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Prediction of forming limit diagram with mixed anisotropic kinematic–isotropic hardening plastic constitutive model based on stress criteria
Journal of Materials Processing Technology 133 (2003) 304±310
Prediction of forming limit diagram with mixed anisotropic kinematic±isotropic hardening plastic constitutive model based on stress criteria C.L. Chow*, X.J. Yang1 Department of Mechanical Engineering, School of Engineering, University of Michigan-Dearborn, 4901 Evergreen Road, Dearborn, MI 48128, USA Received 11 June 2001; received in revised form 7 February 2002; accepted 16 October 2002
Abstract This paper presents an investigation of predicting the forming limit diagram (FLD) based on a stress criterion. The prediction is achieved by employing an anisotropic plastic model with a mixed kinematic±isotropic hardening formulation recently developed by the authors. The loading conditions considered include both proportional and non-proportional loading. The applicability of the stress FLD (SFLD) under the loading conditions and different rolling directions is examined and discussed for AL2008-T4. # 2002 Elsevier Science B.V. All rights reserved. Keywords: FLD; SFLD; Constitutive modeling; Nonproportional loading; Strain paths; Anisotropic kinematic hardening; AL2008-T4
1. Introduction Changes in the direction of strain path are often observed in stamped automotive panels during a manufacturing operation. It is well accepted that the change in strain path has a de®nite effect on the formability of a stamped panel. In other words, loading history can signi®cantly affect the forming limit diagram (FLD) [1±5]. The observation also implies that the applicability of the conventional FLD constructed under proportional loading is limited and should not be indiscriminately used in practical manufacturing process. In order to circumvent the path-dependency of the FLD, several investigators have recently proposed an alternative FLD in stress-space rather than in strainspace. Sing and Rao [6,7] demonstrated that the FLD under proportional loading can be predicted using instead the SFLD. The disadvantage of the SFLD is that the limit stress cannot be readily measured under nonproportional loading. Both Arrieux [8,9] and Sing and Rao [7] determined the forming limit stresses with a numerical
*
Corresponding author. E-mail address: [email protected] (C.L. Chow). 1 On leave from Southwest Jiaotong University, Chengdu, China.
simulation of a constitutive model based on isotropic hardening. With certain assumptions, Arrieux calculated the limit stresses from the measured FLD subjected to different loading paths and plotted the stresses in the space of principal stresses. The resulting SFLD appeared to merge as a single FLD. Quadratic yield criterion [10] for anisotropic sheet metal is widely used in the computer simulation of the FLD. The simulation often employs an isotropic hardening rule. The rule leads to continuous expanding yield surfaces isotropically as plastic deformation progresses. Another well-known yield criterion is the kinematic hardening model used to describe Bauschinger's effect [11±14]. Under complicated loading histories, such as those involving unloading, substantial deviation of actual material behavior from that predicted separately by these two models has been observed [15]. This is because materials often exhibit a mixed or combined isotropic and kinematic hardening rather than a single hardening behavior. In this paper, a mixed kinematic±isotropic hardening model based on the Hill's yield criterion recently developed by the authors [16] is employed for the prediction of the SFLD. The prediction is carried out under unloading and subsequent reloading in different rolling directions. A new stress criterion for localized necking under arbitrary loading
0924-0136/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 2 ) 0 1 0 0 6 - 3
C.L. Chow, X.J. Yang / Journal of Materials Processing Technology 133 (2003) 304±310
Nomenclature ai , c i E F, G, H, N Fp p R0, R45, R90 T(p) W
material constants Young's modulus material constants in Hill's 1948 yield function plastic yield function accumulated effective plastic strain ratios of transverse to through thickness strains under uniaxial tension at 08, 458, 908 to the rolling direction isotropic plastic strain hardening material matrix for orthotropic plastic material
Greek letters a back stress tensor {a} back stress vector {ai} part i of the back stress vector {a} aij back stress components in material coordinates aijk part k of total back stress components aij bi material constants eRD, eTD normal strain components along RD and TD directions epij plastic strain components {dep} incremental plastic strain vector r cauchy stress tensor {r} stress vector s equivalent stress sij stress components sRD, sTD normal stress components along RD and TD directions s0 initial yield stress u Poisson's ratio
305
coordinates based on the rolling direction (RD)±transverse direction (TD) plane are adopted, where RD and TD refer to xaxis and y-axis, respectively. The yield function then becomes q Fp r; a; T p 12 fr agT : W : fr ag T p 0 (2) where [W] is a symmetric tensor of 4th order that may be expressed as a matrix for orthotropic plastic material as 2 3 GH H 0 6 7 W 4 H HF 0 5 (3) 0
0
2N
where G, F, H and N are material constants given by Barata da Rocha et al. [17]. The relation between of the equivalent stress and equivalent strain may be expressed as T p s
3 X i1
bi 1
exp ci p
(4)
P where T p s0 3i1 ai 1 exp ci p; s0 the initial yield stress; p the accumulated inelastic strain; and ai, ci and bi the material constants. A kinematic hardening equation is proposed 8 9 8 9 2 38 p 9 H11i H12i 0 > > > < da11i > = < de11 > = < a11i > = 6 7 da22i ci 4 H21i H22i 0 5 dep22 ci a22i dp > > > > > : ; : p > ; : ; 0 0 H33i da12i a12i de12 (5)
where ci and H values are materials constants. The evolution equations of kinematic hardening can ®nally be given as fdai g ci bi Bfdep g where
ci fai g dp
(6)
2
paths is also proposed. The material chosen for the investigation of SFLD is AL2008-T4.
FH H 2 6 F H H =2 F H H 2 =2 6 6 H 2 B 6 6 F H H 2 =2 F H H 2 =2 6 4 0 0
2. Plastic model mixed isotropic±kinematic hardening
Let
An anisotropic plastic model coupled with a mixed kinematic±isotropic hardening formulation has recently been proposed but yet to be published [16]. As the model is used for the prediction of FLDs based on the stress criterion, a brief description of the formulation is made in this section. In the material coordinate system, the plastic potential function may be expressed as Fp Fp sij ; aij ; T p 0
(1)
where sij denote stresses, aij denote back stresses and T(p) denotes isotropic plastic strain hardening. For this investigation, Hill's yield criterion in the plane stress condition [10] is introduced in the plastic potential function. The material
@Fp @r
9 8 @Fp > > > > > > > @s11 > > > > = < @F > p
> @s22 > > > > > > > > > @F > > p ; : @s12
9 8 > = < a11 i > fai g a22 i ; > > ; : a33 i
0
3
7 7 7 07 7 7 5 2 N
(7)
8 9 > = < c1 > fcg c2 ; > ; : > c3
;
@Fp @a
9 8 @Fp > > > > > @a > > 11 > > > > = < @F > p
> @a22 > > > > > > > > > > @Fp > ; : @a12
(8)
306
C.L. Chow, X.J. Yang / Journal of Materials Processing Technology 133 (2003) 304±310
The associated plastic ¯ow rule can thus be derived as e_ pij
f@Fp =@rgT : frg @Fp P T 3 T @sij @T=@p f@Fp =@ag Afcg i1 ci bi f@Fp =@ag Bf@Fp =@rg
where
2
a11 1
6 A 4 a22 1 a12 1
a11 2
a11 3
(9)
3
a22 2
a22 3 7 5
a12 2
a12 3
(10)
The elastic modulus and Poisson's ratio were measured from standard tensile test specimens. The anisotropy parameters R0, R45, R90 were determined from the strains along the RD, 458 and TD as reported by Graf and Hosford [3]. The values of ai, bi, ci were determined by ®tting the curves of stress± strain under uniaxial tension and uniaxial cyclic loading. The measured material parameters for AL2008-T4 alloy are summarized in Table 1. Fig. 1. SFLDs of AL2008-T4 alloy from six measured FLDs.
3. SFLDs of AL2008-T4 alloy Knowledge of the localized necking is required to de®ne the threshold condition of crack initiation in a stamped panel. The limit stress at localized necking is dependent upon the plastic strain history, stress loading state and other processing parameters. Therefore, an analytical simulation model becomes necessary to determine localized necking for the SFLD. Graf and Hosford [3] reported the effects of strain-path on the FLD for AL2008-T4 prestrained in uniaxial or equibiaxial tension. The measured FLDs are used to perform numerical prediction of the limit stresses by means of the constitutive model described in Section 2. For the investigation, a total of six measured FLDs was used to calculate the SFLDs. The six cases include FLD: (i) of the as-received material loaded normal to the TDs, (ii) of the as-received material loaded normal to the RDs, (iii) with uniaxial prestrains of eTD 0:18 loaded normal to the RD, (iv) with uniaxial prestrain of eTD 0:18, loaded normal to the TD, (v) with the near plane prestrain eRD 0:14 loaded normal to the RD, and (vi) with equibiaxial prestrain eRD eTD 0:17 loaded normal to the RD.
The calculated SFLDs from the six FLDs are depicted in Fig. 1. It can be observed from the ®gure that the SFLDs under four different prestrains do not appear to converge as a single curve as advocated by Arrieux. The maximum variability of the limiting stresses exceeds 40 MPa or about 10%. 4. Prediction of FLD based on stress criteria From the preceding constitutive simulation, it is evident that the SFLD subjected to varying strain histories is unable to converge to a single diagram without a scatter band. In effect, the validity of the forming limit curves under nonproportional loading to merge, as a single diagram in the stress-space, has not been proven experimentally. This is primarily due to the dif®culty in directly measuring the limit stress during a forming process. On the other hand, the forming limit strain diagram (FLD) has been routinely measured. An attempt is made in the following sections to construct the FLD based on a stress limit criterion. The stress criterion is established using the SFLD under proportional loading and then employ it to predict FLD with varying prestrains for AL2008-T4. The predicted FLDs are then used to compare with those measured experimentally under nonproportional loading [3]. As outlined in Section 3, the SFLD can be constructed using the measured FLD of AL2008-T4 alloy under
Table 1 Material parameters of AL2008-T4 R0
R45
R90
E (MPa)
u
s0 (MPa)
a1 (MPa)
a2 (MPa)
a3 (MPa)
b1 (MPa)
b2 (MPa)
b3 (MPa) c1
0.58
0.485
0.78
70121
0.33
154
79.2
398.16
0
8.8
44.24
0
c2
c3
39.1 1.443 0
C.L. Chow, X.J. Yang / Journal of Materials Processing Technology 133 (2003) 304±310
proportional loading without prestrain. From Fig. 1, the maximum limit stress from the as-received material produced normal to the RD is observed to be 40 MPa higher than that of the material normal to the TD. For the present analysis, all the measured FLD results from both the TD and RD directions were included numerically to develop a stress criterion for the prediction of the SFLD. The stress criterion of localized necking developed in terms of MPa is formulated relative to two different sets of RD as For sRD sTD, sTD
372:072
0:497sRD 0:00074s2RD 0
(11a)
and sTD
460:511 0:11541sRD 0:000339s2RD 0
(11b)
For sRD sTD, sRD
415:2 0:008854sTD 0:00044165s2TD 0 (12a)
and sRD
340:246
0:458112sTD 0:0007154s2TD 0 (12b)
A material element is said to have yielded the localized necking when one of the above stress criteria, Eqs. (11a) and (11b) or (12a) and (12b), have been satis®ed. Eqs. (11a) and (11b) refers to the condition of rolling when sRD sTD while Eqs. (12a) and (12b), sRD sTD . In other words, the localized necking would occur when the stress state in a stamped panel exceeds the bonded area of the solid curves in Fig. 1. This implies that both equations of (a) and (b) must be met to yield localized necking.
307
4.1. FLDs with equibiaxial prestrain Fig. 2 depicts the experimental and predicted FLDs normal to the RD with three different equibiaxial prestrains. The prestrains chosen are: (i) eRD 0:04, (ii) eRD 0:07, and (iii) eRD 0:17. The major loading direction after prestrain was made normal to the RD. It can be observed from the ®gure that the predicted FLDs coincide well with the test results. Upon unloading, the load at a given prestrain was reduced to zero. On the other hand, in the numerical simulation, if the unloading is not considered, a recent investigation reported failure in prediction of the left side of FLD for the case of equibiaxial prestrain of 17% [18]. Failure for the prediction may be attributed to two assumptions made. Firstly, the simulation was based on rigid± plasticity. Secondly, unloading after prestrain was not considered. These two assumptions caused the stress to exceed its limiting value when the direction of strain increments after the prestrain changes abruptly. The elasto-plastic constitutive equation introduced in the present investigation allows the developing stress in a sheet to follow a different stress±strain path upon unloading from that based on the rigid±plasticity theory. 4.2. FLDs with uniaxial prestrain Three different strain path combinations are considered, i.e. (1) both the imposed prestrains and the loading direction normal to RD, (2) the uniaxial prestrain parallel to RD followed by the loading direction normal to RD, and (3) the prestrain parallel to TD followed by the loading direction normal to TD. In the second and third cases, the rotation of the principal strain direction is expected to produce a large amount of strain reversal.
Fig. 2. Predicted and measured FLDs normal to the RD after equibiaxial prestrains.
308
C.L. Chow, X.J. Yang / Journal of Materials Processing Technology 133 (2003) 304±310
Fig. 3. Predicted and measured FLDs normal to the RD after uniaxial prestrain normal to the RD.
Fig. 3 shows the experimental and predicted FLDs for the ®rst case (1) where the uniaxial prestrains prescribed are (i) eTD 0:05, (ii) eTD 0:12 and (iii) eTD 0:18. The predicted FLDs agree well with the experimental results for cases (i) and (ii), while the maximum discrepancy of 0.039 is observed for case (iii) at the minimum limit strain. Also shown in the ®gure is a gradual upward shift of FLD to the left with increasing prestrain, a phenomenon that can also be successfully predicted by the stress model. Fig. 4 depicts the experimental and predicted FLDs for three cases: (i) eRD 0:04, (ii) eRD 0:125, and
(iii) eRD 0:18 for the second case (2). The experimental FLDs show that with increasing prestrains parallel to RD, the FLDs are shifted downward to the right. The predicted FLDs demonstrate that the proposed model can accurately characterize this experimental observation. Fig. 5 summarizes the experimental and predicted FLDs for three cases: (i) eTD 0:05, (ii) eTD 0:12, and (iii) eTD 0:18 for the third case (3). The experimental FLDs show that with increasing prestrains parallel to TD, the FLDs are shifted downward to the right. The predicted
Fig. 4. Predicted and measured FLDs normal to the RD after uniaxial prestrains parallel to the RD.
C.L. Chow, X.J. Yang / Journal of Materials Processing Technology 133 (2003) 304±310
309
Fig. 5. Predicted and measured FLDs parallel to the RD after uniaxial prestrain normal the RD.
Fig. 6. Predicted and measured FLDs normal to RD after plane prestrain parallel to the RD.
FLDs demonstrate that the proposed model can also successfully characterize this experimental observation. 4.3. FLDs with pre-plane strains Fig. 6 depicts the experimental and predicted FLDs for two cases of near plane prestrain parallel to RD: (i) eRD 0:08, (2), eRD 0:14. The prestrained specimens are then loaded normal to RD. It can be observed from the ®gure that the model can once again successfully predict
the FLDs as well as the shifting of the FLDs downward to the right with increasing prestrains. 5. Conclusions Abrupt changes in strain path during metal forming can signi®cantly affect the forming limits. A stress criterion for localized necking is developed such that the limit strain subjected to arbitrary loading history can be satisfactorily
310
C.L. Chow, X.J. Yang / Journal of Materials Processing Technology 133 (2003) 304±310
predicted. The following conclusions can be drawn from the investigation on the applicability and validity of SFLD for AL2008-T4 alloy under nonproportional loading. (i) For the SFLD of AL2008-T4 alloy, the convergence of the limit stresses to a single curve has not been observed. The uniqueness of the SFLD is found to be dependent significantly on the strain path. (ii) Based on the mixed anisotropic kinematic±isotropic plastic model and a stress criterion, satisfactory prediction of FLDs under non-proportional loading can be achieved from the measured FLDs in an open literature. Acknowledgements The authors wish to acknowledge a partial support of from the Research Excellent and Economic Development Fund (REEDF) within the College of Engineering and Computer Science at the University of Michigan-Dearborn. References [1] C.L. Chow, W. Tai, M. Demeri, Numerical simulation of the Ubending process for VDIF steel sheets using damage analysis and LSDYNA, Sheet Met. Form. Technol. (1999) 169±178. [2] A.F. Graf, W. Hosford, Calculation of the forming limit diagrams for changing strain paths, Metall. Trans. A 24 (1993) 2497±2501. [3] A.F. Graf, W. Hosford, Effect of changing strain paths on forming limit diagrams of AL2008-T4, Metall. Trans. A 24 (1993) 2503± 2512. [4] A.F. Graf, W. Hosford, The influence of strain-path changes on forming limit diagram of AL6111-T4, Int. J. Mech. Sci. 36 (1994) 897±910.
[5] J.V. Laukonis, A.K. Ghosh, Effect of strain path changes on the formability of sheet metals, Metall. Trans. A 9 (1978) 1849±1856. [6] K.P. Rao, V.R. Mohan Emani, Simple prediction of sheet metal forming limit stresses and strains, in: M.Y. Demeri (Ed.), Sheet Metal Forming Technology, The Minerals, Metals and Materials Society, 1999, pp. 205±231. [7] W.M. Sing, K.P. Rao, Study of sheet metal failure mechanisms based on stress state conditions, J. Mater. Process. Technol. 67 (1997) 201± 206. [8] R. Arrieux, Determination and use of the forming limit stress diagrams, J. Mater. Process. Technol. 53 (1995) 47±56. [9] R. Arrieux, M. Boivin, Determination of the forming limit stress curve for anisotropic sheets, Ann. CIRP 36 (1987) 195±198. [10] R. Hill, A theory of yield and plastic flow of anisotropic metals, in: Proceedings of the Royal Society A, vol. 193, London, 1948, pp. 281± 297. [11] W. Prager, The theory of plasticity: a survey of recent achievements, in: James Clayton Lecture, Proceedings of the institute of Mechanical Engineers, vol. 169, 1955, pp. 41±57. [12] J.L. Chaboche, O. Jung, Application of a kinematic hardening viscoplasticity model with thresholds to the residual stress relaxation, Int. J. Plast. 13 (1998) 785±807. [13] J.C. Moosbrugger, Anisotropic nonlinear hardening rule parameters from reversed proportional axial-torsional cycling, ASME J. Eng. Mater. Technol. 122 (2000) 18±28. [14] P.J. Armstrong, C.O. Frederick, A mathematical representation of the multiaxial Bauschinger effect, CEGB Report RD/B/N73 1, 1966, Central Electricity Generating Board. [15] H.C. Wu, H.K. Hong, Y.P. Shiao, Anisotropic plasticity with application to sheet metals, Int. J. Mech. Sci. 41 (1999) 703±724. [16] C.L. Chow, X.J. Yang, E. Chu, Prediction of forming limit diagram based on a damage coupled kinematic±isotropic hardening model under nonproportional loading, J. Mater. Eng. Technol. 124 (2002) 219±226. [17] A. Barata da Rocha, F. Barlat, J.M. Jalinier, Prediction of the forming limit diagram of anisotropic sheets in linear and nonlinear loading, Mater. Sci. Eng. 68 (1984) 151±164. [18] T.B. Stoughton, A general forming limit criterion for sheet metal forming, Int. J. Mech. Sci. 42 (2000) 1±27.
Prediction of forming limit diagram with mixed anisotropic kinematic±isotropic hardening plastic constitutive model based on stress criteria C.L. Chow*, X.J. Yang1 Department of Mechanical Engineering, School of Engineering, University of Michigan-Dearborn, 4901 Evergreen Road, Dearborn, MI 48128, USA Received 11 June 2001; received in revised form 7 February 2002; accepted 16 October 2002
Abstract This paper presents an investigation of predicting the forming limit diagram (FLD) based on a stress criterion. The prediction is achieved by employing an anisotropic plastic model with a mixed kinematic±isotropic hardening formulation recently developed by the authors. The loading conditions considered include both proportional and non-proportional loading. The applicability of the stress FLD (SFLD) under the loading conditions and different rolling directions is examined and discussed for AL2008-T4. # 2002 Elsevier Science B.V. All rights reserved. Keywords: FLD; SFLD; Constitutive modeling; Nonproportional loading; Strain paths; Anisotropic kinematic hardening; AL2008-T4
1. Introduction Changes in the direction of strain path are often observed in stamped automotive panels during a manufacturing operation. It is well accepted that the change in strain path has a de®nite effect on the formability of a stamped panel. In other words, loading history can signi®cantly affect the forming limit diagram (FLD) [1±5]. The observation also implies that the applicability of the conventional FLD constructed under proportional loading is limited and should not be indiscriminately used in practical manufacturing process. In order to circumvent the path-dependency of the FLD, several investigators have recently proposed an alternative FLD in stress-space rather than in strainspace. Sing and Rao [6,7] demonstrated that the FLD under proportional loading can be predicted using instead the SFLD. The disadvantage of the SFLD is that the limit stress cannot be readily measured under nonproportional loading. Both Arrieux [8,9] and Sing and Rao [7] determined the forming limit stresses with a numerical
*
Corresponding author. E-mail address: [email protected] (C.L. Chow). 1 On leave from Southwest Jiaotong University, Chengdu, China.
simulation of a constitutive model based on isotropic hardening. With certain assumptions, Arrieux calculated the limit stresses from the measured FLD subjected to different loading paths and plotted the stresses in the space of principal stresses. The resulting SFLD appeared to merge as a single FLD. Quadratic yield criterion [10] for anisotropic sheet metal is widely used in the computer simulation of the FLD. The simulation often employs an isotropic hardening rule. The rule leads to continuous expanding yield surfaces isotropically as plastic deformation progresses. Another well-known yield criterion is the kinematic hardening model used to describe Bauschinger's effect [11±14]. Under complicated loading histories, such as those involving unloading, substantial deviation of actual material behavior from that predicted separately by these two models has been observed [15]. This is because materials often exhibit a mixed or combined isotropic and kinematic hardening rather than a single hardening behavior. In this paper, a mixed kinematic±isotropic hardening model based on the Hill's yield criterion recently developed by the authors [16] is employed for the prediction of the SFLD. The prediction is carried out under unloading and subsequent reloading in different rolling directions. A new stress criterion for localized necking under arbitrary loading
0924-0136/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 2 ) 0 1 0 0 6 - 3
C.L. Chow, X.J. Yang / Journal of Materials Processing Technology 133 (2003) 304±310
Nomenclature ai , c i E F, G, H, N Fp p R0, R45, R90 T(p) W
material constants Young's modulus material constants in Hill's 1948 yield function plastic yield function accumulated effective plastic strain ratios of transverse to through thickness strains under uniaxial tension at 08, 458, 908 to the rolling direction isotropic plastic strain hardening material matrix for orthotropic plastic material
Greek letters a back stress tensor {a} back stress vector {ai} part i of the back stress vector {a} aij back stress components in material coordinates aijk part k of total back stress components aij bi material constants eRD, eTD normal strain components along RD and TD directions epij plastic strain components {dep} incremental plastic strain vector r cauchy stress tensor {r} stress vector s equivalent stress sij stress components sRD, sTD normal stress components along RD and TD directions s0 initial yield stress u Poisson's ratio
305
coordinates based on the rolling direction (RD)±transverse direction (TD) plane are adopted, where RD and TD refer to xaxis and y-axis, respectively. The yield function then becomes q Fp r; a; T p 12 fr agT : W : fr ag T p 0 (2) where [W] is a symmetric tensor of 4th order that may be expressed as a matrix for orthotropic plastic material as 2 3 GH H 0 6 7 W 4 H HF 0 5 (3) 0
0
2N
where G, F, H and N are material constants given by Barata da Rocha et al. [17]. The relation between of the equivalent stress and equivalent strain may be expressed as T p s
3 X i1
bi 1
exp ci p
(4)
P where T p s0 3i1 ai 1 exp ci p; s0 the initial yield stress; p the accumulated inelastic strain; and ai, ci and bi the material constants. A kinematic hardening equation is proposed 8 9 8 9 2 38 p 9 H11i H12i 0 > > > < da11i > = < de11 > = < a11i > = 6 7 da22i ci 4 H21i H22i 0 5 dep22 ci a22i dp > > > > > : ; : p > ; : ; 0 0 H33i da12i a12i de12 (5)
where ci and H values are materials constants. The evolution equations of kinematic hardening can ®nally be given as fdai g ci bi Bfdep g where
ci fai g dp
(6)
2
paths is also proposed. The material chosen for the investigation of SFLD is AL2008-T4.
