Static response of functionally graded multilayered two-dimensional quasicrystal plates with mixed boundary conditions

doi:10.1007/s10483-021-2783-9

Applied Mathematics and Mechanics (English Edition) ›› 2021, Vol. 42 ›› Issue (11): 1599-1618.doi: https://doi.org/10.1007/s10483-021-2783-9

Static response of functionally graded multilayered two-dimensional quasicrystal plates with mixed boundary conditions

Xin FENG¹, Liangliang ZHANG², Yuxuan WANG², Jinming ZHANG², Han ZHANG³, Yang GAO²

1. College of Engineering, China Agricultural University, Beijing 100083, China;
2. College of Science, China Agricultural University, Beijing 100083, China;
3. China State Key Laboratory of Acoustics, Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190, China

收稿日期:2021-07-14 修回日期:2021-09-02 发布日期:2021-10-23
通讯作者: Yang GAO, E-mail:gaoyangg@gmail.com
基金资助:
the National Natural Science Foundation of China (Nos. 11972354, 11972365, and 12102458) and the China Agricultural University Education Foundation (No. 1101-2412001)

Static response of functionally graded multilayered two-dimensional quasicrystal plates with mixed boundary conditions

Xin FENG¹, Liangliang ZHANG², Yuxuan WANG², Jinming ZHANG², Han ZHANG³, Yang GAO²

1. College of Engineering, China Agricultural University, Beijing 100083, China;
2. College of Science, China Agricultural University, Beijing 100083, China;
3. China State Key Laboratory of Acoustics, Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190, China

Received:2021-07-14 Revised:2021-09-02 Published:2021-10-23
Contact: Yang GAO, E-mail:gaoyangg@gmail.com
Supported by:
the National Natural Science Foundation of China (Nos. 11972354, 11972365, and 12102458) and the China Agricultural University Education Foundation (No. 1101-2412001)

摘要/Abstract

摘要： The unusual properties of quasicrystals (QCs) have attracted tremendous attention from researchers. In this paper, a semi-analytical solution is presented for the static response of a functionally graded (FG) multilayered two-dimensional (2D) decagonal QC rectangular plate with mixed boundary conditions. Based on the elastic theory of FG 2D QCs, the state-space method is used to derive the state equations composed of partial differential along the thickness direction. Besides, the Fourier series expansion and the differential quadrature technique are utilized to simulate the simply supported boundary conditions and the mixed boundary conditions, respectively. Then, the propagator matrix which connects the field variables at the upper interface to those at the lower interface of any homogeneous layer can be derived based on the state equations. Combined with the interface continuity condition, the static response can be obtained by imposing the sinusoidal load on the top surfaces of laminates. Finally, the numerical examples are presented to verify the effectiveness of this method, and the results are very useful for the design and understanding of the characterization of FG QC materials in their applications to multilayered systems.

关键词: two-dimensional (2D) quasicrystal (QC) laminate, functionally graded material (FGM), mixed boundary condition, static response, differential quadrature technique

Abstract: The unusual properties of quasicrystals (QCs) have attracted tremendous attention from researchers. In this paper, a semi-analytical solution is presented for the static response of a functionally graded (FG) multilayered two-dimensional (2D) decagonal QC rectangular plate with mixed boundary conditions. Based on the elastic theory of FG 2D QCs, the state-space method is used to derive the state equations composed of partial differential along the thickness direction. Besides, the Fourier series expansion and the differential quadrature technique are utilized to simulate the simply supported boundary conditions and the mixed boundary conditions, respectively. Then, the propagator matrix which connects the field variables at the upper interface to those at the lower interface of any homogeneous layer can be derived based on the state equations. Combined with the interface continuity condition, the static response can be obtained by imposing the sinusoidal load on the top surfaces of laminates. Finally, the numerical examples are presented to verify the effectiveness of this method, and the results are very useful for the design and understanding of the characterization of FG QC materials in their applications to multilayered systems.

Key words: two-dimensional (2D) quasicrystal (QC) laminate, functionally graded material (FGM), mixed boundary condition, static response, differential quadrature technique

中图分类号:

Xin FENG, Liangliang ZHANG, Yuxuan WANG, Jinming ZHANG, Han ZHANG, Yang GAO. Static response of functionally graded multilayered two-dimensional quasicrystal plates with mixed boundary conditions[J]. Applied Mathematics and Mechanics (English Edition), 2021, 42(11): 1599-1618.

