0
Research Papers

Nonlocal Analytical Solutions for Multilayered One-Dimensional Quasicrystal Nanoplates

[+] Author and Article Information
Natalie Waksmanski

Department of Civil Engineering,
The University of Akron,
Akron, OH 44325
e-mail: npw5@zips.uakron.edu

Ernian Pan

Fellow ASME
Department of Civil Engineering,
The University of Akron,
Akron, OH 44325
e-mail: pan2@uakron.edu

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received August 8, 2016; final manuscript received September 28, 2016; published online February 3, 2017. Assoc. Editor: Matthew Brake.

J. Vib. Acoust 139(2), 021006 (Feb 03, 2017) (16 pages) Paper No: VIB-16-1391; doi: 10.1115/1.4035106 History: Received August 08, 2016; Revised September 28, 2016

An exact closed-form solution for the three-dimensional static deformation and free vibrational response of a simply supported and multilayered quasicrystal (QC) nanoplate with the nonlocal effect is derived. Numerical examples are presented for a homogeneous crystal nanoplate, homogenous QC nanoplate, and sandwich nanoplates with various stacking sequences. Induced by traction boundary conditions, extended displacements and stresses reveal the important role that the nonlocal parameter plays in the structural analysis of nanoquasicrystals (nano-QCs). The natural frequencies and the corresponding mode shapes of the nanoplates further show the influence of stacking sequence and phonon–phason coupling effect. This exact solution is useful for it provides benchmark results to assess the accuracy of finite element nano-QC models and can assist engineers in tuning their quasicrystal nanoplate design.

FIGURES IN THIS ARTICLE
<>
Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.

