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Article(id=1241029730112164234, tenantId=1146029695717560320, journalId=1227999626482147330, issueId=1241029724655383270, articleNumber=null, orderNo=null, doi=10.16579/j.issn.1001.9669.2025.02.016, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1678377600000, receivedDateStr=2023-03-10, revisedDate=1683388800000, revisedDateStr=2023-05-07, acceptedDate=null, acceptedDateStr=null, onlineDate=1773814144876, onlineDateStr=2026-03-18, pubDate=1739548800000, pubDateStr=2025-02-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1773814144876, onlineIssueDateStr=2026-03-18, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1773814144876, creator=13701087609, updateTime=1773814144876, updator=13701087609, issue=Issue{id=1241029724655383270, tenantId=1146029695717560320, journalId=1227999626482147330, year='2025', volume='47', issue='2', pageStart='1', pageEnd='158', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1773814143575, creator=13701087609, updateTime=1773840259815, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1241139264176574915, tenantId=1146029695717560320, journalId=1227999626482147330, issueId=1241029724655383270, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1241139264176574916, tenantId=1146029695717560320, journalId=1227999626482147330, issueId=1241029724655383270, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=130, endPage=137, ext={EN=ArticleExt(id=1241029730774864270, articleId=1241029730112164234, tenantId=1146029695717560320, journalId=1227999626482147330, language=EN, title=Fracture behavior study of the Octet-truss lattice structures and structural toughening design, columnId=1241029728077935367, journalTitle=Journal of Mechanical Strength, columnName=Optimization∙Reliability, runingTitle=null, highlight=null, articleAbstract=

Octet-truss lattice structure is one of the preferred materials in engineering field with light, high strength and high toughness properties.Three-point bending experiments were carried out on Ti6Al4V Octet-truss lattice structure with shallow pre-crack, and the fracture process was observed by digital image correlation (DIC) technique.To further study the fracture toughness of Octet-truss lattice structure, the fracture behavior of the structure under three-point bending load was analyzed by finite element method.The mechanical response of the truss member for the structure was characterized by the isotropic elastoplastic mechanical model.The validity of the finite element model was verified by experiments.Based on the elastic-plastic fracture behavior of the structure, the J-integral method was used to calculate the fracture toughness of the Octet-truss lattice structures.The results show that the fracture toughness of the Octet-truss lattice structure increases linearly with the relative density and the square root of the truss length.For the truss member at the crack tip, truss in different planes have different stress states and deformation modes with the same cross-sectional size.According to the failure mode of the structure,the stress state of truss from different planes is balanced by optimizing the proportion of the cross-section size, which can optimize toughness of the lattice structure with same relative density.

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WANG Zhiyong, E-mail：
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Octet-truss点阵结构具有高强、高韧以及轻质化的特征，是工程领域的优选材料之一。对具有浅预制裂纹的Ti6Al4V Octet-truss点阵结构进行了三点弯曲试验，并结合数字图像相关（Digital Image Correlation，DIC）技术对断裂过程进行观测。为了深入研究点阵结构的断裂韧性，针对结构的三点弯曲断裂行为进行有限元分析，利用各向同性弹塑性力学模型表征结构杆件组元的力学响应，试验验证了模型的有效性。基于结构的弹塑性断裂行为，利用J积分法计算Octet-truss点阵结构的断裂韧性。结果表明，Octet-truss点阵结构的断裂韧性与其相对密度、杆件长度的平方根呈正相关。对于裂纹尖端处的杆件组元，不同平面的杆件在相同截面尺寸下具有不同的应力状态和变形模式。针对结构的失效模式，通过优化不同平面杆件之间的截面尺寸比例平衡各杆件的应力状态，实现相同相对密度下的结构增韧。

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王志勇（通信作者），男，1982年生，山西太原人，博士，教授；主要研究方向为断裂与损伤力学；E-mail：。
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何婉青，女，1998年生，湖北襄阳人，硕士研究生；主要研究方向为复合材料断裂力学；E-mail：。

