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Experimental and Numerical Formability Analysis of AZ31 and ZE10 Sheets
Available online at www.sciencedirect.com
ScienceDirect Materials Today: Proceedings 2S (2015) S125 – S130
Conference MEFORM 2015, Light Metals – Forming Technologies and Further Processing
Experimental and numerical formability analysis of AZ31 and ZE10 sheets D. Steglicha,*, M. N. Mekonena, J. Bohlenb a
Helmholtz-Zentrum Geesthacht, Institute of Materials Research, Materials Mechanics, Max-Planck-Str. 1, D-21502 Geesthacht, Germany b Helmholtz-Zentrum Geesthacht, Institute of Materials Research, Magnesium Innovation Center MagIC, Max-Planck-Str. 1, D-21502 Geesthacht, Germany
Abstract Experimentally and numerically obtained mechanical responses of Nakazima-type sheet forming tests for the magnesium alloys ZE10 and AZ31 at elevated temperature (200°C) are explored. The results from the experiments reveal sufficient ductility at the prescribed test temperature allowing sheet forming processes. The material's anisotropy as well as the distortional character of hardening is confirmed. Differences in terms of strain paths during the forming experiments between the two materials are quantified. The corresponding numerical responses are obtained employing a suitable plasticity model based on tensor transforms. In addition, for predicting limit conditions of the forming process, the localisation criterion by Marciniak and Kuczynski is adopted. The constitutive model together with the localisation criterion is implemented in a finite element framework based on a fully implicit time integration scheme. The observed differential hardening during plastic deformation is predicted. The reasonably good agreement between the responses of the model and the respective experiments indicate the predictive capabilities of the implemented model for the considered magnesium alloys. The influence of the model parameter identification strategy is demonstrated. © 2014 The Authors. Published by Elsevier Ltd. © 2015 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND Selection and peer-review under responsibility of the Conference Committee of license Conference MEFORM 2015, Light Metals – (http://creativecommons.org/licenses/by-nc-nd/3.0/). Formingand Technologies and Further. This of is the an open accessCommittee article under the CC BY-NC-ND license responsibility Conference of Conference MEFORM 2015, Light Metals – Forming Selection peer-review under Technologies and Further (http://creativecommons.org/licenses/by-nc-nd/3.0/). Keywords: Magnesium alloys; Sheet forming; Finite element simulation; Plastic anisotropy; Localisation
* Corresponding author. Tel.: +49-4152-872543; fax: +49-4152-8742543 E-mail address: [email protected]
2214-7853 © 2015 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/). Selection and peer-review under responsibility of the Conference Committee of Conference MEFORM 2015, Light Metals – Forming Technologies and Further doi:10.1016/j.matpr.2015.05.029
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1. Introduction Forming involves many procedures with a variety of strain paths, e.g. stretch forming or deep drawing to name two of them with distinctly different contribution of material flow from the sheets thickness or from the sheet plane. A comprehensive way to assess the formability of sheets accounting for these different aspects is based on experiments prescribing different strain paths and determining major strains in dependence of the minor strains [14]. A hemispherical punch is applied on a work-piece clamped in place by dies until strain localisation or fracture is observed. The results of such formability tests are commonly represented by the forming limit diagram (FLD), as proposed by Keeler [5]. From these tests a forming limit curve (FLC) results which represents strains of limiting uniform deformation as a function of the (linear) strain path. Limit strains are the boundary between safe and failed zones, the region above the forming-limit strain curve represents the failure zone. This measure of formability can then be applied in the design of forming processes. Besides, there is a need for accurate predictive simulation techniques for metal forming based on finite element (FE) analyses, which represent the current state-of-the-art in virtual prototyping. For realistic finite element predictions, it is vital to use accurate plasticity and localisation models. As the quality of the prediction strongly depends on the constitutive model used, its selection and its calibration becomes a key issue. Over the years, several authors have proposed a number of such models describing the yielding behavior in terms of macroscopic yield