Influence of size on the fractal dimension of dislocation microstructure

Abstract

Three-dimensional (3D) discrete dislocation dynamics simulations are used to analyze the size effect on the fractal dimension of two-dimensional (2D) and 3D dislocation microstructure. 2D dislocation structures are analyzed first, and the calculated fractal dimension (n2) is found to be consistent with experimental results gleaned from transmission electron microscopy images. The value of n2 is found to be close to unity for sizes smaller than 300 nm, and increases to a saturation value of ≈1.8 for sizes above approximately 10 microns. It is discovered that reducing the sample size leads to a decrease in the fractal dimension because of the decrease in the likelihood of forming strong tangles at small scales. Dislocation ensembles are found to exist in a more isolated way at the nano- and micro-scales. Fractal analysis is carried out on 3D dislocation structures and the 3D fractal dimension (n3) is determined. The analysis here shows that (n3) is significantly smaller than (n2 + 1) of 2D projected dislocations in all considered sizes.