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Article(id=1246046393496879377, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246046388547596731, articleNumber=null, orderNo=null, doi=10.19636/j.cnki.cjsm42-1250/o3.2024.032, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1720540800000, receivedDateStr=2024-07-10, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1775010210631, onlineDateStr=2026-04-01, pubDate=1735056000000, pubDateStr=2024-12-25, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1775010210631, onlineIssueDateStr=2026-04-01, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1775010210631, creator=13701087609, updateTime=1775010210631, updator=13701087609, issue=Issue{id=1246046388547596731, tenantId=1146029695717560320, journalId=1241755870837649424, year='2024', volume='45', issue='6', pageStart='709', pageEnd='856', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1775010209451, creator=13701087609, updateTime=1775010293319, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1246046740390986227, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246046388547596731, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1246046740390986228, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246046388547596731, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=808, endPage=819, ext={EN=ArticleExt(id=1246046393735954711, articleId=1246046393496879377, tenantId=1146029695717560320, journalId=1241755870837649424, language=EN, title=Study on Free Vibration and Bending Deformation of Functionally Graded One-Dimensional Hexagonal Quasicrystal Laminated Beams, columnId=1244229834482757770, journalTitle=Chinese Journal of Solid Mechanics, columnName=Research Paper, runingTitle=null, highlight=null, articleAbstract=

The pseudo-Stroh formulism can transform the governing equations of multi-field coupling materials such as quasicrystals into a linear eigensystem, enabling the exact solution of multilayered structures with simply-supported boundary conditions. This provides an important reference for various numerical and experimental methods of quasicrystal beams in engineering practice. In this paper, the free vibration and bending problems of one-dimensional (1D) hexagonal quasicrystal (QC) laminated beams with functional gradients are investigated using the pseudo-Stroh formula. A simply-supported QC laminated beam is modeled, and the transfer matrix method is used to derive the exact solutions for natural frequency of free vibration and bending deformation displacements of the beam under simply-supported boundary conditions. The obtained results are compared with the existing ones to verify the accuracy and precision of the presented model. Numerical examples are provided to show the effects of high span ratio, layer thickness ratio, and functional gradient coefficient on the natural frequency, bending deformation, and mode shape of simply-supported 1D QC laminated beams under two different stacking sequences. The results show that natural frequency increases with the increase of functional gradient coefficient. Phonon displacement decreases while phason displacement increases with the increase of functional gradient coefficient under the two stacking sequences. Functional gradient coefficient and stacking sequence minimally affect phonon displacement modes but significantly impact phason displacement modes. Moreover, they notably affect phason stresses compared to phonon stresses in QC laminated beams. Thus, the optimal natural frequency and deformation displacement of a QC beam can be achieved by adjusting geometric size, stacking sequence, and functional gradient coefficient of the layered beam. These findings can provide theoretical references for various numerical methods and experimental studies on QC beams.

, correspAuthors=Junhong Guo, 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=Dewang Gao, Junhong Guo, Haotian Wang, Tuoya Sun), CN=ArticleExt(id=1246046406964789729, articleId=1246046393496879377, tenantId=1146029695717560320, journalId=1241755870837649424, language=CN, title=功能梯度一维六方准晶层合梁的自由振动和弯曲变形研究, columnId=1241831201896469478, journalTitle=固体力学学报, columnName=研究论文, runingTitle=null, highlight=null, articleAbstract=

本文利用伪Stroh公式，研究具有功能梯度的一维六方准晶层合梁的自由振动和弯曲问题. 建立准晶层合简支梁模型，采用传递矩阵法，推导出简支边界条件下梁自由振动的固有频率及弯曲变形位移的精确解，并与已有结果对比，验证本文模型和方法的准确性和有效性. 通过数值算例，分析两种不同叠层顺序下层合梁的高跨比、层厚比和功能梯度系数对一维准晶层合简支梁固有频率和弯曲变形的影响规律. 结果表明，固有频率随功能梯度系数增大而增大；两种叠层顺序下随功能梯度系数的增大，声子场位移逐渐减小而相位子场位移逐渐增大；功能梯度系数和叠层顺序对层合梁的应力有很大影响，其中对准晶特有的相位子场应力影响更大. 本文给出的精确解可为研究准晶梁的各种数值方法和实验结果提供理论参考.