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何婉青，女，1998年生，湖北襄阳人，硕士研究生；主要研究方向为复合材料断裂力学；E-mail：。

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何婉青，女，1998年生，湖北襄阳人，硕士研究生；主要研究方向为复合材料断裂力学；E-mail：。

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displacement-load curve of specimens when l=8.3 mm, figureFileSmall=OzQhRRlNGRxBD7cWxo8wtw==, figureFileBig=Fi8CxUdq2A1PSIyTlLJq4Q==, tableContent=null), ArticleFig(id=1241029747036180487, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029730112164234, language=CN, label=图12, caption=l=8.3 mm时试件的中力点位移-载荷曲线, figureFileSmall=OzQhRRlNGRxBD7cWxo8wtw==, figureFileBig=Fi8CxUdq2A1PSIyTlLJq4Q==, tableContent=null), ArticleFig(id=1241029747162009612, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029730112164234, language=EN, label=Fig.13, caption=Effect of the truss length on the normalized fracture toughness, figureFileSmall=daaqfkPpnY0u8pNSMGG5lA==, figureFileBig=wzTsaEYHD7exdYdukDwyiQ==, tableContent=null), ArticleFig(id=1241029747250090001, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029730112164234, language=CN, label=图13, caption=杆件长度对归一化断裂韧性的影响, figureFileSmall=daaqfkPpnY0u8pNSMGG5lA==, figureFileBig=wzTsaEYHD7exdYdukDwyiQ==, tableContent=null), ArticleFig(id=1241029747333976086, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029730112164234, language=EN, label=Fig.14, caption=Effect of the relative density on the normalized fracture toughness, figureFileSmall=vw23Li/GI3zWvpeLiQ7shg==, figureFileBig=ndxEp6NuRVkVm5Yk7H9rug==, tableContent=null), ArticleFig(id=1241029747459805214, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029730112164234, language=CN, label=图14, caption=相对密度对归一化断裂韧性的影响, figureFileSmall=vw23Li/GI3zWvpeLiQ7shg==, figureFileBig=ndxEp6NuRVkVm5Yk7H9rug==, tableContent=null), ArticleFig(id=1241029747556274209, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029730112164234, language=EN, label=Fig.15, caption=Unit cell size of the model a, model b and model c, figureFileSmall=7tEEbbexdZO6X+dQdeVpeQ==, figureFileBig=Gi1QTKdoo7aYEFHQNP1Xmw==, tableContent=null), 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journalId=1227999626482147330, articleId=1241029730112164234, language=EN, label=Fig.17, caption=Stress nephogram at the pre-crack, figureFileSmall=4uQEPCWyxyPO4dgCMVvI9A==, figureFileBig=NbJjea99UBysejjd7qoSZA==, tableContent=null), ArticleFig(id=1241029747954733114, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029730112164234, language=CN, label=图17, caption=预制裂纹处的应力云图, figureFileSmall=4uQEPCWyxyPO4dgCMVvI9A==, figureFileBig=NbJjea99UBysejjd7qoSZA==, tableContent=null), ArticleFig(id=1241029748034424894, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029730112164234, language=EN, label=Tab.1, caption=

Material parameters of Ti6Al4V

, figureFileSmall=null, figureFileBig=null, tableContent=
密度
Density/(kg/m3)		泊松比
Poisson ratio		弹性模量
Modulus of elasticity/GPa		屈服强度
Yield strength/MPa		断裂应变
Fracture strain
4 5000.351101 0100.14
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Ti6Al4V的材料参数

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密度
Density/(kg/m3)		泊松比
Poisson ratio		弹性模量
Modulus of elasticity/GPa		屈服强度
Yield strength/MPa		断裂应变
Fracture strain
4 5000.351101 0100.14
), ArticleFig(id=1241029748231557191, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029730112164234, language=EN, label=Tab.2, caption=