functions for various alloys, e.g. [6-9]. Cazacu and Barlat introduced a yield function (CaBa2004) as a modification of the Drucker model accounting for the material anisotropy as well as for the stress-differential effect observed in textured magnesium alloys [10]. This plasticity model has been successfully applied to predict the deformation of magnesium alloys [11] and is therefore employed in the following. The application of the aforementioned model requires a criterion for the determination of limiting conditions. In this work, an approach proposed by Marciniak and Kuczynski [12] is employed (M-K model). These authors showed that the presence of an imperfection in the sheet can lead to unstable deformation in the weaker region and to subsequent localised necking and failure. Although the M-K model is one of the early imperfection theories, its concept can still be regarded as useful due to its flexibility and simplicity. In an attempt to determine the formability of the two magnesium sheet alloys ZE10 and AZ31 at elevated temperature, formability tests were conducted. The respective results are reported [13] and replicated in the following. Numerical predictions are presented in the subsequent section and compared with the test results in terms of both, macroscopic quantities and local strain paths. The application of an advanced plasticity model in conjunction with a localisation criterion allowed a deeper insight into the interaction of field quantities and the impact of model parameter calibration. 2. Materials and methods Formability tests were conducted for two different magnesium alloy sheets (sheet thickness 1.3 mm), namely ZE10 (Mg-1Zn-Ce based mischmetal) and AZ31 (Mg-3Al-1Zn-Mn) in a heat treated condition (O-temper). ZE10 shows improved ductility at room temperature compared to AZ31 which is associated with the included rare earth elements which weaken the crystallographic texture, resulting in higher work hardening and increased ductility. The mechanical characterisation of these materials under uniaxial tensile loading conditions has been detailed previously [11]. Contrary to the common assumption, the computed r-values show a strain dependence. A positive strain rate effect on the yield strength is recorded while the opposite effect was observed on the ductility. The strain rate influence on the materials anisotropy is small. The reasonably large amount of achieved deformation for both alloys suggests the test temperature to be suitable for sheet forming processes. 2.1. Nakazima-type sheet forming tests The Nakazima-type forming limit tests were carried out based on the ISO 12004 standard using a universal sheet metal testing machine. The tests constitute seven circular work-pieces with an outer diameter of 200 mm and recesses of different radii, namely 80 mm, 72.5 mm, 65 mm, 57.5 mm, 50 mm, 40 mm, and 0 mm. The recesses on the work-pieces are all made perpendicular to the rolling direction. The tests cover the negative as well as the
D. Steglich et al. / Materials Today: Proceedings 2S (2015) S125 – S130
positive domain of the FLD, i.e. negative and positive minor principal strains. The test temperature was maintained at 200 °C by heating up the tools, i.e. the die, holder and punch, along with the work-piece in a separate furnace prior to forming. As recommended by the ISO standard, the punch speed was set to 1 mm/s. The clamping force applied to hold the work-piece in place without causing edge fracture was set to 300 kN. The lubrication between the punch and the work-piece was established by the use of PTFE foil placed between two layers of lubrication oil. For checking the reproducibility of the test results, at least two work-pieces per geometry were tested. In addition to the force response and punch displacement, the history of the deformation field was recorded using a digital camera connected to the optical strain measuring system ARAMIS. Details on the necessary surface treatment are reported elsewhere [13]. 2.2. Finite element model The finite element (FE) modeling of the forming process aimed at matching the numerically obtained strain history to the one extracted from the experiments. Thus, the FE model followed directly from the experimental setup. The model was composed of three rigid tools (die, holder and punch) together with the work-piece, see Fig. 1. The tools and the work-pieces were assembled in a 3D modeling space. The work-piece was discretised with 4noded linear shell elements with reduced integration. In order to further reduce the computational cost, an orthogonal symmetry was assumed. Isothermal conditions were assumed within the simulations. Fig. 1 shows the FE-setup for the forming experiment leading to a “close uniaxial” strain path.
