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Article(id=1241833157671059531, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1241833154382725178, articleNumber=null, orderNo=null, doi=10.19636/j.cnki.cjsm42-1250/o3.2025.015, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1749398400000, receivedDateStr=2025-06-09, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1774005696932, onlineDateStr=2026-03-20, pubDate=1756224000000, pubDateStr=2025-08-27, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1774005696932, onlineIssueDateStr=2026-03-20, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1774005696932, creator=13701087609, updateTime=1774005696932, updator=13701087609, issue=Issue{id=1241833154382725178, tenantId=1146029695717560320, journalId=1241755870837649424, year='2025', volume='46', issue='4', pageStart='437', pageEnd='570', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1774005696148, creator=13701087609, updateTime=1774005738977, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1241833334083490628, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1241833154382725178, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1241833334087684933, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1241833154382725178, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=462, endPage=472, ext={EN=ArticleExt(id=1241833157926912081, articleId=1241833157671059531, tenantId=1146029695717560320, journalId=1241755870837649424, language=EN, title=Fatigue-Free Calibration Cohesive Zone Model: a Novel Approach for Predicting Interface Crack Growth Rates under Fatigue Loading, columnId=1241831201674171363, journalTitle=Chinese Journal of Solid Mechanics, columnName=Research Papers, runingTitle=null, highlight=null, articleAbstract=

Fatigue failure, recognized as one of the most prevalent failure modes in engineering structures, remains inadequately understood in terms of its fundamental mechanical mechanisms. Existing fatigue crack growth models are highly dependent on experimental fatigue data while lacking a universal theoretical framework. To overcome these limitations, we develop a Fatigue-Free Calibration Cohesive Zone Model (F-free model), which can efficiently predict fatigue crack growth rates without the need for fatigue data. Through the definition of cohesive endurance limit and its associated separation displacement, a cyclic damage increment triggering criterion is established. The concept of conditional yield stress in elastoplastic materials is extended to the framework of the cohesive zone model. The cohesive endurance limit is determined as the intersection point between the actual traction-separation curve and a straight line parallel to its initial linear segment. The proposed F-free model is validated by comparing its simulated fatigue crack growth rates with experimental data from two key test scenarios: interlaminar delamination in composite laminates and face-core debonding in sandwich structures. The prediction range of this model can effectively encompass the experimental observation results, accurately capturing both the crack growth rates and the Paris' exponent values for mode I interfacial fatigue cracking. The applicability of the F-free model is further evaluated. The fatigue crack growth rates of interlaminar delamination in double cantilever beam (DCB) specimens under different cohesive endurance limits are simulated. The results indicate that the F-free model can provide a prediction region for interfacial fatigue crack growth rates and a prediction range for the Paris' exponent between 0.99 and 6.3. The proposed F-free model is applicable for predicting the fatigue crack growth of elastoplastic materials or at ductile fracture interfaces. This advancement provides a novel theoretical framework for fatigue damage analysis, effectively bridging the gap between empirical observations and mechanical modeling. The proposed F-free model is able to significantly improve the computational efficiency of fatigue damage tolerance analysis.

, correspAuthors=Jian Xiong, 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=Jian Xiong, Pengcheng Xue), CN=ArticleExt(id=1241833172510507501, articleId=1241833157671059531, tenantId=1146029695717560320, journalId=1241755870837649424, language=CN, title=免疲劳实验标定内聚力模型：疲劳载荷下界面裂纹扩展速率预估新方法, columnId=1241831201896469478, journalTitle=固体力学学报, columnName=研究论文, runingTitle=null, highlight=null, articleAbstract=

疲劳失效是工程结构中最常见的失效形式，其力学机制尚未得到充分阐释. 现有疲劳裂纹扩展模型高度依赖疲劳实验数据，缺乏普适性理论框架. 本文基于指数型内聚力模型特性，通过定义内聚疲劳极限及其对应的分离位移，构建循环损伤增量触发准则，建立了免疲劳实验标定内聚力模型（Fatigue-Free Calibration Cohesive Zone Model，简称F-free model）. 借鉴弹塑性材料条件屈服应力概念，确定模型的内聚疲劳极限. 理论模型预测结果与复合材料层间分层及夹芯结构面芯脱粘实验数据高度吻合，实验拟合的Paris指数均位于模型预测区间内. 对无量纲疲劳极限参数分析表明，模型可揭示疲劳裂纹扩展速率分布范围，Paris指数预测范围为0.99～6.3，为疲劳损伤分析提供新型理论工具.