, correspAuthors=郭俊宏, authorNote=null, correspAuthorsNote=
** E-mail：.
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Material properties of Al-Ni-Co quasicrystal (QC1) and QC2 (Cij, Ki and Ri in 109 N/m2 and ρ in 103 kg/m3)

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C11C13C33C44R1R2=R3K1K2ρ
QC1234.3366.63232.2270.198.8468.846122244.186
QC215090130501.51.20.30.184.186
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Al-Ni-Co准晶（QC1）和QC2的材料系数（Cij，Ki和Ri：109 N/m2以及ρ：103 kg/m3）[13]

, figureFileSmall=null, figureFileBig=null, tableContent=
C11C13C33C44R1R2=R3K1K2ρ
QC1234.3366.63232.2270.198.8468.846122244.186
QC215090130501.51.20.30.184.186
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The first four natural frequencies of the quasicrystal uniform simple support beam

, figureFileSmall=null, figureFileBig=null, tableContent=
阶数本文解精确解[14]SSDQM[13]
10.26780.26720.2676
20.98230.98230.9823
31.01641.01471.0147
41.96871.96871.9687
), ArticleFig(id=1246046414879441576, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246046393496879377, language=CN, label=表2, caption=

准晶均匀简支梁的前四阶固有频率

, figureFileSmall=null, figureFileBig=null, tableContent=
阶数本文解精确解[14]SSDQM[13]
10.26780.26720.2676
20.98230.98230.9823
31.01641.01471.0147
41.96871.96871.9687
), ArticleFig(id=1246046414992687789, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246046393496879377, language=EN, label=Table 3, caption=

The first four natural frequencies of the quasicrystal simple supported beam under two stacking sequences (h/L=0.1)

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ηQC1/QC2/QC1QC2/QC1/QC2
12341234
100.27040.81861.01521.59560.17890.52320.69010.9603
00.26410.79660.99681.55130.17550.55320.67911.0082
-100.25740.77420.97541.50650.17200.58100.66691.0464
), ArticleFig(id=1246046415126905519, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246046393496879377, language=CN, label=表3, caption=

两种叠层顺序下准晶简支梁的前四阶固有频率（h/L=0.1）

, figureFileSmall=null, figureFileBig=null, tableContent=
ηQC1/QC2/QC1QC2/QC1/QC2
12341234
100.27040.81861.01521.59560.17890.52320.69010.9603
00.26410.79660.99681.55130.17550.55320.67911.0082
-100.25740.77420.97541.50650.17200.58100.66691.0464
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功能梯度一维六方准晶层合梁的自由振动和弯曲变形研究
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高得旺 1 , 郭俊宏 1, 2, ** , 王浩田 1 , 孙托娅 2
固体力学学报 | 研究论文 2024,45(6): 808-819
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固体力学学报 | 研究论文 2024, 45(6): 808-819
功能梯度一维六方准晶层合梁的自由振动和弯曲变形研究
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高得旺1, 郭俊宏1, 2, ** , 王浩田1, 孙托娅2
作者信息
  • 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

通讯作者:

** E-mail：.
Study on Free Vibration and Bending Deformation of Functionally Graded One-Dimensional Hexagonal Quasicrystal Laminated Beams
Dewang Gao1, Junhong Guo1, 2, ** , Haotian Wang1, Tuoya Sun2
Affiliations
  • 1Department of Mechanics, College of Science, Inner Mongolia University of Technology, Hohhot, 010051
  • 2School of Aeronautics, Inner Mongolia University of Technology, Hohhot, 010051
出版时间: 2024-12-25 doi: 10.19636/j.cnki.cjsm42-1250/o3.2024.032
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摘要
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本文利用伪Stroh公式，研究具有功能梯度的一维六方准晶层合梁的自由振动和弯曲问题. 建立准晶层合简支梁模型，采用传递矩阵法，推导出简支边界条件下梁自由振动的固有频率及弯曲变形位移的精确解，并与已有结果对比，验证本文模型和方法的准确性和有效性. 通过数值算例，分析两种不同叠层顺序下层合梁的高跨比、层厚比和功能梯度系数对一维准晶层合简支梁固有频率和弯曲变形的影响规律. 结果表明，固有频率随功能梯度系数增大而增大；两种叠层顺序下随功能梯度系数的增大，声子场位移逐渐减小而相位子场位移逐渐增大；功能梯度系数和叠层顺序对层合梁的应力有很大影响，其中对准晶特有的相位子场应力影响更大. 本文给出的精确解可为研究准晶梁的各种数值方法和实验结果提供理论参考.

准晶  /  功能梯度  /  简支梁  /  自由振动  /  弯曲变形

The pseudo-Stroh formulism can transform the governing equations of multi-field coupling materials such as quasicrystals into a linear eigensystem, enabling the exact solution of multilayered structures with simply-supported boundary conditions. This provides an important reference for various numerical and experimental methods of quasicrystal beams in engineering practice. In this paper, the free vibration and bending problems of one-dimensional (1D) hexagonal quasicrystal (QC) laminated beams with functional gradients are investigated using the pseudo-Stroh formula. A simply-supported QC laminated beam is modeled, and the transfer matrix method is used to derive the exact solutions for natural frequency of free vibration and bending deformation displacements of the beam under simply-supported boundary conditions. The obtained results are compared with the existing ones to verify the accuracy and precision of the presented model. Numerical examples are provided to show the effects of high span ratio, layer thickness ratio, and functional gradient coefficient on the natural frequency, bending deformation, and mode shape of simply-supported 1D QC laminated beams under two different stacking sequences. The results show that natural frequency increases with the increase of functional gradient coefficient. Phonon displacement decreases while phason displacement increases with the increase of functional gradient coefficient under the two stacking sequences. Functional gradient coefficient and stacking sequence minimally affect phonon displacement modes but significantly impact phason displacement modes. Moreover, they notably affect phason stresses compared to phonon stresses in QC laminated beams. Thus, the optimal natural frequency and deformation displacement of a QC beam can be achieved by adjusting geometric size, stacking sequence, and functional gradient coefficient of the layered beam. These findings can provide theoretical references for various numerical methods and experimental studies on QC beams.

