Bending and vibration of two-dimensional decagonal quasicrystal nanoplates via modified couple-stress theory

doi:10.1007/s10483-022-2818-6

Applied Mathematics and Mechanics (English Edition) ›› 2022, Vol. 43 ›› Issue (3): 371-388.doi: https://doi.org/10.1007/s10483-022-2818-6

Bending and vibration of two-dimensional decagonal quasicrystal nanoplates via modified couple-stress theory

Miao ZHANG¹, Junhong GUO^1,2, Yansong LI³

1. Department of Mechanics, Inner Mongolia University of Technology, Hohhot 010051, China;
2. School of Aeronautics, Inner Mongolia University of Technology, Hohhot 010051, China;
3. College of Architecture and Civil Engineering, Guangdong University of Petrochemical Technology, Maoming 525000, Guangdong Province, China

收稿日期:2021-08-04 修回日期:2021-11-09 发布日期:2022-02-22
通讯作者: Junhong GUO, E-mail: jhguo@imut.edu.cn
基金资助:
the National Natural Science Foundation of China (Nos. 11862021 and 12072166), the Program for Science and Technology of Inner Mongolia Autonomous Region of China (No. 2021GG0254), and the Natural Science Foundation of Inner Mongolia Autonomous Region of China (No. 2020MS01006)

Bending and vibration of two-dimensional decagonal quasicrystal nanoplates via modified couple-stress theory

Miao ZHANG¹, Junhong GUO^1,2, Yansong LI³

1. Department of Mechanics, Inner Mongolia University of Technology, Hohhot 010051, China;
2. School of Aeronautics, Inner Mongolia University of Technology, Hohhot 010051, China;
3. College of Architecture and Civil Engineering, Guangdong University of Petrochemical Technology, Maoming 525000, Guangdong Province, China

Received:2021-08-04 Revised:2021-11-09 Published:2022-02-22
Contact: Junhong GUO, E-mail: jhguo@imut.edu.cn
Supported by:
the National Natural Science Foundation of China (Nos. 11862021 and 12072166), the Program for Science and Technology of Inner Mongolia Autonomous Region of China (No. 2021GG0254), and the Natural Science Foundation of Inner Mongolia Autonomous Region of China (No. 2020MS01006)

摘要/Abstract

摘要： Based on the modified couple-stress theory, the three-dimensional (3D) bending deformation and vibration responses of simply-supported and multilayered two-dimensional (2D) decagonal quasicrystal (QC) nanoplates are investigated. The surface loading is assumed to be applied on the top surface in the bending analysis, the traction-free boundary conditions on both the top and bottom surfaces of the nanoplates are used in the free vibration analysis, and a harmonic concentrated point loading is applied on the top surfaces of the nanoplates in the harmonic response analysis. The general solutions of the extended displacement and traction vectors for the homogeneous QC nanoplates are derived by solving the eigenvalue problem reduced from the final governing equations of motion with the modified couple-stress effect. By utilizing the propagator matrix method, the analytical solutions of the displacements of the phonon and phason fields for bending deformation, the natural frequency of free vibration, and the displacements of the phonon and phason fields for the harmonic responses are obtained. Numerical examples are illustrated to show the effects of the quasiperiodic direction, the material length scale parameter, and the the stacking sequence of the nanoplates on the bending deformation and vibration responses of two sandwich nanoplates made of QC and crystal materials.

