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Article(id=1228048668683596293, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228048667874095618, articleNumber=null, orderNo=null, doi=10.16385/j.cnki.issn.1004-4523.2024.03.003, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1650470400000, receivedDateStr=2022-04-21, revisedDate=1657987200000, revisedDateStr=2022-07-17, acceptedDate=null, acceptedDateStr=null, onlineDate=1770719218654, onlineDateStr=2026-02-10, pubDate=1711555200000, pubDateStr=2024-03-28, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1770719218654, onlineIssueDateStr=2026-02-10, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1770719218654, creator=13701087609, updateTime=1770719218654, updator=13701087609, issue=Issue{id=1228048667874095618, tenantId=1146029695717560320, journalId=1225147924628267009, year='2024', volume='37', issue='3', pageStart='365', pageEnd='538', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1770719218462, creator=13701087609, updateTime=1770795476854, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1228368518803030940, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228048667874095618, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1228368518803030941, tenantId=1146029695717560320, journalId=1225147924628267009, issueId=1228048667874095618, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=384, endPage=393, ext={EN=ArticleExt(id=1228048668884922888, articleId=1228048668683596293, tenantId=1146029695717560320, journalId=1225147924628267009, language=EN, title=A simplified model for free vibration analysis of functionally graded plates based on higher-order shear deformation theory, columnId=null, journalTitle=Journal of Vibration Engineering, columnName=null, runingTitle=null, highlight=null, articleAbstract=

A simplified model for free vibration analysis of functionally graded plates is proposed based on higher-order shear deformation theory，the most significant feature of which is that it applies for the vibration analysis of functionally graded plates without any shear corrections. Compared with other shear deformation theories that contain more unknown variables，this model contains only one control equation，and thus greatly reduces the computational cost. Based on this simplified model，the free vibration of functionally graded rectangular plates with simple support boundary conditions is investigated and compared with other existing literature. The results show that the simplified model proposed in this paper is simple and accurate in solving the free vibration behavior of functional gradient plates. In addition，the effects of different gradient indices，aspect ratios，and length-thickness ratios on the free vibration behavior of functionally gradient plates are analytically discussed in the paper by several numerical arithmetic examples.

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基于高阶剪切变形理论提出了一种功能梯度板自由振动分析的简化模型，该简化模型最显著的特点是适用于功能梯度板的振动分析，且不需要剪切修正。相比于其他具有更多未知变量的剪切变形理论，本文提出的简化模型只包含一个控制方程，极大地减少了计算量。基于该简化模型研究了功能梯度矩形板在简支边界条件下的自由振动，并与其他已有文献进行了比较。结果表明，本文提出的简化模型在分析功能梯度板的自由振动行为时简单且精确。此外，文中还通过多个数值算例分析讨论了不同的梯度指数、长宽比和边厚比对功能梯度板自由振动行为的影响。

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马连生（1963―），男，博士，教授。E-mail： 。
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王壮壮（1996—），男，硕士研究生。E-mail： 。

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王壮壮（1996—），男，硕士研究生。E-mail： 。

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王壮壮（1996—），男，硕士研究生。E-mail： 。

