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Article(id=1241838629316792334, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1241838624241681037, articleNumber=null, orderNo=null, doi=10.19636/j.cnki.cjsm42-1250/o3.2024.051, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1730649600000, receivedDateStr=2024-11-04, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1774007001474, onlineDateStr=2026-03-20, pubDate=1740672000000, pubDateStr=2025-02-28, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1774007001474, onlineIssueDateStr=2026-03-20, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1774007001474, creator=13701087609, updateTime=1774007001474, updator=13701087609, issue=Issue{id=1241838624241681037, tenantId=1146029695717560320, journalId=1241755870837649424, year='2025', volume='46', issue='1', pageStart='1', pageEnd='148', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1774007000265, creator=13701087609, updateTime=1774007104623, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1241839062017966445, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1241838624241681037, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1241839062017966446, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1241838624241681037, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=54, endPage=66, ext={EN=ArticleExt(id=1241838630520557602, articleId=1241838629316792334, tenantId=1146029695717560320, journalId=1241755870837649424, language=EN, title=Finite Element Simulation of Ductile Damage in Notched Specimens Based on a Neural Network Surrogate Model, columnId=1241831201674171363, journalTitle=Chinese Journal of Solid Mechanics, columnName=Research Papers, runingTitle=null, highlight=null, articleAbstract=

Under mechanical loading, metallic materials can fail in various ways, including yielding, fracture, buckling, wear, fatigue, and so on, with fracture being particularly destructive. Ductile fracture, characterized by dimples on the fracture surface, is commonly observed in pure metals and alloys. From the microscopic point of view, the ductile fracture of metals and alloys is closely associated with the nucleation, propagation, and coalescence of voids, influenced by factors such as stress state, void size, void volume fraction, void shape, and temperature. Micromechanics-based models developed for ductile damage considering the void evolution, such as the Gurson model and its extensions, usually presume spherical voids, but creating models that consider realistic void shapes and their evolution presents significant challenges. Moreover, conducting mechanical analyses of ductile failure across specimen and component scales requires addressing cross-scale issues. This study first constructed representative volume element models incorporating isolated voids of different initial shapes. Finite element simulations were carried out based on the representative volume elements by adopting a J2 plasticity model for the matrix, systematically analyzing how initial void shape affects stress-strain responses and ductile damage under triaxial tensile and shear loading conditions. A neural network-based surrogate model was trained with the numerical data generated by the simulations to approximate stress-strain responses and damage evolution. This model effectively predicted how initial void shape influences ductile damage. Subsequently, a user-defined material subroutine was developed and integrated into a commercial finite element code to simulate the impact of initial void shapes on the ductile failure process in notched specimens. Results indicated that a reduced aspect ratio for the voids decreased the damage rate, leading to delayed softening at the specimen level. This work demonstrates the potential of using surrogate models to predict ductile damage involving complex microstructural features.

, correspAuthors=Chao Ling, 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=Qingzhuo Pan, Chao Ling), CN=ArticleExt(id=1241838652439990950, articleId=1241838629316792334, tenantId=1146029695717560320, journalId=1241755870837649424, language=CN, title=基于神经网络代理模型的缺口试样韧性损伤有限元模拟, columnId=1241831201896469478, journalTitle=固体力学学报, columnName=研究论文, runingTitle=null, highlight=null, articleAbstract=

金属材料在外力和环境作用下，可能发生变形、断裂、腐蚀、磨损等不同形式的失效破坏，其中断裂是其中危害性和破坏性最为显著的一种. 韧性断裂是金属中常见的一种断裂形式. 从材料内部来看，金属的韧性断裂与微孔洞的形核、扩展和聚合的过程有关. 这个过程受应力状态、孔洞体积分数、孔洞形状及温度等众多因素的影响. 目前，考虑孔洞演化的韧性损伤模型多采用球形孔洞假设，在理论模型推导中考虑非球形孔洞及其演化对材料韧性损伤过程的影响具有较大难度. 此外，在试样和构件尺度上开展韧性损伤的力学分析需要解决跨尺度关联难题. 针对这些问题，本文建立了含不同初始形状孔洞的弹塑性代表体元模型，通过有限元模拟系统地分析了初始孔洞形状和应力状态对代表体元应力应变响应和韧性损伤演化的影响. 基于代表体元模拟数据和神经网络模型，发展了考虑初始孔洞形状影响的韧性损伤本构代理模型. 在此基础上开发了基于神经网络本构模型的有限元计算程序，并分析了初始孔洞形状对带缺口试样韧性损伤过程的影响.

