A new method for estimating the Hurst exponent H for 3D objects

Abstract

Mathematics and computer science are very useful in many other sciences. We use a mathematical method, fractal geometry, in engineering, specifically in laser techniques. Characterization of the surface and the interfacial morphology of robot-laser-hardened material is crucial to understand its properties. The surface microstructure of robot-laser-hardened material is rough. We aimed to estimate its surface roughness using the Hurst parameter H, which is directly related to the fractal dimension. We researched how the parameters of the robot-laser cell impact on the surface roughness of the hardened specimen. The Hurst exponent is understood as the correlation between the random steps X1 and X2, which are followed by time for the time difference t. In our research we understood the Hurst exponent H to be the correlation between the random steps X1 and X2, which are followed by the space for the space difference d. We also have a space component. We made test patterns of a standard label on the point robot-laser-hardened materials of DIN standard GGG 60, GGG 60 L, GGG 70, GGG 70 L and 1.7225. We wanted to know how the temperature of point robot-laser hardening impacts on the surface roughness. We developed a new method to estimate the Hurst exponent H of a 3D-object. This method we use to calculate the fractal dimension of a 3D-object with the equation D = 3 - H.