Vibration and Buckling Characteristics of 2D Functionally Graded Microbeams With Variable Cross Sections

LEI Jian; XIE Yuyang; YAO Mingge; HE Yuming

doi:10.21656/1000-0887.420323

Volume 43 Issue 10

Oct. 2022

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Article Navigation > Applied Mathematics and Mechanics > 2022 > 43(10): 1133-1145

LEI Jian, XIE Yuyang, YAO Mingge, HE Yuming. Vibration and Buckling Characteristics of 2D Functionally Graded Microbeams With Variable Cross Sections[J]. Applied Mathematics and Mechanics, 2022, 43(10): 1133-1145. doi: 10.21656/1000-0887.420323

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Vibration and Buckling Characteristics of 2D Functionally Graded Microbeams With Variable Cross Sections

doi: 10.21656/1000-0887.420323

LEI Jian^1
,,
XIE Yuyang¹,
YAO Mingge²,
HE Yuming^{1
,
,}

1.
School of Aerospace Engineering, Huazhong University of Science and Technology, Wuhan 430074, P.R.China
2.
Tianjin Aerospace Reliability Technology Co. Ltd. , Wuhan 430056, P.R.China

Received Date: 2021-10-26
Rev Recd Date: 2022-03-19

Available Online: 2022-09-09

Publish Date: 2022-10-31

Abstract

Abstract

Based on the modified couple stress theory and the Timoshenko beam theory, the free vibration and buckling mechanics model for 2D functionally graded microbeams with variable cross sections was established by means of the variational principle. The model contains the intrinsic material length scale parameters of the metal and ceramic components, which can predict the size effects of microbeams. The Ritz method was used to obtain the numerical solution of the vibration frequencies and critical buckling loads of the microbeams under arbitrary boundary conditions. Numerical examples reveal that, when the thickness of the microbeam decreases, the dimensionless 1st-order frequency and the dimensionless critical buckling load will increase, and the scale effect will grow larger. The effect of the taper ratio on the dimensionless 1st-order frequency of the microbeam is closely related to the boundary conditions. At the same time, the effects of the taper ratios of the thickness and the width are also significantly different. The dimensionless 1st-order frequencies of microbeams increase with the material length scale parameter ratios of ceramic and metal, and the degrees of increase are different under different boundary conditions. The thick-direction and axial material gradient indexes also have significant influences on the free vibration and buckling behavior of the microbeam.
- modified couple stress theory,
- functionally graded material,
- variable-cross-section beam,
- size effect,
- vibration and buckling

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References(35)