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Applied Mathematical Modelling， 2010， 34（5）： 1276-1291., articleTitle=Free vibration of functionally graded rectangular plates using first-order shear deformation plate theory, refAbstract=null), Reference(id=1228048698916139555, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228048668683596293, doi=null, pmid=null, pmcid=null, year=2008, volume=318, issue=1-2, pageStart=176, pageEnd=192, url=null, language=null, rfNumber=［36］, rfOrder=39, authorNames=Pradyumna S, Bandyopadhyay J N, journalName=Journal of Sound and Vibration, refType=null, unstructuredReference=Pradyumna S， Bandyopadhyay J N. Free vibration analysis of functionally graded curved panels using a higher-order finite element formulation［J］. 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Composites Part B： Engineering， 2011， 42（6）： 1386-1394., articleTitle=A four variable refined plate theory for free vibrations of functionally graded plates with arbitrary gradient, refAbstract=null)], funds=[Fund(id=1228048693924917625, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228048668683596293, awardId=11862012, language=CN, fundingSource=国家自然科学基金资助项目(11862012), fundOrder=null, country=null), Fund(id=1228048694021386622, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228048668683596293, awardId=12062010, language=CN, fundingSource=国家自然科学基金资助项目(12062010), fundOrder=null, country=null), Fund(id=1228048694113661313, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228048668683596293, awardId=ZR2020KA006, language=CN, fundingSource=山东省自然科学基金资助项目(ZR2020KA006), fundOrder=null, country=null)], companyList=[AuthorCompany(id=1228048688803672217, tenantId=1146029695717560320, journalId=1225147924628267009, 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figureFileSmall=gPxZNZUf5s76sLJl7UPS0A==, figureFileBig=NiXSbZl1asvX8jpWie2USQ==, tableContent=null), ArticleFig(id=1228048691035042094, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228048668683596293, language=CN, label=图1, caption=功能梯度矩形板示意图, figureFileSmall=gPxZNZUf5s76sLJl7UPS0A==, figureFileBig=NiXSbZl1asvX8jpWie2USQ==, tableContent=null), ArticleFig(id=1228048691253145914, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228048668683596293, language=EN, label=Fig.2, caption=The effects of gradient index p and a/h ratio on the first four order dimensionless vibration natural frequencies of the functionally graded square plates, figureFileSmall=FlVsxTPM+Fkts4xwdGMCbg==, figureFileBig=4V6mbzyHLD8yVC2W+wHqTg==, tableContent=null), ArticleFig(id=1228048691337031999, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228048668683596293, language=CN, label=图2, caption=梯度指数p和边厚比a/h对功能梯度方板前四阶无量纲自由振动固有频率的影响, figureFileSmall=FlVsxTPM+Fkts4xwdGMCbg==, figureFileBig=4V6mbzyHLD8yVC2W+wHqTg==, tableContent=null), ArticleFig(id=1228048691416723779, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228048668683596293, language=EN, label=Fig.3, caption=Effects of gradient index p and aspect ratio b/a on the dimensionless free vibration fundamental frequency of functionally graded square plates, figureFileSmall=8CtAg0Krh1lqKL40Aw+bVA==, figureFileBig=1lukC26amSq4QxvwApDOOQ==, tableContent=null), ArticleFig(id=1228048691529969990, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228048668683596293, language=CN, label=图3, caption=梯度指数p和长宽比b/a对功能梯度方板无量纲自由振动基频的影响, figureFileSmall=8CtAg0Krh1lqKL40Aw+bVA==, figureFileBig=1lukC26amSq4QxvwApDOOQ==, tableContent=null), ArticleFig(id=1228048691601273161, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228048668683596293, language=EN, label=Fig.4, caption=First four mode shapes of functionally graded square plates, figureFileSmall=/qf/8aQcBdBZW5t1Q6bI4Q==, figureFileBig=x1PxsrMAsrPuotsALcqHMg==, tableContent=null), ArticleFig(id=1228048691697742159, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228048668683596293, language=CN, label=图4, caption=功能梯度方板的前四阶模态, figureFileSmall=/qf/8aQcBdBZW5t1Q6bI4Q==, figureFileBig=x1PxsrMAsrPuotsALcqHMg==, tableContent=null), ArticleFig(id=1228048693073473877, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228048668683596293, language=EN, label=Fig.5, caption=First four mode shapes of rectangular functionally graded plates, figureFileSmall=9+dMi/wK2P4uRy9c7/jBkw==, figureFileBig=pKoHKGEXaUETVTb4J3EtDQ==, tableContent=null), ArticleFig(id=1228048693178331482, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228048668683596293, language=CN, label=图5, caption=功能梯度矩形板的前四阶模态, figureFileSmall=9+dMi/wK2P4uRy9c7/jBkw==, figureFileBig=pKoHKGEXaUETVTb4J3EtDQ==, tableContent=null), ArticleFig(id=1228048693283189085, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228048668683596293, language=EN, label=Tab.1, caption=

Dimensionless fundamental frequency of the Mori-Tanaka distribution-based functionally graded square plates

, figureFileSmall=null, figureFileBig=null, tableContent=
板理论p=0p=1a/h=5
a/h=10a/h=5a/h=10a/h=20p=2p=3p=5
文献[31]0.46580.05780.21920.05960.01530.21970.22110.2225
文献[12]——0.21930.05960.01530.21980.22120.2225
文献[13]——0.21930.05960.01530.22010.22160.2230
文献[14]——0.2193——0.22000.22150.2230
文献[32]——0.21880.05920.01470.21880.22020.2215
文献[33]0.46230.05770.21690.05920.01520.21780.21930.2206
Proposed0.46180.05770.21660.05920.01520.21740.21890.2202
), ArticleFig(id=1228048693400629605, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228048668683596293, language=CN, label=表1, caption=

