内聚力模型的分析及有限元子程序开发
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摘要
内聚力模型是弹塑性断裂力学中一个被广泛应用与研究的计算模型。内聚力模型避免了线弹性断裂力学中的裂纹尖端应力奇异性,计算了开裂过程中应力以及断裂能。内聚力模型提出在裂纹尖端存在微小的内聚力区域,该区域内应力为开裂位移值的函数,即为张力位移关系。在对内聚力模型不断研究中,发展出了不同张力位移关系,使其能被广泛应用于各种韧性开裂、复合材料界面脱层以及粘接界面开裂等研究中。
指数内聚力模型具有非线性连续的张力位移关系,考虑了开裂过程中各向位移对应力的耦合关系。研究证实,只有在其参数q等于1的条件下,各向应力关系才能完全耦合。本文提出了指数模型单向断裂能计算公式,在各种参数条件下进行计算测试。其结果表明,在复合开裂条件下,当任一向应力首先减小为零时,其对应的单向断裂能亦同时达到其临界的最大值。
有限元软件ABAQUS中内聚力单元仅包含双线性内聚力模型,其他类型内聚力模型应用于ABAQUS时需通过子程序编译来实现。本文基于内聚力单元,提出采用自定义材料子程序VUMAT编译内聚力模型。推导了张力位移关系在VUMAT中的应用方法,并分别考虑线性与非线性内聚力模型,设计了内聚力模型VUMAT子程序的程序结构。
编译单向开裂条件下指数内聚力模型VUMAT子程序,并与内聚力单元包含的双线性内聚力模型计算结果进行比较,证明其能准确模拟界面单元的承载以及开裂失效过程。在对指数模型耦合关系研究的基础上,采用单向断裂能作为单元失效的判据,编译复合开裂条件下VUMAT子程序。将其应用于复合材料层板脱层模拟,与实验测试值进行比较,结果表明,通过设置合理的模型参数,内聚力模型VUMAT子程序能准确模拟计算裂纹的开裂萌生与扩展过程。
本文提出内聚力模型VUMAT子程序方法,能为有限元分析中提供更丰富的内聚力模型,并避免了其他子程序方法中计算收敛问题；通过提出单向断裂能公式,拓展了指数模型在复合开裂模式中应用范围,在此基础上编译其VUMAT子程序,实现了在ABAQUS中应用非线性的指数内聚力模型计算界面开裂过程。
感谢国家自然基金对本文研究工作的资助,本文研究课题作为国家自然基金项目各向异性导电胶膜粘接可靠性及界面损伤研究的子课题,编译的指数模型VUMAT子程序以及基于内聚力单元编译内聚力模型的VUMAT子程序方法,将进一步应用到其粘接界面及损伤的界面模型研究中。
Cohesive zone model (CZM) is a computational model based on elastic-plastic fracture mechanics. CZM has been extensively researched and used because the stress singularity at crack tip in linear elastic fracture mechanics can be avoided and both the crack interface stress and fracture energy are taken in calculation. A tiny cohesive zone near the crack tip region has been proposed in CZM. The stress in cohesive zone is a function of the crack opening displacement, and this function is usually called traction-separate law (T-S law). As the research and development of CZM, different T-S laws have been proposed, and have been used in ductile cracking and two-phase material interface cracking.
The relationship of stress and displacement is a continuous function and coupling of different dimensionality in exponential cohesive zone model. Considering the exponential CZM in two-dimensional plane model, the complete coupling of normal and shear stress can be achieved only when the parameter q equal to 1. The one-dimensional fracture energy formulation has been proposed in this paper. In the mixed cracking mode, the fracture energy calculating by one-dimensional fracture energy formulation reaches its critical maximum value while the stress reduced to zero first in this dimensional.
The cohesive element in finite element software ABAQUS has been studied, and the user-define subroutine which can be applied for CZM has also been analyzed. Based on the research, user-defined material subroutine VUMAT combined with cohesive element has been taken to compile the exponential CZM implemented in ABAQUS. The application of T-S law in VUMAT has been derived. By the analysis of damage and fracture of the linear and nonlinear CZM, the program flow of VUMAT for CZM has been designed respectively.