References

Guo, D. , Xie, G. , and Lou, J. , 2013, “ Mechanical Properties of Nanoparticles: Basics and Applications,” J. Phys. D: Appl. Phys., 47(1), p. 013001. [CrossRef]
Eringen, A. C. , 1972, “ Linear Theory of Nonlocal Elasticity and Dispersion of Plane Waves,” Int. J. Eng. Sci., 10(5), pp. 425–435. [CrossRef]
Yang, F. , Chong, A. C. M. , Lam, D. C. C. , and Tong, P. , 2002, “ Couple Stress Based Strain Gradient Theory for Elasticity,” Int. J. Solids Struct., 39(10), pp. 2731–2743. [CrossRef]
Lim, C. W. , Zhang, G. , and Reddy, J. N. , 2015, “ A Higher-Order Nonlocal Elasticity and Strain Gradient Theory and Its Applications in Wave Propagation,” J. Mech. Phys. Solids, 78, pp. 298–313. [CrossRef]
Arash, B. , and Wang, Q. , 2012, “ A Review on the Application of Nonlocal Elastic Models in Modelling of Carbon Nanotubes and Graphenes,” Comput. Mater. Sci., 51(1), pp. 303–313. [CrossRef]
Di Paola, M. , Failla, G. , Pirrotta, A. , Sofi, A. , and Zingales, M. , 2013, “ The Mechanically Based Non-Local Elasticity: An Overview of Main Results and Future Challenges,” Philos. Trans. R. Soc., A, 371(1993), pp. 1–16. [CrossRef]
Eringen, A. C. , 1984, “ Theory of Nonlocal Piezoelectricity,” J. Math. Phys., 25(3), pp. 717–727. [CrossRef]
Eringen, A. C. , 1973, “ Theory of Nonlocal Electromagnetic Elastic Solids,” J. Math. Phys., 14(6), pp. 733–740. [CrossRef]
Alaimo, A. , Bruno, M. , Milazzo, A. , and Orlando, C. , 2013, “ Nonlocal Model for a Magneto-Electro-Elastic Nanoplate,” AIP Conference Proceedings, Rhodes, Greece, Sept. 21–27, Vol. 1558, pp. 1208–1211.
Nan, C. W. , Bichurin, M. I. , Dong, S. X. , Viehland, D. , and Srinivasan, G. , 2008, “ Multiferroic Magnetoelectric Composites: Historical Perspective, Status, and Future Directions,” J. Appl. Phys., 103(3), p. 031101. [CrossRef]
Louzguine-Luzgin, D. V. , and Inoue, A. , 2008, “ Formation and Properties of Quasicrystals,” Annu. Rev. Mater. Res., 38(1), pp. 403–423. [CrossRef]
Trebin, H. R. , 2003, Quasicrystals: Structure and Physical Properties, Wiley-VCH Verlag GmbH and Co. KGaA, Weinheim, Germany.
Fan, T. Y. , 2011, The Mathematical Theory of Elasticity of Quasicrystals and Its Applications, Springer, Heidelberg, Germany.
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), pp. 1951–1953. [CrossRef]
Bindi, L. , Steinhardt, P. J. , Yao, N. , and Lu, P. J. , 2009, “ Natural Quasicrystals,” Science, 324(5932), pp. 1306–1309. [CrossRef] [PubMed]
Inoue, A. , and Takeuchi, A. , 2004, “ Recent Progress in Bulk Glassy, Nanoquasicrystalline and Nanocrystalline Alloys,” Mater. Sci. Eng. A, 375–377(1–2), pp. 16–30. [CrossRef]
Fournee, V. , Sharma, H. R. , Shimoda, M. , Tsai, A. P. , Unal, B. , Ross, A. R. , Lograsso, T. A. , and Thiel, P. A. , 2005, “ Quantum Size Effects in Metal Thin Films Grown on Quasicrystalline Substrates,” Phys. Rev. Lett., 95(15), p. 155504. [CrossRef] [PubMed]
Lefaix, H. , Prima, F. , Zanna, S. , Vermaut, P. , Dubot, P. , Marcus, P. , Janickovic, D. , and Svec, P. , 2007, “ Surface Properties of a Nano-Quasicrystalline Forming Ti Based System,” Mater. Trans., 48(3), pp. 278–286. [CrossRef]
Zhang, J. , Pei, L. , Du, H. , Liang, W. , Xu, C. , and Lu, B. , 2008, “ Effect of Mg-Based Spherical Quasicrystals on Microstructure and Mechanical Properties of AZ91 Alloys,” J. Alloys Compd., 453(1–2), pp. 309–315. [CrossRef]
Wang, Z. , Zhao, W. , Qin, C. , Cui, Y. , Fan, S. , and Jia, J. , 2012, “ Direct Preparation of Nano-Quasicrystals Via a Water-Cooled Wedge-Shaped Copper Mould,” J. Nanomater., 2012, p. 708240.
Inouea, A. , Kongb, F. , Zhua, S. , Liud, C. T. , and Al-Marzoukic, F. , 2015, “ Development and Applications of Highly Functional Al-Based Materials by Use of Metastable Phases,” Mater. Res., 18(6), pp. 1414–1425. [CrossRef]
Wang, Y. Z. , Li, F. M. , and Kishimoto, K. , 2012, “ Effects of Axial Load and Elastic Matrix on Flexural Wave Propagation in Nanotube With Nonlocal Timoshenko Beam Model,” ASME J. Vib. Acoust., 134(3), p. 031011. [CrossRef]
Lu, P. , Zhang, P. Q. , Lee, H. P. , Wang, C. M. , and Reddy, J. N. , 2007, “ Non-Local Elastic Plate Theories,” Proc. R. Soc. A, 463(2088), pp. 3225–3240. [CrossRef]
Wang, Q. , and Varadan, V . K. , 2007, “ Application of Nonlocal Elastic Shell Theory in Wave Propagation Analysis of Carbon nanotubes,” Smart Mater. Struct., 16(1), pp. 178–190. [CrossRef]
Waksmanski, N. , Pan, E. , Yang, L. Z. , and Gao, Y. , 2014, “ Free Vibration of a Multilayered One-Dimensional Quasi-Crystal Plate,” ASME J. Vib. Acoust., 136(4), p. 041019. [CrossRef]
Pagano, N. J. , 1970, “ Exact Solutions for Rectangular Bidirectional Composites and Sandwich Plates,” J. Compos. Mater., 4(1), pp. 20–34. [CrossRef]
Pan, E. , 2001, “ Exact Solution for Simply Supported and Multilayered Magneto-Electro-Elastic Plates,” ASME J. Appl. Mech., 68(4), pp. 608–618. [CrossRef]
Pan, E. , and Heyliger, P. R. , 2002, “ Free Vibrations of Simply Supported and Multilayered Magneto-Electro-Elastic Plates,” J. Sound Vib., 252(3), pp. 429–442. [CrossRef]
Ding, D. H. , Yang, W. G. , Hu, C. Z. , and Wang, R. H. , 1993, “ Generalized Elasticity Theory of Quasicrystals,” Phys. Rev. B, 48(10), pp. 7003–7010. [CrossRef]
Bak, P. , 1985, “ Symmetry, Stability and Elastic Properties of Icosahedral Incommensurate Crystals,” Phys. Rev. B, 32(9), pp. 5764–5772. [CrossRef]
Yang, L. Z. , Gao, Y. , Pan, E. , and Waksmanski, N. , 2015, “ An Exact Closed-Form Solution for a Multilayered One-Dimensional Orthorhombic Quasicrystal Plate,” Acta Mech., 226(11), pp. 3611–3621. [CrossRef]
Narendar, S. , and Gopalakrishnan, S. , 2009, “ Nonlocal Scale Effects on Wave Propagation in Multi-Walled Carbon Nanotubes,” Comput. Mater. Sci., 47(2), pp. 526–538. [CrossRef]
Pan, E. , and Waksmanski, N. , 2016, “ Deformation of a Layered Magnetoelectroelastic Plate With Nonlocal Effect, an Analytical Three-Dimensional Solution,” Smart Mater. Struct., 25(9), p. 095013. [CrossRef]
Fan, T. Y. , Xie, L. Y. , Fan, L. , and Wang, Q. Z. , 2011, “ Interface of Quasicrystals and Crystal,” Chin. Phys. B, 20(7), p. 076102. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Schematic configuration of a QC nanoplate with J layers

