Bertram, A.

Brüggemann, C.

Böhlke, T.

Estrin, Y.

Lebyodkin, M.

English

KITopen

Modeling and Simulation of the Portevin-Le Chatelier EffectAlbrecht Bertram1, Thomas Böhlke1,2, Carina Brüggemann∗1, Yuri Estrin3, and Mikhail Lebyodkin31 Institut für Mechanik, Otto-von-Guericke-Universität Magdeburg, Postfach 4120, D-39016 Magdeburg, Germany2 Institut für Mechanik, Fachbereich Maschinenbau, Universität Kassel, D-34109 Kassel, Germany3 Institut für Werkstoffkunde und Werkstofftechnik, Technische Universität Clausthal, D-38678 Clausthal-Zellerfeld, Ger-manyDuring deformation of an Al-Mg alloy (AA5754) dynamic strain aging occurs in a certain range of temperatures and strain-rates. An extreme manifestation of this phenomenon, usually referred to as the Portevin-Le Chatelier (PLC) effect, consistsin the occurrence of strain localisation bands accompanied with discontinuous yielding. The PLC effect stems from dynamicdislocation-solute interactions and results in negative strain-rate sensitivity of the flow stress. The PLC effect is detrimentalto the surface quality of sheet metals and also affects the ductility of the material. Since the appearance of the effect stronglydepends on the triaxiality of the stress state, three-dimensional finite element simulations are necessary in order to optimizemetal forming operations. We present a geometrically nonlinear material model which reproduces the main features of thePLC effect. The material parameters were identified based on experimental data from tensile tests. Special emphasis was puton the critical strain for the onset of PLC effect, εc, and the statistical characteristics of the stress drop distribution.1 Experimental findingsPolycrystalline flat specimens (gage length 21mm, width 5.1mm, thickness 1.55mm) were manufactured from an Al-Mg alloy(AA5754). All samples were cut from sheet materials with the tensile axis chosen in the rolling direction. After grinding andpolishing, the specimens were annealed for two hours at 400°C and quenched in water. Tensile tests were performed at roomtemperature with four different imposed strain rates between 2 · 10−5s−1 and 6 · 10−3s−1. The experimental stress-straincurves are shown in Fig. 1 (left).2 Modeling of the PLC effectIn the following, an isotropic elastic viscoplastic material model is formulated. For small elastic deformations Hooke’s law isgiven byτ = C[EAe ], C ∼= λI ⊗ I + 2GIS , EAe =12(I − B−1e ) ∼=12(Be − I), (1)where the Kirchhoff stress tensor τ is connected to the elastic Almansi strain tensor EAe by the isotropic stiffness tensor C. λand G denote the Lamé coefficients and Be = FeFTe the elastic left Cauchy Green tensor.The viscoplastic behavior is given by a von Mises type flow rule for large plastic but small elastic deformations−12Lv(Be) = sym(LpBe), sym(LpBe) ≈ Dp, Dp = 32ε̇pστ ′, Lv(Be) = Ḃe −LBe −BeLT, (2)where ε̇p =√23 ‖ Dp ‖ and σ =√32 ‖ τ ′ ‖. L = ḞF−1 is the velocity gradient, where F is the defomation gradient. Theprime denotes the symmetric and traceless part of a tensor.On the microscale the PLC effect is produced by additional pinning of dislocations temporarily arrested at localized ob-stacles (forest dislocation junctions) by diffusing solutes. This mechanism is represented by the following scalar flow rule(Zhang et al., 2001)ε̇p = ε̇o exp(σ − σdS− P1Cs), σd = d1 + d2(1 − exp(− εpd3)), S = s1 + s2√εp. (3)σd describes the strain hardening and S the strain rate sensitivity of the flow stress. Cs is the concentration of solute atoms atpinned dislocations and depends on the accumulated plastic strain εp and the aging time taCs =(1 − exp(−P2εαp tna))Cm, ṫa = 1 − tatw, tw =Ωε̇p, Ω = ω1 + ω2εβp . (4)∗ Corresponding author: e-mail: carina.brueggemann@mb.uni-magdeburg.de, Phone: +49 (0)391 67-12076 Fax: +49 (0)391 