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Article(id=1246023205710840727, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246023204117005194, articleNumber=null, orderNo=null, doi=10.19636/j.cnki.cjsm42-1250/o3.2023.052, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1697644800000, receivedDateStr=2023-10-19, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1775004682232, onlineDateStr=2026-04-01, pubDate=1713974400000, pubDateStr=2024-04-25, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1775004682232, onlineIssueDateStr=2026-04-01, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1775004682232, creator=13701087609, updateTime=1775004682232, updator=13701087609, issue=Issue{id=1246023204117005194, tenantId=1146029695717560320, journalId=1241755870837649424, year='2024', volume='45', issue='2', pageStart='145', pageEnd='288', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1775004681852, creator=13701087609, updateTime=1775004747143, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1246023478026027853, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246023204117005194, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1246023478026027854, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246023204117005194, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=213, endPage=224, ext={EN=ArticleExt(id=1246023205907973017, articleId=1246023205710840727, tenantId=1146029695717560320, journalId=1241755870837649424, language=EN, title=Buckling Analysis of Functionally Graded Graphene-reinforced Plates Based on Moving Kriging and Third-order Deformation Theory, columnId=1244229834482757770, journalTitle=Chinese Journal of Solid Mechanics, columnName=Research Paper, runingTitle=null, highlight=null, articleAbstract=

The emergence of graphene nanoplatelets (GPLs) has enabled the development of lightweight and high-strength plates, making it a prominent area of research in science and engineering. Therefore, it is essential to study the buckling performance of functionally graded graphene-reinforced composite (FG-GRC) plates. This paper presents a new meshless model to solve the buckling behavior problem of FG-GRC plates. The model is based on an improved Reddy-type third-order shear deformation theory (TSDT) with seven degrees of freedom and a moving Kriging (MK) interpolation method, which can overcome the challenge of implementing the second-type boundary conditions in meshless methods and eliminate the need for shear correction factors. The model is applicable to thin/medium/thick plate problems and has high computational accuracy. The Halpin-Tsai model is used to predict the effective Young's modulus of the FG-GRC plate, and the effective Poisson's ratio is determined using the mixture law. The meshless governing equation for the buckling of the FG-GRC plate with seven unknowns is derived based on the principle of minimum potential energy. The convergence and effectiveness of the proposed method are verified by comparing it with literature results. The numerical results demonstrate that when the total number of layers (NL) of the FG-GRC plate is less than 10-15, the critical buckling load of the FG-O-type and FG-X-type plates changes more drastically than that of the epoxy pure plate, indicating that the stiffness of the graphene-reinforced plate decreases (or increases) rapidly in this stage, as opposed to the epoxy pure plate. However, when NL exceeds 10-15, the change rate of the critical buckling load for the FG-GRC plate becomes smoother. Furthermore, the critical buckling load of the FG-GRC plate increases sharply when the length-thickness ratio of the GPLs reaches around 1000. Once the length-thickness ratio of GPLs surpasses 2000, the critical buckling load of the FG-GRC plate tends to stabilize, and the length-width ratio and length-thickness ratio of the GPLs have no significant effect on it. Overall, the research findings of this study not only contribute to the understanding of FG-GRC plates but also offer practical and insightful recommendations for their design and theoretical research.

, correspAuthors=Linxin Peng, 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=Wei Chen, Yaochu Fang, Linxin Peng), CN=ArticleExt(id=1246023215412265286, articleId=1246023205710840727, tenantId=1146029695717560320, journalId=1241755870837649424, language=CN, title=基于MK和TSDT的功能梯度石墨烯增强复合材料板屈曲分析, columnId=1241831201896469478, journalTitle=固体力学学报, columnName=研究论文, runingTitle=null, highlight=null, articleAbstract=

针对功能梯度石墨烯增强复合材料（FG-GRC）板屈曲行为问题，提出一种含7个自由度变量改进Reddy型三阶剪切变形理论（TSDT）和移动克里金（MK）插值的无网格模型. 该模型不但能避免无网格法中第二类边界条件难以施加的问题，且不需人工引入剪切修正因子，适用于薄/中厚/厚板问题，同时具有较高的计算精度. 通过Halpin-Tsai模型来预测FG-GRC板的有效杨氏模量，并根据混合定律来确定其有效泊松比. 利用最小势能原理推导了含7个未知量FG-GRC板屈曲的无网格控制方程. 通过与文献结果对比验证了方法的收敛性及有效性. 数值结果表明：当FG-GRC板的总层数NL小于10～15时，FG-O和FG-X型的FG-GRC板临界屈曲荷载变化率较为剧烈，说明该阶段相较于环氧树脂板，GPLs增强板的刚度降低（或增加）较快；当FG-GRC板的总层数NL>10～15时，临界屈曲荷载变化率较为平缓；随着GPLs的长厚比lGPL/hGPL增加到1000左右，FG-GRC板临界屈曲荷载急剧增加. 当GPLs的长厚比lGPL/hGPL增加到2000以上时，FG-GRC板临界屈曲荷载趋向于稳定且GPLs的长宽比lGPL/wGPL和长厚比lGPL/hGPL对FG-GRC板临界屈曲荷载影响不再明显.

