This work presents the main strategies and algorithms adopted in the DD3MAT in-house code, specifically developed for identifying the anisotropy parameters. The algorithm adopted is based on the minimization of an error function, using a downhill simplex method. The set of experimental values can consider yield stresses and r - values obtained from in-plane tension, for different angles with the rolling direction (RD), yield stress and r - value obtained for biaxial stress state, and yield stresses from shear tests performed also for different angles to RD. All these values can be defined for a specific value of plastic work. Moreover, it can also include the yield stresses obtained from in-plane compression tests. The anisotropy parameters are identified for an AA2090-T3 aluminium alloy, highlighting the importance of the user intervention to improve the numerical fit.The authors gratefully acknowledge the financial support of the Portuguese Foundation for Science and Technology (FCT) via the projects PTDC/EME-TME/118420/2010 and UID/EMS/00285/2013. The first author is also grateful to the FCT for the PhD grant SFRH/BD/98545/2013.info:eu-repo/semantics/publishedVersio

Alves, J. L.

Barros, P. D.

Carvalho, P. D.

Menezes, L. F.

Oliveira, M. C.

Journal of Physics Conference Series

Universidade do Minho: RepositoriUM

      DD3MAT – a code for yield criteria anisotropy parameters identification. P D Barrosa, P D Carvalhoa, J L Alvesb, M C Oliveiraa and L F Menezesa aCEMUC, Mechanical Engineering Department, University of Coimbra, Polo II, Rua Luís Reis Santos, Pinhal de Marrocos, Coimbra, 3030-788, Portugal bCMEMS, Microelectromechanical Systems Research Unit, University of Minho, Campus de Azurém, 4800-058 Guimarães, Portugal pedro.barros@dem.uc.pt Abstract. This work presents the main strategies and algorithms adopted in the DD3MAT in-house code, specifically developed for identifying the anisotropy parameters. The algorithm adopted is based on the minimization of an error function, using a downhill simplex method. The set of experimental values can consider yield stresses and r -values obtained from in-plane tension, for different angles with the rolling direction (RD), yield stress and r -value obtained for biaxial stress state, and yield stresses from shear tests performed also for different angles to RD. All these values can be defined for a specific value of plastic work. Moreover, it can also include the yield stresses obtained from in-plane compression tests. The anisotropy parameters are identified for an AA2090-T3 aluminium alloy, highlighting the importance of the user intervention to improve the numerical fit. 1. Introduction Sheet metal forming processes are nowadays designed and optimized virtually using finite element analysis (FEA), which is consensually accepted as the main reason for the huge decrease in time-to-market life cycle and for notable savings in terms of money, time and effort in the design, production and process set-up of new formed parts. However, the success of finite elements solvers on the design and optimization of sheet metal formed parts is strongly dependent on their ability to accurately describe the material’s mechanical behaviour. Sheet metals generally exhibit anisotropic mechanical behaviour due to the rolling process, characterized by the symmetry of the mechanical properties with respect to three orthogonal planes, i.e. orthotropic behaviour. Thus, different mechanical behaviours are expected for different loading directions and conditions. Phenomenological models are the most widely used approach to describe the elastoplastic response of metallic sheets, since they are computationally efficient when compared with microscopic models. The material’s orthotropic behaviour is modelled by the yield surface, used to describe the yielding and the plastic flow of the material, based on the hardening law selected. This dual role of the yield surface requires a particular care and accuracy in its modelling and numerical implementation. The material anisotropy parameters identification must be performed in a way that a given yield criterion should reproduce the materials mechanical behaviour as close as possible. Moreover, since sheet metal forming processes are carried out with inhomogeneous deformation and under multiaxial strain paths it is important to use as much information as possible. In this work DD3MAT in-house code is used to identify the anisotropy parameters for an AA2090-T3 aluminium alloy [1], considering three yield criteria YLD91 [2], CB2001 [3] and CPB06 [4]. The predicted in-plane distribution of the yield stresses and r -values evolution, as well as the yield surfaces,       are discussed to show that a more or less accurate fit of the experimental data is achieved, depending on the selected yield criterion and on the user intervention, through the selection of the weighting factor for each test result. 