, correspAuthors=熊健, authorNote=null, correspAuthorsNote=
**E-mail：.
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Dimensional parameters and interlaminar fracture toughness of composite laminates and sandwich structures

, figureFileSmall=null, figureFileBig=null, tableContent=
试件类型长度（mm）宽度（mm）上子梁/面板厚度（mm）上子梁/面板厚度（mm）芯子厚度（mm）预分层长度（mm）层间断裂韧性（J/m2）
双悬臂梁层合板[30]200252.252.25—30664
双悬臂梁层合板[31]140201.51.5—20 ϕ0（Δa）=64.93+247.15Δa0.23
双悬臂梁夹芯结构[32]1521021.51.59.5201180
单悬臂梁夹芯结构[33]20025.43.83.82050.4722
), ArticleFig(id=1241833177753387661, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241833157671059531, language=CN, label=表1, caption=

复合材料层合板和夹芯结构的尺寸参数与层间断裂韧性

, figureFileSmall=null, figureFileBig=null, tableContent=
试件类型长度（mm）宽度（mm）上子梁/面板厚度（mm）上子梁/面板厚度（mm）芯子厚度（mm）预分层长度（mm）层间断裂韧性（J/m2）
双悬臂梁层合板[30]200252.252.25—30664
双悬臂梁层合板[31]140201.51.5—20 ϕ0（Δa）=64.93+247.15Δa0.23
双悬臂梁夹芯结构[32]1521021.51.59.5201180
单悬臂梁夹芯结构[33]20025.43.83.82050.4722
), ArticleFig(id=1241833177845662351, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241833157671059531, language=EN, label=Table 2, caption=

Dimensionless parameters and Paris' exponent for different values of plastic deformation

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塑性变形量01%5%10%20%30%40%50%60%
δf/δ000.10260.23700.34380.50470.63690.75480.86420.9677
σf/σmax，000.25160.50830.66260.82820.91570.96460.98990.9995
Paris指数0.991.62.22.53.85.26.36.26.0
), ArticleFig(id=1241833177921159826, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241833157671059531, language=CN, label=表2, caption=

不同塑性变形量对应的无量纲参数取值以及Paris指数

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塑性变形量01%5%10%20%30%40%50%60%
δf/δ000.10260.23700.34380.50470.63690.75480.86420.9677
σf/σmax，000.25160.50830.66260.82820.91570.96460.98990.9995
Paris指数0.991.62.22.53.85.26.36.26.0
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免疲劳实验标定内聚力模型：疲劳载荷下界面裂纹扩展速率预估新方法
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熊健 1, 2, ** , 薛鹏程 1, 2
固体力学学报 | 研究论文 2025,46(4): 462-472
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固体力学学报 | 研究论文 2025, 46(4): 462-472
免疲劳实验标定内聚力模型：疲劳载荷下界面裂纹扩展速率预估新方法
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熊健1, 2, ** , 薛鹏程1, 2
作者信息
  • 1哈尔滨工业大学航天学院复合材料与结构研究所，哈尔滨，150001
  • 2哈尔滨工业大学特种环境复合材料技术国家级重点实验室，哈尔滨，150001