, correspAuthors=凌超, authorNote=null, correspAuthorsNote=
**E-mail：.
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Keyword(id=1241838653501149900, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838629316792334, language=EN, orderNo=2, keyword=void shape), Keyword(id=1241838653597618894, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838629316792334, language=EN, orderNo=3, keyword=void evolution), Keyword(id=1241838653710865106, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838629316792334, language=EN, orderNo=4, keyword=artificial neural network), Keyword(id=1241838653832499924, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838629316792334, language=CN, orderNo=1, keyword=韧性损伤), Keyword(id=1241838653907997398, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838629316792334, language=CN, orderNo=2, keyword=孔洞形状), Keyword(id=1241838653996077785, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838629316792334, language=CN, orderNo=3, keyword=孔洞演化), Keyword(id=1241838654113518298, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838629316792334, language=CN, orderNo=4, keyword=神经网络)], refs=[Reference(id=1241838658114884365, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838629316792334, doi=null, pmid=null, pmcid=null, year=2010, volume=44, issue=null, pageStart=169, pageEnd=305, url=null, language=null, rfNumber=[1], rfOrder=0, authorNames=Benzerga A A, Leblond J B, journalName=Advances in Applied Mechanics, refType=null, unstructuredReference=Benzerga A A, Leblond J B. 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European Journal of Mechanics-A/Solids, 2017, 61: 433-442., articleTitle=Prediction of shear-dominated ductile fracture in a butterfly specimen using a model of plastic porous solids including void shape effects, refAbstract=null)], funds=[Fund(id=1241838658018415370, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838629316792334, awardId=12002105, language=CN, fundingSource=国家自然科学基金项目(12002105), fundOrder=null, country=null)], companyList=[AuthorCompany(id=1241838652674871979, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838629316792334, xref=null, ext=[AuthorCompanyExt(id=1241838652683260589, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838629316792334, companyId=1241838652674871979, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=School of Science, Harbin Institute of Technology, Shenzhen, 518055), 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language=CN, label=图10, caption= u/L0=0.006时，不同材料本构下试样的εyy云图：（a）弹塑性；（b）w0=1，（c）w0=0.2；（d）w0=5, figureFileSmall=YnA1hWXCF90KfNBKjBJJSg==, figureFileBig=DYotgiOoL8ASdvxWgbg/TQ==, tableContent=null), ArticleFig(id=1241838657338938111, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838629316792334, language=EN, label=Fig.11, caption=Contour plot of local distribution of f at u/L0=0.002 using the surrogate model with (a) w0=1, (b) w0=0.2, and (c) w0=5, figureFileSmall=FVQ2UyglNXQdUY0FD5YI/w==, figureFileBig=NglEk7TchaqirdFJ1m3qZQ==, tableContent=null), ArticleFig(id=1241838657431212800, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838629316792334, language=CN, label=图11, caption= u/L0=0.002时，不同孔洞形状下试样的f云图：（a）w0=1；（b）w0=0.2；（c）w0=5, figureFileSmall=FVQ2UyglNXQdUY0FD5YI/w==, figureFileBig=NglEk7TchaqirdFJ1m3qZQ==, tableContent=null), ArticleFig(id=1241838657506710273, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838629316792334, language=EN, label=Fig.12, caption=Contour plot of local distribution of f at u/L0=0.006 using the surrogate model with (a) w0=1, (b) w0=0.2, and (c) w0=5, figureFileSmall=AzehPSX3J+LnW6edqWeIUw==, figureFileBig=tG0GwZTiXcH1da/DVW6mGw==, tableContent=null), ArticleFig(id=1241838657590596354, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838629316792334, language=CN, label=图12, caption= u/L0=0.006时，不同孔洞形状下试样的f云图：（a）w0=1；（b）w0=0.2；（c）w0=5, figureFileSmall=AzehPSX3J+LnW6edqWeIUw==, figureFileBig=tG0GwZTiXcH1da/DVW6mGw==, tableContent=null), ArticleFig(id=1241838657657705219, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838629316792334, language=EN, label=Table 1, caption=

