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Article(id=1246045606267957756, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246045605047415288, articleNumber=null, orderNo=null, doi=10.19636/j.cnki.cjsm42-1250/o3.2024.021, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1716393600000, receivedDateStr=2024-05-23, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1775010022942, onlineDateStr=2026-04-01, pubDate=1729785600000, pubDateStr=2024-10-25, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1775010022942, onlineIssueDateStr=2026-04-01, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1775010022942, creator=13701087609, updateTime=1775010022942, updator=13701087609, issue=Issue{id=1246045605047415288, tenantId=1146029695717560320, journalId=1241755870837649424, year='2024', volume='45', issue='5', pageStart='565', pageEnd='708', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1775010022651, creator=13701087609, updateTime=1775010094858, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1246045907972636933, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246045605047415288, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1246045907972636934, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246045605047415288, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=652, endPage=664, ext={EN=ArticleExt(id=1246045606477672958, articleId=1246045606267957756, tenantId=1146029695717560320, journalId=1241755870837649424, language=EN, title=Sensitivity of Structures to Random Defects in Film-substrate Systems, columnId=1244229834482757770, journalTitle=Chinese Journal of Solid Mechanics, columnName=Research Paper, runingTitle=null, highlight=null, articleAbstract=

Random defects due to differences in raw materials and the complexity of the manufacturing process are inevitable in engineering structures. Based on the inherent characteristics of sensitivity to defects in the film-substrate system, the Monte Carlo method is applied in the study of the stability of structures with random defects, coupled with numerical simulations to investigate the morphological evolution and post-buckling equilibrium path of film-substrate systems with random defects during instability. The numerical results show that the critical load of the structure with random defects is unstable. The defects significantly reduce the critical load of the structure, and the random defects destroy the symmetry of the structure, leading to a transformation from an ordered checkerboard pattern to a disordered fold nuclear pattern, hence affecting the subsequent morphological trend. This analysis assesses the potential risks and effects of random defects in thin-film structures, aiming to improve the reliability and performance of thin-film devices, coatings and surface treatments, and to bridge the gap between theoretical stability research findings and practical design applications.

, correspAuthors=Guo Fu, 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=Andi Lai, Jun Liao, Di Ou, Guo Fu), CN=ArticleExt(id=1246045611888325237, articleId=1246045606267957756, tenantId=1146029695717560320, journalId=1241755870837649424, language=CN, title=薄膜/衬底系统中随机缺陷的敏感性研究, columnId=1241831201896469478, journalTitle=固体力学学报, columnName=研究论文, runingTitle=null, highlight=null, articleAbstract=

由于原材料的差异和制造工艺的复杂性，工程结构中难以避免地存在随机缺陷. 考虑到薄膜/衬底系统对缺陷的敏感性，本文采用Monte-Carlo法对随机缺陷结构的稳定性进行统计分析，并结合数值模拟探讨含随机缺陷的薄膜/衬底系统在失稳下的形貌演化和后屈曲平衡路径. 数值结果表明，随机缺陷结构的临界载荷呈现不稳定性，其中缺陷显著降低了结构的临界载荷，随机缺陷破坏了结构的对称性，使有序的棋盘状图案转化成无序的褶皱核图案并影响了后续的形貌走势. 该研究评估了薄膜结构中随机缺陷的潜在风险和影响，旨在提高薄膜器件、涂层和表面处理的可靠性和性能，同时缩小稳定性理论研究成果与实际设计应用之间的差距.