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**E-mail：.
Fatigue-Free Calibration Cohesive Zone Model: a Novel Approach for Predicting Interface Crack Growth Rates under Fatigue Loading
Jian Xiong1, 2, ** , Pengcheng Xue1, 2
Affiliations
  • 1Center for Composite Materials and Structures, Harbin Institute of Technology, Harbin, 150001
  • 2National Key Laboratory of Science and Technology on Advanced Composites in Special Environments, Harbin Institute of Technology, Harbin, 150001
出版时间: 2025-08-27 doi: 10.19636/j.cnki.cjsm42-1250/o3.2025.015
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摘要
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疲劳失效是工程结构中最常见的失效形式，其力学机制尚未得到充分阐释. 现有疲劳裂纹扩展模型高度依赖疲劳实验数据，缺乏普适性理论框架. 本文基于指数型内聚力模型特性，通过定义内聚疲劳极限及其对应的分离位移，构建循环损伤增量触发准则，建立了免疲劳实验标定内聚力模型（Fatigue-Free Calibration Cohesive Zone Model，简称F-free model）. 借鉴弹塑性材料条件屈服应力概念，确定模型的内聚疲劳极限. 理论模型预测结果与复合材料层间分层及夹芯结构面芯脱粘实验数据高度吻合，实验拟合的Paris指数均位于模型预测区间内. 对无量纲疲劳极限参数分析表明，模型可揭示疲劳裂纹扩展速率分布范围，Paris指数预测范围为0.99～6.3，为疲劳损伤分析提供新型理论工具.

免疲劳实验标定内聚力模型  /  疲劳裂纹扩展速率  /  疲劳极限  /  Paris定律  /  层合结构

Fatigue failure, recognized as one of the most prevalent failure modes in engineering structures, remains inadequately understood in terms of its fundamental mechanical mechanisms. Existing fatigue crack growth models are highly dependent on experimental fatigue data while lacking a universal theoretical framework. To overcome these limitations, we develop a Fatigue-Free Calibration Cohesive Zone Model (F-free model), which can efficiently predict fatigue crack growth rates without the need for fatigue data. Through the definition of cohesive endurance limit and its associated separation displacement, a cyclic damage increment triggering criterion is established. The concept of conditional yield stress in elastoplastic materials is extended to the framework of the cohesive zone model. The cohesive endurance limit is determined as the intersection point between the actual traction-separation curve and a straight line parallel to its initial linear segment. The proposed F-free model is validated by comparing its simulated fatigue crack growth rates with experimental data from two key test scenarios: interlaminar delamination in composite laminates and face-core debonding in sandwich structures. The prediction range of this model can effectively encompass the experimental observation results, accurately capturing both the crack growth rates and the Paris' exponent values for mode I interfacial fatigue cracking. The applicability of the F-free model is further evaluated. The fatigue crack growth rates of interlaminar delamination in double cantilever beam (DCB) specimens under different cohesive endurance limits are simulated. The results indicate that the F-free model can provide a prediction region for interfacial fatigue crack growth rates and a prediction range for the Paris' exponent between 0.99 and 6.3. The proposed F-free model is applicable for predicting the fatigue crack growth of elastoplastic materials or at ductile fracture interfaces. This advancement provides a novel theoretical framework for fatigue damage analysis, effectively bridging the gap between empirical observations and mechanical modeling. The proposed F-free model is able to significantly improve the computational efficiency of fatigue damage tolerance analysis.