Initial void shape parameters

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初始孔洞形状 w0
圆球1
扁平椭球10.5
扁平椭球20.2
扁长椭球12
扁长椭球25
), ArticleFig(id=1241838657733202693, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838629316792334, language=CN, label=表1, caption=

初始孔洞形状参数

, figureFileSmall=null, figureFileBig=null, tableContent=
初始孔洞形状 w0
圆球1
扁平椭球10.5
扁平椭球20.2
扁长椭球12
扁长椭球25
), ArticleFig(id=1241838657808700167, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1241838629316792334, language=EN, label=Table 2, caption=

Parameters for typical testing cases

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算例编号 w0 r q
11-0.4720.810
22-0.5110.21
30.2-0.3941.753
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典型测试算例的相关参数

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算例编号 w0 r q
11-0.4720.810
22-0.5110.21
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基于神经网络代理模型的缺口试样韧性损伤有限元模拟
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潘清卓 , 凌超 **
固体力学学报 | 研究论文 2025,46(1): 54-66
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固体力学学报 | 研究论文 2025, 46(1): 54-66
基于神经网络代理模型的缺口试样韧性损伤有限元模拟
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潘清卓, 凌超**
作者信息
  • 哈尔滨工业大学（深圳），理学院，深圳，518055

通讯作者:

Finite Element Simulation of Ductile Damage in Notched Specimens Based on a Neural Network Surrogate Model
Qingzhuo Pan, Chao Ling**
Affiliations
  • School of Science, Harbin Institute of Technology, Shenzhen, 518055
出版时间: 2025-02-28 doi: 10.19636/j.cnki.cjsm42-1250/o3.2024.051
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摘要
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金属材料在外力和环境作用下，可能发生变形、断裂、腐蚀、磨损等不同形式的失效破坏，其中断裂是其中危害性和破坏性最为显著的一种. 韧性断裂是金属中常见的一种断裂形式. 从材料内部来看，金属的韧性断裂与微孔洞的形核、扩展和聚合的过程有关. 这个过程受应力状态、孔洞体积分数、孔洞形状及温度等众多因素的影响. 目前，考虑孔洞演化的韧性损伤模型多采用球形孔洞假设，在理论模型推导中考虑非球形孔洞及其演化对材料韧性损伤过程的影响具有较大难度. 此外，在试样和构件尺度上开展韧性损伤的力学分析需要解决跨尺度关联难题. 针对这些问题，本文建立了含不同初始形状孔洞的弹塑性代表体元模型，通过有限元模拟系统地分析了初始孔洞形状和应力状态对代表体元应力应变响应和韧性损伤演化的影响. 基于代表体元模拟数据和神经网络模型，发展了考虑初始孔洞形状影响的韧性损伤本构代理模型. 在此基础上开发了基于神经网络本构模型的有限元计算程序，并分析了初始孔洞形状对带缺口试样韧性损伤过程的影响.

