二维准晶平面问题中的Hamilton体系求解方法

李彤; 屈建龙; 王炜; 王晨龙; 徐新生

doi:10.21656/1000-0887.450204

二维准晶平面问题中的Hamilton体系求解方法

doi: 10.21656/1000-0887.450204

大连理工大学力学与航空航天学院，辽宁大连 116023

详细信息

作者简介:
李彤(1998—)，女，博士(E-mail: litong1998@mail.dlut.edu.cn)

通讯作者:
徐新生(1957—)，男，教授，博士，博士生导师(通讯作者. E-mail: xsxu@dlut.edu.cn)

中图分类号: O343.1
计量
- 文章访问数: 91
- HTML全文浏览量: 36
- PDF下载量: 33
- 被引次数: 0
出版历程
- 收稿日期: 2024-07-10
- 修回日期: 2024-08-16
- 刊出日期: 2024-11-01

A Hamiltonian System Solution Method for Planar Problems of 2D Quasicrystals

School of Mechanics and Aerospace Engineering, Dalian University of Technology, Dalian, Liaoning 116023, P.R.China

摘要

摘要: 针对二维准晶平面问题，该文通过导入Hamilton体系，将问题转化为Hamilton体系下的辛本征值和辛本征解问题，即问题的解可由辛本征解组成的级数表示. 利用辛本征解之间的辛共轭正交关系，可将满足边界条件的解问题归结为代数方程组的求解问题，从而形成一种解析求解方法. 这种方法可直接推广到求解混合边界条件及分段边界条件问题中.
- 二维准晶 /
- Hamilton体系 /
- 辛共轭正交 /
- 辛方法 /
- 平面问题
Abstract: Aimed at the planar problem of 2D quasicrystals, the problem was transformed into one of symplectic eigenvalues and symplectic eigensolutions through introduction of the Hamiltonian system. In the Hamiltonian system, the solution to this problem was expressed by a series of symplectic eigensolutions. With the symplectic conjugate orthogonality relationship between symplectic eigensolutions, the solving problem satisfying boundary conditions can be reduced to a problem of solving algebraic equations, thus to form an analytical solution method. The proposed method can be directly extended to solve the problems of mixed boundary conditions and segmented boundary conditions.
- 2D quasicrystal /
- Hamiltonian system /
- symplectic conjugate orthogonality /
- symplectic method /
- planar problem

HTML全文

图 1 收敛性分析

Figure 1. The convergence study

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图 2 声子场位移对比

Figure 2. Comparison of phonon field displacements

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图 3 广义应力分布

Figure 3. Generalized stress distributions

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图 4 广义位移分布

注为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.

Figure 4. Generalized displacement distributions

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图 5 相关广义应力分布

Figure 5. Correlative generalized stress distributions

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图 6 零本征值本征解和非零本征值本征解的作用

Figure 6. The effects of zero-eigenvalue eigensolutions and non zero-eigenvalue eigensolutions

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参考文献(30)

