Concentrated ring loadings in a full space or half space: solutions for transverse isotropy and isotropy

Abstract

In this analysis solutions for concentrated ring loading in a transversely isotropic full space or half space are found. The elastic field is derived by integrating the known fundamental point force solutions along a circular ring. Three cases of ring loading are considered. Taking the z axis along the material axis (the z = 0 plane is the isotropic plane), the first case studied is when the ring load is in the z direction, referred to as normal loading. The other two cases are shear ring loads directed in the plane of isotropy. In the first instance, the ring load is unidirectional and independently applied in either the x or y direction. The second case considers the axisymmetric radial and axisymmetric torsional ring loads. The solution for ring loading applied to the surface of a half space is first obtained. Subsequently the solutions for ring loading in a full space and buried ring loads in a half space are found. In all cases the elastic displacement and stress fields are evaluated in terms of closed form expressions containing complete elliptic integrals of the first, second and third kinds. An interesting feature of the full space solution is that the potential function and its radial derivatives exhibit a cylindrical discontinuity for negative z values. However, it is shown that these discontinuous functions do provide continuous displacement and stress fields. A limiting form of the solutions for transverse isotropy also provides the corresponding results for isotropic materials.