The response of an anisotropic elastic half-space to a rolling cylinder

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Abstract
The problem of the steady motion of a heavy cylinder, of infinite length, over the surface of an elastic half-space has been examined by Craggs and Roberts(1), Clements(2) and Roberts (3). In papers (1) and (2) the surfaces of both the cylinder and the half-space are assumed to be smooth so that the shear stress over the region of contact is zero. With this assumption Craggs and Roberts analysed the motion for an isotropic half-space while Clements considered the case of an anisotropic half-space. In the paper by Roberts (3) various problems concerning the rolling of rigid and non-rigid cylinders over an isotropic elastic half-space are discussed. In this paper we consider a similar class of rolling cylinder problems for an anisotropic half-space. We begin, in section 2, by deriving the relevant basic equations for the stress and displacement while in section 3 some properties of certain constants occurring in the equations of section 2 are derived for use in later sections. The problem of the steady motion of an inflated tyre is discussed in section 4 under the assumptions that the tyre exerts a constant pressure over the contact region which is instantaneously at rest. The solution shows that, with these assumptions, the shear stress is infinite near the ends of the region of contact. This indicates that there will be slipping of the tyre over the half-space at both ends of the contact region. If these zones of slip are large compared with the total contact area then the assumption of a contact region which is instantaneously at rest is invalidated so that the solution may only be regarded as being applicable when the zones of slip are small. An expression to determine the width of the zones of slip is derived and this expression is used, together with the numerical results of section 5, to obtain information about the influence of anisotropy on the width of the zones of slip. In section 6 the problem of the inflated tyre is discussed with allowance made for a zone of slip at both ends of the region of contact. The results of this section could be expected, particularly when the zones of slipping are wide, to more accurately predict the response of the anisotropic half-space to the rolling tyre than the results of section 4. Unfortunately, in order to satisfactorily complete the analysis, it is necessary to restrict the class of anisotropic materials for which the results are applicable. As a consequence, the work of section 4 is in many ways preferable to that of section 6 in indicating the influence which elastic anisotropy has upon the results.

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