ANALYSIS OF THE ANTIPLANE DEFORMATION OF A CRACKED QUASICRYSTAL WITH CONSIDERATION FOR SURFACE EFFECTS
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...
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Technical Mechanics| _version_ | 1861590258501550080 |
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| author | KLETSKOV, O. M. LOBODA, V. V. |
| author_facet | KLETSKOV, O. M. LOBODA, V. V. |
| author_sort | KLETSKOV, O. M. |
| baseUrl_str | https://journal-itm.dp.ua/ojs/index.php/ITM_j1/oai |
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| datestamp_date | 2026-04-04T20:10:29Z |
| description | 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. The obtained results are important for the advancement of modern engineering mechanics, particularly fracture mechanics and the mechanics of nanostructured materials, as they extend classical approaches by incorporating surface and size-dependent effects and contribute to improving the reliability of strength predictions for structural components.
REFERENCES
1. Gurtin M. E., Murdoch A. A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 1975. V. 57. Pp. 291-323.https://doi.org/10.1007/BF00261375
2. Kim C. I., Schiavone P., Ru C. Q. The effects of surface elasticity on an elastic solid with mode-III crack: complete solution. ASME J. Appl. Mech. 2010. V. 77. 021011.https://doi.org/10.1115/1.3177000
3. Wang X., Zhou K. A crack with surface effects in a piezoelectric material. Math. Mech. Solids. 2015. V. 20. Pp. 1131-1146.
4. Gurtin M. E., Weissmuller J., Larche F. A general theory of curved deformable interface in solids at equilibrium. Philos. Mag. A. 1998. V. 78. Pp. 1093-1109.https://doi.org/10.1080/014186198253138
5. Fan T. Y. Mathematical Theory of Elasticity of Quasicrystals and Its Applications. Beijing : Science Press, 2011. 363 pp.https://doi.org/10.1007/978-3-642-14643-5
6. Chen T. Exact size-dependent connections between effective moduli of fibrous piezoelectric nanocomposites with interface effects. Acta Mechanica. 2008. V. 196. No. 3-4. Pp. 205-217.https://doi.org/10.1007/s00707-007-0477-1
7. Huang G. Y., Yu S. W. Effect of surface piezoelectricity on the electromechanical behaviour of a piezoelectric ring. Phys. Stat. Solidi B. 2006. V. 243. Pp. R22-R24.https://doi.org/10.1002/pssb.200541521
8. Dai S., Gharbi M., Sharma P., Park H. S. Surface piezoelectricity: size effects in nanostructures and the emergence of piezoelectricity in non-piezoelectric materials. J. Appl. Phys. 2011. V. 110. 104305.https://doi.org/10.1063/1.3660431
9. Pan X., Yu S., Feng X. A continuum theory of surface piezoelectricity for nanodielectrics. Sci. China. 2011. V. 54. Pp. 564-573.https://doi.org/10.1007/s11433-011-4275-3
10. Ru C. Q. Simple geometrical explanation of Gurtin-Murdoch model of surface elasticity with clarification of its related versions. Sci. China. 2010. V. 53. Pp. 536-544.https://doi.org/10.1007/s11433-010-0144-8
11. Zhao X. F., Ma Y. Y., Lu S. N. Anti-plane problem of nano-cracks emanating from a regular triangular nano-hole in one dimensional hexagonal quasicrystals. Science Technology and Engineering. 2023. V. 23. No. 7. Pp. 2727-2733.
12. Xin Y. Y., Xiao J. H. Fracture mechanics of an arbitrary position crack emanating from a nano-hole in one-dimensional hexagonal piezoelectric quasicrystals. Acta Mechanica. 2023. V. 234. No. 4. Pp. 1409-1420.https://doi.org/10.1007/s00707-022-03424-y
13. Xin Y. Y., Xiao J. H. An analytic solution of an arbitrary location through-crack emanating from a nano-circular hole in one-dimensional hexagonal piezoelectric quasicrystals. Mathematics and Mechanics of Solids. 2024. V. 29. No. 1. Pp. 71-82.https://doi.org/10.1177/10812865231186341
14. Chakrabarti A., Hamsapriye. Numerical solution of a singular integro-differential equation . Journal of Applied Mathematics and Mechanics. 1999. V. 79. No. 4. Pp. 233-241.https://doi.org/10.1002/(SICI)1521-4001(199904)79:4<233::AID-ZAMM233>3.3.CO;2-Y |
