Abstract
Accurate prediction of ductile fracture requires determining the material properties, including the parameters of the constitutive and ductile fracture model, which represent the true material response. Conventional calibration of material parameters often relies on a trial-and-error approach, in which the parameters are manually adjusted until the corresponding finite element model results in a response matching the experimental global response. The parameter estimates are often subjective. To address this issue, in this paper we treat the identification of material parameters as an optimization problem and introduce the particle swarm optimization (PSO) algorithm as the optimization approach. We provide material parameters of two uncoupled ductile fracture models—the Rice and Tracey void growth model (RT-VGM) and the micro-mechanical void growth model (MM-VGM), and a coupled model—the Gurson-Tvergaard-Needleman (GTN) model for ASTM A36, A572 Gr. 50, and A992 structural steels using an automated PSO method. By minimizing the difference between the experimental results and finite element simulations of the load-displacement curves for a set of tests of circumferentially notched tensile (CNT) bars, the calibration procedure automatically determines the parameters of the strain hardening law as well as the uncoupled models and the coupled GTN constitutive model. Validation studies show accurate prediction of the load-displacement response and ductile fracture initiation in V-notch specimens, and confirm the PSO algorithm as an effective and robust algorithm for seeking ductile fracture model parameters. PSO has excellent potential for identifying other fracture models (e.g., shear modified GTN) with many parameters that can give rise to more accurate predictions of ductile fracture. Limitations of the PSO algorithm and the current calibrated ductile fracture models are also discussed in this paper.
概要
目的
准确预测延性断裂需要确定材料参数(包括本构参数和延性断裂模型参数), 以反映真实的材料响应. 传统的材料参数标定方法往往依赖于试错法, 需手动调整参数, 直到相应的有限元模型得到与实验结果相匹配的材料力学响应. 参数估计的过程通常是主观的. 为了解决这一问题, 本文将材料断裂参数辨识问题转化为优化问题, 并引入粒子群优化(PSO)算法作为优化方法.
创新点
1. 基于粒子群优化算法, 给出了自动识别钢材应变硬化参数的方法;2. 建立了ASTM结构钢材Gurson-Tvergaard-Needleman(GTN)损伤模型的参数识别方法.
方法
1. 通过圆形缺口杆件的拉伸试验, 以试验和有限元模拟的载荷-位移曲线差值为目标方程, 采用PSO优化算法及参数自动校准程序, 以最小化目标方程确定应变硬化准则和非耦合断裂模型的参数;2. 基于文献调研的结果, 确定GTN模型各参数的合理取值范围, 以此确定PSO算法中参数的取值, 从而能够高效、 准确地确定GTN参数.
结论
1. PSO算法能够准确地预测V形缺口试件的载荷-位移响应和延性断裂萌发, 是一种识别延性断裂模型参数的有效算法;2. PSO在识别其他具有更多参数的断裂模型(如剪切修正GTN模型)方面具有很好的潜力, 这些模型可以更准确地预测延性断裂.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abbasi M, Shafaat MA, Ketabchi M, et al., 2012a. Application of the GTN model to predict the forming limit diagram of IF-steel. Journal of Mechanical Science and Technology, 26(2):345–352. https://doi.org/10.1007/s12206-011-1038-z
Abbasi M, Bagheri B, Ketabchi M, et al., 2012b. Application of response surface methodology to drive GTN model parameters and determine the FLD of tailor welded blank. Computational Materials Science, 53(1):368–376. https://doi.org/10.1016/j.commatsci.2011.08.020
Abbassi F, Mistou S, Zghal A, 2013. Failure analysis based on microvoid growth for sheet metal during uniaxial and biaxial tensile tests. Materials & Design, 49:638–646. https://doi.org/10.1016/j.matdes.2013.02.020
Abendroth M, Kuna M, 2006. Identification of ductile damage and fracture parameters from the small punch test using neural networks. Engineering Fracture Mechanics, 73(6):710–725. https://doi.org/10.1016/j.engfracmech.2005.10.007
