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Article(id=1246028559760868345, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246028557319783390, articleNumber=null, orderNo=null, doi=10.19636/j.cnki.cjsm42-1250/o3.2023.057, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1699027200000, receivedDateStr=2023-11-04, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1775005958738, onlineDateStr=2026-04-01, pubDate=1719244800000, pubDateStr=2024-06-25, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1775005958738, onlineIssueDateStr=2026-04-01, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1775005958738, creator=13701087609, updateTime=1775005958738, updator=13701087609, issue=Issue{id=1246028557319783390, tenantId=1146029695717560320, journalId=1241755870837649424, year='2024', volume='45', issue='3', pageStart='289', pageEnd='426', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1775005958156, creator=13701087609, updateTime=1775006058227, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1246028977123471371, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246028557319783390, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1246028977123471372, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246028557319783390, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=379, endPage=391, ext={EN=ArticleExt(id=1246028560142550013, articleId=1246028559760868345, tenantId=1146029695717560320, journalId=1241755870837649424, language=EN, title=Vibration Characteristics of Graphene-reinforced Porous Cylindrical Shells with Arbitrary Boundary Conditions, columnId=1244229834482757770, journalTitle=Chinese Journal of Solid Mechanics, columnName=Research Paper, runingTitle=null, highlight=null, articleAbstract=

To investigate the vibration characteristics of graphene-platelet-reinforced porous composite (GPLRPC) cylindrical shells under arbitrary boundary conditions, a semi-analytical method using Gegenbauer polynomials as admissible functions is proposed in this paper. First, the effective material properties of the GPLRPC cylindrical shell are derived based on the Halpin-Tsai micromechanical model and open-cell body theory. The artificial spring technique is utilized to simulate the boundary conditions at both ends of the shell and continuous coupling conditions between shell segments. Then, based on the first-order shear deformation shell theory, the motion equations of the structure are derived and its dimensionless frequencies are obtained with the Rayleigh-Ritz method. Numerical calculations are performed to analyze the effects of boundary conditions, porosity coefficients, porosity types, graphene distribution patterns, graphene mass fractions, boundary spring stiffness, and geometric parameters on the vibration characteristics of the shell structure. The results show that Gegenbauer polynomials have excellent convergence and accuracy as admissible functions. It is also found that boundary conditions have different effects on the frequency of cylindrical shells, and the GPL-A distribution pattern and Type-II pore distribution exhibit the best stiffness enhancement effect. Additionally, it is observed that the influence of translational springs on frequency is greater than rotational springs, and the effect of cylindrical shell length-to-diameter ratio is greater, but the effect of diameter-to-thickness ratio is less. Overall, applying graphene to cylindrical shells has a wide range of applications, and the research results can provide data support and theoretical reference for the engineering design.

, correspAuthors=Yu Wang, authorNote=null, correspAuthorsNote=null, copyrightStatement=null, copyrightOwner=null, extLink=null, articleAbsUrl=null, sourceXml=null, magXml=null, pdfUrl=null, pdf=null, pdfFileSize=null, pdfExtLink=null, richHtmlUrl=null, mobilePdfUrl=null, reviewReport=null, pdfFirstPage=null, abstractGraph=null, abstractGraphContent=null, abstractVideo=null, citation=null, cebUrl=null, magXmlContent=null, mapNumber=null, authorCompany=null, fund=null, authors=null, authorsList=Hongda Xu, Yu Wang, Ying Zhang, Xiaoyu Jia, Xuehui Li), CN=ArticleExt(id=1246028580749165329, articleId=1246028559760868345, tenantId=1146029695717560320, journalId=1241755870837649424, language=CN, title=任意边界下石墨烯增强多孔圆柱壳的振动特性, columnId=1241831201896469478, journalTitle=固体力学学报, columnName=研究论文, runingTitle=null, highlight=null, articleAbstract=