Fig. 1. Sample geometries I-VII used to obtain the forming limits (left); numerically predicted distribution of plastic equivalent strain for specimen geometry I (right)
All simulations reported in the following were conducted using the implicit solver of the commercial code ABAQUS. The underlying constitutive model was implemented as a user-defined subroutine (UMAT). 3. Constitutive model and localisation criterion A yield function suitable for describing the plastic deformation of magnesium alloys should capture the stress differential effect and the anisotropy of the material. A function based on tensor transforms complying with such requirements was presented in a series of papers by Cazacu and Barlat, e.g. [10], which was adopted in the present paper. The yield function is formulated using the modified second and third invariants ܬଶι and ଷι , respectively, of the transformed stress tensor
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D. Steglich et al. / Materials Today: Proceedings 2S (2015) S125 – S130
ª¬ J 2o Σ1 º¼
32
J 3o Σ3 W Y3
H1Σ, Σ2
0; Σ1
H 2 Σ .
(1)
By exploiting material symmetries and pressure independence, the number of coefficients in the linear transforms H1 and H2 can be reduced. They read in Voigt notation
Hi
§ c2i c3i ¨ 3 ¨ i ¨ c3 ¨ 3 ¨ i ¨ c2 3 ¨ ¨ 0 ¨ 0 ¨ ¨ 0 ©
c3i
c
i 1
3
c
i 3
c1i
c2i
3
3
c
i 3
c
i 1
3 3
c2i
3
0
0
0
0
0
0
0
0
c4i
0
0
0
0
c5i
0
0
0
0
· 0¸ ¸ ¸ 0¸ ¸ , i=1,2. 0¸ ¸ 0¸ ¸ 0¸ c6i ¸¹
(2)
Exponential functions were introduced for the model coefficients in Eq. 2 [14, 15], allowing to account for distortional hardening (distortion of the yield surface). The viscoplastic rate effect is captured by using a ratedependent hardening variable τY in Eq. 1, 3
W Y3
m ª §H · º 3 «W H ¨ ¸ » W 0 . E © ¹ ¼» ¬«
(3)
Here, τ0, β and m are model parameters, H represents the equivalent plastic strain. According to this choice, a rate effect is only accounted for the isotropic hardening part. The shape of the yield function (distortional hardening) is thus rate-independent. For the prediction of strain localisation the M-K model was adopted. Localisation is assumed to occur once the ratio of the strains in a band of reduced thickness to that in the parent region exceeds a prescribed critical value of 10. The most critical orientation of the band is that leading to the strongest thickness reduction, which was numerically probed in steps of 1 degree. 4. Results The mechanical responses obtained from the forming limit tests and the respective simulations are presented here in terms of limit and failure strain curves for both materials. Experimental results are indicated by solid symbols. Strain paths included in Fig. 2 were obtained from the optical system. In order to avoid overloaded graphs, only the strain paths resulting from geometries I, III, V and VII are discussed. The strain history at the pole of each specimen basically follows linear paths. Differences in the principal strain paths between the two materials were observed during the forming of the fully circular work-piece (geometry VII), for which an equi-biaxial loading condition is generally assumed. While for geometry VII the strain path appears as a straight line in case of ZE10, the major strain is evolving more rapidly than the minor strain in case of AZ31.
