Abstract
This study introduces a novel implementation of exponential finite element (EFE) shape functions within the phase field model for predicting fracture responses in functionally graded materials (FGMs). The proposed approach utilizes an effective fracture toughness concept to analyze load–displacement responses and crack paths in various examples of FGMs. To optimize computational efficiency, a mixed scheme combining both linear finite element (LFE) and EFE shape functions is employed. Specifically, only the critical elements are corrected using EFE shape functions. Comparative analysis of simulation results against converged outcomes demonstrates the superiority of the EFE scheme, even when employing coarser meshes. The EFE approach accurately predicts load–displacement responses and crack paths for the evaluated examples. The proposed implementation offers a reliable and efficient tool for studying fracture behavior in FGMs. Despite the higher integration schemes and orientation requirements associated with EFE shape functions, the additional computational effort is found to be negligible. Implementation aspects include a staggered iteration scheme, a hybrid tension–compression splitting scheme, and automatic orientation of EFE shape functions.
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