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Crack initiation and propagation analysis in brittle two-dimensional isotropic materials are conducted using the damage Phase Field Method (PFM) following the variational approach. Here, we study two-dimensional (2D) structures that contain material heterogeneities (e.g., interfaces and inclusions) as well as geometric ones (e.g., cracks and voids) when subjected to quasi-static uniaxial tensile loading. The adopted methodology combines Extended Finite Element Method (XFEM) and Level-Set couple with PFM in order to investigate both the case of heterogeneous materials (composite or porous) as well as the case of interfacial cracks between two different materials for their interest and relevance as practical examples. Among many others, one can, for example, argue that: (a) the regular hexagonal arrangement of heterogeneities leads to a significantly higher strength than any random arrangement, for both the composite material (containing fibers) and the porous material (containing voids), (b) the effects of contrasting stiffness and toughness between two materials along with their respective impacts on the interfacial crack trajectory, on energy balance and on reaction force are in favor of toughness.

The present work investigates the fatigue life of a functionally graded material (FGM) made of aluminum alloy and alumina (ceramic) under cyclic mixed mode loading. Both element free Galerkin method (EFGM) and extended finite element method (XFEM) are employed to simulate and compare the fatigue crack growth. Partition of unity is used to track the crack path in XFEM while a new enrichment criterion is proposed to track the crack path in EFGM. The fatigue lives of aluminum alloy, FGM and an equivalent composite (having the same composition as of FGM) are compared for a major edge crack and center crack in a rectangular domain. The proposed enrichment criterion not only simulates the crack propagation but it also extends the applicability and robustness of EFGM for accurate estimation of fatigue life of component.

Macroscale mesh sensitivity and RVE size dependence are the two major issues that make the conventional homogenization techniques incapable of modeling the softening behavior of quasi-brittle materials. In this paper, a new continuous–discontinuous multiscale modeling approach to failure is presented. Inspired by the classical crack band model of Bazant and Oh (1983), this approach is built upon an extended computational homogenization (CH) scheme for representing the macroscale crack behavior. During the multiscale computation, once a macroscale material point loses its stability with the XFEM, a new crack segment represented is inserted for which cohesive RVE models using the extended CH and with copied initial states are coupled to crack integration points. In the extended CH, the macroscale strain applied to the boundary of the cohesive RVE model is enriched with a macroscale discontinuity related term regularized with the effective length of the microscale localization band. This helps alleviate the RVE size dependency of the homogenized cohesive response. The weakly periodic BCs that are aligned with the localization direction are employed to minimize spurious boundary effects. Several numerical examples are provided to demonstrate the effectiveness of this framework, with a comparison against direct numerical simulations.

In the framework of the extended finite element method, a two-dimensional four-node quadrilateral element enriched with only the Heaviside step function is formulated for stationary and propagating crack analyses. In the proposed method, two types of signed distance functions are used to implicitly express crack geometry, and finite elements, which interact with the crack, are appropriately partitioned according to the level set values and are then integrated numerically for derivation of the stiffness matrix and internal force vectors. The proposed method was verified by evaluating stress intensity factors, performing crack propagation analyses and comparing the obtained results with reference solutions.