Mori-Tanaka分布模式下功能梯度方板的无量纲基频

, figureFileSmall=null, figureFileBig=null, tableContent=
板理论p=0p=1a/h=5
a/h=10a/h=5a/h=10a/h=20p=2p=3p=5
文献[31]0.46580.05780.21920.05960.01530.21970.22110.2225
文献[12]——0.21930.05960.01530.21980.22120.2225
文献[13]——0.21930.05960.01530.22010.22160.2230
文献[14]——0.2193——0.22000.22150.2230
文献[32]——0.21880.05920.01470.21880.22020.2215
文献[33]0.46230.05770.21690.05920.01520.21780.21930.2206
Proposed0.46180.05770.21660.05920.01520.21740.21890.2202
), ArticleFig(id=1228048693518070121, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228048668683596293, language=EN, label=Tab.2, caption=

Dimensionless fundamental frequency of the functionally graded square plates based on power-law distribution

, figureFileSmall=null, figureFileBig=null, tableContent=
板理论p=0p=1a/h=5
a/h=10a/h=5a/h=10a/h=20p=2p=3p=5
文献[36]0.46190.05770.23230.06330.01620.23250.23340.2334
文献[35]0.46180.05760.22700.06110.01580.22490.22540.2265
文献[34]0.46580.05780.22850.06180.01580.22640.22700.2281
文献[37]0.46220.05760.22700.06180.01580.22490.22550.2266
Proposed0.46180.05770.22740.06180.01580.22530.22600.2270
), ArticleFig(id=1228048693601956202, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228048668683596293, language=CN, label=表2, caption=

幂律分布模式下功能梯度方板的无量纲基频

, figureFileSmall=null, figureFileBig=null, tableContent=
板理论p=0p=1a/h=5
a/h=10a/h=5a/h=10a/h=20p=2p=3p=5
文献[36]0.46190.05770.23230.06330.01620.23250.23340.2334
文献[35]0.46180.05760.22700.06110.01580.22490.22540.2265
文献[34]0.46580.05780.22850.06180.01580.22640.22700.2281
文献[37]0.46220.05760.22700.06180.01580.22490.22550.2266
Proposed0.46180.05770.22740.06180.01580.22530.22600.2270
), ArticleFig(id=1228048693702619503, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228048668683596293, language=EN, label=Tab.3, caption=

The first four order dimensionless natural frequencies of functionally graded rectangular plates based on the power-law distribution model

, figureFileSmall=null, figureFileBig=null, tableContent=
a/h模态阶数(m,n)板理论p
00.5125810
51(1,1)文献[35]3.44092.93222.64732.40172.25282.19852.1677
文献[29]3.44122.93472.64752.39492.22722.16972.1407
Proposed3.44092.93682.65072.39992.23532.17482.1431
2(1,2)文献[35]5.28024.51224.07733.69533.44923.35873.3094
文献[29]5.28134.5184.07813.68053.39383.29643.2514
Proposed5.28024.52254.08453.69053.40933.30533.2544
3(1,3)文献[35]8.07106.92316.26365.66955.25795.10455.0253
文献[29]8.07496.93666.26635.63905.14254.97584.9055
Proposed8.07106.94556.27815.65635.16864.98764.9054
4(2,1)文献[35]9.74168.69267.87117.11896.57495.90625.7518
文献[29]10.11648.71387.87627.07516.40746.18466.0954
Proposed10.10898.72597.89147.09626.43896.19526.0888
101(1,1)文献[35]3.65183.09832.79372.53862.39982.35042.3197
文献[29]3.65183.09902.79372.53642.39162.34112.3110
Proposed3.65183.09972.79472.53822.39462.34322.3123
2(1,2)文献[35]5.76934.89974.41924.01423.78813.70723.6580
文献[29]5.76944.90144.41924.00903.76823.68463.6368
Proposed5.76934.90314.42184.01323.77533.68953.6396
3(1,3)文献[35]9.18767.81457.05126.40156.02475.88875.8086
文献[29]9.18807.81897.05156.38865.97655.83415.7575
Proposed9.18777.82307.05766.39865.99285.84505.7633
4(2,1)文献[35]11.831010.07409.09288.25157.75057.56887.4639
文献[29]11.831510.08109.09338.23097.67317.48137.3821
Proposed11.830710.08759.10298.24657.69847.49767.3903
201(1,1)文献[35]3.71233.14562.83522.57772.44252.39482.3642
文献[29]3.71233.14582.83522.57712.44032.39232.3619
Proposed3.71233.14602.83552.57762.44112.39292.3623
2(1,2)文献[35]5.91985.01754.52284.11153.89393.81703.7681
文献[29]5.91995.01804.52284.11003.88843.81073.7622
Proposed5.91995.01844.52354.11133.89053.81223.7631
3(1,3)文献[35]9.56688.11217.31326.64716.29036.16396.0843
文献[29]9.56698.11337.31326.64336.27606.14766.0690
Proposed9.56698.11457.31516.64646.28146.15146.0713
4(2,1)文献[35]12.456010.56609.52618.65728.18758.02077.9166
文献[29]12.456210.56779.52618.65098.16367.99347.8909
Proposed12.456210.56989.52938.65628.17248.00007.8947
), ArticleFig(id=1228048693807477106, tenantId=1146029695717560320, journalId=1225147924628267009, articleId=1228048668683596293, language=CN, label=表3, caption=