, correspAuthors=彭林欣, authorNote=null, correspAuthorsNote=
** E-mail：
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figureFileBig=jTTVOvKxuI9mOJXj/UGr1w==, tableContent=null), ArticleFig(id=1246023218465718724, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246023205710840727, language=CN, label=图5, caption=总层数NL对单压下四边简支FG-GRC板无量纲临界屈曲荷载变化率RN的影响, figureFileSmall=3E44U7OlZi4nQt0TJLu6yg==, figureFileBig=jTTVOvKxuI9mOJXj/UGr1w==, tableContent=null), ArticleFig(id=1246023218557993417, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246023205710840727, language=EN, label=Fig.6, caption=Effect of width-to-thickness ratio b/h on the dimensionless critical buckling load of simply supported FG-GRC plates under uniaxial compression, figureFileSmall=2s5WIX0sOyy7NWtMXBJN/Q==, figureFileBig=56QNeYePrV6ZghWmgn3/Jg==, tableContent=null), ArticleFig(id=1246023218658656715, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246023205710840727, language=CN, label=图6, caption=宽厚比b/h对单压下四边简支FG-GRC板无量纲临界屈曲荷载的影响, figureFileSmall=2s5WIX0sOyy7NWtMXBJN/Q==, figureFileBig=56QNeYePrV6ZghWmgn3/Jg==, tableContent=null), ArticleFig(id=1246023218738348492, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246023205710840727, language=EN, label=Fig.7, caption=Effect of GPLs geometry on the dimensionless critical buckling load of simply supported FG-GRC plates under uniaxial or biaxial compression, figureFileSmall=i0appSMiV1VR6kTgv0889w==, figureFileBig=ry8cupM+2Khq8P+o9ULEmg==, tableContent=null), ArticleFig(id=1246023218813845967, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246023205710840727, language=CN, label=图7, caption=GPLs几何尺寸对单压或双压下四边简支FG-GRC板无量纲临界屈曲荷载的影响, figureFileSmall=i0appSMiV1VR6kTgv0889w==, figureFileBig=ry8cupM+2Khq8P+o9ULEmg==, tableContent=null), ArticleFig(id=1246023218897732049, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246023205710840727, language=EN, label=Fig.8, caption=The variation of dimensionless critical buckling load of FG-GRC plate with gGPL under different boundary conditions, figureFileSmall=z8ngSXV+fEgz1UKJWT5Yag==, figureFileBig=mqJMwLjDJwhMiopsRDtr8w==, tableContent=null), ArticleFig(id=1246023218969035219, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246023205710840727, language=CN, label=图8, caption=不同边界条件下FG-GRC板无量纲临界屈曲荷载随重量分数gGPL的变化, figureFileSmall=z8ngSXV+fEgz1UKJWT5Yag==, figureFileBig=mqJMwLjDJwhMiopsRDtr8w==, tableContent=null), ArticleFig(id=1246023219086475734, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246023205710840727, language=EN, label=Table 1, caption=

Material properties of FG-GRC plate

, figureFileSmall=null, figureFileBig=null, tableContent=
Material propertiesEpoxyGPL
Young's modulus（GPa）EM=3EGPL=1010
Poisson's ratioνM=0.36νGPL=0.186
Density（kg/m3）ρM=1200ρGPL=1060
), ArticleFig(id=1246023219199721946, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246023205710840727, language=CN, label=表1, caption=

FG-GRC板材料属性

, figureFileSmall=null, figureFileBig=null, tableContent=
Material propertiesEpoxyGPL
Young's modulus（GPa）EM=3EGPL=1010
Poisson's ratioνM=0.36νGPL=0.186
Density（kg/m3）ρM=1200ρGPL=1060
), ArticleFig(id=1246023219304579547, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246023205710840727, language=EN, label=Table 2, caption=

Dimensionless critical buckling loads of simply supported FG-GRC plate under uniaxial and biaxial compression

, figureFileSmall=null, figureFileBig=null, tableContent=
Load typ eMethodsUDFG-OFG-X
Uniaxial compressionTSDT[12]0.01520.01080.0191
FSDT[7]0.01520.01070.0195
present0.01520.01080.0192
Biaxial compressionTSDT[12]0.00760.00540.0096
FSDT[7]0.00760.00530.0097
present0.00760.00540.0096
), ArticleFig(id=1246023219375882718, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246023205710840727, language=CN, label=表2, caption=

单压或双压下四边简支FG-GRC板无量纲临界屈曲荷载

, figureFileSmall=null, figureFileBig=null, tableContent=
Load typ eMethodsUDFG-OFG-X
Uniaxial compressionTSDT[12]0.01520.01080.0191
FSDT[7]0.01520.01070.0195
present0.01520.01080.0192
Biaxial compressionTSDT[12]0.00760.00540.0096
FSDT[7]0.00760.00530.0097
present0.00760.00540.0096
), ArticleFig(id=1246023219522683361, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246023205710840727, language=EN, label=Table 3, caption=

Comparative analysis of critical buckling loads of sandwich panels under short edge single compression

, figureFileSmall=null, figureFileBig=null, tableContent=
Core materialExperimental（kN）[27]FEM（kN）[27]Present（kN）
Balsa334365369
Foam267347343
), ArticleFig(id=1246023219640123877, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246023205710840727, language=CN, label=表3, caption=