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| spelling | oai:ojs2.journal-itm.dp.ua:article-1732026-04-04T20:10:29Z ANALYSIS OF THE ANTIPLANE DEFORMATION OF A CRACKED QUASICRYSTAL WITH CONSIDERATION FOR SURFACE EFFECTS KLETSKOV, O. M. LOBODA, V. V. stress, quasicrystal, crack, antiplane deformation, phonon–phason coupling, surface elasticity, singular integro-differential equations. 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. The obtained results are important for the advancement of modern engineering mechanics, particularly fracture mechanics and the mechanics of nanostructured materials, as they extend classical approaches by incorporating surface and size-dependent effects and contribute to improving the reliability of strength predictions for structural components. REFERENCES 1. Gurtin M. E., Murdoch A. A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 1975. V. 57. Pp. 291-323.https://doi.org/10.1007/BF00261375 2. Kim C. I., Schiavone P., Ru C. Q. The effects of surface elasticity on an elastic solid with mode-III crack: complete solution. ASME J. Appl. Mech. 2010. V. 77. 021011.https://doi.org/10.1115/1.3177000 3. Wang X., Zhou K. A crack with surface effects in a piezoelectric material. Math. Mech. Solids. 2015. V. 20. Pp. 1131-1146. 4. Gurtin M. E., Weissmuller J., Larche F. A general theory of curved deformable interface in solids at equilibrium. Philos. Mag. A. 1998. V. 78. Pp. 1093-1109.https://doi.org/10.1080/014186198253138 5. Fan T. Y. Mathematical Theory of Elasticity of Quasicrystals and Its Applications. Beijing : Science Press, 2011. 363 pp.https://doi.org/10.1007/978-3-642-14643-5 6. Chen T. Exact size-dependent connections between effective moduli of fibrous piezoelectric nanocomposites with interface effects. Acta Mechanica. 2008. V. 196. No. 3-4. Pp. 205-217.https://doi.org/10.1007/s00707-007-0477-1 7. Huang G. Y., Yu S. W. Effect of surface piezoelectricity on the electromechanical behaviour of a piezoelectric ring. Phys. Stat. Solidi B. 2006. V. 243. Pp. R22-R24.https://doi.org/10.1002/pssb.200541521 8. Dai S., Gharbi M., Sharma P., Park H. S. Surface piezoelectricity: size effects in nanostructures and the emergence of piezoelectricity in non-piezoelectric materials. J. Appl. Phys. 2011. V. 110. 104305.https://doi.org/10.1063/1.3660431 9. Pan X., Yu S., Feng X. A continuum theory of surface piezoelectricity for nanodielectrics. Sci. China. 2011. V. 54. Pp. 564-573.https://doi.org/10.1007/s11433-011-4275-3 10. Ru C. Q. Simple geometrical explanation of Gurtin-Murdoch model of surface elasticity with clarification of its related versions. Sci. China. 2010. V. 53. Pp. 536-544.https://doi.org/10.1007/s11433-010-0144-8 11. Zhao X. F., Ma Y. Y., Lu S. N. Anti-plane problem of nano-cracks emanating from a regular triangular nano-hole in one dimensional hexagonal quasicrystals. Science Technology and Engineering. 2023. V. 23. No. 7. Pp. 2727-2733. 12. Xin Y. Y., Xiao J. H. Fracture mechanics of an arbitrary position crack emanating from a nano-hole in one-dimensional hexagonal piezoelectric quasicrystals. Acta Mechanica. 2023. V. 234. No. 4. Pp. 1409-1420.https://doi.org/10.1007/s00707-022-03424-y 13. Xin Y. Y., Xiao J. H. An analytic solution of an arbitrary location through-crack emanating from a nano-circular hole in one-dimensional hexagonal piezoelectric quasicrystals. Mathematics and Mechanics of Solids. 2024. V. 29. No. 1. Pp. 71-82.https://doi.org/10.1177/10812865231186341 14. Chakrabarti A., Hamsapriye. Numerical solution of a singular integro-differential equation . Journal of Applied Mathematics and Mechanics. 1999. V. 79. No. 4. Pp. 233-241.https://doi.org/10.1002/(SICI)1521-4001(199904)79:4<233::AID-ZAMM233>3.3.CO;2-Y текст 3 2026-03-31 Article Article https://journal-itm.dp.ua/ojs/index.php/ITM_j1/article/view/173 Technical Mechanics; No. 1 (2026): Technical Mechanics; 62-72 Институт технической механики Национальной академии наук Украины и Государственного космического агентства Украины; № 1 (2026): Technical Mechanics; 62-72 ТЕХНІЧНА МЕХАНІКА; № 1 (2026): ТЕХНІЧНА МЕХАНІКА; 62-72 Copyright (c) 2026 Technical Mechanics |
| spellingShingle | KLETSKOV, O. M. LOBODA, V. V. ANALYSIS OF THE ANTIPLANE DEFORMATION OF A CRACKED QUASICRYSTAL WITH CONSIDERATION FOR SURFACE EFFECTS |
| title | ANALYSIS OF THE ANTIPLANE DEFORMATION OF A CRACKED QUASICRYSTAL WITH CONSIDERATION FOR SURFACE EFFECTS |
| title_full | ANALYSIS OF THE ANTIPLANE DEFORMATION OF A CRACKED QUASICRYSTAL WITH CONSIDERATION FOR SURFACE EFFECTS |
| title_fullStr | ANALYSIS OF THE ANTIPLANE DEFORMATION OF A CRACKED QUASICRYSTAL WITH CONSIDERATION FOR SURFACE EFFECTS |
| title_full_unstemmed | ANALYSIS OF THE ANTIPLANE DEFORMATION OF A CRACKED QUASICRYSTAL WITH CONSIDERATION FOR SURFACE EFFECTS |
| title_short | ANALYSIS OF THE ANTIPLANE DEFORMATION OF A CRACKED QUASICRYSTAL WITH CONSIDERATION FOR SURFACE EFFECTS |
| title_sort | analysis of the antiplane deformation of a cracked quasicrystal with consideration for surface effects |
| topic_facet | stress quasicrystal crack antiplane deformation phonon–phason coupling surface elasticity singular integro-differential equations. |
| url | https://journal-itm.dp.ua/ojs/index.php/ITM_j1/article/view/173 |
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