幂律分布模式下功能梯度矩形板的前四阶无量纲固有频率

, figureFileSmall=null, figureFileBig=null, tableContent=
a/h模态阶数(m,n)板理论p
00.5125810
51(1,1)文献[35]3.44092.93222.64732.40172.25282.19852.1677
文献[29]3.44122.93472.64752.39492.22722.16972.1407
Proposed3.44092.93682.65072.39992.23532.17482.1431
2(1,2)文献[35]5.28024.51224.07733.69533.44923.35873.3094
文献[29]5.28134.5184.07813.68053.39383.29643.2514
Proposed5.28024.52254.08453.69053.40933.30533.2544
3(1,3)文献[35]8.07106.92316.26365.66955.25795.10455.0253
文献[29]8.07496.93666.26635.63905.14254.97584.9055
Proposed8.07106.94556.27815.65635.16864.98764.9054
4(2,1)文献[35]9.74168.69267.87117.11896.57495.90625.7518
文献[29]10.11648.71387.87627.07516.40746.18466.0954
Proposed10.10898.72597.89147.09626.43896.19526.0888
101(1,1)文献[35]3.65183.09832.79372.53862.39982.35042.3197
文献[29]3.65183.09902.79372.53642.39162.34112.3110
Proposed3.65183.09972.79472.53822.39462.34322.3123
2(1,2)文献[35]5.76934.89974.41924.01423.78813.70723.6580
文献[29]5.76944.90144.41924.00903.76823.68463.6368
Proposed5.76934.90314.42184.01323.77533.68953.6396
3(1,3)文献[35]9.18767.81457.05126.40156.02475.88875.8086
文献[29]9.18807.81897.05156.38865.97655.83415.7575
Proposed9.18777.82307.05766.39865.99285.84505.7633
4(2,1)文献[35]11.831010.07409.09288.25157.75057.56887.4639
文献[29]11.831510.08109.09338.23097.67317.48137.3821
Proposed11.830710.08759.10298.24657.69847.49767.3903
201(1,1)文献[35]3.71233.14562.83522.57772.44252.39482.3642
文献[29]3.71233.14582.83522.57712.44032.39232.3619
Proposed3.71233.14602.83552.57762.44112.39292.3623
2(1,2)文献[35]5.91985.01754.52284.11153.89393.81703.7681
文献[29]5.91995.01804.52284.11003.88843.81073.7622
Proposed5.91995.01844.52354.11133.89053.81223.7631
3(1,3)文献[35]9.56688.11217.31326.64716.29036.16396.0843
文献[29]9.56698.11337.31326.64336.27606.14766.0690
Proposed9.56698.11457.31516.64646.28146.15146.0713
4(2,1)文献[35]12.456010.56609.52618.65728.18758.02077.9166
文献[29]12.456210.56779.52618.65098.16367.99347.8909
Proposed12.456210.56989.52938.65628.17248.00007.8947
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基于高阶剪切变形理论的功能梯度板自由振动分析简化模型
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王壮壮 , 王腾 , 丁艳梅 , 马连生
振动工程学报 | 2024,37(3): 384-393
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振动工程学报 | 2024, 37(3): 384-393
基于高阶剪切变形理论的功能梯度板自由振动分析简化模型
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王壮壮 , 王腾, 丁艳梅, 马连生
作者信息
  • 潍坊科技学院建筑工程学院，山东　潍坊　262700

    • 王壮壮（1996—），男，硕士研究生。E-mail： 。

通讯作者:

马连生（1963―），男，博士，教授。E-mail： 。
A simplified model for free vibration analysis of functionally graded plates based on higher-order shear deformation theory
Zhuang-zhuang WANG , Teng WANG, Yan-mei DING, Lian-sheng MA
Affiliations
  • School of Architecture and Engineering，Weifang University of Science and Technology，Weifang 262700，China
出版时间: 2024-03-28 doi: 10.16385/j.cnki.issn.1004-4523.2024.03.003
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摘要
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基于高阶剪切变形理论提出了一种功能梯度板自由振动分析的简化模型，该简化模型最显著的特点是适用于功能梯度板的振动分析，且不需要剪切修正。相比于其他具有更多未知变量的剪切变形理论，本文提出的简化模型只包含一个控制方程，极大地减少了计算量。基于该简化模型研究了功能梯度矩形板在简支边界条件下的自由振动，并与其他已有文献进行了比较。结果表明，本文提出的简化模型在分析功能梯度板的自由振动行为时简单且精确。此外，文中还通过多个数值算例分析讨论了不同的梯度指数、长宽比和边厚比对功能梯度板自由振动行为的影响。

自由振动  /  功能梯度材料  /  简化模型  /  精化板理论  /  固有频率

A simplified model for free vibration analysis of functionally graded plates is proposed based on higher-order shear deformation theory，the most significant feature of which is that it applies for the vibration analysis of functionally graded plates without any shear corrections. Compared with other shear deformation theories that contain more unknown variables，this model contains only one control equation，and thus greatly reduces the computational cost. Based on this simplified model，the free vibration of functionally graded rectangular plates with simple support boundary conditions is investigated and compared with other existing literature. The results show that the simplified model proposed in this paper is simple and accurate in solving the free vibration behavior of functional gradient plates. In addition，the effects of different gradient indices，aspect ratios，and length-thickness ratios on the free vibration behavior of functionally gradient plates are analytically discussed in the paper by several numerical arithmetic examples.