关键词: quasicrystal (QC), bending, vibration, layered nanoplate

Abstract: Based on the modified couple-stress theory, the three-dimensional (3D) bending deformation and vibration responses of simply-supported and multilayered two-dimensional (2D) decagonal quasicrystal (QC) nanoplates are investigated. The surface loading is assumed to be applied on the top surface in the bending analysis, the traction-free boundary conditions on both the top and bottom surfaces of the nanoplates are used in the free vibration analysis, and a harmonic concentrated point loading is applied on the top surfaces of the nanoplates in the harmonic response analysis. The general solutions of the extended displacement and traction vectors for the homogeneous QC nanoplates are derived by solving the eigenvalue problem reduced from the final governing equations of motion with the modified couple-stress effect. By utilizing the propagator matrix method, the analytical solutions of the displacements of the phonon and phason fields for bending deformation, the natural frequency of free vibration, and the displacements of the phonon and phason fields for the harmonic responses are obtained. Numerical examples are illustrated to show the effects of the quasiperiodic direction, the material length scale parameter, and the the stacking sequence of the nanoplates on the bending deformation and vibration responses of two sandwich nanoplates made of QC and crystal materials.

Key words: quasicrystal (QC), bending, vibration, layered nanoplate

中图分类号:

Miao ZHANG, Junhong GUO, Yansong LI. Bending and vibration of two-dimensional decagonal quasicrystal nanoplates via modified couple-stress theory[J]. Applied Mathematics and Mechanics (English Edition), 2022, 43(3): 371-388.