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** E-mail：.
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International Journal of Solids and Structures, 2014, 51(21-22): 3619-3632., articleTitle=3D finite element modeling for instabilities in thin films on soft substrates, refAbstract=null)], funds=[Fund(id=1246045620729918369, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045606267957756, awardId=11ZDXK11, language=CN, fundingSource=长沙理工大学桥梁与隧道工程创新性项目(11ZDXK11), fundOrder=null, country=null), Fund(id=1246045620826387365, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045606267957756, awardId=CSLGCX23146, language=CN, fundingSource=湖南省研究生科研创新项目(CSLGCX23146), fundOrder=null, country=null)], companyList=[AuthorCompany(id=1246045612110623357, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045606267957756, xref=null, ext=[AuthorCompanyExt(id=1246045612123206270, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045606267957756, companyId=1246045612110623357, language=EN, country=null, 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caption=含随机缺陷的薄膜/衬底系统模型, figureFileSmall=m8WQ9n7oRyejK349Pm9c/w==, figureFileBig=9xp3liKoLpX7Ro6qstpKyw==, tableContent=null), ArticleFig(id=1246045616204264209, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045606267957756, language=EN, label=Fig.2, caption=Probability density of random defect amplitude, figureFileSmall=lGjbTl0/LcM9wAz8d13Sug==, figureFileBig=pNJkp1f9SetORilv4oN4vQ==, tableContent=null), ArticleFig(id=1246045616313316116, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045606267957756, language=CN, label=图2, caption=随机缺陷幅值的概率密度, figureFileSmall=lGjbTl0/LcM9wAz8d13Sug==, figureFileBig=pNJkp1f9SetORilv4oN4vQ==, tableContent=null), ArticleFig(id=1246045616393007896, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045606267957756, language=EN, label=Fig.3, caption=(a) Disordered fold core morphology; (b) Herringbone fold morphology; (c)-(d) Zigzag fold appearance; (e) Synoptic table of the hybrid modeling results for films on flat (glass) substrates: numerical (in silico FEM simulations) vs. experiments (right)[27], figureFileSmall=WiDDpXRnAmMmuMNtl3rG9A==, figureFileBig=qoy7ZhQk2W6t6OOEoFFshA==, tableContent=null), ArticleFig(id=1246045616485282588, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045606267957756, language=CN, label=图3, caption=（a）无序褶皱核形貌；（b）人字形褶皱形貌；（c）-（d）锯齿状褶皱形貌；（e）平面基板上薄膜混合建模结果：数值（有限元模拟）与实验（右）[27], figureFileSmall=WiDDpXRnAmMmuMNtl3rG9A==, figureFileBig=qoy7ZhQk2W6t6OOEoFFshA==, tableContent=null), ArticleFig(id=1246045616585945892, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045606267957756, language=EN, label=Fig.4, caption=Comparison of instability morphologies of film/substrate systems without defects under uniaxial compression, figureFileSmall=tYwdDZ0p2ItmdgGw6x5OBw==, figureFileBig=JM+czrV0XIRywzBvQceD4A==, tableContent=null), ArticleFig(id=1246045616690803499, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045606267957756, language=CN, label=图4, caption=在单轴作用下，无缺陷的薄膜/衬底系统的失稳形貌对比, figureFileSmall=tYwdDZ0p2ItmdgGw6x5OBw==, figureFileBig=JM+czrV0XIRywzBvQceD4A==, tableContent=null), ArticleFig(id=1246045616783078192, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045606267957756, language=EN, label=Fig.5, caption=The post-buckling equilibrium path under different distributions of the same defect amplitude, in which φ0=0 represents the ideal model, and φ1=φ2=φ3=φ4=0.3 stands for equal amplitude η(x, y), which is randomly distributed in different locations (x, y), figureFileSmall=alWJpOFf3t0RBHtf+E8JYA==, figureFileBig=aelyF33tn5vEJp1a/pxlDw==, tableContent=null), ArticleFig(id=1246045616921490231, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045606267957756, language=CN, label=图5, caption=相同缺陷幅值不同分布下的后屈曲平衡路径，其中φ0=0为理想模型，φ1=φ2=φ3=φ4=0.3为相等缺陷幅值η（x，y），其中随机刚度缺陷分布在不同位置（x，y）, figureFileSmall=alWJpOFf3t0RBHtf+E8JYA==, figureFileBig=aelyF33tn5vEJp1a/pxlDw==, tableContent=null), ArticleFig(id=1246045618460799805, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045606267957756, language=EN, label=Fig.6, caption=Morphology evolution of film-substrate system with random defects.in which φ0=0 represents the ideal mode, and φ1=φ2=φ3=φ4=0.3 stands for equal amplitude η(x, y), which is randomly distributed in different locations (x, y), figureFileSmall=7mypnKhoxPbeOO5k8jOR0Q==, figureFileBig=g6VPPTf1WrC4XmO4ynOU2Q==, tableContent=null), ArticleFig(id=1246045618582434628, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045606267957756, language=CN, label=图6, caption=具有随机缺陷的薄膜/衬底系统的形貌演变. 其中φ0=0表示理想模型，φ1=φ2=φ3=φ4=0.3表示相等缺陷幅值η（x，y），其中随机刚度缺陷分布在不同位置（x，y）, figureFileSmall=7mypnKhoxPbeOO5k8jOR0Q==, figureFileBig=g6VPPTf1WrC4XmO4ynOU2Q==, tableContent=null), ArticleFig(id=1246045618687292232, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045606267957756, language=EN, label=Fig.7, caption=The post-buckling equilibrium path of different defect amplitudes in Weibull distribution, figureFileSmall=Kt6P/zGLVJ8TbWQVid9qqg==, figureFileBig=wmFNqiSxAA8yfBRn3hY80A==, tableContent=null), ArticleFig(id=1246045618800538446, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045606267957756, language=CN, label=图7, caption=不同缺陷幅值在Weibull分布下的后屈曲平衡路径, figureFileSmall=Kt6P/zGLVJ8TbWQVid9qqg==, figureFileBig=wmFNqiSxAA8yfBRn3hY80A==, tableContent=null), ArticleFig(id=1246045618909590355, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045606267957756, language=EN, label=Fig.8, caption=Effects of different defect amplitudes on the morphology evolution of the system under Weibull distribution, figureFileSmall=UIAFt4D3sANKrndMml/yAA==, figureFileBig=eaqkP9o6PQD0ZN+0NfabEw==, tableContent=null), ArticleFig(id=1246045619001865046, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045606267957756, language=CN, label=图8, caption=Weibull分布下不同缺陷幅值对系统形貌演化的影响, figureFileSmall=UIAFt4D3sANKrndMml/yAA==, figureFileBig=eaqkP9o6PQD0ZN+0NfabEw==, tableContent=null), ArticleFig(id=1246045619123499867, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045606267957756, language=EN, label=Fig.9, caption=Effects of defect types and defect distribution type functions with different distribution parameters on the post-buckling equilibrium path: (a)-(b) Table 3 and 4 show the post-buckling equilibrium path of the system with corresponding parameters, figureFileSmall=ct6oIjoWekdkLSv9Tpn/3g==, figureFileBig=GFqC/G1Hq5yRyn1rVKC6FA==, tableContent=null), ArticleFig(id=1246045619219968865, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045606267957756, language=CN, label=图9, caption=不同分布参数下缺陷类型和缺陷分布类型函数对后屈曲平衡路径的影响：（a）-（b）为表3-4相应参数下系统的后屈曲平衡路径, figureFileSmall=ct6oIjoWekdkLSv9Tpn/3g==, figureFileBig=GFqC/G1Hq5yRyn1rVKC6FA==, tableContent=null), ArticleFig(id=1246045619354186596, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045606267957756, language=EN, label=Fig.10, caption=The post-buckling equilibrium path of the system under different boundary conditions, figureFileSmall=LB5rA9STvn/WHg+bEHEwyg==, figureFileBig=4g194+QnaXnz2PFWAUyDWA==, tableContent=null), ArticleFig(id=1246045619471627115, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045606267957756, language=CN, label=图10, caption=不同边界条件下系统后屈曲平衡路径, figureFileSmall=LB5rA9STvn/WHg+bEHEwyg==, figureFileBig=4g194+QnaXnz2PFWAUyDWA==, tableContent=null), ArticleFig(id=1246045619555513199, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045606267957756, language=EN, label=Fig.11, caption=Influence of random defects under free boundaries on post-buckling equilibrium path, figureFileSmall=bJxpD0uPzSqj8cUuESpE4w==, figureFileBig=GE3NwvP0oTeY+0ZqyBFrmA==, tableContent=null), ArticleFig(id=1246045619660370803, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045606267957756, language=CN, label=图11, caption=自由边界下随机缺陷对后屈曲平衡路径的影响, figureFileSmall=bJxpD0uPzSqj8cUuESpE4w==, figureFileBig=GE3NwvP0oTeY+0ZqyBFrmA==, tableContent=null), ArticleFig(id=1246045619773617016, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045606267957756, language=EN, label=Table 1, caption=

