Two-surface plasticity Model and Its Application to Springback Simulation of Automotive Advanced High Strength Steel Sheets Taejoon Parka, Dong-Yoon Seoka, Chul-Hwan Leea, Nobuyasu Nomab, Toshihiko Kuwabarab, Thomas B. Stoughtonc and Kwansoo Chunga a
Department of Materials Science and Engineering, Research Institute of Advanced Materials, Seoul National University, Daehak-dong, Gwanak-gu, Seoul 151-744, Republic of Korea b Division of Advanced Mechanical Systems Engineering, Institute of Engineering, Tokyo University of Agriculture and Technology, 2-24-16, Nakacho, Koganei-shi, Tokyo 184-8588, JAPAN c General Motors Global Research and Development Center MC 480-106-244, Warren, MI, 48090-9055, USA
Abstract. A two-surface isotropic-kinematic hardening law was developed based on a two-surface plasticity model previously proposed by Lee et al., (2007, Int. J. Plast. 23, 1189-1212). In order to properly represent the Bauschinger and transient behaviors as well as permanent softening during reverse loading with various pre-strains, both the inner yield surface and the outer bounding surface expand (isotropic hardening) and translate (kinematic hardening) in this twosurface model. As for the permanent softening, both the isotropic hardening and the kinematic hardening evolution of the outer bounding surface were modified by introducing softening parameters. The numerical formulation was also developed based on the incremental plasticity theory and the developed constitutive law was implemented into the commercial finite element program, ABAQUS/Explicit and ABAQUS/Standard using the user-defined material subroutines. In this work, a dual phase (DP) steel was considered as an advanced high strength steel sheet and uni-axial tension tests and uni-axial tension-compression-tension tests were performed for the characterization of the material property. For a validation purpose, the developed two-surface plasticity model was applied to the 2-D draw bending test proposed as a benchmark problem of the NUMISHEET 2011 conference and successfully validated with experiments. Keywords: Isotropic-kinematic hardening, Constitutive behavior, Bauschinger effect, Transient behavior, Permanent softening, Advanced high strength steels (AHSS), Spring-back PACS: 62.20.F-
INTRODUCTION Major efforts has been made to replace conventional steel sheets with light weight sheets and advanced high strength steel sheets in the automotive industry. However, there are several technical obstacles to overcome for their automotive applications such as inferior formability and larger spring-back. One of the effective ways to resolve such drawbacks is to optimize sheet forming processes utilizing computational methods, which can properly predict failure and spring-back in the process design stage. Since the spring-back of advanced high strength steel sheets is generally larger than that of conventional steel sheets and its computational prediction needs more sophisticated material characterization and numerical formulations. In order to properly account for mechanical properties of advanced high strength steel sheets, a constitutive model was developed based on a two-surface plasticity model previously proposed by Lee et al. [1]. Both the inner yield surface and the outer bounding surface expand (isotropic hardening) and translate (kinematic hardening) in this two-surface model. As for the permanent softening, both the isotropic hardening and the kinematic hardening evolution of the outer bounding surface were modified by introducing softening parameters. In this work, a dual
The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 1175-1183 (2011); doi: 10.1063/1.3623736 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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phase (DP) steel was considered as an advanced high strength steel (AHSS) sheet. Besides uni-axial tension tests, disk compression tests and uni-axial tension-compression-tension tests with various pre-strains were performed for the characterization of the DP sheet. For a validation purpose, the developed model was applied to the 2-D draw bending test proposed as a benchmark problem of the NUMISHEET 2011 conference.
THEORY In order to represent the Bauschinger and transient behavior during reverse loading, the combined type isotropic– kinematic hardening constitutive law based on the modified Chaboche model (Chung et al. [2], Chaboche [3]) is given by M f ( iso 0
(1)
where is the Cauchy stress, is the back-stress for the kinematic hardening and the effective stress (related to the isotropic hardening), iso , is the size of the yield surface as a function of the accumulative effective strain,
( d ) . Note here that the effective strain increment, d , is defined to be a conjugate value of the effective
stress, iso , under the following work equivalence principle:
dwiso ( ) : d p iso d
(2)
In the two-surface plasticity model, the back-stress for the kinematic hardening of the (inner) yield surface consists of its relative kinematic motion, , (with respect to the bounding surface kinematic hardening) and the back-stress evolution for the kinematic hardening of the (outer) bounding surface; i.e.,
(3)
where and are the back-stress for kinematic hardening of the yield surface and the bounding surface, respectively. As for the evolution of , the following evolution rule is applied: ( ) d d h iso iso iso
f
1 M
(4)
( )
where is the difference between the size of the bounding surface, iso , and the size of the inner yield surface, iso , while d is defined in Eq. (2). Note that the parameter h controls the rate of the relative kinematic motion. As for isotropic hardening of the inner yield surface to describe the expansion of the yield surface size, d iso is obtained from simple tension/compression tests. As for the description of the bounding surface, the evolution of the isotropic hardening is defined as, d iso Kbound nbound ( 0,bound )nbound 1 d
(5)
which is the Swift type law. As for the kinematic hardening of the bounding surface, h ( ) d h2,bound 1,bound d h2,bound iso
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(6)
In order to describe the permanent softening behavior, the hardening evolution of the bounding surface was modified: s
iso * * iso bound iso ( , n ) * s bound h1,bound h1,bound kine ( , n* )
(7)
bound bound where iso and kine are the softening parameters of the bounding surface for isotropic hardening and kinematic
hardening, respectively. Here, * is the accumulative effective strain measured during the n* -th (current) reverse 0.0 iso ( * , n* ) 1.0 , 0.0 kine ( * , n* ) 1.0 , iso ( * 0.0, n* ) 1.0 loading and while and
kine ( * 0.0, n* ) 1.0 . With 1.0 in Eq. (7) for all reverse loading, reverse loading curves without permanent softening are recovered.
MATERIAL CHARACTERIZATION In this work, a dual phase (DP) steel, DP980, produced by Arcelor-Mittal and supplied by GM was considered. Uni-axial tensile, uni-axial T-C-T and disk compression tests were performed for the characterization of the DP980 sheet.
Uni-axial Tensile Behavior Uni-axial tension tests were carried out based on ASTM E 8M standard procedure with specimens prepared by a wire-cutting process along the rolling direction. Tensile grip speed was 0.05 mm/sec, which corresponds to approximately 0.001/s in the engineering strain rage considering the 50 mm gauge length. For anisotropy, uni-axial tensile tests along the rolling, 45ȋ off and transverse directions were performed to measure the hardening curves and R-values based on ASTM E 8M and ASTM E 517, respectively. The measured thicknesses, Young’s moduli (E), 0.2% offset yield strength (YS), ultimate tensile strength (UTS), uniform and total elongations as well as plastic strain ratios (R-values), averaged for three tests per each, are summarized respectively in Table 1 and the engineering stress-engineering strain curves for each direction are comparatively shown in Fig. 1.
Engineering stress(MPa)
1200
DP980-550
1000 800 600 400 RD 45D TD
200 0 0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Engineering strain
FIGURE 1. Uni-axial tension curves of the DP980 steel
Thickness (mm) 1.188
TABLE 1). Elastic and plastic properties of the DP980 steel %Elongation Dir. E YS UTS (MPa) (MPa) (degree) (GPa) Uniform Total 0º 45º 90º
204.095 212.998 208.755
833.69 788.12 751.00
1177
1104.70 1098.33 1098.24
7.11 5.31 5.73
11.53 9.69 9.77
R-value 0.985 0.835 1.121
Uni-axial Tension-Unloading-Tension Behavior In order to measure the reductions of Young’s modulus according to the equivalent plastic strain increase, uniaxial tension-loading-tension tests were performed. Tensile tests were performed following the ASTM E 8M standard procedure at a constant tensile speed of 0.05 mm/s, which corresponds to approximately 0.001/s in the engineering strain rate with the gauge length of 50 mm. Using the specimens prepared along the rolling direction, Young’s moduli during loading and unloading for each pre-strain increment, 1% and 3% in engineering strain, were measured as shown in Fig. 2. The measured Young’s moduli were fit to the following equation (Yoshida et al., [4]). E E0 ( E0 ESAT )(1 eCSAT )
(8)
Here, E0 is the initial Young’s modulus measured from the initial true stress-true strain curves of the uni-axial tension tests, ESAT is the saturated Young’s modulus when the effective strain, , goes to infinity and CSAT is the convergent speed factor. Determined parameters, E0 , ESAT and CSAT are 204.095 GPa, 176.337 GPa and 67.743, respectively.
Engineering stress (MPa)
1200
DP980-550
1000 800 600 400 200 0 0.00
1% pre-strain interval 3% pre-strain interval
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Engineering strain
FIGURE 2. Uni-axial tension-unloading-tension curves of the DP980 steel
Disk Compression Tests The Rb value (the ratio of transverse to rolling direction strains in balanced biaxial tension) was obtained from the disk compression tests. Assuming that hydrostatic pressure has no influence on yielding, through-thickness disk compression tests were performed on a Gleeble Tester 3500. Several specimens, with a diameter of 6.0 mm were compressed through the sheet normal direction. For each specimen and each material, compression was performed with increments of 4kN from 20kN to 40kN. After each compression increment, diameters in the rolling and transverse directions were measured. Based on these measurements, assuming incompressibility Rb value and the 0.2% offset yield stress under the balanced biaxial stress condition, b , were calculated as 1.1657 and 713.199 MPa, respectively.
Uni-axial Tension-Compression-Tension Behavior In-plane tension-compression-tension tests were performed using the testing apparatus developed by Kuwabara et al. [5]. Since buckling of the specimen is a major problem while carrying out compression tests on thin samples, this testing apparatus uses specially designed comb-shaped dies for applying a small blank holding pressure of 5 MPa to a sheet specimen to prevent buckling of the specimen. In order to prevent the specimen from galling the dies, it was lubricated on both sides with Vaseline and Teflon sheets. A run without the sample is carried out to measure the friction in the die assembly, which is later compensated in the results by subtracting (or adding) it from measured load values. Tests were carried out under a displacement controlled mode and two different speeds were used: in the elastic regions, the test speed was 0.5 mm/min whereas, in the plastic region, it was 1.5 mm/min. As shown in Fig. 3, using the specimens prepared along the rolling direction, T-C-T tests were carried out for the DP980 steel under the following true strain histories:
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(a) 0.02 ˧ 0 ˧ 0.02, 0.04 ˧ 0 ˧ 0.04 and 0.06 ˧ 0 ˧ 0.06 (b) 0.04 ˧ -0.04 ˧ 0.04 and 0.06 ˧ -0.06 ˧ 0.06
DP980-550
1000
True Stress (MPa)
True Stress (MPa)
1000 500 0 -500 0.02 pre-strain 0.04 pre-strain 0.06 pre-strain
-1000
DP980-550
500 0 -500 -1000
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07
-0.06 -0.04 -0.02
True Strain
0.00
0.02
0.04
0.06
True Strain
(a)
(b)
FIGURE 3. Uni-axial tension-compression-tension curves of the DP980 steel Experiment Fitting (Swift type) Fitting (Voce type) Fitting (Combined type)
1500
DP980-550
True stress (MPa)
1000 500 0 -500 -1000 -1500 -0.06 -0.04 -0.02
0.00
0.02
0.04
0.06
True strain
FIGURE 4. Calibrated and measured uni-axial tension-compression-tension curves of the DP980 steel
From the measured uni-axial tension-compression curves, at each unloading position, the effective yield stress ( ) is calculated first as
iso
f r
(9)
2
Here, f is the yield stresses at the start of unloading and r is the initial yield stress during reverse loading. Then, the measured effective stresses, iso , at each unloading position were fit to the following Swift type hardening law as summarized in Table 2:
iso Kinner ( 0,inner )n
inner
(10)
After the characterization of the isotropic hardening, iso , of the (inner) yield surface, the isotropic hardening and kinematic hardening motions of the bounding surface were calibrated before the characterization of the kinematic hardening motion of the inner surface. Under the uni-axial tension/compression in the rolling direction, the following relation is derived from Eq. (6): d h1,bound h2,bound d
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(11)
As shown in Fig. 4, the DP980 steel does not show the permanent offset (softening) during the first reverse loading of the uni-axial tension-compression-tension test, compared to the reverse loading curves calculated from the Voce type, Swift type and combined type hardening coefficients. However, the DP980 steel shows permanent softening during the second reverse loading. In summary, for the DP980 steel, no permanent softening was assumed for the first reverse loading, while permanent softening was assumed for the second loading. After the characterization of the effective stresses, iso , of the (inner) yield surface, the effective size of the bounding surface, , for each unloading position was fit to the following Swift type hardening law: total
nbound total Kbound ( 0,total bound )
(12)
For the uni-axial deformation, the following relationship is derived from Eqs. (5) and (6):
iso iso (0) iso iso (0)
(13)
where iso (0) is the initial effective size of the bounding surface. Also, d d iso d d (1 )d , 0
1
(14)
Here, the parameter, , determines the ratio of the isotropic hardening increment with respect to the effective size of the bounding surface increment, d iso d , and the ratio of the kinematic hardening increment with respect to the effective size of the bounding surface increment, d (1 )d . Assuming the parameter,
, to be a specific
constant, the isotropic hardening, iso , and the kinematic back-stress movement, , of the bounding surface can be determined from the measured effective size of the bounding surface, . In this work, the parameter, assumed to be 0.5 for simplicity; i.e., iso iso (0) d iso (0) 0.5d
, was
(15)
(0) d 0.5d
where (0) iso (0) and (0) 0 . The isotropic hardening of the bounding surface was fit to the Swift type hardening law as summarized in Table 2. Now, the general solution of Eq. (11) for the kinematic hardening becomes,
( ) e h
2,bound
( 0* )
h * e 2,bound ( 0 ) ! h1,bound ( )d ( 0* ) * 0
(16)
when h2,bound is a constant as assumed here. Then, h1,bound is determined by the measured kinematic hardening, , and the assumed constant h2,bound . In order to describe the change of the parameter, h1,bound , the following equation was used: h1,bound ( , n* , n* ) a1 g1 (1 e d1 ) b1ec1 ( 0 ) b1ec1 ( 1 ) b1ec1 ( 2 ) *
*
*
b1ec1 ( n ) *
(17)
Here, 0* and n* denote the initial effective stain and the effective strain at the n* -th reverse loading, respectively. If the saturation rate parameter, c1 , is large enough and the reverse loading number, n* is zero, Eq. (17) becomes h1,bound ( ) a1 g1 (1 e d1 )
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(18)
After the characterization of parameters for the kinematic movement of the bounding surface, the following softening parameter was introduced only to the kinematic hardening of the bounding surface in order to describe the permanent softening of the second reverse loading curves: bound kine ( n* , n* ) (1.0 a4 ) a4 eb
* 4 n
(19)
Finally, in order to control the different rate of the relative kinematic hardening motion, , of the inner surface during initial loading and reverse loading, the following effective strain dependent values of h was used: h( n* , n* ) h0 (hsat h0 )(1 eCh ( n n1 ) ) *
*
(20)
The measured parameters were summarized in Table 2, while the measured and calibrated true stress-true strain curves during the uni-axial tension-compression-tension curves are compared in Fig. 5.
Yld2000-2d Yield function The eight anisotropic coefficients of the plane stress yield stress function Yld2000-2d [6] were determined based on eight mechanical measurements 0 , 45 , 90 , R0 , R45 , R90 , b and Rb are listed in Table 3.
K (MPa) 803.842 a1 (MPa)
iso h1,bound
40881.6
bound kine
TABLE 2). Elastic and plastic properties of the DP980 steel 0,bound 0 Kbound (MPa) n diso 0.2225 0.4953 1019.37 0.000012996 g1 (MPa) d1 b1 (MPa) c1 (MPa) h2,bound 100000 500 21941.0 10.4831 a4 b4 0.2057
n* 0 h
500 (constant)
# c11
# c22
0.8199
1.3428
h0
Ch
h0
161.5685
63.4993
32.8226
186.7992
0.9822
0.1141
0.1853
1.054
185
Ch
23.2201
TABLE 3). Anisotropic coefficients of Yld2000-2d ## ## ## ## c11 c12 c21 c22
Experiments Two-yield surf.
15.1693
# c66
## c66
1.060
0.9365
Experiments Two-yield surf.
1000
1000
True Stress (MPa)
True Stress (MPa)
34.5485 n* " 2 hsat
n* 1 hsat
nbound
0.0436
500 0 -500
500 0 -500 -1000
-1000 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07
-0.06 -0.04 -0.02
0.00
0.02
0.04
0.06
True Strain
True Strain
(a)
(b)
FIGURE 5. Comparison of the calibrated and measured uni-axial tension-compression-tension curves of the DP980 steel
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NUMERICAL SIMULATION OF THE 2-D DRAW BENDING TEST For validation purposes, the developed constitutive model was applied for the simulation of the 2-D draw bending test proposed as a benchmark problem of the NUMISHEET 2011 conference. Two different specimens were prepared for the 2-D draw bending tests: one with base material (without pre-strain) and the other with prestretching by 0.04 in true strain. A schematic view of tools and their dimensions are shown in Fig. 6 and Table 4. The hardening model was incorporated into the commercial finite element programs, ABAQUS/Explicit and ABAQUS/Standard, utilizing the characterized mechanical properties of the DP980 steel sheet. Forming process was simulated using the explicit finite element code, ABAQUS/Explicit, with the user-defined subroutines VUMAT, while springback behavior was simulated using the implicit finite element code, ABAQUS/Standard, with the userdefined subroutines UMAT. The 4-node three-dimensional rigid body element (R3D4) was used for tools and the 4node shell reduced volume integration element (S4R) with 9 integration points through thickness was employed for the blank. The element size of the blank was approximately 1.0mm Ý 1.0mm throughout the whole blank and only the quarter of the whole blank was simulated for the computational efficiency. Blank holding force of 300 kgf (2.94 kN) was maintained throughout the test and the friction coefficient was assumed to be a constant of 0.1. The simulation punch speed was 100mm/sec, which is 100 times higher than the actual speed. Fig. 7 shows the measured and simulated spring-back profiles. The simulation results reasonably well represent the spring-back behavior of the DP980 grade advanced high strength steel sheet. Plane of symmetry
W1 G1 Thickness
W3
R1 Holder
Punch
Holder
Die
Sheet Die
R2
W4 Stroke
W2
(a) W
Rolling direction
L
(b) Lgrip Grip Position
Lgrip Rolling direction
L0
Grip Position
Cutting
Grip Position
W0
Grip displacement
Rolling direction
L
Grip Position
Cutting
(c) FIGURE 6. Schematic view of specimens and tools and their dimensions for the 2-D draw bending test: (a) tools, (b) base specimens (without pre-strain) and (c) 4% pre-strained specimens
Parameters Dimensions Parameters Dimensions
W1 50.0 L 300.0
TABLE 4). Dimensions for the 2-D draw bending test (unit: mm) W2 W3 W4 R1 R2 G1 54.0 89.0 89.0 5.0 7.0 2.0 W L0 W0 Lgrip Grip displacement 30.0 300.0 30.0 34.705 9.411
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Stroke 70.0 L 240.0
Experiments (w/o pre-strain) Simulation
50
50 40 Z coord.(mm)
Z coord.(mm)
40 30 20
30 20
10
10
0
0 0
10
20
30
40
50
60
70
80
90
100
Experiments (with 4% pre-strain) Simulation
0
10
20
X coord.(mm)
30
40
50
60
70
80
90
100
X coord.(mm)
(a)
(b)
FIGURE 7. Measured and simulated spring-back profiles for the 2-D draw bending test: (a) without pre-strain, (b) with 4% prestrain
CONCLUSION A two-surface isotropic-kinematic hardening law was developed to account for the springback of automotive advanced high strength steel sheets. A dual phase (DP) steel sheet, DP980 steel, produced by Arcelor-Mittal was considered as an advanced high strength steel sheet in this work. Besides uni-axial tension, disk compression and uni-axial T-C tests, T-C-T tests were performed with various pre-stains for the characterization of the DP980 sheet. The numerical formulation was also developed based on the incremental plasticity theory and the developed constitutive law was implemented into the commercial finite element program, ABAQUS/Explicit and ABAQUS/Standard using the user-defined material subroutines. For a validation purpose, the developed two-surface plasticity model was applied to the 2-D draw bending test proposed as a benchmark problem of the NUMISHEET 2011 conference and successfully validated with experiments.
ACKNOWLEDGMENTS The work was performed under the collaboration between GM and SNU. This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (R11-2005-065)/the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2010-220-D00037)/the Engineering Research Institute at Seoul National University. The authors are also grateful to Prof. Myoung-Gyu Lee at POSTECH, Dr. Kanghwan Ahn and Dr. Kyung-Hwan Chung at POSCO for performing the disk compression tests, uni-axial tension tests and 2-D draw bend tests.
REFERENCES 1. 2. 3. 4. 5. 6.
M.-G. Lee, D. Kim, C. Kim, M. L. Wenner, R. H. Wagoner, K. Chung., K., Int. J. Plast. 23 (7), 1189-1212 (2007) K. Chung, M.-G. Lee, D. Kim, C. Kim, M. L. Wenner, F. Barlat, Int. J. Plast. 21 (5), 861-882 (2005) J. L. Chaboche, Int. J. Plast. 2 (2), 149-188 (1986) F. Yoshida, T. Uemori, Int. J. Plast. 18, 661-686 (2002) T. Kuwabara, Y. Kumano, J. Ziegelheim, I. Kurosaki, Int. J. Plast. 25, 1769-1776 F. Barlat, J. C. Brem, J. W. Yoon, K. Chung, R. E. Dick, S-H Choi, F. Pourboghrat, E. Chu, D. J. Lege, Int. J. Plast. 19 (9), 1297-1319 (2003)
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Paradigm Change: Alternate Approaches to Constitutive and Necking Models for Sheet Metal Forming Thomas B. Stoughtona and Jeong Whan Yoonb a
General Motors Global Research and Development Center MC 480-106-244, Warren, MI, 48090-9055, USA b Faculty of Engineering and Industrial Sciences, Swinburne University of Technology PO Box 218, Hawthorn, VIC 3122, Australia Abstract. This paper reviews recent work proposing paradigm changes for the currently popular approach to constitutive and failure modeling, focusing on the use of non-associated flow rules to enable greater flexibility to capture the anisotropic yield and flow behavior of metals using less complex functions than those needed under associated flow to achieve that same level of fidelity to experiment, and on the use of stress-based metrics to more reliably predict necking limits under complex conditions of non-linear forming. The paper discusses motivating factors and benefits in favor of both associated and non-associated flow models for metal forming, including experimental, theoretical, and practical aspects. This review is followed by a discussion of the topic of the forming limits, the limitations of strain analysis, the evidence in favor of stress analysis, the effects of curvature, bending/unbending cycles, triaxial stress conditions, and the motivation for the development of a new type of forming limit diagram based on the effective plastic strain or equivalent plastic work in combination with a directional parameter that accounts for the current stress condition. Keywords: Non-Associated Flow Rule, Plastic Dilatancy, Pressure Sensitivity, Distortional Hardening, Necking Limits. PACS: 62.20.F-,62.20.mm
INTRODUCTION This paper reviews recent work that proposes paradigm changes for constitutive and necking models in sheet metal forming. The first section reviews the motivating factors and benefits in favor of both associated and nonassociated flow models for metal forming in the view of experimental, theoretical, and practical aspects. This section completes with a review of three examples of non-associated flow, one allowing capture of the most common documented anisotropies of the material response in both plastic flow and yield behavior using quadratic function, another a fully anisotropic treatment of pressure-dependent yielding in the absence of plastic dilatancy, and finally, one allowing treatment of four degrees of anisotropic hardening. The topic of constitutive models will be followed by a section reviewing a paradigm change of metal forming limits from strain to stress metrics. This section will review the topic of the forming limits, the limitations of strain analysis, the evidence in favor of stress analysis, and motivation for the development of a new type of forming limit diagram based on the effective plastic strain or equivalent plastic work in combination with a directional parameter that accounts for the current stress condition. Particular attention is paid to common misunderstanding in the application of stress metrics, which is contributing to the continued use of strain metrics for the analysis of formability under the violated conditions. The section will also review the role of stress and strain gradients through the sheet thickness, the effect of non-zero normal stress, and cyclic bending and unbending, all of which affect not only the interpretation of the stress and strain conditions during forming, but also the interpretation of the forming limit curve itself.
The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 15-34 (2011); doi: 10.1063/1.3623589 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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PARADIGM CHANGE IN CONSTITUTIVE MODEL FLOW RULES The associated flow rule (AFR) is most commonly used as a basis in constitutive models for metals for several reasons. One reason is simplicity; By equating the plastic potential, which is used to define the direction of the plastic strain rate, to the yield function, one need only define a single function, which ostensibly is simpler and easier to calibrate. Another reason is theoretical; Bishop and Hill (1951) derived a proof that the AFR applies to a material whose deformation is restricted to simple slip mechanisms characterized by reaching a critical shear stress on a slip plane known as the Schmid criterion, which is understood to be the primary mechanism of plastic deformation. A third reason is physical and numerical stability; Drucker (1951) proved that use of the associated flow rule guarantees positive plastic work and uniqueness of solution to the equations of classical plasticity, and further established a class of stable metals, known as non-energetic materials in Drucker (1959). Additional questions of stability under non-AFR have been raised by Sandler and Rubin (1987), who derived an indeterminate (non-unique) solution to the elastic-plastic wave equation under non-AFR, which they argued proved that non-AFR is an intrinsically unusable concept in plasticity modeling. While the AFR is well established in plasticity modeling of metals, this popularity is brought under serious challenge by the experimental work of Richmond (1980), and Spitzig and Richmond (1984), who studied pressure sensitivity of yielding and plastic dilatancy of steel and aluminum alloys. They found a small but finite pressure sensitivity of the yield behavior for all steel and aluminum alloys that they studied. Under the AFR, this pressure sensitivity would be expected to result in an associated small but finite plastic dilatancy, or permanent volume change. However, they observed no plastic dilatancy in their experiments within their experimental resolution, which was more than 10 times smaller than the amount of dilatancy that was expected under the AFR. This led them to conclude that the AFR does not generally apply to metals, a conclusion that has not yet had much influence on the selection of constitutive modeling for metals. The experimental evidence against the AFR raises doubts about the significance of the theoretical arguments that suggest it is necessary. One possibility that undermines Bishop and Hill’s theoretical foundation for the AFR, for example, is that it does not consider the effect of dislocation interactions or other effects that could lead to deviations from the Schmid criterion. Qin and Bassani (1992), and Bassani et al.(2001) describe dislocation interactions and structures that lead to non-associated flow effects. Also, Bulatov et al. (1999) describes details on a specific dislocation mechanism derived from first principles that predicts the magnitude of the AFR violations in Richmond’s work. Stoughton and Yoon (2006) derived a set of four conditions that apply to both AFR and non-AFR models in response to the stability issues raised by Drucker. The constraints, which are described later in this paper, ensure uniqueness and positive-definite plastic work when applied to a class of non-AFR models. The conditions include generalizations of convexity requirements applied individually to the plastic potential and yield functions, a requirement of positive work hardening (for uniqueness under applied stress changes), and a fourth requirement that is a generalization of Drucker’s postulate. The nature of that latter requirement, which is automatically satisfied under the AFR, can also be satisfied under a class of non-AFR. This work proves that satisfaction of Drucker’s postulate is a sufficient, but not necessary condition of uniqueness of solution and positive-definite plastic work. Stoughton and Yoon (2006) also discuss the intrinsic energetic properties of non-associated flow, showing that the characteristics of an energetic material are in a different class from questions of uniqueness and positive plastic work. It is pointed out that the amount of work that can be productively used in a closed strain cycle comes from the stored elastic energy, is a small fracture of that energy, is uniquely defined in terms of the load state and the size and direction of the strain cycle, and is only released during the first cycle since the entire strain cycle is within the yield function after the initial loading. In those terms, its nature is not comparable to the type of instabilities that would occur if the plasticity equations had non-unique solutions or permitted negative plastic work. Stoughton and Yoon (2008) also reviewed the derivation of the indeterminate solution to the elastic-plastic wave equations that was derived by Sandler and Rubin (1987), showing that this solution does not explain the equations of motion unless the plastic wave speed is strictly constant. They then showed that the curvatures of the yield function, plastic potential, and stress-strain response that are typically used to characterized metal deformation, generally work to prevent the plastic modulus and plastic wave speed from reaching a static value at any time during plastic deformation.
16
While the interpretation of experiments, such as the implications of Spitzig and Richmond’s work, and the theoretical arguments in favor of either the AFR or the non-AFR, will continue to be debated for some time, the third factor that plays a role in an engineer’s selection of material model are simplicity, the practical issues of accuracy, complexity, calibration, and cost. As implied earlier, there may be a wrong assumption that non-AFR is intrinsically more complex, involves more parameters that require calibration, and leads to higher computational cost due to the fact that there are two functions involved in the constitutive equations, instead of one. However, it can be argued the other way, that with the constraint of the AFR, it becomes a greater challenge to simultaneously capture the anisotropy of both yielding and plastic flow with a single function, which often appear to be unrelated, forcing one to introduce functions of increasing complexity. The most simple model for anisotropic metals under the AFR is the quadratic yield function proposed by Hill (1948). The limitations of this yield function under the AFR are well known. Bramley and Mellor (1966) were one of the first to show that this model is not able to predict the in-plane anisotropy of the yield behavior of steels when the parameters were calibrated from the experimental r values. They noted that if one ignores the in-plane anisotropy and restricts the model to consider only normal anisotropy, or average r value in the plane of the sheet, then the model gives an acceptable fit to the ratio of the yield in equal biaxial tension and the average yield in uniaxial tension for the four steel alloys that the considered in their study. With the state of the art of material modeling of anisotropic materials in its infancy, Bramley and Mellor proposed using the special case of the normal anisotropic version of Hill’s quadratic function for steels. Pierce (1968) expanded on Bramley and Mellor’s idea by applying Hill’s normal anisotropic model to nine metals with average r values ranging broadly from 0.18 to 3.80. He found that the normal anisotropic model did not correlate well for materials with average r value less than one. This paper may have been the first to refer to this discrepancy between experiment and the quadratic model as the ―anomalous yield‖ behavior in equal-biaxial tension associated with metals with r values less than one. Not surprisingly, researchers for aluminum alloy sheets played a leading role in advancing constitutive models for anisotropic materials in the next several decades. Even today, the dominance of the use of Hill’s normal anisotropic model, and the primacy of the consideration of r value in decisions to use more advanced models, can probably trace its origins to the influence of the work of Bramley and Mellor (1966) and Pierce (1968), respectively. However, the role of the AFR was not considered in these earlier works. If non-AFR had been considered, the conclusions would have been that Hill’s fully anisotropic quadratic function has the capability to exactly match all the experimental r values and yield stress at 0, 45, and 90 degrees to the rolling direction, as well as the yield stress in equal-biaxial tension, regardless of the r value. In that case, at least within the context of the data presented, the conclusion would have been that Hill’s fully anisotropic model was good choice of function to use under non-AFR for the 4 steel alloys in Bramley and Mellor’s study and all 9 metals in Pearce’s study, including the zinc alloy with average r equal to 0.18. These conclusions were reported in Stoughton (2002a), who proposed a simple non-AFR model based on two independent quadratic functions of the form proposed by Hill. This model will be reviewed in the next subsection, followed by a series of developments including a pressure-sensitive anisotropic yield function that is consistent with Spitzig and Richmond’s work published by Stoughton and Yoon (2004), and a simple, but powerful distortional hardening model that takes advantage of the quadratic form to analytically incorporate the anisotropy in the strain hardening functions at 0, 45, and 90 degrees to the rolling direction, as well as in equalbiaxial tension. The final subsection concludes with a discussion of potential directions in material model development, including higher order functions and kinematic models, which introduce a higher level of flexibility with minimal complexity under non-AFR.
Fully Anisotropic Non-Associated Flow Rule The yield criterion for an anisotropic metal under isotropic strain hardening is defined as follows,
y 11 , 22 , 23 f p
17
(1)
where σf(εp) is the strain hardening function along the rolling direction of the coil, in the normalization adopted in the yield function. Using the model proposed by Hill(1948) with normalization to the rolling direction along the 1axis, Stoughton (2002a) defined a non-AFR model with yield function parameters as follows 2 y 112 y 22 2 y 11 22 2 y 122
y 0 90
2
1 2 y 0 2 45
2
2 2 1 0 0 y 1 2 90 B 2 0 B
(2)
where σ0, σ45, and σ90, are the initial yield stress in uniaxial tension along the rolling, diagonal, and transverse directions and σB, is the initial yield stress in equal biaxial tension. To generalize this model using the flow rule,
p E ij( p ) p ij ,
(3)
the plastic potential is given by an identical quadratic form with different coefficients under a non-AFR. The definition is the same as that used by industry in most calibrations of Hill’s model under the AFR, 2 p 112 p 22 2 p 11 22 2 p 122
p
1 1 / r90 1 1 / r0
p
1 1/ r 1/ r
1 1 1 / r0
(4)
90 1 2r45 p 0 2 1 1 / r0
where r0, r45, and r90, are the r values in uniaxial tension along the rolling, diagonal, and transverse directions. As noted in Stoughton and Yoon (2004), it is convenient for expressing the constitutive equations to use lower case symbols p and y to define the gradient tensor components of the plastic potential and yield functions respectively,
p11 1 p p 22 p p12
11 p 22 p 22 p 11 2 p 12
y11 1 y y 22 y y12
11 y 22 y 22 y 11 2 y 12
(5)
and use upper case P and S to define the tensor product of the elastic matrix C and, respectively, the gradient of the plastic potential p and the total strain e=[E11,E22,2E12] ,
S C : e
P C:p
(6)
where
1 E C 1 1 2 0 0
0 1 2 1 0
18
(7)
and E and ν are Young’s Modulus and Poisson Ratio, respectively. This leads to the following definitions for this particular model
1 p 11 p p 22 E p p 22 p 11 P 2 1 - p 1 p 12
S
E 1 - 2
E11 E 22 E 22 E11 . 1 E12
(8)
Then the rate of change of the effective plastic strain is given by,
p
MAX 0, y11S11 y 22 S 22 y12 S12 h y11 P11 y 22 P22 y12 P12
(9)
where h is the derivative of the stress-strain relation along the rolling direction, which is used in the yield condition defined in Eq. 1. The constitutive law relating the stress rates to the total strain rates is then given by,
11 S11 P11 S P 22 22 22 p 12 S12 P12
(10)
It should be noted that the only difference in computational costs using this non-AFR is the three extra calculations given in Eq. 4, since all the subsequent calculations leading to the solution in Eq. 10 are identical to those used in an AFR in the case p=y. This small extra computation cost is offset by the advantage that the model now exactly matches the initial uniaxial and equal biaxial yield stresses and the uniaxial strain ratios at 0, 45 and 90 degrees.
Non-AFR with Pressure Sensitive Yielding One of the motivations for non-AFR discussed in the introduction was the experimental evidence published by Spitzig and Richmond’s of pressure-sensitive yield behavior in the absence of plastic dilatancy. In the general full stress version of the non-AFR model in the previous section, the plastic potential and yield functions are both invariant under application of hydrostatic stress. As a result, the model constrains the plastic dilatancy to zero, in agreement with the experimental results, but also results in zero pressure sensitivity. In order to obtain a physically meaningful description of the pressure sensitivity, we consider the analysis reported in Spitzig, Sober, and Richmond (1975). This study reported the yield behavior for a high strength steel subjected to combined tension with hydrostratic pressures from 0 to 8 GPa. They found the best fit to the data is described in terms of the J2 and I1 invariants for an isotropic material in the following form,
3J 2 1 I1 f p
(11)
where α is a small material constant of inverse stress such that αI1 is on the order of a few percent. This is the term that gives rise to a small pressure sensitivity. This yield function also predicts a strength differential effect (SDE) between tension and compression, which agreed with experiment. Stoughton and Yoon (2004) generalized this yield criterion for full anisotropy in both the invariant contributions for plane stress conditions to obtain the following component of a non-AFR for plane-stress conditions,
y ij 1 1 11 2 22 f p
19
(12)
where 2 y 1 112 2 22 2 y 11 22 2 y 122
1
C 0 T0 C 02 T02
1 1 1 1T0
2
2
C 90 T90 2 C 90 T902
T 1 2 0 T 1 T 90 2 90
T 1 1 y 1 2 0 2 TB 1 1 2 TB
2
2
2T 2T 2T 2 T45 4TB 1 2T y 0 0 0 0 1 T0 2 T45 TB T45 TB 2 T T 45 B 1 12 1 2 T45 2TB
1 12 1 2 T45 2 1 1 2 TB 2
(13)
where T0, T45, and T90, are the initial yield stress in uniaxial tension along the rolling, diagonal, and transverse directions, TB, is the initial yield stress in equal biaxial tension, and C0, and C90, are the initial yield stress in uniaxial compression along the rolling and transverse directions. Note that while the equations for some of the 6 constants of this model are complex, they are all explicit functions of easily measured 6 experimental yield stresses. Although we could also generalize the plastic potential, there may be no need for metals characterized by zero plastic dilatancy. Consequently, the plastic potential described by Eq. 4 was adopted in Stoughton and Yoon (2004). While the material parameter definitions for the yield function are more complex, this does not affect the computational speed. Consequently, the ability to account for SDE and pressure sensitivity in the yield stress may make the modest increase in complexity of this constitutive law worth the effort. In this case the gradient of the yield function y becomes
1 11 y 22 1 1 1 11 2 22 y 2 22 y 11 2 y y 2 y 12 0
(14)
Surprisingly, no other changes or additional computations are required to find the solution using Eqs. 6-10.
Non-AFR with Distortional Hardening One of the disadvantages of a yield criterion in which the coefficients of the yield function are constants is that it is not possible to capture the anisotropy in the rate of hardening that is actually observed in most metals. While the rate of hardening in uniaxial tension in the plane of the sheet appears to be nearly isotropic to a good approximation, this is not generally true in comparison of the rate of hardening in uniaxial to equal biaxial tension. Consequently, a noticeable increase in accuracy can be obtained over the range of biaxial stress conditions that occur in metal forming processes by allowing for distortional hardening, as is noted in Stoughton and Yoon (2009).
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One way to capture this differential hardening is to treat the coefficients of the yield function as explicit functions of the plastic strain, as described by Zamiri and Pourboghrat (2007). However, for high exponent yield functions, the fidelity of the predicted hardening is affected by the interpolation function used to define the evolution of the coefficients. Furthermore, it may be difficult to closely match the hardening behavior and control the shape of the yield function without excessive discretization of the interpolation function when using higher exponent functions for fully anisotropic materials. This difficulty is a result of the convoluted relationship between the coefficients and the hardening behavior that do not allow analytical forms to directly control the shape of the stress-strain response. The dependence on interpolation function does not occur for a quadratic yield function because the coefficients can be defined explcitly in terms of the stress-strain relations at the four critical experimental conditions, i.e., at 0, 45, and 90 degrees in uniaxial tension as well as in equal biaxial tension. In this case, no parameter interpolation functions are required. The accuracy of the distortional hardening is limited only by the accuracy of the analytic relationship between stress and strain along the directions of the model calibration. Such a simple approach to distortional hardening would not be attractive under an AFR because, not only would one loose correlation of the predicted plastic strain rates, which as mentioned earlier, occurs under isotropic hardening models, but the distortion of the yield surface would introduce a deformation dependence in the direction of plastic strain under linear loading conditions, which is not typically observed. However, in a non-AFR model, the plastic potential can remain as defined in Eq. 4, allowing the model to exactly match the primary experimental stress-strain responses in uniaxial tension at 0, 45, and 90 degrees and in equal biaxial tension, without affecting the direction of the plastic strain components. The first step in the construction the distortional hardening model described in Stoughton and Yoon (2009) is to redefine the yield criterion in Eq. 1 as follows:
122 4 2 f y 2 11 2 22 11 22 11 222 2 12 1 B p 45 p 90 p 0 p
(15)
where the four functions σx(εp) are the explicit stress-strain relations that are experimentally determined along the directions x=0, 45, 90 in uniaxial tension, and in equal biaxial tension for x=B. [We discuss calibration of these relations later in this subsection.] Although perhaps not obvious, Eq. 15 is mathematically equivalent to Eqs. 1 and 2 in the limiting case when all four stress-strain relations are equal to the function on the right hand side of Eq. 1. Unlike most other models, the yield function above is dimensionless, and although it also defines an implicit yield surface for a given value of εp, and the yield function serves the same purpose as the yield stress in the constitutive equations, there is no explicit scalar function that can be called the yield stress. Due to the dimensionless character of the yield function, the gradient tensor of the yield function y has dimensions of inverse stress.
2 11 22 22 0 22 1 1 1 y 2 11 2 2 22 11 2 0 2 11 0 p 90 p 45 p B p 8 12 2 12 0 0 1
(16)
The only other major change to the model is the definition of the function h, in Eq. 9 which is now defined by requiring the yield function fy be invariant under changes to εp during plastic deformation in the same manner as is done in conventional models. This constraint leads to the following definition, which is again dimensionless,
122 4 2 h 2 2 11 h0 p 2 22 h90 p 11 22 11 222 hB p 2 12 h45 p (17) 0 p 90 p B p 45 p
21
where the four dimensionless functions hx(εp) for X=[0,45,90,B] are given by
h X p
1
X p
d X p d p
.
(18)
The yield criterion given by Eq. 15 is defined in a way that the stresses in the four stress-strain responses are equal to the actual true stresses in uniaxial and equal-biaxial tensions. In other words, they do not need to be converted to an effective stress to account for anisotropy, as is commonly the case when comparing experiment to predictions of anisotropic models. However, as discussed in more detail in Stoughton and Yoon (2009), the effective plastic strain in these four functions is not equivalent to the measured plastic strain, so that calibration of the functions used in the model is necessary. For calibration, the measured plastic strain must be scaled using a formula derived from the flow rule and plastic potential, Eqs. 3 and 4. The scale factor for true strain in uniaxial tension at 0, 45 and 90 degrees and for the sum of the principal true strains in equal biaxial tension are given, respectively, by
1
1 2
1 p 2 p p
p
1 p 2 p .
(19)
This model was compared in Stoughton and Yoon (2009) to conventional isotropic models for three aluminum and two steel alloys for which equal-biaxial and uniaxial tension data was available in 15 degree increments in the plane of the sheet. Using the additional uniaxial data at 15, 30, 60, and 75 degrees, it is possible to obtain an estimate of the error between experiment and this model, as well as other popular models that do not consider anisotropic hardening. As expected, with calibration of the isotropic hardening models to the initial stress, the root mean square error of most of the models studied, including the model described in Barlat et al. (2003) and the isotropic hardening non-AFR model described in a previous subsection of this paper, was found to be under 1%. The small error in these models at the onset of yielding arises from differences in the behavior at uniaxial tension angles of 15, 30, 60, and 75 degrees, whose data was not used in the parameter definitions. However, what is more interesting is that as the plastic strain increases to 10% strain or higher, the error for all the isotropic hardening models increased quickly to levels comparable to the error found using von Mises and Hill quadratic model under the AFR, even for the aluminum allows using the most sophisticated high exponent yield functions. In contrast, the error for the distortional hardening model is found to remain static over all levels of plastic strain at under 1%. This result suggests that unless distortional hardening is included in the constitutive model, there may not be as much benefit to use advanced constitutive models with isotropic hardening for analysis of problems with a large mixture of biaxial stress conditions.
Future Developments Using Non-AFR We close this section with some comments about the fertile ground for research in the application and extension of non-AFR models. We have considered above only quadratic forms of the plastic potential and the deviatoric component of the yield function. However, it is well known that the curvature of the yield function in plane strain conditions is relatively flat for aluminum, leading to higher exponent functions. It appears to be more convenient to use a simpler higher exponent function to describe the yielding combined under non-AFR with a lower exponent or quadratic function for the plastic potential. Furthermore, a quadratic plastic potential has advantages because there is an explicit analytic expression for the effective plastic strain, as well as the relation between the stress and plastic strain tensor directions. The higher convexity of plastic potentials also means that the solution is less sensitive to small variations in stress near the plane strain condition, allowing more easily converged solutions even using low convexity high exponent yield functions, including those with zero curvature and vertices. These more sophisticated non-AFR models may have numerical and perhaps physical advantages over traditional higher exponent functions under AFR. In addition, the expansion of non-AFR into the realm of kinematic hardening is also wide open with opportunities adding another dimension to consider in coupling the kinematic properties of the yield function and plastic potential (for example, Taherizadeh et al.(2010)). All considered, the flexibility of non-AFR models to capture a higher degree of complex material behaviors is one of its most attractive features. However, while the authors have
22
addressed the issue in Stoughton and Yoon (2006, 2008) for the class of non-AFR model in which the above examples are included, the issue of stability generally needs to be addressed when expanding non-AFR beyond the class of models considered in this and previous papers. In this class of scalar hardening models, the following set of constraints were proven to be necessary and sufficient to ensure desired characteristics of plastic stability, including unique solution and positive changes to the plastic work,
11 P : 22 0 12
11 Y : 22 0 12
h0
hP:y 0
(20)
The first, third and fourth constraints are necessary to ensure positive increments to the plastic work. The third and fourth constraints are also necessary to avoid non-uniqueness or singularity under stress-rate or strain-rate boundary conditions, respectively. The second constraint is the least important of the four, and was shown to be only necessary to ensure that the second order work under proportional loading is positive definite, a characteristic which may be desired to avoid the possibility if yield point elongation, for example. Further details of the meaning of these constraints can be found in Stoughton and Yoon (2006). The first two models described in the previous subsections satisfy all four constraints, but it is important to consider that these stability conditions may come into play with any new variant of non-AFR. For example, an interesting violation of the third condition was found in application of the distortional hardening law described in the previous subsection. This violation arises from the stress-dependent weighting function of the biaxial stress-strain term in the explicit definition of h in Eq. 17, which becomes negative between uniaxial and pure shear stress conditions. It was found in the empirical fits for all three aluminum alloys considered in Stoughton and Yoon (2009), with the relative hardening rate differences between uniaxial and equal biaxial tension, that the h parameter falls below zero in pure shear loading at a strain of 0.449 to 0.600, depending on the alloy. This occurs even though the strain-hardening in all four stress-strain relations is positive. While h=0 leads to non-uniqueness in stress-controlled processes at this critical strain level, the fourth condition, which would lead to non-uniqueness in strain-controlled processes, was not observed in the data because the produce P:y was sufficiently large for all stress conditions. It is interesting that the third stability conditions remain satisfied up to these very high strains for aluminum, which suggests the possibility that they may be physically associated with a strain localization mechanism and metal failure. So in this case, the violation of the third stability may not be evident of a flaw or limitation of the application of this non-AFR, but could be representative of a physical instability or characteristic of these aluminum alloys. And, while there is no direct connection in the topics of non-AFR and necking models in this paper, the question of the role of instability on localization mechanism is a good segue into the topic of forming limits.
PARADIGM CHANGE IN FORMING LIMIT MODELS Continuing along the common thread of paradigm change, we now review developments in modeling failure in sheet metal forming. Many practitioners of the metal forming community remain faithful to the idea that strain metrics are useful for formability assessment, while many others who are aware of at least some of the limitations of strain metrics are nevertheless reluctant to use stress metrics because of concerns about their viability. In this paper we will consider a broad set of these concerns and point to evidence in the literature that may help to resolve the issues. Finally, a new type of forming limit diagram is described that has advantages of both stress and strain metrics. Perhaps the most serious limitation of the strain FLC is shown in Fig 1A. However, it may not be immediately obvious how serious the problem is because it is often argued that the complex pattern of strain FLCs shown here do not apply when strain paths are linear. But that argument is seriously misinterpreting what the data in Fig 1A means. The FLC’s in this figure are in actuality all characterization of the strain FLC for linear paths. To understand this meaning, consider, for example, the following three FLC’s in the figure: 1) the BLACK FLC with FLDo at a longitudinal strain of about 0.19 for the zero-prestrained condition, 2) the BLUE FLC with a cusp close to the horizontal axis at a transverse strain of about 0.13, and 3) the RED FLC with a cusp just below the horizontal axis.
23
The set of all three of these curves define the ―single‖ FLC for a linear strain path corresponding to uniaxial strain along the transverse direction. What this means is the strain FLC for linear strain path is not static, but is an intrinsically dynamic limit. This is not semantic difference. It has serious consequences on the use and utility of the strain FLD. It means that the strain FLC for the as-received condition gives absolutely no clue about the margin of safety of a given process, regardless of whether the path is linear or non-linear. Once the implications of this data are understood, it cannot be rationally argued that the strain FLD can be used in conditions where the strain path is ―almost‖ linear. In other words, the only point on the original FLC, the conventional FLC for the as-received material, is the endpoint of the single strain path on which a linear strain path sets its initial course. In view of that, few measures are more limited in application than the strain-FLC for materials in the as-received condition, Experimental FLC's of 2008 T4 Al After Prestrain Stress-Based FLC's of 2008 T4 Al Before and After Selected Prestrains
0.50
500 0.40
450 400
0.20
0.10
P1
U2
350
Major True Stress (MPa)
LongitudinalTrue Strain
0.30
E2 300 250 Yield Surface after E2 prestrain
200 150
P4 Initial Yield Surface
100
-0.30
-0.20
-0.10
0.00 0.00
0.10
0.20
50
0.30
U4
0 0
-0.10
50
100
150
200
250
300
350
400
450
500
Minor True Stress (MPa)
TransverseTrue Strain
FIGURE 1. A) Net-strain FLC’s of 2008 T4 AA reported by Graf and Hosford (1993) for the as-received and for 12 prestrain conditons. B) Corresponding stress FLC(s) for the as-received and 5 prestrain conditions selected from shown in (A) that result in prestress to points U2, P1, E2, P4, and U4 as calculated and reported in Stoughton (2000). The solid black curve shows the expansion of the initial yield surface to E2, which is the starting point for plastic flow in the second strain increment for that case.
It continues to be argued explicitly or implicitly that strain paths in most metal forming applications and particularly in the first draw die, are sufficiently linear that the strain FLC for the as-received condition can be used without serious risk of error in formability assessment. In fact, with the exception of axisymmetric and uniform sectioned parts, studies of industrial applications show that most of the strains in automotive applications that reach levels of plastic strain that would be considered at risk, also include a substantial nonlinear component in the strain history. Nonlinear strain paths occur in every region of late contact with 3D tooling surfaces, as well as in areas in which the metal flows from one region of the tool geometry to another. While nature tends to take the shortest and therefore linear path, the complex 3D geometries we attempt to form metal to, do not allow metal to be stretched to high strains by linear paths, except in very rare circumstances.. Arrieux, Bedrin, and Boivin (1982) were among the first to publish evidence that the dynamic nature of the strain FLC for steel arises from a single path-independent stress limit in combination with the non-unique relationship between the loading stress and net plastic strain tensor components. They were among the first to propose using the stress FLD in formability analysis. This discovery has been shown by others to apply to all metals, including the 2008 T4 AA using in Graf and Hosford (1993) shown in Fig 1A and B. The stresses reported here are calculated using the method described in Stoughton (2000) based on the Graf’s reported material properties and either following a linear loading condition from zero strain to the as-received FLC, or linear loading to the prestrain, followed by stress unloading and reloading in different directions to the final net strain on the corresponding strain FLC. The final stress conditions on each of these strain FLCs appears to be consistent with a single stress FLC, within experimental uncertainty. Based on these and similar experimental studies, the stress condition appears to be the casual factor in the initiation of necking. More importantly, it support a simple practical solution to the
24
ambiguity of strain forming limits in manufacturing, as originally proposed by Arrieux, et al.; As long as the stress condition is maintained below this single stress FLC, the forming process will be safe from necking. So the stress FLC satisfies the need of industry for a simple diagram to clearly separate safe and non-safe forming conditions, much as the strain FLC is currently misused today. Although this FLC is apparently path-independent, the stresses calculated in the finite element (FE) analysis are path dependent variables. Therefore, it is necessary to monitor the stress state at each step of the computation and determine if it is below or above the stress-based FLC. For this purpose, it is convenient to use a single parameter to monitor the formability margin as described in Stoughton and Yoon (2005). The formability parameter γC, defined in terms of critical stress ratios, as well as an alternate formability parameter γ’ defined in terms of critical effective plastic strain ratios proposed by Zeng et al. (2008) are defined as follows,
C
y ij Y FLC 1
FLC pij
C
p11 p22
p ij
p FLC
e1FLC
(21)
p11 p22 p11 p22
p11 p22 2 p122 p11 p22 2 p122
where σY(ε) is the stress-strain relation, εp is the effective plastic strain, σy(σij) is the yield function, σp(σij) is the plastic potential, β is the direction of the current increment to the plastic strain, and e 1FLC is the major strain at the point on the strain FLC in the direction defined by β in the strain FLD. In Eq. 21, γC is a single stress scaling parameter that represents the degree of formability according to the criteria γC 1 (neck). In the next subsection, when we review the effect of curvature, we will show that this determination must be made at each integration point, and that necking occurs only if the γ C values at all integration points simultaneously exceed the value of 1. Despite the strong experimental evidence supporting a path-independent stress FLC, among of the factors that have slowed the acceptance of stress-based metrics are the decades of theoretical work and development of models to explain the strain FLC. These theoretical works include bifurcation analysis conducted by pioneers in the field of plasticity, including Hill (1952), Storen and Rice (1975), Hutchinsen and Neale (1977), as well as necking models proposed by Marciniak and Kuczynski(1967). This strong theoretical basis for the strain FLC has contributed to overconfidence in strain analysis, and perhaps a lack of appreciation for its limitations. In contrast to what is perceived by some to be a proof of the validity of a strain forming limit criterion, careful study of these models show that they are in reality a proof of the validity of a path-independent stress forming limit criterion. As pointed out in Stoughton and Zhu (2004) in a review of these bifurcation models, at some point in the derivation of the strain limit, the authors of these models imposed an assumption of linear strain. Prior to imposing this constraint, the equations defining the instability condition were shown by Stoughton and Zhu to be actually in the form of a relationship between the stress variables. These relations were shown to have no dependence on a history variable that isn’t uniquely defined by the current stress state. Furthermore, Zeng et al. (2008) reported the results of a study of bilinear strain paths using MK Analysis for a model with isotropic hardening, predicting the pattern of behaviors such as seen in Fig 1A. However, they also found no difference in the stress condition within the matrix when the strain rate inside the defect reached the critical level associated with onset of necking. This work shows that MK Analysis also naturally leads to a path-independent stress limit. In considering the application of the stress FLC, it is important to understand that although it is possible in principle to measure stresses using X-rays or in specially equipped tube hydro-forming experiments, such stress limits would actually not be helpful for assessing formability in finite element simulation. This is because our material models are not sophisticated enough to predict the stresses with sufficient accuracy to compare them to actual measured forming limit stresses. Seemingly, that fact may discourage some from using the stress FLC for formability analysis. But in fact, this is also not a justified position. To understand why this is so, for the sake of argument, let us refer to the ―stresses‖ we calculate using any given model as ―process-control-variables‖ or PCV’s. We would like to think of these PCVs as the real stresses, but in fact, their actual function in the metal forming
25
simulation is to define the force equilibrium conditions with sufficient accuracy that we can predict the strain history and strain distributions in the part. If we think of these PCVs as strain-control parameters used in the forming simulation and admit that the correlation with real stresses need not be perfect to obtain good prediction of the strains and strain history, we clearly put ourselves at risk when we use them to predict springback, since then we rely on the accuracy of the material model to allow us to treat these PCVs as the real stress tensors. Now the question is, ―Do we also put ourselves at risk in using these PCVs to predict forming severity?‖ The answer to the latter question is perhaps surprising, ―No‖. To demonstrate that, Stoughton (2002b) calculated the limit conditions for several sets of PCVs calculated from a variety of divergent material models, including models developed by von Mises, Logan-Hosford (1980), and Barlat (1997), and the non-AFR proposed by Stoughton (2000), all calculated based on the same strain FLC’s shown in Fig 1. As expected, the shape and location of the FLCs defined in terms of these sets of PCVs in the space of the PCVs was shown to be significantly different for each material model. However, it was also shown that the PCV-space FLC’s collapsed to the same limit condition for linear and nonlinear strain paths, such as is shown in Fig 1B using Hill’s model. What this means is that for formability analysis, it does not matter if the PCVs are real stresses or proxies for the real stress; As long as we translate the strain FLC to the space of the PCVs using the same material model as we use in simulation, we can use these variables to deal with the path dependent nature of strain metrics. However, for convenience, we can continue to refer to these variables as stresses.
Role of Curvature One of the primary factors that cause confusion in understanding forming limits is the role of the stress and strain gradients through the sheet thickness. These gradients are intrinsic to curved sheet and therefore critical to understanding and applying forming limit criterion based on stress or strain. For example, stretching a 1 mm thick sheet over a 2 mm radius will introduce a difference in the true strain between the top and bottom side of the sheet of up to ln(1.5)=0.405, depending on the amount of in-plane tension that thins the metal. That strain difference is on the order of the FLDo value of most steels and twice the limit of aluminum. So this raises the question, ―What layer do you use to define the stress (or strain) that will be compared the stress (or strain FLC) in the formability analysis?‖ When industry first started to implement the FLD in the 1960’s, it was quickly discovered that strains measured most conveniently on the convex side of the sheet were commonly found to be well above the FLC with no sign of necking. Remarkably, without any experimental evidence to justify the decision, the metal forming industry adopted the approach of using the membrane strains in making comparison to the strain FLC. This assumption has continued unchecked in both physical tryout and analysis of numerical simulations for nearly two decades, and continues to be the dominant practice used in industry today. Unfortunately, the assumption is wrong, and the truth has serious consequences in both the interpretation of forming limits and their application in analysis. Tharrett conducted a series of simple bending under tension tests at General Motors on strips of different thickness of steel, aluminum, and brass and different punch tip radii with the objective to determine what strains through the thickness are the cause of necking. He discovered that necking initiated not when the membrane strains exceeded the strain FLC, as was previously thought, but much later in the forming process, when the strains on the concave side of the sheet rose to the level of the FLC. While the tests were limited to plane strain conditions, the results were confirmed in all materials and tooling geometry. The details of the experiments for steel were later published by Tharrett and Stoughton (2003), and the results for one test geometry are shown in Fig 2. There are two necks observed in this specimen on either side of the center punch tip radius at the location where the strains on the concave side, shown by the enlarged circles, rose to the level of the FLC for this material. Considering the importance of stress metrics, Stoughton (2008) noted that Tharrett’s results are also understood to apply to the stress conditions, so that this important factor can be applied to both linear and nonlinear deformation processes. In other words, for a neck to initiate, the stress on all layers through the thickness must exceed the stress FLC. This generalization has interesting consequences because often the stress field is more complex than the strain field, due for example, to a history of cycling bending/unbending. Furthermore, it is important to note that use of membrane values will result in overly conservative predictions of necking on curved sheet. This mistake will undermine correlation with experiment, but also, because the level of the conservative estimate is proportional to the strain gradient through the thickness, it will result in a proportional bias in the safety margin towards regions of higher curvature, while providing no additional margin of safely in regions of zero curvature or through-thickness
26
stress gradient. Since failures most often occur away from curved areas of the product for this very reason, the bias of using membrane strains in formability assessment provides no real benefit to producing robust processes. There is another insidious effect of Tharrett’s results, which is on the interpretation or definition of the experimental FLC which is used as the basis of both strain and stress analysis. Many forming limit curves are based on data obtained directly from the Nakajima Test where the necking strains are measured on the convex side of the sheet stretched over a 200 mm diameter hemispherical punch. In view of Tharrett’s results, the limit strains should be measured on the concave side. But by using the strains on the convex side, the shape of the FLC is biased in a complex manner that affects both the major and minor strain of each point. Furthermore, it introduces a thicknesses effect on the FLC that is not real. The bias on these strain measurement becomes even more severe if sub-size tooling smaller than a 200mm diameter punch is used to generate the FLCs. The effect of using this biased FLC generated from the convex surface strains without compensation for Tharrett’s effect will proportionally undermine correlation with reality and may exacerbate the controversy over correlation between prediction and tryout.
FIGURE 2. A) Sum of the principal strains for a 50 wide strip of 1008 AK steel stretch-bent over a punch wedge with a ¼ inch radius to the depth at which onset of necking occurs, as reported in Tharrett and Stoughton (2003). The forming limit is characterized as a simple limit on the sum of the principals because the minor strain was less than or equal to zero at all points along the strip in a region of the FLD characterized by a limit on thinning strain for this metal. The FLC and FLDo was obtained from standard FLD tests independent of the stretch-bend test.
Role of Cyclic Bending/Unbending In the late 1990’s, Chrysler engineers brought attention to the observation that strains in the die wall within the ―bead affected zone‖ (BAZ), i.e. metal that has undergone cyclic bending/unbending through drawbeads prior to drawing into the die wall, were frequently found to be higher than the FLC with no sign of necking. Since the FLC is used in decisions to approve the die and forming process as production ready, the engineers struggled with resolving the question whether it was really necessary to continue to work the process to reduce strains further in the BAZ, especially since no problems with formability were observed in tryout arising from these high strains. In order to come to an understanding and solution to this problem, the US Auto-Steel Partnership launched the Enhanced
27
Forming Limit Project, which was completed in 2003 and published in a Technology Report (2003) and associated documents. The project involved an extensive experimental program involving 6 sheet steels and generation of about 70 FLC’s for the as-received materials and after a large number of prestrain conditions obtained entirely through cyclic bending and unbending. The cyclic prestrain was obtained in a large draw die that pulled 12 inch wide strips of metal into a vertical channel die wall to a depth sufficient to expose a 12 inch square of the material in the die wall to uniform cyclic deformation. The amount of net strain from the cyclic deformation was varied from a few percent up to 40% through a variable die radius (3, 6, and 12 mm) and zero, one, or two insertable square or round beads of variable radii bolted in slots located near to the die opening. Another insertable bead, located more than 12 inches outboard of the two inner bead locations, was used as an option to add backforce, which increases the stretch by cyclic deformation from the inner beads and die radius through the exponential effect of the product of friction and angle of contact with the bead and die radii. After the prestrain, the material from the 12 inch square region in the die wall was cut and stretched in a Marciniak Test to obtain the FLC, defined in terms of the net strain, including the prestrain through cyclic near-plane-strain conditions. These FLCs were compared to the measured FLC for the as-received materials. This test geometry and process is also familiar to the global industrial community because it was also used in Benchmark #3 in the Numisheet 2005 Benchmark Study as described by Stoughton et al. (2005). Forming Limit Diagram Forming Limit Diagram AKDQ-CR 0.7 mm
Major Strain (%)
AKDQ-CR 0.7 mm - Prestrain Condition # 403
Major Strain (%)
100
100
DFLC0 = + 10.4 %
90
90
80 80
70 70
60 60
50 50
40 40
30 30
As-received
After prestrain
As-received Theoretical FLC0: 33.7 % Experimental FLC0: 32.3 % DFLC0: -1.4%
-20
-10
Experimental FLC0:
0
10
20
30
20
42.7 %
Back tension ratio: 0.000 Square double beads: 4 mm Penetration: 50 %
10 0
-30
Experimental FLC0: 32.3 %
20
10 0
-20
-15
-10
-5
0
5
10
15
20
Minor Strain (%)
Minor Strain (%)
FIGURE 3. TypicalKeeler-Brazier results from theofUS LimitsFLC offorCyclic Deformation (2003). Data Experimental as-received material prediction FLC Auto-Steel Partnership Study on Forming Experimental FLC after Experimental (A) for the as-received AKDQFLCsteel and (B) the same material after passing through a double set prestrain of square 4mm radius beads at Not necked necked 50% penetration andNotacross a 6 mm die radius, resulting in a net major (engineering) prestrain of 14.8% in plane-strain In the neck field In the neck field conditions. Necked Necked
An example of the experimental results that illustrates the objective of the study is shown in Fig 3B for an AKDQ cold-rolled steel. Figure 3A shows the FLC for the as-received material, with FLDo of 32.3% engineering strain. Figure 3B shows the FLC after prestrain in cyclic bending/unbending process using two square beads with 4mm radii at half penetration. The cyclic prestrain results in a net engineering stretch of 14.8%. (Recall that because this net principal strain is from cyclic deformation, the effective plastic strain (EPS) and strain hardening on the surface layers is much higher, and in this 2 square bead geometry is expected to be in excess of 100%.) Remarkably, despite the excessive amount of plastic work on the surface layers and the high net principal strain under plane-strain conditions, the FLC for the net strain limit is shifted upward, with a new FLDo at 42.7% engineering strain. This confirms the original observations made by Chrysler engineers that the effect is real and significant. The absolute shift of +10.4% is about 70% of the engineering prestrain of 14.8% (0.72 if using true strain metrics). It is as though 70% of the prestrain causes no damage to the material with respect to necking. This behavior was found in all 6 steels, including a DP 600 alloy, although the shift was found to average to a smaller percentage, 60% over all bead and process configurations.
28
The primary purpose of the A-SP study was to improve the interpretation of forming severity in die tryout. For this purpose, the A-SP developed a practical solution for die tryout, leading to techniques to estimate the component of the net strain due to cyclic bending/unbending by measuring the change in thickness on either side of the impact line (boundary of the BAZ). The measured principal strains within the BAZ are then corrected by subtracting 40% of the cyclic strain from the measured major strain before plotting the data on the conventional as-received FLC (for example, before plotting in Fig 3A). With this technique, it is not necessary to measure or use other FLCs such as the one shown in Fig 3B. In addition to the application of the effect of enhanced formability from cyclic bending in physical tryout, the effect also impacts the interpretation of results from simulation. However, since the most significant impact is in areas subjected to the largest history of cyclic deformations, which are in regions where the metal pulled through bead geometries, the effect is not as important to consider in simulation as it is in physical tryout if drawbeads are approximated by a line force model, as is often done in industrial applications for reasons of efficiency. Nevertheless, whether or not physical beads are included in the tool geometry used in simulation, the effect of enhanced formability from cyclic bending impacts other areas in which the metal bends and unbends over tool radii. Although the resulting shifts in the FLC are usually smaller than what occurs in the BAZ because they usually stem from only one cycle of bending and unbending, the shift in the forming limit can be enough to lead to mistakes in formability assessment in numerical simulation, especially for metal drawing across a sharp feature radius. Therefore, it is generally important to account for this effect in numerical simulation and as explained earlier, also to account for it using stress metrics to handle nonlinear deformation processes. Stoughton and Green (2005) analyzed the A-SP Project database and found that depending on the use of kinematic and/or distortional hardening models, the effect can be described with a simple stress limit. However, an easier method of implementing the stress analysis using isotropic hardening models is to apply the correction to the accumulated effective plastic strain (EPS) computed in the simulation in the same manner as proposed by the A-SP method for physical tryout. To show that this is effective, Fig. 4A shows the calculated stress for the data points shown in Fig. 3A that were classified as necked, in order to define the stress FLC for the as-received condition. That FLC is reproduced in Fig. 4B, which overlays the calculated stress conditions for the necked data for the prestrained condition shown in Fig 3B. However, to obtain the correlation of the prestrained data with the as-received FLC, the effective plastic strain from the prestrain was reduced by a factor of 0.60, from a value of 0.170 to a value of 0.102.
Stress FLD data for As-Received FLC
Cyclic Prestrain to EPS 0.170 Annealed to 0.102
700 700
600 600
500
Major True Stress
Major True Stress
500
400
As-Received Fit
300
400
As-Received FLC
300
Initial Yield
Initial Yield 200
Necked
200
Necked
Isotropic Yield
100
100
0
0
Annealed Yield Prestrain Path
0
100
200
300
400
500
600
700
0
Minor True Stress
100
200
300
400
500
600
700
Minor True Stress
FIGURE 4. Stress conditions calculated from the measured strain conditions in areas classified as necked in the test results shown in Fig 3, where A) shows the as-received forming limit behavior, and B) shows results based on an ―annealed‖ or reduced EPS from prestrain by a factor of 0.60 from a value of 0.170 to a value 0.102. All calculations are based on Hill’s isotropic hardening model for a material with normal anisotropy.
29
To apply this correction to simulation with isotropic hardening, it is necessary to keep a separate record of the contributions to the effective plastic strain, EPS, from compression, EPSC, and tension, EPST, at each integration point. The smallest of these two components is equal to half the contribution from cyclic bending and unbending, and the difference is the non-cyclic component. The cyclic contribution is then reduced by the empirical factor fEFL representing the ―annealing‖ of the ―Enhanced Formability Effect‖ of cyclic deformation. And finally, this annealed value is added to the non-cyclic component to obtain an annealed or corrected value of the EPS for the integration point, resulting in the following relation,
EPS max EPS T , EPS C 2 f EFL 1 min EPS T , EPS C .
(22)
This modified value that can be used to correct or calculate the forming stress components by projecting the predicted stresses onto the ―annealed‖ yield surface defined by the stress-strain relation at this value of EPS. This reduction is an empirical correction, similar to the one proposed by the A-SP for the strain FLD. Results would be different, and a correction may not be necessary at all, if a kinematic model is used in the analysis. While it is not common practice to consider lab tests involving cycles of compression and tension to measure forming limits or provide data to calibrate strain FLC’s, care should be taken to consider the beneficial effect of compression on formability to avoid serious mischaracterization of forming limits and formability prediction.
Role of Through-Thickness and Hydrostatic Stress Another area of interest in forming limits is the role of through-thickness and hydrostatic stress, which has resulted in several recent papers regarding the impact of these variables on strain limits. Since stress metrics have more general application to nonlinear processes, we will focus in this paper on the effect of these variables on stress limits, and will only mention the consequences on strain limits as it helps to understand the effect on stress. The through-thickness and hydrostatic stress does not arise in simulation based on shell elements because by assumption, the through-thickness stress is zero, and whatever effect there is to the hydrostatic component would be build into the definition of the forming limit for plane-stress conditions. To understand the impact of non-zero through-thickness and more generally, hydrostatic stress, we first imagine that we have employed MK Analysis to define the strain limit for any specific (linear or nonlinear) strain path under plane-stress conditions. If this is a linear path, we expect to obtain a point on the conventional FLC such as shown in Fig 3A and if it’s nonlinear, we would expect to obtain a point on an FLC with a shape like one of those in Fig 1A. Similarly, we expect this same analysis will result in a single stress FLC, regardless of whether the strain path was linear or not, as has been confirmed in a numerical study by Zeng et al. (2008). So the question is, ―What is the effect on this calculated necking limit if the through-thickness stress was nonzero in our MK Analysis?‖ To obtain the answer to this question, it is helpful to first ask, ―What is the effect on this calculated necking limit if we add a hydrostatic stress to the deformation history in our calculation?‖ We can answer this question without actually doing the MK Analysis, through the following stream of logic. If the material model is based on a pressure-insensitive constitutive relation, as we most often use in modeling metal deformation, then the answer is simple. Adding a constant stress to all three normal stress components has no effect in either the matrix or defect of the MK Analysis on yielding, on the direction of plastic flow, and as well as on the stress-strain relation. Consequently, the strain history within the defect, which is used to define the onset of necking, would be unaffected by hydrostatic pressure. It follows that the strain FLC obtained using MK Analysis under hydrostatic pressure will be identical to the strain FLC obtained under plane-stress conditions. Perhaps surprisingly, the independence of the strain FLC with respect to hydrostatic pressure has an important consequence on the application of the conventional stress FLC as previously described in the literature and as shown in Fig 1B and Fig 4. The FLC’s shown in these figures as functions of the in-plane principal stresses do not apply under conditions when the through-thickness stress is non-zero. The reason for this is that if the triaxial stress condition (σ1, σ2, σ3=0) is a point of necking instability on the conventional stress FLC, then, as a result of the invariance with respect to hydrostatic stress, all stress states (σ1+c, σ2+c, σ3=c) will also be unstable against necking. So as a result, all points in the entire (σ1, σ2) space used in the conventional stress FLC could be safe or necked,
30
depending on the magnitude of σ3. This ambiguity in the 2D stess FLD can be eliminated using the two in-plane deviatoric components of stress, but this dramatically changes the shape of the stress FLC. A simpler solution, proposed in Stoughton (2008) and Stoughton and Yoon (2011), which retains the shape of the conventional stress FLC for plane-stress conditions, is to simply subtract the current value of σ3 from all three principal stresses to obtain a plane-stress equivalent state, (σ1-σ3, σ2-σ3, 0). Plotting this data in modified stress FLC in the 2D space, (σ1σ3, σ2-σ3) accounts for the effects of both the invariance of the necking limit under changes in the hydrostatic stress and the dependence on the through-thickness stress component. Under the plane stress condition, Stoughton and Yoon (2011) combined the necking limit with a model for the fracture limit in the principal stress space by employing a stress-based forming limit curve (FLC) and the maximum shear stress (MSS) criterion. A new metal failure criterion for in-plane isotropic metals is described, based on and validated by a set of critical experiments. This criterion also takes into consideration of the stress distribution through the thickness of the sheet metal to identify the mode of failure, including localized necking prior to fracture, surface cracking, and through-thickness fracture, with or without a preceding neck.
The Polar Effective Plastic Strain Diagram One of the difficulties of the stress FLD is due the reduction of the slope of the true stress-strain relation. Due to this effect, larger changes in strain occur at stress levels close to the necking limit compared to at stress levels further below the limit stress. This makes it difficult to visually see or quantify the margin of safety without a magnifying glass or overlay of the contours of equivalent strain in the stress FLD. To remedy this difficulty, we will consider using the effective plastic strain as one of the metrics to assess formability following a modification of the idea proposed by Zeng et al. (2008). Although effective plastic strain is described as a type of strain, it is not directly linked to the principal or tensor components of the strain tensor. It is however, uniquely linked to the stress tensor through the yield function and stress-strain relation, and therefore falls under the category of a stress metric. To understand the idea proposed by Zeng et al.(2008), it is helpful to begin with the equations for translating the strain FLC into the stress FLC. In the translation of the principal strains to principal stress for a point (e 1, e2) on the conventional (as-received) strain FLC, for a material model defined by a in-plane isotropic plastic potential, σp(σ1,σ2), yield function, σy(σ1,σ2), and monotonic stress-strain relation, σY(εp), we can use the following set of relations for calculating the corresponding point (σ1,σ2) on the stress FLC,
e2 e1
p 1
e1 e2 p 1,
,
Y p y 1,
(23)
2 1
where β is the strain ratio, α is the stress ratio, and α(β) is the inverse function or solution to the following explicit relation,
p 1 , 2
2 p 1 , 2 1
.
(24)
1 , 2 1,
The definitions of the function α(β) and εp have more simple analytic forms than those given above in the case of a quadratic plastic potential, but the above equations are more general and apply to both quadratic and non-quadratic forms of σp(σ1, σ2). Generalization of this translation to anisotropic material models, including anisotropic forming limit data, is described in Stoughton and Yoon (2005).
31
It is useful to point out that that the accounting of nonlinear paths comes into play at only two places in Eq. 23. The first is in the definition of β, which for non-linear strain paths is more generally defined in terms of the ratio of the current strain rates. Nonlinear paths also play a role in the definition of the effective plastic strain ε p, which is more generally defined by the time integral of a function of the strain rates, rather than the net strains as shown in Eq. 23. This integration is also further complicated by the fact that α and β, which is used to define α, are also changing in time. No other relation in Eq. 23 depends explicitly on deformation history. What this means is that the forming limit for linear and nonlinear deformations can also be characterized as a simple limit on the accumulated effective plastic strain, εp , as a function of β, or as a function of α. The idea of an FLD based on the variables (εp ,β) was proposed by Zeng et al. (2008) as a better way to account for formability. As seen in Eq. 23, it is mathematically equivalent to the stress FLC for models with positive work hardening, in other words, as long as the function is monotonic, since each point in (εp,β)-space maps to a unique value in the (σ1,σ2)-space through these equations. Although it is equivalent, the new metric has several important practical advantages; First it scales with the magnitude of strain, so it is easier to visualize safety margins for conditions that are near to the necking limit. Second, it does not depend on the stress-strain relation at all. In one sense, that means it is less complex to use; But this independence with respect to the stress-strain relation has an even bigger advantage in that the forming limit criterion can be extended to material models that allow for hardening stagnations, or in which there is softening in the relation between true stress and true strain, such as in models in which temperature effects are considered. If for any reason, the stress-strain relation, σY(εp) is not monotonic, the effective plastic strain may continue to be a good metric for formability under nonlinear conditions, whereas the saturating or decreasing stress variables that arise from a non-monotonic stress-strain relation, would no longer be practical for assessment of formability. Although monotonic stress-strain relations ensure stability and uniqueness as discussed in the third requirement for stability given in Eq. 3, it is not necessary to forbid hardening behaviors that might lead to physical instabilities such as necking. So in these cases, the effective plastic strain is an attractive solution. Experimental FLC's in Conventional Strain Diagram
Experimental FLC's in Polar EPS Diagram
0.50
0.50
0.45 0.40
0.40
0.35
0.30
EPS*COS(θ)
LongitudinalTrue Strain
0.30
0.20
0.25
0.20
0.15
0.10 0.10
-0.30
-0.20
-0.10
0.00 0.00
0.05 0.10
0.20
0.30 -0.30
-0.20
-0.10
0.00 0.00
0.10
0.20
0.30
-0.10 TransverseTrue Strain
EPS*SIN(θ)
FIGURE 5. A) Strain FLC’s of the 5 prestrain and as-received condition corresponding to the 6 Stress FLC’s shown in Fig 1B. B) Calculated EPS FLC’s shown in the PEPS Diagram with the direction given by the angle defined in Eq.2 4. Note that the scatter in the EPS strain is clearly on the order of the experimental uncertainty, which is more difficult to argue from Fig 1B.
Another alternate and still equivalent solution to using the (εp, β) variables as proposed by Zeng et al. (2008) may be more compelling to industrial engineering applications. In this generalization, we propose to plot the data in a polar diagram of the εp variable with the angle defined as the arctangent of the ratio of the principal strain rates,
tan 1 e1 , e2 .
32
(25)
In a Cartesian equivalent system, the variables of the proposed diagram become (v1, v2)= (εpcos(θ), εpsin(θ)). The reason the polar diagram is compelling is seen in comparison of the shape of the conventional strain-FLD and the polar diagram in Fig 5. As is the case for the as-received condition in the strain-FLD, all the FLC’s in the polar diagram have cusps or low points in the plane-strain condition. Furthermore, since the effective plastic strain is actually a stress metric and the radial direction in this diagram uniquely defines the stress state, it is not surprising that we see in Fig 5B that the FLCs for these nonlinear strain paths are insensitive to the remarkably different strain histories. Interestingly, the shape of the FLC in the polar diagram is very similar to the shape of the strain FLC for the as-received condition. Most importantly, the radial directions in the polar diagram that correspond to uniaxial, plane-strain, equal-biaxial, etc. are parallel to the corresponding directions in the strain FLC. As long as engineers understand what this means, and do not make the mistake of plotting net strain tensor component in the polar diagram or plot EPS values without consideration of strain path, the polar diagram could be a convenient tool for more reliable formability assessments in manufacturing. For this reason, we propose to refer to it as the PEPS FLD, in reference to its polar nature and its radial variable defined by the effective plastic strain.
SUMMARY In this paper we review recent work that raises questions about the applicability of associated flow rules and use of strain metrics in the assessment of necking limits. We describe some simple models based on non-associated flow that provide less complex solutions to capture a wide range of initial and evolving material anisotropy. We also review discussions of stability under associated and non-associated flow models. For formability, we discuss the importance of strain path, through-thickness stress-gradients and strain-gradients caused by curvature, cyclic bending/unbending effects, and through-thickness or normal stress effects, including practical solutions to deal with these effects through stress analysis. Finally, we proposed a new ―stress-based‖ forming limit criterion based on a polar diagram of the effective plastic strain with the direction defined by the arctangent of the ratio of the current plastic strain rates, both of which are defined by the current stress condition, or the most recent stress condition while the material was undergoing plastic deformation. This diagram, which we refer to as the PEPS FLD in reference to its polar nature and its radial variable defined by the effective plastic strain, has advantages over the previously proposed Stress FLDs based on the principal stresses or in case of non-zero normal stress, the difference between the in-plane and through-thickness principal stresses. The primary advantages of the PEPS FLD over other diagrams based on variables with stress dimensions is the lack of dependence on the stress-strain relation and any saturation or softening that may be included in the material model. These advantages are shared by the model described by Zeng et al. (2008) in which the limit on the effective plastic strain is characterized by an unfamiliar parameter, the ratio of the plastic strain rates. The primary distinction and potential appeal of the proposed PEPS FLD is that the FLC is similar in shape to the strain FLC, and the directions in the diagram corresponding to uniaxial, plane-strain, equal-biaxial, etc. are parallel to the corresponding directions in the conventional strain FLD.
ACKNOWLEDGMENTS The authors would like to thank Daniel Green from University of Windsor, Xinhai Zhu from Livermore Software Technology Corporation, Ming F. Shi from United States Steel Corporation, Cedric Xia from Ford Motor Company, John Siekirk from Chrysler, Ching-Kuo Hsiung, Siguang Xu and Lumin Geng from General Motors, all for stimulating discussion and contributions to our understanding metal forming limits. The authors would also like to thank John Bassani of University of Pennsylvania, John Hutchinson from Harvard, Rich Becker from US Army Research Laboratory, Robert Wagoner from the Ohio State University, Rebecca Brannon from the University of Utah, and especially Owen Richmond (deceased) formerly from Alcoa Technical Center, for stimulating discussion and contributions to our understanding of the issues regarding non-associated flow. Finally, the authors are particularly indebted to the reviewers of our papers for their thorough and critical reviews to challenge our understanding and improve our presentation of the topics covered in this and prior related papers.
REFERENCES Arrieux, R., Bedrin, C., Boivin, M., 1982. Determination of an intrinsic forming limit stress diagram for isotropic metal sheets. In: Proceedings of the 12th Biennial Congress of the IDDRG, pp. 61–71. Auto-Steel Partnership, Enhanced Formability Project Report, 2003, www.a-sp.org, Publications.
33
Barlat, et al., 1997,―Yield function development for aluminum alloy sheet,‖ J. Mech. Phys. Solids, 45, p. 1727. Barlat, F., Brem, J.C., Yoon, J.W., Chung, K., Dicke, R.E., Legea, D.J., Pourboghrat, F., Choif, S.H., Chuh, E., 2003. Plane stress yield function for aluminum alloy sheets—Part 1: Theory. Int. J. Plasticity 19, 1297. Bassani, J.L., Ito, K., Vitek, V., 2001. Complex macroscopic plastic flow arising from non-planar dislocation core structures. Mater. Sci. Eng. A319-321, 97–101. Bishop, J.F.W., Hill, R., 1951. A theory of plastic distortion of a polycrystalline aggregate under combined stresses. Philosophical Magazine 42, 414. Bramley, A.N., Mellor, P.B., 1966. Plastic Flow in Stabilized Sheet Steels. Int. J. Mech. Sci. 8, 101–114. Bulatov, V.V., Richmond, O., Glazov, M.V., 1999. An atomistic dislocation mechanism of pressure-dependent plastic flow in aluminum. Acta Mater. 47, 3507–3514. Drucker, D.C., 1951. A more fundamental approach to plastic stress–strain relations. In: Proc. of the First US National Congress of Applied Mechanics, ASME, New York, pp. 487–491. Drucker, D.C., 1959. A definition of stable inelastic material. ASME, J. Appl. Mech. 26, 101–106. Graf, A., Hosford, W.F., 1993. Effect of changing strain paths on forming limit diagrams of aluminum 2008-T 4. Metall. Trans. A24, 2503–2512. Hill, R., 1948. A theory of the yielding and plastic flow of anisotropic metals. Proc. Roy. Soc. London A 193, 281–297. Hill, R., 1952. On discontinuous plastic states with special reference to localized necking in thin sheets. J. Mech. Phys. Solids 1, 19–30. Hutchinson, J. and Neale, K., 1977, Sheet necking-II:time-independent behavior, Proceedings of the Symposium of Mechanics of Sheet Metal Forming, Edit by D.P. Koistinen and N.M. Wang, 127-153. Logan, R. and Hosford, W. F., 1980, Int J. of Mech. Sci.,22, p. 419. Marciniak, Z. and Kuczynski, K., 1967, Limit strains in the processes of stretch forming sheet steel, J. Mech. Phys. Solids 1, 609– 620. Pearce, R., 1968. Some aspects of anisotropic plasticity in sheet metals. Int. J. Mech. Sci. 10, 995–1004. Qin, Q., Bassani, J.L., 1992. Non-associated plastic flow in single crystals. J. Mech. Phys. Solids 40, 835. Richmond, O., 1980. Plastic dilatancy in metals. In: Lee, E.H., Mallet, R.L. (Eds.), Plasticity of Metals at Finite Strain: Theory, Experiment and Computation. p. 343. Sandler, I.S., Rubin, D., 1987. The consequences of non-associated plasticity in dynamic problems. In: Desai, C.S. (Ed.), Constitutive Laws for Engineering Materials: Theory and Applications. Elsevier Sci. Pub.Co., Inc., Amsterdam, pp. 345–353. Spitzig, W.A., Richmond, O., 1984. The effect of pressure on the flow stress of metals. Acta Metall. 32,457. Spitzig, W.A., Sober, R.J., Richmond, O., 1975. Pressure dependence of yielding and associated volume expansion in tempered martinsite. Acta Metall. 23, 885–983. Storen, S., Rice, J.R., 1975. Localized necking in thin sheets. J. Mech. Phys. Solids 23, 421–441. Stoughton, T.B., 2000. A general forming limit criterion for sheet metal forming. Int. J. Mech. Sci. 42, 1–42. Stoughton, T.B., 2002a, A non-associated flow rule for sheet metal forming. Int. J. Plasticity 18, 687–714. Stoughton T.B., 2002b,The influence of the material model on the stress-based forming limit criterion. SAE 2002-01-0157. Stoughton, T.B., Yoon, J.W., 2004. A pressure-sensitive yield criterion under a non-associated flow rule for sheet metal forming. Int. J. Plasticity 20, 705–731. Stoughton, T.B., Zhu, X., 2004. Review of theoretical models of the strain-based FLD and their relevance to the stress-based FLD. Int. J. Plasticity 20, 1463–1486. Stoughton, T.B., Yoon, J.W., 2005. Sheet metal formability analysis for anisotropic materials under non-proportional loading. Int. J. Mech. Sci. 47, 1972-2002. Stoughton, T.B., Green, D., and Iadicola, M., 2005, Specification for BM3: Two-stage channel/cup draw, Numisheet 2005 Conference Benchmark Study, AIP Conference Proceedings 778 B, pp. 905-918. Stoughton, T.B., Yoon, J.W., 2006. Review of Drucker’s postulate and the issue of plastic stability in metal forming. Int. J. Plasticity 22, 391–433. Stoughton, T.B., 2008. Generalized Metal Failure Criterion,. Numisheet 2008 Conference Proceedings, Ed. P. Hora, 241-246. Stoughton, T.B., Yoon, J.W., 2008. On the existence of indeterminate solutions to the equations of motion under non-associated flow. Int. J. Plasticity 24, 583–613. Stoughton, T.B., Yoon, J.W., 2009. Anisotropic hardening and non-associated flow in proportional loading of sheet metals. Int. J. Plasticity 25, 1777-1817. Stoughton, T.B., Yoon, J.W., 2011. A new approach for failure criterion for sheet metals. Int. J. Plasticity 27, 440-459. Taherizadeh, A., Green, D.E., Ghaei, A., Yoon, J.W.,2010. A Non-associated Constitutive Model with Mixed Iso-Kinematic Hardening for Finite Element Simulation of Sheet Metal Forming. Int. J. of Plasticity, 25, 288-309. Tharrett, M., Stoughton, T., 2003. Stretch-bend forming limits of 1008 AK steel. SAE 2003-01-1157. Zamiri, A., Pourboghrat, F., 2007, Characterization and development of an evolutionary yield function for the superconducting niobium sheet, Int. J. of Solids and Structures, 44, 8627-8647. Zeng, D. Chappuis, L., Xia, Z. C., and Zhu, X. 2008, A Path Independent Forming Limit Criterion for Sheet Metal Forming Simulations, SAE 2008-01-1445.
34
Zero Failure Production Methods Based on a Process Integrated Virtual Control P. Horaa*, J. Heingärtnera, N. Manopuloa, L. Tonga a
IVP, ETH Zurich,Tannenstrasse 3, 8092 Zurich, Switzerland
Abstract. Although the virtual methods are nowadays fully established as a widely used tool in the planning and optimization of forming processes they are still completely omitted for a direct, “intelligent” process control in the later production of the parts. The paper presents a proposal for a Process-Integrated-Virtual-Control (PIVC) considering the real process perturbations induced by deviations of material properties as well as by further time dependent process parameters like the tool temperature. For the detection of the material deviations an in-line eddy-current measurement method and the appropriate evaluation method for the definition of the stochastic yield curves will be presented. The paper closes with a virtually taught control system modifying the blank-holder forces in dependency of thermal conditions and material deviations. The goal of this PIVC coupled to in-line process controls is to achieve a Zero Failure production even under alternative time dependent process conditions Keywords: Forming technology, zero defect production, modelling of perturbation effects, virtual robustness control, intelligent process. PACS: 46.80; 46.65.+g, 46.55.+d, 46.50.+a
INTRODUCTION To establish a ZD-production1 became in last years a more and more challenging task. The reasons are different: x The complexity of the parts increases continuously. The requested reduction of weight can be only achieved by the use of new light weight and high strength materials. Especially the widely used TRIP and HSS show significant temperature dependency and a high spring-back potential and are not easy to control. x To achieve crash resistance, new fabrication methods like press-hardening have been introduced. The PHprocess requires additional thermal parameters. x The requirements of quality as well as the requirements of close tolerances continuously increase. x New and more complex assembly methods will have to be applied if body in white with a complex material mix has to be realized.
Methods for an In-Process-Integrated Virtual-Control The new virtual based strategy to achieve the Zero Failure productions goals bases on three pillars as described in Table 1. Their interaction is plotted in FIGURE 1 left. For their realization following concepts have been applied:
1
ZD „Zero Defects“ synonym for „Zero Failure”
The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 35-47 (2011); doi: 10.1063/1.3623590 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
35
TABLE 1.
FIGURE 1.
“Pillars” of the Zero Defect Production Control
Technical concepts
Process Integrated Virtual Control (PIVC)
Mapping of the system behavior based on virtual tryouts under stochastic conditions. Description with metamodels
In-line material control
In-line, non-destructive measurement of constitutive parameters (eddy current methods)
“Intelligent” Tool and Press Equipment
In-line control of process parameters with sensors
Modules of the Virtually Controlled Zero-Defect Production Strategy.
x Process Integrated Virtual Control The goal of the PIVC is to establish a virtual process control based on the FEM-sensitivity analysis of the process. To achieve the needed very fast response for the on-line process regulation the system behavior will be described with metamodels. The models can base on the FEA - especially on their sensitivity analysis - as well as be taught directly on the process (“Self Teaching Neuronal Networks”). x “Intelligent” Tool and Press Equipment Until now the deep drawing processes were not directly controlled. If any changes of the process settings are needed, they are done by the operating staff based on the check of some specific parameters like the draw in of the sheet and an optical quality control. For those reasons it is not surprising that for an identical problem different “correction strategies” will be applied. The goal of the “Intelligent Tool” is to detect the critical parameters automatically and so to allow a close loop regulation of the process. Methods for the realization of such principals are discussed by Annen et al., [1]. x On-line Material Control A further significant process parameter may be the non-constancy of the material properties. They can be essential as well for the possible influence on the process robustness as for the on-line actions to correct the process parameters. The applicability of the non-destructive measurement methods of the relevant parameters was recently presented by Heingärtner et al., [2]. The paper will show how the connection among those components can be realized and how they are coupled. The theoretical background is described more profoundly in the different papers of the “Zero Failure Production Methods” conference proceedings, [3].
DETECTION OF SIGNIFICANT PERTURBATION PARAMETERS AND A PROPOSAL FOR THEIR APPROPRIATE NUMERICAL MODELLING As mentioned above the process unreliability may be induced by fluctuation of different process parameters. One of the significant sources of uncertainty in the process is the fluctuation of the material properties, [4, 5]. For those reasons it is indispensable to detect this behavior. In the next section a distortion free monitoring method will be presented.
Eddy-current method for the detection of material changes The applied eddy-current measurement method (EC) bases on the detection of the magnetic response at different frequencies. The relation to the physical value will be described by a correlation function (Metamodel) between the classically evaluated tensile test data and the eddy current signals, FIGURE 2.
36
FIGURE 2.
Eddy current measurement equipment and the principal of the model.
With the eddy-current system realized by Heingärtner [2] and Born [6], the material properties can be measured with a frequency of about 24 to 28 blanks per minute. FIGURE 3 shows the blank cutting unit, where the ECmeasurement was installed. Per coil usually about 1’500 blanks will be evaluated. The diagram in FIGURE 3 demonstrates the detected changes of Rp0,2, Rm and Ag-values as measured by Born, [6].
FIGURE 3.
On-line measurement system installed in the blank cutting unit.
In FIGURE 4 those data are re-plotted as distribution functions.
FIGURE 4.
Material DC06. Distribution of material properties as measured on about 6’000 specimens of different coils.
37
For the behavior of the material and its modelling the relation between the values are significant. FIGURE 5 shows the correlation among the values Rp0,2 and Rm as well the influence of the hardening ratio Rm/Rp0,2 on the development of the uniform elongation Ag.
FIGURE 5.
Material DC06. Correlations between the parameters Rp0,2, Rm and Ag.
For the investigated material DC06 the distribution functions in FIGURE 4 prove that the variation of the mechanical properties is very small inside a single coil. Under the consideration of the correlations of FIGURE 5 the deviations became even smaller. If different coils are considered, FIGURE 6, the changes of the properties are more evident. Nonetheless the variations for the investigated material remain moderate.
a) b) FIGURE 6.
Material DC06. Changes over different coils a) Correlation Rm-Rp0,2. b) Uniform strain distribution
Generation of Yield curves based on the Rp0,2, Rm and Ag Data The stochastic yield curves for the investigated material DC06 will be described based on a “slope fixed”, material specific Hockett-Sherby model k HS f
V sat (V sat V 0, 2 ) exp( M const M n )
(1)
In contrast to the normal H-S-approximation [7] the parameter Mconst is considered as a predefined material constant. As demonstrated in FIGURE 7 Mconst influences, independently of the other parameters, the yield curve slope in the extrapolated area. The idea to keep this parameter constant bases on the assumption, that for a specific material the hardening type will not change significantly.
38
HC220YD 1200 exp M = 0.5 1000
M = 1.0 M = 2.0
kf [MPa]
800
600
400
200
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
[-]
FIGURE 7.
Material HC220YD. Determination of a material specific constant M.
The value of Mconst can be defined by a “full” Hockett-Sherby approximation for all 4 parameters using the combined tensile-bulge yield curve. If the bulge date are not available, the diffuse necking condition
dk f
kf
dM
(2) can be applied as an additional constrain condition. Both methods are described in [9] and [10]. For the material DC06 Mconst= 1.70 was identified. Because the EC system delivers the Rp0,2, Rm and Ag-values and not directly the yield curves, the H-S parameters have to be generated based on those data. This can be done in two different ways. Method 1: A direct way is to specify the stochastic yield curve in dependency of the data Rp0,2, Rm and Ag using eq.1. In this approach the parameters V0,2, Vsat and n of eq. 1 will be specified as follows: x Eq. strain at diffused necking
M DN
ln(1 Ag )
(3)
R p 0, 2
(4)
x Yield stress V0,2
V 0,2 x Stress DN at strain Ag
V DN
R m exp(M DN )
(5)
x Parameter sat and n of the HS-equation will be evaluated using the both conditions k HS f (M DN ) V DN ˆ 0 wk HS f (M DN
)
V DN ˆ 0 wM This procedure was used to generate the stochastic yield curves in FIGURE 11.
(6) (7)
Method 2: An alternative method for the description of the stochastic influence is to overload a mean yield curve with a deviation function 'k stoch . In this case the specific yield curve can be written as f
k HS f
39
~ kf
k HS 'k stoch f f
(8)
where 'k stoch can be described with a linear interpolation of the deviated values V0,2 (=Rp0,2) and DN corresponf ding to Rm · § § M gl M · ¸ 'V gl ¨ M ¸ 'V 0, 2 ¨ ¨ M gl ¸ ¨ M gl ¸ ¹ ¹ © © 'V 0,2 (1 - m) 'V gl m
'k stoch f
(9)
The value m is restricted to the range 0 d m d 1 and will be assumed m=1 in the extrapolated range.
Impact of strain rate and temperature Industrial deep drawing processes achieve strain rates up to 10; after a short production time the tools warm up to 50qC and the temperature of the sheet reaches locally 80qC ore more. For those reasons it should be checked if even for the normal “cold” deep drawing processes the temperature and the strain rate influence are not relevant either [8]. This is definitely the case for special materials like the TRIP steel. Their hardening has then to be described by more complex hardening laws [10], [11], [12].
kf
f (M , M , T ) 'k f (TRIP)
(10)
But even for the investigated DC06 material the temperature influence is of importance, s. FIGURE 12, and will be taken into consideration in the present study.
Impact of yield locus shape changes One of the recent yield locus models is the so called YLD2000 description proposed by Barlat et al. [13]
I where the functions
I 'I ' ' 2V a
(11)
I ' and I ' ' are defined as I'
X 1' X 2'
a
a
I ' ' 2 X 2'' X 1'' 2 X 1'' X 2''
(12)
a
Both linear transformations of YLD2000 are given with
X'
C' s
C' T
L'
(13.1)
X ''
C '' s
C '' T V
L'' V
(13.2)
where L’ and L’’ can be defined as ªLc º 2/3 0 0º « 11 » ª c » « 1 / 3 « L12 0 0»» ªD1 º « » « » « c » « 0 1 / 3 0» «D 2 » « L21 » «Lc » « 0 2 / 3 0» ««D »» ¬ 7¼ « 22 » « 0 1»¼ « L c » «¬ 0 ¬ 66 ¼
ª L cc º « 11 » cc » « L12 « » cc » « L21 « L cc » « 22 » « L cc » ¬ 66 ¼
40
8 2 ª 2 2 « 1 4 4 4 1« « 4 4 4 1 9« 2 2 « 2 8 «¬ 0 0 0 0
0º ª«D 3 º» 0»» «D 4 » « » 0» «D 5 » » 0» «D » « 6» 9»¼ «D » ¬ 8¼
(14)
As specified in Table 4, for the material HC220YD, the yield stresses and so the yield locus increase non- homogenously.
Experimental stress and r-values at different strain levels for the material HC220YD.
TABLE 2.
strain
V0
V45
V90
Vb
r0
r45
r90
rb
0.00 0.05 0.10 0.15 0.20
339 407 440 465
326 388 421 446
343 410 444 468
366 454 502 532
1.28 1.28 1.28 1.28
2.08 2.08 2.08 2.08
1.86 1.86 1.86 1.86
1.0 1.0 1.0 1.0
For the description of this type of anisotropic hardening Hora et al. [9] proposed to describe the Dk-parameters as linear functions of the equivalent strain:
D k (H ) D k0 D k' H
(14)
Based on the data of Table 2 the parameters of eq. 14 get the values specified below. The corresponding linear functions are given in FIGURE 8; the yield loci are plotted in FIGURE 9.
FIGURE 8.
FIGURE 9.
k
D k0
D k'
1
0.9631
-0.3039
2
1.0801
0.2157
3
1.0323
-0.9148
4
0.9372
-0.1600
5
0.9608
-0.1342
6
0.8470
-0.8739
7
1.0877
0.0033
8
0.9886
0.7053
Material HC220YD. Dk-parameters of the YLD2000 function, if evaluated under the constraint of linear strain dependency
Material HC220YD. Yield locus evolution under the consideration of the linear changes of the Dk-parameters as in eq. 14.
41
How significant the continuous yield locus change with an increasing Vb/VY-ratio is, demonstrates the comparison of the thickness distributions in FIGURE 10 calculated in Case A) without and in Case B) with consideration of the anisotropic yield locus evolution. Case A: isotropic hardening
Case B: Anisotropic hardening
tmin = 0.47
tmin = 0.59
FIGURE 10.
Influence of the YLD2000 description on the strain distribution without (left) and with the consideration of the anisotropic yield locus change as given in FIGURE 9.
MODELLING OF THE REAL PROCESS BEHAVIOR UNDER CONSIDERATION OF PERTURBATIONS Besides the correct “deterministic” modeling, as mentioned earlier in this paper, the real production process is perturbed by the fluctuation of different process parameters. The influence of those changes on the process behavior will be demonstrated again on the cross deep drawing part. For the investigation following parameters have been considered: x x x x
blank specific (stochastic) changes of hardening behaviour (yield curve) thermal influences on hardening thermal influences on friction dependency of the blank holder force FN on above parameters
FIGURE 11 shows the yield curve evaluated by the method 1 given by eq. 3 to 7 for a single coil. The light lines in the plot correspond to the slope wk f / wI of the hardening curve and indicate at the cross point the position of the uniform strain Ig. FIGURE 12 specifies the impact of temperature increases on the yield curve. FIGURE 13 describes the temperature dependency of the friction coefficient for one specific lubricant.
42
FIGURE 11.
Case
V0,2
Vsat
Mconst
n
RT max RT min
142 134
579 540
1.7 1.7
0.542 0.537
DC06 - Yield curve distribution evaluated by the EC-method for one coil.
Case
V0,2
Vsat
Mconst
n
RT mean heated 60qC heated 100qC
117 110 105.05
566 491 449.55
1.7 1.7 1.7
0.562 0.576 0.5747
FIGURE 12.
FIGURE 13.
DC06 Thermal influences on the yield curve.
Temperature dependency of the friction coefficient P as given by Krauer [12].
43
Sensitivity ranking of the process parameters The sensitivity analysis in FIGURE 14 demonstrates that moderate material fluctuations, as seen in FIGURE 11 for the DC06 grades, will have no significant impact on the process behavior. This may be the most surprising result of the process analysis based on the real stochastic data. For those reasons – for the prediction of the system behavior in FIGURE 16 - the yield locus will be considered constant. The major influence comes clearly from the friction, which is in turn temperature dependent.
Parameter AF-Sigma
Min.
Yield Curve DC06 Rp0,2 deviations Yield Locus R0 R90 Biaxial factor Friction P Pressure
kf_mean at RT -10 MPa Banabic 1.45 1.5 0.9 0.05 3 MPa
FIGURE 14.
Max 10 MPa 1.70 2.8 1.2 0.25 15 MPa
Sensitivity analysis of the process investigated for the cross deep drawing part.
FIGURE 15 demonstrates this behaviour for the extremes of the stochastic yield curves as well as for two thermal cases. The strong temperature sensitivity shows, that the self-heating effect, induced by plastic work, will have a significant influence on the process behaviour. Already a warming up to 60qC, which is not unrealistic under continuous production conditions, will provoke necking2.
Case RT max
Case RT min
2
For the temperature dependent description of the yield curves the data of FIGURE 12 have been used.
44
Case Temp. 60q
Case Temp. 100q
FIGURE 15.
DC06 material. Deep drawing behaviour under consideration of stochastic and thermal influences of the yield curve as given in figures 16 and 17.
ZERO FAILURE CONCEPTS BASED ON PROCESS INTEGRATED VIRTUAL CONTROL The goal of the PIVC is the detection of admissible parameter fields and their integration in an in-line process control.
Temperature dependent process control In the simulation below the influence of the parameters P(T), kf(T) and FN(T) have been investigated. FIGURE 16 visualizes the impact of increasing temperature on the admissible blank holder force. The changes are driven by the thermal dependency of the friction, FIGURE 13, and by the thermal change of the hardening as specified in FIGURE 12. The comparison of the both diagrams in FIGURE 16 proves that the temperature dependency of the yield curve significantly strengthen the T-dependency of the process as a whole, s. Fig. 16 b.
a) FIGURE 16.
b) Influence of the temperature changes on the admissable FN-settings. a) Consideration of the P(T)-influence. b) Consideration of both P(T) and kf(T) influences.
45
Material dependent process control The GUI showed in FIGURE 17 is directly coupled to the eddy-current measurement system. Based on a “virtual” knowledge of the deep drawing behavior, predicted in the same way as for the thermal dependency and mapped with a metamodel the system can immediately detect if the specific blank can be formed without necking.
FIGURE 17.
GUI for the visualization of the characteristic parameters coupled with the analysis for the applicability for a specific part.
If the material reaches critical values, a ZD-production can be achieved only after an adaption of the process parameters. One possibility for their modification is demonstrated in FIGURE 16 by the adaption of the blank holder forces. Another approach is to influence the friction by increasing the amount of lubricant.
FIGURE 18.
Layout of an integrated “Zero-Defect-Process Control”.
46
Integration of the control system in a production line The introduced modules are the pillars of an integrated process control unit as shown in FIGURE 18. The “intelligent” process monitoring makes it then possible to avoid part defects induced by material fluctuations or long-term heating effects. In the demonstrated example, the process modification was done by adaption of the blank holder forces. For complex parts the integral BH-force have to be split into single column forces. The basic idea of the approach stays the same. The last “pillar” of the concept, the sensors of the “intelligent” tool, has not been discussed. The detection of temperature changes can be done with commercial sensors. In many cases it is helpful to control the process over the draw-in behavior of the sheet, too. This can be done directly in the tool or in-directly following the forming step. For the direct measurement different mechanical and laser sensors are under development. The indirect measurements base usually on optical evaluation methods.