free vibration  /  functionally graded materials  /  simplified model  /  refined plate theory  /  natural frequency
王壮壮, 王腾, 丁艳梅, 马连生. 基于高阶剪切变形理论的功能梯度板自由振动分析简化模型. 振动工程学报, 2024 , 37 (3) : 384 -393 . DOI: 10.16385/j.cnki.issn.1004-4523.2024.03.003
Zhuang-zhuang WANG, Teng WANG, Yan-mei DING, Lian-sheng MA. A simplified model for free vibration analysis of functionally graded plates based on higher-order shear deformation theory[J]. Journal of Vibration Engineering, 2024 , 37 (3) : 384 -393 . DOI: 10.16385/j.cnki.issn.1004-4523.2024.03.003
正文
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引言
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功能梯度材料通常由金属和陶瓷材料制成，但这两种材料并不是简单地堆叠层合，而是沿厚度方向呈梯度分布。这种特殊的构造给予了功能梯度材料不同于其他层合材料的优秀性能，因此功能梯度材料在核电、航空、土木工程、机械制造等领域被广泛使用。
诸多板理论可用于功能梯度材料的力学行为分析，其中最简单的是经典板理论［1-4］，它只包含3个变量，但是由于忽略了剪切变形和法向变形，所以经典板理论只适用于分析特别薄的板。为了考虑剪切变形效应，Mindlin［5］开发了一阶剪切变形板理论。但一阶剪切变形板理论局限于“横向切应力沿厚度方向为常量”假设，不能够满足板顶面和底面上的零剪应力边界条件，因此在使用时需引入一个取决于几何参数、载荷和边界条件等因素的剪切修正系数［6-7］。对于功能梯度材料这种性质不均匀的材料，确定最适合的剪切修正系数仍旧是一项挑战。基于改善经典板理论和一阶剪切变形板理论的目的，众多包含不同数目未知数的高阶剪切变形理论［8-14］被提出。这些高阶剪切变形板理论不需要剪切修正，与三维解的结果相近，但这些理论中包含的未知量数目太多，所以计算量非常大。为了减少板理论中的未知量数目，Senthilnathan等［15］通过将横向位移分为弯曲部分和剪切部分，并作出进一步假设，提出了精化板理论（RPT）。把位移分成弯曲和剪切部分不仅减少了未知数，而且有助于分辨出剪切和弯曲对总变形的贡献。RPT已经被广泛应用于功能梯度梁［16］、功能梯度板［17-18］、石墨烯纳米片增强板［19-20］和碳纳米管增强板［21］的力学分析，大量的研究已经验证了RPT的便捷性和准确性。用于自由振动分析时RPT的位移场中只包含2个未知变量，且不需要剪切修正。
为了进一步减少未知量，Shimpi［22］首先对RPT做了进一步简化，引入了“剪切位移对弯曲力矩没有贡献”的假设，将位移公式中的未知量缩减到了一个。Endo等［23-24］、Senjanovic等［25-26］以及Shimpi等［27］基于“剪切位移对弯曲力矩没有贡献”的假设，各自提出了单变量剪切变形板理论。然而，他们提出的位移场中的惯性相关项来自于一阶剪切变形板理论，也就是说这些单变量理论［23-27］与一阶剪切变形板理论一样，仍旧存在“横向切应力沿厚度方向为常量”假设，需要剪切修正。Thai等［28］和Vinh等［29］基于RPT对各向同性板以及功能梯度板使用剪切应变函数提出了单变量板理论，这样就避免了“横向切应力沿厚度方向为常量”的问题，不须引入剪切修正系数，但他们的理论仍旧基于“剪切位移对弯曲力矩没有贡献”的假设，而这一基本假设是在缺乏解释的情况下作出的［30］。为了解决这个问题，Nguyen等［30］基于RPT提出了一种单变量理论，用于各向同性板和功能梯度板的静力分析。该单变量板理论并未采用“剪切位移对力矩没有贡献”假设，且通过引入剪切应变函数避免了“横向切应力沿厚度方向为常量”的问题，因此无需剪切修正。
本文沿用Nguyen等［30］的思路，在位移场中包含2个未知量的RPT［15］基础上使用三维弹性理论将RPT位移场中的未知量缩减为1个，并将该简化模型应用于FGM板的自由振动分析。本文提出的简化模型优于其他单变量理论之处在于引入了剪切应变函数以避免剪切修正系数的选取，且摒弃了“剪切位移对弯曲力矩没有贡献”这一假设，并考虑了面内应力与横向位移之间的耦合。通过与已有文献的比较，验证了本文所提出的简化模型的简便性和准确性。此外，本文还通过多个数值算例分析讨论了梯度指数、长宽比和边厚比对简支功能梯度板自由振动固有频率的影响。
1　材料性质
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考虑如图1所示的功能梯度矩形板，顶部是陶瓷，底部是金属，长、宽和高分别为a，b和h。功能梯度材料的性质沿厚度方向梯度变化。陶瓷的体积分数Vc和金属的体积分数Vm定义如下：
式中　 p为梯度指数。
功能梯度材料的弹性模量E（z）采用两种分布方式来描述，一种是幂律分布［29］：
式中　 Em为金属材料的弹性模量；Ec为陶瓷材料的弹性模量。