Model size of Octet-truss lattice structures and the simulation results

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编号
Number		相对密度
Relative density/%		杆件长度
Truss length/mm		极限载荷
Limit load
Pq/N		断裂韧性
Fracture toughness/(MPa·m1/2)
12010.319 70026.8
2208.313 50023.6
3206.27 91021.6
4288.321 50038.7
5128.37 74312.52
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Octet-truss点阵结构的模型尺寸及数值模拟结果

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编号
Number		相对密度
Relative density/%		杆件长度
Truss length/mm		极限载荷
Limit load
Pq/N		断裂韧性
Fracture toughness/(MPa·m1/2)
12010.319 70026.8
2208.313 50023.6
3206.27 91021.6
4288.321 50038.7
5128.37 74312.52
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Octet-truss点阵结构的断裂行为研究及结构增韧设计
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何婉青 1, 2 , 王宇航 1, 2 , 罗伟洪 1, 2 , 王志勇 1, 2
机械强度 | 设计·计算 2025,47(2): 130-137
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机械强度 | 设计·计算 2025, 47(2): 130-137
Octet-truss点阵结构的断裂行为研究及结构增韧设计
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何婉青1, 2 , 王宇航1, 2, 罗伟洪1, 2, 王志勇1, 2
作者信息
  • 1.太原理工大学 航空航天学院 应用力学研究所，太原　030024
  • 2.太原理工大学 材料强度与结构冲击山西省重点实验室，太原　030024

    • 何婉青，女，1998年生，湖北襄阳人，硕士研究生；主要研究方向为复合材料断裂力学；E-mail：。

通讯作者:

王志勇（通信作者），男，1982年生，山西太原人，博士，教授；主要研究方向为断裂与损伤力学；E-mail：。
Fracture behavior study of the Octet-truss lattice structures and structural toughening design
Wanqing HE1, 2 , Yuhang WANG1, 2, Weihong LUO1, 2, Zhiyong WANG1, 2
Affiliations
  • 1.Institute of Applied Mechanics, College of Aeronautics and Astronautics, Taiyuan University of Technology, Taiyuan 030024, China
  • 2.Shanxi Key Laboratory of Material Strength & Structural Impact, Taiyuan University of Technology, Taiyuan 030024, China
出版时间: 2025-02-15 doi: 10.16579/j.issn.1001.9669.2025.02.016
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摘要
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Octet-truss点阵结构具有高强、高韧以及轻质化的特征，是工程领域的优选材料之一。对具有浅预制裂纹的Ti6Al4V Octet-truss点阵结构进行了三点弯曲试验，并结合数字图像相关（Digital Image Correlation，DIC）技术对断裂过程进行观测。为了深入研究点阵结构的断裂韧性，针对结构的三点弯曲断裂行为进行有限元分析，利用各向同性弹塑性力学模型表征结构杆件组元的力学响应，试验验证了模型的有效性。基于结构的弹塑性断裂行为，利用J积分法计算Octet-truss点阵结构的断裂韧性。结果表明，Octet-truss点阵结构的断裂韧性与其相对密度、杆件长度的平方根呈正相关。对于裂纹尖端处的杆件组元，不同平面的杆件在相同截面尺寸下具有不同的应力状态和变形模式。针对结构的失效模式，通过优化不同平面杆件之间的截面尺寸比例平衡各杆件的应力状态，实现相同相对密度下的结构增韧。

Octet-truss点阵结构  /  数字图像相关  /  J积分  /  数值模拟  /  结构增韧设计

Octet-truss lattice structure is one of the preferred materials in engineering field with light, high strength and high toughness properties.Three-point bending experiments were carried out on Ti6Al4V Octet-truss lattice structure with shallow pre-crack, and the fracture process was observed by digital image correlation (DIC) technique.To further study the fracture toughness of Octet-truss lattice structure, the fracture behavior of the structure under three-point bending load was analyzed by finite element method.The mechanical response of the truss member for the structure was characterized by the isotropic elastoplastic mechanical model.The validity of the finite element model was verified by experiments.Based on the elastic-plastic fracture behavior of the structure, the J-integral method was used to calculate the fracture toughness of the Octet-truss lattice structures.The results show that the fracture toughness of the Octet-truss lattice structure increases linearly with the relative density and the square root of the truss length.For the truss member at the crack tip, truss in different planes have different stress states and deformation modes with the same cross-sectional size.According to the failure mode of the structure,the stress state of truss from different planes is balanced by optimizing the proportion of the cross-section size, which can optimize toughness of the lattice structure with same relative density.