The VUMAT of exponential CZM for the pure cracking mode has been complied. The results calculated by VUMAT subroutine of exponential CZM and bilinear CZM contained in cohesive element have been compared. It has confirmed that the crack initiation and evolution can be accurately simulated by VUMAT subroutine. By the analysis of stress coupling and fracture energy, one-dimensional fracture energy has been taken as the failure criterion of cohesive element in the mixed mode. The VUMAT of exponential CZM for mixed mode has been used to simulate the delamination of composite laminates, and has been compared to the experimental test. The result showed that, the VUMAT can accurately simulated crack initiation and propagation process.
The VUMAT subroutine of CZM developed in this paper is applicable for various forms of CZM in finite element analysis, and the convergence problem has been solved in the calculation of VUMAT. The application of exponential CZM for mixed mode has been expanded by one-dimensional fracture energy formulation. By the VUMAT subroutine of exponential CZM, Non-linear CZM can be applied to solve the crack opening simulation in ABAQUS. As a part of National Natural Science Foundation projects (the study of the bonding reliability of anisotropic conductive film and the interface damage), the VUMAT subroutine of exponential CZM and compiling method of VUMAT subroutine combined with cohesive element will be applied in further research of interface model for the study of bonding interface and interface damage.
指数内聚力模型具有非线性连续的张力位移关系,考虑了开裂过程中各向位移对应力的耦合关系。研究证实,只有在其参数q等于1的条件下,各向应力关系才能完全耦合。本文提出了指数模型单向断裂能计算公式,在各种参数条件下进行计算测试。其结果表明,在复合开裂条件下,当任一向应力首先减小为零时,其对应的单向断裂能亦同时达到其临界的最大值。
有限元软件ABAQUS中内聚力单元仅包含双线性内聚力模型,其他类型内聚力模型应用于ABAQUS时需通过子程序编译来实现。本文基于内聚力单元,提出采用自定义材料子程序VUMAT编译内聚力模型。推导了张力位移关系在VUMAT中的应用方法,并分别考虑线性与非线性内聚力模型,设计了内聚力模型VUMAT子程序的程序结构。
编译单向开裂条件下指数内聚力模型VUMAT子程序,并与内聚力单元包含的双线性内聚力模型计算结果进行比较,证明其能准确模拟界面单元的承载以及开裂失效过程。在对指数模型耦合关系研究的基础上,采用单向断裂能作为单元失效的判据,编译复合开裂条件下VUMAT子程序。将其应用于复合材料层板脱层模拟,与实验测试值进行比较,结果表明,通过设置合理的模型参数,内聚力模型VUMAT子程序能准确模拟计算裂纹的开裂萌生与扩展过程。
本文提出内聚力模型VUMAT子程序方法,能为有限元分析中提供更丰富的内聚力模型,并避免了其他子程序方法中计算收敛问题；通过提出单向断裂能公式,拓展了指数模型在复合开裂模式中应用范围,在此基础上编译其VUMAT子程序,实现了在ABAQUS中应用非线性的指数内聚力模型计算界面开裂过程。
感谢国家自然基金对本文研究工作的资助,本文研究课题作为国家自然基金项目各向异性导电胶膜粘接可靠性及界面损伤研究的子课题,编译的指数模型VUMAT子程序以及基于内聚力单元编译内聚力模型的VUMAT子程序方法,将进一步应用到其粘接界面及损伤的界面模型研究中。
Cohesive zone model (CZM) is a computational model based on elastic-plastic fracture mechanics. CZM has been extensively researched and used because the stress singularity at crack tip in linear elastic fracture mechanics can be avoided and both the crack interface stress and fracture energy are taken in calculation. A tiny cohesive zone near the crack tip region has been proposed in CZM. The stress in cohesive zone is a function of the crack opening displacement, and this function is usually called traction-separate law (T-S law). As the research and development of CZM, different T-S laws have been proposed, and have been used in ductile cracking and two-phase material interface cracking.
