ANALYSIS OF THE ANTIPLANE DEFORMATION OF A CRACKED QUASICRYSTAL WITH CONSIDERATION FOR SURFACE EFFECTS

• Дата: 2026
• Автори: KLETSKOV, O. M., LOBODA, V. V.
• Формат: Стаття
• Опубліковано: текст 3 2026
• Онлайн доступ: https://journal-itm.dp.ua/ojs/index.php/ITM_j1/article/view/173
• Назва журналу: Technical Mechanics

The antiplane shear deformation (Mode III) of a linear hexagonal quasicrystal containing an isolated crack under remote uniform loading is investigated taking into account phonon–phason coupling and surface effects within the framework of the Gurtin–Murdoch surface elasticity model. The crack faces are modeled as elastic membranes possessing surface phonon, phason, and coupling constants of their own, which makes it possible to adequately describe size effects at the nano- and submicron scales. Based on the equations of quasicrystal elasticity theory incorporating both phonon and phason fields, a mathematical model of the problem is developed. The boundary conditions on the crack faces, modified by surface energy, lead to a system of singular integro-differential equations with a Cauchy-type kernel. To solve this system, the collocation method with Chebyshev polynomials is applied, thus ensuring a high accuracy and good convergence of the numerical procedure. The numerical analysis performed for a one-dimensional hexagonal quasicrystal shows that accounting for surface elasticity significantly alters the stress and strain field near the crack tip. In contrast to classical fracture mechanics, where a square-root stress singularity occurs, the proposed model predicts finite stresses and strains. Surface elasticity acts as a regularizing mechanism that “smooths” the singularity and introduces a size-dependent response. It is shown that as the crack length increases, the stresses at the crack tip and in its vicinity increase, while the normalized crack opening displacement changes only slightly. For small cracks, surface effects are dominant, whereas with increasing defect size the behavior gradually approaches the classical solution, although it does not coincide with it completely. The results can be used to assess the strength and fracture toughness of quasicrystalline materials taking into account nanoscale effects. 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