Octet-truss lattice structure  /  Digital image correlation  /  J-integral  /  Numerical simulation  /  Structural toughening design
何婉青, 王宇航, 罗伟洪, 王志勇. Octet-truss点阵结构的断裂行为研究及结构增韧设计. 机械强度, 2025 , 47 (2) : 130 -137 . DOI: 10.16579/j.issn.1001.9669.2025.02.016
Wanqing HE, Yuhang WANG, Weihong LUO, Zhiyong WANG. Fracture behavior study of the Octet-truss lattice structures and structural toughening design[J]. Journal of Mechanical Strength, 2025 , 47 (2) : 130 -137 . DOI: 10.16579/j.issn.1001.9669.2025.02.016
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0　引言
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点阵结构是一种轻质、高强、高韧的多孔材料[1-3]，在航空航天，自动化和汽车领域已有了广泛应用。由于材料或结构的断裂韧性与其安全直接相关，所以，进一步开展对点阵结构的断裂特性研究，对于指导实际工程设计有重要意义。不同于连续体，多孔材料的多孔特性使材料的断裂韧性研究充满挑战。迄今为止，已有许多成熟的理论被用于多孔材料的断裂研究。1983年，ASHBY等[4]将多孔蜂窝结构视为等效连续体并提出孔壁的峰值强度到达材料破坏应力时启裂。基于此，GIBSON等[5]282-283讨论了点阵结构的I型断裂韧性与材料屈服强度及相对密度之间的递增关系，并提出了断裂韧性近似公式。以上研究均假定材料发生的是脆性断裂，但在实际工程中，中、低强度材料的断裂更多是屈服断裂，为此必须充分考虑裂纹体的弹塑性行为。J积分法是弹塑性断裂力学的一种基本方法，O'MASTA等[6]271-289利用J积分法计算了单边缺口弯曲（Single Edge Notched Bend，SENB）点阵结构试样的I型断裂韧性。研究发现，结点处的塑性变形引起耗散，导致裂纹发展过程中的试件抗断裂能力适度上升。
有限元模拟是一种研究多孔材料断裂的有效方法，在研究其单胞失效模式、构件应力分布状态等细观力学性能领域有极大优势。学者们建立了各种多孔材料的有限元模型，探究了裂纹几何特征，单胞结构参数等对材料或结构断裂行为的影响。其中，CHOI等[7]利用宏观等效连续体本构模型来模拟开孔碳泡沫在I型裂纹下的断裂行为，并依据单边试件的四点弯曲断裂试验验证了有限元模型的有效性。LEE等[8]建立了梯度碳泡沫的三维细观力学有限元模型，研究了其在I型裂纹下的断裂行为，指出断裂韧性随着裂尖附近相对密度的增大而提高。此外，对于多孔材料的弹塑性断裂，SCHMIDT等[9]基于有限元法中的双线性单轴应力-应变规律描述了细胞壁的弹塑性响应，并利用有限元法中的单元删除方法模拟了胞状结构在小范围屈服条件下的裂纹扩展。
点阵结构的断裂韧性与其单胞构型密切相关[10]，因此进行单胞结构的优化设计是提高点阵结构断裂韧性的有效手段，多种方法已被提出，包括单胞的节点、杆件组元、双相结构、梯度结构以及多层级结构等多角度优化设计[11]。例如，MOESTOPO等[12]通过使用消除固定节点的螺旋编织结构来降低节点应力集中、增强裂纹扩展阻力，提高了晶格结构的断裂韧性。LI等[13]通过改变金属点阵结构杆件形态降低了裂纹尖端附近的应力集中，达到阻碍裂纹扩展、大幅提升断裂韧性的目的。MINH-SON等[14]设计了一种多晶点阵超结构来抑制剪切带的形成及快速扩展，具有更好的裂纹扩展抑制能力。