为了研究任意边界石墨烯增强多孔复合材料（GPLRPC）圆柱壳的振动特性，提出了一种以盖根堡多项式作为容许函数的半解析法. 基于Halpin-Tsai微观力学模型和开胞体理论得到GPLRPC圆柱壳的有效材料参数. 利用人工弹簧技术模拟了壳体两端的边界条件和壳段之间的连续耦合. 然后，基于一阶剪切变形理论，采用Rayleigh-Ritz方法推导出壳体结构的运动方程，求得GPLRPC圆柱壳的无量纲频率. 数值算例验证了该方法的有效性和准确性，并讨论了边界条件、孔隙系数、孔隙类型、石墨烯分布模式、石墨烯质量分数、边界弹簧刚度和几何参数等因素对壳体结构振动特性的影响. 结果表明：盖根堡多项式作为容许函数具有良好的收敛性和准确性；边界条件对圆柱壳的频率有不同的影响，GPL-A分布模式和孔隙类型Ⅱ的刚度增强效果最好；此外，平移弹簧对频率的影响大于旋转弹簧，圆柱壳长径比的影响较大，但径厚比的影响较小. 研究结论可为此类结构的工程设计提供理论参考.

, correspAuthors=王宇, authorNote=null, correspAuthorsNote=
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the vibration characteristics of cylindrical shells, figureFileSmall=FMcoprfP7r/ukgv+1ROAAQ==, figureFileBig=ua7P3MRvLuLfAFyahThhdw==, tableContent=null), ArticleFig(id=1246028584800863053, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246028559760868345, language=CN, label=图11, caption=圆柱壳几何参数对圆柱壳振动特性的影响, figureFileSmall=FMcoprfP7r/ukgv+1ROAAQ==, figureFileBig=ua7P3MRvLuLfAFyahThhdw==, tableContent=null), ArticleFig(id=1246028584872166222, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246028559760868345, language=EN, label=Table 1, caption=

Material properties of cylindrical shells

, figureFileSmall=null, figureFileBig=null, tableContent=
参数石墨烯片泡沫金属铜
弹性模量（GPa）1010130
密度（kg/m3）1062.58960
泊松比0.1860.34
), ArticleFig(id=1246028584951857999, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246028559760868345, language=CN, label=表1, caption=

圆柱壳的材料属性

, figureFileSmall=null, figureFileBig=null, tableContent=
参数石墨烯片泡沫金属铜
弹性模量（GPa）1010130
密度（kg/m3）1062.58960
泊松比0.1860.34
), ArticleFig(id=1246028585018966864, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246028559760868345, language=EN, label=Table 2, caption=

Spring stiffness values with different boundary conditions

, figureFileSmall=null, figureFileBig=null, tableContent=
边界条件，，，，，
F00000
S01012101200
C10121012101210121012
E11091012101210121012
E21012109101210121012
E3109109101210121012
), ArticleFig(id=1246028585086075729, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246028559760868345, language=CN, label=表2, caption=

不同边界条件的弹簧刚度值

, figureFileSmall=null, figureFileBig=null, tableContent=
边界条件，，，，，
F00000
S01012101200
C10121012101210121012
E11091012101210121012
E21012109101210121012
E3109109101210121012
), ArticleFig(id=1246028585153184594, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246028559760868345, language=EN, label=Table 3, caption=

Effect of the number of shell segments on the vibration characteristics of cylindrical shells

, figureFileSmall=null, figureFileBig=null, tableContent=
孔隙分布类型壳段数NP
23456
类型Ⅰ0.29650.29650.29650.29650.2965
类型Ⅱ0.29670.29670.29670.29670.2967
类型Ⅲ0.29640.29640.29640.29640.2964
), ArticleFig(id=1246028585249653587, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246028559760868345, language=CN, label=表3, caption=

壳段数对圆柱壳振动特性的影响

, figureFileSmall=null, figureFileBig=null, tableContent=
孔隙分布类型壳段数NP
23456
类型Ⅰ0.29650.29650.29650.29650.2965
类型Ⅱ0.29670.29670.29670.29670.2967
类型Ⅲ0.29640.29640.29640.29640.2964
), ArticleFig(id=1246028585325151060, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246028559760868345, language=EN, label=Table 4, caption=

Dimensionless frequency comparison of GPLRPC cylindrical shells

, figureFileSmall=null, figureFileBig=null, tableContent=
孔隙分布类型e1文献[1]本文误差（%）
类型Ⅰ0.20.22720.22730.04
0.40.21890.21900.05
0.60.20990.21000.05
类型Ⅱ0.20.22740.22760.09
0.40.21980.22050.32
0.6——
类型Ⅲ0.20.22710.22720.04
0.40.21800.21830.14
0.60.20740.20770.14
), ArticleFig(id=1246028585400648533, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246028559760868345, language=CN, label=表4, caption=

