PSI - Issue 68

Roman Kushnir et al. / Procedia Structural Integrity 68 (2025) 32–38 R. Kushnir et al. / Structural Integrity Procedia 00 (2025) 000–000

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novel and efficient Kutt-type quadratures for fast and accurate evaluation of hypersingular integrals, advanced techniques for the precise evaluation of kernel functions, and specialized shape functions for the accurate determination of phonon and phason stress intensity factors. The developed numerical method was applied to study a planar penny-shaped crack problem in a 2D hexagonal quasicrystal, for which an analytical solution is available (Li et al., 2017, 2019). The crack faces were discretized using only 12 quadrilateral boundary elements. The extended phonon-phason stress intensity factors were calculated along the entire crack front. The maximum deviation from the analytical solution was less than 0.5%, validating the accuracy and efficiency of the proposed approach. This close agreement with the known solution confirms the method’s robustness in solving crack problems in quasicrystals, even with a relatively coarse mesh. 6. Conclusion In this work, a novel boundary element method is developed for analyzing 3D cracks in thermoelastic quasicrystal solids. A new formalism is introduced to handle thermoelasticity in quasicrystals, simplifying the formulation by using extended vectors and tensors to represent phonon and phason fields. This formalism, combined with an extended version of Green's second identity, enables the derivation of boundary integral formulae and equations without the need for volume discretization, focusing solely on boundary elements. Efficient numerical techniques, such as Kutt type quadratures and special quadrilateral discontinuous boundary elements, are employed to solve these equations accurately. The method is validated through several crack problems. This approach offers significant computational efficiency and accuracy for analyzing fracture behavior in quasicrystals. References Deans, S.R., 1983. The Radon transform and some of its applications. New York: Wiley-Interscience Publication. Fan, C.Y., Yuan, Y.P., Pan, Y.B., Zhao, M.H., 2017. Analysis of cracks in one-dimensional hexagonal quasicrystals with the heat effect. International Journal of Solids and Structures 120, 146–156. https://doi.org/10.1016/j.ijsolstr.2017.04.036 Fan, T.-Y., 2016. Mathematical Theory of Elasticity of Quasicrystals and Its Applications. Second Edition. Springer, 2016. https://doi.org/10.1007/978-981-10-1984-5 Fan, T.Y., Mai, Y.W., 2004. Elasticity theory, fracture mechanics and some relevant thermal properties of quasicrystalline materials. Applied Mechanics Reviews 57(5), 325-344. https://doi.org/10.1115/1.1763591 Fan, T.Y., Yang, W., Cheng, H., Sun, X.H., 2022. Generalized Dynamics of Soft-Matter Quasicrystals: Mathematical Models, Solutions and Applications. Second Edition. Singapore: Springer, 2022. https://doi.org/10.1007/978-981-16-6628-5 Li, X.-Y., Wang, Y.-W., Li, P.-D., Kang, G.-Z., Müller, R., 2017. Three-dimensional fundamental thermo-elastic field in an infinite space of two dimensional hexagonal quasi-crystal with a penny-shaped/halfinfinite plane crack. Theoretical and Applied Fracture Mechanics 88, 18–30. http://dx.doi.org/10.1016/j.tafmec.2016.11.005 Li, Y., Zhao, M.H., Qin, Q.-H., Fan, C.Y., 2019. Analysis solution method for 3D planar crack problems of two-dimensional hexagonal quasicrystals with thermal effects. Applied Mathematical Modelling 69, 648–664. https://doi.org/10.1016/j.apm.2019.01.004 Long, F., Li, X.-F., 2022. Thermal stresses of a cubic quasicrystal circular disc. Mechanics Research Communications 124, 103913. https://doi.org/10.1016/j.mechrescom.2022.103913 Pasternak, Ia., Pasternak, R., Pasternak, V., Sulym, H., 2017. Boundary element analysis of 3D cracks in anisotropic thermomagnetoelectroelastic solids. Engineering Analysis with Boundary Elements 74, 70-78. https://doi.org/10.1016/j.enganabound.2016.10.009 Pasternak, I., Pasternak, R. Sulym, H., 2016. A comprehensive study on Green׳s functions and boundary integral equations for 3D anisotropic thermomagnetoelectroelasticity. Engineering Analysis with Boundary Elements 64, 222-229. https://doi.org/10.1016/j.enganabound.2015.12.004 Pasternak, V., Sulym, H., Pasternak, Ia.M., Hotsyk, I., 2024. Extended Stroh formalism for plane problems of thermoelasticity of quasicrystals with applications to Green’s functions and fracture mechanics. International Journal of Engineering Science 203. 104124. https://doi.org/10.1016/j.ijengsci.2024.104124 Pi., J., Zhao, Y., Li, L., 2022. Interaction between a Screw Dislocation and Two Unequal Interface Cracks Emanating from an Elliptical Hole in One Dimensional Hexagonal Piezoelectric Quasicrystal Bi-Material. Crystals 12, 314. https://doi.org/10.3390/cryst12030314 Ting, T.C.T., 1996. Anisotropic elasticity: theory and applications. New York: Oxford University Press, 1996.