韧性损伤  /  孔洞形状  /  孔洞演化  /  神经网络

Under mechanical loading, metallic materials can fail in various ways, including yielding, fracture, buckling, wear, fatigue, and so on, with fracture being particularly destructive. Ductile fracture, characterized by dimples on the fracture surface, is commonly observed in pure metals and alloys. From the microscopic point of view, the ductile fracture of metals and alloys is closely associated with the nucleation, propagation, and coalescence of voids, influenced by factors such as stress state, void size, void volume fraction, void shape, and temperature. Micromechanics-based models developed for ductile damage considering the void evolution, such as the Gurson model and its extensions, usually presume spherical voids, but creating models that consider realistic void shapes and their evolution presents significant challenges. Moreover, conducting mechanical analyses of ductile failure across specimen and component scales requires addressing cross-scale issues. This study first constructed representative volume element models incorporating isolated voids of different initial shapes. Finite element simulations were carried out based on the representative volume elements by adopting a J2 plasticity model for the matrix, systematically analyzing how initial void shape affects stress-strain responses and ductile damage under triaxial tensile and shear loading conditions. A neural network-based surrogate model was trained with the numerical data generated by the simulations to approximate stress-strain responses and damage evolution. This model effectively predicted how initial void shape influences ductile damage. Subsequently, a user-defined material subroutine was developed and integrated into a commercial finite element code to simulate the impact of initial void shapes on the ductile failure process in notched specimens. Results indicated that a reduced aspect ratio for the voids decreased the damage rate, leading to delayed softening at the specimen level. This work demonstrates the potential of using surrogate models to predict ductile damage involving complex microstructural features.

