Abstract
For plate bending and stretching problems in two-dimensional (2D) dodecagonal quasi-crystal (QC) media, the reciprocal theorem and the general solution for QCs are applied in a novel way to obtain the appropriate stress and mixed boundary conditions accurate to all order. The method developed by Gregory and Wan is used to generate necessary conditions which the prescribed data on the edge of the plate must satisfy in order that it should generate a decaying state within the plate; these decaying state conditions are obtained explicitly for axisymmetric bending and stretching of a circular plate when stress or mixed conditions are imposed on the plate edge. They are then used for the correct formulation of boundary conditions for the interior solution. For the stress data, our boundary conditions coincide with those obtained in conventional forms of plate theories. More importantly, appropriate boundary conditions with a set of mixed edge-data are obtained for the first time. Furthermore, the corresponding necessary conditions for transversely isotropic elastic plate are obtained directly, and their isotropic elastic counterparts are also obtained.
Similar content being viewed by others
References
D Shechtman, I Blech, D Gratias and J W Cahn, Phys. Rev. Lett. 53, 1951 (1984)
I A Ovid’ko, Mater. Sci. Eng. A154, 29 (1992)
M Wollgarten, M Beyss, K Urban, H Liebertz and U Koster, Phys. Rev. Lett. 71, 549 (1993)
T Y Fan and Y W Mai, Appl. Mech. Rev. 57, 325 (2004)
D H Ding, W G Yang, C Z Hu and R H Wang, Phys. Rev. B48, 7003 (1993)
W G Yang, R H Wang, D H Ding and C Z Hu, Phys. Rev. B48, 6999 (1993)
R H Wang, W G Yang, C Z Hu and D H Ding, J. Phys.: Condens. Matter 9, 2411 (1997)
C Z Hu, W G Yang, R H Wang and D H Ding, Acta Crystallogr. 52, 251 (1996)
R D Gregory and F Y M Wan, J. Elast. 14, 27 (1984)
R D Gregory and F Y M Wan, Int. J. Solids Struct. 21, 1005 (1985)
R D Gregory and F Y M Wan, Edge effect in the stretching of plates. Local effects in the analysis of structures edited by P Ladeveze (Elsevier Science Publishers, Amsterdam, 1985) pp. 35–54
R D Gregory and F Y M Wan, ASME J. Appl. Mech. 55, 551 (1988)
Y H Lin and F Y M Wan, Stud. Appl. Math. 79, 93 (1988)
F Y M Wan, Int. J. Solids Struct. 40, 4107 (2003)
Y Gao and M Z Wang, Int. J. Solids Struct. 46, 1854 (2006)
F Y M Wan, Int. J. Solids Struct. 46, 1856 (2006)
K Tanaka, Y Mitarai and M Koiwa, Phil. Mag. A73, 1715 (1996)
H C Jeong and P J Steinhardt, Phys. Rev. B48, 9394 (1993)
M A Chernikov, H R Ott, A Bianchi, A Migliori and T W Darling, Phys. Rev. Lett. 80, 321 (1998)
H A Elliott, Proc. Camb. Phil. Soc. 44, 522 (1948)
A S Lodge, Q. J. Mech. Appl. Math. 8, 211 (1955)
S G Lekhnitskii, Theory of elasticity of an anisotropic body (Mir Publishers, Moscow, 1981)
G R Kirchhoff, J. Reine Angew. Math. (Crelle) 40, 51 (1850)
E Reissner, ASME J. Appl. Mech. 12, 69 (1945)
S P Timoshenko and J C Goodier, Theory of elasticity (McGraw-Hill, New York, 1970)
E Reissner, Proc. Roy. Soc. (London) A276, 178 (1963)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gao, Y., Xu, SP. & Zhao, BS. Stress and mixed boundary conditions for two-dimensional dodecagonal quasi-crystal plates. Pramana - J Phys 68, 803–817 (2007). https://doi.org/10.1007/s12043-007-0079-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12043-007-0079-4
Keywords
- Boundary conditions
- plates
- axisymmetric deformation
- two-dimensional dodecagonal quasi-crystals
- decaying states