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http://dx.doi.org/10.12989/sem.2022.83.4.435

Evolution of post-peak localized strain field of steel under quasi-static uniaxial tension: Analytical study  

Altai, Saif L. (Department of Civil and Environmental Engineering, University of Missouri)
Orton, Sarah L. (Department of Civil and Environmental Engineering, University of Missouri)
Chen, Zhen (Department of Civil and Environmental Engineering, University of Missouri)
Publication Information
Structural Engineering and Mechanics / v.83, no.4, 2022 , pp. 435-449 More about this Journal
Abstract
Constitutive modeling that could reasonably predict and effectively evaluate the post-peak structural behavior while eliminating the mesh-dependency in numerical simulation remains to be developed for general engineering applications. Based on the previous work, a simple one-dimensional modeling procedure is proposed to predict and evaluate the post-peak response, as characterized by the evolution of localized strain field, of a steel member to monotonically uniaxial tension. The proposed model extends the classic one-dimensional softening with localization model as introduced by (Schreyer and Chen 1986) to account for the localization length, and bifurcation and rupture points. The new findings of this research are as follows. Two types of strain-softening functions (bilinear and nonlinear) are proposed for comparison. The new failure criterion corresponding to the constitutive modeling is formulated based on the engineering strain inside the localization zone at rupture. Furthermore, a new mathematical expression is developed, based on the strain rate inside and outside the localization zone, to describe the displacement field at which bifurcation occurs. The model solutions are compared with the experimental data on four low-carbon cylindrical steel bars of different lengths. For engineering applications, the model solutions are also compared to the experimental data of a cylindrical steel bar system (three steel bars arranged in series). It is shown that the bilinear and nonlinear softening models can predict the energy dissipation in the post-peak regime with an average difference of only 4%.
Keywords
bifurcation; constitutive modeling; low carbon steel; size effect; softening; strain localization;
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1 Altai, S. (2019), "Experimental and analytical investigation of localization and post-peak behavior of steel members in tension", Ph.D. Dissertation, University of Missouri-Columbia, Columbia.
2 Wackerfuss, J. (2008), "Efficient finite element formulation for the analysis of localized failure in beam structures", Int. J. Numer. Meth. Eng., 73(9), 1217-1250. https://doi.org/10.1002/nme.2116.   DOI
3 Wu, S. and Wang, X. (2010), "Mesh dependence and nonlocal regularization of one-dimensional strain softening plasticity", J. Eng. Mech., 136(11), 1354-1365. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000184.   DOI
4 Yalcinkaya, T. and Lancioni, G. (2014), "Energy-based modeling of localization and necking in plasticity", Procedia Mater. Sci., 3, 1618-1625. https://doi.org/10.1016/j.mspro.2014.06.261.   DOI
5 Zhang, G. and Khandelwal, K. (2016), "Modeling of nonlocal damage-plasticity in beams using isogeometric analysis", Computer. Struct., 165, 76-95. https://doi.org/10.1016/j.compstruc.2015.12.006.   DOI
6 Altai, S., Orton, S. and Chen, Z. (2020), "Evolution of localization length during postpeak response of steel in tension: Experimental study", J. Eng. Mech., 146(7), 04020069. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001804.   DOI
7 Bao, C., Francois, M. and Le Joncour, L. (2016), "A closer look at the diffuse and localised necking of a metallic thin sheet: Evolution of the two bands pattern", Strain, 52(3), 244-260. https://doi.org/10.1111/str.12184.   DOI
8 Antolovich, S.D. and Armstrong, R.W. (2014), "Plastic strain localization in metals: Origins and consequences", Prog. Mater. Sci., 59(1), 1-160. https://doi.org/10.1016/J.PMATSCI.2013.06.001.   DOI
9 Armero, F. and Ehrlich, D. (2006), "Numerical modeling of softening hinges in thin Euler-Bernoulli beams", Comput. Struct., 84(10-11), 641-656. https://doi.org/10.1016/J.COMPSTRUC.2005.11.010.   DOI