ductile damage  /  void shape  /  void evolution  /  artificial neural network
潘清卓, 凌超. 基于神经网络代理模型的缺口试样韧性损伤有限元模拟. 固体力学学报, 2025 , 46 (1) : 54 -66 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2024.051
Qingzhuo Pan, Chao Ling. Finite Element Simulation of Ductile Damage in Notched Specimens Based on a Neural Network Surrogate Model[J]. Chinese Journal of Solid Mechanics, 2025 , 46 (1) : 54 -66 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2024.051
正文
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0　引言
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金属材料与人类文明的发展密不可分，凭借其独特的性能，被广泛用于工业领域. 金属结构在外部载荷和环境的作用下，可能发生变形、断裂、腐蚀、磨损等形式的失效破坏，其中断裂的危害性和破坏性最大. 韧性断裂是金属中常见的一种断裂形式. 从金属材料内部来看，韧性断裂是一种由材料中已有或随形变新生成的孔洞扩展直到合并形成连续的断裂轨迹的断裂模式. 通过准确预测孔洞扩展来预测金属韧性断裂对于控制金属损伤演化和设计抗损伤工程材料具有重要意义. 导致材料韧性断裂的主要原因是金属材料内部的孔洞由于颗粒-基体界面的脱粘或颗粒开裂在夹杂物和第二相颗粒处萌生[1,2]，在周围基体塑性变形的驱动下逐渐扩展和聚合，最终形成裂纹[3]. 因此，考虑孔洞演化相关的微观机理的韧性损伤模型，如GTN模型[4-6]及其修正模型[7-9]，在学术和工业界取得了大量关注，广泛用于试样和构件在宏观尺度上的失效行为. 由于材料析出相的形状多样性，圆球孔洞的假设难以满足实际需求，而目前针对初始孔洞形状建立的韧性损伤本构模型，由于细观结构信息的复杂性仍存在一定不足[10]. 近些年来，基于数据和机器学习的建模方法在各个工程领域引起了越来越多的关注，包括固体材料本构和多尺度建模的领域[11-13]. 这为金属材料的韧性失效建模，引入复杂的微结构信息和失效机理，提供了新的视角和方法. 本文为探究孔洞形状对韧性失效的影响，发展了一个数据驱动和基于神经网络的韧性损伤本构代理模型，并对带缺口试样的拉伸行为进行了有限元模拟和分析.
1　含孔洞固体代表性体元和基体的本构模型
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1.1　代表性体元模型
本文所采用的代表体元如图1所示. 代表体元为边长D0=1 mm的立方体，所含孔洞假设为椭球形，其中心与立方体中心重合，主轴分别与坐标系三轴平行，孔洞的三个半主轴初始长度为a1、a2、a3，孔洞的初始体积分数f0由下式得出：
在本文中，f0设为1%，与真实材料中析出相的体积比相当.
根据以往研究，孔洞扩展受多个因素影响，例如外部应力状态、内部微观结构和孔洞特征. 为考虑初始孔洞形状对孔洞扩展的影响，本文建立了圆球形、扁平椭球、扁长椭球三类不同初始孔洞形状的代表体元几何模型，如图2所示. 其中，扁平椭球a2<a1=a3，扁长椭球a2>a1=a3. 参考已有的关于初始孔洞形状影响的研究[14,15]，孔洞纵横比的影响最为显著，因此引入描述孔洞形状的参数纵横比w1=a2/a1和w2=a2/a3，扁平椭球形孔洞和扁长椭球形孔洞均采用了两种不同纵横比，共建立了五种代表体元几何模型，具体数值见表1. 由于本文所建模型的两个纵横比相等，将采用w1=w2=w0一个参数来表征模型的初始孔洞形状.
1.2　基体的本构模型
假设代表体元模型基体材料为各向同性材料，其本构行为用von Mises弹塑性模型进行描述. 本课题采用有限变形弹塑性理论框架，Euler变形率张量D分解为弹性部分De和塑性部分Dp：
Euler塑性变形率张量Dp满足关联流动法则：
其中，p为累计塑性应变，f为屈服函数，σ为柯西应力张量. 这里为累积塑性应变率由下式给出
屈服函数f表示为
其中，σeq为von Mises等效应力，σy为屈服应力. 硬化规律由下式给出：
这里，σ0为初始屈服应力，E为杨氏模量，N为硬化指数. 为满足客观性原理，同时与Abaqus/Standard求解器中实体单元所采用的客观率一致[16]，本文采用Jaumann率建立本构关系，假设柯西应力的Jaumann率与Euler弹性变形率张量De间存在线性关系：
其中，为柯西应力的Jaumann率，C为四阶弹性张量. 对于各向同性材料，C可由杨氏模量E和泊松比ν表示.
本文所采用的模型参数参考Cao等的工作[10]，其中杨氏模量E=210 GPa，泊松比ν=0.3，材料屈服应力σy=835 MPa，硬化指数N=0.05.
1.3　周期性边界条件与宏观应力应变
基于代表性体积元的研究方法假设材料由大量完全相同的代表体元堆积而成[3]. 因此，本文对代表体元采用周期性边界条件，在代表体元的变形过程中，代表体元的外表面应始终保持对面两两平行. 在模型应力加载的三个方向设置位移参考点，建立位移参考点与垂直该方向的两个表面上的节点之间的位移约束来实现.
以x=0和x=D两个表面为例，设置位移参考点1，则该点的位移u1为
其中，e1，e2，e3为笛卡尔坐标系三坐标轴方向的单位矢量，u1x，u1y，u1z分别为点1在三个方向上的位移分量，则对于边界x=0和x=D上的节点有
式中ux，uy，uz为体元左右侧表面沿直角坐标系三个方向的位移场. 其他四个表面上的周期性边界条件，以相似的方式进行施加.
代表体元的宏观变形梯度可由参考点的位移确定，其分量为：
在此引入格林应变张量和右柯西-格林变形张量，两者的关系如下：
其中I为单位矩阵，可由变形梯度求得：
由式（15）可推出的特征主值为[17]
其中为的特征主值，即主应变，为线元伸长比. 体元总体的真应变张量可由下式得到：
其中为的主特征矩阵. 本文引入等效应变来描述代表体元的变形程度，其定义为：
其中，为代表体元总体的Euler变形率张量.
本文使用均匀化方法来表征体元的总体柯西应力张量为：
其中，f为体元孔隙率，Vmatrix为单元总体积，Vtotal为体元总体积.
本文对体元施加了两种不同的载荷. 其一，参照Guo等的研究[18]，进行宏观应力三轴度T和洛德参数L控制的加载. 体元的宏观应力状态由应力三轴度T，和洛德参数L定义为：
其中，为体元总体静水应力，为体元总体von Mises应力，η1和η2为主应力比
本文以y方向为主加载轴，即假设. 本文通过位移加载进行应力分量的控制，进而实现应力三轴度和洛德参数控制的加载. 如图1（a）所示，为了控制应力状态，本文将一弹性杆AB与代表体元连接，杆A端与单元绑定，并在B端施加一定的位移. 根据所设定的应力三轴度和洛德参数值，使用B端在加载过程中沿特定轨迹移动. 由于本文所采用的加载方式，本质上是位移控制的加载，因而适用于代表性体元失稳前后的分析. 该应力控制方法广泛应用于含孔洞代表体元的模拟分析，包括对代表体元的失稳分析，如文献[19，20]等.
其二，为了充分考虑剪切对含孔洞体元的影响，对体元施加如下的位移载荷：
代表体元模拟在达到以下其中一个条件时终止计算：（1）代表体元宏观变形梯度分量；（2）代表体元宏观变形梯度分量或；（3）孔洞内壁产生接触；（4）计算不收敛.
2　基于神经网络的韧性损伤代理模型构建
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参考Guo及其合作者的研究[13,18]，本文所采用的数据驱动韧性损伤预测方法如图3所示. 本文基于代表体元有限元计算结果，使用神经网络构建用以描述含孔洞弹塑性体本构行为的代理模型，具体分为以下几个步骤：
（1）构建神经网络数据库. 本文所构建的本构代理模型可表示为如下映射关系gNN：
其中，X为神经网络模型的输入向量，Y为输出向量. 为了预测含孔洞弹塑性体的应力应变响应和韧性损伤行为，参考Guo等的研究[18]，本文将输入和输出向量表示为
这里，为体元的宏观应变分量，w0为孔洞纵横比；为体元的宏观应力分量，f为体元的孔隙率.
为训练神经网络，本文通过对代表体元施加两种不同载荷，生成了两类数据. 第一类数据，针对应力三轴度T和洛德参数L控制的计算，参考Ibragimova等[11]的工作，对于不同代表体元模拟中主应力比η1和η2的取值，通过Sobol序列来生成500个不同应力状态，如图4所示. 第二类数据，通过Sobol序列在-1≤r≤0，0≤q≤2范围内随机选取500个加载路径组合进行有限元模拟计算. 本文将一个代表体元模拟所获得的结果，称为一组数据. 通过针对5个不同的初始孔洞形状代表体元分别进行1000个载荷条件下有限元分析，获得了代表体元宏观应力应变响应数据，以及体元的孔隙率f的演化数据，共计5000组数据. 根据深度学习研究流程，将体元计算所生成的数据集划分为三个部分：80%的算例作为训练集，10%的算例作为验证集，剩下10%的算例作为测试集. 训练集用于神经网络模型的训练，在不断最小化损失函数的同时更新模型的权重和偏置，根据验证集的结果调整神经网络模型的超参数，最后用测试集来评估最终模型的预测能力.
