SIMP Phase-field topology optimization framework to maximize fracture resistance in FGMs

In this study, we explore the use of SIMP topology optimization and the phase field approach to fracture to maximize fracture resistance in functionally graded materials (FGMs) in the presence of a second phase. We derive a mathematical formulation using a consistent derivation of the second law of thermodynamics to maximize the external work under the constraints of volume fraction. We also demonstrate that, for every distribution of the density function, the topology optimization problem Gamma-Converges. We highlight the significant difference between the fracture resistance in FGMs and homogeneous materials. We investigate the crack propagation path along with the optimum topology for the FGM under different grading profiles, elastic mismatch ratio, strength mismatch ratio, and inclusion mismatch ratio. We present several numerical examples to demonstrate the predictive capability of the presented model. A comparison between the initial design guess and the final optimized design is also provided for each example, to further assess the model capability.