GPLRPC圆柱壳无量纲频率对比

, figureFileSmall=null, figureFileBig=null, tableContent=
孔隙分布类型e1文献[1]本文误差（%）
类型Ⅰ0.20.22720.22730.04
0.40.21890.21900.05
0.60.20990.21000.05
类型Ⅱ0.20.22740.22760.09
0.40.21980.22050.32
0.6——
类型Ⅲ0.20.22710.22720.04
0.40.21800.21830.14
0.60.20740.20770.14
), ArticleFig(id=1246028585471951702, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246028559760868345, language=EN, label=Table 5, caption=

Comparison of natural frequency finite elements of GPLRPC cylindrical shells

, figureFileSmall=null, figureFileBig=null, tableContent=
边界条件（m，n）固有频率（Hz）误差（%）
有限元法本文方法
S-S（1，3）48.9047.832.19
（1，2）63.2062.371.31
（1，4）72.4171.700.98
（2，4）101.8799.861.97
C-C（1，3）69.1069.040.09
（1，4）78.7878.760.03
（1，2）102.44102.300.14
（1，5）114.37114.280.08
C-S（1，3）58.1057.431.15
（1，4）75.0874.580.67
（1，2）82.9682.460.60
（2，4）109.35108.290.97
C-F（1，2）26.0026.030.11
（1，3）37.3337.370.12
（1，1）60.6160.540.12
（1，4）68.3168.300.01
), ArticleFig(id=1246028585555837783, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246028559760868345, language=CN, label=表5, caption=

GPLRPC圆柱壳固有频率有限元对比

, figureFileSmall=null, figureFileBig=null, tableContent=
边界条件（m，n）固有频率（Hz）误差（%）
有限元法本文方法
S-S（1，3）48.9047.832.19
（1，2）63.2062.371.31
（1，4）72.4171.700.98
（2，4）101.8799.861.97
C-C（1，3）69.1069.040.09
（1，4）78.7878.760.03
（1，2）102.44102.300.14
（1，5）114.37114.280.08
C-S（1，3）58.1057.431.15
（1，4）75.0874.580.67
（1，2）82.9682.460.60
（2，4）109.35108.290.97
C-F（1，2）26.0026.030.11
（1，3）37.3337.370.12
（1，1）60.6160.540.12
（1，4）68.3168.300.01
), ArticleFig(id=1246028585639723864, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246028559760868345, language=EN, label=Table 6, caption=

Dimensionless frequencies of GPLRPC cylindrical shells with different boundary conditions

, figureFileSmall=null, figureFileBig=null, tableContent=
边界条件孔隙分布类型ω*
e1=0.1e1=0.3e1=0.5
S-S类型Ⅰ0.23120.22330.2146
类型Ⅱ0.23130.22400.2175
类型Ⅲ0.23120.22290.2133
C-C类型Ⅰ0.29650.28690.2762
类型Ⅱ0.29650.28760.2794
类型Ⅲ0.29650.28650.2747
C-S类型Ⅰ0.26170.25300.2434
类型Ⅱ0.26170.25370.2464
类型Ⅲ0.26160.25260.2420
C-F类型Ⅰ0.09390.09080.0873
类型Ⅱ0.09390.09100.0884
类型Ⅲ0.09390.09060.0867
E1-E1类型Ⅰ0.25280.24640.2394
类型Ⅱ0.25280.24700.2420
类型Ⅲ0.25270.24600.2382
E2-E2类型Ⅰ0.23070.22790.2247
类型Ⅱ0.23040.22720.2247
类型Ⅲ0.23070.22740.2233
E3-E3类型Ⅰ0.20840.20600.2039
类型Ⅱ0.20850.20590.2041
类型Ⅲ0.20840.20580.2027
), ArticleFig(id=1246028585723609945, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246028559760868345, language=CN, label=表6, caption=