FH H 2 6 F H H =2 F H H 2 =2 6 6 H 2 B 6 6 F H H 2 =2 F H H 2 =2 6 4 0 0
2. Plastic model mixed isotropic±kinematic hardening
Let
An anisotropic plastic model coupled with a mixed kinematic±isotropic hardening formulation has recently been proposed but yet to be published [16]. As the model is used for the prediction of FLDs based on the stress criterion, a brief description of the formulation is made in this section. In the material coordinate system, the plastic potential function may be expressed as Fp Fp sij ; aij ; T p 0
(1)
where sij denote stresses, aij denote back stresses and T(p) denotes isotropic plastic strain hardening. For this investigation, Hill's yield criterion in the plane stress condition [10] is introduced in the plastic potential function. The material
@Fp @r
9 8 @Fp > > > > > > > @s11 > > > > = < @F > p
> @s22 > > > > > > > > > @F > > p ; : @s12
9 8 > = < a11 i > fai g a22 i ; > > ; : a33 i
0
3
7 7 7 07 7 7 5 2 N
(7)
8 9 > = < c1 > fcg c2 ; > ; : > c3
;
@Fp @a
9 8 @Fp > > > > > @a > > 11 > > > > = < @F > p
> @a22 > > > > > > > > > > @Fp > ; : @a12
(8)
306
C.L. Chow, X.J. Yang / Journal of Materials Processing Technology 133 (2003) 304±310
The associated plastic ¯ow rule can thus be derived as e_ pij
f@Fp =@rgT : frg @Fp P T 3 T @sij @T=@p f@Fp =@ag Afcg i1 ci bi f@Fp =@ag Bf@Fp =@rg
where
2
a11 1
6 A 4 a22 1 a12 1
a11 2
a11 3
(9)
3
a22 2
a22 3 7 5
a12 2
a12 3
(10)
The elastic modulus and Poisson's ratio were measured from standard tensile test specimens. The anisotropy parameters R0, R45, R90 were determined from the strains along the RD, 458 and TD as reported by Graf and Hosford [3]. The values of ai, bi, ci were determined by ®tting the curves of stress± strain under uniaxial tension and uniaxial cyclic loading. The measured material parameters for AL2008-T4 alloy are summarized in Table 1. Fig. 1. SFLDs of AL2008-T4 alloy from six measured FLDs.
3. SFLDs of AL2008-T4 alloy Knowledge of the localized necking is required to de®ne the threshold condition of crack initiation in a stamped panel. The limit stress at localized necking is dependent upon the plastic strain history, stress loading state and other processing parameters. Therefore, an analytical simulation model becomes necessary to determine localized necking for the SFLD. Graf and Hosford [3] reported the effects of strain-path on the FLD for AL2008-T4 prestrained in uniaxial or equibiaxial tension. The measured FLDs are used to perform numerical prediction of the limit stresses by means of the constitutive model described in Section 2. For the investigation, a total of six measured FLDs was used to calculate the SFLDs. The six cases include FLD: (i) of the as-received material loaded normal to the TDs, (ii) of the as-received material loaded normal to the RDs, (iii) with uniaxial prestrains of eTD 0:18 loaded normal to the RD, (iv) with uniaxial prestrain of eTD 0:18, loaded normal to the TD, (v) with the near plane prestrain eRD 0:14 loaded normal to the RD, and (vi) with equibiaxial prestrain eRD eTD 0:17 loaded normal to the RD.
