一维纳米准晶层合梁的非局部振动、屈曲与弯曲研究

原庆丹; 郭俊宏

doi:10.21656/1000-0887.440260

一维纳米准晶层合梁的非局部振动、屈曲与弯曲研究

doi: 10.21656/1000-0887.440260

原庆丹^1,,
郭俊宏^{1, 2, ,}

1.
内蒙古工业大学理学院力学系，呼和浩特 010051
2.
内蒙古工业大学航空学院，呼和浩特 010051

基金项目:

国家自然科学基金 12072166

内蒙古自治区科技计划项目 2021GG0254

内蒙古自治区直属高校基本科研业务费 JY20220075

详细信息

作者简介:
原庆丹(1999—)，女，硕士生(E-mail: 783694647@qq.com)

通讯作者:
郭俊宏(1981—)，男，教授，博士，博士生导师(通讯作者. E-mail: jhguo@imut.edu.cn)

中图分类号: O343
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- 文章访问数: 293
- HTML全文浏览量: 96
- PDF下载量: 61
- 被引次数: 0
出版历程
- 收稿日期: 2023-08-28
- 修回日期: 2023-10-20
- 刊出日期: 2024-02-01

Nonlocal Vibration, Buckling and Bending of 1D Layered Quasicrystal Nanobeams

YUAN Qingdan^1
,,
GUO Junhong^{1, 2
, ,}

1.
Department of Mechanics, College of Science, Inner Mongolia University of Technology, Hohhot 010051, P.R.China
2.
School of Aeronautics, Inner Mongolia University of Technology, Hohhot 010051, P.R.China

摘要

摘要: 基于非局部理论，建立了一维纳米准晶层合简支深梁模型，研究了其自由振动、屈曲行为及其弯曲变形问题. 采用伪Stroh型公式，导出了纳米梁的控制方程，并通过传递矩阵法获得简支边界条件下纳米准晶层合梁固有频率、临界屈曲载荷及弯曲变形广义位移和广义应力的精确解. 通过数值算例，分析了高跨比、层厚比、叠层顺序及非局部效应对一维纳米准晶层合简支梁固有频率、临界屈曲载荷和弯曲变形的影响. 结果表明：固有频率和临界屈曲载荷随着非局部参数增大而减小；外层准晶弹性常数更高时，固有频率和临界屈曲载荷更大；叠层顺序对纳米准晶梁的力学行为有较大影响. 所得的精确解可为纳米尺度下梁结构的各种数值方法和实验结果提供参考.
- 纳米准晶 /
- 简支梁 /
- 自由振动 /
- 屈曲 /
- 弯曲 /
- 非局部效应
Abstract: Based on the nonlocal theory, a 1D layered nano-quasicrystal (QC) simply supported beam model was established to investigate the free vibration, buckling behavior, and bending deformation of nano-QC beams. The pseudo-Stroh formula was used to derive the governing equations for the nanobeam. Using the transfer matrix method, exact solutions of the natural frequency, the critical buckling load, the generalized displacement and the generalized stress for bending problems of layered nano-QC beams was obtained under simply supported boundary conditions. The effects of the height-span ratio, the layer thickness ratio, the stacking sequence, and the nonlocal effect on the natural frequency, the critical buckling load and the bending deformation of layered nano-QC simply supported beams were analyzed. The results show that, the natural frequency and the critical buckling load decrease with increasing nonlocal parameter. The bigger the outer-layer quasicrystal elastic constant is, the higher the natural frequency and the buckling critical load will be. The stacking sequence has a significant effect on the mechanical behavior of nano-QC beams. The obtained exact solution provides a reference for various numerical methods and experimental results of nanoscale beam structures.
- nano-quasicrystal /
- simply supported beam /
- free vibration /
- buckling /
- bending /
- nonlocal effect

HTML全文

图 1 一维纳米准晶层合二维简支梁模型

Figure 1. A 2D layered simply-supported beam for 1D nano-quasicrystals

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图 2 两种叠层顺序下准晶层合简支梁的第一阶固有频率随h/L的变化

Figure 2. Variations of the 1st-order natural frequency of quasicrystal layered simply supported beams with h/L under two different stacking sequences

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图 3 两种叠层顺序下准晶层合简支梁的第一阶固有频率随h₁/h₂的变化

Figure 3. Variations of the 1st-order natural frequency of quasicrystal layered simply supported beams with h₁/h₂ under two different stacking sequences

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图 4 不同非局部参数下准晶层合简支梁一阶模态沿厚度方向的变化