Ali AN, Huang SJ, 2019. Ductile fracture behavior of ECAP deformed AZ61 magnesium alloy based on response surface methodology and finite element simulation. Materials Science and Engineering: A, 746:197–210. https://doi.org/10.1016/j.msea.2019.01.036
Amaral R, Teixeira P, Azinpour E, et al., 2016. Evaluation of ductile failure models in sheet metal forming. MATEC Web of Conferences, 80:03004. https://doi.org/10.1051/matecconf/20168003004
Anderson TL, 2017. Fracture Mechanics: Fundamentals and Applications, 3rd Edition. CRC Press, Boca Raton, USA. https://doi.org/10.1201/9781315370293
Benzerga AA, Leblond JB, 2010. Ductile fracture by void growth to coalescence. Advances in Applied Mechanics, 44:169–305. https://doi.org/10.1016/S0065-2156(10)44003-X
Bernauer G, Brocks W, 2002. Micro-mechanical modelling of ductile damage and tearing-results of a European numerical round robin. Fatigue & Fracture of Engineering Materials & Structures, 25(4):363–384. https://doi.org/10.1046/j.1460-2695.2002.00468.x
Bridgman PW, 1964. Studies in Large Plastic Flow and Fracture. Harvard University Press, London, UK. https://doi.org/10.4159/harvard.9780674731349
Brinnel V, Langenberg J, Kordtomeikel F, et al., 2015. Numerical derivation of strain-based criteria for ductile failure: discussions on sensitivity and validity. Engineering Fracture Mechanics, 148:421–440.
Cha WG, Kim N, 2014. Quantification of micro-cracks on the bending surface of roll formed products using the GTN model. Metals and Materials International, 20(5):841–850. https://doi.org/10.1007/s12540-014-5008-8
Chu CC, Needleman A, 1980. Void nucleation effects in biaxially stretched sheets. Journal of Engineering Materials and Technology, 102(3):249–256. https://doi.org/10.1115/1.3224807
Cuesta II, Alegre JM, Lacalle R, 2010. Determination of the Gurson-Tvergaard damage model parameters for simulating small punch tests. Fatigue & Fracture of Engineering Materials & Structures, 33(11):703–713. https://doi.org/10.1111/j.1460-2695.2010.01481.x
Dassault Systèmes, 2016. ABAQUS Analysis User’s Manual. Dassault Systèmes Simulia, Providence, USA.
Defaisse C, Mazière M, Marcin L, et al., 2018. Ductile fracture of an ultra-high strength steel under low to moderate stress triaxiality. Engineering Fracture Mechanics, 194:301–318. https://doi.org/10.1016/j.engfracmech.2017.12.035
Eberhart R, Kennedy J, 1995. A new optimizer using particle swarm theory. Proceedings of the 6th International Symposium on Micro Machine and Human Science, p.39–43. https://doi.org/10.1109/MHS.1995.494215
Eberle A, Klingbeil D, Schicker J, 2000. The calculation of dynamic JR-curves from the finite element analysis of a Charpy test using a rate-dependent damage model. Nuclear Engineering and Design, 198(1–2):75–87. https://doi.org/10.1016/S0029-5493(99)00281-2
Gatea S, Lu B, Ou HG, et al., 2015. Numerical simulation and experimental investigation of ductile fracture in SPIF using modified GTN model. MATEC Web of Conferences, 21:04013. https://doi.org/10.1051/matecconf/20152104013
Gurson AL, 1977. Continuum theory of ductile rupture by void nucleation and growth: part I—yield criteria and flow rules for porous ductile media. Journal of Engineering Materials and Technology, 99(1):2–15. https://doi.org/10.1115/1.3443401
Hancock JW, Brown DK, 1983. On the role of strain and stress state in ductile failure. Journal of the Mechanics and Physics of Solids, 31(1):1–24. https://doi.org/10.1016/0022-5096(83)90017-0
Haušild P, Nedbal I, Berdin C, et al., 2002. The influence of ductile tearing on fracture energy in the ductile-to-brittle transition temperature range. Materials Science and Engineering: A, 335(1–2):164–174. https://doi.org/10.1016/S0921-5093(01)01913-X