Normal distribution

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随机变量分布类型均值ξ标准差σ缺陷幅值φ
Ef正态分布2508.60.3
), ArticleFig(id=1246045619899446140, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045606267957756, language=CN, label=表1, caption=

正态分布

, figureFileSmall=null, figureFileBig=null, tableContent=
随机变量分布类型均值ξ标准差σ缺陷幅值φ
Ef正态分布2508.60.3
), ArticleFig(id=1246045620029469570, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045606267957756, language=EN, label=Table 2, caption=

Weibull distribution

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随机变量分布类型均值ξ非均匀系数k
hfWeibull分布1100
hfWeibull分布120
hfWeibull分布15
), ArticleFig(id=1246045620125938565, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045606267957756, language=CN, label=表2, caption=

Weibull分布

, figureFileSmall=null, figureFileBig=null, tableContent=
随机变量分布类型均值ξ非均匀系数k
hfWeibull分布1100
hfWeibull分布120
hfWeibull分布15
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Different distribution parameters

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随机变量分布类型均值ξ标准差
hf正态分布10.01
Ef正态分布2508.6
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不同分布参数

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随机变量分布类型均值ξ标准差
hf正态分布10.01
Ef正态分布2508.6
), ArticleFig(id=1246045620457288596, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045606267957756, language=EN, label=Table 4, caption=

Different defect distribution function types

, figureFileSmall=null, figureFileBig=null, tableContent=
随机变量分布类型均值ξ标准差
hf正态分布10.01
hf均匀分布0.9-1.1
hfweibull分布1100
), ArticleFig(id=1246045620566340505, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045606267957756, language=CN, label=表4, caption=

不同分布函数类型

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随机变量分布类型均值ξ标准差
hf正态分布10.01
hf均匀分布0.9-1.1
hfweibull分布1100
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薄膜/衬底系统中随机缺陷的敏感性研究
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赖安迪 , 廖军 , 欧迪 , 付果 **
固体力学学报 | 研究论文 2024,45(5): 652-664
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固体力学学报 | 研究论文 2024, 45(5): 652-664
薄膜/衬底系统中随机缺陷的敏感性研究
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赖安迪, 廖军, 欧迪, 付果**
作者信息
  • 长沙理工大学土木工程学院力学系，长沙，410114

通讯作者:

** E-mail：.
Sensitivity of Structures to Random Defects in Film-substrate Systems
Andi Lai, Jun Liao, Di Ou, Guo Fu**
Affiliations
  • Department of Mechanics, School of Civil Engineering, Changsha University of Science and Technology, Changsha, 410114
出版时间: 2024-10-25 doi: 10.19636/j.cnki.cjsm42-1250/o3.2024.021
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摘要
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由于原材料的差异和制造工艺的复杂性，工程结构中难以避免地存在随机缺陷. 考虑到薄膜/衬底系统对缺陷的敏感性，本文采用Monte-Carlo法对随机缺陷结构的稳定性进行统计分析，并结合数值模拟探讨含随机缺陷的薄膜/衬底系统在失稳下的形貌演化和后屈曲平衡路径. 数值结果表明，随机缺陷结构的临界载荷呈现不稳定性，其中缺陷显著降低了结构的临界载荷，随机缺陷破坏了结构的对称性，使有序的棋盘状图案转化成无序的褶皱核图案并影响了后续的形貌走势. 该研究评估了薄膜结构中随机缺陷的潜在风险和影响，旨在提高薄膜器件、涂层和表面处理的可靠性和性能，同时缩小稳定性理论研究成果与实际设计应用之间的差距.