F-free model  /  fatigue crack growth rates  /  endurance limit  /  Paris' law  /  laminated structure
熊健, 薛鹏程. 免疲劳实验标定内聚力模型：疲劳载荷下界面裂纹扩展速率预估新方法. 固体力学学报, 2025 , 46 (4) : 462 -472 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2025.015
Jian Xiong, Pengcheng Xue. Fatigue-Free Calibration Cohesive Zone Model: a Novel Approach for Predicting Interface Crack Growth Rates under Fatigue Loading[J]. Chinese Journal of Solid Mechanics, 2025 , 46 (4) : 462 -472 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2025.015
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0　引言
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疲劳失效是材料最常见的失效模式，占服役期间结构材料失效的80%以上，其内在机制研究与理论建模始终是该领域的核心挑战. 疲劳断裂现象普遍存在于工程结构中，如航空航天领域应用广泛的金属和复合材料板壳结构[1,2]，船舶领域的复合材料夹层结构[3]，风电领域的复合材料叶片[4]等，但当前对疲劳断裂力学机制的理解尚显不足，基于力学理论的建模分析研究较为有限. 目前，疲劳裂纹扩展行为的分析与评估仍高度依赖经验公式[5]及疲劳实验[6]，缺乏普适性理论模型的指导.
过去几十年间，研究者相继提出Paris定律、裂纹尖端张开位移准则及内聚力模型等经验公式与数值分析方法，用于预测材料的疲劳裂纹扩展特性. 其中，Paris定律作为经典理论，通过构建疲劳裂纹扩展速率与应力强度因子或能量释放率的单一幂律关系[7-9]，实现了对疲劳裂纹扩展规律的宏观描述.
学者还提出修正的Paris定律[10]，通过解析公式直接预测疲劳裂纹扩展速率. Sch ö n[11]通过实验数据确定相关输入参数，开发了计算不同模式比和载荷比下Paris定律中常数的方法. 姚辽军等人[12]基于疲劳裂纹扩展相似性原理，结合J积分、能量耗散分析和损伤失效机理分析，提出了新的Paris形式的疲劳分层扩展准则，实现了纤维桥联作用下复合材料Ⅰ型疲劳分层扩展的有效分析和表征. Jia等[13]基于裂纹扩展阻力与驱动力之间的竞争机制，提出了修正的Paris定律，根据疲劳数据拟合经验常数，可预测任意试件深度、初始裂纹长深比和疲劳荷载水平下混凝土的疲劳寿命. Miarka等[14]提出了一种从S-N曲线直接获得Paris常数的方法，通过Paris定律方程的积分重新计算了疲劳失效周数，验证了方法的正确性. Shivakumar等[15]提出了用于复合材料层合结构的疲劳寿命模型，通过疲劳实验预先确定能量释放率阈值. 尽管给出了Paris定律的解析公式，但仍需疲劳实验数据校准公式参数.
由于疲劳实验实施困难且成本高昂，学者开发数值方法模拟疲劳裂纹扩展. 赵翔等[16]提出了一种描述复合型裂纹扩展路径的等效修正模型，引入修正因子，修正了由于裂纹等效过程引起的裂纹角偏差，并将模型嵌入扩展有限元模块，实现了疲劳裂纹扩展的仿真模拟. Sérgio等[17]使用节点释放技术进行了裂纹扩展模拟，提出了一种以累积塑性应变和多孔损伤的平行传播准则，然而所采用的节点释放准则需要实验中测量的裂纹扩展速率来确定. Hosseini等[18]通过单轴循环塑性实验获得节点释放的塑性耗散能临界值，采用节点释放技术提出了理论模型来预测疲劳裂纹扩展速率.