短边单压下夹芯板临界屈曲荷载对比分析

, figureFileSmall=null, figureFileBig=null, tableContent=
Core materialExperimental（kN）[27]FEM（kN）[27]Present（kN）
Balsa334365369
Foam267347343
), ArticleFig(id=1246023219736592872, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246023205710840727, language=EN, label=Table 4, caption=

Dimensionless critical buckling loads and RN of simply supported FG-GRC plate under uniaxial compression

, figureFileSmall=null, figureFileBig=null, tableContent=
GPLs typeMethodgGPL
0.0%0.2%0.4%0.6%0.8%1.0%1.2%
UDFSDT[7]0.00350.0058（165.7%）0.0082（234.3%）0.0105（300.0%）0.0128（365.7%）0.0152（434.3%）0.0175（500.0%）
TSDT[12]0.00350.0058（165.7%）0.0082（234.3%）0.0105（300.0%）0.0128（365.7%）0.0152（434.3%）0.0175（500.0%）
I-TSDT0.003520.00587（166.8%）0.00821（233.2%）0.01055（299.7%）0.01289（366.2%）0.01523（432.7%）0.01757（499.1%）
FG-OFSDT[7]0.00350.0050（142.9%）0.0064（182.9%）0.0078（222.9%）0.0093（265.7%）0.0107（305.7%）0.0121（345.7%）
TSDPT[12]0.00350.0050（142.9%）0.0064（182.9%）0.0079（225.7%）0.0093（265.7%）0.0108（308.6%）0.0122（348.6%）
I-TSDT0.003520.00500（142.0%）0.00645（183.2%）0.00791（224.7%）0.00936（265.9%）0.01081（307.1%）0.01226（348.3%）
FG-XFSDT[7]0.00350.0067（191.4%）0.0099（282.9%）0.0131（374.3%）0.0163（465.7%）0.0195（557.1%）0.0227（648.6%）
TSDT[12]0.00350.0067（191.4%）0.0098（280.0%）0.0129（368.6%）0.0160（457.1%）0.0191（545.7%）0.0222（634.3%）
I-TSDT0.003520.00669（190.1%）0.00982（279.0%）0.01294（367.6%）0.01605（456.0%）0.01916（544.3%）0.02226（632.4%）
), ArticleFig(id=1246023219837256170, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246023205710840727, language=CN, label=表4, caption=

单压下四边简支FG-GRC板无量纲临界屈曲荷载和RN

, figureFileSmall=null, figureFileBig=null, tableContent=
GPLs typeMethodgGPL
0.0%0.2%0.4%0.6%0.8%1.0%1.2%
UDFSDT[7]0.00350.0058（165.7%）0.0082（234.3%）0.0105（300.0%）0.0128（365.7%）0.0152（434.3%）0.0175（500.0%）
TSDT[12]0.00350.0058（165.7%）0.0082（234.3%）0.0105（300.0%）0.0128（365.7%）0.0152（434.3%）0.0175（500.0%）
I-TSDT0.003520.00587（166.8%）0.00821（233.2%）0.01055（299.7%）0.01289（366.2%）0.01523（432.7%）0.01757（499.1%）
FG-OFSDT[7]0.00350.0050（142.9%）0.0064（182.9%）0.0078（222.9%）0.0093（265.7%）0.0107（305.7%）0.0121（345.7%）
TSDPT[12]0.00350.0050（142.9%）0.0064（182.9%）0.0079（225.7%）0.0093（265.7%）0.0108（308.6%）0.0122（348.6%）
I-TSDT0.003520.00500（142.0%）0.00645（183.2%）0.00791（224.7%）0.00936（265.9%）0.01081（307.1%）0.01226（348.3%）
FG-XFSDT[7]0.00350.0067（191.4%）0.0099（282.9%）0.0131（374.3%）0.0163（465.7%）0.0195（557.1%）0.0227（648.6%）
TSDT[12]0.00350.0067（191.4%）0.0098（280.0%）0.0129（368.6%）0.0160（457.1%）0.0191（545.7%）0.0222（634.3%）
I-TSDT0.003520.00669（190.1%）0.00982（279.0%）0.01294（367.6%）0.01605（456.0%）0.01916（544.3%）0.02226（632.4%）
), ArticleFig(id=1246023219925336557, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246023205710840727, language=EN, label=Table 5, caption=

Dimensionless critical buckling loads and RN of simply supported FG-GRC plate under biaxial compression