The relationship of stress and displacement is a continuous function and coupling of different dimensionality in exponential cohesive zone model. Considering the exponential CZM in two-dimensional plane model, the complete coupling of normal and shear stress can be achieved only when the parameter q equal to 1. The one-dimensional fracture energy formulation has been proposed in this paper. In the mixed cracking mode, the fracture energy calculating by one-dimensional fracture energy formulation reaches its critical maximum value while the stress reduced to zero first in this dimensional.
The cohesive element in finite element software ABAQUS has been studied, and the user-define subroutine which can be applied for CZM has also been analyzed. Based on the research, user-defined material subroutine VUMAT combined with cohesive element has been taken to compile the exponential CZM implemented in ABAQUS. The application of T-S law in VUMAT has been derived. By the analysis of damage and fracture of the linear and nonlinear CZM, the program flow of VUMAT for CZM has been designed respectively.
The VUMAT of exponential CZM for the pure cracking mode has been complied. The results calculated by VUMAT subroutine of exponential CZM and bilinear CZM contained in cohesive element have been compared. It has confirmed that the crack initiation and evolution can be accurately simulated by VUMAT subroutine. By the analysis of stress coupling and fracture energy, one-dimensional fracture energy has been taken as the failure criterion of cohesive element in the mixed mode. The VUMAT of exponential CZM for mixed mode has been used to simulate the delamination of composite laminates, and has been compared to the experimental test. The result showed that, the VUMAT can accurately simulated crack initiation and propagation process.
The VUMAT subroutine of CZM developed in this paper is applicable for various forms of CZM in finite element analysis, and the convergence problem has been solved in the calculation of VUMAT. The application of exponential CZM for mixed mode has been expanded by one-dimensional fracture energy formulation. By the VUMAT subroutine of exponential CZM, Non-linear CZM can be applied to solve the crack opening simulation in ABAQUS. As a part of National Natural Science Foundation projects (the study of the bonding reliability of anisotropic conductive film and the interface damage), the VUMAT subroutine of exponential CZM and compiling method of VUMAT subroutine combined with cohesive element will be applied in further research of interface model for the study of bonding interface and interface damage.
引文
[1]范天佑.断裂力学基础[M].北京：科学出版社,2003.189～197
[2]康颖安.断裂力学的发展与研究现状[J].湖南工程学院学报.2006.16(1)：39～42
[3]王自强,陈少华.高等断裂力学[M].北京：科学出版社,2009.14～23
[4]M. Elices G. V. Guinea, J. Gomez. The cohesive zone model:advantages, limitations and challenges [J]. Engineering Fracture Mechanics.2002.69(2): 137-163
[5]刘红霞.复合材料分层损伤的数值模拟[D].[硕士学位论文].西北工业大学.2006
[6]虞青俊.内聚单元模型在混凝土断裂数值模拟中的应用[D].[硕士学位论文].西北工业大学.2003
[7]吴治辉.韧性材料损伤断裂过程的数值模拟[D].[硕士学位论文].西北工业大学.2006
[8]解德,钱勤,李长安.断裂力学中的数值计算方法及工程应用[M].北京：科学出版社,2009.7～10
[9]陈普会,柴亚南.整体复合材料结构失效分析的粘聚区模型[J].南京航空航天大学学报[J].2008.40(4)：442～446
[10]张军,陈旭.粘接界面本构模型研究进展[J].科技导报.2007.25(13)：75～79
[11]K. Y. Volokh. Comparison between cohesive zone models[J]. Communications In Numerical Methods In Engineering.2004.20:845-856
[12]Giulio Alfano. On the influence of the shape of the interface law on the application of cohesive-zone models[J]. Composites Science and Technology.2006.66:723-730
[13]MI Y, CRISFIELD M A, DAVIES G A O. Progressive delamination composite using interface elements [J]. Journal of composite materials.1998.32:1246-1272.