ACKNOWLEDGMENTS The presented paper bases on results which have been realized in close cooperation with Daimler AG and the steel supplier VoestAlpine. We would like to thank all of those partners for their support, especially Prof. K. Roll and Dr. D. Hortig, who have promoted this project and Ms. A. Neumann for her support with the experiments.
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10. 11.
12. 13.
Ch. Annen , F. Schwarz, M. Wahl, P. Hora, “Virtual layout of adaptive deep drawing processes”, in Zero Failure Production Methods, Conference Proceedings of the Forming Technology Forum FTF-2011, edited by P. Hora, 2011, ETH Zürich. Heingärtner J., Hora P, “Application of the eddy-current methods in industrial car body production”, in Zero Failure Production Methods, Conference Proceedings of the Forming Technology Forum FTF-2011, edited by P. Hora, 2011, ETH Zürich. P. Hora, “Zero Failure Production Methods”, Conference Proceedings of the Forming Technology Forum FTF2011, edited by P. Hora, 2011, ETH Zürich. K. Grossenbacher, “Virtuelle Planung der Prozessrobustheit in der Blechumformung“, Ph.D. Thesis, ETH Zurich, Nr. 17470, VDI-Verlag, VDI Rh. 2, Nr. 667, 2008. M.H.A. Bonte, “Optimization Strategies for Metal Forming Processes”, Print Partners Ipskamp, ISBN 978-90365-2523-7, Enshede, 2007. M. Born, "Integration des Wirbelstromsystems zur Erfassung der Werkstoffkennwerte in der Produktion“, M.Sc. Thesis, ETH Zurich, 2009, ETH Zürich. J.E. Hockett and O.D. Sherby, Large strain deformation of polycrystalline metals at low homologous temperatures. J. Mech. Phys. Solid, 23-2 (1975) pp. 87-98. P. Larour, "Strain rate sensitivity of automotive sheet steels: influence of plastic strain, strain rate, temperature, microstructure, bake hardening and pre-strain", Ph.D. Thesis, RWTH Aachen, Publishe in Berichte aus dem Institut für Eisenhüttenwesen, Shaker Verlag, Band 1/2010, 2010. P. Hora, B. Hochholdinger, A. Mutrux, L. Tong: “Modelling of anisotropic hardening behaviour based on Barlat 2000 yield locus description”, in Constitutive Modelling of Kinematic and Anisotropic Hardening Effects for Ductile Materials, Conference Proceedings of the Forming Technology Forum FTF-2009, edited by P. Hora, 2009, ETH Zürich, pp. 21-29. P. Hora: "Modellierung des Kaltverfestigungsverhaltens bei metallischen Werkstoffen“, Conference Proceedings MEFORM 2011, edited by R. Kawalla, Freiberg, 2011. A. Hänsel: "Nicht-isothermes Werkstoffmodell für die FE-Simulation von Blechumformprozessen mit metastabilen austenitischen CrNi-Stählen", Ph.D. Thesis, ETH Zurich, Nr. 12672, published in VDI Reihe 2 Nr. 491. J. Krauer, "Erweitere Werkstoffmodelle zur Beschreibung des thermischen Umformverhaltens metastabiler Stähle", Ph.D. Thesis, ETH Zurich, Nr. 19070; published in VDI Reihe Nr. 676. F. Barlat F., J.C. Brem, J.W. Woo, K. Chung, R.E. Dick, D.J. Lege, F. Purboghrat, S.-H. Choi, E. Chu: "Plane stress yield function for aluminium alloy sheets – part I – theory“, Int. J. of Plasticity, 19(2003), pp. 1297-1319.
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New Forming Technologies for Autobody at POSCO Hong-Woo Leea, Myung-Hwan Chaa, Byung-Keun Choia, Gyo-Sung Kima, Sung-Ho Parkb a
Products Application Research Center, POSCO, 180-1 Songdo-dong, Yeonsu-gu, Incheon 406-840, Korea Steel Technology Strategy Department, POSCO, 892, Daechi4-dong, Gangnam-gu, Seoul, 135-777, Korea
b
Abstract. As development of car body with light weight and enhanced safety became one of the hottest issues in the auto industry, the advanced high strength steels have been broadly applied to various automotive parts over the last few years. Corresponding to this trend, POSCO has developed various types of cold and hot rolled AHSSs such as DP, TRIP, CP and FB, and continues to develop new types of steel in order to satisfy the requirement of OEMs more extensively. To provide optimal technical supports to customers, POSCO has developed more advanced EVI concept, which includes the concept of CE/SE, VA/VE, VI and PP as well as the conventional EVI strategy. To realize this concept, POSCO not only supports customers with material data, process guideline, and evaluation of formability, weld-ability, paint-ability and performance but also provides parts or sub-assemblies which demand highly advanced technologies. Especially, to accelerate adoption of AHSS in autobody, POSCO has tried to come up with optimal solutions to AHSS forming. Application of conventional forming technologies has been restricted more and more by relatively low formability of AHSS with high tensile-strength. To overcome the limitation in the forming, POSCO has recently developed new forming technologies such as hydro-forming, hot press forming, roll forming and form forming. In this paper, tailored strength HPF, hydroformed torsion beam axle and multi-directional roll forming are introduced as examples of new forming technologies. Keywords: Hydro-forming, Hot Press Forming, Roll forming, Form forming, EVI
INTRODUCTION Steel, typically used in vehicle structures, enabled the economic mass production of millions of units over the past several decades [1]. It has been argued during last decade that steel would be replaced as body structure materials by more expensive light materials such as Al, Ti and CFRP (Carbon Fiber Reinforced Plastics) for weight reduction. However, the majority of vehicle bodies are still manufactured from sheet steels. This means that steel products are able to provide not only the excellent physical properties (in strength, formability, weld-ability and corrosion resistance) but also the cost-effectiveness. However, application of conventional steels to vehicle bodies has been restricted more and more by the regulations of environment and safety. As is well known, weight reduction and safety enhancement are the most important goals of automotive industry. However, generally, the increase of strength for certain metallurgy is accompanied with the loss of ductility. To overcome above mentioned problem, steel makers have been developing new grades of high performance steel called as advanced high strength steel(AHSS). DP, TRIP, FB, CP steels are the recent examples of AHSS. These steels have expanded in application for weight reduction and improvement of safety. The principal difference between conventional HSS and AHSS is their microstructure[2]. By appropriate balance of microstructure, AHSS can have their own unique mechanical properties. For example, some types of AHSS have a higher formability than conventional HSS within same strength range and other types show very good stretch flange-ability with high strength. As mentioned above, good performance of AHSS enables to increase in the use of ultra high strength steels for autobody with allowing weight reduction. In order to expand utilization of AHSS more and more, various steel application technologies, i.e. integrated steel solutions are highly needed. Those include not only evaluation of mechanical properties but also alternative forming technologies such as hydroforming, hot press forming, roll forming and form forming. However, application solutions of AHSS do not mean a radical change compared to solutions of conventional HSS. But, in order to secure
The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 48-53 (2011); doi: 10.1063/1.3623591 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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target performance, several process requirements should be considered in addition to conventional forming technologies. The present paper describes the current status of the forming technologies at POSCO.
CUSTOMER SUPPORT TECHNOLOGIES Auto makers continuously pursue not only high performance and quality but also low costs of production. As these technological strategies, auto makers devote their best efforts to apply the optimal material such as AHSS and other light weight material in each part of a new car model. Also, various new part manufacturing processes have been adopted for weight reduction, cost saving and safety enhancement of car body. Tailor-welded blank (TWB) is now widely accepted, and hydro-forming and hot press forming are extended to various applications for auto part manufacture. Steel makers are able to be involved in the early stage of car designing by contributing materials data as well as developed materials for new car project or phase shift. Moreover, to intensify this supporting activity, steel makers often propose alternative designs for individual parts and develop the new concept for the whole body structure. The typical example of this supporting activity, ULSAB (Ultra Light Steel Auto Body) as shown in Fig.1 and ULSABAVC (advance vehicle concepts), which are designed and tested by WAS (WorldAutoSteel) in collaboration with a consortium of worldwide steel companies[3]. ULSAB/-AVC proved to be lightweight, structurally sound, safe, executable and affordable. One of the major contributors to the success of the ULSAB/-AVC was a group of new steel types and grades called AHSS and new forming technologies such as TWB and hydro-forming. FSV (Future Steel Vehicle) program [4] is recently progressing. This program focuses on the achievement of future safety requirements coupled with the demonstration of low CO2 emissions, and affordability of a steel intensive vehicle body using optimized design, AHSS usage and advanced power train technologies. The phase 2 results for electric vehicle design is shown in Fig.1 Even individual companies have attempted to develop their own concept models utilizing their high performance steels together with new ideas of design. These include NSB (New Steel Body Concept) by TKS, ABC (Arcelor Body Concept) by Arcelor and ATLAS (Advanced Technology for Lightweight Auto- bodies in Steel) by Salzgitter. POSCO also developing electric vehicle body by applying newly developed steel and forming technologies. The above supporting process by steel suppliers is called EVI which covers the activities involved in the vehicle concept design and tooling/ prototyping stages. POSCO has developed a more advanced EVI concept to serve auto makers better. In POSCO EVI program, the supportive activities cover beyond the tooling/prototyping stage, being responsible for part manufacturing and sub-assembling too. Advanced EVI program consists of the following four strategies: CE (Concurrent Engineering) / SE (Simultaneous Engineering), VA (Value Analysis) / VE (Value Engineering), VI (Value Innovation) and PP (Part Proposal), as shown in Fig.2.
(a) ULSAB body[3]
(b) FSV Phase 2 design for electric vehicle[4]
FIGURE 1. Concept car models utilizing high performance steels together with new design ideas.
FIGURE 2. Advanced EVI concept of POSCO
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CE/SE means that a workflow of POSCO carries out a number of tasks in parallel with customers’ process. In order to establish win-win strategy with auto makers, POSCO provides the optimal steel solutions concurrently and simultaneously in each stage from styling to mass production. On the base of VA/VE strategy, POSCO proposes the best fit materials from cost viewpoint including degradation, thinning, logistics, packing and so on. From this, customers can obtain optimal material information with weight reduction and cost saving or performance improvement. VI is the POSCO EVI activity to reduce or remove unnecessary values to save the prime cost in order to strengthen customers’ competitiveness. VI activity doesn’t need high technology because it aims to maximize the efficiencies of the properties on what customers already have or use. For example, through cost saving program such as substituting GI for EG product, POSCO gives customers maximum profit without modifying the existing process. In PP activity, POSCO supports customers with new forming technologies including new design concept, prototyping and manufacturing. In order to implement CE/SE, VA/VE, VI and PP programs, various steel application technologies are highly needed. POSCO has developed integrated steel solutions which are fully utilized through whole car manufacturing process. To realize this concept, POSCO not only supports customers with material data, process guideline, and evaluation of formability, weld-ability, paint-ability and performance but also provides parts or sub-assemblies which demand highly advanced technologies. Especially, to accelerate adoption of AHSS in autobody, POSCO have tried to come up with optimal solutions to AHSS forming.
NEW FORMING TECHNOLOGIES As development of car body with light weight and enhanced safety became one of the hottest issues in the auto industry, the advanced high strength steels have been broadly applied to various automotive parts over the last few years. Corresponding to this trend, POSCO has developed various types of cold and hot rolled AHSSs such as DP, TRIP, CP and FB, and continues to develop new types of steel in order to satisfy the requirement of OEMs more extensively. Meanwhile, application of conventional forming technologies has been restricted more and more by relatively low formability of AHSS with high tensile-strength. To overcome the limitation in the forming, POSCO has recently developed new forming technologies such as hydro-forming, hot press forming, roll forming and form forming. From this point of view, POSCO’s technical activities will be shown in the following sections.
Tailored Strength Hot Press Forming As a way to acquire high strength and good formability in manufacturing of automotive parts, an advanced forming technique, often called hot press forming (HPF) was developed. During this process, sheet metal is heated to austenizing temperature, formed rapidly and then quenched in a tool. Soft austenite transforms into very hard martensite due to high hardenability of HPF material, so HPF can be used to fabricate complex and crash-resistant parts with ultra high strength. Recently, two innovative approaches, named tailored strength HPF in this article which has two different graded strength levels with 1500MPa and 600MPa in the formed parts, have been implemented. The main purpose of these approaches is to enhance the ductility in some localized areas on the hot press formed part in order to obtain both anti-intrusion by harder strength area and energy absorption with the deflection of ductile strength zone at the same time during crash accidents.
(a) (b) (c) FIGURE 3. Conceptual definition of tailored HPF for (a) MS-HPF part, (b) process conditions and (c) TWB-HPF part.
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The first approach, named Multi-Strength HPF (MS-HPF) defined in Fig. 3(a), is carried out by using different heating temperature path of a monolithic blank in the special furnace before forming and quenching. The harder strength can be given by conventional condition of HPF temperature and resulted from martensitic phase transformation. However, lower heating temperature history leads to diffusional phase transformation, so we can get a local area of ductile strength. The second approach, named TWB-HPF described in Fig. 3(b), can be obtained by originally different hardenability of two materials at the same temperature condition. Tailor Welded Blank (TWB) sheet used for this process is prepared by laser welding of HSLA material and HPF material. Ductile strength of TWB-HPF is provided due to low hardenability of HSLA material. The characteristic of mechanical properties and different microstructure shown in Fig. 4 for MS- and TWB-HPF result in tailored strength of hot press formed parts. Our newly designed special oven for MS-HPF was already installed and suitable partner material for TWB sheet was also designed to get lower grade strength even after rapid quenching. We have studied optimization of process parameters and evaluated the performance of tailored strength HPF parts to apply automotive components to our customers’ vehicles to increase crash performance and improve safety.
(a) (b) FIGURE 4. Hardness distribution and miscrostructure after heat treatment for (a) MS-HPF and (b) TWB-HPF.
Tube Hydroforming The main goals of Tube hydroforming are to reduce the part weight and improve structural strength of the part at the same time. Therefore the choice of the material which is used for the hydroforming application is essential and depends on several parameters. Formability of the material, material cost, requirement of part performance and weld quality are only a few of them. High strength hydroformed structural parts including engine cradles, suspension parts, etc. can help achieve the high goals of the automotive industry. Parts can be designed with thinner wall thicknesses therefore, the part will be lighter and at the same time the physical properties of AHSS grades like DP590 or DP780 will increase the stiffness of the part. The Torsion Beam Axle (TBA) developed by POSCO combines the advantages of hydroforming such as high geometrical repeatability, increased wall thickness at the feeding area due to axial feeding, and the material properties of DP780. Fig.5 shows the TBA and the connection nodes with other suspension components. To achieve the requested part performance, the choice of the material thickness for this TBA design was very important because roll stiffness and maximum stress of the part depend on the wall thickness of the part. Simulations were carried out to determine the correlation between the wall thickness, roll stiffness and maximum stress of the part. From the simulation results, it was observed that the higher the TBA wall thickness, the higher the roll stiffness will be and the lower the maximum stress will be. This leads to a conflict between roll stiffness, maximum stress and wall thickness. Choosing a wall thickness to satisfy the allowable stress means that the roll stiffness would be too high and likewise, choosing a wall thickness to achieve the required roll stiffness would mean that maximum stress would be too high. Fig.6 shows the stress contour at rolling condition of the TBA and it can be seen that the areas where the highest stress occurred are located in the axial feeding zone during hydroforming. To satisfy the requested roll stiffness and the allowable stress of the TBA, 3 major criterions had to be met: • The original tube thickness could not change in the middle section of the TBA. • Due to axial feeding during hydroforming, the wall thickness needed to be increased in the area where the highest stress occurred.
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• The original design needed to be modified and a bead designed and implemented into the affected area. Based on the above design concept, the final part and manufacturing process were designed by the FEA simulation. The final design and hydroforming tool of TBA are shown in Fig.7 and Fig.8 respectively.
FIGURE 5. Torsion beam axle(TBA)
FIGURE 6. Stress contour of TBA
Bead
Green color indicates increase of wall thickness due to axial feeding FIGURE 7. Hydroformed TBA design to satisfy the required performance.
FIGURE 8. Hydroforming Tool of TBA
Roll Forming Roll forming is a process producing long-shaped profile from sheet metal through roll sets located serially. It has an advantage of high productivity such as high speed production and little scrap amount. Especially, the technology is being magnified recently in the automotive industry because of relative suitability for forming of UHSS which has a tensile strength over than 1 GPa as its application is getting increased. Roll design has been dependent on designer’s experience more than simulation because of long calculation time required in addition to low accuracy of simulation. So POSCO has placed more emphasis on shortening of analysis time to practical level which leads to a prompt technical assistance to customers. For the achievement, POSCO uses new developed analysis software designed for roll forming only as well as various common analysis tools. One of major restrictions in roll forming process is that it can produce only uniform sectional product. To overcome this limitation, POSCO has been developing MD-RF(Multi-Directional Roll Forming) technology. MDRF is a process which can produce variable sectional profile by linear and rotational movement of roll stands. It is expected by developing this process to provide more flexibility in designing automotive structure and extend application of roll forming process. For the research of MD-RF technology, which is expected to be one of POSCO future EVI ways, POSCO has set up an in-house test facility and developed analysis software designed for MD-RF only.
(a) Roll Forming Simulation (b) MD-RF Simulation FIGURE 9 POSCO EVI activities for roll forming and MD-RF.
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CONCLUDING REMARK Steel makers have made a lot of efforts to develop advanced high strength steels for the weight reduction. But, AHSS is not easy to be applied by conventional forming technology due to relatively poor formability. To overcome the limitation in the forming, POSCO has recently developed new forming technologies such as hot press forming, hydro-forming, roll forming and form forming. Hot press forming is a forming technique to acquire high strength and good formability by heating and quenching in manufacturing of automotive parts. Recently, POSCO has developed MS- and TWB-HPF to enhance the ductility in some localized areas on the hot press formed part in order to obtain both anti-intrusion by harder strength area and energy absorption with the deflection of ductile strength zone. Hydroforming is able to reduce the part weight and improve structural strength of the part at the same time. In the hydroforming, AHSS such as DP590 or DP780 is generally applied on hydroformed parts as other forming techniques. In this paper, a torsion beam axle is introduced as an example for AHSS and a self-designed part. The torsion beam axle (TBA) developed by POSCO combines the advantages of hydroforming such as high geometrical repeatability, increased wall thickness at the feeding area due to axial feeding, and the material properties of DP780. AHSS, especially steels with tensile strength more than 1 GPa are difficult to be formed by stamping. For such steels with poor formability, it is useful to make a part by bending. Roll forming and form forming which have studied in POSCO, are the typical forming techniques by bending. MD-RF is a special process which can produce variable sectional profile by linear and rotational movement of roll stands.
REFERENCES 1. P. Prasad and J. E. Belwafa, Vehicle Crashworthiness and Occupant Protection, AISI, 2004. 2. WorldAutoSteel, Advanced High Strength Steel (AHSS) Application Guidelines, Online at www.worldautosteel.org, 2009. 3. S.H. Park, “Application of ULSAB Technology to manufacture High Safety Low Weight Vehicle”, KATECH Seminar, Korea, 1999, p.83. 4. Future Steel Vehicle Phase 2, Report by World Auto Steel, 2011.
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Drawability Prediction Method using Continuous Texture Evolution Model Toshiharu Morimotoa and Jun Yanagimotob a
Graduate student,the university of Tokyo,4-6-1 Komaba Meguro-ku,Tokyo,153-8505,Japan b Professor,the university of Tokyo,4-6-1 Komaba Meguro-ku,Tokyo,153-8505,Japan
Abstract. Drawability is one of steel strip properties which control press forming. Many predicted method for the Lankford value have been proposed. First, we predict recystallization texture based on the idea that total amount of microscopic crystal slips is proportional to accumulated dislocation density in grain boundaries. Next, we can predict Lankford value of ultra low carbon strips and ferritic stainless steel strips using Sachs model. Our method is very practical to use in hot and cold steel rolling industry. Keywords: drawability,Lankford value,recrystallization texture,Sachs model PACS: 81
INTRODUCTION Drawability is one of steel strip properties which control press forming. Drawablity of a steel strip is affected by density of {111} preferred orientations. Thus, to increase {111} preferred orientations, an ultra low-carbon steel strip was developed, which are generally used for automobile bodies. An ultra low-carbon steel strip has as low carbon and nitrogen contents as possible. Cold rolled by more than 80% reduction ratio and annealed above recrystallization temperature, an ultra low-carbon steel has superior deep drawability. While, being much alloyed with about 16 wt% chrome, a ferritic stainless-steel strip has less {111} preferred orientations and lower drawability than an ultra-low carbon strip does. It is capable of estimating Lankford value from measured orientations of final cold rolled and annealed strips[1]. But, few research have been reported to estimate drawability of steel strips considered hot rolling, cold rolling and annealing pattern because it’s difficult to predict recrystallization texture. We made a mathematical modeling to predict rolling texture and transformation texture[2][3]. Furthermore, we developed a method to predict recystallization texture combined with our rolling texture prediction model[4].This method was based on the idea that total amount of crystal slips is proportional to accumulated strains in grain boundaries. By this method, we can predict recrystallization texture of an ultra low-carbon strip and a ferritic stainless-steel strip precisely considering hot and cold rolling condition. This method is very practical for use in the hot and cold steel rolling industry. Finally, we calculated Lankford value using our deformation texture prediction model easily[3].
RECRYSTALLIZATION FORMATION MODEL Many mechanism for the formation of the recrystallization texture in steels have been proposed, including the oriented nucleation model[5], the oriented grain growth model[6] .In other models, such as the dislocation passed model[7], the dislocation build-up model[8], the crystal rotation model[9], the dislocation twist area model[10] and the encroachment model[11], recrystallized grains encroach on deformed non-recrystallized grains. However, no definite conclusions have been reached regarding the validity of each model. In the dislocation twist area model[12], it was predicted that the recrystallization texture of ultra low-carbon steels was inherited from the cold rolling texture. Dillamore proposed a prediction model using the Taylor factor of the pre cold rolling texture[13]. However, the dislocation twist area model was a different type of model because crystal slips and rotations that occurred during cold rolling were taken into account. In this study, we developed a new prediction method expanded The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 57-62 (2011); doi: 10.1063/1.3623592 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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from the dislocation twist theory. First, we predicted the rolling texture using the crystal plasticity by applying Lee’s elastic-plastic decomposition rule[14]. Second, we predicted the recrystallization texture from the predicted cold rolling texture. There are two classical models of deformation texture prediction. One is the Taylor model[15], in which macro strains are equal to micro strains, and the other is the Sachs model[16], in which macro stresses are equal to micro stresses. The prediction accuracy of the steel deformation texture has been reported to be good for the Taylor model[17]. Since nucleation during the recrystallization of polycrystals is generally not only in the transition bands but also in the grain boundaries[18], within which the range of orientations mostly exists. Our method of using the total numbers of crystal slip to predict the recrystallization and grain growth texture has not been previously reported. Our method has many advantageous features. For example, even though it is based on simple assumptions, it enables us to predict the recrystallization texture accurately. Moreover, it can predict the recrystallization texture of polycrystals with few parameters and in a short time. Furthermore, we explain the mechanism of recrystallization texture formation used by our prediction method. By the way, it was too trouble to calculate Lankford value from measured orientations, using the way to determine the minimum Taylor factor[1]. Thus, we used the Sachs deformation texture prediction model[16] to calculate Lankford value more easily.
PREDICTION ACCURACY OF RECRYSTALLIZATION TEXTURE Rolling Texture Analysis Model Here, J , x , a and b are the slip rate, the position, the unit vector normal to the slipping plane and the unit vector in the slipping direction, respectively. Thus, the plastic strain rate D P
P
and plastic spin W
P
can be defined using
Equations (1) and (2), respectively. The plastic strain rate D is calculated using the Orowan calculation model[3]. A b.c.c. crystal lattice has 12 slip systems of the {110} type, 12 slip systems of the{112} type and 24 slip systems of the{123} type. Strain equilibrium, the shear stress of each slip system and the rate-dependent rule are expressed using Equations.(5)-(7), respectively. Equation (5) implies that the sum of the slip values J for each slip system in a crystal is equal to the sum of the micro strains dž . Equation (6) gives the relationship between micro stress ǔ and the resolved shear stress W acting on the slip plane. Equation (7) is the Asaro rate-dependent rule and m is a material parameter[19]. Here, J0 , W y and dt are the reference slip strain rate, the critical shear stress and the time increment, respectively. The critical shear stress W y is determined from the equivalent flow stress V , which is calculated using the macro-micro combined model for microstructure evolution [2] using Equations(8). Rotation tensor R is calculated using Equation (9). I is unit tensor.
G Dij
p
W ij
P
where Pij
Qij dH i G W j G G dJ j
Wy
1 J a i b j a j bi Pij J .G (1) 2 1 J a i b j a j bi Qij J .G (2) 2 1 ai b j a j bi .G (3) 2 1 a i b j a j bi . G (4) 2 ¦ Pij dJ j .G G (5)
¦P V ij
i
. (6)
J0 * dt * W j / W y
1/ m
* sign(W j ) G G (7)
V / TaylorFactor (8)
58
Rij where
: ij
I ij dt: ij 1 / 2dt 2 : ij : ij )
Wij Wij and ƺ is lattice rotation spin and W is total spin. P
Prediction Accuracy of Rolling Texture Obtained the Taylor model with the Asaro rate-dependent rule (hereafter referred to as the Taylor model), Figures 1 and 2 respectively show the predicted and observed cold rolling textures of ultra low-carbon steel and hot rolling texture of ferritic stainless-steel. Here, J0 is 100 and m is 0.05. The prediction accuracy for ultra low-
Observed Analyzed
Cold strip
carbon cold rolled texture and ferritic stainless-steel hot rolled texture is high. We consider that because ferritic stainless-steels hardly recrystallize during hot rolling in the finisher, we can accurately simulate hot rolling texture of ferritic stainless-steel strip using the Taylor model.
Observed Analyzed
Hot strip
FIGURE 1. Texture prediction accuracy of ultra low carbon cold strip.
FIGURE 2. Texture prediction accuracy of ferritic stainless-steel hot strip.
Drawability Prediction of Ultra Low-Carbon Steel and Ferritic Stainless-Steel Here we suggested together the conventional assumptions in the above model. The recrystallized grains have a range of orientations in the deformed state that satisfy the preferential orientation model. Nucleation occurs from piled-up dislocations at grain boundaries. The total number of microscopic slips is estimated as the total number of displaced dislocations. The magnitude of slips of crystals is calculated from the Schmid factor. In other words, crystals with a low Schmid factors slip a longer distance than crystals with a high Schmid factors. Needless to say, the Taylor factor is approximately inversed to the Schmid factor. We introduce one further assumption here. The total number of microscopic slips is estimated as the total number of dislocations piled up at grain boundary[20]. This might be a self-evident assumption. Figure 3 shows the case when we selected crystals with the highest slip value as those having the preferential orientation. We can predict the recrystallization texture of cold rolled and annealed ultra low-carbon steel strip precisely because this analysis simulated that the gamma fiber structure encroached the alpha fiber structure. Figure 4 shows the case when we selected crystals with the lowest slip value as those having preferential orientation. We can predict the recrystallization texture of hot rolled and annealed ferritic stainless-steel strips precisely because this analysis simulated the existing bow out into the grains which have the
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Cold strip after annealing Analyzed Observed
higher stored energy otherwise nuclei develop from subgrains close to that grain boundaries. Figure 5 shows the accuracy of predicting the texture of cold rolled and annealed ferritic stainless-steel strip whose grains were classified as the highest number of slips from the predicted texture of the hot rolled and annealed ferritic stainlesssteel strip in Fig. 4 by continuous analysis. We believe that it is possible to analyze the continuous change in the texture from hot rolled strips to cold rolled and annealed strips by considering both hot rolling and cold rolling conditions.
Observed Analyzed
Hot strip after annealing
FIGURE 3. Texture prediction accuracy of ultra low carbon annealing strip.
Observed Observed Analyzed
Cold strip after annealing
Analyzed
Cold strip
FIGURE 4. Texture prediction accuracy of ferritic stainless-steel annealing strip.
FIGURE 5. Texture prediction accuracy of cold rolled and annealed ferritic stainless-steel.
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DISCUSSION According to the recrystallization textures of ultra low-carbon steel and ferritic stainless-steel whose orientations were classified by total number of slips using the Taylor model, the orientation of the recrystallization grains is the same as that of the rolled grains. In other words, this result proves that the nucleation preferred grain model is correct. However, why is the rolled texture classified by the total number of slips using the Taylor model the same as the observed recrystallization texture of ultra low-carbon steel and ferritic stainless-steel? The orientation of ultra low-carbon steel cold rolled at reduction ratio of 80% or of ferritic stainless-steel hot rolled in the nonrecrystallized region is considered to be almost stable. Two recrystallization mechanisms of ultra low-carbon steel have been proposed[11]. In mechanism A, the gamma fiber grains with dislocations undergo self-recrystallization. In mechanism B, the gamma fiber grains with fewer dislocations make encroach on alpha fiber grains, which have more dislocations. The final texture of cold rolled and annealed texture of ultra low-carbon steel is determined by mechanism B. As gamma fiber grains have a less stable orientation than alpha fiber grains, subgrains are formed at grain boundaries and gamma fiber grains initially undergo self-recrystallization. Next, alpha fiber grains that are elongated in the rolling direction and located next to gamma fiber grains are encroached upon by the gamma fiber grains. For the above-mentioned reasons, a texture whose orientations classified by the total number of slips with the highest order using the Taylor model become a final recrystallization texture of cold rolled and annealed ultra low-carbon steel strip. In contrast, as hot rolled and annealed ferritic stainless-steel strip recrystallizes by mechanism A, the orientation classified by the total number of slips with the lowest order using the Taylor model is the same as that in the observed recrystallization texture.. Mechanism A has been argued to contradict to mechanism B by many researchers, but we consider the both mechanisms are valid. Thus, mechanism A should be referred to as recrystallization and mechanism B should be referred to as grain growth, which means that coalescence and shrinkage occurs between recrystallized grains and nonrecrystallized grains. However, more investigation is required to clarify these differences.
LANKFORD VALUE CALCULATION MODEL We estimated accuracy of our Lankford value prediction method. First, we measured crystal orientation of ultra low-carbon cold rolled and annealed strip by Electron Back Scatter Diffraction Patterns, whose cold reduction was 80% and heated at 800 degrees for 90 seconds. Next, we predicted texture of a 15% stretched ultra low-carbon strip and Lankford value using Sachs deformation prediction model with the Asaro rate-dependent[3],[16]. Here, J0 is 1.0 10-3 and m is 0.05. Figure 6 shows difference between calculated and measured Lankford values in longitudinal, diagonal and cross direction. Prediction accuracy was good because grains of stretched strip were controlled by the law of large numbers. And Figure 7 shows measured texture, predicted texture using Sachs model and predicted texture using Taylor model of 15% stretched cold rolled and annealed ultra low carbon strip. Our method makes us predict Lankford value of steel strips easily.
Lankford value,r
2.5 2.0 1.5 1.0
Calculated
0.5
Measured
0.0 0
15
30
45
60
75
90
Angle from rolling direction,/° FIGURE 6. Lankford value prediction accuracy of ultra low carbon strip using Sachs deformation prediction model.
61
FIGURE 7. Measued texture, predicted texture using Sachs model and predicted texture using Taylor model streched by 15 % at thickness center of ultra low carbon cold rolled and annealed strip.
CONCLUSION The prediction accuracy for the textures of cold rolled ultra low-carbon strip and hot rolled ferritic stainless-steel strip using the Taylor model is high. Moreover, the prediction accuracy of the recrystallization texture classified by the total number of slips using the Taylor model is also high. The reason for this is that ultra low-carbon steels and ferritic stainless-steels recrystallize in accordance with the oriented nucleation model. The hot rolled and annealed texture is classified by the lowest total number of slips because nuclei develop from subgrains close to the grain boundaries. In contrast, the cold rolled and annealed texture satisfies the highest total number of slips because gamma fiber grains encroach on alpha fiber grains in the latter recrystallization period. Next we can predict Lankford value of ultra low carbon strips and ferritic stainless steel using the Sachs model. Though the Taylor model and the Sachs model are classical deformation texture analysis methods, it is practical to use in hot and cold steel rolling industry.
REFERENCES 1. H.Inoue and N.Inakazu:J. Jpn. Inst. Met.,58 (1994),pp.892-898. 2. J.Yanagimoto:Modeling and Simulation in Materials Science and Engineering,10(2002),pp.R1-R24. 3. T.Morimoto,F.Yoshida,Y.Kusumoto and A.Osamu:ISIJ Int.,50 (2010),pp.1683-1688. 4. T.Morimoto,F.Yoshida,Y.Kusumoto and A.Yanagida:Current advances in materials and processes,24(2011),pp.393. 5. W.G.Burgers and T.J.Tiedema:Acta Metallurgica,1(1953),pp.234-238. 6. J.J.Jonas and L.S.Toth:Scripta Metallurgica,27(1992),pp.1575-1580. 7. B.F.Decker and D.Harker:J.Appl.Phys.,22(1951),pp.900-904. 8. C.G.Dunn and P.K.Koh:Trans. AIME,206(1956),pp.1017-1024. 9. T.Urabe and J.J.Jonas:ISIJ international,34(1994),pp.435-442. 10. O. Akisue:J. Jpn. Inst. Met.40(1976)206-210. 11. D.Vanderschueren,N.Yoshinaga and K.Koyama:ISIJ international,39(1996),pp.1046-1054. 12. O. Akisue:Tetsu-to-Hagane,72(1986),pp.1320-1327. 13. I.L,Dillamore and H.Katoh,Metals Sci.,8(1974),pp.21-27. 14. E.H.Lee:J.Appl.Mech.,36(1969),pp.1-6. 15. G.I.Taylor:Roy.Soc.Proc.,A,116(1927),pp.16-39. 16. G.Sachs:Z.Ver.Deut.Ing.72(1928),pp.734-736. 17. K.Sekine:Recrystallization texture and their application to structural control,ISIJ,Tokyo(1999),pp.164-168. 18. I.L.Dillamore,P.L.Morris,C,J,E.Smith and W.B.Hutchinson:Proc.R.Soc.Lond A,329(1972),pp.405-420. 19. R.J.Asaro and A.Needleman:Acta metal,33(1985),pp.923-953. 20. W.B.Hutchinson:Metal Science,8(1974),pp.185-196.
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A General Yield Function within the Framework of Linear Transformations of Stress Tensors for the Description of Plastic-strain-induced Anisotropy Shun-lai Zanga ,Myoung-gyu Leeb School of Mechanical Engineering, Xi’an Jiaotong University, No.28, Xianning Road, Xi’an, Shaanxi, P.R. China b Graduate Institute of Ferrous Technology, Pohang University of Science and Technology, San 31, Hyoja-dong, Nam-gu, Pohang, Geongbuk 790-784, Republic of Korea a
Abstract. A general yield function based on linear transformations of stress tensors for orthotropic sheet metals is presented to describe plastic strain-induced anisotropy. The subsequent anisotropy is considered by introducing a set of tensorial state variables in the linear transformations of stress tensors, and defining the tensorial state variables evolved with plastic deformation. The evolution of the tensorial state variables is based on eigen decomposition. The eigenvectors are normalized and used to represent the rotation of orthotropy axes of polycrystalline materials which relates to the texture rotation. The eigenvalues are used to capture the local reorientation of the grain preferred directions. Taking the anisotropic yield function Yld2000-2d as an example, the evolution of eigenvalues and eigenvectors of linear transformation tensors is investigated for AA3003-O aluminum alloy sheet. Based on the observations, a modified Yld2000-2d yield function incorporating with isotropic hardening model was developed to predict the anisotropy and subsequent yield surfaces in uni-axial tensions to confirm the validity of current yield function. Keywords: Anisotropic material; Yield function; Linear transformations; Plastic strain-induced anisotropy PACS: 83.60.La; 81.40.Jj
INTRODUCTION In the last two decades, the so-called isotropic plasticity equivalent (IPE) theory generalized by Karafillis[1] has gained some popularity in developing new anisotropic yield functions. In the IPE theory, plastic anisotropy is represented by one or more linear transformation tensors on the stress tensor, further the principal values of these linear transformation tensors are substituted into an isotropic yield function to get a new anisotropic yield function. Within the IPE framework, many anisotropic yield functions have been proposed to describe initial anisotropy of metallic sheets with phenomenological constitutive models[2-6], such as Yld2000-2d and Yld2004-13p/18p yield functions by Barlat[3,7] and Bron's anisotropic yield function [4]. The material coefficients of these anisotropic yield functions are usually constant and identified from the experimental initial tensile, shear or bi-axial yield stresses and/or r-values. However, Hu pointed out that it is difficult to get exact initial yield stress since the anisotropy feature at initial yield state is too sensitive to the definition of yield point[6]. Therefore, the plastic strain-induced anisotropy seems to be more important in relation to the constitutive models. Wagoner and Suh assumed that the coefficients characterizing anisotropy do not change with plastic deformation, while the stress exponent M associated with the shape of yield surface in most anisotropic yield functions, is a function of equivalent plastic strain[8-9]. But, since changing exponent M is not a general concept, it is difficult to apply this method to other anisotropic yield functions[6]. To obtain a general model which can cover plastic strain-induced anisotropy, a dynamic yield function was recently proposed, which assumes that the material coefficients of anisotropic yield function are not constant and identified from the current flow stresses and r-values[6]. However, the dynamic yield function is not easy to be implemented for non-quadratic yield functions because iterative solution procedures of non-linear is inevitably involved at each increment of plastic deformation. In addition, only successive strainhardening process can be considered, which means that the plastic deformation is continuous without abrupt
The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 63-70 (2011); doi: 10.1063/1.3623593 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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loading-path change and the stress path of loading-unloading-reloading is not included as well. Recently, Wang also presented a yield function to consider the plastic strain-induced anisotropy by assuming the material parameters are polynomial functions of equivalent plastic strain [10]. In this paper, a general yield function theory based on linear transformations of stress tensors is firstly presented to consider plastic strain-induced anisotropy. In order to understand the evolution of subsequent anisotropy, the eigen decompositions for the linear transformation tensors of anisotropic yield function Yld2000-2d of the AA3003O sheet at different equivalent plastic strains are carried out. Thereafter, the evolution of eigenvalues and eigenvectors is investigated and discussed. Based on the observations, a simple evolution law was proposed and applied to predict the flow stresses and r-values in uni-axial tensions to validate the accuracy of current general yield function.
THEORY According to the classical plasticity theory, an isotropic stress yield function can be expressed with three invariants of the stress tensor [11]. Although many sets of three invariants have been introduced, the principal values are usually adopted to represent the isotropic yield function.
f ( ij ) F 1 , 2 , 3 where f is an isotropic stress yield function with respect to the Cauchy stress tensor of stress tensor ij , and F is an isotropic function of its arguments, i.e.,
F I , II , III F 1 , 2 , 3
(1)
ij , i
are principal values
(2)
where (I,II,III) are permutations of (1,2,3). When pressure-independent plastic deformation is considered, the general formulation of the yield function can be expressed as a function of the three invariants of the deviatoric stress tensor,
g (sij ) G S1 , S2 , S3
(3)
where g is an isotropic stress yield function in the deviatoric stress space, G is an isotropic function of its arguments as well, Si represents the principal values of the deviatoric stress tensor sij . To formulate an anisotropic yield function, a reference frame associated with the material symmetry axes must be defined. In the isotropic plasticity equivalent approach for pressure-independent polycrystalline materials, a stress tensor sij is introduced as a linear transformation of deviatoric stress tensor sij , i.e.,
s = Cs = CT = L
(4)
where C and L are 4th-order tensors, which contain the anisotropy coefficients expressed in the material frame, T is a 4th-order tensor as well, which transforms the Cauchy stress tensor to its deviator. Substituting the principal values Si of the linearly transformed stress tensor s into Eq. (3), an anisotropic yield function can be obtained,
sij g sij G S1 , S2 , S3 Note that tensor s .
(5)
is an anisotropic yield function of s , while g is still an isotropic yield function with respect to stress
64
Considering the symmetry of the stress tensor s and the assumed symmetry of s , C can be represented as a 6 6 matrix. The symmetry of the material and the zero trace of s further reduce the number of independent coefficients in matrix C ; for instance, 9 for an orthotropic material. Therefore, C can be written as,
0 C12 C 21 0 C C32 C = 31 0 0 0 0 0 0
C13
0
0
C23 0
0 0
0 0
0
C44
0
0 0
0 0
C55 0
0
0 0 0 0 C66
(6)
From a mathematical point of view, the linear transformation tensor C makes an initial isotropic yield surface homogeneously deform to an anisotropic yield surface. In most of phenomenological constitutive models, the matrix C is assumed to be constant. Recently, the importance of the description of plastic strain-induced anisotropy has been recognized and emphasized[6,10,12]. Within the isotropy plasticity equivalent framework, the matrix C should be evolved with plastic deformation in order to consider plastic strain-induced anisotropy. Similarly as deformation kinematics, the eigen decomposition can also be applied to the matrix C ,
C = VAV 1
(7)
where V includes the eigenvectors representing the rotation of orthotropy axes of polycrystalline materials, and A contains the eigenvalues which can be used to capture the local reorientation of the grain preferred directions. If we divide the plastic deformation into N steps, then the matrix C at k-step can be written as,
Ck = Vk Ak Vk1
(8)
In Eq. (8), A k is
1k 0 0 Ak 0 0 0
0
0
0
0
0
0
0
0
0
0
0
0
k 4
0
0
0
0
5k
0
0
0
0
k 2
k 3
0
0 0 0 0 6k
(9)
The matrix C at (k+1)-step can be written as,
Ck 1 = Vk 1Ak 1Vk11
(10)
In Eq.(9), considering the evolution of A k , its current eigenvalues can be defined as
ik 1 ik i
65
(11)
Since the local reorientation of the grain preferred directions usually influences the intensity of texture, the k eigenvalues i could be used to capture texture reorientation by defining proper evolution laws of i , i.e.,
i i p , p ,
(12)
In terms of the texture rotation, if the incremental rotation between k-step and (k+1)-step is could be expressed as
Q , then the Vk 1
Vk 1 QT Vk Q
(13)
To consider the influence of plastic deformation on texture rotation, deformation as well;
Q should be related to the plastic
Q Q p , p ,
(14)
Considering the physical aspects of texture evolution, and then defining proper evolution laws of Eq.(12) and (14), the plastic strain-induced anisotropy is expected to be accurately described.
EVOLUTION OF LINEAR TRANSFORMATION TENSOR In this section, taking the anisotropic yield function Yld2000-2d as an example, the evolution of eigenvectors and eigenvalues of linear transformation tensors is investigated. For this yield function, two linear transformation tensors are used to represent the anisotropy in plane stress condition, and total eight anisotropy coefficients, 1 8 , are usually adopted to transform the Cauchy stress to stress tensor sij , as shown in Eq. (15)[11].
1 0 0
C = 0 2 0 0 0 7
and
4 5 3 1 C = 2 3 2 5 3 0
2 6 2 4 4 4 6 0
0
0 3 8
(15)
where these eight anisotropy coefficients are usually calculated from the measured experimental data, 0 , 45 , 90 , b , r0 , r45 , r90 and rb . Here and r are yield stresses and r-values along different angles to rolling direction, b and rb are bi-axial yield stress and r-value, respectively. To describe plastic strain-induced anisotropy, the subsequent yield surfaces should evolve with plastic deformation. It implies that these experimental data at the same subsequent hardening state should be determined to re-calculate the anisotropy coefficients. Regarding the experimental data, there are two methods to determine the experimental data in relation to a subsequent yield state. One method assumes that the anisotropy of yield function is a function of plastic work, which indicates that the experimental data at the same plastic work should be used to determine the subsequent yield surface. The other is based on the so-called principle of equivalent strain-hardening work, which can well explain the hardening behavior of perfect plastic body in which no further hardening occurs we may still have the response of the increment of plastic strain [6]. Here, the first order plastic work principle is used that can be defined as
dw ij d ijp d p
(16)
In this work, the aluminum alloy 3003-O in literature [6] was used as experimental data, and the Eqs. (17) - (19) were adopted to fit all experimental stress-strain curves and r-values as suggested by Hu [6].
k e p , k e n
66
n
(17)
b kb be bp , b kb be nb
1 n
1 nw
r n k
nw
nb
(18)
1 n 1 nw 1 1 nw n n nw kw n k nw
1
(19)
where k , n , k w , nw are the constants of material parameters of tensile tests at angle to rolling direction, kb , nb material parameters of equibiaxial tension, and e , be are the limit elastic strains of corresponding experiments. All the material parameters are listed in Table 1. Base on the above plastic work principle, the material properties of aluminum 3003-O at different hardening states are calculated as listed in Table 2. [6]
TABLE 1. Material parameters of aluminum 3003-O . Directions: Rolling Transverse Regarding the strain: wp0 0p 90p wp90 k values 195.0 240.0 183.0 240.0 n values 0.214 0.217 0.215 0.227 TABLE 2. Material properties of aluminum 3003-O at different hardening states. Directions: Rolling Transverse Diagonal Equibiaxial Plastic strain: (MPa) (MPa) (MPa) (MPa) Yield stress (0.2%) 51.58 48.1 47.1 49.98 0.0077 72.32 68.26 67.71 75.56 0.0120 78.23 73.91 73.49 81.04 0.0159 82.42 77.91 77.58 84.81 0.0195 85.77 81.10 80.85 87.76 0.0402 99.03 93.73 93.80 99.09 0.0740 112.33 106.38 106.83 110.07 0.1087 121.75 115.34 116.09 117.67 0.1446 129.3 122.53 123.52 123.68 0.1818 135.71 128.63 129.84 128.73 0.2203 141.34 133.99 135.39 133.13
Diagonal
Equibiaxial
45p
wp45
bp
187.0 0.222
232.0 0.229
171.3 0.17
r0
r90
r45
0.704 0.677 0.672 0.668 0.665 0.655 0.646 0.641 0.637 0.633 0.631
0.662 0.576 0.559 0.548 0.54 0.511 0.488 0.473 0.463 0.455 0.449
0.842 0.769 0.754 0.745 0.737 0.712 0.690 0.677 0.667 0.660 0.654
As shown in Table 2, four stresses and three r-values are available for AA3003-O aluminum alloy sheet. With these data, only seven anisotropy coefficients can be identified. Therefore, 3 6 is assumed for the anisotropic yield function Yld2000-2d. Fig. 1 shows the evolution of eigenvalues of linear transformation tensors with plastic deformation. It can be seen from Fig. 1(a), the eigenvalues of linear transformation tensor C gradually saturate to the constants after large plastic deformation. However, for tensor C , the obvious abrupt changes exist for all eigenvalues when plastic deformation is small. It might result from that the deformation at early stage is quite unstable due to the fact that the early plastic range may not be completely plastic, or may be elastic-plastic transient regions. Fig. 2 shows the evolution of eigenvectors of linear transformation tensor C with plastic deformation. Here, the evolution of eigenvectors is respectively represented by the angles between current eigenvector and the initial one as shown in Fig. 2(a), and the angles between current eigenvector and previous one as shown in Fig. 2(b). Supposed n1k , n 2k , and n 3k are the current eigenvectors of linear transformation tensor C , n1k 1 , n 2k 1 , and n3k 1 are the 1 2 3 previous eigenvectors, n 0 , n 0 , and n 0 are initial one, the angles used here are respectively defined as
ni ni k 0 nik ni0
i arcos
and
ni ni d i arcos i k k i 1 n k n k 1
(20)
Fig. 2(a) indicates that the eigenvector evolved with plastic deformation. When the traditional initial yield stresses and r-values, i.e., assuming the equivalent plastic strain is zero, are selected to determine the initial eigenvectors, the
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subsequent eigenvectors have abrupt changes. However, when the eigenvectors which correspond to equivalent plastic strain of 0.0077 and 0.0120 are used to represent the initial state, the evolution of subsequent eigenvectors is continuous and almost linear. Fig. 2(b) shows that the eigenvectors have a rapid change rate at small plastic deformation, and after that the rates of i to equivalent plastic strain are almost constant. It implies that the evolution of directions of eigenvectors has a linear relation to equivalent plastic strain.
(a) Linear transformation tensor C (b) Linear transformation tensor C FIGURE 1. Evolution of eigenvalues of linear transformation tensors with plastic deformation.
(a) Angles between current eigenvectors and initial one (b) Angles between current eigenvectors and previous one FIGURE 2. Evolution of eigenvectors of linear transformation tensor C with plastic deformation.
PREDICTION OF THE ANISOTROPY AND SUBSEQUENT YIELD SURFACES As shown in Fig. 1, the eigenvalues of linear transformation tensors tend to saturate towards constants after large deformation. Two possible reasons result in such behavior. One is that only monotonic loading paths are considered in the present work. In such cases, the evolution of anisotropy and hardening behavior is continuous, and thus no abrupt changes exist. It implies that the evolution of the eigenvalues should be continuous too. The other reason is that, for most polycrystalline metallic materials, the hardening rate decreases with plastic deformation. Therefore, it can be inferred that the change rate of anisotropy and hardening behavior should decrease with plastic deformation as well. In this work, for simplicity the eigenvectors of linear transformation tensor C are assumed to be fixed during plastic deformation as a first attempt and Eq. (21) is used to describe the evolution of eigenvalues.
i i0 Qi 1 eb
68
i
p
(21)
where Qi and bi are material parameters, equivalent plastic strain, the plus-minus denotes that the eigenvalues increase or decrease. This sign might be related to the plastic strain rate when strain path changes exist. In our case, for simplicity, the signs are directly defined based on the evolution of eigenvalues as shown in Fig. 1. The important part of the current general yield function is to define the initial eigenvalues and eigenvectors of linear transformation tensors. As mentioned above, the definitions of initial yield stresses and r-values might be very difficult and inaccurate with traditional methods. To overcome it, the early stable anisotropy at small plastic deformation should be chosen to represent the initial anisotropy. In this work, the experimental data from equivalent plastic strain of 0.0077 was used to fit the initial eigenvalues and eigenvectors. For the other material parameters of anisotropy-evolved yield function Yld2000-2d, inverse identification is carried out by optimization using uni-axial tensile tests with a developed Matlab toolbox SMAT. A cost function is defined in the least square sense and minimized, starting from an initial guess of parameters. Finally, the identified material parameters are summarized in Table 3. The initial eigenvectors are shown in Eq. (22). p
TABLE 3. Material parameters of anisotropy-evolved yield function Yld2000-2d for aluminum 3003-O. Transformation tensors: b Q Q b 0 0 0
C C
3
Q3
b3
0.9033
0.0507
17.435
1.0351
0.1005
17.006
1.0204
0.0268
16.796
0.9524
0.0
0.0
1.1922
0.0
0.0
1.2522
0.0
0.0
1
1
1
2
2
2
0
0.8402 0.6546 V0 = 0.5423 0.7560 0 0 0 1.0
(22)
Fig. 3 is the simulated flow stresses and r-values compared with experimental data. It is found that the stressstrain and r-value curves predicted by the current yield function are in good agreement with the experimental curves in all three directions, especially, for the evolution of r-values. For the case of constant material coefficients of yield function, r-values are usually kept when isotropic hardening model is adopted. However, the current method gives a good correlations with simplified evolution equation, Eq. (21).
(a) Cauchy stress (b) R-value FIGURE 3. Predicted Cauchy stresses and r-values in uni-axial tensions.
When a hardening state is selected, its corresponding yield surface could be predicted by the current yield function. Choosing the hardening states presented in Table 2, the initial and subsequent yield surfaces are plotted as shown in Fig. 4 when the shear stress is zero. For comparisons, the yield surfaces calculated by the constant and variable material coefficients of yield function are expressed in the same graph, marked with ‘anisotropy-unevolved’ and ‘anisotropy-evolved’, respectively. The results show that the subsequent yield surfaces predicted by the two methods are rather different, especially, for the bi-axial tensile stress states.
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FIGURE 4. Initial and subsequent yield surfaces with anisotropy-unevolved and anisotropy-evolved yield functions.
CONCLUSIONS A general yield function based on linear transformations of stress tensors for orthotropic sheet metals is presented to model plastic strain-induced anisotropy. The subsequent anisotropy is considered by defining proper evolution laws for the eigenvalues and eigenvectors of linear transformation tensors. Taking the anisotropic yield function Yld2000-2d and AA3003-O aluminum alloy sheet, the evolution of eigenvalues and eigenvectors of linear transformation tensors is investigated. To confirm the validity of the present method, the flow stresses and r-values of uni-axial tensions are predicted by a modified Yld2000-2d yield function which can cover plastic strain-induced anisotropy under successively strain-hardening process. The results show that the present model can accurately describe the anisotropy and hardening behavior, especially, the evolution of r-values. As a future work, the model will be applied to a real sheet metal forming process to verify the importance of the plastic induced anisotropy.
ACKNOWLEDGMENTS Shun-lai Zang would like acknowledge financial support by the Specialized Research Fund for the Doctoral Program of Higher Education (No.200806981025), and by the National Natural Science Foundation of China (No.11002105) and by the Opening Project of Key Laboratory of Testing Technology for Manufacturing Process (Southwest University of Science and Technology), Ministry of Education (No.10ZXZK03).
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
A.P. Karafillis and M.C. Boyce, J. Mech. Phys. Solids 41, 1859-1886 (1993). D. Banabic et al, Int. J. Plast. 21, 493-512 (2005). F. Barlat et al, Int. J. Plast. 19, 1297-1319 (2003). F. Bron and J. Besson, Int. J. Plast. 20, 937-963 (2004). O. Cazacu et al, Int. J. Plast. 22, 1171-1194 (2006). W. Hu et al, Int. J. Plast. 23, 620-639 (2007). F. Barlat et al, Int. J. Plast. 21, 1009-1039 (2005). R.H. Wagoner, Metallurgical and Materials Transactions A 11, 165-175 (1980). Y.S. Suh et al, Int. J. Plast. 12, 417-438 (1996). H.B. Wang et al, Comput. Mater. Sci. 47, 12-22 (2009). F. Barlat et al, Int. J. Plast. 23, 876-896 (2007). J.H. Hahm and K.H. Kim, Int. J. Plast. 24, 1097-1127 (2008).
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Experimental Investigations on Anisotropic Evolution of 304 Stainless Sheets under Tensile Pre-Strains Lai Tenga and Cheng Guob School of Mechanical Engineering, Xi’an Jiaotong university, No.28 Xianning West Road.Xi’an.Shaanxi, Xi’an, 710049, China Abstract. The anisotropic evolution of cold rolled 304 stainless steel sheets under pre-strains is investigated experimentally. Uni-axial tensile yield stress and r-value are measured in experiments to represent the anisotropy. The tensile pre-strains under plane stress are achieved by cutting large specimens into small one at different angles to rolling direction. Then the uni-axial tensile tests are performed on the small specimens to investigate the anisotropic evolution. It is found that the yield stress increases with the increase of the pre-strains and decreases with the increase of the angles. However, the changes of r-value are hardly affected by the pre-strains, the small changes of r-value show that the material may remember the rolling direction even after the pre-strains. The sigmoidal shape can be observed in the tensile curves, and its shape depends on the pre-strains and angles. The change of hardening rate can be divided into three stages, and is the most significant at 90° to the rolling direction in the three stages, at the same time the pre-strains cause noncoincidence of the hardening rate curves at the same angle. Moreover, the hardening rate depends on the directions of tensile loading. Second derivative of the hardening rate also can be divided into three stages, and the differences of three stages may mainly be controlled by the different volume fraction of martensite. Keywords: SUS304, austenitic stainless steel, martensitic transformation, anisotropic hardening. PACS: 81.40.Ef, 81.30.Kf.
INTRODUCTION The 304 stainless steel sheets are widely used as structural components in machines and structures because of good toughness, ductility and resistance to corrosion. The anisotropy of cold rolled materials is a well-known phenomenon and is of practical interest for the formability of sheets. So a great deal of efforts has been made to understand the yield and hardening behavior of cold rolled anisotropic materials experimentally [1, 2] as well as theoretically [3, 4]. Austenitic phase in 304 stainless steels is metastable, and it is known to develop the straininduced martensitic transformation caused by lower stability of austenitic phase when it is subjected to plastic deformation. Many reports refer to the response of these evaluations in terms of the effects of the martensite contents in the cold rolled 304 stainless steel [5, 6]. Type 304 steel is one of the typical austenitic stainless steels and exhibits remarkable work hardening [7], thus pre-strained type 304 steel will be expected to behave in different manner from unpre-strained one. This paper utilized the method of Kim and Yin [8]. In their previous work, tensile deformation was applied at some angles to the rolling direction. Kim and Yin suggested that the orientations of the orthotropy axes are altered by the tensile pre-strain at an angle to the rolling direction. In this study, the change of yield stress, r-value, true stress-strain and hardening rate will be presented for the better description of the hardening behavior. The main purpose of this work was to report the experimental observations and to attempt to understand the observed plastic yield and hardening behavior. Moreover, the work did not pay attention to the change of the orthotropy axes and microscopic structure.
EXPERIMENTAL SCHEME An experimental method for the study of anisotropy of cold rolled 304 stainless steel sheets is to perform tensile tests at angles to the rolling direction. The initial anisotropy is enhanced by stretching 12 full size sheets along the The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 71-77 (2011); doi: 10.1063/1.3623594 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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rolling direction by 6%, 11%, 16% and 21% tensile pre-strains. After the pre-strains, uni-axial tensile tests are performed at every 22.5º interval to rolling direction for 5 small specimens (FIGURE 1). The full size sheets were loaded different pre-strains with the help of huge chucks, and digital image correlation system (XJTUDIC) and photogrammetry system (XJTUSM) were used to confirm homogeneous deformed zone. Center lines were drawn in the huge chucks and full size sheets in order to ensure stretch axis coincided with center axis of the full size sheets, and many spots were corroded at the back of the full size sheets, the photogrammetry system (XJTUSM) is used to measure strain after spring back (FIGURE 2).
FIGURE 1. Schematic illustration of experimental procedure.
(a) Digital image correlation system (XJTUDIC). (b) Photogrammetry system (XJTUSM). FIGURE 2. The huge chucks, full size sheet and strain measurement system.
(a) Before pre-strain (b) After pre-strain FIGURE 3. Strain measurement of the full size sheet before and after the pre-strain.
FIGURE 3 shows the strain measurement before and after the pre-strain, and the homogeneous deformation can be observed. However, the strain is negligible before the pre-strain. So it could cut the small specimens from the deformed zone to investigate the anisotropic evolution. The true stress, longitudinal and transverse strain ( l and w ) have been measured in tensile tests, and yield points have been measured at 0.2% offset strain. The incremental plastic strain ratio ( r ) is defined by
r Here,
tp
d wp d wp . d t p d lp d wp
(1)
means the through-thickness plastic strain. The r-value is determined from the longitudinal and
72
transverse strains under the assumption of plastic incompressibility. Measurement noise in the strain channels and uncertainties in the elastic properties of the material tend to disrupt the estimation of r-value in the small strain range. Therefore r-value has been measured at a sufficiently large longitudinal strain up to 10%, 20% [8-11].
EXPERIMENTAL RESULTS AND DISCUSSION The strain rate is about 4.78 10 for the full size sheets and 3 10 for the small specimens at room temperature. The following figures show the results of tensile tests. Figure 4 shows the variations of the uni-axial yield stress for the small specimens applied at 0º, 22.5º, 45º, 67.5º and 90º to the rolling direction. The yield stress of the small specimens increases with the increase of the pre-strains and decreases with the increase of the angles. In contrast to the variations of the yield stress, the changes of r-value are found to be hardly affected by the pre-strains in Figure 5, the small changes of r-value show that the material may remember the rolling direction even after the pre-strains. 4
3
FIGURE 4. Uni-axial yield stress (at 0.2% offset strain) for the small specimens applied at Ψ=0°，22.5°，45°，67.5°，90°, respectively.
(a) Longitudinal plastic strains
lp =10%
(b) Longitudinal plastic strains
lp =20%
FIGURE 5. Incremental r-value distributions at 10% and 20% of longitudinal plastic strains l for the pre-strain of 6%, 11%, p 16% and 21%: (a), (b) Longitudinal plastic strains l =10%, 20%, respectively. p
It was mentioned in many literature that deformation could induce martensitic transformation from austenite in the austenitic stainless steels [12, 13, 14]. The volume fraction of transformed martensite increased with the increase of deformation, and this was explained from transformation kinetics in [15]. Some researchers have shown that martensitic transformation can increase the strength and decrease the toughness of the austenitic stainless steels, and the results of tests are in accordance with the previously reported results (FIGURE 6). For further study change of the hardening rate, true stress-strain and strain-hardening rate curves for each pre-
73
strain condition need to be examined in detail. Figure 6 shows the true stress-strain curves of the small specimens. As mentioned above, for the different pre-strains, it can be observed that strength increases and elongation decreases with the increase of the pre-strains, it may mainly due to the pre-strains martensite in its matrix, and the effect is similar to that of martensite in the dual-phase steel. The yield strengths are slightly different at different angles to rolling direction in the same pre-strains, while the hardening rate is essentially similar in each direction except for the rolling direction. The sigmoidal shape can be observed in the tensile curves, and the larger the pre-strain is, the more obvious its trend is. Moreover, such trend is more obvious at other directions than at the rolling direction, for it may transfer more mastentics at other directions [16, 17, 18].
(a) Small specimens after 6% pre-strain
(b) Small specimens after 11% pre-strain
(c) Small specimens after 16% pre-strain (d) Small specimens after 21% pre-strain FIGURE 6. True stress-strain of the small specimens at angles to the rolling direction for different pre-strains: (a), (b), (c), (d) after 6%, 11%, 16%, 21% pre-strains along the rolling direction, respectively.
FIGURE 7 shows the hardening rate curves of the small specimens, and it cuts off the curves for better expression. The curves can be divided into three stages in the FIGURE 7. In the first stage, the hardening rate decreases with the increase of true strain, but the initial trends are similar to that of other materials, for the content of transformed martensite in the austenitic matrix may not result in difference of initial hardening rate. In the second stage, the hardening rate exhibits an opposite trend compared to the first stage. In the final stage, the hardening rate decreases with the increase of true strain again. The change of hardening rate is the most significant at 90° to the rolling direction in the three stages, at the same time the pre-strains can cause noncoincidence of the hardening rate curves at the same angle. Moreover, the hardening rate depends on the directions of tensile loading. Due to the different hardening rate in each of loading directions, the yield stress depends on the offset plastic strain level. For further study, second derivative of the hardening rate has been gotten from FIGURE 7. The pre-strains have significant effects on initial and final stage, but it makes very little difference in intermediate stage. The change s of the three stages can be explained by martensitic phase transformation, for the volume fraction of transformed martensite increases with the increase of the pre-strains when the full size sheets undergo the different pre-strains. Moreover, the differences of initial and final stage in FIGURE 8 may mainly controlled by the martensite which was
74
induced by the previous pre-strains. However, the curves trend to be uniform in the intermediate stage, this is because the volume fraction of martensite may be same after the following deformation.
(a) Small specimens along Ψ=0°
(b) Small specimens along Ψ=22.5°
(c) Small specimens along Ψ=45°
(d) Small specimens along Ψ=67.5°
(e) Small specimens along Ψ=90° FIGURE 7. Strain-hardening rate at angles to the rolling direction for different pre-strains: (a), (b), (c), (d), (e) the small specimens applied at Ψ=0°, 22.5°, 45°, 67.5°, 90°, respectively.
For further study on microstructure evolution of 304 stainless steel under the pre-strains and the tensile loading, in the next stage, light optical microscope will be used to determine microscopic structural change, and X-ray
75
diffraction will be used to measure the relative amounts of different phases formed during pre-prestrain and tensile loading in cold rolled 304 stainless steel sheets.
(a) Small specimens along Ψ=0°
(b) Small specimens along Ψ=22.5°
(c) Small specimens along Ψ=45°
(d) Small specimens along Ψ=67.5°
(e) Small specimens along Ψ=90° FIGURE 8. Strain-second derivative of hardening rate at angles to the rolling direction for different pre-strains: (a), (b), (c), (d), (e) the small specimens applied at Ψ=0°, 22.5°, 45°, 67.5°, 90°, respectively.
76
CONCLUSION The yield stress increases with the increase of the pre-strains and decreases with the increase of the angles. The change of r-value is affected by the pre-strains. The sigmoidal shape can be observed in the tensile curves. The changes of the hardening rate and second derivative of the hardening rate curves can be divided into three stages.
ACKNOWLEDGMENTS Lai Teng is grateful to Dr. Zang for his valuable advice and comments and to Professor Liang at Xi’an Jiaotong University for his experimental devices and encouragement during the progress of this work. Support from Yang Zhang and Xiang Guo of Research Institute of Mould and advanced forming technique with the experiment process and from Hua Zhang with the specimen preparation are greatly appreciated.
REFERENCES 1. Mallick, K., Sanabta, S.K., Kumar, A., “An experimental study of the evolution of yield loci for anisotropic materials subjected to finite shear deformation”, J. Eng. Mat. Tech. Trans. ASME 113, 1991, pp. 192. 2. Truong Qui, H.P., Lippmann, H., “Plastic spin and evolution of an anisotropic yield condition”, Int. J.Mech.Sci. 43, 2001, pp. 1969–1983. 3. Hill, R., “Continuum micro-mechanics of elastoplastic polycrystals”, J. Mech. Phys. Solids 13, 1965, pp. 89–101. 4. Berveiller, M., Zaoui, A., “An extension of the self-consistent scheme to plastically-flowing polycrystals”, J.Mech. Phys. Solids 26, 1979, pp. 325–344. 5. T. Yamaski, S. Yamamoto, M. Hiro, NDT&E Int. 29, 263 (1996). 6. D.Q. Sullivan, M. Cottrell, I. Meszaros, NDT&E Int. 37,265 (2004). 7. Schuster G, Altstetter C. “Fatigue of annealed and cold worked stable and unstable stainless steels”, Metall Trans A 14A,1983, pp. 2077-2084. 8. K. H. Kim, J. J. Yin, “Evolution of anisotropy under plane stress”, Journal of the Mechanics and Physics of Solids 45, 1997, pp. 841-851. 9. J.H. Hahm, K.H. Kim, “Anisotropic work hardening of steel sheets under plane stress”, International Journal of Plasticity 24, 2008, pp. 1097-1127. 10. Magnus Harrysson, Matti Ristinmaa, “Description of evolving anisotropy at large strains”, Mechanics of Materials 39, 2007, pp. 267-282. 11. Z. J. Li, G. Winther, N. Hansen, “Anisotropy of plastic deformation in rolled aluminum”, Materials Science and Engineering A 387-389, 2004, pp. 199-202. 12. P.L. Mongonon, G. Thomas, “Structure and properties of thermal-mechanically treated 304 stainless steel”, Metall Trans 1, 1970, pp. 1587-1594. 13. F.D. Fischer, Q.P. Sun, K. Tanaka, “Transformation Induced Plasticity”, Appl. Mech. Rev 49, 1996, pp. 317-364. 14. D. G. Park, D. W. Kim, C. S. Angani, “Measurement of the magnetic moment in a cold worked 304 stainless steel using HTS SQUID”, Journal of Magnetism and Magnetic Material 320, 2008, pp. 571-574. 15. Smaga M, Walther F, Eifler D, “Deformation Induced Martensitic Transformation in Metastable Austenitic Steels”, Materials Science and Engineering A, 2008, pp.483-484. 16. K. H. Lo, D. Zeng, C. T. Kwok, “Effects of sensitisation-induced martensitic transformation on the tensile behaviour of 304 austenitic stanless steel”, Materials Science and Engineering A 528, 2011, pp. 1003-1007. 17. K. Spencer,M. Ve´ron, K. Yu-Zhang, “The strain induced martensite transformation in sustenitic stainless steels Part1Influence of temperature and strain history”, Materials Science and Technology 25(1), 2009, pp. 7-17. 18. XUE Zong-yu, ZHOU Sheng, WEI Xi cheng, “Influence of Pre-transformed Martensite on Work-Hardening Behavior of SUS 304 Metastable Austenitic Stainless Steel”, International Journal of Iron and Steel Research 17(3), 2010. pp. 51-55.
77
([SHULPHQWDO&KDUDFWHUL]DWLRQDQG&RQVWLWXWLYH0RGHOLQJRI 7$90HFKDQLFDO%HKDYLRULQ3ODQH6WUDLQ6WDWH DW5RRP7HPSHUDWXUH G. Gillesa,b, V. Tuninettia, M. Ben Bettaïeba, O. Cazacuc, A.M. Habrakena,b and L. Duchênea,b a
Department ArGEnCo, Division MS²F, University of Liège, Chemin des Chevreuils 1, 4000 Liège, Belgium b Fonds de la Recherche Scientifique-FNRS, Belgium c Department of Mechanical and Aerospace Engineering, University of Florida REEF, 1350 N Poquito Road, Shalimar, FL 32579, USA $EVWUDFWThis paper presents an experimental and theoretical study of the quasi-static behavior of TA6V titanium alloy in plane strain state. In order to quantify the anisotropy of the material, tests were carried out at room temperature on specimens cut out from a sheet along three loading directions. The initial yield locus is described by the phenomenological CPB06ex3 criterion and Voce’s type isotropic hardening is used. Finite element simulations are performed and compared with the experiments. .H\ZRUGV TA6V titanium alloy, material anisotropy, strength differential effect, anisotropic yield criterion, 3$&6 46.35.+z, 62.20.F-
,1752'8&7,21 TA6V titanium alloy displays mechanical properties for which industrial sectors such as aerospace, biomedical or transportation have shown a high interest. Indeed, owing to its high-strength-to-weight ratio, good formability and biocompatibility, it is used in the design of aircraft engines and structures, surgical implants, valves, connecting rods, etc. [1,2]. Many investigations have already been conducted in order to characterize and model the mechanical behavior of TA6V [3,4,5]. This hexagonal-closed packed (hcp) material exhibits both a pronounced anisotropy as well as a strength asymmetry in tension and compression. Due to these specific characteristics, classical plasticity models (Hill 1948, J 2 -flow theory) are not adapted to describe the mechanical response of this alloy [6]. New phenomenological yield criteria [7,8] have been proposed to take into account both the tension-compression asymmetry and the anisotropy of hcp materials. Recently, an experimental study on the quasi-static mechanical behavior of TA6V in tension and compression has been performed [9]. The yield criterion CPB06ex3 [8] was used in order to describe yielding. However, tensile and compressive tests are not enough to identify the material parameters with accuracy. In this study, quasi-static plane strain tests in three different directions at room temperature have been performed on 0.6 mm thick samples. A description of the methodology used to analyze the experimental results and inspired by [10] is presented. The observed behavior is next modeled with CPB06ex3 yield criterion and an isotropic hardening. The model identification is improved by taking into account the onset of plasticity for the plane strain loading. The ability of the modeling to predict the material behavior is then examined by comparing the experimental and simulation stress-strain curves in plane strain tension until large strains, as well as the experimental and numerical results obtained from a simple shear test.
The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 78-85 (2011); doi: 10.1063/1.3623595 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
78
(;3(5,0(17$/&+$5$&7(5,=$7,21 0DWHULDO The investigated TA6V is a commercial alloy produced by TIMET (France). The material is in sheet form with a 0.6 mm thickness. It presents an average grain size of 11 µm for the hcp α-phase and 1 µm for the bcc β-phase, respectively. The chemical composition of the sheet is given in 7$%/(.
Top Bottom %DO EDODQFH
7L Bal. Bal.
7$%/(Chemical composition of the TA6V sheet alloy [in %].
$O 6.22 6.27
9 3.93 4.00
2 0.19 0.20
1 0.006 0.006
& 0.008 0.009
+ 0.01 0.0086
)H 0.16 0.16
&KDUDFWHUL]DWLRQRIWKH$QLVRWURSLF%HKDYLRULQ3ODQH6WUDLQ The experimental device used in this work is shown in ),*85( . The machine includes a vertical and a horizontal actuators (denoted V1 and V2 in ),*85( , respectively) which enable to carry out successive or simultaneous plane strain and simple shear tests. This biaxial machine was developed and validated at the Materials and Structures Mechanics Laboratory of the University of Liège [11].
9
9
),*85(Biaxial machine: the white arrows indicate the displacement of the actuators.
In order to characterize the material anisotropy in plane strain tensile state, tests were performed at a velocity of the actuators of 5x10-3 mm/s (corresponding to an initial strain rate of 1.67x10-3 s-1 when neglecting any slip of the specimen under the grips) and room temperature along three in-plane directions, namely the rolling (RD), transverse (TD) and 45° directions. Each test was duplicated four times in order to check the experimental reproducibility. The specimens are rectangular, with notches at mid-length (see ),*85( D ). The overall length and width are respectively 120 mm and 30 mm, while the gage zone is 25 mm wide and 3 mm high.
79
(a)
(b)
),*85( D Geometry and dimensions of the plane strain specimen. E Configuration of the specimen inside the grips. ),*85( E shows the configuration of a specimen inside the grips of the biaxial machine. A stochastic pattern is previously painted on the sample’s surface in order to determine the strain field of the gage zone by using the optical measurement system Aramis®. ),*85( shows the typical distribution of the axial (x1 direction) and transversal (x2 direction) strains along the width of the gage zone for different stages of the deformation process. It can be distinguished two different areas: - a homogeneous central zonein plane strain state (the transversal strain is low compared to the axial strain); - a heterogeneous zone at the edges of the specimen.
),*85(Evolution of the axial and transversal strain fields along the width of the gage zone
Since the strain field is not uniform, the edge effects must be taken into account in the stress computation. The methodology used in this work is similar to the one proposed by [10]. The homogeneous strain field zone is first identified at the different stages of the deformation. Theoretically, the strain distribution is uniform when: ∂ε11 (1) =0 ∂x2
80
where ε11 denotes the axial strain. In practice, a range ξ is defined at each stage and the condition (1) is expressed as: ∂ε −ξ ≤ 11 ≤ ξ (2) ∂x2 In this study, the boundary ξ was chosen as the average deviation of the derivative from zero at a 5 mm wide central zone of the specimen. The homogeneous strain field width WH is then computed as follows:
WH = x2max − x2min
(3)
with x2 verifying (2), i.e.:
§ ∂ε · § ∂ε · x2min = x2 ¨ 11 = −ξ ¸ , x2max = x2 ¨ 11 = ξ ¸ (4) ∂ ∂ x x © 2 ¹ © 2 ¹ Based on a numerical analysis of a plane strain tensile test for different materials and specimen geometries, [10] showed that the axial stress in the homogeneous zone can be computed by: F F σ 11 = α T + (1 − α ) T (5) WT tH WH t H where FT is the force measured by the load cell of the machine and WT is the total width of the gage zone, while t H is the actual thickness of the homogeneous area computed thanks to the initial thickness t0 and the average strain at the specimen’s center ε 11c as follows:
t H = t0 exp ( −ε11c )
(6)
It has to be noted that, in equation (6), the volume conservation is assumed. The coefficient α in equation (5) is a correction factor which was assessed at 0.98 in [10]. ),*85( shows the average axial stress-axial strain curves along RD, TD and 45°-direction obtained with the methodology described above, while 7$%/( lists the plane strain yield stress in each orientation. The latter were determined using the work-equivalence principle [12] and the experimental results in tension and compression reported in [9]. It can be observed that the material displays moderate in-plane anisotropy in plane strain, the strongest behavior being in TD.
),*85( Plane strain test results along the rolling (RD), transverse (TD) and 45° directions. 7$%/( Plane strain yield stresses determined by the work-equivalence principle
/RDGLQJGLUHFWLRQ>GHJUHHV@ 0 45 90
81
03D@ 1139 1185 1199
02'(/,1* ,QLWLDO03D@ 971
R6$7 >03D@ 212
C R >@ 12.6
5HVXOWV ),*85( displays the theoretical yield surface for TA6V in the biaxial plane ( σ 3 = 0 ) according to Von Mises and CPB06ex3 criteria in comparison with the experimental data corresponding to the plastic work levels associated to the onset of plasticity ( W p = 1.95 MPa) and a plastic strain of 5% in tension along RD ( W p = 51.3 MPa). It is
worth noting that, since only one component of the stress tensor can be determined from the plane strain tensile tests, the condition imposed on the shape of the yield locus is in this case to be tangent to a straight line. It can be noticed that the initial anisotropy and tension-compression asymmetry exhibited by the material are well predicted by CPB06ex3 yield criterion. In addition, it has to be noted that, when observing the location of the experimental points at both plastic work levels, the hardening is slightly anisotropic.
),*85(Initial yield locus according to Von Mises criterion (blue line) and CPB06ex3 criterion (red line) in comparison with experimental data in tension, compression, simple shear and plane strain (symbols and straight lines) at two different plastic work levels.
Plane strain tensile tests were simulated by using a single 8-node 3D brick element with one integration point. ),*85( compares the experimental stress-strain curves in RD and TD with the results of the simulations.
83
Although the onset of plasticity is pretty well captured (see ),*85( E ), the flow stress for plastic strain is underestimated in both loading directions. Nevertheless, the work-hardening rate is in quite good agreement after about a strain of 2%.
(a)
(b)
),*85((a) Comparison between experimental and simulated stress-strain curves in plane strain tension for the rolling and transverse directions; (b) zoom at the onset of plasticity.
In order to test the phenomenological modeling, a simple shear test (),*85( D ) was performed on the biaxial machine and next simulated by using the finite element mesh shown in ),*85( E . During the experiment, the applied force was measured by the load cell of the horizontal actuator and the shear strain at the center of the specimen was computed thanks to Aramis®. ),*85( F compares the experimental and numerical curves. It can be observed that, although the results do not perfectly match, they are however in good agreement.
(a)
(b)
(c)
),*85((a) Simple shear test diagram; (b) finite element mesh; (c) force applied to the specimen as a function of the shear strain at its center and comparison with the simulated curve.
&21&/86,21 In addition to a previous study on TA6V in tension and compression, plane strain tests were carried out in order to obtain new data on the quasi-static mechanical behavior of the alloy. From the experimental results, it can be observed that the material displays a moderate anisotropy in plane strain state. The initial yielding was described by using CPB06ex3 yield criterion adapted to hcp metals and the hardening was assumed to be isotropic, even if the experiments show a slightly anisotropic hardening. Finite element simulations of the plane strain tests and a simple shear test were performed in order to examine the ability of the model to predict the material behavior. The experimental and numerical results are not in good agreements in plane strain. However the simulations show that efforts must be focused on the description of the hardening. In order to improve the modeling, hardening depending on the strain path is required.
84
$&.12:/('*0(176 The authors acknowledge the Belgian National Fund for Scientific Research F.R.S.-FNRS and the Belgian Federal Science Policy Office (Contract P6/24) for their support.
5()(5(1&(6 1. R. Boyer, G. Welsch and E.W. Collings, Materials Properties Handbook: Titanium Alloys, Materials Park, OH: ASM International, 1994. 2. G. Lütjering and J.C. Williams, Titanium, Berlin Heidelberg: Springer, 2007. 3. A.S. Khan, R. Kazmi and B. Farrokh, Int. J. Plast. , 931-950 (2007). 4. A. Majorell, S. Srivatsa and R.C. Picu, Mater. Sci. Eng. A , 297-305, 2002. 5. R.C. Picu and A. Majorell, Mater. Sci. Eng. A , 306-316, 2002. 6. T. Kuwabara, C. Katami, M. Kikuchi, T. Shindo and T. Ohwue, Proceedings of the 7th International Conference on Numerical Methods in Industrial Forming Processes, Toyohashi, Japan, 2001, pp. 781-787. 7. O. Cazacu, B. Plunkett and F. Barlat, Int. J. Plast. , 1171-1194 (2006). 8. B. Plunkett, O. Cazacu and F. Barlat, Int. J. Plasticity , 847-866 (2008). 9. G. Gilles, W. Hammami, V. Libertiaux, O. Cazacu, J.H. Yoon, T. Kuwabara, A.M. Habraken and L. Duchêne, Int. J. Solids Struct. , 1277-1289 (2011). 10. P. Florès, V. Tuninetti, G. Gilles, P. Gonry, L. Duchêne and A.M. Habraken, J. Mater. Process. Tech. , 1772-1779 (2010). 11. P. Florès, “Development of Experimental Equipment and Identification Procedures for Sheet Metal Constitutive Laws”, Ph.D. Thesis, University of Liege, 2005. 12. R. Hill, J. Mech. Phys. Solids , 23-33 (1987). 13. E. Aarts and J. Korst, Simulated annealing and Boltzmann machines: a stochastic approach to combinatorial optimization and neural computing, New York: John Wiley & Sons, Inc., 1989.
85
Anisotropic yield function capable of predicting eight ears J.H. Yoona, O. Cazacua* a
Department of Mechanical and Aerospace Engineering, University of Florida, REEF, Shalimar, FL 32579, USA.
Abstract Deep drawing of a cylindrical cup from a rolled sheet is one of the typical forming operations where the effect of this anisotropy is most evident. Indeed, it is well documented in the literature that the number of ears and the shape of the earing pattern correlate with the r-values profile. For the strongly textured aluminum alloy AA 5042 (Numisheet Benchmark 2011), the experimental r-value distribution has two minima between the rolling and transverse direction data provided for this show that the r-value along the transverse direction (TD) is five times larger than the value corresponding to the rolling direction. Therefore, it is expected that there are more that the earing profile has more than four ears. The main objective of this paper is to assess whether a new form of CPB06ex2 yield function (Plunkett et al. (2008)) tailored for metals with no tension-compression asymmetry is capable of predicting more than four ears for this material.
Keywords: Anisotropic yield function; AA5042; Orthotropy; Deep drawing.
INTRODUCTION The main objective of this paper is to assess the ability of the form of the CPB06ex2 yield function for metals with cubic crystal structure to predict more than four ears in single-stage cup drawing for a strongly anisotropic aluminum. The paper is organized as follows. We begin by presenting the anisotropic yield function that will be used in the FE simulations along with the identification procedure used for determination of the anisotropy coefficients based on rvalues and tensile flow stresses. Comparison between simulation results and data are presented for two sets of anisotropy coefficients and different friction conditions. Irrespective of the sets of parameters used and friction conditions, the CPB06ex2 yield function predicts the eight ears.
ANISOTROPIC YIELD FUNCTION Cazacu et al. (2006) proposed a pressure-insensitive yield criterion that accounts for material anisotropy as well as yielding asymmetry between tension and compression associated with deformation twinning or non-Schmidt effects at single crystal level. This criterion can be applied to represent plastic behavior of various materials with FCC, BCC, and HCP structures and is defined as G ( 1 , 2 ,3 ,k , a) = 61 k 61 6 2 k 6 2 63 k 63 , a
a
a
(1)
To improve representation of the anisotropy, linear transformations are incorporated into the CPB06 criterion. The orthotropic yield criterion with two linear transformations, CPB06ex2 (Plunkett et al., 2008), is of the form:
F 6, 6 c, k,k c ,a = G ( 1 , 2 ,3 ,k , a) + G ( 1c ,c2 ,3c ,k c,a) = 61 k 61 6 2 k 6 2 63 k 63 61c k c61c 6c2 k c6c2 6c3 k c6c3 a
a
a
a
a
a
(2)
where a is the degree of homogeneity, and k and k are material parameters describing strength differential effects; 61 , 6 2 , 6 3 and 61c , 6c2 , 6c3 are the principal values of the tensors and c , respectively. These tensors are defined as The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 86-91 (2011); doi: 10.1063/1.3623596 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
86
c Cc : s
C:s ,
(3)
where s indicates the stress deviator while C and C’ are orthotropic fourth-order symmetric tensors. With respect to the axes of orthotropy, anisotropic coefficients are represented by
C
ªC11 «C « 12 «C13 « « 0 « 0 « ¬« 0
C12
C13
0
0
C 22
C23
0
0
C23 0
C33 0
0
0 0
0
0
C 44 0
0
0
0
C55 0
c ªC11 «C c « 12 «C13 c « « 0 « 0 « ¬« 0
0 º 0 »» 0 » », Cc 0 » 0 » » C66 ¼»
c C12
c C13
0
0
c C 22 c C 23
c C23 c C33
0
0 0
0
0
0 c C 44
0
0
0
0 c C55
0
0
0
0
0 º 0 »» 0 » ». 0 » 0 » » c ¼» C66
(4)
Note that when C = C and k = kƍ, CPB06ex2 yield criterion reduces to the CPB06 yield criterion (see Eq. (2)). For materials with cubic crystal structure for which the yield in tension is equal to the yield in compression, the effective stress according to CPB06ex2 becomes:
V eff
1
a
+ 2 + 3 + 1c + c2 + c3 a
a
a
a
a 1/a
(5)
c , are related to the in-plane properties of the sheet. Without The parameters Cij , Cijc , with i, j = 1…3 and C66 , C66
c = 1. The remaining Cij , Cijc with i, j = 1…3 (10 coefficients) and the loss in generality, we can set C11 = 1 and C11 c can be determined based on the experimental normalized flow stress values shear coefficients C66 and C66
VT
V T / V 0 and Lankford coefficients rT (T represents the angle between the uniaxial loading direction)
experimental biaxial tension and experimental rb. To simplify the equations, we introduce the following notations:
1 2 C11 - C12 - C13 , 3 1 2 = 2C12 -C22 -C23 , 3
1 =
3 =
1 =
1 -C11 + 2C12 - C13 3
(6)
1 -C12 +2C22 -C23 3 1 3 = -C13 +2C23 -C33 3
2 =
1 2C13 - C23 - C33 , 3
From Eq. (3), the only non-zero components of the transformed stress tenor 6are
xx = 1V xx + 1V yy , yy = 2V xx + 2V yy , zz = 3V xx + 3V yy and xy = C 66 xy .
(7)
Similarly, the components of the transformed tensor c are expressed in terms of the Cauchy stress components by relations similar to (7) using notations similar to (6) for defining )1c , )c2 , )c3 in terms of the anisotropy coefficients Cijc . The principal values of the transformed tensors and c are given by
1§ 1 = ¨ xx + yy + 2©
xx
2 · - yy + 4 2xy ¸ , ¹
2 1§ 2 = ¨ xx + yy xx - yy + 4 2xy 2© and 3 = zz c3 = czz .
·, ¸ ¹
1§ 1c = ¨ cxx +cyy + 2©
c
2 · -cyy + 4 cxy2 ¸ ¹
1§ c2 = ¨ cxx +cyy 2©
c
2 · -cyy + 4 cxy2 ¸ ¹
xx
xx
(8)
Let denote by VT the yield stress in a direction at angle T with the rolling direction x. According to the criterion, for cubic metals with no tension-compression asymmetry (see (5)):
87
1
° a + 2 a + 3 a + 1c a + c2 a + c3 a ½° a 0 ® 1 a a a a a a ¾ ¯° 1 + 2 + 3 + 1c + c2 + c3 ¿°
(9)
where V0 is the tensile yield stress in the rolling direction (i.e. for = 0$ ) and
§ + cos 2 + 1 + 2 sin 2 · 1 ¨ 1 2 ¸ 1 = 2 ¨¨ + - cos 2 + - sin 2 2 + 4 C 2sin 2 cos 2 ¸¸ 1 2 1 2 66 © ¹ 2 2 § + cos + 1 + 2 sin · 1 1 2 ¸ 2 = ¨ 2 ¨¨ - cos 2 + - sin 2 2 + 4 C 2sin 2 cos 2 ¸¸ 1 2 1 2 66 © ¹ 2 2 3 = 3cos + 3sin ,
(10)
and )1 , ) 2 , ) 3 and 0 region shown in Fig. 8: FLCs for (a) Zircaloy-4 and (b) Zirlo
97
0.3
CONCLUSIONS In this study we obtained theoretical forming limit diagram in order to predict the occurrence of local necking during the stamping process of fuel rod spacer grid. For Zircaloy-4 sheet, the predicted forming-limit strains based on Swift model with the Hill's criterion are in good agreement with the experimental results. For Zirlo sheet, the experimental FLC has a better agreement with the curve predicted by the Storen-Rice model and the Hosford yield criterion having the exponent a is 6. It would be beneficial to formability prediction in stamping process of fuel rod spacer grid.
ACKNOWLEDGMENTS This research was supported by National Nuclear R&D Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2010-0020425).
REFERENCES 1. S. P. Keeler and W. A. Backofen, ASM Transaction Quarterly 56, 25-48 (1963). 2. G. M. Goodwin, “Application of Strain Analysis to Sheet Metal Forming Problems in the Press Shop” in SAE Automotive Engineering Congress, Detroit, 1968, pp. 380-387. 3. T. B. Stoughton, International Journal of Mechanical Sciences 42, 1-27 (2000). 4. H. W. Swift, Journal of the Mechanics and Physics of Solids 1, 1-16 (1952). 5. R. Hill, Journal of the Mechanics and Physics of Solids 1, 19-30 (1952). 6. Z. Marciniak and K. Kuczynski, International Journal of Mechanical Sciences 9, 609-620 (1967). 7. S. Storen and J. R. Rice, Journal of the Mechanics and Physics of Solids 23, 421-441 (1975). 8. R. Hill, Proceedings of Royal Society of London A 193, 281-297 (1948). 9. W. F. Hosford, “On Yield Loci of Anisotropic Cubic Metals” in Seventh North American Metalworking Research Conference, Dearborn, Michigan, 1979, pp. 191-196.
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Evaluation of Experimental Forming Limit Curves and Investigation of Strain Rate Sensitivity for the Start of Local Necking Wolfram Volka and Pavel Horab a
Prof. Dr.-Ing. Wolfram Volk, Institute of Metal Forming and Casting (utg), Technische Universität München, Walther-Meißner Str. 4, 85748 Garching near Munich, Germany
b
Prof. Dr.-Ing.Pavel Hora, Institute of Virtual Manufacturing, ETH Zürich, Tannenstr. 3, 8092 Zurich, Switzerland Abstract. The failure prediction in sheet metal forming is typically realized by evaluating the so called forming limit curves (FLC). Up to now, the FLC determination was performed with failed specimens of Nakajima or Marciniak test setups. Standard methods determine the failure by considering the occurrence of cracking and do not consider the possibility of time continuous recording of the Nakajima test. Consequently forming limit curves which have been evaluated in such way are often “laboratory dependent” and deviate for identical materials significantly. This contribution presents an algorithm for a fully automatic and time-dependent determination of the beginning plastic instability based on physical effects. The algorithm is based on the evaluation of the strain distribution based on the displacement field which is evaluated by optical measurement and treated as a mesh of a finite element calculation. The critical deformation states are then defined by 2D-consideration of the strain distribution and their time derivatives using a numerical evaluation procedure for detecting the beginning of the localization. The effectiveness of the proposed algorithm will be presented for different materials used for the Numisheet’08 Benchmark-1 with Nakajima test. Additionally the effect of strain rate sensitivity on the beginning instability of local necking is discussed. It can be shown that the strain rate sensitivity is from major importance and should not be neglected for the forming simulation of sheet metal materials. Keywords: Sheet Metal Forming, Forming Limit Curves, Strain Rate Sensitivity. PACS: 62.20 Mechanical Properties of Solids
INTRODUCTION The prediction and evaluation of failure is one of the main tasks in sheet metal forming simulation in industrial practice. The current standard is a so-called a posteriori interpretation. The results of the simulation are strains and stresses and the calculated strains are compared in a post processing step with theoretical or experimental reference curves, the so-called Forming Limit Curves (FLC). The curves of experimental methods are normally obtained with Nakajima or Marciniak [1] tests and have to be accomplished under laboratory-like conditions. The ISO standardization is given in [1]. The established evaluation method of the ISO standardization is the intersection line method. Therein the failed specimen is analysed and the strains of beginning instability are calculated with a mathematical approach using the 2nd derivative of the thinning distribution. This method is robust but sometimes inaccurate because the strains of start of necking are re-calculated. With the development of photogrammetric equipment it is nowadays possible to record the whole experiment and identify directly the start of necking, see Figure 1. This procedure is called the time continuous evaluation method. Feldmann & Schatz [2] proposed to analyse directly the strain gradient around the necking zone. Volk [3] suggested the evaluation of the thinning gradient around the necking zone over time. This method is mechanically motivated, and the identification of the start of necking was realized by a so-called frequency diagram. Based on this approach Eberle et al. [4] and Eberle [5] made further examinations and additional proposals. Volk & Hora [6] summarized the ideas of [3], [4] and [5] in combination with a simple and fully automated evaluation algorithm. Merklein et al. [7] proposed the application of the 2nd derivative of the strain rate in an evaluation field around the necking zone. All of these time continuous evaluation methods are based on experimental series, where gridded samples are being deformed and simultaneously The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 99-106 (2011); doi: 10.1063/1.3623598 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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the position of the intersection of the grid lines is being recorded with photogrammetric methods (figure 1). Typical boundary conditions are punch velocity of 1 mm/s and recording frequency of 10 pictures per second.
FIGURE 1. Principle sketch and experimental specimen of Nakajima test equipment with camera system
In the present paper the basic ideas of [6] are used for the identification of the start of necking. With this evaluation method some additional investigations are presented. It will be shown, that for a 5182-Aluminum alloy, it is possible to see the local instabilities of the Portevin-Le-Chatelier effect beside the identification of the start of necking. Finally some basic assessments of the effect of strain rate sensitivity on the beginning instability are presented.
EVALUATION OF STRAINS AND STRAINRATE DISTRIBUTION Input Data and Finite Element Approach The only required input data for the new method is a full set of global coordinates of the grid line crosspoints, which can be handled like global coordinates of a finite element mesh, see Figure 2 and [3].
FIGURE 2. Interpretation of measurement grid as finite element mesh
Following this idea all interesting physical quantities (strains, strain rate etc.) can be calculated by using well established methods of FE theory. If the test data is obtained with a regular grid pattern the crosspoints of the grid can be directly used as the vertices of the FE mesh. If crosspoints of the optical measurement are missing the coordinates of the voids can be calculated by suitable mathematical methods, e.g. [3] or [4]. If stochastic spray structures are applied for the optical measurement an additional preparation step to generate the grid pattern before starting the evaluation process is necessary.
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Calculation of Strains For the determination of the forming limit strains a Total Lagrange description is used, where the initial mesh is taken as reference configuration Xk and the current picture as present configuration xk. The strains are evaluated using classical FEM 4-node membrane element theory, described for example by [8]. Following this the linear Cauchyian strain tensor can be described with:
1 2
ij ( H ij ( H ij )T )
(1)
where
H ij
ui N N J 1 i ( x j X j ) J 1 i u j x j
(2)
Therein ui is the displacement vector, J-1 is the inverse of the Jacobian and Ni the typical isoparametric functions [8]. The plane principal strains result from the characteristic equation with:
det( ij ij ) 0 1 2 1 2 ( 11 22 ( 11 22 ) 2 4 122 2
2 ( 11 22 ( 11 22 ) 2 4 122
(3)
Ultimately the Henckyian strain gauge is applied due to large strain components because of the Total Lagrange description:
1 ln(1 1 ) 2 ln(1 2 ) 3 (1 2 )
(4)
With these strain components, the points in the FLC are determined.
Calculation of Strain Rates For the determination of the strain rates and especially of the thinning rate, it is necessary to take the numerical derivative of the deformation gradient with respect to time into account. The determination of the deformation gradient tensor on the basis of the displacement gradient is performed as follows:
Fij ( I H ij )
(5)
Subsequently, with the derivative of the deformation gradient, the velocity gradient tensor is figured out and the deformation rate tensor can be computed
Lij
Fij
1 1 1 T Fij Fij Fij ; Dij ( Lij Lij ) t 2
101
(6)
Finally, the plane principal components of the deformation rate are evaluated by computing the eigenvalues D1 and D2 of the deformation rate tensor Dij. To calculate the necessary thinning rate, the deviatoric character of D is used:
: D3 ( D1 D2 )
(8)
The thinning rate is the main physical quantity to identify the picture with beginning instability and calculating the strain values for the entries of the FLC. The numerical derivative of the deformation gradient will be very un-accurate, if no smoothing over the time steps is applied. For those reasons in [3] a quadratic interpolation over time with typically seven time history points is proposed for the deformation gradient Fij(t), see Figure 3.
FIGURE 3. Determination of
Fij (t ) by using quadratic least square interpolation with seven time history points
Detection of the Forming Limit States From a continuum mechanical point of view, the plastic instability is the local concentration of the remaining plastic deformation in small (shear-) bands and a fall-back of the other areas in the elastic range, Figure 4.
FIGURE 4. Development of the localized necking by the reduction of the plastic zone to the shear band(s).
Hence, two effects can be noticed, namely the strain inside this band increases on the one hand and on the other hand it has to remain roughly constant outside. This is displayed in Figure 4. It is remarkable in that figure that the characteristics of thinning over time seem to grow relatively continuously, which can cause problems for an automatic detection of beginning plastic instability. In [3] – [6] it has been shown that the thinning rate is a suitable physical quantity to identify the start of necking for the time continuous evaluation method. The motivation can be also seen in Figure 4. The concentration of the remaining plastic deformation in small shear bands will lead to high thinning rates where the thinning rates outside the shear bands will stay nearly constant. The algorithm to detect the start of necking proposed by Volk & Hora [6] makes use of this effect. In a first step a sufficient number of grid points are identified which are sure in the necking area. Then, only for these points the mean value of thinning rates are examined over time. In Figure 5 the typical behavior is shown.
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FIGURE 5. Development of the localized necking by the reduction of the plastic zone to the shear band(s) [6].
At the beginning one can see the stable nearly homogeneous deformation and the localized necking as instable deformation at the end until the crack occurs. The stable and instable areas are fitted with two linear curves by using the least square method. The picture which is next to the intersection of these two straight lines will be defined as beginning instability, see Figure 5. This method was applied to the different materials of the Numisheet 2008 Benchmark problems [9] in two different laboratories. HC220YD, 0.8 mm
HC220YD, 1.6 mm
HC260LAD, 1.1 mm
FIGURE 6. Detection of the FLC. Comparison of different materials, laboratories and mesh sizes [6].
Visualization of Local Instabilities In addition to the evaluation of Forming Limit Curves the analysis of the thinning rates deliver interesting results. It is well known that some Aluminum alloys tends to transient local instabilities due to crystalline stick slip effects, the so-called Portevin-Le-Chatelier effect [10]. For the visualization of the effect a so-called two color plot is applied. In the evaluation field of 20 mm x 20 mm and a grid size of 1 mm the 25 elements with the highest thinning rate are marked with red rectangles. All other elements are displayed with blue diamonds. Therewith the concentration of plastic deformation in each time step can be visualized. The material is an AA5182 alloy from Numisheet 2008 Benchmark [9]. In Figure 7 the two color plots of a Nakajima 100 mm specimen with a punch velocity of 1 mm/s are displayed. The time step is set to zero for a punch height of 3 mm before the specimen failed. Interestingly one can see the jump of the highest thinning rates within the evaluation field for the different time steps. Finally, one of the transient localizations ends in the local necking and causes the failure of the Nakajima specimen.
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FIGURE 7. Visualization of transient local instabilities (Portevin-Le-Chatelier effect) for AA5182
EFFECT OF STRAIN RATE SENSITIVITY ON START OF LOCAL NECKING The strain rate sensitivity of materials influences significantly the material behavior for strain values larger the tensile strain. The explanation is the non-homogeneous deformation which causes also different strain rates. If a material has a positive strain rate sensitivity like most of the steel materials this leads to a regularization of the postcritical behavior for strains larger than the tensile strain. This effect can be easily examined by an artificial boundary value problem with the FE method, see Figure 8 and Volk & Charvet [11].
FIGURE 8. Boundary value problem of ideal plain strain test
The problem is now, that for simulations without strain rate sensitivity the finite elements become numerically instable before reaching strain values of the FLC. It can be seen that without strain rate sensitivity the failure prediction is too conservative in comparison to the experimental FLC, and feasible geometry proposals would have been rejected, see Figure 9.
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FIGURE 9. Simulation results without strain rate sensitivity
Even with a small strain rate sensitivity which can be lower than measured material parameters this effect nearly vanishes, see Figure 10.
FIGURE 10. Simulation results with strain rate sensitivity
It is recommended especially for dynamic explicit codes to always perform a simulation with small strain rate sensitivity if no accurate experimental values exist. In [11] it has been shown, that this strategy can be necessary for the simulation of complex geometries in automotive industry.
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FIGURE 11. Simulation results without (left) and with (right) strain rate sensitivity for an industrial application .
Additional assessments on strain rate sensitivity in combination with temperature dependence on the yield curve and material failure can be found in Hora [12].
CONCLUSIONS AND OUTLOOK The prediction of material failure is an essential task in sheet metal forming simulation. In the present paper the discussion of new approaches for the evaluation of experimental FLC are discussed. It has been shown that the thinning rates are the interesting physical quantity for the assessment of beginning instabilities. Beside an accurate determination of experimental FLC this concept can be used to demonstrate the appearance of transient local instabilities for an Aluminium alloy AA5182. It is emphasized, that the proposed method of detecting the start of necking is not restricted to any loading or strain path. The problem of non-proportional loadings with non-linear strain paths can also be evaluated with the presented algorithm. Finally the importance of the strain rate on start of necking and beginning material instability has been proven with a simple numerical boundary value problem.
REFERENCES 1. ISO copyright office, “Metallic materials -Sheet and strip – Determination of forming limit curves – Part 2: Determination of forming limit curves in laboratory” in ISO/DIS 12004-2, 2006. 2. Feldmann, P. and Schatz M., “Effective Evaluation of FLC-Tests with the Optical In-Process Strain Analysis System AUTOGRID®” in Proceedings of FLC-Zurich 06, edited by P. Hora, Zurich, 2006. 3. Volk, W.,“New Experimental and Numerical Approach in the Evaluation of the FLD with the FE-Method” in Proceedings of FLC-Zurich 06, edited by P. Hora, Zurich, 2006. 4. B. Eberle, W. Volk and P. Hora, “Automatic approach in the evaluation of the experimental FLC with a full 2d approach based on a time depending method” in NUMISHEET’08 - Proceedings of the 7th Int. Conf. and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes, Part A, pages 279-284, Switzerland, ISBN 978-3-909386-80-2. 5. B. Eberle, “Entwicklung eines robusten numerischen Verfahrens zur automatischen Bestimmung experimenteller FLCKurven“, Master thesis, Inst. of Virtual Manufacturing, ETH-Zurich, 2008. 6. Volk, W. and Hora P.,“New Algorithm for a Robust User-Independent Evaluation of Beginning Instability for the Experimental FLC Determination” in Int. J. Mater. Form., DOI 10.1007/s12289-010.1012-9, 2010. 7. M. Merklein, A. Kuppert, S. Mütze and A. Geffert, “New Time Dependent Method for Determination of Forming Limit Curves Applied to SZBS800” in Proc. of 50th Conference of IDDRG 2010 edited by R. Kolleck, Verlag der Technischen Universität Graz, p. 489-498, Graz 2010. 8. K.-J. Bathe, “Finite Element Procedures”, Springer Verlag (1995), ISBN 0133014584. 9. W. Volk, “Benchmark 1 – Virtual Forming Limit Curve”, NUMISHEET’08 - Benchmark Proceedings of the 7th Int. Conf. and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes, Part B, pages 3-42, Zurich 2008, ISBN 9783-909386-80-2. 10. A. Portevin and H. Le Chatelier, “Heat treatment of aluminium-copper alloys” in Transactions of the American Society of Steel Treating, Vol. 5, p. 457-478, 1924. 11. W. Volk and P. Charvet, “Virtual Engineering and Planning Process in Sheet Metal Forming” in Proceedings of the 7th European LS-Dyna Conference, Salzburg 2009. 12. P. Hora, “Modellierung des Kaltverfestigungsverhaltens von metallischen Werkstoffen” in Proceedings of MEFORM Conference, Freiberg 2011.
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1XPHULFDO9DOLGDWLRQRI$QDO\WLFDO%LD[LDO7UXH6WUHVV±7UXH 6WUDLQ&XUYHVIURPWKH%XOJH7HVW M. Vucetica, A. Bouguechaa, I. Peshekhodova, T. Götzea, T. Huininka, H. Friebeb, T. Möllerb and B.-A. Behrensa a
Institute of Metal Forming and Metal-Forming Machines (IFUM), Leibniz Universität Hannover An der Universität 2, 30823 Garbsen, Germany
[email protected] b
GOM mbH Mittelweg 7-8, 38106 Braunschweig, Germany E-mail:
[email protected], Web page: www.gom.com $EVWUDFW The present investigation deals with the validation of the experimentally obtained biaxial true stress - true strain curves of the HCT 780 C sheet material from the bulge test with the help of the FEA. Furthermore the investigation handles the consideration of the bending influence via the blank curvature evaluation with an optical measurement system Gom ARAMIS. .H\ZRUGV material characterization, bulge test, yield curve, FEA, optical strain measurement. 3$&6 02.70.Dh, 81.70.-q
,1752'8&7,21 Experimentally determined yield curves from tensile tests are useful only for values within the range of uniform elongation, due to the reason of the uniaxial deformation behavior of tensile specimens under tension load. Beyond the point of the uniform elongation a common approach is to extrapolate the yield curve using mathematical models after e.g. Ludwik, Nadai, Swift or Gosh. Due to used mathematical algorithms flow curve values are extrapolated differently with the result of varying curve shapes. A more accurate and reliable extrapolation of the yield curve beyond the uniform elongation is archived by using the bulge test which enables an experimental determination of the biaxial true stress – true strain curve at much larger strain values (figure 1). This paper discusses and analyzes the latest improvements in the determination of the biaxial yield curve utilizing the bulge test.