quasicrystal  /  functional gradient  /  simply-supported beam  /  free vibration  /  bending deformation
高得旺, 郭俊宏, 王浩田, 孙托娅. 功能梯度一维六方准晶层合梁的自由振动和弯曲变形研究. 固体力学学报, 2024 , 45 (6) : 808 -819 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2024.032
Dewang Gao, Junhong Guo, Haotian Wang, Tuoya Sun. Study on Free Vibration and Bending Deformation of Functionally Graded One-Dimensional Hexagonal Quasicrystal Laminated Beams[J]. Chinese Journal of Solid Mechanics, 2024 , 45 (6) : 808 -819 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2024.032
正文
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0　引言
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1984年，以色列科学家Shechtman[1]在急冷凝固的Al-Mn合金中发现了二十面体准晶体，它具有五重旋转对称性. 准晶被首次报导后，引起了晶体学、材料学、物理学等各界的强烈反响. 准晶是一种新型固体结构[2]，具有宏观对称性但不具有普通晶体的平移对称性，其独特的原子排列方式使准晶具有比传统的合金更优异的物理性能和力学性能[3]，比如高强度、高硬度和高摩擦系数等，因此，准晶常用作表面改性材料如不粘锅炊具表面、隔热材料、颗粒增强相等[4].
在实际的工程试验中，梁板[5]结构是重要的组成部分，受到大量学者广泛关注. 对于准晶板的研究，Gao等[6,7]对二维正交准晶材料梁和板的混合边界条件进行研究，导出准晶梁弯曲下的衰减状态条件. Pan[8]提出的伪Stroh公式与传递矩阵法，在解决多层板和梁问题上具有推导过程简单，并具有高精度优点. Sun等[9]利用伪Stroh型公式和传递矩阵法，得到了二维准晶层状纳米板非局部振动频率和屈曲临界载荷的精确解. Wang和Guo[10]研究了具有非理想界面的一维六方准晶板的弯曲变形和自由振动，分析了非理想参数和叠层顺序对准晶梁板弯曲变形和振动的影响. Zhang等[11]研究多层一维六方压电准晶板受表面电、弹性载荷作用下的静态弯曲问题. 对于准晶梁的研究，黄孚等[12]利用准晶二维本构关系和物理方程以及层间连续条件和边界条件，得出一维六方准晶悬臂梁在自由端受到集中力作用下声子场和相位子场的位移解析解. Sun和Guo[13]利用状态空间-微分求积法（SSDQM），研究了一维准晶层合梁在各种边界条件下的振动响应和弯曲变形. 陈韬等[14]利用伪Stroh型公式和传递矩阵法，给出了一维六方准晶层合梁自由振动和屈曲问题的精确解. Li等[15]引入修正偶应力理论，研究了一维压电准晶纳米梁的自由振动.
功能梯度材料是一种新型复合材料，其成分和结构呈连续梯度变化[16]. 功能梯度材料的最大优点在于其可以根据实际服役条件进行灵活设计，以满足不同部位对材料性能的特定要求[17]. 通过调整组成和结构，功能梯度材料能够实现性能的连续梯度变化，从而优化整体性能[18,19]. 实际生产生活中对称结构广泛应用各个领域，如机械设备滑轮等、建筑领域梁板结构等，对称结构具有设计简单，高效率和易于加工和制造等特点[20]. 功能梯度梁板材料中也存在对称结构，在梁板的顶层和底层形成功能梯度过渡层，使材料性能缓慢变化，代替原有的突变界面，从而提高梁板结构的整体力学性能. 随着制备技术的不断进步和新型材料的不断开发，功能梯度可以与准晶深度结合，这种功能梯度准晶材料在多个领域都具有巨大的应用潜力，如在航空航天领域用于制造耐高温和压力的结构部件，在能源领域可以用于开发高效稳定的存储材料，在生物医学领域可用于人造假骨材料等. 