The objective of this letter is to report a new development in the understanding of the fractal behaviour in fatigue of materials. The main results set up in the present investigation include the scaled Fokker-Planck equation and corresponding scaling solution for the crack size distribution (CSD) function.The starting point of the present investigation is the work by Ding et al. [1], which, starting from the theoretical framework of non-equilibrium statistical mechanics, discusses problems of microcrack system evolution and explores the physical origin of the statistically self-similarity property in fatigue of materials.However, unfortunately we found that the scaled 9) is incomplete and questionable, since the average crack size calculated from this P(a, N) does not increase with time, but decreases. In fact, all of the existing experimental results show that the average crack size increases with time.In an effort to elucidate this problem, further studies have been performed, the details of which are given below. The letter is organized as follows. Firstly, we discuss [1] as a basic premise. We then set up the scaled Fokker-Planck equation to rederive the scaled CSD function P(a, N). Finally, we point out possible developments and summarize our discussions.The problem in [1] was treated through the addition of fluctuation of "internal noise" dependent propagation terms to the deterministic fatigue crack growth (FCG) law (i.e. Paris formula [2]: many experiments have shown that for a wide range of materials the FCG rate can be described by the Paris formula [3][4][5][6]); that is, to take the set of cracks and randomize them in space to make them statistically self-similar. Thus:
