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Characterisation of kinematic hardening and yield surface evolution from uniaxial to biaxial tension with continuous strain path change
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CIRP-1150; No. of Pages 4 CIRP Annals - Manufacturing Technology xxx (2014) xxx–xxx
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CIRP Annals - Manufacturing Technology jou rnal homep age : ht t p: // ees .e lse vi er . com /ci r p/ def a ult . asp
Characterisation of kinematic hardening and yield surface evolution from uniaxial to biaxial tension with continuous strain path change Marion Merklein (2)a,*, Sebastian Suttner a, Alexander Brosius b a b
Institute of Manufacturing Technology LFT, Friedrich-Alexander-Universita¨t Erlangen-Nu¨rnberg, Germany Chair of Forming and Machining Processes FF, TU Dresden, Germany
A R T I C L E I N F O
A B S T R A C T
Keywords: Sheet metal Stress Kinematic hardening
With respect to multistage forming processes the material behaviour and the history of the strain path during the process is of special interest for the improvement of the numerical prediction of forming processes. While different researchers investigated the Bauschinger effect during a load reversal and biaxial loading with pre-strained specimens, in this paper the yield locus evolution in the first quadrant of the principles stress space under biaxial loading of a modified cruciform specimen without test interruption is presented. The movement of the yield surface centre caused by kinematic hardening is approximated by an alternative approach based on experimental results. ß 2014 CIRP.
1. Introduction
initiated stress along the yield surface. These characteristics at strain path changes are of special interest for the material description in the numerical prediction of a multistage forming operation. Continuous strain path change was investigated by Ha et al. [9] that gives an overview of the topic and focuses on continuous strain path change in a numerical process from plane strain tension to simple shear and compared it to experimental observations of van Riel and van den Boogard [10]. Characterising the material with online strain path change is needed due to the effect of early relaxation after pre-straining as investigated by Meyers et al. [11]. Moreover, the results showed a significant relaxation of the specimen in the first few minutes after pre-straining. This is confirmed by Hannula et al. [12] and can be prevented by online switching the strain path during the test without unloading the specimen for secondary observations.
Increasing demands on weight reduction and the complexity of products lead to the development of new materials. To enhance the numerical prediction of forming operations, a proper material characterisation is essential. Therefore, an accurate analytical modelling requires yield stress data of the workpiece material [1]. Considering the anisotropic behaviour of sheet metals is important for realistic finite-element-simulation results [2]. The determination of the material’s characteristic only at a uniaxial stress state often leads to a falsified approximation of the material behaviour in the numerical model [3]. An investigation of the material at different states of stress, e.g. simple shear and biaxial tension improve the understanding of the material behaviour for numerical modelling. Banabic and Sester [4] depicted the influence of the chosen material model on the accuracy of the numerical process design. Besides, Sillekens et al. [5] described the history of the strain before altering as pre-strain and referred to changes in the plastic material behaviour during a change from a straight strain path as existing in a uniaxial tensile test. Tensile tests with subsequent torsion tests and torsion tests with following tensile tests of a rod clarified a dependency of the material properties to the strain path [6]. Larsson et al. [7] used large tensile specimen for a uniaxial prestraining with subsequent extraction of secondary specimen for further investigation. Although the material behaviour under prestraining can be investigated, the unclamping after pre-straining leads to a removal of the elastic strain and a change in the stress state. Kuwabara et al. [8] proposed a method for analysing the material at abrupt strain path changes without unloading in a servocontrolled biaxial tensile test machine for a movement of the * Corresponding author. +49 9131 8527140. E-mail address: [email protected] (M. Merklein).
2. Experimental setup and methodology 2.1. Choice of material Since deep drawing steels are frequently applied in sheet metal forming processes, the investigated material is chosen to the coldrolled deep drawing steel DC06 with a specified yield strength YS between 120 and 170 MPa and an initial sheet thickness t0 = 1.0 mm. 2.2. Biaxial tensile test setup To realise a better application of the material behaviour in a numerical forming process, the biaxial yield stress sb and the biaxial r-value rb are needed for the parameter identification of complex yield criteria. The benefit of a displacement driven experiment is that rb remains constantly 1.0. The biaxial tensile test machine is based on the displacement driven stand-alone machine developed by Merklein and Biasutti [13].
http://dx.doi.org/10.1016/j.cirp.2014.03.039 0007-8506/ß 2014 CIRP.
Please cite this article in press as: Merklein M, et al. Characterisation of kinematic hardening and yield surface evolution from uniaxial to biaxial tension with continuous strain path change. CIRP Annals - Manufacturing Technology (2014), http://dx.doi.org/10.1016/ j.cirp.2014.03.039
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CIRP-1150; No. of Pages 4 M. Merklein et al. / CIRP Annals - Manufacturing Technology xxx (2014) xxx–xxx
2
Table 1 Mechanical parameters of DC06 at the yield stress s in uniaxial and biaxial tension and after strain path change. Parameter
s (MPa)
r
sbx (MPa)
sby (MPa)
08 R.D. 458 R.D. 908 R.D. epre = 0.05 epre = 0.10 epre = 0.12
138.0 0.96 139.4 0.39 139.9 1.21 139.7 2.70 135.5 2.58 141.8 2.22
1.98 0.09 1.72 0.04 2.36 0.13 1.97 0.08 1.90 0.02 2.01 0.02
– – – 292.3 0.87 358.9 1.16 372.9 2.92
– – – 222.6 4.75 261.9 2.57 269.6 4.14
Biaxial
Fig. 1. Schematic sectional view of the x-direction of the biaxial tensile test machine (left) with clamped cruciform specimen in the centre (right).
The tensile load is introduced by a link mechanism and the motion of the movable table in z-direction. The mounting of the bearing to an angular adjustment converts the load Fz of the lifting gear into the in-plane tensile forces Fx and Fy in x- and y-direction, as can be seen schematically for one side of the x-direction in Fig. 1. The specimen consists of a cruciform shaped geometry with a maximum length of 120.0 mm and a width of 19.8 mm. Moreover, seven slits with a length of 20.0 mm and a width of 0.2 mm are realised in each arm by laser cutting in order to homogenise the plastic deformation. A thickness reduction of the sheet metal to 0.5 mm is attained by milling the centre of the specimen to a gauge section of 16.4 16.4 mm2. The local strains are measured using a two-dimensional CCD-camera and an optical strain measurement system (ARAMIS, GOM mbH). 2.3. Methodology of online strain path variation An approach is presented to realise an online change of the stress state from uniaxial tension to biaxial tension without an extensive control system by modification of the cruciform specimen. The modification includes the variation of the specimen length ly in ydirection between 122.5 mm and 127.5 mm in a variation step of 2.5 mm. The length lx in x-direction remains constant at 120.0 mm. Tests with three different length ratios ly/lx are carried out. Through the length ratio higher than 1.0 only a load in the rolling direction (R.D.) is applied at the beginning of the test because of the earlier contact of the bearing in x-direction to the movable table. During the test the adjusting contact of the bearing in y-direction leads to the application of a further tensile force perpendicular to the R.D. (ydirection). Thus, a continuous change of the stress state from uniaxial tension to biaxial tension is induced. The machine speed is set to 3 mm/min for all tests. This involves an equivalent strain rate of 0.0029 1/s. Tests at 0.05, 0.10 and 0.12 uniaxial pre-strain in R.D. are carried out for the material DC06 (Fig. 2). The true stress is defined by the ratio of the current load to the instantaneous cross section in each principal direction. The true
Biaxial tensile test setup Material: DC06 (t0 = 0.5 mm, N = 4)
500
True stress
MPa 300
pre-strain 0.12 pre-strain 0.10 pre-strain 0.05 experiment x experiment y uniaxial 0° R.D.
200
100 0 0.00
0.05
0.10 0.15 True strain x
-
0.25
Fig. 2. Principal true stress-true strain curves of DC06 in x- and y-direction with online strain path changes at different levels of uniaxial pre-straining (epre = 0.05, 0.10 and 0.12).
rb
sbx (MPa)
sby (MPa)
1.00
160.1 0.98
157.3 0.91
stress vs. strain curve of a uniaxial tensile test in R.D. is also demonstrated (red dashed line). In case of the pre-strained cruciform specimen, the progress of the principal true stresses is displayed. The comparison of the experimental results of the modified test setup features a good reproducibility for defined levels of pre-straining. The progression of the hardening behaviour has a close overlap to the uniaxial tensile test results until the onset of strain path changing. Regarding the beginning of plastic yielding and the continuing work hardening behaviour, especially at the investigated levels of pre-straining, the difference is negligible. As listed in Table 1, the comparison of the beginning of plastic yielding s0 in R.D. with 138.0 MPa 0.96 MPa for the uniaxial tensile test and the yield strength 139.7 MPa 2.70 MPa, 135.5 MPa 2.58 MPa and 141.8 MPa 2.22 MPa for uniaxial pre-straining of the modified cruciform specimen, demonstrates the analogy of the test setups. Besides, the components of the beginning of plastic yielding sbx, sby for the standard configuration of the biaxial tensile test are determined with the assumption of constant plastic work. The subsequent biaxial yield locus (Table 1) is represented through the principal yield stresses sbx and sby at 0.05, 0.10 and 0.12 of uniaxial pre-strain. It can be seen that pre-straining leads to an increase of the subsequent plastic yield stress with a higher value of the yield stress component sbx in R.D., the direction of uniaxial pre-straining. 3. Numerical modelling and experimental results 3.1. Material modelling and Bauschinger coefficient In order to analyse the yield surface movement regarding the influence of isotropic and kinematic hardening, the comparison of yield loci of the experimental results to a material model is shown in Fig. 4. To describe the plastic material behaviour of DC06 the complex yield criterion Yld2000-2d [14] is chosen for investigating the subsequent yield loci. Because of better fitting results, the exponent m is set to 8.0. In case of the conventional criteria of von Mises or Hill’48 [15], no differentiation of the material behaviour at biaxial or shear stress state is made. This can lead to a wrong approximation of the biaxial stress state and to inaccurate results of the numerical calculation. The work hardening of the material is mapped with the isotropic hardening law of Hockett–Sherby [16] based on the approximation of the flow curve under uniaxial tension in 08 to R.D. (Table 2). To investigate the influence of kinematic hardening, the Bauschinger effect is analysed and the Bauschinger coefficient b, that describes the relationship between the beginning of plastic yielding sr under subsequent compressive Table 2 Identified material parameter of the yield criterion Yld2000-2d and the isotropic hardening law Hockett–Sherby for DC06. Yield criterion Yld2000-2d (m = 8)
a1
a2
a3
a4
a5
a6
a7
a8
1.027
1.026
0.842
0.908
0.909
0.737
1.011
1.062
Hardening law Hockett–Sherby (s
m HS = A Bexp(Cepl ))
A
B
C
m
415.505
277.505
6.508
0.863
Please cite this article in press as: Merklein M, et al. Characterisation of kinematic hardening and yield surface evolution from uniaxial to biaxial tension with continuous strain path change. CIRP Annals - Manufacturing Technology (2014), http://dx.doi.org/10.1016/ j.cirp.2014.03.039
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600
0° R.D. 90° R.D.
True stress
MPa
3
Modified setup vs. FEM models Material: DC06 (t0 = 0.5 mm)
400 300 experiment Hill'48 Swift isotropic Yld2000-2d HS isotropic Yld2000-2d HS =0.73
200 100 0 0.12
0
0.14 0.16 True strain
Fig. 3. Tension–compression test setup according to [17].
load and the maximum yield stress sm under tension, is determined by tension-compression tests according to the test setup of Staud et al. [17]. Moreover, distortional hardening is not explicitly investigated in this work. The specimen consists of an effective range of 2.0 mm 2.0 mm in the centre (Fig. 3). The strain was detected by an optical measurement system. Due to the effect of buckling under compressive load, the Bauschinger coefficient was determined at a maximum pre-strain of 0.14. In case of the investigated material the Bauschinger coefficient b was estimated with a constant value of 0.73 and included in the material model. The initial yield surface is calculated from the experimental results of the uniaxial tensile tests in 08, 458 and 908 to R.D. and the biaxial tensile test. The evolution of the initial yield surface for the investigated levels of plastic pre-strain is shown in Fig. 4. The comparison of subsequent yield loci under uniaxial pre-straining and the analytically identified material model with pure isotropic (Yld2000-2d, isotropic) and isotropic-kinematic hardening (Yld2000-2d, b = 0.73) reveals the translation of the experimental yield locus caused by the uniaxial pre-strain. Comparing the equibiaxal yield stress of the pure isotropic material model with the experimentally determined subsequent yield loci, a large deviation between the pure isotropic model and the experiment can be observed (Fig. 4). Using an isotropic-kinematic modelling, a reduced deviation can be seen, leading to the conclusion that a kinematic hardening component is present in the material.
Fig. 4. Subsequent yield loci of DC06 with online strain path changes at different levels of uniaxial pre-straining (epre = 0.05, 0.10 and 0.12) compared to the yield criterion Yld2000-2d with a pure isotropic and an isotropic-kinematic (b = 0.73) hardening law.
-
0.22
Fig. 5. Comparison of the true stresses vs. strain curves of the subsequent material behaviour of the investigated material DC06 to numerical simulations (isotropic and isotropic-kinematic hardening).
plastic yielding in comparison to the experimental results is affected by disregarding the biaxial stress state in the analytical model formulation. Analogously to the analytical prediction of the evolution of yield loci (Fig. 4), the material model Yld2000-2d with isotropic and kinematic hardening component leads to a better application of the subsequent hardening behaviour under strain path change than the pure isotropic model (Fig. 5). According to the results of modelling the material behaviour under online strain path change with a kinematic hardening component, the translation of the yield surface in Fig. 4 also leads to a change of the tangent’s gradient in the uniaxial stress direction. Therefore, a new approach is presented to use the change of the gradient during plastic deformation. 4. Alternative estimation of kinematic hardening behaviour The aim of this alternative approach is the estimation of the isotropic and kinematic portions of hardening based on experimental results from the proposed combined uniaxial-biaxial tensile test without unloading. During the uniaxial tensile test, the r-values remain not on a constant level, but rather have a non-linear progress [18]. Therefore, the r-values are usually selected at a defined and material dependent pre-strain level in order to fix the shape of the yield locus [19]. This procedure is meaningful, when the derived material parameters are used with an isotropic hardening law within a finite-element-analysis. But the described effect of r-value evolution is generated by the combination of isotropic and kinematic hardening, i.e. the growth and translation of the yield surface in stress space. To illustrate the effect of isotropic-kinematic hardening behaviour on the evolution of the yield locus and the observed r-value, a one element finite-element-analysis was done using the parameters of DC06 and Barlat’ yield criterion (Table 2). Fig. 6 shows the translated and grown yield locus for different combinations of isotropic-kinematic hardening with b-values of 1.0 and 0.73 respectively. Furthermore, Fig. 6 shows the corresponding evolution of the r0-value, which can be derived from the tangent on the intersection of the yield surface with the s1-axis (compare Fig. 4). During pure isotropic hardening behaviour (b = 1.0) this point remains at the same position on the yield surface during plastic deformation and no change of the
2.3
500
3.2. Comparison to the numerical simulation model
2.2
r0-value
An implicit one-element numerical simulation is established in LS-DYNA v.9.7.1 (LSTC) with a pure isotropic and an isotropickinematic material model. The identified criterion Yld2000-2d with the isotropic hardening law of Hockett–Sherby (HS) is implemented. The kinematic behaviour is expressed through the Bauschinger coefficient (b = 0.73). The progress of the principal stresses is exemplarily shown in case of the maximum pre-strain level 0.12 in Fig. 5. In order to investigate the influence of abstracting away from the biaxial stress, a third simulation model is shown in Fig. 5 that consists of the yield criterion Hill’48. The strong deviation of the subsequent beginning of
0.18 x
2.1
500
-500
-500
2
0
0.1 0.2 0.3 Equivalent plastic strain
Material: DC06, t0 = 0.5 mm Initial yield surface = 1.0 (isotropic)
= 0.73
Fig. 6. Evolution of yield locus during tensile test and resulting evolution of the r0value.
Please cite this article in press as: Merklein M, et al. Characterisation of kinematic hardening and yield surface evolution from uniaxial to biaxial tension with continuous strain path change. CIRP Annals - Manufacturing Technology (2014), http://dx.doi.org/10.1016/ j.cirp.2014.03.039
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tangent will occur. If isotropic-kinematic hardening takes place (0 < b < 1.0), the evolving back stress tensor X causes a movement of the yield locus. This leads to a continuously change of intersection points, which results in the change of the r0-value. The calculation procedure to calculate the back stress tensor X based on one load path change without unloading, is described in the following. The r0-value gives the information about the relationship between the components of the incremental principal plastic strain de2 and de3, as well as de1 by using the law of volume constancy. Using the associated flow rule, the ratio between the components of dei and the corresponding deviatoric stress components is constant for every deformation step. Regarding the change of the back stress tensor components dXi it is assumed, that they are proportional to the (anisotropic) deviatoric stress components. This relationship can be expressed by the change of the r0-values dr0, as given by Eqs. (1) and (2). dX 1 ð1 þ dr 0 Þ ¼ dr 0 dX 2
(1)
dX 1 ¼ ð1 þ dr 0 Þ dX 3
(2)
Because de1 is continuously measured during the test, all other components of the principal strain tensor, and the ratio between all components of the back stress tensor X can be directly calculated for every measured deformation step. To identify not only the ratios but also the values of X, two different stress states have to be generated without unloading to keep the back stress tensor constant. The first considered stress state (I: s tensile) is related to the last uniaxial tensile stress and the second stress state (II: s biaxial) is given by the first biaxial stress situation. If the specimen is continuously in a loading situation that fulfils the yield condition, the assumption of a constant back stress tensor X during the load path change is possible. Therefore, the yield condition F for the two stress states s tensile and s biaxial is equal and can be written as shown in Eq. (3).
F ¼ f ðs tensile XÞ ¼ f ðs biaxial XÞ F ¼ f ðs compression XÞ
(3)
Because the relationship of all components from the back stress tensor Xi can be derived using the mentioned equations and the online measurement data, a direct identification procedure with a Newton algorithm for the root-finding is possible. Only the first principal back stress X1 has to be considered in order to equalise the two stress states and fulfil the yield condition in Eq. (3). The theoretical Bauschinger coefficient b can be directly calculated using Eq. (3) with the identified back stress tensor X. The yield condition as well as the back stress tensor will be kept constant and the uniaxial compression stress state scompression as well as the Bauschinger coefficient can be directly calculated. The application of the described procedure on the experimental data from the biaxial test with uniaxial pre-straining leads to the results shown in Table 3. Thereby, b0.02 and b0.05 indicate the Bauschinger coefficient related to a stress condition, where a replastification of De = 0.02 respectively De = 0.05 is present. The result shows an evolution of the Bauschinger coefficient similar to the evolution of the r0-value. The deformation starts with a nearly isotropic expansion of the yield locus (b nearby 1). With larger pre-strain values, kinematic hardening takes place (0 < b < 1.0). Obviously, the Bauschinger coefficients b0.02, b0.05 and b are different to the initially observed value of b = 0.73, which can be explained by two reasons: Table 3 Determined Bauschinger coefficients. Pre-strain epre Bauschinger coefficient
b b 0.02 b 0.05
0.05
0.10
0.12
0.93 0.99 1.00
0.54 0.69 0.77
0.54 0.66 0.77
(a) During the initially mentioned identification procedure using a tension–compression test, a very smooth transition between elastic and elastic-plastic region is observed under load reversal nearby the compression stress state scompression. This effect cannot be approximated by the described identification strategy, due to the used yield criterion. To overcome this, a more complex kinematic hardening model like Chaboche has to be used or a fixed level of re-plastification De has to be defined. (b) A direct, abrupt change from uniaxial to biaxial stress state is needed for the assumption of a constant back stress tensor during load path change. This is not exactly achievable in any experiment, which causes an additional deviation.
5. Conclusion A new experimental approach is presented in this paper, to investigate the evolution of the yield locus after online strain path change. Moreover, numerical simulations are carried out to assess the accuracy of the material model. Concerning the influence of kinematic hardening an alternative approach is proposed to estimate the kinematic hardening behaviour by analysing the rvalue progression during plastic deformation. Acknowledgements The authors are grateful to the German Research Foundation (DFG) for supporting the research project ‘‘Identification and modelling of material characteristics for Finite-Element-Analysis of sheet metal forming processes’’ ME (2043/11-2) and BR (3500/6-2).
References [1] Shatla M, Kerk C, Altan T (2001) Process modeling in machining. Part I: determination of flow stress data. Int J Mach Tool Manu 41(10):1511–1534. [2] Gu¨ner A, Soyarslan C, Brosius A, Tekkaya AE (2012) Characterization of the anisotropy of sheet metals employing inhomogneneous strain fields for Yld2000-2d yield function. Int J Solids Struct 49(25):3517–3527. [3] Hannon A, Tiernan P (2008) A review of planar biaxial tensile test systems for sheet metal. J Mater Process Technol 198:1–13. [4] Banabic D, Sester M (2012) Influence of material models on the accuracy of the sheet forming simulation. Mater Manuf Process 27:304–308. [5] Sillekens WH, Dautzenberg JH, Kals JAG (1988) Flow curves for C45 steel at abrupt changes in the strain path. CIRP Ann Manuf Technol 37(1):213–216. [6] Sillekens WH, Dautzenberg JH, Kals JAG (1991) Strain path dependence of flow curves. CIRP Ann Manuf Technol 40(1):255–258. [7] Larsson R, Bjo¨rklund O, Nilsson L, Simonsson K (2011) A study of high strength steels undergoing non-linear strain paths – experiments and modelling. J Mater Process Technol 211(1):122–132. [8] Kuwabara T, Van Bael A, Iizuka E (2002) Measurement and analysis of yield locus and work hardening characteristics of steel sheets with different rvalues. Acta Mater 50(14):3717–3729. [9] Ha J, Kim J-H, Barlat F, Lee M-G (2014) Continuous strain path change simulations for sheet metal. Comp Mater Sci 82:286–292. [10] van Riel M, van den Boogaard AH (2007) Stress–strain responses for continuous orthogonal strain path changes with increasing sharpness. Scripta Mater 57:5381–5384. [11] Meyers MA, Guimaraes JRC, Avillez RR (1979) On stress-relaxation experiments and their significance under strain-aging conditions. Metall Trans A 10(1):33–40. [12] Hannula S-P, Korhonen MA, Li C-Y (1986) Strain aging and load relaxation behavior of type 316 stainless steel at room temperature. Metall Mater Trans A 17(10):1757–1767. [13] Merklein M, Biasutti M (2013) Development of a biaxial tensile machine for characterization of sheet metals. J Mater Process Technol 213(6):939–946. [14] Barlat F, Brem JC, Yoon JW, Chung K, Dick RE, Lege DJ, Pourboghrat F, Choi S-H, Chu E (2003) Plane stress yield function for aluminum alloy sheets – part 1: theory. Int J Plasticity 19(9):1297–1319. [15] Hill R (1948) A theory of the yielding and plastic flow of anisotropic metals. Proc R Soc Lond A: Math Phys Sci 193(1033):281–297. [16] Hockett JE, Sherby OD (1975) Large strain deformation of polycrystalline metals at low homologous temperatures. J Mech Phys Solids 23:87–98. [17] Staud D, Merklein M (2009) Zug-Druck-Versuche an Miniaturproben zur Erfassung von Parametern fu¨r kinematische Verfestigungsmodelle. Borsutzki M, Geisler S, (Eds.) Tagungsband Werkstoffpru¨fung, vol. 2. 211–218. [18] An YG, Vegter H, Melzer S, Triguero PR (2013) Evolution of the plastic anisotropy with straining and its implication on formability for sheet metals. J Mater Process Technol 213(8):1419–1425. [19] Tekkaya AE (2000) State-of-the-art of simulation of sheet metal forming. J Mater Process Technol 103(1):14–22.
Please cite this article in press as: Merklein M, et al. Characterisation of kinematic hardening and yield surface evolution from uniaxial to biaxial tension with continuous strain path change. CIRP Annals - Manufacturing Technology (2014), http://dx.doi.org/10.1016/ j.cirp.2014.03.039
CIRP-1150; No. of Pages 4 CIRP Annals - Manufacturing Technology xxx (2014) xxx–xxx
Contents lists available at ScienceDirect
CIRP Annals - Manufacturing Technology jou rnal homep age : ht t p: // ees .e lse vi er . com /ci r p/ def a ult . asp
Characterisation of kinematic hardening and yield surface evolution from uniaxial to biaxial tension with continuous strain path change Marion Merklein (2)a,*, Sebastian Suttner a, Alexander Brosius b a b
Institute of Manufacturing Technology LFT, Friedrich-Alexander-Universita¨t Erlangen-Nu¨rnberg, Germany Chair of Forming and Machining Processes FF, TU Dresden, Germany
A R T I C L E I N F O
A B S T R A C T
Keywords: Sheet metal Stress Kinematic hardening
With respect to multistage forming processes the material behaviour and the history of the strain path during the process is of special interest for the improvement of the numerical prediction of forming processes. While different researchers investigated the Bauschinger effect during a load reversal and biaxial loading with pre-strained specimens, in this paper the yield locus evolution in the first quadrant of the principles stress space under biaxial loading of a modified cruciform specimen without test interruption is presented. The movement of the yield surface centre caused by kinematic hardening is approximated by an alternative approach based on experimental results. ß 2014 CIRP.
1. Introduction
initiated stress along the yield surface. These characteristics at strain path changes are of special interest for the material description in the numerical prediction of a multistage forming operation. Continuous strain path change was investigated by Ha et al. [9] that gives an overview of the topic and focuses on continuous strain path change in a numerical process from plane strain tension to simple shear and compared it to experimental observations of van Riel and van den Boogard [10]. Characterising the material with online strain path change is needed due to the effect of early relaxation after pre-straining as investigated by Meyers et al. [11]. Moreover, the results showed a significant relaxation of the specimen in the first few minutes after pre-straining. This is confirmed by Hannula et al. [12] and can be prevented by online switching the strain path during the test without unloading the specimen for secondary observations.
Increasing demands on weight reduction and the complexity of products lead to the development of new materials. To enhance the numerical prediction of forming operations, a proper material characterisation is essential. Therefore, an accurate analytical modelling requires yield stress data of the workpiece material [1]. Considering the anisotropic behaviour of sheet metals is important for realistic finite-element-simulation results [2]. The determination of the material’s characteristic only at a uniaxial stress state often leads to a falsified approximation of the material behaviour in the numerical model [3]. An investigation of the material at different states of stress, e.g. simple shear and biaxial tension improve the understanding of the material behaviour for numerical modelling. Banabic and Sester [4] depicted the influence of the chosen material model on the accuracy of the numerical process design. Besides, Sillekens et al. [5] described the history of the strain before altering as pre-strain and referred to changes in the plastic material behaviour during a change from a straight strain path as existing in a uniaxial tensile test. Tensile tests with subsequent torsion tests and torsion tests with following tensile tests of a rod clarified a dependency of the material properties to the strain path [6]. Larsson et al. [7] used large tensile specimen for a uniaxial prestraining with subsequent extraction of secondary specimen for further investigation. Although the material behaviour under prestraining can be investigated, the unclamping after pre-straining leads to a removal of the elastic strain and a change in the stress state. Kuwabara et al. [8] proposed a method for analysing the material at abrupt strain path changes without unloading in a servocontrolled biaxial tensile test machine for a movement of the * Corresponding author. +49 9131 8527140. E-mail address: [email protected] (M. Merklein).
2. Experimental setup and methodology 2.1. Choice of material Since deep drawing steels are frequently applied in sheet metal forming processes, the investigated material is chosen to the coldrolled deep drawing steel DC06 with a specified yield strength YS between 120 and 170 MPa and an initial sheet thickness t0 = 1.0 mm. 2.2. Biaxial tensile test setup To realise a better application of the material behaviour in a numerical forming process, the biaxial yield stress sb and the biaxial r-value rb are needed for the parameter identification of complex yield criteria. The benefit of a displacement driven experiment is that rb remains constantly 1.0. The biaxial tensile test machine is based on the displacement driven stand-alone machine developed by Merklein and Biasutti [13].
http://dx.doi.org/10.1016/j.cirp.2014.03.039 0007-8506/ß 2014 CIRP.
Please cite this article in press as: Merklein M, et al. Characterisation of kinematic hardening and yield surface evolution from uniaxial to biaxial tension with continuous strain path change. CIRP Annals - Manufacturing Technology (2014), http://dx.doi.org/10.1016/ j.cirp.2014.03.039
G Model
CIRP-1150; No. of Pages 4 M. Merklein et al. / CIRP Annals - Manufacturing Technology xxx (2014) xxx–xxx
2
Table 1 Mechanical parameters of DC06 at the yield stress s in uniaxial and biaxial tension and after strain path change. Parameter
s (MPa)
r
sbx (MPa)
sby (MPa)
08 R.D. 458 R.D. 908 R.D. epre = 0.05 epre = 0.10 epre = 0.12
138.0 0.96 139.4 0.39 139.9 1.21 139.7 2.70 135.5 2.58 141.8 2.22
1.98 0.09 1.72 0.04 2.36 0.13 1.97 0.08 1.90 0.02 2.01 0.02
– – – 292.3 0.87 358.9 1.16 372.9 2.92
– – – 222.6 4.75 261.9 2.57 269.6 4.14
Biaxial
Fig. 1. Schematic sectional view of the x-direction of the biaxial tensile test machine (left) with clamped cruciform specimen in the centre (right).
The tensile load is introduced by a link mechanism and the motion of the movable table in z-direction. The mounting of the bearing to an angular adjustment converts the load Fz of the lifting gear into the in-plane tensile forces Fx and Fy in x- and y-direction, as can be seen schematically for one side of the x-direction in Fig. 1. The specimen consists of a cruciform shaped geometry with a maximum length of 120.0 mm and a width of 19.8 mm. Moreover, seven slits with a length of 20.0 mm and a width of 0.2 mm are realised in each arm by laser cutting in order to homogenise the plastic deformation. A thickness reduction of the sheet metal to 0.5 mm is attained by milling the centre of the specimen to a gauge section of 16.4 16.4 mm2. The local strains are measured using a two-dimensional CCD-camera and an optical strain measurement system (ARAMIS, GOM mbH). 2.3. Methodology of online strain path variation An approach is presented to realise an online change of the stress state from uniaxial tension to biaxial tension without an extensive control system by modification of the cruciform specimen. The modification includes the variation of the specimen length ly in ydirection between 122.5 mm and 127.5 mm in a variation step of 2.5 mm. The length lx in x-direction remains constant at 120.0 mm. Tests with three different length ratios ly/lx are carried out. Through the length ratio higher than 1.0 only a load in the rolling direction (R.D.) is applied at the beginning of the test because of the earlier contact of the bearing in x-direction to the movable table. During the test the adjusting contact of the bearing in y-direction leads to the application of a further tensile force perpendicular to the R.D. (ydirection). Thus, a continuous change of the stress state from uniaxial tension to biaxial tension is induced. The machine speed is set to 3 mm/min for all tests. This involves an equivalent strain rate of 0.0029 1/s. Tests at 0.05, 0.10 and 0.12 uniaxial pre-strain in R.D. are carried out for the material DC06 (Fig. 2). The true stress is defined by the ratio of the current load to the instantaneous cross section in each principal direction. The true
Biaxial tensile test setup Material: DC06 (t0 = 0.5 mm, N = 4)
500
True stress
MPa 300
pre-strain 0.12 pre-strain 0.10 pre-strain 0.05 experiment x experiment y uniaxial 0° R.D.
200
100 0 0.00
0.05
0.10 0.15 True strain x
-
0.25
Fig. 2. Principal true stress-true strain curves of DC06 in x- and y-direction with online strain path changes at different levels of uniaxial pre-straining (epre = 0.05, 0.10 and 0.12).
rb
sbx (MPa)
sby (MPa)
1.00
160.1 0.98
157.3 0.91
stress vs. strain curve of a uniaxial tensile test in R.D. is also demonstrated (red dashed line). In case of the pre-strained cruciform specimen, the progress of the principal true stresses is displayed. The comparison of the experimental results of the modified test setup features a good reproducibility for defined levels of pre-straining. The progression of the hardening behaviour has a close overlap to the uniaxial tensile test results until the onset of strain path changing. Regarding the beginning of plastic yielding and the continuing work hardening behaviour, especially at the investigated levels of pre-straining, the difference is negligible. As listed in Table 1, the comparison of the beginning of plastic yielding s0 in R.D. with 138.0 MPa 0.96 MPa for the uniaxial tensile test and the yield strength 139.7 MPa 2.70 MPa, 135.5 MPa 2.58 MPa and 141.8 MPa 2.22 MPa for uniaxial pre-straining of the modified cruciform specimen, demonstrates the analogy of the test setups. Besides, the components of the beginning of plastic yielding sbx, sby for the standard configuration of the biaxial tensile test are determined with the assumption of constant plastic work. The subsequent biaxial yield locus (Table 1) is represented through the principal yield stresses sbx and sby at 0.05, 0.10 and 0.12 of uniaxial pre-strain. It can be seen that pre-straining leads to an increase of the subsequent plastic yield stress with a higher value of the yield stress component sbx in R.D., the direction of uniaxial pre-straining. 3. Numerical modelling and experimental results 3.1. Material modelling and Bauschinger coefficient In order to analyse the yield surface movement regarding the influence of isotropic and kinematic hardening, the comparison of yield loci of the experimental results to a material model is shown in Fig. 4. To describe the plastic material behaviour of DC06 the complex yield criterion Yld2000-2d [14] is chosen for investigating the subsequent yield loci. Because of better fitting results, the exponent m is set to 8.0. In case of the conventional criteria of von Mises or Hill’48 [15], no differentiation of the material behaviour at biaxial or shear stress state is made. This can lead to a wrong approximation of the biaxial stress state and to inaccurate results of the numerical calculation. The work hardening of the material is mapped with the isotropic hardening law of Hockett–Sherby [16] based on the approximation of the flow curve under uniaxial tension in 08 to R.D. (Table 2). To investigate the influence of kinematic hardening, the Bauschinger effect is analysed and the Bauschinger coefficient b, that describes the relationship between the beginning of plastic yielding sr under subsequent compressive Table 2 Identified material parameter of the yield criterion Yld2000-2d and the isotropic hardening law Hockett–Sherby for DC06. Yield criterion Yld2000-2d (m = 8)
a1
a2
a3
a4
a5
a6
a7
a8
1.027
1.026
0.842
0.908
0.909
0.737
1.011
1.062
Hardening law Hockett–Sherby (s
m HS = A Bexp(Cepl ))
A
B
C
m
415.505
277.505
6.508
0.863
Please cite this article in press as: Merklein M, et al. Characterisation of kinematic hardening and yield surface evolution from uniaxial to biaxial tension with continuous strain path change. CIRP Annals - Manufacturing Technology (2014), http://dx.doi.org/10.1016/ j.cirp.2014.03.039
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600
0° R.D. 90° R.D.
True stress
MPa
3
Modified setup vs. FEM models Material: DC06 (t0 = 0.5 mm)
400 300 experiment Hill'48 Swift isotropic Yld2000-2d HS isotropic Yld2000-2d HS =0.73
200 100 0 0.12
0
0.14 0.16 True strain
Fig. 3. Tension–compression test setup according to [17].
load and the maximum yield stress sm under tension, is determined by tension-compression tests according to the test setup of Staud et al. [17]. Moreover, distortional hardening is not explicitly investigated in this work. The specimen consists of an effective range of 2.0 mm 2.0 mm in the centre (Fig. 3). The strain was detected by an optical measurement system. Due to the effect of buckling under compressive load, the Bauschinger coefficient was determined at a maximum pre-strain of 0.14. In case of the investigated material the Bauschinger coefficient b was estimated with a constant value of 0.73 and included in the material model. The initial yield surface is calculated from the experimental results of the uniaxial tensile tests in 08, 458 and 908 to R.D. and the biaxial tensile test. The evolution of the initial yield surface for the investigated levels of plastic pre-strain is shown in Fig. 4. The comparison of subsequent yield loci under uniaxial pre-straining and the analytically identified material model with pure isotropic (Yld2000-2d, isotropic) and isotropic-kinematic hardening (Yld2000-2d, b = 0.73) reveals the translation of the experimental yield locus caused by the uniaxial pre-strain. Comparing the equibiaxal yield stress of the pure isotropic material model with the experimentally determined subsequent yield loci, a large deviation between the pure isotropic model and the experiment can be observed (Fig. 4). Using an isotropic-kinematic modelling, a reduced deviation can be seen, leading to the conclusion that a kinematic hardening component is present in the material.
Fig. 4. Subsequent yield loci of DC06 with online strain path changes at different levels of uniaxial pre-straining (epre = 0.05, 0.10 and 0.12) compared to the yield criterion Yld2000-2d with a pure isotropic and an isotropic-kinematic (b = 0.73) hardening law.
-
0.22
Fig. 5. Comparison of the true stresses vs. strain curves of the subsequent material behaviour of the investigated material DC06 to numerical simulations (isotropic and isotropic-kinematic hardening).
plastic yielding in comparison to the experimental results is affected by disregarding the biaxial stress state in the analytical model formulation. Analogously to the analytical prediction of the evolution of yield loci (Fig. 4), the material model Yld2000-2d with isotropic and kinematic hardening component leads to a better application of the subsequent hardening behaviour under strain path change than the pure isotropic model (Fig. 5). According to the results of modelling the material behaviour under online strain path change with a kinematic hardening component, the translation of the yield surface in Fig. 4 also leads to a change of the tangent’s gradient in the uniaxial stress direction. Therefore, a new approach is presented to use the change of the gradient during plastic deformation. 4. Alternative estimation of kinematic hardening behaviour The aim of this alternative approach is the estimation of the isotropic and kinematic portions of hardening based on experimental results from the proposed combined uniaxial-biaxial tensile test without unloading. During the uniaxial tensile test, the r-values remain not on a constant level, but rather have a non-linear progress [18]. Therefore, the r-values are usually selected at a defined and material dependent pre-strain level in order to fix the shape of the yield locus [19]. This procedure is meaningful, when the derived material parameters are used with an isotropic hardening law within a finite-element-analysis. But the described effect of r-value evolution is generated by the combination of isotropic and kinematic hardening, i.e. the growth and translation of the yield surface in stress space. To illustrate the effect of isotropic-kinematic hardening behaviour on the evolution of the yield locus and the observed r-value, a one element finite-element-analysis was done using the parameters of DC06 and Barlat’ yield criterion (Table 2). Fig. 6 shows the translated and grown yield locus for different combinations of isotropic-kinematic hardening with b-values of 1.0 and 0.73 respectively. Furthermore, Fig. 6 shows the corresponding evolution of the r0-value, which can be derived from the tangent on the intersection of the yield surface with the s1-axis (compare Fig. 4). During pure isotropic hardening behaviour (b = 1.0) this point remains at the same position on the yield surface during plastic deformation and no change of the
2.3
500
3.2. Comparison to the numerical simulation model
2.2
r0-value
An implicit one-element numerical simulation is established in LS-DYNA v.9.7.1 (LSTC) with a pure isotropic and an isotropickinematic material model. The identified criterion Yld2000-2d with the isotropic hardening law of Hockett–Sherby (HS) is implemented. The kinematic behaviour is expressed through the Bauschinger coefficient (b = 0.73). The progress of the principal stresses is exemplarily shown in case of the maximum pre-strain level 0.12 in Fig. 5. In order to investigate the influence of abstracting away from the biaxial stress, a third simulation model is shown in Fig. 5 that consists of the yield criterion Hill’48. The strong deviation of the subsequent beginning of
0.18 x
2.1
500
-500
-500
2
0
0.1 0.2 0.3 Equivalent plastic strain
Material: DC06, t0 = 0.5 mm Initial yield surface = 1.0 (isotropic)
= 0.73
Fig. 6. Evolution of yield locus during tensile test and resulting evolution of the r0value.
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tangent will occur. If isotropic-kinematic hardening takes place (0 < b < 1.0), the evolving back stress tensor X causes a movement of the yield locus. This leads to a continuously change of intersection points, which results in the change of the r0-value. The calculation procedure to calculate the back stress tensor X based on one load path change without unloading, is described in the following. The r0-value gives the information about the relationship between the components of the incremental principal plastic strain de2 and de3, as well as de1 by using the law of volume constancy. Using the associated flow rule, the ratio between the components of dei and the corresponding deviatoric stress components is constant for every deformation step. Regarding the change of the back stress tensor components dXi it is assumed, that they are proportional to the (anisotropic) deviatoric stress components. This relationship can be expressed by the change of the r0-values dr0, as given by Eqs. (1) and (2). dX 1 ð1 þ dr 0 Þ ¼ dr 0 dX 2
(1)
dX 1 ¼ ð1 þ dr 0 Þ dX 3
(2)
Because de1 is continuously measured during the test, all other components of the principal strain tensor, and the ratio between all components of the back stress tensor X can be directly calculated for every measured deformation step. To identify not only the ratios but also the values of X, two different stress states have to be generated without unloading to keep the back stress tensor constant. The first considered stress state (I: s tensile) is related to the last uniaxial tensile stress and the second stress state (II: s biaxial) is given by the first biaxial stress situation. If the specimen is continuously in a loading situation that fulfils the yield condition, the assumption of a constant back stress tensor X during the load path change is possible. Therefore, the yield condition F for the two stress states s tensile and s biaxial is equal and can be written as shown in Eq. (3).
F ¼ f ðs tensile XÞ ¼ f ðs biaxial XÞ F ¼ f ðs compression XÞ
(3)
Because the relationship of all components from the back stress tensor Xi can be derived using the mentioned equations and the online measurement data, a direct identification procedure with a Newton algorithm for the root-finding is possible. Only the first principal back stress X1 has to be considered in order to equalise the two stress states and fulfil the yield condition in Eq. (3). The theoretical Bauschinger coefficient b can be directly calculated using Eq. (3) with the identified back stress tensor X. The yield condition as well as the back stress tensor will be kept constant and the uniaxial compression stress state scompression as well as the Bauschinger coefficient can be directly calculated. The application of the described procedure on the experimental data from the biaxial test with uniaxial pre-straining leads to the results shown in Table 3. Thereby, b0.02 and b0.05 indicate the Bauschinger coefficient related to a stress condition, where a replastification of De = 0.02 respectively De = 0.05 is present. The result shows an evolution of the Bauschinger coefficient similar to the evolution of the r0-value. The deformation starts with a nearly isotropic expansion of the yield locus (b nearby 1). With larger pre-strain values, kinematic hardening takes place (0 < b < 1.0). Obviously, the Bauschinger coefficients b0.02, b0.05 and b are different to the initially observed value of b = 0.73, which can be explained by two reasons: Table 3 Determined Bauschinger coefficients. Pre-strain epre Bauschinger coefficient
b b 0.02 b 0.05
0.05
0.10
0.12
0.93 0.99 1.00
0.54 0.69 0.77
0.54 0.66 0.77
(a) During the initially mentioned identification procedure using a tension–compression test, a very smooth transition between elastic and elastic-plastic region is observed under load reversal nearby the compression stress state scompression. This effect cannot be approximated by the described identification strategy, due to the used yield criterion. To overcome this, a more complex kinematic hardening model like Chaboche has to be used or a fixed level of re-plastification De has to be defined. (b) A direct, abrupt change from uniaxial to biaxial stress state is needed for the assumption of a constant back stress tensor during load path change. This is not exactly achievable in any experiment, which causes an additional deviation.
5. Conclusion A new experimental approach is presented in this paper, to investigate the evolution of the yield locus after online strain path change. Moreover, numerical simulations are carried out to assess the accuracy of the material model. Concerning the influence of kinematic hardening an alternative approach is proposed to estimate the kinematic hardening behaviour by analysing the rvalue progression during plastic deformation. Acknowledgements The authors are grateful to the German Research Foundation (DFG) for supporting the research project ‘‘Identification and modelling of material characteristics for Finite-Element-Analysis of sheet metal forming processes’’ ME (2043/11-2) and BR (3500/6-2).
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Please cite this article in press as: Merklein M, et al. Characterisation of kinematic hardening and yield surface evolution from uniaxial to biaxial tension with continuous strain path change. CIRP Annals - Manufacturing Technology (2014), http://dx.doi.org/10.1016/ j.cirp.2014.03.039