Incomplete Self-Similarity of Fatigue in the Linear Range of Crack Growth

Abstract

The application of dimensional analysis and similarity methods to the study of the speed of fatigue crack growth is considered. It is shown that the Paris range of the crack propagation diagram is an intermediate-asymptotic stage of the crack growth process. Over this stage the influence of the initial conditions on the process of fatigue crack growth has disappeared but the influence of the instability has not yet intruded. So-called incomplete self-similarity prevails at this stage with respect to a basic similarity parameter, equal to the ratio of the stress intensity factor amplitude to the fracture toughness. It is shown that for a certain material under fixed external loading conditions the exponent in the Paris power law is a universal function of the ratio of specimen thickness to the ultimate size of the cyclic plastic zone. Processing of available experimental data confirmed the results obtained by this approach. Nomenclaturegrain size hK range of the stress intensity factor specimen thickness N number of loading cycles length of a fatigue crack static yield strength C, m constants in the Paris equation cyclic yield strength z similarity parameter cyclic fracture toughness t time plane strain fracture toughness ultimate size of cyclic plastic zone dlldN rate of fatigue cmck growth R asymmetry ratio of the loading cycle

Incomplete self-similarity and fatigue-crack growth

“…In this less restrictive form, it is apparent that considerations of similarity imply that both the C and m constants in the Paris law are not simply a function of the material properties and the nature of the applied loading, but also on specimen size, h, through the basic similarity parameter Z. Indeed, comparison with Heiser and Mortimer data (Heiser and Mortimer, 1972) on 4340 steel revealed such a relationship between m and Z, as predicted by Equations (5) and (6) (Barenblatt and Botvina, 1981). One aim of the present study is to explain such a result in terms of physical mechanisms.…”

“…Since in the original analysis of Barenblatt and Botvina (1981) only a single set of experimental data was considered, we first examine a large body of fatigue-crack growth rate results in order to interrogate relationships between the Paris exponent m and specimen dimensions. To facilitate this approach, we use the compilation of ambient-temperature data from the work of Ritchie and Knott (1973), based on experimental fatigue-crack propagation studies at ambient temperatures on a wide range of steels from (Carman and Katlin, 1966;Wei et al, 1967;Crooker et al, 1968;Miller, 1968;Bates et al, 1969;Crooker and Lange, 1970;Clark and Wessel, 1970;Evans et al, 1971;Ritchie and Knott, 1973).…”

Scaling Phenomena in Fatigue and Fracture

“…L.R. Botvina, processing the data by Heiser and Mortimer [33] (see [32]), and R.O. Ritchie [34], processing the data by Knott and Ritchie [35] (Figures 7 and 8), showed that the dependence of m upon Z, i.e., upon the specimen size can be substantial.…”

“…For the vast majority of cases m is substantially larger than 2. Assuming incomplete similarity we obtain [32] (see also [16]):…”

“…Besides the asymmetry of the cycle R they should depend on the specimen length scale. The dependence of the exponent in Paris law (17) on the similarity parameter Z for 4340 steel (Botvina, see [32]). …”