[1]	范天佑. 准晶数学弹性理论和某些有关研究的进展(上)[J]. 力学进展, 2012, 42 (5): 501-521. FAN Tianyou. Development on mathematical theory of elasticity of quasicrystals and some relevant topics (Ⅰ)[J]. Advances in Mechanics, 2012, 42 (5): 501-521. (in Chinese)
[2]	范天佑. 准晶数学弹性理论和某些有关研究的进展(下)[J]. 力学进展, 2012, 42 (6): 675-691. FAN Tianyou. Development on mathematical theory of elasticity of quasicrystals and some relevant topics (Ⅱ)[J]. Advances in Mechanics, 2012, 42 (6): 675-691. (in Chinese)
[3]	DUBOIS J M. New prospects from potential applications of quasicrystalline materials[J]. Materials Science and Engineering: A, 2000, 294/296 : 4-9. doi: 10.1016/S0921-5093(00)01305-8
[4]	AMINI M, RAHIMIPOUR M R, TAYEBIFARD S A, et al. Towards physical and mechanical properties of the novel Al-Cr-Ni-Fe decagonal quasicrystal and crystalline approximants[J]. Advanced Powder Technology, 2022, 33 (2): 103383. doi: 10.1016/j.apt.2021.12.002
[5]	TAKAGIWA Y, MAEDA R, OHHASHI S, et al. Reduction of thermal conductivity for icosahedral Al-Cu-Fe quasicrystal through heavy element substitution[J]. Materials, 2021, 14 (18): 5238. doi: 10.3390/ma14185238
[6]	STROUD R M, VIANO A M, GIBBONS P C, et al. Stable Ti-based quasicrystal offers prospect for improved hydrogen storage[J]. Applied Physics Letters, 1996, 69 (20): 2998-3000. doi: 10.1063/1.117756
[7]	康国政, 陈义甫, 黄伟洋. 介电高弹体的力-电耦合循环变形和疲劳失效行为研究[J]. 力学进展, 2023, 53 (3): 592-625. KANG Guozheng, CHEN Yifu, HUANG Weiyang. Review on electro-mechanically coupled cyclic deformation and fatigue failure behavior of dielectric elastomers[J]. Advances in Mechanics, 2023, 53 (3): 592-625. (in Chinese)
[8]	JARIĆ M V, NELSON D R. Diffuse scattering from quasicrystals[J]. Physical Review B, 1988. DOI: 10.1103/PhysRevB.37.4458.
[9]	FAN Tianyou. Mathematical Theory of Elasticity of Quasicrystals and Its Applications[M]. Berlin: Springer, 2011.
[10]	DING D H, YANG W G, HU C Z, et al. Generalized elasticity theory of quasicrystals[J]. Physical Review B: Covering Condensed Matter and Materials Physics, 1993, 48 (10): 7003-7010. doi: 10.1103/PhysRevB.48.7003
[11]	FAN T Y, GUO L H. The final governing equation and fundamental solution of plane elasticity of icosahedral quasicrystals[J]. Physics Letters A, 2005, 341 (1/4): 235-239.
[12]	GAO Y, SHANG L G. Governing equations and general solutions of plane elasticity of two-dimensional decagonal quasicrystals[J]. International Journal of Modern Physics B, 2011, 25 (20): 2769-2778. doi: 10.1142/S0217979211101065
[13]	ZHANG Liangliang, YANG Lianzhi, YU Lianying, et al. General solutions of thermoelastic plane problems of two-dimensional quasicrystals[J]. Transactions of Nanjing University of Aeronautics and Astronautics, 2014, 31 (2): 142-146.
[14]	ZHAO X F, LI X, DING S H. Two kinds of contact problems in three-dimensional icosahedral quasicrystals[J]. Applied Mathematics and Mechanics (English Edition), 2015, 36 (12): 1569-1580. doi: 10.1007/s10483-015-2006-6
[15]	李光芳, 刘昉昉, 于静, 等. 立方准晶压电材料的半空间问题[J]. 应用数学和力学, 2023, 44 (7): 825-833. doi: 10.21656/1000-0887.430221 LI Guangfang, LIU Fangfang, YU Jing, et al. The half space problem of cubic quasicrystal piezoelectric materials[J]. Applied Mathematics and Mechanics, 2023, 44 (7): 825-833. (in Chinese) doi: 10.21656/1000-0887.430221
[16]	杨震霆, 王雅静, 聂雪阳, 等. 含切口的压电准晶组合结构界面断裂分析的辛-等几何耦合方法[J]. 应用数学和力学, 2024, 45 (2): 144-154. doi: 10.21656/1000-0887.440247 YANG Zhenting, WANG Yajing, NIE Xueyang, et al. Symplectic isogeometric analysis coupling method for interfacial fracture of piezoelectric quasicrystal composites with notches[J]. Applied Mathematics and Mechanics, 2024, 45 (2): 144-154. (in Chinese) doi: 10.21656/1000-0887.440247
