading of the surface of a homogeneous elastic half-space is extended to cover the case in which the half-space is reinforced with a fully bonded rigid inclusion of finite radius. The problem is reduced to the solution of two coupled Fredholm integral equations of the second kind. Numerical results presented in the paper illustrate the manner in which the displacement of the rigid disk inclusion is influenced by the depth of the embedment/inclusion radius ratio.
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References
1.
[1] Boussinesq, J.: Application des potentials a l'etude de l'equilibre et due mouvement des solides elastique, Gauthier-Villars, Paris, 1885.
2.
[2] Michell, J. H.: Some elementary distributions of stress in three dimensions. Proc. Lond. Math. Soc., B2, 23-35 (1900).
3.
[3] Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity, Cambridge University Press, Cambridge, UK, 1927.
4.
[4] Mindlin, R. D.: Force at a point in the interior of a semi-infinite solid. Physics, 7, 195-202 (1936).
5.
[5] Cerruti, V.: Sulla deformazione di uno strato isotropo indefinite limitato da due piani paralleli. Atti. Acad. Nazl. Lincei. Rend, 4(1), 521-522 (1884).
6.
[6] Mura, T.: Micromechanics of Defects in Solids, Sijthoff and Noordhoff, Martinus Nijhoff Publications, The Hague, the Netherlands, 1981.
7.
[7] Green, A. E. and Zerna, W.: Theoretical Elasticity, Clarendon, Oxford, UK, 1968.
8.
[8] Atkinson, K. E.: A Survey of Numerical Methods for the Solutions of Fredholm Integral Equations of the SecondKind,Soc. Ind. Appl. Math., Philadelphia, 1976.
9.
[9] Baker, C.TH.: The Numerical Treatment of Integral Equations, Clarendon, Oxford, UK, 1977.
10.
[10] Delves, L. M. and Mohamed, J. L.: Computational Methods for Integral Equations, Cambridge University Press, Cambridge, UK, 1985.
11.
[11] Ufliand, I. S.: The contact problem of the theory of elasticity for a die, circular in its plane, in the presence of adhesion. Prikl. Mekh., 20, 578-587 (1956).
12.
[12] Mossakovskii, V. I.: The fundamental mixed boundary value problem of the theory of elasticity for a halfspace with a circular line separating the boundary conditions. Prikl. Math. Mekh., 18, 187-196 (1954).
13.
[13] Gladwell, G.M.L.: Contact Problems in the Classical Theory of Elasticity, Sijthoff and Noordhoff, Aalphen aan den Rijn, the Netherlands, 1980.
14.
[14] Willis, J. R.The penny-shaped crack on an interface. Q.J. Mech. Appl. Math., 25, 367-385 (1972).
15.
[15] Keer, L. M.: Mixed boundary value problems for a penny-shaped cut. J. Elasticity, 5, 89-98 (1975).
16.
[16] England, A. H.: On stress singularities in linear elasticity. Int. J Engng. Sci., 9, 571-585 (1971).
17.
[17] Selvadurai, A.P.S.: Influence of boundary fracture on the elastic stiffness of a deeply embedded anchor plate. Int. J. Num. Analytical Meth. Geomech., 13, 159-170 (1989).
18.
[18] Selvadurai, A.P.S.: Analytical methods for embedded flat anchor problems in geomechanics. Computer Methods and Advances in Geomechanics, 1, 305-321 (1994).
19.
[19] Collins, W. D.: Some axially symmetric stress distributions in elastic solids containing penny-shaped cracks: I. Cracks in an infinite solid and a thick plate. Proc. Roy. Soc., A203, 359-386 (1962).
20.
[20] Kassir, M. K. and Sih, G. C.: Some three-dimensional inclusion problems in elasticity theory. Int. J. Solids Struct., 4, 225-241 (1968).
21.
[21] Selvadurai, A.P.S.: Load-deflexion characteristics of a deep anchor in an elastic medium. Geotechnique, 26, 405-418 (1976).
22.
[22] Selvadurai, A.P.S.: The axial loading of a rigid circular anchor plate embedded in an elastic halfspace. Int. J. Num. Analytical Meth. Geomech., 17, 343-353 (1993).
