The elastic half-space

Non-conforming elastic bodies in contact whose deformation is sufficiently small for the linear small strain theory of elasticity to be applicable inevitably make contact over an area whose dimensions are small compared with the radii of curvature of the undeformed surfaces. The contact stresses are highly concentrated close to the contact region and decrease rapidly in intensity with distance from the point of contact, so that the region of practical interest lies close to the contact interface. Thus, provided the dimensions of the bodies themselves are large compared with the dimensions of the contact area, the stresses in this region are not critically dependent upon the shape of the bodies distant from the contact area, nor upon the precise way in which they are supported. The stresses may be calculated to good approximation by considering each body as a semi-infinite elastic solid bounded by a plane surface: i.e. an elastic half-space. This idealisation, in which bodies of arbitrary surface profile are regarded as semi-infinite in extent and having a plane surface, is made almost universally in elastic contact stress theory. It simplifies the boundary conditions and makes available the large body of elasticity theory which has been developed for the elastic half-space.

In this chapter, therefore, we shall study the stresses and deformations in an elastic half-space loaded one-dimensionally over a narrow strip (‘line loading’). In our frame of reference the boundary surface is the x–y plane and the z-axis is directed into the solid.

The assumptions and restrictions made in the Hertz theory of elastic contact were outlined in the previous chapter: parabolic profiles, frictionless surfaces, elastic half-space theory. In this chapter some problems of normal elastic contact are considered in which we relax one or more of these restrictions. Before looking at particular situations, however, it is instructive to examine the stress conditions which may arise close to the edge of contact.

Stress conditions at the edge of contact

We have seen in Chapter 4 that, when two non-conforming elastic bodies having continuous profiles are pressed into contact, the pressure distribution between them is not determined uniquely by the profiles of the bodies within the contact area. Two further conditions have to be satisfied: (i) that the interface should not carry any tension and (ii) that the surfaces should not interfere outside the contact area. These conditions eliminate terms in the pressure distribution of the form C(1 –x2/a2)-1/2 which give rise to an infinite tension or compression at the edge of the contact area (x = ±a) (see equation (4.41)). The resulting pressure distribution was found to be semi-ellipsoidal, i.e. of the form p0(1 –x2/a2)1/2, which falls to zero at x = ±a.

If we now recall the stresses produced in line loading by a uniform distribution of pressure (§2.5), they are everywhere finite, but the gradient of the surface is infinite at the edge of the contact (eq. (230b) and Fig. 2.8).

Introduction

Classical elastic contact stress theory concerns bodies whose temperature is uniform. Variation in temperature within the bodies may, of itself, give rise to thermal stresses but may also change the contact conditions through thermal distortion of their surface profiles. For example if two non-conforming bodies, in contact over a small area, are maintained at different temperatures, heat will flow from the hot body to the cold one through the ‘constriction’ presented by their contact area. The gap between their surfaces where they do not touch will act more or less as an insulator. The interface will develop an intermediate temperature which will lie above that of the cold body, so that thermal expansion will cause its profile to become more convex in the contact region. Conversely the interface temperature will lie below that of the hot body, so that thermal contraction will lead to a less convex or a concave profile. Only if the material of the two bodies is similar, both elastically and thermally, will the expansion of the one exactly match the contraction of the other; otherwise the thermal distortion will lead to a change in the contact area and contact pressure distribution. This problem will be examined in §4 below.

A somewhat different situation arises when heat is generated at or near to the interface of bodies in contact. An obvious example of practical importance is provided by frictional heating at sliding contacts.

Real and apparent contact

It has been tacitly assumed so far in this book that the surfaces of contacting bodies are topographically smooth; that the actual surfaces follow precisely the gently curving nominal profiles discussed in Chapters 1 and 4. In consequence contact between them is continuous within the nominal contact area and absent outside it. In reality such circumstances are extremely rare. Mica can be cleaved along atomic planes to give an atomically smooth surface and two such surfaces have been used to obtain perfect contact under laboratory conditions. The asperities on the surface of very compliant solids such as soft rubber, if sufficiently small, may be squashed flat elastically by the contact pressure, so that perfect contact is obtained throughout the nominal contact area. In general, however, contact between solid surfaces is discontinuous and the real area of contact is a small fraction of the nominal contact area. Nor is it easy to flatten initially rough surfaces by plastic deformation of the asperities. For example the serrations produced by a lathe tool in the nominally flat ends of a ductile compression specimen will be crushed plastically by the hard flat platens of the testing machine. They will behave like plastic wedges (§6.2(c)) and deform plastically at a contact pressure ≈3Y where Y is the yield strength of the material. The specimen as a whole will yield in bulk at a nominal pressure of Y.

Micro-slip and creep

In Chapter 1 rolling was defined as a relative angular motion between two bodies in contact about an axis parallel to their common tangent plane (see Fig. 1.1). In a frame of reference which moves with the point of contact the surfaces ‘flow’ through the contact zone with tangential velocities V1 and V2. The bodies may also have angular velocities ωZ1 and ωZ2 about their common normal. If v1 and v2 are unequal the rolling motion is accompanied by sliding and if ωz1 and ωz2 are unequal it is accompanied by spin. When rolling occurs without sliding or spin the motion is often referred to as ‘pure rolling’. This term is ambiguous, however, since absence of apparent sliding does not exclude the transmission of a tangential force Q, of magnitude less than limiting friction, as exemplified by the driving wheels of a vehicle. The terms free rolling and tractive rolling will be used therefore to describe motions in which the tangential force Q is zero and non-zero respectively.

We must now consider the influence of elastic deformation on rolling contact. First the normal load produces contact over a finite area determined by the Hertz theory. The specification of ‘sliding’ is now not so straightforward since some contacting points at the interface may ‘slip’ while others may ‘stick’. From the discussion of incipient sliding in §7.2, we might expect this state of affairs to occur if the interface is called upon to transmit tangential tractions.