本文结合试验及数值模拟方法研究了具有浅预制裂纹的Ti6Al4V Octet-truss点阵结构在三点弯曲荷载下I型裂纹的弹塑性断裂行为，并将数字图像相关（Digital Image Correlation，DIC）技术引入到点阵结构的三点弯曲试验中。考虑到结构的弹塑性行为，利用J积分法计算结构的断裂韧性，并考虑了预制裂纹尺寸的影响。针对结构的失效模式，提出一种截面尺寸比例优化设计方法，并利用数值模拟方法阐明了此方法的结构增韧效果，为提升点阵结构的断裂韧性提供了依据。
1　Octet-truss点阵结构试验与模拟
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1.1　Octet-truss点阵结构三点弯曲试验
对含预制裂纹的Octet-truss点阵结构进行三点弯曲断裂试验。试件制备方法为电子束选区熔融技术[15]，基体材料选用Ti6Al4V。为了获得Ti6Al4V的材料参数，使用SHIMADZUAG-25TB/SFL-5AG型万能试验机对5组哑铃型标准试件进行拉伸试验，以50 mm/min的测试速度（恒定应变速率）测试，直至试件发生断裂破坏，材料参数如表1所示。如图1所示，Octet单元晶胞的杆件组元分布在X-Y、Y-Z和X-Z这3个平面，同平面杆件夹角为45°，结点连通度为12。Octet-truss点阵结构是由Octet单元晶胞在三维空间阵列而成的一种点阵结构。试件的相对密度，组元杆件长度l=8.3 mm，试件总长为175.5 mm，宽度W=35.1 mm，厚度B=17.55 mm，宏观尺寸符合ASTM E1820[16]标准，如图2所示。O'MASTA等[6]271-289指出，裂纹发展方向的晶格排列数为3的试件能够在三点弯曲中建立足够的KI场，因此，试件满足试验条件。预制裂纹的尺寸对材料的断裂行为和韧性具有重要的影响，因此点阵结构的裂纹尺寸设置尤为关键。然而目前尚未有明确的标准来确定多孔材料的裂纹尺寸。考虑到Octet-truss点阵结构的单胞构型及尺寸，设置试件的裂纹长度a=1/6W。其中预制裂纹止于单元格的缝隙中，而不是止于结构材料（杆件或结点）中。DUGDALE[17]提出两条从裂尖出发夹角为90°的直线与裂纹左右表面交点之间距离可作为裂尖张开位移（Crack Tip Opening Displacement，CTOD），如图2所示。
本文试件由西安赛隆增材技术股份有限公司利用电子束选区熔融技术制备而成，该技术是一种高精确度的增材制造技术，具有效率高、热应力小等特点，适用于钛合金、钛铝基合金等高性能金属材料的成型制造。成品经过多组力学性能验证，结果证明试件的力学性能稳定，试验结果的重复性高。采用SHIMADZUAG-25TB/SFL-5AG型万能试验机对Ti6Al4V Octet-truss试件进行静态三点弯曲试验。试件跨度S=4W，上、下圆柱压头的直径为10 mm。试件以1.2 mm/min的恒定速率进行测试，直到试件完全失效。在试验方面，相对于传统的接触式传感器，DIC方法具有更高的精度和更广泛的应用范围[18]，因此本文利用DIC方法获取试件的位移-载荷曲线、裂纹扩展路径等断裂特性的相关数据。该DIC测量系统包括摄像机、图像采集卡和计算机。为了获得高对比度的散斑图案，在试件最外表面的杆件上喷涂哑光白漆，其中散斑位于结点处，如图3所示。
1.2　Octet-truss点阵结构的连续体模型
利用有限元方法研究Octet-truss点阵结构在三点弯曲荷载下的断裂行为。ZHANG等[19]的研究表明，对于杆件径长比d/l>0.1的点阵结构，实体单元网格划分明显优于梁单元网格划分，即连续体模型优于框架模型。因此，本文采用C3D4实体单元对点阵结构进行离散化，建立如图4所示的宏观连续体模型。以A点、B点和C点3个参考点为主节点，分别与上、下压头平面上的网格进行运动学耦合，上、下压头直径均为10 mm。B点和C点在所有平移自由度上都受到约束，A点能够在Z方向上移动，直到模型完全失效。