另一种是Mori-Tanaka分布［12-14］：
式中　 ν为金属和陶瓷的泊松比，取0.3。
功能梯度材料的质量密度ρ（z）定义如下：
式中　 ρm为金属材料的密度；ρc为陶瓷材料的密度。
2　位移场的推导
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本文提出的简化模型是通过对RPT［15］的优化而来，其中RPT［15］在位移场中忽略了几何中面的面内位移，如下所示：
式中　（u，v，w）分别表示板在（x，y，z）方向上的位移；wb和ws分别表示横向位移的弯曲部分和剪切部分。f（z）为剪切应变函数，表达式如下：
与式（6）位移场相关的线性应变为：
应力应变关系如下所示：
式中　 C1，C2和C3为弹性常数：
将式（8）和（10）代入式（9），并沿板厚度积分，可得板内力和内力矩的表达式如下所示：
其中：
为了简化RPT［15］，考虑三维弹性理论的运动方程如下式所示：
上式中（X，Y，Z）分别表示体力在（x，y，z）方向上的分量。体力可以作为外力处理，文献［22］表明这种处理方式并不会显著降低计算精度。式（13）简化为如下形式：
将式（6）RPT位移场的表达式代入式（14），可得：
其中上标“·”表示该变量相对于时间t的导数。
通过对式（15a）和（15b）沿厚度进行积分会得到控制板的面内运动的两个方程（16a）和（16b）。通过对式（15c）沿厚度积分可以得到控制板横向运动的方程（16c）。将方程（15a）和（15b）乘以z并沿厚度进行积分，可以得到控制力矩平衡的两个方程（16d）和（16e）：
式中　 为横向载荷，自由振动时令。
惯性系数的定义为：
将式（11）代入式（16），得到：
式中　 ，表示功能梯度板的中性面到几何中面的距离。式（18）中z0是考虑到面内应力与横向位移之间的耦合而引入的量［30］。用式（18a）减去式（18d），式（18b）减去式（18e），然后分别对x和y积分，可以得到如下关系式：
正如Senjanovic等［25-26］所描述，式（19a）和（19b）必须相等，因此，F0是代表刚体运动的常数，可以略去。将式（18c）代入式（19）然后忽略其中剪切位移ws的高阶项，可以得到：
式中：
式（20）即为根据三维弹性理论得到的RPT中弯曲位移wb和剪切位移ws的关系式。该关系式的获得考虑了面内应力与横向位移之间的耦合，且未引入精化板理论［22］“剪切位移对弯曲力矩没有贡献”这一假设。需要注意的是：在简化过程中首先在式（6）中忽略了几何中面的面内位移，其次是忽略了式（19）右端ws的高阶项。这两次简化均会对频率计算结果产生影响。
将式（20）代入式（6）可以得到本文简化模型的位移场：
该位移场最大的特点是引入了惯性相关项。显然，若忽略位移场中的惯性相关项就可以得到用于分析静态问题的位移场，如文献［30］所示。
3　控制方程及解析解
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对应于简化模型位移场式（22）的剪切力表达式如下所示：
将式（23）代入式（16c），即可得到控制方程：
由式（24）的控制方程可知位移场式（22）中的唯一未知量wb可以通过求解式（24）中的四阶微分方程得到。简化模型中控制方程式（24）的推导类似于经典板理论，与通过变分原理推导出的基于高阶剪切理论的板的控制方程相比，缺少等效内力和内力矩的高阶量。
对于四边简支矩形板的自由振动，可取满足边界条件的Navier解为［27］：
式中　 ；；Wbmn为待定系数，m和n分别表示板在自由振动时x方向和y方向的正弦半波数，与振动的模态阶数相关；ωmn为自由振动固有频率。
将式（25）代入式（24），得到如下等式：
若式（26）有非平凡解，Wbmn的系数必须等于零。因此自由振动固有频率ωmn可以通过求解下式获得：
需要注意的是，式（27）的实正根代表具有不同模态阶数时的固有频率，最小值即为自由振动的基频。
4　数值结果和讨论
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4.1　验证研究
在本节中，将几个数值算例的计算结果与已有文献的结果进行比较，验证本文提出的简化模型在功能梯度板自由振动分析中的准确性和有效性。
例1：选择Al/ZrO2简支功能梯度矩形板进行自由振动分析，金属和陶瓷的材料性能为：
为了与已有文献比较，在表1和2中，当p=0时，无量纲基频表示为，p≠0时，无量纲基频表示为。
表1给出了功能梯度方板的无量纲基频。其中功能梯度板的弹性模量E（z）使用式（4）所示的Mori-Tanaka分布计算。计算结果分别与三维精确解［31］、准三维解［12-14］、三阶剪切变形板理论［32］和RPT［33］的结果作了比较。结果表明：对于p=0的均匀材料而言，简化模型在边厚比a/h=10的时候与三维精确解［31］和RPT［33］的结果几乎相同；p=1时，简化模型的结果与准三维解［12-14］、三阶剪切变形板理论［32］和RPT［33］的结果几乎相同，均随着边厚比a/h的增大而接近三维精确解［31］；边厚比a/h=5时，各种梯度指数下简化模型的计算结果与三维精确解［31］的最大相对误差在1%左右，而与RPT［33］结果的最大相对误差仅在0.1%左右。需要注意的是，准三维解［12-14］包含9个未知量，三阶剪切变形板理论［32］有5个未知量，RPT［33］包含2个未知量，而简化模型只有1个。由于未知量数目与计算量正相关，因此简化模型的计算量要远小于其他多变量理论。