, figureFileSmall=null, figureFileBig=null, tableContent=
GPLs typeMethodgGPL
0.0%0.2%0.4%0.6%0.8%1.0%1.2%
UDFSDT[7]0.00180.0029（161.1%）0.0041（227.8%）0.0053（294.4%）0.0064（355.6%）0.0076（422.2%）0.0088（488.9%）
TSDT[12]0.00180.0029（161.1%）0.0041（227.8%）0.0053（294.4%）0.0064（355.6%）0.0076（422.2%）0.0088（488.9%）
I-TSDT0.001760.00293（166.5%）0.00410（233.0%）0.00527（299.4%）0.00644（365.9%）0.00761（432.4%）0.00878（498.9%）
FG-OFSDT[7]0.00180.0025（138.9%）0.0032（177.8%）0.0039（216.7%）0.0046（255.6%）0.0053（294.4%）0.0061（338.9%）
TSDT[12]0.00180.0025（138.9%）0.0032（177.8%）0.0039（216.7%）0.0047（255.6%）0.0054（294.4%）0.0061（338.9%）
I-TSDT0.001760.00250（142.0%）0.00323（183.5%）0.00395（224.4%）0.00468（265.9%）0.00540（306.8%）0.00613（348.3%）
FG-XFSDT[7]0.00180.0034（188.9%）0.005（277.8%）0.0066（366.7%）0.0082（455.6%）0.0097（538.9%）0.0113（627.8%）
TSDPT[12]0.00180.0033（188.9%）0.0049（277.8%）0.0065（366.7%）0.0080（455.6%）0.0096（538.9%）0.0111（627.8%）
I-TSDT0.001760.00334（189.8%）0.00491（279.0%）0.00647（367.6%）0.00803（456.3%）0.00958（544.3%）0.01113（632.4%）
), ArticleFig(id=1246023220013416945, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246023205710840727, language=CN, label=表5, caption=

双压下四边简支FG-GRC板无量纲临界屈曲荷载和RN

, figureFileSmall=null, figureFileBig=null, tableContent=
GPLs typeMethodgGPL
0.0%0.2%0.4%0.6%0.8%1.0%1.2%
UDFSDT[7]0.00180.0029（161.1%）0.0041（227.8%）0.0053（294.4%）0.0064（355.6%）0.0076（422.2%）0.0088（488.9%）
TSDT[12]0.00180.0029（161.1%）0.0041（227.8%）0.0053（294.4%）0.0064（355.6%）0.0076（422.2%）0.0088（488.9%）
I-TSDT0.001760.00293（166.5%）0.00410（233.0%）0.00527（299.4%）0.00644（365.9%）0.00761（432.4%）0.00878（498.9%）
FG-OFSDT[7]0.00180.0025（138.9%）0.0032（177.8%）0.0039（216.7%）0.0046（255.6%）0.0053（294.4%）0.0061（338.9%）
TSDT[12]0.00180.0025（138.9%）0.0032（177.8%）0.0039（216.7%）0.0047（255.6%）0.0054（294.4%）0.0061（338.9%）
I-TSDT0.001760.00250（142.0%）0.00323（183.5%）0.00395（224.4%）0.00468（265.9%）0.00540（306.8%）0.00613（348.3%）
FG-XFSDT[7]0.00180.0034（188.9%）0.005（277.8%）0.0066（366.7%）0.0082（455.6%）0.0097（538.9%）0.0113（627.8%）
TSDPT[12]0.00180.0033（188.9%）0.0049（277.8%）0.0065（366.7%）0.0080（455.6%）0.0096（538.9%）0.0111（627.8%）
I-TSDT0.001760.00334（189.8%）0.00491（279.0%）0.00647（367.6%）0.00803（456.3%）0.00958（544.3%）0.01113（632.4%）
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基于MK和TSDT的功能梯度石墨烯增强复合材料板屈曲分析
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陈卫 1 , 方耀楚 1 , 彭林欣 2, 3, **
固体力学学报 | 研究论文 2024,45(2): 213-224
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固体力学学报 | 研究论文 2024, 45(2): 213-224
基于MK和TSDT的功能梯度石墨烯增强复合材料板屈曲分析
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陈卫1, 方耀楚1, 彭林欣2, 3, **
作者信息
  • 1南华大学土木工程学院，衡阳，421001
  • 2广西大学土木建筑工程学院，南宁，530004
  • 3广西防灾减灾与工程安全重点实验室，工程防灾与结构安全教育部重点实验室，南宁，530004

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Buckling Analysis of Functionally Graded Graphene-reinforced Plates Based on Moving Kriging and Third-order Deformation Theory
Wei Chen1, Yaochu Fang1, Linxin Peng2, 3, **
Affiliations
  • 1School of Civil Engineering, University of South China, Hengyang, 421001
  • 2School of Civil Engineering and Architecture, Guangxi University, Nanning, 530004
  • 3Key Laboratory of Disaster Prevention and Structural Safety of Ministry of Education, Guangxi Key Laboratory of Disaster Prevention and Engineering Safety, Guangxi University, Nanning, 530004
出版时间: 2024-04-25 doi: 10.19636/j.cnki.cjsm42-1250/o3.2023.052
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摘要
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针对功能梯度石墨烯增强复合材料（FG-GRC）板屈曲行为问题，提出一种含7个自由度变量改进Reddy型三阶剪切变形理论（TSDT）和移动克里金（MK）插值的无网格模型. 该模型不但能避免无网格法中第二类边界条件难以施加的问题，且不需人工引入剪切修正因子，适用于薄/中厚/厚板问题，同时具有较高的计算精度. 通过Halpin-Tsai模型来预测FG-GRC板的有效杨氏模量，并根据混合定律来确定其有效泊松比. 利用最小势能原理推导了含7个未知量FG-GRC板屈曲的无网格控制方程. 通过与文献结果对比验证了方法的收敛性及有效性. 数值结果表明：当FG-GRC板的总层数NL小于10～15时，FG-O和FG-X型的FG-GRC板临界屈曲荷载变化率较为剧烈，说明该阶段相较于环氧树脂板，GPLs增强板的刚度降低（或增加）较快；当FG-GRC板的总层数NL>10～15时，临界屈曲荷载变化率较为平缓；随着GPLs的长厚比lGPL/hGPL增加到1000左右，FG-GRC板临界屈曲荷载急剧增加. 当GPLs的长厚比lGPL/hGPL增加到2000以上时，FG-GRC板临界屈曲荷载趋向于稳定且GPLs的长宽比lGPL/wGPL和长厚比lGPL/hGPL对FG-GRC板临界屈曲荷载影响不再明显.