采用各向同性弹塑性模型[20]来描述基体材料的力学响应，材料遵循Mises屈服准则。本文利用单元删除法模拟裂纹的扩展行为[21]，当裂纹尖端支柱的最大应力超过支柱材料屈服强度时单元被删除。韧性损伤材料模型[22]可用于预测材料中由于微裂纹生长和合并而引起的损伤。在此有限元模型中，韧性损伤材料模型中覆盖的塑性损伤参数由基体材料参数确定，断裂应变为0.14，应力三轴度参数为0.33，准静态载荷应变率为0.001 s-1。
1.3　有限元模型的有效性
为了验证模型的有效性，将数值模拟得出的试件中力点位移-载荷及裂纹扩展路径与试验结果进行对比。
图5所示为试验结果与模拟结果的中力点位移-载荷曲线。对比分析可知，位移-载荷曲线趋势基本一致，试验所得峰值荷载为13 000 N，有限元模拟所得峰值荷载为13 800 N，相对误差为不超过10%。图6所示为试验和模拟的试件三点弯裂纹扩展路径对比。由图6可知，试验和模拟的裂纹萌生和裂纹路径基本一致，当试件的中力点位移D=5 mm时，预制裂纹尖端附近存在明显的应力集中，随后，预制裂纹开始发展。随着中力点位移的增加，裂纹在试件中向加载点层级扩展，各个阶段的总体裂纹方向都平行于载荷线方向。
对比图5、图6可知，试验与模拟的位移-载荷曲线及裂纹扩展路径吻合较好，证明该有限元模型能可靠地预测Octet-truss点阵结构在三点弯曲荷载下的断裂行为。
2　结果讨论
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2.1　DIC试验结果
试验中上压头对试件的加载速率为1.2 mm/min，试件的中力点位移如图7所示。为了验证DIC技术在本试验中的准确性和可靠性，对比DIC方法测得的中力点位移曲线与设备定义的中力点位移曲线，结果表明，DIC测得的曲线斜率与设备定义的曲线几乎一致，DIC方法可靠。
图8、图9所示分别为试件的中力点位移（D）-载荷（P）曲线和裂尖张开位移（δ）-载荷（P）曲线。图10为试件在三点弯曲载荷下的横向位移云图。在峰前加载初期，试件的变形响应是弹性的，D和δ均随载荷近乎线性增加，但裂纹尖端区域并未发生损伤。当荷载达到峰值荷载的70% ~ 80%时，结点和杆件对裂缝产生和发展的阻力使曲线进入屈服阶段，直至达到峰值荷载。在试件到达峰值载荷A处后，预制裂纹尖端附近的材料断裂，失效位置为结点。裂纹随后在单胞孔隙中快速扩展，对应于A-B阶段荷载快速下降，同时裂尖张开位移随之增大。如图10所示，裂纹层级扩展，失效位置主要集中在结点及靠近结点的杆件处，裂尖延伸至下一层级的结构材料处。受到结点及杆件对裂纹发展的阻力，D和δ均增大，而载荷减幅不大，如B-C阶段。同理，在C-D阶段试件裂纹在单胞孔隙中快速发展。
如图10所示，裂纹的扩展路径曲折，随着裂尖张开位移的增加，裂纹在试件中向加载点层级扩展，结构呈I型断裂。试样的最大裂缝张开位移为10 mm。由于点阵结构的表面不连续，并不能通过DIC手段体现出结构中的微裂纹演化情况，所以，需要结合数值模拟结果分析结构的细观力学行为。
2.2　数值模拟结果
2.2.1　试样三点弯曲裂纹扩展过程
根据标准ASTM E1820，设计Octet-truss点阵结构的有限元模型的尺寸，具体参数如表2所示。为了探究结构的极限荷载与其相对密度和杆件长度的依赖关系，绘制如图11和如图12的中力点位移-载荷曲线。
