PSI - Issue 58

P.C. Sidharth et al. / Procedia Structural Integrity 58 (2024) 115–121 P.C. Sidharth and B.N. Rao/ Structural Integrity Procedia 00 (2019) 000–000

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Therefore,  , represented as / h l   , serves as an indicator of the mesh density within the fracture zone. These shape functions possess the unique characteristic of adaptability to both the mesh size and the regularization length. a b

Fig. 1. (a) 1D analytical solution of a stationary crack; (b) Plot of EFE shape functions for two 1D elements having common node at x=0.

This adaptability arises from their dependence on both the element size, h, and the regularization parameter, l . For maintaining accuracy, it is imperative to correctly align the exponential finite element (EFE) shape functions with respect to the nodal values of the phase-field variable within all elements. In contrast, linear finite element (LFE) shape functions are symmetric and do not rely on the phase-field variable  . However, this limitation can be overcome by aligning the exponential finite element (EFE) shape functions with the nodal values of  , leading to

enhanced accuracy in simulations. 2.2. Finite element implementation

MOOSE is specifically designed to be programmed in Linux environments, offering a robust and stable platform for computational tasks. The input file is structured to house essential finite element mesh data, including nodal coordinates and element connectivity information. The heart of the executable is composed of C++ objects and classes, allowing for modular and object-oriented programming, which enhances the software's flexibility and maintainability. Output files are obtained in Exodus and CSV formats, making it adaptable to various post processing and data analysis needs. The results are visually represented and analyzed using Paraview. Additionally, the software supports massive parallelization with scalability up to 30,000 cores. Built-in physics modules and natural multi-physics support are employed to simplify complex simulations with one kernel object per PDE. Additionally, it offers time and mesh adaptivity for dynamic adjustments during runtime. 3. Numerical examples and discussion This study aims to assess the accuracy of exponential finite element (EFE) shape functions in modeling fracture responses in functionally graded materials (FGMs), given their variable material properties. Phase-field simulations are conducted using EFE and linear finite element (LFE) shape functions, with a focus on smaller length scale parameters. Results are compared, and a refined mesh with increased elements is used. Different length scale values correspond to distinct load responses, and EFE shape functions require additional quadrature points for integration. This study considers four examples, including tension and shear loading cases. 3.1. Cracking of alumina/zirconia FG plate under tension Fracture analysis in functionally graded alumina/zirconia plates is a benchmark case for Functionally Graded Materials (FGMs). This FGM comprises varying volume fractions of alumina (Al2O3) and zirconia (ZrO2), and the material properties for these constituents are detailed in Table 1. The plate contains a mode-I edge crack under external uniaxial tension, assuming plane strain conditions. We investigate four configurations with variations along