13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- has mechanical equivalence with several actual atoms. Under the boundary condition of (1), an array of these super atoms was considered and a convenient mechanical analysis which obtains mechanical equilibrium positions of super atoms was conducted. (3) Furthermore, these super atoms were sequentially dispersed under the mechanically equivalent state. Consequentially, the number of these atoms was increased and at each stage of the corresponding scale of these atoms, the mechanical equilibrium positions of these atoms were numerically analyzed. (4) The characteristic of self-similarity of atom arrays was investigated based on the fractal analysis through each stage of the scale of super atoms. Super atoms were sequentially dispersed and the mechanical equilibrium positions of these atoms were numerically analyzed. (5) The numerical analyses were conducted up to the scale stage of the super atomic array which shows the characteristic of self-similarity. From this scale stage of super atoms, disordered region of atomic arrays around the crack tip or dislocation with the actual scale of atomic array was predictively calculated and this region was defined as the process region of fracture. Using this theory, the characterization of disordered region in the macroscopic local stress field around the crack tip becomes possible conveniently and with high accuracy. It enables us to theoretical prediction of the process region which is the dominant region of brittle fracture without the aid of experimental results. 2. The Model and Method of Analysis 2.1. The Model of Analysis The concept of super atoms and dislocations which represent mechanically equivalent several actual atoms and dislocations, that is, the concept of image atoms and dislocations were used. Since stress field around a dislocation and interactive force between atoms are conservative, mechanical similarity on scale will be valid. Therefore, similar equations of stress field and interactive forces of dislocation and atoms of which values of intensity are different are adopted for super atoms and dislocations. At each scale stage of super atoms, using equations of interactive forces which have corresponding intensity, the equilibrium positions of super atoms were numerically analyzed. The flow of analysis and the concept of mechanical equivalence between super and actual atoms were shown in Fig. 1. With increment increase in the number of atoms, interactive forces between these atoms were dispersed and the equilibrium positions of atoms in the array were numerically analyzed respectively. The fractal analyses were conducted for the morphology of distribution of atomic density in the array at each scale stage of super atoms and the existence of self-similarity of the atoms array was investigated through each scale stage. When the characteristic of the self-similarity appears through each scale, the projection method is applied to the sequential changing characteristics of the morphology of distribution of atomic density to predict the equilibrium distribution of actual atomic arrays by considering similarity of the changing characteristic being kept due to the existence of the self-similarity. The dominating local region of brittle fracture was predicted by this method and results obtained was compared with previous results [8].
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