不同边界条件下GPLRPC圆柱壳的无量纲频率

, figureFileSmall=null, figureFileBig=null, tableContent=
边界条件孔隙分布类型ω*
e1=0.1e1=0.3e1=0.5
S-S类型Ⅰ0.23120.22330.2146
类型Ⅱ0.23130.22400.2175
类型Ⅲ0.23120.22290.2133
C-C类型Ⅰ0.29650.28690.2762
类型Ⅱ0.29650.28760.2794
类型Ⅲ0.29650.28650.2747
C-S类型Ⅰ0.26170.25300.2434
类型Ⅱ0.26170.25370.2464
类型Ⅲ0.26160.25260.2420
C-F类型Ⅰ0.09390.09080.0873
类型Ⅱ0.09390.09100.0884
类型Ⅲ0.09390.09060.0867
E1-E1类型Ⅰ0.25280.24640.2394
类型Ⅱ0.25280.24700.2420
类型Ⅲ0.25270.24600.2382
E2-E2类型Ⅰ0.23070.22790.2247
类型Ⅱ0.23040.22720.2247
类型Ⅲ0.23070.22740.2233
E3-E3类型Ⅰ0.20840.20600.2039
类型Ⅱ0.20850.20590.2041
类型Ⅲ0.20840.20580.2027
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任意边界下石墨烯增强多孔圆柱壳的振动特性
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徐宏达 , 王宇 ** , 张颖 , 贾小羽 , 李学辉
固体力学学报 | 研究论文 2024,45(3): 379-391
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固体力学学报 | 研究论文 2024, 45(3): 379-391
任意边界下石墨烯增强多孔圆柱壳的振动特性
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徐宏达, 王宇** , 张颖, 贾小羽, 李学辉
作者信息
  • 辽宁科技大学机械工程与自动化学院，鞍山，114051

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** E-mail：.
Vibration Characteristics of Graphene-reinforced Porous Cylindrical Shells with Arbitrary Boundary Conditions
Hongda Xu, Yu Wang** , Ying Zhang, Xiaoyu Jia, Xuehui Li
Affiliations
  • School of Mechanical Engineering and Automation, University of Science and Technology Liaoning, Anshan, 114051
出版时间: 2024-06-25 doi: 10.19636/j.cnki.cjsm42-1250/o3.2023.057
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摘要
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为了研究任意边界石墨烯增强多孔复合材料（GPLRPC）圆柱壳的振动特性，提出了一种以盖根堡多项式作为容许函数的半解析法. 基于Halpin-Tsai微观力学模型和开胞体理论得到GPLRPC圆柱壳的有效材料参数. 利用人工弹簧技术模拟了壳体两端的边界条件和壳段之间的连续耦合. 然后，基于一阶剪切变形理论，采用Rayleigh-Ritz方法推导出壳体结构的运动方程，求得GPLRPC圆柱壳的无量纲频率. 数值算例验证了该方法的有效性和准确性，并讨论了边界条件、孔隙系数、孔隙类型、石墨烯分布模式、石墨烯质量分数、边界弹簧刚度和几何参数等因素对壳体结构振动特性的影响. 结果表明：盖根堡多项式作为容许函数具有良好的收敛性和准确性；边界条件对圆柱壳的频率有不同的影响，GPL-A分布模式和孔隙类型Ⅱ的刚度增强效果最好；此外，平移弹簧对频率的影响大于旋转弹簧，圆柱壳长径比的影响较大，但径厚比的影响较小. 研究结论可为此类结构的工程设计提供理论参考.

石墨烯增强  /  多孔复合材料  /  任意边界  /  盖根堡多项式  /  人工弹簧技术

To investigate the vibration characteristics of graphene-platelet-reinforced porous composite (GPLRPC) cylindrical shells under arbitrary boundary conditions, a semi-analytical method using Gegenbauer polynomials as admissible functions is proposed in this paper. First, the effective material properties of the GPLRPC cylindrical shell are derived based on the Halpin-Tsai micromechanical model and open-cell body theory. The artificial spring technique is utilized to simulate the boundary conditions at both ends of the shell and continuous coupling conditions between shell segments. Then, based on the first-order shear deformation shell theory, the motion equations of the structure are derived and its dimensionless frequencies are obtained with the Rayleigh-Ritz method. Numerical calculations are performed to analyze the effects of boundary conditions, porosity coefficients, porosity types, graphene distribution patterns, graphene mass fractions, boundary spring stiffness, and geometric parameters on the vibration characteristics of the shell structure. The results show that Gegenbauer polynomials have excellent convergence and accuracy as admissible functions. It is also found that boundary conditions have different effects on the frequency of cylindrical shells, and the GPL-A distribution pattern and Type-II pore distribution exhibit the best stiffness enhancement effect. Additionally, it is observed that the influence of translational springs on frequency is greater than rotational springs, and the effect of cylindrical shell length-to-diameter ratio is greater, but the effect of diameter-to-thickness ratio is less. Overall, applying graphene to cylindrical shells has a wide range of applications, and the research results can provide data support and theoretical reference for the engineering design.

