The stress state of an elastic composite cone with centre of rotation at the apex

Abstract

The torsion of a composite cone that has a centre of rotation at its apex is investigated in a spherical system of coordinates. A composite cone is a cone with one shear modulus, inserted into a conical funnel having another shear modulus and with ideal mechanical contact between its surface and the inner surface of the conical funnel. The auxiliary problem of a composite cone with its apex truncated by a spherical surface is considered first. The outer surface of such a conical body is not loaded, but a load that reduces to a torque is applied to its spherical surface. The auxiliary problem is reduced to a one-dimensional discontinuous boundary-value problem using a specially constructed integral transformation. The exact solution of this boundary-value problem is constructed. The limit is then taken in the solution obtained as the radius of the spherical surface tends to zero for the purpose of obtaining an exact solution of the problem of the torsion of a composite cone that has a centre of rotation at the apex.