功能梯度石墨烯增强复合材料板  /  改进Reddy型三阶剪切变形理论  /  移动克里金  /  临界屈曲荷载

The emergence of graphene nanoplatelets (GPLs) has enabled the development of lightweight and high-strength plates, making it a prominent area of research in science and engineering. Therefore, it is essential to study the buckling performance of functionally graded graphene-reinforced composite (FG-GRC) plates. This paper presents a new meshless model to solve the buckling behavior problem of FG-GRC plates. The model is based on an improved Reddy-type third-order shear deformation theory (TSDT) with seven degrees of freedom and a moving Kriging (MK) interpolation method, which can overcome the challenge of implementing the second-type boundary conditions in meshless methods and eliminate the need for shear correction factors. The model is applicable to thin/medium/thick plate problems and has high computational accuracy. The Halpin-Tsai model is used to predict the effective Young's modulus of the FG-GRC plate, and the effective Poisson's ratio is determined using the mixture law. The meshless governing equation for the buckling of the FG-GRC plate with seven unknowns is derived based on the principle of minimum potential energy. The convergence and effectiveness of the proposed method are verified by comparing it with literature results. The numerical results demonstrate that when the total number of layers (NL) of the FG-GRC plate is less than 10-15, the critical buckling load of the FG-O-type and FG-X-type plates changes more drastically than that of the epoxy pure plate, indicating that the stiffness of the graphene-reinforced plate decreases (or increases) rapidly in this stage, as opposed to the epoxy pure plate. However, when NL exceeds 10-15, the change rate of the critical buckling load for the FG-GRC plate becomes smoother. Furthermore, the critical buckling load of the FG-GRC plate increases sharply when the length-thickness ratio of the GPLs reaches around 1000. Once the length-thickness ratio of GPLs surpasses 2000, the critical buckling load of the FG-GRC plate tends to stabilize, and the length-width ratio and length-thickness ratio of the GPLs have no significant effect on it. Overall, the research findings of this study not only contribute to the understanding of FG-GRC plates but also offer practical and insightful recommendations for their design and theoretical research.