),*85( Flow curve from the tensile, its extrapolations Ludwik, Gosch and a biaxial stress strain curve of the bulge test for the HCT 780 C 1.5 mm
The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 107-114 (2011); doi: 10.1063/1.3623599 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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(;3(5,0(17$/,19(67,*$7,212)7+(%8/*(7(67 The bulge test is carried out with a tool, which consists of a blank holder, die and a fluid chamber. A circular blank is completely clamped with the lock bead between the die and blank holder. A bulge is formed by pressing a fluid against the blank until final fracture occurs. When the pressure of the fluid p, the radius of curvature , the local thickness t and the local strains from a bulge test are know, it is possible to determine the corresponding biaxial membrane stress using the equations given in figure 2 [3, 4, 5].
),*85( Test principle of the bulge test [5]
The bulge test at the Institute of Metal Forming and Metal Forming Machine (IFUM) is realized in a modular tool system in a hydraulic press. The hydraulic press can apply a blank holder force up to 2500 kN. The diameter d = 100 mm or d = 200 mm of the die can be modularly exchanged. A lock bead is realized to avoid any material flow during the test. Furthermore the tool is equipped with a measurement system in the middle of the fluid chamber which enables to measure the pressure of the fluid during the test. The pressure increase is realized with a hydraulic unit with a maximum pressure of 280 bars [5].
),*85( Modular tool system for the bulge test with a optical measurement system Gom ARAMIS
For the measurement of the curvature radius and the strain values the measuring system ARAMIS [6] was used. A typical sensor (two cameras and a light source) is shown in figure 3. To protect the measuring system from splashing oil at failure, a glass plate is in front of the lenses of the optical measuring system. The calibration of the system was done with inserted protection. A stochastic pattern was applied to the specimen's surface and synchronized stereo images of the pattern are recorded at different loading stages. Figure 4 shows the undeformed and the deformed situation for a specimen. In this case the image of the undeformed situation of the left camera is divided into a large amount of subsets (facets). The center of each facet is shown as a cross, five of the facets are shown as quadrangle. For each face the
108
corresponding area is calculated for the right camera also for all loading stages. Based on the well known geometry of the optical setup for each center of a facet (each cross in figure 4) a 3D point is calculated for all loading stages and for each point following result values are e.g. derived: 3D coordinates, displacements and speeds Surface strain tensor (strain x, -y, xy, major, minor, thinning) and strain rate Using an optical measurement system typically all results are derived from the surface and are typically directly used for the calculation of the yield curve.
),*85( Typical measured images: left side full image size, right side enlarged local area with mathematical subsets (facets) for the calculation of surface point.
&203$5,6212)(;3(5,0(17$/5(68/76 By defining a general rule to calculate biaxial true stress – true strain curves from bulge tests by a IDDRG working group a large number of influences were discussed. Comparing the results of two different test departments for the same material with the current state of the rule still some differences can be recognized. In figure 5 the results for HCT780C with an initial thickness of 1.5 mm are shown. Both departments have performed three similar tests to show their reproducibility. The continuous curves where calculated from a test device with a die diameter of 100 mm by Voest Alpine, the dashed curves with 200 mm were carried out at the IFUM. At a true strain value of 0.53 for the tests with 100 mm show true stress values around 1100 MPa, the tests with 200mm true stress values around 1065 MPa. This is a difference of approx. 3%.
),*85( True stress true strain curves for the HCT780C for different blank holder diameter (continuous =100 mm, dashed = 200 mm), (a) image with complete curve, (b) image with reduced range for the y-axis to point out the differences
For the calculation, the simplified equation for biaxial membrane stress was used as given in figure 2. An obvious difference between these two tests was the die diameter. This formula was defined based on the assumptions that the thickness is much smaller than the diameter. The results of the optical measuring system (strains, curvature and thickness) were calculated from the surface (outer layer) of the specimen and directly used for the simplified formula. If these assumptions are not valid, the influence of the material thickness has to be taken into account in more detail:
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x
Strains in the material are a mixture of stretching and bending influences. In thickness direction the strains of the outer layer are bigger than in the middle of the material or as the average over the thickness. This means that o the average strains are overestimated (true strain too large) o the actual thickness is underestimated (thickness strain too small = true stress too large) x The radius for the surface curvature Us is bigger than the radius of the middle layer Um (true stress too large) x The pressure affects are only at the inner surface of the material [9] (true stress too large) So typically in the standard procedure the calculated true stress values will be overestimated. All these single influences are leading to higher values. The compensation of the thickness influence is an obvious idea and can be used to explain the higher values for the smaller die diameter in figure 5.
)(02'(/62)7+(%8/*(7(676 To verify the experimental results and to better understand the stress and strain distribution in the blank, finite element simulations of the bulge test were carried out. Two 3D models were set up as shown in figure 6 with the help of the HyperMesh pre-processor from Altair. Similar to the experiment, two die diameters d = 100 mm and d = 200 mm were investigated while the other geometry and material parameters were kept constant. To reduce the computational costs, the die was modelled as a discrete rigid body. The deformable blank was meshed with linear full-integration elements with four elements along the z axis. Linear reduced-integration elements, which are usually used for metal plasticity problems, tend to underpredict bending stiffness due to hourglassing and were therefore not used for the present study [7]. Linear full integration elements are free from this drawback. Though leading to an overly stiff material behaviour in bending, these elements were found to be more suitable for the present investigation. According to the test set-up, the implemented boundary condition at the outer blank edge as well as the corresponding symmetry boundary conditions along the x and y axes were applied as shown in figure 6 (a). To simulate sheet bulging, an equally distributed pressure p was gradually applied to the blank bottom surface corresponding to the test conditions as presented exemplarily for d = 200 mm in figure 6 (b). The constant friction coefficient of 0.15 was assumed between the die and blank. The sheet material was modelled with the von Mises yield criterion, the associated flow rule, and isotropic hardening with the material data determined from the uniaxial tensile test carried out at Voest Alpine. Beyond the uniform elongation, the hardening curve was extrapolated with the Gosh approach [8]. The analysis was carried out with the explicit solver LS-Dyna from Livermore Software Technology Corporation.
),*85(Finite element model of the bulge test with the applied boundary conditions
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(;3(5,0(17$/$1'6,08/$7,215(68/76 To verify the hardening curve extrapolation from the uniaxial tensile test, the simulated dome height was compared with its experimental counterpart. The results presented in figure 7 show a good agreement between the simulation and experiment and prove a good extrapolation quality.
),*85(Simulated (a), (b) and experimental (c), (d) dome height
To assess the accuracy of the analytical evaluation of the biaxial stress strain curve, the so determined curve was compared with the equivalent stress vs. equivalent strain curve estimated numerically in the upper element at the pole centre. As it can be seen in figure 8, both in experiment and simulation the equivalent strain – equivalent stress curve in case of the smaller die diameter is higher than the one estimated for the larger die diameter. The steeper hardening for the smaller die diameter can be attributed to higher bending and stretching. Furthermore, the comparison of the experimental and simulated curves also shows higher discrepances between the two in the range of the equivalent plastic strain from 0 to approx. 0.15.
),*85(Simulated and experimental biaxial true strain – true stress curves
A further interesting issue obtained with the help of the numerical simulation is shown in figure 9. Here, the level of the equivalent stress in the top and bottom elements of the blank for two locations X and Y as a function of the test time is presented. As it can be seen, the top and bottom layer of the blank undergo different extent of plastic deformation. The discrepancies in the yield stress between the layers are more pronounced close to the die but gradually decrease with increasing pole curvature.
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),*85( Development of the equivalent stress in the upper and bottom layer of the blank during the test.
The level of the equivalent stress in the top and the bottom layer of the blank depending on the radial distance from the pole centre for the time frame t = 1/3 ttest is presented in figure 10. The figure shows that the differences in the equivalent stress through the blank thickness are considerably higher for the die diameter d of 100 mm. From the relative radial distance of approx. 0.4 (d = 100 mm) and approx. 0.9 (d = 200 mm) to 1.0 the bending effects cause the curves to deviate further from each other.
),*85(Equivalent stress in the upper and bottom layer of the blank depending on the radial distance from the pole centre.
The results of the analytical and numerical investigations conclude that a consideration of stretching and bending influences with the aid of the material thickness compensation in the analytical calculation of the biaxial stress strain curve of bulge test is necessary.
0DWKHPDWLFDOH[WHQVLRQIRUFDOFXODWLQJELD[LDOVWUHVVVWUDLQFXUYH By using the ARAMIS system for the optical measurement a material thickness compensation can be activated for the post-processing. This eliminates the bending influence in the strain determination by calculating the strain values and coordinates directly for the middle layer of the blank (engineer strains M1m, M2m and true strains H1m, H2m). By further investigations it was approved that the real actual thickness t is similar to a simplified computed thickness tm (based on the middle layer strains) and can be calculated by the following equation
t
1
³ 1 M (t ) 1 M (t ) dt
t0
1
| tm
2
t0
1 1 M1m 1 M 2 m
(1)
Therefore only the following assumptions must be valid: x Linear strain distribution (M1(t), M2(t)) in material thickness direction x The initial material thickness (t0) has to be known x The volume of the material does not change during the deformation For the determining the radius Um* of the curvature in the middle layer a first idea was the following equation:
U m*
Us
tm 2
(2)
However, a comparison with a real fit in the coordinates of the middle layer pointed out significant deviations. Due to the local thinning at the apex of the dome, equation (2) does not give reasonable results for the real Um for
112
higher true strain values. Figure 11 shows the principle difference between the real Um and Um*. In this figure the radius Us of the complete outer layer is printed with a constant value. In the case of a homogeneous thinning of the material (left side) equation (2) is valid. On the right side the case of a localized thinning at the apex of the dome is shown and Um is much smaller than Um* from the left side.
),*85( Real Um for the case of a localized thinning defined by a best fit in middle layer coordinates (right image side) in comparison to simple approach by equation (2), which is only valid for a homogenous thinning (left image side)
The equation for the biaxial membrane stress (figure 11) is based on the balance of forces introduced by the pressure on the inner surface and the tangential stresses in the material can be written as:
V 2S U t
p S U 2
(3) For taking the thickness influences into account this equation can be modified by the related different layer values:
V 2S U m t m
V
t · § p S ¨ Um m ¸ 2¹ ©
2
§ t · p U m ¨¨1 m ¸¸ © 2U m ¹ 2t m
2
(4)
This new equation was used to calculate a modified true stress true strain curve based on the same test data as used in figure 5. Figure 12 shows the change of the results for one sample of both tests (diameter 100 mm and 200 mm). Additionally the influence of a modification only for single values is shown (only strain of middle layer, radius of curvature determined for the middle layer or pressure on inner surface). As discussed above all influences are reducing the true stress values. In this case the calculation of the real radius Um of the curvature with the middle layer coordinates has the biggest influence and legitimates the additional effort. The sum of all influences compensated by equation (4) shows a significant reduction of the true stress values. At a true strain value of 0.53 for the tests with 100 mm the curve is reduced from 1100 MPa to 1035 MPa (approx. 6% reduced). For the tests with 200 mm the values at the end of the curve is reduced from 1065 MPa to 1035 MPa (approx 3% reduced). Because of the higher curvature, the compensation is stronger for the smaller die diameter.
),*85( True Stress true strain curves based on “surface data” with the overestimated values; with different compensations and the full compensation by using “middle layer data” with equation (4)
113
The comparison of the new results is shown in figure 13 for both diameters. The approach of reducing the true stress values lead to a better correlation of the curves for 100 mm and 200 mm diameter (thin continuous and dashed curves). The differences are small and at a true strain level of 0.53 there is no significant difference (the variation of the repetitions for the same diameter is larger than differences for 100 mm and 200 mm).
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Earing Prediction in Cup Drawing using the BBC2008 Yield &riterion Marko Vrha, Miroslav Halilovia, Bojan Starmana, Boris Štoka, Dan-Sorin Comsab, and Dorel Banabicb, a
Laboratory for Numerical Modelling & Simulation; Faculty of Mechanical Engineering, University of Ljubljana; Ljubljana – Slovenia b Technical University of Cluj Napoca, CERTETA Research Centre, 103-105 B-dul Muncii, 400641 Cluj Napoca, Romania Abstract. The paper deals with constitutive modelling of highly anisotropic sheet metals. It presents FEM based earing predictions in cup drawing simulation of highly anisotropic aluminium alloys where more than four ears occur. For that purpose the BBC2008 yield criterion, which is a plane-stress yield criterion formulated in the form of a finite series, is used. Thus defined criterion can be expanded to retain more or less terms, depending on the amount of given experimental data. In order to use the model in sheet metal forming simulations we have implemented it in a general purpose finite element code ABAQUS/Explicit via VUMAT subroutine, considering alternatively eight or sixteen parameters (8p and 16p version). For the integration of the constitutive model the explicit NICE (Next Increment Corrects Error) integration scheme has been used. Due to the scheme effectiveness the CPU time consumption for a simulation is comparable to the time consumption of built-in constitutive models. Two aluminium alloys, namely AA5042-H2 and AA2090-T3, have been used for a validation of the model. For both alloys the parameters of the BBC2008 model have been identified with a developed numerical procedure, based on a minimization of the developed cost function. For both materials, the predictions of the BBC2008 model prove to be in very good agreement with the experimental results. The flexibility and the accuracy of the model together with the identification and integration procedure guarantee the applicability of the BBC2008 yield criterion in industrial applications. Keywords: Anisotropy, Yield criteria, BBC2008 model, Earing prediction, Numerical integration, NICE scheme, Simulation. PACS: 46.35.+z
INTRODUCTION Numerical simulation of the manufacturing processes is an important way of reducing the overall production costs and the failure rate. During the last decades, this computational instrument has extended its applicability in various industrial fields. The earing phenomenon has been noticed in the early applications of the rolled metallic sheets for the deep-drawing of cylindrical parts. The number of ears is usually four, but there are also situations when six or eight ears may occur at the upper edge of the drawn cups. Which particular evolution of the earing process will take place depends generally on the anisotropy of the sheet metal and the lubrication conditions. The development of the numerical techniques during the last three decades allows the earing prediction by finite element simulation. The quantitative analyses related to the accuracy of the numerical results show that the constitutive models used in the simulation have a significant influence on the predicted earing profile. As a consequence, many researchers have focused their interest on the investigation of the possibilities to improve the quality of the numerical predictions by adopting more realistic material models. The anisotropic yield criterion proposed by Hill in 1948 [1] was extensively used for the simulation of the cup drawing processes with the aim of analyzing the earing profile. But, since earing prediction based on the Hill 1948 yield criterion proved to be rather limited, more sophisticated yield criteria have been developed [2-14] during the last two decades. Some of these material models have been implemented in commercial finite element codes and used for a prediction of the earing profile. The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 142-149 (2011); doi: 10.1063/1.3623604 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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As mentioned above, cylindrical cups obtained by deep-drawing usually exhibit four ears. In the case of some materials having a more specific anisotropy, six or even eight ears may develop during the drawing process. The experimental and theoretical studies have proved the existence of direct relationship between the number of ears and the variation of the anisotropy coefficient in the plane of the sheet metal. The occurrence of more than four ears can be predicted only by those yield criteria that use at least eight material parameters associated to different planar directions. In fact, such plasticity models have been developed mainly as a response to this challenge. An analysis of the number of mechanical parameters needed to correspondingly describe the anisotropy and their influence on the predicted earing profile has been performed by Soare and Banabic [15] using the Poly 8 yield criterion. The first yield criterion that uses general stress state and more than eight mechanical characteristics in the identification procedure has been developed by Barlat et al. [7]. The capability of this model to predict the occurrence of six or eight ears in the drawing of cylindrical cups has been proved by Yoon et al. in a series of papers [16-21]. Aiming at improving the accuracy of the earing predictions Yoon et al. [20] have taken in their simulations also the evolution of the anisotropy during the forming process into account. Comsa and Banabic [22] have recently proposed a simple plane-stress yield criterion (BBC 2008) defined as an extension of the BBC 2005 model [12]. As an alternative of using linear transformations of the stress tensor and absolute value function, the model is very convenient for finite element implementation. Due to the fact that in the identification procedure of the new constitutive model more than eight mechanical parameters can be used, a more accurate description of the anisotropy is possible. The capability of the BBC 2008 yield criterion to predict the occurrence of more than four ears will be proved in the next sections of this paper by performing corresponding numerical simulations of the cylindrical cup deep-drawing with two aluminium alloys considered.
BBC 2008 YIELD CRITERION The BBC 2008 yield criterion is a plane-stress criterion developed to describe the plastic behaviour of highly orthotropic sheet metals [22]. The yield surface defined by this model results from the implicit equation ) (V 11 , V 22 , V 12
V 21 , Y ) : V (V 11 , V 22 , V 12
V 21 ) Y
(1)
0
where V (V 11 , V 22 , V 12 V 21 ) t 0 is the equivalent stress (see below), Y ! 0 is the yield stress, while V 11 , V 22 , and V 12 V 21 are planar components of the stress tensor expressed in an orthonormal basis superimposed to the local axes of plastic orthotropy. One assumes that the third unit vector of the local basis is always perpendicular to the mid-surface of the sheet metal. In the particular case of the BBC 2008 yield criterion, the equivalent stress is defined as follows [22]:
V 2k w 1 L( r )
¦ ^w s
r 1
r 1
^ª¬ L
(r )
2k
M ( r ) º¼ ª¬ L( r ) M ( r ) º¼
2k
` w ^ª¬M sr
(r )
2k
N ( r ) º¼ ª¬ M ( r ) N ( r ) º¼
2k
``,
A1( r )V 11 A (2r )V 22 ,
M (r )
ª¬ m1( r )V 11 m2( r )V 22 º¼ ª¬ m3( r ) V 12 V 21 º¼ ,
N (r )
ª¬ n1( r )V 11 n2( r )V 22 º¼ ª¬ n3( r ) V 12 V 21 º¼ ,
2
k , s ` \ ^0` , w
2
2
(2)
2
(3 / 2)1/ s ! 1, A1( r ) , A (2r ) , m1( r ) , m2( r ) , m3( r ) , n1( r ) , n2( r ) , n3( r ) \.
The quantities denoted as k , A1( r ) , A (2r ) , m1( r ) , m2( r ) , m3( r ) , n1( r ) , n2( r ) and n3( r ) (r 1,! , s ) are material parameters. It is easily noticeable that the equivalent stress defined by Eq. (2) reduces to the isotropic formulation proposed by Barlat and Richmond [23] if A1( r ) A (2r ) m1( r ) m2( r ) m3( r ) n1( r ) n2( r ) n3( r ) 1/ 2, r 1,! , s. (3) Under these circumstances, the value of the integer exponent k may be adopted according to the crystallographic structure of the sheet metal, as in Barlat and Richmond’s model: k 3 for BCC alloys, and k 4 for FCC alloys. The other material parameters involved in Eq. (2) are evaluated upon a corresponding identification procedure. Their number is n p 8s , where s ` \ ^0` is the summation limit. Let ne be the number of experimental values
describing the plastic anisotropy of the sheet metal. If ne t 8, the summation limit s must be chosen according to the constraint s d ne / 8. When ne 8, the minimum value s 1 must be adopted. In this case, the identification
143
constraints obtained from experiments should be accompanied by at least 8 ne artificial conditions involving the material parameters. For example, if ne 6, one may enforce m1(1) n1(1) and m2(1) n2(1) . Due to the expandable structure of the yield criterion, many identification strategies can be devised. Comsa and Banabic developed a numerical procedure based on the minimisation of an error-function operating only with normalized yield stresses and r-coefficients obtained from uniaxial / biaxial tensile tests [22].
IMPLEMENTATION OF THE BBC 2008 YIELD CRITERION IN ABAQUS The above presented constitutive model has been implemented in a general purpose finite element code ABAQUS via VUMAT subroutine. For the integration of the constitutive model the NICE (Next Increment Corrects Error) explicit integration scheme, developed recently by some co-authors, is used. Its task is to find a proper increment of the plastic multiplier 'O from a given total strain increment 'H ij . The basic ideas of the NICE scheme are presented in Halilovic et al. [24], whereas in Vrh et al. [25] the reader can find its theoretical background adequately elaborated. The comparison studies and the proof given in the appendix of the latter paper show, that the accuracy of the new scheme is comparable to the accuracy of the backward-Euler scheme, while it is computationally far more (up to ten times) efficient in explicit dynamics simulations. Implementation of the constitutive model via user subroutine requires integration of the constitutive model along a known strain path, which is mathematically reflected in known total strain increments 'H ij . Although deduction of the NICE integration scheme is general, its implementation for shell applications needs a particular care. Namely, in order to satisfy the zero normal stress condition during the whole integration a throughthickness strain increment has to be adequately chosen in each integration step.
Treatment of zero normal stress constraint in shell applications General approach for calculation of the through-thickness strain increment is derived in Vrh et al [25]. In classical elasto-plastic plane stress applications the user of the NICE scheme can calculate the through-thickness strain increment with the following simple equation § · w) C33kl 'H kl* ¨ ) Cijkl 'H kl* ¸ 4E ¨ ¸ wV ij © ¹ (4) 'H 33 C3333 4 2 E
where 1
w) § · V ij ¸ ¨ ; i j 3 wV ij °0 Y w) w) w) w * ¸ , 4 w) C (5) Cijkl E ¨ ® ii 33 , H ij p H ¨ wV ij ¸ Y wV kl wY wH eq wV ii °¯ ij ; otherwise ¨ ¸ © ¹ and ) is the plastic potential (see Eq. (13)). In the above equations derivative w) / wV 33 is required which defines the influence of the through-thickness stress on yielding. The most elegant way to overcome this enigma is upgrading of the yield criterion with a simple substitution of the stress state using the deviatoric expression ) (V 11 , V 22 , V 12 , Y ) ) (V 11 V 33 , V 22 V 33 , V 12 , Y ) (6) Note that with such a manipulation the plane stress problem solution is not influenced since V 33 0 , whereas the plastic incompressibility condition w) / wV 11 w) / wV 22 w) / wV 33 0 (7) is preserved.
Integration of BBC 2008 with NICE scheme Assuming, that in each increment the total strain increments are available from the computed increments of the displacement field (note that strain increments 'H kl* are then given as input in VUMAT subroutine for further
144
processing, whereas 'H 33 can be calculated by considering (4)), the stress state at the end of the increment must be calculated using the following system of differential and algebraic equations:
) V ij , Y
'V ij
Cijkl 'H kl 'H klp
'H ijp
w) 'O wV ij
V kl 'H eqp 'Y
wY wH eqp
(8)
0
w) wV kl 'O Y
(9)
'H eqp
The plastic deformation is constrained by the consistency condition 0 , Eq. 8, which must be respected through all the integration process during the evolution of plastic strains. According to the NICE scheme, this is achieved by expanding the consistency condition into Taylor series, where higher order differentials are neglected. The numerical scheme is thus based, provided the values of the state variables are known at the beginning of the considered increment, on imposing d 0 (10) to be fulfilled in the considered increment, which leads to w w V ij Y 0 (11) wV ij wY With regard to the forward-Euler approach which uses a differential form of the consistency condition, i.e. d 0 , our approach considers the additional term . Though this term should be zero, because it represents a function whose value should obey the consistency condition 0 , numerically this is usually not true. This small difference between the two explicit schemes, NICE and forward-Euler, is the key reason for a considerable improvement in stability of the numerical integration. The remaining equations in the above DAE system, which accomplish the integration procedure, are the evolution equations, here expressed in the incremental form (9). Considering those equations and expanded consistency condition (11) yields the increment of the plastic multiplier w Cijkl H kl wV ij (12) 'O w V kl wV kl w w w dY Cijkl Y wV ij wV kl wY dH eqp With the increment of the plastic multiplier 'O calculated the incremental solution of the elasto-plastic model, defined by the evolution equations (9), can be considered solved. The numerical integration procedure requires a calculation of the yield function and its respective derivatives in every increment. For implementation purposes, the yield function can be rewritten in the following form: V 2 k V ij Vˆ V ij ) 1 0 (13) 1 Y 2k Y 2k with the derivatives used in the numerical scheme being: 1 w) Vˆ w) 2k 2 k 1 ; wY wVˆ Y 2 k Y (14) s ª r r r w) w) wV⏳ wL wV wM wV? wN º « » r wV ij wVˆ r 1 «¬ wL r wV ij wM r wV ij wN wV ij »¼ where
¦
145
wVˆ
r wL
wVˆ wN
r
ª r r 2k w 1 wr 1 « L M ¬
2 k 1
ª r r 2k w 1 ws r « M N ¬
r r L M
2 k 1
2 k 1 º
» ¼
r r M N
2 k 1 º
(15)
» ¼
2 k 1 2 k 1 º ½ r 1 ª r r r r L M °w « L M » ° ° ¬ ¼ ° 2k w 1 ® ¾ ° ws r ª M r N r 2 k 1 M r N r 2 k 1 º ° « »° ° ¬ ¼¿ ¯
wVˆ
r wM
and wL w r
r wM w
^A
M
r
r N
r
r
, A 2 , A1 A 2
1
1
wN w r M
r r
^M m , M m , M m m , m 2V ` ^N n , N n , N n n , n 2V `
r
1 r N
, 0`
r r 1
r r 1
r r
r
2
r r 2
r
r 2
r 2
r
1
r
r 3
1
r 3
12
12
(16)
r r m1 V 11 V 33 m2 V 22 V 33
r
n1
V11 V 33 n2 r V 22 V 33 w, w r
Equation (16) is due to simplicity written in Voigt notation, where
° w, r w, r w, r w, r ½° , , , ® ¾ for wV wV 22 wV 33 wV12 ° ¯° 11 ¿
r 1, !, s . Let us remind that according to the explicit approach all the state variables appearing in the above equations and expressions are written at the beginning of the considered increment. Once the increment of the plastic multiplier 'O is calculated, the respective increments of the other state variables can be readily calculated using Equations (9). The integration procedure is presented in Figure 1.
FIGURE 1. Integration (implementation) of constitutive model
146
EARING PREDICTION USING THE BBC2008 YIELD CRITERION In this section a validation of the presented approach is discussed, based on two performed round cup deep drawing simulations. Due to orthotropic material model only a quarter section of the cup with the corresponding symmetric boundary conditions applied is analyzed. A total of 2560 shell elements with reduced integration (ABAQUS S4R) and 21 section points through the sheet thickness are used for the simulation. To model the elastoplastic sheet metal response the BBC2008 model is employed. Its implementation with respective parameters identified for each of the sheet metals considered is enabled via user material subroutine VUMAT. The material parameters of the equivalent stress were calculated using experimental values of the normalized yield stresses and rcoefficients in uniaxial and biaxial tension. A detailed description of the BBC2008 identification procedure is given in [22].
Earing prediction for AA5042-H2 The simulation of deep drawing of aluminium AA5042-H2 follows the one, presented in Yoon et al. [20]. The tool geometry is as follows: blank diameter is 76.07mm, die opening diameter is 46.74 mm, punch diameter is 45.72 mm, die-profile radius and punch-profile radius are 2.28 mm. The initial thickness of the blank is 0.274 mm and holder force is 10 kN. Displacement of the punch is set to be large enough to pull the whole blank into the die. The friction is considered with the Coulomb model, the coefficient of friction being 0.008 for all contact surfaces. In this 18.416 H p
eq case the Voce hardening law Y 404.16 107.17 e is assumed to model the work hardening behaviour of the sheet metal. Material data (normalized yield stresses and r-coefficients) are given in [20], whereas the identified parameters of the BBC2008 model are given in Table 1.
TABLE 1. Parameters for 16 parameters BBC2008 version for AA5042-H2
k
s
w
A1(1)
A (1) 2
m1(1)
m2(1)
m3(1)
4
2
1.224744
0.719915
0.350253
0.573905
0.606047
0.718855 x 10-4
n1(1)
n2(1)
n3(1)
l1(2)
l2(2)
m1(2)
m2(2)
m3(2)
0.306974
0.492406
0.15355 x 10-5
0.564817
0.322344
0.374385
0.468435
0.737118
n1(2)
n2(2)
n3(2)
0.320818
0.641638
0.
Figure 2a shows the final geometry of deep drawn cup, as simulated, with the equivalent plastic strain field distribution displayed, whereas in Figure 2b a comparison between the predicted ears and the experimental ones [20] is given where h(D ) denotes the height of the deep drawn cup as a function of angle D (angle from the rolling direction). Also, for the sake of comparison, the calculated ears profile with the CPB06ex2 model [20] is included.
FIGURE 2. Earing prediction for aluminium AA5042-H2, a) Simulation b) Ears profile
147
Prediction of ears profile with the BBC2008 model is in good agreement with results of CPB06ex2 model, proposed by Yoon et al. [20]. The predicted ears are in same location, but are predicted to be little more intense. Eight ears are predicted by the model and agreement with experimental results is, at least qualitatively, very good.
Earing prediction for AA2090-T3 BBC2008 is in this section validated also on aluminium AA2090-T3. The round cup drawing experiment and a simulation with Yld2004 model is presented in Yoon et al. [18]. In this case the tool geometry is as follows: blank diameter is 158.76 mm, die opening diameter is 101.48 mm, punch diameter is 97.46 mm, die-profile radius and punch-profile radius are 12.70 mm. The initial thickness of blank is 1.6 mm and holder force is 22.2 kN. The friction is considered with the Coulomb model, the coefficient of friction being 0.1 for all contact surfaces. Hardening is modelled with the following stress-strain curve Y
646 0.025 H eqp
0.227
. The identified parameters of the
BBC2008 model for 8 and 16 parameters versions are given in Table 2 and Table 3, respectively. TABLE 2. Parameters for 8 parameters BBC2008 version for AA2090-T3
k
s
w
A1(1)
A (1) 2
m1(1)
m2(1)
m3(1)
n1(1)
n2(1)
n3(1)
4
1
1.500000
0.449938
0.513218
0.630315
0.601445
0.727299
0.153818
0.479391
0.499818
TABLE 3. Parameters for 16 parameters BBC2008 version for AA2090-T3
k
s
w
A1(1)
A (1) 2
m1(1)
m2(1)
m3(1)
n1(1)
n2(1)
4
2
1.22474
0.130866
0.621742
0.783422
0.660402
0.79 x 10-4
0.110991
0.048245
n3(1)
l1(2)
l2(2)
m1(2)
m2(2)
m3(2)
n1(2)
n2(2)
n3(2)
0.307522
1.033922
-0.071963
1.13 x 10-4
0. 77 x 10-4
0.538047
0.055764
1.018603
0.778150
Figure 3a displays the simulated final geometry of deep drawn cup (for 16 parameters model) and the corresponding equivalent plastic strain distribution, whereas in Figure 3b a comparison between predicted and experimental ears is given. Also, for the sake of comparison, ears profile calculated with the Yld2004 model [18] is included.
FIGURE 3. Earing prediction for aluminium AA2090-T3, a) Simulation b) Ears profile
Predictions of the BBC2008 model are in good agreement with a prediction of the Yld2004 model and also with the experiment [18]. As expected, the 8 parameters version was unable to predict six ears, which were experimentally observed. On the contrary, the 16 parameters version predicts 6 ears and their location, and at least qualitatively, the results are in good agreement with the experiment.
148
CONCLUSION In the paper the authors have shown an approach to modelling of plastic anisotropy in numerical simulation, which includes the BBC2008 plane-stress yield criterion, an efficient procedure for its implementation and a procedure for the identification of the respective model parameters. The constitutive model is derived in a form of a finite series that can be correspondingly expanded to retain more or less terms, depending on the available experimental data. The simulations of deep drawing of a round cup, which were performed with ABAQUS/Explicit with shell S4R elements, prove the capability of the model to predict more than four ears occurrence. Further, considering a comparison with the experimentally obtained ears profiles we can conclude that the presented approach is physically objective and is able to predict complex plastic anisotropic behaviour of sheet metal. Since the BBC2008 model does not use linear transformations of the stress tensor and absolute value function in the yield criterion formulation it is very convenient for implementation in finite element programs. Due to the fact, that it is in this paper explicitly integrated, the computational efficiency of the approach should be superior in the simulation of sheet metal forming processes comparing to existing techniques.
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Formability study of magnesium alloy AZ31B Z.G Liua, P. Lasneb, E. Massonia a CEMEF (Centre of Material Forming), Mines-Paristech 1, rue Claude Daunesse, BP 207, 06904, Sophia-Antipolis Cedex, France E-mail:
[email protected], elisabeth.massoni@ mines-paristech.fr b TRANSVALOR company 694, av du Dr Maurice Donat, 06255, Mougins Cedex, France E-mail:
[email protected] Abstract. The main aim of this paper is to study the formability of the AZ31B magnesium alloy at various temperature and strain rates. The tensile tests are performed to describe the rheological behavior of material, and the constitutive law is identified with Voce law [1], which contains a softening item. The law is proved effectiveness by fitting the equation with the experimental data. Nakazima experiments with hemispherical punch have been performed at CEMEF on a hydraulic testing machine. Six strain paths are selected by performing various sample geometries [2]. The Aramis© Optical strain measurement system has been used to obtain principle forming limit strain. The Forming Limit Diagram (FLD) is obtained by the critical point on the specimen surface at various temperatures. It is shown that the forming limit curve is higher at high temperature. Based on the Voce law model, finite element simulations of deep drawing test have been done with the commercial finite element code FORGE® in order to investigate the feasibility of hot stamping process for AZ31. In the simulation, the punch load and the thickness distributions have been studied. Meanwhile, the cross-shaped cup deep drawing simulations have been conducted with the data provided in the conference Website. The similar conclusion are obtained that the formability of AZ31 improve at high temperature and the simulation is effective in hot stamping processing. The study results are helpful for the application of the stamping technology for the magnesium alloy sheet [3]. Keywords: Formability, AZ31B Mg alloy, FLD, Finite element method. PACS: 62.20.F
1. INTRODUCTION Magnesium alloys are the lightest structural alloys at present, its density is about two-third that of aluminum alloy and one-fourth that of steel [4]. It has been widely used for structural components in the aerospace, automobile and civil industry in order to reduce fuel consumption. However the magnesium-alloy usually exhibits limited ductility at room temperature due to their hexagonal close-packed (HCP) crystal structure [5]. But the mechanical properties of magnesium-alloy can be improved at elevated temperatures. The AZ31 is one of magnesium-based alloy which include 3%Al and 1%Zn, and it has been reported that AZ31 sheets have good formability at elevated temperature between 150 and 3000C [6]. Deep drawing process is an important and wide-used process in assessment of formability of sheet metal, and it has been studied using the finite element method and experiments [4, 7, 8]. However, there are still few studies to profoundly investigate the forming limits of Mg sheet at elevated temperatures experimentally and theoretically. In the present study, the deformation behaviors of the AZ31B magnesium sheet are analyzed by tensile test at various temperatures and strain rates. The relationship between stress and strain is studied and an advanced constitutive model containing a softening item is developed to describe the deformation behavior of AZ31. The Nakazima experiments with hemisphere punch have been performed at various temperatures to obtain the experimental forming limit strains. Meanwhile, the finite element simulations of deep drawing test have been performed with the commercial finite element code FORGE®. In simulation, the punch load, thickness distribution and temperature distribution have been gotten, the forming limit strain have been obtained and compared with experiment.
The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 150-157 (2011); doi: 10.1063/1.3623605 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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2. THE MECHANICAL PROPERTY The material used in this study is a commercial AZ31B-O magnesium alloy sheet with thickness of 1.2mm. The average grain size is about 10µm. The tensile tests are selected to obtain the mechanical data of material, because it is the basic and simple test to study mechanical property of material in industry.
2.1 The tensile test The tensile specimens are made from the sheet with a gauge of 40mm length and 30mm width. The tensile tests are carried out in an Instron tensile machine between room temperature and 300 0C under three constant tensile velocities (0.04, 0.4, 4mm/s). Figure 1 shows the deformed tensile test specimens of AZ31 Magnesium alloy sheet at various temperature and velocities. It is shown that the elongation increase remarkably with increasing temperature and decreasing velocity. It is evident that the ductility is substantially enhanced.
(a) Specimens at various temperatures (v=0.4mm/s) (b) Specimens at various strain rate (T=3000C) FIGURE 1. Deformed specimens at different temperatures and strain rates
The true stress–true strain curves obtained at various temperatures and strain rates are shown in Figure 2. These curves are plotted up to appearance of the fracture phenomenon. It can be seen that the flow stress decreases with increasing temperature while the strain to fracture increases with increasing temperature, and the yield stress is higher at the lower strain rate. At room temperature, the strain hardening phenomenon is clear for each strain rate. But at higher temperatures, the flow stress is practically independent of strain, no significant work hardening is observed for any specimens. The softening phenomenon can be seen at high temperature, especially at 300 0C.
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2.2 The Equation for flow stress The Voce Law including a softening item is used to describe the flow behavior of AZ31. The equation is following [1]:
K m ,
K (1 W ) K ecr W * K sat
(1)
K ecr : Strain hardening function, K sat : saturation function, W : softening function For the AZ31 magnesium alloy sheet, it has been shown clear softening character, but the saturation stage is not evident. So, the saturation function K sat has not been considered. The power law has been used to describe the
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strain hardening. The softening phenomenon is different at various temperatures, so the polynomial expressions which depend on temperature have been used in softening function. The strain hardening and softening expressions are following: (2) K ecr k ( 0 ) n * exp( / T )
W 1 exp( r * ) , with r r0 r1 *T
(3)
Finally, the Voce law is following:
k ( 0 ) n * exp( / T ) * exp((r0 r1T ) * ) * ( m m *T ) 0
(4)
1
In order to get a proper global constitutive equation to describe the flow behavior of AZ31 magnesium alloy sheet at various temperatures and strain rate, the genetic algorithm (GA) in Matlab toolbox® has been used. The objective function is following: n
O( f )
(
exp i
i )2 cal
(5)
i
n
(
exp 2 i
)
i
n is the number of data point, i is the stress at the data point “i”. The genetic algorithm is sensitive to the initial value of parameters, the parameter value ranges have been set by fitting the individual stress-strain curves, and then getting the maximum and minimum value of parameters. Finally, according the Voce law and the setting objective function, the global optimal fitting parameters have been obtained (table 1). Table 1. optimal fitting parameters for Voce constitutive equation Parameters
k
n
r0
r1
m0
m1
0
Optimal evaluation
235.15
0.13787
86.343
-0.999
0.0107
0.062
0.00043
1.00E-06
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200
200
150 100 50 0 0.0
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strain
(a) =0.01s (b) =0.001s-1 FIGURE 3. Comparison between calculated and experimental flow curves at various temperatures.
The comparison between calculated and experimental flow curves at various temperatures has been shown in figure 3. It can be seen that the constitutive equation can describe correctly the flow properties of AZ31, especially the softening behaviors. From the optimal fitting equation, the sensitivity of process parameters can be analyzed. The choice of this law has undoubtedly improved the modeling of magnesium at every temperature, but it is clear that there is still some dissatisfaction, especially during the phase of softening. To do so would undoubtedly go to a finer coupling taking into account the evolution of microstructure.
3. EXPERIMENTAL FORMABILITY STUDY It is known that the Nakazima and the Marciniak tests are two common experimental tests which provide information on formability of sheet material. The main difference between these tests is the shape of the punch which is respectively hemispherical or flat. The Nakazima set-up has been initially selected for a straight-forward reason that it is much simpler to perform than Marciniak test, and it does not need any driving blank.
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3.1 Nakazima test set-up The Nakazima set-up is made of a hemispherical punch, a die, a blank-holder with drawbeads to prevent any sliding motion. A furnace has been installed close to the press so that the sheet can be transferred readily into the stamping device (figure 4). An AZ31 magnesium alloy sheet of 1.2mm thick is used for the experiment. The device and these tests have been performed at CEMEF on a tension-compression hydraulic testing machine Dartec (maximum 300mm/s punch velocity and 300kN punch load).The advantages of this set-up are to support high temperature and to allow a rapid insertion of the hot blank into the press (between the die and the blank holder) thanks to the pneumatic cylinder. In stamping test, a rectangular sheet specimen is clamped firmly, and stretched by a semi-spherical headed punch. In order to prevent wrinkle and keep the sheet surface smooth, the drawbead have been used in the blank holder zone. The blank holder plays an important role in stamping process, it keeps the material flowing smoothly and makes the blank and die contacting well. In this study, the constant blank holder force has been used.
(a) (b) Figure 4. set-up for Nakazima tests (a) and the temperature controller (b)
3.2 Experimental FLD The selected set-up has to allow varying the strain paths in order to get fracture in different strain states and strain history configurations: from a uniaxial tensile mode to pure bi-expansion mode. This test technique requires the use of different sample geometries and lubrication condition to generate all possible strain states. Each sample represents one specific strain path on the FLD [2]. In this study, the BN (Boron Nitride) lubricants have been used. Generally, two optical strain measurement systems, called ASAME© (automated strain analysis and measurement environment) and Aramis©, are used to measure forming limit strains. These measurement systems are widely used to analyze formability issues for many industrial applications. In this work, The Aramis strain measurement system has been used to obtain principal strain. Figure 5 shows the deformed samples after Nakazima tests and the strain distribution by Aramis. The samples have appeared necking or fracture phenomenon ultimately.
(a) Samples after test (b) Aramis strain measurement Figure 5. Samples after Nakazima tests and strain measurement
It has been shown that the localized necking of AZ31 is followed closely by the diffused necking at room temperature from stress-strain curves. But the plasticity is improved obviously with temperature increasing, the
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plastic deformation can still occur after diffused necking at high temperature, and then up to localized necking. So, generally, the forming limits diagram has three types: the safe FLD, the necking FLD and the broken FLD. The broken FLD of AZ31 has been shown in figure 6. It is illustrated that the strain paths of experimental FLC have discrepancies with theoretical FLC, because the strain measurement and lubricant is complicated in experiment. And the more important reason is a sudden change of strain occurs after necking, this phenomenon is obvious on the deformation images. It must be defined a certain safety margin if the broken FLD used in the actual application. The tensile tests have demonstrated that the AZ31 sheet still have some plasticity when necking occurs, especially at high temperature. So, the necking FLC is lower than the broken FLC in the forming limit diagram. This result is also published in the reference [3]. Nakazima_experiment 0.7 0.6
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Figure 6. FLD determined by Nakazima experiment
4. SIMULATION 4.1 hemisphere punch deep drawing The Nakazima stamping (hemispherical punch drawing) simulations are conducted at various temperatures. In these simulations, the geometry is just quarter model about the Z-axis due to axisymmetric deformation mode. The blank is meshed with two elements through the thickness. Smaller meshing is used in forming zone, and coarse meshing is used in flange area. A drawbead is used in order to compare with real Nakazima hot stamping test. The flow stress model of AZ31 sheet is based on Voce law with parameters identified and shown in table 1. The friction coefficient used in the simulations is obtained from reference [8] and it is assumed not to vary locally with temperature and pressure. The heat transfer coefficient is assumed to be uniform for all surfaces and the value is selected based on the results published in the literature [9]. The blank holder force also plays an important role in simulation. In this work, constant blank holder force is used. The inputs to the FE simulation are shown in table 2. Table 2. Process parameters used in simulation
Tooling setup Punch diameter (mm): 60 Blank-holder and die corner radius (mm): 10 Thickness (mm): 1.2 Friction coefficient, μ: 0.10
Mechanical properties Density (kg/dm3): 1.78 Young’s modulus, E (GPa): 45 Poisson’s ratio, ν: 0.3
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The punch loads obtained from the simulation at various temperatures are compared with experiments in figure 7. The maximum punch load obtained at all the simulated temperature is higher than the load in experiment. It is shown that the tendency is obvious, when the temperature increase, the punch load decrease. And the results match well at small displacement. Nevertheless, the discrepancies appear especially at large displacement. Many reasons could explain these deviations. Firstly, the friction coefficient is overestimated especially at low temperature. Secondly, the behavior model well represents the flow behavior at high temperature from stress-strain curves, but there is deviation at low temperature. Thirdly, in the simulation, the material is considered isotropic. However, magnesium alloy sheets exhibit some normal anisotropy, although it is not obvious. However, the best anisotropic yield criterion that represents yielding in Mg alloy may still have to be determined. 25
35 experiment,100deg
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(a) T=1000C (b) T=2000C (c) T=3000C FIGURE 7. Punch load obtained from simulation and experiment at various temperatures.
Thickness distribution from experiments is available at various temperatures from Aramis system. A comparison of the simulated and experimental results along a section of blank is shown in figure 8. The tendency of thickness distribution at 2000C predicted by simulation fit well with corresponding experiment result for all three geometry samples. However, less thickening and more thinning have been observed in the simulation comparing to the experiments. Maximum thinning has been observed in punch radius zone for both simulation and experiment. It is probably because the strength of the punch radius zone in warm forming is not uniform and depends on the temperature distribution. Exp. (w=200mm)
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FIGURE 8. Thickness distribution obtained from simulation and experiment at 2000C.
4.2 cross-shaped cup deep drawing The AZ31 cross-shaped cup deep drawing simulations are performed using FORGE software. The stress-strain data using uniaxial tension test and the tool setting are provided by conference committee. The surface temperature of the die and the blank-holder are 2500C, the surface temperature of punch and pad are 100 0C. The punch velocity is 0.15mm/s. the blank holder force increase linearly from 1.8KN to 4.8KN, the pad force increase linearly from 0 to
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4KN. The lubricant is barrierta L55/2. A cross-shaped cup sample and the simulation model are shown in figure 9. The simulation geometry is just quarter model about Z-axis thanks to axisymmetric deformation. The thickness of blank material is 0.5mm and meshed with two elements through the thickness. The Hill 48 anisotropic yield criterion is used to describe material anisotropy. The inputs to the FE simulation are shown in table 3. Table 3. Process parameters used in simulation
Mechanical properties Young’s modulus (GPa): 45 Poisson’s ratio, ν: 0.3 Friction coefficient, μ: 0.1 Density (kg/mm3): 1.770
Thermal properties Thermal conductivity (W/(m 0C): 96 Heat capacity (J/(kg 0C): 1000 Interface heat transfer coefficient (W/(m2 0C) : 4500
(a) A cross-shaped deep drawing cup (b) the quarter of model in FORGE FIGURE 9. A cross-shaped cup sample (a) and the quarter of simulation model (b)
The punch loads obtained from simulation are shown in figure 10(a). The punch loads have not much difference with various meshing size, and there are slight fluctuation at each punch force curve especially at high displacement. The possible reason is the temperature does not distribute uniformly during forming processing, the force distribution is fluctuated correspondingly. And other possible reasons are the boundary condition and contact which play an important role in and change during simulation. Figure 10(b) shows the temperature distribution at the punch displacement of 25mm. As the stamping is non-isothermal, the temperature distributes gradually along the wall. And similar phenomena are observed during all of processing, there are no obvious temperature changes at the punch radius and die radius zone. 16
Punch Load (KN)
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(a) Punch load and displacement curves (b) Temperature distribution at deep height 25mm FIGURE 10. Punch load and displacement curves at various meshing (a) and temperature distribution of the blank (b)
Thickness distributions are available in simulation at the punch displacement of 10mm and 25mm (figure 11). It is obviously shown that the thickening occurs at flange zone and thinning at cup wall zone. The maximum thinning is observed at the die corner radius for both punch displacement which is contrary to the conventional stamping where the maximum thinning is obtained at punch radius. It is probably due the strength of the cup wall in this stamping forming is not uniform as in conventional isothermal stamping. As in figure 10(b), the initial die and blank holder temperature (2500C) are higher than the punch and pad temperature (100 0C). Therefore, the yield strength of the material at the die corner radius is lower compared to the material at punch corner caused thinning in the cup
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wall. The thickness distribution is sensitive with the temperature, and the plasticity improves dramatically with temperature increasing.
(a) Deep height=10mm (b) Deep height=25mm FIGURE 11. Thickness distribution of the blank along the center line of X-axis
CONCLUSION AZ31 magnesium sheets exhibit higher yield stress and smaller elongation at room temperature, but its formability improve significantly with temperature increasing and strain rate decreasing. The tensile experiments have been performed to describe the flow behavior of AZ31 magnesium alloy sheet. The softening phenomenon is obvious at higher temperature. A new model containing softening items is developed based on mathematical fitting, and the genetic algorithm is used to get a global optimal equation. It is proved that this Voce law can fit the softening phenomenon well, but the discrepancies still exist and the model must be improved. The Nakazima hot stamping experiments have been performed to investigate the formability of material. The broken FLD are obtained using image correlation system. Although the strain paths of experimental FLC have discrepancies with theoretical FLC, It is clearly illustrated that the formability is better at higher temperature. The hot Nakazima simulation based on Voce law is performed with FORGE. From the comparison between simulation and experiment, the punch load predicted by simulation overestimate the experimental results, but the trends match well with experiment. The distribution of thickness thinning observed in the simulation match with experiment and more thinning are obtained in the simulation. Meanwhile, the cross-shaped cup drawing simulations have been studied using conference data. The similar conclusion are obtained that the formability of AZ31 improve at high temperature, the simulation is effective despite occurring dissatisfaction. And the study results are helpful to conduct the magnesium alloy sheet application in industry.
REFERENCES 1. A. Gavrus, E. Massoni, J.L. Chenot. An inverse analysis using a finite element model for identification of rheological parameters. Journal of Materials Processing Technology, 60 (1996) 447-454. 2. Ozturk F, Lee D. Analysis of forming limits using ductile fracture criteria. Journal of Mater Process Technology, 147 (2004) 397–404. 3. Fuh-Kuo Chen, Tyng-Bin Huang. Formability of stamping magnesium-alloy AZ31 sheets. Journal of Materials Processing Technology, 142 (2003) 643-647. 4. K.F. Zhang, D.L. Yin, D.Z. Wu. Formability of AZ31 magnesium alloy sheets at warm working conditions. International Journal of Machine Tools & Manufacture, 46 (2006) 1276–1280. 5. Yong Qi Cheng, Hui Zhang, Zhen Hua Chen, Kui Feng Xian. Flow stress equation of AZ31 magnesium alloy sheet during warm tensile deformation. Journal of materials processing technology, 208 (2008) 29-34. 6. E. Doege, K. Droder. Sheet metal forming of magnesium wrought alloys - formability and process technology. Journal of Materials Processing Technology, 115 (2001) 14-19. 7. S.H. Zhang, K. Zhang, Y.C. Xu, Z.T. Wang, Y. Xu, Z.G. Wang. Deep-drawing of magnesium alloy sheets at warm temperatures. Journal of Materials Processing Technology, 185 (2007) 147–151. 8. K. Droder. Analysis on forming of thin magnesium sheets. Ph.D.Dissertation, IFUM, University of Hanover, 1999. 9. S.L. Semiatin, E.W. Collings, V.E. Wood, T. Altan. Determination of the interface heat transfer coefficient for nonisothermal bulk forming process. Journal of Industrial Engineering, 109 (1987) 49–57.
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Crystal Plasticity Simulation of Forming Limit Strains for Fcc Polycrystalline Sheets with Different r-values Kengo Yoshidaa and Mitsutoshi Kurodaa a
Graduate School of Science and Engineering, Yamagata University, 4-3-16 Jonan, Yonezawa, Yamagata, 992-8510 Japan
Abstract. Plastic deformation characteristics and limit strains are simulated for textured face-centered cubic polycrystalline sheets using a generalized Taylor-type crystal plasticity model. The r-values are predicted to be 1.04, 7.74 and 0.17 for the random, {111} and {001} textures, respectively. The {111} texture gives limit strains as large as the random texture, whereas the {001} texture yields limit strains evidently higher than the other two even though its r-value is extremely low. Thus, the r-value cannot act as an indicator to the stretchability of sheet metals. For the {001} texture, a superior strain-hardening ability under plane-strain stretching mode is found to be responsible for the increase in the limit strains under plane-strain and equi-biaxial stretching modes. We conclude that the enhancement of the strain-hardening ability for plane strain mode is one of the key factors for high stretchability sheets. Keywords: Forming limit, r-value, stretchability, polycrystalline sheet, crystal plasticity PACS: 46.15.-x, 81.05.Bx
INTRODUCTION Improvement of formability of polycrystalline metal sheets, such as drawability, stretchability and bendability, by controlling microstructure is an attractive and considerable challenge. Intensive researches have been conducted to clarify an essential factor affecting the formability. Lankford et al. [1] experimentally investigated the formability in automotive fender draws for various low carbon steels having almost the same work-hardening exponent (nvalue), and demonstrated that the drawing performance has good correlation with the r-value, which is a ratio of width strain to thickness strain under uniaxial tension. The positive correlation between the r-value and drawability was again experimentally observed for various sheets, such as steels, aluminum alloys, copper and brass, which have the r-value ranging from 0.6 to 1.6 [2]. In addition to the experimental studies, a numerical simulation supporting this correlation has also been reported [3]. Therefore, the r-value is widely known to be a key parameter representing the drawability of sheet materials. The r-value has close connection with the preferred crystallographic orientations of polycrystalline metal [4], [5], [6], [7], [8]. Analyses based on a crystalline plasticity model revealed that the high r-value with small in-plane variation is obtained for the {111} and {111} orientations [4], [5], [7]. In a fully annealed low carbon steel sheet, the {111} texture, which includes the {111} and {111} orientations, is generated, and this results in high r-values. On the other hand, in a fully annealed aluminum alloy sheet, texture develops around {001} orientation. The r-value of this orientation is unity for the rolling and transverse directions and zero for 45 ° direction from the rolling direction. Thus, the r-value of aluminum alloy sheet is generally smaller than unity. It is known that the out of plane shear deformation could generate the shear texture characterized by the {001}, {111} and {111} orientations in aluminum alloy sheets [9], [10], [11]. Thus, the r-value could be increased by creating the strong shear texture. Asymmetric rolling, in which circumferential velocities of
The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 158-164 (2011); doi: 10.1063/1.3623606 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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the top and bottom rolls are different, has been developed in order to generate shear strain throughout the thickness and to improve the r-value. In the past decade, asymmetric rolling has been widely investigated [11] , [12] , [13]. It is often presumed, without evidential experimental data, that the high r-value provides good stretchability, which refers the formability in biaxial stretching mode, based on the knowledge of the significance of the r-value in the deep drawing. As is mentioned, the texture components that raise the r-value have been known, but the role of the r-value in the stretchability is not fully understood. Ayres et al. [14], for instance, concluded based on their experiments that the stretchability is not be related to the r-value. Sowerby and Duncan [15] carried out numerical studies on forming limit curves for sheets having different r-values by means of the Hill’s quadratic yield function[16] in conjunction with the Marciniak and Kuczynski (M-K) model [17], and showed that the limit strains decrease with increasing the r-value. On the other hand, Lian et al. [18] and Graf and Hosford [19] showed that the forming limit strains do not depend on the r-value, provided that a higher-order phenomenological yield function is employed in the analysis. Lian et al. [18] also demonstrated that a ratio of the major stress at plane strain stretching to that at equi-biaxial stretching on a yield surface can be an indicator that governs the limit strains in biaxial stretching range. Wu et al. [20] used a generalized Taylor-type model of polycrystalline plasticity for computations of forming limit curve of the cube texture ({001}), whose r-value is rather low. It was shown that the cube texture with some orientation scatter improves limit strains near equi-biaxial stretching mode. Yoshida et al. [21] reported that the forming limit curve for the cube texture is significantly increased when the sheet is stretched along 45 ° direction relative to the rolling direction, although the r-value in this direction is almost zero. The computation carried out by Yoshida et al. [21] shows that the {001} uvw texture, which has quite low r-values, enhances the forming limit curve for the full range of linear strain paths. These results show that the stretchability does not have any dependence on the r-value, and that conflict with the belief that a high r-value gives high stretchability. In the present study, plastic deformation characteristics and stretchability of textured aluminum alloy sheets are simulated based on a crystal plasticity model in order to re-examine the influence of the texture and the r-value on stretchability. For this goal, attention is directed to the {111} uvw and {001} uvw textures, because these texture give extreme cases of very high and low r-values, respectively. First, the plastic deformation characteristics such as the r-values, yield stresses and yield loci of these textures are illustrated using a generalized Taylor-type polycrystalline plasticity model. Then, limits to localized necking under plane-stress condition are computed by means of the M-K-type model. In the program of investigation, how the texture and r-value relate to the limit strains are discussed, and we clarify a key factor for superior stretchability of sheet metals other than the r-value. It is noted that the present paper is a part of our recent work [22].
PROCEDURE OF NUMERICAL SIMULATION
Crystal Plasticity Model A finite strain crystal plasticity model used here is along the lines presented in Peirce et al. [23]. The slip rate
( ) on the th slip system is given by a power law dependence on the resolved shear stress ( ) , 1/ m ( ) ( ) 0 sgn( ( ) ) ( ) g
(1)
where 0 is a reference slip rate, m is a strain rate sensitivity exponent, and g ( ) is a slip system hardness. The evolution law for g ( ) is specified by g ( ) h ( ) ,
h h h0 1 0 a 0n
n 1
,
a
t
0
( )
dt , dt
(2)
where 0 is the initial value of g ( ) , h0 is the initial slip hardening modulus, n is a power law hardening exponent, and t is time. An isotropic elasticity is assumed in the present applications, which is determined with the Young’s modulus E and Poisson’s ratio . As a model for a polycrystal, a Taylor type approach [24] is adopted. The deformation in each grain is taken to be identical to the macroscopic deformation of the aggregate. Taking the volume fraction of each grain to be identical, the macroscopic stress σ and macroscopic plastic strain rate D p are respectively obtained from averaging
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the Cauchy stress σ and plastic strain rate D p in each grain over the total number of grains. A macroscopic measure of plastic strain for a polycrystal is defined by t
eq (2 / 3)Dp : Dp dt .
(3)
0
Texture generation The procedure for generating textures is equivalent to one used in the previous studies [8] [21], [25]. To represent crystal orientations, we introduce Bunge’s Euler angles (1 , ,2 ) . Euler angles and 2 are set to be 54.7° and 2 45° for the {111} uvw texture and 0° and 2 0° for the {001} uvw texture, and 1 is randomly selected from the range between 0° and 90° . A misorientation from the ideal orientation is given such that misorientation angles follow a Gaussian distribution with the standard deviation 0 . In addition to these two textures, random texture is generated for comparison purposes. The three textures constructed by 2000 grains with 0 15 ° . These three textures conceptually have in-plane isotropic property. As a matter of fact, small in-plane anisotropy arises because the textures consist of a finite number of discrete crystallographic orientations.
RESULTS In-plane distribution of r-value and 0.2% proof stress In this subsection, the r-value and 0.2% proof stress are analyzed for the random, 111 uvw and 001 uvw textures by means of a generalized Taylor-type ppolycrystalline plasticity model described above. Material y parameters are taken to be E / 0 1400, 0.3,, 0 0.002,, m 0.002, h0 / 0 30, n 0.2 and 0 50MPa. Prescribed quantities for uniaxial tension are 111 0 , ij 0 for any other i, j, L11 0 (prescribed), and L21 L31 L32 0 . The six unknown values, 11 , L22 , L33 , L23 , L13 and L32 , are solved with the rate-form constitutive relationships in each increment. The x1 -, x2 - and x3 -axes are firstly coincide with the RD, TD and ND directions. When the uniaxial tension is applied along a ° direction from the RD, the textured sheet is rotated
° about x3 -axis and the boundary conditions described above are imposed. The direction of 0°ޓand 90° correspond to the uniaxial tension along the RD and TD, respectively. The r-value is defined by the ratio of plastic strain rates D22p / D33p at eq 0.1, and the 0.2% proof stress, denoted by 0.2 , is defined as a tensile flow stress 11 at eq 0.002. Figure 1 shows the in-plane distribution of r-value and 0.2% proof stress for the random, 111 uvw and 001 uvw textures, where values in the parenthesis are arithmetic averages of the predictions from 0 ° to 3.50
10 9
random (3.23)
8
3.25
6
{111} (7.74)
5
0.2 / 0
r-value
7 eq = 0.1
4
random (1.04)
3
{001} (0.17)
3.00
{111} (3.12) {001} (2.65)
2.75
2 1 0
0
15
30
45
60
75
2.50
90
0
15
30
45
60
75
90
(°) (a) (b) FIGURE 1. In-plane distribution of r-value (a) and 0.2% proof stress (b). is an angle between tensile and rolling directions. A value in parenthesis is an arithmetic average of predictions from 0 ° to 90 ° . (°)
160
90 ° . Since the three textures have conceptually in-plane isotropy, the computed r-values do not have apparent orientation dependence. The highest r-value of 7.74 is obtained for the 111 uvw texture, and the lowest r-value about 0.17 is predicted for the 001 uvw texture. The 111 uvw texture involves the 111 110 and 111 112 orientations, which are known to have high r-value, and the 001 uvw texture consists of the 001 001 orientation rotating about ND, whose r-values are in between zero and unity. Thus, Fig. 1 is consistent with the well-known crystallographic orientation dependence of the r-value. The r-values for the 111 uvw and 001 uvw textures are much larger and much less than the random texture, respectively. On the other hand, the 0.2% proof stress, shown in Fig. 1(b), is the highest for the random texture and those for the other two textures are lower. Both the textures lead to softer response than the random texture under the uniaxial tension.
Sheet necking analysis
Problem formulation of sheet necking analysis Sheet necking analysis is carried out based on the Marciniak–Kuczyński (M-K)-type model [17] in conjunction with the generalized Taylor-type model. Limit strains are expressed as a forming limit curve in strain space. A sheet specimen in the x1 x2 plane with an initial inhomogeneity of a reduced thickness band under the plane-stress condition is considered (Fig.2). The RD and TD axes have an angle I relative to the x1 and x2 directions, respectively. The same texture model is assigned to the regions both inside and outside the imperfection band. In all computations performed here, linear strain paths, L D 22 22 , (4) L11 D11 are assumed outside the band. The range of the strain ratio is taken to be -0.5 1. The other velocity gradient components outside the band are set to be L12 L21 L23 L32 L13 L31 0. g is judged byy the occurrence of a much higher principal strain rate inside the band The onset of localized necking than outside the band, 1b / 1 ! 105 , where 1b and 1 respectively denote the maximum principal values of the rate of deformation tensors inside and outside the band. Localization strains 11L for a certain strain ratio are computed for various initial band angles I with 5 ° interval. The minimum value of localization strains is defined as the forming * limit strain, denoted by 11* and 22 , and the corresponding initial band angle is defined as the critical initial band angle. The initial imperfection value, i.e., the ratio of the thickness inside the band to that outside the band, is taken to be 0.999 for all computations. Forming limit curves The material parameters used here are the same as ones in the previous section. Initial orientation of RD and TD is taken to be I 0° , and the initial band angle ranges from I 0° to 90° . Forming limit curves for the random, 111 uvw and 001 uvw textures are shown in Fig. 3. The forming limit strains of the 111 uvw texture are almost the same as those for the random texture over most strain range, with a local increase at 0.25. The 111 uvw texture exhibits extremely high r-value as shown in Fig. 1. The forming limit curve is, however,
FIGURE 2. Textured thin sheet with imperfection band initially inclined at angle I .
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almost the same as the random texture. In the case of 001 uvw texture, the limit strain is almost identical to that of the random texture at -0.5, while in the other strain range, the limit strains are clearly higher than the random texture. At 1 the limit strain is about double the others. The forming limit curve of the 001 uvw texture, exhibiting low r-value, is the highest among those three textures. The limit strains at 0 and 1 are definitely dependent on the texture, and the mechanism behind it is examined below. For the plane-strain stretching ( 0), the critical initial band angle is calculated to be 0° for all textures, thereby the deformation inside the band is also the plane-strain stretching. The strain-hardening ability of the material under this strain mode is one of the most influential factors that govern the strain level at the onset of localized necking. Instantaneous hardening rates for the plane-strain stretching mode ( 0) are shown in Fig. 4. Here, the instantaneous hardening rate H is obtained as a slope of the true stress-logarithmic strain curve, i.e., H d11 / d11 , and 0 is a flow stress 11 at eq 0.002 for each texture. The developments of hardening rate for the random and 111 uvw textures almost coincide with each other, and this leads to the same level of the forming limit strains at 0 as is seen in Fig. 3. The instantaneous hardening rate is clearly greater for the 001 uvw texture than for the other two after 11 0.1. The superior strain-hardening ability delays the occurrence of the localized neck. Therefore, the forming limit strain is increased for this texture. The high hardening 0.6 0.5 0.4
{001}
11*
0.3
{111}
0.2
random
0.1 0.0 -0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
22* FIGURE 3. Forming limit curve for the three textures 8
= 0
7 6
H / 0
5
{001}
4
random
3 2 1 0 0.0
{111} 0.1
0.2
0.3
0.4
0.5
"" FIGURE 4. Development of instantaneous hardening rate under plane-strain stretching mode. H d11 / d11 and 0 represents a flow stress 11 at eq 0.002.
162
rate of this texture is attributed to the geometrical hardening and this topic has already been discussed in detail by Yoshida et al. [25]. Next, the limit strain for the equi-biaxial stretching is considered. For the equi-biaxial stretching mode, localization strains 11L are identical for any initial band angles, provided that the material has in-plane isotropy. The textures considered here all have in-plane isotropy. (We neglect the in-plane anisotropy arose from the finite number of discrete orientations. As a matter of fact, the variation of localization strains 11L for the initial band angles between 0° and 90° is about 0.02 for the random and 111 uvw textures and is about 0.07 for the 001 uvw texture.) The investigation is proceeded considering the case that the initial band angle is I 0° . When the initial band angle is I 0° , only normal components of stress and strain arise inside the band and no shear components appear. The mode of straining inside the band is almost the equi-biaxial stretching at the beginning of loading, and it gradually changes toward the plane-strain stretching mode as the strain localization proceeds. Thus, the modes of strain outside and inside the band are not identical for the equi-biaxial stretching mode, unlike for the plane-strain mode considered above. The strain hardening characteristic is examined for the following two strain paths: (i) proportional strain path for the equi-biaxial stretching ( 1), (ii) non-proportional strain path in which the ratio of strain rate is unity at the beginning of loading program and then gradually changes to zero, that is, strain path changes form equi-biaxial stretching to plane strain stretching mode. The change in is described in Fig. 5(a). A resultant non-proportional strain path fairly coincides with the strain paths inside the band for the random and 111 uvw textures. Developments of instantaneous hardening rate for the three textures under the proportional and non-proportional strain paths are depicted in Fig. 5(b). The results for the proportional strain path are shown by thin lines. The development of hardening rate of the 001 uvw texture is almost the same as that of the random texture, and the hardening rate of the 111 uvw texture is lower than the other two. Thus, no systematic correlation between the degree of the hardening rate and the forming limit strain is seen. The thick lines indicate development of hardening rate for non-proportional strain path. For the case of the random and 111 uvw textures, the hardening rates for the non-proportional strain path are close to those for the proportional strain paths. This means that the strain hardening behaviors are almost the same for both inside and outside the band in the sheet necking analysis. On the other hand, for the 001 uvw texture, the instantaneous hardening rate becomes greater for the non-proportional strain path than for the proportional strain path as the strain path deviates from equi-biaxial stretching at 11 0.15. Thus, for this texture, an additional hardening occurs for the non-proportional strain path. In the sheet necking analysis, this increase in hardening rate can restrain the growth of a neck. Therefore, the strain localization has not predicted for the 001 uvw texture at the strain level where the strain localization occurred for the other two textures. The increase in the strain hardening rate consists of the superior strain hardening behavior for the plane strain stretching mode shown in Fig.4. Therefore, a key factor for the high formability is again found to be the strain 8
thick: non-proportional strain path thin: proportional strain path with = 1
7
1.0
0.5
4 3
1
0.15
11
{001}
2
0.33
0.0 0.0
random
5
H / 0
(=D22 / D11)
6
0 0.00
0.35
{111} 0.05
0.10
0.15
0.20
0.25
0.30
0.35
11
(a) (b) FIGURE 5. Variation of the ratio of strain rate in non-proportional strain path (a) and development of instantaneous hardening rate for proportional strain path in equi-biaxial stretching and non-proportional strain path described in (a). 0 represents a flow stress 11 at eq 0.002 for equi-biaxial stretching mode
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hardening ability under the plane strain mode.
CONCLUSIONS In this study the influence of the texture on the plastic deformation characteristics and stretchability has been investigated for the random, 111 uvw and 001 uvw textures. It has been confirmed that there is no positive correlation between the r-value and stretchability. Therefore, the high r-value is not a necessary condition for the high stretchability. The superior hardening behavior for plane strain mode yields high forming limit strains at the plane strain mode as well as at the equi-biaxial stretching mode for the 001 uvw texture. The enhancement of the strain-hardening ability for plane strain stretching mode is a key factor for high stretchability sheets.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.
W.T. Lankford, S.C. Snyder and J.A. Bauscher, Trans. ASM 42, 1197-1232 (1950). R.L. Whiteley, D.E. Wise and D.J. Blickwede, Sheet Metal Industries 38, 349-358 (1961). W.F. Hosford and R.M. Canddell, Metal Forming, Second ed., Prentice-Hall, 1993. P.H. Lequeu, P. Gilormini, F. Montheillet, B. Bacroix and J.J. Jonas, Acta Metall. 35, 1159-1174 (1987). P.H. Lequeu and J.J. Jonas, Metall. Trans. A 19, 105-120 (1988). F. Barlat and O. Richmond, Mater. Sci. Eng. 95, 15-29 (1987). J. Hu, K. Ikeda and T. Murakami, Mater. Trans. JIM 36, 1363-1371 (1995). M. Kuroda and S. Ikawa, Mater. Sci. Eng. A 385, 235-244 (2004). J.-H. Han, J.-Y. Suh, K.-K. Jee and J.-C. Lee, Mater. Sci. Eng. A 477, 107-120 (2008). O. Engler, M.-Y. Huh and C.N. Tomé, Metall. Mater. Trans. A 31, 2299-2315 (2000). K.-H. Kim and D.N. Lee, Acta Mater. 49, 2583-2595 (2001). H. Jin and D.J. Lloyd, Mater. Sci. Eng. A 465, 267-273 (2007). S.-B. Kang, B.-K. Min, H.-W. Kim, D.S. Wilkinson and J. Kang, Metall. Mater. Trans. A 36, 3141-3149 (2005). R.A. Ayres, W.G. Brazier and V.F. Sajewski, Appl. Metal Working 1, 41-49 (1979). R. Sowerby and J.L. Duncan, Int. J. Mech. Sci. 13, 217-229 (1971). R. Hill, Proc. Roy. Soc. London A193, 281-297 (1948). Z. Marciniak and K. Kuczy?ski, Int. J. Mech. Sci. 9, 609-620 (1967). J. Lian, F. Barlat and B. Baudelet, Int. J. Plasticity 5, 131-147 (1989). A. Graf and W. F. Hosford, Metall. Trans. A 24A, 2497-2501 (1993). P.D. Wu, S.R. MacEwen, D.J. Lloyd and K.W. Neale, Mater. Sci. Eng. A 364, 182-187 (2004). K. Yoshida, T. Ishizaka, M. Kuroda and S. Ikawa, Acta Mater. 55, 4499-4506 (2007). K. Yoshida and M. Kuroda, Numerical investigation on a key factor for superior stretchability for face-centered cubic polycrystalline sheets, in preparation. D. Peirce, R.J. Asaro and A. Needleman, Acta Metall. 31, 1951-1976 (1983). R. J. Asaro and A. Needleman, Acta Metall. 33, 923-953 (1985). K. Yoshida, Y. Tadano and M. Kuroda, Comp. Mater. Sci. 46, 459-468 (2009).
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Numerical Investigation into the Effect of Uniaxial and Biaxial Pre-Strain on Forming Limit Diagram of 5083 Aluminum Alloy F.Zhalehfar, S.J. Hosseinipour, S. Nourouzi and A.H. Gorji Metal Forming Research Group, Faculty of Mechanical Engineering, Babol Noshirvani University of Technology, Babol, Iran (
[email protected])
Abstract. In this paper, numerical simulations are carried out to determine the forming limit diagram (FLD) of 5083 aluminum alloy. The aim is to predict the effect of strain path change on the forming limit curve (FLC) of this alloy. For this purpose, out-of-plane formability test method with hemispherical punch was simulated by using commercial finite element software, ABAQUSE 6.9. In the first stage, square blanks were modeled and then some of them were prestrained uniaxially by tension test and some others were pre-strained biaxially by stretching over the hemispherical punch. In the second stage, the formability test specimens’ models were prepared by trimming the pre-strained blanks with the longitudinal axis parallel and perpendicular to the rolling direction. For trimming, a program was written in MATLAB 7.6 which could determine the new elements and introduce their properties to the FEM model. Ductile fracture criteria were used to predict failure. Furthermore, forming limit stress diagram (FLSD) was determined. The numerical results were compared with the experimental findings. Uniaxial pre-straining increased and shifted the FLC to the left hand side of the diagram for both parallel and perpendicular to the rolling direction. Biaxial pre-straining shifted the FLC to the right hand side of the diagram for both directions, and also decreased the FLC for the specimens parallel to the rolling direction. Keywords: Aluminum alloy 5083, FEM simulation, Forming limit diagram, Pre-Strain. PACS: 81.20Hy
INTRODUCTION Non-heat treatable 5XXX series aluminum alloys are used in various industries e.g. marine, TV towers and cryogenics. Because of their properties such as corrosion resistance as well as reasonable strength, ductility and weldability, they have primary potential for lightweight structural application in automotive and aerospace industries. It is believed that identification of forming capability in these alloys will enhance the potential for such applications. Forming limit diagram (FLD) is a graph of the major strain (1) versus the minor strain (2) at the onset of localized necking. However it is a well-known method to describe the formability of sheet metals, researches indicate that the FLD is dependent of the strain path. In many sheet metal forming industries, sheets are deformed under various dies and experience different strain paths. Therefore, it is important to investigate the influence of strain path change on the FLD in various alloys. Stoughton [1] showed that the forming limit for both proportional loading and non-proportional loading can be explained from a single criterion which is based on the state of stress rather than the state of strain. Graf and Hosford [2,3] studied the influence of strain path change on forming limits of AA2008 and AA6111-T4 by using uniaxial, plane strain and biaxial pre-strains parallel and perpendicular to the rolling direction. Dariani [4] investigated the effect of non proportional loading path on forming limit of St-12 steel alloy. Kuwabara et al [5] studied the effect of abrupt strain path change on the yield surface of AA6XXXT-4 by using a biaxial tensile testing machine. Kuroda and Tvergaard [6] considered the effect of different non-proportional strain paths on the FLD of anisotropic metal sheets. Reyes and Hopperstad [7] investigated numerically the effects of non-proportional loading on the Forming limit diagrams. Chow et al. [8] extended an anisotropic damage model for predicting FLDs under non-proportional loading.
The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 165-172 (2011); doi: 10.1063/1.3623607 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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In this paper, numerical simulations are carried out by using commercial finite element software, ABAQUSE 6.8 and programming in MATLAB 7.6 to predict the effect of strain path change on the forming limit curve (FLC) of 5083 aluminum alloy sheet. The numerical results are compared with experiments which discussed in the previous woks [9].
FINITE ELEMENT SIMULATION Simulations were performed by using commercial finite element software ABAQUS/explicit 6.9. The out-ofplane formability test (dome test) with a hemispherical punch was modeled to determining the forming limit diagram according to the Figure 1. It consists of three main tools, hemispherical punch, blank-holder and die with draw-bead. In this method, eight specimens with different geometries, as shown in Figure 2, are required to complete a forming limit diagram.
FIGURE 1. Out-of-plane formability test setup [10].
FIGURE 2. The geometry of the FLD specimens [10].
Due to symmetry, only one-quarter of the geometry was modeled, as shown in Figure 3. The tools were assumed rigid parts and the specimens were modeled as elastoplastic and meshed by S4R shell element. The size of the element 0.0015 m was obtained by comparison of different mesh size for the specimens. Commercial 5083 aluminum alloy sheet with 1 mm thickness was examined in this research. Tensile tests, according to the ASTM B557m, were carried out at 0, 45 and 90 degree to the rolling direction. The mechanical properties of the material are summarized in Table 1. The stress-strain curve was approximated by the Hollomon potential law. The anisotropic properties were described by the Hill quadratic yield criterion. The model was assumed to be rate independent. Coulomb friction law of 0.16 was assumed for contact surfaces [11,12].
166
FIGURE 3. FEM simulation model of out-of-plane formability test. TABLE (1). Mechanical properties of the material
Young’s modulus of elasticity (GPa) Poisson’s ratio Density (kg/m3) Yield strength (Mpa) (0.2%) Strength coefficient (MPa) Strain hardening coefficient (n) Normal anisotropy ( r )
70 0.3 3 27 00 28 4 52 2 0.1 1 0.7 4
For introducing biaxial pre-strain, in the first stage square blanks with 200 mm length in each dimension were stretched by the hemispherical punch to 8.5mm and 10.8 mm stroke. Then, the stretched blanks were trimmed according to the FLD specimens shown in Figure 2, with the longitudinal axis parallel (RD) and perpendicular (TD) to the rolling direction, as shown in Figure 4. In the second stage, these specimens were tested to obtaining the FLD.
FIGURE 4. The simulation sequence for biaxial pre-straining and FLD test.
167
For introducing near uniaxial pre-strains, in the first stage square blanks with 200 mm length in each dimension were notched with a special radius according to the Figure 5. These blanks were pre-strained uniaxially to 5.8 mm and 6.5 mm stroke. Then the FLD specimens were prepared by trimming these blanks according to the FLD specimens shown in Figure 2, with the longitudinal axis parallel (RD) and perpendicular (TD) to the rolling direction. In the second stage, these specimens were tested to obtaining the FLD. For trimming the pre-strained specimens, a program was written in MATLAB 7.6 which could determine the new elements and introduce their properties to the FEM model. Clift and Cockcroft ductile fracture criteria were used to predict localized necking [12-15]. The total procedure was summarized in the flowchart of Figure 6. Some experiments were carried out for comparison [9].
First stage of uniaxial tension Preparing FLD specimens by trimming Deforming the FLD specimens FIGURE 5. The simulation sequence for uniaxial pre-straining and FLD test.
168
Simulation of the Pre-straining stage of the initial blanks in ABAQUS
Take a report from the elements properties of the prestrained blanks and importing to MATALB
Preparing the FLD specimens by trimming the prestrained blanks in ABAQUS
Transferring the elements properties from the prestrained blanks to the FLD specimens in MATLAB
Building a new INP file in MATLAB and executing in ABAQUS
Simulation of the dome test stage of the FLD specimens in ABAQUS
Taking a report from the elements properties at each iteration of the dome test and importing to MATALB
Calculating the Clift et al and the Cockroft and Latham fracture criteria for whole elements at each iteration
Checking the elements which satisfy the fracture criteria earlier
Recording the strains and stresses of the elements.
FIGURE 6. Description of the total simulation procedure.
RESULTS AND DISCUSSION Figure 7 shows the experimental and numerical forming limit curves in as-received condition at rolling (RD) and transverse (TD) directions. It is seen that there are an acceptable agreement between the numerical results and experimental findings. The differences could be due to differences in predicting the necking (FLCN) and fracture (FLCF) [15,16]. The FLD0 is about 0.085 at RD and it is reduced to about 0.07 at TD.
169
(b)
(a)
FIGURE 7. The FLD in as-received condition, a) at RD, b) at TD.
Figure 8 shows the forming limit curves at RD and TD for biaxial pre-strained conditions. It is seen that the FLD0 is reduced to about 0.07 at RD and also is shifted to the right hand side of the diagram. The FLD0 is not changed significantly at TD, but it is shifted to the right hand side of the diagram.
(b)
(a)
FIGURE 8. The FLD in biaxial pre-strained condition, a) at RD, b) at TD.
Figure 9 shows the forming limit curves at RD and TD for uniaxial pre-strained conditions. It is seen that for RD the FLD0 is slightly increased and shifted to the left hand side of the diagram. The FLD0 for TD is increased significantly, and also shifted to the left hand side of the diagram. These results are similar to the experimental data of Graf and Hosford [2,3]. Figure 10 shows the FLSD diagram for biaxial pre-strained conditions at RD directions. It is seen that there is not any significant change between them. This indicates that the FLSD is independent of pre-straining effect and is suitable to use for non proportional loading path.
170
(b)
(a)
FIGURE 9. The FLD in unaxial pre-strained condition, a) at RD, b) at TD.
FIGURE 10. The FLDS in biaxial pre-strained condition at RD.
CONCLUSION The effect of biaxial and uniaxial pre-strains on the forming limit diagram of commercial 5083 aluminum alloy has been investigated numerically. Simulations have been carried out at two stage; pre-straining and dome test, by using commercial finite element software, ABAQUSE 6.9. Clift and Cockcroft ductile fracture criteria have been used to predict failure. MATLAB 7.6 has been used for preparing the FLD specimens from the pre-strained blanks and also for determining the ductile fracture criteria. Acceptable agreement between the numerical results and experimental findings has been achieved. The results showed that uniaxial pre-straining increased and shifted the FLC to the left hand side of the diagram for both parallel and perpendicular to the rolling direction. Biaxial prestraining shifted the FLC to the right hand side of the diagram for both directions, and also decreased the FLC for the specimens parallel to the rolling direction.
ACKNOWLEDGEMENT The authors would like to appreciate the office of the Vice President for Research of Babol Noshirvani University of Technology for its financial support.
171
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10. 11. 12. 13. 14. 15. 16.
T. B. Stoughton, Int. J. of Mech. Sci. 42, 1-27 (2000). A. Graf and W. Hosford, Int. J. Mech. Sci. 36, 897-910 (1994). A. Graf and W. Hosford, Metall. Trans. A 24A, 65-75 (1993). B. Dariani, " Effect of non proportional loading path on forming limit of sheet metals ", Ph.D. Thesis, Amirkabir University of Tech., 1999. T. Kuwabara, M. Kuroda, V. Tvergaard and K. Nomura, Acta Materialia 48, 2071-2079 (2000). M. Kuroda1 and V. Tvergaard, Int. J. of Mech. Sci. 42, 867-887 (2000). A. Reyes and O. S. Hopperstad, 7th European LS-DYNA conference 2009. C. L. Chow, L. G. Yu, W. H. Tai and M. Y. Demeri, Int. J. Mech. Sci. 43, 471-486 (2001). F. Zhalehfar, S. J. Hosseinipour, S. Nourouzi and A.H. Gorji, “Effect of Equal Biaxial Pre-Strain on Forming Limit Diagram of AA5083” in Advances in materials and processing Technologies-AMPT 2010, edited by F. Chinesta et al., AIP Conference Proceedings 1315, American Institute of Physics, 2010, pp. 353-358. F. Ozturk and D. Lee, J. Mater. Proce. Technol. 170, 247–253 (2005). K. S. Ragavan, Metall. Trans. A 26A, 2075–2084 (1995). T. Yoshida, T. Katayama and M. Usuda, J. Mater. Proce. Technol. 50, 226-237 (1995). M. G. Cockcroft and D. J. Latham, J. Inst. Met. 96, 33-39 (1968). S. E. Clift, P. Hartley, C. E. N. Sturgess and G. W. Rowe, Int. J. Mech. Sci. 32, 1-17 (1990). H. N. Han and K. H. Kim, J. Mater. Proce. Technol. 142, 231–238 (2003). F. Ozturk and D. Lee, J. Mater. Proce. Technol. 147, 397–404 (2004).
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A Comparison between Three Numerical Criteria for Prediction the Forming Limit Diagram of St14 Steel M. Moslemi, S.J. Hosseinipour, M.E. Hosseini and A.H. Gorji Metal Forming Research Group, Faculty of Mechanical Engineering, Babol Noshirvani University of Technology, Babol, Iran (
[email protected])
Abstract. In this paper, the forming limit diagram (FLD) of a low carbon steel St14 (DIN 1623) is investigated experimentally and numerically. The objective of this study is to find a numerical criterion which enables a simple and reliable determination of the FLD. For this purpose, Out-of-plane stretching test method with hemispherical punch was simulated by using commercial finite element software, ABAQUSE 6.9. One-quarter of the geometry was used due to symmetry. The material was modeled as elastoplastic and the anisotropic properties were described by the Hill quadratic yield criterion. The model was assumed to be rate independent. Coulomb friction law was defined for all contact surfaces. The simulation process was performed in two steps. In the first step the blank-holder moves down and deforms the blank into the draw-bead. Then the punch moves up at 20 increments and deforms the specimen to a specified displacement. Three numerical criteria including maximum second thickness strain derivative (CRIT1), equivalent plastic strain increment ratio (CRIT2), and total equivalent plastic strain ratio (CRIT3) were evaluated and the forming limit curve (FLC) were obtained. The predicted FLC were compared with experimental data. Unlike the CRIT2, the CRIT1 and CRIT3 were in good agreements with were experimental data. The CRIT1 criterion predicted the lower bound of the experimental results. While by using the CRIT3 criterion both the lower and upper bounds of the experimental results were predicted. Keywords: FEM simulation, Forming limit diagram, Numerical criteria, St14 steel. PACS: 81.20Hy
INTRODUCTION A forming limit diagram (FLD) defines the locus of the limit strains corresponding to various strain ratios. It is conventionally described as a curve in a plot of major strain vs. minor strain. It is a useful measure to evaluate the formability of sheet metal. As long as the principal strains are below the curve in the diagram, that region of the metal will be safe from necking and tearing [1-4]. One of the earliest experimental works on forming limit diagram is reported by Keeler and Backhofen [5]. They developed the right side of the FLD. Goodwin [6] extended this concept for the left side. Marciniak et al. [7] used an experimental method with flat cylindrical punch to determine all the strain ratios of the FLD. Nakazima et al. [8] and Hecker [9] introduced a similar test with a hemispherical punch. Generally, the experimental methods to determine FLD are categorized in two main types, in-plane stretching test using a flat cylindrical punch and out-of-plane stretching test using a hemispherical punch. Both techniques give nearly identical FLD, but the results from the hemispherical punch are slightly larger. Because of surface effect and friction, the out-of-plan test has been used much more extensively than the in-plan test [3, 10]. In practice, experimental determination of a FLD is very time consuming procedure and requires special expensive equipment. Finite element simulation is an alternative method to predict FLD. For numerical prediction of a FLD, the significant problem is the definition of the onset of localized necking under various sheet metal forming conditions. Many efforts have been spent on developing more accurate and reliable numerical procedure. For instance, Takuda et al. [11-13] utilized the ductile fracture criteria to prediction of forming limit of sheet metal with the finite element simulation of in-plane cylindrical test. Yoshida et al. [14] simulated the hemispherical punch stretching using an elastic-plastic three-dimensional finite element model. They predicted the limiting cup height and
The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 173-179 (2011); doi: 10.1063/1.3623608 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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rupture location for several materials. Ozturk and Lee [10] investigated the application of the ductile fracture criteria for prediction of the forming limit diagram of sheet metal with using finite element simulation of out-of-plane stretching test. Brun et al. [15] simulated the out-of-plan stretching test and predicted the localization by introducing the idea of the second time derivative of the thickness strain. Similar analysis was carried out on the out-of-plan stretching test by Geiger and Merklein [16]. They determined the FLD base on this concept that the materials necking are related to the gradient of major strain versus time. They described that the strain gradient during the forming process changes rapidly at the moment the localization occurs. Petek et al. [17] and Pepelnjak et al. [18] introduced an alternative method based on the Brun et al. [15] and Geiger and Merklein [16] concepts, which the second time derivative of the thickness strain is related to the single node or element and defines the time and area of localized necking. They simulated the Marciniak in-plane test with this methodology. In this paper, the forming limit diagram of a low carbon steel St14 (DIN 1623) is investigated experimentally and numerically. The objective of this study is to find a numerical criterion which enables a simple and reliable determination of the FLD. Simulation is performed by using the commercial finite element software ABAQUS 6.9. Different numerical procedures are used to predict the onset of localized necking. The accuracy of the predicted FLCs are compared with experiments.
EXPERIMENTAL PROCEDURE Commercial low carbon steel St14 (DIN 1623) sheet with 0.9 mm thickness was used. The mechanical properties of the material are shown in Table 1, which were determined by tensile tests according to ASTM A370. The plastic strain ratio (r-value) was evaluated in three directions with the tensile axis at 0, 45 and 90 degree to the rolling direction of the sheet according to ASTM E 517. Out-of-plane stretching test was used to determine forming limit diagram. Figure 1 shows the tooling and geometry. It is included of a hemispherical punch, a blank-holder, and a die with draw bead. In this test, various strain ratios are generated by forming specimens with different geometries. The dimension of each specimen is shown in Figure 2. The specimens were prepared in the rolling direction of the sheet metal and were marked by circular grids with 4.4 mm diameter, which were printed on the specimens by a gelatin stamp.
Direction 0 45 90
rm
TABLE (1). Mechanical properties of St14 mild steel sheet. Yield strength Tensile strength Elongation (MPa) (MPa) (%) 201 312 45 218 333 40 206 311 40
(r0 2r45 r90 ) / 4 1.51
Punch
Die
Blank-holder FIGURE 1. Out-of-plane stretching test tools.
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r-values 1.77 1.16 1.94
D 10 0
W 20
W 15
0
0 75 50 40
17
12
5
5
15
10
0
0
12
75
5 FIGURE 2. The specimens for out-of-plane stretching test [19].
The experiments were performed by DMG 600 kN capacity universal hydraulic press equipped with a computerized control system. The specimens were clamped by pushing the material in the draw bead, using a springloaded blank holder. The punch displacement rate was 10 mm/min and the tests were conducted up to the failure. After experiments, the circular grids on the specimens were deformed to elliptic shapes. Mayler tape was used to determine the major and minor limited strains by measuring the major and minor diameters of the ellipses that was located at the nearest distance to the failure zone.
FINITE ELEMENT SIMULATION Simulations were performed by using the commercial finite element software ABAQUS/explicit 6.9. Out-ofplane stretching test was modeled according to the experiments. One-quarter of the geometry was used due to symmetry. Figure 3 shows the simulation model of the die, punch head and blank-holder. The specimens were meshed by S4R shell element and to increase the accuracy, the mesh size was reduced from the edge to the center area of the specimens under the punch, as shown in Figure 4. The material properties which were used in the simulations are presented in Table 2. The material was modeled as elastoplastic where the stress-strain curve was approximated by Hollomon equation. The anisotropic properties were described by the Hill quadratic yield criterion. The model was assumed to be rate independent. Coulomb friction 0.16 was assumed for all contact surfaces. The simulation process was performed in two steps. In the first step the blank-holder moves down and deforms the blank into the draw bead with 40 KN force to prevent the specimen sliding between the blank holder and the die. Then the punch moves up in 20 increments and deforms the specimen to a specified displacement.
FIGURE 4. Finite element meshing of specimens
FIGURE 3. One-quarter model of out-of-plane stretching test.
TABLE (2). The material properties.
Young’s modulus of elasticity (GPa) Poisson’s ratio Density (kg/m3) Yield strength (Mpa) (0.2%) Strength coefficient (MPa) Strain hardening coefficient (n)
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210 0.3 7850 201 548 0.23
FAILURE CRITERION In the finite element simulation approach of determining the FLD it is necessary to detect at which element on the specimen and when the localized necking occurs. For this purpose, three numerical criteria are evaluated: maximum second thickness strain derivative (CRIT1), equivalent plastic strain increment ratio (CRIT2), and total equivalent plastic strain ratio (CRIT3). For CRIT1, according to Petek et al. [17] and Pepelnjak et al. [18], the localized necking was specified by tracing the thickness strain and its first and second derivatives versus time at thinnest nodes as shown in Figure 5 When necking appears, a sharp change in the strain behavior is observed on the curve. The second derivative of thickness strain versus time represents a peak in time which assumed as the onset of localized necking. Three nodes were selected for each specimen. The limit strains have been determined by evaluating the major and minor strains for the selected nodes. For CRIT2, according to Marciniak and Kuczynski (M–K) theoretical model [20], at first the equivalent plastic strain increment ( 'H ) for the nodes in the localized necking zone (zone A) and its adjacent zone (zone B) was determined (Figure 6). Then the ratio of equivalent plastic strain increment in zone A to zone B was calculated. The localized necking time was specified when the equivalent plastic strain increment ratio reaches to 10, ( 'H A / 'H B t 10 ) [21]. The major and minor strains in Zone B are noted as the limit strains. Three nodes were selected for each zone. CRIT3 is similar to the CRIT2, except that at first the total equivalent plastic strain ( H ) for the nodes in the localized necking zone (zone A) and the node in zone C was determined (Figure 6). Then the ratio of total equivalent plastic strain in zone A to zone C was calculated. The localized necking time was specified when the total equivalent plastic strain ratio reaches to 10, ( H A / H C t 10 ). The major and minor strains in Zone B (CRIT3-B) and zone C (CRIT3-C) are noted as the limit strains.
Thickness strain First derivative
0.0 -0.5
-3
Knee zone
-1.0 -1.5 -2.0
Second derivative
0 -6 -9 -12
Sharp change in strain rate
-15
600 300 0 -300 -600 -900 -1200 0.00
Onset of necking 0.25
0.50
0.75
1.00
Simulation time / Total time
FIGURE 5. Thickness strain as a function of time.
FIGURE 6. Finite element simulation after stretching
RESULTS AND DISCUSSION Figure 7 shows the specimens after stretching tests. The failure zone appears as a groove in the deformed region of the sheet metal. Figure 8 shows the FLD that obtained from experiments for St14 steel. A fourth order polynomial curve has been used for fitting the data. As seen the FLD0 (the plane-strain state) is about 0.25, which is near to the material strain hardening coefficient.
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FIGURE 7. The experimental specimens after stretching tests. 0.6
Major strain
0.5
0.4
0.3
0.2 0.1
Exp Data Poly Fit
0.0 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30
Minor strain
FIGURE 8. The experimental FLD of the St14 steel.
In Figure 9 the experimental specimens are compared with the simulation models. It is seen simulations are in good agreement with experiment.
FIGURE 9. Comparison of the experimental specimens and the simulation models.
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0.6
0.6
0.5
0.5
Major strain
Major strain
In Figure 10 the numerical results of the FLC for St14 steel are compared with the experimental data. Figure 10a shows the numerical criterion CRIT1. It indicates that the overall shape of curve based on the numerical points is in good agreement with the experimental curve. But it seems the CRIT1 criterion shows the lower bond of the experimental results. This numerical approach predicted the onset of necking on the specimen. It is noted that finding the critical nodes and determination of the FLC based on CRIT1 criterion is complicated. In Figure 10b the predicted FLC based on CRIT2 criterion is compared with the experimental data. It indicates that the overall shape of curve based on the simulation is not in good agreement with the experimental curve. The numerical obtained points of the FLC based on CRIT3 criterion are shown in Figure 10c. It indicates that the overall shape of curve based on the numerical points is in good agreement with the experimental curve. The CRIT3-C criterion shows the lower bond of the experimental points and the CRIT3-B criterion shows the upper bond of the experimental points. It seems the lower line on the diagram represents the localization, whereas the upper line represents fracture limit curve. It is noted that the calculation in this criterion is simple.
0.4
0.3
0.4
0.3
0.2
0.2
0.1
0.1
Exp Data FEM CRIT1
Exp Data FEM CRIT2
0.0 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30
0.0 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30
Minor strain
Minor strain
(a)
(b) 0.6
Major strain
0.5
0.4
0.3
0.2
Exp Data FEM CRIT3-B FEM CRIT3-C
0.1
0.0 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30
Minor strain
(c) FIGURE 10. Comparison of the experiment data with the predicted FLC based on;
a) CRIT1 criterion, b) CRIT2 criterion and c) CRIT3 criterion
CONCLUSION The forming limit diagram (FLD) of a low carbon steel St14 (DIN 1623) is investigated experimentally and numerically. Out-of-plane stretching test method was simulated by using commercial finite element software, ABAQUSE 6.9. Three numerical criteria including maximum second thickness strain derivative (CRIT1), equivalent plastic strain increment ratio (CRIT2), and total equivalent plastic strain ratio (CRIT3) were evaluated and the forming limit curve (FLC) were obtained. While the CRIT1 criterion was in good agreement with experimental data,
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it predicted the onset of necking on the specimen and its determination was complicated. The CRIT2 criterion was not in good agreement with experimental data. By using the CRIT3 criterion both the lower and upper bounds of the experimental results were predicted. Since this numerical approach for determination of the FLC is simple and the numerical curves were placed in the critical area of the experimental points, it seems it can be used for determination of the material formability.
ACKNOWLEDGEMENT The authors would like to appreciate the office of the Vice President for Research of Babol Noshirvani University of Technology for its financial support.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
T. B. Stoughton, Int. J. of Mech. Sci. 42, 1-27 (2000). T. B. Stoughton and X. Zhu, Int. J. of Plasticity 20, 1463-1486 (2004). V. M. Nandedkar and K. Narasimhan, J. of Mater. Process. Technol. 89, 24-29 (1999). P. D. Wu, A. Graf, S. R. MacEwen, D. J. Lloyd, M. Jain and K.W. Neale, Int. J. of Solids and Structures 42, 2225–2241 (2005) S.P. Keeler and W.A. Backhofen, ASM Transactions Quarterly 56, 25-48 (1964). G. M. Goodwin, SAE paper No. 680093 (1968). Z. Marciniak, K. Kuczynski and T. Pokora, Int. J. Mech. Sci. 15, 789-805 (1973). K. Nakazima, T. Kikuma and K. Asuka, Yawata Technical report 264, 678-680 (1971). S. S. Hecker, Sheet Met. Ind. 52, 671-676 (1975). F. Ozturk and D. Lee, J. of Mater. Process. Technol. 147, 397–404 (2004). H. Takuda, K. Mori, H. Fujimoto and N. Hatta, J. Mater. Process. Technol. 60, 291–296 (1996). H. Takuda, K. Mori and N. Hatta, J. Mater. Process. Technol. 95, 116–121 (1999). H. Takuda, K. Mori, N. Takakura and K. Yamaguchi, Int. J. Mech. Sci. 42, 785–798 (2000). T. Yoshida, T. Katayama and M. Usuda, J. Mater. Process. Technol. 50, 226–237 (1995). R. Brun, A. Chambard, M. Lai and P. de Luca, Proc. NUMISHEET 99, Besançon, France 393-398 (1999). M. Geiger and M. Merklein, Annals of the CIRP 52, 1-9 (2003). A. Petek, T. Pepelnjak and K. Kuzman, J. of Mech. Eng. Slovenia 51, 330-345 (2005). T. Pepelnjak, A. Petek and K. Kuzman, Advanced Material Research 6, 697-704 (2005). B. Dariani, " Effect of non proportional loading path on forming limit of sheet metals ", Ph.D. Thesis, Amirkabir University of Tech., 1999. C. Zhang, L. Leotoing, D. Guines and E. Ragneau, J. of Mater. Process. Technol. 209, 3849–3858 (2009) W. F. Hosford and R. M. Caddell, Metal Forming; Mechanics and Metallurgy, 3rd Edition, Cambridge university press, 2007.
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Possibility of Expressing Anisotropic Yield Functions of Metals using the Invariants of Stress Tensor M.H.Parsa1, K. A. Shirvanedeh2 and P. H. Matin3 1
Professor, School of Metallurgy and Materials Engineering, University College of Engineering, University of Tehran, Iran, P.O. Box 11155/4563,
[email protected] 2 Graduate Student, School of Metallurgy and Materials Engineering, University College of Engineering, University of Tehran, Iran, P.O. Box 11155/4563,
[email protected] 3 Assistant Professor, Department of Engineering and Aviation Sciences, University of Maryland Eastern Shore, Princess Anne, MD 21853, USA,
[email protected] Abstract. Most of existing yield function have been developed base on the curve fitting of experimental data and also some theoretical effort have been done for deriving yield functions. Framework of invariant theory prepared valuable opportunity for modeling non-linear constitutive equation that it is shown in the works of Rivlin. It has been tried using invariant theory for representation of yield function for isotropic material and it is shown that invariant theory could cover all yield functions needed for isotropic metals using a general equation. Therefore, in the present paper it has been tried to extend invariant theory for anisotropic materials. Results have shown that invariant theory have the capacity for deriving a general equation for representation of anisotropic yield function for orthotropic materials. Keywords: Yield function, invariant theory, anisotropy PACS: 62.20.fq
INTRODUCTION Development of metal forming technology accompanies of fast progress in using Finite Element Method (FEM). One of the most important parameters which affect the accuracy of FEM is the yield criterion. This criterion defined the boundary between elastic and plastic behaviors of metals which defines a closed surface in the stress space. Therefore defining the yield surface is one of the main aspects of plastic behavior of metals which should be described appropriately. One of main features of metals is anisotropic behavior due to their structures. This anisotropic behavior is very important in the case of sheet metals and it also affects the yield function. There are many anisotropic yield functions which are proposed by different researchers. First anisotropic yield function has been proposed by Hill [1] in 1948. This criterion was extension of Von Mises isotropic yield function. Simplicity of Hill’s quadratic yield function caused to make it as most popular anisotropic yield function. It has one major problem. This yield function cannot predict anomalous behaviors often observed in aluminum alloy sheets. Therefore Hill tried to improve his criterion [2-4]. Hosford [5] has been used another approach for developing phenomenological yield function that was based on polycrystal aggregate calculations. Hosford’s anisotropic yield function turns out to be a base for other developments in anisotropic yield functions. In recent two decades many different kind of yield functions have been developed by researchers. Barlat and co-workers had major role in extending Hosford yield function and they proposed series of yield functions for anisotropic sheet metals [6-9]. Majority of yield functions are proposed for orthotropic sheet metals. Most of mentioned yield functions have proposed base on the experimental observations and in order to fit the experimental results. In an attempt to express the yield function of metal based on the mathematical foundation, authors have been tried to use invariant theory. Ronald Rivlin was pioneer of using invariant theory in solid mechanics of isotropic and anisotropic materials. In 1940s, Rivlin established the basic of the finite elasticity for nonlinear continuum mechanics by invariant theory for isotropic materials [10]. Also Ericksen and Rivlin [11] for first time studied strain energy function for transversely isotropic materials. In the previous work, authors showed that invariant theory could be used for representation of yield surface of isotropic metals [12]. In the present work the ability of invariant theory for expressing yield surface for orthotropic metals are investigated. The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 180-186 (2011); doi: 10.1063/1.3623609 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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Deriving framework for orthotropic sheets yield function In a Cartesian coordinate defined by x, y and z directions, an orthotropic anisotropy has been considered. In such case, the material symmetry Group S can be shown by: S={±I, S 1 , S 2 , S 3 }
(1)
Where I is Identity matrix and S 1 , S 2 , and S 3 are the reflections with respect to the basic planes consisting of (y, z), (z,x), and (x,y) directions of orthonormal frame 0f x, y and z. Two series of symmetric second order tensors are needed for defining the framework of yield surface of anisotropic media using invariant theory. In First step it is necessary to find structural tensors that account for the material internal structure symmetries. In the second step, tensors that are corresponding for plastic behaviors of material are defined. Three symmetric second order tensors have been considered as structural tensors [13]: 1 0 0 0 0 0 0 0 0 𝑀1 = 0 0 0, 𝑀2 = 0 1 0, and 𝑀3 = 0 0 0 0 0 0 0 0 0 0 0 1
(2)
Selection appropriate stress tensor is very important for representing mechanical behaviors of material such as yield function. Two stress tensors, stress tensor in the principle direction and deviatoric stress tensor, have been chosen for investigating the possibility expressing anisotropic yield function using invariant theory. These two symmetric second order tensors are as shown in equation (3) and (4). 𝜎11 𝜎𝑖𝑗 = � 0 0 ⎛ 𝜎𝚤𝚥́ = ⎜ ⎝
0 𝜎22 0
0 0 � 𝜎33
(2𝜎11 − 𝜎22 − 𝜎33 )� 3 0 0
(3)
0
(2𝜎22 − 𝜎11 − 𝜎33 )� 3 0
0
0
(4) ⎞ ⎟
(2𝜎33 − 𝜎11 − 𝜎22 )� 3⎠
Where subscripts 1~3 are referred to the directions of the Cartesian coordinates. Integrity bases are essential part for establishing new formulation of yield function using invariant theory. Most of integrity base for different kind of tensors are established and they can be found in handbooks [13]. Integrity bases of an arbitrary symmetric second order tenor A that is tried to use in yield function expressed in polynomial form considering orthotropic anisotropy should be consist of terms such trM 1 A, trM 2 A, trM 3 A, trM 1 A2, trM 2 A2, trM 3 A2, trA3. The General yield function F can be expressed using seven mentioned integrity bases [13]. The general yield function F is considered to be a scalar function of the seven scalar invariants of three structural tensors and one stress related tensor related. Therefore, the general yield function F can be expressed as shown in equation (5). F(trM 1 A, trM 2 A, trM 3 A, trM 1 A2, trM 2 A2, trM 3 A2, trA3, k)=0 (5) Various traces of arbitrary symmetric second order tenor A are presented in equations (6) to (8) that can be used to express the various invariants of the mechanical tensor A. 𝑡𝑟 𝐴 = (𝐴11 + 𝐴22 + 𝐴33 ) 2 𝑡𝑟𝐴2 = (𝐴11 + 𝐴222 + 𝐴233 ) 3 3 𝑡𝑟𝐴 = (𝐴11 + 𝐴322 + 𝐴333 )
(6) (7) (8)
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The general form of equation (5) can be expressed using a polynomial function of the seven integrity bases as shown in equation (9). This the most general polynomial forms for yield surface of orthotropic anisotropic materials. a 0 (trM 1 A)b0+ a 1 (trM 1 A2)b1+a 2 ( trA3)b2+ a 3 (trM 2 A)b3+ a 4 (trM 2 A2)b4+ a 5 (trM 3 A)b5+ a 6 ( trM 3 A2)b6 +a 7 (trM 1 A)b7(trM 1 A2)b8+ a 8 (trM 1 A)b9( trA3)b10+ a 9 (trM 1 A)b11(trM 2 A)b12+ a 10 (trM 1 A)b13(trM 2 A2)b14+ a 11 (trM 1 A)b15(trM 3 A)b16+ a 12 (trM 1 A)b17( trM 3 A2)b18 + a 13 (trM 1 A2)b19( trA3)b20+ a 14 (trM 1 A2)b21 (trM 2 A)b22+ a 15 (trM 1 A2)b23 (trM 2 A2)b24 +a 16 (trM 1 A2)b25(trM 3 A)b26+ a 17 (trM 1 A2)b27 ( trM 3 A2)b28+ a 18 ( trA3)b29 (trM 2 A)b30+ a 19 ( trA3)b31 (trM 2 A2)b32 + a 20 ( trA3)b33 (trM 3 A)b34+ a 21 ( trA3)b35 ( trM 3 A2)b36 + a 22 (trM 2 A)b37 (trM 2 A2)b38 + a 23 (trM 2 A)b39 (trM 3 A)b40+ a 24 (trM 2 A)b41 ( trM 3 A2)b42+ a 25 (trM 2 A2)b43 (trM 3 A)b44 + a 26 (trM 2 A2)b45 ( trM 3 A2)b46+ a 27 (trM 3 A)b47 ( trM 3 A2)b48=k
(9)
where a i (i=0,1 … 27) and b j (j=0,1 … 48) are material constants and k is related to yield strength of material. In special case, when mechanical tensor A is diagonal, some equalities will be arose such as (trM 1 A)2=trM 1 A2, (trM 2 A)2=trM 2 A2, and (trM 3 A)2=trM 3 A2. Therefore equation (9) could be reduced to the form presented by equation (10) a 0 (trM 1 A)b0+ a 1 (trM 2 A)b1+ a 2 (trM 3 A)b2 +a 3 ( trA3)b3+a 4 (trM 1 A)b4(trM 2 A)b5 + + a 6 (trM 3 A)b8(trM 1 A)b9 + a 7 (trM 1 A)b10( trA3)b11+ a 5 (trM 2 A)b6(trM 3 A)b7 b12 3 b13 b14 3 b15 a 8 (trM 2 A) ( trA ) + a 9 (trM 3 A) ( trA ) =k
(10)
Where a i (i=0,1 … 9) and bj (j=0,1 … 15) are limited to ninth and fifth values. Therefore, diagonalization of mechanical tensor A, reduces the number of required in comparison to the general case.