Grahic Jump Location
Fig. 2

Variation of phonon stress σ33 along the thickness direction in homogeneous crystal nanoplate made of BaTiO3 with different nonlocal parameters l. Results in the homogeneous QC nanoplate made of Al–Ni–Co, sandwich C/Q/C nanoplate, and sandwich Q/C/Q nanoplate overlap the phonon stress σ33.

Grahic Jump Location
Fig. 3

Through-thickness variation of the normalized phonon (a) displacement u1, (b) displacement u3, (c) normal stress σ11, and (d) shear stress σ13 in the homogeneous crystal nanoplate with different nonlocal parameters l

Grahic Jump Location
Fig. 4

Through-thickness variation of the normalized (a) phonon displacement u1, (b) phonon displacement u3, (c) phonon normal stress σ11, (d) phonon shear stress σ13, (e) phason displacement w3, and (f) phason stress H33 in the homogeneous QC nanoplate with different nonlocal parameters l

Grahic Jump Location
Fig. 5

Through-thickness variation of the normalized (a) phonon displacement u1, (b) phonon displacement u3, (c) phonon normal stress σ11, (d) phonon shear stress σ13, (e) phason displacement w3, and (f) phason stress H33 in the sandwich C/Q/C nanoplate with different nonlocal parameters l

Grahic Jump Location
Fig. 6

Through-thickness variation of the normalized (a) phonon displacement u1, (b) phonon displacement u3, (c) phonon normal stress σ11, (d) phonon shear stress σ13, (e) phason displacement w3, and (f) phason stress H33 in the sandwich Q/C/Q nanoplate with different nonlocal parameters l

Grahic Jump Location
Fig. 7

Nonlocal effect on the (a) phonon displacement u1 and (b) phonon displacement u3 along the thickness direction of the homogeneous crystal nanoplate on the first modal frequency

Grahic Jump Location
Fig. 8

Nonlocal effect on the (a) phonon displacement u3 and (b) phason displacement w3 along the thickness direction of the homogeneous QC nanoplate on the first modal frequency

Grahic Jump Location
Fig. 9

Nonlocal effect on the (a) phonon displacement u1, (b) phonon displacement u3, and (c) phason displacement w3 along the thickness direction of the C/Q/C sandwich nanoplate on the first modal frequency

Grahic Jump Location
Fig. 10

Nonlocal effect on the (a) phonon displacement u1, (b) phonon displacement u3, and (c) phason displacement w3 along the thickness direction of the Q/C/Q sandwich nanoplate on the first modal frequency

Grahic Jump Location
Fig. 11

Nonlocal effect on the (a) phonon displacement u1 and (b) phonon displacement u3 along the thickness direction of the homogenous crystal nanoplate on the third modal frequency

Grahic Jump Location
Fig. 12

Nonlocal effect on the (a) phonon displacement u1 and (b) phason displacement w3 along the thickness direction of the homogenous QC on the third modal frequency

Grahic Jump Location
Fig. 13

Nonlocal effect on the (a) phonon displacement u1, (b) phonon displacement u3, and (c) phason displacement w3 along the thickness direction of the C/Q/C sandwich nanoplate on the third modal frequency

Grahic Jump Location
Fig. 14

Nonlocal effect on the (a) phonon displacement u1, (b) phonon displacement u3, and (c) phason displacement w3 along the thickness direction of the C/Q/C sandwich nanoplate on the third modal frequency

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In