对于功能梯度材料，Li等[21]利用传递矩阵法推导出功能梯度二维热弹性多层板的解析解，分析了热弹性、功能梯度因子对准晶板的影响. Pan等[22]利用伪Stroh型公式对功能梯度各向异性线性磁电弹性多层板进行了研究，并得出其精确解. Guo等[23]基于修正偶应力理论对各向异性功能梯度复合材料层合板的尺寸依赖性进行了研究，得到了其弹性场的精确解. Daneshmehr等[24]基于非局部弹性理论和高阶板理论，研究了双轴功能梯度纳米板的稳定性问题，分析了功能梯度系数等对屈曲载荷的影响. Jandaghian等[25]利用非局部弹性理论对功能梯度压电纳米板的自由振动问题进行了分析，并得到其解析解. Huang等[26]利用状态向量法和传递矩阵法，研究了功能梯度压电准晶层合板的静态响应，分析了功能梯度系数对声子场、相位子场和电场的影响.
迄今为止，关于功能梯度准晶层合梁的相关研究尚未报道，而功能梯度准晶层合梁的研究为材料科学家研究其制造工艺、性能表征等以及工程师对其在实际工程的应用情况提供重要的理论依据，所以有必要开展功能梯度层合准晶梁的力学性能研究. 本文通过建立准晶层合简支梁二维模型，研究具有功能梯度一维六方准晶层合梁的自由振动和弯曲问题. 利用伪Stroh型公式得出单层功能梯度梁的广义位移和广义应力，再通过传递矩阵法，获得三明治型功能梯度准晶梁自由振动和弯曲问题的精确解，最后，通过数值算例分析高跨比、层厚比、叠层顺序和功能梯度系数对固有频率和弯曲变形的影响规律，为功能梯度准晶在作为材料增强相、航空航天结构件的应用提供科学依据，也为研究功能梯度准晶材料的各种数值计算方法和实验结果提供理论参考.
1　问题描述和基本方程
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本文研究两边简支的功能梯度一维六方准晶层合梁的振动和弯曲特性，梁模型如图1所示，中间为均匀层，上下为功能梯度层，其原子排列在x1方向为周期排列，在x3方向为准周期排列. 梁的跨度为L，梁的总厚度为h，三明治型梁的底部和顶部分别为和，宽度为b.
对于简支的多层准晶梁，其声子场和相位子场的边界条件为
其中u3为声子场位移，w3为相位子场位移，σ11为声子场应力.
假设不同层之间在x3方向上的声子场和相位子场的位移和应力是连续的，满足连续性边界条件
其中u（j）={u1，u3，w3}T，t（j）={σ13，σ33，H33}T分别表示层j的广义位移和广义应力，上标T表示矩阵或向量的转置.
基于一维六方准晶的线弹性理论，其应变-位移关系为
其中εij和ωij为声子场和相位子场的应变张量，下标逗号表示偏微分.
在平面应力假设下，一维六方准晶梁的本构关系式可简化为[14]
其中σ11，σ13，σ31，σ33为声子场应力，C11，C13，C33，C44为声子场弹性常数，H31，H33为相位子场应力，K1，K2为相位子场弹性常数，R1，R2，R3为声子场-相位子场耦合弹性常数.
单层功能梯度一维六方准晶常数沿厚度x3方向按指数率变化，则式（4）中材料常数为
式中η是沿x3方向的功能梯度指数因子，上标0表示均匀材料（即η=0）时的材料常数.
平衡方程为
对于自由振动问题，fi=ρui，tt，gi=ρwi，tt；对于弯曲问题，fi=0，gi=0. 式中下标t表示时间，ρ表示密度，沿厚度方向也呈指数率变化，即.
2　功能梯度准晶层合梁的一般解
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为了满足简支梁的边界条件，功能梯度一维准晶层合梁的广义位移通解设为
其中a1~a3是待定常数，p=nπ/L，n为正整数. 对于准晶层合梁的自由振动问题，ϕ=sx3+iωt；对于准晶层合梁的弯曲问题，ϕ=sx3，其中s是待确定的特征值，i代表的是虚数，ω是角频率.
将式（7）代入式（3）和（4），可得广义应力的一般解为
引入两个向量和，得到a和b的关系为
其中
把式（8）代入平衡方程（6），化简得到如下控制方程
注意，这里弯曲问题和自由振动问题的控制方程形式虽相同，但是Q的表达式不同，对于自由振动问题
对于弯曲问题
根据式（9）和（11），得到线性特征系统为
其中矩阵N为
其中I为3×3的单位矩阵.
通过求解式（14），可得到特征值s和特征向量a、b，从而获得广义位移和广义应力的通解表达式为
其中