模拟结果表明，相同相对密度试件的承载力随杆件长度的增加而增加，相同杆件长度试件的承载力随相对密度的增加而增加。试件加载初期，时曲线斜率几乎一致，而l=8.3 mm时，曲线斜率与相对密度值成正比。载荷继续增加到达峰值载荷70%～80%时，由于预制裂纹尖端应力集中作用，在尖端附近的X-Z平面杆及X-Y平面杆件开始出现离散的单元失效，对应于实体试件的微裂纹发生。A点后，裂纹尖端的X-Z平面杆件应力最先达到εf。虽然仅仅X-Z平面杆件断裂不会导致裂纹向上扩展，但是应力会重新分布，因此连接于同一结点处的其他杆件应力都会相应增加，处于临界断裂应变状态的X-Y平面杆件迅速断裂，故当X-Z平面杆中峰值强度达到εf时即认为启裂发生。在A-B阶段，裂纹在结构空隙中迅速扩展。B-C阶段体现了节点和杆件对裂纹发展的阻力作用，此变形模式与试验结果一致。如图6所示，在整个裂纹扩展过程中，应力集中发生在裂纹尖端处。如图12所示，，l=8.3 mm的Octet-truss点阵结构的模拟结果表明，裂纹尖端处同一结点连接的杆件断裂顺序为：X-Z平面杆件先断裂，X-Y平面杆件随后断裂。在整个加载过程中，Y-Z平面杆件未观测到失效行为。
2.2.2　基于J积分的断裂韧性
材料在裂纹稳定扩展时有一定的塑性变形，研究裂纹扩展判据时应充分考虑塑性变形的影响。定义结构的预制裂纹满足a/W<0.4时为浅裂纹，本文利用J积分方法研究具有浅裂纹试样的断裂韧性。J积分由1个弹性分量Je和1个塑性分量Jp组成，在加载状态下，瞬时J（i）的值由求和得到[6]271-289
对于弹性分量Je（i）[23]，即
式中，EXX为理想Octet晶格的弹性模量。DESHPANDE等[24]研究表明，，E为结构基体材料的弹性模量。层级卸载开始时的Ki的值为[25]
周康年等[26]给出了三点弯曲浅裂纹的J积分解析式，可知
式中，Pi为卸载开始时的载荷值；S、B和W分别为试件的跨度、厚度和高度；试件的因子f（ai/W）取决于当前裂纹长度与试件高度的比值[6]271-289。
对于塑性分量J p(i)[27]，有
式中，Up(i)为塑性部分应变能；ΔP(i)为加载点塑性位移，ΔP(i)=D-Pi[(dP/dδ)](i)；ηp=1.9。最后，KJ(i)=(J(i)EXX)1/2[6]271-289。Octet-truss点阵结构的断裂韧性如表2所示。
图13、图14总结了零裂纹扩展时的I型断裂韧性KIC规律：断裂韧性KIC随着相对密度和杆件长度的增加而增加。对KIC归一化后，重新绘制了这两种依赖关系，在所考虑的相对密度范围内，Octet-truss点阵结构的断裂韧性与其相对密度和杆件长度的平方根都近似符合线性关系。无量纲断裂韧性与相对密度和杆件长度的平方根之间存在斜率一致的线性关系，此无量纲变量为0.43。认为基体材料的断裂应力Rf等于屈服应力Re，基于文献[5]282-283的近似公式，给出Octet-truss点阵结构的断裂公式为
3　结构的增韧设计
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由模拟结果可知，Octet单胞中的不同杆件组元之间应力分布及变形模式不一致。在裂纹尖端附近，连接于同一结点处的其余平面杆件发生断裂时，Y-Z平面杆件并未达到材料的屈服应力。基于此，可以通过优化单元晶胞的不同平面杆件的截面尺寸比例进行单胞的细观杆件力学性能调控，均匀各杆件之间的应力分布状态，从而实现宏观点阵结构的增韧设计。将模型a（试件2）作为对照组，相对密度一致时，减弱Y-Z平面杆件的承载力，增强其余平面杆件的承载力。具体方法如图15所示，模型b和模型c的Y-Z平面杆件半径分别为0.7、0.6 mm，其余平面杆件截面尺寸分别为1.7、1.77 mm。
图16所示为模型a、模型b和模型c的中力点位移-载荷曲线。图17为3种模型的预制裂纹附近杆件的应力云图，其中dY-Z / dX-Y为Y-Z平面和X-Y平面杆件的直径比例。如图16所示，相同密度下，模型b极限载荷高于模型a，相差18%；而模型c的极限载荷低于模型a，相差16%。基于模型的细观杆件力学性能分析其原因，在弹性变形阶段，模型b中的Y-Z平面杆件比模型a中的更快到达峰值应力，而模型c中的Y-Z平面杆件很快就达到整体屈服应力；在X-Y平面杆件失效时，就Y-Z平面杆件而言，模型a中的平面杆件处于弹性变形阶段，模型b中的杆件已到达屈服应力，但未失效，模型c中的Y-Z平面杆件已屈曲失效。即整个变形过程中，模型b有最大的结构效率。模拟结果表明：相同密度下，通过改变结构杆件组元的横截面尺寸可以有效地均匀不同平面杆件的应力分布状态，提升其结构效率达到增韧效果。