Jia LJ, Ikai T, Shinohara K, et al., 2016. Ductile crack initiation and propagation of structural steels under cyclic combined shear and normal stress loading. Construction and Building Materials, 112:69–83. https://doi.org/10.1016/j.conbuildmat.2016.02.171
Kami A, Dariani BM, Vanini AS, et al., 2015. Numerical determination of the forming limit curves of anisotropic sheet metals using GTN damage model. Journal of Materials Processing Technology, 216:472–483. https://doi.org/10.1016/j.jmatprotec.2014.10.017
Kanvinde AM, Deierlein GG, 2006. The void growth model and the stress modified critical strain model to predict ductile fracture in structural steels. Journal of Structural Engineering, 132(12):1907–1918. https://doi.org/10.1061/(ASCE)0733-9445(2006)132:12
Khandelwal K, El-Tawil S, 2007. Collapse behavior of steel special moment resisting frame connections. Journal of Structural Engineering, 133(5):646–655. https://doi.org/10.1061/(ASCE)0733-9445(2007)133:5(646)
Kiran R, Khandelwal K, 2013. A micromechanical model for ductile fracture prediction in ASTM A992 steels. Engineering Fracture Mechanics, 102:101–117. https://doi.org/10.1016/j.engfracmech.2013.02.021
Kiran R, Khandelwal K, 2014a. Experimental studies and models for ductile fracture in ASTM A992 steels at high triaxiality. Journal of Structural Engineering, 140(2):04013044. https://doi.org/10.1061/(ASCE)ST.1943-541X.0000828
Kiran R, Khandelwal K, 2014b. Fast-to-compute weakly coupled ductile fracture model for structural steels. Journal of Structural Engineering, 140(6):04014018. https://doi.org/10.1061/(ASCE)ST.1943-541X.0001025
Kiran R, Khandelwal K, 2014c. Gurson model parameters for ductile fracture simulation in ASTM A992 steels. Fatigue & Fracture of Engineering Materials & Structures, 37(2):171–183. https://doi.org/10.1111/ffe.12097
Kong DY, Yang B, 2020. Enhanced voids growth model for ductile fracture prediction of high-strength steel Q690D under monotonic tension: experiments and numerical simulation. Journal of Structural Engineering, 146(6):04020107. https://doi.org/10.1061/(ASCE)ST.1943-541X.0002658
Kossakowski PG, 2012. Prediction of ductile fracture for S235JR steel using the stress modified critical strain and Gurson-Tvergaard-Needleman models. Journal of Materials in Civil Engineering, 24(12):1492–1500. https://doi.org/10.1061/(ASCE)MT.1943-5533.0000546
Kulawinski D, Iding K, Schornstein R, et al., 2020. Improvement of the inverse finite element analysis approach for tensile and toughness predictions by means of small punch technique. ASME Turbo Expo: Turbomachinery Technical Conference and Exposition. https://doi.org/10.1115/GT2020-16054
Kumar P, Dutta BK, Chattopadhyay J, 2017. Fracture toughness prediction of reactor grade materials using pre-notched small punch test specimens. Journal of Nuclear Materials, 495:351–362. https://doi.org/10.1016/j.jnucmat.2017.08.035
Lemaitre J, 1985. A continuous damage mechanics model for ductile fracture. Journal of Engineering Materials and Technology, 107(1):83–89. https://doi.org/10.1115/1.3225775
Li H, Pan XF, Yuan H, 2015. A nonlocal treatment technique based on the background cell concept for micro-mechanical damage modeling. Acta Mechanica, 226(5):1529–1547. https://doi.org/10.1007/s00707-014-1268-0
Li JC, Li SP, Xie ZY, et al., 2015. Numerical simulation of incremental sheet forming based on GTN damage model. The International Journal of Advanced Manufacturing Technology, 81(9):2053–2065. https://doi.org/10.1007/s00170-015-7333-6