参考文献

[1] SHECHTMAN, D., BLECH, I., GRATIAS, D., and CAHN, J. W. Metallic phase with long-range orientational order and no translational symmetry. Physical Review Letters, 53, 1951–1953(1984)
[2] FAN, T. Y. The Mathematical Theory of Elasticity of Quasicrystals and Its Application, Springer, New York (2016)
[3] LOUZGUINE-LUZGIN, D. V. and INOUE, A. Formation and properties of quasicrystals. Annual Review of Materials Research, 38, 403–423(2008)
[4] BINDI, L., YAO, N., LIN, C., HOLLISTER, L. S., ANDRONICOS, C. L., DISTLER, V. V., EDDY, M. P., KOSTIN, A., KRYACHKO, V., MACPHERSON, G. J., STEINHARDT, W. M., YUDOVSKAYA, M., and STEINHARDT, P. J. Natural quasicrystal with decagonal symmetry. Scientific Reprts, 5, 9111(2015)
[5] INOUE, A., KIMURA, H., and AMIYA, K. Recent progress in bulk glassy, nano-quasicrystalline and nanocrystalline alloys. Materials Science and Engineering A, 375, 16–30(2004)
[6] USTINOV, A. I., MOVCHAN, B. A., and POLISHCHUK, S. S. Formation of nanoquasicrystalline Al-Cu-Fe coatings at electron beam physical vapour deposition. Scripta Materialia, 50, 533–537(2004)
[7] GALANO, M., MARSH, A., AUDEBERT, F., XU, W., and RAMUNDO, M. Nanoquasicrystalline Al-based matrix/γ-Al₂O₃ nanocomposites. Journal of Alloys Compounds, 643, S99(2015)
[8] INOUE, A., KONG, F., ZHU, S., LIU, C. T., and AL-MARZOUKI, F. Development and applications of highly functional Al-based materials by use of metastable phases. Materials Research, 18(6), 1414–1425(2015)
[9] HUANG, H., KATO, H., CHEN, C. L., WANG, Z. C., and YUAN, G. Y. The effect of nanoquasicrystals on mechanical properties of as-extruded Mg-Zn-Gd alloy. Materials Letters, 79, 281–283(2012)
[10] YANG, L. Z., GAO, Y., PAN, E., and WAKSMANSK, N. An exact closed-form solution for a multilayered one-dimensional orthorhombic quasicrystal plate. Acta Mechanica, 226, 3611–3621(2015)
[11] YANG, L. Z., GAO, Y., PAN, E., and WAKSMANSK, N. An exact solution for a multilayered two-dimensional decagonal quasicrystal plate. International Journal of Solids and Structures, 51, 1737–1749(2014)
[12] WAKSMANSK, N., PAN, E., YANG, L. Z., and GAO, Y. Free vibration of a multilayered onedimensional quasi-crystal plate. Journal of Vibration and Acoustics, 136, 041019(2014)
[13] ERINGEN, A. C. Nonlocal Continuum Field Theories, Springer, New York (2002)
[14] AIFANTIS, E. C. Strain gradient interpretation of size effects. International Journal of Fracture, 95, 299–314(1999)
[15] YANG, F., CHONG, A. C. M., LAM, D. C. C., and TONG, P. Couple stress based strain gradient theory for elasticity. International Journal of Solids and Structures, 39, 2731-43(2002)
[16] WAKSMANSK, N. and PAN, E. Nonlocal analytical solutions for multilayered one-dimensional quasicrystal nanoplates. Journal of Vibration and Acoustics, 139(2), 021006(2017)
[17] ZHANG, L., GUO, J. H., and XING, Y. M. Bending deformation of multilayered one-dimensional hexagonal piezoelectric quasicrystal nanoplates with nonlocal effect. International Journal of Solids and Structures, 132-133, 278–302(2018)
[18] LI, Y., YANG, L. Y., ZHANG, L. L., and GAO, Y. Size-dependent effect on functionally graded multilayered two-dimensional quasicrystal nanoplates under patch/uniform loading. Acta Mechanica, 229(8), 3501–3515(2018)
[19] ZHANG, L., GUO, J. H., and XING, Y. M. Nonlocal analytical solution of functionally graded multilayered one-dimensional hexagonal piezoelectric quasicrystal nanoplates. Acta Mechanica, 230(5), 1781–1810(2019)
[20] GUO, J. H., SUN, T. Y., and PAN, E. Three-dimensional nonlocal buckling of composite nanoplates with coated one-dimensional quasicrystal in an elastic medium. International Journal of Solids and Structures, 185-186, 272–280(2020)
[21] LI, Y., YANG, L. Y., ZHANG, L. L., and GAO, Y. Nonlocal free and forced vibration of multilayered two-dimensional quasicrystal nanoplates. Mechanics of Advanced Materials and Structures, 28(12), 1216–1226(2019)
[22] MAZUR, O., KURPA, L., and AWREJCEWICZ, J. Vibrations and buckling of orthotropic smallscale plates with complex shape based on modified couple stress theory. Journal of Applied Mathematics and Mechanics, 100(11), 3–14(2020)
[23] TSIATAS, G. C. and YIOTIS, A. J. Size effect on the static, dynamic and buckling analysis of orthotropic Kirchhoff-type skew micro-plates based on a modified couple stress theory: comparison with the nonlocal elasticity theory. Acta Mechanica, 226(4), 1267–1281(2014)
[24] MIANDOAB, M. E., PISHKENARI, N. H., YOUSEFI-KOMA, A., and HOORZAD, H. Polysilicon nano-beam model based on modified couple stress and Eringen’s nonlocal elasticity theories. Physica E, 63, 223–228(2014)