functionally graded graphene-reinforced composite plates  /  improved Reddy-type third-order shear deformation theory  /  moving Kriging  /  critical buckling load
陈卫, 方耀楚, 彭林欣. 基于MK和TSDT的功能梯度石墨烯增强复合材料板屈曲分析. 固体力学学报, 2024 , 45 (2) : 213 -224 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2023.052
Wei Chen, Yaochu Fang, Linxin Peng. Buckling Analysis of Functionally Graded Graphene-reinforced Plates Based on Moving Kriging and Third-order Deformation Theory[J]. Chinese Journal of Solid Mechanics, 2024 , 45 (2) : 213 -224 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2023.052
正文
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0　引言
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石墨烯[1]是一种二维、单原子厚的碳层，因其优异的热、电、机械性能[2-4]而迅速成为学术界和工业界的关注热点. 2009年，Rafiee等[2]开创性地用实验证明了通过只添加0.1重量分数（gGPL%）的石墨烯纳米片（graphene nanoplatelets，GPLs），增强聚合物复合材料的刚度和强度可以达到添加1.0 gGPL%碳纳米管（carbon nanotubes，CNTs）所能达到的程度，并且弹性模量增加了约31%. 这是因为GPLs在聚合物基体中的分布因其二维属性而得到了极大改善，相比于一维各向异性的CNTs，其团聚现象更少. 这为基于GPLs增强复合材料的轻量化高强结构的开发创造了巨大机遇. 因此，板作为工程结构使用的基本元件，对FG-GRC板力学行为研究非常重要.
2017年，Song等[5-7]基于一阶剪切变形理论（first-order deformation theory，FSDT），采用Navier解答，给出了四边简支FG-GRC板的静态弯曲、屈曲、自由振动等数值结果. 研究发现：GPLs的分布模式、重量分数、几何参数等对FG-GRC板的弯曲、屈曲、振动等力学行为均具有显著的影响. 2018年，Lei等[8]基于FSDT，根据扩展Halpin-Tsai模型，采用kp-Ritz法研究了热环境下FG-GRC板的屈曲行为. 2019年，Thai等[9]基于四变量的精细化板理论（refined plate theory，RPT），采用等几何分析法（isogeometric analysis，IGA）研究了FG-GRC板的自由振动、弯曲和屈曲力学行为. 2020年，庞有卿等[10]基于双曲剪切变形理论（hyperbolic shear deformation theory，HSDT），利用Navier技术求出了四边简支FG-GRC板的自振频率. 2021年，黄小林等[11]考虑粘弹性地基的影响，基于经典板理论（classical plate theory，CPT），采用Galerkin法和龙格库塔法求解了四边简支或固支下FG-GRC板自振频率和动力响应问题. 2022年，王壮壮等[12]基于正弦剪切变形理论（sinusoidal shear deformation theory，SSDT）和TS-DT，采用Navier解答，研究了四边简支FG-GRC板静态弯曲和屈曲力学行为. 2022年，Zhou等[13]基于TS-DT，同样采用Navier解答，求解了winkler地基上四边简支FG-GRC板临界屈曲荷载.
综上可知，目前大多数文献基于FSDT或者TSDT框架下，利用Navier解答技术给出了FG-GRC板静动力学行为的解析解. 然而解析解只能针对特定的边界条件和简单的载荷，与实际工况具有一定的差异，因而数值算法显得尤为重要. 无网格伽辽金法作为众多数值算法的一种，因其具有列式简单、稳定性好及计算精度高等优点而被广泛应用. 但因第二类边界条件施加困难问题，采用无网格伽辽金法研究各类板问题的文献均是基于FSDT[14-16]. 2016年，Liew等[17,18]将TSDT中的高阶项设为一个独立变量，将每节点含5个自由度变量扩展到7个自由度变量，避免了第二类边界条件的施加，结合无网格法分析了功能梯度碳纳米管增强复合材料（functionally graded carbon nanotube-reinforced composite，FG-CNTRC）板的自由振动与受迫振动问题.
本文基于改进Reddy型TSDT和移动克里金插值，建立了分析FG-GRC板屈曲的无网格模型. 相比FSDT下的无网格模型，该模型在不需要人工引入的前提下能同时考虑横向剪切变形效应和满足零牵引力边界条件，适用于薄/中厚/厚板问题. 文中首先给出了改进Reddy型TSDT下FG-GRC板屈曲的无网格列式，接着通过基准算例验证本文方法的收敛性和有效性. 最后数值讨论了GPLs分布模式，重量分数、总层数、宽厚比和边界条件等对FG-GRC板临界屈曲荷载的影响.
1　材料性质
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图1给出了总层数为NL=6，长、宽、高分别为a、b、h的FG-GRC板. FG-GRC板由NL层等厚度h/NL石墨烯纳米增强片组成，每层含均匀分散的GPLs. GPL重量分数（gGPL%）呈逐层变化以形成功能梯度结构. 由于拉-弯耦合效应，屈曲分析时，即使最小的面内荷载对非对称层合板也会产生的挠度和弯矩[19]. 因此，本文只考虑如图2所示三种不同的GPLs分布模式，包括均匀分布（UD型）和功能梯度分布（FG-O型，FG-X型），对于GPLs非对称分布的FG-Λ型或FG-V型不考虑.