由于神经网络模型的输入和输出变量的数值范围差异较大，如应力分量和孔隙率变化，需要对数据进行归一化来消除尺度效应，使所有特征具有相似的尺度. 归一化处理也可以加快模型训练的收敛速度，并均匀化不同尺度数据对模型训练的影响. 考虑到本文所用数据集均来自有限元模拟，在此假设数据集中不存在异常数据，因此采用最大最小值归一化方法来实现对部分原始数据的等比例缩放，最大程度的保留同类数据的数量关系. 令Z为处理后的数据变量，对不同分量的具体处理方式如下：
其中min和max分别表示训练集中变量的最小值和最大值，表示所有应力分量.
（2）神经网络模型训练与超参数调整. 本文采用全连接的前馈神经网络模型进行代理模型的建立，以tanh作为隐藏层的激活函数. 使用均方误差（MSE）损失函数，其表达式如下：
其中，Nx为训练样本数，表示第i个样本归一化后的目标值，）表示神经网络对归一化后的输入变量的预测值. 使用代表体元的计算结果对神经网络模型进行训练，并对神经网络模型结构、学习率等超参数进行调整，以获得较好的模型预测结果. 本文采用Adam优化器进行神经网络的训练. 训练过程中，损失函数值随迭代次数的变化如图5所示. 优化所得的神经网络隐藏层的结构为[4-50-100-50-4].
（3）预测代表体元的应力应变响应及损伤变量演化. 神经网络模型训练完成后可用于预测含不同初始孔洞形状孔洞的弹塑性代表体元的应力应变响应及孔隙率变化. 训练好的本构代理模型对典型条件下含孔洞固体的应力应变响应和孔洞演化的预测结果如图5所示，可见代理模型的预测结果和代表体元模拟结果符合较好. 为了进一步将基于神经网络的本构代理模型用于有限元计算，本文参考Gulikers[22]的工作，将训练好的神经网络及其参数编入ABAQUS的用户子程序中. 为此，本文使用EORTRAN语言将训练所得神经网络的前馈计算过程编入UMAT，用以更新应力和状态变量（SDV）孔洞体积比f. 此外，UMAT子程序还需计算单元刚度矩阵（DDSDDE）. 依据Gulikers，获得DDSDDE常见的方法包括常刚度法、切线刚度法、割线刚度法. 为了简单起见，本文采用常刚度法，即以弹性刚度矩阵作为DDSDDE.
3　基于韧性损伤代理模型的缺口试样拉伸试验有限元模拟
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3.1　缺口试样拉伸试验有限元模型
应力状态会影响孔洞的演化行为进而影响韧性断裂行为，在实验中，一般使用带缺口试样来研究应力状态对韧性断裂的影响. 参考Laiarinandrasana等[23]的工作，图6给出了本文有限元模拟所用带圆弧缺口的拉伸试样的主要特征尺寸，总长度为65 mm，宽度为7.2 mm，缺口中心间距为3.5 mm，缺口半径R=4 mm.
由于宏观试样的对称性，只需对四分之一试样进行有限元模拟，建立了如图7所示有限元模型. 采用采用460个二次缩减积分单元（C3D20R）进行网格划分. 对试样左侧和底面施加对称边界，前后两面施加固定边界以实现平面应变状态，在试样右侧面施加位移载荷进行加载.
根据以上有限元模型，试样初始长度L0=32.5 mm，初始截面积S0=0.0875 mm2，当位移加载为u，试样的外载荷由下式计算得到：
其中n为截面处节点总数，表示截面第i个节点所受平行于拉伸方向的支反力.
3.2　有限元模拟结果
图8出了圆球、扁长椭球、扁平椭球（w0=1、5、0.2）三种初始形状孔洞的代理模型和使用含孔洞体元基体相同的弹塑性本构下的E/S0随u/L0变化曲线. 可以看出，对于不同本构，试样载荷在初始阶段均随着位移的增加而增大，当载荷达到最大值之后，试样承载力开始下降. 不同的是，由于von Mises弹塑性本构没有考虑材料损伤导致的承载力下降，因此弹塑性本构的试样载荷仅随位移缓慢下降，而考虑损伤的本构在加载后期载荷迅速下降，这是因为随着加载进行，孔洞发生扩展，材料迅速软化，引起试样失效，与Madi等[24]的实验结果一致.
从不同初始孔洞形状来看，扁长椭球材料本构下的试样能承受更大的载荷，进入快速软化的阶段更晚. 而扁平椭球本构下的试样能承受的最大载荷更小，会更早进入快速软化阶段. 这是因为，相同初始孔隙率条件下，垂直于主加载方向投影面积越小的初始孔洞所能承受的最大应力越大，孔洞聚合发生越晚.
为了更好的对比不同本构的计算结果，图9和图10分别给出了弹性阶段和局部变形阶段不同本构的加载方向应变εyy云图. 对于弹性阶段，不同本构的εyy分布和大小差异很小，对应图8在弹性阶段不同曲线基本一致，孔洞对材料弹性阶段的影响很小. 由于缺口存在，εyy集中于对称截面的边缘位置，离边缘越近的单元等效应变越大. 随着加载的进行，由于泊松效应和截面中心位置的多轴载荷作用，应变集中区域向截面中心转移. 从图11可以看出，在局部变形阶段，不同本构的εyy分布仍比较相似，εyy集中于对称截面的中心位置，考虑孔洞损伤的模型由于材料软化，在应变集中区域的应变更大，且随着w0的减少，影响更加明显，说明w0越小的含椭球孔洞材料的抗变形能力越弱.