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I III V V II
major strain
0,6
F LC
strain path
failure exp. local. exp. M -K 1% M -K 0.5% M -K 0.1%
0,4
0,2
0,8
failure exp. local. exp. M -K 8% M -K 5% M -K 4%
0,4
0,2
Z E 10 0,0
F LC
I III V V II
0,6
major strain
strain path 0,8
-0,4
-0,2
0,0
minor strain
0,2
0,4
0,0
A Z 31 -0,4
-0,2
0,0
0,2
0,4
minor strain
Fig. 2. Comparison of experimentally determined and numerically predicted forming limits of ZE10 (left) and AZ31 (right); reprinted from [13], with permission from Elsevier
The numerical prediction of the forming limit strains is presented by employing the M-K localisation criterion. The required imperfection was obtained here by performing simulations with different imperfection magnitudes. The responses obtained are presented in Fig. 2. It can be seen that, the higher the initial imperfection, the earlier the localisation occurs. The range of imperfection values required strongly depends on the material considered. In the case of ZE10, the required imperfection for the M-K model is about 0.1 %. In the case of AZ31 with the distortional hardening model, imperfection values between 4.0 % and 5.0 % lead to limit strains within the range of the reference values observed in the respective experiments. These values are surprisingly high. This fact, together with apparent mismatch of the strain path for geometries III and V, calls to reconcile the plasticity model used in the simulations of the forming behavior of AZ31. In case of ZE10 the numerical predictions are regarded to be reasonably accurate. The constitutive law allows for a description of distortional hardening of the material. The respective model parameters were calibrated based on the evolution of r-values taken from tensile tests along three independent orientations, i.e. 0°, 45°, and 90° with respect to the rolling direction of the sheet. In the experiments reported, the rvalues were continuously determined over the whole range of deformation using optical strain field measurements as a local quantity. Although still a valid number, its significance is diminished in the necking regime compared to the previous (uniform) stage. In order to investigate the influence of this particular input, simulations based on the re-calibrated constitutive model assuming a constant r-value taken at a strain of 0.12 were performed for AZ31. Based on this input, the complete identification procedure was repeated, assuming constant parameters in the transformation tensors Hi, see Eq. 2. The distortional character of hardening is suppressed in this approach, while the orthotropic anisotropy of the material is still considered. The resulting punch force-displacement responses, presented in Fig. 3, generally show a reduced punch force compared to the distortional hardening model. The peak values for geometries I and III, however, are still underestimated. Nevertheless, the forming limits depicted in Fig. 3 are more realistic than the previously obtained ones. For geometries II and V, the strain path exactly follows the experimentally obtained ones, and an assumed imperfection of 1 % in the M-K model reveals the forming limit. The strain path of geometry I is still not captured to its full extent, but an improvement could be achieved for AZ31.
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40
punch force [kN]
30 25 20
strain path
S im. I II III IV V VI V II
0,8
15 10
F LC
I III V V II
0,6
major strain
E xp. 35
failure exp. local. exp. M -K 2% M -K 1.5% M -K 1%
0,4
0,2
5 0
0,0
0
10
20 30 40 punch displacement [mm]
50
-0,4
-0,2
0,0
0,2
0,4
minor strain
Fig. 3. Punch force-displacement record (left) and forming limits (right) for AZ31 from the recalibrated model in comparison to experiments; reprinted from [13], with permission from Elsevier
5. Conclusions The mechanical responses of the forming experiments revealed sufficient formability at 200°C. This was illustrated by a 0.5 limit major strain in the uniaxial state and a 0.3 limit strain in the equi-biaxial state of deformation. Moreover, and consistent with the large amount of non-uniform deformation recorded during the uniaxial tensile test, a large difference between the failure strain and the forming limit strain was observed in the domain of negative minor strain of the FLD. The analysis of strain paths revealed distortional hardening, which was accounted for in the FE-simulations. Two major influences on the M-K localisation criterion were confirmed here. The criterion is sensitive to the underlying plastic potential and the hardening law used. The latter is commonly based on an extrapolation of uniaxial tensile tests beyond the onset of diffuse necking and thus not directly measured. Although the planar anisotropy evidenced by the test has been captured very well by the distortional hardening model used here, the extrapolation of the r-value to strains beyond the localisation strain remains arguable. The implicitly assumed increase of the r-value causes an overprediction of localisation. This can successfully be circumvented by fitting the plastic potential to constant r-values, and it remains the choice of the user as to which strain level (and thus r-value) data will be considered. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
S.P. Keeler, in, Society of Automotive Engineers, 1965. K. Nakazima, T. Kikuma, K. Hasuka, in: F.R. Laboratory (Ed.), 1971, pp. 141-154. V.V. Hasek, Blech Rohre Profile, 25 (1978) 213- 220. G.M. Goodwin, in, Society of Automotive Engineers, 1968. S.P. Keeler, ASM Trans., 56 (1964) 25-48. R. Hill, Proc. Roy. Soc. London A, 193 (1948) 281-297. R. Hill, J. Mech. Phys. Solids, 38 (1990) 405-417. F. Barlat, D.J. Lege, J.C. Brem, International Journal of Plasticity, 7 (1991) 693-712. O. Cazacu, F. Barlat, Mathematics and Mechanics of Solids, 6 (2001) 513-630. O. Cazacu, F. Barlat, Int. J. Plast., 20 (2004) 2027-2045. M.N. Mekonen, D. Steglich, J. Bohlen, D. Letzig, J. Mosler, Materials Science and Engineering: A, 540 (2012) 174-186. Z. Marciniak, K. Kuczynski, International Journal of Mechanical Sciences, 9 (1967) 609-620. M.N. Mekonen, D. Steglich, J. Bohlen, L. Stutz, D. Letzig, J. Mosler, Materials Science and Engineering: A, 586 (2013) 204-214. S. Ertürk, W. Brocks, J. Bohlen, D. Letzig, D. Steglich, Int. J. Material Forming, 5 (2011) 325-339. D. Steglich, W. Brocks, J. Bohlen, F. Barlat, International Journal of Material Forming, 4 (2011) 243-253.