[25] TILMANS, H. A. and LEGTENBERG, R. Electrostatically driven vacuum-encapsulated polysilicon resonators, Part II: theory and performance. Sensors and Actuators A, 45, 67–84(1994)
[26] LEI, J., HE, Y. M., GUO, S., LI, Z. K., and LIU, D. B. Size-dependent vibration of nickel cantilever microbeams: experiment and gradient elasticity. AIP Advances, 6, 1050202(2016)
[27] LI, Z. K., HE, Y. M., LEI, J., GUO, S., LIU, D. B., and WANG, L. A standard experimental method for determining the material length scale based on modified couple stress theory. International Journal of Mechanical Sciences, 141, 198(2018)
[28] LI, X, F., GUO, J. H., and SUN, T. Y. Bending deformation of multilayered one-dimensional quasicrystal nanoplates based on the modified couple stress theory. Acta Mechanica Solida Sinica, 32(6), 785–802(2019)
[29] GUO, J. H., ZHANG, M., CHEN, W. Q., and ZHANG, X. Y. Free and forced vibration of layered one-dimensional quasicrystal nanoplates with modified couple-stress effect. SCIENCE CHINA Physics, Mechanics & Astronomy, 63(7), 274621(2020)
[30] DING, D. H., YANG, W. G., HU, C. A., and WANG, R. H. Generalized elasticity theory of quasicrystals. Physical Review B, 48(10), 7003–7010(1993)
[31] SUN, T. Y., GUO, J. H., and PAN. E. Nonlocal vibration and buckling of two-dimensional layered quasicrystal nanoplates embedded in an elastic medium. Applied Mathematics and Mechanics (English Edition), 42(8), 1077–1094(2021) https://doi.org/10.1007/s10483-021-2743-6
[32] ZHANG, L., GUO, J. H., and XING, Y. M. Bending analysis of functionally graded onedimensional hexagonal piezoelectric quasicrystal multilayered simply supported nanoplates based on nonlocal strain gradient theory. Acta Mechanica Solida Sinica, 34(2), 237–251(2021)
[33] LI, Y. S. and XIAO, T. Free vibration of the one-dimensional piezoelectric quasicrystal microbeams based on modified couple stress theory. Applied Mathematical Modelling, 96, 733–750(2021)
[34] CHEN, W. Q. and DING, H. J. On free vibration of a functionally graded piezoelectric rectangular plate. Acta Mechanica, 153(3-4), 207–216(2002)
[35] GUO, J. H., CHEN, J. Y., and PAN, E. Free vibration of three-dimensional anisotropic layered composite nanoplates based on modified couple-stress theory. Physica E, 87, 98–106(2017)
[36] LUBENSKY, T. C., RAMASWAMY, S., and JONER, J. Hydrodynamics of icosahedral quasicrystals. Physical Review B, 32(11), 7444–7452(1985)
[37] LAM, D. C. C., YANG, F., CHONG, A. C. M., WANG, J., and TONG, P. Experiments and theory in strain gradient elasticity. Journal of the Mechanics and Physics of Solids, 51, 1477–1508(2003)
[38] LIEBOLD, C. and MLLER, W. H. Comparison of gradient elasticity models for the bending of micromaterials. Computational Materials Science, 116, 52–61(2016)
[39] PARK, S. K. and GAO, X. L. Bernoulli-Euler beam model based on a modified couple stress theory. Journal of Micromechanics and Microengineering, 16, 2355–2359(2006)
[40] KHORSHIDI, M. A. The material length scale parameter used in couple stress theories is not a material constant. International Journal of Engineering Science, 133, 15–25(2018)
[41] JOMEHZADEH, E., NOORI, H. R., and SAIDI, A.R. The size-dependent vibration analysis of micro-plates based on a modified couple stress theory. Physica E, 43, 877–883(2011)
[42] TSIATAS, G. C. A new Kirchhoff plate model based on a modified couple stress theory. International Journal of Solids and Structures, 46, 2757–2764(2009)
[43] ZHAO, Z. N. and GUO, J. H. Surface effects on a mode-III reinforced nano-elliptical hole embedded in one-dimensional hexagonal piezoelectric quasicrystals. Applied Mathematics and Mechanics (English Edition), 42(5), 625–640(2021) https://doi.org/10.1007/s10483-021-2721-5
[44] REDDY, J. N. Theory and Analysis of Elastic Plates, Taylor and Francis, Texas (1999)
[45] AJRI, M. and FAKHRABADI, M. M. S. Nonlinear free vibration of viscoelastic nanoplates based on modified couple stress theory. Journal of Computional Applied Mechanics, 49(1), 44–53(2018)
[46] WAKSMANSK, N., PAN, E., YANG, L. Z., and GAO, Y. Harmonic response of multilayered one-dimensional quasicrystal plates subjected to patch loading. Journal of Sound and Vibration, 375, 237–253(2016)

Bending and vibration of two-dimensional decagonal quasicrystal nanoplates via modified couple-stress theory

Bending and vibration of two-dimensional decagonal quasicrystal nanoplates via modified couple-stress theory

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