Fractal Geometric Characterization of Functionally Graded Materials

Abstract

The complex structure and mechanics of elastoplastic functionally graded materials (FGM) is studied from the standpoint of fractal geometry. First, upon introducing the fineness as the number of grains of either phase across the FGM, the two-phase FGM is characterized using fractals, and an interfacial fractal dimension is estimated for varying degrees of fineness. A variation in local fractal dimension is considered across or along the FGM domain, and it is characterized by Fourier series and Beta function fits. Assuming the FGM is made of locally isotropic Titanium (Ti) and Titanium Monoboride (TiB), pure shear tests are simulated using ABAQUS for fineness levels of 50, 100, and 200 under the uniform kinematic boundary condition (UKBC) and the uniform static boundary condition (USBC). The material response observed under these BCs shows high sensitivity of these systems to loading conditions. Furthermore, plastic evolution of Ti grains, assuming isotropic plastic hardening, displays a fractal, partially plane-filling behavior. Fractal dimensions of sets of plastic grains are calculated using the box-counting method, validating the mechanical results, thus again showing high sensitivity of this material system to loading conditions.