Structural Engineering and Mechanics

  • 제53권3호
  • /
  • Pages.411-427
  • /
  • 2015
  • /
  • 1225-4568(pISSN)
  • /
  • 1598-6217(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

The refined theory of 2D quasicrystal deep beams based on elasticity of quasicrystals

  • Gao, Yang (College of Science, China Agricultural University) ;
  • Yu, Lian-Ying (College of Science, China Agricultural University) ;
  • Yang, Lian-Zhi (College of Science, China Agricultural University) ;
  • Zhang, Liang-Liang (College of Science, China Agricultural University)
  • 투고 : 2013.07.09
  • 심사 : 2014.06.20
  • 발행 : 2015.02.10
⟨ 이전 논문 다음 논문 ⟩

초록

Based on linear elastic theory of quasicrystals, various equations and solutions for quasicrystal beams are deduced systematically and directly from plane problem of two-dimensional quasicrystals. Without employing ad hoc stress or deformation assumptions, the refined theory of beams is explicitly established from the general solution of quasicrystals and the Lur'e symbolic method. In the case of homogeneous boundary conditions, the exact equations and exact solutions for beams are derived, which consist of the fourth-order part and transcendental part. In the case of non-homogeneous boundary conditions, the exact governing differential equations and solutions under normal loadings only and shear loadings only are derived directly from the refined beam theory, respectively. In two illustrative examples of quasicrystal beams, it is shown that the exact or accurate analytical solutions can be obtained in use of the refined theory.