Figure 4. Variations of the 1st mode shape of quasicrystal layered simply supported beams along the thickness direction under different nonlocal parameters

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图 5 两种纳米准晶层合简支梁的临界屈曲载荷随高跨比h/L的变化

Figure 5. Variations of the critical buckling loads of nano-quasicrystal layered simply supported beams with h/L

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图 6 两种纳米准晶层合简支梁的临界屈曲载荷随层厚比h₁/h₂的变化

Figure 6. Variations of the critical buckling loads of nano-quasicrystal layered simply supported beams with h₁/h₂

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图 7 声子场位移u₃和相位子场位移w₃沿层合梁厚度方向的变化

Figure 7. Variations of phonon displacement u₃ and phason displacement w₃ along the thickness direction of layered beams

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图 8 相位子场应力沿层合梁厚度方向的变化

Figure 8. Variations of the phason stress along the thickness direction of layered beams

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表 1 Al-Ni-Co准晶(QC1)和QC2的材料系数

Table 1. Material properties of Al-Ni-Co quasicrystals QC1 and QC2

	C₁₁/(10⁹ N/m²)	C₁₃/(10⁹ N/m²)	C₃₃/(10⁹ N/m²)	C₄₄/(10⁹ N/m²)	R₁/(10⁹ N/m²)
QC1	234.33	66.63	232.22	70.19	8.846
QC2	150	90	130	50	1.5

	R₂=R₃/(10⁹ N/m²)	K₁/(10⁹ N/m²)	K₂/(10⁹ N/m²)	ρ/(10³ kg/m³)
QC1	8.846	122	24	4.186
QC2	1.2	0.3	0.18	4.186

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表 2 准晶均匀简支梁的前四阶固有频率

Table 2. The first four natural frequencies of the quasicrystal homogenous simply supported beam

h/L	mode	present frequency	SSDQM^[23]	Stroh formula^[24]
0.1	1	0.267 7	0.267 6	0.267 2
	2	0.982 3	0.982 3	0.982 3
	3	1.016 4	1.014 7	1.014 7
	4	1.968 7	1.968 7	1.968 7
0.15	1	0.392 7	0.392 3	0.392 6
	2	0.983 1	0.983 1	0.983 1
	3	1.412 1	1.407 9	1.412 1
	4	1.979 0	7.978 8	1.979 0
0.2	1	0.508 2	0.507 4	0.508 2
	2	0.984 4	0.984 4	0.984 4
	3	1.707 3	1.700 7	1.707 3
	4	2.009 8	2.008 3	2.009 8

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表 3 两种叠层顺序下准晶简支梁的前四阶固有频率

Table 3. The first four natural frequencies of quasicrystal simply supported beams under two different stacking sequences

l/L	QC1/QC2/QC1				QC2/QC1/QC2
l/L	1	2	3	4	1	2	3	4
0	0.264 1	0.796 6	0.996 8	1.551 3	0.175 5	0.553 2	0.679 1	1.008 2
0.015	0.263 9	0.794 8	0.993 8	1.497 3	0.175 1	0.551 4	0.672 6	0.976 8
0.03	0.263 3	0.787 0	0.985 6	1.087 7	0.173 8	0.545 4	0.653 6	0.860 2

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表 4 均匀准晶简支梁的临界屈曲载荷(N_cr)

Table 4. Critical buckling loads (N_cr) of the homogenous simply supported beam

h/L	0.1	0.05	0.02
present results	10.091 1	10.336 0	10.406 6
exact solutions^[24]	10.090 0	10.340 0	10.410 0

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表 5 两种叠层顺序下纳米准晶层合简支梁的临界屈曲载荷(N_cr)

Table 5. Critical buckling loads (N_cr) of nano-quasicrystal layered simply supported beams under two different stacking sequences

h/L	QC1/QC2/QC1			QC2/QC1/QC2
h/L	h/L=0.1	h/L=0.15	h/L=0.2	h/L=0.1	h/L=0.15	h/L=0.2
0	7.068 0×10^-3	1.512 79×10^-2	2.520 23×10^-2	3.123 3×10^-3	6.848 0×10^-3	1.176 00×10^-2
0.015	7.057 4×10^-3	1.510 49×10^-2	2.516 37×10^-2	3.107 9×10^-3	6.815 2×10^-3	1.170 49×10^-2
0.03	7.026 3×10^-3	1.503 79×10^-2	2.505 06×10^-2	3.062 8×10^-3	6.717 7×10^-3	1.154 15×10^-2

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