10 Audoly, B. and Hutchinson, J.W. (2016), "Analysis of necking based on a one-dimensional model", J. Mech. Phys. Solid., 97, 68-91. https://doi.org/10.1016/j.jmps.2015.12.018.   DOI
11 Bazant, Z.P. (2003a), "Asymptotic matching analysis of structural failure due to softening hinges. 2: Implications", ASCE J. Eng. Mech., 129, 651-654.   DOI
12 di. Luzio, G. (2007), "A symmetric over-nonlocal microplane model M4 for fracture in concrete", Int. J. Solid. Struct., 44(13), 4418-4441. https://doi.org/10.1016/j.ijsolstr.2006.11.030.   DOI
13 Engelen, R.A.B., Fleck, N.A., Peerlings, R.H.J. and Geers, M.G.D. (2006), "An evaluation of higher-order plasticity theories for predicting size effects and localisation", Int. J. Solid. Struct., 43(7-8), 1857-1877. https://doi.org/10.1016/j.ijsolstr.2004.05.072.   DOI
14 Engelen, R.A.B., Geers, M.G.D. and Baaijens, F.P.T. (2002), "Nonlocal implicit gradient-enhanced elasto-plasticity for the modelling of softening behaviour", Int. J. Plast., 19(4), 403-433. https://doi.org/10.1016/S0749-6419(01)00042-0.   DOI
15 Schreyer, H.L. and Chen, Z. (1986), "One-dimensional softening with localization", J. Appl. Mech., 53(4), 791. https://doi.org/10.1115/1.3171860.   DOI
16 Li, Y. and Karr, D.G. (2009), "Prediction of ductile fracture in tension by bifurcation, localization, and imperfection analyses", Int. J. Plast., 25(6), 1128-1153. https://doi.org/10.1016/j.ijplas.2008.07.001.   DOI
17 Lu, X., Bardet, J.P. and Huang, M. (2009), "Numerical solutions of strain localization with nonlocal softening plasticity", Comput. Meth. Appl. Mech. Eng., 198(47-48), 3702-3711. https://doi.org/10.1016/J.CMA.2009.08.002.   DOI
18 Salehi, M. and Sideris, P. (2017), "Refined gradient inelastic flexibility-based formulation for members subjected to arbitrary loading", J. Eng. Mech., 143(9), 1-18. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001288.   DOI
19 Sideris, P. and Salehi, M. (2016), "A gradient inelastic flexibilitybased frame element formulation", J. Eng. Mech., 142(7), 1-14. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001083.   DOI
20 Altai, S., Orton, S. and Chen, Z. (2019), "Effect of the post-peak behavior on collapse of structural systems", Structures Congress 2019: Blast, Impact Loading, and Research and Education, American Society of Civil Engineers, Reston, VA. https://doi.org/10.1061/9780784482247.011.   DOI
21 Jacques, N. and Rodriguez-Martinez, J. A. (2021), "Influence on strain-rate history effects on the development of necking instabilities under dynamic loading conditions", Int. J. Solid. Struct., 230-231, 111152. https://doi.org/10.1016/J.IJSOLSTR.2021.111152.   DOI
22 Jirasek, M. (2004), "Nonlocal theories in continuum mechanics", Acta Polytechnica, 44(5), 16-34. https://doi.org/10.14311/610.   DOI
23 Jirasek, M. and Rolshoven, S. (2003), "Comparison of integraltype nonlocal plasticity models for strain-softening materials", Int. J. Eng. Sci., 41(13-14), 1553-1602. https://doi.org/10.1016/S0020-7225(03)00027-2.   DOI
24 Jirsek, M. (1997), "Analytical and numerical solutions for frames with softening hinges", J. Eng. Mech., 123(1), 8-14. https://doi.org/10.1061/(ASCE)0733-9399(1997)123:1(8).   DOI
25 Valipour, H.R. and Foster, S.J. (2009), "Nonlocal damage formulation for a flexibility-based frame element", J. Struct. Eng., 135(10), 1213-1221. https://doi.org/10.1061/(ASCE)ST.1943-541X.0000054.   DOI
26 Huang, M., Qu, X. and Lu, X. (2017), "Regularized finite element modeling of progressive failure in soils within nonlocal softening plasticity", Comput. Mech., 62(3), 347-358. https://doi.org/10.1007/S00466-017-1500-6.   DOI
27 Lancioni, G., Yalcinkaya, T. and Cocks, A. (2015b), "Energybased non-local plasticity models for deformation patterning, localization and fracture", Proc. Roy. Soc. A: Math. Phys. Eng. Sci., 471(2180), 20150275. https://doi.org/10.1098/rspa.2015.0275.   DOI
28 Bigoni, D. (2012), Nonlinear Solid Mechanics: Bifurcation Theory and Material Instability, Cambridge University Press.
29 Bazant, Z.P. (2003b), "Asymptotic matching analysis of scaling of structural failure due to softening hinges. II: Implications", J. Eng. Mech., 129(6), 651-654. https://doi.org/10.1061/(ASCE)0733-9399(2003)129:6(651).   DOI
30 Bazant, Z.P. and Cedolin, L. (2010), "Stability of structures", Elastic, Inelastic, Fracture and Damage Theories, 1st Edition, Oxford University Press, New York.