随机缺陷  /  缺陷敏感性  /  形貌演化  /  后屈曲  /  稳定性

Random defects due to differences in raw materials and the complexity of the manufacturing process are inevitable in engineering structures. Based on the inherent characteristics of sensitivity to defects in the film-substrate system, the Monte Carlo method is applied in the study of the stability of structures with random defects, coupled with numerical simulations to investigate the morphological evolution and post-buckling equilibrium path of film-substrate systems with random defects during instability. The numerical results show that the critical load of the structure with random defects is unstable. The defects significantly reduce the critical load of the structure, and the random defects destroy the symmetry of the structure, leading to a transformation from an ordered checkerboard pattern to a disordered fold nuclear pattern, hence affecting the subsequent morphological trend. This analysis assesses the potential risks and effects of random defects in thin-film structures, aiming to improve the reliability and performance of thin-film devices, coatings and surface treatments, and to bridge the gap between theoretical stability research findings and practical design applications.

random defects  /  defect sensitivity  /  morphology evolution  /  post-buckling  /  stability
赖安迪, 廖军, 欧迪, 付果. 薄膜/衬底系统中随机缺陷的敏感性研究. 固体力学学报, 2024 , 45 (5) : 652 -664 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2024.021
Andi Lai, Jun Liao, Di Ou, Guo Fu. Sensitivity of Structures to Random Defects in Film-substrate Systems[J]. Chinese Journal of Solid Mechanics, 2024 , 45 (5) : 652 -664 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2024.021
正文
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0　引言
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硬薄膜和柔软基底构成的复合材料系统表面在外部环境刺激下容易发生较大的变形和各种不稳定性行为，并呈现丰富的屈曲褶皱形貌[1-10]. 这种表面不稳定性在自然界和日常生活中被广泛观察. 如血细胞的球形表面褶皱[4]，大脑皮层组织弯曲成团形成褶皱[5]，水果在干燥后果肉失水收缩导致果皮出现褶皱[6]，荷叶的超疏水表面呈现褶皱状结构[7,8]，以及人体皮肤和弹性体表面也存在各种规则或不规则的褶皱纹理[9,10]. 在软材料结构领域，研究材料的结构稳定性和缺陷敏感性[11]对于设计和制造柔性电子器件[12,13]、功能表面的制作[14-16]、纳米反应器和生物支架[17,18]具有重要的应用意义.