Linse T, Kuna M, Viehrig HW, 2014. Quantification of brittle-ductile failure behavior of ferritic reactor pressure vessel steels using the small-punch-test and micromechanical damage models. Materials Science and Engineering: A, 614:136–147. https://doi.org/10.1016/j.msea.2014.05.095
Mahnken R, 2002. Theoretical, numerical and identification aspects of a new model class for ductile damage. International Journal of Plasticity, 18(7):801–831. https://doi.org/10.1016/S0749-6419(00)00105-4
Malcher L, Pires FMA, de Sá JMAC, 2014. An extended GTN model for ductile fracture under high and low stress triaxiality. International Journal of Plasticity, 54:193–228. https://doi.org/10.1016/j.ijplas.2013.08.015
Mansouri LZ, Chalal H, Abed-Meraim F, 2014. Ductility limit prediction using a GTN damage model coupled with localization bifurcation analysis. Mechanics of Materials, 76:64–92. https://doi.org/10.1016/j.mechmat.2014.06.005
McClintock FA, 1968. A criterion for ductile fracture by the growth of holes. Journal of Applied Mechanics, 35(2):363–371. https://doi.org/10.1115/1.3601204
Neimitz A, Galkiewicz J, Dzioba I, 2018. Calibration of constitutive equations under conditions of large strains and stress triaxiality. Archives of Civil and Mechanical Engineering, 18(4):1123–1135. https://doi.org/10.1016/j.acme.2018.02.013
Nguyen HH, Nguyen TN, Vu HC, 2018. Ductile fracture prediction and forming assessment of AA6061-T6 aluminum alloy sheets. International Journal of Fracture, 209(1–2):143–162. https://doi.org/10.1007/s10704-017-0249-4
Pal S, Wathugala GW, Kundu S, 1996. Calibration of a constitutive model using genetic algorithms. Computers and Geotechnics, 19(4):325–348. https://doi.org/10.1016/S0266-352X(96)00006-7
Pineau A, Benzerga AA, Pardoen T, 2016. Failure of metals I: brittle and ductile fracture. Acta Materialia, 107:424–483. https://doi.org/10.1016/j.actamat.2015.12.034
Rice JR, Tracey DM, 1969. On the ductile enlargement of voids in triaxial stress fields. Journal of the Mechanics and Physics of Solids, 17(3):201–217. https://doi.org/10.1016/0022-5096(69)90033-7
Rossoll A, Berdin C, Prioul C, 2002. Determination of the fracture toughness of a low alloy steel by the instrumented Charpy impact test. International Journal of Fracture, 115(3):205–226. https://doi.org/10.1023/A:1016323522441
Rousselier G, 1987. Ductile fracture models and their potential in local approach of fracture. Nuclear Engineering and Design, 105(1):97–111. https://doi.org/10.1016/0029-5493(87)90234-2
Sajid HU, Kiran R, 2018. Influence of high stress triaxiality on mechanical strength of ASTM A36, ASTM A572 and ASTM A992 steels. Construction and Building Materials, 176:129–134. https://doi.org/10.1016/j.conbuildmat.2018.05.018
Seupel A, Kuna M, 2019. A gradient-enhanced damage model motivated by engineering approaches to ductile failure of steels. International Journal of Damage Mechanics, 28(8):1261–1296. https://doi.org/10.1177/1056789518823879
Seupel A, Hütter G, Kuna M, 2020. On the identification and uniqueness of constitutive parameters for a non-local GTN-model. Engineering Fracture Mechanics, 229:106817. https://doi.org/10.1016/j.engfracmech.2019.106817
Smith C, Kanvinde A, Deierlein G, 2017. Calibration of continuum cyclic constitutive models for structural steel using particle swarm optimization. Journal of Engineering Mechanics, 143(5):04017012. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001214
Soyarslan C, Gülçimen B, Bargmann S, et al., 2016. Modeling of fracture in small punch tests for small- and large-scale yielding conditions at various temperatures. International Journal of Mechanical Sciences, 106:266–285. https://doi.org/10.1016/j.ijmecsci.2015.12.007