根据修正的Halpin-Tsai模型[20]，第k层GRC有效杨氏模量为：
其中：
式中：下标C、GPL和M分别表示石墨烯增强复合材料，GPLs及基体材料；EGPL和EM分别为GPLs和基体的杨氏模量；lGPL，wGPL和hGPL是GPLs的长、宽、厚. 此外，第k层GRC等效泊松比可由混合定律确定如下：
第k层GRC的体积分数为：
式中：ρGPL和ρM分别为GPLs和基体材料的质量密度；为第k层GPLs的重量分数，可由GPLs的分布模式确定如下[5]：
2　移动克里金插值
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根据文献[21]，在MK插值技术中，子域Ωx中的函数u（x）可由uh（x）近似为：
或
式中，n是子域Ωx内的节点数，NI（x）是MK的形函数，可定义为：
式（8）中的矩阵A和B为：
式中，I为n×n阶单位矩阵，向量P（x）是阶数为m的多项式基函数：
对于二维问题常采用二次基函数PT（x）=[1x y x2 xy y2]，（m=6）. 为了消除节点间距较少导致插值不稳定，Tu等[22]于2019年提出了归一化多项式基函数P（x）：
式中，xe或（xe，ye）是子域Ωx内任意点xe的坐标，dm是子域Ωx的半径：dm=αdave，dave为平均节点间距，α为比例因子，本文采用方形影响域α=3.
式（12）中大小为n×m的矩阵P包含了子域Ωx内n个节点处多项式基函数的值，表达式如下：
式（8）中r（x）定义为：
式中，R[R（xi，xj）]是相关矩阵，而R（xi，xj）是n个节点xi和xj之间的相关函数，它表示u（x）的协方差：R（xi，xj）=cov[u（xi）u　（xj）]和R（xi，x）=cov[u（xi）　 u（x）]. 通常采用如下高斯函数：
式中，rij=‖xi-xj‖，a0为子域Ωx内节点之间的最大距离，θ>0为依赖于子域Ωx节点坐标的参数. 根据文献[23]取θ=5.0.
式（15）中矩阵R[R（xi，xj）]n×n可以写为：
MK形函数的一阶导为
移动克里金形函数满足Kronecker delta特性，可以像有限元那样施加边界条件：
3　FG-GRC板的无网格控制方程
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3.1　总能量泛函
根据Liew等[17,18]提出的改进Reddy型TSDT，含7个自由度变量的板位移场可表示为：
式中：（u0，v0，w0）T为板中面任意一点在x，y，z方向的位移；φx，φy分别为绕y轴和x轴的转动；c=-4/3h2.
根据几何方程，板的面内应变为：
其中：
而板的剪切应变为：
其中：
第k层FG-GRC板的应力-应变关系为：
其中：
式中，和分别为第k层GPLs的有效弹性模量及泊松比.
FG-GRC板的应变能为：
其中：
如图3所示，对FG-GRC板施加面内荷载作用，它的势能为：
将式（27）和式（31）进行叠加，则得FG-GRC板屈曲时的势能泛函为：
3.2　控制列式
利用式（9）对FG-GRC板节点进行离散可得：
将式（33）代入式（32），根据最小势能原理，δΠ=0，可得FG-GRC板屈曲无网格控制方程：
其中：
式中：
设Rx=c1Ry，Rx=c2Rxy（c1、c2为常数），将Rx从G中提出来，则有：
通过解上述特征值方程，可得FG-GRC板临界屈曲荷载.
4　数值结果与分析
收起
文中所有算例采用无网格均布节点，高斯节点数为4×4. 考虑单轴压缩（Rx=-1，Ry=0，Rxy=0），双轴压缩（Rx=-1，Ry=-1，Rxy=0），面内剪切（Rx=0，Ry=0，Rxy=-1）三种荷载类型. FG-GRC板和GPLs的几何尺寸分别为a×b×h=0.45 m×0.45 m×0.045 m，lGPL×wGPL×hGPL=2.5 μm×1.5 μm×1.5 nm. 环氧树脂基体和GPLs的材料属性见表1[2,24,25]. 如无特别说明，默认取a/b=1，NL=10，gGPL=1.0%. 临界屈曲荷载无量纲公式为：Ncr=Ncra2/EMh3.
FG-GRC板结构考虑如下常见3种边界条件：
（1）固支边界（C）：当x=0和a时，v=w=φx=φy=ψx=ψy=0. 当y=0和b时，u=w=φx=φy=ψx=ψy=0.
（2）简支边界（S）：当x=0和a时，v=w=φy=ψy=0. 当y=0和b时，u=w=φx=ψx=0.
（3）自由边界（F）：没有限制.
4.1　收敛性及有效性分析
图4给出了单压下四边简支FG-GRC板归一化无量纲临界屈曲荷载随均布离散点数的收敛情况. 由图4可知：随着无网格节点数的增多，数值结果越加逼近解析解；在无网格节点数在50左右时，计算结果已基本收敛. 二者说明了本文方法收敛性好，易趋于稳定. 后续算例中均采用11×11均布节点进行离散.
表2计算了四边简支FG-GRC板在单轴压缩和双轴压缩下无量纲临界屈曲荷载，并将计算结果与Song[7]、王壮壮[12]分别基于FSDT和TSDT，采用Navier解答得到的解析解进行对比. 结果发现，采用本文方法所得计算结果与已有文献的解析解吻合良好，证明了本文方法求解FG-GRC板屈曲问题的有效性.
为了将本文方法与文献中夹芯板的试验数据结果进行对比，把本文FG-GRC板物理关系退化到各向异性材料，应力应变关系见文献[26]. 板芯考虑轻木和泡沫两种材料，上下板面为纤维增强材料，三者材料属性见文献[27]中的表1. 轻木夹芯板和泡沫夹芯板的长宽分别为183 cm×92 cm和122 cm×92 cm，板芯厚度均为1.27 cm，板面厚度为0.32 cm. 边界条件为长边简支，短边固定. 表3给出了短边单压下夹芯板临界屈曲荷载试验结果、有限元解及本文解，对比发现本文所获得数值结果更接近于有限元解. 数值解与试验解的误差来源[27]主要有两点：一是试验时无法真正获得跟数值分析时同等固定及简支边界条件；二是试验过程可能会伴随着材料非线性发生.
4.2　参数分析
