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An exact closed-form solution for a multilayered one-dimensional orthorhombic quasicrystal plate

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Abstract

By extending the pseudo-Stroh formalism to multilayered one-dimensional orthorhombic quasicrystal plates, we derive an exact closed-form solution for simply supported plates under surface loadings. The propagator matrix method is introduced to efficiently and accurately treat the multilayered cases. As a numerical example, a sandwich plate made of quasicrystals and crystals with two different stacking sequences is investigated. The displacement and stress fields for these two stacking sequences are presented, which clearly demonstrate the importance of the stacking sequences on the induced physical quantities. Our exact closed-form solution should be of particular interest to the design of one-dimensional quasicrystal laminated plates. The numerical results can be further used as benchmarks to various numerical methods, such as the finite element and finite difference methods, on the analysis of laminated composites made of one-dimensional quasicrystals.

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References

  1. Shechtman D., Blech I., Gratias D., Cahn J.W.: Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53, 1951–1953 (1984)

    Article  Google Scholar 

  2. Ovid’ko I.A.: Plastic deformation and decay of dislocations in quasi-crystals. Mater. Sci. Eng. A 154, 29–33 (1992)

    Article  Google Scholar 

  3. Wollgarten M., Beyss M., Urban K., Liebertz H., Köster U.: Direct evidence for plastic deformation of quasicrystals by means of a dislocation mechanism. Phys. Rev. Lett. 71, 549–552 (1993)

    Article  Google Scholar 

  4. Levine D., Steinhardt P.J.: Quasicrystals: a new class of ordered structures. Phys. Rev. Lett. 53, 2477–2480 (1984)

    Article  Google Scholar 

  5. Fan T.Y.: The Mathematical Elasticity of Quasicrystals and its Applications. Springer, Heidelberg (2011)

    Book  Google Scholar 

  6. Ishimasa T., Nissen H.U., Fukano Y.: New ordered state between crystalline and amorphous in Ni–Cr particles. Phys. Rev. Lett. 55, 511–513 (1985)

    Article  Google Scholar 

  7. Merlin R., Bajema K., Clarke R., Juang F.Y., Bhattacharya P.K.: Quasiperiodic GaAs–AlAs heterostructures. Phys. Rev. Lett. 55, 1768–1770 (1985)

    Article  Google Scholar 

  8. Balbyshev V.N., King D.J., Khramov A.N., Kasten L.S., Donley M.S.: Investigation of quaternary Al-based quasicrystal thin films for corrosion protection. Thin Solid Films 447–448, 558–563 (2004)

    Article  Google Scholar 

  9. Xu D.K., Han E.H.: Effects of icosahedral phase formation on the microstructure and mechanical improvement of Mg alloys: a review. Prog. Nat. Sci.: Mat. Int. 22, 364–385 (2012)

    Article  Google Scholar 

  10. Bak P.: Phenomenological theory of icosahedral incommensurate (“quasiperiodic”) order in Mn–Al alloys. Phys. Rev. Lett. 54, 1517–1519 (1985)

    Article  Google Scholar 

  11. Bak P.: Symmetry, stability, and elastic properties of icosahedral incommensurate crystals. Phys. Rev. B 32, 5764–5772 (1985)

    Article  MathSciNet  Google Scholar 

  12. Levine D., Clubensky T., Qstlund S., Ramaswamy S., Steinhardt P.J., Toner J.: Elasticity and dislocations in pentagonal and icosahedral quasicrystals. Phys. Rev. Lett. 54, 1520–1523 (1985)

    Article  Google Scholar 

  13. Ding D.H., Yang W.G., Hu C.Z., Wang R.H.: Generalized elasticity theory of quasicrystals. Phys. Rev. B 48, 7003–7010 (1993)

    Article  Google Scholar 

  14. Jeong H.C., Steinhardt P.J.: Finite-temper elasticity phase transition in decagonal quasicrystals. Phys. Rev. B 48, 9394–9403 (1993)

    Article  Google Scholar 

  15. Letoublon A., de Boissien M., Boudard M., Mancini L., Gastaldi J., Hennion B., Caudron R., Bellissent R.: Phason elastic constants of the icosahedral Al–Pd–Mn phase derived from diffuse scattering measurements. Phil. Mag. Lett. 81, 273–283 (2001)

    Article  Google Scholar 

  16. Edagawa K., So G.Y.: Experimental evaluation of phonon–phason coupling in icosahedral quasicrystals. Phil. Mag. 87, 77–95 (2007)

    Article  Google Scholar 

  17. Hu C.Z., Ding R.H., Ding D.H.: Symmetry groups, physical property tensors, elasticity and dislocations in quasicrystals. Rep. Prog. Phys. 63, 1–39 (2000)

    Article  MathSciNet  Google Scholar 

  18. Fan T.Y.: Mathematical theory and methods of mechanics of quasicrystalline materials. Engineering 5, 407–448 (2013)

    Article  Google Scholar 

  19. Ding D.H., Wang R.H., Yang W.G., Hu C.Z., Qin Y.L.: Elasticity theory of straight dislocations in quasicrystals. Philos. Mag. Lett. 72, 353–359 (1995)

    Article  Google Scholar 

  20. Li X.F., Fan T.Y.: A straight dislocation in one-dimensional hexagonal quasicrystals. Phys. Stat. Sol. (b) 212, 19–26 (1999)