31 Challamel, N., Lanos, C. and Casandjian, C. (2008), "Plastic failure of nonlocal beams", Phys. Rev. E-Statist. Nonlin. Soft Matter. Phys., 78(2), 026604. https://doi.org/10.1103/PhysRevE.78.026604.   DOI
32 Chen, Z. and Schreyer, H.L.L. (1990), "A numerical solution scheme for softening problems involving total strain control", Comput. Struct., 37(6), 1043-1050. https://doi.org/10.1016/0045-7949(90)90016-U.   DOI
33 Chen, Z., Gan, Y. and Labuz, J.F. (2008), "Analytical and numerical study of the size effect on the failure response of hierarchical structures", Int. J. Multisc. Comput. Eng., 6(4), 339-348.   DOI
34 Coleman, J. and Spacone, E. (2001), "Localization issues in forcebased frame elements", J. Struct. Eng., 127(11), 1257-1265. https://doi.org/10.1061/(ASCE)0733-9445(2001)127:11(1257).   DOI
35 Dai, H.H., Zhu, X. and Chen, Z. (2011), "An analytical study on the post-peak structural response", J. Appl. Mech., 78(4), 044501. https://doi.org/10.1115/1.4003740.   DOI
36 del Piero, G., Lancioni, G. and March, R. (2013), "A diffuse cohesive energy approach to fracture and plasticity: The onedimensional case", J. Mech. Mater. Struct., 8(2-4), 109-151. https://doi.org/10.2140/jomms.2013.8.109.   DOI
37 Kolwankar, S., Kanvinde, A., Kenawy, M. and Kunnath, S. (2017), "Uniaxial nonlocal formulation for geometric nonlinearity-induced necking and buckling localization in a steel bar", J. Struct. Eng., 143(9), 1-13. https://doi.org/10.1061/(ASCE)ST.1943-541X.0001827.   DOI
38 Kolwankar, S., Kanvinde, A., Kenawy, M., Lignos, D. and Kunnath, S. (2018), "Simulating local buckling-induced softening in steel members using an equivalent nonlocal material model in displacement-based fiber elements", J. Struct. Eng., 144(10), 04018192. https://doi.org/10.1061/(ASCE)ST.1943-541X.0002189.   DOI
39 Lancioni, G. (2015), "Modeling the response of tensile steel bars by means of incremental energy minimization", J. Elast., 121(1), 25-54. https://doi.org/10.1007/s10659-015-9515-8.   DOI
40 Lancioni, G. and Corinaldesi, V. (2016), "Localized versus diffuse damage: A variational approach", AIP Conf. Proc., 1769, 160013. https://doi.org/10.1063/1.4963556.   DOI
41 Lancioni, G., Yalcinkaya, T. and Cocks, A. (2015a), "Energybased non-local plasticity models for deformation patterning, localization and fracture", Proc. Roy. Soc. A: Math. Phys. Eng. Sci., 471(2180), 20150275. https://doi.org/10.1098/rspa.2015.0275.   DOI
42 Lestringant, C. and Audoly, B. (2020), "A one-dimensional model for elasto-capillary necking", Proc. Roy. Soc. A, 476(2240), 20200337. https://doi.org/10.1098/RSPA.2020.0337.   DOI
43 Pijaudier-Cabot, G. and Bazant, Z.P. (1988), "Nonlocal damage theory", J. Eng. Mech., 113(10), 1512-1533.   DOI
44 Rolshoven, S. and Jirasek, M. (2002), "Nonlocal formulations of softening plasticity", WCCMV, Fifth World Congress on Computational Mechanics, 1-10.
45 Salehi, M., Sideris, P. and Liel, A.B. (2020), "Assessing damage and collapse capacity of reinforced concrete structures using the gradient inelastic beam element formulation", Eng. Struct., 225, 111290. https://doi.org/10.1016/J.ENGSTRUCT.2020.111290.   DOI
46 Stromberg, L. (2008), "A special case of equivalence between nonlocal plasticity and gradient plasticity in a one-dimensional formulation", Int. J. Eng. Sci., 46(8), 835-841. https://doi.org/10.1016/j.ijengsci.2008.01.019.   DOI
47 Zhu, Y., Kanvinde, A. and Pan, Z. (2019), "Analysis of postnecking behavior in structural steels using a one-dimensional nonlocal model", Eng. Struct., 180, 321-331. https://doi.org/10.1016/j.engstruct.2018.11.050.   DOI
48 Lu, X., Bardet, J.P. and Huang, M. (2010), "Length scales interaction in nonlocal plastic strain localization of bars of varying section", J. Eng. Mech., 136(8), 1036-1042. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000145.   DOI
49 Peerlings, R.H.J., Geers, M.G.D., de Borst, R. and Brekelmans, W.A.M. (2001), "A critical comparison of nonlocal and gradient-enhanced softening continua", Int. J. Solid. Struct., 38(44-45), 7723-7746. https://doi.org/10.1016/S0020-7683(01)00087-7.   DOI