近年来，内聚力模型已广泛应用于材料和结构的疲劳裂纹扩展分析中. 内聚力模型将疲劳裂纹扩展视为内聚区材料的逐渐退化，是预测疲劳裂纹扩展的一种有前景的方法. 内聚力模型的概念首先由Dugdale[19]和Barenblatt[20]提出，随后应用于各种断裂问题，可预测层间分离以及双材料界面的裂纹萌生和扩展. 在传统内聚力模型中，界面材料的损伤仅与施加载荷的大小有关. 因此只能描述界面中裂纹扩展的单调过程. 面对疲劳问题时，界面材料的损伤还应与载荷历史相关. Roe和Siegmund[21]将内聚力模型与连续损伤力学相结合，在内聚力模型中引入损伤变量，考虑循环载荷下内聚牵引力的退化，建立了一个循环内聚力模型. Zhang等[22]提出了一个加载速率相关的内聚力模型，将临界牵引力和断裂韧性建模为分离速率的函数，从而考虑材料损伤和开裂的速率依赖性. 然而，模型中的损伤参数仍需疲劳实验数据进行校准. Wang等[23]通过引入等效分离位移量化沿裂纹路径损伤的不均匀分布，提出了一种基于裂纹闭合实验的循环内聚力模型损伤参数获取方法. Allegri[24]提出了统一的内聚力模型，用于描述准脆性材料中的疲劳驱动裂纹萌生和扩展. 该模型无需额外参数或校准常数，可以直接从S-N曲线中获得疲劳裂纹扩展的Paris定律常数. Leciñana等[25]通过多目标优化开发了一种疲劳参数确定方法，直接从疲劳实验数据中获取疲劳模型的输入参数.
综上所述，当前研究疲劳裂纹扩展的方法和预测模型本质上仍依赖于疲劳实验数据. Paris定律是基于疲劳实验数据的经验拟合公式，难以揭示疲劳裂纹扩展的物理本质. 基于节点释放技术的循环裂纹尖端张开位移方法，若缺乏疲劳实验数据进行参数校准，其预测精度将受到显著影响. 针对循环载荷条件的疲劳内聚力模型同样依赖疲劳实验数据以获取模型输入参数或实施校准. 因此，发展不依赖疲劳实验数据标定的理论预测模型，具有重要的理论价值与工程意义.
本文提出免疲劳实验标定内聚力模型（Fatigue-Free Calibration Cohesive Zone Model，简称F-free model），该模型实现了对疲劳裂纹扩展速率的直接预测，无需通过疲劳实验数据获得模型输入参数或实施校准. 免疲劳实验标定内聚力模型通过定义内聚疲劳极限及其对应的分离位移，构建循环损伤增量触发的物理准则. 进一步借鉴弹塑性材料的条件屈服应力概念，确定模型的内聚疲劳极限. 通过将理论模型预测的疲劳裂纹扩展速率与复合材料层合板层间分层和夹芯结构面芯脱粘实验的测试数据进行系统对比验证，充分证明了免疲劳实验标定内聚力模型的理论可靠性. 本文提出的免疲劳实验标定内聚力模型仅基于准静态测试数据，即可实现对材料或结构疲劳裂纹扩展速率的预测，显著减少了疲劳实验所需的工作量.
1　免疲劳实验标定内聚力模型
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传统断裂力学框架（如线弹性断裂力学理论体系）在分析延性断裂行为、复合材料层间开裂及粘接界面失效问题时，存在裂纹尖端应力场奇异和附近塑性区的不可忽视性. 为克服上述理论缺陷，研究者[19,20]基于弹塑性断裂力学理论框架，提出了内聚力模型. 通过在裂纹尖端引入微小内聚区域，构建裂纹表面牵引力与位移场的本构关系（即张力-位移准则），为复杂断裂行为研究提供了突破性的理论方法.
Volokh和Needleman[26]提出的指数内聚力模型是连续函数且考虑耦合关系的内聚力模型，其张力-位移关系表达式为：
为构建能够捕捉结构有限寿命的疲劳内聚力模型，考虑内聚力性能的退化，在内聚力模型中引入损伤力学概念. 在指数型内聚力模型的张力-位移表达式中引入损伤变量D，其形式如下[21]：
其中，ϕt，0和ϕn，0分别为切向和法向的断裂韧性；ut和un分别表示切向和法向位移间隙；δ0是内聚区特征长度.
根据疲劳内聚力模型[21]，循环载荷和单调载荷引起的损伤演化增量分别为：
其中，σmax，0是未产生损伤时的初始法向内聚强度，σmax是在特定载荷循环次数下的法向内聚强度，其值由σmax=σmax，0（1-D）确定；是总内聚张力，定义为；σf表示内聚疲劳极限；δΣ是使内聚区失效所需的总位移量；和分别表示在一个载荷循环期间的累积位移量及其增量；H代表Heaviside函数.
在疲劳载荷加载过程中，循环损伤增量和单调损伤增量均可能存在，这两个损伤增量中的较大者确定为当前循环的损伤增量，整体累积的损伤变量D为：
