Abstract
The cohesive zone model (CZM) has been used widely and successfully in fracture propagation, but some basic problems are still to be solved. In this paper, artificial compliance and discontinuous force in CZM are investigated. First, theories about the cohesive element (local coordinate system, stiffness matrix, and internal nodal force) are presented. The local coordinate system is defined to obtain local separation; the stiffness matrix for an eight-node cohesive element is derived from the calculation of strain energy; internal nodal force between the cohesive element and bulk element is obtained from the principle of virtual work. Second, the reason for artificial compliance is explained by the effective stiffnesses of zero-thickness and finite-thickness cohesive elements. Based on the effective stiffness, artificial compliance can be completely removed by adjusting the stiffness of the finite-thickness cohesive element. This conclusion is verified from 1D and 3D simulations. Third, three damage evolution methods (monotonically increasing effective separation, damage factor, and both effective separation and damage factor) are analyzed. Under constant unloading and reloading conditions, the monotonically increasing damage factor method without discontinuous force and healing effect is a better choice than the other two methods. The proposed improvements are coded in LS-DYNA user-defined material, and a drop weight tear test verifies the improvements.
摘要
黏结区模型在断裂扩展方面被广泛而成功的应用, 但仍然存在一些基本问题被忽视需要解决. 本文研究了黏结区模型中的人工 柔度和不连续力. 首先, 介绍了关于黏结单元(局部坐标系、刚度矩阵和内部节点力)的理论. 局部坐标系被定义用于获得局部分离. 八 节点黏结单元的刚度矩阵是从应变能的计算中导出的. 黏结单元与体积单元之间的内部节点力是根据虚功原理得出的. 其次, 从零厚 度和有限厚度的黏结单元的有效刚度角度解释了人工柔度的原因. 基于有限厚度黏结单元的有效刚度, 可以通过调整黏结单元的刚度 来完全消除人工柔度. 1D和3D模拟验证了这一结论. 第三, 分析了三种损伤演化方法(单调增加的有效分离、损伤因子以及有效分离和 损伤因子的结合). 在恒定卸载和重新加载条件下, 没有不连续力和愈合效应的单调增加的损伤因子方法显示出比其他两种方法更好的 选择. 所提出的改进方法已经编码到LS-DYNA用户定义材料中, 并且通过落锤撕裂试验获得的模拟结果验证了这些改进.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G. I. Barenblatt, The formation of equilibrium cracks during brittle fracture. General ideas and hypotheses. Axially-symmetric cracks, J. Appl. Math. Mech. 23, 622 (1959).
D. S. Dugdale, Yielding of steel sheets containing slits, J. Mech. Phys. Solids 8, 100 (1960).
L. Meng, and A. Tabiei, An irreversible bilinear cohesive law considering the effects of strain rate and plastic strain and enabling reciprocating load, Eng. Fract. Mech. 252, 107855 (2021).
M. May, O. Hesebeck, S. Marzi, W. Böhme, J. Lienhard, S. Kilchert, M. Brede, and S. Hiermaier, Rate dependent behavior of crash-optimized adhesives–Experimental characterization, model development, and simulation, Eng. Fract. Mech. 133, 112 (2015).
Y. Freed, and L. Banks-Sills, A new cohesive zone model for mixed mode interface fracture in bimaterials, Eng. Fract. Mech. 75, 4583 (2008).
P. F. Liu, and M. M. Islam, A nonlinear cohesive model for mixed-mode delamination of composite laminates, Compos. Struct. 106, 47 (2013).
S. Abrate, J. F. Ferrero, and P. Navarro, Cohesive zone models and impact damage predictions for composite structures, Meccanica 50, 2587 (2015).
N. V. De Carvalho, M. W. Czabaj, and J. G. Ratcliffe, Piecewise-linear generalizable cohesive element approach for simulating mixed-mode delamination, Eng. Fract. Mech. 242, 107484 (2021).
M. F. S. F. de Moura, J. P. M. Gonçalves, and F. G. A. Silva, A new energy based mixed-mode cohesive zone model, Int. J. Solids Struct. 102-103, 112 (2016).
T. Yamaguchi, T. Okabe, and S. Yashiro, Fatigue simulation for titanium/CFRP hybrid laminates using cohesive elements, Compos. Sci. Tech. 69, 1968 (2009).
D. Kumar, R. Roy, J. H. Kweon, and J. Choi, Numerical modeling of combined matrix cracking and delamination in composite laminates using cohesive elements, Appl. Compos. Mater. 23, 397 (2016).
C. Sarrado, F. A. Leone, and A. Turon, Finite-thickness cohesive elements for modeling thick adhesives, Eng. Fract. Mech. 168, 105 (2016).
W. Trawiński, J. Bobiński, and J. Tejchman, Two-dimensional simulations of concrete fracture at aggregate level with cohesive elements based on X-ray uCT images, Eng. Fract. Mech. 168, 204 (2016).
W. Trawiński, J. Tejchman, and J. Bobiński, A three-dimensional meso-scale modelling of concrete fracture, based on cohesive elements and X-ray μCT images, Eng. Fract. Mech. 189, 27 (2018).
K. Park, and G. H. Paulino, Computational implementation of the PPR potential-based cohesive model in ABAQUS: Educational perspective, Eng. Fract. Mech. 93, 239 (2012).