在方程（16），K1和K2是两个待确定的3×1常数列向量，可由边界条件确定. 功能梯度准晶梁任意j层上表面的广义应力和广义位移可由j层下表面所表示，即
其中.
由于假设广义位移和广义应力在层与层之间是连续的，所以通过重复使用传递关系式（18）使物理量从功能梯度准晶层合梁的底层x3=0传递到顶层x3=h，得到如下的传递关系式为
其中Pj（hj）为传递矩阵，即
2.1　自由振动
对于梁的自由振动问题，梁的上下表面自由，从而有
其中C1~C4是梁从底层到顶层的传递矩阵的子矩阵. 当C3的行列式等于零时，可求出自由振动的固有频率.
2.2　静态弯曲
对于梁的静态弯曲问题，考虑上表面仅受到声子场机械载荷σ33，下表面应力自由，其边界条件为
从而式（19）变为
将式（22）代入式（23）中，得到上下表层的位移为
最终可得功能梯度准晶层合梁任意位置的广义应力和广义位移为
3　数值算例
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本节考虑两种叠层顺序QC1/QC2/QC1和QC2/QC1/QC2下高跨比、层厚比、功能梯度系数和叠层顺序对自由振动固有频率和对弯曲位移和应力的影响. QC1和QC2的材料属性如表1所示. 设三明治梁的顶层和底层为功能梯度材料，中间层为均匀材料，为了符合实际且计算相对简单，令顶层和底层的材料系数关于中间层对称变化. 图2为材料功能梯度系数变化图，以下算例中选取η=10，0，-10三种功能梯度系数.
3.1　自由振动
为了验证本文功能梯度准晶梁模型的准确性，首先将本文所得结果与已有三明治准晶层合梁的精确解[14]和应用状态空间-微分求积法（SSDQM）[13]的三明治准晶层合梁进行对比，见表2，其中材料为QC1准晶，高跨比为h/L=0.1. 从表中发现，本文所得结果与已有精确解和数值结果吻合很好，从而验证本文模型的正确性和有效性.
表3给出两种叠层顺序下功能梯度一维六方准晶层合梁的前四阶固有频率，从表中随着功能梯度系数的增大，QC1/QC2/QC1层合梁的前四阶固有频率呈增大趋势；QC2/QC1/QC2层合梁的第一阶和第三阶固有频率呈增大趋势，第二阶和第四阶固有频率呈递减趋势. 当功能梯度系数增大时，材料的属性分布会发生变化，这些变化会直接影响结构的自由振动特性. 功能梯度系数增大导致第二种叠层梁的某区域的刚度和强度发生变化进而导致其不同阶的固有频率发生不同的变化. 对比两种叠层顺序的层合梁固有频率，发现QC1/QC2/QC1层合梁的固有频率比QC2/QC1/QC2层合梁高，所以高弹性模量的QC1准晶适用于涂层材料.
图3显示两种叠层顺序功能梯度准晶层合梁的第一阶固有频率随高跨比h/L的变化. 由图发现，两种叠层顺序的准晶梁的第一阶固有频率随着高跨比的增加而增大，随着功能梯度系数的增加而增大，表明可以通过增加准晶层合梁厚度或功能梯度因子来提高其固有频率.
图4显示两种叠层顺序功能梯度准晶层合梁的第一阶固有频率随层厚比h1/h2的变化. 由图发现，随着h1/h2的增加，QC1/QC2/QC1层合梁的固有频率逐渐增大，QC2/QC1/QC2层合梁逐渐减小，最后均趋于稳定. 随着h1/h2的增加，功能梯度对固有频率的影响更加明显.
图5为不同功能梯度系数下两种叠层顺序准晶层合梁沿厚度方向变化的振型图，其中umax为u1，u3，w3中的最大值. 由图可知，随着功能梯度系数的增加，两种叠层顺序下的声子场位移u1振型在增大. QC1/QC2/QC1层合梁的相位子场位移w3振型在顶层随功能梯度系数增加而减小，在底层则相反；而对于QC2/QC1/QC2层合梁的相位子场位移w3振型则随功能梯度系数增加而增大. 可见，功能梯度因子对准晶的相位子场的振型影响更为明显.
3.2　弯曲
假设在简支层合梁的上表面（x3=h）仅作用声子场应力σ33=σ0sinpx1，在本模型中，应力载荷幅值可以设置为任意值以满足实际要求，为了与已有文献结果进行对比，本文应力载荷幅值取为σ0=1 N/m2[23]，其他方向上的应力分量为零. 梁的尺寸高跨比为h/L=0.1，考虑两种叠层顺序QC1/QC2/QC1和QC2/QC1/QC2，梁的每层厚度相同. 图6显示两种叠层顺序下声子场和相位子场位移沿厚度方向的变化. 由图可知，声子场位移u1在两种叠层顺序下具有相同的变化趋势，在整个板厚方向声子场位移u1的绝对值随功能梯度系数的增大而减小. 在两种叠层顺序下随功能梯度系数的增大，声子场位移u3逐渐减小，而相位子场位移w3逐渐增大，表明功能梯度对准晶材料的普通弹性场（声子场）和特有的相位子场会产生不同的影响.
