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Boussinesq's problem for a rigid cone

  • Ian N. Sneddon (a1)

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1. The problem of determining the distribution of stress in a semi-infinite solid medium when its plane boundary is deformed by the pressure against it of a perfectly rigid cone is of considerable importance in various branches of applied mechanics. It arises in soil mechanics where the cone is the base of a conical-headed cylindrical pillar and the semi-infinite medium is the soil upon which it rests (1). In this instance the distribution of stress in the soil is known to be more or less similar to that calculated on the assumption that the soil is a perfectly elastic, isotropic and homogeneous medium, at least if the factor of safety of a mass of soil with respect to failure by plastic flow exceeds a value of about three (2). The same problem occurs in the theory of indentation tests in which a ductile material is indected by cylindrical punches with conical heads (3).

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(1)Krynine, D. F.Soil mechanics (New York, 1941), Chap. iv.
(2)Terzaghi, K.Theoretical soil mechanics (New York, 1943), p. 367.
(3)Bishop, R. F., Hill, R. and Mott, N. F.Proc. Phys. Soc. 57 (1945), 147.
(4)v. Kármán, Th. and Duwez, P.Comptes rendus du 6econgres de mecan. appl. (in course of publication).
(5)Prandtl, L.Z. angew. Math. Mech. 3 (1923), 6, 401.
(6)Hill, R., Lee, E. H. and Tupper, S. J.Proc. Roy. Soc. A, 188 (1947), 273.
(7)Boussinesq, J.Applications des potentiels (Paris, 1885).
(8)Love, A. E. H.Quart. J. Math. 10 (1939), 161.
(9)Harding, J. W. and Sneddon, I. N.Proc. Cambridge Phil. Soc. 41 (1945), 16.
(10)Sneddon, I. N.Proc. Cambridge Phil. Soc. 42 (1946), 29.
(11)Watson, G. N.The theory of Bessel functions (2nd ed., Cambridge, 1944).
(12)Sneddon, I. N.Proc. Roy. Soc. A, 187 (1946), 229.
(13)Rayleigh, Lord. Phil. Mag. (6), 11 (1906), 283.
(14)Hencky, H.Z. angew. Math. Mech. 4 (1924), 323.
For further references see Nadai, A.Handbuch der Physik, 6, p. 470.

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