薄膜/衬底系统自身的不稳定性和对外部刺激高度敏感性的特性是导致其表面复杂性的主要原因. 尽管近年来在形貌不稳定性的模拟方面取得了显著的进展，但仍存在许多引人关注的问题需要进一步的实验和理论研究. 现实中的薄膜通常会存在各种缺陷，如杂质、表面粗糙度或厚度的不规则性. 这些缺陷可能有各种来源，包括制造工艺、材料沉积或环境条件等. 在简单的载荷作用下，这些缺陷在薄膜内充当局部扰动，导致应力集中并影响褶皱的初始形成和后续演化. 以往已经进行了一些关于含缺陷薄膜/衬底系统表面失稳的研究，如在制造和加工过程中出现的初始褶皱作为一种结构缺陷，可显著影响薄膜的局部几何特征和力学性质[19]. Wang等[20]通过设计界面上的周期性厚度变化来控制薄膜的变形，产生各种褶皱形态. 之后Wang等[21]还介绍了一种利用周期性硬层控制软材料起皱微图案的方法. 通过改变表面硬度，从三个方面控制表面起皱. Li等[22]通过改变薄膜/衬底系统中的周期性表面刚度来调节起皱模式. Xie等[23]采用仿生微结构来调节表面起皱. 通过在微结构上使用金字塔形凹槽和突起可以预测表面起伏，实现可控性. Budday等[24]使用计算模型模拟了大脑在发育过程中的机械行为. 研究发现，不均匀的刚度和生长会导致脑组织的机械不稳定性，进而导致折叠和起皱，最终导致生长发育异常. 在曲面系统中，Xu等[25]研究了不同曲率下软材料表面皱纹的演变模式，揭示了表面曲率及其各向异性在皱纹形态转变中的关键作用.
现阶段该领域的研究主要集中在周期性缺陷和局部缺陷对褶皱形貌演变的影响，其中周期性缺陷由于其简单性和易于实验和理论分析而常被采用. 尽管这些研究为形貌演化的基本原理和行为提供了宝贵的见解，但它们可能无法充分反映现实薄膜的复杂性和真实性，例如，生物组织和器官的复杂形态、工业原材料制造过程中真实表面的特性. 在实际应用中，真实薄膜往往表现出随机和不规则的缺陷分布，以及材料的异质性等不均匀性特征. 忽略这些真实特征可能会限制对真实场景中褶皱形貌行为和演变的理解. 因此，考虑薄膜的真实性对于弥合基础研究和实际应用之间的差距至关重要.
本研究将采用Monte-Carlo法[26]建立含随机缺陷的薄膜/衬底模型，我们将定义随机概率模型的缺陷敏感系数指标，并结合数值模拟方法分析含随机缺陷结构在等双轴作用下的屈曲行为，通过将其与理想无缺陷结构的后屈曲形貌进行比较，以研究随机缺陷结构的稳定性，并对比不同模型的合理性和适用性. 同时探讨其有关物理机制和随机缺陷对系统的敏感性影响，为柔性电子、微尺度器件和功能涂层等领域的进步提供理论和实验基础.
1　材料与方法
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1.1　含随机缺陷的薄膜/衬底模型
假设软基底上的矩形薄膜，长Lx宽Ly厚度hf. 采用直角坐标系x，y，z，其中u，v，w为对应的位移分量.
如图1所示，我们引入了一个具有随机缺陷、边界条件和压缩位移方向的薄膜/衬底系统示意图. 其中和分别表示薄膜的拉伸刚度和弯曲刚度，薄膜材料的泊松比和弹性模量用υf和Ef表示.
为了深入理解薄膜中随机缺陷的内部机理并有效预测其形态发生，我们假设hf=h（x，y）表示薄膜的实际厚度，其具体表达式为：
其中h（x，y）是薄膜在每个位置的实际值，ξ是薄膜的平均厚度，η（x，y）表示薄膜在每个位置（x，y）随机偏离平均值的缺陷幅度. 图1（b）中的随机缺陷由随机厚度hf（x，y）或随机刚度Ef（x，y）表征.
针对薄膜表面材料力学或几何特性的非均匀分布，不同的概率分布函数无法提供统一的方法来描述随机分布的分散性. 因此，定义缺陷的非均匀系数并使用合适的概率分布函数是数值模拟研究的基础. 本文仅讨论正态分布、威布尔分布和对数正态分布下的数值模拟结果.
正态分布是一种常用的统计分布，它可以用期望值ξ和方差S. 来表征. 为了便于记录，这里引入了缺陷不均匀系数φ来衡量正态分布的离散程度：
与威布尔分布不同，正态分布的参数值可能小于0，因此在数值计算中需要特殊处理.
威布尔分布的密度函数为：
分布函数为：
其中，1/k可视为缺陷的不均匀系数，k越大，薄膜表面越均匀，分散程度越小.
1.2　数值计算
含随机缺陷的薄膜/衬底的屈曲控制方程，其求解方法一般只能采用数值方法，如有限元法、能量法等. 将缺陷以随机分布的方式引入到材料中，这使得硬膜和软基底的本构关系在统计理论的基础上应用存在一些限制. 推导这种关系的过程相当复杂，而且难以满足复杂的边界条件，也难以具有广泛适用性. 与此相比，数值计算方法提供了更为方便和强大的分析能力.
本文研究将概率统计学的思想应用到缺陷随机性问题中，假定薄膜的细观尺度单元的几何形状和力学性质服从某种既定的分布，并通过二次开发将这一假设纳入到模型中，通过薄膜弹性模量、力学性质和厚度等几何性质的随机分布来描述单元的缺陷，引入简化的本构关系，来分析随机缺陷薄膜/衬底的形貌演化过程.
为了追踪整个屈曲后的形貌演变过程，本文使用有限元软件ABAQUS对含有随机缺陷的薄膜/衬底系统进行了表面屈曲的数值分析. 通过蒙特卡罗方法将缺陷随机引入模型网格节点. 假设薄膜微尺度单元的几何形状和力学性能都遵循特定的既定分布，系统的缺陷就可以用薄膜几何特性的随机分布来描述，如弹性模量、力学性能和厚度. 对于基底，基底底部的垂直位移受到约束，而薄膜的上表面则是无约束的，薄膜/衬底系统的四端受到水平位移的压缩. 在该模拟中，通过引入与速度相关的阻尼和惯性项，实现了一种伪动态算法，将其自然地视为引发表面屈曲的扰动.