[17]	FENG X, ZHANG L L, LI Y, et al. On the propagation of plane waves in cubic quasicrystal plates with surface effects[J]. Physics Letters A, 2023, 473 : 128807. doi: 10.1016/j.physleta.2023.128807
[18]	原庆丹, 郭俊宏. 一维纳米准晶层合梁的非局部振动、屈曲与弯曲研究[J]. 应用数学和力学, 2024, 45 (2): 208-219. doi: 10.21656/1000-0887.440260 YUAN Qingdan, GUO Junhong. Nonlocal vibration, buckling and bending of 1D layered quasicrystal nanobeams[J]. Applied Mathematics and Mechanics, 2024, 45 (2): 208-219. (in Chinese) doi: 10.21656/1000-0887.440260
[19]	范俊杰, 李联和, 阿拉坦仓. 对边简支十次对称二维准晶板弯曲问题的辛分析[J]. 应用数学和力学, 2023, 44 (7): 834-846. doi: 10.21656/1000-0887.430267 FAN Junjie, LI Lianhe, ALATANCANG. Symplectic analysis on the bending problem of decagonal symmetric 2D quasicrystal plates with 2 opposite edges simply supported[J]. Applied Mathematics and Mechanics, 2023, 44 (7): 834-846. (in Chinese) doi: 10.21656/1000-0887.430267
[20]	王会苹, 王桂霞, 陈德财. 含椭圆孔有限大二十面体准晶板平面弹性问题的边界元分析[J]. 应用数学和力学, 2024, 45 (4): 400-415. doi: 10.21656/1000-0887.440241 WANG Huiping, WANG Guixia, CHEN Decai. Boundary element analysis for the plane elasticity problems of finite icosahedral quasicrystal plates containing elliptical holes[J]. Applied Mathematics and Mechanics, 2024, 45 (4): 400-415. (in Chinese) doi: 10.21656/1000-0887.440241
[21]	ZHU S B, TONG Z Z, LI Y Q, et al. Post-buckling of two-dimensional decagonal piezoelectric quasicrystal cylindrical shells under compression[J]. International Journal of Mechanical Sciences, 2022, 235 : 107720. doi: 10.1016/j.ijmecsci.2022.107720
[22]	ZHONG W X. Duality System in Applied Mechanics and Optimal Control[M]. Boston: Kluwer Academic Publishers, 2004.
[23]	WANG H, LI L H, HUANG J J, et al. Symplectic approach for the plane elasticity problem of quasicrystals with point group 10 mm[J]. Applied Mathematical Modelling, 2015, 39 (12): 3306-3316. doi: 10.1016/j.apm.2014.10.060
[24]	QIAO Y F, HOU G L, CHEN A. Symplectic approach for plane elasticity problems of two-dimensional octagonal quasicrystals[J]. Applied Mathematics and Computation, 2021, 400 : 126043. doi: 10.1016/j.amc.2021.126043
[25]	SUN Z Q, HOU G L, QIAO Y F, et al. Hamiltonian system for the inhomogeneous plane elasticity of dodecagonal quasicrystal plates and its analytical solutions[J]. Chinese Physics B, 2024, 33 (1): 016107. doi: 10.1088/1674-1056/acfaf3
[26]	LI G F, LI L H. An analysis method of symplectic dual system for decagonal quasicrystal plane elasticity and application[J]. Crystals, 2022, 12 (5): 636. doi: 10.3390/cryst12050636
[27]	郭丽辉, 范天佑. 准晶弹性理论边值问题的可解性[J]. 应用数学和力学, 2007, 28 (8): 949-957. http://www.applmathmech.cn/article/id/952 GUO Lihui, FAN Tianyou. Solvability on boundary-value problems of elasyicity of three-dimensional quasicrystals[J]. Applied Mathematics and Mechanics, 2007, 28 (8): 949-957. (in Chinese) http://www.applmathmech.cn/article/id/952
[28]	CAO H B, SHI Y Q, LI W. Analytic solutions to two-dimensional decagonal quasicrystals with defects using complex potential theory[J]. Crystals, 2019, 9 (4): 209. doi: 10.3390/cryst9040209
[29]	LI W, FAN T Y. Plastic analysis of the crack problem in two-dimensional decagonal Al-Ni-Co quasicrystalline materials of point group[J]. Chinese Physics B, 2011, 20 (3): 036101. doi: 10.1088/1674-1056/20/3/036101
[30]	LI T, YANG Z T, XU C H, et al. A phase field approach to two-dimensional quasicrystals with mixed mode cracks[J]. Materials, 2023, 16 (10): 3628. doi: 10.3390/ma16103628