在公式（5）和（6）中，损伤增量演化增量同时由Heaviside阶跃函数控制，并在总位移大于特征长度时产生，因此由循环载荷和单调载荷引起的损伤演化增量和同时产生. 然而通过对损伤演化增量公式的详细考察，有如下不等式关系：
公式（9）给出的不等式关系导致单调损伤增量恒大于循环损伤增量，即只有模型中的单调损伤增量被成功触发，而循环损伤增量并未计入疲劳损伤演化增量计算体系中.
不仅如此，根据疲劳损伤的定义，材料在承受低于强度的周期循环应力下，即会产生损伤的累积导致最终的断裂破坏. 因此，对于疲劳载荷下的裂纹扩展问题，采用特征长度δ0作为循环损伤增量的触发判据存在理论缺陷. 此外，公式（5）中的内聚疲劳极限σf需依赖疲劳实验数据进行标定，导致高昂的时间与经济成本.
针对现有研究的局限性，借鉴弹塑性材料的屈服应力概念，当材料所受应力大于屈服应力时，产生不可恢复的塑性变形；低于屈服应力时，材料产生弹性变形，不产生任何损伤. 对比疲劳极限的定义，当经过无穷多次应力循环而不发生破坏时的最大应力值即为疲劳极限. 由于屈服应力与疲劳极限的定义具有相同的物理意义，因此采用屈服应力来定义界面的内聚疲劳极限点（δf，σf），并将内聚疲劳极限所对应的分离位移δf作为循环损伤增量的触发判据. 当一个载荷循环期间的累积位移量小于分离位移δf时，界面不产生损伤. 同时，为使单调损伤增量和循环损伤增量对整体累积损伤变量D的贡献具有相同的权重，损伤增量表达式设置相同的比例基准δΣ.
所提出模型的循环损伤增量和单调损伤增量分别为：
此外，整体累积的损伤变量D乘以一个修正系数k，其推荐值为0.01.
本研究提出采用内聚疲劳极限σf对应的分离位移δf作为控制参量，构建循环损伤增量触发的物理准则，不仅改进了现有模型无法将循环损伤增量计入疲劳损伤演化增量计算体系的根本性问题，并且使模型能够真实反映疲劳损伤累积的物理意义.
由于疲劳内聚力模型呈现非线性指数软化特性，无明确屈服特征点，通常将以发生微量的塑性变形时的应力作为该材料的屈服应力，称为条件屈服应力（proof stress）[27]，本文采用条件屈服应力代替屈服应力，对内聚疲劳极限点（δf，σf）进行定义. 具体通过定义一条张力-位移曲线弹性响应阶段的平行线，按指定的塑性变形量d偏移，条件屈服应力值通过该偏移线（图1中红色平行线）与张力-位移曲线（图1中虚线）的相交点处的两个无量纲参数（δf/δ0和σf/σmax，0）联合确定，如图1所示. 然而对于不同的界面材料体系，并没有明确的塑性变形量d的数值来确定条件屈服应力. 因此，在没有疲劳实验数据进行模型参数标定的情况下，本文模型采用不同取值的塑性变形量d计算疲劳裂纹扩展速率以及Paris指数，以此来评估塑性变形量d对结果的影响.
由内聚力模型的张力-位移曲线图可以看到，如图1所示，当分离位移达到10δ0时，对应的界面张力退化至初始内聚强度的0.1%，可以认为界面破坏，失去承载能力，因此，设定内聚区失效总位移量δΣ为10δ0.
基于上述分析，通过条件屈服应力概念确定了内聚疲劳极限及其相应的分离位移，提出了免疲劳实验标定内聚力模型，其本构关系示意图如图2所示. 当时，内聚区同时产生单调损伤和疲劳损伤，两种损伤增量同时存在，较大者为当前循环的最终损伤增量，卸载路径由当前状态点线性卸载返回张力零点；当时，内聚张力超过了内聚疲劳极限但小于内聚强度，内聚区仅产生疲劳损伤，卸载路径由当前状态点线性卸载返回张力零点；当时，内聚张力处于弹性段，未产生损伤，卸载路径沿弹性段返回原点.
2　结果与讨论
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复合材料层合板和夹芯结构是两类典型的层合结构，在疲劳载荷作用下，结构内部的层间界面易发生分层和脱粘失效. 以这两种典型结构为例，验证所提出内聚力模型的准确性及有效性.
2.1　层合结构理论模型
复合材料层合板由分层界面分为上、下两个子梁，夹芯结构由上、下面板和芯子组成，局部坐标系下建模如图3所示. 层合结构的长度和宽度分别为L和b，上、下子梁/面板的厚度分别为dt、db，夹芯结构芯子厚度为dc.
上、下子梁及面板采用Timoshenko梁理论建模，沿x方向和z方向的中面位移、及中面相对于法向的转角φr（x）表示为：