graphene-reinforced  /  porous composites  /  arbitrary boundary conditions  /  Gegenbauer polynomial  /  artificial spring technology
徐宏达, 王宇, 张颖, 贾小羽, 李学辉. 任意边界下石墨烯增强多孔圆柱壳的振动特性. 固体力学学报, 2024 , 45 (3) : 379 -391 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2023.057
Hongda Xu, Yu Wang, Ying Zhang, Xiaoyu Jia, Xuehui Li. Vibration Characteristics of Graphene-reinforced Porous Cylindrical Shells with Arbitrary Boundary Conditions[J]. Chinese Journal of Solid Mechanics, 2024 , 45 (3) : 379 -391 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2023.057
正文
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0　引言
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将石墨烯片作为纳米填充物添加到泡沫金属中，形成了石墨烯增强多孔复合材料（Graphene Platelet Reinforced Porous Composite，GPLRPC），兼具石墨烯片的优异力学性能和多孔材料的轻质性[1]，因此受到广大学者的关注. 圆柱壳应用于实际工程中，例如汽轮机和航空发动机机匣等[2,3]，将GPLRPC应用到圆柱壳结构中会有广泛的应用前景，同时近年来，复杂边界下的壳体振动特性是研究热点，因此对任意边界GPLRPC圆柱壳的振动特性展开研究具有重要的意义.
目前，大量学者对功能梯度石墨烯增强复合材料板壳结构的振动特性展开了研究，例如Qin[4]、Feng[5]、沈慧申[6]和徐巍[7]等. 关于GPLRPC相关结构的动力学研究，Kitipornchai等[8]率先对GPLRPC开展了研究，分析了GPLRPC梁结构的自由振动和弹性屈曲行为. Yang等[9]通过基于一阶剪切变形理论来考虑横向剪切应变，采用切比雪夫-里兹法研究了GPLRPC板结构的屈曲和自由振动行为，并对石墨烯的质量分数、几何参数以及孔隙系数的影响进行了参数分析. Dong等[10]采用一阶剪切变形理论，研究了GPLRPC旋转圆柱壳的非线性自由振动. Wang等[11]考虑了不同的孔隙分布类型和石墨烯分布模式，对石墨烯增强金属泡沫圆柱壳进行了非线性振动分析. 黄小林等[12]基于Donnell经典薄壳理论研究了弹性介质中含内部孔隙的石墨烯增强功能截锥壳的振动特性.
通过以上所述可知，目前对于GPLRPC圆柱壳的动力学研究较多，但关于讨论边界条件影响的文献较少；此外，现有研究中采用的Rayleigh-Ritz法，将盖根堡多项式作为容许函数的文献较少涉及，因此开展此壳体结构的振动特性研究具有重要意义和应用前景. 本文基于一阶剪切变形理论，利用人工弹簧技术模拟任意边界条件和壳段间的连续条件，将盖根堡多项式作为容许函数求解其圆柱壳的振动频率，验证所提出方法的有效性，讨论边界条件、孔隙系数、孔隙类型、石墨烯分布类型、石墨烯质量分数、边界弹簧刚度和几何参数等参数对壳体结构振动特性的影响.
1　理论方法
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1.1　GPLRPC圆柱壳力学模型
如图1（a）建立GPLRPC圆柱壳的力学模型，在壳体中面定义柱坐标系（x，θ，z），O为坐标原点，轴向x、切向θ和径向z三个方向的位移分量分别由u、v和w表示，壳体中面半径为R，壳体总长度为L，如图1（b）所示，采用罚参数法[13]，每个壳段长度为L1、L2…、，每个壳段厚度为h1、h2…、. 为了便于文中数学计算和表达，定义无量纲坐标η=x/L，因此∂/∂x=（1/L）∂/∂η.