4　结论
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基于数值模拟和试验方法研究了具有浅预制裂纹的Ti6Al4V Octet-truss点阵结构在三点弯曲荷载下的断裂行为，并采用J积分方法计算了试件的I型断裂韧性。基于试件的失效模式，提出了一种结构增韧方法。详细结论如下：
1）在三点弯曲试验中引入DIC方法获得了点阵结构裂纹扩展时的裂尖张开位移，证明DIC技术研究多孔点阵结构断裂行为的有效性，降低了试验难度。
2）对点阵结构的I型断裂韧性进行归一化参数拟合，给出了具有浅预制裂纹的Octet-truss点阵结构的近似断裂韧性公式。结果表明，结构的断裂韧性与相其对密度和杆件长度平方根呈正相关。
3） Octet-truss点阵结构的I型失效模式是结点处的杆件断裂，裂纹向上层级扩展。对于具有相同截面尺寸杆件的Octet-truss点阵结构而言，X-Z平面杆件先于X-Y平面杆件发生断裂，而Y-Z平面杆件未被观测到断裂失效行为。
4）相同密度下，合理地减小Y-Z平面杆件的截面尺寸，增大其余平面杆件的截面尺寸，能提高杆件组元的材料效率，优化宏观结构的结构效率和承载能力，实现相同相对密度下的结构增韧。
基金
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  • 国家自然科学基金面上项目(12272257)
  • 山西省基础研究计划项目(202203021211169)
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2025年第47卷第2期
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文章信息
doi: 10.16579/j.issn.1001.9669.2025.02.016
  • 接收时间：2023-03-10
  • 首发时间：2026-03-18
  • 出版时间：2025-02-15
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  • 收稿日期：2023-03-10
  • 修回日期：2023-05-07
基金
National Natural Science Foundation of China Project(12272257)
国家自然科学基金面上项目(12272257)
Basic Research Program of Shanxi Province(202203021211169)
山西省基础研究计划项目(202203021211169)
作者信息
    1.太原理工大学 航空航天学院 应用力学研究所，太原　030024
    2.太原理工大学 材料强度与结构冲击山西省重点实验室，太原　030024

通讯作者:

王志勇（通信作者），男，1982年生，山西太原人，博士，教授；主要研究方向为断裂与损伤力学；E-mail：。
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https://castjournals.cast.org.cn/joweb/jxqd/CN/10.16579/j.issn.1001.9669.2025.02.016
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2种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
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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