Steglich D, Brocks W, 1998. Micromechanical modelling of damage and fracture of ductile materials. Fatigue & Fracture of Engineering Materials & Structures, 21(10):1175–1188. https://doi.org/10.1046/j.1460-2695.1998.00078.x
Sun GQ, Sun FY, Cao FL, et al., 2015. Numerical simulation of tension properties for Al-Cu alloy friction stir-welded joints with GTN damage model. Journal of Materials Engineering and Performance, 24(11):4358–4363. https://doi.org/10.1007/s11665-015-1715-7
Sun Q, Lu YB, Chen JJ, 2020. Identification of material parameters of a shear modified GTN damage model by small punch test. International Journal of Fracture, 222(1–2):25–35. https://doi.org/10.1007/s10704-020-00428-4
Teng BA, Wang WN, Liu YQ, et al., 2014. Bursting prediction of hydroforming aluminium alloy tube based on Gurson-Tvergaard-Needleman damage model. Procedia Engineering, 81:2211–2216. https://doi.org/10.1016/j.proeng.2014.10.310
Teng BG, Wang WN, Xu YC, 2017. Ductile fracture prediction in aluminium alloy 5A06 sheet forming based on GTN damage model. Engineering Fracture Mechanics, 186:242–254. https://doi.org/10.1016/j.engfracmech.2017.10.014
Tvergaard V, 1981. Influence of voids on shear band instabilities under plane strain conditions. International Journal of Fracture, 17(4):389–407. https://doi.org/10.1007/BF00036191
Tvergaard V, Needleman A, 1984. Analysis of the cup-cone fracture in a round tensile bar. Acta Metallurgica, 32(1):157–169. https://doi.org/10.1016/0001-6160(84)90213-X
VazJr M, Muñoz-Rojas PA, Cardoso EL, et al., 2016. Considerations on parameter identification and material response for Gurson-type and Lemaitre-type constitutive models. International Journal of Mechanical Sciences, 106:254–265. https://doi.org/10.1016/j.ijmecsci.2015.12.014
Wang LY, Li L, 2017. Parameter identification of GTN model using response surface methodology for high-strength steel BR1500HS. Journal of Materials Engineering and Performance, 26(8):3831–3838. https://doi.org/10.1007/s11665-017-2806-4
Wen HJ, Mahmoud H, 2016. New model for ductile fracture of metal alloys. I: monotonic loading. Journal of Engineering Mechanics, 142(2):04015088. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001009
Yan S, Zhao XZ, 2018. A fracture criterion for fracture simulation of ductile metals based on micro-mechanisms. Theoretical and Applied Fracture Mechanics, 95:127–142. https://doi.org/10.1016/j.tafmec.2018.02.005
Yan S, Zhao XZ, Wu AH, 2018. Ductile fracture simulation of constructional steels based on yield-to-fracture stressstrain relationship and micromechanism-based fracture criterion. Journal of Structural Engineering, 144(3):04018004. https://doi.org/10.1061/(ASCE)ST.1943-541X.0001970
Yan YX, Sun Q, Chen JJ, et al., 2013. The initiation and propagation of edge cracks of silicon steel during tandem cold rolling process based on the Gurson-Tvergaard-Needleman damage model. Journal of Materials Processing Technology, 213(4):598–605. https://doi.org/10.1016/j.jmatprotec.2012.11.006
Ying L, Wang DT, Liu WQ, et al., 2018. On the numerical implementation of a shear modified GTN damage model and its application to small punch test. International Journal of Material Forming, 11(4):527–539. https://doi.org/10.1007/s12289-017-1362-7
Yoshida F, Urabe M, Hino R, et al., 2003. Inverse approach to identification of material parameters of cyclic elastoplasticity for component layers of a bimetallic sheet. International Journal of Plasticity, 19(12):2149–2170. https://doi.org/10.1016/S0749-6419(03)00063-9