采用人工弹簧技术，壳体两端将定义五组弹簧，即三组平动弹簧以及两组旋转弹簧，如图1（c）所示，壳体左端（η=0）以及右端（η=1）的弹簧刚度系数分别表示为、、、、和、、、、，调整壳体两端的弹簧刚度系数可以得到任意边界条件.
1.2　GPLRPC圆柱壳的材料属性
沿厚度方向考虑3种孔隙分布类型[1]，类型Ⅰ和类型Ⅱ为对称分布，类型Ⅲ为均匀分布，如图2所示.
用孔隙系数e1、e2和e3来表示孔隙的分布情况，根据开胞体理论[14]，GPLRPC圆柱壳3种孔隙分布类型的有效弹性模量Ei（z）、密度ρi（z）和泊松比μi（z）依次表示为：
式中：上标i表示第i壳段，Ei、ρi和μi是无孔隙GPLRPC圆柱壳的弹性模量、密度和泊松比；e1、e2和e3分别表示孔隙分布类型Ⅰ、类型Ⅱ和类型Ⅲ的孔隙系数；em1、em2和em3分别表示孔隙分布类型Ⅰ、类型Ⅱ和类型Ⅲ的密度系数.
基于Halpin-Tsai微观力学模型[15]，GPLRPC圆柱壳的弹性模量Ei为：
式中：Em表示基体材料的弹性模量；表示石墨烯的体积分数；ηL和ηW表示与石墨烯以及基体材料弹性模量相关的参数；ξL和ξW表示与石墨烯长度LGPL、宽度wGPL和厚度hGPL相关的参数；分别表示为：
GPLRPC圆柱壳密度ρi和泊松比μi表示为：
GPLRPC圆柱壳的弹性模量和密度的关系如下式表示：
通过式（9）可以建立不同孔隙分布类型的密度系数和孔隙系数之间的关系，表示为：
通过设置具有不同孔隙系数和石墨烯分布的壳体质量相等，得出如下式表示：
此外，本文还考虑了3种石墨烯的分布模式，如图3所示. GPL-A在壳的外表面和内表面都表现出高含量的石墨烯；GPL-B在中表面表现出高含量的石墨烯，并逐渐向两个表面递减；GPL-C是均匀分布的. 3种分布模式的石墨烯体积分数分别为[11]：
式中：si1，si2和si3为石墨烯的体积分数系数，i=1，2，3分别对应类型Ⅰ，类型Ⅱ和类型Ⅲ.
石墨烯的总体积分数为：
式中：WGPL表示石墨烯的质量分数.
式（12）中的体积分数系数由下式求得：
式中：e1、e2和e3分别表示孔隙分布类型Ⅰ、类型Ⅱ和类型Ⅲ的孔隙系数；em1、em2和em3分别表示孔隙分布类型Ⅰ、类型Ⅱ和类型Ⅲ的密度系数.
2　GPLRPC圆柱壳振动控制方程
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基于一阶剪切变形理论，圆柱壳上任意一点沿着x、y和z轴方向的位移表示如下：
式中：、和为壳体中面位移，和分别表示壳体任意点处横截面绕θ轴和x轴的转角.
应变分量表示为：
式中：、、、、表示中面应变分量；、、、、表示中面曲率，分别表示为：
利用广义胡克定律可以得到以下关系：
式中：、表示壳体内任意一点沿x、θ方向上的正应力，、、分别表示xθ、xz、θz平面内的切应力；Qij为包含材料参数的缩减刚度系数，表达式为：
沿x和θ方向中面上的内力、内力矩和横向剪切力由式（20）表示：
式中，κ为修正系数，取值5/6.
沿x和θ方向中面上的内力、力矩和横向剪切力与曲率的关系如下：
式中：、和分别是拉伸刚度、耦合刚度和弯曲刚度，表示为：
GPLRPC圆柱壳的第i壳段动能Ti表示为：
式中：惯性项，，分别表示为：
第i壳段应变能表示为：
边界弹簧所储存的弹性势能Ub为：
相邻壳段之间的附加势能表示为：
采用统一的盖根堡多项式作为容许函数，将GPLRPC圆柱壳的位移场展开为：
式中：，，，，为未知系数，m和n分别为轴向半波数和周向波数，ω为固有角频率，M为最大截断数，为盖根堡多项式，在区间[-1，1]之间的递推关系[17]如式（29）（α>-1/2，α≠0），因为η∈[0，1]，使其t区间控制在[-1，1]，因此令t=2η-1. 定义α=1.
取各能量表达式的最大值，采用Rayleigh-Ritz法得到拉格朗日能量方程表示为：
式中，Np表示壳段数.
对式（30）未知系数q求导，由下式表示：
其中，
通过式（31）得到GPLRPC圆柱壳振动控制方程为：
式中：Kε、Kb、Ks和M分别表示为势能矩阵、连接刚度矩阵、边界刚度矩阵和质量矩阵.