[14]Geubelle, P.H., Baylor, J., Impact-induced delamination of laminated composites: a 2D simulation [J]. Composites Part B Engineering,1998,29(5):589-602.
[15]Tvergaard, V, Hutchinson, J.W. The relation between crack growth resistance and fracture process parameters in elastic-plastic solids [J]. Journal of the Mechanics and Physics of Solids,1992,40(6):1377-1397
[16]Needleman A. Micromechanical modeling of interfacial decohesion. Ultramicroscopy [J].1992.40(3):203-214
[17]赵海峰.反分析确定金属薄膜与陶瓷间界面的力学性能参数[J].工程力学.2008.25(10)：80～84
[18]N. Chandra a, H. Li a. Some issues in the application of cohesive zone models for metal-ceramic interfaces[J]. International Journal of Solids and Structures.2002.39(2002):2827-2855
[19]Kent Salomonsson. Mixed mode modeling of a thin adhesive layer using a meso-mechanical model[J]. Mechanics of Materials.2008.40 (2008):665-672
[20]赵宁,欧阳海彬.内聚力模型在结构胶接强度分析中的应用[J].现代制造工程.2009.11：128～131
[21]周储伟,杨卫.内聚力界面单元与复合材料的界面损伤分析[J].力学学报,1999,31(3)：372～377
[22]张彦.纤维增强复合材料层合结构冲击损伤预测研究[D].[博士学位论文].上海交通大学机械与动力工程学院.2007
[23]闫亚宾,尚福林.PZT薄膜界面分层破坏的内聚力模拟[J].中国科学,2009,39(7)：1007～1017
[24]Rene de Borst. Numerical aspects of cohesive-zone models[J]. Engineering Fracture Mechanics.2003.70(2003):1743-1757
[25]De Xie, Anthony M. Waas. Discrete cohesive zone model for mixed-mode fractureusing finite element analysis[J]. Engineering Fracture Mechanics.2006, 73(2006):1783-1796
[26]Needleman, A. An analysis of tensile decohesion along an interface. Journal of the Mechanics and Physics of Solids [J].1990.38:289-324.
[27]X.P. XU, A. NEEDLEMAN. NUMERICAL SIMULATIONS OF FAST CRACK GROWTH IN BRITTLE SOLIDS [J]. J. Mech. Plys. Solid..1994.42(9):1397-1434,
[28]M.J. van den Bosch. P.J.G. Schreurs. An improved description of the exponential Xu and Needleman cohesive zone law for mixed mode decohesion[J]. Engineering Fracture Mechanics.2006.73(2006):1220-1234
[29]Y F Gao, A F Bower. A simple technique for avoiding convergence problems in finite element simulations of crack nucleation and growth on cohesive interfaces[J]. Modelling Simul. Mater. Sci. Eng.2004.12(2004):453-463
[30]Abdul.Baqi A, Van der Giessen E. Numerical analysis of indentation induced cracking of brittle coatings on ductile substrates[J]. Int J Solids Struct 2002.39: 1427-1442
[31]Hattiangadi A, Siegmund T.. An analysis of the delamination of an environmental protection coating under cyclic heat loads[J]. Eur JMech Solids.2005.24:361-370.
[32]张军,陈旭,贾宏.粘接界面的损伤研究[J].郑州大学学报(工学版).2006.27(2)：48～51
[33]Seong Hyeok Song, Glaucio H. Paulino. A bilinear cohesive zone model tailored for fracture of asphalt concrete considering viscoelastic bulk material[J]. Engineering Fracture Mechanics.2006.73(18):2829-2848
[34]Y. Wang, J. Chen, H.B. Li. Improved Cohesive Zone Model and Its Application In Interface Contact Analysis[J]. Acta Metall. Sin.(Engl. Lett.).2008.21(4): 295-302
[35]Yan Zhang, Ping Zhu, Xinmin Lai. Finite element analysis of low-velocity impact damage in composite laminated plates[J]. Materials and Design.2006.27:513-519
[36]Yong WANG, Jun CHEN, Bing-tao TANG. Finite element analysis for delamination of laminated vibration damping steel sheet[J]. Transactions of Nonferrous Metals Society of China.2007.17(3):455-460
[37]杨曼娟.ABAQUS用户材料子程序开发及应用[D].[硕士学位论文].华中科技大学.2005
[38]张全福,王里青.基于ABAQUS的混凝土材料VUMAT开发与应用[G].中国建筑学会主编.第二十届全国高层建筑结构学术交流会论文集.大连：中国建筑科学研究院出版社,2008：724-729
[39]Yupeng Li, S. Sridharan. Performance of two distinct cohesive layer models for tracking composite delamination[J]. International Journal of Fracture.2005.136: 99-131
[40]L. Hamitouche, M. Tarfaoui. An interface debonding law subject to viscous regularization for avoiding instability:Application to the delamination problems [J]. Engineering Fracture Mechanics.2008.75:3084-3100
[41]C. Balzani, W. Wagner. An interface element for the simulation of delamination in unidirectional fiber-reinforced composite laminates[J]. Engineering Fracture Mechanics.2008.75:2597-2615
[42]张彦,朱平,来新民,等.低速冲击作用下碳纤维复合材料铺层板的损伤分析[J].复合材料学报.2006.3(20)：150～157
[43]Alfano G, Crisfield MA. Solution strategies for the delamination analysis based on a combination of local-control arc length and line searches[J]. Int J Numer Meth Eng.2003.58(7):999-1048
[44]Camanho P, Davila C. Mixed-mode decohesion finite elements for the simulation of delamination in composite materials[J]. NASA/TM.2002, 2117372002:1-37.
[45]A.M. Elmarakbi, N. Hub, H. Fukunag. Finite element simulation of delamination growth in composite materials using LS-DYNA[J]. Composites Science and Technology.2009.78(2009):150～157
[2]康颖安.断裂力学的发展与研究现状[J].湖南工程学院学报.2006.16(1)：39～42
[3]王自强,陈少华.高等断裂力学[M].北京：科学出版社,2009.14～23
[4]M. Elices G. V. Guinea, J. Gomez. The cohesive zone model:advantages, limitations and challenges [J]. Engineering Fracture Mechanics.2002.69(2): 137-163
[5]刘红霞.复合材料分层损伤的数值模拟[D].[硕士学位论文].西北工业大学.2006
[6]虞青俊.内聚单元模型在混凝土断裂数值模拟中的应用[D].[硕士学位论文].西北工业大学.2003
[7]吴治辉.韧性材料损伤断裂过程的数值模拟[D].[硕士学位论文].西北工业大学.2006
[8]解德,钱勤,李长安.断裂力学中的数值计算方法及工程应用[M].北京：科学出版社,2009.7～10
[9]陈普会,柴亚南.整体复合材料结构失效分析的粘聚区模型[J].南京航空航天大学学报[J].2008.40(4)：442～446
[10]张军,陈旭.粘接界面本构模型研究进展[J].科技导报.2007.25(13)：75～79
[11]K. Y. Volokh. Comparison between cohesive zone models[J]. Communications In Numerical Methods In Engineering.2004.20:845-856
[12]Giulio Alfano. On the influence of the shape of the interface law on the application of cohesive-zone models[J]. Composites Science and Technology.2006.66:723-730
[13]MI Y, CRISFIELD M A, DAVIES G A O. Progressive delamination composite using interface elements [J]. Journal of composite materials.1998.32:1246-1272.
[14]Geubelle, P.H., Baylor, J., Impact-induced delamination of laminated composites: a 2D simulation [J]. Composites Part B Engineering,1998,29(5):589-602.
[15]Tvergaard, V, Hutchinson, J.W. The relation between crack growth resistance and fracture process parameters in elastic-plastic solids [J]. Journal of the Mechanics and Physics of Solids,1992,40(6):1377-1397
[16]Needleman A. Micromechanical modeling of interfacial decohesion. Ultramicroscopy [J].1992.40(3):203-214
[17]赵海峰.反分析确定金属薄膜与陶瓷间界面的力学性能参数[J].工程力学.2008.25(10)：80～84
[18]N. Chandra a, H. Li a. Some issues in the application of cohesive zone models for metal-ceramic interfaces[J]. International Journal of Solids and Structures.2002.39(2002):2827-2855
[19]Kent Salomonsson. Mixed mode modeling of a thin adhesive layer using a meso-mechanical model[J]. Mechanics of Materials.2008.40 (2008):665-672
[20]赵宁,欧阳海彬.内聚力模型在结构胶接强度分析中的应用[J].现代制造工程.2009.11：128～131
[21]周储伟,杨卫.内聚力界面单元与复合材料的界面损伤分析[J].力学学报,1999,31(3)：372～377
[22]张彦.纤维增强复合材料层合结构冲击损伤预测研究[D].[博士学位论文].上海交通大学机械与动力工程学院.2007
[23]闫亚宾,尚福林.PZT薄膜界面分层破坏的内聚力模拟[J].中国科学,2009,39(7)：1007～1017
[24]Rene de Borst. Numerical aspects of cohesive-zone models[J]. Engineering Fracture Mechanics.2003.70(2003):1743-1757
[25]De Xie, Anthony M. Waas. Discrete cohesive zone model for mixed-mode fractureusing finite element analysis[J]. Engineering Fracture Mechanics.2006, 73(2006):1783-1796
[26]Needleman, A. An analysis of tensile decohesion along an interface. Journal of the Mechanics and Physics of Solids [J].1990.38:289-324.
[27]X.P. XU, A. NEEDLEMAN. NUMERICAL SIMULATIONS OF FAST CRACK GROWTH IN BRITTLE SOLIDS [J]. J. Mech. Plys. Solid..1994.42(9):1397-1434,
[28]M.J. van den Bosch. P.J.G. Schreurs. An improved description of the exponential Xu and Needleman cohesive zone law for mixed mode decohesion[J]. Engineering Fracture Mechanics.2006.73(2006):1220-1234
[29]Y F Gao, A F Bower. A simple technique for avoiding convergence problems in finite element simulations of crack nucleation and growth on cohesive interfaces[J]. Modelling Simul. Mater. Sci. Eng.2004.12(2004):453-463
[30]Abdul.Baqi A, Van der Giessen E. Numerical analysis of indentation induced cracking of brittle coatings on ductile substrates[J]. Int J Solids Struct 2002.39: 1427-1442
[31]Hattiangadi A, Siegmund T.. An analysis of the delamination of an environmental protection coating under cyclic heat loads[J]. Eur JMech Solids.2005.24:361-370.
[32]张军,陈旭,贾宏.粘接界面的损伤研究[J].郑州大学学报(工学版).2006.27(2)：48～51
[33]Seong Hyeok Song, Glaucio H. Paulino. A bilinear cohesive zone model tailored for fracture of asphalt concrete considering viscoelastic bulk material[J]. Engineering Fracture Mechanics.2006.73(18):2829-2848
[34]Y. Wang, J. Chen, H.B. Li. Improved Cohesive Zone Model and Its Application In Interface Contact Analysis[J]. Acta Metall. Sin.(Engl. Lett.).2008.21(4): 295-302
[35]Yan Zhang, Ping Zhu, Xinmin Lai. Finite element analysis of low-velocity impact damage in composite laminated plates[J]. Materials and Design.2006.27:513-519
[36]Yong WANG, Jun CHEN, Bing-tao TANG. Finite element analysis for delamination of laminated vibration damping steel sheet[J]. Transactions of Nonferrous Metals Society of China.2007.17(3):455-460
[37]杨曼娟.ABAQUS用户材料子程序开发及应用[D].[硕士学位论文].华中科技大学.2005
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