67-12863© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimPAMM · Proc. Appl. Math. Mech. 6, 353–354 (2006) / DOI 10.1002/pamm.200610158 © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimFirst published in:EVA-STAR (Elektronisches Volltextarchiv – Scientific Articles Repository) http://digbib.ubka.uni-karlsruhe.de/volltexte/1000014423  The aging time ta is the time available to solute atoms to diffuse to dislocations. The rate of change of ta depends on ta andthe waiting time tw, the time a dislocation spends at localized obstacles before it unpins.The material parameters used are as follows: d1 = 75.0 MPa, d2 = 165.0 MPa, d3 = 7.0 · 10−2, P1 = 25.0, P2 = 3.3s1/3, n = 1/3, α = 0.44, β = 0.68, ω1 = 3.60 · 10−7, ω2 = 7.5 · 10−4, s1 = 0.41 MPa, s2 = 2.91 MPa, E = 70000 MPa,ν = 0.3, ε̇o = 2.30 · 10−7 s−1. In order to solve the differential equations, the backward Euler method was used.3 Comparison of experimental findings with numerical resultsIn Fig. 1 (left) the experimental findings are compared to the numerical results. The hardening behavior is reproduced well.The different types of serrations observed experimentally are recovered by the simulation. Detailed information about astatistical analysis of the PLC effect can be found in Lebyodkin et al. (2001) and Kubin et al. (2002). Without going intodetail, it is only pointed out that the comparison of the statistical behavior, such as the stress drop distribution function andthe correlation function, shows good agreement. Another characteristic feature is the critical strain εc, which is shown in Fig.1 (right). In the high strain rate regime εc increases (normal behavior), while in the low strain rate regime it decreases withincreasing ε̇ (inverse behavior). The inverse branch is not recovered in the present version of the model using a monotonicstrain dependence of Ω, but this can be rectified by using a non-monotonic function.801001201401601802002202402600 2 4 6 8 10 12 14 16 18experimentsimulationε[%]σ[MPa]01234560.0001 0.001 0.01experimentsimulationε̇[s−1]ε c[%]Fig. 1 Comparison of experimental findings with numerical results: σ vs. ε. The curves are deliberately shifted alongthe ordinate axis for clarity. The strain rate from the upper curve downwards are as follows: 2 · 10−5s−1[+30MPa], 2 ·10−4s−1[+20MPa] , 8 · 10−4s−1[+10MPa], 6 · 10−3s−1[+0MPa] (left) and εc vs. ε̇ (right)4 ConclusionsFE simulations were performed with a 3D isotropic, geometrically nonlinear PLC model. Material parameters were identified,and the results of the simulations were compared with experimental findings. The comparison of experiment with simulationshows good agreement for the strain hardening behavior and the statistical behavior of the stress drops for all strain rates.Acknowledgements Financial support of Deutsche Forschungsgemeinschaft (DFG) is gratefully acknowledged. This project was alsofunded by the Graduiertenkolleg 828.References[1] Lebyodkin, M., Dunin-Barkowskii, L., and Lebedkina, T., Statistical and multifractal analysis of collective dislocation processes inthe Portevin-Le Chatelier effect Physical, Mesomechanics 4, 9 (2001).[2] Kubin, L., Fressengeas, C., and Ananthakrishna, G., In Nabarro, F. and Duesbery, M., Collective Behaviour of Dislocations in Plastic-ity, Editors, Dislocations in Solids, 102 (2002).[3] Zhang, S., McCormick, P., and Estrin, Y., The morphology of Portevin-Le Chatelier bands: finite element simulation for Al-Mg-Si,Acta. Mater. 49, 1087 (2001).© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimSection 6 354

Modeling and simulation of the Portevin-Le Chatellier effect

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Modeling and simulation of the Portevin-Le Chatellier effect

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