The axisymmetric shrink fit problem subjected to torsion
Introduction
The problem of a cylinder shrink fitted into an undersized cavity arises in a a number of engineering problems such as when a disk (pulley, gear etc) is shrink fitted onto a shaft, or in the domestic environment of, for example, a cork or stopper closure. The relationship between the degree of interference and the contact pressure generated in a plane problem has been known for a very long time and is associated with the name Lamé, but the presence of a free surface, where the contact terminates, produces interesting complicating three dimensional effects which can be solved in closed form (see Paynter, Hills, and Barber (2009)). The gradients along the axis of the shaft which the free surface induces means that axial shear tractions arise, and these have been shown by Paynter et al. (2009) to be sufficiently severe for slip always to be present when the contact is formed, for realistic coefficients of friction.
The object of the present paper is to extend this solution to problems in which a torque is inserted at the free surface itself, and to see how this modifies the interfacial shear tractions present. Although the solution has relevance to engineering assemblies it also has relevance to the cork closure refereed to: a corkscrew is normally twisted and released before an axial force is applied, and this paper shows how a self-equilibrating residual interface shear traction is developed by the twisting action. Although the axial force part of the problem is not considered, it is is clear that the torque pre-load will reduce the value of the axial force needed to achieve extraction of the cork.
The formulation presented in this work relies on two idealisations that might lead to gross aproximations when applied to practical applications but have to be made due to the current state of knowledge of the tools for analytical modelling of shrink fit under partial slip.
The method used to introduce interfacial slip relies on the knowledge of the state of stress induced in a half-space by a ring dislocation lying in a plane parallel with the surface (kernel or core of the solution). In practical applications, the hub and the shaft might be made of different materials and this would imply knowing the kernel for a mismatched half-space, which is not available in the literature. We assume, therefore, that the hub and the shaft are made of the same material, and consider the the case where the half-space is homogeneous (Paynter & Hills, 2009).
In addition, the application of a torque is usually distributed over the surface of the shaft. However, a point torque was chosen to be applied because; (a) the solution was available in the literature in an elegant format for a homogeneous half-space (Chowdhury, 1983), (b) the only mathematical singularity in the stress fields appears at the origin (away from the contact interface), (c) and it presents no intrinsic length dimension, i.e. it introduces no extra parameters in the model.
Section snippets
Shrink fit assembly
The stresses due to the shrink fit assembly (shown in Fig. 1) are given by Paynter et al. (2009) and are reproduced here. Upon assembly, an infinitely long oversized shaft of radius is inserted in a hub with a cylindrical hole of radius a, present in an elastic half-space. Both the shaft and the hub have Poisson’s ratio ν and Young’s modulus E. The assembly can be achieved by cooling the oversized shaft through a temperature differential ΔT until it fits in the hub and then letting it warm
Formulation of twist problem
After assembling the shrink fit, a point torque T is applied acting into the elastic half-space. In the cylindrical coordinate set of Fig. 2(a) and (b), the state of stress within the half-space is given by Chowdhury (1983), Chow and Yang (1990), and the non-zero tractions arising on any cylindrical cut (an constant surface), are given by
A normalised variable γ is chosen as a measure of the shrink fit to applied torque
Results
The problem was coded using the numerical processor MATLAB. Convergence was obtained using . A pseudo-time step of was used in all calculations. It was noted that changes in stresses and displacements for large γ was negligible when Δγ was made smaller than 0.10. All the results presented in this paper are for .
Figs. 3 and 4 show the tractions at the contact interface for and 1.0 for 1.0, 3.0 and 5.0, normalised by the reference stress σ0/2. The stresses at the
Conclusions
A solution was obtained for the tractions, displacements and geometric parameters for an oversized shaft shrink fitted to an elastically identical hub and subjected to torsion. Due to the torque, there are in-plane and anti-plane components of shear and, consequently, an incremental solution was needed.
It was shown that the problem exhibits considerable frictional coupling. Even though the torsion of the shaft does not explicitly induce σrz and σrr stresses, the solution shows that the contact
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
J. L. gratefully acknowledges the financial support of Christ Church Oxford, Rolls Royce PLC and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES.
References (16)
- et al.
An elastic strip in plane rolling contact
International Journal of Mechanical Sciences
(1968) - et al.
On the solution of a concentrated torque on an orthotropic half-space
International Journal of Engineering Science
(1990) - et al.
Frictional slip between a layer and a substrate caused by a normal load
International Journal of Engineering Science
(1980) - et al.
Ring cracks at the surface of a half-space
Engineering Fracture Mechanics
(2018) - et al.
The effect of path cut on Somigliana ring dislocations in a half-space
International Journal of Solids and Structures
(2009) - et al.
The effect of path cut on Somigliana ring dislocation elastic fields
International Journal of Solids and Structures
(2007) - et al.
Torsional contact of an elastic flat-ended cylinder
Journal of the Mechanics and Physics of Solids
(2008) - et al.
An investigation of convection effects in complete and almost complete contact problems
European Journal of Mechanics-A/Solids
(2009)
Cited by (1)
Effect of Form Defect in Thin-Walled Cylinder Assembly
2023, Journal of The Institution of Engineers (India): Series C