3　模型收敛性和有效性验证
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3.1　模型参数
首先验证本文方法在求解GPLRPC圆柱壳振动特性方面的收敛性和合理性. 连接弹簧刚度值取值为[16]，石墨烯片的长度LGPL=2.5 μm，宽度wGPL=1.5 μm，厚度hGPL=1.5 nm，材料参数取值如表1所示.
GPLRPC圆柱壳尺寸为L/R=5，R/h1=50，h1=h2，L1=L2. 定义壳体无量纲频率为.
3.2　收敛性和有效性验证
本文引入了人工弹簧技术，可以随意改变边界弹簧的刚度值，图4为壳体无量纲频率随边界弹簧刚度值的变化. 从图4可以看出，当弹簧刚度值小于105时，壳体的无量纲频率几乎没有变化，可以定义为自由边界条件；当弹簧刚度值在105-1012之间时，无量纲频率迅速增加，这一段可看作弹性边界条件；当弹簧刚度值高于1012时，无量纲频率几乎没有变化，被认为是固支边界条件. 为了在后续研究中更方便地表达，用不同的字母表示不同的边界条件，F代表自由边界，S代表简支边界，C是固支边界，E1、E2和E3是三种不同的弹性边界. 不同的边界条件和相应的刚度值如表2所示.
采用盖根堡多项式作为容许函数，需要对振型函数展开项截断数进行分析，不同边界下截断数对壳体无量纲频率收敛性的分析如图5所示. 由图5可以看出，当截断数M=12时，频率计算结果已经达到收敛，说明盖根堡多项式具有良好的收敛性. 文中截断数均取M=12.
本文采用罚参数法分析GPLRPC圆柱壳，需要考虑壳段数对收敛性的影响，不同边界下3种孔隙分布类型不同壳段数对GPLRPC圆柱壳无量纲频率的影响如表3所示，可以看出，壳段数在NP=2就具有较好的收敛性，因此下文研究取壳段数NP=2.
为了验证本文方法的有效性，对S-S边界条件下GPLRPC圆柱壳的文献[1]进行验证，其中h1=h2，（m，n）=（1，1），通过文献[1]的验证如表4所示，材料参数如表1所示. 可以发现，分析结果与文献结果最大误差为0.32%，具有良好的一致性，适于分析GPLRPC圆柱壳的振动特性.
对于GPLRPC圆柱壳，对4种经典边界下的前4阶固有频率和有限元法进行验证，如表5所示. 从中看出，两者最大误差为2.19%，误差在合理范围内，说明此方法适合GPLRPC圆柱壳的振动特性分析. 为了更加清晰地反映GPLRPC圆柱壳的振动特性，取C-C边界下的前3阶振型图如图6所示，可以发现，振型图以周向模态振型变化为主，可以清晰地看出前三阶的周向波数分别为3、4和2.
4　数值结果与讨论
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4.1　边界条件的影响
不同边界条件下GPLRPC圆柱壳频率随周向波数的变化如图7所示. 可以看出，7种边界条件都随着周向波数的增大呈现先减小后增大的趋势，表明m=1，n=1不是所有模态振型中频率最小的，最低频率集中在n=2或n=3时；随着周向波数的增大，不同边界下的壳体频率有逐渐接近的趋势，说明边界条件对壳体频率的影响逐渐减小. 由图7还可以发现，当n≤3时，C-C边界下的壳体频率最高，而当n>3时，从局部放大图中可以看出，E2-E2弹性边界具有最大的壳体频率.
4.2　孔隙系数和分布模式的影响
给定石墨烯分布模式为GPL-C，不同边界条件下各孔隙分布类型和不同孔隙系数的壳体无量纲频率如表6所示. 可以发现，不同边界条件下的壳体频率有所不同，C-C边界条件下频率最大；3种孔隙分布类型中，类型Ⅱ的频率最大，具有最大结构刚度；同时壳体频率随着孔隙系数的增大而减小，这是由于较高的孔隙系数表示壳体中有较密集和较大的孔隙，削弱了圆柱壳结构的刚度.
给定孔隙分布类型为类型Ⅰ，C-C边界下3种石墨烯分布类型下壳体频率随孔隙系数的变化如图8所示，由图8看出，GPL-A分布模式下壳体频率最高，具有最佳的刚度增强效果，其次为GPL-C、GPL-B分布模式，造成3种不同石墨烯分布模式壳体频率产生差异的主要原因是石墨烯片的弹性模量不同造成的. 工程中可以选取外壁和内壁石墨烯含量多的圆柱壳作为最佳刚度增强效果的结构.
4.3　石墨烯质量分数的影响