Result and discussion Selection of stress tensor have significant role on number of parameters and coefficients. It is shown that by selection of principle stress tensor, the general form of equation (9) can be reduced to equation (10) which has very fewer coefficients than equation (9). In this research, two special types of diagonal tensor have been considered for representing yield function of materials. Therefore, the equation (10) can be used for examining the effect of different parameters on the capacity of this equation for forming the closed surfaces in the stress space that could be used to represent yield surface of materials. In the first step, stress tensor in the principle form (equation (3)) has been used and general form of yield function has been derived (equation (11)). It is shown that equation (11) is combination of single and multiple terms of integrity bases of principle stresses tensor. a 0 𝛔 1 b0+ a 1 𝛔 2 b1+ a 2 𝛔 3 b2 +a 3 (𝛔 1 3+𝛔 2 3+𝛔 3 3)b3+a 4 𝛔 1 b4 𝛔 2 b5 + a 5 𝛔 2 b6 𝛔 3 b7 + a 6 𝛔 3 b8 𝛔 1 b9 + a 7 𝛔 1 b10(𝛔 1 3+𝛔 2 3+𝛔 3 3)b11+ a 8 𝛔 2 b12(𝛔 1 3+𝛔 2 3+𝛔 3 3)b13+ a 9 𝛔 3 b14(𝛔 1 3+𝛔 2 3+𝛔 3 3)b15=k
(11)
It has been tried to evaluate the resultant closed surfaces that can be established using equation (11) in the stress space. Commercial software has been used to exhibit graphical contours that can be established using different combination of single and multiple factors in stress space [14]. Calculation shows that none of the mentioned terms in equation (11) could produce proper closed-surface to representation of yield surface in stress space. Therefore, it is revealed that first chosen stress tensor couldn’t represent yield surface of materials. In the next step, it has been tried to use second stress tensor (equation (4) to construct integrity bases required for generating general yield function. Established general yield function is shown as equation (12). a 0 (2𝛔 1 - 𝛔 2 - 𝛔 3 )b0+ a 1 (2𝛔 2 - 𝛔 1 - 𝛔 3 )b1+ a 2 (2𝛔 3 - 𝛔 1 - 𝛔 2 )b2 +a 3 ((2𝛔 1 - 𝛔 2 - 𝛔 3 )3+(2𝛔 2 𝛔 3 )3+(2𝛔 3 - 𝛔 1 - 𝛔 2 )3)b3+a 4 (2𝛔 1 - 𝛔 2 - 𝛔 3 )b4 (2𝛔 2 - 𝛔 1 - 𝛔 3 )b5 + a 5 (2𝛔 2 - 𝛔 1 - 𝛔 3 )b6 (2𝛔 3 𝛔 2 )b7 + a 6 (2𝛔 3 - 𝛔 1 - 𝛔 2 )b8 (2𝛔 1 - 𝛔 2 - 𝛔 3 )b9 + a 7 (2𝛔 1 - 𝛔 2 - 𝛔 3 )b10(((2𝛔 1 - 𝛔 2 - 𝛔 3 )3+(2𝛔 2 𝛔 3 )3+(2𝛔 3 - 𝛔 1 - 𝛔 2 )3)b11+ a 8 (2𝛔 2 - 𝛔 1 - 𝛔 3 )b12(((2𝛔 1 - 𝛔 2 - 𝛔 3 )3+(2𝛔 2 - 𝛔 1 - 𝛔 3 )3+(2𝛔 3 𝛔 2 )3)b13+ a 9 (2𝛔 3 - 𝛔 1 - 𝛔 2 )b14(((2𝛔 1 - 𝛔 2 - 𝛔 3 )3+(2𝛔 2 - 𝛔 1 - 𝛔 3 )3+(2𝛔 3 - 𝛔 1 - 𝛔 2 )3)b15=k
𝛔1𝛔1𝛔1𝛔1-
(12)
This equation like equation (11) composed of single and multiple terms of integrity bases of deviatoric stress tensor. The effect of the single and multiple terms in individual or in combination with other terms have evaluated by using computational software for representing closed surface in the stress space. For evaluating the effect of elements of
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equation (12), first the effect of single terms has been examined individually. Next the effect of combined terms has been evaluated for establishing closed surface in the stress space.
Examination the effect of terms in forming closed-surface All the exponents of equation (12) can only be integer numbers. Corresponding of a 0 (trM 1 A)b0 in equation (10) is a 0 (2x-y-z)b0 in equation (12). Established shape of surface using a 0 (trM 1 A)b0 term mainly depends on values of exponent b0. If odd numbers are attributed to b0, these cause to build of flat surfaces in stress space. If even numbers are attributed to b0, these cause to formation of two flat parallel surfaces. Evaluations shows that two other single terms of a 1 (trM 2 A)b1 and a 2 (trM 3 A)b2 have same results by this difference that the directions of formed surfaces are different and they cut each other in the stress space. In the next step the first three single terms has been investigated for possibility of making closed surface. Exponents b 0 , b 1 , and b 2 and coefficients a 0 , a 1 , and a 2 could be changed separately. Examining of exponents variations show that each exponent term that have been attributed bigger value becomes dominant term and eliminates other terms effect in the equation and it is not possible to form closed-surface. Therefore, b 0 , b 1 , and b 2 must get identical values (b 0 =b 1 =b 2 ). Further studies show that when the exponents are attributed odd numbers, no closed-surface can be built. But when even numbers are chose for exponents, it is possible to build closed-surface in the stress space. Therefore by choosing even numbers for exponents in the three first single terms (a 0 (trM 1 A)b0, a 1 (trM 2 A)b1, and a 2 (trM 3 A)b2) different cylindrical closed-surface could be achieved by changing coefficients (a 0 , a 1 , and a 2 ). Figure 1 shows closed surface variation by changing of coefficients when exponents b0, b1 and b2 are equated to 4. Curve number one, two and three show intersections of π-plane by closed-surfaces result from assumption of coefficients as a 0 =2, a 1 =1, and a 2 =1, a 0 =1, a 1 =3, and a 2 =1, a 0 =1, a 1 =1, and a 2 =4, respectively.
Figure 1. effect of changing constant coefficient in first three single terms
Fourth single term, a 3 ( trA3)b3, has different behavior from other three first single terms, so it has been examined separately. In this term like other single terms exponent b3 play major role in capacity of forming closed-surface. When b3 is attributed odd numbers it can’t form closed-surface and only several intersected flat surfaces are formed. But when b3 is attributed even numbers, this term can build closed-surfaces. By adding this single term to other first three single terms, it becomes possible to bring new capacity to equation and increase flexibility of equation on showing yield surface in stress space. Figure 2 shows this case. Curve number one shows closed-surface formed from first three single terms when b 0 , b 1 , and b 2 are equated to 6 and a 0 , a 1 and a 2 are equated to one. Curves
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number two and three show the effect of adding fourth term to three first single terms when is b3 equated to 2 and a 3 is attributed 1 and 4, respectively. It is clear that adding of fourth term to other single term increase ability of equation in forming variety of closed-surfaces and yield surfaces of anisotropic materials.
Figure 2. effect of changing constant coefficient in forth single terms by adding to other three single terms
In the next step, the effects of multiple terms have been examined. Three first multiple terms a 4 (trM 1 A)b4(trM 2 A)b5, a 5 (trM 2 A)b6(trM 3 A)b7, and a 6 (trM 3 A)b8(trM 1 A)b9 have similar behavior individually and also in combinations. Therefore, all of them have been examined in one case. Exponents b4, b5, b6, b7, b8, and b 9 have main role in forming of closed-surface and defining suitable exponents is very important. Computational results show that b 4 , b 6 , and b 8 must be identical (b4=b6=b8) and also b5, b7, and b9 should be equated (b5=b7=b9). Consequently, only two groups of exponents must be selected to be odd or even numbers. Only when even numbers are attributed to the exponents of multiple terms, combinations of them could form closed-surfaces that stand for anisotropic yield functions of materials. Also when these three terms are added to other first three single terms, it causes that resultant equation get more ability in prediction of yield function and increase the flexibility of yield function. This case is shown in the Fig. 3. The three first curves show the intersection of the closed surface formed from the combination of first three multiple terms with the π-plane when b4, b6, and b8 are equated to 10, b5, b7 and b9 equated to 2 for curve one, b4, b6 and b8 equated to 8 and b5, b7 and b9 equated to 4 for curve two, and b4, b6, b8, b5, b7 and b9 are equated to 6 for curve three. Figure 3 shows that change in selection of exponents cause rotation of closed surface π-plane. Curves four and five show When exponents b4, b6 and b8 are equated to 11, b5 b7 and b9 are equated 1 and coefficients a 4 , a 5 and a 6 are equated 1, curve four resulted. But when exponents b4, b6 and b8 are equated 10, b5, b7 and b9 are equated 2, and coefficients a 4 , a 5 and a 6 are equated to 1 curve five resulted. In both cases the three first single terms assuming exponents of b0, b1and b2 which equated to 12 and coefficients of a 0 , a 1 and a 2 that equated to 1 are added to the multiple terms.
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Figure 3. Effect of changing exponential coefficients in first three multiple terms and also adding of them to first three single terms.
The effect of three first multiple terms have been examined and it is seen that they have very similarity in behavior and they cause reduction in amount of coefficients. The three last multiple terms a 7 (trM 1 A)b10( trA3)b11, a 8 (trM 2 A)b12( trA3)b13, anda 9 (trM 3 A)b14( trA3)b15 have same trend mentioned for the three first multiple terms and they have similar behavior. Computational results show that exponential coefficients b10, b11, b12, b13, b14, and b15 can be divided in to two category, in first category b10, b12, and b14 that must be identical (b10=b12=b14) and second category b11, b13, and b15 that should be attributed same values (b11=b13=b15). Examinations of equation (10) show that most of exponents are depend on each other. Computational results prove that some of exponents must be equal and consequently numbers of exponents in equation (10) reduced from 15 to 5. The result can be shown in equation (13). a 0 (trM 1 A)b0+ a 1 (trM 2 A)b0+ a 2 (trM 3 A)b0 +a 3 ( trA3)b1+a 4 (trM 1 A)b2(trM 2 A)b3 + b2 b3 b2 b3 a 5 (trM 2 A) (trM 3 A) + a 6 (trM 3 A) (trM 1 A) + a 7 (trM 1 A)b4( trA3)b5+ a 8 (trM 2 A)b4( trA3)b5+ a 9 (trM 3 A)b4( trA3)b5=k
(13)
Conclusion Selection of tensor that can be used to represent the yield behavior of materials, play very important role on ability of invariant theory in expressing anisotropic yield functions. It is seen that by selection of principle stresses tensor, invariant theory couldn’t represent any closed-surface to stand for anisotropic yield functions. But by choosing deviatory stresses tensor, it is seen that invariant theory could be a reliable theory for deriving yield functions for anisotropic materials. Invariant theory generates a very general equation for expressing yield function of orthotropic materials that it is shown by equation (9). But by selecting a diagonal tensor it is possible to reform equation (9) into equation (10). This results in reduction the number of coefficients and exponents from 27 and 48 in equation (9) to 9 and 15 in equation (10), respectively. The effects of each terms of equation (10) have been analyzed in order to form closed surface and the relationship between coefficients of equation (10) have been revealed. It is observed that exponents have major role in forming closed surface. Therefore, by appropriate choice of them it is possible to build a range of closed-surface that could represent yield function of orthotropic materials. The results show that exponent of equation (10) have relationship with each other and some of them must be identical until they could form closed surface. Therefore equation (10) can be reformed into equation (13) which reduces the number of exponents from 15 to 5.
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10. 11. 12. 13. 14.
R. Hill, A theory of the yielding and plastic flow of anisotropic metals, Proc. R. Soc. Lond. A 193, p.p 281–297, 1948 R. Hill, Theoretical plasticity of textured aggregates, Math. Proc. Cambridge Philos. Soc. 85, p.p 179–191, 1979 R. Hill, Constitutive modeling of orthotropic plasticity in sheet metals, J. Mech. Phys. Solids 38, p.p 405– 417, 1990 R. Hill, A user-friendly theory of orthotropic plasticity in sheet metals, Int. J. Mech. Sci. 15, p.p 19–25, 1993 W. F. Hosford, A generalized isotropic yield criterion, J. Appl. Mech. 39, p.p 607-609, 1972 F. Barlat, O. Richmond, Prediction of tricomponent plane stress yield surfaces and associated flow and failure behaviour of strongly textured F.C.C. polycrystalline sheets, Mat. Sci. Eng. 91, p.p 15–29, 1987 F. Barlat, J. Lian, Plastic behaviour and stretchability of sheet metals – Part I: A yield function for orthotropic sheets under plane stress condition, Int. J. Plasticity 5, p.p 51–66, 1989 F. Barlat, D.J. Lege, J.C. Brem, A six-component yield function for anisotropic materials, Int. J. Plasticity 7, p.p 693–712, 1991 F. Barlat, J.C. Brem, J.W. Yoon, K. Chung, R.E. Dick, S.H. Choi, F. Pourboghrat, E. Chu, D.J. Lege, Plane stress yield function for aluminium alloy sheets – part 1: theory, Int. J. Plasticity 19, p.p 1297–1319, 2003 R.S. Rivlin, Large elastic deformations of isotropic materials IV. Further development of the general theory, Philos. Trans. Royal Soc. London A241, p.p 379–397, 1948 J.L. Ericksen, R.S. Rivlin, Large elastic deformations of homogeneous anisotropic materials, J. Rational Mech. Anal. 3, p.p 281–301, 1954 M.H. Parsa, K.A. Shirvanedeh, P. H. Matin, An attempt to express yield functions of isotropic metals based on the invariants of stress tensor, TMS 2011 Conference J. P. Boehler, “Applications of tensor functions in solid mechanics”, Springer-Verlag, Wien-New York, 1987 Wolfram Mathematica7, 2008.
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The Effect of Anisotropy on Thin-Walled Tube Bending K. Hasanpoura*, B. Aminib, M. Poursinaa, M. Baratic a
Department of mechanical engineering, Faculty of engineering, University of Isfahan, P.O.Box 81746-73441, Isfahan, Iran. b
Department of Mechanical Engineering, Islamic Azad University, Khomeinishahr Branch, Iran. c
Department of Mechanical Engineering, K. N. Toosi, University of Technology, Iran.
*
Corresponding author: +98-3117934021, Fax: +98-3117932746,
[email protected].
Abstract. Thin-walled tube bending has found many of its applications in the automobile and aerospace industries. The rotary-drawbending method which is a complex physical process with multi-factor interactive effects is one of the advanced tube forming processes with high efficiency, high forming precision, low consumption and good flexibility for bending angle changes. However it may produce a wrinkling phenomenon, over thinning and cross-section distortion if the process parameters are inappropriate. Wrinkles propagate permanently in thin-walled tube, but finally, localize in a finite zone and lead to failure. The prediction of wrinkling in thin-walled tube bending processes has been a challenging topic. In this paper, firstly, the plastic deforming behavior and wrinkling mechanism for a thin-walled tube is simulated and the results will compare with the available experimental ones. Then, the effect of anisotropy on ovalization, thickness and wrinkling of tube will be investigated using FEM. Extensive numerical results are presented showing the effects of the various kinds of materials and geometric parameters on wrinkling using anisotropic yield function. Keywords: Thin-walled tube; Anisotropy; Wrinkling.
INTRODUCTION Thin- walled tubes find wide applications in many branches of engineering. Examples include aircraft, spacecraft, cooling towers, steel silos and tanks for bulk solid and liquid storage, pressure vessels, pipelines and offshore platforms. The rotary-draw bending process of thin-walled tubes has been attracting more and more applications due to its high forming precision advantage and satisfying the increasing needs for high strength per weight ratio products. However, the inner side of thin-walled tubes may experiences a wrinkling phenomenon if the process parameters are inappropriate especially for tubes with large diameter and thin wall thickness. It is because of the compressive stress during the bending process, which will leads to the process failure or even die damage if wrinkling is severe. Possible wrinkling at two locations, before the material enters the bending die and at the bending intrados, are identified. How to predict this phenomenon rapidly and accurately is one of the urgent key problems to be solved for the development of this process at present. It is well known that the problem resolution relies heavily on experience and involves repeated trial-and-errors in practice, which spends excessive manpower, raw material, and time in designing and adjusting the process and dies, and moreover makes the production efficiency abate drastically. Even so, sometimes contented outcomes may not be obtained. Li et al. developed a complete 3D elastic-plastic FEM model of the process using ABAQUS/Explicit code based on the analysis of the forming characteristics by analytical and experimental methods, and thus the plastic deformation characteristics with small bending radius were investigated [1]. Yang et al. have proposed a method to obtain the wrinkling limit of thin-walled tube with large diameter under different loading paths by the numerical control (NC) bending process analytically [2]. Zhan et al. have established one analytical model of the mandrel (including mandrel shank and balls) and have deduced some reference formulas in order to select the mandrel parameters preliminarily, then carried out experiment to verify the analytical model [3]. Gu et al. proposed and investigated the interactive effects of the wrinkling and other two defects in the process by using the developed Explicit FE model based on the ABAQUS/Explicit code. then a strategy for selecting the positive parameters was proposed based on the understanding of the interactive effects of wrinkling, thinning and section distortion [4]. Kumar was simulated rotary draw bending process focusing more on wrinkle formation [5]. The effects of various process parameters on the
The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 187-193 (2011); doi: 10.1063/1.3623610 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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formation of wrinkles have been analyzed and optimum process parameters were selected to get wrinkle-free bend. Gu et al. revealed the spring back mechanism and rule of thin-walled tube NC bending based on the numerical simulation of the whole process [6]. Yang et al. established a 3D elastic–plastic finite element model and a wrinkling energy prediction model under multi-die constraints considering the characteristics of the bending processes of aluminum alloy thin-walled tubes with large diameters[7]. Li et al. developed an energy-based wrinkling prediction model for thinwalled tube bending by using the energy principle, combined with analytical and finite element (FE) numerical methods [8]. Yan et al. developed an analytical model for thin-walled tube NC bending and then based on ABAQUS platform, a series of 3D-FE models are developed to simulate the bending process with large diameter and small bending radiuses[9]. Also, numerical study on the deformation behaviors of bending process was conducted. Kou et al. aimed at revealing the deformation behaviors of thin-walled tube in rotary draw bending under push assistant loading conditions with multiple defects. They presented an analytical description of push assistant loading conditions firstly and proposed some indices for evaluation of push assistant loading roles, then the interactive effects of push assistant loading conditions on wall thinning, cross-section deformation and wrinkling investigated extensively using 3D-FE simulation considering push assistant loading functions combined with experiment [10]. In this paper, firstly, the plastic deforming behavior and wrinkling mechanism for thin-walled tube is simulated and the results will compare with the available experimental ones. Then, the effect of anisotropy on ovalization, thickness and wrinkling of tube will be investigated using FEM. Extensive numerical results are presented showing the effects of the various kinds of materials and geometric parameters on wrinkling using anisotropic yield function. THE ROTARY-DRAW BENDING PROCESS The whole process of thin-walled tube bending includes three processes: bending tube, retracting mandrel and spring back. In the thin-walled tube rotary-draw bending process, as shown in Figure 1, both sides of the tube are subjected to various tooling’s strictly contacting force, such as bend die, clamp die, pressure die, wiper die and mandrel. The tube is clamped against the bend die; drawn by the bend die and the clamp die, the tube goes past the tangent point and rotates along the groove of the bend die to the desired bending degree and the bending radius. So the process needs precise coordination of various dies and strictly controlling of forming parameters. Among the above tooling, mandrel is positioned inside the hollow tube to provide the rigid support.
FIGURE 2. Illustration of FEM model for rotary-draw bending process
FIGURE 1.Forming principle of rotary draw bending method
Figure 2 shows FEM model for rotary-draw bending process. The four-node doubly curved thin shell is applied to describe three-dimensional deformable tube. Five integration points are selected across the thickness to describe the tube bending deformation better. The external and internal rigid tools are modeled as rigid bodies using 4-node 3D bilinear quadrilateral rigid element to describe smooth contact geometry curved faces. In this paper, the results will be compared with the experimental data of Li et al. [3]. The geometry parameters are reported in Table 1. The material is aluminum alloy LF2M, which its mechanical properties are shown in Table 2. The material model is the commonly used Swift’s power-law plastic model as in Eq. (1):
V
KH
n
188
(1)
Where K is the strength coefficient, n the work-hardening exponent [7]. TABLE 1. Geometrical parameters of simulations
Parameters Tube outside diameter(D) Wall thickness(t) Bending radius(R) Mandrel diameter(d) Nose radius(r) Extension length(e) Number of balls(n) Balls thickness(k) Space length between mandrel Shank and ball(p) Mandrel length Pressure die length Clamp die length Wiper die length
Value 38mm 1mm 57mm 35.60mm 6mm 6mm 1 12mm 15mm 153mm 250mm 115mm 120mm
In the tube bending process, five contact interfaces between tube and dies are as following: tube/mandrel (ball), tube/wiper die, tube/bend die, tube/clamp die and tube/pressure die. The classical Coulomb model has been chosen to represent the interfaces’ friction conditions:
Vf Where
P Vn
(2)
V f is the frictional stress, P the friction coefficient (0 P 0.5) and V n the stress on the contact surface.
The tube’s bending deformation depends on the contact and friction between various tube portions and different dies. According to the different contact conditions, the friction coefficient can be classified into 4 kinds: 0.05, 0.1, 0.25 and ‘‘Rough’’, in which ‘‘Rough’’ type refers to no relative slipping when nodes contact each other and suitable for the tube/clamp die friction conditions. The other friction coefficients have been assigned to the different contact interfaces as shown in Table 3. TABLE 2. Mechanical properties of tube
Material parameter Ultimate tension strength (MPa) Extensibility (%) Poisson’s ratio Initial yield stress (MPa) Hardening exponent, n Strength coefficient K Young’s modulus E (GPa) Density (kg/m3)
TABLE 3. Friction conditions in various contact interfaces
Contact interface Tube/wiper die Tube/pressure die Tube/clamp die Tube/bend die Tube/mandrel Tube/balls
190 22 0.34 90 0.262 398 56 2700
Friction coefficients 0.05 0.25 Rough 0.1 0.1 0.1
HILL ANISOTROPY CRITERIA To express the behavior of anisotropic materials, different equations have been introduced. Among these equations, quadratic equation of Hill is more popular in the papers and its constants can be easily calculated by performing tensile test. In this paper, the Hill yield function was used to study the effect of anisotropy which is as follows:
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V2
F (V yy V zz )2 G(V zz V xx )2 H (V xx V yy )2 2LV 2 yz 2MV 2 zx 2 NV 2 xy
(3)
The constants in this equation can be obtained using the following relations:
Vzzy
2 y y Vxx ; Vyy H F
2 y y Vxx ;Vxy G F
1 y y Vxx ;Vyz L
1 y y Vxx ; Vxz M
1 y Vxx N
(4)
In these formulas the yield stress in the extruding direction ( V xx ), has been selected as a reference yield stress. It must y
be noted that in this finite element model, the x axes is in the extruding direction, the z one is in the circumferential direction and the y direction represents the radial direction. A convenient method for determining the coefficients in equation (3) is conducting three tensile tests, using specimens cut from the tube at the angles 0°, 45° and 90° with respect to the extruding direction, with the additional conditions: (5) H G 2; N=L=M Lankford coefficients are defined as follows[12]: H H 2N F G (6) r0 ; r90 ; r45 G F 2(G F ) To define the anisotropic behavior of the material, the six ratios of yield stresses to the reference yield stress in extruding direction i.e. x direction ( V 0
R11
V xxy V0
V xxy ) are used. These ratios are:
1; R22
The normal anisotropy index
V yyy ; R33 V0
V zzy ;R V 0 12
3
V xyy ;R V 0 13
3
V yzy ; R23 V0
3
V xzy V0
(7)
(r ) and the planar anisotropy ('r ) are calculated using the following formulas:
1 ( r0 2 r45 r90 ) 4 (8) 1 'r (r0 2 r45 r90 ) 2 In normal anisotropy r0=r45=r90 .In this paper, the effect of normal anisotropy on the wrinkling of tube under bending is investigated. Lankford coefficients and Hill yield function values associated with this case is given in Table 4. r
TABLE 4. Normal anisotropic cases
Case 1 2 3 4 5 6 7 8 9 10
r
R11
R22
R33
R12
R13
R23
0.5 0.6 0.7 0.8 1.1 1.2 1.3 1.4 1.5 2
1 1 1 1 1 1 1 1 1 1
0.866 0.894 0.921 0.948 1.024 1.048 1.072 1.095 1.118 1.224
1 1 1 1 1 1 1 1 1 1
1.060 1.044 1.030 1.019 0.992 0.985 0.978 0.973 0.968 0.948
1.060 1.044 1.030 1.019 0.992 0.985 0.978 0.973 0.968 0.948
1.060 1.044 1.030 1.019 0.992 0.985 0.978 0.973 0.968 0.948
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RESULTS At first the validity of results is verified comparing with ones in reference [3]. Simulations show that the initial tube circular cross section deforms to the oval, through the arc of bended tube, which is in agreement with the experiments. The parameter of ovalization in tube cross-section is defined as: Major Diameter - Minor Diameter (9) Ovalization Ratio u 100 % Initial Tube Diameter
As can be seen in Figure 3, parameter of ovalization in tube cross-section for the middle of the bending angle was used to compare the results, which indicates good agreement between the results of two researches. Von mises stress contour in bending angle of 104° is shown in Figure 4.
FIGURE 4. Von mises Stress distribution in the bending angle of 104 º
FIGURE 3. Comparison of the ovalization parameter in this research and reference [3]
Now, the normal anisotropy effect on wrinkling and tube deformation in rotary-draw bending process is studied. In Figure 5, the effect of normal anisotropy on ovalization of tube cross-section is shown. In this case, the bending angle is 90° and the ovalization of tube cross-section is calculated in angle 45°. This diagram shows that with increasing in normal anisotropy coefficient, ovalization of tube cross-section will be increased. Based on experimental findings, depending on the applied conditions, possible wrinkling at two locations, before the material enters the bending die and at the bending intrados, will be occurred as is shown in Figure 6. The simulated process in this paper, leads the tube to be wrinkled in both locations.
Ovalization Ratio at 45
o
15
10
5 0.5
1
1.5
2
r-value
FIGURE 5. Anisotropy effect on ovalization of tube crosssection
FIGURE 6. Two wrinkling locations occurred in rotary-draw bending process
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To study of anisotropy effects on wrinkling, the regions and II are considered. Figure 7 illustrates the node numbers 1 to 35 on the tube edge which will be entered to the intrados of bended tube.
FIGURE 7. Node numbers and definition of Y value to study of anisotropy effects on wrinkling
The Y value is considered for the study of anisotropy effects on wrinkling. According to Figure 7, for nodes were located in left hand side of line AB, the value of Y is equal to the distance of nodes from the line CD and for nodes were located in right hand of line AB, the value of Y is equal to the distance of nodes from the point O (which is the center of bending die). When the tube does not wrinkle, Y value is almost equal to 38 mm, considdering the distance of point O from axis of tube is 57mm and tube radius is 19 mm. If the tube wrinkles, the value of Y will be more or less which this deviation is a characteristic of wrinkling. In this section, all geometrical features are similar to previous case except tube thickness. For having a great wrinkle wavelength and observing wrinkles on the tube, tube thickness is considered 0.5 mm. The anisotropy effects on wrinkling are shown in the Figures 8 and 9. In Figure 8, Anisotropy coefficients are between 0.5 to 1 and Figure 9 relating to the anisotropy coefficients between 1 and 1.5. 40.5
40
40
39
Y value
Y value
39.5 39
38.5
38.5
38
38
37.5
37.5 0
r = 1.1 r=2
39.5
r = 0.5 r = 0.8
5
10
15
20
25
30
37 0
35
Node Number
5
10
15
20
25
30
35
Node Number
FIGURE 8. Wrinkling of intrados for r 1
In Figure 10, the contour of tube thickness, for four cases with normal anisotropy coefficients 0.5, 0.8, 1.1 and 2 are shown. It is seen that with increasing the normal anisotropy coefficient, the maximum value of wall thickness is increased.
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FIGURE 10. Tube thickness contours for different r values.
CONCLUSION The thin-walled tube bending process was simulated with the aim of investigating the effect of tube anisotropy on ovalization, thickness and wrinkling of tube which is exposed to the most severe conditions. An anisotropic tube material with Hill quadratic yield criterion was employed and normal anisotropy was considered in this study. Based on this investigation, the following conclusions can be drawn: 1- The larger normal anisotropy index (r) leads to the higher the ovalization ratio. 2- The maximum value of tube wall thickness is increased by increasing of the normal anisotropy index (r), for r γ ref
315
(5)
with τ Y( ) = τ 0 at the beginning, and α
γ = ∑ ∫ γ (α ) dt.
(6)
α
In this study, three families of slip systems–basal slip , prismatic slip , and pyramidal-2 slip – and one family of {10 1 2} twinning systems are used to model the mechanical behavior of a rolled AZ31 Mg alloy sheet. There are three basal, three prismatic, six pyramidal-2, and six twinning systems. The linear hardening (eq. (3)) is assumed for the basal slip, and Voce hardening (eq. (4)) is assumed for the prismatic slip and pyramidal-2 slip. Twinning is assumed to have a polar character in which each system is activated only by tension of the c-axis. Because shear strain that arises in a grain due to twinning can be assumed to be similar to the shear caused by an activated slip system [11, 19], the shear strain induced by twinning is also calculated using eq. (2). The combined hardening (eq.(5)) is assumed for the twinning, with γref denoting the shear strain due to twinning arising in a grain twined entirely. For the sake of simplicity, the lattice rotation due to twinning and the so-called untwinning is not taken into consideration. Because the deformation considered in this study is at most plastic strain of 0.01 in the first quadrant of the stress space and the activation of twinning is negligible as explained below, the above simplification may not affect the simulation result. The above rate-dependent crystal-plasticity model is incorporated into each Gauss point in the static finiteelement method [11, 16, 20]. The rate tangent modulus method [17] is used for the explicit time integration of the constitutive model. To prevent excessive increase of non-equilibrium between external and internal forces, the generalized rmin-strategy is employed. The plastic work Wp is given in the form ns
W p = ∫ σ :D p dt, with D p = ∑ γ (α ) α =1
(
1 (α ) s ⊗ m (α ) + m (α ) ⊗ s (α ) 2
)
(7)
where Dp is the plastic strain rate tensor, the unit vectors s(α) and m(α) are the slip direction and slip plane normal, respectively, and ns is the number of slip and twinning systems in a grain. A contour of plastic work was calculated using the following procedure proposed in the literature [12]. A uniaxial tensile test in RD was selected as a reference condition. The uniaxial stress σRD and the plastic work Wp given by eq.(7) are calculated at various p which is given by integrating the plastic strain rate DP. Then the biaxial stresses σRD and uniaxial plastic strains ε RD σTD obtained for various preset biaxial-stress ratios σTD/σRD are determined at the same plastic work. Consequently p by plotting the obtained stresses in the principal stress the contours of plastic work can be obtained for various ε RD p space. The contour of plastic work was calculated up to ε RD =1.0% in this study. It should be noted that the plastic work Wp may not be zero even during the elastic range in the stress-strain curve because of the rate-dependent model of crystal-plasticity.
MATERIAL MODELLING A finite-element model used in the present study was a cube which had seven uniform eight-node isoparametric brick elements using selective reduced integration in each direction, as shown in Fig. 1 (a). The total number of elements was 343. In this study, the same initial crystallographic orientation was assigned to all eight Gauss integration points in an element, thus the present model had 343 initial crystallographic orientations. Figure 1 (b) shows the initial (0001) pole figure used in the simulation. The initial crystallographic orientations were artificially created using a procedure proposed by Mayama et al. [20] to simulate the rolling texture of Mg alloy sheets [7, 8]. In the simulation, the x-, y-, and z-axes in Fig. 1 (a) were defined to be the rolling direction (RD), transverse direction (TD), and normal direction (ND), respectively, thus the majority of c-axes tended to align in the z direction (ND). The following boundary conditions were used to simulate the contour of plastic work: the planes x = 0, y = 0, and z = 0 were fixed in the x, y, and z directions, respectively, due to the assumption of plane symmetries, and the small displacement increments were given to the planes x = l and y = l such that pseudo-proportional stress paths were achieved.
316
RD (b)
(a)
l
TD
z
y
x
l
l
FIGURE 1. Material modelling used in the simulation. (a) Finite-element model and (b) (0001) pole figure. TABLE 1. Latent hardening parameters qαβ used.
300
Prismatic Pyramidal-2 Twinning 0.5 0.5 0.5 0.2 0.2 0.5 1.0 0.2 0.25 1.0 0.2 0.25
200 True stress /MPa
Basal Prismatic Pyramidal-2 Twinning
Basal 0.2 0.2 1.0 1.0
TABLE 2. Calibrated material parameters in equations (3), (4), and (5) used in the present simulation.
τ0 τ∞ h0 n
Basal 10 20 -
Prismatic Pyramidal-2 Twinning 84 160 48 260 360 680 750 150 10
Simulation
100 0 -100 Experiment
-200 -300 -0.1
-0.05
0
0.05
0.1
Logarithmic strain ε FIGURE 2. Stress-strain curves obtained by uniaxial tensile and compressive tests.
From an experiment of a uniaxial tensile test of the AZ31 Mg alloy sheet with a thickness of 0.8 mm, Young’s modulus E = 42 GPa and Poisson ratio ν = 0.3 were assumed [6, 11]. The rate sensitivity exponent was set to m = 0.02 and the reference strain rate to γ0 = 0.001 s-1. The choice of m has been explained in the literature [11]. The self-hardening parameters were set to 1, while the latent-hardening parameters qαβ shown in Table 1 were adopted based on the literature [11, 16]. γref in equation (5) was 0.027. Other parameters used in equations (2), (3), (4), and (5) were identified by trial and error to achieve a reasonable fit with experimental stress–strain curves obtained by the uniaxial tensile and compressive tests of the AZ31 Mg alloy sheet, as shown in Fig. 2. The determined parameters are shown in Table 2.
RESULTS AND DISCUSSION Contour of plastic work To examine a linearity of stress path during deformations in the simulations, Fig. 3 shows the evolutions of σRD and σTD for various preset biaxial-stress ratios. Clearly fairly-linear stress paths are achieved in the simulations regardless of the stress ratios, indicating that the simulation procedure adopted in this study is acceptable to calculate a contour of plastic work. p p . From the beginning of plastic deformation to ε RD Figure 4 shows the contours of plastic work at various ε RD
317
300
300 250
250
200
200
σTD /MPa
σTD /MPa
p ε RD
150 100
0.08% 0.14% 0.16% 0.2% 0.24% 0.3% 0.45% 0.64% 1.0%
150 100
50
50
0
0 0
50
100
150
200
250
300
0
σRD /MPa
50
100
150
200
σRD /MPa
250
300
FIGURE 4. Contours of plastic work obtained using p simulation. The numbers shown in the legend are ε RD .
FIGURE 3. Stress paths during deformation up to p ε RD = 1.0%.
=0.16 %, the contour is flattened in the vicinity of σTD/σRD =1. Subsequently, in the uniaxial plastic-strain range p ≤ 0.3 %, the work-hardening rates in the vicinity of σTD/σRD =1 become larger than those in the 0.16 % ≤ ε RD vicinity of σTD/σRD =0 and the contour bulges severely. Then the work-hardening rates in the vicinity of σTD/σRD =1 p p higher than 0.3%. Clearly the shapes of the contours change as ε RD increases, exhibiting become small again at ε RD the differential work-hardening behavior in the simulation result. The trends observed in the simulation results are qualitatively in good agreement with those of the experimental results reported by Andar et al. [12], demonstrating that the present simulation results are qualitatively acceptable. Because the definition of plastic strain is different between the experiment and the simulation, a quantitative comparison between the experiment and the simulation is not conducted here.
Relative Activity of Slip Systems In the following, a variation of the relative activity of each family of slip and twinning systems is used to investigate the mechanism of the differential work-hardening behavior. The relative activity is the relative contribution of each family of slip and twinning systems to plastic deformation. The relative activity of each family of slip and twinning systems i is given in the form [11, 21]
∑ ri =
∑
(
p k a family of slip and twinning systems , i
∑ p
(
∑
j all families of slip and twinning systems
)
)
Δγ ( Δγ (
p,k )
p, j )
(8)
where the numerator is the plastic strain increment contributed by the family of slip or twinning systems i, summed over all grains. The denominator is the plastic strain increment contributed by all the families, summed over all the grains. The variations of the relative activity as a function of the plastic work obtained from uniaxial and equi-biaxial p is also shown in Fig. 5 (a). The tension are shown in Fig. 5. The correlation between the plastic work and ε RD variation of the relative activity during the uniaxial tension (σTD/σRD =0) (Fig. 5 (a)) is as follows. The activity of the basal slip systems is predominant at the very beginning of plastic deformation. Shortly after that the prismatic slip systems start to be activated rapidly and at the same time the relative activity of the basal slip systems decreases. During the middle stage of plastic deformation, both the prismatic slip and the basal slip systems are activated but the relative activity of the prismatic slip systems is more predominant. The relative activities of the pyramidal-2 slip
318
1
Relative activity
0.8
0.008
0.6
0.006
(b)
Prismatic
0.4
0.004 Basal
0.2
0.002
Uniaxial plastic strain
0.01
Relative activity
(a)
1 0.8 Basal
0.6 0.4
Pyramidal
0.2
Prismatic
Twinning
0
0 0
0.2
0.4 0.6 0.8 Plastic work / MJ*m-3
0
1
0
0.2
0.4
0.6
0.8
1
Plastic work / MJ*m-3
FIGURE 5. Variations of relative activities as a function of plastic work. (a) Uniaxial tension, and (b) equi-biaxial tension.
and twinning are negligible throughout the deformation simulated in this study. This result shows that the work hardening during the uniaxial tension is determined mainly by the basal slip at the very beginning, while by the prismatic slip and the basal slip after that. The variation of relative activity during the equi-biaxial tension (σTD/σRD =1) (Fig. 5 (b)) is different from that of the uniaxial tension as below. The activity of the basal slip systems is predominant at the beginning as well as the uniaxial tension, but the strain range where this trend is kept is larger than that in the uniaxial tension. Subsequently the prismatic slip and pyramidal-2 slip systems start to be activated gradually. Because the relative activities of the prismatic slip and pyramidal-2 slip systems are much smaller than those in the uniaxial tension during the middle stage, the relative activity of the basal slip systems is kept being predominant. Interestingly the relative activity of the pyramidal-2 slip systems is larger than that of the prismatic slip systems. The relative activity of the twinning systems is negligible throughout the deformation in this study. From this result, the mechanism of the work hardening during the equi-biaxial tension may be explained as follows. The stresses are small in a small plasticstrain range because only the basal slip systems are activated at the beginning, while after that the work-hardening rate increases during the middle stage because not only the basal slip systems but also, to some extent, the pyramidal-2 slip and prismatic slip systems start to be activated gradually. The overall transitions of relative activity are qualitatively similar to those of the uniaxial tension, but the strain range where only the basal slip is activated at the beginning is larger and the increases in the relative activities of the prismatic slip and pyramidal-2 slip are more gradual than the uniaxial tension, thus emphasizing the change in the work-hardening rate from the initial to middle stages. Summarizing the above results, we conclude that the differential work-hardening in a rolled Mg alloy sheet is due to the facts that the slip systems that govern the work hardening are notably different depending on the biaxial stress ratio and that the variation of the relative activity is also different depending on the biaxial stress ratio.
Mechanism of Differential Work-Hardening p Figure 6 shows the correlations between the relative activities and the biaxial stress ratio at ε RD =0.2 % and 1.0 %. In the both results, the relative activity of the prismatic slip systems is the largest at σTD/σRD =0 (uniaxial tension). This decreases as the biaxial stress ratio σTD/σRD increases, whereas the relative activities of other slip systems, i.e. p p the basal slip systems for ε RD =0.2 % and the basal slip and pyramidal-2 slip systems for ε RD =1.0 %, increase with σTD/σRD. It is presumed from these results that one of the main reasons of the differential work-hardening behavior is the relative activity of the prismatic slip systems decreases as the biaxial stress ratio increases. A mechanism of the change in the activity of the prismatic slip systems is examined using the following simple analytical model. The directions of c-axes in rolled Mg alloy sheets in fact exhibit some variation from the sheet normal direction as shown in Fig. 1 (b). However, to simplify the model, an idealized HCP crystal structure whose c-axis trends exactly in the sheet normal direction is considered as shown in Fig. 7. Referring to the x-y coordinate system in Fig. 7, the unit vectors s and m of an arbitrary prismatic slip can be given as
⎧cos θ ⎫ ⎧cos ϕ ⎫ m=⎨ ⎬, s = ⎨ ⎬. ⎩sin θ ⎭ ⎩sin ϕ ⎭
319
(9)
1
(b)
0.8
Basal Prismatic Pyramidal Twinning
0.6 0.4
1
Basal Prismatic Pyramidal Twinning
0.8 Relative activity
Relative activity
(a)
0.2 0
0.6 0.4 0.2 0
0
0.2
0.4 0.6 σTD/σRD
0.8
1
0
0.2
0.4
0.6 σTD/σRD
0.8
1
p p FIGURE 6. Correlations between relative activity and biaxial stress ratio. (a) ε RD =0.2 %, and (b) ε RD =1.0 %.
σTD
σRD
y
ϕ
σTD
m
θ
m
σRD
ϕ − θ = 90D s
s
x
FIGURE 7. A simple analytical model of HCP crystal structure on which biaxial stresses are acting.
Assuming that the biaxial stresses σRD and σTD are acting on the crystal structure in the x and y directions respectively, the resolved shear stress of the prismatic slip τpris can be depicted in the form
τ pris = m ⋅ σ ⋅ s ⎡σ τ pris = {cos θ sin θ } ⎢ RD
τ pris
0 ⎤ ⎧cos ϕ ⎫ ⎬ ⎥⎨ 0 σ ⎣ TD ⎦ ⎩sin ϕ ⎭ = (σ RD − σ TD ) sin ϕ cos ϕ .
(10)
Equation (10) shows that τpris becomes smaller as the biaxial stress ratio σTD/σRD approaches to one, yielding the activity of the prismatic slip systems decreases. The similar trend would arise also in real rolled Mg alloy sheets with some variation of the c-axes from the sheet normal direction. Because the prismatic slip systems are more difficult to be activated for the equi-biaxial tension (σTD/σRD =1) than the uniaxial tension (σTD/σRD =0), the strain range where only the basal slip is activated at the beginning is larger for σTD/σRD =1 (Fig. 5 (b)) than σTD/σRD =0. Furthermore, from the same reason, the relative activity of the prismatic slip systems increases gradually for σTD/σRD =1, but is still smaller during the middle stage than σTD/σRD =0. The relative activities of the basal slip and pyramidal-2 slip systems also change with the change of the activity of the prismatic slip systems. We conclude that such changes in the activities of the slip systems eventually result in the differential work-hardening behavior.
320
CONCLUSIONS The contour of plastic work for an AZ31 magnesium alloy sheet was predicted using a rate-dependent crystalplasticity finite-element method. The mechanism of the differential work-hardening behavior was examined in detail using the relative activity of each family of slip systems. The results obtained in this study are as follows. (1) The contour of plastic work obtained with the simulation exhibits clearly a differential work-hardening behavior; the contour is initially rather flattened in the vicinity of equi-biaxial tension, but severely bulges at the middle stage. This trend is qualitatively in good agreement with the experimental result reported in the literature. (2) The variation of the relative activities during the uniaxial tension shows that the work hardening under the uniaxial tension is determined solely by the basal slip in the very beginning, while by both the prismatic slip and the basal slip in the following deformation. (3) From the variation of the relative activity during the equi-biaxial tension, the mechanism of the work hardening under the equi-biaxial tension may be explained as follows. The stresses are small in a small plastic-strain range because the deformation during which only the basal slip systems are activated in the beginning is kept longer than that in the uniaxial tension. Then the work-hardening rate increases during the middle stage because not only the basal slip systems but also, to some extent, the pyramidal-2 slip and prismatic slip systems start to be activated gradually. Nevertheless, the activity of the basal slip systems is kept being predominant throughout the deformation because the relative activities of the prismatic slip and pyramidal-2 slip systems are smaller than those in the uniaxial tension. (4) The activity of the prismatic slip systems decreases as the biaxial stress ratio increases. A simple analysis reveals that the prismatic slip systems are more difficult to be activated for the equi-biaxial tension than the uniaxial tension because the two biaxial stresses are cancelled with each other. From the same reason, the activity of the prismatic slip systems increases gradually during the middle stage for the equi-biaxial tension, but is still smaller than that of the uniaxial tension. We conclude that such changes in the activities of the slip systems eventually result in the differential work-hardening behavior of the contour of plastic work.
REFERENCES 1. Japan Society of Technology of Plasticity, Magnesium Processing Technology, Tokyo: Corona Publishing Co., Ltd., 2004, pp. 12-33. (in Japanese) 2. B.L. Mordike and T. Ebert, Mater. Sci. Eng. A 302, 37-45 (2001). 3. H.J. Kim, S.C. Choi, K.T. Lee and H.Y. Kim, Mater. Trans. 49, 1112-1119 (2008). 4. J. Kaneko and M. Sugamata, J. JILM 54, 484-492 (2004). (in Japanese) 5. Y.S. Lee, M.C. Kim, S.W. Kim, Y.N. Kwon, S.W. Choi and J.H. Lee, J. Mater. Process. Technol. 187-188, 103-107 (2007). 6. T. Hama, Y. Kariyazaki, K. Ochi, H. Fujimoto and H. Takuda, Mater. Trans. 51, 685-693 (2010). 7. X.Y. Lou, M. Li, R.K. Boger, S.R. Agnew and R.H. Wagoner, Int. J. Plast. 23, 44-86 (2007). 8. Y. Chino, J.S. Lee, K. Sassa, A. Kamiya and M. Mabuchi, Mater. Lett. 60, 173-176 (2006). 9. C.H. Cáceres, T. Sumitomo and M. Veidt, Acta Mater. 51, 6211-6218 (2003). 10. G.E. Mann, T. Sumitomo, C.H. Cáceres and J.R. Griffiths, Mater. Sci. Eng. A 456, 138-146 (2007). 11. T. Hama, H. Takuda, Int. J. Plast. in press (2010). 12. M. O. Andar, D. Steglich, T. Kuwabara, Proceedings of AMPT 2010, 75-80 (2010). 13. Y. Tadano, Int. J. Mech. Sci. 52, 257-265 (2010). 14. M. Kawka, A. Makinouchi, J. Mater. Process. Technol. 50, 105-115 (1995). 15. T. Hama, T. Nagata, C. Teodosiu, A. Makinouchi, H. Takuda, Int. J. Mech. Sci. 50, 175-192 (2008). 16. S. Graff, W. Brocks and D. Steglich, Int. J. Plast. 23, 1957-1978 (2007). 17. D. Pierce, R.J. Asaro and A. Needleman, Acta Metall. 31, 1951-1976 (1983). 18. R.J. Asaro, A. Needleman, Acta Metall. 33, 923-953 (1985). 19. P. Van Houtte, Acta Metall. 26, 591-604 (1978). 20. T. Mayama, K. Aizawa, Y. Tadano and M. Kuroda, Comput. Mater. Sci. 47, 448-455 (2009). 21. C. N. Tomé, R.A. Lebensohn and U.F. Kocks, Acta Metall. Mater. 39, 2667-2680 (1991).
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3D Crystal Plasticity Modelling of Complex Microstructures in Extruded Products S. Dumoulina, J. Friisa, S. Gouttebrozea, B. Holmedalb, K. Marthinsenb a
b
SINTEF Materials and Chemistry, NO-7465 Trondheim, Norway Department of Materials Science and Engineering, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway Abstract. The current study relates to the modelling of plastic anisotropy in aluminium alloy AA6063-W using a ratedependent crystal plasticity finite element approach. A virtual microstructure, in which grains are explicitly resolved, was first generated based on experimental data using a grain growth algorithm. The texture was obtained by sampling the experimental ODF. The microstructure was then meshed using three different methods and appropriate boundary conditions were used in order to simulate uniaxial tensile testing in different directions. The results obtained from the different meshes were finally compared with experimental true stress-true strain curves and discussed. Keywords: Finite Element Method; Texture; Microstructure; Anisotropy; Aluminium. PACS: 02.70.Dh; 81.40.-z; 62.20.F
INTRODUCTION In the extrusion process of 6xxx aluminium alloys the microstructure and texture strongly evolves with the deformation. This leads to strong anisotropy in all mechanical properties, and especially in strain-hardening [1,2]. Furthermore, through-thickness variations in microstructure and texture may also occur during processing. The material thus exhibits an even more complex anisotropic behaviour [3]. The source of this anisotropy is therefore attributed to these through-thickness heterogeneities but has not yet been clearly identified. Attempts in modelling the anisotropic behaviour of extruded alloys using phenomenological approaches, based on advanced anisotropic yield criteria, have usually proven successful when no through-thickness heterogeneity is present [2]. However, difficulties appear when through-thickness variations are present [1,3]. In these approaches, the complex microstructure is considered as homogenous which is suspected to be a too crude approximation. A composite shell element based modelling approach has also been suggested; each integration point of the shell element was given different material parameters, or initial yield surfaces, identified using polycrystalline – Taylortype and visco-plastic self consistent (VPSC) – models and experimental textures [3]. Nevertheless, no improvements were obtained. This was attributed to the inability of shell elements in reproducing through-thickness variations in stresses and strains, and the use of brick elements was therefore recommended. An alternative to the approach described above is the use of crystal plasticity combined with the finite element method (CP-FEM). CP-FEM has proven good capability in predicting strain heterogeneity and anisotropy in polycrystalline materials. Furthermore it has regained interest over the last ten to fifteen years owing to the increase in computational power. This approach explicitly accounts for both microstructure and texture. Evolution of texture and thereby anisotropic hardening are inherently part of the approach [4,5,6]. The current study thus aims at describing plastic anisotropy in AA6063-W by use of CP-FEM. The next section introduces the material, its microstructure and the uniaxial tensile curves. The third section describes the modelling approach: building of the microstructure, meshing, single crystal model, calibration of model parameters and modelling of tensile tests. Then, before the conclusion, the results are presented including comparisons between experiments and simulations.
The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 322-329 (2011); doi: 10.1063/1.3623627 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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MATERIAL DATA The material studied here is the heat-treatable wrought alloy AA6063, containing magnesium and silicon, solution heat treated to W temper, i.e. alloying elements are in solid solution. It is widely used as extruded-andformed part in automotive applications [3]. The material was received as 3-mm thick strip profiles. Figure 1 shows the through-thickness microstructure with micrographs at different positions along the thickness together with corresponding pole figures. It clearly appears that both microstructure and texture vary across the thickness.
(a) (b) FIGURE 1. (a) Micrographs of the microstructure at different positions through the sample thickness with (b) corresponding pole figures [3].
FIGURE 2. True stress-true strain curves until necking for uniaxial tensile tests.
Uniaxial tensile tests were carried out along three different directions, 0°, 45° and 90° with respect to the extrusion direction (ED), at an initial strain rate of 10-3 s-1. The specimens, with a gauge length of 35 mm and a width of 24 mm, were cut out from the strip. The true stress-true strain curves, shown in Fig. 2, were obtained using digital image correlation (DIC) [7]. Note that the accuracy of DIC for these tests was about 0.5% strain thus not
323
allowing computing accurately stresses at small strains. Portevin-Le Châtelier (PLC) bands were observed during the tests as attested by the serrations on the curves. Anisotropy in strength, elongation to necking and hardening is observed.
MODELLING The modelling approach consists of three steps: i) generation of a microstructure and texture, ii) building of the FE model and iii) run of FE simulations.
Generation of Microstructure and Texture First, a representative volume element (RVE) of the material was determined based on micrographs of the microstructure; its volume was estimated to be 1.5×1.5×1.5 mm3. Then a 3D voxel periodic microstructure of volume 1.5×1.5×1.5 mm3 was generated using a Potts Monte Carlo [8] grain growth algorithm based on [9,10,11]. Nuclei are initially positioned in space in order to obtain a microstructure similar to the experimental one (see Fig. 1). The RVE contained then 364 grains. Figure 3.a shows cross-sections of the generated microstructure. Variations in grain size are obtained through the thickness although they are not as large as in the experiments. The texture for the RVE was obtained by sampling a representative set of 364 orientations from the experimental texture using the approach proposed in [12]. Figure 3.b shows selected sections from the global, i.e. throughthickness, experimental and reconstructed orientation distribution functions (ODF). The experimental ODF reveals a sharp rotated cube texture with some Goss component that is smoothened in the reconstructed one due to the low number of orientations. Finally, the reconstructed texture is randomly assigned to the grains. Surface
ND
Experimental ODF
TD ED
ED Surface
ND
Reconstructed ODF TD
(a) (b) FIGURE 3. (a) Cross-sections of the microstructure generated; (b) selected sections from experimental and reconstructed orientation distribution functions.
Finite Element Model The 3D voxel microstructure is converted into a finite element model using three different methods. The first one consists in mapping the microstructure onto the integration points of an N q N q N solid element mesh with N
324
equals to 10 or 20 (hereafter called RVE-IPN); the grain boundaries are not explicitly resolved and several grains can then share the same element. The second method is similar to the previous one, only that the mapping is done onto the elements, i.e. all integration points of each element have the same initial grain orientation (hereafter called RVE-ELN); the grain boundaries are then resolved in a stair-like fashion (Figure 4.a). The last method consists in meshing the microstructure based on the grain boundaries using tetrahedron elements (hereafter called RVE-GB) using the CGAL library [13]; the grains are thus explicitly resolved (Figure 4.b) and about 45390 elements are generated. For the latter, note that some grains may not be convex since they are not convex in the generated microstructure. In all RVEs each grain has a unique initial orientation. Note that 8-node brick elements with 8 integration points are used for the two first RVEs and 4-node tetrahedron elements with 1 integration point are used for the third RVE.
ND
TD ED
(a) (b) FIGURE 4. (a) Element-based mesh (RVE-ELN) and (b) grain boundary-based mesh (RVE-GB).
In all simulations, periodic boundary conditions (e.g. [14]) are applied along the extrusion (ED) and transverse (TD) directions. In the normal direction (ND), the plane representing the surface of the material is unconstrained; however symmetric boundary conditions are applied in the opposite plane representing the mid-thickness of the material. Appropriate nodal boundary conditions are applied in order to avoid any rigid body rotation during the simulations. Tensile conditions are generated by imposing a velocity in the required direction. While this is straightforward for the 0° and 90° directions, this is not the case for the 45° direction. For the latter the texture was then rotated about the ND. Mass scaling was used in all simulations; kinetic effects, which were monitored, were negligible.
Crystal Plasticity Model The crystal plasticity model is defined within the framework of finite deformations, i.e. based on the multiplicative decomposition of the total deformation gradient into elastic and slip-induced plastic parts. It follows a classical approach as e.g. in [15,16]. For FCC metals, slip occurs on the twelve \111^ 110 slip systems. The constitutive model is as follows. Shearing rate on slip system D is defined by the power-law D
D 0 D c
1 m
sgn D
(1)
where 0 is the reference shearing rate, cD the slip resistance, or critical resolved shear stress, and m the microscopic strain rate sensitivity. The slip resistance is a function of the plastic straining
WcD
n
T * ¦ qDE J E
(2)
E 1
where T defines the hardening rate for a given accumulated plastic strain, qDE is the latent hardening matrix, being equal to 1 for self-hardening and q for latent hardening, and * is the accumulated plastic shear strain. The work-
325
hardening rate T is defined as the derivative with respect to * of a modified Voce hardening model in the form (e.g. [17]) N
§
§ Tk
©
© Wk
W c * W 0 ¦W k ¨¨1 exp ¨ k 1
·· *¸¸ ¸ ¹¹
(3)
where W 0 is the initial critical resolved shear stress, assumed to be the same for all slip systems, while W k and T k are the hardening rate and saturated value, respectively, of hardening term k. In the current work, only two terms are used. Note that this model has proven good capability in describing strain-hardening in FCC metals and has furthermore physical grounds [17]. The model has been implemented into the explicit non-linear finite element code LS-DYNA [18] via a user material subroutine developed for solid elements [19]. The model parameters were calibrated against the experimental curve for the 90° direction with the design optimisation tool LS-OPT [20] using RVE-IP10 described in the previous section. In order to improve the efficiency of the calibration m , 0 and q were taken from the literature (e.g. [16]). These parameters were then fixed in the calibration while the others were free. The values obtained after calibration are given in Table 1. TABLE 1. Model parameters used in the simulations. q m 0 (/s) W 0 (MPa) 0.001
0.01
1.4
12.6
T1 (MPa)
W 1 (MPa)
T 2 (MPa)
W 2 (MPa)
41.4
25.6
148.7
13.9
RESULTS AND DISCUSSION Figure 5 shows the comparison between experiments and simulations using RVE-IP10 for the three directions of interest i.e. 0°, 45° and 90°. A perfect fit is obtained at 90° thus validating the parameter identification. However, even though the model predicts an anisotropic material response, the trend observed experimentally is not reproduced. The simulations give similar strain-hardening in both 0° and 90° directions while strain-hardening is much higher in 0° direction in the experiments up to a strain of about 0.08-0.09. The maximum difference in flow strength between the experiment and the simulation in 0° direction is 18% at a strain of about 0.06. Furthermore, in 45° direction the strain-hardening obtained in the simulation is higher than in the experiment leading to differences in strength up to 14% at a strain of about 0.095.
FIGURE 5. Comparison between experimental and numerical (RVE-IP10, RVE-IP20) true stress-true strain curves.
In order to check whether mesh sensitivity has an effect, all simulations were rerun using a finer mesh, RVEIP20 (Figure 5). While a finer mesh does not affect the results in neither 0° nor 90° directions, an minor improvement is observed in 45° direction: the maximum difference in flow strength between the experiment and the simulation is now about 11% i.e. an improvement of about 2%.
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Next, simulations were run using different meshes, i.e. based on elements (RVE-EL20) and grain boundaries (RVE-GB). Figure 6 shows the true stress-true strain curves obtained for the different meshes, including the one based on integration points using a fine mesh (RVE-IP20). Meshes based on integration points (RVE-IP20) and elements (RVE-EL20) give identical results with differences less than 0.7%. However, simulations run with a mesh based on grain boundaries (RVE-GB) show a slightly higher strain-hardening; maximum differences between RVEGB and RVE-EL20 are about 3% for all directions. Finally, in order to assess whether accounting for the microstructure, i.e. grain size and shape, has any influence on the material response, simulations were run using the Taylor full constraint model. The Taylor model used here was built within the framework described previously, i.e. based on finite deformations and a rate-dependent formulation. The results are included in Fig. 6. The Taylor model gives results identical to those obtained with RVEGB.
(a)
(b)
(c) FIGURE 6. Comparison between true stress-true strain curves obtained experimentally and using different RVEs, for each direction: (a) 0°, (b) 45° and (c) 90°.
RVE-IP20, RVE-EL20 and RVE-GB give similar results at the macro-scale, i.e. true stress-true strain curves, with RVE-GB generating slightly higher strain-hardening. In order to assess whether these meshes also provide similar results at the micro-scale, i.e. at the grain level, the plastic strain in each RVE in uniaxial tensile testing in 90° direction is plotted in Fig. 7. While RVE-IP20 and RVE-EL20 give almost identical results, RVE-GB generates about 10% lower plastic strain. This figure also emphasises the main difference between CP-FEM and high grainmatrix interaction models such as Taylor, i.e. CP-FEM provides inter- and intra-granular resolution. The above analysis thus implies that, in the current study, the microstructure does not play any role in the anisotropy of the material while texture is of prime importance.
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ND TD ED (a)
(b)
(c)
FIGURE 7. Plastic strain field in (a) RVE-IP20, (b) RVE-EL20 and (b) RVE-GB, for tensile testing in 90° direction.
As suggested in [3], the anisotropic hardening of AA6063-W is mainly attributed to the ED-rotated cube texture. It is suggested that the microstructure has little if no effect since AA6063-W has equi-axed grains with a size, even though varying through the thickness, expected to be above any value giving strength contribution. In materials having symmetric cube texture, uniaxial tensile responses in 0° and 90° directions are identical while higher strainhardening is observed in the 45° direction [21]. In the current study, 0° and 45° directions show similar strainhardening while higher strain-hardening is observed in the 0° direction which is attributed to the ED-rotated texture. Furthermore, different width contractions were observed through the thickness of the specimens, depending on the tensile direction, i.e. plastic strain ratios (R-values) varied through the thickness [3]. This effect was attributed to heterogeneities in both texture and microstructure across the thickness. Variations in R-values through the thickness will lead to strain incompatibilities and thereby change the stress state in the sheet which in turn may affect the anisotropy. In general, the contribution from these through-thickness variations on anisotropic hardening is difficult to assess due to the strong correlations between the different variables involved. Based on the above analysis, weaknesses in the modelling approach can be identified. The main drawback resides in texture modelling i.e. the through-thickness texture variation was not accounted for in the current modelling. A second source of errors related to texture lies in the ODF measurement: the latter was obtained based on data from X-ray diffraction which, for heterogeneous microstructures, may lead to overestimation of some texture components. Furthermore, in the modelling approach, an homogeneous distribution of dislocations was assumed: the initial slip resistance W 0 was identical for all slip systems, which is reasonable for annealed materials but not for work-hardened materials, i.e. anisotropy due to prior deformation (extrusion) was ignored. This may however have a significant role as shown in [22]. Finally, the size of the RVE investigated in this study was not validated; this may also affect the response of the material.
CONCLUSION In the current study modelling of plastic anisotropy in aluminium alloy AA6063-W was investigated using a multiscale approach with a strong experimental background. A RVE 3D periodic microstructure was generated based on micrographs of the microstructure using a Potts Monte Carlo grain growth algorithm. The texture for the RVE was obtained by sampling the experimental ODF. This microstructural and textural RVE was then meshed using three different methods. A rate-dependent CP-FEM modelling approach was then used together with a modified Voce hardening model. Simulations of uniaxial tensile tests in three different directions were then performed. Comparison between numerical and experimental results revealed that the modelling approach had difficulties in reproducing the anisotropic material response observed experimentally. Comparison with simulations performed with the Taylor FC model showed that, for the approach chosen here, the microstructure had no effect on the anisotropy and all meshes gave similar responses. These difficulties were attributed to various sources relating to heterogeneities in texture and material properties: both variation of texture through the thickness and heterogeneity in dislocation distributions, or material history, were not accounted for.
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Ongoing work aims at accounting for through-thickness texture variation using experimental ODF at different positions across the thickness (see Fig. 1.b). In addition, the effect of the size of the RVE on the material response will also be investigated.
ACKNOWLEDGMENTS The authors would like to thank Dr. S.K. Fjeldbo for providing all experimental data. The financial support of this work from the “Through Process Modelling” Hydro-Fund project is gratefully acknowledged.
REFERENCES 1. K. Pedersen, O.-G. Lademo, T. Berstad, T. Furu and O.S. Hopperstad, J. Mater. Process. Technol. 200, 77-93 (2008). 2. D. Achani, O.S. Hopperstad and O.-G. Lademo, J. Mater. Process. Technol. 209, 4750-4764 (2009). 3. S.K. Fjeldbo, “Studies of the influence of strong through thickness variation on the plastic floe properties of extruded AA6063-W aluminium strips and tubes, with application to the modelling and simulation of tube hydroforming of T-shapes”, Ph.D. Thesis, Norwegian University of Science and Technology, 2008. 4. Z. Zhao, S. Kuchnicki, R. Radovitzky and A. Cuitiño, Acta Mater. 55, 2361-2373 (2007). 5. K. Inal, R.K. Mishra and O. Cazacu, Int. J. Solids Structures 47, 2223-2233 (2010). 6. F. Roters, P. Eisenlohr, L. Hantcherli, D.D. Tjahjanto, T.R. Bieler and D. Raabe, Acta Mater. 58, 1152-1211 (2010). 7. S. Dumoulin, L. Tabourot, C. Chappuis, P. Vacher and R. Arrieux, J. Mater. Process. Technol. 133, 79-83 (2003). 8. R.B. Potts, Proc. Cambridge Philos. Soc. 48, 106-109 (1952). 9. A. Brahme, M.H. Alvi, D. Saylor, J. Friday and A.D. Rollett, Scr. Mater. 55, 75-80 (2006). 10. A. Brahme, J. Fridy, H. Weiland and A.D. Rollett, Modell. Simul. Mater. Sci. Eng. 17, 015005 (2009). 11. E. Fjeldberg and K. Marthinsen, Comput. Mater. Sci. 48, 267-281 (2009). 12. P. Eisenlohr and F. Roters, Comput. Mater. Sci. 42, 670-678 (2008). 13. The CGAL project homepage, www.cgal.org (2010). 14. A. Prakash, S.M. Weygand and H. Riedel, Comput. Mater. Sci. 45, 744-750 (2009). 15. D. Peirce, R.J. Asaro and A. Needleman, Acta Metall. 30, 1087-1119 (1982). 16. S.R. Kalidindi, C.A. Bronkhorst and L. Anand, J. Mech. Phys. Solids 40, 537-569 (1992). 17. Y. Estrin, J. Mater. Process. Technol. 80-81, 33-39 (1998). 18. J.O. Hallquist, LS-DYNA keyword user’s manual, Version 971, California: Livermore Software Technology Corporation (2007). 19. S. Dumoulin, O.S. Hopperstad and T. Berstad, Comput. Mater. Sci. 46, 785-799 (2009). 20. N. Stander, W. Roux, T. Goel, T. Eggleston and K. Craig, LS-OPT user’s manual, Version 4.1, California: Livermore Software Technology Corporation (2010). 21. A.B. Lopes, F. Barlat, J.J. Gracio, J.F. Ferreira Duarte and E.F. Rauch, Int. J. Plast. 19, 1-22 (2003). 22. S.R. Kalidindi and S.E. Schoenfeld, Mater. Sci. Eng. A 293, 120-129 (2000).
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Anisotropic Sheet Forming Simulations Based on the ALAMEL Model: Application on Cup Deep Drawing and Ironing P. Eyckensa,b*, J. Gawadc,d, Q. Xiea,b, A. Van Baela,e, D. Roosec, G. Samaeyc, J. Moermanf, H. Vegterf and P. Van Houttea a
Materials innovation institute, Mekelweg 2, 2600 GA Delft, the Netherlands Department of Metallurgy and Materials Engineering (MTM), Katholieke Universiteit Leuven, Kasteelpark Arenberg 44, 3001 Heverlee, Belgium c Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200A, 3001 Leuven, Belgium d Department of Applied Computer Science and Modeling, AGH University of Science and Technology, al. Mickiewicza 30, Krakow, 30-059, Poland e Department IWT, Limburg Catholic University College (KHLim), Campus Diepenbeek, Agoralaan Gebouw B, bus 3, 3590 Diepenbeek, Belgium f Tata Steel Research, Development & Technology, IJmuiden Technology Centre, PO Box 10.000, 1970 CA, IJmuiden, The Netherlands *Corresponding author: tel (+32) 16 321305, fax (+32) 16 321990,
[email protected] b
Abstract. The grain interaction ALAMEL model [1] allows predicting the evolution of the crystallographic texture and the accompanying evolution in plastic anisotropy. A FE constitutive law, based on this multilevel model, is presented and assessed for a cup deep drawing process followed by an ironing process. A Numisheet2011 benchmark (BM-1) is used for the application. The FE material model makes use of the Facet plastic potential [2] for a relatively fast evaluation of the yield locus. A multi-scale approach [3] has been recently developed in order to adaptively update the constitutive law by accommodating it to the evolution of the crystallographic texture. The identification procedure of the Facet coefficients, which describe instantaneous plastic anisotropy, is accomplished through virtual testing by means of the ALAMEL model, as described in more detail in the accompanying conference paper [4]. Texture evolution during deformation is included explicitly by re-identification of Facet coefficients in the course of the FE simulation. The focus of this paper lies on the texture-induced anisotropy and the resulting earing profile during both stages of the forming process. For the considered AKDQ steel material, it is seen that texture evolution during deep drawing is such that the anisotropic plastic flow evolves towards a more isotropic flow in the course of deformation. Texture evolution only slightly influences the obtained cup height for this material. The ironing step enlarges the earing height. Keywords: Anisotropy, Texture Evolution, Crystal Plasticity, Multi-Scale Modeling, Finite Element Method. PACS: 46.70.De, 81.40.Ef, 83.50.-v, 62.20.F-
INTRODUCTION It is well recognized that the crystallographic texture of sheet metal largely determines the anisotropic characteristics of plastic flow in pressing operations. For axisymmetric deep drawing, it may lead to a macroscopic shape effect called earing. In terms of plastic anisotropy properties, not only the directional anisotropy in the flow (often characterized by r-value) may contribute significantly to earing, but also the directionality in flow stress [5]. It is even the case for yield strength differences as low as 2MPa [5], which are close to the measurement accuracy for many tensile test devices. Besides phenomenological yield loci, which rely on experimental measurements, another valuable approach consists of multi-level modeling of the texture-induced anisotropy. This last approach will be followed in this paper, by adopting a scheme that considers not only the initial texture, but also the deformation textures obtained by the ALAMEL multilevel model, as proposed in [3]. The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 330-336 (2011); doi: 10.1063/1.3623628 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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DESCRIPTION OF EXPERIMENT AND SIMULATIONS Experimental set-up In the axisymmetric cup drawing and ironing experiment of the NumiSheet2011 Benchmark-1 (BM-1), a circular blank (38.062mm radius) is deep drawn under the action of a punch with external radius of 22.860mm and fillet radius of 2.229mm. Punch speed is set to 70.0mm/s. A constant blank holding force of 8.9 kN is applied throughout the process. The die has an orifice with a radius of 23.368 and a fillet radius of 1.905mm. After the circular cup is formed, the punch passes through an ironing ring with internal radius of 23.025mm and an entrance angle of 8°. Two materials are described in the benchmark, one of which (AKDQ steel with initial thickness of 0.229mm) is reported on in this paper. For more details on the experimental set-up, the reader is referred to the BM-1 description of these proceedings.
Finite Element Model The forming process is modeled by explicit Finite Element (FE) simulations, using the commercial FE software Abaqus. The blank mesh consists of three layers of first-order solid brick elements with reduced integration. Exploiting the orthorhombic sample symmetry of the initial sheet produced by rolling, only one quarter of the blank is considered in the FE model, which is divided in 3375 solid elements in total. The initial mesh can be seen in Figure 2(a). All tools are modeled as analytic rigid surfaces. Punch speed is set to 5000.0mm/s in the explicit simulation (time scaling) as a compromise between calculation time efficiency and approximation of the quasi-static forming conditions. The adopted kinematic contact algorithm avoids any penetration of blank nodes into the (analytically defined) tool surfaces (‘hard contact’). Coulomb friction is assumed between all contacting surfaces (friction coefficient: 0.05).
Multi-scale Model The multi-scale modeling scheme adopted comprises a macro-scale description of the plastic anisotropy in the form of a Facet plastic potential expressed in strain rate space [2]. As shown in [2], its convexity can be assured. A polynomial of order 6 is currently chosen. It is furthermore assumed that the material is plastically incompressible and rate-insensitive. The Facet potential, being an analytic expression, is relatively efficient to evaluate whenever required in the course of the FE simulation (compared to a multilevel model). Starting from a measured initial texture (or calculated deformation texture), the grain-interaction ALAMEL model [1] is invoked in order to identify the coefficients of the Facet potential. This identification procedure is detailed further in [2,4]. The ALAMEL model is also adopted to make predictions of deformation textures after a pre-determined amount of straining (an interval of 0.1 in equivalent strain is currently chosen). Since this scheme uses the local deformation history (i.e. as obtained from a FE integration point), the texture evolves differently for sections of the sheet that are strained differently. Also the plastic anisotropic properties will then evolve accordingly. Isotropic hardening of the sheet is furthermore assumed (besides the textural hardening described above) by specification of an equivalent stress-strain relationship. The multi-scale modeling scheme thus considers the mutual effect between the local history of plastic deformation and the local plastic anisotropy. More details on the adopted multi-scale models can be found in the accompanying conference proceedings [4] and in papers [6,7].
Material Data The sheet considered is an aluminum-killed drawing quality (AKDQ) steel of 0.229mm thickness. Table (1) mentions all the material data used: density , Young’s modulus E, Poisson coefficient (isotropic elasticity is assumed), and the parameters K, 0 and n of the isotropic Swift hardening law: V eq
K H H 0 , in which and n
are the equivalent stress and strain, respectively. The Swift parameters are derived from uniaxial tensile tests along the rolling direction. The parameter 0 is set in order to let the initial yield stress of the material law equal the experimental yield stress Y=315.7MPa.
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[kg/m³] 7800.0
TABLE (1). Material data used for AKDQ steel E K 0 [/] [GPa] [/] [MPa] 210.0 0.3 599.7 9.89E-3
n [/] 0.139
It should be pointed out that all other mechanical test data that is available from the benchmark (tensile test data in other directions, equal biaxial tension test data, disk compression test data) is not required to obtain the results shown in this paper (and so they have not been utilized). The (initial) anisotropy of the sheet is derived exclusively from measurement of the crystallographic texture, which has been provided in the BM-1 description and is shown in Figure 1.
FIGURE 1. The 1=0°, 2=45° and 1=90°sections of the ODF measured on the as-received sheet, assuming orthorhombic sample symmetry. Texture index: 1.93.
Figure 1 shows three sections of the initial ODF in which the main texture components of ferritic steels are found. The 2=45° section contains the -fibre as well as the -fibre. The -fibre contains all orientations having a direction parallel to the rolling direction RD, such as the H (001)[110], J (114)[1-10], I (112)[1-10] and E (111)[110] components. The -fibre consists of all orientations with a direction parallel to the normal direction ND, such as E (111)[1-10], E’ (111)[0-11], F (111)[1-21] and F’ (111)[-1-12]. This particular steel texture consists of the complete -fibre and partial -fibre, having a similar intensity. In addition, also a relatively strong fibre from rotated H-component to Cube (001)[100] is present.
RESULTS Deformed shape The deformed meshes after respectively deep drawing and ironing stages are shown in Figure 2(b-c), while the corresponding cup profiles are shown in Figure 3. After drawing a profile with 4 ears, each at about 45° with respect to the rolling direction, is obtained. The ironing process obviously increases the average cup height. Besides this, the earing height is also more pronounced: it increases from about 0.35mm after drawing to 0.65mm after ironing. The consideration of texture evolution is seen to have only a very limited effect on the earing behaviour. It leads to a local change in cup height of about 0.1mm at most after the ironing stage. After the drawing stage, the difference is neglectable. In a recent study using the same modelling approach for the deep drawing process [8], it is seen that texture evolution does not improve earing predictions for a low carbon steel and Mn-Fe-alloyed aluminium, both
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having 4 ears, while a noticeable improvement of accounting for texture evolution is seen for an IF steel with 6 ears and a commercially pure aluminium sheet with 8 ears.
(a)
(b)
(c)
A90 B90 A45 B45
TD RD
B0 A0
FIGURE 2. (a) mesh of one quarter of blank (undeformed state). Deformed mesh after (b) deep drawing stage and (c) ironing stage. Some specific elements are highlighted and labeled.
Cup height [mm] After ironing
After drawing Angle to RD FIGURE 3. Prediction of cup height profiles without texture evolution (full line) and with texture evolution (dashed line).
Texture evolution The Figures 4-5 show the evolution of texture, as predicted by the ALAMEL model, for the labeled elements of Figure 2(a). In each case, texture is shown for the central of three solid elements used in the sheet thickness direction. It can be seen that, after the deep drawing process, the original fibrous texture is mostly broken up into individual texture components with a particular spread around them, close to the ideal components I (112)[1-10], E’ (111)[011] and Cube (001)[100]. Although these components are quite similar for all different locations, relative intensities and the shape of the texture spread around ideal orientations are clearly distinct for different positions with respect to the original RD. The texture evolution is very similar for the corresponding elements from A- and B-zones that are located on the same angular position, while larger qualitative differences are seen between the three different angular positions under consideration. The ironing process appears to decrease the orientations near Cube in favour of components near the ideal Iorientation. Also the E’-component is seen to be strengthened. In addition, a change in shape of textural spread around E’ within the shown sections of ODF is systematically seen during the ironing process. The ‘vertically’ oriented spread at the end of deep drawing (cf. for instance the dotted lines in Fig. 4(a)) appears to be rotated after ironing (cf. for instance the dotted lines in Fig. 4(b)). It implies that that the ironing causes these orientations near the E’-orientation to rotate (on average) towards the perfect -fibre, i.e. aligning a {111} plane with the sheet plane. The texture evolution during ironing is more pronounced as the location is closer to the cup rim (compare Fig. 4(b)
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to Fig. 5b)). Sheet thickness increases towards the cup rim in the intermediate state after deep drawing, and so significantly more plastic straining takes place during the ironing stage for the A-elements compared to the Belements.
(a)
(b)
(c)
(d)
(e)
(f)
FIGURE 4. 2=45°-sections of the calculated ODFs in (a-b) A0-, (c-d) A45- and (e-f) A90-elements. (a),(c),(e): at = 0.5 (approximately at the end of drawing stage). (b),(d),(f): final textures (after the ironing stage). ODF intensity lines: 1.0, 1.4, 2.0, 2.8, 4.0, 5.6, 8.0, 11.0, 16.0, 22.0.
(a)
(b)
(c)
(d)
(e)
(f)
FIGURE 5. 2=45°-sections of the calculated ODFs in (a-b) B0-, (c-d) B45- and (e-f) B90-elements. (a),(c),(e): at = 0.5 (approximately at the end of drawing stage). (b),(d),(f): final textures (after the ironing stage). ODF intensity lines: 1.0, 1.4, 2.0, 2.8, 4.0, 5.6, 8.0, 11.0, 16.0, 22.0.
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Evolution of Mechanical Properties Figure 6 presents evolution of (a part of) the anisotropic plastic behavior in the form of q-values. It is defined as the ratio of plastic contraction in the sheet width direction per unit tensile elongation in uniaxial deformation. The qvalue is bounded in the interval [0;1] and is related to the r-value (bounded in [0;]) by q=r/(r+1). Reported values are calculated by application of the Full Constraints (FC) Taylor model on the experimentally measured texture (for = 0) and textures predicted by the ALAMEL model (for > 0). Both the local radial (‘q0°’) as the local circumferential (‘q90°’) directions are considered as the deformation axes.
q-value
q-value
(a)
drawing
ironing
drawing
(b)
ironing
FIGURE 6. Evolution of FC Taylor-based q-value predictions for uniaxial tension/compression along the local radial direction (‘q0°’) and along the local circumferential direction (‘q90°’), for different elements (in (a): ‘A0’,’A45’ and ’A90’, in (b): ‘B0’,’B45’ and ’B90’,), cf Fig. 2(a). The horizontal axis shows the (local) equivalent plastic strain . Transition from drawing to ironing stage occurs in both (a-b) at 0.5 < < 0.6.
For all the considered elements, the evolution of q90° is seen to converge to about 0.5 (isotropic behavior) at the end of deep drawing. Near the rim of the blank, the local anisotropic yielding during a deep drawing process is actually described very well by q90°, as the deformation is approximately uniaxial compression1 along the local circumferential direction. So as deformation progresses, the texture is evolving in such a way that the anisotropy associated to the current deep drawing mode of deformation, is diminishing. This explains the rather limited effect of the incorporation of texture evolution on the earing profile for this material (Figure 3). On the other hand, the q0°values do not converge but show a decreasing trend during the deep drawing process. Other directions besides 0° and 90° have also been studied; only q90°-values from different locations showed the tendency to converge to each other. So it appears that the properties of anisotropic flow that are not ‘active’ during the deep drawing generally do not evolve towards isotropy in the course of deformation. The change of deformation mode occurring in the transition from drawing to ironing results in a noticeable change in evolution of the q-values. An effect of the intermediate deformation textures is seen. For instance the q0°values evolve somewhat differently for the elements A0, A45 and A90 during ironing. Correspondingly, the change in cup height as a result of texture evolution after ironing (cf. Figure 3) depends on the angle with respect to RD.
CONCLUSIONS This paper demonstrated the feasibility of a multi-scale modeling approach on the subsequent processes of deep drawing and ironing. Simulation results for the investigated AKDQ steel sheet show a 4-earing pattern after drawing that is hardly affected by consideration of texture evolution. After an ironing step, the earing profile is more pronounced and slightly more dependent on deformation texture. A detailed analysis of the deformation texture predicted by the ALAMEL model and corresponding evolution in flow properties leads to the conclusion that the 1
The uniaxial tension and compression deformation modes along a particular direction have the same q-value for the FC Taylor model with critical resolved shear stresses that are independent on the sense of plastic slip [1].
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plastic flow during deep drawing becomes increasingly isotropic. This increase of isotropy refers exclusively to the active deformation mode during deep drawing. The simulated cup height after deep drawing complies with previous findings of the same authors [8]. In this paper, earing profiles for 4-eared materials appear to have only marginal benefits from consideration of texture evolution, while clear improvements are on the other hand seen for materials that show either 6 or 8 ears, i.e. materials exhibiting more complex anisotropic behavior. Future research will be focused on those materials, more in particular by assessment of possible further improvements through increase of the mesh density in the FE model, and/or refinements in the identification procedure of the Facet plastic potential in the multi-scale model.
ACKNOWLEDGMENTS This research was carried out under the project number M41.2.08307a/M41.10.08307b in the framework of the Research Program of the Materials innovation institute M2i (www.m2i.nl). The authors gratefully acknowledge the financial support from the project IDO/08/09, funded by K.U.Leuven, and from the Interuniversity Attraction Poles Program from the Belgian State through the Belgian Science Policy agency, contracts IAP6/24 and IAP6/4. GS is Postdoctoral Fellow of the Research Foundation – Flanders (FWO).
REFERENCES 1. Van Houtte, P., Li, S., Seefeldt, M., Delannay, L., 2005. Deformation texture prediction: from the Taylor model to the advanced Lamel model. Int. J. Plasticity 21, 589-624. 2. Van Houtte, P., Yerra, S.K., Van Bael, A., 2009. The Facet method: A hierarchical multilevel modelling scheme for anisotropic convex plastic potentials. Int. J. Plasticity 25, 332-360. 3. Gawad, J., Van Bael, A., Yerra, S., Samaey, G., Van Houtte, P. and Roose, D. A coupled multiscale model of texture evolution and plastic anisotropy. In Barlat, F. (Ed.), Moon, Y. (Ed.), Lee, M. (Ed.), AIP Conference Proceedings: Vol. 1252 (1). NUMIFORM 2010. Pohang, Korea, pp. 770-777. 4. Gawad, J. et al.: Proc. Numisheet 2011 conference (submitted). 5. Mulder, J. and Vegter, H.,2011. An analytical approach for earing in cylindrical deep drawing based on uniaxial tensile test results. Proc. Esaform2011 (accepted). 6. Gawad J., Van Bael A., Eyckens P., Van Houtte P., Samaey G., Roose D., Effect of texture evolution in cup drawing predictions by multiscale model. Steel Research International, Supplement Metal Forming, 81/9 (2010) 1430-1433. 7. Van Bael A., Eyckens P., Gawad J., Samaey G., Roose D., Van Houtte P., Evolution of crystallographic texture and mechanical anisotropy during cup drawing. Steel Research International, Supplement Metal Forming, 81/9 (2010), 1392-1395. 8. Van Bael, A., Gawad, J, Eyckens, P., Samaey, G., Roose, D., Van Houtte, P., Modelling texture anisotropy and its evolution in sheet forming processes, Proc. ICTP2011 (accepted).
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Microstructure Evolution and Mechanical Properties of Al/Al-Mg/Al composite sheet metals Jaehyung Cho∗ , Su-Hyeon Kim∗ , Hyoung-Wook Kim∗ , Cha-Yong Lim∗ , Eun-Young Kim† and Shi-Hoon Choi† ∗
Korea Institute of Materials Science, 797 Changwondaero, Seongsan-gu, Changwon, Gyeongnam, 642-831, South Korea † Department of Materials Science and Metallurgical Engineering, Sunchon National University, Sunchon, 540-742, South Korea
Abstract. Two different types of aluminum alloys of AA1050 and AA5182 were used to manufacture Al/Al-Mg/Al composite sheet metals by roll bonding technology at room temperature. The composite sheet metals were annealed at 400 o C and carried out uniaxial tension tests to investigate mechanical properties. Macroscopic mechanical properties are strongly dependent on the volume (or thickness) fraction of two component layers. Microstructure and texture evolution were also investigated during roll bonding process. The AA1050 sheets located in the outer layer mainly consist of shear texture components and the AA5182 sheet located in the center layer consists of plane strain texture components. With differential speeds of the top and bottom rolls, roll bonding was also carried out. Elongation along the RD and TD was improved at a speed difference of approximately 10% − 20%. Keywords: cladding, asymmetric rolling, texture and microstructure, aluminum alloys PACS: 62
INTRODUCTION One of the most frequently used clad processes is a roll bonding. During roll bonding, surface of the sheets should be cleaned and degreased, and then be carried out scratch brushing with fine wires. The surface layers on the metals usually contain non metallic films such as oxide films, which often prevent virgin metals from being tightly bonded. Scratches on the surface increase the contact area and increase bonding strength. In addition to the surface preparation, rolling temperature and speed, reduction in area, contact friction, and geometry of sheets and rolls would also play an important role in successful fabrication of the clad materials [1, 2, 3, 4, 5, 6]. There exists some critical or threshold reduction in area to break surface oxide films, below which no bonding occurs. The threshold reduction is associated with threshold strain and strength [7]. The former indicated bonding of base metals occurs when strain is greater than the threshold value. The latter implies that the bonded interface would not be separated when bonded strength is larger than the threshold. Increased reduction in area usually resulted in increased bonding strength. Elevated temperature of sheets at the roll entry also helps to make further reduction in area and increases bonding strength. It is necessary to optimize reduction in area, and annealing time and temperature to avoid undesirable intermetallic compound layer or properties [8, 9, 10, 11]. Roll bonding of dissimilar materials is a naturally asymmetric rolling process, because their plastic responses were different [12, 13]. Sometimes, the shearing effect was intentionally used to fabricate clad materials [14, 15]. The rolling process was designed with a hard material contacting with a high speed roll, and a soft materials contacting with a slow speed roll. The asymmetrical bonding made a uniform metal flow of both metals [6]. This is because cross shear zone occurred and decreased friction hill during asymmetric rolling. Our main goal is to better understand cladding process and the resultant mechanical properties, and to finally fabricate high strength structural clads with enhanced surface treatment capability. The clad system of interest is AA1050/AA5182/AA1050, which seems to contain both high strength and elongation, and good anodizing properties. It is known that shear deformation is favorable to enhancement of the mechanical properties of aluminum alloys [16, 17, 18, 19]. Therefore, asymmetric rolling process was applied to roll bonding process in order to fabricate three-layer clads of Al/Al-Mg/Al with enhanced mechanical properties. Variation of microstructure and mechanical properties was investigated with speed ratio.
The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 337-342 (2011); doi: 10.1063/1.3623629 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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TABLE 1. Chemical composition of clad materials Si
Fe
Cu
Mn
Mg
Cr
Zn
Ti
AA5182
0.08
0.18
0.05
0.45
4.68
AA1050
0.04
0.37
0.07
-
-
Al
0.02
0.03
0.01
Bal.
-
0.01
-
Bal.
EXPERIMENTS Commercial aluminum alloys of AA1050 and AA5182 were used for fabrication of cladding composite. The thicknesses of the AA1050 and AA5182 sheets were 0.5 mm and 2 mm, respectively. Chemical composition of them was given in Table 1. Sample preparation of the aluminum sheets was carried out by degreasing with acetone, and followed by surface scratch-brushing using the wire brush. The diameter of the brusher is 200 mm, and each stainless steel brusher wire is 0.3 mm in diameter. Rotational speed of the brusher was 300 rpm (revolution per minute) and sample feeding speed was 0.2 mpm (meter per minute). The clad materials were stacked in a sequence of top AA1050/center AA5182/bottom AA1050, and they were clamped with steel wire to avoid their separation during roll bonding. The top and bottom sheets of AA1050 imply that they contact the top and bottom rolls, respectively. The AA5182 sheet was located between the AA1050 sheets. The diameter of the rolling mill was 280 mm and the rotational speeds of the top and bottom rolls were separately operated. Overall reduction in area was 50% per pass at the room temperature. The velocity of the top roll Vtop was held constant at 5 mpm, and that of the bottom roll Vbot was varied intentionally at 5, 5.5, 6, 6.5, and 7 mpm, and thus the speed ratio of the top to bottom rolls was varied from 1:1 to 1:1.4. Texture and microstructure were characterized using XRD (X-ray diffraction), OM (optical microscope), and EBSD (electron backscatter diffraction)/SEM. Macroscopic texture measurements were carried out by XRD on the surface region of the bonded sheets. The AA1050 sheet contacting the top roll was measured both sides, the one contacting the top roll, and the other contacting the AA5182. The center AA5182 sheet was measured on the side contacting the top AA1050 sheet. The XRD system used was Xpert Pro (PANalytical) with Cu-Kα radiation operated at 40 kV and 30 mA. Three incomplete (111), (200), and (220) pole figures were collected on a 5◦ grid up to a sample tilt of 70◦ . The obtained data were analyzed using the WIMV method with an automated conditional ghost correction [20, 21], assuming cubic crystal and triclinic sample symmetries. Microstructure and microtexture analyses were carried out using an automated high-resolution EBSD (JEOL7001F) with HKL Channel5 and a generalized EBSD data analysis code, REDS [22]. EBSD samples were mechanically polished and then electropolished using a solution of ethanol (60 ml), and perchloric acid (10 ml) at a voltage of 10 V and a temperature of −15◦ C to −20◦ C. Samples observed by OM were prepared by mechanical polishing followed by subsequent electro-chemical etching with a solution of fluoroboric acid (5 ml), and distilled water (95 ml). Annealing heat treatments were carried out with clad samples before tensile tests. Tensile specimens with a gauge length of 12.5 mm and a width of 3 mm were machined according to the ASTM E8 standard, with the tensile axis at 0◦ , and 90◦ from the rolling direction (RD). The tensile tests were conducted on a standard universal testing machine (Instron 4206) at ambient temperature with a head speed of 1.25 mm/min.
RESULTS AND DISCUSSION Figure 1 displays schematic sequence of roll bonding, heating and mechanical tests of AA1050/AA5182/AA1050 clads. Texture and microstructure evolution of as-rolled clad sheets during clad were investigated using both XRD and EBSD. Mechanical properties were measured through tensile tests after annealing at elevated temperature of 400◦ C for 1 hr. Initial microstructure and texture of AA1050 and AA5182 sheets were presented in Fig. 2. The AA1050 sheet was an as-rolled state and elongated grains were observed. The complete (111) pole figures showed the typical fcc rolling fiber or plane strain compression (PSC) texture, α(Goss{110}001 ∼ Brass{110}112) and β (Brass{110}112 ∼ S{123}634 ∼ Copper{112}111). The complete (111) pole figures of AA1050 were computed from both EBSD and XRD experiments based on the cubic crystal and triclinic sample symmetries. There existed some minor difference between pole figures computed from EBSD and XRD. The EBSD results usually provide more localized information, and both α and β fibers looked well-developed. While, the XRD results provide more bulky information, and the Copper texture component looked stronger than others in the AA1050 sheet. The overall microstructure of the AA5182 sheet showed equiaxed grain shapes and revealed that it was fully-recrystallized. The (111) pole figure computed from
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FIGURE 1.
Schematic sequence of clad fabrication and heat treatments followed by mechanical tests.
FIGURE 2. Initial microstructure (inverse pole figure maps) and texture (111 pole figures) of AA1050 and AA5182 sheets. Contours for XRD PFs: 1, 2, 3, 5, 7, 10.
EBSD showed both Cube{100}001 and retained Brass type texture. The (111) pole figure by XRD clearly presented strong Cube texture component. Some minor components also existed. Figure 3 illustrated bent-shaped clad sheets and optical micrographs through thickness. From the optical micrographs, the outer layers of AA1050 were distinguished from the center (core) layer of AA5182. The elongated grains in the as-rolled AA1050 sheets were much more elongated during roll bonding. The equiaxed grains in the AA5182 sheets were also elongated. It is unclear to what extent the speed ratio affected the microstructure from the optical micrographs. The thickness of each layer had some variation from position to position. Figure 4 exhibited EBSD results of roll bonded clads (outer layer of AA1050) with speed ratio. The Outer region is the surface contacting rolls and the Inner is the surface contacting AA5182. The Outer region experience more deformation and grains in the Outer region were more refined than those inside the sheets. The metal flow near Inner region revealed inhomogeneous deformation due to hard materials of AA5182. Overall, elongated grains looked similar to the optical micrographs found in Fig. 3. The grain size of AA5182 layer looks greater than those of AA1050 layers shown in Fig. 4. Grain size and aspect ratio were summarized in Table 2. Figure 5 showed (111) pole figures computed from EBSD and XRD results. The (111) pole figures of the core
339
FIGURE 3. Optical micrographs of clads fabricated by asymmetric roll bonding.
FIGURE 4.
Inverse pole figure maps of clads fabricated by asymmetric roll bonding. TABLE 2. ratio
Grain size (GS) and aspect ratio (AR) with speed Ini.
1:1.0
1:1.05
1:1.10
1:1.20
AA5182 (GS)
6.4
7.8
6.1
6.0
7.7
AA5182 (AR)
1.3
2.4
2.3
2.2
2.5
AA1050 (GS)
3.0
1.2
1.3
1.4
1.5
AA1050 (AR)
2.5
2.8
2.7
2.7
2.7
340
FIGURE 5. (111) pole figures of clads fabricated by asymmetric roll bonding. Contours for EBSD PFs: 0.92, 1.42, 1.92, 2.42, 2.92, 3.42, 3.92, 4.42, 4.92, 5.42, 5.92, 6.42, 6.92, 7.42. Contours for XRD PFs: 1, 2, 3, 5, 7, 10.
FIGURE 6.
Variation of mechanical properties of clad sheets along the RD and TD.
AA5182 and outer layer AA1050 sheets were computed from EBSD maps measured from the whole thickness. In addition, XRD results taken from the top surface were also presented for the outer layer of AA1050. The core AA5182 sheet showed well-developed PSC texture of α and β fibers. Overall intensities looked similar to each speed ratio. The outer layer of AA1050 sheets displayed shearing texture, major rotated cube component, {100}011 and minor {111}//ND. The minor {111}//ND components found in the XRD were weaker than those in EBSD results. Results of the uniaxial tension tests were presented in Fig. 6. Variation of the tensile and yield strength was negligible with velocity ratio. Tensile and uniform elongation have some peak around speed ratio of 1.15. These trends were found both the RD and TD samples.
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SUMMARY AA1050/AA5182/AA1050 clad sheets were fabricated with various rolling conditions at room temperature. Initial microstructure of AA1050 was as-rolled, and that of AA5182 was fully recrystallized. The outer layer of AA1050 sheets experienced more shear deformation than the core of AA5182. Plane strain compression texture was observed in the core, and rotated cube type texture was found in the outer layer. The variation of texture and microstructure with speed ratio from 1.0 to 1.2 was not evident during cladding. It was found that elongation was improved at a speed ratio of about 1.1 to 1.2.
ACKNOWLEDGMENTS The authors would like to thank the financial supports of Korea Institute of Materials Science (KIMS).
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Y.-M. Hwang, H.-H. Hsu, and Y.-L. Hwang, International Journal of Mechanical Sciences 42, 2417–2437 (2000). H. Manesh, and A. Taheri, Mechanics of Materials 37, 531–542 (2005). H. Manesh, and A. Taheri, Journal of Materials Processing Technology 166, 163–172 (2005). H. Yan, Trans. Nonferrous Met. Soc. China 16, 84–90 (2006). M. Nezhad, and A. Ardakani, Materials and Design 30, 1103–1109 (2009). X. Li, G. Zu, and Q. Deng, Light Metals 2011 "(TMS2011)" 1, 615–619 (2011). X. Zhang, T. Yang, S. Castagne, C. Gu, and J. Wang, Materials and Design 32, 2239–2245 (2011). H. Manesh, and A. Taheri, Materials and Design 24, 617–622 (2003). H. D. Manesh, and A. K. Taheri, Journal of Alloys and Compounds 361, 138–143 (2003). J. Lee, D. Bae, W. Chung, K. Kim, J. Lee, and Y. Cho, Journal of Materials Processing Technology 188, 546–549 (2007). J. S. Yoon, S. H. Lee, and M. S. Kim, Journal of Materials Processing Technology 111, 85–89 (2001). A. Segawa, and T. Kawanami, Journal of Materials Processing Technology 47, 375–384 (1995). A. Segawa, and T. Kawanami, Journal of Materials Processing Technology 47, 544–551 (1995). N. Bay, H. Bjerregaard, S. Petersen, and C. dos Santos, Journal of Materials Processing Technology 45, 1–6 (1994). Y. Hwang, T. Chen, and H. Hsu, International Journal of Mechanical Science 38, 443–460 (1996). T. Sakai, S. Hamada, and Y. Saito, Scripta mater. 44, 2569–2573 (2001). H. Jin, and D. Lloyd, Materials Science and Engineering A 399, 358–367 (2005). S. B. Kang, B. K. Min, and W. D. S. J. K. Kim, H. W., Metall. Mater. Trans. A 36, 3141–3149 (2005). H. Jin, and D. Lloyd, Materials Science and Engineering A 465, 267–273 (2007). S. Matthies, and G. Vinel, Phys. Stat. Sol.(b) 112, k111–k114 (1982). S. Matthies, S. Vinel, and K. Helming, Standard Distribution in Texture Analysis, Akademie-Verlag, Berlin, 1987. J. H. Cho, A. D. Rollett, and K. H. Oh, Metall. Mater. Trans. A 36A, 3427–3438 (2005).
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Influence of Hydrostatic Pressure on FLDs for AZ31B Sheets H. Wanga, Y. Wub, P.D. Wua, K.W. Nealec a
Department of Mechanical Engineering, McMaster University, Hamilton, Ontario L8S 4L7, Canada b
State Key Laboratory for Geomechanics and Deep Underground Engineering China University of Mining and Technology, Xuzhou, Jiangsu 221008, China
c
Faculty of Engineering, University of Sherbrooke, Sherbrooke, Quebec J1K 2R1, Canada
Abstract. Sheet metal formability in terms of the Forming Limit Diagram (FLD) for magnesium alloy AZ31B sheets is studied by using the recently developed Elastic Visco-Plastic Self-Consistent (EVPSC) model, in conjunction with the classical M-K approach. The effect of superimposed hydrostatic pressure on FLDs is numerically assessed. It is found that the superimposed pressure delays the initiation of necking for any strain path. The increment of formability becomes significant when the pressure is larger than about half the initial tensile yield stress of the sheet metal. The difference in predicted FLDs between that with a superimposed hydrostatic pressure and that with a stress component normal to the sheet plane is also discussed. Keywords: crystal plasticity; hydrostatic pressure; formability; magnesium alloy PACS: 62.20.fk
1. INTRODUCTION Conventionally rolled or extruded magnesium alloys inevitably result in a dominant basal texture and thus show low formability due to the limited number of plastic deformation modes available at room temperature. However, recent experimental works have revealed that the formability of magnesium sheets can be significantly improved through texture optimization by mainly re-orientating the basal plane through various shear processes [1-5]. In a recently study, Wang et al. [6, 7] investigated the influence of basal texture on the uniform strain under uniaxial tension and the limit strain under biaxial stretching. This study suggested that formability can be significantly improved by controlling texture even without grain refinement. Another possible way to increase formability for magnesium alloys is by superimposing hydrostatic pressure. It has been generally accepted that the superimposed hydrostatic pressure significantly enhances ductility for various monolithic metals and composites [8-11]. Based on the classic isotropic rate-independent plasticity theory, Wu et al. [12] have studied the effects of superimposed hydrostatic pressure on the sheet metal formability in terms of the FLD. It has been observed that the superimposed hydrostatic pressure increases sheet metal limit strains for any strain path. In the present study, the influence of superimposed pressure on FLDs is numerically studied for the magnesium alloy sheet AZ31B. All simulations are based on the recently developed Elastic Visco-Plastic SelfConsistent (EVPSC) model [13], in conjunction with the M-K approach developed by Marciniak and Kuczynski [14].
2. CONSTITUTIVE MODEL The elastic constitutive equation for a crystal is:
The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 343-350 (2011); doi: 10.1063/1.3623630 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
343
*
L : d e tr (d e )
(1) *
where L is the fourth order elastic stiffness tensor, d is the elastic strain rate tensor and is the Jaumann rate of e
the Cauchy stress based on the lattice spin tensor L through the crystal elastic constants Cij .
w e . The single crystal elastic anisotropy is included in
FIGURE 1. Plastic deformation modes for hexagonal structure: (a) basal slip systems, (b) prismatic slip systems, (c) pyramidal slip systems, and (d) tensile twin.
Plastic deformation of a crystal is assumed to be due to crystallographic slip and twinning on systems ( s D , nD ) . Here, s D and nD are the slip/twinning direction and the direction normal to the slip/twinning plane for system D , respectively. In the present study we consider slip in the Basal ( {0001} 1120 ! ), Prismatic ( {10 1 0} 11 2 0 ! ) and Pyramidal ( {1 1 22} 1 1 23 ! ) slip systems, and twinning in the
{10 1 2} 1 011 ! tensile twin system (see Fig. 1). The following equation gives the grain (crystal) level plastic strain rate:
dp where
PD
J0
WD J0 ¦ P D D W cr D
1 1 m
WD W crD
(2)
is a reference value for the slip/twinning rate, m is the slip/twinning rate sensitivity,
( s D nD nD s D ) 2 is the Schmid tensor for system D , and W D
: P D and W crD are the resolved shear
stress and critical resolved shear stress (CRSS) for system D , respectively. The self-consistent approach is applied to obtain the response of a polycrystal comprised of many grains. In a self-consistent approach each grain is treated as an ellipsoidal inclusion embedded in a Homogeneous Effective Medium (HEM), which is an aggregate of all the grains. Interactions between each grain and the HEM are described using the Eshelby inclusion formalism. During each deformation step, the single crystal constitutive rule (which describes the grain-level response) and the self-consistency criteria are solved simultaneously. This ensures that the grain-level stresses and strain rates are consistent with the boundary conditions imposed on the HEM. To date, various self-consistent schemes have been proposed. The Elastic Visco-Plastic Self-Consistent (EVPSC) model for polycrystals recently developed by Wang et al. [13] is a completely general elastic-viscoplastic, fully anisotropic, self-consistent polycrystal model, applicable to large strains and to any crystal symmetry.Very recently, Wang et al. [15-17] evaluated various linearization self-consistent schemes employed in self-consistent modeling by applying them to the large strain behavior of magnesium alloys under different deformation processes, including torsion. It was found that the Affine self-consistent scheme gave the best overall performance among the self-consistent approaches examined. Therefore, the EVPSC model with the Affine self-consistent scheme is applied in the present study. For a detailed description of the EVPSC model we refer to Wang et al. [13].