ScienceDirect Materials Today: Proceedings 2S (2015) S125 – S130
Conference MEFORM 2015, Light Metals – Forming Technologies and Further Processing
Experimental and numerical formability analysis of AZ31 and ZE10 sheets D. Steglicha,*, M. N. Mekonena, J. Bohlenb a
Helmholtz-Zentrum Geesthacht, Institute of Materials Research, Materials Mechanics, Max-Planck-Str. 1, D-21502 Geesthacht, Germany b Helmholtz-Zentrum Geesthacht, Institute of Materials Research, Magnesium Innovation Center MagIC, Max-Planck-Str. 1, D-21502 Geesthacht, Germany
Abstract Experimentally and numerically obtained mechanical responses of Nakazima-type sheet forming tests for the magnesium alloys ZE10 and AZ31 at elevated temperature (200°C) are explored. The results from the experiments reveal sufficient ductility at the prescribed test temperature allowing sheet forming processes. The material's anisotropy as well as the distortional character of hardening is confirmed. Differences in terms of strain paths during the forming experiments between the two materials are quantified. The corresponding numerical responses are obtained employing a suitable plasticity model based on tensor transforms. In addition, for predicting limit conditions of the forming process, the localisation criterion by Marciniak and Kuczynski is adopted. The constitutive model together with the localisation criterion is implemented in a finite element framework based on a fully implicit time integration scheme. The observed differential hardening during plastic deformation is predicted. The reasonably good agreement between the responses of the model and the respective experiments indicate the predictive capabilities of the implemented model for the considered magnesium alloys. The influence of the model parameter identification strategy is demonstrated. © 2014 The Authors. Published by Elsevier Ltd. © 2015 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND Selection and peer-review under responsibility of the Conference Committee of license Conference MEFORM 2015, Light Metals – (http://creativecommons.org/licenses/by-nc-nd/3.0/). Formingand Technologies and Further. This of is the an open accessCommittee article under the CC BY-NC-ND license responsibility Conference of Conference MEFORM 2015, Light Metals – Forming Selection peer-review under Technologies and Further (http://creativecommons.org/licenses/by-nc-nd/3.0/). Keywords: Magnesium alloys; Sheet forming; Finite element simulation; Plastic anisotropy; Localisation
* Corresponding author. Tel.: +49-4152-872543; fax: +49-4152-8742543 E-mail address: [email protected]
2214-7853 © 2015 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/). Selection and peer-review under responsibility of the Conference Committee of Conference MEFORM 2015, Light Metals – Forming Technologies and Further doi:10.1016/j.matpr.2015.05.029
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D. Steglich et al. / Materials Today: Proceedings 2S (2015) S125 – S130
1. Introduction Forming involves many procedures with a variety of strain paths, e.g. stretch forming or deep drawing to name two of them with distinctly different contribution of material flow from the sheets thickness or from the sheet plane. A comprehensive way to assess the formability of sheets accounting for these different aspects is based on experiments prescribing different strain paths and determining major strains in dependence of the minor strains [14]. A hemispherical punch is applied on a work-piece clamped in place by dies until strain localisation or fracture is observed. The results of such formability tests are commonly represented by the forming limit diagram (FLD), as proposed by Keeler [5]. From these tests a forming limit curve (FLC) results which represents strains of limiting uniform deformation as a function of the (linear) strain path. Limit strains are the boundary between safe and failed zones, the region above the forming-limit strain curve represents the failure zone. This measure of formability can then be applied in the design of forming processes. Besides, there is a need for accurate predictive simulation techniques for metal forming based on finite element (FE) analyses, which represent the current state-of-the-art in virtual prototyping. For realistic finite element predictions, it is vital to use accurate plasticity and localisation models. As the quality of the prediction strongly depends on the constitutive model used, its selection and its calibration becomes a key issue. Over the years, several authors have proposed a number of such models describing the yielding behavior in terms of macroscopic yield functions for various alloys, e.g. [6-9]. Cazacu and