2　数值结果及分析
收起
2.1　对比实验
本文对不同制造工艺中不同结构的缺陷进行统计分析，并利用Monte-Carlo法研究随机缺陷结构的敏感性. 为了保证随机性不影响最终结果，进行了多次计算模拟. 每次选取的元素材料属性是由程序随机生成的，其缺陷分布结果不同. 通过高斯分布和有限元模拟得到了无随机缺陷和含有随机缺陷的薄膜/衬底系统的临界载荷和屈曲模态以及后屈曲平衡路径和后屈曲形貌演化规律. 在计算中，除另有说明外，薄膜/衬底系统的四端均为简支边界条件，材料和几何参数选择为：Ef=250 MPa，υf=0.4，Es=10 MPa，υs=0.3，Lx=200 mm，Ly=200 mm.
图3为有关文献实验和本文有限元仿真模拟的对比实验. 该文献[27]实验考虑了薄膜在沉积过程中引起薄膜厚度实际不均匀变化的情况，在完全平整的薄膜上随机引入初始厚度为1 μm、振幅为Δti的膜厚度缺陷分布建立了平面基底上随机厚度薄膜的模型并得到了其最终的褶皱图案. 图3（e）显示了相应的褶皱波长和形态分布图. 本文实验在数值模拟中采用了伪动力学（大变形），其基底上的薄膜引入了缺陷幅值为0.3的正态分布的随机刚度缺陷. 数值模拟所得到的褶皱形貌演化可以与文献中的实验结果进行比较. 如图3（a）-（d）所示，可以发现本文数值跟踪的结果和文献实验跟踪的褶皱分布路径基本一致. 在小应变状态下，如图3（a）所示，当薄膜存在随机刚度缺陷时，模量的不均匀性会导致在应力施加下出现不规则的屈曲形变. 这种无序褶皱的形成可能是因为在不同位置上的模量变化导致了应力场的不均匀性，进而引起了局部的屈曲和褶皱形成. 这种情况下，褶皱的形态和分布可能缺乏明显的规律性，呈现出一种无序的特征. 随着应变的增大，无序的褶皱核停止分离，薄膜上的刚度差导致局部的应力集中，从而促使褶皱核的边界开始发生较大的屈曲导致其紧密区域相交形成人字形褶皱见图3（b）. 受随机刚度缺陷的影响，薄膜在不同位置上受到了不同程度的拉伸或压缩应力，进而导致褶皱形貌最终呈现出波动变化的特征. 在这种情况下形成了类似锯齿状的褶皱如图3（c）-（d）. 结合本文3.3节的研究可以发现，该文献是在缺陷幅值较大的情况下的研究结果. 含随机缺陷的薄膜/衬底系统出现了无序褶皱核形貌，其受缺陷位置分布影响的初始褶皱对后续褶皱的形态起着决定性作用，整个过程可以发现随机缺陷在破坏对称性和引发面外弯曲变形方面起着至关重要的作用.
图4为无缺陷薄膜/衬底系统在失稳状态下的理论预测与实验对比. 用传统的数值方法很难检测经典模型的所有分岔，Xu等[28]将渐近数值方法（ANM）应用于无缺陷的薄膜/衬底系统的典型模型中，解决了这一难题. ANM方法能够有效地检测到二次分岔并计算非线性响应曲线上的分岔模式. 在图4（b）中可以清晰地观察到在单轴作用下，失稳后阶段形成的正弦褶皱模式. 其理论预测结果与我们本文采用的数值追踪的结果高度一致，从而验证了所采用方法的准确性和可靠性. 虽然该研究已经成功的应用了ANM方法来研究无缺陷/衬底系统的失稳行为，但对于更复杂的系统，例如存在随机缺陷的系统，其相关理论可能需要进一步深入研究.
2.2　薄膜/衬底系统的缺陷敏感性
2.2.1　随机缺陷对系统后屈曲平衡路径及形貌演化的影响
薄膜中的随机刚度缺陷呈正态分布，表1为相应的参数值. 本节对4组具有相同缺陷幅值的薄膜/衬底系统的屈曲行为进行了统计分析.
图5是含随机缺陷的薄膜/衬底系统在等双轴压缩作用下的荷载-位移曲线以及相应的形貌相图. 其中φ0表示的是理想无缺陷的薄膜/衬底模型，φ1=φ2=φ3=φ4表示相同缺陷幅值η（x，y）但缺陷分布的位置各不相同的四个正态分布随机刚度缺陷薄膜/衬底模型. 结果表明，不同的随机缺陷分布会在导致系统中存在不同程度的不均匀性和局部扰动，从而打破系统的对称性. 这种对称性的打破会影响系统在屈曲过程中的形貌演化路径，从而导致不同的后屈曲平衡路径. 具体而言，随着压缩位移的增大，随机缺陷打破了形貌演化的对称性，原本对称的棋盘形貌受缺陷分布的影响转变为无序的褶皱核形貌且影响后续的形貌走势. 与理想无缺陷模型相比，存在随机缺陷的系统屈曲临界载荷降低，这是因为随机缺陷会导致系统中存在局部的应力集中和不均匀性，从而降低了系统的整体稳定性和承载能力.
图6所示为理想无缺陷和含随机缺陷的薄膜/衬底系统在等双轴压缩下沿z轴的形貌演变. 通过观察图中的结果可以发现，与理想无缺陷模型相比，受缺陷分布的影响，当压缩位移为U=4.05时不同的随机分布缺陷模型其产生的第一分岔形貌是不同的无序褶皱核形貌，当U=5.10时受缺陷分布位置的影响其人字形的走势各不相同. 随着压缩的逐渐增加，其无序褶皱核的边界增加，弯曲区域相交形成各不相同的有序不对称褶皱形貌.