P. Rahul-Kumar, A. Jagota, S. J. Bennison, S. Saigal, and S. Muralidhar, Polymer interfacial fracture simulations using cohesive elements, Acta Mater. 47, 4161 (1999).
B. L. V. Bak, E. Lindgaard, and E. Lund, Analysis of the integration of cohesive elements in regard to utilization of coarse mesh in laminated composite materials, Numer. Meth. Eng. 99, 566 (2014).
A. Tabiei, and L. Meng, Improved cohesive zone model: Integrating strain rate, plastic strain, variable damping, and enhanced constitutive law for fracture propagation, Int. J. Fract 244, 125 (2023).
S. H. Song, G. H. Paulino, and W. G. Buttlar, A bilinear cohesive zone model tailored for fracture of asphalt concrete considering viscoelastic bulk material, Eng. Fract. Mech. 73, 2829 (2006).
N. Blal, L. Daridon, Y. Monerie, and S. Pagano, Micromechanical-based criteria for the calibration of cohesive zone parameters, J. Comput. Appl. Math. 246, 206 (2013).
V. P. Nguyen, Discontinuous Galerkin/extrinsic cohesive zone modeling: Implementation caveats and applications in computational fracture mechanics, Eng. Fract. Mech. 128, 37 (2014).
A. Tabiei, and W. Zhang, Cohesive element approach for dynamic crack propagation: Artificial compliance and mesh dependency, Eng. Fract. Mech. 180, 23 (2017).
W. Zhang, and A. Tabiei, Improvement of an exponential cohesive zone model for fatigue analysis, J. Fail. Anal. Preven. 18, 607 (2018).
V. Tomar, J. Zhai, and M. Zhou, Bounds for element size in a variable stiffness cohesive finite element model, Numer. Meth. Eng. 61, 1894 (2004).
S. Kozinov, M. Kuna, and S. Roth, A cohesive zone model for the electromechanical damage of piezoelectric/ferroelectric materials, Smart Mater. Struct. 23, 055024 (2014).
K. Park, and G. H. Paulino, Cohesive zone models: A critical review of traction-separation relationships across fracture surfaces, Appl. Mech. Rev. 64, 060802 (2011).
W. Gao, J. Xiang, S. Chen, S. Yin, M. Zang, and X. Zheng, Intrinsic cohesive modeling of impact fracture behavior of laminated glass, Mater. Des. 127, 321 (2017).
F. Hirsch, and M. Kästner, Microscale simulation of adhesive and cohesive failure in rough interfaces, Eng. Fract. Mech. 178, 416 (2017).
A. Turon, P. P. Camanho, J. Costa, and J. Renart, Accurate simulation of delamination growth under mixed-mode loading using cohesive elements: Definition of interlaminar strengths and elastic stiffness, Compos. Struct. 92, 1857 (2010).
C. Sarrado, A. Turon, J. Renart, and I. Urresti, Assessment of energy dissipation during mixed-mode delamination growth using cohesive zone models, Compos. Part A-Appl. Sci. Manufact. 43, 2128 (2012).
A. Turon, E. V. González, C. Sarrado, G. Guillamet, and P. Maimí, Accurate simulation of delamination under mixed-mode loading using a cohesive model with a mode-dependent penalty stiffness, Compos. Struct. 184, 506 (2018).
L. A. de Oliveira, and M. V. Donadon, Delamination analysis using cohesive zone model: A discussion on traction-separation law and mixed-mode criteria, Eng. Fract. Mech. 228, 106922 (2020).
N. H. Kim, Introduction to Nonlinear Finite Element Analysis (Springer, New York, 2014).
P. Rahulkumar, A. Jagota, S. J. Bennison, and S. Saigal, Cohesive element modeling of viscoelastic fracture: Application to peel testing of polymers, Int. J. Solids Struct. 37, 1873 (2000).
D. W. Spring, and G. H. Paulino, A growing library of three-dimensional cohesive elements for use in ABAQUS, Eng. Fract. Mech. 126, 190 (2014).
P. A. Klein, J. W. Foulk, E. P. Chen, S. A. Wimmer, and H. J. Gao, Physics-based modeling of brittle fracture: Cohesive formulations and the application ofmeshfree methods, Theor. Appl. Fract. Mech. 37, 99 (2001).
Livermore Software Technology Corporation, LS-DYNA Keyword User’s Manual, Vol. II Material Models (LSTC, Livermore, 2019).
Z. J. Ren, and C. Q. Ru, Numerical investigation of speed dependent dynamic fracture toughness of line pipe steels, Eng. Fract. Mech. 99, 214 (2013).
Author information
Authors and Affiliations
Contributions
Author contributions Ala Tabiei: Software, Data curation, Resources, Writing–review and editing, Supervision, Project administration. Li Meng: Conceptualization, Methodology, Investigation, Software, Data curation, Validation, Writing–original draft, Writing–review and editing.
Corresponding author
Ethics declarations
Conflict of interest On behalf of all authors, the corresponding author states that there is no conflict of interest.
Rights and permissions
About this article
Cite this article
Tabiei, A., Meng, L. Improvements of cohesive zone model on artificial compliance and discontinuous force. Acta Mech. Sin. 40, 423345 (2024). https://doi.org/10.1007/s10409-023-23345-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10409-023-23345-x