References

[1]	SINA S, NAVAZI H M, HADDADPOUR H. An analytical method for free vibration analysis of functionally graded beams[J]. Materials and Design, 2009, 30(3): 741-747. doi: 10.1016/j.matdes.2008.05.015
[2]	ALSHORBAGY A E, ELTAHER M A, MAHMOUD F. Free vibration characteristics of a functionally graded beam by finite element method[J]. Applied Mathematical Modelling, 2011, 35(1): 412-425. doi: 10.1016/j.apm.2010.07.006
[3]	ŞIMŞEK M. Vibration analysis of a functionally graded beam under a moving mass by using different beam theories[J]. Composite Structures, 2010, 92(4): 904-917. doi: 10.1016/j.compstruct.2009.09.030
[4]	AYDOGDU M. Semi-inverse method for vibration and buckling of axially functionally graded beams[J]. Journal of Reinforced Plastics and Composites, 2008, 27(7): 683-691. doi: 10.1177/0731684407081369
[5]	王伟斌, 杨文秀, 滕兆春. 多孔功能梯度材料Timoshenko梁的自由振动分析[J]. 计算力学学报, 2021, 5: 1-13 doi: 10.7511/jslx20200311001 WANG Weibin, YANG Wenxiu, TENG Zhaochun. Free vibration analysis of porous functionally graded materials Timoshenko beam[J]. Chinese Journal of Computational Mechanics, 2021, 5: 1-13.(in Chinese) doi: 10.7511/jslx20200311001
[6]	蒲育, 周凤玺. FGM梁临界屈曲载荷的改进型GDQ法分析[J]. 应用基础与工程科学学报, 2019, 27(6): 1308-1320 doi: 10.16058/j.issn.1005-0930.2019.06.011 PU Yu, ZHOU Fengxi. Critical buckling loads analysis of FGM beams by a modified generalized differential quadrature method[J]. Journal of Basic Science Engineering, 2019, 27(6): 1308-1320.(in Chinese) doi: 10.16058/j.issn.1005-0930.2019.06.011
[7]	马连生, 贾金政. 机械载荷作用下功能梯度梁的过屈曲分析[J]. 兰州理工大学学报, 2020, 46(3): 160-164 doi: 10.3969/j.issn.1673-5196.2020.03.029 MA Liansheng, JIA Jinzheng. Analysis of mechanical postbuckling behavior of functionally graded beams[J]. Journal of Lanzhou University of Technology, 2020, 46(3): 160-164.(in Chinese) doi: 10.3969/j.issn.1673-5196.2020.03.029
[8]	葛仁余, 张金轮, 韩有民, 等. 功能梯度变截面梁自由振动和稳定性研究[J]. 应用力学学报, 2017, 34(5): 875-880, 1012 GE Renyu, ZHANG Jinlun, HAN Youmin, et al. Free vibration and stability of axially functionally graded beams with variable cross-section[J]. Chinese Journal of Applied Mechanics, 2017, 34(5): 875-880, 1012.(in Chinese)
[9]	杜运兴, 程鹏, 周芬. 变截面功能梯度Timoshenko梁的自由振动分析[J]. 湖南大学学报(自然科学版), 2021, 48(5): 55-62. DU Yunxing, CHENG Peng, ZHOU Fen. Free vibration analysis of functionally graded Timoshenko beams with variable section[J]. Journal of Hunan University(Natural Sciences), 2021, 48(5): 55-62.(in Chinese)
[10]	LÜ C, LIM C W, CHEN W. Size-dependent elastic behavior of FGM ultra-thin films based on generalized refined theory[J]. International Journal of Solids and Structures, 2009, 46(5): 1176-1185. doi: 10.1016/j.ijsolstr.2008.10.012
[11]	SHAAT M, MAHMOUD F, ALSHORBAGY A E, et al. Finite element analysis of functionally graded nano-scale films[J]. Finite Elements in Analysis and Design, 2013, 74: 41-52. doi: 10.1016/j.finel.2013.05.012
[12]	FLECK N A, MULLER G M, ASHBY M F, et al. Strain gradient plasticity: theory and experiment[J]. Acta Metallurgica et Materialia, 1994, 42(2): 475-487. doi: 10.1016/0956-7151(94)90502-9
[13]	LAM D C C, MULLER G M, ASHBY M F, et al. Experiments and theory in strain gradient elasticity[J]. Journal of the Mechanics and Physics of Solids, 2003, 51(8): 1477-1508. doi: 10.1016/S0022-5096(03)00053-X
[14]	LEI J, HE Y, GUO S, et al. Size-dependent vibration of nickel cantilever microbeams: experiment and gradient elasticity[J]. AIP Advances, 2016, 6(10): 105202. doi: 10.1063/1.4964660
[15]	LI Z, HE Y, GUO S, et al. A standard experimental method for determining the material length scale based on modified couple stress theory[J]. International Journal of Mechanical Sciences, 2018, 141: 198-205. doi: 10.1016/j.ijmecsci.2018.03.035
[16]	ERINGEN A C, EDELEN D. On nonlocal elasticity[J]. International Journal of Engineering Science, 1972, 10(3): 233-248. doi: 10.1016/0020-7225(72)90039-0
[17]	YANG F, CHONG A C M, LAM D C C, et al. Couple stress based strain gradient theory for elasticity[J]. International Journal of Solids and Structures, 2002, 39(10): 2731-2743. doi: 10.1016/S0020-7683(02)00152-X