其中，r=t，b分别表示上、下子梁或面板.
基于小变形假设，结构应变表示为：
面内应力合力、弯矩和剪应力合力表达式为：
其中，A11，B11，D11，A55分别为拉伸、拉弯耦合、弯曲和剪切刚度.
芯子位移场采用扩展高阶夹芯板理论表示，x方向和z方向的位移表示为坐标z的二阶和三阶多项式：
其中，和为轴向和横向方向上的待定函数.
碳纤维复合材料蜂窝和泡沫芯子的本构方程和应力合力如文献[28]和[29]中所述.
层间界面采用所建立的疲劳内聚力模型进行建模，定义层间界面切向位移间隙Δut和法向位移间隙Δun. 相应地，层间界面存在切向张力Tt和法向张力Tn，如图4所示.
基于位移场方程（13）和（16），层间界面位移间隙可表示为：
其中，i=b，c分别表示复合材料层合板的下子梁和夹芯结构的芯子.
基于最小势能原理，推导层合结构平衡方程：
层合结构总应变能表达式如下：
外载荷势能表达式为：
其中，Pt，Nt和Mt分别表示法向和面内集中力载荷及弯矩载荷；δD为狄拉克δ函数；xe为载荷施加位置.
结合最小势能原理方程、本构方程及界面位移间隙方程，推导控制方程，具体表达式见文献[28].
2.2　模型准确性验证
通过对比复合材料层合板和夹芯结构的层间疲劳裂纹扩展速率，验证所提出的免疲劳实验标定内聚力模型. 实验验证所用层合结构尺寸参数和层间断裂韧性如表1所示，表中的ϕ0（Δa）表示层间断裂韧性与裂纹扩展长度的关系函数. 由于复合材料层合板在分层扩展过程中，相邻层间纤维在分层面上存在桥接效应，起到阻裂作用，提高了分层阻力和层间断裂韧性. 纤维桥接区影响分层扩展的速度，可能导致裂纹减速甚至裂纹止裂，故采用层间断裂韧性与裂纹扩展长度的关系函数来考虑纤维桥接的影响.
复合材料层合板疲劳裂纹扩展速率对比如图5所示，免疲劳实验标定内聚力模型预测的疲劳裂纹扩展速率能够覆盖实验数据. 为了评估所提出模型的有效性，分别采用Paris公式对实验数据和理论模型预测数据进行拟合. 如图5（a）所示，当塑性变形量d取值为10%至30%时，理论模型预测数据拟合得到的Paris指数分别为2.5和5.2，实验数据拟合得到的Paris指数为3.9，位于理论模型预测结果之间；在图5（b）中，塑性变形量d取值为10%至40%时，理论模型预测数据拟合得到的Paris指数分别为3.2和10.9，实验数据拟合得到的Paris指数为6.0，实验结果同样位于理论模型预测结果之间；对比结果表明，所提出的免疲劳实验标定内聚力模型给出了疲劳裂纹扩展速率的分布范围以及Paris指数的预测范围，能够有效预测复合材料层合板的疲劳裂纹扩展速率.
夹芯结构面芯脱粘疲劳裂纹扩展速率对比如图6所示，理论模型计算得到的疲劳裂纹扩展速率数据能够包括实验数据，给出了夹芯结构面芯脱粘疲劳裂纹扩展速率的上、下限. 采用Paris公式对实验数据和理论模型预测数据进行拟合. 如图6（a）所示，当塑性变形量d取值为10%至30%时，理论模型预测数据拟合得到的Paris指数分别为2.4和6.1，实验数据拟合得到的Paris指数为3.2，实验Paris指数位于理论模型预测结果之间；在图6（b）中，理论模型预测数据拟合得到的Paris指数分别为2.3和5.5，实验数据拟合得到的Paris指数为5.1，实验Paris指数同样位于理论模型预测结果之间；结果表明，所提出的免疲劳实验标定内聚力模型同样能够有效预测夹芯结构的疲劳裂纹扩展速率及Paris指数，进一步验证了模型的有效性.
用于确定条件屈服应力的塑性变形量d是预测疲劳裂纹扩展速率的关键参数，决定了内聚疲劳极限点（δf，σf）的位置. 在没有疲劳实验数据进行模型参数标定的情况下，采用不同取值的塑性变形量d计算疲劳裂纹扩展速率以及Paris指数. 采用与文献[30]中相同的尺寸参数和力学参数，计算得到的疲劳裂纹扩展速率如图7所示. 不同取值塑性变形量d对应的两个无量纲参数（δf/δ0和σf/σmax，0）值列于表2中. 采用Paris定律拟合的疲劳裂纹扩展速率曲线的Paris指数也列于表2中.
由图7可以发现，疲劳裂纹扩展速率存在上下限，采用不同取值的塑性变形量d可以预测疲劳裂纹扩展速率的分布范围以及Paris指数的范围，对工程实际中疲劳裂纹扩展速率的预测同样具有一定的指导意义. 然而，传统方法或理论模型，如果没有疲劳实验数据的辅助，甚至无法给出预测范围，这也证实了该模型的实际应用价值与工程适用性.