키워드

과제정보

연구 과제 주관 기관 : National Natural Science Foundation

참고문헌

  1. Bak, P. (1985), "Phenomenological theory of icosahedron incommensurate (quasiperiodic) order in Mn-Al alloys", Phys. Rev. Lett., 54(14), 1517-1519. https://doi.org/10.1103/PhysRevLett.54.1517
  2. Barrett, K.E. and Ellis, S. (1988), "An exact theory of elastic plates", Int. J. Solid. Struct., 24(9), 859-880. https://doi.org/10.1016/0020-7683(88)90038-8
  3. Cheng, S. (1979), "Elasticity theory of plates and a refined theory", ASME J. Appl. Mech., 46(3), 644-650. https://doi.org/10.1115/1.3424620
  4. Chernikov, M.A., Ott, H.R., Bianchi, A., Migliori, A. and Darling, T.W. (1998), "Elastic moduli of a single quasicrystal of decagonal Al-Ni-Co: Evidence for transverse elastic isotropy", Phys. Rev. Lett., 80(2), 321-324. https://doi.org/10.1103/PhysRevLett.80.321
  5. Ding, D.H., Yang, W.G., Hu, C.Z. and Wang, R.H. (1993), "Generalized elasticity theory of quasicrystals", Phys. Rev. B, 48(10), 7003-7010. https://doi.org/10.1103/PhysRevB.48.7003
  6. Edagawa, K. (2007), "Phonon-phason coupling in decagonal quasicrystals", Philos. Mag., 87(18-21), 2789-2798. https://doi.org/10.1080/14786430701264178
  7. Fan, T.Y. (2011), The Mathematical Elasticity of Quasicrystals and Its Applications, Springer, Heidelberg.
  8. Fan, T.Y. and Mai, Y.W. (2004), "Elasticity theory, fracture mechanics, and some relevant thermal properties of quasi-crystalline materials", Appl. Mech. Rev., 57(5), 325-343. https://doi.org/10.1115/1.1763591
  9. Gao, Y. (2009), "The appropriate edge conditions for two-dimensional quasicrystal semi-infinite strips with mixed edge-data", Int. J. Solid. Struct., 46(9), 1849-1855. https://doi.org/10.1016/j.ijsolstr.2008.12.017
  10. Gao, Y. and Ricoeur, A. (2011), "The refined theory of one-dimensional quasicrystals in thick plate structures", ASME J. Appl. Mech., 78(3), 031021. https://doi.org/10.1115/1.4003367
  11. Gao, Y. and Wang, M.Z. (2004), "The refined theory of magnetoelastic rectangular beams", Acta Mech., 173(1-4), 147-161. https://doi.org/10.1007/s00707-004-0171-5
  12. Gao, Y. and Wang, M.Z. (2005), "A refined beam theory based on the refined plate theory", Acta Mech., 177(1-4), 191-197. https://doi.org/10.1007/s00707-005-0223-5
  13. Gao, Y., Xu, B.X. and Zhao, B.S. (2007), "The refined theory of beams for a transversely isotropic body", Acta Mech., 191(1-2), 109-122. https://doi.org/10.1007/s00707-006-0436-2
  14. Gao, Y. and Zhao, B.S. (2007), "A note on the nonuniqueness of the Boussinesq-Galerkin solution in elastic theory", Int. J. Solid. Struct., 44(5), 1685-1689. https://doi.org/10.1016/j.ijsolstr.2006.06.024
  15. Gao, Y. and Zhao, B.S. (2009), "General solutions of three-dimensional problems for two-dimensional quasicrystals", Appl. Math. Model., 33(8), 3382-3391. https://doi.org/10.1016/j.apm.2008.11.001
  16. Hu, C.Z., Wang, R.H. and Ding, D.H. (2000), "Symmetry groups, physical property tensors, elasticity and dislocations in quasicrystals", Rep. Prog. Phys., 63(1), 1-39. https://doi.org/10.1088/0034-4885/63/1/201
  17. Hu, C.Z., Wang, R.H., Yang, W.G. and Ding, D.H. (1996), "Point groups and elastic properties of two-dimensional quasicrystals", Acta Crystallogr. Sect. A, 52, 251-256.
  18. Janssen, T. (1992), "The symmetry operations for n-dimensional periodic and quasi-periodic structures", Z. Kristall., 198, 17-32. https://doi.org/10.1524/zkri.1992.198.1-2.17
  19. Jeong, H.C. and Steinhardt, P.J. (1993), "Finite-temperature elasticity phase transition in decagonal quasicrystals", Phys. Rev. B, 48(13), 9394-9403. https://doi.org/10.1103/PhysRevB.48.9394
  20. Levine, D. and Steinhardt, P.J. (1986), "Quasicrystals. 1. definition and structure", Phys. Rev. B, 34(2), 596-616.
  21. Levinson, M. (1981), "A new rectangular beam theory", J. Sound Vib., 74(1), 81-87. https://doi.org/10.1016/0022-460X(81)90493-4
  22. Lubensky, T.C., Ramaswamy, S. and Joner, J. (1985), "Hydrodynamics of icosahedral quasicrystals", Phys. Rev. B, 32(11), 7444-7452. https://doi.org/10.1103/PhysRevB.32.7444
  23. Lur'e, A.I. (1964), Three-dimensional problems of the theory of elasticity, Interscience, New York.
  24. Ovidko, I.A. (1992), "Plastic deformation and decay of dislocations in quasicrystals", Mater. Sci. Eng. A, 154(1), 29-33. https://doi.org/10.1016/0921-5093(92)90359-9
  25. Ronchetti, M. (1987), "Quasicrystals, an introduction overview", Philos. Mag. B, 56(2), 237-249. https://doi.org/10.1080/13642818708208530
  26. Shechtman, D., Blech, I., Gratias, D. and Cahn, J.W. (1984), "Metallic phase with long-range orientational order and no translational symmetry", Phys. Rev. Lett., 53(20), 1951-1953. https://doi.org/10.1103/PhysRevLett.53.1951
  27. Socolar, J.E.S., Lubensky, T.C. and Steinhardt, P.J. (1986), "Phonons, phasons and dislocations in quasi-crystals", Phys. Rev. B, 34(5), 3345-3360.
  28. Stadnik, Z. (1999), Physical Properties of Quasicrystals, Springer Series in Solid State Sciences, Springer, Berlin.
  29. Takeuchi, S. and Edagawa, K. (2007), Elasticity and plastic properties of quasicrystals, Elsevier, Amsterdam.
  30. Tanaka, K., Mitarai, Y. and Koiwa, M. (1996), "Elastic constants of Al-based icosahedral quasicrystals", Philos. Mag. A, 73(6), 1715-1723. https://doi.org/10.1080/01418619608243008
  31. Timoshenko, S.P. (1921), "On the correction for shear of the differential equation for transverse vibration of prismatic bars", Philos. Mag., 41(245), 744-746. https://doi.org/10.1080/14786442108636264
  32. Timoshenko, S.P. and Goodier, J.C. (1970), Theory of Elasticity, McGraw-Hill, New York.
  33. Wang, C.M., Reddy, J.N. and H., L.K. (2000), Shear Deformable Beams and Plates: Relationships with Classical Solutions, Elsevier, Amsterdam.
  34. Wang, F.Y. (1990), "Two-dimensional theories deduced from three-dimensional theory for a transversely isotropic body. I. Plate problems", Int. J. Solid. Struct., 26(4), 455-470. https://doi.org/10.1016/0020-7683(90)90068-7
  35. Wollgarten, M., Beyss, M., Urban, K., Liebertz, H. and Koster, U. (1993), "Direct evidence for plastic deformation of quasicrystals by means of a dislocation mechanism", Phys. Rev. Lett., 71(4), 549-552. https://doi.org/10.1103/PhysRevLett.71.549
  36. Zhao, B.S., Wu, D. and Wang, M.Z. (2013), "The refined theory and the decomposed theorem of a transversely isotropic elastic plate", Eur. J. Mech. A Solid, 39, 243-250. https://doi.org/10.1016/j.euromechsol.2012.12.002

피인용 문헌

  1. Size-dependent effect on functionally graded multilayered two-dimensional quasicrystal nanoplates under patch/uniform loading vol.229, pp.8, 2018, https://doi.org/10.1007/s00707-018-2177-4
  2. An efficient and simple refined theory for free vibration of functionally graded plates under various boundary conditions vol.16, pp.1, 2015, https://doi.org/10.12989/gae.2018.16.1.001
  3. Electric-elastic analysis of multilayered two-dimensional decagonal quasicrystal circular plates with simply supported or clamped boundary conditions vol.26, pp.9, 2021, https://doi.org/10.1177/1081286520981618