Elucidating the Fractal Nature of Powder Bed in Selective Laser Melting of Metallic Components

Abstract

In this work, the fractal nature of Selective Laser Melting (SLM) additive manufacturing process (AM) is elucidated. Fractal dimension and lacunarity of metallic powders are calculated from Scanning Electron Microscopy (SEM) images adapted from literature. The complexity and homogeneity of the textures of the powder beds are also studied through the comparison of fractal dimension and lacunarity. It is found that better densification results are obtained when the powder bed’s fractal dimension is closer to the golden mean number of 1.618. Furthermore, this finding is extended to expressions for predicting the component’s bulk density produced via SLM by setting the [Formula: see text] exponent equal to the golden mean value and finding the proportionality constant, [Formula: see text], using a nonlinear least squares method. The proposed approach works well since theoretical prediction and experimental data compare well with root-mean-square-error (RMSE) values that do not exceed [Formula: see text]. This work sheds new light on enhancing additive manufacturing technologies considering the fractal nature of SLM since its process mathematical models are constructed around Euclidian space-time with continuous smooth assumptions that should be adapted to include the fractal nature of the manufacturing process aiming to improve their precision. The underlying interweaving of SLM, as a fractal process, and the golden mean number is revealed.

Soft Tissue Hybrid Model for Real-Time Simulations

“…The mathematical treatment of Equation (23) can be done considering the solutions procedures discussed in [ 57 , 58 , 59 , 60 ]. Recently, other fractal theory definitions [ 61 , 62 ], and solution methods have been proposed [ 63 , 64 , 65 , 66 ]. However, the discussion of our findings using fractal material constitutive equations shall be addressed in a forthcoming paper to be published somewhere else.…”