C-C边界条件下石墨烯的质量分数对壳体频率的影响如图9所示，频率随着石墨烯质量分数的增大而增大，当石墨烯质量分数为1%时频率最大，说明石墨烯对壳体结构的刚度具有明显的增强效果. 这是因为石墨烯质量分数的增大导致GPLRPC弹性模量的增大，从而增加了壳体结构的固有频率和刚度.
4.4　边界弹簧刚度的影响
为了研究边界弹簧刚度值的影响，设置平动和旋转弹簧刚度值从0增加到1014，当平移或旋转弹簧刚度值增加时，另一组弹簧的刚度保持不变，计算各个刚度值的GPLRPC圆柱壳无量纲频率如图10所示. 从图10看出，频率变化经历三个阶段，在开始阶段，频率保持不变且频率较低；第二阶段频率明显增加；最后阶段，频率再次保持稳定不变. 此外还可以发现，平动弹簧对频率的影响较大，旋转弹簧对频率的影响较小.
4.5　几何参数的影响
图11为C-C边界条件下圆柱壳几何参数对壳体振动特性的影响. 由图11可知，随着圆柱壳长径比的增大，壳体频率降低，长径比对圆柱壳的振动特性具有较大影响；径厚比在长径比L/R≤2时对壳体频率有所影响，但当长径比L/R>2，径厚比对圆柱壳振动特性影响较小.
5　结论
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基于开胞体理论和一阶剪切变形壳理论，以GPLRPC圆柱壳为研究对象，提出了一种以盖根堡多项式为容许函数的半解析法，该方法具有良好的收敛性和准确性，通过确定不同边界下的边界弹簧刚度值，从而得到任意边界下的壳体频率，主要结论如下：
（1）不同边界条件下壳体频率都随着周向波数的增大呈现先减小后增大的趋势，且随着周向波数的增大频率有逐渐接近的趋势；不同边界条件下的GPLRPC圆柱壳的频率有所不同，C-C边界条件下频率最大；GPL-A分布模式和孔隙分布类型Ⅱ具有最佳的刚度增强效果.
（2）孔隙系数对壳体频率的影响显著，孔隙系数越大，频率越小；石墨烯质量分数越大，频率越大，微量的石墨烯能对壳体刚度具有较大的增强效果.
（3）边界弹簧刚度对壳体频率具有明显影响，平动弹簧对频率的影响较大，旋转弹簧对频率的影响较小.
（4）圆柱壳频率随着长径比的增大而减小，而径厚比对圆柱壳振动特性的整体影响较小. 本文研究可提供数据支撑和理论参考，工程中对此类结构可进行主动设计.
基金
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  • 国家自然科学基金项目(51775257; 51775258)
  • 辽宁省教育厅基金项目(LJKMZ20220637)
文献
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参考文献 引证文献
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[1]
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2024年第45卷第3期
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doi: 10.19636/j.cnki.cjsm42-1250/o3.2023.057
  • 接收时间：2023-11-04
  • 首发时间：2026-04-01
  • 出版时间：2024-06-25
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  • 收稿日期：2023-11-04
基金
国家自然科学基金项目(51775257; 51775258)
辽宁省教育厅基金项目(LJKMZ20220637)
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    辽宁科技大学机械工程与自动化学院，鞍山，114051

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2种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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