344
3. PROBLEM FORMULATION AND METHOD OF SOLUTION Following the numerical procedure developed by Wu et al. [18], the EVPSC model, in conjunction with the M-K approach, has been implemented into a numerical code for constructing FLDs by Wang et al. [7]. The basic assumption of the M-K approach is the existence of material imperfections, which are in the form of a groove on the surface of the sheet. Marciniak and Kuczynski [14] showed that a slight intrinsic inhomogeneity in load bearing capacity throughout a deforming sheet can lead to unstable growth of strain in the region of imperfection, and subsequently cause localised necking and failure. The influence of the superimposed hydrostatic pressure on formability is studied by considering a nonproportional history using combinations of two linear deformation paths. The first loading path is to apply hydrostatic pressure to materials both outside and inside the imperfection (3) V11 V 22 V 33 p until the desired level of superimposed hydrostatic pressure p is reached. It is noted that the deformations inside and outside the band remain very small during this almost purely elastic loading path even when p is very high. In the subsequent linear strain path, the deformation outside the band is assumed to be such that
D22 D11 where H22
D22 and H11
H22 H11
D12
0, W12
0
(4)
D11 are the (principal) logarithmic strain rates and the Wij values are components of the
spin tensor. It is further assumed that D13 condition V 33
U,
D 23 W13 W23
0 . The thickness strain rate D33 is specified by the
0 ( V 33 { p ). For the orthotropic textures considered here, these boundary conditions imply that
V 13 V 23
0. By enforcing the compatibility at the band interface, the equilibrium balance on each side of the interface, and the assumption V 33 0 inside the band, the deformation inside the band
the average stress components
can be completely determined. The onset of sheet necking is defined by the occurrence of a much higher maximum b b 4 principal logarithmic strain rate inside the band than outside, taken here as the condition H D11 t 10 , where H
represents the strain rate inside the band. The corresponding principal logarithmic strains H11 and H 22 outside the band are the limit strains. The entire FLD of a sheet is determined by repeating the procedure for different strain paths outside the band as prescribed by the strain ratio U . A detailed description on how superimposed hydrostatic pressure is included in the FLD calculation can be found in Wu et al. [12]. *
*
4. RESULTS AND DISCUSSION The material studied in the present paper is the magnesium alloy AZ31B sheet; its mechanical behavior at room temperature has been experimentally studied by Jain and Agnew [19]. The initial crystallographic texture of the sheet is represented by 2160 orientations. The measured texture shows that the grains tend to have their basal planes oriented perpendicular to the normal direction of the sheet. The reference slip/twinning rate, J0 , and the rate sensitivity, m , are prescribed to be the same for all slip/twinning systems: J0
0.001s 1 and m 0.05 , respectively. The room temperature elastic constants of the magnesium single crystal are assumed to be C11 58.0 , C12 25.0 , C13 20.8 , C33 61.2 and C44 16.6 (unit of GPa) [20]. Values of the hardening parameters for
each mode are estimated by fitting numerical simulations of uniaxial tension and compression along the rolling direction (RD) to the corresponding experimental flow curves. The uniaxial tension and compression true stresstrue strain curves along the RD are presented in Fig. 2. The characteristic S-shape of the compressive flow curve clearly reveals the importance of twinning in compression. The macroscopic yield stresses for uniaxial compression and tension are about 100 MPa and 160 MPa, respectively, showing the strong tension/compression asymmetry associated with twinning. It is clear that the EVPSC model with the Affine scheme fits the experimental curves quite well. The values of these parameters will be used in the subsequent simulations. In addition, the initial imperfection in the M-K approach is assumed to be f 0 0.99 .
345
500
V 11
Compression 400
300
Tension
200 Experiment Fitting
100
0
0
0.05
0.1
0.15
0.2
H 11p
0.25
FIGURE 2. True stress and plastic strain curves under uniaxial tension/compression along the RD. The experimental data are taken from Jain and Agnew [19]. 600
V 11
500
p 0MPa
40 400
80 120 160
300
200
100
0
0
0.1
0.2
0.3
H11
0.4
FIGURE 3. Uniaxial tensile stress responses under superimposed hydrostatic pressure represented in terms of V 11 .
We first consider in-plane plane strain tension ( U Figure 3 presents the true stress
V 11
and true strain
0 in Eq. 4) under a superimposed hydrostatic pressure p.
H11 curves for the material point outside the imperfection band.
346
The calculations are stopped at necking; i.e., when H b D11 t 10 4 . It is found that the superimposed hydrostatic pressure p lowers the
V 11 ~ H11
curve vertically.
600
V 11
120
40
500
400
p 0MPa
80
160
300
200
100
0
0
0.1
0.2
0.3
H11
0.4
FIGURE 4. Uniaxial tensile stress responses under superimposed hydrostatic pressure represented in terms of
V 11 V 11 p .
0.7
H*
0.6
0.5
0.4
0.3
0.2
U U U
0.1
0
0
40
80
120
160
200
p (MPa )
FIGURE 5. Effect of superimposed hydrostatic pressure p on major limit strains for strain paths uniaxial tension U plane plane strain tension
U
0 and equi-biaxial tension U 1 .
347
0.5 , in-
The V 11 ~ H11 curves in Fig. 3 are converted into the V 11 V 11 p ~ H11 curves in Fig. 4. The symbols represent the necking points. It is clear that the superimposed hydrostatic pressure p does not affect work-hardening but lowers the V 11 ~ H11 curve vertically by the amount of p. However, it is also clear that the superimposed hydrostatic pressure increases the limit strain.
0.8
H
* 11
0.6
200
0.4
160 120 80 40 0.2
p= 0MPa
-0.4
-0.2
0
0
0.2
H
0.4
* 22 FIGURE 6. Effect of superimposed hydrostatic pressure p on FLDs.
Figure 5 shows the effect of superimposed hydrostatic pressure on the predicted major limit strains under uniaxial tension ( U 0.5 ), in-plane plane strain tension ( U 0 ) and equi-biaxial tension ( U 1 ). It is found that the major limit strain increases monotonically with the superimposed hydrostatic pressure for all three strain paths. Figure 6 presents the predicted full FLDs under various superimposed pressures. It is clear that the superimposed hydrostatic pressure increases the sheet metal limit strain for any strain path. Finally, it is worth mentioning that we have analyzed the effect of hydrostatic pressure on FLDs. It is noted that the effect of the stress component V 33 (normal to the sheet plane) on FLDs has been reported by Banabic and Soare [21]. They showed that the effect of the normal stress on FLDs was similar to that of a hydrostatic pressure. We have numerically assessed the effect of the normal stress on FLDs for the AZ31B sheet. Figure 7 shows the influence of the normal stress/pressure on the predicted FLDs. It is found that, in the range of p d 200 MPa , the effect of the normal stress on FLDs is very similar to that of the hydrostatic pressure. Figure 8 presents effects of superimposed hydrostatic pressure and normal pressure/stress on the major limit strain for in-plane plane strain tension U 0 . It is demonstrated that the difference in the predicted FLDs between superimposed hydrostatic pressure and normal pressure is negligible even at pressures up to 320 MPa.
348
0.8
H 11* 0.6
200
0.4
160 120 80 40 0.2
-0.4
0
-0.2
V 33
0
0MPa
0.2
FIGURE 7. Effect normal pressure
V 33
0.4
* H 22
p on FLDs.
0.8
H* 0.6
0.4
p V 33
0.2
0
0
100
200
300
V 33 ( p)(MPa )
400
FIGURE 8. Effect of superimposed pressure and normal stress on major limit strain for in-plane plane strain tension U = 0.
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4. CONCLUSIONS The effect of superimposed hydrostatic pressure has been included in the analysis of the forming limit diagram for the magnesium alloy AZ31B sheet. All the simulations have been based on the recently developed elastic-viscoplastic self-consistent (EVPSC) model, in conjunction with the M-K approach. It has been numerically demonstrated that the superimposed hydrostatic pressure increases sheet metal limit strains for any strain path. It has been also found that, in the range considered, the effect of the normal stress on FLDs is very similar to that of a hydrostatic pressure.
ACKNOWLEDGEMENTS H. Wang and P.D. Wu were supported by funding from the NSERC Magnesium Strategic Research Network. More information on the Network can be found at www.MagNET.ubc.ca. Y. Wu was supported by the National Basic Research Program of China (2007CB209400) and the 111 Project of China (B07028).
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T. Mukai, M. Yamanoi, H. Watanabe and K. Higashi, Scripta Mater., 45, 89-94 (2001). S. R. Agnew, J. A. Horton, T. M. Lillo and D. W. Brown, Scripta Mater., 50, 377-381 (2004). X. S. Huang, K. Suzuki, A. Watazu, I. Shigematsu and N. Saito, Mater. Sci. Eng., A488, 214-220 (2008). Y. Chino, K. Sassa and M. Mabuchi, Scripta Mater., 59, 399-402 (2008). B. Song, G. S. Huang, H. C. Li, L. Zhang, G. J. Huang and F. S. Pan, J. Alloy Compd.,489, 475-481 (2010). H. Wang, P. D. Wu and M. A. Gharghouri, Mater. Sci. Eng., A527, 3588-3594 (2010). H. Wang, P. D. Wu, K. P. Boyle and K. W. Neale, Int. J. Solids Struct., 48, 1000-1010 (2011). P. W. Bridgman, Studies in Large Plastic Flow and Fracture - With Special Emphasis on the Effects of Hydrostatic Pressure. McGraw-Hill, New York. 1952. W. A. Spitzig and O. Richmond, Acta Metall., 32, 457-463 (1984). A. S. Kao, H. A. Kuhn, O. Richmond and W. A. Spitzig, Metall. Trans.,20A,1735-1741 (1989). J. J. Lewandowski and P. Lowhaphandu, Int. Mater. Rev., 43, 145-187 (1998). P. D. Wu, J. D. Embury, D. J. Lloyd, Y. Huang and K. W. Neale. Int. J. Plasticity, 25, 1711-1725 (2009). H. Wang, P. D. Wu, C. N. Tomé and Y. Huang, J. Mech. Phys. Solids, 58, 594-612 (2010). Z. Marciniak and K. Kuczynski, Int. J. Mech. Sci., 9, 609-620 (1967). H. Wang, B. Raeisinia, P. D. Wu, S. R. Agnew and C. N. Tomé, Int. J. Solids Struct., 47, 2905-2917 (2010). H. Wang, P. D. Wu and K. W. Neale. J. Zhejiang Univ.-SC. A, 11,744-755 (2010). H. Wang, Y. Wu, P. D. Wu and K. W. Neale. CMC-Comput. Mater. Con., 19, 255-284 (2010). P. D. Wu, K. W. Neale and E. Van der Giessen, P. R. Soc. London, A453, 1831-1848 (1997). A. Jain and S. R. Agnew, Mater. Sci. Eng., A462, 29-36 (2007). G. Simmons and H. Wang, Single crystal elastic constants and calculated polycrystal properties. Cambridge (MA): MIT Press, 137 (1971). D. Banabic and S. Soare, On the effect of the normal pressure upon the forming limit strains. In: Hora, P. (Ed.), Numisheet 2008 - Proceedings of the 7th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes, 199-204 (2008).
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Application of a Reduced Texture Methodology to Model the Plasticity of Anisotropic Extruded Aluminum Sheets Meng Luoa, Gilles Rousselierb and Dirk Mohra,c a
Impact and Crashworthiness Laboratory, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge MA, USA b MINES ParisTech, Centre des Matériaux, CNRS UMR 7633, BP 87, 91003 Evry Cedex, France c Ecole Polytechnique ParisTech, Solid Mechanics Laboratory, CNRS UMR 7649, 91120 Palaiseau, France
Abstract. A recently developed Reduced Texture Methodology (RTM) featuring (i) a significantly reduced number of crystallographic orientations, (ii) a special experiment-based parameter calibration procedure, and (iii) reasonable computational time for industrial applications is adopted to model the anisotropic plastic behavior of a 2mm-thick extruded aluminum 6260-T6 sheet. Firstly, the full-thickness sheet is modeled with twelve crystallographic orientations, and the model parameters are identified through an optimization procedure based on uniaxial tensile tests with seven different material orientations. The calibrated model describes well the stress-strain curves and Lankford ratios for all directions, while the optimized grain orientations are in good agreement with EBSD measurements. However, the EBSD results also reveal that the present sheet exhibits a strong heterogeneity through the thickness as far as crystallographic orientations and grain sizes are concerned. To account for this heterogeneity, eight grain orientations are selected out of the total twelve for the full-thickness sheet to model the 0.7mm-thick central layer of the sheet based on the EBSD measurements. It is found that the reduced eight-grain model provides good predictions of the macroscopic responses in uniaxial tensile tests on reduced-thickness specimens, even without further calibration. A combined calibration is also performed to determine the final set of parameters which provide excellent modeling for both the full-thickness sheet (twelve-grain model) and its central layer (eight-grain model). Keywords: Polycrystalline model, anisotropy, aluminum extrusion, heterogeneity, finite element method, sheet forming PACS: 62.20.-x; 46.35.+z; 61.66.Dk
INTRODUCTION Extensive research has been conducted in the field of phenomenological plasticity modeling of anisotropic metals. As a result, the initial anisotropy of their plastic response can be described with good accuracy using advanced macroscopic models, in particular, with the use of linearly transformed stress tensors in isotropic yield functions, e.g. Refs.1, 2. However, phenomenological plasticity models cannot capture the complex evolution of anisotropy throughout loading3 or the cross-hardening behavior in non-proportional loading4, unless more sophisticated rotational or distortional yield surfaces are employed5. Polycrystalline metal plasticity models have intrinsic advantages in modeling the anisotropy and complex hardening behaviors under multi-axial and multi-path loadings. The main limitation of the polycrystalline models is the large computational effort involved for structural analysis and material model parameter identification through inverse analysis. The tremendous CPU times required for a single simulation limit the number of parameters that can be calibrated and the number of iterations for optimization. As a result, the accuracy of polycrystalline models at the macroscopic level is often compromised, despite their sound mechanism-based constitutive description at the microscopic level. Even with advanced selfconsistent models and an experimentally-measured anisotropic texture, it can be difficult to obtain a good evaluation of flow stresses and directions simultaneously3. This is a huge drawback for the accurate finite element analysis of sheet metal forming processes. Reasonable CPU time can be achieved through a significant reduction of the number of crystallographic orientations6, 7. Rousselier et al.8, 9 have used a small number of crystallographic orientations with a self-consistent polycrystalline model. Their parameter calibration procedure (which includes the texture
The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 351-359 (2011); doi: 10.1063/1.3623631 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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parameters) relies on a comprehensive experimental database to suppress irregularities in the predicted stress-strain response associated with the limited number of crystallographic orientations. With this approach, they managed to obtain an accurate description of both the flow stresses and strain rates of a highly anisotropic 2090-T3 aluminum sheet; the CPU time of their model on a deep drawing FE simulation is only twice as long as that of an advanced phenomenological model8. In this paper, a Reduced Texture Methodology (RTM) is applied to model the anisotropic plastic behavior of a 6260-T6 aluminum extrusion of 2mm wall thickness. Our objective is not the modeling of the actual initial texture and its evolution, but the accurate description of the mechanical response at the macroscopic scale. One particular challenge in this study is the through-thickness heterogeneity present in the extruded aluminum sheets. Our experimental data base therefore contains results for both full-thickness and reduced-thickness specimens 10. It is shown that reduced texture methodology allows for a complete identification of all model parameters. The resulting RTM based polycrystal plasticity model is able to describe the plastic behavior of both full-thickness and reducedthickness specimens with good accuracy.
POLYCRYSTALLINE PLASTICITY MODELING In the following, , ⃗, represent a scalar, a vector, a second order tensor and a fourth order tensor, respectively. A dot denotes differentiation with respect to time. Upper-case symbols (Σ, , , … ) and lower-case symbols ( , , … ) correspond to mechanical variables at the macroscopic and microscopic scales, respectively. All the following equations are written in a material embedded co-rotational frame8. Due to the length limit, only key ingredients of the constitutive equations are presented in this section. Readers are referred to Refs.8, 9 for details on the model formulation.
The Self-consistent Polycrystalline Model The Reduced Texture Methodology (RTM) considers a very small number of crystallographic orientations only. These orientations are determined from mechanical experiments through inverse analysis instead of crystallographic measurements. The RTM framework can be applied to any polycrystalline model. In this paper, a particular viscoplastic self-consistent model is employed. Each of the “grains” of the model represent a set of physical grains with close orientations. In each “grain” g of volume fraction , the stress and strain = + are assumed to be homogeneous. In the case of isotropic elasticity, with Young modulus and Poisson’s ratio , we have
Σ=
with
=
= 1
(1)
(2)
1+ 1 − 1 ⊗ 1 ∶ Σ
(3)
≡ =
A so-called -rule11, 12 which captures the elasto-plastic accommodation of intergranular deformations is adopted in the present work:
= Σ + − with =
$
̇ = ̇ − !: " ̇ " where " ̇ " = # ̇ : ̇ %
(4) (5)
where is an intermediate deviatoric strain tensor which evolves non-linearly. The scalar parameter of the initial elastic accommodation is close to the elastic shear modulus & = /2(1 + ) . The fourth-order tensor ! is
352
proposed by Sai12 for anisotropic materials. With Voigt the notation, ! has 10 independent elements !'* , as it has orthotropic symmetry and it is a linear relation between two deviatoric tensors (see details in ref. 8 ).
The Single Crystal Formulation This section briefly presents the constitutive equations at the level of a single crystal. The single crystal model relates the slip rate -.̇ and the resolved shear stress 0. of each slip system (s = 1 to M). The orientation tensor 3. attributed to the slip system number 4 is defined as 3. = 56⃗7 ⊗ 8⃗7 + 8⃗. ⊗ 6⃗. 9/ 2
(6)
8⃗. and 6⃗. being the “slip plane” normal vector and the “slip direction” vector in this plane, respectively. The socalled resolved shear stress is given by 0. = : 37
(7)
In the case of the fcc crystallographic structures, the ; = 12 glides considered in this paper are the octahedral systems {111}(100). A phenomenological viscoplastic model11 is used as the constitutive equations for each slip system: D
|0. − 1, failure should not occur in the weld zone and this is the case with most of the steel TWBs. The ‘nr’ and ‘YSr’ values considered were: nr = 0.56, 0.83, 1.1, 1.3, 1.7 and YSr = 3, 4, 5. All other properties of the weld zone were assumed to be same as that of base metal. Initially, weld region was placed at the blank centre (for both longitudinal and transverse weld orientations) and its width was kept constant as 5 mm throughout the analysis. The limit strains of TWB will be compared to that of un-welded sheet. In the case of transverse weld orientation, the effect of weld location was also studied by placing the weld at some critical offset locations from the centre. For each weld location, weld properties (nw, YSw) were changed to investigate their influence. The FLCs of un-welded and welded sheets (with changing weld locations) were compared. 7$%/(Base material properties assumed for the simulations [8]
nb
Kb (MPa)
YSb* (MPa)
R0
R45
R90
0.18
561
188
1.2
1.3
1.5
* Assuming flow equation, V = K H n
432
Young’s Modulus (GPa) 206
Poisson’s Ratio
Thickness (mm)
0.33
1.2
)DLOXUH/LPLW6WUDLQ&ULWHULRQ A novel thickness gradient-based necking criterion was used to obtain FLC. This criterion basically perceives the localized necking as the critical, local thickness gradient developed in the sheet [9]. A local critical thickness gradient (Rc) must exist at the onset of a local visible neck. In the simulations, the thickness gradient developing in the adjoining elements was monitored and when this gradient (Ractual) drops below Rc, then it was concluded that the local neck has initiated in the element where thickness is lower. The major and minor strain in the adjoining thicker element was then reported as the forming limit strain of that particular deformation condition. It was found from the earlier experiments that necking occurs in the sheet metal when the thickness gradient (ratio of thickness of neighboring elements or circles) falls below 0.92 [10]. Hence in the present FE simulations performed, element pairs where the thickness ratio equals or falls below 0.92 were considered as necked elements. It was observed that the necking criterion is satisfied between few element pairs at the same punch travel. Hence the major strains in all the thicker elements of such element pairs were recorded. The largest major strain and the corresponding minor strain of such elements were treated as the forming limit strain. This procedure of obtaining the limit strain was followed for both un-welded and welded blanks.
5(68/76$1'',6&866,21 /LPLW6WUDLQVRI7:%LQ7UDQVYHUVH:HOG2ULHQWDWLRQ Figure 1 shows the predicted Forming Limit Curve (FLC) of un-welded sheet and TWB with transverse and longitudinal weld orientations, when weld is placed at the centre. The limit strains of the TWB for the different strain paths, for the various weld conditions (properties, orientations) are shown within ellipses/circles. Limit strains of TWB are shown for three strains paths – tensile, near plane strain and bi-axial stretching, only. The TWB limit strains are grouped as ellipses within which limit strains vary with the ‘nr’ and ‘YSr’ values. It is observed that, in the case of transverse weld orientation, weld has insignificant effect on the limit strains developed in TWB. All the three strain paths – tensile, near plane strain and bi-axial stretching conditions of TWB have limit strains similar to that of un-welded sheet. The possible reasons behind this behavior are: (i) less severe ‘YSr’ and ‘nr’ values, (ii) smaller weld width, and (iii) weld located in non-critical region. Since weld exhibits higher yield strength in comparison with the base metal (more than 3 times that of the base metal), it is conceivable that the weld is not contributing to the deformation. Also failure occurs in the base metal, weld being at centre of the sheet, and the failure location is near punch nose region as in LDH testing of un-welded sheet. This failure location is at 30-40 mm from the weld zone. Since weld is placed at the noncritical location in TWB, the effect is insignificant. In order to analyze further, the weld width and weld-base metal property combinations are chosen as before and the weld was placed in the critical regions or severely strained regions (near punch nose region), i.e., at some offset from the centre. Here critical regions are considered as regions of the sheet where peak strains and strong thickness gradient develop. Tensile, near plane strain and bi-axial stretching conditions are studied in this case. Once again the weld zone properties are changed as described earlier and the limit strains are predicted as before. In all the three strain paths, weld is placed at two different critical locations from the centre of the blank. In the case of tensile strain path, the weld is placed at 20 mm and 35 mm from the centre of the blank. In the case of near plane strain condition, the weld is located at 10 mm and 25 mm from the centre. In bi-axial stretching mode, the weld is located at 15 mm and 30 mm from the centre of the blank. Figure 2 shows the effect of weld when it is placed at severely strained regions for tensile, near plane strain and bi-axial stretching conditions. In the case of near plane strain stretching condition, limit strains have decreased significantly and are nearly zero (~ 0.005). This limit strain value is very low and is unexpected. When placed at different locations (10 mm and 25 mm from the centre), the effect is same. Failure occurs at the interface of weld and the base metal, when weld is at an offset position. It is observed that the weld zone, which had an initial thickness of 1.2 mm before deformation, has a thickness of 1.18 mm when limit strains are measured. So it is clear that weld is not deforming much but it just changes the limit strains of the TWB. In the case of bi-axial stretching conditions, decrease in limit strain is observed when compared to that of TWB with weld at centre of the sheet. But the decrease in limit strains in this case is not to the extent as observed in near plane strain condition. The different weld locations, 15 mm and 30 mm
433
from the centre of the blank have nearly equal limit strains like in near plane strain condition. Since yield strength of the weld (YSw) is higher than the base metal, it behaves like a rigid body and the deformation is insignificant in the weld region. Since weld is harder, it concentrates strain or deformation in neighboring regions at a faster rate as compared to that of un-welded blank (or centre weld case) and hence the limit strains of TWB are much lower than that of un-welded sheet. The above argument is supported by observing the thickness distribution under different conditions.
Tensile stretching condition with longitudinal weld
0.5 Major strain 0.4
Tensile stretching condition with transverse weld
0.3
Bi-axial stretching condition (Transverse, Longitudinal)
0.2
Near plane strain condition with longitudinal weld
0.1
Near plane strain condition with transverse weld
0 -0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
Minor strain
),*85( Forming limit of base metal, TWB with longitudinal and transverse weld orientations
Figure 3 compares the thickness distribution of TWB with weld located at centre to weld located at critical offset location from the centre of the blank. TWB with weld at centre exhibits lesser thinning when compared to that of TWB with weld at critical location. Though thickness distribution is same in the base metal region, considerable difference is observed near the weld and failure region. This is because of strong thickness gradients developing near the weld region. Also in the case of bi-axial stretching condition, the difference in thinning is lesser when compared to that of plane strain condition. This is the reason why not much of limit strain difference is seen in this case. Also, decrease in limit strains is not to the extent as observed in near plane strain condition. It is quite clear from the discussion that not only the weld orientation and weld properties affect the limit strains, but weld location also plays a vital role in affecting the limit strains. In the case of tensile strain path, weld is located at 20 mm and 35 mm from the centre of the blank. It is observed that limit strains have increased when weld is placed at 20 mm from the centre when compared to that of un-welded blank (or weld at centre case). When the weld is located at 35 mm from the centre, the limit strains are decreasing as expected (shown in Fig. 2). In the case of TWB with weld at 20 mm from the centre, the failure occurs nearly 20 mm from the weld region, but in between the weld and clamping region. This behavior of increase in limit strains of TWB to that of un-welded blank is due to the reduction in effective area of deformation (X) (see Fig. 4). In the case of 20 mm weld offset, the effective area of deformation is reduced, unlike previous cases. After a particular punch travel, strain distribution will be non-uniform and also concentrated on the effective deformation area. This will generate a local tensile mode of deformation in the small effective deformation area. The rigid weld during the deformation process can
434
just locally stretch the region, because of which both major and minor strain increases significantly till the limit strain is reached. In order to demonstrate the existence of tensile mode of deformation, major strain is monitored with increasing punch travel till failure is reached. As mentioned earlier, the thickness gradient criterion will be satisfied between two successive elements. The major strain and corresponding minor strain in the thicker element will be identified as forming limit strains.
Major strain 0.5
,QFUHDVHLQOLPLWVWUDLQV
Limit strain of un-welded blank in tensile 0.4 strain path 0.3
Tensile stretching – weld at 20mm
FLC of un-welded blank
0.2
Tensile stretching – weld at 35mm
Bi-axial stretching – weld at 15mm and 30mm
0.1 Near plane strain – weld at 10mm and 25mm 0 -0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
Minor strain ),*85(Limit strains of transverse TWB with weld at critical locations of the blank Figure 4 shows the variation of major strain with punch travel in thinner and thicker elements where thickness criterion is satisfied. This is shown for both TWB with weld at centre and at 20 mm offset. It can be understood from the figure that tensile mode of deformation exists at the effective deformation area (X) in 20 mm weld offset case. As a result the major strain is increasing and non-linear. It is higher when weld is placed at 20 mm offset as compared to weld at centre case. Though the change is major strain in the thinner element finally reaches approximately same peak major strain, the change in thicker element is considerable. Also, one can clearly observe the increase in limiting major strain in weld at 20 mm offset case, when compared to that of weld at centre. Not only limiting major strain, but the entire change in major strain is different in these two cases, with weld at 20 mm offset case showing higher strains till failure. This resulted in increasing forming limit strains in the case of weld at 20 mm offset location. In whole, when weld is at center, limit strains of TWB are same as that of un-welded blanks. TWB limit strains are found to decrease when weld region is placed at critical locations of the blank. This particular inference on the weld location effect will be significant during TWB product design. It is difficult to locate the weld line position in actual TWB parts before forming. This depends on the TWB forming characteristics, structural design and TWB part compatibility with its mating sheet parts. Since weld is not deforming, irrespective of weld location, the influence of weld properties (nr, YSr) on the forming limit strains of TWB with transverse weld is found to be insignificant.
435
1.21
Plane strain More difference
1.19 1.17
Thickness (mm)
1.15 1.13 1.11 Near plane strain - w eld at 25mm from centre Near plane strain - w eld at centre Bi-axial stretching - w eld is at 30 mm from centre Bi-axial stretching - w eld at centre
1.09 1.07
Bi-axial - Less difference
1.05 1.03 0
25
50
75
100
125
150
175
200
225
Distance from one end of sam ple (m m )
),*85( Comparison of thickness distribution of TWBs with weld at centre to weld at critical locations (Weld properties: nr = 0.83 and YSr = 3) 0.6 Weld Weld Weld Weld
0.5
at at at at
center - Thinner element (2) center - thicker element (4) 20 mm of f set - thinner element (1) 20 mm of f set - thicker element (3)
1 2 3
Major strain
0.4 Thinner element
0.3
Necked region
;
4 Thicker element
0.2
Specimen geometric center
0.1 0 0
5
10
15
20
25
30
35
Form ing he ight (m m )
),*85( Variation of major strain with forming height at necked region
436
40
/LPLWVWUDLQVRI7:%LQORQJLWXGLQDOZHOGRULHQWDWLRQ In the case of longitudinal weld orientation, as shown in Fig. 1, limit strains are considerably different when compared to that of un-welded sheet in all the three strain paths. Weld is placed only at the centre location. In this case, failure has initiated in the weld and propagated to the base metal and hence the weld conditions affect limit strains of TWB significantly. As shown in Fig. 1, in the case of tensile and near plane strain conditions, the limit strains of TWB are higher or lower than that of un-welded blank depending on the weld properties. In the case of bi-axial stretching condition, the limit strains are lower than that of un-welded blank. The increase or decrease of limit strains of TWBs in comparison to that of un-welded blank depends on the base metal-weld property combination. It was hypothesized that the increase in limit strain observed for a particular weld-base metal property combination may not only be in tensile and plane strain deformation mode, but could also be occurring for all other strain paths in between. If this is the case, then one can expect a higher FLC for TWBs when compared to that of un-welded blank for that weld-base metal combination. In order to verify this hypothesis, few weld-base metal property combinations, nr = 1.67, YSr = 3; nr = 0.56, YSr = 5; nr = 1.11, YSr = 5, are selected for analyses randomly. All the strain paths are simulated using LDH set up for these weld-base metal property combinations and FLCs obtained are compared to that of un-welded blank. Figure 5 compares the FLCs of TWB with that of un-welded blank. It is quite clear from the figure that the overall FLCs obtained for TWBs are similar to that of un-welded blank. Negligible difference is observed in all the cases. So the increase in limit strains observed for some weld-base metal property combinations is only in few strain paths.
0.5
Major strain
0.4 Un-welded blank
0.3
FLC of un-welded blank
nr1.67ysr3 nr0.56ysr5
0.2
nr1.11ysr5
0.1
0 -0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
Minor strain
),*85(Comparison of FLCs of TWB having longitudinal weld with un-welded blank
&21&/86,216 The following are the conclusions obtained from the present work. x Simulation results show that transverse weld located at the centre of the blank has no effect on the limit strains developed in the forming of TWB. The limit strains of TWB are nearly equivalent to that of un-welded blank. x Weld location has significant effect on the limit strains of TWB in transverse weld orientation. Decrease in limit strains is observed when the weld is placed at the critical locations of the blank, where strain has tendency to concentrate. In the case of TWB with weld at 20 mm offset in tensile strain path, anomalous behavior of increase in
437
x x
limit strain is witnessed. Weld tensile properties, viz., yield strength and strain hardening exponent, are found to show insignificant effect on the forming limit strains of transverse welded blanks. In longitudinal weld orientation, the limit strains are lower or higher than that of un-welded blank depending on the weld properties and strain path. Though increase and decrease of limit strains is seen in few selected strain paths because of weld zone properties, the overall forming limit is same for un-welded and longitudinal welded blanks. The relative location of failure with respect to weld region plays a vital role in deciding the forming limit strains of TWB. In transverse weld orientation, drastic decrease in limit strain is witnessed when failure location is very near to the weld region. Increase in limiting major strain is also witnessed in a particular case because of the change in failure location. In TWBs with longitudinal weld orientation, the location of elements between which thickness gradient criterion is satisfied, i.e., in the weld region or in base material, decides the change in forming limit strains.
5()(5(1&(6 Y. Choi, Y. Heo, H. Y. Kim, and D. Seo, Journal of Materials Processing Technology , 1-7 (2000) L. C. Chan, C. H. Chang, S. M. Chan, T. C. Lee, and C. L. Chow, Journal of Manufacturing Science and Engineering , 743-751 (2005) 3. A. Reis, P. Teixeira, J. Ferreira Duarte, A. Santos, A. Barata da Rocha, and A. A. Fernandes, Computers and Structures , 1435-1442 (2004) 4. R. W. Davies, M. T. Smith, H. E. Oliver, M. A. Khaleel, and S. G. Pitman, Metallurgical and Materials Transactions A $, 2755-2763 (2000) 5. R. Ganesh Narayanan, and K. Narasimhan, Journal of Strain Analysis for Engineering Design , 551-563 (2008) 6. M. Jie, C. H. Cheng, L. C. Chan, C. L. Chow, and C. Y. Tang, Journal of Engineering Materials and Technology , 151-158 (2007) 7. R. Ganesh Narayanan, and K. Narasimhan, International Journal of Forming Processes , 154-178 (2007) 8. K. Abdullah, P. M. Wild, J. J. Jeswiet, and A. Ghasempoor, Journal of Materials Processing Technology , 91-97 (2001) 9. V. M. Nandedkar, “Formability Studies on a Deep Drawing Quality Steel”, PhD Thesis, IIT Bombay, India, 2000 10. S. Chaudhary, and K. Narasimhan, “Prediction and Validation of Forming Limit Strains” in The 5th International conference and workshop on Numerical simulation of 3D sheet forming process, edited by D. Y. Yong et al., Jeju Island, Korea, 2002, pp. 481-486 1. 2.
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Shear Fracture of Dual Phase AHSS in the Process of Stamping: Macroscopic Failure Mode and Micro-level Metallographical Observation Wurong Wanga , Xicheng Weia, Jun Yangb, Gang Shib a
School of Materials Science and Engineering, Shanghai University (Room 607, Rixin Building, No. 149 Yanchang Road, Shanghai 200072, China) b SAIC Motor Company (No.201 Anyan Rd. Jiading Shanghai 201804, China) Abstract. Due to its excellent strength and formability combinations, dual phase (DP) steels offer the potential to improve the vehicle crashworthiness performance without increasing car body weight and have been increasingly used into new vehicles. However, a new type of crack mode termed as shear fracture is accompanied with the application of these high strength DP steel sheets. With the cup drawing experiment to identify the limit drawing ratio (LDR) of three DP AHSS with strength level from 600 MPa to 1000 MPa, the study compared and categorized the macroscopic failure mode of these three types of materials. The metallographical observation along the direction of crack was conducted for the DP steels to discover the micro-level propagation mechanism of the fracture. Keywords: DP AHSS; Shear fracture; Stamping. PACS: 81.05.Bx; 81.20.Hy; 81.40.Lm; 83.85.Ei.
INTRODUCTION The application of high-strength steel (AHSS) has been one of the major ways to reduce the vehicle weight and fuel consumption, while improving vehicle crashworthiness. Dual Phase (DP) steel sheets, as the most common type of AHSS, has been increasingly used into vehicles and served as structural parts due to the combination of special mechanical properties such as high tensile strength, high work hardening rate at early stages of plastic deformation as well as very good ductility. One of key issues associated with DP AHSS is the springback occurring in the process of stamping. To reduce springback, small radii are often recommended [1]. However, a new type of crack mode which is termed as “shear fracture” is periodically seen along small bending radii in the stamping with DP AHSS [2-4]. Shear fracture is termed because the fracture cracks with limited localized necking, and cracks on alternating 45 planes, through thickness. Due to the nature of limited necking in shear fracture of DP, the traditional Forming Limit Diagram (FLD) proves insufficient to predict such fractures because the theory based FLD is mainly derived on the basis of concentrated necking. A significant effort has been made to address this challenge. Microstructure changes was taken into account to model stress and strain relationship for DP steel to analyze its strain hardening properties [5]. Sun et al. [6] suggested considering plastic strain localization, resulting from the incompatible deformation between the harder martensite phase and the softer ferrite matrix, to predict ductile failure of dual phase steels. Wierzbicki et al. [7] developed a Modified Mohr–Coulomb (MMC) ductile fracture criterion and applied it to analyze the failure behavior of a Dual Phase (DP) steel sheet during a simple quasi-static stretch-bending operation. In our previous study [8], different hardening formulas and yield functions were compared in the numerical prediction of onset crack for DP1000 sheet metal drawing. The investigation showed that a Swift and Hockett-Sherby combined formula was in good agreement with the flow curve of the tensile test and Batlat-89 yield model successfully predicted the onset shear crack of DP AHSS.
Corresponding Author: Wurong Wang; Email:
[email protected]; Tel: +86 21 56331377; Fax: +86 21 56331466
The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 439-445 (2011); doi: 10.1063/1.3623642 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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DP steels contain two major phases, ferrite and martensite. Such ferrite–martensite structure is called dual phase microstructure in which, soft ferritic network provides good ductility; while, hard particles and martensitic phase play the load-bearing role [9]. Many researches have compared the martensite morphology produced by some variations of two basic heat treatments, the quenching process from austenite region, or the intercritical annealing [10-12]. Martensite morphology and volume has a significant influence in the void nucleation and crack propagation. G. Avramovic-Cingara et al. [13] investigated the effect of martensite morphology and distribution in a ferrite matrix on the mechanical properties and the damage accumulation in uniaxial tension of two different automotivegrade dual phase DP600 steels. And they found that voids nucleation occurred by martensite cracking, separation of adjacent martensite regions, or by decohesion at the ferrite/martensite interface. In the SEM observation of DP780 fracture by 3-point bending, H.C. Shih et al. [14] revealed that (i) ductile fracture initiated from void nucleation and growth in ferrite phases and (ii) zigzag crack path bypassing martensitic grains. However, According to Kim and Thomas [15], cleavage fracture occurred at ferrite in a coarse martensite structure, whereas voids initiated at ferrite– martensite interfaces in a fine martensite structure. The cause why different damage mechanisms of particular dual phase steels were reported may be due to their chemical compositions, heat treatment history, and differences in their final microstructure. In the present work, three commercial DP steel sheets of strength level from 600MPa to 1000MPa are studied. Firstly, their chemical compositions are analyzed. Martensite distribution and volume are also identified. Then cup drawing experiments are conducted for these three types of materials to identify their limit drawing ratios (LDR). Macroscopic failure modes are determined for the LDR failure parts. Finally with the help of scanning electron microscope (SEM), the fracture propagation mechanism are observed and compared with each other.
MATERIALS AND EXPERIMENTAL PROCEDURE The three commercial DP AHSS sheets, 1.7 mm thick DP600, 1.2 mm thick DP800 and 1.0 mm thick DP1000 were studied in the as-received condition. The chemical compositions of the three steel are shown in Table 1. With increasing strength level, steels contain more C, Si and P. Steel DP600 DP800 DP1000
C 0.110 0.134 0.161
TABLE 1. The chemical compositions of steels in wt.% Si Mn P Cr Ni Al 0.193 1.400 0.005 0.170 0.017 0.0457 0.202 1.490 0.034 0.016 0.037 0.0461 0.501 1.480 0.037 0.021 0.036 0.060
Co 0.0066 0.0186 0.0189
Fe balanced balanced balanced
To observe the martensite distribution and volume of three DP steels, three samples were cut from the original sheets by electrical discharge machining (EDM). Then the samples were mechanically polished and etched in a 4% Nital solution for 10-15 seconds. The revealed microstructure in JSM-6700F SEM shows a dark ferrite phase with embedded white martensite. From FIGURE 1, one can see that the martensite phase in DP600 and DP800 is in island shape; while in DP1000, the martensite phase shows a continuous mesh shape. For the particular DP600, DP800 and DP1000 sheets in this study, the martensite volume is approximately 18%, 32% and 50%, respectively. As mentioned in introduction, the martensite distribution and volume play a dominant role in load bearing and will have a significant effect on the fracture propagation.
(a)
(b)
FIGURE 1. Microstructure of three DP steels.
(a) DP600, (b) DP800, (c) DP1000
440
(c)
LDR is an indicator of material formability, defined as the ratio of the maximum blank diameter that can be safely drawn into a cup without flange to the punch diameter. A punch of I32 mm diameter recommended by the International Deep-Drawing Research Group (IDDRG) was selected in this study. FIGURE 2 shows the press and the die tools. A group of DP1000 circular blanks which were prepared by EDM are shown in FIGURE 2. The diameter of the critical level that is safely drawn is the maximum diameter Dmax , and the next level is identified as the onset crack level. The LDR is then calculated by the ratio of Dmax to 32.
FIGURE 2. The press and die tools.
FIGURE 3. The group of DP1000 circular blanks.
RESULTS AND DISCUSSION LDR and Macroscopic Failure Mode Through the LDR cup drawing test, the maximum blank diameter that can be safely drawn into the cup without flange for 1.7 mm thick DP600, 1.2 mm thick DP800 and 1.0 mm thick DP1000 is 70 mm, 68.75 mm and 66.25 mm respectively, which gives a LDR of 2.19, 2.15, and 2.07 for the three materials. FIGURE 4 shows the crack level for each material. It can be seen that all three cups crack near the punch radius. In view of drawing depth, three DP steels crack at the initial stage of the cup drawing, which indicates they crack under stretch. Focusing on the fracture morphology, they all underwent shear fracture as they cracked on alternating 45 planes through thickness and there were very limited necking phenomena. Therefore, shear fracture is not only observed on DP800 and above steel grade, but also observed on DP600 in the deep drawing process. Moreover, DP600 only cracked with latitudinal fracture (square marked in FIGURE 4 (a)). However, not only latitudinal fracture (square
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marked in FIGURE 4 (b) and (c)), but also longitudinal fracture (circular marked in FIGURE 4 (b) and (c)) were found on DP800 and above DP grade. This indicates that higher the DP strength, the more brittle the crack initiates.
(a)
(b)
(c)
FIGURE 4. The LDR crack level.
(a) DP600, (b) DP800, (c) DP1000
Micro-level Metallographical Observation To identify the surface morphology of the latitudinal fracture and the longitudinal fracture, samples were cut from the cracked cup and observed by SEM. The SEM result of the latitudinal fracture and the longitudinal fracture is shown in FIGURE 5 and FIGURE 6, respectively. By observation, both the latitudinal fracture and the longitudinal fracture are of mainly ductile fracture with dimple appearance. Through FIGURE 5, one can see that the higher the strength of the DP sheets, the smaller diameter and the shallower depth of the dimples. This corresponds well with formability of three DP steels. Moreover, the dimples in FIGURE 6 (a) is smaller and shallower than those in FIGURE 5 (b), as is the case in FIGURE 6 (b) and FIGURE 5 (c), which tells ductility of the latitudinal fractures is better than that of the longitudinal fracture.
(a)
(b)
(c)
FIGURE 5. Surface morphology of the latitudinal fracture.
(a) DP600, (b) DP800, (c) DP1000
(a)
(b)
FIGURE 6. Surface morphology of the longitudinal fracture.
(a) DP800, (b) DP1000
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The fracture propagation mechanism is analyzed through the longitudinal fractures in DP800 and DP1000 cracked cups as the longitudinal fracture provides a relatively smooth plane for SEM sampling. Two samples were cut from the locations of the longitudinal fracture. Then the samples were also mechanically polished and etched in a 4% Nital solution for 10-15 seconds. FIGURE 7 shows an x32 image of the fracture. The SEM result of the fracture end for DP800 and DP1000 is shown in FIGURE 8 and FIGURE 9, respectively. It can be seen from FIGURE 8 that the fractures of DP800 propagate along the ferrite–martensite interfaces. Concerning the DP1000 fractures, there are two propagation modes that are (i) propagation along the ferrite–martensite interfaces (marked as A in FIGURE 9) and (ii) propagation across the martensite grains (marked as B in FIGURE 9). However, it is difficult to determine where and how fracture is initiated. Potential causes can be (i) inside the ferrite phases (void nucleation and growth), (ii) at the ferrite-martensite interface (interfacial debonding), (iii) inside the Martensite particles (cleavage fracture) and (iv) separation between two martensite particles [16].
FIGURE 7. The zoom-in of the longitudinal fracture.
END
FIGURE 8. The SEM result of the longitudinal fracture (DP800)
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A
B
END
FIGURE 9. The SEM result of the longitudinal fracture (DP1000)
The difference in the fracture propagation mode of DP800 and DP1000 is majorly due to the ferrite/martensite volume and their distribution in each sheet metal. For DP800, soft ferrite is still the dominant phase and hard martensite is the second phase and distributes in island shape. Therefore, DP 800 has a better ductility and higher LDR than DP1000. The finding that the DP800 fractures propagate along the ferrite–martensite interfaces is likely related to the void nucleation along the interfaces. While for DP1000, the volume of ferrite and martensite is balanced and hard martensite is in continuous mesh shape. Thus, the fracture must bypass the martensite grain and forms a mixed propagation mode.
CONCLUSION Three commercial DP steel sheets of strength level from 600 MPa to 1000 MPa were studied in this work. The study focused on the recognition of their macroscopic failure mode and understanding of their micro-level fracture propagation mechanism. Major conclusions can be drawn as follows. (1) For the particular DP600, DP800 and DP1000 sheets in this study, the martensite volume is approximately 18%, 32% and 50%, respectively. The martensite phase in DP600 and DP800 is in island shape; while in DP1000, the martensite shows a continuous mesh shape. (2) A LDR of 2.19, 2.15, and 2.07 has been achieved for the three materials. And they all encounter consistent shear crack. Not only latitudinal but also longitudinal cracks are observed on DP800 and above steel grades. (3) SEM result shows the fractures of the three DP sheets are ductile failures. DP800 sheets crack and propagate along the ferrite–martensite interfaces. As for the DP1000 fractures, there is a mixed propagation mode that contains (i) propagation along the ferrite–martensite interfaces and (ii) propagation across the martensite grains.
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ACKNOWLEDGMENTS The joint support from National Science Foundation of China (Grand No. 51005144), Innovation Program of Shanghai Municipal Education Commission and Shanghai Automotive Industry Science and Technology Development Foundation (Grand Nos. 0910 and 1009) is greatly acknowledged. The author also wishes to thank the guidance from Dr. Z Cedric Xia through several communications and the sheet materials provided by SSAB, Sweden.
REFERENCES 1. P. Chen, M. Koç, M.L. Wenner, Trans ASME J Manu Sci Eng 130, 0410061-8 (2008). 2. D. Zeng, Z.C. Xia, H.C. Shih, M. Shi, SAE Paper No. 2009-01-1172 (2009). 3. Wagoner RH, Kim JH, Sung JH. Int J Mater Form 2, 359-362 (2009). 4. A.W. Hudgins, D.K. Matlock, J.G. Speer, C.J. Van Tyne, J Mater Proc Technol 210,741-750 (2010). 5. F.G. Caballero, M.J. Santofimia, C. García-Mateo, J. Chao, C. García de Andrés, Mater Des 30, 2077-2083 (2009). 6. X. Sun, K.S. Choi, W.N. Liu, M.A. Khaleel, Int J Plasticity 25, 1888-1909 (2009). 7. M. Luo, T. Wierzbicki, Int. J. Solids Struct 47, 3084–3102 (2010). 8. W.R. Wang, C.W. He, Z.H. Zhao, X.C. Wei, Mater Des 32, 3320-3327 (2011). 9. T. Baudin, C. Quesnea, J. Jura, R. Penelle, Mater Charac 47, 365-373 (2001). 10. H.S. Lee, B. Hwang, S. Lee, C.G. Lee, S.J. Kim, Metall. Trans. 35A, 2371-2382 (2004). 11. X.J. He, N. Terao, A. Berghezan, Metal Sci. 18, 367-373 (1984). 12. D.A. Korzekwa, R.D. Lawson, D.K. Matlock, G. Krauss, Scripta Metall. 14, 1023-1028 (1980). 13. G. Avramovic-Cingara, Y. Ososkova, M.K. Jain, D.S. Wilkinson, Mat Sci Eng 516, 7-16 (2009). 14. H.C. Shih, M. F. Shi, “An Experimental Study on Shear Fracture of Advanced High Strength Steels,” in 2009 AISI Great Design in Steel Seminar, May 2009, Livonia, MI. 15. N.J. Kim, G. Thomas, Metall Trans. 12A, 483-489, (1981). 16. Z.C. Xia (private communication).
445
'HWHUPLQLQJ([SHULPHQWDO3DUDPHWHUVIRU7KHUPDO 0HFKDQLFDO)RUPLQJ6LPXODWLRQFRQVLGHULQJ0DUWHQVLWH )RUPDWLRQLQ$XVWHQLWLF6WDLQOHVV6WHHO Philipp Schmida, Mathias Liewaldb a Forschungsgesellschaft Umformtechnik mbH Stuttgart, Kornbergstraße 23, 70176 Stuttgart, Germany Institue for Metal Forming Technology, University of Stuttgart, Holzgartenstraße 17, 70174 Stuttgart, Germany
b
$EVWUDFW The forming behavior of metastable austenitic stainless steel is mainly dominated by the temperaturedependent TRIP effect (transformation induced plasticity). Of course, the high dependency of material properties on the temperature level during forming means the temperature must be considered during the FE analysis. The strain-induced formation of '-martensite from austenite can be represented by using finite element programs utilizing suitable models such as the Haensel-model. This paper discusses the determination of parameters for a completely thermal-mechanical forming simulation in LS-DYNA based on the material model of Haensel. The measurement of the martensite evolution in non-isothermal tensile tests was performed with metastable austenitic stainless steel EN 1.4301 at different rolling directions between 0° and 90 °. This allows an estimation of the influence of the rolling direction to the martensite formation. Of specific importance is the accuracy of the martensite content measured by magnetic induction methods (Feritscope). The observation of different factors, such as stress dependence of the magnetisation, blank thickness and numerous calibration curves discloses a substantial important influence on the parameter determination for the material models. The parameters obtained for use of Haensel model and temperature-dependent friction coefficients are used to simulate forming process of a real component and to validate its implementation in the commercial code LS-DYNA. .H\ZRUGV stainless steel, martensite formation, Haensel model, TRIP, thermal-mechanical forming simulation 3$&6 64.60.Ej; 64.60.My; 64.60.qe
,1752'8&7,21 The trend of utilizing FEA for sheet metal forming objectives of austenitic stainless steel has been increased in the last few years. Temperature and the martensite content must be taken into account, in respect to forming history [1]. Highly temperature-dependent TRIP-effect (Transformation Induced Plasticity Effect) hereby plays an important role with regard to material modelling and model parameter determination. There are many efforts spent during last decades on modelling strain-induced martensite formation. One of the most discussed models is the nonisothermal model by Haensel [1]. It is important to determine the actual martensite and temperature evolution in austenitic stainless steel accurately, independent from preciseness of the used model. Measurements of the martensite and temperature evolution for determination of model parameters during tensile tests in austenitic steel as proposed by Haensel have been conducted by various authors [4, 5, 6]. Typically, the martensite content is measured by magnetic induction methods (feritscope), but it is challenging to calculate the hardening behaviour influencing ’-martensite content. This paper validates the parameter determination process including implementation of Haensel material model into LS-DYNA code and shows the possibility to put thermalmechanical forming simulations of austenitic stainless steel into practice.
0(7$67$%/(67((/6$1'0$7(5,$/02'(/ Modelling of metastable austenitic stainless steel is mainly based on feasibility and alloy composition [2] or on the effect of martensite formation at shear band crossings [3]. As a high temperature dependence of martensite formation is proven by Angel [7], attempts by Tsuta [8] include influence of raising temperature level. Various
The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 446-452 (2011); doi: 10.1063/1.3623643 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
446
complex matrix or thermodynamically-bases material models for strain induced martensite formation are known from Hallberg [9] or Galleé [11]. Incremental descriptions for non-isothermal forming operations were formulated by Haensel [1] and Springhub [12]. Haensel model for that reason until now is one of the most discussed models and showed sufficient correlation with real components in [13, 4, and 5]. Schedin [14] implemented the Haensel model into commercial code of LS-DYNA, but commented found misinterpretation of yield surface based on von Mises theory and the lack of temperature-dependent forming limit curves. Attempts to clarify this issue and to overcome found weaknesses were conducted by Krauer [5]. Parameter determination was done by Hansel’s approach of using non-isothermal tensile tests in this work with the aim of validating the martensite and temperature-measurement methods, parameter calculation, implementation in LS-Dyna and the model itself. The Haensel model is separated into two equations, one describing the martensite formation rate, the other describes occurring work hardening.
dVM d
Q
B T e A
§ 1 VM ¨¨ © VM
· ¸¸ ¹
1 B B
VM ^0,5 >1 tanh C D T @` p
(1)
The work hardening is based on the above calculated martensite content Vm
k ges. f
>B
HS
@
BHS AHS exp m n (K1 K 2T) kfo VM '
(2) o' f
There is a total of 13 material parameters in equations 1 and 2: A, B, Q, p, C, D, BHS, AHS, m, n, K1, K2, and k . Another parameter 0 was implemented by Schedin [13] to avoid numerical problems; the initial martensite content must be set in advance and is also a parameter. The fundamental idea is to determine model parameters in ordinary tensile tests.
(;3(5,0(176 6KHHWPDWHULDO The metastable austenitic stainless steel EN 1.4301 with sheet metal thickness of 0.8 mm was supplied by ThyssenKrupp Nirosta / Germany. Its chemical composition is shown in table 1. The Md30-temperature can be calculated as 15.8°C using the formula of Nohara [10]. 7$%/( chemical composition of the investigated steel grade EN 1.4301 [%] PDWHULDO& (1
0.036
6L
0Q
3
6
&U
1L
1
0.39
1.29
0.029
0.005
18.1
8.01
0.051
1RQLVRWKHUPDOWHQVLOHWHVWLQJRYHUYLHZ The procedure was adopted by Haensel at different test velocities of 0,01 s-1, 0,005 s-1 and 0,002 s-1 to keep temperature level low and the martensite evolution relatively high. Tests were performed according to DIN ISO 10002 with a minimum of 5 valid test runs per velocity and rolling direction. Strain, temperature and martensite content were continuously measured and recorded during the uniaxial tensile test. In order to determine martensite evolution, a Feritscope MP30 (Fischer Comp. / Germany) was used. It was directly assembled to specimen surface and fixed with an elastic cord. One must ensure, that function of lateral contraction gauge as used in common tensile test machines (here Roell & Korthaus RKM 100/ 20) was not influenced by the elastic cord. Temperature itself was measured by a non-contact pyrometer. Emissivity factor fluctuates during the tensile test because of roughening effects of the surface. Calibration with thermocouples before and after the test was used to adjust this effect by linear interpolation.
447
Measurement and calculation of the martensite evolution Measuring procedure of the martensite content basically turned out to play an important key role in parameter determination procedure, because of its direct influence on the hardening behaviour. To calculate the actual martensite content, three main effects must be included in the procedure as follows: (1) Calibration factor between Fertiscope indication and the true ’-martensite content. To simplify the determination, Fischer [15] recommends a constant factor of 1.7, while calibration curves are recommended by Greisert [16] and other authors, see figure 1. Because Greisert used x-ray diffractometer and pre-electro-polished surface of test specimen to eliminate residual martensite influences, his calibration curve is used in this work. It must be noted, that no data is available for Feritscope values below 2%, which leads to unknown failure in this relevant range. 30
Greisert Hünicke Peterson Hecker Fischer
Feritscope %
25 20 15 10 5 0 0
10
20
30
40
50
60
'-martensite; %
),*85( Calibration curves of different authors in comparison to constant factor 1.7 by Fischer
(2) Villari-effect which describes the stress level influence on the magnetic behaviour of steel. The correction factor can be calculated as
VM VM0
(3)
0
In equation 3, VM is the indicated martensite content during the test and VM the measured martensite content value after load relief. all tests were carried out at 0°, 45°, 90° rolling directions, calculated Villari factors show positive values for 0° and 45° and 90° neutral. A precise determination is difficult because of the limited Feritscope measurement accuracy. The Villari correction factor for high stress levels was extrapolated using the existing data.
Villari correction
1,02 1,01
0° 45° 90°
1,00 0,99 0,98 150
200
250
300
350
400
2
stress[N/mm ]
),*85( Calculated Villari-effect correction factors with consideration of rolling direction
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(3) Blank thickness influences the measurement results and can be corrected by correction tables of Fischer or the equation calculated by Haensel [1]:
1 a0
a 1 In VM s 2 (4) s
Where a0 equals 0.10435 and a1 = 0.01976. The compensated martensite content can be calculated using equation (5). It must be noted here, that blank thickness is not a constant factor and that thickness reduces during the test from 0.79 mm to approximately 0.65mm.
VM
VMV
(5)
A further point which was not taken into consideration is the residual ferromagnetic phase. In as-delivered condition, EN 1.4301 grade has 0.2-1% magnetic indication, usually caused by pre-formed martensite during cold rolling operations, but presence of residual ferrite can not be excluded either.
7HQVLOHWHVWUHVXOWV Tensile test are performed under the conditions described above. Figure 3 (a) shows the flow curve of EN 1.4301 in the rolling direction at different velocities. The velocity indirectly influences the temperature development (figure 3 (b)) because of lower heat conduction und convection loss, coupled with a faster test. Change in temperature mainly influences the elongation at fracture and tensile strength within this temperature range. Maximum temperature was measured at 32°C in the zone without necking immediately before fracture. (a) flow curves
(b) temperature development
1200
35
30
800
tem perature [°C]
yield stress [N/mm 2]
1000
= 0,01 1/s 600
25
= 0,005 1/s = 0,002 1/s
400
= 0,01 1/s
20
= 0,005 1/s
200
= 0,002 1/s 15
0 0
10
20
30 strain [%]
40
50
60
0
20
strain [%]
40
60
),*85( Flow curve of EN 1.4301 in rolling direction and measured temperature development
Low test temperatures promoted high martensite formation. Figure 4 shows the final calculated martensite evolution for EN 1.4301 at different rolling directions over the strain. Martensite formation starts at a certain martensite start temperature Ms. This temperature must not be reached at the same strain level; in this case it differs between 16.8% for = 0.01 to 20.8% for = 0.002. The total maximum martensite content is approximately the same at the 0° and 45° rolling direction, but on average 1.05 times higher at 90°. Anisotropy of martensite formation is low, but cannot be excluded.
449
(b) 45° direction of rolling
25 20 15 10 5 0 0
20 40 strain [%]
(c) 90° direction of rolling
35
calc. martensite content [%]
= 0,01 1/s = 0,005 1/s = 0,002 1/s
30
calc. martensite content [%]
calc. martensite content [%]
(a) 0° direction of rolling 35
= 0,01 1/s = 0,005 1/s = 0,002 1/s
30 25 20 15 10 5
35
25 20 15 10
0
60
= 0,01 1/s = 0,005 1/s = 0,002 1/s
30
5 0
0
20 40 strain [%]
60
0
20 40 strain [%]
60
),*85( Calculated martensite evolution in EN 1.4301 for strain rates of 0.01, 0.005 and 0.002 1/s and rolling directions
'HWHUPLQLQJWKHSDUDPHWHUVRIWKH+DHQVHOPRGHO Six parameters A, B, Q, p, C, D of the Haensel material model mentioned above were determined using the least square method and fitting the model parameter to the data. The martensite formation rate dVm/d can be calculated from the results in figure 4 (from its derivation). The strain , the temperature T and the martensite content Vm must be adapted. It became apparent that a proper fit of the model to the measured data cannot be found if the complete strain range was implicated. The model was only fit to the measured data in a range between approximately 18 and 45% strain. Comparison of measured martensite formation rate and the model fit is show in figure 5. Due to complexity of Haensel model, the variation of starting values, solver adjustments, chosen measured values and range influence the result of the least square method operations. As noted by other authors [4, 5], it can still not be determined whether the found result is a global or local maximum, which can lead to many correct parameter combinations.
martensite forming rate
0,25 0,20 0,15 0,01 model
0,10
0,002 model 0,05
0,01 measured 0,002 measured
0,00 0
2
4
6
8
martensite content [%]
),*85( Comparison between model fit and measured martensite formation rate
However, the model parameter could be calculated for EN 1.4301 and all three rolling directions as shown in table 1. Heinemann [13] attempted to minimize the number of parameters and proposed to set the B, C, D and Q constant. This work also attempted to fit a curve on the measured values using the parameters from Heinemann. A satisfying correlation could not be found for any test series; in addition, constant parameters could not be determined in any case. It must be stated that the constant factors determined by Heinemann are not valid for any austenitic stainless steel grades or types.
450
7$%/( Parameter for Haensel material model in reference to rolling direction of sheet A
B
C
D
p
Q
0°
8.986
0.340
-45.012
0.1452
4.413
1384.35
45°
1.929
0.148
-47.870
0.0641
8.443
1376.15
90°
5.591
0.249
-47.870
0.0641
5.544
1375.27
7$%/( Parameter for Haensel hardening model in reference to rolling direction of sheet AHS
BHS
m
n
k
dkf -’
0°
334.9
994.6
4.3243
0.962
0
702.3
45°
352.1
1703.0
1.2937
0.9041
0
320.9
90°
279.0
979.6
5.324
1.149
0
857.8
6,08/$7,215(68/76 The parameters determined were used for the Haensel-model implementation (mat113) in LS-DYNA and the tensile test was simulated under adiabatic conditions. Former attempts done by Hochholdinger [17] or Krauer [18] show good correlations between trial and simulation. Figure 6 shows the distribution of the martensitic content along the specimen - the qualitative correlation is sufficient, but because of the disregarded temperature loss, the total martensitic contents did not match well. Because temperature is an important input parameter to the model, temperature development in tools must be simulated as well. Therefore, thermal boundary conditions (heat conductance, heat radiation, convection and heat transfer) must be know and must be incorporated.
60 start martensite 1.4301 0º 0,01 1.4301 0º 0,005 1.4301 0º 0,002
martensite content [%].
50 40
fracture
30 20 10 0 0
2
4
6 8 10 12 14 16 18 20 22 24 measurement position [cm]
),*85( Simulated tensile test and measured martensite contents along the specimen
In further simulations a circular cup was simulated with the TRIP model in LS-Dyna and with regard of the thermal development in the forming tool. It was possible to show the influence of higher temperatures on the martensite content during start-up processes, which varies in a forming process relevant range.
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',6&866,21 An accurate simulation of the TRIP-effect during sheet metal forming processes according to state of the art looks possible, but challenging. The performed measurements of the ’-martensite content in austenitic stainless steel by magnetic methods proved to be influenced by various factors which are not adequate as of yet. The influence of rolling direction on the martensite evolution looks fairly low, but cannot be excluded. Determining parameters for the Haensel model is difficult because of its high complexity. The calibration of measured temperature and martensite distributions make parameter identification unstable. Even different parameter combinations result from the rolling direction; the influence of marginal alloy variations has not yet been investigated and will be a part of further investigations. First tests showed a high influence of alloy variations on the martensite evolution and are part of current investigations. Further FEM-simulation will be planned with the inclusion of thermal boundary conditions and especially the heat development, which influences the whole process and is needed as an input parameter for forming simulations with regard to the TRIP-effect. It is no longer sufficient to measure the temperature in real process and set this temperature as a constant tool temperature; a fully thermal-mechanical forming simulation will be done in future work.
$&.12:/('*0(176 This work was supported by the research consortium “Heat Development in Thin Sheets of Stainless Steel during Forming Operations“ of the Forschungsgesellschaft Umformtechnik mbH in Stuttgart / Germany in collaboration with the Institute for Forming Technology (IFU) at University Stuttgart from May 2009 to October 2011.
5()(5(1&(6 1. A. Haensel, „Nichtisothermes Werkstoffmodell für die FE-Simulation von Blechumformprozessen mit metastabilen austenitischen CrNi-Stählen“, Dr.-Ing. Thesis, ETH Zürich, 1998. 2. DC. Ludwigson, JA Berger: “Plastic behaviour of metastable austenitic stainless steels”, Journal of the Iron and Steel Institute, 1969 3. G.B. Olsen and Cohen, Kinetics of Strain Induced Martensitic Nucleation,Metallurgical Transactiona 6A, p.791-794, 1975 4. Next Generation Vehicle, “Stainless Steel Automotive Engineering Guide Lines”, p.30, 2007 5. J. Krauer, „Erweiterte Werkstoffmodelle zur Beschreibung des thermischen Umformverhaltens metastabiler Stähle“, Dr.-Ing. Thesis, ETH Zürich 2010 6. E. Schedin and J. Kajberg, Real Behaviour and FEA Modelling of Stainless Steels, Proceedings of the FLC, Zürich, 2006 7. T. Angel, “Formation of Martensite in Austenitic Stainless Steels, Effect of deformation, Temperature and Composition”, Journal of the Iron and Steel Institute, p.165, 1954 8. T. Tsuta, JAC. Ramirez, Y. Mitani, K. Osakada: Flow Stress and Phase Transformation Analyses in the Austenitic Stainless Steel under Cold, JSME Int. Journal, Vol. 35, No. 2, 1992 9. H. Hallberg, Paul Håkansson, Matti Ristinmaa, “Thermo-mechanically coupled model of diffusionless phase transformation in austenitic steel“, International Journal of Solids and Structures 47, p.1580–1591, 2010 10. K. Nohara, Y. Ono, N. Ohashi, Tetsu-to-Hagane 63 (1977) p. 5 11. S. Gallée, P. Pilvin, “Deep drawing simulation of a metastable austenitic stainless steel using a two-phase model”, Journal of Materials Processing Technology 210, p.835–843, 2010 12. B. Springhub, „Semi-analytische Betrachtung des Tiefziehens rotationssymmetrischer Bauteile unter Berücksichtigung der Martensite evolution“, Dr.-Ing. Thesis, IFUM Universität Hannover, 2006 13. G. Heinemann, “Virtuelle Bestimmung des Verfestigungsverhaltens von Bändern und Blechen durch verformungsinduzierte Martensitbildung bei metastabilen rostfreien austenitischen Stählen“, Dr.-Ing. Thesis, ETH Zürich, 2004 14. E. Schedin, D. Hilding, “Finite Element Simulation of the TRIP-effect in Austenitic Stainless Steel”, SAE International 04M3, 2003 15. H. Fischer GmbH, FERITSCOPE® FMP30 Bedienungsanleitung, 2008 16. C. Greisert, “Strain-induced formation of martensite during forming and springback behaviour of annealed and hard coldrolled stainless steel grades EN 1.4301 and EN 1.4318”, Dr.-Ing. Thesis, 2004 17. Hochholdinger, “Simulation of Austenitic Stainless Steel in LS-DYNA”, 7.Anwenderformum LS-DYNA, Frankenthal, 2007 18. P. Hora, J.Krauer, Thermoactive process modelling of deep drawing with austenitic stainless steel, IDDRG international conference, p.17-28, Golden Co USA, 2009
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Efficient and Robust Prediction of Localized Necking in Sheet Metals Holger Aretz1and Olaf Engler Hydro Aluminium Rolled Products GmbH, Research & Development, Georg-von-Boeselager-Str. 21, D-53117 Bonn, Germany Abstract. The recently proposed Critical Specific Tension (CST) model is jointly used with the well-known Marciniak and Kuczy´nski (M-K) model to predict localized necking in anisotropic sheet metals in the regime of negative and positive minor in-plane strains, respectively. A significantly simplified method is presented to calculate the critical tensile stress required in the CST model, without the need of iterative computations. In the present work the CST/M-K model is used along with a rate-independent phenomenological elasto-plastic constitutive model as well as the known visco-plastic self-consistent (VPSC) crystal plasticity model developed by Tomé and Lebensohn. A comparison between experimental data and the limit strains predicted by means of the phenomenological constitutive model reveals a very good agreement. In order to validate the correctness of the non-trivial computational implementation of the VPSC-based CST/M-K model the predicted necking strains are compared with results obtained by using the phenomenological constitutive model. It is shown that the results of both approaches are in good agreement. Keywords: Sheet metal forming, Formability, Localized necking, Limit strains PACS: 81.05.Bx, 81.20.Hy, 81.40.Ef, 81.40.Lm, 83.10.Ff, 83.10.Gr, 83.50.Uv, 81.70.Bt, 83.60.La, 83.80.Ab, 89.20.Bb, 89.20.Kk
INTRODUCTION The most popular model to predict localized necking in sheet metals is the one proposed by Marciniak and Kuczy´nski [14], referred to as the M-K model in the following. The M-K model assumes a pre-existing initial imperfection in form of a narrow band or groove whose orientation is assumed of being normal to the major stress axis. The original M-K model was intended to predict necking strains in the regime of positve minor in-plane strains. Due to the presence of the imperfection the strain will localize there. In other words the M-K model assumes that a neck develops normal to the major stress axis. In order to predict necking strains in the regime of negative minor in-plane strains Hutchinson and Neale [9, 10] extended the M-K model. It is well-known that for negative minor in-plane strains the assumption of a neck that is normal to the major stress axis is not valid. For example, in uniaxial tension of sheet specimens a neck develops at an angle that is inclined to the tensile direction. In the Hutchinson-Neale extension [9, 10] different orientations of the imperfection are tested and the one leading to the most conservative (i.e. lowest) limit strain is finally selected. Obviously, anticipating an acceptable resolution of imperfection orientations to be tested is a task left to the user and is a compromise between accuracy and computational effort. In a typical computational implementation of the M-K model the considered (imperfect) sheet is incrementally stretched for different imposed strain-modes, usually ranging from uniaxial over plane-strain to equibiaxial tension. Necking occurs when the strain increment inside the imperfection becomes several times larger than in the region outside, see e.g. [2]. The in-plane strain at necking associated with the region outside the imperfection constitute a single point of the forming limit curve (FLC), which is the central part of the well-known forming limit diagram (FLD). The imposed strain increments used in an M-K analysis must be small in order to capture the instant of necking accurately. Usually, the strain increments are of the order 1 · 10−4 to 1 · 10−3 . For most metallic materials this results in a large number of increments until necking is detected. In each increment the field quantities inside the imperfection must be calculated, which is achieved by incrementally solving the force equilibrium equation in the direction normal to the imperfection. Due to the non-linearity of the governing boundary value problem the associated numerical solution process is costly, especially in the regime of negative minor in-plane strains. In particular, the solution process becomes very time demanding when computationally expensive texture-based crystal plasticity models are being used
1
Corresponding author; e-mail:
[email protected] The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 453-460 (2011); doi: 10.1063/1.3623644 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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to describe the mechanical response of the material at hand. In this contribution the recently proposed critical specific tension (CST) model is jointly used with the M-K model to predict localized necking in anisotropic sheet metals [3] in the regime of negative and positive minor in-plane strains, respectively. The CST model avoids all of the previously mentioned pitfalls of the M-K model and is, therefore, very appealing. In the aforementioned reference it was already demonstrated that the predictions of the CST model agree very well with predictions of the M-K model. The critical tensile stress required in the CST model can be easily computed using the major strain at plane-strain tension predicted by the M-K model; compared to the original paper [3] a significantly simplified procedure will be presented without the need of iterative computations. In the present work the combined CST/M-K approach is applied to predict localized necking of different materials. Phenomenological constitutive models as well as the known visco-plastic self-consistent (VPSC) crystal plasticity model developed by Lebensohn and Tomé [12] are utilized to describe the material behavior.
MODELLING OF LOCALIZED NECKING In the present work the M-K model [14] is combined with an extended version of Hill’s localized necking model [6]. Due to space limitations only the extended Hill model will be described in the following. For further details the reader is referred to [2, 3]. In Hill’s model of localized necking rigid-plastic material behavior is assumed, see [6]. An equivalent, but more general, derivation of Hill’s model can be found in the book of Marciniak et al. [15, p. 68]. Their derivation is actually independent of the material behavior, i.e. it holds for rigid-plastic as well as for elastic-plastic materials, no matter whether the response is rate-dependent or rate-independent. Therefore, the derivation of Marciniak et al. [15, p. 68] is followed in the present paper. Accordingly, necking is assumed to set in when the major force per unit width, f1 , attains or passes a maximum, i.e. d f1 = d(σ11 · t) = d σ11 · t + σ11 · dt ≤ 0
⇔
d σ11 + σ11 · d ε33 ≤ 0
(1)
with t being the current sheet thickness, σ11 the 11-component of the tensor of the Cauchy stress and d ε33 = dt/t the thickness strain increment. Due to the fact that necking is assumed to coincide with the maximum of f1 , Hill’s criterion for localized necking can be referred to as the Maximum Specific Tension (MST) criterion. The motivation for extending the MST model stems from the following idea, see also [3]. The MST model can only be applied to negative minor in-plane strains while the Marciniak and Kuczy´nski [14] (M-K) model can be applied to both the regime of negative and positive minor in-plane strains. Since the MST model is significantly more efficient than the M-K model it is appealing to combine the MST model with the M-K model, especially when computationally demanding material models are being used (e.g. texture-based crystal plasticity models). However, for plane-strain tension the M-K model generally predicts lower limit strains than the MST model does, which is due to the imperfection used in the M-K model. From a geometrical perspective there will always be a ‘gap’ between the limit strains predicted by both models. This shows that a combination of the M-K model and the MST model is not straightforward. For this reason the MST model was slightly modified based on the following assumption: Necking sets in when f1 attains a critical value, which is not necessarily coincident with the maximum of f1 . Accordingly, the extended MST criterion reads as follows: d f1 = d(σ11 · t) = d σ11 · t + σ11 · dt ≤ f1∗ f1∗
⇔
∗ d σ11 + σ11 · d ε33 ≤ σ11
∗ , σ11
(2)
The critical values are denoted as and respectively. In the following the inequalities (2) are referred to as the Critical Specific Tension (CST) model. ∗ can be considered as material dependent parameters. Thus, one may think of The critical values f1∗ and σ11 determining them experimentally. This is, however, not in the scope of the present work. It is recalled that the motivation for the CST model was to combine it with the M-K model. By means of a numerical procedure the ∗ can be adapted so that the predictions of the CST model and the M-K model agree for planecritical stress σ11 ∗ is used for the other strain-modes being considered. An iterative strain tension. Thereafter, the same value of σ11 ∗ was proposed in [3]. Meanwhile, it was realized that such an iterative procedure can be procedure to calculate σ11 entirely avoided. The new procedure works as follows. Provided that the prediction of the M-K model at plane-strain ∗ for the CST model can be readily computed from tension is available, the critical stress σ11 plane-strain ∗ σ11 := d σ11 + σ11 · d ε33 (3) M-K model
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The bracketed quantities are to be computed by the M-K model for plane-strain tension. In the computational implementation differentials are replaced by finite differences, see [3]: plane-strain ∗ k+1 k+1 k+1 σ11 := Δσ11 + σ11 · Δε33 ,
with
k+1 k+1 k Δσ11 := σ11 − σ11 ,
k = 0, 1, 2, . . .
(4)
M-K model
The superscripts k and k + 1 are used to label consecutive time steps during the solution process. The combination of the CST model with the M-K model will be referred to as the CST/M-K model in the remainder of this paper. For more details it is referred to [3].
CONSTITUTIVE MODELLING The same rate-independent elastic-plastic CST/M-K model as in [3] was used in the present work. For a detailed description it is referred to [3]. In addition to the elastic-plastic model, in the present work the visco-plastic self-consistent (VPSC) model (version 7) developed by Tomé and Lebensohn [12, 17] is used in combination with the CST/M-K model to predict localized necking. Further details of the VPSC model may be found in the book [11, pp. 466-509]. Compared to the elasticplastic approach used in [3] there are no conceptional differences when the VPSC model is used. This is due to the fact that both material models repond in the same way, i.e. a strain increment is prescribed/imposed and the corresponding stress tensor is returned. Hence, the numerical implementation of the VPSC-based CST/M-K model follows exactly the same principles as for the elastic-plastic version described in [3]. For the VPSC-based necking prediction the prescribed strain increment is 5 · 10−4 , which is somewhat larger than the value used in the elastic-plastic model described in [3]. Signorelli et al. [16] implemented the VPSC model in an M-K model including the HutchinsonNeale extension for the regime of negative minor in-plane strains. The VPSC model is a rate-dependent plasticity model. Since the rate-dependence can not be entirely switched off a comparison with results obtained by means of a rate-independent elastic-plastic model is a priori difficult and a perfect agreement can not be expected. As a workaround the rate-dependence of the VPSC model can be reduced to a low level by using an appropriate value for the parameter that controls the rate-sensitivity of the mechanical response. However, due to reasons of numerical stability the rate-sensitivity parameter can not be arbitrarily chosen, see below. Furthermore, it is known that the limit strain for plane-strain tension predicted by the M-K model is shifted to larger values when the rate-sensitivity has a positive sign (see e.g. [15]). Hence, in order to obtain the same limit strains for plane-strain tension different imperfection sizes must be used in the VPSC-based and in the elastic-plastic M-K model. In general, in order to obtain comparable plane-strain predictions the imperfection in the VPSC-based M-K model must be larger in size than for a rate-independent model. Since the VPSC code was not designed for the present purpose some minor but important modifications had to be employed which are described in detail in [13]. The major modification refers to all history-dependent variables (hardening, texture). In the M-K model the force equilibrium between the imperfection and the surrounding matrix must be established. Due to the non-linearity of the governing equation an iterative procedure is required, preferably using Newton’s method for non-linear equations. However, unless appropriate modifications are made the VPSC code updates the history-dependent variables each time it is called. Thus, the major modification being made was to store the history variables at the beginning and the end of the current time increment. Before entering a Newton iteration the history variables are initialized by their values at the beginning of the time increment. An update of the history variables is only employed when convergence of the force equilibrium iterations is achieved. Moreover, one has to ensure that the input data is read only at the very beginning of the considered strain-mode, otherwise the history variables will always be overwritten by their initial values. It is finally noted that in the present work solely plane-stress problems are considered. The VPSC code [17] can handle plane-stress problems by properly specifying the boundary conditions. In particular, the user can specify a zero stress in the sheet normal direction, which was done in the present work.
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EXAMPLE NO. 1: ALUMINIUM ALLOY AA6016-T4 Objectives The objective of this application example is to demonstrate that the predictions of the elastic-plastic CST/M-K model are in close agreement with the Hutchinson-Neale extension of the M-K model. For this purpose, the simulation results of Butuc et al. [5] were used as a reference. The same example was already considered in [3], but in the said reference ∗ -value was computed by means of an iterative procedure. The new method outlined above is used in the present the σ11 work.
Elastic-Plastic CST/M-K Model The material data used in the present work correspond to the aluminium alloy AA6016-T4 considered in the work of Butuc et al. [5]. Isotropic hardening is assumed. The reference strain-hardening curve is given as follows [5]: Yref (ε¯ ) = 318.103 − 191.103 · exp(−8.706 · ε¯ ) MPa
(5)
According to [5] the directional r-values, uniaxial yield stresses as well as the equibiaxial yield stress are as follows: Uniaxial tension, 0◦ : yield stress 127 MPa, r-value 0.80. • Uniaxial tension, 45◦ : yield stress 127 MPa, r-value 0.43. • Uniaxial tension, 90◦ : yield stress 114 MPa, r-value 0.61. • Equibiaxial tension: 123.44 MPa. •
In order to account for plastic anisotropy the yield function ‘Yld96’ [4] was used, in accordance with [5]. The calibration of Yld96 involved the anisotropy data listed above. The parameters of the yield function were adopted from [5] and read as follows: c1 = 1.0474, c2 = 0.7752, c3 = 1.0720, c4 = 1.0000, c5 = 1.0000, c6 = 0.9288, αx0 = αx1 = 2.0, αy0 = αy1 = 3.5, αz0 = 1.0, αz1 = 1.05, m = 8. They correspond to Yref = 127 MPa. With these parameters the yield function Yld96 fits the input data listed above exactly. A Young’s modulus of E = 70 GPa and a Poisson’s ratio of ν = 0.33 were used, along with the assumption of isotropic elasticity. Furthermore, the M-K imperfection used in the present work had a value of 0.998, in agreement with [5]. Linear strain-paths were assumed in the necking predictions.
Results The calculated forming limit curve using the elastic-plastic CST/M-K model is shown in Fig. 1. One may notice the good agreement of the predictions of the M-K model of Butuc et al. [5] (including the Hutchinson-Neale extension) ∗ = 0.65977 · 10−2 MPa was calculated. and the present elastic-plastic CST/M-K model. In the CST/M-K model σ11 The total computation time took 10 s for 10 points of the forming limit curve. The good agreement of the computed limit strains with the experimental data shown in Fig. 1 is remarkable.
EXAMPLE NO. 2: ALUMINIUM ALLOY AA6016-T4 WITH PRE-STRAINS Objectives The objective of this application example is to demonstrate that the elastic-plastic CST/M-K model can be applied to non-linear strain-paths as well, which is a fundamental requirement for the general applicability of necking models. The same material as in the previous section (i.e. AA6016-T4 according to Butuc et al. [5]) is considered. Only qualitative results are presented, i.e. a comparison with experimental data is not given.
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0.4
Log. major in-plane strain [-]
0.35 0.3 0.25 0.2 0.15 0.1 0.05
Elastic-Plastic CST/M-K Model Rigid-Plastic Simulation Butuc et al. Experiment Butuc et al.
0 -0.2
-0.1
0
0.1
0.2
0.3
0.4
Log. minor in-plane strain [-]
FIGURE 1. Predicted forming limit curve for AA6016-T4 using the elastic-plastic CST/M-K model. For comparison, the curve predicted by Butuc et al. [5] is shown as well.
Elastic-Plastic CST/M-K Model Exactly the same material data that was used in the first application example on AA6016-T4 is used here. The value of the M-K imperfection was again 0.998. Three cases were generated involving the following types of pre-straining: 1. Case 1: No pre-strains. 2. Case 2: Pre-deformation under equibiaxial straining with a pre-strain of 0.05. 3. Case 3: Pre-deformation under uniaxial tension with a pre-strain of 0.1.
Results Figure 2 shows the predicted forming limit curves (FLC) for the generated cases described above. Compared to case 1 the minimum of the FLC corresponding to case 2 is shifted to the lower right, which is typical for predeformation under equibiaxial straining. In contrast, the minimum of the FLC corresponding to case 3 is shifted to the upper left, which is typical for pre-straining under uniaxial tension. The results confirm that the present model can be used for forming limit strain prediction under linear as well as non-linear strain-paths.
EXAMPLE NO. 3: ISOTROPIC ALUMINIUM Objectives The primary objective of the application example was to validate the non-trivial computational implementation of the VPSC-based CST/M-K model. For this purpose the intensively validated elastic-plastic CST/M-K model described in [3] (and also used in the previous application examples of the present paper) was applied whose predicted limit strains serve as a reference.
VPSC-Based CST/M-K Model The imperfection factor used in the VPSC-based M-K model was 0.992. The ‘secant’ inclusion-matrix interaction type was used along with a rate sensitivity parameter of n = 80 (see [17] for details). Note that larger values for
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0.4
Log. major in-plane strain [-]
0.35 0.3 0.25 0.2 0.15 0.1 Case 1 Case 2 Case 3
0.05 0 -0.2
-0.1
0
0.1
0.2
0.3
0.4
Log. minor in-plane strain [-]
FIGURE 2. Predicted forming limits curves for AA6016-T4 using the elastic-plastic CST/M-K model. Different types of prestraining are imposed.
the rate-sensitivity parameter resulted in an error termination. The hardening on the slip systems is described by the following extended Voce approach [17]:
τ (Γ) = τ0 + (τ1 + θ1 · Γ) · (1 − exp(−Γ · |θ0 /τ1 |)) MPa
(6)
Herein, Γ denotes the accumulated shear strain in a grain and τ the shear yield stress. The hardening parameters used in the present work are as follows [13]: τ0 = 14.67, τ1 = 12.67, θ0 = 62.00, θ1 = 8.78 (units omitted). The considered material is isotropic aluminium with fcc lattice structure. Accordingly, a random orientation distribution of the grains was used. For a parametric study the number of grains was varied as follows: in the first case 100 and in the second case 500 grains were used. The crystallographic texture was not updated, i.e. the initial texture was used throughout deformation. This ensures that the predictions of the VPSC-based model are comparable with the predictions of the elastic-plastic model that uses isotropic hardening, see below.
Elastic-Plastic CST/M-K Model In the elastic-plastic CST/M-K model the considered material is isotropic aluminium with a Young’s modulus of E = 70 GPa and a Poisson’s ratio of ν = 0.33. The yield surface is described by means of the non-quadratic plane-stress yield function ‘Yld2003’ [1]. Accordingly, the equivalent stress is defined as follows: 1/m 1 |σ1 |m + |σ2 |m + |σ1 − σ2 |m σ¯ := (7) 2 The generalized principal stresses σ1 , σ2 , σ1 , σ2 are defined as follows: a8 σ11 + a1 σ22 (a2 σ11 − a3 σ22 )2 σ1 ± + (a4 σ12 )2 := σ2 2 4 σ11 + σ22 (a5 σ11 − a6 σ22 )2 σ1 ± + (a7 σ12 )2 := σ2 2 4 The dimensionless parameters a1 , . . . , a8 can be calibrated to experimental data. In the present work a1 = . . . = a8 = 1 is used, which reduces the yield function to the one proposed by Hosford [7]. The latter one describes the yield surface of isotropic fcc metals accurately. Furthermore, according to the recommendations concerning fcc materials given in [8] the exponent was set to m = 8.
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1
200
VPSC (100 Orientations) VPSC (500 Orientations) Elastic-Plastic
0.8 Log. Major Strain [-]
True Stress [MPa]
150
100
0.6
0.4
50 0.2 VPSC (100 Orientations) VPSC (500 Orientations) Elastic-Plastic
0 0
0.2
0.4
0.6
0.8
0 1
-0.4
FIGURE 3. curves.
-0.2
0
0.2
0.4
Log. Minor Strain [-]
True Strain [-]
Left: comparison of macroscopic strain-hardening curves under uniaxial tension. Right: computed forming limit
In the elastic-plastic model isotropic strain-hardening was used. The utilized reference hardening curve is analogous to that implemented in the VPSC code and is given as follows: Yref (ε¯ ) = A + (B +C · ε¯ ) · (1 − exp(−D/B · ε¯ )) MPa
(8)
When the elastic-plastic CST/M-K model is compared with its VPSC-based counterpart it is very important that the same hardening curve is used in both approaches. An important difference between both models is the following: the VPSC model uses a hardening curve in form of a shear stress that depends on the accumulated shear strain for each slip system, while the elastic-plastic model uses a macroscopic hardening description. The transformation of both hardening models is not straightforward. Thus, in addition to the prediction of forming limit strains the VPSCbased model was also used to compute a stress-strain curve for uniaxial tension. Thereafter, the material-dependent parameters A, B, C, D of the macroscopic hardening description used in the elastic-plastic model were calibrated by means of a least squares solver so that (8) agrees with the hardening curve predicted by the VPSC model. They read as follows: A = 43.7844, B = 37.9214, C = 76.7851, D = 538.94 (units omitted). The imperfection factor used in the elastic-plastic M-K model was 0.996, which represents a somewhat smaller imperfection than the one used in the VPSC-based model, see above.
Results In Fig. 3 the macroscopic hardening curves predicted by the VPSC-based model are compared with the macroscopic description (8) that was utilized in the elasto-plastic model. As one may see the hardening curves are practically coincident. Hence, the fundamental requirement of equal hardening curves in both models is satisfied. Furthermore, there is no notable difference between the prediction that used 100 grains and the one which used 500 grains. Figure 3 also shows the predicted forming limit curves. As can be seen the difference between the prediction that used 100 grains and the one that used 500 grains is negligible. Furthermore, the differences between the VPSCbased model and the elastic-plastic model are small for positive minor in-plane strains, whereas larger differences can be observed for the regime of negative minor in-plane strains. The most significant difference can be observed for uniaxial tension. However, the agreement between both approaches is very promising. The agreement at plane-strain tension can be improved by adapting the imperfection being used in the M-K model, but this was not attempted in the present work. The computation time consumed by the elastic-plastic model was 12 s (for 30 points of the forming limit curve) while the VPSC-based model required 24 min and 50 min (for 10 points of the forming limit curve) using 100 and 500 grains, respectively. All three simulations were run on the same computer using the same compiler options. The
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high computation times associated with the VPSC-based model indicate that the use of an efficient necking model ∗ -values are as follows: (i) σ ∗ = 0.16238 · 10−2 MPa for the VPSC-based can be very beneficial. The calculated σ11 11 ∗ ∗ = model with 100 grains, (ii) σ11 = 0.15783 · 10−2 MPa for the VPSC-based model with 500 grains and (iii) σ11 −2 0.21491 · 10 MPa for the elastic-plastic model.
CONCLUSIONS The combined CST/M-K model is a very efficient tool to predict forming limit curves for sheet metals. This becomes very beneficial when the utilized material model is time consuming, like the VPSC model used in the present work. The results in the present paper represent a snapshot of ongoing development work. At present the VPSC code is being transferred from FORTRAN 77 to Fortran 90/95 programming language, mainly to make programming safer (by using Module Procedures available in Fortran 90/95), to simplify maintenance and to reduce its execution time.
ACKNOWLEDGMENTS Parts of the presented VPSC-based CST/M-K model were implemented by G. Liu during his MSc thesis, see [13].
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
H. Aretz: Applications of a new plane stress yield function to orthotropic steel and aluminium sheet metals, Modelling and Simulation in Materials Science and Engineering 12 (2004) 491-509 H. Aretz: Numerical analysis of diffuse and localized necking in orthotropic sheet metals, International Journal of Plasticity 23 (2007) 798-840 H. Aretz: An extension of Hill’s localized necking model, International Journal of Engineering Science 48 (2010) 312-331 F. Barlat, Y. Maeda, K. Chung, M. Yanagawa, J.C. Brem, Y. Hayashida, D.J. Lege, K. Matsui, S.J. Murtha, S. Hattori, R.C. Becker, S. Makosey: Yield function development for aluminium alloy sheets, Journal of the Mechanics and Physics of Solids 45 (1997) 1727-1763 M.C. Butuc, J.J. Gracio, A. Barata da Rocha: A theoretical study on forming limit diagrams prediction, Journal of Materials Processing Technology 142 (2003) 714-724 R. Hill: On discontinous plastic states with special reference to localized necking in thin sheets, Journal of the Mechanics and Physics of Solids 1 (1952) 19-30 W.F. Hosford: Yield locus of randomly oriented fcc metals according to the Bishop-Hill theory, Metallurgical Transactions 4 (1973) 1416-1417 W.F. Hosford, R.M. Caddell: Metal Forming – Mechanics and Metallurgy, 2nd edition, Prentice Hall, 1993 J.W. Hutchinson, K.W. Neale: Sheet necking-II. Time independent behavior, In: D.P. Koistinen, N.-M. Wang (Eds.), Mechanics of Sheet Metal Forming, Plenum Publ. Corp, pp. 127-153 J.W. Hutchinson, K.W. Neale: Sheet necking-III. Strain-rate effects, In: D.P. Koistinen, N.-M. Wang (Eds.), Mechanics of Sheet Metal Forming, Plenum Publ. Corp, pp. 269-285 U.F. Kocks, C.N. Tomé, H.-R. Wenk: Texture and Anisotropy, Cambridge University Press, 1998 R.A. Lebensohn, C.N. Tomé: A self-consistent anisotropic approach for the simulation of plastic deformation and texture development of polycrystals: application to zirconium alloys, Acta Metall. Mater. 41 (1993) 2611-2624 G. Liu: Formability simulation from texture data by means of the VPSC polycrystal plasticity code, Master thesis, RWTH Aachen University, 2010 Z. Marciniak, K. Kuczy´nski: Limit strains in the process of stretch-forming sheet metal, International Journal of Mechanical Sciences 9 (1967) 609-620 Z. Marciniak, J.L. Duncan, S.J. Hu: Mechanics of Sheet Metal Forming, 2nd ed., Butterworth-Heinemann, 2002 J.W. Signorelli, M.A. Bertinetti: On the role of constitutive model in the forming limit of FCC sheet metal with cube orientations, International Journal of Mechanical Sciences 51 (2009) 473-480 C.N. Tomé, R.A. Lebensohn: Manual for Code Visco-Plastic Self-Consistent (VPSC) Version 7a, updated July 2006
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Fracture prediction in hydraulic bulging of AISI 304 austenitic steel sheets based on a modified ductile fracture criterion Y. Xua, H.W. Songa*, S.H. Zhanga, M. Chenga a
Institute of Metal Research, Chinese Academy of Sciences, Shenyang 110016, China
Abstract. The demand for weight reduction in modern vehicle construction has resulted in an increase in the application of hydroforming processes for the manufacture of automotive lightweight components. This trend led to the research of evaluation on formability of the sheet or tube hydroforming to be noted, particularly the prediction of fracture. In this study, a new proposed approach based on damage theory for fracture prediction considering the deformation history was introduced. And the modified ductile fracture criterion was applied to predict the failure for hydraulic bulging of AISI 304 austenitic steel sheets. The material parameters in terms of the function of strain rate in the failure criterion were determined from the equivalent fracture strains corresponding tensile tests under different stress conditions. Then, in the finite element simulation the effect of strain rates and their distribution as well during practical sheet metal forming process was considered. The hydraulic bulging tests were carried out to identify the fracture behavior predicted from FE analysis. A comparison between the prediction and experimental results showed that the proposed approach with a modified ductile fracture criteria can give better fracture predictions than traditional ways. Keywords: Hydraulic bulging; Austenitic steel; Ductile fracture criteria; Deformation history PACS: 62.20.mm; 81.20.hy; 83.50.uv
INTRODUCTION To save energy and reduce exhaust gas, hydroforming technologies as a modern advanced technology have got wide applications for manufacturing lightweight structural parts in automotive, aeronautic and aerospace industries [1, 2]. By using different blanks, hydroforming technologies can be divided into three types: tube hydroforming, shell hydroforming and sheet hydroforming. For the past two decades, a lot of research works have been carried out in key technologies and solution about the main defects such as wrinkling, buckling and bursting during the hydroforming process. Basically, buckling failure, which prevails when the axial compressive stress on a part exceeds the strength of the material, takes place during the beginning stage of the forming process and causes localized necking due to excessive tensile forces. Wrinkling failure, on the other hand, occurs during both the initial and intermediate stages of forming and can sometimes be eliminated by increasing the internal pressure during the final stage of the hydroforming process. In contrast with buckling and wrinkling, bursting failure is an irrecoverable failure mode. Hence, in order to obtain sound hydroformed products, it is necessary to study the effects of the process parameters on bursting failure in the hydroforming processes[3]. Among the various forming limit criteria suggested for sheet metal forming processes, FLD (forming limit diagram) has been most widely utilized following the pioneering work of Keeler and Backofen [4]. FLD, however, has a strong strain path dependent nature, such that it needs a tremendous amount of data acquisition to depict various in-plane strain modes involved in sheet forming processes[5]. As alternatives to FLD, ductile fracture criteria and continuum damage models have also been used[6]. With recent advances in numerical analysis, particularly in the finite element method, the use of ductile fracture criteria to predict the occurrence of fracture has become popular. Moreover, in order to make the prediction close to the practical situation for various materials and process conditions, the ductile fracture criteria have been constantly modified. In the present study, a new approach was proposed for extending the applications of existing ductile fracture criteria to sheet metal forming
The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 461-468 (2011); doi: 10.1063/1.3623645 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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with the effect of deformation history considered. And then its applications in hydraulic bulging of AISI 304 austenitic steel sheets were discussed.
NEW APPORACH FOR FRACTURE PREDICITON WITH A MODIFIED DUCTILE The Background of Ductile Fracture The idea of ductile fracture criteria to predict failure in a material has been applied for the past fifty years since Freudenthal [7] derived a generalized plastic work criterion based on von Misses stress and strain. There have been many ductile fracture criteria proposed over the years, several of which are based on the void growth relation reported by Rice and Tracey [8]. Cockroft and Latham [9] observed that ductile fractures in metal forming processes tend to occur in the region of largest tensile stress, and they proposed a criterion that is based on the total plastic work per unit volume. Brozzo et al. [10] created a criterion which is a modified version of Cockcroft and Latham and takes hydrostatic pressure into account. Oyane et al. [11] have derived a ductile fracture criterion from the equations of plasticity theory for porous materials. Clift et al. [12] have carried out some plane strain experiments and some finite element calculations on an aluminum alloy. It was found that the total plastic-work criterion was the successful criterion for the prediction of site of fracture. Fracture criteria based on continuum damage mechanics have also been proposed by Lemaitre [13]. Takuda et al. [14] applied various fracture criteria to predict fracture in axisymmetric deep drawing of aluminum alloys and mild steel sheets. Hambli [15] simulated damage evolution and fracture failure during bulk metal forming including stamping and extrusion using damage fracture criterion based on continuum damage mechanics.
New Proposal for Fracture prediction Fracture criteria for metal forming processes as mentioned above usually involve a function of the form:
C where f is a certain function of
Vh V
Vh
³ f(V
)dH
and C is an integration when
(1)
H
is the equivalent strain at fracture. It is
considered that a fracture occurs when the C value exceeds the critical value. Although a wide variety of fracture criteria exist, it is still difficult to find one suitable for all forming processes and it should be noted that all the aforementioned fracture criteria were proposed and used for predicting the fractures in cold metal forming with no consideration of effect of deformation parameters, i.e. temperature and strain rate on the accumulative fracture strain. However, the inhomogeneity of deformation leads to varied strain rates at different areas of the workpiece. In other words, there has been a deformation history which has an influence on the occurrence of the facture in the process of deformation and the critical value C considered to be characteristic of material constant was inaccurate to predict the forming process that has a complicated deformation history. Fracture criteria in terms of the influence of temperature and strain rates on deformation history were firstly proposed by the author in 2007 [16, 17] and used for fracture prediction in Non-isothermal forming for magnesium alloys sheet and hot forming of titanium alloy. In our early work [16], only the deformation temperature history was considered for sheet metal forming, and there is a lack of the influence of the strain rates in the criterion. Recently, Kim et al [3] applied the approach for magnesium alloy sheet forming and considered both temperature and strain rate effects. However, just the influence of deformation history on the critical fracture strain that was a function of Zenerholloman parameter was considered in their criteria without the effect of deformation history on the process of damage evolution. In the proposed approach by considering the effect of deformation history on the process of damage evolution, two hypothesizes are proposed: (1) the strain rate is regarded constant when a small part of material is undergoing a finite strain in a short interval; (2) the contribution of the finite strain to final fracture can be expressed with a parameter 'D which can be calculated with the following equation:
'D
f(
Vh ) 'H / C (T , H ) V
462
(2)
H
f (V h / V ) are the average value of the strain rate and f function in the incremental step that the 'H , C (T , H ) is the critical value of fracture at the deformation condition with temperature T and strain rate H . The integral form of expression (2) can be written as: Hf ª V º (3) D ³ « f ( h ) / C (T , H )»dH 0 ¬ V ¼ where f and C (H ) have the same meanings as in equations (1) and (2); D is named fracture danger index which can where
and
material experienced a finite strain
be regarded as the accumulation on the contribution of finite strain at different deformation conditions to final fracture. When fracture danger index D exceeds 1, a fracture is implied. Then equations (3) is the proposed modified fracture criterion corresponding to the traditional one shown in equations (1).
MATERIALS AND RELATED PARAMETERS Materials Austenitic stainless steel is one kind of widely used engineering materials, due to their high corrosion resistance and versatile mechanical properties. These metastable alloys belong to a kind of typical transformation induced plasticity (TRIP) steels [18] whose high strength and excellent ductility are expected for their strain-induced martensitic transformation [19-21]. They can usually be manufactured by cold working including cold rolling [22], drawing [23] and hydroforming [24]. Therefore prediction on the forming limit for the process of hydraulic bulging of AISI 304 austenitic steel sheets was carried out.
Determination of the Material Parameters in the modified criterion In the present study, the hydraulic bulge tests were carried out at room temperature without considering little internal thermal lift. Since the applied stress and fracture strain of AISI 304 austenitic steel are strongly depended on strain rates, the present work can be considered as an isothermal process and just the influence of the strain rates on accumulated damage value was considered. In addition, the modified Oyane’s ductile fracture criterion is based on damage evolution where fracture occurs when the damage level, D reaches unity. According to the approach in preceding section, f (V h / V ) can be expressed as following:
f (V h / V )
Vh A(H ) V
where A(H ) is a correction factor including the influence of the strain rate Therefore, equation (3) can be rewritten as follows:
Vh A(H )) / C (H )]dH V where C (H ) is the critical damage value with strain rate H . D
³
Hf
0
[(
(4)
H . (5)
To determine the material parameters in ductile fracture criterion, experiments of different stress triaxialities [25] or uniaxial tension combined with numerical simulations [26] were generally carried out. However, in the present study considering that it is inaccurate to determine the material parameters from the results of numerical simulation, since when fracture occurs can not be estimated by the commercial finite element software before inducing the ductile fracture criterion in them. So a threshold value of some deformation parameter needs to be pre-set. It is considered that fracture occurs if the simulation results surpass the threshold value, which has inaccuracy compared with the real process. In consequence, two groups of mechanical experiments of different stress state for AISI 304 stainless steel were investigated to determine the modified parameters A(H ) and C (H ) in the present research. Data measurement and collection need to be as simple as possible. Bar notched specimens are widely used to investigate tensile fracture under different stress trixialities. According to analysis by Bridgman [28], relationship between stress trixiality and geometry of specimens is as following:
463
Vh V
§ a 2 2aR r 2 1 ln¨¨ 3 2aR ©
· ¸¸ ¹
(6)
This ratio is 1/3 at the notch root, r=a, while for r=0 the ratio achieves the maximum value:
Vh V where
V
is equivalent stress and
§ a · 1 ln¨¨1 0 ¸¸ 3 2 R0 ¹ ©
(7)
V h is the hydrostatic pressure or the mean normal stress, a and R are the radius of
the minimum cross-section and the radius of the circumferential, notch, a0 is the initial value of a. The geometries of smooth and notched bar tensile specimens which were selected for the present study are shown in Fig. 1. The stress triaxiality calculated using the initial geometry for both smooth and notched bar tensile specimens were 0.33 and 0.68 respectively. However, it should be noted that the change of the radius of curvature of the neck due to the deformation was not considered in calculating stress and strain components using the Bridgman solution.
FIGURE 1. Smooth and notched bar specimens
A series of tensile tests has been carried out at room temperature under various strain rates within the range for metal forming. Fig. 2 shows the true stress-strain curves of smooth and notched specimens respectively. Fracture strain of AISI 304 stainless steel at various strain rates for different shape specimens was shown in Fig. 3. According to the results of tensile tests, it is obviously noted that the deformation behavior of AISI 304 stainless steel is strain rates-dependent. Fracture strain grew down with strain rates increasing. 1200
1200
(a)
1000
1000
32
800 4 76 5
600 400 200 0 0.0
0.1
0.2
0.3
0.4
0.5
1 1--0.001/s 2--0.003/s 3--0.005/s 4--0.008/s 5--0.01/s 6--0.05/s 7--0.1/s 0.6
True strain
True stress
True stress /MPa
1400
(b)
1
800 7
600
6
5
400 200 0 0.00
0.05
0.10
0.15
True strain
FIGURE 2. The true stress-strain curves at various strain rates for (a) smooth specimens, (b) notched specimens
464
32 4 1--0.001/s 2--0.003/s 3--0.005/s 4--0.008/s 5--0.01/s 6--0.05/s 7--0.1/s 0.20 0.25
0.72
smooth specimens notched specimens
0.5
calculated value fitted curve
0.68 0.64
0.4
C
Fracture strain
0.6
0.3
0.60 0.56
0.2
0.52
0.1 0.00
0.02
0.04
0.06
0.08
0.10
Strain rates FIGURE 3. Fracture strain at various strain rates for different specimens
0.00
0.02
0.04
0.06
0.08
0.10
Strain rates FIGURE 4. Calculated C- H relation through tensile tests and equation (5)
Combined with equation (5) and the data from the tensile tests, the modified material parameters
A(H ) and
C (H ) could be calculated. Fig. 4 shows the relation of C values with strain rates obtained. From curve fitting, the material parameters can be expressed as: (8) A 0.05402 exp( H 0.00645) 0.17955 C 0.44453 0.12951 exp( H 0.25367) 0.09115 exp( H 0.00438) 0.07327 exp( H 0.00437)
(9)
FAILURE PREDICTION IN SHEET HYDRAULIC BULGING Sheet Hydraulic Bulge Test Hydraulic bulge tests were carried out on rectangular specimens of dimension 200 mmÝ200 mmÝ0.46 mm. In this experiment, the die set was installed in a hydraulic driven type universal sheet forming testing machine. Firstly, the sheet metal was clamped at its edges by a blank holder. When the lower bar began to rise and push the lower chamber, oil used as a pressure carrying medium in the lower chamber was pressurized. Then the sheet was bulged into the cavity of the upper die. The structure of hydraulic bulge test setup was shown in Fig. 5.
