A pure complex variable enrichment method for modeling progressive fracture of orthotropic functionally gradient materials

Reasonably reducing degrees of freedom (DOFs) to achieve higher numerical efficiency and robustness is a long-standing challenge in all aspects of computational mechanics and this is especially true in the case of computational fracture analysis. In this work, a novel technique pursuing the further reduction of the number of the general singular enriched terms for fracture modelling is developed by means of the complex variable basis system inspirited by the Euler's identity. By introducing the complex solution, the desirable enrichment to capture the crack-tip field can be constructed as a complex variable form with fewer terms. Thereby, the necessary number of nodes, as well as the DOFs, can be reasonably reduced for crack-tip modelling. The proposed novel pure complex variable enriched basis system, which is jointed with a standard complex variable meshless scheme, is applied to analyze the crack propagation problems in orthotropic functional gradient materials (FGM). The numerical results, of both the stress fields and the crack path, demonstrated that the proposed novel pure complex variable enrichment can effectively model fractures in orthotropic FGM with fewer DOFs compared with the element-free Galerkin method and the extended finite element method.