表2也给出了功能梯度方板的无量纲基频。其中功能梯度板的弹性模量E（z）使用式（3）所示的幂律分布。将简化模型的结果分别与二维高阶近似板理论［34］、一阶剪切变形板理论［35］，高阶剪切变形板理论［36］和RPT［37］的结果进行了比较。结果表明：不同梯度指数p和边厚比a/h时，简化模型的结果与上述各理论的结果均吻合良好。可以看出简化模型的结果与RPT［37］的差别仍旧很小，两者最大相对误差在0.2%左右。
由表1和2可以得出，无论功能梯度材料性质采用何种方法描述，本文提出的简化模型均能准确地预测功能梯度板的自由振动基频。值得注意的是：简化模型与RPT［33，37］结果的差异是由于在式（6）中忽略了几何中面的面内位移且在式（19）处忽略了等式右端ws的高阶项导致的。
例2：本例研究Al/Al2O3功能梯度板的前四阶无量纲固有频率，并将结果与已有文献结果进行比较。分别考虑功能梯度方形板和b/a=2的矩形板，金属和陶瓷的材料性能分别为：
弹性模量E（z）根据式（3）所示的幂律分布确定。表3中的无量纲频率表示为。
表3给出了幂律分布模式下功能梯度矩形板的前四阶无量纲固有频率。将简化模型的结果分别与一阶剪切变形板理论［35］和三阶剪切变形板理论［29］进行了比较。结果表明：不同的梯度指数p和边厚比a/h下简化模型与上述理论对前四阶自由振动固有频率的计算结果均非常接近，且随着板的边厚比a/h的增大，三种理论结果的差异也在减小。其中，一阶剪切变形板理论［35］和三阶剪切变形板理论［29］均含有5个未知变量，而本文提出的简化模型只有1个未知量。
例1和2中，表1~3的结果充分显示了本文提出的简化模型在计算功能梯度板的自由振动固有频率时的准确性和有效性，并凸显了其计算量小这一非常重要的优点。
4.2　参数研究
4.1节的分析结果验证了本文所提出简化模型的高效和正确性。本节基于该简化模型，研究边厚比a/h、长宽比b/a、梯度指数p对功能梯度矩形板自由振动的影响。以下研究对象为4.1节例2中的Al/Al2O3功能梯度方板，无量纲固有频率表示为。
图2给出了不同边厚比a/h和梯度指数p对功能梯度板前四阶无量纲自由振动固有频率的影响。功能梯度板的前四阶频率在板的边厚比a/h从2增大到5的过程中增长幅度很大，a/h继续增大时振动频率的增长则趋于平缓。当梯度指数p增大时，前四阶无量纲振动固有频率普遍减小。当梯度指数p由0增大到2时，固有频率降低的幅度非常大，p继续增加则固有频率降低的速度变慢。原因是当梯度指数p增加时，功能梯度板中金属成分逐渐增多，陶瓷成分占比降低，而陶瓷的弹性模量大于金属，因此整个板的弹性模量随着p增大而降低，导致无量纲固有频率降低。边厚比a/h和梯度指数p对功能梯度板的影响在前四阶模态几乎相同。
图3给出了不同梯度指数p下功能梯度板的无量纲自由振动基频与长宽比b/a的关系。可以看到，当b/a增加时，板的无量纲基频会降低。当b/a在1~2的范围内变化时，无量纲基频的变化幅度最大。长宽比大于4以后，无量纲基频趋于稳定。从图3还可以看出，纯陶瓷板的频率大于纯金属板的频率，这是由于陶瓷的弹性模量大于金属的弹性模量。
图4和5分别给出了a/h=20，p=1时功能梯度方板和矩形板（b/a=2）的前四阶自由振动模态。可以看到，模态阶数（m和n）和长宽比b/a的不同都会影响到功能梯度板的振动模态。
5　结论
收起
本文基于高阶剪切变形理论发展了一种简化模型，用于分析材料性质沿着厚度方向不均匀分布的功能梯度板的自由振动。该模型通过引入剪切应变函数，给出了沿板厚度方向横向剪应力按抛物线分布的描述，并满足了板顶面和底面的零剪应力边界条件，因此不需要剪切修正。位移场的推导过程中摒弃了“剪切位移对弯曲力矩没有贡献”的假设，并考虑了面内应力与横向位移之间的耦合，这是本文简化模型与其他单变量理论最大的不同。本文提出的简化模型在分析功能梯度板的自由振动行为时，只需要一个控制方程，这与其他剪切理论相比大大减少了计算量。所有数值结果均表明，本模型在减少未知数的同时保证了计算的准确度。基于该简化模型研究了功能梯度板的梯度指数、边厚比和长宽比对无量纲自由振动固有频率的影响。可以预见到目前的简化模型不止适用于功能梯度材料，还可适用于其他的性质分布不均匀的材料板的自由振动分析。
基金
收起
  • 国家自然科学基金资助项目(11862012)
  • 国家自然科学基金资助项目(12062010)
  • 山东省自然科学基金资助项目(ZR2020KA006)
文献
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2024年第37卷第3期
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doi: 10.16385/j.cnki.issn.1004-4523.2024.03.003
  • 接收时间：2022-04-21
  • 首发时间：2026-02-10
  • 出版时间：2024-03-28
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  • 收稿日期：2022-04-21
  • 修回日期：2022-07-17
基金
国家自然科学基金资助项目(11862012)
国家自然科学基金资助项目(12062010)
山东省自然科学基金资助项目(ZR2020KA006)
作者信息
    潍坊科技学院建筑工程学院，山东　潍坊　262700

通讯作者:

马连生（1963―），男，博士，教授。E-mail： 。
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