上述结果表明，随机缺陷对后屈曲平衡路径的显著影响表明系统对随机缺陷非常敏感. 随机缺陷的存在会引入局部的不均匀性和应力集中，从而影响系统的整体稳定性和形貌演化的趋势，导致系统的屈曲行为出现复杂的特征. 缺陷的随机性不仅降低了系统后屈曲平衡路径的屈曲临界载荷，而且由于缺陷分布的不同，导致系统的后屈曲平衡路径分散. 在形貌演化过程中，褶皱的初始形貌很容易受到缺陷分布的影响，有序的规则棋盘格形状变成无序的不规则褶皱核形状. 此外，随机缺陷的分布对屈曲后的形貌相图起着决定性的作用. 在棋盘形和人字形的过渡过程中，缺陷的分布也会影响人字形的走向和最终的锯齿形.
2.2.2　缺陷幅值和缺陷分布对系统形貌演变的影响
本节研究理想模型和不同幅值的随机厚度缺陷薄膜/衬底系统在等双轴压缩过程中的形貌演化. 其理想无缺陷薄膜/衬底系统的薄膜厚度为固定值hf=1 mm，而随机厚度缺陷薄膜中的缺陷幅值不同，以下将对缺陷幅值逐渐增加的随机厚度缺陷模型依次进行仿真模拟.
表2为薄膜表面随机厚度缺陷Weibull分布的参数. 图7为不同随机缺陷幅值下系统失稳后的平衡路径. 可以看出，随着缺陷幅值的增大，随机缺陷系统的后屈曲平衡路径显著减小. 这也表明其会导致系统在屈曲时的行为变得更加不稳定，屈曲临界载荷也随之减小. 具体而言，当k=5时，其缺陷幅值为所研究的最大幅值，系统的后屈曲平衡路径明显偏离原有的轨迹，而系统的屈曲临界载荷从Pcr=4.05 kN降低到Pcr=3.38 kN. 这意味着随机缺陷的增大会使系统更容易在较低的荷载下发生屈曲，降低了系统的稳定性和承载能力. 无论非均匀系数k大小如何，缺陷的存在都会较大幅度降低系统的临界荷载和影响后屈曲平衡路径. 屈曲后平衡路径变得越来越平缓，系统的屈曲承载能力大大降低.
这些结果强调了在设计和工程中考虑缺陷敏感性的重要性. 在实际应用中，需要谨慎考虑系统的缺陷，并采取适当的措施来减轻随机缺陷对系统性能的不利影响，以确保系统的可靠性和稳定性.
图8展示了不同缺陷幅值在Weibull分布下的形貌演化. 从图中可以看出，随着压缩位移的逐渐增大，较大缺陷幅值的系统可能会出现更为集中的屈曲变形，这是因为这些区域刚度差过大导致应力集中更为显著，从而导致了屈曲形态的集中化. 由于局部的应力集中和不均匀性在初期阶段影响了其系统的第一失稳形貌，随着缺陷幅值的增大，受缺陷分布影响的初始形貌对随后的屈曲形态有显著影响. 其在双轴压缩下的形貌分岔逐渐消失，所形成的第一失稳形貌逐渐占据主导地位. 此外，从加载过程中可以发现较大缺陷幅值的系统会更早地出现不稳定模式，这是因为这些区域的屈曲形态更为集中和突出，从而更容易受到外部扰动的影响而产生失稳. 相比2.2.1节的结果分析，也进一步验证了随机缺陷的存在会导致褶皱核形态的无序，而这种无序所导致的初始形貌对后续的失稳形态有着显著影响.
上述结果表明，随机缺陷振幅的增加会显著降低系统的临界载荷和后屈曲平衡路径，从而降低系统的稳定性和承载能力. 此外，缺陷幅值的增加也会明显改变系统的屈曲模式和形貌演化过程，当缺陷幅值过大时，系统对缺陷的敏感度会随着增大，初始形貌对系统的后屈曲形貌占主导作用，其后屈曲形貌过渡会逐渐消失导致不出现二次分岔. 表明系统对随机缺陷幅值的大小非常敏感.
2.2.3　缺陷分布对系统平衡路径和形貌演化的影响
本小节分析不同分布参数的缺陷类型和不同缺陷分布类型函数对系统后屈曲平衡路径和形貌演化的影响.
表3为不同的分布参数：随机厚度缺陷hf（x，y）和随机刚度缺陷Ef（x，y）. 表4为不同的缺陷分布函数类型：正态分布、威布尔分布、均匀分布.
由抗弯刚度可以知道薄膜厚度是薄膜弹性模量的三次方量级，图9（a）可以得到随机刚度缺陷与随机厚度缺陷对结果具有相似的影响程度. 这是因为两种类型的缺陷都会在系统中引入局部的应力集中和不均匀性，从而影响系统的屈曲行为. 虽然具体的影响机理可能有所不同，但它们都导致系统在后屈曲过程中出现复杂的形貌演化和失稳行为. 结合不同分布参数对系统形貌演化的结果可以发现，在随机缺陷研究下改变弹性模量和改变厚度对系统后屈曲平衡路径与形貌演化的影响是类似的.
不同类型的概率分布函数对结果产生的影响方式具有一定的相似性. 在统计学和概率论中，不同类型的概率分布函数（比如正态分布、泊松分布、指数分布等）对数据的分布和结果的影响有着特定的模式. 然而，图8（b）表明尽管不同分布函数的形式和特性不同，但它们在影响结果方面的作用方式是相似的. 它们的主要差异在于数据偏离均值的程度，一些分布形式可能会导致形貌演化路径更加复杂和不规则，而另一些分布形式可能导致形貌演化路径相对较为简单和规则. 由于不同分布形式引起的局部扰动程度和应力集中程度不同，从而影响系统的整体稳定性和承载能力.
2.3　边界条件的影响
2.3.1　系统在不同边界条件下的后屈曲平衡路径
由图10（a）结果发现在薄膜/衬底系统失稳的情况下，无论采用固支边界条件还是简支边界条件，在模拟结果方面会呈现相似的效果. 这可能是由于系统的特性使得两种边界条件下的影响方式相同，导致模拟结果相同. 由于边界约束的差异，自由边界条件和简支边界条件对系统的约束程度不同. 在简支边界条件下，系统的边界被允许在一个平面上自由移动，但在自由边界条件下，边界完全没有被固定. 这导致了系统的固有失稳模式可能会有根本性的不同. 另外在自由边界条件下，系统可能会经历较大的形变，形成的几何形状和不同分布的随机缺陷会在系统中引入不同的应力集中区域和几何非均匀性. 在失稳过程中起到局部的扰动作用，从而影响系统的失稳行为和形貌演化. 图10（b）所示，在自由边界条件下含随机缺陷薄膜/衬底系统在受到等双轴作用时，沿x轴和y轴方向的荷载位移曲线表现出两种不同的失稳特征或者仅呈现分岔失稳.