2. DD3MAT – yield criteria parameters identification code. The anisotropy parameters should be determined such that the yield criterion reproduces the material’s mechanical behavior as close as possible. The most used experimental results for the identification of anisotropy parameters are the yield stresses and r -values obtained from in-plane tension, for different angles ( )  with the RD. In order to improve the description of the yield surface, it is also recommended to experimentally determine the biaxial yield stress and the biaxial anisotropy coefficient [5–7]. For the CPB06 yield criterion, uniaxial compression experimental results are also necessary for describing the strength differential effects. However, when performing compression tests for thin metallic sheets it is necessary to avoid buckling effects. This requires the use of small specimens, leading to supplementary difficulties in the acquisition and analysis of experimental results, particularly for high strain values [8]. The anisotropy parameters for the three selected yield criteria were obtained with the DD3MAT in-house code. The procedure adopted is based in an optimization problem regarding the minimization of an error function, evaluating the difference between the estimated values and the experimental ones, as follow                 T C902 2 2T T C C02 21 1 11 1b brb b r b bF w w w r rw w r r                     A A A AA A (1) where A  represents the set of parameters associated with the selected yield criterion. T , C  and r  are the experimental yield stresses in tension, compression and r -values determined in uniaxial tension, respectively, obtained from the uniaxial tests for a specific orientation ( )  with respect to RD. b  is the experimental yield stress obtained from the equibiaxial tensile test, br  is the experimental r -value obtained from the disc compression test, and  T A ,  C A   r A ,  b A  and  br A  are the correspondent values predicted from the adopted yield criterion. Such procedure can be considered a generalization of the one proposed by Banabic et al., 2005 [9]. The weighting factors, Tw, Cw, rw  , bw  and brw  are used to balance the influence of the experimental data. Nevertheless, the selection of the weighting factors is normally a manual procedure, strongly dependent on users’ expertise and knowledge. The identification procedure, defined in (1) also implies the pre-selection of an initial yield stress or the hardening law parameters, if a specific value of plastic work is defined to select the experimental values for the yield stresses. 3. Results and discussion The anisotropy parameters for the three previously mentioned yield criteria were identified for an AA2090-T3 aluminium alloy, for which the experimental values of T , C , r  (at each 15º with RD), b  and br  are available [1]. Figure 1 shows the evolution of the yield stress and r -values with the angle from RD, for the three yield criteria, obtained considering all weighting factors equal to 1.0 (labelled “Set 1”). As expected, YLD91 shows a less accurate fit when compared with the other yield criteria, mainly for the yield stresses. CPB06 presents a better fit for the yield stresses, when compared with the YLD91, with the added benefit of also describing the materials’ behavior for compression stress states. Regarding the CB2001, both yield stresses and r -values are globally well described. Nevertheless, it is possible to see that all yield criteria fail to capture the r -value at 45º and tend to overestimate the yield stresses for angles closer to the transverse direction (TD). Therefore, a new set of parameters was identified, through the selection of the weighting factors (shown in Table 1), i.e. with user intervention, labelled “Set 2”.  Figure 2 shows the evolution of the yield stress and r -values with the angle from the RD, for the three yield criteria, considering the user intervention. For the CB2001, the description is enhanced mainly for yield stresses for angles closer to TD and the predicted r -values at 45º and 90º are closer to the experimental ones. For the YLD91, the main improvement occurs for yield stresses closer to the rolling direction and the r -value at 45º. The CPB06 also benefited from the user’s input in the identification       procedure since the yield stresses evolution in tension and compression are globally closer to the experimental values.  Figure 3 presents the yield surfaces for the three considered yield criteria in the 1 2   plane considering, or not, user intervention. These results, together with Table 2, allows analyzing the material behavior for stress states other than uniaxial tensile stress. YLD91 presents lower experimental b  values, for both sets, slightly more accurate for Set 2. The CB2001 yield criterion predicts accurate b  and br  values for both stets, but Set 1 presents a br  value lower than the experimental, while it is slightly higher for Set 2. Globally, CPB06 presents the b  and br  values least accurate, since the introduction of the compression yield stress in the optimization procedure reduces the relative importance of these values in the objective function. A summary of the parameters identified for the three yield criteria, considering both sets, is presented in Table 3, highlighting the parameters’ interdependence.   (a) (b) Figure 1. Experimental and predicted (a) yield stresses and (b) r -values (Set 1).   (a) (b) Figure 2. Experimental and predicted (a) yield stresses and (b) r -values (Set 2).   (a) (b) Figure 3. Predicted yield surfaces in the 1 2   plane (a) Set 1 and (b) Set 2. 2002402803200 15 30 45 60 75 90Yield stress [MPa]Angle with rolling direction [º]Exp. T Exp. CCPB06 T CPB06 CYLD91 CB20010.00.51.01.52.00 15 30 45 60 75 90r-valueAngle with rolling direction [º]Exp. CPB06YLD91 CPB06-CCB20012002402803200 15 30 45 60 75 90Yield stress [MPa]Angle with rolling direction [º]Exp. T Exp. CCPB06 T CPB06 CYLD91 CB20010.00.51.01.52.00 15 30 45 60 75 90r-valueAngle with rolling direction [º]Exp. CPB06YLD91 CPB06-CCB2001-400-2000200400-400 -200 0 200 400σ2 [MPa]σ1 [MPa]Exp.CPB06YLD91CB2001-400-2000200400-400 -200 0 200 400σ2 [MPa]σ1 [MPa]Exp.CPB06YLD91CB2001      Table 1. Experimental and predicted yield stress and r -values for the biaxial test. CB2001 45 5rw   75 5rw   90 5rw   20bw   T6010w   T7510w   T9010w   YLD91 45 5rw   20bw   T020w   T1520w   T7520w   T9020w   CPB06 45 7rw   20bw   T9030w   C1540w   C30 5w   C455w   C60 5w   C75 5w   C9040w   Table 2. Experimental and predicted yield stress and r -values for the biaxial test.   Set 1 Set 2  Exp. CB2001 YLD91 CPB06 CB2001 YLD91 CPB06 b  289.4 289.6 (0.06%) 235.1 (-18.7%) 232.8 (-19.6%) 285.9 (-1.2%) 239.4 (-17.3%) 229.5 (-20.7%) br  0.67 0.66 (-0.74%) 0.57 (-15.7%) 0.78 (+15.8%) 0.66 (-1.5%) 0.76 (+12.7%) 0.89 (+33.3%) Table 3. Anisotropy parameters identified for the considered yield criteria.  Set a1  a2  a3  a4  b1  b2  b3  b4  b5  b10  c  CB2001 1 0.728 1.402 1.334 1.525 3.576 -0.838 -1.983 -0.581 -4.580 -0.567 0.608 2 1.358 1.848 1.075 1.709 5.357 -0.623 -4.386 -3.654 -6.046 -0.882 0.857  Set c1  c2  c3  c6  m  YLD91 1 1.069 1.300 0.856 1.213 8 2 1.110 1.224 0.835 1.238 8  Set C11  C22  C33  C66  C23  C13  C12  k  CPB06 1 -1.010 0.475 1.199 -1.082 -0.102 0.104 -0.012 -0.057 2 -0.818 0.449 1.232 -1.224 -0.088 -0.060 -0.819 -0.050 Note that, for metal sheets, the off plane parameters cannot be experimentally evaluated. Thus, parameters 5 6,  a a  and kb   6,7,8,9,11k  , for the CB2001, 4 5,  c c  for the YLD91 and 44 55,  C C  for the CPB06 take the corresponding isotropic values, i.e. 1.0. 4. Conclusions The results presented show the ability of DD3MAT in-house to perform the identification of the anisotropy parameters for different yield criteria. The classical identification procedure adopted allows the user to control the importance of each test through the weighting factors. However, the users’ input and knowledge is only as good as the flexibility allowed by the yield criterion considered.  Acknowledgments The authors gratefully acknowledge the financial support of the Portuguese Foundation for Science and Technology (FCT) via the projects PTDC/EME-TME/118420/2010 and UID/EMS/00285/2013.The first author is also grateful to the FCT for the PhD grant SFRH/BD/98545/2013. References [1] J.W. Yoon, F. Barlat, K. Chung, F. Pourboghrat, D.Y. Yang, Int. J. Plast. 16 (2000) 1075–1104. [2] F. Barlat, D.J. Lege, J.C. Brem, Int. J. Plast. 7 (1991) 693–712. [3] O. Cazacu, F. Barlat, Math. Mech. Solids 6 (2001) 613–630. [4] O. Cazacu, B. Plunkett, F. Barlat, Int. J. Plast. 22 (2006) 1171–1194. [5] D.J. Lege, F. Barlat, J.C. Brem, Int. J. Mech. Sci. 31 (1989) 549–563. [6] K. Pöhlandt, D. Banabic, K. Lange, in:, ESAFORM 2002, Krakow, Poland., 2002, pp. 723–727. [7] F. Barlat, J.C. Brem, J.W. Yoon, K. Chung, R.E. Dick, D.J. Lege, F. Pourboghrat, S.H. Choi, E. Chu, Int. J. Plast. 19 (2003) 1297–1319. [8] M. Tritschler, A. Butz, D. Helm, G. Falkinger, J. Kiese, Int. J. Mater. Form. 7 (2014) 259–273. [9] D. Banabic, H. Aretz, D.S. Comsa, L. Paraianu, Int. J. Plast. 21 (2005) 493–512. 

DD3MAT: a code for yield criteria anisotropy parameters identification

This work presents the main strategies and algorithms adopted in the DD3MAT inhouse code, specifically developed for identifying the anisotropy parameters. The algorithm adopted is based on the minimization of an error function, using a downhill simplex method. The set of experimental values can consider yield stresses and r -values obtained from in-plane tension, for different angles with the rolling direction (RD), yield stress and r -value obtained for biaxial stress state, and yield stresses from shear tests performed also for different angles to RD. All these values can be defined for a specific value of plastic work. Moreover, it can also include the yield stresses obtained from in-plane compression tests. The anisotropy parameters are identified for an AA2090-T3 aluminium alloy, highlighting the importance of the user intervention to improve the numerical fit

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DD3MAT - a code for yield criteria anisotropy parameters identification

P D Barros

P D Carvalho

J L Alves

M C Oliveira

L F Menezes

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DD3MAT - a code for yield criteria anisotropy parameters identification.

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DD3MAT: a code for yield criteria anisotropy parameters identification

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Universidade do Minho: RepositoriUM

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