为了进一步探究孔洞损伤和孔洞形状对材料的影响，图11和图12给出了弹性阶段和局部变形阶段不同形状下的孔隙率分布云图. 从总体来看，孔隙率分布与εyy非常相似，弹性阶段，不同孔洞形状的孔隙率有差异但是数值差距很小；局部变形阶段，在εyy越大的区域，孔隙率越大，同样集中于截面的中心位置，这与Laiarinandrasana[23]的实验结果相符. 从细观角度看，当体元处于拉伸状态时，起初不同孔洞的孔隙率变化较小且比较接近，随着体元变形增加而不断扩展，表现为孔隙率的快速增加. 对于不同初始形状孔洞，可以看出，在相同的位移载荷下，对于相同区域，扁平椭球孔洞材料试样的孔隙率最大，扁长椭球孔洞材料试样的孔隙率最小，说明在相同外载荷下，w0越小的椭球孔洞孔隙率增长越快.
4　结论
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本文发展了一个基于神经网络的韧性损伤本构代理模型，并对带缺口试样的拉伸行为进行了有限元模拟，研究了孔洞形状对于韧性损伤行为的影响. 本文所得主要结果如下：
（1）通过代表体元模拟设计神经网络训练数据库的算例，可构建能直接用于有限元计算的神经网络韧性损伤本构代理模型. 通过与文献中实验结果进行对比，证明该本构模型可以考虑孔洞损伤引起的材料软化，并可以预测不同区域的孔洞损伤演化.
（2）基于多种椭球形状孔洞体元模拟结果构建的代理模型可以体现初始孔洞形状对材料承载能力和损伤演化的影响. 通过对比不同初始孔洞形状参数下的有限元模拟发现，相同初始孔隙率下，椭球孔洞纵横比越小，能承受的载荷越大，孔隙率增长越慢，进入快速软化的阶段越晚.
由于真实材料中的孔洞形状具有随机性，且在材料变形过程中材料孔洞形状会发生变化，对于本文所构建的代理模型尚可以进行以下几方面的改进：
（1）在使用代表体元生成数据时，依据材料中孔洞的真实形状特征，考虑更多的孔洞形状. 但考虑到理论上孔洞形状有无穷多种可能性，以及数据生成的成本问题，需要对孔洞形状的影响进行进一步的分析，例如可进行聚类分析，以确定充分考虑孔洞形状效应所必需算例.
（2）对于孔洞形状的演化，可在代表体元计算时，提取出孔洞形状的特征参数，如孔洞纵横比w1和w2，随应变的演化数据. 在代理模型的构建时，进而将孔洞形状的特征参数演化数据，作为神经网络输出的一部分. 如此便可以通过训练，使得代理模型具备预测孔洞形状演化的能力.
（3）在使用代理模型进行缺口试样损伤过程的模拟时，可首先给试样赋予随机的孔洞形状，再使用考虑孔洞形状演化的代理模型进行计算. 由于孔洞形状的随机性，在计算结果分析时，可以进一步进行不确定性分析.
此外，从相关文献报道知道，在低应力三轴度和剪切主导的情况中，未考虑孔洞形状效应的GTN韧性损伤模型可能会低估实验试样中韧性损伤[25]；而考虑孔洞形状效应的韧性损伤模型能够更好地预测实验结果. 因此，为了进一步验证本文所提出代理模型对韧性损伤演化行为预测的准确性，相关的实验研究工作正在进行中.
基金
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  • 国家自然科学基金项目(12002105)
文献
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参考文献 引证文献
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doi: 10.19636/j.cnki.cjsm42-1250/o3.2024.051
  • 接收时间：2024-11-04
  • 首发时间：2026-03-20
  • 出版时间：2025-02-28
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  • 收稿日期：2024-11-04
基金
国家自然科学基金项目(12002105)
作者信息
    哈尔滨工业大学（深圳），理学院，深圳，518055

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2种不同金属材料的力学参数
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Number of
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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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