Yu HL, Tieu K, Lu C, et al., 2014. Tensile fracture of ultrafine grained aluminum 6061 sheets by asymmetric cryorolling for microforming. International Journal of Damage Mechanics, 23(8):1077–1095. https://doi.org/10.1177/1056789514538083
Yuenyong J, Suthon M, Kingklang S, et al., 2018. Formability prediction for tube hydroforming of stainless steel 304 using damage mechanics model. Journal of Manufacturing Science and Engineering, 140(1):011006. https://doi.org/10.1115/1.4038208
Yun GJ, Shang S, 2011. A self-optimizing inverse analysis method for estimation of cyclic elasto-plasticity model parameters. International Journal of Plasticity, 27(4):576–595. https://doi.org/10.1016/j.ijplas.2010.08.003
Zhang TR, Lu K, Mano A, et al., 2021. A novel method to uniquely determine the parameters in Gurson — Tvergaard — Needleman model. Fatigue & Fracture of Engineering Materials & Structures, 44(12):3399–3415. https://doi.org/10.1111/ffe.13568
Zhang WW, Cong S, 2016. Failure analysis of SUS304 sheet during hydro-bulging based on GTN ductile damage model. The International Journal of Advanced Manufacturing Technology, 86(1):427–435. https://doi.org/10.1007/s00170-015-8199-3
Zhang Y, Lorentz E, Besson J, 2018. Ductile damage modelling with locking-free regularised GTN model. International Journal for Numerical Methods in Engineering, 113(13):1871–1903. https://doi.org/10.1002/nme.5722
Zhao PJ, Chen ZH, Dong CF, 2016. Failure analysis based on microvoids damage model for DP600 steel on in-situ tensile tests. Engineering Fracture Mechanics, 154:152–168. https://doi.org/10.1016/j.engfracmech.2015.11.017
Zhong J, Xu T, Guan K, et al., 2016. Determination of ductile damage parameters using hybrid particle swarm optimization. Experimental Mechanics, 56(6):945–955. https://doi.org/10.1007/s11340-016-0141-6
Zhu YZ, Engelhardt MD, 2018a. A nonlocal triaxiality and shear dependent continuum damage model for finite strain elastoplasticity. European Journal of Mechanics-A/Solids, 71:16–33. https://doi.org/10.1016/j.euromechsol.2018.03.012
Zhu YZ, Engelhardt MD, 2018b. Prediction of ductile fracture for metal alloys using a shear modified void growth model. Engineering Fracture Mechanics, 190:491–513. https://doi.org/10.1016/j.engfracmech.2017.12.042
Acknowledgments
This work is supported by the National Natural Science Foundation of China (No. 51908416), the Shanghai Pujiang Program (No. 19PJ1409500), and the Fundamental Research Funds for the Central Universities, China. The authors would like to particularly thank Ravi KIRAN (North Dakota State University, USA) for sharing test data.
Author information
Authors and Affiliations
Corresponding author
Additional information
Author contributions
Ya-zhi ZHU and Shi-ping HUANG designed the research. Hao HONG processed the corresponding data. Ya-zhi ZHU wrote the first draft of the manuscript. Shi-ping HUANG helped to organize the manuscript. Hao HONG revised and edited the final version.
Conflict of interest
Ya-zhi ZHU, Shi-ping HUANG, and Hao HONG declare that they have no conflict of interest.
Rights and permissions
About this article
Cite this article
Zhu, Yz., Huang, Sp. & Hong, H. Identification of ductile fracture model parameters for three ASTM structural steels using particle swarm optimization. J. Zhejiang Univ. Sci. A 23, 421–442 (2022). https://doi.org/10.1631/jzus.A2100369
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1631/jzus.A2100369
Key words
- Parameter calibration
- Void growth model (VGM)
- Gurson-Tvergaard-Needleman (GTN) model
- A36 steel
- A572 Gr. 50 steel
- A992 steel
- Particle swarm optimization (PSO)