FIGURE 5. Tools used in hydraulic bulge test
Fracture Prediction Based on the Modified Fracture Criterion In order to numerically predict the bursting which appeared in the previous experimental tests, the currently developed fracture criteria were implemented into the user subroutine of finite element code MSC. Marc, and the damage values were evaluated at each step to monitor the fracture initiation. In addition, the sheet blank was meshed by four-node 3-D shell elements. The die and blank holder were defined rigid. Due to the symmetry of the model, 1/4 part was applied as shown in Fig. 6. Six loading paths were applied in the simulation as shown in Fig. 7. The maximum hydraulic pressure was uniform, while the loading time was different for the six loading paths. In other
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words, the loading rates varied. The results of simulation using the modified fracture criterion are compared with the traditional one in Fig. 8. It can be seen that the predicted expanding height using the modified fracture criterion is more close to the experimental results than using traditional one. And it is also obviously noted that the limit expanding height of bulged sheet decreases with increase of loading rate by using the modified fracture criterion. However, the limit expanding height rarely fluctuates for different loading paths in the simulation by using tradition ways. From the results of tensile tests, it is known that it has a great influence of strain rates on the deformation behavior of AISI 304 stainless steel. That is to say, the modified fracture criterion makes the fracture prediction more accurate and close to the real situation. In addition, as shown in Fig. 9, the simulation results based on the modified fracture criterion show that the fracture position occurs near the dome top of the bugled sheet instead of at the center of the dome top, which agreed well with the experimental observation.
FIGURE 6. Finite element model in simulation
Expanding height /mm
Hydraulic pressure /MPa
18 16 14 12 10
Path 1 Path 2 Path 3 Path 4 Path 5 Path 6
8 6 4 2 0
2
4
6
8
10
200 400 600 800 1000
Time /s
FIGURE 7. Loading paths applied in the simulation 36 prediciton by new approach prediciton by traditional approach 34 experimental result
32 30 28 26 24 22 -2
0
2
4
6
8
10
12
14
16
18
Loading rate /MPa*s-1 FIGURE 8. Comparison of the limit expanding height of bulged sheet obtained from the simulation
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FIGURE 9. Comparison of fracture predicted by finite element simulation coupled (a) with the modified fracture criterion, (b) without the traditional fracture criterion and (c) the experimentally observed fracture near the dome top of bulged specimen
CONCLUSIONS (1) A new approach for fracture prediction based on damage theory by considering the influence of strain rate history is proposed. (2) Uniaxial tensile testes for AISI 304 steel specimens under different stress conditions are carried out at different strain rates, and the material parameters (modification factor A and critical damage value C) in the proposed fracture criterion are determined to be function of strain rate. (3) The application of the new fracture criterion in the fracture prediction of hydraulic bulging of AISI steel sheets shows a good agreement with the experimental results, which validates the proposed method.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
F. Dohmann and C. Hartl, J. Mater. Process. Technol 60. 669-676 (1996). M. Koc and T. Altan, J. Mater. Process. Technol 108. 384-393 (2001). M. Koc and T. Altan, Int. J. Mach. Tools Manuf. 42:123-138 (2002) H. K. Kim and W. J. Kim, Mech. Mater 42. 293-303 (2010). T. B. Stoughton and X. Zhu, Int. J. Plasticity 20, 1463-1486 (2004). J. Kim et al., Int. J. Adv. Manuf. Technol 22. 357ˀ 362 (2003). A.M. Freudenthal, The Inelastic Behaviour of Engineering Materials and Structures, Publisher City: John Wiley & Sons, New York, 1950. J. R. Rice and D.M. Tracey, J. Mech. Phys. Solids 17. 201–217 (1969). M. G. Cockroft and D.J. Latham, J. Inst. Met 96. 33-39 (1968). P. Brozzo et al., Proceedings of the Seventh Biennial Conference of IDDRG, Amsterdam, The Netherlands, 1972. M. Oyane et al., J. Mech. Work. Tech 4. 65 (1980). S. E. Clift et al., Int. J. Mech. Sci 32. 1-17 (1990). J. Lemaitre, Trans. ASME: J. Engng. Mater. Technol 107. 83–89 (1985). H. Takuda et al., J. Mater. Process. Technol 95. 116-121 (1999). R. Hambli and D. Badie-Levet, Comput. Methods Appl. Mech. Engrg 186. 109-120 (2000). W. T. Zheng et al., Proc. IMechE, B: J. Eng. Manu 221. 981-986 (2007).
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17. 18. 19. 20. 21. 22. 23. 24. 25. 26.
S. H. Zhang et al., Proceedings of the Plasticity of Conventional and Emerging Materials: Theory and Applications, Alaska, 2007. F. D. Fischer et al., Int. J. Plasticity 16. 723–748 (2000). S. S. Hecker et al., Metall. Trans 13A. 619 (1982). M. R. Rocha et al., Mater. Sci. Eng. A 517. 281–285 (2009). Y. Xu et al., Mater. Lett 65. 1545–1547 (2011). S. Gallée and P. Pilvin, J. Mater. Process. Technol 210. 835–843 (2010). M. Ahmetoglu et al., J. Mater. Process. Technol 98. 25–33 (2000). D. Raabe, Acta. Mater 45. 1137-1151 (1997). M. S. Mirza et al., J. Mater. Sci 31. 453-461 (1996). P.W. Bridgman, Studies in Large Plastic Flow and Fracture, Publisher City: McGraw-Hill, New York, 1952.
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Identification of Deformation Mechanisms Responsible for Failure in Incremental Forming using a Damage Based Fracture Model Rajiv Malhotra, Liang Xue, Jian Cao* and Ted Belytschko
Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA Abstract. Single Point Incremental forming (SPIF) has generated significant interest recently due to its increased formability and greater process flexibility. However, the complicated deformation mechanisms involved in SPIF have prevented conclusive identification of the primary mechanisms responsible for failure. This work successfully predicts the forming forces and occurrence of failure in SPIF using explicit FEA with a damage based fracture model in which failure envelope depends on the hydrostatic pressure and the Lode angle. Furthermore it is shown that through-thethickness shear is primarily responsible for failure in SPIF. Simulations are also performed to form the same component using SPIF and using a conventional punch and die, and a comparison is made between the dominant mechanisms of failure in the two processes. Furthermore, it is shown that reduction of tool-sheet friction can delay fracture in SPIF and the mechanism behind this effect is discussed as well. Keywords: Incremental Forming, Fracture, Through-the-thickness shear, Friction PACS: 81.20.Hy
INTRODUCTION Single Point Incremental Forming (SPIF) is a method of forming sheet metal into a desired shape without using a conventional die and punch set and has envisioned considerable interest due to its greater process flexibility, higher formability and significant cost savings for low batch volume production. However, a significant challenge in this process has been the prediction of fracture during the forming process and a better understanding of the physical mechanisms governing fracture in SPIF is of significant importance to the engineering community. In the recent past Silva et al. [1] compared a membrane based analytical solution of the stress and strain fields with a shell based simulation in LS-DYNA and found good agreement between the principal stresses. Malhotra et al. [2] investigated the use of various material models to simulate SPIF using FEA and showed that a damage based material model can predict forces in the process quite well. Cerro at al. [3] simulated SPIF of a pyramid with a 75° wall angle with shell elements and obtained a 5% difference between the maximum values of the measured and calculated tool z forces. Ambrogio et.al [4] simulated SPIF with one linear brick element through the thickness of the sheet and reported that a compressive mean stress was observed in the local zone of contact near the tool. Van Bael et al. [5] extended a Marciniak-Kuczyisnki type of localized necking prediction model with three different material models and showed that while the forming limit predictions were higher than that for monotonic loading, their models still underestimated the forming limits obtained experimentally in SPIF. What has been lacking in most of these efforts has been the lack of a means to accurately predict occurrence of fracture in SPIF. This work deals with the prediction of fracture and localization in SPIF using a damage based material model in FEA in which damage accumulation is a function of the hydrostatic pressure and shear (via the Lode angle). SPIF experiments were performed to form a cone and a funnel shape and the measured forces and fracture locations are found to match well with corresponding FEA predictions. The same material model is used to simulate the forming of the cone shape using a larger punch and die and the differences between conventional forming and SPIF in terms of the mechanisms governing fracture are discussed. Moreover, it is shown that a reduction in friction can increase the fracture depth in SPIF and the mechanisms responsible for this effect are discussed as well.
The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 469-476 (2011); doi: 10.1063/1.3623646 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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MATERIAL MODEL In the present work, the mechanical response of the blank material is modeled using the damage plasticity model proposed by Xue [6]. The constitutive relationship consists of a damage coupled yield function, the evolution laws for the plastic strain and a reference fracture envelope. In particular, the stress-based fracture envelope is chosen. The damage coupled yield function is expressed as shown in Eqn. 1
) V eq w( D )V M d 0
(1)
where eq is the equivalent stress, M is the matrix resistance and w is a scalar weakening factor used to describe the material deterioration. The factor w is related nonlinearly to the damage variable D as w(D)=1-D where is a material constant. In Eqn. 1 the undamaged matrix resistance M is expressed as a function of the plastic strain p using the Swift hardening relationship i.e. M=y0 (1+p/0 )n where y0 is the initial yield stress, 0 is the initial yield strain and n is the hardening exponent. The associated flow rule is enforced. The damage accumulation is described using a power law function of the plastic strain p and a limiting fracture strain f and the evolution of damage is expressed in the rate form as x
D
§Hp m¨ ¨H © f
· ¸ ¸ ¹
m 1
x
Hp Hf
(2)
where m is a material constant and f is a function of the hydrostatic pressure and the Lode angle of the current stress state. The fracture envelope is given in the stress space envelope Mf [7], which takes the form of a modified Tresca type of pyramid as Mf=f0(1+kpp)1/n(3/2cosL)1/n where f0 is the fracture stress under uniaxial tension, kp is a material constant related to pressure sensitivity, p = - (kk /3) is the hydrostatic pressure and L is the Lode angle. The limiting fracture strain f is then obtained from the inverse of the matrix stress as shown in Eqn. 3. An element completely loses its load carrying capacity and is removed when the value of the damage variable D reaches 1.0. Additionally, closed form solutions for the onset of the diffused necking, i.e. a maximum power criterion, which is a three dimensional generalization of the one-dimenional Considere condition [8], and the onset of localized necking, i.e. occurrence of local shear bands, in a Hadamard-Hill sense [9] are included in the material model as well. 1/ n ½ § V ·1 / n ª 3 º ° °¨ f 0 ¸ H f ( p,T L ) H 0 ®¨ (1 k p p ) 1¾ « » ¸ 2 cosT L ¼ °¿ °¯© V y 0 ¹ ¬
(3)
EXPERIMENTS AND FINITE ELEMENT ANALYSIS SPIF experiments were performed to form a 70° wall angle cone and a variable angle funnel (Figs. 1a and 1b) on 1 mm thick AA5052 sheet with tool of diameter 9.525 mm, feed rate 150 mm/min, using a helical toolpath with incremental depths of 1.0 mm and 0.5 mm respectively. To examine the effect of reduction in tool sheet friction coefficient, experiments were also performed for SPIF of the funnel with a Teflon sheet wrapped around the tool and Teflon lubricant was used on the sheet in addition to the regular lubricant. The Z forces on the tool were measured throughout the SPIF process using a Kistler dynamometer mounted below the fixture and the tool tip depths at which fracture occurred were recorded in each case. FEA was performed in LS-DYNA to simulate SPIF of the cone and the funnel shape with coefficient of friction = 0.10 at the tool sheet interface (Fig. 1c), SPIF of the funnel with = 0.0 (corresponding to the reduced friction funnel experiment) and for the 70° wall angle cone formed with a larger punch (Fig. 1d) using four linear brick elements through the thickness of the sheet. The tools were modeled as rigid bodies and the fixture was modeled using a top and a bottom clamping plate with coefficient of friction of 0.15 between the blank and the fixture. A user subroutine was coded to implement the aforementioned material model in LS-DYNA. The Young’s Modulus E, yield stress y0, hardening coefficient n and initial yield strain 0 for the blank material were obtained from tensile tests (Table 1). The material constants f0, kp , m and cannot be obtained using tensile tests and were calibrated manually by matching the tool Z forces from 3 trial simulations with those from experiments for SPIF of the 70° cone (Table 1). To show the quality of calibration of the damage model parameters, the tool Z forces from experiments and simulation for the 70° cone are compared in Fig. 2a. Furthermore, the
470
maximum thinning just before fracture was measured to be 64% from experiments and 64.1% from simulation. The fracture depths measured from experiments and predicted by corresponding simulations for SPIF are shown in Table 2. The predicted and experimental tool Z forces for SPIF of the funnel with the regular lubricant are shown in Fig. 2b. Both predicted tool Z forces and the fracture locations in SPIF are found to match quite well with experiments. For the simulation of forming the cone with the large punch the predicted fracture depth was 14.5 mm, i.e. the formed depth in SPIF was greater than in the punch forming case. The deformation mechanics in SPIF and the large punch case were examined by looking at the deformation history from FEA, of four elements through the thickness of the sheet at four sections A, B, C, and D, along the wall of the formed component (Fig. 2c). Section D is the zone where the through-the-thickness crack first appears. At each of these sections the four elements through the thickness are labeled ‘1’ to ‘4’ where ‘1’ is the element on the side of the sheet which is in contact with the tool (i.e. the inner side) and ‘4’ is the element on the other side of the sheet (i.e. the outer side). 25 mm
50
30
70°
SPIF tool Top clamp (green)
20 mm
Blank (Red)
Large punch Blank holder (green)
Blank (Red)
Bottom clamp (yellow) Bottom die (yellow)
85°
(a) (b) (c) (d) FIGURE 1. (a) CAD model of 70° cone (b) CAD model of variable angle funnel (c) FEA model for SPIF (d) FEA model for 70° cone formed with large punch TABLE (1). Calibrated material parameters. (kg/m3) 2680
E (GPa) 68.6
0.3
y0 (MPa) 117
n 0.22
0 0.0045
fo(MPa) 490
kp (Mpa-1) 0.0001
m 2.0
3.0
TABLE (2). Comparison of SPIF tool-tip depths at fracture measured from experiments and predicted by FEA Component Experiment (in mm) FEA (in mm) 70° cone 16.9 16.1 Funnel (regular lubricant) 15.02 15.03 Funnel (regular lubricant & Teflon) 15.56 15.45
Section A Section B Section C Section D
(a) (b) (c) FIGURE 2. (a) Comparison of tool z-forces for 70° cone (b) Comparison of tool z-forces for variable angle funnel with =0.10 (c) Sections along component wall at which deformation history is examined
Material Instability and Failure This section examines the reasons for the increase in the formed depth in SPIF as compared to the punch forming case in terms of occurrence of material localization. Figures 3a and 3b show that the plastic strain p for SPIF is always higher at any section, than it is for the punch case. Moreover, the plastic strain before failure in SPIF is about
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20% higher than that before failure in the punch case. The reasons for this trend can be found by comparing the occurrence of diffused and localized necking in both cases (Figs. 4 and 5). In Figs. 4 and 5, for the localization flag contour plots, the localization flag has a value of 1.0 (blue colour) in regions when the material is in a diffused necking state and a value of 3.0 (red colour) when the material is in a localized necking state (i.e. shear bands have formed). Furthermore the regions with diffused and localized necking are marked as D and L respectively.
(a) (b) FIGURE 3. Evolution of plastic strain along sections A,B,C,D for 70° cone formed using (a) SPIF (b) Large punch
FIGURE 4. For SPIF (a) at onset of diffused necking (b) at onset of localized necking (c) just before fracture
FIGURE 5. For Punch forming case (a) at onset of diffused necking (b) at onset of localized necking (c) just before fracture
Figures 4a and 5a show that in SPIF diffused necking starts out at an earlier Z depth (5.6 mm) as compared to the punch case (8.6 mm). In both cases the material deformed after this point is in a diffused necking state for the next 5 mm of Z depth before localized necking starts (Figs. 4b and 5b). Once localized necking begins the component fractures very quickly in the punch case (within 1.5 mm Z depth) as compared to SPIF where material for about 6 mm in Z depth undergoes localized necking before fracture. Since deformation in the case of the punch forming is global in nature, after initial localized necking occurs this unstable material is still being actively stretched by the
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punch resulting in rapid growth of the shear bands which leads to rapid fracture. In contrast, in SPIF the tool is constantly moving and while the newly deformed regions of the blank in the tool contact zone are becoming unstable, the previously formed unstable region is not being actively deformed. Therefore the shear bands already present in the previously formed unstable regions do not grow as quickly leading to a larger region which is in a state of localized necking before fracture occurs. The effect of this previously formed material becoming unstable enough to undergo localized necking but not reaching the point of fracture, is that the some of the deformation of newly formed material in the subsequent passes of the tool is taken up by this previously formed localized region. This allows the newly formed material to undergo greater amount of plastic strain and reach a greater Z depth without fracture. This effect is also seen by examining the plastic strain contours for both cases which show that in the punch case (Fig. 5c) the plastic strain becomes concentrated very quickly into the shear bands in the localized necking region before fracture. However, in SPIF (Fig. 4c) the plastic strain is distributed over a larger localized necking region and has greater magnitude before fracture, as compared to the punch case. Further evidence for this is obtained by observing that in the punch case (Fig. 3b) the plastic strain is very highly concentrated at sections C and D. In fact at section D after a certain point (at which point this section is in a state of localized necking) the plastic strain rate increases dramatically. No such effect can be seen in SPIF (Fig. 3a) where the increase in plastic strain from sections A to D is fairly regular. The occurrence of material localization was further confirmed by the observation of regions along the circumference of the formed sheet on the outer surface showing distinctive bands that indicated material localization in these regions (Fig. 6). At this point it is useful to keep in mind that occurrence of diffused and subsequent localized necking is influenced by damage accumulation as well as by material weakening [8,9]. As will be shown in the next section, through-the-thickness shear plays a significant role in damage evolution in SPIF and therefore indirectly influences occurrence of material instability as well as the resulting effects mentioned above.
FIGURE 6. Regions along the outer surface of SPIF components indicating material localization
Deformation Mechanisms affecting Damage Evolution
(a) (b) FIGURE 7. evolution of damage index (D) along sections A,B,C,D for 70° cone formed using (a) SPIF (b) Large punch
In SPIF the damage index D (Fig. 7a) evolves faster for element 4 as compared to element 1. On the other hand in the case of the punch forming (Fig. 7b) the difference in D between elements 1 and 4 is negligible. Therefore, in
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SPIF element 4 is the first element to be removed and the crack propagates from outside to the inside of the sheet, whereas for the large punch all the elements through the thickness of the sheet are removed simultaneously. In SPIF the hydrostatic pressure (Fig. 8a) is negative on the inner side (element 1) and positive on the outer side of the sheet (element 4). This is due to local bending of the sheet around the small hemispherical SPIF tool. When the tool has passed over a particular region, that region undergoes local springback as a result of which the hydrostatic pressure becomes negative on the inner side and positive on the outer side of the sheet. This means that for SPIF the term (1+kpp)1/n in Eqn. 3 is greater than 1.0 for element 4 and lesser than 1.0 for element 1 which, in the absence of a significant shear effect, should cause the limiting fracture strain f to be higher for element 4 than for element 1. For the punch forming case (Fig. 8b), since the material is continuously being stretched without any local springback, the hydrostatic pressures are always equally negative on either side of the sheet. Therefore the aforementioned pressure term will also be equally lesser than 1.0 on either side of the sheet, and will not cause a difference in f between the two sides of the sheet. To pinpoint the effect of shear on f the product term [(1+kp.p). (3/2.cos L)](1/n) in Eqn. 3 is split into a pressure term (1+kpp)1/n and a Lode angle term (3/2.cos L) (1/n) and the evolution of all these terms at section D is plotted against p (Fig. 9). In SPIF (Fig. 9a), inspite of the pressure term following the aforementioned trends, the product term and therefore f is much lower for element 4 due to the Lode angle term being much lower for element 4. Therefore, in SPIF the effect of shear on f dominates over the effect of the hydrostatic pressure. In the case of the punch the the Lode angle terms are similar on either side of the sheet (Fig. 9b) resulting in the product term and therefore the fracture strain f being similar on either side as well. In SPIF since p is higher (Fig. 3a) and f is lower (Fig. 9a) on the outer side of the sheet, damage is higher on the outer side (from Eqn. 2) causing the crack to begin on the outer side of the sheet. In the case of the punch since the difference between p and f on either side of the sheet is negligible, so is the difference in damage accumulation and the crack begins simultaneously through the thickness of the sheet. Therefore in SPIF the dominant mechanisms governing the evolution of damage and the nature of fracture are local bending (causing higher p for element 4) and through-thethickness shear (causing lower f for element4). In contrast, in the case of the large punch the deformation is dominated by uniaxial tension along the component wall and failure is by a continuous circumferential crack.
(a) (b) FIGURE 8. Evolution of Hydrostatic Pressure along sections A,B,C,D for 70° cone formed using (a) SPIF (b) Large punch
A further effect of shear on fracture in SPIF can be seen by examining the deformation mechanisms responsible for the increase in fracture depth for SPIF of the funnel when the friction at the tool sheet interface is reduced. Figures 10a and 10b show that when =0.0, as compared to when =0.10, the reduction in damage accumulation is much larger for element 1 than it is for element 4. Therefore the increase in final fracture depth of the component is mainly because element 1 takes longer to be removed when is reduced. Figure 10d shows that the reduction in plastic strain when =0.0 is again much larger for element 1 than it is for element 4 which by itself implies larger reduction in damage accumulation for element 1 than for element 4 (Eqn. 2). Comparing the tensile strains along the component wall (Fig. 10e) and the shear strains along and perpendicular to the tool motion (Figs. 10f and 10g respectively) shows that the reduction in shear strain along the toolpath for element 1 is not just greater than that for element 4, it is also the most significant reduction among these three major components of the strain. Therefore the reduction in shear along the toolpath motion, is the most significant factor responsible for the significantly greater reduction in plastic strain for element 1 as compared to element 4 (Fig. 10d), when is reduced.
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(a) (b) FIGURE 9. Comparison of product term, pressure term and Lode angle term for 70° cone with (a) SPIF (b) larger punch
(a)
(b)
(c)
(d)
(e) (f) (g) FIGURE 10. Comparison of (a) Damage evolution (b) Reduction in damage evolution (c) Plastic strain (d) Reduction in plastic strain (e) Tensile strain along component wall (f) Through-the-thickness shear along component wall (g) Through-the-thickness shear along tool motion, between =0.10 and =0.0 case, for elements 1 and 4.
.
(a) (b) FIGURE 11. Comparison of Pressure, Lode angle and Product terms of Eqn .3 between =0.10 and =0.0 cases for (a) Element 4 (b) Element 1.
Comparing the individual components of the fracture envelope f at section D for the =0.10 and =0.0 cases (Fig. 11) shows that for both elements 1 and 4 the pressure term does not differ significantly with a reduction in and for element 4 the Lode angle terms and therefore the product terms are quite similar irrespective of the value of
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(Fig. 11b). However, for element 1 (Fig. 11a) the Lode angle term is slightly reduced with reduced , resulting in a reduction in the product term. Therefore the reduced through-the-thickness shear due to reduced friction reduces f and therefore tends to accelerate damage accumulation on the inner side of the sheet. Since damage accumulation depends on the ratio p/ f (Eqn. 4) and the reduction in p is much larger (Fig. 10b) than the reduction in f (Fig. 11a), the net effect is a larger reduction in damage accumulation for element 1 as compared to element 4 when is reduced (Figs. 10a and 10b). Physically this means that reduced tool sheet friction causes the crack to initiate on the outer side of the sheet at pretty much the same time but the crack takes longer to propagate all the way to the inner side and cause a complete through-the-thickness crack. This results in the observed increase in fracture depth.
SUMMARY AND CONCLUSIONS This work uses a damage based fracture and localization prediction model in which damage evolution is dependent on the hydrostatic pressure as well the shear (via the Lode angle), to simulate SPIF of a cone and a funnel shape as well as forming of the same cone with a larger punch and die. Good agreement between experiments and simulations is obtained for tool forces and fracture locations in SPIF. It is shown that in SPIF, the evolution of damage is controlled by local bending around the tool as well as by through-the-thickness shear whereas in the punch case it is influenced mainly by tension along the component wall. The evolution of damage also affects occurrence of diffused and localized necking [8,9]. In SPIF the local nature of the deformation causes diffused and localized necking to develop faster than in the punch case, which is characterized by a more global deformation. However, this very feature of SPIF also prevents the localized region of the sheet from going to failure very quickly. As a result a significant portion of the sheet undergoes localized necking before failure allowing this region to take some of the plastic strain in subsequent passes of the tool. As a resulting the plastic strain and the Z depth before before fracture is greater in SPIF than in the punch forming case. In terms of failure in SPIF, any combination of operational parameters that can reduce the rate of damage accumulation and therefore delay localization, or increase the region of the sheet in a state of localized necking before fracture, can increase formability. One such operational parameter is friction at the tool sheet interface. This work shows both experimentally and via simulations for SPIF of a funnel shape that a reduction in friction reduces damage accumulation on the inner side of the sheet, primarily via a reduction in through-the-thickness shear and consequently in plastic strain. The result is an increase in fracture depth.
ACKNOWLEDGMENTS The authors gratefully acknowledge the support provided by the National Science Foundation Grant #CMMI0758607, for this work.
REFERENCES 1. 2.
3. 4.
5.
6. 7. 8. 9.
M.B. Silva, M. Skjoedt, N. Bay, P.A.F. Martins, “Revisiting single-point incremental forming and formability/failure diagrams by means of finite elements and experimentation”, Journal of strain analysis 44, 221-234 (2009). R. Malhotra, Y. Huang, L. Xue, J. Cao, T. Belytschko, “An Investigation on the Accuracy of Numerical Simulations for Single Point Incremental Forming with Continuum Elements” in Proceedings of the 10th International Conference on Numerical Methods in Industrial Forming Processes NUMIFORM 2010, AIP Conference Proceedings 1252, 221-227 (2010). I. Cerro, E. Maidagan, J. Arana, A. Rivero, P.P. Rodríguez, “Theoretical and experimental analysis of the dieless incremental sheet forming process” in Journal of Materials Processing Technology 177(1-3), 404-408, (2006). G. Ambrogio, L. Filice, L. Fratini, F. Micari, “Process Mechanics Analysis in Single Point Incremental Forming” in Proceedings of the 8th International Conference on Numerical Methods in Industrial Forming Processes NUMIFORM 2004, AIP Conference Proceedings 712, 922-927 (2004). A. Van Bael, P. Eyckens, S. He, C. Bouffioux, C. Henrard, A.M. Habraken, J. Dulfou, P. Van Houtte, “Forming limit predictions for single point incremental sheet metal forming” in Proceedings of the 10th ESAFORM Conference on Material Forming, AIP Conference Proceedings 907, 309-314 (2007). L. Xue,“Damage accumulation and fracture initiation in uncracked ductile solids subjected to triaxial loading” in International Journal Solids and Structures 44(16), 5163-5181 (2007). L. Xue “Stress based fracture envelope for damage plastic solids” in Engineering Fracture Mechanics 76(3), 419-438, (2009). L. Xue, “Localization conditions and diffused necking for damage plastic solids” in Engineering Fracture Mechanics 77, 1275-1297, (2010). L. Xue, T. Belytschko, “Fast methods for determining instabilities of elastic-plastic damage models through closed-form expressions” in Internationl Journal for Numerical Methods in Engineering 84, 1490-1518, (2010).
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Prediction of Fracture in Deep Drawing Process, Using Ductile Damage Criteria Hassan Nejatbakhsha , Mohamad Khataeib*, Mehrdad Poursinac a,
Faculty of Engineering, University of shahrekord, 8118634141 shahrekord, Iran b* Department of Mechanical Eng., Isfahan University of Technology, Iran. c Faculty of Engineering, University of Isfahan, 8174473441 Isfahan, Iran
Abstract. In the deep drawing process, determination of the drawing depth and prediction of the time and the place that fracture occurs has been one of the important case studies which engineers tend to take into account. Because of a drastic reduction in design and manufacturing expenditures, numerical methods are extended to calculate the drawing depth during the process. In this paper, ductile damage model in which the stress triaxiality and equivalent strain are the most effective parameters on the damage growth and fracture of the material is used to predict fracture. For prediction the place and time of ductile fracture, according to ductile damage criteria, the fracture strain for various stress triaxiality values should be determined. To obtain the parameters of ductile damage model for St12 steel, some tensile tests have been performed on the notched specimens. Numerical simulation of deep drawing was performed using commercial finite element ABAQUS. Results obtained from simulation are in good agreement with the experimental ones and emphasize that using ductile damage model is appropriate to anticipate the place and time of the fracture during the deep drawing process. Keywords: Fracture; Ductile damage; Deep drawing; Experiment; St 12 steel. PACS: 62.20.me
INTRODUCTION Exceeding the usages of finite element software in forming processes simulations, several failure and damage models have been implemented in finite element software and researchers have investigated intensely about different damage models as well. Rice and Tracey [1] have shown that fracture of the ductile metals is strongly dependent on hydrostatic stress by studying growth of spherical voids. Hancock and Mackenzie [2] performed experiments on notched specimens and observed that ductile damage depends severely on stress triaxiality value applied to the material. Johnson and Cook [3] in 1985 presented a damage model that depending to stress triaxiality, strain rate and temperature by performing practical experiments. Their offered model included some material constants which had to be determined via specific experiments. Many studies were carried out defining the parameters of Johnson and Cook’s model. One of the most appropriate approaches was presented by Holmquist [4] in 1991 which was based on investigating the effects of each parameter separately on materials failure. In 1998, Rule [5] reported a method for specifying the constants of the model with high values of strain rates. Following that, Juthras [6] presented an almost perfect and accurate technique for Johnson and Cook’s model in 2008, using torsion tests. After that, several investigations were carried out to find an equation to relate fracture strain to stress triaxiality for various materials. Hooputra et al. [9] from BMW R&D center calculated the damage parameter according to an equation for stress triaxiality. With the aim of their suggested equation, it is possible to evaluate damage growth numerically and predict the probable arias of fracture. In the present paper, the tension tests were carried out initially on notched specimens and constant values of Hooputra model for St12 steel were obtained. Following that, deep drawing process was modeled and simulated commercial finite element ABAQUS, applying ductile damage model for predicting the failure zones in the simulation. Finally, real deep drawing tests were performed and simulation results from Hooputra ductile damage model were compared with experimental results.
The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 477-483 (2011); doi: 10.1063/1.3623647 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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DUCTILE DAMAGE MECHANIC Ductile damage is a proceeding physical phenomenon which causes the fracture of the materials. This phenomenon occurs in three steps which are void nucleation, void growth and void coalescence. To analysis the growth of ductile damage in every step of loading, a parameter is defined as ductile damage parameter (D) which is the ratio of the void area in intersection to the total area of the intersection. Obviously this parameter is always between zero and one values (0( , ;, ?) = @ A + 2B' ? C cosh @ B 2 ; ;
(5)
In the Nielsen-Tvergaard model, the empirical modification permitting to predict fracture under shear loading conditions affects only the differential equation describing the evolution of the void volume fraction, which reads: ?G = (1 ?)8G : I + JGK + ;L ?MN ( )
: 8G OK
(6)
where the first term accounts for the growth of voids due to hydrostatic stress. The second term accounts for the nucleation of voids. The third term has been introduced to account for material deterioration due to shear loading. ;L is a model parameter and MN ( ) a stress-state dependent weighting function:
MN ( ) = M( )P()
where
M( ) = 1
and
1 for < ' ) P() = for ' $ R(' ) 0 for > Q S (
(7)
It is emphasized that the original physical meaning of f is lost due to the addition of the empirical third term in Eq. (6). In other words, ? can no longer be seen as the void volume fraction and is interpreted as an empirical damage parameter. The reader is refered to [4] for a complete overview of the constitutive equations. The fourteen material parameters of the Nielsen-Tvergaard model are determined from four experiments: (1) uniaxial tension, (2) punch experiment: (3) notched tension with U = 20VV, and (4) shear-dominated butterfly experiment ( = 0°). A Monte-Carlo based inverse method is used to identify most model parameters.
Comparison of model predictions and experimental results The fracture models are evaluated by comparing the predicted and experimentally-measured displacements to fracture for nine distinct experiments. FE simulations are run of all experiments with the measured displacements imposed as boundary conditions. The predicted displacement to fracture corresponds to the applied displacement at the instant when = 1 for the MMC model and ? = 0.9?X for the modified Gurson model. Figure 5 depicts the ratios of the predicted and measured displacements to fracture for all nine experiments. Error bars are included to represent the experimental scatter on the measured fracture displacements (when applicable). Recall that four fracture experiments (three butterfly fracture experiments and the punch test) have been used to identify the parameters of the MMC model, and three for the Nielsen-Tvergaard model. It is found that the MMC model (in red in Fig. 5) predicts the displacements to fracture with great accuracy for all experiments. The relative error is smaller than 2% for eight of nine experiments. The largest relative error of 3.2% is observed for the tensile specimen with a central hole. Fracture displacements predicted by the Nielsen-Tvergaard model are shown in blue in Fig. 5. For the three experiments used for calibration, the fracture displacements are predicted with less than 0.5% error. The fracture displacement of the 10mm notched specimen is very accurately predicted, with a relative error of 0.2% which is less than the experimental scatter. For the two other full thickness tensile experiments (6.67mm notch and specimen with a central hole), the relative error affecting the predicted displacement to fracture is less than 10% (respectively 6.9% and 4.5%). However, the shear-modified Gurson model is clearly inaccurate for three butterfly experiments (more than 10% error): it underestimates the displacement to fracture by 11% for the shear-dominated combined loading
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FIGURE 5. Ratio of predicted and experimental fracture displacements. Experiments used for calibration have their name underlined
( = 25°) and by 17.7% for tension-dominated combined loading ( = 63°); conversely, the fracture displacement is overestimated by 10.6% for the transverse plane strain tension experiment ( = 90°).
DISCUSSION Effect of damage on the elasto-plastic response can be neglected The accumulation of damage affects the elasto-plastic material response prior to fracture when using the Gurson model. Conversely, the predictions of the standard plasticity model which is used in conjunction with the MMC model remain unaffected by the damage accumulation. The comparison of the force-displacement curves predicted by the standard plasticity model and the Nielsen-Tvergaard model (respectively in red and blue lines in Figs. 2 and 3) show that both models predict the same global response of the specimens at the early stages of loading. Before nucleation occurs, material damage is negligible and the Gurson yield function, given in Eq. (5), reduces to a standard quadratic yield function. The predicted force-displacement curves slightly differ only when material damage becomes significant.. It is interesting to see that the Nielsen-Tvergaard model does not always predict the force prior to fracture with greater accuracy; for tension with a central hole (Fig. 2b) and the butterfly experiment at = 0° (Fig. 3b), the uncoupled plasticity model predicts the force-displacement curves more accurately than the Gurson model. Based on the above observations, it is tentatively concluded that the modeling of the effect of damage on the elasto-plastic behavior does not yield any improvement of the accuracy of force-displacement curve predictions.
Modeling of shear-induced material deterioration inaccurate According to the Nielsen-Tvergaard model, the shear-induced damage rate for low stress triaxialities reads fGYZ[\] = ;L ?M( )
: 8G :
(8)
where ? is the damage parameter, ;L is a material parameter and M = 1 measures the deviatoric state of stress. G The quadratic dependence of ?^_` b on the third stress invariant (as proposed in [1]) is a purely empirical assumption. Among the above experiments, the results from shear-dominated butterfly experiments ( = 0° and = 25°) can be used to assess the validity of this choice. In these experiments, fracture occurs at stress triaxialities close to zero, while the corresponding third stress invariants are significantly different (Fig. 4). Recall that the experiment for = 0° has been used for calibration; however, the inability of the Nielsen-Tvergaard model to predict the fracture displacement for = 25° suggests that the empirical choice of the quadratic relationship G and needs to be revisited. between ?^_` b
489
(a)
(b)
(c) (d) FIGURE 6. Fracture loci for proportional loading conditions in the (, , ) plane of the MMC fracture criterion (in red) and Nieslen-Tvergaard model (in blue). Cross sections at constant stress triaxiality: (a) = 0.1, (b) = 0.5; and constant third invariants: (c) = 0, (d) = 1.
Fracture locus for proportional loading Proportional loadings are characterized by constant stress triaxiality g and normalized third invariant j throughout loading. In this specific case, the equivalent plastic strain to fracture, or fracture strain k l , can be represented as a function of g and j to create the so-called fracture locus in the (g, j, k m ) space. Figures 6a and 6b depict the fracture strains for proportional loading as a function of the third stress invariant, at a fixed stress triaxiality ( = 0.1 in Fig. 6a, = 0.5 in Fig. 6b), while Figs. 6c and 6d depicts the fracture strains for proportional loading as a function of the stress triaxiality, at fixed third invariants ( = 0 in Fig. 6c, = 1 in Fig. 6d). The MMC fracture strain decreases in stress triaxiality and exhibits a U-shaped dependence on the third invariant, resulting in a continuous fracture locus shaped as a half-tube in the (g, j, k m ) space [2]. Note that the MMC fracture locus is asymmetric with respect to j = 0 and that, at a fixed stress triaxiality, the minimum ductility does not correspond to the generalized shear stress state (characterized by j = 0). The Nielsen-Tvergaard fracture locus shows a more complex dependence on the stress state. For low stress triaxialities (0 < < g' ), the fracture locus is a U-shaped function of the third invariant. The fracture strain is smallest for generalized shear stress states (j = 0). For axisymmetric loading states (j = ±1), there is no shear-induced damage and the fracture strain increases asymptotically as the stress triaxiality decreases to 0. For intermediate stress triaxialities (g' < < g ), the fracture strain increases in stress triaxiality for loadings states with third invariants close to 0, as the shear-induced damage slows down. For high stress triaxialities (g > g ), the shear damage term in Eq. 6 is inactive: the fracture strain
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decays almost exponentially in stress triaxiality and is independent of the third invariant. Note that the NielsenTvergaard fracture locus is symmetric with respect to j = 0. The calibrated MMC and Nielsen-Tvergaard fracture loci are remarkably different over the entire range of stress states considered, except within the vicinity of the experimental data that have been used to calibrate the NielsenTvergaard model: for p 0.5 and p 0.7 (average stress state in the 20mm notch tensile experiment) and for p 0.7 and p 1 (average stress state in the punch experiment), the two fracture loci intersect.
CONCLUSIONS The predictive capabilities of the shear-modified Gurson model by Nielsen and Tvergaard and the phenomenological MMC fracture criterion are evaluated over a wide range of stress states (shear to equibiaxial tension). The MMC model can predict the onset of fracture with great accuracy for all experiments. The predictions of the shear-modified Gurson model are found to be less accurate. It is shown that the Lode angle dependence of the shear-induced damage mechanism needs to be modified further to improve the accuracy of the Nielsen-Tvergaard model. Aside from the small differences in the prediction accuracies, it is surprising to see that two fundamentally different fracture models (stress-based criterion versus void growth model) are both able to predict the fracture displacement over a wide range of stress states. This observation suggests that the underlying physical assumptions are less important than the models’ mathematical flexibility to be fitted to a wide range of experimental data. Another key fundamental difference between the MMC model and the Nielsen-Tvergaard model lies in the coupling of plasticity and damage. The MMC model makes use of a damage indicator function which has no effect on the elasto-plastic behavior, while damage reduces the load carrying capacity and changes the shape of the yield surface in the case of the shear-modified Gurson model. However for the TRIP780 material, both models show equally good predictions of the force-displacement curves. The MMC model is recommended for practical applications because of the greater computational stability of uncoupled damage models, the smaller number of parameters to be identified and the ease of their identification based on experiments. However, it is important to identify the MMC model parameters based on experiments that cover a wide range of stress states.
ACKNOWLEDGMENTS POSCO Steel (Korea) is thanked for providing the material. The financial support of the AHSS Fracture Consortium at MIT (AISI, Hyundai, Nissan, Posco, ThyssenKrupp, Volkswagen) is gratefully acknowledged.
REFERENCES 1. 2. 3. 4. 5. 6. 7.
K. Nahshon and J. Hutchinson, Eur. J. Mech. A-Solids 27, 1-17 (2008). Y. Bai and T. Wierzbicki, Int. J. Fract. 161, 1-20 (2010). M. Dunand and D. Mohr, J. Mech. Phys. Solids 59, 1374-1394 (2011). K. Nielsen and V. Tvergaard, Eng. Fract. Mech. 77, 1031-1047 (2010). M. Dunand and D. Mohr, Int. J. Solids. Struct. 47, 1130-1143 (2010). M. Dunand and D. Mohr, Eng. Fract. Mech, In Revision. D. Mohr, M. Dunand, K.H. Kim, Int. J. Plasticity 26, 939-956 (2010).
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An Ellipsoidal Void Model for the Evaluation of Ductile Fracture in Sheet Metal Forming Kazutake Komoria* *Department of Integrated Mechanical Engineering, School of Engineering, Daido University 10-3, Takiharu-town, Minami-ward, Nagoya-city, Aichi-prefecture, 457-8530, Japan Abstract. We propose a void model that can be used for the evaluation of ductile fracture in the simulation of metal forming processes. This void model is an extension of the Thomason model of void coalescence, which is based on the internal necking of intervoid matrix ligaments. The validity of the new void model for bulk metal forming processes was confirmed by experiments and simulations. In this paper, we confirm the validity of the new void model in the simulation of a bore expanding test, which is one of the material tests used in sheet metal forming. Keywords: Ductile fracture, Thomason model, Ellipsoidal void, Multi-scale simulation. PACS: 46.35.+z, 46.50.+a, 62.20.fk.
INTRODUCTION Ductile fracture [1], which occurs when a material is subjected to a large plastic deformation, is a troublesome problem in metal-forming processes. Many fracture criteria for various materials have been proposed. However, no fracture criterion that is applicable to all metal-forming processes has been found. Fracture criteria which are widely used for metal-forming processes are fracture criteria proposed by Freudenthal [2], Cockcroft and Latham [3], Brozzo et al. [4], and Oyane [5]. All these fracture criteria are expressed in terms of a definite integral of an integrand that is a function of stress and strain components, while, according to Lemaitre [6], the damage variable D is a function of stress and strain components. Since stress and strain are defined macroscopically, each fracture criterion is derived from a macroscopic point of view. However, since ductile fracture occurs through nucleation, growth, and coalescence of voids, it is a microscopic phenomenon. Hence, it is difficult to improve the accuracy of microscopic ductile fracture predictions using a macroscopic fracture criterion. We have been investigating the prediction of ductile fracture in metal-forming processes from a microscopic point of view. In our previous study, using a void model that we had previously proposed, we simulated the inner fracture defects in drawing of carbon steels for machine structural use [7] and high-carbon steel wire rods [8], while we simulated the blanking process of sheet metals [9]. Thomason proposed a model of void coalescence based on the internal necking of intervoid matrix ligaments [10]. The model was derived from the upper bound method, in which the material is assumed to fracture when the energy required to coalesce voids by internal necking is less than the energy required to deform the material homogeneously. The model of void coalescence is useful, since it has definite physical meaning. Our void model is derived from the Thomason model. Since the Thomason model assumes that voids are rectangular, our void model assumes that voids are parallelogrammic. However, parallelogrammic voids are unrealistic in shape. We recently proposed an improved void model [11] in which voids are assumed to be ellipsoidal. Ellipsoidal voids are more realistic in shape and thus the ellipsoidal void model is an improvement on the parallelogrammic void model. In the present study, we use the ellipsoidal void model in the simulation of a bore expanding test, which is one of the material tests used in sheet metal forming. The simulation results are compared with the experimental results. This comparison confirms the validity of the ellipsoidal void model.
The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 492-497 (2011); doi: 10.1063/1.3623649 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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SIMULATION METHOD The simulation method is described in our previous paper [9] in which parallelogrammic voids are assumed. The present study assumes ellipsoidal voids. Hence, we describe the void shape in the simulation in detail below.
Outline Figure 1 shows the outline of the whole simulation. Our simulation is a multiscale simulation. The macroscopic simulation models the deformation of the material by the rigid-plastic finite-element method, while the microscopic simulation evaluates the fracture of the material using our void model. Since the deformation of the material after the material fracture is not required in this study, the simulation is performed until the material fractures. Macroscopic Simulation (Simulate Deformation of Material byRigid-Plastic Finite-Element Method) DisplacementGradient Rate VoidVolumeFraction Rate
Determine Whether Material Fractures
Microscopic Simulation (EvaluateFracture of Material by our Void Model) FIGURE 1. Outline of whole simulation.
Outline of Macroscopic Simulation The simulation of the deformation of the material is performed using the conventional axisymmetric rigid-plastic finite-element method [12]. The yield function proposed by Gurson [13] is adopted: §V 3 V ijc V ijc 2 2 f cosh¨¨ kk 2 VM © 2V M
)
· ¸ 1 f 2 ¸ ¹
0,
(1)
where V M is the tensile yield stress of the matrix and f is the void volume fraction of the material. Since the yield function ) is not a function of the second power of stress, it is not easy to perform a rigid-plastic simulation using Equation (1). Hence, cosh x is approximated to be 1 x 2 2 [14]. The approximated yield function )c used in this study is )c
3 V ijc V ijc f 2 V M2 4
§V 2 ¨¨ kk2 ©VM
· ¸ 1 f 2 ¸ ¹
0.
(2)
On the other hand, the following evolution equation, which denotes the change in the void volume fraction, is assumed: f
1 f Hkk A
V kk B H , 3V
(3)
where V is the equivalent stress, H is the equivalent strain rate and A and B are material constants. Here, x in Equation (3) denotes Macauley’s bracket. In other words, x is equal to x when x is positive, while x is equal to zero when x is negative. The first and second terms of the right-hand side of Equation (3) denote void
493
growth and void nucleation, respectively. The second term indicates that a void nucleates when the stress triaxiality V kk 3V is larger than B .
Outline of Microscopic Simulation The outline of the microscopic simulation in each step, from the calculations of the void volume fraction and the average deformation gradient to the determination of whether or not the material fractures, is shown below: (1) The void volume fraction f and the average deformation gradient wx wX are calculated from the void volume fraction rate f and the displacement gradient rate wu wX , which are calculated by the macroscopic rigidplastic finite-element simulation. (2) The void configuration and the void shape are calculated. (3) The ratio of the energy-dissipation rate of internal necking to the energy-dissipation rate of homogeneous deformation E is calculated. (4) Whether or not the material fractures is determined. For simplicity, the deformation gradient of the material in the macroscopic simulation is assumed to be identical with the deformation gradient of the void in the microscopic simulation. In this study, the experiment of the bore expanding test using a prestrained material is performed. The void before the bore expanding test for a non-prestrained material is circular. However the void before the bore expanding test for a prestrained material is not circular but ellipsoidal, so that the deformation gradient of the void coincides with that of the material.
Evaluation of Material Fracture Figure 2 shows the coordinates in the macroscopic simulation and the microscopic simulation. In the simulation of blanking [9], the cylindrical coordinates r and z are used in the macroscopic simulation due to the axisymmetric deformation. Furthermore, the cylindrical coordinates r and z are used in the microscopic simulation, since the normal stress in the circumferential direction V T becomes the intermediate principal stress. In this study, the cylindrical coordinates r and z are used in the macroscopic simulation due to the axisymmetric deformation. However, the cylindrical coordinates T and z are used in the microscopic simulation, since the normal stress in the circumferential direction V T becomes the maximum principal stress [15].
z
0
z
Macroscopic Simulation
r
0
Microscopic Simulation
T
FIGURE 2. Coordinates in macroscopic simulation and microscopic simulation.
Figure 3 shows the two neighboring voids and the velocity field. Figure 3(a) shows the two neighboring voids. The surfaces of the velocity discontinuity shown by broken lines are assumed. Here, v denotes the material velocity in the direction of the maximum principal stress. T1 and T 2 denote the angles between the direction of the maximum principal stress and the direction of the surface of the velocity discontinuity. 2l1 and 2l2 denote the lengths of the surface of the velocity discontinuity. L denotes the distance between two neighboring voids. Figure 3(b) shows the velocity field denoted as a hodograph. Here, 'v1 and 'v2 denote the amounts of velocity discontinuity. First, the rate of energy dissipation on the surface of the velocity discontinuity when two neighboring voids coalesce by internal necking, i.e., the energy dissipation rate of internal necking, is expressed as
494
k 'v1 2l2 k 'v2 2l1
l1 sin T 2 l2 sin T1 . sin T1 T 2
4kv
(4)
Next, the rate of energy dissipation by homogeneous plastic deformation of the material, i.e., the energy dissipation rate of homogeneous deformation, is expressed as L 2 k 2v
4kvL.
(5)
Since void coalescence is assumed to occur when the energy dissipation rate of internal necking is less than the energy dissipation rate of homogeneous deformation, the criterion of void coalescence is expressed as Lt
l1 sin T 2 l2 sin T1 . sin T1 T 2
(6)
The ratio of the energy dissipation rate of internal necking to the energy dissipation rate of homogeneous deformation E is defined as E
l1 sin T 2 l2 sin T1 . L sin T1 T 2
(7)
When E is less than one, the criterion of void coalescence is satisfied. Moreover, the material is assumed to fracture when E is less than one.
L v
v
ȟ v1 T2
T 2 T1
ȟ v2
T1
0
l2 l1 v
v
(a) Two neighboring voids (b) Velocity field FIGURE 3. Two neighboring voids and velocity field.
SIMULATION RESULT Experiment and Simulation of Tensile Test The tensile test was performed prior to the bore expanding test. The uniaxial tensile specimen was made of JIS SPCC, which is equivalent to ISO CR1. The thickness of the material was 1 mm. The following stress-strain relationship was obtained in the experiment using the uniaxial tensile specimen, and applied in the simulation:
VM
460 H M 0.005
0.14
MPa ,
(8)
where H M is the strain of the matrix. The simulation using the uniaxial tensile specimen was performed, in which plane stress is assumed. The material constants A and B in Equation (3) were selected such that the reduction in thickness of the material calculated in the simulation was almost equal to that obtained in the experiment. Here, B is assumed to be zero to simplify the simulation.
495
A
0.06, B
(9)
0
Experiment and Simulation of Bore Expanding Test Table 1 shows the experimental condition for the bore expanding test. To produce a prestrained sheet metal, a thick sheet metal was rolled to obtain a thin sheet metal with a thickness equal to the thickness of a non-prestrained sheet metal. Two rolling directions were used in the laboratory: rolling in the rolling direction of the as-rolled sheet and rolling in the width direction of the as-rolled sheet. Thickness of as-rolled sheet (mm) 1 2 3
TABLE 1. Experimental condition. Thickness in bore expanding test (mm) 1 1 1
Magnitude of prestrain 0 0.69 1.10
Figure 4 shows the illustration and photograph of the specimen in the bore expanding test. The outer and inner diameters of the specimen were 85 mm and 9 mm, respectively. The die for the bore expanding test was shaped like a cone with a corner angle of 45 degrees. The bore expanding ratio is defined as d d 0 1 , where d is the fractured bore diameter and d 0 is the pierced bore diameter.
Beforetest
After test
d0
d
(a) Illustration (b) Photograph FIGURE 4. Illustration and photograph of specimen.
Figure 5 shows the finite element meshes of the specimen before and after the bore expanding test in the case of no prestrain. Axisymmetry of the specimen was assumed.
(a) Before bore expanding test (b) After bore expanding test FIGURE 5. Finite element meshes of specimen in case of no prestrain.
Figure 6 shows the relationship between the prestrain and the bore expanding ratio. In the experiment, the bore expanding ratio for the specimen rolled in the rolling direction of the as-rolled sheet is almost the same as that rolled in the width direction of the as-rolled sheet. As the prestrain increases, the bore expanding ratio decreases. Although the bore expanding ratio calculated from the simulation is slightly larger than that obtained in the experiment, the simulation result agrees with the experimental result qualitatively.
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Bore expanding ratio
2.5 2.0
Experiment (rolling direction in laboratory = rolling direction of as-rolled sheet) Experiment (rolling direction in laboratory = width direction of as-rolled sheet) Simulation
1.5 1.0 0.5 0.0 0.0
0.3
0.6
0.9
1.2
Prestrain FIGURE 6. Relationship between prestrain and bore expanding ratio.
CONCLUSION We performed a simulation of the bore expanding test to confirm the validity of the new model of void coalescence. Since the simulation results and the experimental results generally agree, the method for considering the prestrain in the new model was proved to be effective.
REFERENCES B. Dodd and Y. Bai, Ductile Fracture and Ductility, London: Academic Press, 1987. A. M. Freudenthal, The Inelastic Behavior of Engineering Materials and Structures, New York: John Wiley & Sons, 1950. M. G. Cockcroft and D. J. Latham, Journal of the Institute of Metals 96, 33-39 (1968). P. Brozzo, B. De Luca and R. Rendina, Proceedings of the Seventh Biennial Congress of the International Deep Drawing Research Group, (1972). 5. M. Oyane, Bulletin of JSME 15, 1507-1513 (1972). 6. J. Lemaitre, A Course on Damage Mechanics, 2nd Edition, Berlin: Springer-Verlag, 1996. 7. K. Komori, Acta Materialia 54, 4351-4364 (2006). 8. K. Komori, Theoretical and Applied Fracture Mechanics 50, 157-166 (2008). 9. K. Komori, Materials Science and Engineering A 421, 226-237 (2006). 10. P. F. Thomason, Journal of the Institute of Metals 96, 360-365 (1968). 11 K. Komori, AIP Conference Proceedings 1252, 97-102 (2010). 12. S. Kobayashi, S.-I. Oh and T. Altan, Metal Forming and the Finite-Element Method, New York: Oxford University Press, 1989. 13. A. L. Gurson, Transactions of the ASME Journal of Engineering Materials and Technology 99, 2-15 (1977). 14. Y. Tomita, Numerical Elasticity and Plasticity, Tokyo: Yokendo, 1990 (in Japanese). 15. Y. K. Ko, J. S. Lee, H. Huh, H. K. Kim and S. H. Park, Journal of Materials Processing Technology 187-188, 358-362 (2007). 1. 2. 3. 4.
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Formability Prediction of Advanced High Strength Steel with a New Ductile Fracture Criterion Yanshan Loua, Sungjun Lima, Jeehyang Huha, and Hoon Huha a
School of Mechanical, Aerospace and Systems Engineering, KAIST, 335, Gwahanno, Daedoek Science Town, Daejeon, 305-701, Republic of Korea
Abstract. A ductile fracture criterion is newly proposed to accurately predict forming limit diagrams (FLD) of sheet metals. The new ductile fracture criterion is based on the effect of the non-dimensional stress triaxiality, the stress concentration factor and the effective plastic strain on the nucleation, growth and coalescence of voids. The new ductile fracture criterion has been applied to estimate the formability of four kind advanced high strength steels (AHSS): DP780, DP980, TRIP590, and TWIP980. FLDs predicted are compared with experimental results and those predicted by other ductile fracture criteria. The comparison demonstrates that FLDs predicted by the new ductile fracture criterion are in better agreement with experimental FLDs than those predicted by other ductile fracture criteria. The better agreement of FLDs predicted by the new ductile fracture criterion is because conventional ductile fracture criteria were proposed for fracture prediction in bulk metal forming while the new one is proposed to predict the onset of fracture in sheet metal forming processes. Keywords: Ductile fracture criterion, Advanced high strength steel, Forming limit diagram, Sheet metal forming. PACS: 46.50.+d
INTRODUCTION A forming limit diagram (FLD), first introduced by Keeler and Backofen [1] and Goodwin [2], is the most popular tool to characterize the formability of sheet metals. An FLD can be constructed by experimental methods such as hemispherical punch-stretch tests and Marciniak cup tests [3]. Experimental methods, however, requires intensive effort and tremendous time. For more efficient construction of FLDs, many analytical models were proposed based on different hypotheses such as the Hill’s localized necking model [4], the Swift’s diffuse necking model [5], the Marciniak-Kuczynski model [6], the vertex theory [7, 8] and the modified maximum force criterion (MMFC) [9]. Recently, various ductile fracture criteria [10-15] were utilized in finite element analysis (FEA) to predict fracture initiation in sheet metal forming processes [16-21]. The problem of these conventional ductile fracture criteria is that these criteria were proposed for the fracture prediction in bulk metal forming processes. Conventional ductile fracture criteria could not describe the formability of sheet metals with higher accuracy [22, 23]. It is because conventional ductile fracture criteria were derived based on the mechanism of void growth while the fracture in sheet metal forming processes is the combination of the void growth inside sheet metals and necking outside [24]. A new ductile fracture criterion is proposed based on the effect of the non-dimensional stress concentration factor, the stress triaxiality and the effective plastic strain on nucleation, growth and coalescence of voids. The new ductile fracture criterion is then applied to predict FLDs of AHSS steels: DP780, DP980, TRIP590 and TWIP980. The AHSS steels are characterized by the Swift’s strain hardening rule and R-values obtained from uniaxial tensile tests and the FLDs from hemispherical punch-stretch tests. The Yld89 anisotropic yield function is employed to construct the yield surfaces. FLDs of these AHSS steels are predicted by the new ductile fracture criterion and conventional ones and compared with the experimental results in order to evaluate the performance of the new ductile fracture criterion.
The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 498-505 (2011); doi: 10.1063/1.3623650 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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THEORETICAL BASIS Ductile Fracture Criteria The forming limit of metal forming depends severely on the deformation history. The history of plastic deformation is the main concern in ductile fracture criteria. Generally, ductile fracture criteria have a form of
f
f σ d C
0
(1)
where f denotes a weight function considering the stress state, σ is a stress state, is the equivalent plastic strain, f is the equivalent plastic strain at fracture and C is the material constant. Review of Conventional Ductile Fracture Criteria Various ductile fracture criteria have been proposed empirically as well as theoretically based on different hypotheses. Cockcroft and Latham [10] postulated that fracture was controlled by the maximum principal tensile stress integrated over the plastic strain as below:
f
0
*d C1
(2)
Brozzo et al. [11] proposed a modified criterion in which the effect of hydrostatic stress was introduced such as:
f
0
2 * d C2 3 * m
(3)
Oh et al. [12] introduced the effect of the non-dimensional stress concentration factor into ductile fracture criteria as below:
f
0
* d C3
(4)
Oyane et al. [13] derived a ductile fracture criterion from the equations of the plasticity theory for porous materials in a form of
f
0
m C4 d C5
(5)
Clift et al. [14] substituted the equivalent stress for the maximum stress of the Cockcroft–Latham criterion:
f
0
d C6
(6)
Ko et al. [15] proposed another modification of the Cockcroft-Latham criterion to consider the influence of the hydrostatic stress in a form of
f
0
* 1 3 m d C7
x , when x 0 x x , when x 0
499
(7)
1.3
0.70
(a)
(b)
0.65 1.2
0.60 Stress triaxiality
Stress concentration factor
Plane strain
1.1
1.0
Equibiaxial tension
0.55 Plane strain
0.50 0.45 0.40 0.35
0.9 -0.75 -0.50 -0.25
Uniaxial tension
Equibiaxial tension
Uniaxial tension 0.00
0.25
0.50
0.75
1.00
0.30 -0.75 -0.50 -0.25
1.25
Strain Path
0.00
0.25
0.50
0.75
1.00
1.25
Strain Path
FIGURE 1. Effect of strain paths on weight functions in ductile fracture criteria: (a) the stress concentration factor; (b) the stress triaxiality.
In Eqs. (2)-(7), * is the maximum principal stress, m is the mean or hydrostatic stress, is the effective stress, and C1 C7 are the material constants in ductile fracture criteria. New Ductile Fracture Criterion for Accurate Prediction of FLDs The ductile fracture criteria reviewed above were proposed to predict the onset of fracture in various bulk metal forming processes such as extrusion, rolling, upsetting drawing and hub-hole expanding [10-15]. These conventional ductile fracture criteria cannot predict FLDs accurately especially in the biaxial stress region [22, 23]. Ductile fracture is preceded by severe plastic deformation involving nucleation, growth and coalescence of voids in metals and alloys. These steps are influenced mainly by three factors: the hydrostatic stress, the maximum stress and the equivalent plastic strain. Three factors should be considered in ductile fracture criteria in order to accurately predict fracture initiation of sheet metals. The hydrostatic stress is normally considered in terms of stress triaxiality m while the effect of the maximum stress is introduced by the stress concentration factor * . The stress concentration factor and the stress triaxiality are functions the strain path d 2 d 1 , as presented in FIGURE 1 (a) and (b). The stress triaxiality increases monotonously as the strain path shifts from uniaxial tension to balanced biaxial tension. The stress concentration factor increases as the strain path moves from uniaxial tension to plane strain tension and then decreases when the strain path shifts to balanced biaxial tension from plane strain tension. The stress triaxiality and the stress concentration factor are utilized as the weight function in the new ductile fracture criterion. To adjust the effect of the stress triaxiality and the stress concentration factor on the accumulated damage of materials, a material constant of an exponent is added to each of these factors. The new ductile fracture criterion is proposed to have a form of
f
0
C8
* 3 m 9 d C10 C
(8)
where C8, C9 and C10 are material constants. The materials constants are evaluated from three experiments: the uniaxial tensile test; the plane strain test; the balanced biaxial tensile test. These three kinds of experiments are carried out to measure the limit strains in the uniaxial tension, plane strain and balanced biaxial tension conditions. From the limit strain in the uniaxial tension, the material constant C10 is obtained as below:
C10 u 0
(9)
where u 0 indicates the limit strain in uniaxial tension. In the plane strain condition, Eq. (8) is reduced to a form of
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psC 1 ps 1 ps 8
C9
ps 0 C10
(10)
Here ps , ps 2 1 and ps 0 denote the stress concentration factor, stress ratio and the limit strain in the plane strain conditions. In the balanced biaxial tension d 1 d 2 , the new ductile fracture criterion has a form of
eqC eq 1 eq 8
C9 1
eq 0 C10
(11)
where eq , eq 2 1 and eq 0 are the stress concentration factor, stress ratio and the limit strain in the balanced biaxial tension conditions. The material constants in the new ductile fracture criterion are calculated by solving Eqs. (9), (10) and (11). Yld89 Yield Function Barlat and Lian [25] generalized the Hosford79 non-quadratic anisotropic yield function [26, 27] to consider the shear stress which has a form of
a K1 K 2 a K1 K 2 c 2 K 2 m
m
m
2 m
(12)
where
K1
xx h yy 2
xx h yy 2 2 , K2 p xy 2 2
(13)
where c, h and p are anisotropic parameters. The exponent m is suggested to be 6 for BCC metals and 8 for FCC metals. The anisotropic parameters c and h are easily obtained as below:
c 2a
R0 R90 R0 1 R90 ,h 1 R0 1 R90 1 R0 R90
(14)
Whereas the anisotropic parameter p is evaluated by the Lankford coefficient in the diagonal direction R45 by solving an implicit equation as below: 2 2 1 h 1 h m 1 m 1 R45 a K1 K 2 2 K 2 1 h a K1 K 2 2 K 2 1 h 2 2 2 1 h m 1 m 1 2 c 2 K 2 1 h a K1 K 2 K 2 1 h p2 4 2 2 1 h m 1 m 1 1 h 2 a K1 K 2 K 2 1 h p 2c 2 K 2 p2 0 4 4
(15)
where
1 h K1 , K 2 4
1 h
2
4
501
4 p2 (16)
TABLE 1. Mechanical properties of AHSS steels (Unit: MPa). Material
Thickness
K
0
n
R
u0
ps 0
eq 0
DP780 (1.0t)
1.0 mm
0.002 0.002
0.179 0.109
0.20
0.105
0.32
1.2 mm
1429 1442
0.930
DP980 (1.2t)
0.658
0.12
0.07
0.19
TRIP590 (1.2t)
1.2 mm
1192
0.015
0.285
0.995
0.39
0.24
0.40
TWIP980 (1.4t)
1.4 mm
2537
0.120
0.726
0.812
0.58
0.46
0.48
TABLE 2. Anisotropic parameters in the Yld89 yield functions of AHSS steels. Material DP780 (1.0t) DP980 (1.2t) TRIP590 (1.2t) TWIP980 (1.4t)
a
c
h
p
1.0363 1.2063 1.0025 1.1038
0.9637 0.7937 0.9975 0.8962
1.0000 1.0000 1.0000 1.0000
1.0000 1.0000 1.0000 1.0000
yy [MPa]
2000
Stress [MPa]
1750
1000 750
1500
500
1250
250
m 6 6 6 6
DP780 (1.0t) DP980 (1.2t) TRIP590 (1.2t) TWIP980 (1.4t)
1000 750
-250 DP780 (1.0t) DP980 (1.2t) TRIP590 (1.2t) TWIP980 (1.4t)
500 250
-500 -750
(a) 0 0.0
0.1
0.2
0.3
0.4
0.5
(b) -1000 -1000 -750 -500 -250
0
250
500
750 1000 xx [MPa]
Plastic strain
FIGURE 2. (a) Stress-strain curves of AHSS steels; (b) yield surfaces of AHSS steels.
MATERIAL PROPERTIES The new ductile fracture criterion is applied to predict FLDs of four AHSS sheets: DP780, DP980, TRIP590 and n TWIP980. The materials are characterized by the Swift’s strain hardening law ( K 0 ) and the R-values obtained by uniaxial tensile tests. The coefficients in the Swift’s strain hardening law and the R-values are summarized in Table 1. The true stress - true strain curves in the rolling direction are fitted with the Swift’s strain hardening law and shown in FIGURE 2 (a). The anisotropic parameters of the Yld89 yield function are calculated with Eqs. (14), (15) and (16) as presented in Table 2. The yield surfaces at yielding are compared in FIGURE 2 (b). Hemispherical punch-stretch tests were conducted to construct FLDs of AHSS steels. Different specimen sizes and lubrication conditions are applied to obtain various strain paths [28]. The width of arc-shaped specimens is selected as 20 mm, 40 mm, 60 mm, 80 mm, and 100 mm while the dimension of square specimens is 200 mm × 200 mm. Teflon and Plasticine are used for the arc-shaped specimens as lubricants while square-shaped specimens are lubricated as dry (D), Teflon (T), Teflon + Vaseline (T+V), and Teflon + Plasticine (T+P). Specimens tested are shown in FIGURE 3 (a), (b), (c) and (d) for DP780, DP980, TRIP590 and TWIP980, respectively. FLDs are constructed in the true strain space by measuring the deformation of circular grids from fractured specimens. The limit strains in the uniaxial, plane strain and balanced biaxial tension conditions are listed in Table 1.
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FIGURE 3. Fractured specimens of hemispherical punch-stretch tests. (a) DP780; (b) DP980; (c) TRIP590; (d) TWIP980. TABLE 3. Material constants in ductile fracture criteria. C2 C3 C1 Material DP780 (1.0t)
145.20
0.18
0.16
C4
C5
C6
C7
C8
C9
C10
-0.03
0.06
140.19
0.37
12.86
-1.51
0.20
DP980 (1.2t)
103.27
0.11
0.10
0.01
0.04
100.13
0.23
12.29
-1.11
0.12
TRIP590 (1.2t)
241.19
0.37
0.34
0.13
0.18
231.67
0.78
8.36
-0.93
0.39
TWIP980 (1.4t)
716.79
0.62
0.57
1.11
0.84
686.28
1.31
4.63
-0.61
0.58
APPLICATION TO FLD PREDICTION OF AHSS SHEETS After careful experiments, material constants from C1 to C7 are calculated with the corresponding equation for each ductile fracture criterion after appropriate experiments of the uniaxial and plane strain tension tests. Material constants C8, C9 and C10 in the new ductile fracture criterion are calculated with Eqs. (9), (10) and (11) after careful experiments of the uniaxial, the plane strain and the equibiaxial tension tests. The material constants calculated are presented in Table 3. These material constants calculated are utilized to predict FLDs of AHSS sheets. The FLDs predicted by various ductile fracture criteria are compared with experimental results in FIGURE 4 (a), (b), (c) and (d) for DP780, DP980, TRIP590 and TWIP980, respectively. For FLD prediction in the left hand side, the Cockcroft, Brozzo, Oh, Clift and Ko-Huh criteria underestimate the limit strain in the uniaxial tension conditions for DP780, DP980 and TRIP590. For TWIP980, the limit strain predicted by the Cockcroft, Oh, Oyane-Sato, Clift and new ductile fracture criteria are close to experimental results while the Brozzo and Ko-Huh criteria overestimate the formability in the left hand side of FLDs. The formability in the left hand side of FLDs can be predicted by the Oyane-Sato and new ductile fracture criteria with higher accuracy than others because these two ductile fracture criteria has more material constants than other ductile fracture criteria.
503
0.4
0.4
Major strain
0.3
Fracture Necking Safe
0.3
0.2
0.2 Cockcroft Brozzo Oh Oyane-Sato Clift Ko-Huh New criterion
0.1
(a) DP780 -0.2
-0.1
Major strain Fracture Necking Safe
0.0
0.1
0.2
0.1
(b) DP980 0.4 -0.2
0.3
Cockcroft Brozzo Oh Oyane-Sato Clift Ko-Huh New criterion
-0.1
0.0
0.1
0.2
0.6
0.5
0.3
0.4
Minor strain
Minor strain
0.7
Major strain Fracture Necking Safe
Major strain Fracture 0.6 Safe 0.5
0.4 0.4 0.3 0.3 Cockcroft Brozzo Oh Oyane-Sato Clift Ko-Huh New criterion
0.2
0.1 (c) TRIP590 -0.3
-0.2
-0.1
0.0
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Cockcroft Brozzo Oh Oyane-Sato Clift Ko-Huh New criterion
0.2 0.1 (d) TWIP980 0.6-0.3
Minor strain
-0.2
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Minor strain
FIGURE 4. Comparison of FLDs predicted by ductile fracture criteria with experimental results. (a) DP780; (b) DP980; (c) TRIP590; (d) TWIP980.
For prediction of the limit strain in the plane strain condition, the Oyane-Sato and the new ductile fracture criteria can accurately describe it because of the larger number of material constants in these two ductile fracture criteria. Other ductile fracture criteria cannot accurately estimate the formability in the plane strain condition because there is only one material constant in these ductile fracture criteria. The only material constant in these ductile fracture criteria are calculated simply by the limit strain in two tests: the uniaxail and the plane strain tension tests. Consequently, based on ductile fracture criteria with only one material constant, the predicted limit strain in the uniaxial tension is greater than the experimental results if the limit strain predicted in plane strain condition is less than the experimental data and vice versa. For prediction of FLDs in the right hand side, conclusions are obviously obtained that FLDs can be accurate predicted by the new ductile fracture criterion while conventional ductile fracture criteria are too conservative in this stress region. It is because the conventional ductile fracture criteria were proposed to predict the onset of fracture in bulk metal forming processes while the new ductile fracture criterion is developed to accurately predict FLDs of sheet metals. In the new ductile fracture criterion, more material constants are introduced to consider the effect of stress triaxiality and the stress concentration factor on the nucleation, growth and coalescence of voids. More material constants make the new ductile fracture criterion more flexible than conventional ductile fracture criteria. For the accurate prediction of FLDs in both the left and right hand sides, the new ductile fracture criterion is recommended for sheet metals. FLDs predicted by the Oyane-Sato ductile fracture criterion are close to the experimental FLDs in
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the left hand side. Moreover, only two tests are required to calculate the material constants in the Oyane-Sato ductile fracture criterion. The Oyane-Sato ductile fracture criterion is qualified if the fracture initiation is expected to occur in FLDs of the left hand side.
CONCLUSIONS A ductile fracture criterion is newly proposed to accurately predict FLDs of sheet metals. The new ductile fracture criterion considers the effect of the stress triaxiality, the stress concentration factor and the plastic strain on the nucleation, growth and coalescence of voids in ductile fracture. The new ductile fracture criterion is applied to predict FLDs of four AHSS steels: DP780, DP980, TWIP590 and TWIP980. FLDs predicted by the new ductile fracture criterion are compared with experimental results and those predicted by conventional ductile fracture criteria. The comparison demonstrates that conventional ductile fracture criteria cannot predict FLDs of sheet metals especially in the right hand side of FLDs. If the onset of fracture is expected to occur in the left hand side of FLDs, the Oyane-Sato ductile fracture criterion is suggested to be utilized. The new ductile fracture criterion is recommended to be utilized to predict fracture initiation from the uniaxial tension to balanced biaxial tension.
REFERENCES 1. S. P. Keeler and W. A. Backofen, Trans. ASM 56, 25-48 (1963). 2. G. M. Goodwin, Trans. Soc. Automot. 25, 1413-1424 (1968). 3. ISO 12004-2, Metallic material – Sheet and strip – Determination of forming limit curves – Part 2: Determination of forminglimit curves in the laboratory, First Edition, (2008). 4. R. Hill, J. Mech. Phys. Solids 1, 19-30 (1952). 5. H. W. Swift, J. Mech. Phys. Solids 1, 1-18 (1952). 6. Z. Marciniak and K. Kuczynski, Int. J. Mech. Sci. 9, 609-620 (1968). 7. S. Storen and J. R. Rice, J. Mech. Phys. Solids 23, 421-441 (1975). 8. X. Zhu, K. Weinmann and A. Chandra, J. Eng. Mater. Technol. 123, 329-333 (2001). 9. P. Hora, L. Tong and J. Reissner, Proc. Numisheet’96 Conf. Dearborn, Michigan, USA, 1996, pp. 252-256. 10. M. G. Cockcroft and D. J. Latham, J. Inst. Met. 96, 33-39 (1968). 11. P. Brozzo, B. Deluca and R. Rendina, Proc. 7th Biennial Conf. IDDRG on Sheet Metal Forming and Formability, 1972. 12. S. I. Oh, C. C. Chen and S. Kobayashi, J. Eng. Ind. 101, 36-44 (1979). 13. M. Oyane, T. Sato, K. Okimoto and S. Shima, J. Mech. Work. Technol. 4, 65-81 (1980). 14. S. E. Clift, P. Hartley, C. E. N. Sturgess and G. W. Rowe, Int. J. Mech. Sci. 32, 1-17 (1990). 15. Y. K. Ko, J. S. Lee, H. Huh, H. K. Kim and S. –H. Park, J. Mater. Process. Technol. 187/188, 358-362 (2007). 16. H. Takuta, K. Mori and N. Hatta, J. Mater. Process. Technol. 95, 116-121 (1999). 17. H. Takuda, K. Mori, H. Fujimoto and N. Hatta, J. Mater. Process. Technol. 92/93, 433-438 (1999). 18. H. Takuda, K. Mori, N. Takakura and K. Yamaguchi, Int. J. Mech. Sci. 42, 785-798 (2000). 19. F. Ozturk and D. Lee, J. Mater. Process. Technol. 147, 397-404 (2004). 20. H. S. Liu, Y. Y. Yang, Z. Q. Yu, Z. Z. Sun and Y. Z. Wang, J. Mater. Process. Technol. 209, 5443-5447 (2009). 21. J. S. Chen, X. B. Zhou and J. Chen, J. Mater. Process. Technol. 210, 315-322 (2010). 22. Y. S. Lou, H. Huh, Y. K. Ko and J. W. Ha, Proc. KSAE Spring Conf. Pusan, Korea, 2010, pp. 40-45. 23. Y. S. Lou, S. B. Kim, Y. K. Ko and H. Huh, Proc. KSTP Fall Conf. Jeju, Korea, 2010, pp. 298-303. 24. T. B. Stoughton and J. W. Yoon, Int. J. Plasticity 27, 440-459 (2011). 25. F. Barlat and K. Lian, Int. J. Plasticity 5, 51-66 (1989). 26. W. F. Hosford, Proc. 7th North American Metalworking Conf., Sme, Dearborn, MI 191-197 (1979). 27. R. W. Logan and W. F. Hosford, Int. J. Mech. Sci. 22, 419-430 (1980). 28. H. Huh, C. H. Lee and J. W. Chung, J. Kor. Soc. Technol. Plast. 7, 81-93 (1998).
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Numerical Modeling for Hole-Edge Cracking of Advanced High-Strength Steels (AHSS) Components in the Static Bend Test Hyunok Kim1*, William Mohr1, Yu-Ping Yang1, Paul Zelenak1, and Menachem Kimchi1 1
Edison Welding Institute, 1250 Arthur E. Adams Drive, Columbus, Ohio 43221-3585, U.S. *Corresponding author: 1-614-688-5239, 1-614-688-5001 and
[email protected] Abstract. Numerical modeling of local formability, such as hole-edge cracking and shear fracture in bending of AHSS, is one of the challenging issues for simulation engineers for prediction and evaluation of stamping and crash performance of materials. This is because continuum-mechanics-based finite element method (FEM) modeling requires additional input data, “failure criteria” to predict the local formability limit of materials, in addition to the material flow stress data input for simulation. This paper presents a numerical modeling approach for predicting hole-edge failures during static bend tests of AHSS structures. A local-strain-based failure criterion and a stress-triaxiality-based failure criterion were developed and implemented in LS-DYNA simulation code to predict hole-edge failures in component bend tests. The holes were prepared using two different methods: mechanical punching and water-jet cutting. In the component bend tests, the water-jet trimmed hole showed delayed fracture at the hole-edges, while the mechanical punched hole showed early fracture as the bending angle increased. In comparing the numerical modeling and test results, the load-displacement curve, the displacement at the onset of cracking, and the final crack shape/length were used. Both failure criteria also enable the numerical model to differentiate between the local formability limit of mechanical-punched and water-jettrimmed holes. The failure criteria and static bend test developed here are useful to evaluate the local formability limit at a structural component level for automotive crash tests. Keywords: Failure analysis, Hole-edge failure, Component bend test. PACS: NUMI_ABS215
INTRODUCTION Hole-edge cracking has been observed during plastic deformations in stamping of advanced high-strength steels (AHSS) and vehicle crash tests. Due to the complex nature of edge fracture behavior, finite-element (FE) modeling with conventional material model/data has had only limited success for predicting edge cracking. There are several important factors which influence on hole-edge cracking, including:
Sheet material properties Edge quality of hole from punching Pre-strains on the edge caused by trimming or punching Burr formation (up or down with respect to a moving punch).
STATE OF THE ART Research is actively ongoing in the area of hole-edge cracking prediction. Two different approaches include more complicated modeling techniques by either correlation of FE simulation results of blanking/piercing with stamping [1], or conducting numerical modeling to explicitly consider the volume fraction of martensite and ferrite in the dual-phase microstructure [2, 3]. However, these approaches may be limited for the direct use in industry forming simulations, due to the higher computation cost and unknown parameters that the FE model cannot take into account, such as hole-edge imperfections (e.g., micro-cracks remained on hole-edge after punching) and microstructure characteristics (banding width and micro-void areas, etc.). The stress-triaxiality-based failure model to predict the failures in deep-drawing TRIP 700 material and bending the automotive structural component [4]. The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 506-513 (2011); doi: 10.1063/1.3623651 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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Compared to these complicated models, simpler prediction models were recently introduced [5, 6, 7]. A regression model was developed to handle all these unknown parameters in the hole-tensile test and hole-expansion test (HET) [5]. A simple ductile damage model was used to predict edge cracking in the bow-tie tensile test and HET [6]. A stress-strain based damage model was used to predict the local shear fracture during draw-bending of AHSS [7]. To account for macro-crack formation in FE simulations of HET, recently an inverse calibration method based on a damage model utilizing a triaxiality-dependent fracture criterion and hardening behavior was developed [8]. Therefore, for a given sheet material (including material properties and microstructure) and trimming process (punching, water-jet cutting, and laser cutting), a unique number or a small range of critical damage values that is the energy consumed to failure can be determined. With this background, it is desirable to develop a simple and practical failure model for industry applications for FE modeling of stamping and vehicle crash tests.
OBJECTIVE The primary objective of this study was to predict the hole-edge failures for AHSS structural components. The specific objectives are to: Develop a component bend test (i.e., the structure design and test conditions) to evaluate the hole-edge failures Predict hole-edge failure by using FE simulations with reliable failure criteria.
THE FAILURE CRITERIA FOR HOLE-EDGE CRACKING The hole-edge failure criteria were determined by using different local formability testing (LFT) methods and FEM model. Fig. 1 illustrates the flow chart to determine the failure map and its application for component bend simulations. Four different LFT methods such as a tensile, hole-tensile and a plane-strain shear and the hole expansion tests were used to characterize the effects of material properties and trimming methods on the hole-edge failures. Experimental results were analyzed by using FEM simulations to calculate the local strain, damage value and stress-triaxiality (i.e., a ratio of equivalent stress and mean stress). From experimental and simulation models, two different failure criteria: a local-strain-based damage model and a stress-triaxiality-based failure model were determined. The failure models were used to design the component bend test and predict the hole-edge failures observed in the component bend test.
Figure 1. Flow chart to determine the failure map
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The failure map was constructed with the local strains and different stress triaxiality calculated from FE simulations of various local-formability tests. The detailed procedure to construct the failure map is available in [4]. The CDV for TRIP 780 material was determined for different trimming conditions. Furthermore, two different failure maps were constructed for the punched-hole and water-jet-cut-hole cases. Both failure criteria were implemented for LS-DYNA simulations with MAT_224 as shown in Fig. 2. In the stress-traxiality based failure criteria, the failure strain varies with the stress triaxiality, while the local-strain based damage model has a constant failure strain as shown in Fig. 2. When the strain of FE model reaches this failure strain, the element at failure strains starts eroding in LS-DYNA, which emulates the crack initiation and propagation in experiments.
Figure 2. Failure map and criteria defined for FEM simulations for bending of the structural part
DESIGN OF THE COMPONENT BEND TEST A component bend test was developed by conducting preliminary FE simulations of various hat-shaped structure designs by bench-marking other designs available in literature [9, 10]. Different locations and sizes of hole and spot welds were investigated to obtain reproducible hole-edge failures in the test.
Figure 3. Schematic of the component structure and a bending configuration
Fig. 4 illustrates the procedure to optimize key design parameters in the hat-structure geometry and bending configuration. A series of spot weld models were defined on the flange of a lower channel as shown in Fig. 5. A weld-pitch of 25-mm was used and individual weld nugget model was defined to have 6-mm diagonal length and 1.65-mm thickness based on previous welding simulation experiences.
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Figure 4. A flow chart of the component design using FEA
Figure 5. FE model of a lower channel part and a spot weld model
EXPERIMENTAL RESULTS Component bending tests were conducted at the MTS machine (300-kN capacity) and Fig. 6 shows the testing setup. During the test, the high-resolution camera recorded the hole-edge failure as the upper ram moved down and the structure is bent.
Figure 6. A 3-pt. bend test for AHSS structures
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The component has a rectangular opening with four corner holes that were trimmed by punching or water-jet trimming. The lower hat-shaped channel was bent by using a press-brake machine. Finally, a top plate and a lower channel section were spot welded (e.g., 6-mm spot weld nugget diameter). The same TRIP 780 material was used for the entire component structure. During the test, the load-displacement curve was measured and synchronized with a high-resolution video camera monitoring the hole-edge region. Testing was conducted up to 120-mm stroke. Test results were summarized in Table 1. The punched structure samples showed the edge cracking in a lower stroke (e.g. 30~40-mm) compared to the water-jet structure samples and this resulted that the punched samples gave 5-10 mm larger crack length compared to the water jet samples at the final stroke (i.e. 120-mm). Fig. 7 shows both punched and water jet samples tested. The overall bend angles were similar between the punched and the water-jettrimmed structures. However, the punched case showed more severe cracks at two trimmed holes compared to the water-jet case. Table 1. Summary of experimental results
Sample No. WJ-1 WJ-2 WJ-3 P-4 P-5 P-6
Stroke at the start of cracking 87 mm 95 mm 100 mm 65 mm 60 mm 57 mm
Maximum Force 19.41 kN 19.27 kN 19.35 kN 19.31 kN 19.33 kN 19.23 kN
Figure 7. Tested component structures
FEM RESULTS Fig. 8 shows the FE model for 3-point bend testing. A commercial FEM code, LS-DYNA was used for simulations. The fine mesh (2-mm) was used for the areas near the hole-edges and the remaining areas were meshed with 5-mm elements. The rollers were meshed with 4-mm elements uniformly. When the local strain reaches the limit strain, the element erosion is initiated as shown in Fig. 9, and they are propagated to neighboring elements.
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Figure 8. Testing configuration of a component bend test
Figure 9. FEM predictions of the initiation of the hole-edge cracking (left) and the final fractured structure with the effective stress contour (right)
FEM simulations were conducted with the local strain-based damage model and the stress-traxiality-based failure map (Fig. 2). Compared to the test results of the punched case, the local-strain-based damage model gave better correlations than the stress triaxiality based failure map in terms of severity of cracking (Fig. 10) as well as the critical stroke at the onset of hole-edge cracking as shown Fig. 11.
Figure 10. Comparison of edge-failure shapes between the experiment and FEM predictions with different failure criteria
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Figure 11. Comparison of the critical stroke at the onset of hole-edge cracking
The stress triaxiality and local strain were compared for two different failure criteria of punched and water-jet cases (Fig. 12). The local strain value (0.74) from the ductile failure of a normal tensile test showed no edge failure and the local strain exceeded the boundary of failure strain limit that were determined by other failure criteria.
Figure 12. FEM predictions of the stress triaxiality and local strain with the different failure criteria for the punched case
DISCUSSSIONS In this study, different trimming methods such as a mechanical punching and a water-jet cutting were used to make the hole of specimens remain different residual stresses and damages at the hole-edge and this difference can result in different behaviour of hole-edge failure in secondary forming and crash tests. A local-strain based failure model was determined by using the CDV for different trimmed edge quality. This model was evaluated for FE simulations of various tests and it showed the better predictions of hole-edge failures compared to the complex failure map that was established by considering the stress triaxiality. Therefore, to be able to reliably apply the complex stress triaxility based model for the large scale crash model, a further study on stress triaxiality based failure model should be followed as a future work.
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CONCLUSIONS The following conclusions can be summarized from this study. A new methodology was able to evaluate the quality of hole edge with different trimming methods and also quantify the damage/local strain remained from those trimming processes. The punched hole showed the hole-edge failure at a lower strain compared to water-jet cutting. The local-strain based failure criterion gave better correlations with experiments than the stress-triaxiality based failure criterion. A failure map based on the stress-triaxiality and local strain is useful to predict the hole-edge failure.
REFERENCES 1. R. Wiedenmann, P. Sartkulvanich, T. Altan, Finite element analysis on the effect of sheared edge quality in blanking upon hole expansion of advanced high strength steel, International Deep Drawing Research Group (IDDRG) 2009 International Conference, Golden, CO, United States. 2. X. Sun, K.S. Choi, Effects of AHSS Microstructure Inhomogeneity on Strain Localization, Shear Fracture Technical Progress Review, Auto/Steel Partnership, Sept., 10, 2008, Southfield, MI, United States. 3. X. Wu, C. Xie, H. Bahmanpour, Flange shear affected zone study, Shear Fracture Technical Progress Review, Auto/Steel Partnership, Sept, 10, 2008, Southfield, Michigan 4. D.Z. Sun, F. Andrieux, M. Feucht, Damage modeling of a TRIP steel for integrated simulation from deep drawing to crash, 7th European LS-DYNA Conference, May 14-15, 2009, Salzburg, Austria. 5. K. Watanabe, M. Tachibana, K. Koyanagi, K. Motomura, Simple Prediction Method for the Edge Fracture of Steel Sheet during Vehicle Collision (1st report) – Evaluation of Fracture Limit from the Edge using Small-Sized Test Pieces, LS-DYNA Conference. 2006. 6. J.S. Lee, Y.K. Ko, H. Huh, H.K. Kim, S.H. Park, Evaluation of Hole Flangeability of Steel Sheet with Respect to the Hole Processing Condition, Engineering Plasticity and its Applications 340-341, 2007, pp. 665-670. 7. H. Kim, A.R. Bandar, Y. Yang, J.H. Sung, R.H. Wagoner, Failure Analysis of Advanced High Strength Steels (AHSS) During Draw Bending, International Deep Drawing Research Group, IDDRG 2009 International Conference, Golden, CO, United States. 8. K. Chung, N. Ma, T. Park, D. Kim, D. Yoo, C. Kim, A modified damage model for advanced high strength steel sheets, International Journal of Plasticity, Article In Press, 2011 9. N. Kojima, K. Fukui, N. Mizui, Improvement in Bending Crashworthiness of Hat-Shape Columns by using High Strength Steel Sheets, 40th Mechanical Working and Steel Processing Conference, 1998, Pittsburgh, PA, United States. 10. K. Koyanagi, K. Motomura, Simple Prediction Method for the Edge Fracture of Steel Sheet during Vehicle Collision (2nd report), LS-DYNA Conference 2006, Anwenderforum, Ulm.
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Numerical Prediction of Microstructure and Mechanical Properties During the Hot Stamping Process Dongbin Kan, Lizhong Liu, Ping Hu , Ning Ma, Guozhe ShenˈXiaoqiang Han, Liang Ying State Key Laboratory of Structural Analysis for Industrial Equipment, School of Automotive Engineering, Dalian University of Technology, Dalian 116024, P.R.China Abstract. Numerical simulation and prediction of microstructures and mechanical properties of products is very important in product development of hot stamping parts. With this method we can easily design changes of hot stamping products’ properties prior to the manufacturing stage and this offers noticeable time and cost savings. In the present work, the hot stamping process of a U-channel with 22MnB5 boron steels is simulated by using a coupled thermo-mechanical FEM program. Then with the temperature evolution results obtained from the simulation, a model is applied to predict the microstructure evolution during the hot stamping process and mechanical properties of this U-channel. The model consists of a phase transformation model and a mechanical properties prediction model. The phase transformation model which is proposed by Li et al is used to predict the austenite decomposition into ferrite, pearlite, and bainite during the cooling process. The diffusionless austenite-martensite transformation is modeled using the Koistinen and Marburger relation. The mechanical properties prediction model is applied to predict the products’ hardness distribution. The numerical simulation is evaluated by comparing simulation results with the U-channel hot stamping experiment. The numerically obtained temperature history is basically in agreement with corresponding experimental observation. The evaluation indicates the feasibility of this set of methods to be used to guide the optimization of hot stamping process parameters and the design of hot stamping tools. Keywords: Hot stamping, Numerical simulation, 22MnB5 steel, Phase transformations, Hardness estimation. PACS: 81.05.Bx, 81.20.Hy
1. INTRODUCTION In order to reduce the weight of automobiles and improve the crash safety, more and more high-strength or ultrahigh-strength steel components are used on the vehicles. Conventional cold stamping of high-strength steel leads to disadvantages such as low formability and sever springback, so hot stamping technology is consequently developed for quenchenable boron steels such as 22MnB5. However, there are many process parameters in the hot stamping process and the tools need to be designed with cooling system, these make the design of hot stamping process and tools really complex. And this problem will become much knottier when we design hot stamping products with tailored properties. Numerical simulation and prediction of microstructures and mechanical properties of products can solve these problems mentioned above. Through this method, we can simply modify hot stamping technical process parameters and the design of tools before the manufacturing stage until the mechanical properties of the hot stamping products meet our expectation. In other words, the hot stamping process can be controlled precisely and quantitatively. This will greatly reduce the development costs and shorten the research cycle time. In this paper, the hot stamping process of a U-channel with 22MnB5 boron steels was simulated by using FE program LS-DYNA. Then with the temperature evolution results obtained from the simulation, models proposed by Li et al [1] were applied to predict the microstructure evolution during the hot stamping process and final
Corresponding author, email address:
[email protected] The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 602-609 (2011); doi: 10.1063/1.3623663 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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mechanical properties of this U-channel. The hot stamping experiment was designed to evaluate the prediction method.
2. EXPERIMENTAL SET-UP The material studied is a low alloyed boron steel called 22MnB5 with a thickness of 1.6mm. The chemical composition of this material is given in Table 1. The tests were performed in a hydraulic press machine. The tool consists of a punch and a die, both with cooling systems, Fig. 1. In the experiment, the blank was heated in a furnace to 950ć, and held for 5 minutes. After having a homogeneous austenitic microstructure, the blank was transferred to the water cooled tools where stamping and quenching took place simultaneously. After deformation, the hot blank remained in the tool to be cooled for 8s. Figure 2 shows the dimensions of the rectangular blank and the location of the temperature measurement point. The temperature evolution of the blank during the whole process was measured at the point on the sheet surface, using type K thermo-couple. C (%) 0.22–0.25
TABLE 1. Chemical composition of the 22MnB5 steel given in wt.% B (%) Si (%) Mn (%) Cr (%) P (%) S (%) 0.002-0.005 0.2–0.3 1.2–1.4 0.11-0.2 0–0.02 0–0.005
Al (%) 0.02-0.05
FIGURE 1. Illustration of hot stamping tools with cooling system.
FIGURE 2. Dimensions of the blank and location of the temperature measurement point, all dimensions are given in mm.
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3. FE MODELING The FE-code LS-DYNA has been used in this simulation. The modeled punch, die and blank as shown in Fig. 3 are corresponding to the experimental set-up. Only half of the blank and tools is modeled and symmetry constraints are applied. The simulation model consists of 9760 Belytschko-Tsay shell elements, with 5 through thickness integration points. Set TSHELL to 1 in the keyword *control_shellˈso that in the thermal calculations of the blank, the shells will be treated as 12-node brick elements to accurately calculate the through thickness temperature gradient, which is very important in the simulation of hot stamping process [2,3]. Material model 106 [3] which is a thermal elastic visco-plastic material model is used. Stress versus strain data is taken from the Numisheet 2008 BM03 benchmark specification [4]. Fig. 4 shows stress-strain curves for 22MnB5 steel for different temperatures from 500ć to 800ć at a strain rate of 0.1s-1. Data for Young’s modulus, Poisson’s ratios, thermal conductivity and heat capacity of the blank as a function of temperature are taken from the literature [5]. And the heat capacity and thermal conductivity for the material used in the tools are extracted from [6].
FIGURE 3. FE-model of blank and tools.
FIGURE 4. Stress–strain curves at temperatures of 800–550ć at a strain rate of 0.1 s1.
A convection heat transfer coefficient of 8 Wm-2K-1 is used and the radiation heat transfer coefficient is set to107 Wm-2K-1. Thermal boundary conditions are turned off for areas in contact with tools. The contact heat transfer coefficient which has a strong dependency on the contact pressure is a most critical parameter controlling cooling of
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the blank. In this work, the contact heat transfer coefficient is defined as a function of contact pressure, and the data given in the Numisheet 2008 BM03 benchmark specification [4] is used. The frictional coefficients is defined as a function of temperature, the data is extract from [7]. According to the experiment, temperature of the blank at the beginning of the die movement is set to 810ć, and the tools are held at 75ć. The punch is given a velocity of 70mm/s which is the actual punch velocity in the experiment. After the stamping procedure, the blank is held in the tools for 8s for the quenching.
4. MODELING OF MICROSTRUCTURAL EVOLUTION The numerical prediction model of microstructure and mechanical properties consists of a phase transformation model to predict the austenite decomposition into ferrite, pearlite, bainite and martensite during the cooling process, and a mechanical properties prediction model to predict the products’ final hardness distribution.
4.1 Phase Transformation Model The diffusion controlled transformations of austenite decomposition into ferrite, pearlite, and bainite for the isothermal condition is described by a set of reaction kinetics equations developed by Li et al [1]. The austenite to ferrite reaction equation is given as:
20.41G Ae3 T exp 27500 RT X F 0.4 1 X F 1 X F 3
dX F dt where
0.4 X F
(1)
FC exp 1.00 6.31C 1.78Mn 0.31Si 1.12 Ni 2.70Cr 4.06 Mo
FC
For the pearlite reaction, the equation is stated as:
20.32G Ae1 T exp 27500 RT X P 0.4 1 3
dX P dt
XP
1 X P
0.4 X P
PC
where
(2)
exp 4.25 4.12C 4.36Mn 0.44Si 1.71Ni 3.33Cr 5.19 Mo
PC
And the bainite reaction equation is expressed as:
dX B dt where
BC
20.29G Bs T exp 27500 RT X B 0.4 1 2
XB
1 X B
0.4 X B
BC
(3)
exp 10.23 10.18C 0.85Mn 0.55 Ni 0.90Cr 0.36 Mo
In Equation (1)-(3), G is the ASTM grain size number for the austenite; Ae3 , Ae1are the critical temperatures from the Fe–C equilibrium diagram; Bs is the bainite start temperature; R is the universal gas constant, T is the current temperature; XF , XP and XB are the volume fraction of ferrite, pearlite and bainite, respectively. The diffusionless austenite-martensite transformation is modeled using the Koistinen and Marburger relation [8], which is expressed as
Xm
X J (1- e-D ( Ms -T ) )
(4)
where Xm is the volume fraction of martensite, X¤ is the volume fraction of austenite available for the reaction, ¢is a constant coefficient set to 0.011, Ms is the martensite start temperature, and T is the current temperature.
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4.2 Mechanical Properties Prediction Model The mechanical properties prediction model is given as [1]:
Hv
X M HvM X B HvB X F X P HvF P
(5)
where XM, XB, XF and XP are the volume fraction of martensite, bainite, ferrite, and pearlite, respectively; and HvM, HvB and HvF+P are the Vickers hardness of martensite, bainite, and the mixture of ferrite and pearlite, respectively. For the calculation of HvM, HvB and HvF+P, empirically based expressions suggested by Maynier et al. [9] are used:
HvM HvB
127 949C 27 Si 11Mn 8 Ni 16Cr 21log Vr
323 185C 330 Si 153Mn 65 Ni 144Cr 191Mo
89 53C 55Si 22Mn 10 Ni 20Cr 33Mo log Vr HvF P
42 223C 53Si 30 Mn 12.6 Ni 7Cr 19 Mo
10 19Si 4 Ni 8Cr 130V log Vr
(6)
(7)
(8)
where Vr is the cooling rate at 70Wć (ć/s).
4.3 Prediction Process The kinetics equations are solved by the RUNGE_KUTTA method. With the application of the additivity rule, Equation (1)-(3) can be used to computing phase transformations under continuous cooling conditions. A fourth order polynomial is used to fit the temperature history curves obtained from the simulation. And then the coefficients of the polynomial are used as initial data in the prediction program. Fig. 5 shows a schematic structure for the prediction program.
FIGURE 5. Schematic structure for the prediction program.
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5. RESULTS AND DISCUSSION 5.1 Temperature History The measured and calculated temperature history of the point (Fig. 2) is compared in Fig. 6. In the experiment, the hot blank is transferred from the furnace to the tools manually. The blank is air cooled to 810ć when the stamping begins. As it is illustrated in Fig. 6, the calculated temperature history is in good agreement with the measured data. The cooling rate of the point is higher than 150ć/s. However, deviation appears in the last few seconds of the quenching process. The main reasons can be summarized as follows: 1) The tools were held at 75ć in the simulation, which is not in correspondence with the actual situation. 2) The phase transformation latent heat is not considered in the simulation.
FIGURE 6. Measured and calculated temperatures for the point defined in Fig. 2.
FIGURE 7. Calculated temperature distribution of the blank after the stamping process.
Figure 7 shows the calculated temperature distribution of the blank just after the stamping process. As we can see from the picture, the lowest temperature is found at the lower fillet of the U-channel. The blank material at this area keeps in contact with the fillet of the punch during the whole stamping process, almost no relative sliding is observed. Thus the contact heat transfer is high efficient because of the high contact pressure. The highest temperature value can be observed in the bottom and flange area, because these areas are the last ones which come in contact with the tools. Figure 8 shows the temperature evolution of four representative nodes (Fig.7) during the whole process. The average cooling rate (between 800 and 300ć) is the highest around the lower fillet of the U-
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channel, about 340ć/s. While the average cooling rate is the lowest at the bottom area and side wall, about 120ć/s. These can be also attributed to the contact conditions at these areas.
FIGURE 8. Temperature evolution of four representative nodes illustrated in Fig.7 during the whole process.
5.2 Microstructure and Hardness The hot stamped U-channel can be divided into five parts: the flange area, the side wall, the bottom area, the upper fillet and the lower fillet. Prediction of microstructures and hardness is conducted by the developed program for selected points from each area. Location of selected points and prediction results are shown in the unfolding drawing of the U-channel FE model, illustrated as Fig.9. The cooling rates at these points are always high enough to drive the martensite transformation, so all the predicted values of martensite volume fraction are above 95%, and the values of hardness above 490. While at the upper and lower fillet, the values of martensite volume fraction and hardness are a little higher than those at other areas, because the cooling rate is higher at the fillet than at other areas.
. FIGURE 9. Location of selected points, predicted Martensite volume fraction (%) and Vickers Hardness(within parenthesis).
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6. CONCLUSIONS The simulation of hot stamping process of a U-channel component with 22MnB5 boron steels was conducted by using FE program LS-DYNA. In addition, a program was developed to predict the final microstructure and mechanical properties. The temperature history of the blank from calculated and measured results are basically in agreement, which indicates that this set of simulation is feasible. According to the simulation, the lower fillet of blank and the fillet of punch keep in contact with high pressure during the stamping process. As a result, the cooling rate is higher at this area, and the predicted values of martensite volume fraction and hardness are a little higher than those at other areas.
ACKNOWLEDGMENTS This work was funded by the Key Project of the National Natural Science Foundation of China (No. 10932003), “973” National Basic Research Project of China (No. 2010CB832700), “863” Project of China (No. 2009AA04Z101) and the Fundamental Research Funds for the Central Universities(No. 893324, DUT11ZD202). These supports are gratefully acknowledged. Many thanks are due to the referees for their valuable comments.
REFERENCE 1. M. Li, et al., "A computational model for the prediction of steel hardenability," Metallurgical and Materials Transactions B, 1998, pp. 661-672. 2. G. Bergman and M. Oldenburg, "A finite element model for thermomechanical analysis of sheet metal forming," International Journal for Numerical Methods in Engineering, 2004, pp. 1167-1186. 3. J. O. Hallquist, "LS-DYNA Keyword User’s Manual - Version 971," Livermore Software Technology Corporation, 2007. 4. Nunmisheet 2008, The Numisheet Benchmark Study, Benchmark Problem BM03, Interlaken, Switzerland, 2008. 5. A. Shapiro, "Finite Element Modeling of Hot Stamping," Steel Research International, 2009, pp. 658-664. 6. P. Akerstrom, "Modelling and simulation of hot stamping," PhD. Thesis, Lulea University of technology, 2006. 7. T. Stöhr, M. Merklein and J. Lechler, "Determination of frictional and thermal characteristics for hot stamping with respect to a numerical process design," 1st International Conference on Hot Sheet Metal Forming of High-Performance Steel, Kassel, Germany, 2008, pp. 293–300. 8. D. P. Koistinen and R. E. Marburger, "A general equation prescribing the extent of the austenite-martensite transformation in pure iron-carbon alloys and plain carbon steels," Acta Metallurgica, 1959, pp. 59-60. 9. P. Maynier, J. Dollet and P. Bastien, "Hardenability Concepts with Applications to Steels," D.V. Doane and J.S. Kirkaldy, eds., AIME, New York, NY, 1978, pp. 518-44.
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Simulation of Tailored Tempering with a ThermoMechanical-Metallurgical Model in AutoFormplus S. Ertürka, M. Sestera, M. Seliga, P. Feuserb, K. Rollb a
AutoForm Development GmbH, Technoparkstrasse 1, CH-8005 Zurich, Switzerland b Daimler AG, Bela-Barenyi-Str. 1, 71063 Sindelfingen, Germany
Abstract. For automotive applications, the hot stamping of ultra-high-strength steels such as 22MnB5 is a wellestablished process providing significant reduction of fuel consumption and improving the component strength and geometrical accuracy due to reduced springback. Tailored tempering is a special type of hot stamping, in which different areas of the component experience different cooling histories leading to different final properties. The potential of manufacturing-optimised components consisting of high-strength and high-ductility regions in harmony with an enhanced crash performance makes tailored tempering very attractive compared with other conventional hot stamping processes. The optimisation of this process, where deformation and cooling take place simultaneously, requires a complete understanding in terms of material behaviour, formability, heat transfer and phase transformation kinetics. To this end, a thermo-mechanical-metallurgical model has been implemented in AutoFormplus in order to capture the material behaviour during the forming and quenching processes. Both radiation and convection are taken into account to describe heat transfer to ambient. Moreover, latent heat is considered and its effect on simulation is discussed. A guideline for parameter identification strategy has been developed and validated by separate experiments. The simulation results of tailored tempering of a B-pillar are presented together with measured tensile strength and elongation at fracture. Keywords: Tailored tempering, phase transformation, high-strength steels PACS: 81.05.Bx; 81.40.Ef
INTRODUCTION The number of automobile structural components produced from ultra-high-strength steels such as 22MnB5 is continuously growing due to the need to reduce the fuel consumption together with improved safety requirements. In hot stamping of 22MnB5, high initial temperatures are used together with high cooling rates leading to a martensitic phase transformation and thus high strength. Besides high strength, the components produced by conventional hot stamping exhibit a high geometrical accuracy due to reduced springback [1]. The benefits of hot stamping mentioned above have been extended to obtain improved crashworthiness by a special type of hot stamping, i.e. tailored tempering where forming and phase transformation can be optimised. The material properties of the parts can be selectively adjusted by introducing different localised temperatures within a tool. This affects the cooling rate, therefore phase transformation. Consequently, combining areas with high strength and areas with high ductility is possible via tailored tempering. This favours production of structural components with improved properties such as an optimised crash resistance [2].
