A dislocation model for fatigue crack growth in metals

Abstract

Abstract A model of fatigue crack growth is presented based on the analysis of plastic yield at a notch in terms of dislocations which has been given by Bilby et al. (1963). This analysis is extended to the case of relaxation of the applied stress, and the reverse plastic displacement at the crack tip is calculated. Reasons are given for equating this quantity with the amount of crack growth, and hence a growth law is obtained. The rate of growth for a specimen of finite width is similarly derived. In both cases, for low stresses, the growth per cycle is proportional to the square of the elastic stress intensity, while for higher stresses, terms involving the fourth and higher powers of stress enter. The model is expected to be applicable to the slip-band type of growth called stage I by Forsyth (1961) and to the ductile component of stage II growth. Although the model is developed only for shear cracks, there are indications that it may be at least approximately applicable to tension-compression fatigue, and we have tested it against several sets of results from this type of experiment. Good agreement for the rate of crack growth per cycle, dc/dn, is obtained over a range of a dozen different materials, provided the value of (1/c)(dc/dn) is greater than about 10−4 (where 2c is the length of the crack). There are no adjustable parameters in the theory. The failure of the theory when (1/c)(dc/dn) is less than 10−5 is associated with the existence of non-propagating cracks, and it is suggested that the two may be due to the formation of a hard oxide layer at the crack tip. A modified growth law is obtained to account for this effect. The model predicts that the rate of crack growth is inversely proportional to the shear modulus G and σ1 which is the maximum attainable resistance to plastic shear. This can reasonably be taken to be the ultimate strength. It also predicts certain qualitative environmental effects: at moderate or high growth rates, a slow adsorption rate, high vacuum, high frequency and good re-welding characteristics of the metal all reduce crack growth. On the other hand, factors which enhance the formation of a hard oxide layer inhibit growth at low stress concentrations, and may prevent it altogether. All through we assume a homogeneous and isotropic material, so that effects of second phase particles, grain boundaries and crystal structure do not appear.