总而言之，上述结果指出含随机缺陷的系统在自由边界条件下的失稳行为相对于简支边界条件更为敏感. 在理想无缺陷的薄膜/衬底模型中，系统在自由边界条件下表现出极值失稳. 然而当系统存在随机缺陷时，其失稳特征变得更加复杂，可能表现出两种不同的失稳特征，或者仅出现分岔失稳的情况.
2.3.2　自由边界对随机缺陷系统后屈曲平衡路径（双端）的影响
图11为自由边界下薄膜/衬底系统在双轴压缩工况下的后屈曲平衡路径及形貌演化，其中RF1、RF2分别为x轴、y轴方向的荷载-位移曲线. 由图11（a）可知，在无缺陷模型中，薄膜/衬底系统在受到等双轴压缩作用时，沿x轴与y轴方向的荷载-位移曲线在双端呈现出极值失稳的特征. 图11（b）-（d）中是三个相同随机分布形式（正态分布）但缺陷位置分布不同的随机刚度缺陷模型（详见2.2.1节的随机缺陷薄膜/衬底模型）. 当薄膜/衬底系统存在随机缺陷时，沿x轴与y轴方向的荷载位移曲线可能呈现两种不同类型的失稳行为，或者仅呈现分岔失稳.
结果表明含随机缺陷的系统对边界条件更加敏感. 系统的随机缺陷可能导致失稳行为的变化和多样性，使得系统对边界条件的响应不同. 相对于理想模型，缺陷可能使得系统在自由边界条件下更容易出现不同类型的失稳特征，这突显了含随机缺陷系统对边界条件的敏感性. 系统的缺陷可能导致失稳行为的不确定性和复杂性，因此需要更加细致和全面地考虑边界条件对系统行为的影响.
3　结论
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本论文的主要结论如下：
（1）探讨了不同类型的缺陷（随机厚度/刚度缺陷）对薄膜/衬底系统后屈曲行为的影响. 结果表明，由于随机缺陷的局部应力集中和应力分布不均匀，这些缺陷显著改变了系统的屈曲模态且大幅度降低了屈曲临界载荷，其随机分布缺陷打破了形貌的对称性，进一步导致了系统屈曲行为的复杂性.
（2）研究还考虑了随机缺陷系统在双轴压缩作用下的屈曲行为. 发现缺陷幅值的增加会显著降低系统的屈曲临界载荷和后屈曲平衡路径. 当缺陷幅值过大时，系统对缺陷的敏感度增加，导致系统不会出现二次分岔现象，其初始形貌对后屈曲形貌的影响占主导作用.
（3）研究发现含随机缺陷的系统在自由边界条件下的失稳行为相对于简支边界条件更为敏感. 当系统存在随机缺陷时，其失稳特征变得更加复杂，可能表现出两种不同的失稳特征，或者仅出现分岔失稳的情况.
（4）微小的几何缺陷可以导致系统的临界荷载和失稳形貌发生显著变化，这使得实验测试结果低于经典理论值. 该研究为理解含随机缺陷的薄膜/衬底系统的屈曲行为提供了重要见解，并为今后的研究工作和工程应用提供了有价值的参考.
基金
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  • 长沙理工大学桥梁与隧道工程创新性项目(11ZDXK11)
  • 湖南省研究生科研创新项目(CSLGCX23146)
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2024年第45卷第5期
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doi: 10.19636/j.cnki.cjsm42-1250/o3.2024.021
  • 接收时间：2024-05-23
  • 首发时间：2026-04-01
  • 出版时间：2024-10-25
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  • 收稿日期：2024-05-23
基金
长沙理工大学桥梁与隧道工程创新性项目(11ZDXK11)
湖南省研究生科研创新项目(CSLGCX23146)
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    长沙理工大学土木工程学院力学系，长沙，410114

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2种不同金属材料的力学参数
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Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
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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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