The calculated SFLDs from the six FLDs are depicted in Fig. 1. It can be observed from the ®gure that the SFLDs under four different prestrains do not appear to converge as a single curve as advocated by Arrieux. The maximum variability of the limiting stresses exceeds 40 MPa or about 10%. 4. Prediction of FLD based on stress criteria From the preceding constitutive simulation, it is evident that the SFLD subjected to varying strain histories is unable to converge to a single diagram without a scatter band. In effect, the validity of the forming limit curves under nonproportional loading to merge, as a single diagram in the stress-space, has not been proven experimentally. This is primarily due to the dif®culty in directly measuring the limit stress during a forming process. On the other hand, the forming limit strain diagram (FLD) has been routinely measured. An attempt is made in the following sections to construct the FLD based on a stress limit criterion. The stress criterion is established using the SFLD under proportional loading and then employ it to predict FLD with varying prestrains for AL2008-T4. The predicted FLDs are then used to compare with those measured experimentally under nonproportional loading [3]. As outlined in Section 3, the SFLD can be constructed using the measured FLD of AL2008-T4 alloy under
Table 1 Material parameters of AL2008-T4 R0
R45
R90
E (MPa)
u
s0 (MPa)
a1 (MPa)
a2 (MPa)
a3 (MPa)
b1 (MPa)
b2 (MPa)
b3 (MPa) c1
0.58
0.485
0.78
70121
0.33
154
79.2
398.16
0
8.8
44.24
0
c2
c3
39.1 1.443 0
C.L. Chow, X.J. Yang / Journal of Materials Processing Technology 133 (2003) 304±310
proportional loading without prestrain. From Fig. 1, the maximum limit stress from the as-received material produced normal to the RD is observed to be 40 MPa higher than that of the material normal to the TD. For the present analysis, all the measured FLD results from both the TD and RD directions were included numerically to develop a stress criterion for the prediction of the SFLD. The stress criterion of localized necking developed in terms of MPa is formulated relative to two different sets of RD as For sRD sTD, sTD
372:072
0:497sRD 0:00074s2RD 0
(11a)
and sTD
460:511 0:11541sRD 0:000339s2RD 0
(11b)
For sRD sTD, sRD
415:2 0:008854sTD 0:00044165s2TD 0 (12a)
and sRD
340:246
0:458112sTD 0:0007154s2TD 0 (12b)
A material element is said to have yielded the localized necking when one of the above stress criteria, Eqs. (11a) and (11b) or (12a) and (12b), have been satis®ed. Eqs. (11a) and (11b) refers to the condition of rolling when sRD sTD while Eqs. (12a) and (12b), sRD sTD . In other words, the localized necking would occur when the stress state in a stamped panel exceeds the bonded area of the solid curves in Fig. 1. This implies that both equations of (a) and (b) must be met to yield localized necking.
307
4.1. FLDs with equibiaxial prestrain Fig. 2 depicts the experimental and predicted FLDs normal to the RD with three different equibiaxial prestrains. The prestrains chosen are: (i) eRD 0:04, (ii) eRD 0:07, and (iii) eRD 0:17. The major loading direction after prestrain was made normal to the RD. It can be observed from the ®gure that the predicted FLDs coincide well with the test results. Upon unloading, the load at a given prestrain was reduced to zero. On the other hand, in the numerical simulation, if the unloading is not considered, a recent investigation reported failure in prediction of the left side of FLD for the case of equibiaxial prestrain of 17% [18]. Failure for the prediction may be attributed to two assumptions made. Firstly, the simulation was based on rigid± plasticity. Secondly, unloading after prestrain was not considered. These two assumptions caused the stress to exceed its limiting value when the direction of strain increments after the prestrain changes abruptly. The elasto-plastic constitutive equation introduced in the present investigation allows the developing stress in a sheet to follow a different stress±strain path upon unloading from that based on the rigid±plasticity theory. 4.2. FLDs with uniaxial prestrain Three different strain path combinations are considered, i.e. (1) both the imposed prestrains and the loading direction normal to RD, (2) the uniaxial prestrain parallel to RD followed by the loading direction normal to RD, and (3) the prestrain parallel to TD followed by the loading direction normal to TD. In the second and third cases, the rotation of the principal strain direction is expected to produce a large amount of strain reversal.
Fig. 2. Predicted and measured FLDs normal to the RD after equibiaxial prestrains.
308
C.L. Chow, X.J. Yang / Journal of Materials Processing Technology 133 (2003) 304±310
Fig. 3. Predicted and measured FLDs normal to the RD after uniaxial prestrain normal to the RD.
Fig. 3 shows the experimental and predicted FLDs for the ®rst case (1) where the uniaxial prestrains prescribed are (i) eTD 0:05, (ii) eTD 0:12 and (iii) eTD 0:18. The predicted FLDs agree well with the experimental results for cases (i) and (ii), while the maximum discrepancy of 0.039 is observed for case (iii) at the minimum limit strain. Also shown in the ®gure is a gradual upward shift of FLD to the left with increasing prestrain, a phenomenon that can also be successfully predicted by the stress model. Fig. 4 depicts the experimental and predicted FLDs for three cases: (i) eRD 0:04, (ii) eRD 0:125, and
(iii) eRD 0:18 for the second case (2). The experimental FLDs show that with increasing prestrains parallel to RD, the FLDs are shifted downward to the right. The predicted FLDs demonstrate that the proposed model can accurately characterize this experimental observation. Fig. 5 summarizes the experimental and predicted FLDs for three cases: (i) eTD 0:05, (ii) eTD 0:12, and (iii) eTD 0:18 for the third case (3). The experimental FLDs show that with increasing prestrains parallel to TD, the FLDs are shifted downward to the right. The predicted
Fig. 4. Predicted and measured FLDs normal to the RD after uniaxial prestrains parallel to the RD.
C.L. Chow, X.J. Yang / Journal of Materials Processing Technology 133 (2003) 304±310
309
Fig. 5. Predicted and measured FLDs parallel to the RD after uniaxial prestrain normal the RD.
