An efficient PU-based approach to model quasi-brittle fracture and interfaces

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Partition of Unity (PU)-based approaches in Nonlinear Fracture Mechanics facilitated and improved the modelling of the fracture behaviour of quasi-brittle materials, such as mortar, concrete, masonry and rock. To this end, the discrete crack approach is assumed to localise the microcracks into the discontinuity surface represented by fictitious crack. The eXtended Finite Element Method (XFEM), as an advantageous modelling technique in the PU context, has been introduced and noticed in recent decades; however, its advantages have accompanied a number of difficulties such as ill-conditioned system, significant growth in the bandwidth of global matrix and inaccurate local solution around the crack path. Furthermore, the designation of reliable criterion, providing information about the strain localisation (i.e. crack initiation) and its orientation, can be considered another difficulty in using the discrete crack approaches. Recently, various attempts have been made to overcome the difficulties in PU-based approaches. Regardless of all efforts, almost related to the improvement of convergence rate, numerical integration scheme and PU satisfaction, the difficulties of increasing additional degrees of freedom and inaccurate local solution at a discontinuity tip have gained little attention, and no efficient treatments have been presented. In this study, a comparison between conventional cracking criteria and modified ones is drawn to obtain the appropriate criterion for different fracture modes. In addition, two innovative formulations are proposed: the XFEM with multi-layered Heaviside enrichment and a polygonal enriched Partition of Unity Method. The main advantages are i) the cracking criterion can be employed in tensile and compressive states without any special consideration, ii) the bandwidth of global matrix and condition number decrease, and iii) the displacement jump and stress field are captured accurately. The capability of the presented formulations is assessed by comparing them with standard XFEM. Itis found that the formulation of XFEM with multi-layered Heaviside enrichment shows a remarkable agreement with standard XFEM locally and globally. The proposed formulation with polygonal enrichment overcomes the spurious behaviour of PU-based elements utilised in standard XFEM and opens the possibility of interface problems analysis. Furthermore, several benchmark tests concerning mode-I, mode-II and mixed-mode fracture are simulated, and the numerical results indicate remarkable similarities with the corresponding experimental data. As a final result of this study, a robust and efficient numerical tool has been introduced to model the crack propagation and interfaces of quasi-brittle materials.
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