Barlat introduced a yield function (CaBa2004) as a modification of the Drucker model accounting for the material anisotropy as well as for the stress-differential effect observed in textured magnesium alloys [10]. This plasticity model has been successfully applied to predict the deformation of magnesium alloys [11] and is therefore employed in the following. The application of the aforementioned model requires a criterion for the determination of limiting conditions. In this work, an approach proposed by Marciniak and Kuczynski [12] is employed (M-K model). These authors showed that the presence of an imperfection in the sheet can lead to unstable deformation in the weaker region and to subsequent localised necking and failure. Although the M-K model is one of the early imperfection theories, its concept can still be regarded as useful due to its flexibility and simplicity. In an attempt to determine the formability of the two magnesium sheet alloys ZE10 and AZ31 at elevated temperature, formability tests were conducted. The respective results are reported [13] and replicated in the following. Numerical predictions are presented in the subsequent section and compared with the test results in terms of both, macroscopic quantities and local strain paths. The application of an advanced plasticity model in conjunction with a localisation criterion allowed a deeper insight into the interaction of field quantities and the impact of model parameter calibration. 2. Materials and methods Formability tests were conducted for two different magnesium alloy sheets (sheet thickness 1.3 mm), namely ZE10 (Mg-1Zn-Ce based mischmetal) and AZ31 (Mg-3Al-1Zn-Mn) in a heat treated condition (O-temper). ZE10 shows improved ductility at room temperature compared to AZ31 which is associated with the included rare earth elements which weaken the crystallographic texture, resulting in higher work hardening and increased ductility. The mechanical characterisation of these materials under uniaxial tensile loading conditions has been detailed previously [11]. Contrary to the common assumption, the computed r-values show a strain dependence. A positive strain rate effect on the yield strength is recorded while the opposite effect was observed on the ductility. The strain rate influence on the materials anisotropy is small. The reasonably large amount of achieved deformation for both alloys suggests the test temperature to be suitable for sheet forming processes. 2.1. Nakazima-type sheet forming tests The Nakazima-type forming limit tests were carried out based on the ISO 12004 standard using a universal sheet metal testing machine. The tests constitute seven circular work-pieces with an outer diameter of 200 mm and recesses of different radii, namely 80 mm, 72.5 mm, 65 mm, 57.5 mm, 50 mm, 40 mm, and 0 mm. The recesses on the work-pieces are all made perpendicular to the rolling direction. The tests cover the negative as well as the
D. Steglich et al. / Materials Today: Proceedings 2S (2015) S125 – S130
positive domain of the FLD, i.e. negative and positive minor principal strains. The test temperature was maintained at 200 °C by heating up the tools, i.e. the die, holder and punch, along with the work-piece in a separate furnace prior to forming. As recommended by the ISO standard, the punch speed was set to 1 mm/s. The clamping force applied to hold the work-piece in place without causing edge fracture was set to 300 kN. The lubrication between the punch and the work-piece was established by the use of PTFE foil placed between two layers of lubrication oil. For checking the reproducibility of the test results, at least two work-pieces per geometry were tested. In addition to the force response and punch displacement, the history of the deformation field was recorded using a digital camera connected to the optical strain measuring system ARAMIS. Details on the necessary surface treatment are reported elsewhere [13]. 2.2. Finite element model The finite element (FE) modeling of the forming process aimed at matching the numerically obtained strain history to the one extracted from the experiments. Thus, the FE model followed directly from the experimental setup. The model was composed of three rigid tools (die, holder and punch) together with the work-piece, see Fig. 1. The tools and the work-pieces were assembled in a 3D modeling space. The work-piece was discretised with 4noded linear shell elements with reduced integration. In order to further reduce the computational cost, an orthogonal symmetry was assumed. Isothermal conditions were assumed within the simulations. Fig. 1 shows the FE-setup for the forming experiment leading to a “close uniaxial” strain path.