Fig. 6. Predicted and measured FLDs normal to RD after plane prestrain parallel to the RD.
FLDs demonstrate that the proposed model can also successfully characterize this experimental observation. 4.3. FLDs with pre-plane strains Fig. 6 depicts the experimental and predicted FLDs for two cases of near plane prestrain parallel to RD: (i) eRD 0:08, (2), eRD 0:14. The prestrained specimens are then loaded normal to RD. It can be observed from the ®gure that the model can once again successfully predict
the FLDs as well as the shifting of the FLDs downward to the right with increasing prestrains. 5. Conclusions Abrupt changes in strain path during metal forming can signi®cantly affect the forming limits. A stress criterion for localized necking is developed such that the limit strain subjected to arbitrary loading history can be satisfactorily
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C.L. Chow, X.J. Yang / Journal of Materials Processing Technology 133 (2003) 304±310
predicted. The following conclusions can be drawn from the investigation on the applicability and validity of SFLD for AL2008-T4 alloy under nonproportional loading. (i) For the SFLD of AL2008-T4 alloy, the convergence of the limit stresses to a single curve has not been observed. The uniqueness of the SFLD is found to be dependent significantly on the strain path. (ii) Based on the mixed anisotropic kinematic±isotropic plastic model and a stress criterion, satisfactory prediction of FLDs under non-proportional loading can be achieved from the measured FLDs in an open literature. Acknowledgements The authors wish to acknowledge a partial support of from the Research Excellent and Economic Development Fund (REEDF) within the College of Engineering and Computer Science at the University of Michigan-Dearborn. References [1] C.L. Chow, W. Tai, M. Demeri, Numerical simulation of the Ubending process for VDIF steel sheets using damage analysis and LSDYNA, Sheet Met. Form. Technol. (1999) 169±178. [2] A.F. Graf, W. Hosford, Calculation of the forming limit diagrams for changing strain paths, Metall. Trans. A 24 (1993) 2497±2501. [3] A.F. Graf, W. Hosford, Effect of changing strain paths on forming limit diagrams of AL2008-T4, Metall. Trans. A 24 (1993) 2503± 2512. [4] A.F. Graf, W. Hosford, The influence of strain-path changes on forming limit diagram of AL6111-T4, Int. J. Mech. Sci. 36 (1994) 897±910.
[5] J.V. Laukonis, A.K. Ghosh, Effect of strain path changes on the formability of sheet metals, Metall. Trans. A 9 (1978) 1849±1856. [6] K.P. Rao, V.R. Mohan Emani, Simple prediction of sheet metal forming limit stresses and strains, in: M.Y. Demeri (Ed.), Sheet Metal Forming Technology, The Minerals, Metals and Materials Society, 1999, pp. 205±231. [7] W.M. Sing, K.P. Rao, Study of sheet metal failure mechanisms based on stress state conditions, J. Mater. Process. Technol. 67 (1997) 201± 206. [8] R. Arrieux, Determination and use of the forming limit stress diagrams, J. Mater. Process. Technol. 53 (1995) 47±56. [9] R. Arrieux, M. Boivin, Determination of the forming limit stress curve for anisotropic sheets, Ann. CIRP 36 (1987) 195±198. [10] R. Hill, A theory of yield and plastic flow of anisotropic metals, in: Proceedings of the Royal Society A, vol. 193, London, 1948, pp. 281± 297. [11] W. Prager, The theory of plasticity: a survey of recent achievements, in: James Clayton Lecture, Proceedings of the institute of Mechanical Engineers, vol. 169, 1955, pp. 41±57. [12] J.L. Chaboche, O. Jung, Application of a kinematic hardening viscoplasticity model with thresholds to the residual stress relaxation, Int. J. Plast. 13 (1998) 785±807. [13] J.C. Moosbrugger, Anisotropic nonlinear hardening rule parameters from reversed proportional axial-torsional cycling, ASME J. Eng. Mater. Technol. 122 (2000) 18±28. [14] P.J. Armstrong, C.O. Frederick, A mathematical representation of the multiaxial Bauschinger effect, CEGB Report RD/B/N73 1, 1966, Central Electricity Generating Board. [15] H.C. Wu, H.K. Hong, Y.P. Shiao, Anisotropic plasticity with application to sheet metals, Int. J. Mech. Sci. 41 (1999) 703±724. [16] C.L. Chow, X.J. Yang, E. Chu, Prediction of forming limit diagram based on a damage coupled kinematic±isotropic hardening model under nonproportional loading, J. Mater. Eng. Technol. 124 (2002) 219±226. [17] A. Barata da Rocha, F. Barlat, J.M. Jalinier, Prediction of the forming limit diagram of anisotropic sheets in linear and nonlinear loading, Mater. Sci. Eng. 68 (1984) 151±164. [18] T.B. Stoughton, A general forming limit criterion for sheet metal forming, Int. J. Mech. Sci. 42 (2000) 1±27.