An Analysis of Fretting Cracks-ii. Unloading and Reloading Phases

An efastic half-plane containing a su~ace-b~ak~og crack normal to the free surface, subjected to loading by uniform tractions over a given length af its surface, is considered. The tractions consist of pressure, constant in time, and a shear load, varying sinu~idaily in time, both applied adjacent to the crack. This geometry approximates the classical fretting problem with a resulting fatigue crack. The faces of the crack are allowed to transmit Coulomb friction. In this paper it is assumed that the pressure has already been applied, and that the shear traction has been increased contin~ousiy from zero to a m~imum value. The effect of varying the shear traction through the rest of one load cycle is considered. Stress intensity factors are computed for various crack Iengths, friction coefftcients, and ratios of applied tractions. The history of stick and slip zones found along the crack faces is monitored. The geometry of the surface-breaking crack is shown in Fi. l(a). The half-space x > 0 is assumed to be loaded by constant normal pressure pi over a small portion (0 < y < t) of its surface. A shear traction q = MO veil ha~onic~Iy with time is then applied over L. This configuration is used to model a fatigue crack emanating from a fretting contact. References included in Part 1 [l] describe ex~~ments which commonly result in this type of failure. It was expected that a severe stress intensity would develop at the crack tip as the shear traction was increased in the positive sense [Fig. l(b)]. This was studied in Part 1. Although the absolute maximum stress intensity factor is certainly important and probably indicates the portion of the loading cycle where most crack growth occurs, it is the range of stress intensity which is required by a growth rate/stress intensity law such as that due to Formanf2J. Therefore, in the present paper we aim to extend the resufts of [lf by following the stress intensity experienced by the crack tip throughout its loading cycle. The four quadrants of one cycie of loading are shown in Fig. I(b) and are denoted by Roman numerals. The vertical crack (0 -C x < c) is located along the y =e 0 axis, and during part of the cycle it may be open in the interval 0 C= x < a. It will be assumed in the present paper that the magnitude of A,,,, the ratio of shear to normal tractive loads, is suffrcientiy great to open the crack to its tip in region I. It is felt that for practical purposes this is not likely to be a restriction, since a signi~cant crack growth inurement will be experienced when the crack is fully open and suffering combined modes I and II loading, and this condition will therefore probably obtain for all cracks which are not experiencing self-arrest. A consequence of this assumption is that ah residual shear tractions developed at the end of each cycle of loading will be relaxed out, and therefore the crack will experience the same interfacial tractions in the first loading cycle as in the steady state. To assist in our description of crack response throu~out a cycle, it is convenient if we classify cracks as one of two types, as shown in Fig. 2. Thus, long cracks with a low coefficient of interfaciat friction f, which fall betow the dividing line shown, are denoted type A, while short cracks with high coefficients of friction will Lie above that line and are hence denoted type B. Norm~ized crack lengths (c/L) are used. In the foIlowing development, reference should be made to Table 1. t Permanent address: event of E~nee~~ Science, Oxford University, Parks Road, Oxford OXf 3PJ, U.K. S. D. SHEPPARD et al.