图7给出两种叠层顺序下声子场和相位子场应力沿厚度方向的变化图，其中Cmax为表1最大值. 由图可知，在两种叠层顺序下声子场应力σ13的最大值均在梁中部位置，且QC2/QC1/QC2的声子场应力σ13的最大值大于QC1/QC2/QC1的声子场应力σ13；声子场应力σ33随着功能梯度系数的增大而减小；此外，相比于声子场应力，相位子场应力H33对功能梯度系数的改变较为敏感，尤其对于高弹性常数准晶作外层的功能梯度层合梁.
4　结论
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利用伪Stroh型公式与传递矩阵法，研究功能梯度一维六方准晶层合梁的自由振动和弯曲问题，获得简支边界条件的功能梯度准晶层合梁自由振动和弯曲问题的精确解. 通过数值算例，分析两种叠层顺序下高跨比、层厚比以及非局部参数对准晶层合简支梁固有频率、一阶模态以及弯曲变形的广义应力和广义位移的影响规律，得到如下结论：
（1）随着功能梯度系数的增大，QC1/QC2/QC1层合梁的前四阶固有频率始终增大，而QC2/QC1/QC2层合梁的第一阶和第三阶固有频率呈增大趋势，但第二阶和第四阶固有频率呈递减趋势.
（2）准晶层合简支梁的自由振动固有频率随着高跨比的增加而增大. 随着层厚比的增大，QC1/QC2/QC1层合梁的固有频率逐渐增大，而QC2/QC1/QC2层合梁逐渐减小，最后均趋于稳定，并且两种准晶层合梁的固有频率随着功能梯度系数的增加而增大.
（3）随功能梯度系数增加，两种叠层顺序下的声子场位移u1振型在增大，而相位子场位移w3振型变化不同.
（4）在两种叠层顺序下随功能梯度系数的增大，声子场位移u1具有相同的变化趋势，声子场位移u3逐渐减小，而相位子场位移w3逐渐增大. 功能梯度系数和叠层顺序对层合梁的应力有很大影响，其中对准晶特有的相位子场应力影响更大.
本文所研究的功能梯度准晶梁二维模型不仅适用于三明治梁，还可以解决多层梁的相关问题. 利用COMSOL和有限元等计算平台研究功能梯度准晶梁将是今后工作的重点方向之一，可进一步验证本文理论计算的正确性和有效性.
基金
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  • 国家自然科学基金项目(12072166; 21462007)
  • 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项目(JY20220075)
  • 内蒙古自然科学基金重点项目(2024ZD21)
  • 内蒙古自治区研究生科研创新项目(B20231101Z)
文献
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2024年第45卷第6期
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doi: 10.19636/j.cnki.cjsm42-1250/o3.2024.032
  • 接收时间：2024-07-10
  • 首发时间：2026-04-01
  • 出版时间：2024-12-25
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  • 收稿日期：2024-07-10
基金
国家自然科学基金项目(12072166; 21462007)
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项目(JY20220075)
内蒙古自然科学基金重点项目(2024ZD21)
内蒙古自治区研究生科研创新项目(B20231101Z)
作者信息
    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

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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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