参考文献

[1] SHECHTMAN, D. G., BLECH, I. A., GRATIAS, D., and CAHN, J. W. Metallic phase with longrange orientational order and no translational symmetry. Physical Review Letters, 53, 1951-1953(1984)
[2] FAN, T. Y. and GUO, Y. C. Mathematical methods for a class of mixed boundary-value problems of planar pentagonal quasicrystal and some solutions. Science in China Series A:Mathematics, 40, 990-1003(1997)
[3] FAN, T. Y. Mathematical theory and methods of mechanics of quasicrystalline materials. Engineering, 5, 407-448(2013)
[4] GUO, J. H. and LIU, G. T. Analytic solutions to problem of elliptic hole with two straight cracks in one-dimensional hexagonal quasicrystals. Applied Mathematics and Mechanics (English Edition), 29(4), 485-493(2008) https://doi.org/10.1007/s10483-008-0406-x
[5] FLEURY, E., LEE, S. M., KIM, W. T., and KIM, D. H. Effects of air plasma spraying parameters on the Al-Cu-Fe quasicrystalline coating layer. Journal of Non-Crystalline Solids, 278, 194-204(2000)
[6] DUBOIS, J. M. Properties and applications of quasicrystals and complex metallic alloys. Chemical Society Reviews, 41, 6760-6777(2012)
[7] TIAN, Y., HUANG, H., YUAN, G. Y., and DING, W. J. Microstructure evolution and mechanical properties of quasicrystal-reinforced Mg-Zn-Gd alloy processed by cyclic extrusion and compression. Journal of Alloys and Compounds, 626, 42-48(2015)
[8] LI, X. Y., WANG, T., ZHENG, R. F., and KANG, G. Z. Fundamental thermo-electro-elastic solutions for 1D hexagonal QC. Journal of Applied Mathematics and Mechanics, 95, 457-468(2015)
[9] PAN, E. Exact solution for functionally graded anisotropic elastic composite laminates. Journal of Composite Materials, 37, 1903-1920(2003)
[10] PAN, E. and HAN, F. Exact solution for functionally graded and layered magneto-electro-elastic plates. International Journal of Engineering Science, 43, 321-339(2005)
[11] YAS, M. H. and RAHIMI, S. Thermal vibration of functionally graded porous nanocomposite beams reinforced by graphene platelets. Applied Mathematics and Mechanics (English Edition), 41(8), 1209-1226(2020) https://doi.org/10.1007/s10483-020-2634-6
[12] GUO, J. H., CHEN, J. Y., and PAN, E. A three-dimensional size-dependent layered model for simply-supported and functionally graded magnetoelectroelastic plates. Acta Mechanica Solida Sinica, 31, 652-671(2018)
[13] GUO, J. H., CHEN, J. Y., and PAN, E. Size-dependent behavior of functionally graded anisotropic composite plates. International Journal of Engineering Science, 106, 110-124(2016)
[14] MATSUNAGA, H. Free vibration and stability of functionally graded shallow shells according to a 2D higher-order deformation theory. Composite Structures, 84, 132-146(2008)
[15] HUANG, Z. Y., LU, C. F., and CHEN, W. Q. Benchmark solutions for functionally graded thick plates resting on Winkler-Pasternak elastic foundations. Composite Structures, 85, 95-104(2008)
[16] LU, C. F., LIM, C. W., and CHEN, W. Q. Exact solutions for free vibrations of functionally graded thick plates on elastic foundations. Mechanics of Composite Materials Structures, 16, 576-584(2009)
[17] YING, J., LU, C. F., and LIM, C. W. 3D thermoelasticity solutions for functionally graded thick plates. Journal of Zhejiang University-SCIENCE A, 10, 327-336(2009)
[18] HUANG, Y. Z., LI, Y., YANG, L. Z., and GAO, Y. Static response of functionally graded multilayered one-dimensional hexagonal piezoelectric quasicrystal plates using the state vector approach. Journal of Zhejiang University-SCIENCE A, 20, 133-147(2019)
[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, 1781-1810(2019)
[20] LI, Y., YANG, L. Z., and GAO, Y. An exact solution for a functionally graded multilayered one-dimensional orthorhombic quasicrystal plate. Acta Mechanica, 230, 1257-1273(2017)
[21] LI, Y., YANG, L. Z., and GAO, Y. Thermo-elastic analysis of functionally graded multilayered two-dimensional decagonal quasicrystal plates. Journal of Applied Mathematics and Mechanics, 98, 1585-1602(2018)
[22] LI, Y., YANG, L. Z., and GAO, Y. Bending analysis of laminated two-dimensional piezoelectric quasicrystal plates with functionally graded material properties. Acta Physica Polonica A, 135, 426-433(2019)
[23] BALUBAID, M., TOUNSI, A., DAKHEL, B., and MAHMOUD, S. R. Free vibration investigation of FG nanoscale plate using nonlocal two variables integral refined plate theory. Computers and Concrete, 24, 579-586(2019)
[24] ZHANG, B., YU, J. G., and ZHANG, X. M. Guided wave propagation in functionally graded one-dimensional hexagonal quasi-crystal plates. Journal of Mechanics, 36, 773-788(2020)
[25] RUOCCO, E. and MINUTOLO, V. Buckling of composite plates with arbitrary boundary conditions by a semi-analytical approach. International Journal of Structural Stability and Dynamics, 12, 347-363(2012)