SPECIAL FEATURES OF TAILORED TEMPERING In the simulation of a conventional direct hot-forming process, temperature field computation is necessary only for the simulation of material flow at elevated temperatures in order to ensure that the part can be produced without splits. The design of a conventional hot-forming process excluding quenching can be accomplished easily with the hot-forming functionality of AutoForm-Incrementalplus. On the other hand, the planning of tailored tempering processes places much higher demands on simulation software because now the quenching process itself has to be modelled in detail. Tailored tempering with heated tools and with cooled tools provides the possibility of adjusting The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 610-617 (2011); doi: 10.1063/1.3623664 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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zones with very high strength and zones with enhanced ductility in the same part, with a gradual transition of properties between these zones. A simulation model that is capable of predicting final part properties must, on the one hand, provide the complete temperature history of a material point with sufficient accuracy and, on the other hand, employ a phase transformation model. Such a model for the formation of martensite and bainite, for generation of latent heat and for prediction of final properties, has been implemented in AutoForm-ThermoSolverplus and is described in the next section.
THERMO-MECHANICAL-METALLURGICAL ANALYSIS In order to predict final part properties with sufficient accuracy, a complete modelling of the relevant phenomena with a coupled thermo-mechanical-metallurgical description is essential. The phenomena observed in tailored tempering can be described by thermal (i.e. heat transfer between hot sheet, tools and environment), mechanical (i.e. plastic deformation of sheet) and metallurgical descriptions (i.e. phase transformation of sheet resulting from cooling). Figure 1 illustrates the interaction between thermal, mechanical and microstructural fields, which is adapted in AutoForm-ThermoSolverplus. A more detailed discussion on the interaction can be found in [3]. Contact
Mechanical
Phase transformation
Thermal
Heat generation
Metallurgical
Latent heat
Mechanical properties
FIGURE 1. Interaction between thermal, mechanical and metallurgical fields in AutoForm-ThermoSolverplus
Mechanical Effects Since metals show temperature-dependent yielding in the temperature regime of tailored tempering, a coupled temperature and displacement analysis is required. Moreover, the strain rate effect on plasticity cannot be neglected for any hot-forming process, since it is pronounced at elevated temperatures. In AutoFormplus, hardening curves can depend on strain, temperature and strain rate. Details regarding temperature- and strain-rate-dependent hardening and the strategies to determine model parameters are explained in [4]. The hardening description can be combined either with the quadratic Hill yield criterion or with the non-quadratic BBC yield criterion. However, for hotforming applications a simple isotropic description is usually sufficient.
Thermal Effects Both mechanical and metallurgical effects strongly depend on temperature. This makes accurate temperature calculation crucial. Temperature calculation according to Newton's cooling law is achieved by taking thermal energy balance into account. For a given volume, the summation of all heat flux applied to the body and the rate of energy generated within the body has to be equal to the internal energy rate.
qtool
Punch
qtool qtool
q amb
qtool
HTCtool (Tblank Ttool )
q amb
effective HTCamb (Tblank
effective HTCamb
Tamb )
radiation convection HTCamb HTCamb
Die
effective HTCamb
Ambient heat transfer coefficient
Blank holder
Heat transfer coefficient [mW/mm2K]
100 90
Effectiveradiation HTCamb Radiation Convection convection
80 70
HTCamb
60 50 40 30 20 10 0 0
plus
FIGURE 2. Left: Heat fluxes between sheet, tools and ambient in AutoForm-ThermoSolver of heat transfer to ambient
611
200
400
600
800
Temperature
Temperature [°C]
1000
Right: Temperature dependency
The generated energy considered in a simulation of AutoForm-ThermoSolverplus can be any arbitrary combination of generated energy due to plastic deformation, phase transformation and friction. The modelling of heat flux is summarised in Figure 2. Heat conduction within the blank can be neglected in a hot-forming simulation but should be taken into account for tailored tempering due to a longer process time. Heat transfer to ambient can be defined as a function of temperature when both convection and radiation contribute to the heat transfer to ambient. Radiation plays a significant role especially at elevated temperatures and causes a strong deviation from the linearity of heat transfer. Heat transfer to tools is modelled based only on convection. The linear heat transfer coefficient regarding heat transfer to tools can be a function of contact pressure as described in [5].
Metallurgical Effects The base material 22MnB5 exhibits a pro-eutectoid ferritic-pearlitic microstructure with a hardness of about 170 HV10. However, as seen in the continuous cooling transformation (CCT) diagram [6] (Figure 3) after quenching with a cooling rate exceeding the critical cooling rate, the component might have a martensitic microstructure with a hardness of about 475 HV10. 900 C 700
A+
Ms 0.2
0.7
228
163
10 8 6 5 3 278
474
475 470
Hardness HV10
100
429 417
200
A+M
30 25 20
Cooling rate K/s
300
100 80
Temperature
B
500 400
A+P
A+
600
50%
F
Source: Arcelor
2h
1h
15 mi n
in 5m
2m
1m
in in
0
A: Austenite F: Ferrite P: Pearlite B: Bainite M: Martensite
Time
FIGURE 3. Continuous cooling transformation (CCT) diagram of 22MnB5
The phase transformation model implemented in AutoForm-ThermoSolverplus is the product of a co-operation with GMS GmbH [7]. The complexities of phase transformation and its kinetics are adapted and captured especially for 22MnB5. This enables the calculation of phase fraction and the resulting material properties, which are hardness, tensile strength and elongation at fracture respectively. Phase transformation and its kinetics are mainly derived from chemical composition. 22MnB5 gains special characteristics due to the presence of carbon (C), manganese (Mn) and boron (B). To this end, the chemical composition of 22MnB5 defined with respect to C, Mn and B is essential input because the corresponding phase transformation temperatures such as Ms are calculated based on chemical composition. Moreover, the energy generated during phase transformation, called latent heat, is also computed by the phase transformation model and taken into account in the temperature calculation of AutoForm-ThermoSolverplus.
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IDENTIFICATION OF ADDITIONAL MODEL PARAMETERS FOR TAILORED TEMPERING Since material flow in tailored tempering takes place in the same temperature regime compared to conventional hot-forming of 22MnB5, mechanical material characterisation as described in the Numisheet 2008 Benchmark 3 “Continuous press hardening” [8] is sufficient, and the corresponding material data preparation is performed as explained in [4]. No additional mechanical material data is required for simulation of tailored tempering. In order to determine the effect of latent heat and the effect of heat transfer to ambient, cooling curves of 22MnB5 sheets and austenitic stainless steel 1.4310 sheets subjected to air cooling were measured [9]. The latent heat was calculated as 75.8 kJ/kg based on the different cooling curves of 1.4310 and 22MnB5. Moreover, a strong temperature dependency of heat transfer to ambient was demonstrated in [9].
VALIDATION In co-operation with Daimler AG, an experimental setup was designed as shown in Figure 4. The designed tool consists of four modules, one pair cooled by cooling fluid and the other pair heated by heating cartridges. Austenised blanks at 950°C were transferred from the furnace to the tool and subjected to contact pressure during closure for controlled cooling. All tests were performed at the Lehrstuhl für Fertigungstechnologie (Prof. M. Merklein) of the University of Erlangen-Nuremberg in Germany. The nominal tool temperature was varied up to 550°C. Apart from tool temperature, contact pressure, transfer time, holding time and blank thickness were chosen as process parameters. During the experiments, the blank temperature was measured by a thermocouple with a diameter of 1.0 mm, which was connected to a small hole in the edge of the blank. In addition, the tool temperature was recorded simultaneously at four measuring points close to the surface of the contact plate. The measured tool surface temperatures are about 25°C higher than nominal values. All experiments were performed with a Schenck-Trebel RM400, which makes it possible to log the loaddisplacement path of the machine. Tensile tests were performed on the specimens prepared from the blank subjected to cooling experiments for the purpose of validation of resulting mechanical properties.
upper mounting plate
bores for water-cooling contact plate pressure pin
COOLED
HEATED
Experimental tool • partially cooled (15°C) • partially heated (variable heating up to 550°C) bores for heating • tests performed with Schenckdevices Trebel RM400 • tool material: 1.1730 positioning bolt
base plate
disc spin support
Measuring of • tool temperature at 4 points • specimen temperature at 1 point • load-displacement path of machine P1 P2
specimen
lower mounting plate
contact plate
FIGURE 4. Experimental tool setup for validation
Simulations of the experiments described before were performed with a pre-release version of AutoFormThermoSolverplus in order to validate the thermo-mechanical-metallurgical model. The material parameters used for the simulations are summarised in Table 1. Including a pressure-dependent heat transfer coefficient between the tool and the blank as described in [5] had only a minor effect on the results.
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Density [kg/m3] 7951.07
TABLE 1. Input data of 22MnB5 for the simulations Heat capacity Thermal conductivity Latent heat Chemical composition [J/(kg*K)] [W/(m*K)] [kJ/kg] [wt%] 550 32 75.8 C: 0.217; Mn: 1.162; B: 0.0029
Table 2 shows the heat transfer coefficients used for the simulations. For the tools, a constant temperature Ttool=Tnominal+25°C was prescribed. TABLE 2. Heat transfer data for the simulations Temperature HTC tool [°C] [mW/(mm2*K)] 1050 20 2100 250 2625 500 3150 750 3500 950
Pressure [MPa] 0 1 2 5 20
HTC amb [mW/(mm2*K)] 20 26.71 41.87 68.39 100
For two experiments with a nominal tool temperature of 400°C and 500°C, the measured and computed temperature curves are shown in Figure 5. A temperature rise due to latent heat is observed after removal, since bainite transformation is not completed at this time. The measured temperatures are described well by the simulation. On the other hand, if latent heat is neglected in the simulation, the temperature during air cooling is underestimated.
1000 Experiments
900
AutoForm^plus with Latent Heat AutoForm^plus without Latent Heat
Temperature [°C]
800
coole d
700
heate d
Nominal tool temp. 500° 500°C
600 500 400
Nominal tool temp. 400° 400°C 300
Transfer
Quenching
Removal
Air cooling
200 0
10
20
30
40
50
60
70
80
Time [s] FIGURE 5. Temperature predictions together with the experimental result for a nominal tool temperature of 400°C and 500°C
Figure 6 shows the computed and measured hardness results for a nominal tool temperature of 500°C. The hardness results are plotted over the distance from the intersection line between the hot and cooled tools. From the hardness measurements, the width of the transition region between hard and soft blank can be defined as 15-20 mm. The width of the transition region is captured well by the simulation results taking heat conduction into account. The tools are idealised to have a constant and homogeneous temperature. In reality, there is a temperature gradient between the tools. This can explain why the hardness between s=0 and s=5mm is not captured exactly by the
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simulation. However, the magnitude of the hardness distribution curve is calculated sufficiently in order to evaluate the dimension of the transition zone. 600
coole d
550
Measured line 1
heate d s
Measured line 2 Measured line 3
Hardness [HV10]
500
AutoForm^plus with Conductivity AutoForm^plus without Conductivity
450 400 350 300 250 200 -10
-5
0
5
10
15
20
25
30
Distance s [mm]
FIGURE 6. Hardness predictions with and without heat conduction in the blank together with the experimental results for a tool temperature of 500°C
Figure 7 shows computed and measured tensile strength results in a measurement series where the temperature of the heated tool was varied keeping all other process parameters constant. The tensile test specimens were taken from the soft zone of the blank, far away from the transition zone. 1550
coole d
Tensile strength [MPa]
1400
heate d
1250 1100 950 800
Experiments AutoForm^plus with Latent Heat
650 500 150
AutoForm^plus without Latent Heat
200
250
300
350
400
450
500
550
600
Tool temperature [°C]
FIGURE 7. Tensile strength predictions with and without latent heat effect together with the experimental results for different tool temperatures
Tensile strength as a product of phase transformation is predicted successfully by the simulations with latent heat for all temperatures. Neglecting the effect of latent heat leads to deviation from the experiments in the case of warmer tool temperatures, i.e. more than 350°C, where bainitic transformation dominates.
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TAILORED TEMPERING RESULTS OF A B-PILLAR Basic knowledge concerning the process window and resulting material properties as a function of the relevant process parameters was developed by tests using a planar and partially tempered tool. The simulation model of AutoForm-ThermoSolverplus was calibrated and validated by measurements from these tests described in the previous section. With the objective of transferring and applying the findings to the production of a real partially press-hardened body-in-white component and in order to review the quality of simulation results under these conditions, a tool for a B-pillar was manufactured by Daimler and tests were performed in Sindelfingen, Germany. The B-pillar comprises three zones of different material characteristics. For the upper area, a tensile strength of about 1500 MPa and a corresponding martensitic microstructure was targeted. The middle area of the part was to have a tensile strength of 850 MPa, and the lower section a tensile strength of 750 MPa combined with an elongation at fracture of about 15%. Different mechanical properties can be adjusted and controlled by the cooling rate and the tool temperature. The process parameters and the part geometry are shown in Figure 7. Using this configuration, a small batch of partially press-hardened parts was produced. In order to validate the simulation results with a focus on the mechanical properties, several A30 tensile specimens (Z1, Z2, Z3 and Z4) were extracted from different areas of the manufactured components.
Tool
Process parameters 500°C 450°C 25°C
Z1
Part 750 MPa 850 MPa 1500 MPa
Z2 Z4
Z3
material: oven temperature: oven time: tool temperature: transfer time: thickness: contact force: closure time:
22MnB5 + Zn 910 °C 320 s variable 9s 1.50 mm 8000 kN 25 s
FIGURE 8. Tailored tempering of a B-pillar with process parameters
The model implemented was applied to simulate the production of a partially hardened B-pillar as described before. The final material properties with respect to tensile strength and elongation at fracture obtained from simulation with and without latent heat effects and corresponding test are compared in Table 3. TABLE 3. Simulation with and without latent heat and test results of final material properties Measured (n=3)
Z1 Z2 Z3 Z4
759±1.4 838±10.0 1441±24.0 1501±6.4
Tensile strength [MPa] AutoFormAutoFormThermoSolverplus ThermoSolverplus with latent heat without latent heat 767 830 869 906 1430 1450 1430 1450
Measured (n=3)
13.0±0.2 10.1±1.2 7.4±0.5 8.0±0.1
Elongation at fracture A30 [%] AutoFormAutoFormThermoSolverplus ThermoSolverplus with latent heat without latent heat 14.1 11.8 10.3 9.0 6.3 6.3 6.3 6.3
By comparing measured and calculated data for tensile strength and elongation at fracture with and without consideration of latent heat, a statement can be established: The disregard of latent heat leads to raised strength values and in the same context to an elongation at fracture that is too low. Under this condition, the elevated specimen’s cooling rate can be named as the reason for this effect. Simulation with latent heat improves the quality of the calculated results significantly. Both tensile strength and elongation at fracture can be calculated quite accurately in AutoForm-ThermoSolverplus. Thus, the consideration of latent heat is indispensible for proper
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simulation of the considered tailored tempering process, where the bainite transformation is used to define the mechanical properties of the produced component.
CONCLUSION In order to capture the relevant phenomena in tailored tempering, a coupled thermo-mechanical-metallurgical description is employed in AutoForm-ThermoSolverplus. The implementation was validated via hybrid studies of experiments and simulations. The predicted final part properties of a B-pillar show satisfactory agreement with results from tensile tests. The additional CPU time for the combined thermo-mechanical-metallurgical model is about 5% for this example. Concerning industrial application, the further development of hot-forming processes is a very important step with regard to process and product quality. On the one hand, this development enables engineers to perform a profound simulation of tailored tempering. On the other hand, the presented feature contributes to a better understanding of conventional hot-forming processes, where the temperature history of produced parts and the related phase transformation as well as resulting mechanical properties are aspects of paramount importance for feasibility studies and for process design.
REFERENCES 1. Merklein, M.; Lechler, J.: Investigation of the thermo-mechanical properties of hot stamping steels. In: Journal of Materials Processing Technology 177 (2006), pp. 452–455 2. Feuser, P.; Schweiker, T.: Tailored Tempered Parts – Presshärtbauteile mit maßgeschneiderten Eigenschaften. In: Merklein, M.: Tagungsband zum 4. Erlanger Workshop Warmblechumformung, Erlangen, Germany (2009) 3. Karbasian, H.; Tekkaya, A.E.: A review on hot stamping. In: Journal of Materials Processing Technology 210 (2010), pp. 2103–2118 4. Sester, M.; Krasovskyy, A.; Kubli, W.: Material data for advanced yield surface and hardening models in AutoForm. In: Proceedings of International Deep Drawing Research Group (2009), pp. 207–218 5. Hoff, C.: Untersuchung der Prozesseinflussgrößen beim Presshärten des höchstfesten Vergütungsstahls 22MnB5. Dissertation, LFT, University of Erlangen-Nuremberg (2007) 6. Lechler, J.: Beschreibung und Modellierung des Werkstoffverhaltens von presshärtbaren Bor-Manganstählen. Dissertation, LFT, University of Erlangen-Nuremberg (2009) 7. Gesellschaft für metallurgische Systeme mbH (GMS) 8. Oberpriller, B.; Burkhardt, L.; Griesbach, B.: Benchmark 3 - Continuous Press Hardening. In: Proceedings of Numisheet (2008) 9. Ertürk, S.; Sester, M.; Selig, M.; Feuser, P.; Roll, K.: Simulation of Tailored Tempering Using AutoFormplus. In: Proceedings of International Conference on Hot Sheet Metal of High-Performance Steel (2011)
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Simulation of the Press Hardening Process and Prediction of the Final Mechanical Material Properties Bernd Hochholdingera, Pavel Horaa, Hannes Grassb and Arnulf Lippb a
Institute of Virtual Manufacturing, ETH Zurich, Tannenstrasse 3, 8092 Zurich, Switzerland b BMW Group, 80788 Munich, Germany
Abstract. Press hardening is a well-established production process in the automotive industry today. The actual trend of this process technology points towards the manufacturing of parts with tailored properties. Since the knowledge of the mechanical properties of a structural part after forming and quenching is essential for the evaluation of for example the crash performance, an accurate as possible virtual assessment of the production process is more than ever necessary. In order to achieve this, the definition of reliable input parameters and boundary conditions for the thermo-mechanically coupled simulation of the process steps is required. One of the most important input parameters, especially regarding the final properties of the quenched material, is the contact heat transfer coefficient (IHTC). The CHTC depends on the effective pressure or the gap distance between part and tool. The CHTC at different contact pressures and gap distances is determined through inverse parameter identification. Furthermore a simulation strategy for the subsequent steps of the press hardening process as well as adequate modeling approaches for part and tools are discussed. For the prediction of the yield curves of the material after press hardening a phenomenological model is presented. This model requires the knowledge of the microstructure within the part. By post processing the nodal temperature history with a CCT diagram the quantitative distribution of the phase fractions martensite, bainite, ferrite and pearlite after press hardening is determined. The model itself is based on a Hockett-Sherby approach with the Hockett-Sherby parameters being defined in function of the phase fractions and a characteristic cooling rate. Keywords: press hardening, thermo-mechanically coupled simulation, 22MnB5, final part properties PACS: 81.40.Gh, 81.40.Cd, 02.70.Bf, 81.40.-z
INTRODUCTION Today the press hardening process is well established for the manufacturing of body-in-white parts in the automotive industry. The number of parts in a new car generation that are manufactured by press hardening is still increasing. In general two variants of the press hardening process have to be distinguished. In the direct press hardening process (fig. 1a) the part is formed and quenched in one process step whereas in the indirect process (fig. 1b) the part is first classically cold-formed and afterwards heated and quenched in a separate operation.
(a) austenitization
forming and quenching
timming
treatment of surface
(b) cold-forming
trimming
austenitization
quenching
treatment of surface
FIGURE 1. Principal process steps for the direct press hardening process (a) and the indirect press hardening process (b) as proposed by the steel company voestalpine AG.
The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 618-625 (2011); doi: 10.1063/1.3623665 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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The steel that is typically used for press hardening is the boron alloyed steel 22MnB5. It is available from different steel suppliers that offer 22MnB5 with different coatings. Depending on the actual chemical composition of the coating, the supplied 22MnB5 is either suited for the direct or the indirect process variant. With the upcoming of press hardening in the automotive industry the focus lay on the production of structural parts with ultra-high strength. Recently the press hardening process is enhanced to produce parts with tailored properties. Still the generation of ultra-high strength properties is the principal objective of press hardening but at the same time it is desired to get areas within a part that have lower strength but increased ductility. There are several ways to achieve this. A standard approach to realize different mechanical properties within a part is to use tailored blanks by joining two or sheets made of different steel. Another way to increase the ductility is to anneal the part after quenching as for example discussed by Laumann [1]. Instead of using different materials or annealing the part, the hardening process itself can be modified to generate different mechanical properties. In the so called tailored tempering process, as for example discussed by Lenze et al. [2], the tools are partially heated. This results in a locally reduced cooling rate. Instead of a solely martensitic microstructure, as it is the case for cooling rates faster than approx. 30 K/s, a mixture of ferrite, pearlite, bainite and martensite forms that has a lower strength but higher ductility than pure martensite. Another way to locally reduce the cooling rate is to insert grooves in the tool surface. Depending on the size and depth of the groove the heat flux from the hot part to the cold tools at the location of the groove is orders of magnitude smaller than in areas where part and tool are in direct contact. By the insertion of grooves the tool surfaces are not continuous anymore and cannot be used for forming a part. Therefore this approach may only be used in the indirect process variant, where the part is formed in a separate operation before the hardening takes place. For the virtual assessment of the press hardening process, thermo-mechanically coupled simulations have to be conducted, which require a variety of input parameters. Depending on which process variant is followed – direct or indirect press hardening – the relevance of the input parameters regarding the process simulation varies. As shown by Burkhardt [3], for the simulation of hot forming as part of the direct press hardening process, the accurate definition of the yield stress depending on strain, strain rate and temperature as well as temperature dependent friction are of major importance for a reliable prediction of the feasibility of the forming step and the geometrical properties, like sheet thickness and draw-in, of the part after forming. An experimental procedure for the determination of the yield stress and a comparison of the applicability of different mathematical models is given in Eriksson [4] and Hochholdinger [5]. Regarding the prediction of the final mechanical properties after the direct or the indirect process, the accurate prediction of the local temperature history within the blank is of major importance. The temperature history depends on various boundary conditions like thermal convection and radiation to the environment, but mainly on the timedependent contact situation between blank and tool. Especially for the generation of tailored properties by the insertion of grooves in the tool surfaces, the correct modeling of the interfacial heat flux is mandatory. Therefore the focus of this work regarding the determination of input parameters is set to the determination of the contact heat transfer coefficient as described in the following chapter.
DETERMINATION OF THE CONTACT HEAT TRANSFER COEFFICIENT Depending on the actual contact situation during the hardening process the effective heat flux from the blank/part to the tools may vary by several orders of magnitude. The input parameter that determines the heat flux due to contact is the contact heat transfer coefficient (CHTC) h. In general the following two contact situations have to be distinguished: 1. Part and tool are in direct contact. The CHTC depends on the effective area in contact. Since the effort to model the effective contact area accurately with finite elements would be much too high, the CHTC is defined as a function of contact pressure h(p). It is obvious that for high contact pressures the effective contact area is larger than for small pressure values. 2. Part and tool are not in direct contact, but there is a small gap between the contact partners. Since part and tool are close to each other, there is heat transfer due to radiation and convection of the fluid in the gap. In this case CHTC is a function of the gap distance h(lgap). With the gap between part and blank getting larger, the heat flux due to the interaction between the contact partners is decreasing. For gaps with lgap>lgap,max the heat loss due to convection and radiation to the environment is dominant and the heat flux between the contact partners can be neglected.
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Experimental Setup In order to determine the CHTC for the contact situations described above, simple experiments at the Chair of Manufacturing Technology of the University of Erlangen-Nuremberg have been conducted. The principle experimental setup is shown in fig. 2a.
(a)
upper tool blank
(b)
lower tool
FIGURE 2. Experimental setup for determining the CHTC (a) and experimental and simulated temperature over time curves for an applied contact pressure of 30 MPa (b).
After heating up the blank it is quenched between upper and lower tool. In the closed configuration the closing force of the press can be adjusted such that the blank is clamped with a predefined pressure. In the experiments pressure values of 0, 5, 10, 20, 30 and 40 MPa have been used. Alternatively the blank can be placed on adjustable pins on the lower tool and the upper tool can be positioned at a specified distance away from the upper side of the blank. Like that the second contact configuration with a small gap between blank and tools is realized. In the experiments gap distances were set to 0.1, 0.3, 0.5, 1.0 and 2.0 mm. The temperature is measured through thermocouples in the middle of the blank thickness and in the upper and lower tool 1 mm underneath the tool surfaces. In order to ensure the same initial temperature of the tools when conducting a series of consecutive experiments, upper and lower tool can be water-cooled. Further details on the experimental setup can be found in Hoff [6].
Inverse Identification of the CHTC For the determination of the CHTC based on the measured temperature over time curves different approaches can be employed. A fairly simple, analytical approach is to use Newton’s law of cooling as e.g. done by Lechler [7]. When using Newton’s law of cooling it is assumed that the blank has a constant temperature over its thickness and that the tool temperature is constant over time. Especially for the test configurations with blank and tool being in direct contact, the measured tool temperatures showed that the assumption of constant tool temperatures is not appropriate (see fig. 2b). Instead of trying to employ an analytical formula, an inverse approach for the identification of the CHTC is chosen. For this purpose a simple simulation model of the experiment is set up, where the measured temperatures in the upper and lower tool are applied as boundary conditions at the nodes 1 mm underneath the tool surfaces. For each of the tested configurations with a specific pressure or gap distance applied, an optimization run was conducted. The objective of these optimization runs is defined as the mean square error (MSE) between simulated and measured blank temperature over time. In order to avoid a falsification of the results by the release of latent heat during the cooling process, only the temperature history prior to the occurrence of phase change was considered for the evaluation of the MSE. The effects of phase change could be included in the optimization process by specifying the latent heat and the phase change temperature as additional optimization variables. Since such an approach led to unstable results, solely the CHTC was defined as a variable within the optimization procedure. The dashed curve in fig. 2b shows the resulting blank temperature over time using the “optimal” CHTC for the configuration with applied contact pressure equal to 30 MPa. In fig. 3 the normalized values of the determined CHTCs over gap distance and pressure are shown.
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FIGURE 3. Normalized values of the (CHTC) over gap distance (left) and contact pressure (right).
SIMULATION OF THE PRESS HARDENING PROCESS Depending on which answers and results are expected from the simulation, the model setup as well as the numerical strategy for the virtual assessment might differ significantly. If just the feasibility of the forming phase during the direct press hardening process is of interest, it is in most cases sufficient to simulate gravity, closing and hot forming of the blank without consideration of heat conduction or cooling within the tools. As can be seen from the results submitted for Benchmark 3 of the Numisheet 2008 Conference (Oberpriller [8]) the sheet thickness and the position of local necking after hot forming can be fairly well predicted by the use of finite element models that are quite similar to the ones employed for standard cold-forming simulations. If on the other hand the objective of the simulation is the prediction of the mechanical properties after hardening, the process steps that have to be considered as well as the requirements regarding the FE model are more comprehensive. In the following two chapters these issues are discussed.
Simulation of the Process Steps during Press Hardening In order to be able to predict the properties of the final part correctly all process steps that influence the thermal history of the part, have to be considered in the simulation. Fig. 4 shows the process steps for the indirect press hardening process.
heating
transfer
gravity
closing
quenching
FIGURE 4. Simulated process steps during the indirect press hardening process.
All process steps are simulated using a weak coupling of the mechanical and the thermal problem. Like that it is possible to use different time integration schemes with different time step sizes for both problems. Table 1 shows the chosen settings for the thermo-mechanical simulation of the process steps. The starting point for the coupled simulation is the trimmed part after cold-forming. During heating from room temperature up to an austenitization temperature of approx. 900 °C the volume of the part increases due to thermal expansion. The correct definition of the temperature dependent coefficient of thermal expansion (CTE) is essential to get the right dimensions after heating. The FE code LS-DYNA® that is used for the simulations utilizes the tangent CTE in contrast to the secant CTE that is e.g. used by the FE program Abaqus®. After heating the part and holding it at a constant temperature for several minutes, it is assumed that residual stresses and the accumulated plastic strain that evolved during cold-forming are cancelled out. During transfer from the oven to the press the part
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cools due to radiation and convection to the surroundings. The temperature drop depends on the effective duration of transfer, usually 3-8 seconds, and the values provided for the convection heat transfer coefficient and the emissivity of the blank surface. Reasonable values for these input parameters are stated in Shapiro [9]. In the subsequent gravity simulation the part is put on the lower tool. In this step there is a heat flux on the lower side of the part due to local contact with the cold tool and on the upper side due to convection and radiation. For the closing step a separate simulation is setup, since a different time integration scheme is used for the solution of the mechanical problem (see table 1). The experience has shown that the use of an explicit time integration scheme is more robust in handling the changing contact situations than the dynamic implicit Newmark integration. During the quenching step the part is tightly clamped between upper and lower tool. As discussed above, the effective heat flux depends on the local pressure or on the gap distance at the part-to-tool interface. Either the explicit or the implicit time integration may be used. If the explicit approach is chosen the application of mass and time scaling is recommended to reduce the computation time. A detailed description of the recommended settings for the explicit time integration scheme can be found in Lorenz [10]. To reduce the cycle times in serial production, the duration that the tools are held closed should be as short as possible. Accordingly the tools are usually opened when the part is still well above room temperature. In order to get the correct geometry at room temperature, it might therefore be necessary to conduct an additional simulation of opening the tools and cooling down of the part. The more or less free thermal contraction during that cooling phase on air might lead to geometrical distortion of the part. TABLE 1. Applied time integration schemes for the thermo-mechanically coupled simulation of the separate process steps. Process Step Mechanical Problem Thermal Problem heating static implicit implicit transfer static implicit implicit gravity dynamic implicit implicit closing explicit implicit quenching explicit or dynamic implicit implicit
FE-Model for the Simulation Press Hardening In general there are several approaches to model blank/part and the tools in a forming simulation. In standard cold-forming simulations of deep drawing processes, the tools are usually modeled as rigid contact surfaces providing the reference geometry for the part to be formed. For the simulation of press hardening, as discussed in this paper, it is as well assumed that the tools do not deform and therefore can be considered as rigid for the mechanical part of the problem. For the thermal problem the assumption of tools having a constant temperature over time is in most cases not adequate. When just the hot forming phase of the direct process has to be evaluated then the error of “thermally rigid” tools is probably acceptable as shown by Lorenz [10]. However if the final mechanical properties of the part are to be predicted, which requires an accurate simulation of quenching and cooling, then the tools have to be modeled with continuum or at least boundary elements. Only if heat conduction within the tools is considered in the simulation, the time dependent distribution of the tool surface temperatures can be captured, which is a prerequisite for the correct evaluation of the heat flux. TABLE 2. Recommended modeling of tools and blank for the thermo-mechanically coupled simulation of press hardening. Process Step Mechanical Problem Thermal Problem modeling of the tools rigid: no stresses, no strains thermal tetra- or hexahedron modeling of the blank/part elasto-viscoplastic shell 12 node thick thermal shell
For the structural (mechanical) problem, the blank/part is modeled with the same fully integrated shell elements that are usually employed for the simulation of cold-forming. Of course the material model that is used for the simulation of press hardening has to be capable of considering the temperature dependency of the mechanical input parameters, like e.g. Youngs modulus E(T) and yield stress Vy(H, dH/dt, T). For the thermal part the blank is modeled with a 12 node thick thermal shell element (red nodes in fig. 5b) as proposed by Bergman and Oldenburg [11]. The shell element is defined by the 4 nodes of the mechanical shell (black nodes in fig. 5b). The nodes defining the upper and lower surface of the thick shell are generated internally by LS-DYNA. The shell uses linear shape functions in plane and quadratic ones in the thickness direction. This offers several advantages. Like that it is possible to apply and model different boundary conditions on the upper and lower side of the shell, as for example during the gravity step (see fig. 4), and to accurately capture the resulting temperature gradient through the thickness.
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(a)
(b)
FIGURE 5. Solid model of the tools (a) and thick thermal shell element used for modeling the blank (b).
PREDICTION OF THE FINAL MECHANICAL MATERIAL PROPERTIES For the prediction of the mechanical properties after press hardening a post processing procedure was developed that automatically evaluates the temperature history of all nodes in the quenched part over all process steps after heating. The CCT diagram shown in figure 6 is used to evaluate the phase fractions that result after hardening at the nodes of the blank. Fast cooling rates result in a purely martensitic microstructure with a yield stress of more than 1500 MPa and Vickers hardness above 480. For intermediate cooling rates the austenite changes to a mixture of martensite, bainite, ferrite and pearlite. Rather slow cooling rates result in a microstructure that consist of ferrite and pearlite which is quite similar to the unquenched state of the 22MnB5.
FIGURE 6. CCT diagram of 22MnB5 from Naderi [12] for an austenitization temperature of 900 °C.
For the prediction of the yield curves after hardening a simple model based on the well-known Hockett-Sherby yield curve approximation is developed.
V y H pl Ax , dT ( Ax , dT Bx , dT ) exp M x , dT H plN
(1)
Where Ax,dT is the saturation stress, Bx,dT is the initial yield stress, Mx,dT is the saturation rate and N the hardening exponent. As indicated by the subscripts, the Hockett-Sherby parameters A, B and M are defined as functions of the phase fractions x and a characteristic cooling rate dT/dt at 700 °C (973 Kelvin), while the same hardening exponent N, which is unequal to one, is used for all configurations, independently of the specific microstructure or cooling rate. As can be seen from equation (2) each of the parameters Ax,dT, Bx,dT and Mx,dT is defined as linear combination of the phase fractions martensite xmart, bainite xbain, ferrite and pealite xferr-perl and the natural logarithm of the cooling rate at 973 K. As inspiration for this approach served the mixture rule developed by Maynier et al. [13], which uses the chemical composition of the steel to predict the Vickers hardness. The coefficients ax, bx and mx in equation (2) are determined by multi-linear regression. For this purpose tension tests of specimens with different cooling rates have been conducted. The resulting stress-strain curves are fitted employing the mentioned Hockett-Sherby approximation. The phase fractions for each tension test specimen are evaluated with the CCT diagram shown in fig. 6.
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Ax , dT Bx , dT M x , dT
§ dT · amart xmart abain xbain a ferr perl x ferr x perl adT ln¨ 973 ¸ © dt ¹ § dT · bmart xmart bbain xbain b ferr perl x ferr x perl bdT ln¨ 973 ¸ © dt ¹
(2)
§ dT · mmart xmart mbain xbain m ferr perl x ferr x perl mdT ln¨ 973 ¸ © dt ¹
Based on this model the following post processing is developed, that allows predicting the yield curves in the part after press hardening: 1. Extract temperature history from all process steps starting with the transfer simulation until the end of the hardening process. 2. Evaluate the resulting microstructure with the CCT diagram shown in fig. 6. 3. Employ the developed model (eq. (2)) to calculate an individual yield curve for each finite element in the simulation model. 4. Sort elements with similar yield curves in different groups. For this purpose the plastic potential of each yield curve is calculated. Calculate representative yield curve for each group of elements. 5. Output of the post-processed part including the determined yield curves. Fig. 7 shows two small sample press hardening setups and the corresponding results using the described post processing procedure. Fig. 7(a) shows a standard hardening tool. In fig. 7(b) the part after hardening is shown. The different colors represent different yield curves that are shown in fig. 7(c). The figures 7(d) to (f) show tools, the part after quenching and normalized yield curves for a setup, where grooves have been inserted in the surfaces of upper and lower tool. At the location of the grooves the cooling rates are slower than in areas where the tools and the part are in full contact. The yield curves at the location of the grooves are accordingly lower than in areas with higher cooling rates.
(a)
(b)
(c)
(d)
(e)
(f)
FIGURE 7. Figure (a) shows the tools, (b) the hardened part with areas having different properties and (c) the corresponding normalized yield curves for standard press hardening setup. In figures (d), (e) and (f) the tools, the quenched part and the corresponding normalized yield curves for a setup, where grooves have been inserted in the tool surfaces are shown.
SUMMARY AND CONCLUSIONS The intention of this work was to discuss the most relevant issues that are necessary to predict the yield curves of a part after press hardening. The preliminary for an accurate simulation is the knowledge of the input parameters for the thermo-mechanically simulation. The basic idea of press hardening is to quench a sheet metal blank or part made of hardenable steel, as for example the boron alloyed steel 22MnB5, by bringing it into contact with cold tools. The simulation of this process requires the definition of a large variety of input parameters. When the objective of the simulation is to predict the final part properties the by far most important parameter is the contact heat transfer coefficient. An inverse approach was chosen to determine the pressure and gap dependent CHTC from the experimental temperature over time curves. Furthermore a strategy for the simulation of the subsequent steps that are
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involved in the press hardening process was presented. Since the mechanical and the thermal solution are just weakly coupled, different time integrations schemes for the mechanical and the thermal part may be used for solving the initial boundary value problem. In order to get the correct spatial and time-dependent temperature distribution the tools are modeled with volume elements. For the part a thick thermal shell is employed that allows capturing the temperature gradient over the sheet thickness and to apply different boundary conditions on top and bottom surface of the blank. For the prediction of the yield curves of the final part a phenomenological model was developed. This model requires the input of the microstructure in form of the phase fractions martensite, bainite, ferrite and pearlite. The phase fractions are determined by employing a CCT diagram from literature. The model itself is based on a Hockett-Sherby approach. The Hockett-Sherby parameters are defined in function of the phase fractions and a characteristic cooling rate. The evaluation and output is automatically done during post processing of the subsequent simulations. In order to determine the parameters for the presented model tension or compression tests of specimen that have been quenched with different cooling rates have to be conducted. As an alternative to the determination of the phase fractions by using the CCT diagram the material model developed by Åkerström and Oldenburg [14] could be used. This model employs transformation kinetics based on Kirkaldy’s rate equation to predict the phase fractions after quenching. Disadvantages of this model are the required input parameters, like the chemical composition, the austenite grain size and the activation energies of the diffusion driven transformation reactions, and the fact that the time-scaling cannot be applied to the simulation.
ACKNOWLEDGEMENTS The kind support of this work by the BMW Group is gratefully acknowledged. Also many thanks to Dr. A. Shapiro from LSTC for the implementation of many new features into the FE software LS-DYNA without which the virtual assessment of the press hardening process would not have been possible.
REFERENCES 1. T. Laumann, “Qualitative und quantitavie Bewertung der Crashtauglichkeit von höchstfesten Stählen”, Doctoral Thesis, , University of Erlangen-Nuremberg, 2009. 2. F.-J. Lenze, J. Banik, S. Sikora and T. Gerber, “Further development of manganese boron steels for the lightweight design of body in white structures”, IDDRG 2010 Conference Proceedings, edited by R. Kolleck, Institute of Tools and Forming, Graz University of Technology, 2010, pp. 17-26. 3. L. Burkhardt, “Eine Methodik zur virtuellen Beherrschung thermo-mechanischer Produktionsprozesse bei der Karosserieherstellung”, Doctoral Thesis, ETH Zurich, 2009. 4. M.C.F. Eriksson, “Modelling of Forming and Quenching of Ultra High Strength Steel Components for Vehicle Structures”, Doctoral Thesis, Luela University of Technology, 2002 5. B. Hochholdinger, H. Grass, A. Lipp, A. Wahlen and P. Hora, “Determination of Flow Curves by Stack Compression Tests and Inverse Analysis for the Simulation of Press Hardening”, Proceedings of the 7th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes, Part A, edited by P. Hora, Institute of Virtual Manufacturing, ETH Zurich, 2008, pp. 633-639. 6. C. Hoff, “Untersuchung der Prozesseinflussgrössen beim Presshärten des höchstfesten Vergütungsstahls 22MnB5”, Doctoral Thesis, University of Erlangen-Nuremberg, 2007. 7. J. Lechler, “Beschreibung und Modellierung des Werkstoffverhaltens von presshärtbaren Bor-Manganstählen”, Doctoral Thesis, University of Erlangen-Nuremberg, 2008. 8. B. Oberpriller, L. Burkhardt and B. Griesbach, “Benchmark 3 – Continuous Press Hardening – Part B: Benchmark Analysis”, Proceedings of the 7th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes, Part B, edited by P. Hora et al., Institute of Virtual Manufacturing, ETH Zurich, 2008, pp. 119-129. 9. A. Shapiro, “Finite Element Modeling of Hot Stamping” in steel research international, Volume 80, Issue 9, Düsseldorf: Verlag Stahleisen, 2009, pp. 658-664. 10. D. Lorenz, “Simulation of Thermo-Mechanical Forming Process with LS-DYNA”, Proceedings of the 9th LS-DYNA Forum, Bamberg, DYNAmore GmbH, 2010, pp. C-I-35-42. 11. G. Bergmann and M. Oldenburg, “A finite element model for thermo-mechanical analysis of sheet metal forming” in International Journal of Numer. Meth. Engng, Volume 59, Issue 9, 2004, pp. 1167-1186. 12. M. Naderi, “Hot Stamping of Ultra High Strength Steels”, Doctoral Thesis, RWTH Aachen University, 2007. 13. P. Maynier, B. Jungmann and J. Dollet, “Creusot-Loire system for the prediction of the mechanical properties of low alloy steel products” in Hardenability concepts with application to steels, edited by D.V. Doane and J.S. Kirkaldy, AIME, New York, 1978, pp. 518-544. 14. P. Åkerström and M. Oldenburg, “Austenite decomposition during press hardening of a boron steel – Computer simulation and test” in Journal of Materials Processing Technology, 174, 2006, pp. 399-406.
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The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 645-652 (2011); doi: 10.1063/1.3623668 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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REFERENCES . . . . . . . . .
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652
Prediction of Forming Limit Diagram for Seamed Tube Hydroforming Based on Thickness Gradient Criterion Xianfeng Chena, Zhongqin Lina,*, Zhongqi Yua, Xinping Chenb, Shuhui Lia a
Shanghai Key Laboratory of Digital Autobody Engineering, Shanghai Jiao Tong University, Shanghai 200240, China b Shanghai Baoshan Iron & Steel Co., Ltd, Shanghai 201900, China * Corresponding author. E-mail:
[email protected] Tel.: +86-21-34206304 Fax: +86-21-34204542 Abstract. This study establishes the forming limit diagram (FLD) for QSTE340 seamed tube hydroforming by finite element method (FEM) simulation. FLD is commonly obtained from experiment, theoretical calculation and FEM simulation. But for tube hydroforming, both of the experimental and theoretical means are restricted in the application due to the equipment costs and the lack of authoritative theoretical knowledge. In this paper, a novel approach of predicting forming limit using thickness gradient criterion (TGC) is presented for seamed tube hydroforming. Firstly, tube bulge tests and uniaxial tensile tests are performed to obtain the stress-strain curve for tube three parts. Then one FE model for a classical tube free hydroforming and another FE model for a novel experimental apparatus by applying the lateral compression force and the internal pressure are constructed. After that, the forming limit strain is calculated based on TGC in the FEM simulation. Good agreement between the simulation and experimental results is indicated. By combining the TGC and FEM, an alternative way of predicting forming limit with enough accuracy and convenience is provided. Keywords:Forming limit diagram (FLD); Seamed tube; Tube hydroforming; Thickness gradient criterion (TGC) PACS: 81.20.Hy
INTRODUCTION In sheet metal forming operations, the sheet can be deformed only to a certain limit that is usually imposed by the onset of localized necking, which eventually leads to fracture. Forming limit diagram (FLD) is a well-known method of describing this limit and predicting the occurrence of necking during sheet metal forming. So it has been widely used to compare the formability of different materials and as the failure criterion for predicting the limit strains in the simulation of sheet metal forming. Similarly, FLD is an important tool for assessing formability of tube hydroforming, which is commonly obtained from experiment, theoretical calculation and finite element method (FEM) simulation. However, the experimental mean is restricted in the application due to the equipment costs and the different experimental methods and subjective measurement approaches. Since 2000, Davis et al. [1] proposed a tooling and experimental apparatus to establish FLD for AA6061 tube based on the free-expansion tube hydroforming. The FLD based on tube free hydroforming is an effective method of proven and authoritative. But in order to obtain the right hand side of FLD, experimental apparatus are required highly and expensive to clamp the two ends of tube when the tube is subjected to the internal pressure. Theoretically, many researchers have developed various theoretical models as an alternative to fill up the deficiency of experiment. Varma et al. [2, 3] analyzed the localized necking in aluminum tubes free hydroforming using the M-K method along with an anisotropic version of the Gurson model. Hwang et al. [4] obtained the FLD via the Swift’s diffused necking criterion and Hill’s localized necking. Kim et al. [5] also used Oyane’s ductile fracture criterion to predict the initiation of necking under combining internal pressure and independent axial feeding. The FLD for tube hydroforming based on these theoretical models are the same as or different from the FLD for sheet metal forming.
The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 653-660 (2011); doi: 10.1063/1.3623669 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
653
With increasing application of computational techniques, numerical predictions of FLD have become more attractive and the FEM has been selected to simulate the Nakazima and Marciniak tests for sheet metal forming. In analyzing the simulation results for the onset of necking, an appropriate failure criterion is a key to numerical determination of FLD. One of the pioneers in this research area was Brun [6] who analysed the material's thinning in order to determine the onset of necking by simulating the Nakazima test. The method is applicable to the entire range of deformation states of the FLD. Basing on the same test, Geiger and Merklein [7] considered that the gradient of major strain changed rapidly when localized necking occurred. Using the limiting dome height (LDH) test, Narasimhan [8] has predicted the onset of necking by the thickness strain gradient across neighboring regions. A novel failure criterion-thickness gradient criterion (TGC) was proposed. The original idea of TGC is closely related to the hypothesis of thickness inhomogeneity in the M-K model. Some researchers put forward the TGC, which combining the M-K model and Storen-Rice analysis to predict the dome height with good accuracy. Zhang et al. [9] simulated the Marciniak test with the commercially available finite element (FE) program ABAQUS. Based on FEM results, the equivalent plastic strain increment ratio (CRIT3) criterion was chosen to analyze the onset of localized necking. Now numerical predictions of FLD have been widely used in sheet metal forming. However, no many numerical studies on the prediction of FLD for tube hydroforming have been presented. In the field of tube hydroforming, only Kim et al. [10] predicted forming limit for tube hydroforming by means of the FEM combined with Oyane’s ductile fracture criterion based on the Hill’s quadratic plastic potential. Based on the TGC, this work predicts the FLD of seamed tube hydroforming with the FEM simulation of tube hydroforming. The critical thickness gradient in the TGC is determined by the FEM simulation of tube bulge test under plane strain state. The numerical result is compared with the experimental data to show the validity and reliability of the means of the FEM combined with the TGC and the research method adopted.
EXPERIMENTS FOR STRESS-STRAIN CURVE OF SEAMED TUBE There are three test methods for obtaining the stress-strain curve of tube, including uniaxial tensile test, ring tensile test and tube bulge test. In order to perform pertinent numerical simulations of tube hydroforming process, it has been demonstrated that material data obtained from tube bulge test were more suitable than the classical tensile test. Fuchizawa et al. [11] have developed an analytical model for tube bulging. Experimental stress-strain curves obtained by tube bulging and uniaxial tensile tests have been compared and the pertinence of tube bulge test has been confirmed especially for tube manufactured by roll forming and welding. Sokolowski et al. [12] and Strano and Altan [13] also performed tube bulge test which is proposed by Fuchizawa. Koc et al. [14] investigated different method to conduct the tube bulge test depending whether the parameters required for the calculation of the stress– strain relationship (longitudinal and circumferential radius of curvature, bulge height, internal pressure and thickness at the top of the dome) are measured on-line or off-line. In this section, tube bulge tests with two ends fixed are performed to obtain the parameters, such as circumferential and longitudinal radius of curvature, bulge height, internal pressure and thickness at the top of the dome. Then the stress-strain curve of tube can be calculated with equations.
Description of the methodology Consider a tube with two ends fixed which is subject to an internal pressure Pi , as illustrated in Fig.1. For an element at the top of the tube dome, the following equilibrium equation can be written.
FIGURE 1. Stresses on an element at the top of the tube dome
V1 V 2 U1 U 2
Pi ti
654
(1)
As shown in Fig.1, Pi is the instantaneous internal pressure, t0 and ti are the initial and the instantaneous tube wall thickness, V 1 and V 2 are stress components in the circumferential and the axial directions, H 1 and H 2 are strain components in the circumferential and the axial directions, U1 and U 2 are the instantaneous circumferential and longitudinal radius, d 0 is the initial tube radius, and A1 , A2 and A3 are the cross-sectional areas of the inner circle at the top of tube dome, the cross-sectional area of the inner circle at the guiding zone and the cross-sectional area at the top point of bugling area. In the case both tube ends were fixed in axial and radial directions, while the central region of the tube will be free to expand. Considering the point at the top of the tube dome, the axial force can be written as: Fo Pi A1 Pi A2 (2) From the cross section geometry shown in Fig.1, one obtains
t 2
A1
S ( U 1 i ) 2
A2
S( A3
Combining Eqs. (2) – (5), one can write
V2
d0 t0 ) 2 2 2SU1ti
>
Pi (2 U1 ti ) 2 (d 0 2t0 ) 2 8U1ti
Fo A3
(3) (4) (5)
@
(6)
Additionally, Eq. (1) can be rewritten as
V1
(
Pi V 2 )U1 ti U 2
(7)
Assume that the tube material obeys the Swift hardening relationship:
V
K (H 0 H ) n
(8)
where K is the strength coefficient, V is the equivalent stress, H 0 is the initial equivalent strain, H is the equivalent strain and n is the strain hardening exponent. According to Hill’s yield criteria, the equivalent stress V and the equivalent strain H can be written as:
V2
H And
V 12
2r V 1V 2 V 22 1 r
(9)
1 r 2r H 12 H 1H 2 H 22 1 r 1 2r U1 H 1 ln
(10) (11)
(d 0 t0 ) / 2
H3
ln
ti t0
(12)
where H 3 is strain component in the radial direction. Assuming volume constancy, one can write:
H2
(H 1 H 3 )
(13)
Conducting several experimental bugle tests using different internal pressure, it is possible to obtain a set of ( V , H ) couples representing the experimental stress–strain relationship of the material during tube bulge test. These couples plotted in a V ,
H
diagram, can be fitted by means of the Swift’s law V
655
K (H 0 H ) n .
Bulge tests for base metal In this study, tube bugle tests were performed for stress-strain curve on the hydroforming machine designed in Auto Body Technology Center of Shanghai Jiaotong University. The maximum allowable working pressure of the machine is 200MPa and the maximum allowable axial force is 1000kN. Fig. 2 and Fig. 3 show the schematic of experimental apparatus and dies for tube bugle test.
FIGURE 2. Schematic of experimental apparatus for tube bugle test
FIGURE 3. Schematic of dies for tube bugle test
The tubes used in the experiments are straight seamed tubes with 58mm outside diameter, 2.5mm wall thickness and 245mm long. Additionally, circular grids with a diameter of 2.5mm are etched on the tube surface before the experiments. During bulge tests of seamed tubes, six levels of internal pressure are scheduled. The hydroformed parts at different internal pressure levels are shown in Fig.4.
FIGURE 4. Hydroformed parts at different internal pressure
After bugle tests, the corresponding U1 , U 2 , ti , H1 and H 2 for each level are measured and recorded. First, a micrometer is used to measure the bulged diameter of the tube at the pole for the instantaneous circumferential radius U1 . Then the thickness ti at a curved surface is measured with an ultrasonic thickness meter. A profile projector is used to measure the major strains e1 and minor strains e2 of the deformed grids on the tube surface. The necking limit strain is 0.199. Laser 3D scanner is used to measure the instantaneous longitudinal radius U 2 . The tube bugle tests results of ti , U1 and U 2 for different internal pressure levels Pi are listed in Table 1. TABLE 1. Experimental results of the bugle test for seamed tube
Pi /MPa
ti /mm
U1 /mm
43 45 48 50 52
2.485 2.324 2.285 2.191 2.012
28.351 39.321 30.319 30.852 32.351
U2
/mm
1825.3 910.65 481.89 387.65 167.42
Once the above experimental data are obtained, the flow stress and the H 0 , K and n values of the tubular materials can be determined numerically. Then the calculated stress–strain relation of base metal for seamed tube is fitted to the flow curve as V 680(0.040 H ) 0.145 .
Tensile tests for weld zone A seamed tube is composed of weld metal, heat affected zone (HAZ) and base metal, as illustrated in Fig.1. Only the material properties of base metal can be obtained from tube bugle test. Hence, the assumption of iso-strain in both weld metal and HAZ used by Saunders and Wagoner [15] is applied in this study. First, the determination of an appropriate method for measuring the width of weld metal and HAZ is crucial and thus micro-hardness profile is used in this work. Fig.5 depicts measuring positions along the circumference direction.
656
Micro-hardness profile of a seamed tube with 2.5mm thickness and 58 mm outer diameter is plotted in Fig.6. This profile indicates that weld metal and HAZ width is approximately 4 and 6 mm.
FIGURE 5. Measuring positions for micro-hardness test
FIGURE 6. Micro-hardness profile of a seamed tube
Then according to the widths, four weld-only specimens and four specimens of mixed metal containing weld metal and HAZ, which are cut from seamed tubes, are prepared. The two type schematic tensile test specimens are illustrated in Fig.7. The two calculated stress–strain relation for the HAZ and weld metal are fitted to the flow curve as V 699(0.023 H ) 0.126 and V 721(0.011 H ) 0.091 .
TGC BASED FLD PREDICTION The original idea of TGC is closely related to the hypothesis of thickness inhomogeneity in the M-K model. Based on many experimental observations of previous studies, the TGC can be expressed as follow: R RC (14) When the current thickness gradient R in the metal material drops below the critical thickness gradient RC , the localized necking is considered to take place. In this section, TGC is used to predict the FLD for seamed tube hydroforming. The critical thickness gradient in the TGC is determined by the FEM simulation of tube bulge test under plane strain state. After that, the FE models for the experimental tools and methods of our innovations in previous research are constructed. Then the forming limit strain is calculated based on TGC in the FEM simulation of tube hydroforming test. The commercial FEM code LS-DYNA is used as the solver. The Hill’s yield function is adopted in the simulation. The Swift hardening rule V K (H 0 H ) n is used to characterize the flow stress.
Determination of critical thickness gradient The simulation of tube bugle test is performed to determine the critical thickness gradient of the test tube under plane strain state. During the deformation, when the major strain reaches the necking limit strain 0.199, the localized necking is assumed to take place. The current thickness gradient value is recognized as the critical thickness gradient RC in formula (14). For the critical thickness gradient RC is independent of strain path, the TGC is established for predicting the localized necking in the next step simulation of tube bugle tests for FLD of seamed tube. Accuracy of the tube bulge test simulation is tested by comparing the simulated with the experiment via apparatus in Fig.2. The hydroformed part under plane strain state is shown in Fig.8 and Fig.9 shows the simulation result under plane strain state.
FIGURE 7. Tensile test specimens cut from seamed tubes
FIGURE 8. Hydroformed part under plane strain state
After the simulation, the length of element, the thickness variation, the major strain and minor strain can be obtained in the postprocessor. The critical thickness gradient RC is defined in the following procedures: when the major strain reaches the necking limit strain, the node A with the lowest thickness value and its neighboring node B
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along the direction of major stress are found out; then the ratio of thickness value of node A and B is divided by the length of them to gain the value of critical thickness gradient. The thickness gradient variation in the direction perpendicular to the neck which is attained in the simulation of tube bugle test is plotted in Fig.10. The thickness gradient value decreases then increases across the neck, achieves its minimum value in the necking area. And then the thickness gradient value between HAZ and weld metal decreases a little. It can be seen that the thickness gradient value between adjoining nodes (approximately 1.00mm apart) across the neck reaches a critical magnitude of 0.784, which is considered as the critical thickness gradient value for predicting FLD in the FEM simulation of tube hydroforming. The TGC can be further expressed as: R 0.784 (15)
FIGURE 9. Simulation result under plane strain state
FIGURE 10. Variation of the thickness gradient across the necking area
FE simulation for FLD prediction In order to numerically predict localized necking during tube hydroforming process of a seamed tube, the FEM simulation combined with TGC has been carried out. The FE model of tube hydroforming is set up in FE program LS-DYNA, which is composed of a rigid die-up, a die-low, two punches and a deformable seamed tube. The contact interaction is modeled using Coulomb’s law: friction coefficient for die-tube and die-punch are both 0.05. Different loading paths in the test represent different linear strain paths. For each simulation, the thickness gradient developing in the adjoining node is monitored. When the thickness gradient drops bellow RC , the localized necking is considered taking place. Then the corresponding limit major and minor under different strain paths are recorder and plotted as the forming limit curve of the seamed tube. For the left hand side of FLD, a classical tube free hydroforming tool set is used. During free expansion, the tube is subject to axial compressive force F and an internal pressure Pi . The principle schematic is shown in Fig.11 and the FE model is shown in Fig.12.
FIGURE 11. Principle schematic for the left hand side of FLD
FIGURE 12. FE model for predicting left hand side of FLD
The forming limit is determined by controlling the material flow along the linear strain paths. Based on Asnafi [16] constructing analytical models, the relationship for generating constant strain ratio paths between the internal pressure and the axial force is determined. Fig.13 shows the simulation result at one strain state.
FIGURE 13. Simulation result of E
0.16
FIGURE 14. Principle schematic of tension -tension strain states
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In previous researches, the left hand side of FLD for tube hydrforming could be easily obtained by free expansion. But for the right hand side of FLD, the difficulty is to clamp both ends of tube in free tube hydroforming under the tension-tension strain states. In our previous studies, a novel hydroforming tool set is designed for the right hand side of FLD for tube hydroforming. The principle is to control the material flow along tension-tension strain paths by applying the lateral compression force and the internal pressure. In order to generate different forming limit states, three type pressing blocks are used. The width w is 50mm, 75mm and 100mm. The principle schematic is shown in Fig.14 and Fig.15 shows the FE model. The linear strain paths in Fig.16 are determined by FEM simulation.
FIGURE 15. FE model for predicting right hand side of FLD
FIGURE 16. Loading paths for tension-tension strain states
RESULTS AND DISCUSSIONS Many experiments have been performed for FLD of tube hydroforming. The results of the products after free hydroforming with axial feeding and internal pressure for different strain ratio are shown in Fig.17. It is known that the bursting lines occur at the base material near to HAZ and the maximum bulge height increases with the absolute value of the strain ratio. Fig.18 displays the products with lateral compression force and internal pressure.
FIGURE 17. Hydroformed parts for E
FIGURE 18. Hydroformed parts for E
0 .2 ~ 0
0 .1 ~ 0 .4
A three dimensional image processing system is used to measure the engineering major strains e1 and minor strains e2 of the deformed grids on the tube surface. And the values of the true strain ( H 2 , H1 ) are transformed from ( e2 , e1 ) and used to construct the experimental forming limit points in Fig.21. In the FEM simulation of tube hydroforming test, the formula (15) is used as the failure criterion for predicting the forming limit. When the thickness gradient of each specimen drops below 0.784, the corresponding forming limit strain is recorded as the solid line plotted in Fig.19.
FIGURE 19. Comparison of the calculated FLD with experimental data
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Fig.19 shows a comparison between the calculated FLD and experimental data for a seamed tube. The calculated FLD matches well with the experimental data provided by seamed tube hydroforming. In the state of plane strain, the calculated FLD is superposed on the experimental point. This is owed to the TGC is established by tube bulge test at plane strain state. In other deformation states, the calculated FLD based on TGC matches well with the experimental data. The combination of TGC and FEM is shown to be a practical method for predicting forming limit diagram for seamed tube hydroforming with enough accuracy and convenience.
CONCLUSIONS Prediction of forming limit diagram in virtual tube hydroforming was performed by means of the FEM combined with the TGC. The following conclusions are obtained. (1) The critical thickness gradient value was successfully determined in the FEM simulation of tube bulge test under plane strain state to establish the TGC, which is used as the failure criterion in the next step simulation. (2) The FLD of QSTE340 seamed tube was calculated based on TGC in the simulation of tube hydroforming test. The comparative evaluations have shown an excellent correlation between numerically and experimentally obtained FLD, which demonstrate the accuracy and validity of the TGC and the research approach adopted. (3) By combining the TGC and FEM, the FLD of seamed tube can be calculated with enough accuracy and convenience when its three parts properties are available by tube bulge test and uniaxial tensile test.
ACKNOWLEDGMENTS The authors gratefully acknowledge the financial support of National Natural Science Foundation of China (Grant No. 50605043) and Baosteel Group the supply of test materials.
REFERENCES 1. R. Davies, G. Grant, D. Herling, M. Smith, B. Evert, S. Nykerk, and J. Shoup, “Formability investigation of aluminum extrusions under hydroforming conditions”, SAE Technical Paper, 2000, 2000–01–2675. 2. N. S. P. Varma, R. Narasimhan, A. A. Luo, and A. K. Sachdev, “An analysis of localized necking in aluminium alloy tubes during hydroforming using a continuum damage model”, International Journal of Mechanical Sciences, 2007, 49, pp. 200– 209. 3. N. S. P. Varma and R. Narasimhan, “A numerical study of the effect of loading conditions on tubular hydroforming”. Journal of Materials Processing Technology, 2008, 196, pp. 174–183. 4. Y. M. Hwang, Y. K. Lin, and H. C. Chuang, “Forming limit diagrams of tubular materials by bulge tests”, Journal of Materials Processing Technology, 2009, 209, pp. 5024-5034. 5. J. Kim, S. W. Kim, W. J. Song, and B. S. Kang, “Analytical and numerical approach to prediction to prediction of forming limit in tube hydroforming”, International Journal of Mechanical Sciences, 2005, 47, pp. 1023–1037. 6. R. Brun, A. Chambard, M. Lai, and P. DeLuca, “Actual and virtual testing techniques for a numerical definition of materials”, NUMISHEET’1999, Besancon, France, 1999, pp. 393-398. 7. M. Geiger and M. Merklein, “Determination of forming limit Diagrams-a new analysis method for characterization of materials’ formability”, Annals of the CIRP, 2003, 52, pp. 213–216. 8. K. Narasimhan, “A novel criterion for predicting forming limit strains”, In: Proceedings of NUMIFORM 2004, Ohio, USA, 2004, pp. 850–855. 9. C. S. Zhang, L. Leotoing, D. Guines, and E. Ragneau, “Theoretical and numerical study of strain rate influence on AA5083 formability”, Journal of Materials Processing Technology, 2009, 209, pp. 3849-3858. 10. J. Kim, Y. W. Kim, B. S. Kang, and S. M. Hwang, “Finite element analysis for bursting failure prediction in bulge forming of a seamed tube”, Finite Elements in Analysis and Design, 2004, 40, pp. 953–966. 11. S. Fuchizawa And M. Narazaki, “Bulge test for determining stress-strain characteristics of thin tubes”, Advanced Technology of Plasticity, 1993, 1, pp. 488–493. 12. T. Sokolowski, K. Gerke, M. Ahmetoglu, and T. Altan, “Evaluation of tube formability and materials characteristics: hydraulic bulge testing of tubes”, Journal of Materials Processing Technology, 2000, 98, pp. 34–40. 13. M. Strano and T. Altan, “An inverse energy approach to determine the flow stress of tubular materials for hydroforming applications”, Journal of Materials Processing Technology, 2004, 146, pp. 92–96. 14. M. Koc, Y. Aue-u-lan, and T. Altan, “On the characteristics of tubular materials for hydroforming-experimentation and analysis”, International Journal of Machine Tools & Manufacture, 2001, 41, pp. 761–772. 15. F. I. Saunders and R. H. Wagoner, “Forming of tailor welded blanks”, Metallurgical and Materials Transactions A, 1996, 27, pp. 2605–2616. 16. N. Asnafi, “Analytical modeling of tube hydroforming”, Thin-Walled Structures, 1999, 34, pp. 295–330.
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Finite Element Analysis of Tube Hydroforming in NonSymmetrical Dies Abhishek V. Nulkara, Randy Gub and Pilaka Murtyc a
EASi Engineering, 1551 E. Lincoln Ave, Madison Hts, MI 48071, USA Department of Mechanical Engineering, Oakland University, Rochester, MI 48306, USA c Department of Mathematics, Physical Science, and Engineering, West Texas A&M University, Canyon, TX, 79016 b
Abstract. Tube hydroforming has been studied intensively using commercial finite element programs. A great deal of the investigations dealt with models with symmetric cross-sections. It is known that additional constraints due to symmetry may be imposed on the model so that it is properly supported. For a non-symmetric model, these constraints become invalid and the model does not have sufficient support resulting in a singular finite element system. Majority of commercial codes have a limited capability in solving models with insufficient supports. Recently, new algorithms using penalty variable and air-like contact element (ALCE) have been developed to solve positive semi-definite finite element systems such as those in contact mechanics. In this study the ALCE algorithm is first validated by comparing its result against a commercial code using a symmetric model in which a circular tube is formed to polygonal dies with symmetric shapes. Then, the study investigates the accuracy and efficiency of using ALCE in analyzing hydroforming of tubes with various cross-sections in non-symmetrical dies in 2-D finite element settings. Keywords: FEA, positive-semi-definite, singularity, hydroforming, non-symmetry, contact.
PACS: 02.70.Dh, 46.55.+d
INTRODUCTION Since the advent of hydroforming in 1950 many research articles have been published on tube hydroforming (THF). The study by Dohmann and Hartl [1] shows hydroforming of tubes with straight axis and pre-bent tubes. The article also includes discussion of failure cases like buckling, bursting, wrinkling and folding back when tubes with a straight longitudinal axis are expanded. In a comprehensive study, Koc and Altan [2] present an overall review of tube hydroforming technology. Hwang and Altan [3] discuss and compare hydraulic expansion alone and crushing process combined with preforming and hydroforming to a rectangular die. Through finite element modeling the article shows that crushing combined with hydroforming can yield better results. Koc and Altan [4] present a study in which finite element simulation of two-dimensional tube hydroforming is given. Kim and Kang [5] use a backward tracing technique and finite element method to determine the initial tube dimensions and intermediate configuration for preform design of square and rectangular dies. Rama et al. [6] develop a new two-dimensional numerical method based on membrane analogy to tube-sheet deformation with the concept of finite elements for sheet discretization. The efficiency of the method is verified using commercially available FEA packages. Keum and Kim [7] present a study showing the finite element simulation of axisymmetric tube hydroforming. Abrantes et al. [8] make an effort to establishing a basic understanding of the THF processing of aluminum and copper tubes by giving finite element simulations of THF. Some experimental results of THF of T-parts are also included in the article. Hwang and Chen [9] propose a mathematical model considering sliding friction between tube and die. The model explores the plastic deformation behavior of the tubes in a square cross-sectional die. FE simulations are given and experiments are carried out for comparison. As it appears, finite element method has become an important tool in THF for assisting in tool design, predicting the final shape, determining manufacturing parameters, and so on. In this study, we will first disclose the difficult challenge encountered in finite element simulations of THF in a two-dimensional setting. A novel contact element whose material behavior is analogous to the air is proposed to address the difficult task. The stiffness matrix is formulated based on the material model. The effects of the air-like contact element are discussed. Two numerical examples involving symmetric dies are solved using the proposed algorithm and the commercial code ANSYS [10]. The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 661-668 (2011); doi: 10.1063/1.3623670 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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The accuracy of the proposed algorithm is compared against the ANSYS code. Finally, two numerical simulations of tube hydroformed to a triangular and pentagonal dies using the proposed algorithm are presented.
PROBLEM STATEMENT Many applications of THF [3~9] reported involve models featuring symmetric characteristics. With symmetric settings, only a portion of the geometry is needed in finite element modeling. Additional boundary conditions are imposed on nodes on the planes of symmetry. As a result of the additional symmetric boundary conditions, the model becomes well-posed even if the original full model is not. The first two models presented later are such examples. For THF, the cross-sections of the final tube are not necessarily symmetrical. In this event, the symmetric boundary conditions are no longer valid. Consequently, the finite element models do not have sufficient supports or are not constrained at all. As illustrated in FIGURE 1 a deformable object does not have sufficient support at the beginning of the static analysis. The finite element system for such a problem often suffers positive semi-definiteness as a result of singular total stiffness matrix. In this study, a methodology is proposed to tackle the challenging problem.
FIGURE 1. A deformable object with insufficient support
MATERIAL MODEL FOR AIR-LIKE CONTACT ELEMENT The air-like contact element (ALCE) closely resembles the physical behavior of the air or ideal gas, thus it is given the name. First we will look at the physics law of an ideal gas and how it evolves into a material model for ALCE.
Law of Air / Ideal Gas It is well known that the air or ideal gas is governed by the physics law below under a constant temperature. pV c , (1) where p is the pressure, V the volume of the gas, and c a constant. FIGURE 2 shows the plot of the physics law in solid line which is a hyperbolic curve. Note that the pressure and volume are both positive numbers in Eqn. (1). If the curve is first translated to the left and then reflected to the negative half plane, it becomes the base law for ALCE. The transformed curve can be described below. p(V V0 ) c , (2) where V0 is the original volume.
FIGURE 2. Physics law of an ideal gas and its transformed versions.
Material Description for ALCE In solid mechanics, the pressure is considered a negative bulk stress Vb, while the bulk strain Hb is 'V/V0, and 'V is the change in volume. Therefore the transformed curve is in analogy to the following. V b (1 H b ) c , (3) which has the two asymptotes given below.
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V b 0; H b 1 . (4) The two asymptotes have the slopes of 0 and f, respectively. This material law does not serve the purpose of solving the current problem. Now the asymptotes are modified slightly as below. V b E x (1 H b ); V b EnH b , (5) where Ex and En are, respectively, a large (maximum) and a small (minimum) numbers. Note that the two lines intersects at a point very close to (Hb, Vb) = (-1, 0). Thus, we propose the following material law for ALCE through the bulk stress Vb and bulk strain Hb. (6) V b EbH b . Here, the bulk modulus Eb is defined in terms of Ex, En in the following way. En , if b t 1 e ° 3 (7) Eb ®( E x E n )(2[ 3[ 2 ) E x , if 1 e ! b t 1 , ° E , if 1 x b ¯ where e is a very small number and, Hb 1 (8) [ , 0 d [ d 1. e
The graphical representation of Eqns. (6) and (7) is given in FIGURE 3. It is seen that for easy numerical treatment, the sharp intersection of the two asymptotic lines given in Eqn. (5) is smoothened by the cubic polynomial as shown in Eqn. (7). In addition to the material law given in Eqns. (6) and (7) which governs the mechanical behavior of the ALCE, it is also assumed that the ALCE is inactive to the shear stress.
FIGURE 3. Material description for ALCE. Solid line is for Eqn (6) while the dashed line is for Eqn (7).
FINITE ELEMENT FORMULATION FOR ALCE According to the material description given above, the strain energy U of ALCE occupying a volume V is defined as follows. U
³ ³V
b dH b dV
.
(9)
V
In the following development, only triangular ALCE is considered. The finite element matrix for such an ALCE is given below.
Stiffness Matrix for ALCE Let the space between the object and the rigid barrier be filled with ACLE as seen in FIGURE 4. It is readily seen that when the area of the triangular element is decreased to smaller than e, the two objects start engaging in contact and the material behavior of ALCE enters the nonlinear zone indicated as BDG in FIGURE 3. We will now formulate the stiffness matrix of this nonlinear elastic contact element.
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FIGURE 4. Area coordinates for ALCE.
The coordinates xi, i = 1, 2, of a generic point within the ALCE can be found as follows. xi N ij X j ,
(10)
where indicial summation convention is implied, Xj, j = 1, 2, …, 6, are the components of the element nodal coordinate vector, and Nij is the component of the shape-function matrix defined as 0 N2 0 N3 0 º ªN . (11) N ij « 1 0 N 0 N 2 0 N 3 »¼ 1 ¬
> @
The shape functions Ni are in fact the area coordinates [13]. 1 Ai Ni ai bi x1 ci x2 . A 2A where, referring to FIGURE 4, A A1 A2 A3
(12)
a1
X3X6 X5X 4,
a2
X 5 X 2 X1 X 6 ,
a3
X1 X 4 X 3 X 2 ,
b1
X 4 X6,
b2
X6 X 2,
b3
X 2 X 4,
.
(13)
c1 X 5 X 3 , c2 X 1 X 5 , c3 X 3 X 1. Note that, as shown in FIGURE 4, the area of an ALCE, A, is positive, when the nodes are defined in the counter-clockwise direction. The shape function matrix given in Eqn. (10) is also employed for interpolating the displacements in the element. Therefore the bulk strain Hb becomes, (14) H b u i ,i N ij ,iU j ,
where ui is the component of the displacement vector at a point within the element, and Uj is the component of the element nodal displacement vector. Thus the exact strain-displacement matrix is given below. 1 (15) N ij ,i ¬b1 c1 b2 c2 b3 c3 ¼ . 2A It is seen that the bulk strain Hb is constant in the element. Recall that ALCE does not support shear stresses. Hence, the exact stiffness matrix of ALCE, when Hb t -1+e, may be formed in the usual fashion and is given below. ªb12 b1c1 b1b2 b1c2 b1b3 b1c3 º « » c12 c1b2 c1c2 c1b3 c1c3 » « tEn « b22 b2 c2 b2b3 b2c3 » (16) « N ij ,i ». 4A « c22 c2b3 c2c3 » « b32 b3c3 »» symm « «¬ c32 »¼
> @
> @
Eqn. (16) represents the stiffness matrix of ALCE at any state provided En is replaced by the bulk modulus Eb given in Eqn. (7). Note that when A = 0 stemming from a contact situation, a small value of A is used, instead. We now examine the numerical characteristics of the ALCE.
Effects of ALCE Let 3 be the discretized total potential energy of the elastic object and ALCE in FIGURE 4. Thus, 3 * 3 tAS A , where A is the area all ALCE occupy, t the thickness of the plane elastic object, and
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(17)
1 (18) K ijU iU j RiU i , 2 which represents a PSD quadratic functional of the elastic object due to insufficient supports. In addition, SA is the strain energy intensity of ALCE which in essence is the area under the Vb-curve in FIGURE 3. For example, when Hb d -1, SA is equal to the sum of the areas of triangle OAB, ABDC and trapezoid CDGF shown in FIGURE 3. Refer to [12] for more details. Following the usual manner, we may form the total stiffness matrix of the entire system with ALCE included. It is seen that when there is no contact, Hb t -1+e, SA has such an insignificant value that 3* | 3. However, it provides a small disturbance to the singular Kij so that the finite element system is no longer PSD. This effect is known as regularization in many ill-posed problems [14, 15]. On the other hand when the elastic object engages in contact with the rigid barrier, SA registers a very large value as indicated by the area of ABDC and CDGF in FIGURE 3. This is in analogy to the penalty function seen in common optimization problems [16]. Therefore, the current ALCE contains both the regularization and penalizing effects for the constrained minimization problem with a PSD finite element system. 3
NUMERICAL SIMULATIONS The effectiveness and accuracy of ALCE are demonstrated using a few tube hydroforming examples in the following section. For comparison purpose, two symmetric dies are included first which are solved by both ANSYS and the current ALCE algorithm. Two non-symmetric dies are solved using ALCE algorithm are given afterwards.
Symmetric Dies Using ALCE and ANSYS The detailed drawing for the first symmetric die and tube is given in FIGURE 5(a), which shows a square die with rounded corners. The outer diameter of the tube is intentionally chosen at 80 mm, slightly smaller than the square so that there is no contact at the beginning of the analysis. To ensure symmetry, both the die and the tube share the same center. FIGURE 5(b) depicts the finite element mesh for calculating the contact between the tube and the die surface using the ALCE. Due to symmetry, nodes on the Y-axis are constrained in the X-direction while nodes on the X-axis are constrained so that they are not allowed to move in the Y-direction. Nodes for the die are fixed against movement in both directions. The model contains 485 nodes and 120 8-noded elements for the tube, 15 nodes and 14 line elements for the die surface and 190 ALCE elements. Note that there are only a few ALCE shown in FIGURE 5(b); in actual calculation there are a lot more ALCE covering the entire die surface. Plane strain model and frictionless contact are assumed throughout the numerical simulations. For comparison, a finite element model is created for ANSYS. The model is shown in FIGURE 5(c) which features 845 nodes and 212 quadratic elements for the steel tube. Note that the block for the die is also meshed in the model.
(c) (b) (a) FIGURE 5. (a) A square die. Unit: mm; (b) a quarter finite element model for the proposed algorithm using ALCE; and (c) finite element model for ANSYS. TABLE 1 Material constants of steel tube and ALCE. Constant Steel ALCE E ( Ex / En ) 200 GPa (1.0×1010 Pa / 1.0×10-2 Pa) 0.3 Q ET 10 GPa 100 MPa VY e 1.0×10-8
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The material properties for the steel tube and ALCE are given in TABLE 1, where E is the modulus of elasticity for the steel tube, Q Poisson’s ratio, ET tangential modulus and VY yield stress. The rate-independent elastoplastic model of bilinear isotropic hardening is assumed for the steel. Refer to [9, 12] for the sensitivity of Ex and En on the numerical accuracy. FIGURE 6 shows the computed distance traveled by point Q, as labeled in FIGURE 5(a), during forming. Note that the maximum distance point Q can travel is 15.26 mm which is indicated by the top horizontal portion of the curves. It is seen that point Q reaches the die surface when the calculated hydro-pressure is ph = 308 MPa using ALCE and 320 MPa using ANSYS. The maximum difference in the distance traveled between the two predictions is 0.73 mm when the hydro-pressure is near 150 MPa. The deformed shapes of the tube as calculated by ANSYS and ALCE are given in FIGURE 5(b). It is seen that the tube has more thinning on the flattened area as predicted by the algorithm using ALCE than by ANSYS. Other values of parameters for ALCE are also tested which provide similar results. Note also that the CPU times used by ANSYS and the algorithm with ALCE are, respectively, 142 seconds and 59 seconds.
(b) (a) FIGURE 6. (a) Distance traveled of point Q, r = 40 mm, T = 45q, as labeled in FIGURE 5 (a); (b) Deformed shapes of the tube at ph = 160 MPa predicted by ANSYS (shaded mesh) and ALCE (clear mesh).
FIGURE 7(a) shows the detailed drawing for the second symmetric die and tube. As indicated, the tube is in contact with the lower surface of the die. The gap size between the tube and the vertical die surfaces on both sides is set equal to achieve symmetry. FIGURE 7(b) depicts the half model finite element mesh with only a few ALCE shown for calculation using the ALCE. Note that there is no contact between the tube and the die surface in the finite element model. As usual, nodes for the die are fixed against movement in both directions, while nodes for the tube on the Y-axis are constrained in the X-direction. As a result, the model has a rigid-body-motion model in the Ydirection. The model contains 303 nodes and 60 8-noded elements for the tube, 26 nodes and 25 line elements for the die surface and 300 ALCE elements. For comparison, a finite element model is created for ANSYS. The model is shown in FIGURE 7(c) which features 2,937 nodes and 770 quadratic elements for the steel tube. Note that the block for the die is also meshed in the model. In order to achieve a converged solution, there are a couple treatments applied on the model. In addition to the usual boundary conditions, the finite element mesh for the tube is initially in contact with the lower die surface. Further, the two nodes in contact are “coupled” together so the two have the same Y-displacement. Consequently, there is no rigid-body-motion mode for the model.
(a)
(b)
(c) FIGURE 7. (a) A hexagonal die with rounded corners with radius 10 mm. Unit: mm; (b) a half finite element model for the proposed algorithm using ALCE; and (c) the finite element model for ANSYS.
The calculated deformed shapes of the tube are depicted in FIGURE 8. It is seen from FIGURE 8(a) that the tube makes contact with the three surfaces at ph = 18.5 MPa as predicted by ANSYS and at ph = 19.6 MPa by the
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ALCE algorithm. The deformed shapes of the tube when the hydro-pressure is 100 MPa are displayed in FIGURE 8(b). The results indicated that the ALCE algorithm exhibits stiffer behavior for the tube, which may be attributed to the much fewer elements used for the tube (only one-tenth the number of elements used in the ANSYS model). Finally, the hydro-pressure for the tube to reach the complete die is ph = 244 MPa as predicted by ANSYS as opposed to ph = 268 MPa by the ALCE algorithm.
(b) (a) FIGURE 8. Deformed shapes of the tube predicted by ANSYS (shaded mesh) and by ALCE (clear mesh) (a) when the tube contacts three surfaces at ph = 18.5 MPa (by ANSYS) and ph = 19.6 MPa (by ALCE); (b) at ph = 100 MPa.
Non-Symmetric Dies Using ALCE In this section, two numerical examples of tube hydroforming with non-symmetric dies are presented. Due to the limited capability of the ANSYS code in producing converged solution involving insufficient supporting conditions, only numerical predictions from the ALCE algorithm are presented. FIGURE 9(a) shows the detailed drawing for the first non-symmetric die and tube. FIGURE 9 (b) depicts the full model finite element mesh without ALCE. Note that there is no contact between the tube and the die surface anywhere in the finite element model. As usual, fixed boundary constraints are defined for nodes for the die, while there is no constraint imposed on the tube. The model contains 600 nodes and 120 8-noded elements for the tube, 27 nodes and 27 line elements for the die surface and 647 ALCE elements (not shown). FIGURE 9 (c) shows the deformed tube at various hydro-pressures as calculated by the ALCE algorithm. The algorithm fails to produce a converged solution corresponding to the final form of the die. Apparently there is not sufficient material for the tube to be stretched to the final form of the die. Numerical simulation for two-stage forming using ALCE algorithm has not been attempted.
(a)
(c) (b) FIGURE 9. (a) A triangular die with rounded corners. Unit: mm; (b) a full finite element model for the proposed algorithm using ALCE; and (c) the calculated deformed shape for the tube using ALCE algorithm.
The second numerical example involves a pentagonal die with rounded corners as shown in FIGURE 10(a). The geometry is not symmetric about any lines. A full finite element model is shown in FIGURE 10(b) which has 600 nodes and 120 8-noded elements for the tube, 49 nodes and 49 line elements for the die surface and 939 ALCE elements (not shown). All the nodes for the die surface are constrained from any movement, while the tube is not constrained at all. FIGURE 10(c) displays the deformed shapes of the tube predicted by ALCE algorithm at two hydro-pressures. It is seen that the tube reaches the corners with larger radius earlier while leaving a large gap to the corner with smaller radius. According to the ALCE algorithm, the final form of the tube can be achieved when the hydro-pressure is 194 MPa.
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(b) (c) (a) FIGURE 10. (a) A pentagonal die with rounded corners; (b) a full finite element model for the proposed algorithm using ALCE; and (c) the calculated deformed shape for the tube using ALCE algorithm.
CONCLUSIONS There are some tube hydroforming applications that involve non-symmetrical dies. When such dies are used, a tube would be “floating” inside the die without any supports. Finite element system of such a problem is often positive semi-definite and could not be solved by many commercial codes. A numerical algorithm employing a novel airlike contact element is presented. The nonlinear element has both regularization and penalty effects to tackle the constrained minimization problems. The algorithm is first examined using symmetric models that can be solved using ANSYS. It is found that the results predicted by the proposed algorithm are compatible with ANSYS. The proposed algorithm is further tested using two THF examples involving unsymmetrical dies. The predicted deformed shapes of the tube at various pressures are presented. It is found that the proposed algorithm is numerically efficient in handling positive semi-definite finite element systems seen in THF.
REFERENCES 1. 2. 3. 4. 5. 6. 7.
8. 9. 10. 11. 12. 13. 14. 15. 16.
F. Dohmann, Ch. Hartl, Tube hydroforming – research and practical applications, 1997, J. Mat. Proc. Technol. 71 (1997) 174- 186. M. Koc, T. Altan, An overall review of the tube hydroforming (THF) technology, J. Mater. Process. Technol. 108 (2001) 384–393. Y-M. Hwang, T. Altan, Finite element analysis of tube hydroforming processes in a rectangular die, Finite Elements in Analysis and Design 39, 1071-1082 (2002). M. Koc, T. Altan, Application of two dimensional (2D) FEA for the tube hydroforming process, Int. Journal of Machine Tools & Manufacture, 42 (2002) 1285-1295. J. Kim, B.S. Kang, Implementation of backward tracing scheme of the FEM for design of initial tubular blank in hydroforming, 2002, J. Mat. Proc. Tech. 125-126 (2002) 839-848. S.C. Rama, K. Ma, L.M. Smith, J.M. Zhang, A two-dimensional approach for simulation of hydroforming expansion of tubular cross-sections without axial feed, J. Mater. Process. Technol. 141, 420-430 (2003). Y.T. Keum, Y.S. Kim, Finite Element Simulation of Axisymmetrical Tube Hydroforming Processes, Materials Science Forum, v 437-438, pp. 387-390, 2003, Advanced Materials Processing II; Proceedings of the 2nd International Conference on Advanced Materials Processing. J.P. Abrantes, A. Szabo-Ponce, G.F. Batalha, Experimental and numerical simulation of tube hydroforming, 2005, J. Mat. Proc. Tech. 164–165 (2005) 1140–1147. Y-M. Hwang, W-C. Chen, Analysis of tube hydroforming in a square cross-sectional die, Int. Journal of Plasticity, 21 (2005) 1815-1833. ANSYS Workbench User’s Guide, Revision v12.1, ANSYS Inc., Canonsburg, PA. R. J. Gu, P. Murty, and Q. Zheng, Use of penalty variable in finite element analysis of contacting objects with insufficient supports, Computers & Structures 80 (31), 2449-2459 (2002). P. Murty, Finite Element Analysis of Positive Semi-Definite Systems, Ph.D. dissertation, Oakland University, 2001. R. D. Cook, D. S. Malkus, M. E. Plesha, R. J. Witt, Concepts and Applications of Finite Element Analysis, 4th Ed., Wiley, 2001. A.N. Tikhonov, and A.V. Arsenin, Solutions of Ill-Posed Problems, Winston, Washington, D.C., 1977. J. V. Beck, B. Blackwell, and C. R. St. Clair, Jr., Inverse Heat Conduction—Ill-Posed Problems, Wiley-Interscience, New York, 1985. Reklaitis GV, Ravindran A, and Ragsdell KM, Engineering Optimization: Methods and Applications, New York: John Wiley, 1983.
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& $%9*2+*=*2+8 2++8$ , &=PL ?=Q7J=' #$ $ % $ $"V q @ . ˆ p (T ) (4) 0 where q q (D ) represents the expansion of the yield locus due to isotropic hardening where hardening strain like variable. We select a saturation type dependence which reads q KD ª¬V f V 0 º¼ ª¬1 exp > GD @º¼ .
is the isotropic (5)
with K , V and G denote material parameters. For the anisotropic yield function YLD2000-2D, the equivalent stress V is defined as a degree one homogeneous function of T as follows f
1/ a
a a a · §1 X 1c X 2c 2 X 2cc X 1cc 2 X 1cc X 2cc ¸ . ¨ ©2 ¹ where X ic and X icc , i 1, 2,3, respectively denote the principal values of tensors Xc
V
(6)
Lc x T and Xcc
Lc and Lcc are defined in terms of material parameters D i , i 1,...,8 as proposed in [4].
751
Lcc x T where
ˆ p and w D Normality postulate and plastic work identity gives the following evolutionary equations for D t ˆ p Jw ˆ ) ˆ , q ), w D J, ˆ p (T D t
T
(7)
where J denotes the plastic multiplier. In solution of local integration algorithms in a strain driven nature of FEM, it is typical to assume that for a typical ˆ , E p ,D and the solution at tn 1 is sought for time step 't tn 1 tn the solution at tn is known as T n n n
^Tˆ
n 1
,E
p n 1
^
`
`
, D n 1 in the current corotational setting. We follow an elastic predictor plastic/damage corrector type
operator split, where the elastic prediction is carried out assuming the trial values, where the trial stress and yield surface expansion respectively read, ˆ trial Ce : ªE E p , trial º , ½ T n 1 n 1 ¬ n 1 ¼ ° (8) ¾, f trial 0 trial ª º ª º ª º qntrial K D V V GD 1 exp . 1 n 1 n 1 ¼ ¼ ° ¬ ¼¬ ¬ ¿ Within the time step, the elastic or plastic character of the status is checked by inserting the trial stresses into the ˆ trial , q trial . Once ) ˆ p , trial d 0 is satisfied, an elastic state at t is defined and the trial ˆ p , trial ) ˆp T yield function, )
n 1
n 1
n 1
n 1
n1
values does not require any correction. Otherwise, plastic flow is due. During flow, together with an implicit backward Euler integration and utilizing 'J n 1 : 'tJn 1 , the return mapping in the residual form becomes, with a time dependent form, ˆ p ,½ R np1 Enp1 E np 'J n 1w Tˆ ) n 1 ° n 1 (9) ¾, ) p ˆ ˆ Rn 1 'J n 1 ) (Tn 1 , qn 1 ). °¿ We recollect this nonlinear equation set in R n1 , where the solution is sought for the unknown vector y n1 using the Newton method ˆ °½ °R p ½° ° T R n 1 ® )n 1 ¾ and y n 1 ® n 1 ¾ . (10) ¯° Rn 1 ¿° ¯°'J n 1 ¿° Using the iteration index as
k , the solution algorithm reads, A (nk)1 x 'y (nk)1 R (nk)1 , ½° ¾, y (nk11) y (nk)1 T ( k ) 'y (nk)1 .°¿
(11)
which will run until R (nk)1 d TOL . T ( k ) 0,1@ is a suitably selected line-search parameter. Where the explicit definition of the matrix case.
A which is derived following the linearization is given as follows for the rate independent
1
ª Ce , 1 'J ( k ) w 2ˆ ˆ ) ˆ p ,( k ) ˆ p ,( k ) º w Tˆ ) n 1 Tn1
Tn1 n 1 n 1 n 1 (k ) « » . (12) A n 1 ,( ) p k p(k ) » « ˆ ˆ w ) w ) n 1 n 1 ¼ 'J n1 » Tˆ n1 ¬« This ends the Backward Euler return mapping scheme for the plastic correction which falls into a general closest point projection type algorithm, see e.g. [16]. This framework has been implemented as a VUMAT subroutine for ABAQUS-Explicit.
PROPOSED SPECIMEN AND FINITE ELEMENT MODEL The needed inhomogeneous deformation field was realized with a specimen having a varying cross-section as shown in Figure 1.a. The specimen can simply be loaded under tension with a tensile test machine. The geometry was modeled in ABAQUS-Explicit using 720 fully integrated shell elements (Element Type S4 in ABAQUS) as presented in the figure. The symmetry of the problem enables the modeling of one half of the geometry. The maximum tool displacement is given as 4 mm as in the experiments. After this point, the occurring strains go beyond the limits of the uniaxial tension test and therefore the flow curve of material should be extrapolated. This is prevented in this study.
752
FIGURE 1. (a) The geometry of the specimen and the finite element model of the problem. (b) Specimen with stochastic pattern on it. The strain state of the specimen can be analyzed with the strain diagram shown in Figure 2. The diagrams present the major and minor strain of the elements predicted by the simulations for the material AA6016 in rolling direction at the maximum tool displacement of 4 mm.
FIGURE 2. Strain diagram generated from the finite element simulations of AA6016 for isotropic and anisotropic cases When the problem is simulated with von Mises yield condition for an isotropic case, it is seen that the deformation is between uniaxial tension and plane strain tension states. There are elements directly on the uniaxial tension line which are located on the symmetry line of specimen. The deformation state changes significantly when the material is simulated with YLD2000-2D with the correct material parameters of AA6016. The points are shifted in the direction of plane strain tension state. This is because of the Lankford`s coefficients (r-values) of the material which are all lower than 1.0. This means that in uniaxial tension case the plastic strain in the thickness direction is larger than the strains in transverse direction. This effect in uniaxial tension test can also be observed in this specimen geometry. As compared to the isotropic case, anisotropic aluminum does not flow much in the width direction and therefore the observed deformation shifts to the plane strain tension. These diagrams point out two issues about the specimen. The specimen can reflect the plastic behavior of anisotropic materials with its strain distribution on the surface. Secondly, at any instance of the test there exist material points with different strain hardening, meaning that they lie on different yield loci. These affects can be detected with an optical strain measurement system using a stochastic pattern on the specimen (Figure 1.b) and this cloud of information will be used to characterize the initial anisotropy of the material.
753
STRUCTURE OF INVERSE PARAMETER IDENTIFICATION For inverse parameter identification it is crucial to compare the experimental and numerical results in an automated manner. However, the discretization of the optical measurement system is different than the one used in finite element simulations, meaning that the location, dimension and the number of elements are different. Therefore a mapping algorithm is needed to obtain the strains at the exact locations of the experimental measurement points. This is performed in two steps. Firstly, the enclosing finite element of each experimental measurement point is found according to the procedure described by Jayadevan and Narasimhan [17]. In this step the local coordinates of the experimental point is also determined in its surrounding element. Secondly, the nodal values of the finite element are interpolated to the desired coordinates of the point with the following interpolation function:
u
1 1 1 1 1 [ 1 K u1 1 [ 1 K u2 1 [ 1 K u3 1 [ 1 K u4 4 4 4 4
(13)
where ȟ and Ș are the local coordinates and the ui’s are the nodal values of the major and minor strains. This procedure is applied at definite tool displacement increments. In this case the 4 mm tool displacement is divided into 8 intervals and for each 0.5 mm of tool displacement the strains from the simulations are obtained. The tool force values at these increments are also recorded from the simulations and experiments. With the known tool forces and strain values at the same material points for each tool displacement increment, the objective function can now be formed. In conventional manner, the YLD2000-2D parameters are defined with the uniaxial tension tests in 0°, 45°, 90° and equi-biaxial tests. It is shown in the previous section that the deformation state of the proposed specimen lies between uniaxial tension and plane strain tension states, meaning that the region around the equi-biaxial state is not covered. Therefore, the information of the biaxial stress state should be included explicitly in the objective function. The objective function should include the major and minor strain differences, tool force differences and equi-biaxial point differences between the simulations and experiments. As these values are totally different in their order of magnitudes the values are normalized according to the initial contribution of the strains to the objective function, with all the parameters of the YLD2000-2D model set to 1.0 for the isotropic case which is the initial guess. With the formed objective function, the problem is non-linear least-square type and the function is minimized using the Levenberg-Marquardt algorithm [18, 19]. With the introduction of a scalar controlling the magnitude and direction of the iterations, the algorithm calculates a search direction that is between the Gauss-Newton direction and steepest descent direction and therefore eliminates the disadvantages of the Gauss-Newton method especially in the proximity of the optimum solution. In order to automate the optimization procedure a MATLAB code is written which starts the ABAQUS simulations, reads the outputs by calling a Python script, compares the results with the experimental measurements, builds and minimizes the objective function.
CHARACTERIZATION OF INITIAL ANISOTROPY In order to check the results of the inverse scheme, firstly the conventional characterization of AA6016 with 1 mm thickness was performed with uniaxial tension tests and a layer compression test. The results of these tests together with the calculated YLD2000-2D parameters can be seen in Table 1. ı0 112.5 Į1 0.979
TABLE 1. Material parameters of AA6016 (All stresses in MPa) ı45 ı90 ıb r0 r45 r90 rb ı0 107.4 110.0 115 0.85 0.48 0.77 1.0 112.5 Į2 Į3 Į4 Į5 Į6 Į7 Į8 K 0.998 0.885 1.008 1.001 0.965 0.953 1.242 308.9
ı 223 į 20.14
The main concern in this study is the applicability of the method to identify the initial anisotropy of the material. Therefore it is assumed that the flow curve of the material is known and solely the parameters of the yield condition are varied throughout the iterations. This assumption allows also to construct a relation between these parameters and one parameter turns out to be dependent to others. Therefore instead of 8 Į’s, only 7 of them are varied. The yield function exponent a was taken to be equal to 8 as recommended for the FCC materials. Two strategies are followed in the characterization procedure as analyzed in [20]. In the first one, only the specimen in 45° orientation is used and in the second one, the specimens in the 0° and 90° orientations are used simultaneously. In the second
754
case, two simulations are run one after another and the objective function is formed in the same way as before but this time containing the differences of both of the cases. For both strategies the initial guess is the isotropic case, where all the material parameters are equal to 1.0.
FIGURE 3. Comparison of analytically and inversely obtained yield loci The obtained yield loci with the two cases can be seen in Figure 3. The white lines represent the analytically fit YLD2000-2D yield loci with the parameters listed in Table 1. The difference in the absolute values of the yield surfaces is less than 5%. In the second and third quadrants of the stress space there is a one to one match between the analytical and inversely obtained results. The only remarkable dissimilarity is observed in the plane strain regions where the slopes are different. Therefore the predicted plane strain tension points on the yield loci are different. When the results of the both strategies are compared, there is not a significant difference. The both cases can predict the initial anisotropy of material. It should be noted that this methodology does not take the r-values or the yield stress of the material at different orientations as input. Nevertheless the yield stress in 90° or the slopes of the yield locus at the axes, which eventually reflect the r-values, are predicted correctly.
FIGURE 4. Experimental and numerical tool force – displacement diagrams for three orientations The force-displacement curves of the experiments and simulations can be seen in Figure 4. The case with the 45° specimen predicts the forces with a maximum error of 10%. The deviations are even lower with the case of specimens in 0° and 90°. Therefore not only the strains, equi-biaxial point and the r-values but also the predicted tool forces are in good agreement with the experiments.
755
EXPERIMENTAL VERIFICATION In order to verify the obtained Yld2000-2D parameters a scaled car hood geometry is used (Figure 5.a). This industry-oriented geometry is interesting for verification purposes since the material in the side walls flow mainly through the drawbeads, where a complex deformation pattern is observed. AA6016 with 1 mm thickness from the same charge as in the characterization section was used in the experiments. The yield locus of the material with respect to two different yield conditions can be seen in Figure 5.b. The main difference between the two yield loci occurs in the vicinity of the equi-biaxial stress state. Since Hill´48 model does not take this stress state into account, the predictions are unsatisfactory. The strains on the surface of the car hood were measured with an optical measurement system and the measurements are compared with the numerical results along the two sections in Figure 6. The section 1 goes across the workpiece diagonally and the section 2 is along the half height of the rear side wall.
FIGURE 5. (a) Scaled car hood geometry and the analyzed sections (b) Yield locus of AA6016 according to Hill´48 and Yld2000-2D For comparison the results with Hill´48 material model are also included, since this model is still widely in use. Since the verification geometry is large in dimensions, the strains are measured stepwise and afterwards the measurements are combined. The gaps in the experimental results in Figure 6 are due to the gaps between these subprojects. The Hill´48 model apparently oversimplifies the plastic behavior of the AA6016 material. Especially in the corner regions of the geometry, the predicted thicknesses with Hill´48 are 12% lower than the measured thicknesses. As a general tendency the results with the Yld2000-2D model are in good agreement with the experiments. In the regions where moderate deformations occur, the both models predict similar thickness distributions with Yld20002D having slightly larger thickness values.
FIGURE 6. The comparison of the simulation results and the experimental measurements for the two selected sections.
756
CONCLUSION The presented work analyses the applicability of the utilization of strain distributions in the identification of the initial anisotropy. It is showed that in addition to the conventionally used integral measures like force and displacement, utilization of an optical measurement system provides the needed information for an inverse parameter identification of yield functions. If the material properties change significantly along different orientations, it is shown that multiple experiments in different directions can also be used simultaneously, of course leading to longer computation times. Another crucial issue is the flexibility of the used yield condition. Therefore the YLD2000-2D model is implemented in this study. If the same study were performed with Hill´48 for instance, the obtained solution would not be the same, since this model oversimplifies the problem and therefore it is too robust to satisfy changing yield stresses and R-Values along different directions. Since the presented framework can function with all yield loci without limitation, more flexible yield functions with more material parameters should also be tested. The assumption of negligible kinematic hardening is also an important aspect of the study. For this reason the current method is not suitable for materials that show a pronounced anisotropic hardening behavior. In that case the obtained solution would be the best compromise between the initial yield locus and subsequent yield loci. The proposed method should therefore be further developed to include combined isotropic-kinematic hardening models.
ACKNOWLEDGMENTS The authors would like to thank the German Research Foundation (DFG) for their financial support of the work reported here, which is acquired within the scope of the research project PAK250. The support of the second author has been provided by DFG under Contract No. TR 73. The authors would also like to thank Dr.-Ing. Stefan Keller and Hydro Aluminum Rolled Products GmbH for supplying the car-hood toolings.
REFERENCES R. Hill, Proc. Roy. Soc. A 193, 281-297 (1948). R. Hill, J. Mech. Phys. Solids 38, 405-417 (1990). F. Barlat, D. J. Lege, J. C. Brem, Int. J. Plast. 7, 693-712 (1991). F. Barlat, J. C. Brem, J. W. Yoon, K. Chung, R. E. Dick, D. J. Lege, F. Pourboghrat, S. H. Choi, E. Chu, Int. J. Plast. 19, 1297-1319 (2003). 5. D. Banabic, T. Kuwabara, T. Balan, D. S. Comsa, D. Julean, Int. J. Mech. Sci. 45, 797-811 (2003). 6. D. Banabic, H. Aretz, D. S. Comsa, L. Paraianu, Int. J. Plast. 21, 493-512 (2005). 7. D. Banabic, D. S. Comsa, M. Sester, M. Selig, W. Kubli, K. Mattiasson, M. Sifvant, Proceedings of the 7th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes (Numisheet 2008), edited by P. Hora, Institute of Virtual Manufacturing, ETH Zurich, 2008, pp. 37-42. 8. H. Vegter, A. H. van den Boogaard, Int. J. Plast. 22, 557-580 (2006). 9. T. Kuwabara, Int. J. Plast. 23, 385-419 (2007). 10. K. Kavanagh, R. Clough, Int. J. Solids Struct. 7, 11-23 (1971). 11. J. Unger, M. Stiemer, A. Brosius, B. Svendsen, H. Blum, M. Kleiner, Int. J. Solids. Struct. 45, 442-459 (2008). 12. M. Kleiner, A. Brosius, CIRP Annals. Manuf. Technol. 55, 267-270 (2006). 13. J. C. Gelin, O. Ghouati, Annals. CIRP 44, 189-192 (1995). 14. J. C. Gelin, O. Ghouati, J. Mater. Process. Tech. 45, 435-440 (1994). 15. J. Kajberg, G. Lindkvist, Int. J. Solids Struct. 41, 3439-3459 (2004). 16. J.C. Simo, T.J.R Hughes, Computational Inelasticity, Springer (1998). 17. K. R. Jayadevan, R. Narasimhan, Comput. Struct. 57, 915-927 (1995). 18. K. Levenberg, Q. Appl. Math. 2, 164–168 (1944). 19. D. Marquardt, SIAM J. Appl. Math. 11, 431–441 (1963). 20. A. Güner, Q. Yin, C. Soyarslan, A. Brosius, A. E. Tekkaya, Int. J. Mater. Form., (online first) DOI 10.1007/s12289-010-1009-4
1. 2. 3. 4.
(2010).
757
Fracture of 1045 steel under complex loading history Yuanli Bai (
[email protected]) Department of Mechanical, Materials and Aerospace Engineering University of Central Florida, Orlando, FL 32816, USA
Abstract. In the paper, a 3D fracture locus of 1045 steel under proportional loading conditions was calibrated using round, plane strain, and tubular specimens. The fracture locus includes both the stress triaxiality and Lode angle dependence. As an extension to the conventional linear damage evolution assumption, a new form of ductile fracture model considering the loading history effect was proposed, in which two weighting functions were introduced to the damage indicator calculation. One considers the non-linear damage evolution under proportional loading, the other accounts for the effect of change in loading directions. A round of comprehensive fracture tests on 1045 steel was conducted to validate the proposed fracture model. These tests include monotonic loading tests for 3D fracture locus calibration and other tests with complex loading histories, for example two-stage-tension test, compression-tension test and torsion-tension test.
1 1.1
Introduction Ductile fractures under complex loading conditions
The effect of loading history on metal sheets necking has been recognized for many years [1-3]. The strain-based forming limit diagram (FLD) is not unique under non-proportional loading conditions. The effect of loading history on FLD was also studied by many researchers [3-7]. However, the effect of loading history on fracture is still not well noted by the community of fracture mechanics. Both the physically-based fracture models (for example Tvergaard and Needleman [8]) and empirical fracture models [9-11] are usually calibrated and validated by tests under monotonic loading conditions. A fracture model calibrated only from monotonic loading usually fails to predict fracture under complex loading conditions. For example, it was pointed by Johnson and Cook [10] that the total damage is always less than unity at fracture (up to 40%) when the material is subjected to torsion followed by tension. Recently, the effect of strain reversal on fracture was studied for 2024-T351 aluminum alloy [12-14], which is a common phenomenon in mechanics. Consider for example three-point bending of a solid section beam. Initially, there are positive strains on the tensile side and negative strains on the compressive side. As the crack propagates towards the center of the beam, it clearly passes through a region that was initially under compression. This interesting phenomenon has been described by Bao and Wierzbicki [12, 13], who also introduced a modified form of damage function to account for the effect of pre-compression. Another practical example in which the strain rate ratio changes sign is an axially compressed prismatic tube [14]. The objective of the present paper is to study the loading history effect on fracture through a round of comprehensive experimental study. Firstly, the 3D fracture locus of 1045 steel is calibrated under monotonic loading conditions. Secondly, tests with complex loading conditions are designed and conducted. These tests includes two-stage-tension test, compression-tension test, and torsion-tension test. A new form of ductile fracture model is proposed and validated based on the experimental results.