表4和表5分别展示了四边简支FG-GRC板在单轴压缩和双轴压缩下的无量纲临界屈曲荷载和无量纲临界屈曲荷载变化率RN的影响，其中，Ncr和分别表示FG-GRC板和环氧树脂板，即有GPLs和无GPLs增强板的无量纲临界屈曲荷载. 由表4-5可知：（a）本文基于改进Reddy型TSDT（I-TSDT），采用无网格法得到的计算结果与王壮壮等[12]基于TSDT，采用Navier解答得到的解析解基本一致，证明了本文方法计算FG-GRC板临界屈曲荷载具有较高的计算精度；（b）与I-TSDT和TSDT相比，FSDT低估了FG-O型FG-GRC板的临界屈曲荷载和临界屈曲荷载变化率，而高估了FG-X型FG-GRC板的临界屈曲荷载和临界屈曲荷载变化率，三种剪切理论下UD型FG-GRC板的计算结果一致，这是由于FSDT假设截面沿厚度方向的应变是线性的，而I-TSDT和TSDT假设截面沿厚度方向的应变是非线性的，且GPLs中的FG-X型和FG-O型分布的不均匀性会增加其差异；（c）环氧树脂（Pure epoxy）板的临界屈曲荷载均小于FG-GRC板，且随着GPLs的重量分数gGPL增大而增大，说明GPLs极高的弹性模量可有效增加FG-GRC板的刚度；（d）三种GPLs分布模式下的FG-GRC板临界屈曲荷载存在较大的差异，其大小依次为：FG-X>UD>FG-O，这是由于GPLs分布模式差异，导致FG-GRC板结构刚度差异所致.
图5给出了总层数NL对单压下四边简支FG-GRC板无量纲临界屈曲荷载变化率RN的影响. 从图5可以看出：当FG-GRC板的总层数NL<10～15时，FG-X和FG-O分布模式下的FG-GRC板临界屈曲荷载变化率较为剧烈，说明该阶段相较于环氧树脂板，GPLs增强板的刚度降低（或增加）较快；当FG-GRC板的总层数NL>10～15时，板临界屈曲荷载变化率较为平缓；由于UD型板中的GPLs是均匀分布，总层数对其无影响.
图6给出了宽厚比b/h对单压下四边简支FG-GRC板无量纲临界屈曲荷载Ncr的影响，可以得到：三种GPLS分布型的FG-GRC板临界屈曲荷载随宽厚比的变化趋势基本一致；FG-GRC板临界屈曲荷载随着宽厚比的增大而增大，即板越薄，临界屈曲荷载越大.
图7讨论了GPLs的长宽比lGPL/wGPL和长厚比lGPL/hGPL对单轴压缩和双轴压缩下四边简支FG-GRC板无量纲临界屈曲荷载Ncr的影响. 数值计算时，固定lGPL=2.5 μm值，变化wGPL和hGPL. 研究发明：随着GPLs的长厚比lGPL/hGPL增加到1000左右，FG-GRC板临界屈曲荷载急剧增加，然后随着lGPL/hGPL的进一步增加而增加非常缓慢. 当长厚比lGPL/hGPL增加到2000以上时，FG-GRC板临界屈曲荷载趋向于稳定且GPLs的长宽比lGPL/wGPL和长厚比lGPL/hGPL对FG-GRC板临界屈曲荷载影响不再明显；方形GPLs（lGPL/wGPL=1.0）增强FG-GRC板的临界屈曲荷载比矩形GPLs（lGPL/wGPL=3.0）增强FG-GRC板临界屈曲荷载大，这是因为方形GPLs（lGPL/wGPL=1.0）有着较高的表面积，GPLs与基体材料之间具有较大的接触面积，致使其具有更高的结构刚度.
图8给出了不同边界条件、面内荷载类型及GPLs分布模式下FG-GRC板无量纲临界屈曲荷载Ncr随重量分数（gGPL%）的变化. 由图8可知：对于不同边界条件和面内荷载类型，FG-X型的FG-GRC板的临界屈曲荷载值始终最大，UD型次之，FG-O型最小；FG-GRC板的临界屈曲荷载随着边界条件约束的削弱而减小；单轴压缩、双轴压缩和面内剪切下三类GPLs分布的FG-GRC板临界屈曲荷载均随着重量分数gGPL增大而增大，且单轴压缩得到的临界屈曲荷载大于双轴压缩临界屈曲荷载，这是因为受双轴压缩的板比单轴压缩的板更容易屈曲，而对于受面内剪切的板，因最难引起板结构的不稳定，所以其临界屈曲荷载值最大.
5　结论
收起
本文基于改进Reddy型TSDT和移动克里金插值，提出了求解FG-GRC板临界屈曲荷载问题的无网格法，并根据最小势能原理推导出屈曲控制方程，通过数值分析得出以下结论：
（1）本文得到的计算结果与解析解吻合较好，证明了该方法求解FG-GRC板临界屈曲荷载问题的有效性和准确性.
（2）当FG-GRC板的总层数NL<10～15时，FG-O和FG-X型的FG-GRC板临界屈曲荷载变化率较为剧烈，说明该阶段相较于环氧树脂板，GPLs增强板的刚度降低（或增加）较快；当FG-GRC板的总层数NL>10～15时，临界屈曲荷载变化率较为平缓.
（3）GPLs的长厚比lGPL/hGPL在0到1000左右区间内，FG-GRC板临界屈曲荷载急剧增加. 当GPLs的长厚比lGPL/hGPL增加到2000以上时，FG-GRC板临界屈曲荷载趋向于稳定且GPLs的长宽比lGPL/wGPL和长厚比lGPL/hGPL对FG-GRC板临界屈曲荷载影响不再明显.
基金
收起
  • 国家自然科学基金项目(12162004; 11562001)
  • 南华大学博士科研启动基金项目(Y00043-13)
文献
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2024年第45卷第2期
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doi: 10.19636/j.cnki.cjsm42-1250/o3.2023.052
  • 接收时间：2023-10-19
  • 首发时间：2026-04-01
  • 出版时间：2024-04-25
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  • 收稿日期：2023-10-19
基金
国家自然科学基金项目(12162004; 11562001)
南华大学博士科研启动基金项目(Y00043-13)
作者信息
    1南华大学土木工程学院，衡阳，421001
    2广西大学土木建筑工程学院，南宁，530004
    3广西防灾减灾与工程安全重点实验室，工程防灾与结构安全教育部重点实验室，南宁，530004

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2种不同金属材料的力学参数
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鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
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