[18]	ASGHARI M, CHONG M C, LAM D C C, et al. On the size-dependent behavior of functionally graded micro-beams[J]. International Journal of Solids and Structure, 2010, 31(5): 2324-2329.
[19]	REDDY J N. Microstructure-dependent couple stress theories of functionally graded beams[J]. Journal of the Mechanics and Physics of Solids, 2011, 59(11): 2382-2399. doi: 10.1016/j.jmps.2011.06.008
[20]	ŞIMŞEK M J, REDDY J N. Bending and vibration of functionally graded microbeams using a new higher order beam theory and the modified couple stress theory[J]. International Journal of Engineering Science, 2013, 64: 37-53. doi: 10.1016/j.ijengsci.2012.12.002
[21]	AL-BASYOUNI K S, TOUNSI A, MAHMOUD S R. Size dependent bending and vibration analysis of functionally graded micro beams based on modified couple stress theory and neutral surface position[J]. Composite Structures, 2015, 125: 621-630. doi: 10.1016/j.compstruct.2014.12.070
[22]	LEI Jian, HE Yuming, ZHANG Bo, et al. Bending and vibration of functionally graded sinusoidal microbeams based on the strain gradient elasticity theory[J]. International Journal of Engineering Science, 2013, 72: 36-52. doi: 10.1016/j.ijengsci.2013.06.012
[23]	LEI J, HE Y, ZHANG B, et al. Thermal buckling and vibration of functionally graded sinusoidal microbeams incorporating nonlinear temperature distribution using DQM[J]. Journal of Thermal Stresses, 2017, 40(6): 665-689. doi: 10.1080/01495739.2016.1258602
[24]	LEI J, HR M. Effect of nonlocal thermoelasticity on buckling of axially functionally graded nanobeams[J]. Journal of Thermal Stresses, 2019, 42(4): 526-539. doi: 10.1080/01495739.2018.1536866
[25]	杨子豪, 贺丹. 考虑尺度依赖的平面正交各向异性功能梯度微梁的自由振动分析[J]. 复合材料学报, 2017, 34(10): 2375-2384 doi: 10.13801/j.cnki.fhclxb.20170228.003 YANG Zihao, HE Dan. Size-dependent free vibration analysis of plane orthotropic functionally graded micro-beams[J]. Acta Materiae Compositae Sinica, 2017, 34(10): 2375-2384.(in Chinese) doi: 10.13801/j.cnki.fhclxb.20170228.003
[26]	EBRAHIMI F, BARATI M R. A nonlocal strain gradient refined beam model for buckling analysis of size-dependent shear-deformable curved FG nanobeams[J]. Composite Structures, 2017, 159: 174-182. doi: 10.1016/j.compstruct.2016.09.058
[27]	LEI J, GUO S, HE M, et al. Postbuckling analysis of bi-directional functionally graded imperfect beams based on a novel third-order shear deformation theory[J]. Composite Structures, 2019, 209: 811-829. doi: 10.1016/j.compstruct.2018.10.106
[28]	TANG Y, DING Q. Nonlinear vibration analysis of a bi-directional functionally graded beam under hygro-thermal loads[J]. Composite Structures, 2019, 225: 111076. doi: 10.1016/j.compstruct.2019.111076
[29]	BARATI A, HADI A, NORROZI R, et al. On vibration of bi-directional functionally graded nanobeams under magnetic field[J]. Mechanics Based Design of Structures and Machines, 2020, 50(2): 468-485.
[30]	HUANG Y, OUYANG Z Y. Exact solution for bending analysis of two-directional functionally graded Timoshenko beams[J]. Archive of Applied Mechanics, 2020, 90: 1005-1023. doi: 10.1007/s00419-019-01655-5
[31]	WATTANASAKULPONG N, UNGBHAKORN V. Linear and nonlinear vibration analysis of elastically restrained ends FGM beams with porosities[J]. Aerospace Science and Technology, 2014, 32(1): 111-120. doi: 10.1016/j.ast.2013.12.002
[32]	SHAFIEI N, MIRJAVADI S S, RABBY B, et al. Vibration of two-dimensional imperfect functionally graded (2D-FG) porous nano-/micro-beams[J]. Computer Methods in Applied Mechanics and Engineering, 2017, 322: 615-632. doi: 10.1016/j.cma.2017.05.007
[33]	ZHANG B, HE M, LIU D, et al. Size-dependent functionally graded beam model based on an improved third-order shear deformation theory[J]. European Journal of Mechanics A: Solids, 2014, 47: 211-230. doi: 10.1016/j.euromechsol.2014.04.009
[34]	ŞIMŞEK M. Fundamental frequency analysis of functionally graded beams by using different higher-order beam theories[J]. Nuclear Engineering and Design, 2010, 240(4): 697-705. doi: 10.1016/j.nucengdes.2009.12.013
[35]	AKGÖZ B, CIVALEK Ö. Free vibration analysis of axially functionally graded tapered Bernoulli-Euler microbeams based on the modified couple stress theory[J]. Composite Structures, 2013, 98: 314-322. doi: 10.1016/j.compstruct.2012.11.020