当疲劳极限σf=0时，疲劳裂纹扩展速率无阈值，该曲线为疲劳裂纹扩展速率的上限. 随着塑性变形量d的增加，疲劳裂纹扩展速率曲线斜率增大，当塑性变形量d为40%时达到最大值. 超过该点，即使塑性变形量d继续增加，斜率基本上保持不变. 对于塑性材料或弹塑性材料，Paris指数一般较小，如金属材料一般在2～4之间，聚合物材料在2～6之间甚至更高. 然而，对于脆性材料，Paris指数明显更高，通常超过15[34]. 因此，严格地说，所提出的免疲劳实验标定内聚力模型适用于弹塑性材料或延性断裂界面的疲劳裂纹扩展预测.
3　结论
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本文提出了预测界面疲劳裂纹扩展的免疲劳实验标定内聚力模型，该模型基于内聚力模型框架，创新性地引入了新的疲劳损伤演化机制. 依据弹塑性本构关系的条件屈服应力概念，定义张力-位移曲线弹性阶段的平行线，并按指定塑性变形量进行偏移，将偏移线与张力-位移曲线的交点定义为疲劳极限，进而将疲劳极限及其对应的分离位移参数融入疲劳损伤演化增量计算体系，系统建立了理论模型框架.
为验证模型有效性，本研究通过对比复合材料层合板层间裂纹扩展实验和夹芯结构面芯脱粘疲劳裂纹扩展实验得到的疲劳裂纹扩展速率，充分验证了免疲劳实验标定内聚力模型的理论可靠性. 研究结果表明，该模型预测的疲劳裂纹扩展速率及Paris指数分布范围能够覆盖界面疲劳裂纹扩展的实验数据. 进一步计算不同塑性变形量下双悬臂梁试样的层间疲劳裂纹扩展速率，表明免疲劳实验标定内聚力模型能够预测界面疲劳裂纹扩展速率的分布范围，Paris指数的预测范围为0.99～6.3，揭示了模型对界面疲劳裂纹扩展速率分布规律的准确预测能力.
本文提出的免疲劳实验标定内聚力模型突破了传统模型对疲劳实验数据的依赖，无需疲劳数据或S-N曲线进行模型参数标定，仅依靠准静态测试获得的应变能释放率和特征长度，即可实现疲劳裂纹扩展速率与Paris指数的有效预测. 该模型为弹塑性材料或界面的疲劳裂纹扩展行为评估提供了一种高效、便捷的理论分析工具，显著提升了疲劳损伤容限分析的计算效率和工程适用性.
基金
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  • 国家自然科学基金项目(12472345)
  • 中央高校青年教师科研创新能力支持项目(U40)
文献
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2025年第46卷第4期
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doi: 10.19636/j.cnki.cjsm42-1250/o3.2025.015
  • 接收时间：2025-06-09
  • 首发时间：2026-03-20
  • 出版时间：2025-08-27
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  • 收稿日期：2025-06-09
基金
国家自然科学基金项目(12472345)
中央高校青年教师科研创新能力支持项目(U40)
作者信息
    1哈尔滨工业大学航天学院复合材料与结构研究所，哈尔滨，150001
    2哈尔滨工业大学特种环境复合材料技术国家级重点实验室，哈尔滨，150001

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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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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
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