[26] CUI, J., LI, Z. C., YE, R. C., JIANG, W. A., and TAO, S. H. A semianalytical three-dimensional elasticity solution for vibrations of orthotropic plates with arbitrary boundary conditions. Shock and Vibration, 2019, 1-20(2019)
[27] LU, C. F., CHEN, W. Q., and SHAO, J. W. Semi-analytical three-dimensional elasticity solutions for generally laminated composite plates. European Journal of Mechanics-A/Solids, 27, 899-917(2008)
[28] WANG, X. W. Differential quadrature in the analysis of structural components. Advances in Mechanics, 25, 232-240(1995)
[29] ZHOU, Y. Y., CHEN, W. Q., LU, C. F., and WANG, J. Free vibration of cross-ply piezoelectric laminates in cylindrical bending with arbitrary edges. Composite Structures, 87, 93-100(2009)
[30] BELLMAN, R., CASTI, J., and KASHEF, B. G. Differential quadrature:a technique for the rapid solution of nonlinear partial differential equations. Journal of Computational Physics, 10, 40-52(1972)
[31] ZHOU, Y. Y., CHEN, W. Q., and LU, C. F. Semi-analytical solution for orthotropic piezoelectric laminates in cylindrical bending with interfacial imperfections. Composite Structures, 92, 1009-1018(2010)
[32] YAS, M. H., JODAEI, A., IRANDOUST, S., and AGHDAM, M. N. Three-dimensional free vibration analysis of functionally graded piezoelectric annular plates on elastic foundations. Meccanica, 47, 1401-1423(2012)
[33] YAS, M. H. and MOLOUDI, N. Three-dimensional free vibration analysis of multi-directional functionally graded piezoelectric annular plates on elastic foundations via state space based differential quadrature method. Applied Mathematics and Mechanics (English Edition), 36(4), 439-464(2015) https://doi.org/10.1007/s10483-015-1923-9
[34] LI, L. H. and YUN, G. H. Elastic fields around a nanosized elliptic hole in decagonal quasicrystals. Chinese Physics B, 23, 1-6(2014)
[35] ZHAO, M. H., FAN, C. Y., LU, C. S., and DANG, H. Y. Analysis of interface cracks in onedimensional hexagonal quasi-crystal coating under in-plane loads. Engineering Fracture Mechanics, 243, 107534(2021)
[36] YANG, D. S. and LIU, G. T. Anti-plane problem of nano-cracks emanating from a regular hexagonal nano-hole in one-dimensional hexagonal piezoelectric quasicrystals. Chinese Physics B, 29, 104601(2020)
[37] YANG, L. Z., GAO, Y., PAN, E., and WAKSMANSKI, N. An exact solution for a multilayered two-dimensional decagonal quasicrystal plate. International Journal of Solids and Structures, 51, 1737-1749(2014)
[38] ZHANG, L. L., ZHANG, Y. M., and GAO, Y. General solutions of plane elasticity of onedimensional orthorhombic quasicrystals with piezoelectric effect. Physics Letters A, 378, 2768-2776(2014)
[39] LI, Y., QIN, Q. H., and ZHAO, M. H. Analysis of 3D planar crack problems of one-dimensional hexagonal piezoelectric quasicrystals with thermal effect, part II:numerical approach. International Journal of Solids and Structures, 188, 223-232(2020)
[40] LI, Y., QIN, Q. H., and ZHAO, M. H. Analysis of 3D planar crack problems in one-dimensional hexagonal piezoelectric quasicrystals with thermal effect, part I:theoretical formulations. International Journal of Solids and Structures, 188, 269-281(2020)
[41] DING, D. H., YANG, W. G., HU, C. Z., and WANG, R. H. Generalized elasticity theory of quasicrystals. Physical Review B, 48, 7003-7010(1993)
[42] HU, C. Z., WANG, R. H., and DING, D. H. Symmetry groups, physical property tensors, elasticity and dislocations in quasicrystals. Reports on Progress in Physics, 63, 1-39(2000)
[43] HU, C. Z., WANG, R. H., DING, D. H., and YANG, W. G. Piezoelectric effects in quasicrystals. Physical Review B, 56, 2463-2468(1997)
[44] LI, Y., YANG, L. Z., ZHANG, L. L., and GAO, Y. Size-dependent effect on functionally graded multilayered two-dimensional quasicrystal nanoplates under patch/uniform loading. Acta Mechanica, 229, 3501-3515(2018)
[45] NIE, G. J. and ZHONG, Z. Semi-analytical solution for three-dimensional vibration of functionally graded circular plates. Computer Methods in Applied Mechanics and Engineering, 196, 4901-4910(2007)
[46] BERT, C. W. and MALIK, M. Differential quadrature method in computational mechanics:a review. Applied Mechanics Reviews, 49, 1-28(1996)
[47] GAO, Y. and ZHAO, B. S. General solutions of three-dimensional problems for two-dimensional quasicrystals. Applied Mathematical Modelling, 33, 3382-3391(2009)
[48] PESTEL, E. C. and LECKIE, F. A. Matrix Methods in Elastomechanics, McGraw-Hill, London (1963)
[49] GILBERT, K. E. A propagator matrix method for periodically stratified media. The Journal of the Acoustical Society of America, 73, 137-142(1983)

Static response of functionally graded multilayered two-dimensional quasicrystal plates with mixed boundary conditions

Static response of functionally graded multilayered two-dimensional quasicrystal plates with mixed boundary conditions

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