    Article  Google Scholar 

  21. Li L.H., Fan T.Y.: Exact solutions of two semi-infinite collinear cracks in a strip of one dimensional hexagonal quasicrystal. Appl. Math. Comput. 196, 1–5 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  22. Guo Y.H., Yu J., Xing Y.M.: Anti-plane analysis on a finite crack in a one-dimensional hexagonal quasicrystal strip. Mech. Res. Commun. 52, 40–45 (2013)

    Article  Google Scholar 

  23. Li X.Y.: Elastic field in an infinite medium of one-dimensional hexagonal quasicrystal with a planar crack. Int. J. Solids Struct. 51, 1442–1455 (2014)

    Article  Google Scholar 

  24. Chen W.Q., Ma Y.L., Ding H.J.: On three-dimensional elastic problems of one-dimensional hexagonal quasicrystal bodies. Mech. Res. Commun. 31, 633–641 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  25. Gao Y., Xu S.P., Zhao B.S.: General solutions of equilibrium equations for 1D hexagonal quasicrystals. Mech. Res. Commun. 36, 302–308 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  26. Gao Y.: The exact theory of one-dimensional quasicrystal deep beams. Acta Mech. 212, 283–292 (2010)

    Article  MATH  Google Scholar 

  27. Gao Y., Ricoeur A.: The refined theory of one-dimensional quasi-crystals in thick plate structures. J. Appl. Mech. ASME 78, 031021 (2011)

    Article  Google Scholar 

  28. Sladek J., Sladek V., Pan E.: Bending analysis of 1D orthorhombic quasicrystal plates. Int. J. Solids struct. 50, 3975–3983 (2013)

    Article  Google Scholar 

  29. Li X.Y.: Fundamental solutions of penny-shaped and half-infinite plane cracks embedded in an infinite space of one-dimensional hexagonal quasi-crystal under thermal loading. Proc. R. Soc. A. 469, 20130023 (2013)

    Article  Google Scholar 

  30. Gao Y., Zhao Y.T., Zhao B.S.: Boundary value problems of holomorphic vector functions in 1D QCs. Phys. B-Condens. Matter 394, 56–61 (2007)

    Article  Google Scholar 

  31. Radi E., Mariano P.M.: Steady-state propagation of dislocations in quasicrystals. Proc. R. Soc. A 467, 3490–3508 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  32. Radi, E., Mariano, P.M.: Dynamic steady-state crack propagation in quasicrystals. Math. Meth. Appl. Sci. 34, 1–23 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  33. Pan E.: Exact solution for simply supported and multilayered magneto-electro-elastic plates. J. Appl. Mech. 68, 608–618 (2001)

    Article  MATH  Google Scholar 

  34. Wang X., Sudak L.J.: Three-dimensional analysis of multi-layered functionally graded anisotropic cylindrical panel under thermomechanical loading. Mech. Mater. 40, 235–254 (2008)

    Article  Google Scholar 

  35. Yang L.Z., Gao Y., Pan E., Waksmanski N.: An exact solution for a multilayered two-dimensional decagonal quasicrystal plate. Int. J. Solids Struct. 51, 1737–1749 (2014)

    Article  Google Scholar 

  36. Wang R.H., Yang W.G., Hu C.Z., Ding D.H.: Point and space groups and elastic behaviors of one dimensional quasicrystals. J. Phys: Condens. Matter 9, 2411–2422 (1997)

    Google Scholar 

  37. Mariano P.M.: Mechanics of quasi-periodic alloys. J. Nonlinear Sci. 6, 45–77 (2006)

    Article  MathSciNet  Google Scholar 

  38. Mariano P.M., Planas J.: Phason self-actions in quasicrystal. Phys. D 249, 46–57 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  39. Colli S., Mariano P.M.: The standard description of quasicrystal linear elasticity may produce non-physical results. Phys. Lett. A 375, 3335–339 (2011)

  40. Stroh A.N.: Dislocations and cracks in anisotropic elasticity. Philos. Mag. 3, 625–646 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  41. Pan E.: A general boundary element analysis of 2-D linear elastic fracture mechanics. Int. J. Fract. 88, 41–59 (1997)

    Article  Google Scholar 

  42. Kausel E., Roesset J.M.: Stiffness matrices for layered soils. Bull. Seismol. Soc. Am. 71, 1743–1761 (1981)

    Google Scholar 

  43. Fan T.Y., Xie L.Y., Fan L., Wang Q.Z.: Interface of quasicrystal and crystal. Chin. Phys. B 20, 076102 (2011)

    Article  Google Scholar 

  44. Pan E.: Static Green’s functions in multilayered half spaces. Appl. Math. Model. 21, 509–521 (1997)

    Article  MATH  Google Scholar 

  45. Lee J.S., Jiang L.Z.: Exact electroelastic analysis of piezoelectric laminae via state space approach. Int. J. Solids Struct. 33, 977–990 (1996)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Civil and Environmental Engineering, University of Science and Technology Beijing, Beijing, 100083, China

    Lian-Zhi Yang

  2. College of Science, China Agricultural University, Beijing, 100083, China

    Lian-Zhi Yang & Yang Gao

  3. Department of Civil Engineering, University of Akron, Akron, OH, 44325-3905, USA

    Ernian Pan & Natalie Waksmanski

Authors
  1. Lian-Zhi Yang
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  2. Yang Gao
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  3. Ernian Pan
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  4. Natalie Waksmanski
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Correspondence to Yang Gao.

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Yang, LZ., Gao, Y., Pan, E. et al. An exact closed-form solution for a multilayered one-dimensional orthorhombic quasicrystal plate. Acta Mech 226, 3611–3621 (2015). https://doi.org/10.1007/s00707-015-1395-2

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  • DOI: https://doi.org/10.1007/s00707-015-1395-2

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