RESULTS AND DISCUSSION
Some issues relevant to the present work have been addressed for discussion in the previous publication due to Farag et al.52 on cracking simulation-based fatigue life assessment due to CAL. Examples are (1) the inability of the stress intensity factor to correlate the behavior of short cracks at high stresses, (2) the utilization of a correlation between fatigue crack growth (FCG) rates and a devised crack tip deformation parameter equal to δ 0.46Δδ 0.5 for both long and short cracks, where δ and Δδ are the maximum and cyclic crack tip opening displacements, (3) the plastic zone interaction between the two opposing tips of adjacent cracks when the two tips are approaching each other, (4) The similar behavior of \(A_{C}\) and the total length of cracks existing on the specimen surface and (5) the extreme difficulty in the present model of considering the evidence of microstructure evolution of crack initiation. In this publication52, the present authors reviewed some works in the literature in relation to the fatigue behavior of long and short cracks in un-notched specimens under CAL. Here, results of the present simulation due to the application of some patterns of VAL are presented and discussed. As demonstrated below, the discussion of the present results is in need of some results due to CAL 52.
The present work utilized one numerical simulation considering one surface finish to have all the present results. Different specimens were randomly configured, virtually tested at different stresses and consequently their \(S/N\) curve was constructed as shown in Fig. 1. A single experimental endurance data point was utilized to adjust the proposed modeling to the experimental data. The necessity of such an adjustment had been expected in advance before the first run of the present simulation. The expressions for the crack tip deformation utilized in the present model 52 are the result of an elastic-plastic finite element analyses FEA previously performed for a trough-thickness crack in a plate 56. The present crack system is different and, thus, the geometrical factor used in the mathematical form of the stress intensity factors was employed. Alternatively, similar 3D FEA have to be performed to analyze curved surface crack from the surface of round bars.
The model considers growing fatigue micro cracks, \(\leq\ \)0.1 µm, originating only from the roughness of a round specimen surface. Many metallurgical and mechanical factors, manufacturing operations, surface treatments, cleaning and handling have strong influences on surface roughness. For example, fine turning with a tool peak radius of 0.5 mm, and feed 0.13 mm can achieve an average peak-to-valley height of 4 µm. Values as small as 0.05 µm can be achieved by honing. In between values are obtainable by appropriate surface finish processes such as grinding and lapping. In order to correlate these surface roughness parameters to the present model, the metrologist and the production engineer can easily measure the surface roughness in terms of maximum peak-to-valley height and mean surface roughness. From those two values the standard deviation of the surface roughness can be determined. In the present model the average surface roughness is identified as an average initial crack depth, and its standard deviation given by a typical statistical analysis of metrological surface measurement. The maximum value of crack length was 0.1 µm according to the relevant experimental data available for the same specimens 3.
Some of the variables utilized to devise the present model are assumed randomly distributed. These include grain size, starting crack size and monotonic and cyclic yield stress of the grains of the two phases of the specimens’ material. The parameters of the mean value and standard deviation of the grain size and material’s straight can be controlled with proper heat treatments whilst those parameters controlling the starting crack size are given by the roughness of the specimen surface. With this assumption, the present simulation is closer to reality. Further, randomization is invoked (1) to distribute the grains of the two phases of the material along the minimum circumference of the specimen and (2) to select one of the virtually configured specimens to run a specific virtual test. This process is uncontrolled.
The present simulation is based on the collective behavior of growing short fatigue cracks originating from the specimen surface. The model utilizes \(A_{C}\) as a measure of fatigue damage. The results of the present work can easily show that the behavior of \(A_{C}\) with the number of cycles is similar to that of the total length of the existing cracks. On the other hand, the cracking-based cumulative fatigue damage models existing in the literature consider the length of a single isolated dominant crack 34, 36, 37, 47-51. This is expected to over-estimate the fatigue lifetime for the interaction and coalescence of existing small cracks shorten the time to failure47 – 51. Further, an individual crack shows large changes in its growth rate. This depends on the features of that crack in terms of its size and position relative to micro­ structural barriers along with other mechanical and metallurgical effects.
Figures 3I-3IV present examples to demonstrate the anomalous growth behavior of four short surface cracks, labeled as crack 4, crack 306, crack 406 and crack 556, individually traced throughout one of the virtual CAL tests with a stress amplitude equal 390 MPa. Fifteen, nineteen, eleven and twenty-three neighboring short cracks are involved in the cracking activities illustrated respectively in Figs. 3I-3IV.
As an example, a description of the cracking activities demonstrated in Fig. 3I follows. The involved fifteen neighboring short cracks are sequentially numbered from left to right. The growth of the crack labelled 4 is assumed continuously monitored throughout the test. The two tips of that crack commence their advance with the start of the test till N = 35200 cycles when its length becomes 24 µm, see Fig. 3I(A). The crack length, then, suddenly achieves a length of 110.9 µm, see Fig. 3I(B), due to its coalescence with a neighboring crack 86.8 µm long located at its right and composed of the two already merged cracks 5 and 6. The resulting crack continues its growth, see Fig. 3I(B), from its two tips to reach 136.4 µm at N = 54647 cycles when its left tip merges crack 3 to result in a crack 136.512 µm long, see Fig. 3I(C), which grows continuously from both tips to achieve 145 µm at N = 54650 cycles before it coalesces with the growing coalesced cracks 2 and 1 to have a crack of a total length equal to 241.88 µm. As N increases, the two tips of the resulting crack advance to achieve a length equal to 246.78 µm at N = 182364 cycles, see Fig. 3I(C), when its right tip merges the left tip of a growing adjacent crack composing of nine coalesced small cracks to suddenly reach a length of 561.1 µm. For a better illustration, Fig. 3I(A-C) are plotted with different scales.