Fig. 1. Sample geometries I-VII used to obtain the forming limits (left); numerically predicted distribution of plastic equivalent strain for specimen geometry I (right)
All simulations reported in the following were conducted using the implicit solver of the commercial code ABAQUS. The underlying constitutive model was implemented as a user-defined subroutine (UMAT). 3. Constitutive model and localisation criterion A yield function suitable for describing the plastic deformation of magnesium alloys should capture the stress differential effect and the anisotropy of the material. A function based on tensor transforms complying with such requirements was presented in a series of papers by Cazacu and Barlat, e.g. [10], which was adopted in the present paper. The yield function is formulated using the modified second and third invariants ܬଶι and ଷι , respectively, of the transformed stress tensor
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ª¬ J 2o Σ1 º¼
32
J 3o Σ3 W Y3
H1Σ, Σ2
0; Σ1
H 2 Σ .
(1)
By exploiting material symmetries and pressure independence, the number of coefficients in the linear transforms H1 and H2 can be reduced. They read in Voigt notation
Hi
§ c2i c3i ¨ 3 ¨ i ¨ c3 ¨ 3 ¨ i ¨ c2 3 ¨ ¨ 0 ¨ 0 ¨ ¨ 0 ©
c3i
c
i 1
3
c
i 3
c1i
c2i
3
3
c
i 3
c
i 1
3 3
c2i
3
0
0
0
0
0
0
0
0
c4i
0
0
0
0
c5i
0
0
0
0
· 0¸ ¸ ¸ 0¸ ¸ , i=1,2. 0¸ ¸ 0¸ ¸ 0¸ c6i ¸¹
(2)
Exponential functions were introduced for the model coefficients in Eq. 2 [14, 15], allowing to account for distortional hardening (distortion of the yield surface). The viscoplastic rate effect is captured by using a ratedependent hardening variable τY in Eq. 1, 3
W Y3
m ª §H · º 3 «W H ¨ ¸ » W 0 . E © ¹ ¼» ¬«
(3)
Here, τ0, β and m are model parameters, H represents the equivalent plastic strain. According to this choice, a rate effect is only accounted for the isotropic hardening part. The shape of the yield function (distortional hardening) is thus rate-independent. For the prediction of strain localisation the M-K model was adopted. Localisation is assumed to occur once the ratio of the strains in a band of reduced thickness to that in the parent region exceeds a prescribed critical value of 10. The most critical orientation of the band is that leading to the strongest thickness reduction, which was numerically probed in steps of 1 degree. 4. Results The mechanical responses obtained from the forming limit tests and the respective simulations are presented here in terms of limit and failure strain curves for both materials. Experimental results are indicated by solid symbols. Strain paths included in Fig. 2 were obtained from the optical system. In order to avoid overloaded graphs, only the strain paths resulting from geometries I, III, V and VII are discussed. The strain history at the pole of each specimen basically follows linear paths. Differences in the principal strain paths between the two materials were observed during the forming of the fully circular work-piece (geometry VII), for which an equi-biaxial loading condition is generally assumed. While for geometry VII the strain path appears as a straight line in case of ZE10, the major strain is evolving more rapidly than the minor strain in case of AZ31.
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I III V V II
major strain
0,6
F LC
strain path
failure exp. local. exp. M -K 1% M -K 0.5% M -K 0.1%
0,4
0,2
0,8
failure exp. local. exp. M -K 8% M -K 5% M -K 4%
0,4
0,2
Z E 10 0,0
F LC
I III V V II
0,6
major strain
strain path 0,8
-0,4
-0,2
0,0
minor strain
0,2
0,4
0,0
A Z 31 -0,4
-0,2
0,0
0,2
0,4
minor strain
Fig. 2. Comparison of experimentally determined and numerically predicted forming limits of ZE10 (left) and AZ31 (right); reprinted from [13], with permission from Elsevier
The numerical prediction of the forming limit strains is presented by employing the M-K localisation criterion. The required imperfection was obtained here by performing simulations with different imperfection magnitudes. The responses obtained are presented in Fig. 2. It can be seen that, the higher the initial imperfection, the earlier the localisation occurs. The range of imperfection values required strongly depends on the material considered. In the case of ZE10, the required imperfection for the M-K model is about 0.1 %. In the case of AZ31 with the distortional hardening model, imperfection values between 4.0 % and 5.0 % lead to limit strains within the range of the reference values observed in the respective experiments. These values are surprisingly high. This fact, together with apparent mismatch of the strain path for geometries III and V, calls to reconcile the plasticity model used in the simulations of the forming behavior of AZ31. In case of ZE10 the numerical predictions are regarded to be reasonably accurate. The constitutive law allows for a description of distortional hardening of the material. The respective model parameters were calibrated based on the evolution of r-values taken from tensile tests along three independent orientations, i.e. 0°, 45°, and 90° with respect to the rolling direction of the sheet. In the experiments reported, the rvalues were continuously determined over the whole range of deformation using optical strain field measurements as a local quantity. Although still a valid number, its significance is diminished in the necking regime compared to the previous (uniform) stage. In order to investigate the influence of this particular input, simulations based on the re-calibrated constitutive model assuming a constant r-value taken at a strain of 0.12 were performed for AZ31. Based on this input, the complete identification procedure was repeated, assuming constant parameters in the transformation tensors Hi, see Eq. 2. The distortional character of hardening is suppressed in this approach, while the orthotropic anisotropy of the material is still considered. The resulting punch force-displacement responses, presented in Fig. 3, generally show a reduced punch force compared to the distortional hardening model. The peak values for geometries I and III, however, are still underestimated. Nevertheless, the forming limits depicted in Fig. 3 are more realistic than the previously obtained ones. For geometries II and V, the strain path exactly follows the experimentally obtained ones, and an assumed imperfection of 1 % in the M-K model reveals the forming limit. The strain path of geometry I is still not captured to its full extent, but an improvement could be achieved for AZ31.
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Fig. 3. Punch force-displacement record (left) and forming limits (right) for AZ31 from the recalibrated model in comparison to experiments; reprinted from [13], with permission from Elsevier
5. Conclusions The mechanical responses of the forming experiments revealed sufficient formability at 200°C. This was illustrated by a 0.5 limit major strain in the uniaxial state and a 0.3 limit strain in the equi-biaxial state of deformation. Moreover, and consistent with the large amount of non-uniform deformation recorded during the uniaxial tensile test, a large difference between the failure strain and the forming limit strain was observed in the domain of negative minor strain of the FLD. The analysis of strain paths revealed distortional hardening, which was accounted for in the FE-simulations. Two major influences on the M-K localisation criterion were confirmed here. The criterion is sensitive to the underlying plastic potential and the hardening law used. The latter is commonly based on an extrapolation of uniaxial tensile tests beyond the onset of diffuse necking and thus not directly measured. Although the planar anisotropy evidenced by the test has been captured very well by the distortional hardening model used here, the extrapolation of the r-value to strains beyond the localisation strain remains arguable. The implicitly assumed increase of the r-value causes an overprediction of localisation. This can successfully be circumvented by fitting the plastic potential to constant r-values, and it remains the choice of the user as to which strain level (and thus r-value) data will be considered. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
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