1.2
Characterization of the stress state
For isotropic materials, 1 , 2 and 3 denote three principal stresses. It is shown that the stress state can be uniquely described by two parameters, stress triaxiality and normalized Lode angle parameter [15], which are defined as follows. (1) = m. 6 2 (2) = 1 = 1 arccos .
The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 758-764 (2011); doi: 10.1063/1.3623682 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
758
where m is the mean stress, is the equivalent stress, is defined as normalized third deviatroic stress 3
invariant, = J 3 . Here J 3 = s1s2 s3 is the third deviatoric stress invariant. The parameter can be further
related to the Lode angle through = cos(3 ) . The range of is 1 1 . All loading conditions can be characterized by the above defined set of parameters ( , ) . In this paper, the ductile fracture model will be reformulated in such a coordinate system.
2 2.1
Constitutive modeling A plasticity model considering combined hardening rule
In order to simulate a test under complex loading paths, an isotropic hardening rule is not enough, so the kinematic hardening of the material has to be introduced. According to the work of Armstrong and Frederick and Chaboche [16, 17], the evolution of the back stress is given by 2 (3) d [ Sback ] = C[d p ] [ Sback ]d p , 3 where C and are material constants and Sback is the back stress tensor. The isotropic hardening takes the form of the asymmetric plasticity model proposed by the author [15], which assumes that a isotropic yield surface is controlled by both hydrostatic pressure and Lode angle parameter. The following isotropic yield surface was proposed, m 1 (4) yld = f p ,[ ] [ Sback ] = iso p 1 c ( 0 ) cs (cax cs ) , m 1 where and cax are two parameters defined by
=
cos( / 6) 1 1 = 6.4641sec( / 6) 1 , 1 cos( / 6) cos( / 6)
(5)
for 0 for 0
c t cax = c c
(6)
The range of is 0 1 , in which = 0 is corresponding to plane strain or generalized shear condition, and = 1 is corresponding to axial symmetry. The parameter m is a non-negative integer. There are four t
s
c
material constants, ct , cs , cc and m , need to be calibrated. The values of c , c , and c are relative, and at least one of them is equal to unity. This depends on which type of reference test is used to calibrate the isotropic strain hardening function iso ( p ) . The isotropic hardening rule is a specific definition used by Chaboche and Lee [17, 18].
iso
p
=
st 1 e b
p
,
(7)
where is the first yield stress, and the stabilized value is st . Here the material is assumed to follow a combined hardening rule, so Eq. (7) will be used. It is shown in Ref. [19] that the material of 1045 steel has neither pressure dependence nor Lode angle dependence on plasticity, so the following parameters of plasticity are assumed: c = 0.0 , = 0.72 ,
cs = ct = cc = 1.0 , and
m = 6.
In this particular case, the proposed plasticity model reduces to a combined
isotropic/kinematic hardening model with von Mises type of flow. Khan and Jackson[20] proposed a method of finding parameters of both isotropic hardening and kinematic hardening of materials. Here, a new type of compression-tension test on notched round specimens with optical measurement was developed to calibration these two types of hardening [21]. Aslo, finite element model using 4-node axisymmetric elements (CAX4R) is built in ABAQUS to simulate the notched round bar subjected to compression-tension loading. Since the experimental data of hardening are only up to p = 0.152 that is far below the maximum equivalent plastic strains occurring in the tests, the hardening parameters are adjusted around the experimentally calibrated values at each iterate run. Finally, a very good correlation of the force-displacement curves between experiments and numerical simulation is achieved, as shown in Fig. 1. The finally calibrated parameters of the plasticity model is listed in Table. 1. A comparison of two hardening curves between the initial calibration and final adjustment is shown in Fig. 2. This group of parameters will be used for all the remaining tests. 759
Fig. 1: Comparison of force-displacement curves under compression-tension loading.
Table 1:
Fig. 2: Measured isotropic hardening and kinematic hardening of 1045 steel
Calibrated parameters of the plasticity model of 1045 steel
st
b
C
754 MPa
500 MPa
0.4
3500 MPa
c
0.3
cs
ct
c
m
0.0
-0.72
1.0
1.0
1.0
6
E 2.2e+5MPa
2.2
c
23
A new form of ductile fracture model considering the loading history effect
A new parameter is introduced to describe the stress state of non-proportionality,
S back = Sback , Sback
(8)
which denotes the difference between the directions of current stress and back stress. It can be proven that the range of is 0 2 . A new parameter , defined by p
p
0
0
10 d p = 10
Sback d p ,
(9)
is introduced to measure the accumulated amount of non-proportionality during the loading process, For convenience, a coefficient 10 is used to make the range of coefficient vary around unity. It is shown that the parameter can distinguish different stages of a non-proportional loading process [21]. If the two stress state parameters( and ) change under a loading path or the parameter increases away from zero, then the loading is called non-proportional loading. Because the fracture locus ˆ f ( , ) is defined and calibrated only under proportional loading conditions, additionally designed experiments with non-proportional loadings are necessary to calibrate and verify the loading history effect on fracture initiation. The proposed ductile fracture model can be written in an incremental form (Eq. (10)), which is easy for the implementation in the material subroutine. It is assumed that material points will fail when D = Dc = 1 . c d p k (10) dD = cg D c g 1 ch D 1 2 . ˆ f , e g 1 Two weighting functions were introduced in Eq. (10) to the calculation of damage indicator ( D ). One function using parameter cg considers the non-linear damage evolution under proportional loading, the other function,
using another set of parameters ( ch , 1 , 2 , k ), accounts for the effect of change in the loading directions. In this paper, the 3D fracture locus ˆ f , is defined as [15]
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1 ˆ() ˆ() f f ˆ (f ) 2 12 ˆ (f ) ˆ (f ) ˆ (f) 2 1 D D D D D D 1 D1e 2 D5e 6 D3e 4 2 D1e 2 D5e 6 D33e 4 . 2 2
ˆ f , =
(11)
d testss with both mo onotonic and non-proportioonal loading conditions c In next sectioon, carefully designed will be ussed to calibratte and verify the t newly propposed ductile fracture modeel.
3
Experim mental and d simulatio n results
3.1 3D frracture locu us calibratiion for 1045 steel A round of fracture f tests consisting off seven types of specimens was perform med to calibraate the 3D fracture locus of 1045 steel. A co ollection of thhese tests is shown in Fig g. 3, which iincludes smoo oth round specimenn and notchedd round specim men in tensionn, flat specimeens with different radii of ggrooves in ten nsion, tube specimenn in tension annd tube specim men in torsionn. Details abou ut test data and FE simulatiion results can n be found in Ref. [[21, 22]. The fracture testss discussed inn this section n are either under u the loadding condition n of axial symmetryy tension ( = 1 ) or plane strain s tension ( = 0 , or callled generalizeed shear). It iss found that th he fracture locus of 1045 steel is strongly s depen ndent on the L Lode angle parrameter ( ) even e the its pllasticity does not n [22]. The data poinnts from the numerical n sim mulations will be used to co onstruct the 3D D fracture loccus. Since nder the loadinng condition of axial symm metric compreession ( = 1 ), the 3D there is nno data pointss available un fracture llocus is assum med to be sym mmetric, ˆ (f ) = ˆ (f ) , or D1 = D5 and D2 = D6 . The caalibrated paraameters of the fractuure locus (Eqq. (11)) is lissted in Table 2. A 3D geometrical rep presentation oof the fracturee locus is illustratedd in Fig. 4. Thhe error of every data pointt is also marked in Fig. 4. It is shown thaat the 3D fracture locus agrees w well with all the t data pointts. Since all tthe data pointts were obtained under moonotonic load dings, this fracture llocus will be used as a reference for thhe correction of loading hiistory effect uunder compleex loading conditionns. Table 2: 2 Calibrated pparameters of the fracture lo ocus of 1045 ssteel
D1 0.77121
D2 1.6968
D3
D4 1.9454
0.51887
Fiig. 3: Specimenns used to calibrrate the fracturee locus of 1045 steel.
D5
D6
0.7121
1.6 6968
Fig. 4: Calibrated 3D fractture locus of 1045 1 steel from monotonic m loaading conditio ons.
3.2 Load ding history y effect on ffracture of 1045 steel The fracture of o 1045 steel has been welll calibrated in n Section 3.1 under u monotoonic loading conditions. In this seection, three types t of tests with complexx loading con nditions will be b used to callibrate and vaalidate the proposedd correction foor the loading history effecct, Eq. (10). Totally, T there are a five materrial parameterrs, cg , ch h changing nootches, reversal loading 1 , 2 , aand k , to be calibrated. Thhose tests incllude two stagees tension with of comprression-tensionn, and the torssion followed by tension. A collection off specimens iss shown in Fig g. 5.
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Fig. 5: List of o specimens for three typees of tests with h complex loading historiess: (a) two-stag ge-tension, (b) compression-tension rreverse loading, (c) torsion-tension tests.
In the first tyype of test, sm mooth round sspecimens werre subjected to pre-tensile lloading. One specimen was pulleed always to fracture. f The other four weere denoted reespectively as group A andd group B baseed on two different amounts of pre-tension. p The specimenss were re-macchined to intro oduce notchess, then they were w tested under tennsion to fractuure. The damaage calculatedd without histo ory effect correction is shoown in Fig. 6. Note that i found thatt the first weeighting functtion (nonlineaar accumulatiion under = 0 inn this type off test, so it is proportioonal loading condition) is negligible n exceept for the preemature fractu ure on specime men 4. So, the parameter p matical singulaarity. cg is sett as cg = 0.0001 . A very smaall value is useed in order to avoid mathem
Fig. 6: Com mparison of damage d accum mulation for th he two-stage-teension tests w without loading g history correection (case 1, for pure tenssion; case 2&3 3, group A; caase 4&5, grouup B).
The second type of test is the comppression-tension reversal loading l test. Different am mounts of pre-comppression weree applied, then n specimens w were subjected to tension until u crack. T The third type of test is torsion-teension test. Tube specimen ns were subjeccted to differeent amounts of o pre-torsionn, then they were w failed under tennsion again. All A the historiees of the param meters ( p , , , and ) at the fract cture initiation n sits were obtained from the finnite element simulations. Without loaading history correction, tthe calculated d damage wn in Fig. 7 and 8 as cirrcles. It is fo ound that the fracture locuus calibrated from the indicates D are show c in monotonnic loading coonditions can not predict thhese two typees of fracture ( Dc 1 ). It iss found that changes loading ddirections (orr the parameteer ) increasses the ductillity of 1045 steel s because the values of o damage indicatorr at fracture are a greater thaan one. Hencee we chose k = 1 in Eq.((10). A Matlaab code was written to calculate the damages accumulation n and optimizze the model parameters: p ch , 1 , and 2 . The objecttive in the 762
optimization is the make all the values of damage accumulation at fracture Dc as close to unity as possible. Finally, a combination of these parameters is achieved. The calibrated parameters of loading history effect is listed in Table 3.
Fig. 7: Correction of the loading history effect on fracture, notched round bars of 1045 steel subjected to compression followed by tension.
Fig. 8: Correction of the loading history effect on fracture, tubing specimens of 1045 steel subjected to torsion followed by tension
Comparisons of the total damages at fracture between without and with correction of loading history effect are shown in Fig. 7 and 8. Now, the total damages with correction at fracture are much closer to unity for all cases (diamond plots). If the damage is assumed to occur at Dc = 1 (as is proposed by the model), then the predicted displacements to fracture will be very close to the fracture displacement in all tests. Therefore, one can see that the proposed loading history functions works well for both the compression-tension loading and the torsion-tension loading. In other words, the test data validated the proposed ductile fracture model under complex loading history. Table 3: Calibrated parameters of the loading history effect on fracture of 1045 steel
4
cg
ch
1
2
k
0.0001
5.5
2.0
2.0
-1
Conclusion
As an extension to the conventional linear damage evolution assumption, a new form of ductile fracture model accounting for the loading history effect was proposed in this paper. A round of comprehensive fracture tests on 1045 steel was conducted to validate the proposed fracture model. Firstly, the combined isotropic/kinematic hardening plasticity model of the material was calibrated using compression-tension tests. Secondly, the 3D fracture locus of 1045 steel was calibrated using smooth/notched round bars, flat-grooved specimens, and tubular specimens. Thirdly, the proposed correction of loading history effect is studied and calibrated by three groups of tests: two-stage-tension test, compression-tension test and torsion-tension test. Two opposite trends of loading history effect on ductile fracture were observed. Changes in loading direction can either decrease the material ductile (for example, 2024-T351 aluminum and OFHC copper) or increase the material ductility (for example 1045 steel). Using the calibrated fracture locus and the loading history correction, the proposed model well predicts ductile fractures of all tests under complex loading conditions. A more comprehensive study on the 3D fracture locus and the loading history effect on ductile fracture are undergoing to further confirm the present findings.
Acknowledgement The work was done while the author at MIT. Thanks due to Professor Tomasz Wierzbicki of MIT for 763
his advising. Partial financial supports from the NSF/Sandia alliance program and the AHSS MIT industry consortium are greatly appreciated. Thanks due to the continuous support of HyperMesh program from the Altair Company. References [1] Muschenborn, W., and Sonne, H.M. Influence of the strain path on the forming limits of sheet metal. Arch. Eisenhuttenwes, 46:597-602, 1975. [2] Graf, Alejandro and Hosford, William. The influence of strain-path changes on forming limit diagrams of A1 6111 T4. International Journal of Mechanical Sciences, 36(10):897--910, 1994. [3] Stoughton, Thomas B. A general forming limit criterion for sheet metal forming. International Journal of Mechanical Sciences, 42(1):1--27, 2000. [4] Atkins, A. G., and Mai, Y. W. Elastic and Plastic Fracture. Chichester:Ellis Horwood, 1985. [5] Cao, Jian and Yao, Hong and Karafillis, Apostolos and Boyce, Mary C. Prediction of localized thinning in sheet metal using a general anisotropic yield criterion. International Journal of Plasticity, 16(9):1105--1129, 2000. [6] Chow, C. L. and Yu, L. G. and Tai, W. H. and Demeri, M. Y. Prediction of forming limit diagrams for AL6111-T4 under non-proportional loading. International Journal of Mechanical Sciences, 43(2):471--486, 2001. [7] Yuanli Bai and Tomasz Wierzbicki. Forming severity concept for predicting sheet necking under complex loading histories. International Journal of Mechanical Sciences, 50(6):1012 - 1022, 2008. [8] Tvergaard, V. and Needleman, A. Analysis of the cup-cone fracture in a round tensile bar. Acta Materialia, 32:157--169, 1984. [9] Wilkins, M L. and Streit, R D. and Reaugh, J E. Cumulative -strain-damage model of ductile fracture: simulation and prediction of engineering fracture tests, UCRL-53058. Technical report, Lawrence Livermore Laboratory, Livermore, California, 1980. [10] Johnson, Gordon R. and Cook, William H. Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures. Engineering Fracture Mechanics, 21(1):31-48, 1985. [11] Hooputra, H. and Gese, H. and Dell, H. and Werner, H. A comprehensive failure model for crashworthiness simulation of aluminium extrusions. International Journal of Crashworthiness, 9:449-464, 2004. [12] Bao, Yingbin. Prediction of ductile crack formation in uncracked bodies. PhD thesis, Massachusetts Institute of Technology, 2003. [13] Bao, Yingbin and Treitler, Roland. Ductile crack formation on notched Al2024-T351 bars under compression-tension loading. Materials Science and Engineering A, 384(1-2):385--394, 2004. [14] Bai, Yuanli and Bao, Yingbin and Wierzbicki, Tomasz. Fracture of prismatic aluminum tubes under reverse straining. International Journal of Impact Engineering, 32(5):671--701, 2006. [15] Yuanli Bai and Tomasz Wierzbicki. A new model of metal plasticity and fracture with pressure and Lode dependence. International Journal of Plasticity, 24(6):1071 - 1096, 2008. [16] Armstrong, P.L., and Frederick, C.O. A mathematical representation of the multiaxial Bauschinger effect, G.E.G.B. Report RD/B/N 731. 1966. [17] Chaboche, J.L. Viscoplastic constitutive equations for the description of cyclic and anisotropic behavior of metals. Bulletin de l'Academie Poloanise des Sciences. Serie des Sciences Techniques, 25:33, 1977. [18] Lee, D., and Zavenl, Jr. F. A generalized strain rate dependent constitutive equation for anisotropic metals. Acta Metallurgica, 29:1771, 1978. [19] Bai, Yuanli and Teng, Xiaoqing and Wierzbicki, Tomasz. Derivation and application of stress triaxiality formula for plane strain fracture testing. Presented at ASME conference of Applied Mechanics of Materials, Austin, TX, June 3-7, 2007. [20] Khan, Akhtar S. and Jackson, Kamili M. On the evolution of isotropic and kinematic hardening with finite plastic deformation Part I: compression/tension loading of OFHC copper cylinders. International Journal of Plasticity, 15(12):1265--1275, 1999. [21] Bai, Yuanli. Effect of Loading History on Necking and Fracture. PhD thesis, Massachusetts Institute of Technology, 2008. [22] Yuanli Bai and Xiaoqing Teng and Tomasz Wierzbicki. On the Application of Stress Triaxiality Formula for Plane Strain Fracture Testing. Journal of Engineering Materials and Technology, 131(2):021002, 2009.
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Effect of Seam Welding on Forming Limits of IF-Steel Sheet D. Sutariya1, H.Raval1, K. Kalaivani2, K. Hariharan2, A. Prabhu3, K. Narasimhan3* 1
Department of Mechanical Engineering, Sardar Vallabhbhai National Institute of Technology, Ichhanath, Surat, Gujarat. 395007, India 2 Advanced Engineering, Ashok Leyland Ltd, Chennai, India 3 Department of Metallurgical Engineering and Materials Science, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India *Corresponding author Phone +91-22-2576-7630, Fax 91-22-2572 3480, Email
[email protected] Abstract. Laser welding is the most commonly used process for producing Tailor Welded Blanks (TWB). Although laser welding is best suited for TWB applications, it is an expensive process. In this work an alternate cheap process, seam welding, is used to produce welded sheets of IF-steel of same grade and thickness. The effect of this welding on forming limit strains is explored in this work. Forming behavior is characterized by Forming Limit Diagram (FLD). The FLDs of welded blanks are compared with that of un-welded blanks. The effect of longitudinal, transverse and 450 weld orientation on formability is studied. Weld location includes both centered and offset weld location in the transverse weld orientation. Numerical simulations are carried out by considering weld as a zone. Experimental results are compared with the predictions carried out by FE method. Keywords: Forming limit strain, Tailor welded blank (TWB), Weld property, Formability for different weld orientations, FEM simulations. PACS: 81.20.Hy
INTRODUCTION Tailor welded blanks comprise sheets of different thickness, strength, coating etc. welded in a single plane prior to stamping. TWBs offer several benefits such as reduced manufacturing cost, reduced scrap, weight reduction etc. [1] and are therefore currently widely used in automotive industries .They are usually fabricated by a laser welding process, which creates a narrow weld zone. The formability of TWBs is determined by (a) thickness and material combination of blanks welded, and (b) by the weld conditions like weld orientation and weld location. The compounding effect of thickness ratio, weld line movement and weld orientation on the forming limit strain has been studied by Chang et al [2]. An inverse relationship between the thickness ratio and TWB formability has been reported in the research on laser welded TWBs by Chang et al [3]. Wonoh Lee et al [4] studied the friction stir welded (FSW) TWBs of aluminum alloys, dual phase steel and magnesium alloy and investigated formability performance by comparing LDH value of base and TWB sample. A lowering in the LDH was reported in the TWBs in comparison to the base metal regardless of the stretching mode. Some of the researcher studied the effects of weld orientation and weld location on forming behavior of laser welded and friction stir welded blank. Garware et al [5] studied the tensile and fatigue behavior of FSW TWBs of aluminum alloy 5754 for different weld orientations with respect to loading direction, observing that both tensile and fatigue failure in the tailor welded blank occurred on the thin gage side irrespective of the weld orientation with respect to the loading direction. Panda et al [6] studied the effect of different weld location and orientation on laser welded AHSS blanks, reporting that the formability LDH and failure location strongly depends upon the weld location. LDH increases when the weld is positioned at the extreme positions away from the center because of more uniform strain distribution on deform dome. Ganesh et al [7] have also studied the effect of weld configuration on The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 765-771 (2011); doi: 10.1063/1.3623683 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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the forming limit of laser welded blanks and compared with that of un-welded sheet, concluding negligible influence on the FLC when the weld was located at center in both longitudinal and transverse direction. They also reported that the FLD of TWB is lower than the FLD of un-welded sheet. Results show that FLD is much lower at the plan strain condition for the transverse weld configuration than the un-welded sheet as well as longitudinal TWB configuration. In this paper the effect of weld configuration on forming limit strain has been experimentally investigated. TWBs were fabricated by seam welding with same materials and 1.2 mm of thickness. FE simulation of LDH test was carried out using PAM STAMP 2G commercial FEM code to construct a FLD. Limiting Dome Height tests (LDH) were conducted for samples of different widths to get a complete FLD and the strain data was recorded by GOM using Argus software.
MATERIAL PROPERTIES IF-steel was used in present study. The tensile properties of un-welded and welded sheets are shown in Table 1. All these parameters were evaluated by tensile testing of standard ASTM geometry in Instron machine. The ASTM standard procedure was followed to evaluate above parameter [8]. The longitudinal weld configuration was considered to evaluate mechanical properties of TWBs and 5.2 mm width of specimen was tested. The table 2 shows tensile parameters of welded sheet. The engineering stress strain curves of un-welded and weld only sheets are shown in Fig. 1. TABLE 1. Mechanical properties of un-welded sheet Rolling direction 0 45 90 Average
Longitudinal
ys (MPa) 138.84 154 163 151.94
UTS (MPa) 281.04 294.63 292.70 289.45
n 0.3018 0.2894 0.2768 0.2893
R 1.8428 1.7142 2.7045 -
TABLE 2. Tensile parameter of weld only specimen ys (MPa) n 312.35 0.124
Weld only Base metal
FIGURE 1. Engineering Stress-Strain curve of un-welded sheet and the weld region.
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K (Avg.)
545.6
K (MPa) 614.5
METHODOLOGY Weld Condition Weld condition, namely weld orientation and weld location, were considered for the study of influence on limit strains in TWBs. Longitudinal, 450 and transverse weld orientations were studied. The longitudinal weld orientation represents a weld parallel to major straining direction, 450 weld orientation represents a weld is 450 to the major straining direction, and transverse weld orientation represents a weld normal to the major straining direction. In the case of longitudinal-welded blanks, the weld zone was kept at the geometric centre of the sample (Fig. 2). In transverse-welded blanks, the weld zone was located at two different positions including one at the geometric centre and others at 20mm offset (Fig. 2).
(a) (b) (c) FIGURE 2. Schematic diagram of weld configuration: (a) Longitudinal (b) Transverse (c) 45 degree
Out of Plane Formability Test To study the effect of weld configuration on FLC of TWBs, limit dome height (LDH) test was performed. Geometry of standard LDH tooling set up is shown in Fig 3. LDH test was conducted on double acting hydraulic press with standard hemispherical punch at constant speed of 1 mm/sec and 4 tons of binder force was applied to avoid draw in or failure in clamping region. Circle grid was marked on all samples using screen printing. All the samples were deformed up to necking. Eight different width samples were tested to obtain different strain paths. GOM system and ARGUS software was used for post deformation analysis, and recording major and minor strains further used to plot the forming limit diagram.
Figure 3. Geometry of out of plane test
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Simulation The PAM STAMP 2G FE code was used to simulate LDH test in order to predict the influence of the weld conditions. During the simulation, the weld region is modeled as a single entity without separate HAZ. Different blank widths ranging from 25mm to 200 mm in steps of 25 mm were simulated to achieve different strain paths. Mesh size of 1mm was kept constant in the weld zone and base metal. Hollowman’s power law was used to describe the strain-hardening behavior. Hill’s 1948 isotropic hardening yield criterion was used as the plasticity model. Friction coefficient was assumed to be 0.12 and 150 kN binder force was chosen and kept constant throughout the study. Limit strains were obtained using the thickness gradient based necking criterion described by Nandedkar [9].
RESULTS AND DISCUSSION Effect of Weld Orientation Fig. 4 shows the experimental FLD, the effect of weld on forming limit strain of TWBs under different weld orientations like longitudinal, transverse, 45 degree with weld at the geometric center and comparison with that of un-welded sheet. It is observed that the FLC of the transverse welded blank is slightly lower than that of the base metal. In this orientation, the lower influence of weld is attributed to the fact that weld zone is not participating in deformation as failure occurs at 20 – 30 mm away from the weld zone. As shown in Fig. 4, in case of longitudinal weld at center, FLD of welded blank is lower than that of un-welded sheet. The higher deviation in limit strain is found in the drawing (in tensile) region and failure occurs normal to the weld zone in punch nose region of TWBs. The strain path in drawing region approaches tensile strain path which is witnessed during standard tensile testing, where in reduced ductility of longitudinal welded blank is observed in comparison to that of un-welded blank. In the plane strain condition limit strains are almost the same for the both cases of longitudinal and transverse weld. In case of 45 degree weld orientation, FLC is significantly lower than the other weld orientation in plane strain and stretching mode.
Figure 4. Experimental FLC of welded and un-welded sheets
Effect of Weld Location in Case of Transverse Weld Orientation In the transverse weld orientation, both weld at center and weld at an offset by 20 mm was studied. Fig 5 shows the forming limit diagram of both the cases and compares it to that of un-welded blank.
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Figure 5. Experimental FLC of transverse weld at center and 20 mm offset
In the case of transverse weld at 20 mm offset position (Fig 5), the experimental FLC is considerably lower than that of un-welded blank and weld at center in plain strain and bi- axial strain condition. This is because of failure has occurred in weld zone for plain strain (100 x 200) and bi- axial (200x200) conditions.
Comparison of Experimental and Predicted FLCs Fig. 6 (a) – 6 (c) shows the comparison between experimental and predicted FLCs of un-welded sheet and TWBs for varied weld orientation. In the case of base metal (Fig. 6-a) predicted limit strains are in good agreement with experimental limit strain values in drawing and plan strain condition, but in case of stretching region of FLD limit strains are deviated. This is possible due to the assumed constant material properties throughout the simulation.
(a)
769
(b)
(c) Figure 6. Comparison between experimental and predicted FLCs: (a) base metal (b) longitudinal (c) transverse
As shown in Fig. 6 (b) – 6 (c), predicted FLCs are significantly lower than experimental FLCs. Modeling of weld zone plays vital role in FEA analysis. During the simulation weld zone is modeled by considering uniform thickness and weld width while it is not the case in experiment. Failure observed near weld zone and weld zone itself in FEM analysis. Above discussed reasons significantly lower the predicted FLC as compared to experimental FLC.
CONCLUSION In this work, effect of weld configuration on FLDs is studied in seam welded TWB sheets of interstitial free steel. Following conclusions are obtained from present study. (1) Forming limit strains of transverse welded sheet shows negligible effect as necking has occurred away from the weld zone and weld has not participated significantly in deformation. (2) In case of 450 oriented welded sheet, deviation was observed in bi- axial stretching. (3) LDH test shows that the forming limit strain of longitudinal and transverse TWBs are almost same when weld is placed at center. But in case of transverse weld at 20 mm off set FLC is considerably lower especially near plan strain condition. (4) Predicted limit strains of un-welded sheet coincide well with experiments. In case of welded sheets FLCs shows similar trend when compared to experimental FLCs, but are significantly lower.
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REFERENCES 1. Auto/Steel Partnership, 1995, “Tailor Welded Blank Design and Manufacturing Manual” Technical Report 2. Chan, L. C., Chang, C. H., Chan, S. M., Lee, T. C., and Chow, C. L. “Formability analysis of tailor welded blanks of different thickness ratios” J. Mfg Sci. Engng, 2005, 127(4), 743–751. 3. Chang S.M., Chang L.C. and Lee, T. C. “Tailor welded blank of different thickness ratios effect on forming limit diagram” J. Mat. Proc. Tech. 132, 2003, 95-101. 4. Wonoh Lee, Kyung-Hwan Chung, Daeyong Kim, Junehyung Kim, Chongmin Kim, Kazutaka Okamoto, R.H. Wagoner, Kwansoo Chung “Experimental and numerical study on formability of friction stir welded TWB sheets based on hemispherical dome stretch tests” International Journal of Plasticity 25 (2009) 1626–1654 5. M. Garware, G.T. Kridli, and P.K. Mallick “Tensile and Fatigue Behavior of Friction-Stir Welded Tailor-Welded blank of Aluminum Alloy 5754” Journal of Materials Engineering and Performance, Volume 19(8) November 2010—1161 6. S.K. Panda, J. Li, V. H. Baltazar Hernandez, Y. Zhou, F. Goodwin “Effect of Weld Location,Orientation, and Strain Path on Forming Behavior of AHSS Tailor Welded Blanks” j. Engg. Mat. & Tech. Oct 2010, Vol 132 7. R Ganesh Narayanan1, K Narasimhan “Influence of the weld conditions on the forming-limit strains of tailor-welded blanks” J. Strain Analysis 2008 Vol. 43 p- 217-227. 8. ASTM E646-98 ”Testing method for tensile strain hardening exponents (n-values) of metallic materials” In Annual book of ASTM standards, section 3, vol. 03.01, 2000 (ASTM International, West Conshohocken, Pennsylvania). 9. Nandedkar V. M. “Formability Studies on a Deep Drawing Quality Steel” PhD Thesis, IIT Bombay, 2000.
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3UHGLFWLRQRI)RUPLQJ/LPLWVRIWKH0XOWLOD\HU0HWDOOLF6KHHW T. Oya1*, C. Jeong2,and J. Yanagimoto2 a
b
Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo, 153-8505, Japan
$EVWUDFW In this paper, the forming limit of a multilayer metallic steel sheet is numerically investigated. Multilayer metallic sheets are an emerging sheet metal that is designed to achieve both high strength and enough formability, which is seriously demanded especially from the automotive industry. Unfortunately, its mechanism of enhanced ductility, fracture behavior, and forming limits have not been sufficiently studied. Therefore, a new stress-strain curve formula is constructed so as to precisely predict the forming limit that is influenced by the unique characteristics of the multilayer metallic sheets. Tensile and stretching experiment on a 15-layer multilayer metallic sheet is conducted, then the results are used to verify the effectiveness of the presented stress-strain expression, and the predicted forming limits are compared to the experimental value, resulted in a good agreement. .H\ZRUGV Multilayer metallic steel sheet, sheet metal, forming limit, ductility enhancement. 3$&6 81.05.Bx, 83.10.Gr, 83.50.Uv.
,1752'8&7,21 The use of high-strength steels is now becoming inevitable in various industrial fields such as the automotive, aerospace, and other heavy industries. High-strength steels are expected to produce lightweight structures without ruining the rigidity and the performance. However, some hurdles, such as poor shape-freezing property and difficulty in predicting forming limits, have been an obstacle to broaden the application field of the high-strength steels. One of the most troublesome properties would be its low ductility in terms of press forming. Recently, multilayer metallic steel sheets are emerging as a next generation steel to urge the utilization of the high-strength steel. This material consists of alternate hard material layer and soft material layer to achieve the high strength and enough formability. Due to the strong interfaces, the ductile soft layers restrain the onset of necking in the hard layers, leading to a sufficient elongation as a sheet metal. However, the forming limit prediction of multilayered material is difficult because of its unique characteristics of layered geometry. Therefore, mechanism of enhanced ductility, fracture behavior, and forming limit prediction should be rigorously explored in order to promote the application of the multilayer metallic sheet. Syn et al. [1] studied the effect of layering on tensile ductility using a composite made of a particle-reinforced aluminum metal matrix composite (MMC) and a highly ductile Mg -9%Li alloy. Due to the enhanced tensile ductility, the laminate exhibited higher ductility than that of the constituent aluminum MMC. Semiatin and Piehler [2] studied on the forming limit of clad laminates based upon the rule of mixture. Manesh and Taheri[3] evaluated the formability of an aluminum-clad steel sheet by the Erichsen cupping test to determine the optimum conditions of heat treatment in the production process. Mori and Kurimoto [4] evaluated the formability of a stainless-steel aluminum clad sheet by tensile and bending tests. Yoshida and Hino [5] examined the forming limit of stainless steel -clad aluminum sheets under plane stress condition. They prepared two- and three-ply laminate and conducted punch-stretching experiments. Their model based upon Hill’s localized necking theory and the M-K theory predicted forming limits. Hino et al. [6] investigated the springback behavior by draw-bending experiments using two-ply sheet metal laminates. Ohashi et al. [7] produced a 35-layer laminated metal made of ultrahigh-carbon steel (UHCS) and brass, and they investigated its ductility and fracture characteristics upon bending. Carreño et al. [8] created a 7layer laminated steel composed of ultrahigh-carbon steel and mild steel, which was used in impact tests to investigate the fracture mechanism. Inoue et al. [9] have already confirmed the effectiveness of the layering concept by using a brittle martensitic steel and a ductile steel. The layered material they created exhibited a good strength-ductility balance and no
The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 772-779 (2011); doi: 10.1063/1.3623684 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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delamination was observed during the tensile test. Nambu et al. [10] used a layered material consisting of highstrength martensitic steel and an austenitic ductile steel to investigate the transition under a large deformation. Nambu et al. [11] also investigated the effect of the strength of the interfacial bonding of multilayered steel composites, where a marked enhancement of tensile ductility was observed with increased bonding strength. In these studies the fundamental characteristics of multilayer metallic steel sheets have been revealed. In Oya et al. [12], the bending formability of multilayer metallic sheets were evaluated through V- and hemming-bending. Monolithic martensitic steel specimens fractured in hemming-bending, on the other hand, the same martensitic steel was successfully bent even in hemming-bending when it was sandwiched by the ductile steel layers. Oya et al. [13] conducted finite element analyses on multilayer metallic sheets; V-bending was modelled and springback angles were used to verify the results of the simulation with the experiments, and good agreements were obtained. In these studies, the stress-strain curves of the constituent brittle materials were predicted by the rule of mixtures because tensile test on a monolithic brittle metal cannot provide an entire curve of the brittle metal layer in a multilayer metallic sheet. Yanagimoto et al. [14] presented a theoretical explanation on why multilayer metallic sheet can undergo a large strain more than the original fracture limit of the constituent brittle metal. Geometrical restriction by the soft material layers would induce a delay of diffuse necking in the hard material layers, leading to the improved elongation of a multilayer metallic sheet. Chen et al. [15] conducted a numerical study on the effect of cladding on necking and fracture in sheet metals. They reported that cladding by a ductile layer enhances the overall work hardening, leading to the increase in the uniform strain that slows down the stress development causing the void nucleation delay. Liu et al. [16] reported that the tensile plasticity of cold-rolled Cu could be enhanced by a layered structure stacked alternately by thicker Cu and thinner Al layers. They explained that a strong interface can delay the development of premature strain localization and local necking in the ductile Cu layer, and crack formation also can be prevented. In this paper, a new stress-strain formula is proposed to predict the formability of the multilayer metallic sheet. The formula is capable of handling the variation of both strength and work-hardening property; therefore, it would be useful to characterize the flow curve of the multilayer metallic sheet in which the deterioration in the workhardenability can be observed. Numerical simulations were conducted to investigate the performance of this expression by using the experiment of a tensile test.
08/7,/$ d ⎪ ⎪ ⎪ λ ⎪ ⎨ λ +d −z (4) σy = C (ε p + ε0 )n with εp = for d −U ≤ z < d ln ⎪ λ ⎪ ⎪ ⎪ λ +U ⎪ ⎪ for z ≤ d −U ⎩ ln λ in which C and n represents specific material characteristics and ε p the logarithmic strain in the asperities. The strain in the asperities is defined as the amount of flattening or rise of asperities relative to the initial height of the asperities λ . In this respect, a definition for the strain can be derived for 1) asperities in contact with the indenter, 2) asperities which will come into contact due to the rise of asperities and 3) asperities which will not come into contact with the indenter (Equation 4). The model described above is based on a normal loading case without additional bulk strain. To account for flattening due to stretching, the model has to be adapted. The change of the fraction of the real contact area as a function of the nominal strain can be presented as:
dαSi l = φ dSi−1 −USi−1 dε E
(5)
with i the iteration number. The subscript S is used for variables that become strain dependent. The contact area ratio is updated incrementally by: αSi = αSi−1 + dαSi (6) The initial values αS0 , dS0 and US0 are obtained from the model without bulk strain. To calculate the change of αS , the value of US and dS needs to be solved simultaneously while ε is incrementally increased. Based on volume conservation and the definition of the fraction of real contact area (Equation 7) US and dS can be obtained.
αS =
∞
φ (z) dz
US (1 − αS ) =
dS −US
∞ dS −US
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(z − dS ) φ (z) dz
(7)
0.2
0.13
0.58 Fraction of real contact area (-)
Fraction of real contact area (-)
0.17
0.7 FEM: elastic ideal-plastic FEM: elastic nonlinear-plastic Analytical solution
λ =∞
0.1
λ =4.0Rt → 2.0Rt → Rt 0.067 0.033 0
0
FIGURE 3.
FEM: elastic ideal-plastic FEM: elastic nonlinear-plastic Analytical solution: ideal-plastic
0.47 2500 asp/mm2 0.35
5000 asp/mm2 9000 asp/mm2
0.23 0.12
20
40 60 80 Nominal contact pressure (MPa)
0
100
0
0.02
0.04 0.06 Logarithmic strain (-)
0.08
0.1
Development real contact area of analysis 1 (a, left) and development contact area of analysis 2 (b, right)
Shear stresses The model of Challen & Oxley [2, 3] takes the combining effect of ploughing and adhesion between a wedge-shaped asperity and a flat surface into account. Westeneng [4] extended the model of Challen & Oxley to describe friction conditions between a flat workpiece material and multiple tool asperities. He assumed that the flattened peaks of the asperities are soft and perfectly flat and the surface of the tool material is rough and rigid. The difference in hardness between the tool and workpiece material and the difference in length scales between the two surfaces is significant in the case of a sheet metal forming process. Therefore, it is valid to make a subdivision in two length scales using a rigid tool and a soft workpiece. The extended version of Challen & Oxley’s model [4] has been implemented in the friction model to describe friction conditions between the tool and workpiece material: Fw = ρt αS Anom
smax
μasp πωβt H φt (s) ds
(8)
δ
with ω the amount of indentation, βt the mean radius of the tool asperities and μasp the coefficient of friction at single asperity scale [3]. ρt represents the asperity density of the tool surface, Anom the nominal contact area and φt the normalized surface height distribution function of the tool surface. The bounds of the integral are described by smax , the maximum height of the tool asperities, and δ , the separation between the workpiece surface and the mean plane of the tool asperities. Since the normal force is known (input parameter), the coefficient of friction can finally be obtained by: Fw μ= (9) FN
VALIDATION The newly developed non-linear load model as well as the ideal-plastic strain model of Westeneng have been validated by means of FE simulations on a two-dimensional rough surface. In the first analysis, a two-dimensional rough surface of 4mm long was deformed by a perfectly flat and rigid tool. The second analysis was focused on flattening a rough surface by a normal load including a bulk strain in the underlying material. Three simulations have been executed for each analysis using different roughness profiles. The roughness profiles equal three roughness measurements of DC04 low-carbon steel. The surface was modeled by 4 node 2D plane-strain elements. The yield surface was described by the Von Mises yield criterion using the Nadai hardening relation to describe work-hardening effects. The surface height distribution used for the analytical model corresponds to the roughness distribution of the FE simulation. The development of the real area of contact has been tracked during the simulation and compared with the analytical solution. Results shown in Figure 3 are the mean values of the three simulations performed per analysis case. The non-linear load model uses the Nadai hardening relation to account for work hardening effects of the flattened asperities. An unknown parameter has been introduced during the derivation of this model: the initial height of
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asperities λ (see Equation 4). Calculations have been performed using different values for this parameter: 1.0, 2.0 and 4.0 times the Rt value (Figure 3a). The Rt value represents the maximum peak to valley distance between asperities. The amount of strain build up in the asperities will be lower when using a higher value for the initial height λ . From the results, it can be concluded that the exact development of real contact area can be found when using an initial height of 2.0 times the Rt value. Increasing the initial height of asperities to infinity represents ideal plastic material behavior, since the strain build up in the material approaches zero. As shown in Figure 3a, the analytical solution describes the results of the ideal-plastic FE simulation well when using a value of λ = ∞. Combined normal loading and stretching the underlying bulk material decreases the effective hardness [9]. A lower hardness results in an increase of the real area of contact. Both the analytical and the FE results of analysis 2, where a rough surface has been flattened by a nominal load and a bulk strain has been applied to the underlying material, are presented in Figure 3b. As for analysis 1, results shown are the mean values of three simulations performed per analysis case. It can be concluded that work-hardening effects have a large influence on the flattening behavior of the asperities. The strain model developed by Westeneng uses the asperity density to obtain the flattening behavior of asperities. A line profile has been used in this section to generate the FE models. The asperity density of this profile has been obtained by the 3-point summit rule (1D version of the 9-point summit rule for surfaces). By taking the square of the asperity density found by the 3-point summit rule, a density of 2500 asp/mm2 was found. Results obtained by the analytical strain model, using an asperity density of 2500, 5000 and 9000 asp/mm2 , are shown in Figure 3b. The trend of the graphs corresponds well, but the development of the real area of contact is significantly higher compared to the ideal-plastic FE results if an asperity density of 2500 asp/mm2 is used. If a higher value of the asperity density is used the amount of real area of contact will be lower. Using an asperity density of 5000 asp/mm2 it is possible to describe the results of the elastic ideal-plastic FE solution well.
IMPLEMENTATION The developed friction model can be implemented in FE codes using two different strategies. One strategy is to implement the code of the friction model into the contact algorithm of the FE code. The friction model is called if a node of the workpiece comes in contact with the tool, resulting a friction coefficient belonging to that specific node. This strategy returns a specific coefficient of friction for each node in contact. Another strategy is to initialize the coefficient of friction for a predefined range of process variables. Process variables are the nominal contact pressure and the bulk strain in the workpiece material. Since a rough guess can be made about the range of these variables a matrix can be constructed including friction values for all possible combinations. Figure 4a shows for example a friction matrix for a nominal contact pressure in between 0 and 100 MPa and a bulk strain in between 0 and 150%. The time required to construct this matrix, using various functions to evaluate the asperity height distribution, is shown in Figure 4b. The Gauss and Weibull distribution functions are the two fastest methods. However, the applicability of these methods is restricted by the relatively simple shape of the functions. Fourier series or B-splines could be used if more complex shapes are desired. The time required to build a friction matrix using Fourier series increases when increasing the number of Fourier expansions. The time required to construct a friction matrix using B-splines depends on the order of the B-spline (in this case cubic lines where used) instead of the number of lines used to construct the B-spline. Concerning the flexibility of the B-spline function and the numerical stability of the friction algorithm, the B-spline function is favorable in describing complex distributions. The Weibull distribution function is favorable in case of normally distributed distributions. For the first strategy the friction routine is called for every node in contact each step of the simulation. Hence, this strategy can result in a huge increase of the simulation time. For the second strategy, the friction matrix has to be constructed only once. When the friction matrix is constructed it can be used during a FE simulation to find nodal friction values. Since only an interpolation scheme is required to find the nodal friction values, this method will hardly increase the simulation time. Difficulties with the second approach are however expected when it is desired to take history dependency of the deformed asperities into account. Loading/unloading and straining/unstraining of the workpiece material will take place during sheet metal forming which requires the use of history dependency. For the first strategy, friction values are calculated per step per node which makes it more straightforward to take the flattening history of previous steps into account.
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30 Nr. expansions: 5 - 10 - 15 - 20
25
Nr. lines: 5 - 10 - 15 - 20
Time [s]
20 15 10 5 0
FIGURE 4.
Gauss
Weibull
Fourier
B-spline
Friction matrix (a, left) and evaluation time using different distribution functions (b, right)
APPLICATION A cross-die product is used to test the numerical performance of the developed friction model in a large-scale FE simulation (Figure 5a). Due to symmetry only a quarter of the workpiece was modeled. The workpiece was meshed with 9000 triangular discrete Kirchhoff shell elements using 3 integration points in plane and 5 integration points in thickness direction. The coefficient of friction used in the contact algorithm was calculated on the basis of the friction model presented in this paper. Simulations have been performed using both implementation strategies discussed in the previous section. A distribution of friction coefficients can be observed from the results presented in Figure 5a. Values of the friction coefficients are found ranging from 0.13 to 0.19. The gray area represents the non-contacting area. It can be observed from Figure 5a that higher values of the coefficient of friction occur at regions where high strains occur (region A, B and C). Region A is purely stretched, region B is compressed which causes thickening of the material and region C is stretched over the die radius. Overall it can be concluded that the distribution of the coefficients of friction lies within the range of expectation. The draw-in pattern of the simulation is compared to the simulation result in which the standard Coulomb friction model has been used with a friction coefficient of 0.13. It can be observed that the draw-in significantly deviates from the draw-in obtained with the Coulomb friction model. This is logical since the maximum obtained friction coefficient by the developed friction model is much higher than the fixed value of 0.13. When a fixed value of 0.19 is used, which is the maximum value found when using the developed friction model, failure of the cross-die will occur. As mentioned before, simulations have been performed using two different strategies to couple the friction model to the FE code. Figure 5a shows the results obtained by using a friction matrix (the second strategy proposed). Results match perfectly with these obtained when using the first strategy (direct implementation of the friction algorithm in the contact algorithm of the FE code). However the simulation time grows significantly when using the first strategy. An increase of 3 times the simulation time required to perform a Coulomb based simulation was observed. On the other hand, an increase of less than 1% was observed when using the second strategy.
CONCLUSIONS A friction model that can be used in large-scale FE simulations is presented. The friction model includes two flattening mechanisms to determine the real area of contact at a microscopic level. The real area of contact is used to determine the influence of ploughing and adhesion effects between contacting asperities on the coefficient of friction. A statistical approach is adapted to translate the microscopic models to a macroscopic level. The friction model has been validated by means of FE simulations at a micro-scale. A good comparison was found between the FE simulations and the results obtained by the newly developed non-linear loading model. It has been shown that the non-linear load model can be used to describe both ideal-plastic and non-linear plastic material behavior. If a nominal strain is applied to the bulk material, the effect of work-hardening becomes significant. The ideal-plastic strain model is able to describe the trend of the FE results. However, it was not possible to accurately describe the
790
120
0.19 A
y (mm)
90
60
Draw-in Coulomb 0.13 Draw-in friction model
30
C 0.13
0
0
B
FIGURE 5.
30
60 x (mm)
90
120
Development coefficient of friction cross die (a, left) and amount of draw-in (b, right)
ideal-plastic FE results, as well as the influence of work-hardening effects. Other, more advanced, models are required to accurately describe the influence of bulk straining on the flattening behavior of asperities. The friction model could be directly implemented into FE codes or being coupled to the FE code by using a friction matrix. The first strategy requires to calculate nodal friction values each step of the simulation, which can result in a huge increase of the simulation time. When a friction matrix is used, only an interpolation scheme is required to find nodal friction values during the FE simulation which will result in a lower increase of the simulation time. The friction model has been applied to a full-scale sheet metal forming simulation to test the numerical performance and feasibility of the developed friction model. The results look very promising. The modest increase in simulation time shows the feasibility of the friction model in large scale sheet metal forming simulations.
ACKNOWLEDGMENTS This research was carried out under the project number MC1.07289 in the framework of the Research Program of the Materials innovation institute M2i (www.m2i.nl).
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
W. Wilson, Americal Society of Mechanical Engineers 10, 13–23 (1988). J. Challen, and P. Oxley, Wear 53, 229–243 (1979). J. Challen, and P. Oxley, International Journal of Mechanical Siences 26, 403–418 (1984). J. Westeneng, Modelling of contact and friction in deep drawing processes, Ph.D. thesis, University of Twente (2001). J. Greenwood, and J. Williamson, Proceedings of the Royal Society of London. Series A, Mathematical and Physical sciences 295, 300–319 (1966). Y. Zhao, and L. Chang, Journal of Tribology 123, 857–864 (2001). D. M. Y. Zhao, and L. Chang, Journal of Tribology 122, 86–93 (2000). J. Pullen, and J. Williamson, Proceedings of the Royal Society of London. Series A, Mathematical and Physical sciences 327, 159–173 (1972). W. Wilson, and S. Sheu, International Journal of Mechanical Science 30, 475–489 (1988). M. Sutcliffe, International Journal of Mechanical Science 30, 847–868 (1988). J. Greenwood, Proceedings of the Royal Society of London. Series A, Mathematical and Physical sciences 393, 133–157 (1984). M. de Rooij, Tribological aspects of unlubricated deepdrawing processes, Ph.D. thesis, University of Twente (1998). J. Hol, M. C. Alfaro, M. de Rooij, and T. Meinders, WEAR (2010, doi:10.1016/j.wear.2011.04.004).
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Pre-Forming Effects on AHSS Edge Cracking Xiaoming Chen a , Ke Chenb and Lorenzo Smithc a
United States Steel Corporation, Automotive Center, 5850 New King Court, Troy, MI, USA b College of Mechanical Engineering, Tongji University, Shanghai, China c School of Engineering and Computer Science, Oakland University, Auburn Hills, MI, USA
Abstract. Edge cracking in advanced high strength steels (AHSS) is a significant failure mode in many sheet metal stamping processes. Edge pre-forming into a wave (or scallop) shape is a common technique used in conventional steels to gather material in high edge stretch regions in preparation for the subsequent edge stretch process. The pre-forms designed for mild steels do not always apply to AHSS because the properties of AHSS can differ greatly from those of conventional steels. This work has studied the effects of pre-forming on AHSS edge cracking. Experiments have been conducted to stretch pre-formed steel strips to failure. Strain distributions of pre-forms with various levels of stretch have been measured using digital image correlation (DIC) technology. Finite element analyses have been performed and compared with the experimental results. Different failure criteria have also been evaluated for use in this type of application. Keywords: Edge crack, Sheet metal forming, DIC, Finite element analysis
INTRODUCTION AHSS have the advantages of combining high strength with enhanced formability and high work hardening capability, which provide an opportunity to make complex geometric automotive parts with strengths far exceeding those of conventional high strength grades of steels. The high strength components provide an opportunity to improve vehicle crash and structural performance while reducing the overall weight of the vehicle. Therefore, AHSS are increasingly used in the automotive industry to meet future fuel economy and safety requirements. However, AHSS also have disadvantages and some forming issues have arisen in the stamping processes. Edge cracking, stretch bending fracture in a tight radius (shear fracture), and springback are some of the major manufacturing issues for AHSS, and these issues are more severe for AHSS than for the conventional high strength steels. Due to the high strength and multi-phase microstructure of AHSS, the edge cracking and shear fracture limits are lower than the conventional forming limit curve. The higher strength and significant Bauschinger effects cause more severe springback after forming. These issues cannot be predicted using the conventional simulation technology and often involve significant trial and error procedures in the tool and stamping process designs. Thus, a computer aided tool would be very useful to shorten the lead time and to reduce the cost of tool and die development processes. In the last several years, significant research has been conducted to solve these AHSS manufacturing issues. Many publications can be found on better understanding of the mechanisms. In the study of edge cracking, hole expansion and tension tests are usually used. Several experiments [1-3] were conducted to investigate the effects of different shearing methods, process parameters, and sheared edge characters on AHSS edge cracking. Steel microstructures were also found to play an important role in the edge cracking process [4,8]. Various simulation techniques were used to integrate the sheared deformation into hole expansion analyses to provide more accurate results. Different failure criteria were used to improve the AHSS edge cracking predictability [5,6]. In the AHSS shear fracture studies, various experiments were conducted to generate the failures for different stretch bending radii and stretch tensions, and empirical shear fracture limits were proposed [7,8]. Alternative failure criteria were also used in simulations to improve the fracture predictions [9]. In springback studies, prediction accuracy was significantly improved by using the kinematic/isotropic hardening material models for AHSS and upgrading simulation technology [10-12]. In flanging of conventional steels, a stretched edge usually forms a wave (or scallop) shape in the previous stage for gathering material. However, this is not applicable for AHSS stamping in many cases. Figure 1 shows an example of the pre-forming, trimming and then flanging processes of dual phase (DP) 780 steel using the procedure employed in forming conventional steels. The edge cracked in the pre-forming stage and the part could not The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 792-799 (2011); doi: 10.1063/1.3623687 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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withstand flanging in the next stage. AHSS pre-forming and stretching are complicated forming processes. The edge experiences bending, unbending and stretching deformations, which involve all three AHSS issues discussed above. Better understanding of this process is important for future AHSS applications. In this work, experimental and simulation studies were conducted on pre-forming and stretching of two AHSS (DP780 and 590R). The mechanical properties of both steels are shown in Table (1). The DIC technique was used to measure the deformation history of both forming stages. Finite element analysis was also conducted to simulate those processes. The objective is to develop a simulation technique for optimal design of AHSS edge stretching processes. TABLE(1). Mechanical Properties Material
590R DP780
Yield Strength (MPa) 440 497
UTS (MPa)
Total Elong (%) 22.9 15.0
636 856
n value
r ave
0.134 0.122
0.77 0.84
FIGURE 1. Edge cracking in pre-forming of DP780
EXPERIMENTS Experiment Setup A laboratory experiment was designed to simulate the pre-forming and edge stretching of a steel strip. Figure 2 shows the tool used for pre-forming. Using this tool, a pre-formed wave can be created from a straight steel strip. Both one and two waves were made, with an example of a one wave pre-form shown in Figure 3.
FIGURE 2. Tools for pre-forming
FIGURE 3. Pre-formed strip
A pre-formed strip was then pulled in a tension test machine as shown in Figure 4. Pulling stopped when fracture occurred. Figure 5 shows a force-displacement curve for DP780 steel. This experiment was designed to use DIC for strain measurement. The steel samples were sprayed with paint to make a speckle surface for DIC strain measurements. For ce- Di spl acement Cur ve
For ce ( KN)
16 12 DP780 TD DP780 RD
8 4 0 0
10
20
30
40
Di spl acement ( mm)
FIGURE 4. Stretching of pre-formed strip
FIGURE 5. Force-displacement curves
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Strain Distributions The strain distributions were measured using DIC technology. Figure 6 shows the color maps of true major and minor strain (Hencky strain) distributions on the top surface of a pre-formed sample. The maximum strain is about 7% on the peak of the pre-form. Figures 7 and 8 show the distributions of strains at fracture for DP780 and 590R, respectively. It can be seen that the DIC technique can obtain the strain distributions clearly and capture the strains at fracture.
(a)
(b)
FIGURE 6. Major (a) and minor (b) strain distributions on a pre-formed sample of DP780
(a)
(b)
FIGURE 7. Major (a) and minor (b) strain distributions at fracture (DP780 transverse)
(a)
(b)
FIGURE 8. Major (a) and minor (b) strain distributions at fracture (590R transverse)
Another advantage using DIC is that the deformation history can be recorded, which provides very useful information for multi-stage forming analyses as in the current case. The above strain mapping figures show that the major and minor strain distributions are reasonably uniform along the width of the specimens. Therefore, the deformation history along the width can be represented in one section. Figures 9 to 12 show the major strains along the longitudinal direction of the specimens. Strains are measured on the top surfaces at various steps. The first step is pre-forming and bending is the dominant deformation. High tension is found at the peak and compression at the valleys near the peak. The specimen is then stretched in the tension machine. As the specimen is stretching,
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unbending occurs at the pre-formed region and tension strain is increasing along the whole specimen. In the specimen tested in the transverse direction, strains reach the failure limit and fracture occurs in the pre-formed regions as show in Figures 9 and 10 for DP780 and 590R steels, respectively. In the specimen tested in the rolling direction, fracture occurs outside the pre-formed regions and fracture strains are significantly higher as shown in Figures 11 and 12 for DP780 and 590R steels, respectively.
0. 08
Fracture
Maj or St r ai n
0. 07 0. 06
Pr ef or m 5mm 10mm 15mm 17. 5mm
0. 05 0. 04 0. 03 0. 02 0. 01 0
FIGURE 9. Major strain of DP780 along transverse direction
0. 3
Fracture
Maj or St r ai n
0. 25
Pr ef or m 5mm 10mm 15mm 20mm 25mm 30mm 35mm 38mm
0. 2 0. 15 0. 1 0. 05 0 - 0. 05
FIGURE 10. Major strain of 590R in transverse direction
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Fracture
0. 18 0. 16 Pr ef or m 5mm 10mm 15mm 20mm 25mm 30mm 37. 5mm
Maj or St r ai n
0. 14 0. 12 0. 1 0. 08 0. 06 0. 04 0. 02 0
FIGURE 11. Major strain of DP780 in the rolling direction.
Fracture
0. 4 Pr ef or m 5mm 10mm 15mm 20mm 25mm 30mm 35mm 40mm 45mm
0. 35 Maj or St r ai n
0. 3 0. 25 0. 2 0. 15 0. 1 0. 05 0 - 0. 05
FIGURE 12. Major strain of 590R in the rolling direction
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Fracture Three types of tensile failures were observed in the tests. Figure 13 shows those for DP780 steel. Type A is a milled edge without machining damage at the edge. Fracture initiated from the center of a specimen. Thinning and necking are observed. Type B is a sheared edge. Minor necking and thinning are observed around the fracture. In Type C, a sheared edge specimen is pre-formed then pulled. Fracture occurs at the pre-formed region. Fracture initiates at the edge and shear fracture is observed with minimum thinning and necking. Figure 14 is the comparison of the fracture strains for the three types. There is a critical strain value for sheared edge pre-forming and stretching. If a material does not fail at this value, failure will occur outside the pre-formed region, and the failure strain will be significantly higher. This critical strain value depends on the pre-forming geometry, edge damage severity, steel grade and material orientation.
0.50
0. 39 0.31
True Strain
0.40 0.30
0.26
0.30 0.15 0.11
0.20 0.10
0s he ar ed 78 0s he ar +p f 59 0m il le d 59 0s he ar ed 59 0s he ar +p f
78
78
0m il l ed
0.00
FIGURE 13. Three types of fractures of DP780
FIGURE 14. Comparison of the fracture strain for all types
SIMULATION Finite element analyses were conducted to simulate the pre-forming and stretching tests. Figure 15 shows the model. Full integration shell elements and solid elements were respectively used and run on Software packages of LS-Dyna and ABAQUS. Only half of the blank width was used due to the symmetry.
FIGURE 15. Finite element model for simulation
Result Comparison Various material models were used in the simulations. Figure 16 shows the comparison of the cases including von Mises, Hill’s and Barlat yield criteria with the isotropic hardening law, Hill’s yield criterion with modified Yoshida isotropic/kinematic hardening laws, and Johnson-Cook’s fracture model[13]. It can be seen that all models can catch the bending and unbending deformations in the pre-forming and stretching processes. The strain
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distribution trends can be predicted. It seems that the results are not sensitive to the different material models and no significant difference is found in the strain prediction. Though the trend can be predicted, it is difficult to obtain a good agreement with experiments in magnitude. The four fracture cases shown in Section 2 were simulated and the results were compared to the experimental data. Only the DP780 transverse case shows a reasonably good agreement with the experimental data, in which the fracture strain is at the lowest level. Figure 17 shows the discrepancies between the strain distributions. The simulations predicted more severe bending/unbending strains in the pre-formed regions. The results using explicit methods and solid elements over-predict, while shell elements under-predict the strain level. The results using implicit methods and shell elements over-predict the bending/unbending effects. Therefore, more studies are needed to improve the prediction of bending and unbending deformation for pre-forming.
FIGURE 16. Result comparison of various material models
FIGURE 17. Discrepancy of simulations in the pre-form region.
Forming Limit Figure 18 shows the forming limit diagram (FLD) strain mapping on the major-minor strain plane when fracture initiates. It can be seen that all strain points at the edge are along the uniaxial tension line and far below the limit curve at failure. The comparison of different forming limits and strain paths is shown in Figure 19. The strain paths at four different locations are shown in the figure. The edge point at the peak (EP) of the pre-forming moves up and down along the uniaxial tension path during bending and unbending. The edge point at the valley (EV) moves along the uniaxial compression line. The center point at the peak (CP) is in plane strain condition and moves along the major strain axis. The center point at the valley (CV) moves along the minor strain axis. The FLD is drawn based on the sheet thickness and n value. It lies far above the failure strain point. The milled edge failure is shown in a horizontal line at 0.3 major strain. It closes to the intersection of the FLD and the uniaxial tension line. The sheared edge failure line is at 0.15 major strain. The failure strain of a sheared edge with pre-forming is at 0.11 major strain. This line can be used to predict failure in this case. 0.3
Major Strain
Milled edge
0.2
Sheared edge
FLC
Sheared edge+preform
EP
0.1
EV CV CP
0 -0.1
-0.05
0
0.05
Minor Strain
FIGURE 18. FLD strain mapping at fracture initiation.
FIGURE 19. Comparison of different forming limit and strain paths.
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CONCLUSIONS AND FUTURE WORK • • • • • •
Pre-forming is important in the prevention of edge cracking of AHSS in flanging and edge stretching processes. The conventional approach needs to be modified for proper pre-forming of AHSS. Surface strain distributions of experimental pre-forming and stretching can be measured with DIC technique, which provides full field and history of strain distribution for pre-form design. There is a critical strain value for pre-forming and edge stretching which is a boundary of failure occurring in or out of the pre-formed region. This strain value depends on the pre-formed geometry, edge damage severity, steel grade and material orientation. The conventional FLD cannot be used to predict edge failure in pre-forming and post stretching since the failure limit is much lower due to sheared edge and stretch bending deterioration. In this case, the limit strain can be measured with the pre-form and stretch testing. A newly designed pre-forming tool will be used to test the critical strain values of different pre-formed geometries and various AHSS. A computer assisted technology is important for optimal designs of pre-forming. Simulation technologies need to be developed to accurately predict the bending and unbending effects of pre-forming and subsequent stretching.
ACKNOWLEDGMENTS The authors would like to acknowledge the tooling engineers of United States Steel Corporation Automotive Center, for inspirational discussions on the AHSS edge cracking issues. The material in this paper is intended for general information only. Any use of it in relation to specific applications should be based on independent examination and verification of its unrestricted availability for such use, and determination of suitability for the application by professionally qualified personnel. No license under any United States Steel Corporation patents or other proprietary interest is implied by the publication of this paper. Those making use of or relying upon this material assume all risks and liabilities arising from such use or reliance.
REFERENCE 1. A. Konieczny and T. Henderson “On Formability Limitations in Stamping Involving Sheared Edge Stretching ", SAE technical paper 2007-01-0340. 2. A J. Chintamani and S. Sriram, “Sheared Edge Characterization of Steel Products used for Closure Panel Applications”, SAE technical paper 2006-01-1589. 3. Jian Wang, Todd Link, Mattew Merwin, “AHSS Edge Formability in Sheared Edge Tension”, International Conference on New Developments in Advanced High-Strength Sheet Steels, 2008. 4. C. Chiriac, “Sheared Edge Performance of DP780 Steels -Conventional vs. Improved Bending and Flanging (IBF) Steels”, Materials Science and Technology (MS&T) 2010 , Houston, Texas. 5. X. M. Chen , C.Du, X. Wu , X. Zhu ,and S-D. Liu, “Sheet Metal Shearing and Edge Characterization of Dual Phase Sheets”, IDDRG International Conference, June 2009. 6. M. F. Shi and X. Chen, “Stretch Flange-ability Limits of Advanced High Strength Steels”, SAE technical paper 2007-01-1693. 7. Hua-Chu Shih and Ming F. Shi, " Experimental Study On Shear Fracture Of Advanced High Strength Steels", MSEC-ICMP 2008-72046. 8. Hudgins A.W. Matlocck D.K. Speer J.G., “Shear failure in bending of advanced high strength steels” Proceedings, Interational Deep Drawing Reasearch Group , Conference, pp. 53-64, Golden, CO, 2009. 9. Xiaoming Chen, Meng Luo, et.al.,“AHSS Shear Fracture Predictions Based on a Recently Developed Fracture Criterion”, SAE technical paper, 2010-01-0988. 10. Ming F.Shi, Xinhai Zhu, Cedric Xia, Thomas Stoughton, “Determination of Non-linear Isotropic/Kinematic Hardening Constitutive Parameters for AHSS using Tension and Compression Tests”, Numisheet 2008. 11. Xiaoming Chen, et. al, “Springback Prediction Improvement Using New Simulation Technologies”, SAE technical paper 2009-01-9881. 12. Ke Chen, Jianping Lin, Maokang Lu, Liying Wang, “Advanced High Strength Steel Sheet Forming and Springback Simulation”, Advanced Materials Research 2010, 97-101, 200-203. 13. LS-Dyna Keyword User Manual .
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Material Modeling of 6000 Series Aluminum Alloy Sheets with Different Density Cube Textures and Effect on the Accuracy of Finite Element Simulation Daisaku Yanagaa, Toshihiko Kuwabarab, Naoyuki Uemac and Mineo Asanoc a
Department of Mechanical Systems Engineering, Graduate School of Engineering, Tokyo University of Agriculture and Technology, 2-24-16, Nakacho, Koganei-shi, Tokyo 184-8588, JAPAN b Division of Advanced Mechanical Systems Engineering, Institute of Engineering, Tokyo University of Agriculture and Technology, 2-24-16, Nakacho, Koganei-shi, Tokyo 184-8588, JAPAN c Research & Development Center, Sumitomo Light Metal Industries, Ltd., 3-1-12, Chitose, Minato-ku, Nagoya 4558670, JAPAN Abstract. Biaxial tensile tests of 6000 series aluminum alloy sheet with different density cube textures were carried out using cruciform specimens similar to that developed by one of the authors [Kuwabara, T. et al., J. Material Process. Technol., 80/81(1998), 517-523.]. The specimens are loaded under linear stress paths in a servo-controlled biaxial tensile testing machine. Plastic orthotropy remained coaxial with the principal stresses throughout every experiment. Successive contours of plastic work in stress space and the directions of plastic strain rates were precisely measured and compared with those calculated using selected yield functions. The Yld2000-2d yield functions with exponents of 12 and 6 [Barlat, F. et al., Int. J. Plasticity 19 (2003), 1297-1319] are capable of reproducing the general trends of the work contours and the directions of plastic strain rates observed for test materials with high and low cube textures, respectively. Hydraulic bulge tests were also conducted and the variation of thickness strain along the meridian direction of the bulged specimen was compared with that calculated using finite element analysis (FEA) based on the Yld2000-2d yield functions with exponents of 12 and 6. The differences of cube texture cause significant differences in the strain distributions of the bulged specimens, and the FEA results calculated using the Yld2000-2d yield functions show good agreement with the measurement results. Keywords: aluminum alloy sheet, biaxial tensile test, yield function, hydraulic bulge test, finite element analysis. PACS: 46.35.+z, 83.60.-a, 83.85.-c
INTRODUCTION The crystallographic textures of rolled sheet metals have significant effects on the plastic deformation behavior [1-3]. In sheet metal forming simulations based on the phenomenological theory of plasticity, the difference in the plastic deformation characteristics of real sheet metals caused by difference in crystallographic texture is expected to be reproduced using appropriate yield functions [4-8]. Therefore, validation of this notion is crucial to deepen our knowledge of the effects and limitations of yield functions on the accuracy of forming simulations based on the phenomenological theory of plasticity. This study aims to clarify the accuracy of yield functions to reproduce differences in plastic deformation behavior caused by different crystallographic textures using two kinds of 6016 aluminum alloy sheets with the same chemical compositions but different cube textures. An appropriate yield function for each test material was firstly determined by biaxial tensile tests with linear stress paths using cruciform specimens. Hydraulic bulge tests were then conducted to apply large biaxial plastic deformation to the test materials and the thickness strain distribution along meridian directions of the bulged specimen was measured to investigate the effect of crystallographic texture difference on the deformation behavior of the aluminum alloys. Finally, finite element analyses (FEA) of the hydraulic bulging tests were conducted using selected yield functions determined from the biaxial tensile tests. The calculated thickness strains were compared to the measured thickness strains to reveal the effect of the yield functions on the accuracy of the FEA for the hydraulic bulge test. The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 800-806 (2011); doi: 10.1063/1.3623688 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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EXPERIMENTAL METHOD Test Material Two types of 6016 aluminum alloy sheet (Al-1.0Si-0.5Mg-0.1Mn (mass%), 0.9mm thick) were prepared with different intensities of the cube orientation: the intensities of the cube orientation at the 1/4 thickness position were 133 and 21 (at random), respectively. The former is referred to as HC (High Cube) and the latter as LC (Low Cube). The work hardening characteristics and r-values in different directions to the rolling direction (RD) are listed in Table 1. In order to reduce the effect of age hardening on the experimental results as much as possible, both materials were heat treated a sufficiently long period ago prior to the experiments. Table 1 Mechanical properties of test material Tensile direction / deg
0.2 /MPa
c* /MPa
n*
*
r -value**
High Cube (HC)
0 45 90
169 152 163
494 469 485
0.24 0.28 0.25
0.008 0.014 0.007
0.54 0.13 0.55
Low Cube (LC)
0 45 90
152 146 145
474 469 466
0.25 0.27 0.26
0.007 0.009 0.007
0.80 0.26 0.70
*Parameters for Swift’s hardening law, c( p ) , for the strain range of 0.002 p Bp , where Bp is the logarithmic plastic strain at the maximum load. **Measured at a uniaxial nominal strain of 0.10.
Biaxial Tensile Test In order to determine an appropriate anisotropic yield function that is able to reproduce the elastic-plastic deformation behavior of the material, biaxial tensile tests were performed using the cruciform specimen shown in Fig. 1. The geometry of the specimen is similar to that used by Kuwabara et al. [2, 9, 10]. Each arm of the specimen has seven slits, 30 mm long and 0.2 mm wide, at 3.75 mm intervals, so as to exclude the geometric constraint on the deformation of the 30×30 mm2 square gauge section. The slits were fabricated by laser-machining. Hereafter, the RD and transverse directions (TD) of the material are defined as the x- and y-axes, respectively. Biaxial tensile forces (Fx, Fy) were applied to the cruciform specimen using a servo-controlled biaxial tensile testing machine [9]. The nominal tensile stress components ( N x , N y ) were in fixed proportions during each test: N x : N y 1:0, 4:1, 2:1, 4:3, 1:1, 3:4, 1:2, 1:4 and 0:1. The biaxial total strain components ( x , y ) were measured using four uniaxial strain gauges (Tokyo Sokki Kenkyujo, YFLA-2) mounted on the centerlines of the specimens at
Strain gauge
Fig. 1 Cruciform specimen for biaxial tensile test
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(x, y)=(±10 mm, 0). The true stress components ( x , y ) were determined by dividing (Fx, Fy) by the current crosssectional area of the gauge section, which was determined from measurements of the plastic strain components ( xp , yp ) with an assumption of constant volume. xy was assumed to be zero, because x and y were measured on the centerlines of the specimen. For N x : N y 1:0 and 0:1, standard uniaxial tensile specimens (JIS 13 B-type) were used. The equivalent plastic strain rate was 1 to 4×10-4 s-1. According to the FEA of the cruciform specimen and the strain measurement position as shown in Fig. 1, the error of the stress measurement was estimated to be less than 2 % using selected isotropic (von Mises) and anisotropic yield functions as material models, [11, 12]. To evaluate the work hardening behavior of a material under biaxial tension, the concept of the contour of plastic work in the stress space was introduced [13, 14]. The stress-strain curve obtained from a uniaxial tensile test along the RD of the material was selected as a reference datum for work hardening; the uniaxial true stresses 0 and the plastic work per unit volume W0 corresponding to particular values of offset logarithmic plastic strains 0p were determined. The uniaxial true stresses 90 obtained from a tensile test in the TD and the biaxial true stress components ( x , y ) obtained from biaxial tensile tests were then determined at the same plastic work as W0. The stress points ( 0 , 0) , (0, 90 ) and ( x , y ) thus plotted in the principal stress space comprise a contour of plastic work corresponding to a particular value of 0p . For a sufficiently small value of 0p the corresponding work contour can be practically viewed as a yield locus.
Hydraulic Bulge Test Hydraulic bulge tests were performed in order to quantitatively evaluate the effect of the crystallographic texture difference on the deformation behavior during biaxial stretch forming. Fig. 2 shows the experimental apparatus used for the hydraulic bulge test. The diameter of the die opening was 150 mm and the blank diameter was 220 mm. The normal strain components along the rolling (RD) and transverse (TD) directions of the original sheet were measured using strain gauges attached at a distance of 5 mm from the center of the blank. The radius of curvature at the top of the bulged specimen was measured along the direction 45º from the RD using a spherometer, as shown in Fig. 2, the gauge length of which was 40 mm. The spherometer can move in the vertical direction and rotate about an axis normal to the plane of Fig. 2, so that it is always in contact with the bulged specimen at three points. The hydraulic pressure was controlled during each test so that the equivalent plastic strain rate was kept approximately constant ( 0.001 s-1). Samples were bulged up to a height of 38 mm. They had an array of grids of 10 mm square for the measurement of the radial and circumferential plastic strains ( rp , p ) , along the meridian directions 0, 45 and 90 from the RD. After the bulge test, the thickness strains zp along the meridian lines were determined as zp rp p . Blank holding force
Spherometer
Pressure
(a)
P 150 150 (b)
190 Fig. 2 Experimental apparatus for the hydraulic bulge test.
Fig. 3 Schematic illustration of FEM model. (a) Tool. (b) Initial mesh division of a blank
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FINITE ELEMENT SIMULATION OF HYDRAULIC BULGE TEST Material forming simulations of the hydraulic bulge test were carried out using Abaqus/Standard Ver.6.9-1 [15]. 오류! 참조 원본을 찾을 수 없습니다. Fig. 3 shows the finite element mesh used for the analysis. One quarter of a circular blank was analyzed because of the orthotropic anisotropy of the material. The blank diameter was 190 mm and the nodal displacement along the periphery of the blank was assumed to be zero, because the radial position of the draw-bead on the experimental die was 95 mm. 4-node shell elements, S4R, were used. The increment of element division was 2.5º in the circumferential direction and 1 mm in the radial direction in the range of radial coordinates 30 mm R 95mm. Mesh division was automatically performed in the range of R 30 mm. A Surface-to-Surface contact condition of the blank to the die was selected with a blank holding force of 30 kN and a coefficient of friction of 0.3.
RESULTS AND DISCUSSION
1.2
1.2
1.0
1.0
0.8
0.8
0.6 0.4 0.2
von Mises Hill '48 Yld2000-2d M=6 p (0 =0.002) M = 12 p (0 =0.040)
High Cube p 0 0.002 0.010 0.020 0.030 0.040
0.0 0.0
0.2
y / 0
y / 0
Fig. 4 shows the measured stress points that comprise the contours of plastic work. All stress values comprising a work contour are normalized by 0 corresponding to a specific value of 0p . The theoretical yield loci based on the von Mises [16], Hill’s quadratic (Hill ‘48) [17], and the Yld2000-2d yield functions [18] are also depicted in the figure. The unknown parameters of the Hill's quadratic yield function were determined using r0, r45, r90 and 0/0, and those of the Yld2000-2d yield function were determined using r0, r45, r90 and rb, and 0/0, 45/0, 90/0 and b/0, where r and
0.4
0.6
x / 0
0.6
Low Cube p 0 0.002 0.010 0.020 0.030 0.045
0.4 0.2
0.8
1.0
0.0 0.0
1.2
0.2
von Mises Hill '48 Yld2000-2d M=4 p (0 =0.002) M=6 p (0 =0.045)
0.4
(a)
0.6
0.8
x / 0
1.0
1.2
(b)
135
Direction of plastic strain rate / °
Direction of plastic strain rate / °
Fig. 4 Measured stress points comprising contours of plastic work, compared with theoretical yield loci. Each symbol corresponds to a contour of plastic work for a particular value of 0p .
High Cube
y
90
45
x p
0
0.002 0.010 0.020 0.030 0.040
0
-45
0
15
30
45
60
von Mises Hill '48 Yld2000-2d M=6 p (0 =0.002) M = 12 p (0 =0.040)
Loading direction / °
75
90
(a)
135 Low Cube
y 90
45
x p
0
0.002 0.010 0.020 0.030 0.045
0
-45
0
15
30
45
60
von Mises Hill '48 Yld2000-2d M=4 p (0 =0.002) M=6 p (0 =0.045)
Loading direction / °
75
90
(b)
Fig. 5 Comparison of the directions of measured plastic strain rates with those of the local outward vectors normal to yield loci.
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are the r-value and tensile flow stress measured at from the RD, respectively, and rb and b are the ratio of plastic strain rates, d yp / d xp , and the flow stress at equibiaxial tension, N x : N y 1:1, respectively. The values of r0, r45 and r90 used were the same as those in Table 1, both for the Hill ’48 and Yld2000-2d yield functions. The values of 45/0, 90/0, b/0 and rb used to determine the Yld2000-2d yield functions correspond to those comprising the work contours for the 0p as referred to in the figure. We find that the variation in shape of the work contours is small for a strain range of 0.002 0p 0.010, while for 0p 0.010 the work contours show a tendency of approaching to the Yld2000-2d yield function with higher exponents with the increase of 0p . The Yld2000-2d yield function with an exponent of M 6 for HC and 4 for LC had the closest agreement with the measured work contour for 0p 0.002, while the M that provided the closest agreement with the measured work contour at a highest strain ( 0p 0.040 for HC and 0p 0.045 for LC) was 12 for HC and 6 for LC, respectively. In order to validate the normality flow rule for the yield functions, the directions of plastic strain rates were measured at each linear stress path, and the results are shown in Fig. 5. The Yld2000-2d yield function provided the closest agreement with the measurement. From the results of Figs. 4 and 5, it is concluded that the Yld2000-2d yield function is capable of reproducing the plastic deformation behavior of the material more accurately than other yield functions. Fig. 6 shows the measured thickness strain along the meridian lines, 0, 45 and 90º directions from RD, of hydraulic bulge specimens with a bulge height of 38 mm, compared with those calculated using FEA with selected yield functions. In Fig. 6a, the equivalent stress-equivalent plastic strain curve used in the FEA was determined by approximating the measured uniaxial x - xp curve using Swift’s power law as shown in Table 1, while in Fig. 6b, it was determined by approximating the b - zp curve obtained from the hydraulic bulge test. Comparison of the experimental results for HC and LC shows that zp at the top of the bulged specimen is larger for LC than for HC, and zp at the periphery (near die profile) is larger for HC than for LC. This difference in thickness strain distribution between HC and LC can be explained from the difference in the shapes of the work contours, as shown in Fig. 4. The difference in flow stresses between the top (equibiaxial tension) and the periphery 0.4
0.3
Yld2000-2d
0.2
von Mises Hill '48 0° 45° 90° Yld2000-2d ( M = 12 ) 0° 45° 90°
0.1
High Cube Swift's power law: Uniaxial 0.0
0.3
10
20
30
40
50
60
70
80
90
0.1
Low Cube Swift's power law: Uniaxial
100
0
10
Initial radial coordinate R / mm
20
30
40
0.3
0.2 Yld2000-2d
Experimental von Mises 0° Hill '48 45° 0° 90° 45° 90° Yld2000-2d ( M = 12 ) 0° 45° 90°
10
20
30
40
50
60
70
80
90
Hill '48
p
High Cube Swift's power law: Biaxial 0
70
80
90
100
(a2)
0.1
0.0
60
0.4
Logarithmic plastic strain |z |
p
Logarithmic plastic strain |z |
Hill '48
50
Initial radial coordinate R / mm
(a1) 0.4
von Mises Hill '48 0° 45° 90° Yld2000-2d (M=6) 0° 45° 90°
Yld2000-2d
0.2
0.0
0
Experimental 0° 45° 90°
Hill '48
p
Experimental 0° 45° 90°
Logarithmic plastic strain |z |
Hill '48
p
Logarithmic plastic strain |z |
0.4
100
0.3
0.2 Yld2000-2d
Experimental von Mises 0° Hill '48 45° 0° 90° 45° 90° Yld2000-2d (M=6) 0° 45° 90°
0.1
Low Cube Swift's power law: Biaxial 0.0
0
10
20
30
40
50
60
70
80
90
100
Initial radial coordinate R / mm
Initial radial coordinate R / mm
(b1) (b2) Fig. 6 Measured thickness strain along the meridian lines of hydraulic bulge specimens, compared with those calculated using finite element analysis with selected yield functions.
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(plane strain tension) is smaller in HC than in LC; therefore, the plane strain elongation at the periphery is accelerated more in HC than in LC, and the reduction in thickness at the periphery becomes accordingly larger in HC than in LC. Thus, it is concluded that the difference in crystallographic texture causes the relative difference in the magnitudes of flow stresses between equibiaxial tension and plane strain tension and, consequently, causes the difference in the thickness distribution along the radial direction for the two materials. Comparison of Fig. 6 (a) and (b), it is clear that the calculated results obtained using the Yld2000-2d yield functions in Fig. 6 (b1) and (b2) provide the closer agreement with the measurement than those in Fig. 6 (a1) and (a2). Thus, it is recommended to carry out the biaxial stress test that closely reproduces the real stress state to be analyzed in order to determine the work hardening equation used in the FEA. Comparison of the experimental and calculated results indicates that the Yld2000-2d yield functions are capable of reproducing the experimental tendency with good accuracy for both materials, and have the closer agreement with the experimental results than other yield functions, regardless of the work hardening laws used in the FEA. In contrast, the von Mises and Hill’s quadratic yield functions overestimate and underestimate the reduction in thickness at the top and periphery, respectively, which results in a large deviation from the experimental results. The results in Fig. 6 are therefore in accordance with those in Fig. 4 and Fig. 5. Thus, it is concluded that the biaxial tensile testing with cruciform specimens is an effective method to determine appropriate anisotropic yield functions for given sheet metals to improve the predictive accuracy of FEA for stretch forming.
CONCLUSIONS Biaxial tensile tests of 6000 series aluminum alloy sheets with different density cube textures (HC and LC) were conducted using cruciform specimens. The difference in crystallographic texture was clearly detected as the difference in the plastic deformation behavior in the biaxial tensile tests; the shapes of the work contours and the directions of the plastic strain rates were in good agreement with those calculated using the Yld2000-2d yield functions. FEA of the hydraulic bulge test showed that the Yld2000-2d yield function, which had better reproducibility of the biaxial tensile test results than other yield functions, also resulted in closer agreement with the hydraulic bulge test results for both materials. It is concluded that the biaxial tensile testing with cruciform specimens is an effective experimental method to determine appropriate yield functions for given sheet metals and is useful to improve the predictive accuracy of FEA for aluminum alloy sheet forming processes.
ACKNOWLEDGMENT The authors are grateful to Professor Jeong Whan Yoon of the Swinburne University of Technology for kindly providing the subroutine program for the Yld2000-2d yield function used in the FEA.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.
F. Barlat, Mater. Sci. Engng. 91, 55-72 (1987) T. Kuwabara, A. Van Bael and E. Iizuka, Acta Mater. 50, 3717-3729 (2002). K. Yoshida, M. Kuroda, and S. Ikawa, Acta Mater. 55, , 4499-4506 (2007)． J.-W. Yoon, F. Barlat, R.E. Dick, K. Chung and T.J. Kang, Int. J. Plasticity 20, 495-522 (2004). J.-W. Yoon, F. Barlat, R.E. Dick, and M.E. Karabin, Int. J. Plasticity 22, 174-193 (2006). T. Kuwabara, Int. J. Plasticity 23, 385-41 (2007). D. Banabic, F. Barlat, O. Cazacu and T. Kuwabara, Int. J. Mater. Form. 3, 165-189 (2010). T. Kuwabara, K. Hashimoto, E. Iizuka and J.-W. Yoon, J. Mater. Processing Technol., 211, 475–481 (2011). T. Kuwabara, S. Ikeda and T. Kuroda, J. Mater. Process Technol. 80-81 517-523 (1998). T. Kuwabara, M. Kuroda, V. Tvergaard and K. Nomura, Acta Mater. 48, 2071-2079 (2000). Y. Hanabusa, H. Takizawa and T. Kuwabara, Steel Research Int. 81, 1376-1379 (2010). Y. Hanabusa, H. Takizawa and T. Kuwabara, J. JSTP 52, 282-287 (2011). (in Japanese) R. Hill, S.S. Hecker, and M.G.. Stout, Int. J. Solids Struct. 31, 2999-3021 (1994). R. Hill and J.W. Hutchinson, J. Appl. Mech. 59, S1-S9 (1992). Abaqus Analysis, User’s Manual, Version 6.9, Dassault Systemes, 2009. R. Von Mises, Göttingen Nachrichten, math.-phys. Klasse, 582-592 (1913). R. Hill, Proc. Roy. Soc. London A193, 281-297 (1948). F. Barlat, J.C. Brem, J.W. Yoon, K Chung, R.E. Dick, D.J. Lege, F. Pourboghrat, S.H. Choi and E. Chu, Int. J. Plasticity 19,
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1297-1319 (2003).
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A User-friendly 3D Yield Function for Steel Sheets and Its Application Fusahito Yoshidaa, Shohei Tamuraa, Takeshi Uemorib and Hiroshi Hamasakia a
Department of Mechanical Science & Engineering, Hiroshima University, 1-4-1, Kagamiyama, Higashi-Hiroshima, 739-8527, Japan b Department of Mechanical Engineering, Kindai University, 1, Umenobe, Higashi-Hiroshima, 739-2116, Japan
Abstract. A 6th-order polynomial type 3D yield function, which has a high flexibility of describing anisotropic behavior of steel sheets is proposed. A user-friendly scheme of material parameter identification for the model, where only limited experimental data of r-values and yield stresses in three directions ( r0 , r45 , r90 and V 0 , V 45 , V 90 ) plus equi-biaxial yield stress ( V x
Vy
V b ) are needed, is presented. The accuracy of this yield function is verified by comparing the
numerical predictions with the corresponding experimental data on planar anisotropy of r-values and flow stress directionality, as well as the shape of yield loci, on several types of steel sheets (high r-value IF steel and high strength steel sheets of 440-980MPa TS levels). The advantage of this model is demonstrated by showing a numerical simulation of hole-expansion test. Keywords: Yield function, anisotropy, sheet metal forming
PACS: 62.20.fq INTRODUCTION For accurate numerical simulation of sheet metal forming and springback, the use of plasticity models that properly describe material behavior is of vital importance. For material modeling, two items should be considered, one is the yield function to express the sheet anisotropy, and the other is the kinematic hardening law to describe the Bauschinger effect and cyclic workhardening. The present authors have already proposed an advanced kinematic hardening law (Yoshida-Uemori model [1-3]), and the accuracy of this model have been verified by many researchers (e.g., Eggertsen & Mattiasson [4], Ghaei & Green [5]). To describe planar r-value anisotropy and flow stress directionality, many types of anisotropic yield functions have been proposed in the past (e.g., Hill [6-8], Gotoh [9], Barlat et al. [10-12], Banabic [13], et al., Vegter & Boogaard [14] etc.). However, still classical Hill48 quadratic function [6] is widely used for FE simulation in the sheet metal forming industry, because its material parameters are easily determined from uniaxial tension experiments in three directions (0, 45 and 90o with respect to rolling direction) of a sheet, and furthermore, its 3D expression is clearly presented. Although most of FE simulations of sheet metal forming are performed with shell elements on the assumption of plane stress condition, in some cases of forming, we have to take into account the 3D stress condition, e.g., sheet forging, bottoming, ironing etc.. The present paper begins with the summary of some existing yield functions, and points out the advantages of these models and also their drawbacks, and then proposes a new 3D yield function of 6th-order polynomial. A userfriendly scheme of material parameter identification for the model is also presented, where only limited experimental data of r-values and yield stresses in three directions ( r0 , r45 , r90 and V 0 , V 45 , V 90 ) plus equi-biaxial yield stress ( V x V y V b ) are needed. The accuracy of this yield function is verified by comparing the numerical predictions with the corresponding experimental data on planar anisotropy of r-values and flow stress directionality, as well as the shape of yield loci, on several types of steel sheets (high r-value IF steel and high strength steel sheets of 440-980MPa TS levels). The advantage of this model is demonstrated by showing a numerical simulation of holeexpansion test. The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 807-814 (2011); doi: 10.1063/1.3623689 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
807
SUMMARY OF SOME EXISTING YIELD FUNCTIONS Polynomial Type Yield Functions The most popular anisotropic yield function is Hill48 model [6], of which plane stress description is given by the following quadratic function:
IH
A1V x2 A2V xV y A3V y2 3 A4W xy2
V 02 ,
(1)
where, A1 ~ A4 are anisotropic parameters which are all unity for the von Mises isotropic material. They can be determined by using r-values r0 , r45 and r90 in three direction with respect to rolling direction of a sheet, as follows:
A1
A1( r )
1, A2
A2( r )
2r0 , A3 1 r0
A3( r )
r0 (1 r90 ) , A4 r90 (1 r0 )
A4( r )
(r0 r90 )(1 2r45 ) . 3r0 (1 r0 )
(2)
Alternatively, if we use the normalized yield stress values V 0 , V 45 and V 90 in the three directions, as well as the equibiaxial flow stress V b : V 0 1 , V 45 V 45 / V 0 , V 90 V 90 / V 0 and V b V b / V 0 , these parameters are determined as
A1
A1(V )
1, A2
A2(V )
1
1
V
2 90
1
V
2 b
, A3
A3(V )
1
V
2 90
, A4
A4(V )
1§ 4 1 · ¨ 2 2 ¸. 3 © V 45 V b ¹
(3)
Hereafter, we call Hill48 model when using parameters A1( r ) ~ A4( r ) ’Hill48-r’, and when using parameters
A1(V ) ~ A4(V ) ’Hill48-V’. As already well known that, Hill48-r predicts the planer r-value anisotropy reasonably well for most of sheet metals, but it describes the flow stress directionality too strongly. Furthermore, the shape of the yield loci calculated by Hill48-r do not fit the experimental observations, e.g., for high r-value sheets it overestimates the equi-biaxial yield stress V b , and for low r-value sheets (e.g., aluminum sheets) underestimates it (so-called ‘anomalous’) . Hill48-V well predicts the stress directionality, as well as the shape of yield loci for steel sheets, but it poorly predicts the r-value planar anisotropy. For aluminum sheets, the shapes of yield loci are very different from its predictions. Despite such drawbacks, it has an advantage over the other models that its material parameters are explicitly identified by using just three r-values (in Hill48-r), and its 3D expression is clearly presented. The quadratic ellipse function always guarantees the convexity of the yield locus. To overcome the above problems in Hill48 model, Gotoh [9] proposed a 4th-order polynomial type yield criterion. It can be expressed by the following equation by using nine material parameters B1 ~ B9 which are all unity for the von Mises isotropic material.
IG
B1V x4 2 B2V x3V y 3B3V x2V y2 2 B4V xV y3 B5V y4 6 B6V x2 B7V xV y B8V y2 W xy2 9 B9W xy4
(4)
V 64 .
Here, material parameters B1 ~ B5 are determined by using three direction r-values r0 , r45 and r90 , as well as V 90 and V b , explicitly, as follows:
B1
A1( r )
1, B2
A2( r )
2r0 , B5 1 r0
1
V 904
, B4
2 B5 r90 , B3 1 r90
· 1§ 1 B1 2 B2 2 B4 B5 ¸ . ¨ 3 © V b4 ¹
(5)
To identify the rest of four material parameters B6 ~ B9 , we need r45 and V 45 , and two more experimental data. Gotoh recommended the use of r22.5 and V 22.5 which are obtained from a uniaxial tension experiment in 22.5o direction. However, when using such a set of material parameters identified by Gotoh’s explicit approach, the overall characterization of the flow stress directionality, V D , and planar r-value anisotropy, rD , can show unbalanced big errors. To overcome this problem, Hu [15] proposed to use the following equations:
808
V 22.5
(V 0 V 45 ) / 2 , V 67.5
(V 45 V 90 ) / 2
(6)
instead of r22.5 and V 22.5 . Furthermore, Hu [15] recently gave a 3D expression for Gotoh’s yield function. The convexity of this yield locus is not generally guaranteed, however, we can check it analytically (see Soare et al. [16]). Soare et al. [16] recently discuss four, six and eight-order polynomial yield functions, where they proposed a scheme of material parameter identification based on the numerical optimization technique which includes the constraint to satisfy the convexity of the yield loci. This approach is sophisticated, however, for the optimization, we need a lot of additional material data other than r0 , r45 , r90 , V 0 , V 45 , V 90 and V b .
Yield Functions Based on Linear Transformation of Stress Tensors Different from the polynomial type, several anisotropic yield functions have been proposed based on the linear transformation of stress tensors (Barlat & Lian [10], Barlat et al. [11], Banabic et al. [12]). Among these, the simplest plane stress model is Yld89 [10], as follows:
IB 89 K1
a K1 K 2
M
a K1 K 2
1 V x hV y , K 2 2
M
2 a 2 K1
M
2V 0M , (7)
2 1 2 V x hV y pV x , 4
where exponent M takes a value of 6 for BCC or 8 for FCC metals. Three anisotropic parameters a, h and p are determined by using three of the six material data of r0 , r45 , r90 , V 45 , V 90 and V b . Hereafter, when the parameters are identified from r0 , r45 and r90 , we call Yld89 model, ‘Yld89-r’. In contrast, when using and
V 45 , V 90
V b , we call it ‘‘Yld89-V’. This model has been extended to have a higher flexibility of description of yield loci,
e.g., Barlat et al’s Yld2000-2d [11] has a form:
IB
I c I cc 2V a
Ic
X 1c X 2c , I cc
a
X c C 's
Lc ,
X ''
a a 2 X 2cc X 1cc 2 X 1cc X 2cc ,
(8)
C '' s = Lcc,
where V and s are the stress tensor and its deviator, and C ' and C '' are linear transformation matrices representing the anisotropy, where their components are all unity for a isotropic material. By using principal values of thus obtained X ' and X '' ,
X 1' , X 2' , X 1'' and X 2'' , two functions I ' and I '' are defined. Since I ' and I '' are
real-valued convex functions, the yield function and it has been verified that
IB
IB
is also real-valued convex. This approach is very sophisticated
has a high capability of expressing the anisotropy of sheet metals (e.g. Barlat et al.
[12]). Although a 3D formulation has also discussed (Barlat et al. [12]), its expression by the stress components is rather complex because it is only obtained from a calculation scheme for 3D principal stress values.
Plane Stress Yield Function by Interpolation of Biaxial Stress States Vegter & Boogaard [14] proposed an anisotropic plane stress yield function based on the interpolation by second-order Bezier curves. In this model, six reference stress points are selected in the principal stress plane V 1 , V 2 , which are two uniaxial stress points, equi-biaxial stress point, two plane-strain stress points, and two pure shear stress points. Based on the normality rule, giving plastic strain ratios U
H2 / H1 at these reference stress points,
the V 1 V 2 yield locus is defined. For full description of the yield locus, we need a set of data of the reference stress points and the corresponding plastic strain ratios (outward normals) in several principal directions defined by the angle between the first principal stress V 1 and the rolling direction, T . For that purpose, Fourier series are used, where their parameters are determined from the V 1 V 2 yield points and the corresponding outward normals
809
for several choices of T -directions. An advantage of this model is a high flexibility of description of the yield locus. On the other hand, it has a drawback in the material parameter identification, i.e., we have to conduct unconventional biaxial experiments by using specimens cut from a sheet in several different directions T . Furthermore, its extension to 3D model is difficult.
PROPOSITION OF SIXTH-ORDER POLYNOMIAL YIELD FUNCTION Yield Function and Its Material Parameter Identification For the plane stress state, we propose the following anisotropic yield function of a 6th-order polynomial.
IY
C1V x6 3C2V x5V y 6C3V x4V y2 7C4V x3V y3 6C5V x2V y4 3C6V xV y5 C7V y6
9 C8V x4 2C9V x3V y 3C10V x2V y2 2C11V xV y3 C12V y4 W xy2 27 C13V x2 C14V xV y C15V y2 W xy4 27C16W xy6
(9)
V 06 .
Here, C1 ~ C16 are anisotropic parameters which are all unity for the von Mises isotropic material. Parameters C1 ~ C7 can be determined explicitly by using two uniaxial flow stresses, V 0 and V 90 , three biaxial stresses: equibiaxial stress V b ; plane strain stresses V a and V c , as schematically shown in FIGURE 1, as well as r-values r0 and r90 , as follows:
A1( r )
1, C2
B2
A2( r )
2r0 , C7 1 r0
1
B1
C3
16 § 1 · 1 § 1 · 8 § 1 · 7 1 7 35 ¨ ¸ ¨ ¸ ¨ ¸ C 2 C6 C7 , 9 © V a6 ¹ 3 © V b6 ¹ 9 © V c6 ¹ 4 2 12 24
V 906
, C6
2C7 r90 , 1 r90
C1
C4
16 § 1 · 5 § 1 · 16 § 1 · 3 3 45 45 ¨ ¸ ¨ ¸ ¨ ¸ C2 C6 C7 , 7 © V a6 ¹ 7 © V b6 ¹ 7 © V c6 ¹ 2 2 28 28
C5
8 § 1 · 1 § 1 · 16 § 1 · 1 7 35 7 ¨ 6 ¸ ¨ 6 ¸ ¨ 6 ¸ C2 C6 C7 . 9 © Va ¹ 3 © Vb ¹ 9 © Vc ¹ 2 4 24 12
(10)
For shear-normal stress coupling parameters C8 ~ C12 , we assume that they can be determined using Gotoh’s parameters B1 ~ B5 , by introducing a coefficient [ as follows:
FIGURE 1. Five reference stress points for determining (V x , V y ) yield locus: uniaxial stresses V 0 and V 90 ; equi-biaxial stress V b ; plane strain stresses V a and V c .
810
Ci 7
[ Bi , i 1, ",5 .
(11)
For a prescribed value of [ , the rest of four coupling parameters C13 ~ C16 are determined explicitly so as to satisfy three flow stresses, V 22.5 , V 45 and V 67.5 , as well as r45 . Coefficient [ is identified by minimizing the difference between experimental and numerical results for overall rD -values. One of the advantages of this model is a high flexibility of description of the yield locus. Especially, in the present model we can directly use the plane-strain yield points V a and V c , similarly to Vegter’s model. The convexity of this yield locus is not generally guaranteed (refer to Soare et al. [16]), however, it will be confirmed numerically by checking the Hessian matrix w 2I / wV i wV j is semi-positive definite. In the present model, from the assumption of Eq. (10), if the biaxial (V x , V y ) yield loci of 4th- and 6th-order polynomial are convex, the convexity of (V x , V y ,W xy ) yield loci in the (V x , V y ) plane for given shear stress
W xy
is always guaranteed, as
shown in FIGURE 2. The plane-stress model Eq. (9) is easily extended to 3D version as follows:
C1 V x V z 3C2 V x V z V y V z 6C3 V x V z V y V z
IY
6
5
4
2
7C4 V x V z V y V z 6C5 V x V z V y V z 3C6 V x V z V y V z C7 V y V z 3
3
^
4
2
5
9 C8 V x V z 2C9 V x V z V y V z 3C10 V x V z V y V z 4
3
2
2C11 V x V z V y V z C12 V y V z 3
^
4
` W
2 xy
W yz2 W zx2
27 C13 V x V z C14 V x V z V y V z C15 V y V z 2
27C16 W xy2 W yz2 W zx2
3
2
` W
2 xy
6
2
(12)
W yz2 W zx2 . 2
V 06
Here note that the expressions for the normal-shear coupling terms are different from those of Soare et al’s 3D polynomial yield function [16]. 1.5
1
Vy㻌V
0.5
0
-0.5
-1 Experiment
-1.5 -1.5
-1
-0.5
0
0.5
1
1.5
Vx㻌㻛V FIGURE 2. (V x , V y ,W xy ) yield loci in the (V x , V y ) plane for several values of shear stress W xy on 590MPa HSS. Experimental
0.43, r45 1.41, r90 0.61, V 45 0.936, V 90 1.047 and V b 1.145 are estimated from the calculation by Yld89-V ( P 0 .in Eq. (13)).
data are from Hashimoto et al. [17]. r0 stresses V a
1.094 and V c
811
1.000 . Plane strain
User Friendly Scheme of Anisotropic Parameter Identification For users’ convenience, we will offer a scheme of anisotropic parameters identification, in which all the parameters are determined using only r0 , r45 , r90 , V 0 , V 45 , V 90 and V b . As mentioned above, besides these seven material data, theoretically we need some additional data, two uniaxial yield stresses, V 22.5 and V 67.5 , two planestrain yield stresses, V a and V c , and furthermore some of rD -values. Instead of conducting biaxial stretching experiment, we will estimate these values by using simple anisotropic yield functions, e.g., Hill48 or Yld89, as follows. Since uniaxial yield stresses V D in D-directions can be estimated by Hill48-V or Yld89-V, and rD -values by Hill48-r or Yld89-r reasonably well (see FIGURE 3), we may use such estimations for material parameter identification. Note that the values of V 22.5 and V 67.5 calculated by Hill48-V and Yld89-V are almost the same as Hu’s assumption given by Eq. (6). For the prediction of two plane-strain yield stresses,
Va
and V c , we may use
either Hill48-V, Yld89-Vpredictions, or their combinations:
Va
PV aHill 48 1 P V aYld 89 , V c
PV cHill 48 1 P V cYld 89 ,
(13)
where P is a weighting coefficient, which may depend on materials. For example, the calculated yield locus on 590 MPa HSS is compared with the experimental result (after Hashimoto et al. [17]) in FIGURE 2, where V a and V c are estimated by Yld89-V ( P
0 in Eq. (13)).
Verification of the Yield Function for Steel Sheets To verify the proposed yield function, uniaxial tension experiments in various directions were conducted on several types of steel sheets such as a high r-value IF steel, high strength steel sheets (HSS) of 440-980MPa tensile strength levels (eighteen different sheets), where data of the flow stress directionality V D and the r-value planar anisotropy rD were obtained. Moreover, (V x , V y ) yield loci were obtained by performing biaxial stretching experiments using a cruciform specimen. Besides our own experimental data, previously published data on steel sheets by Kuwabara et al. [18, 19] and Hu [20] were also checked. The calculated results of V D and rD , as well as the yield loci, agree reasonably well with the corresponding experimental results. For example, FIGURE 3 shows the flow stress directionality V D and the r-value planar anisotropy rD , predicted by the present model (indicated as ‘Yoshida-St model’) and the corresponding experimental results on 590 MPa HSS (experimental data are from Hashimoto et al. [17] which are for the same sheet as shown in FIGURE 2). 1.15
1.5
Normalized flow stress VD
1.1 1.05
Lankford value rD
Experiment Hill'48-r Hill'48-V Yld2000-2d (a=6) Yoshida-St
1 0.95 0.9
1
0.5
Experiment Hill'48-r Hill'48-V Yld2000-2d (a=6) Yoshida-St
0.85 0.8
0 0
15
30
45
60
75
0
90
15
Tensile angle from R.D. D [deg.]
FIGURE 3. Flow stress directionality
VD
30
45
60
75
90
Tensile angle from R.D. D [deg.]
(left) and r-value planar anisotropy rD (right) on 590 MPa HSS. Experimental data are from Hashimoto et al. [17].
812
3
1.06 1.04
Lankford value rD
Normalized flow stress VD
1.08
1.02 1 0.98 Experiment Hill'48-r Gotoh Yoshida-St
0.96 0.94
0
15
30
45
60
75
2.5
2
1.5
1
90
Experiment Hill'48-r Gotoh Yoshida-St 0
Tension angle from R.D. D[deg] FIGURE 4. Flow stress directionality
VD
15
30
45
60
75
90
Tension angle from R.D. D[deg]
(left) and r-value planar anisotropy rD (right) on high r-valued IF steel. Experimental
1.5
y
Normalized yield stress in T.D. V /V
0
data are from Kitayama et al. [21].
1
0.5 Experiment Hill'48-r Gotoh Yoshida-St 0 0
0.5
1
1.5
Normalized yield stress in R.D. Vx㻌/V0
FIGURE 5. Biaxial yield locus of high r-valued IF steel. Experimental data are from Kitayama et al. [21].
For the present material parameter identification, experimental data of employed, together with the estimation for
Va
r0 , r45 , r90 , V 0 , V 45 , V 90 and V b are only
and V c by Yld89-V. Both for V D and rD , the predictions are agree
with the experimental results very well. In FIGURE 4, the flow stress directionality V D and the r-value planar anisotropy rD predicted by the present model and the corresponding experimental results on a high r- value IF steel sheet [21]. The results on the biaxial yield locus are also shown in FIGURE 5. In these figures, calculated results by Hill48-r and Gotoh’s 4th-order yield function are also indicated. It is seen from this, the calculated yield locus by Gotoh’s model is too distorted, in contrast, the newly proposed Yoshida-St model simulates the locus reasonably well.
APPLICATION: SIMULATION OF HOLE EXPANSION Recently, Hashimoto et al. [17] pointed out that the numerical simulation of hole expansion is strongly influenced by a choice of yield functions, especially for the strain localization characterization. We performed FE simulation of hole expansion on 590HSS using the proposed model, as well as some other models such as Hill48-V,
813
Gotoh and Yld2000. The simulation results were compared with the experimental observations, and consequently, it was found that the present model can well simulate the strain localization characterization (details will be presented at the conference).
CONCLUDING REMARKS A 3D yield function of 6th-order polynomial type to describe the anisotropic behavior of steel sheets, YoshidaSt model, has been proposed. The advantages of this model are summarized as follows: 1.
2.
The model has a high flexibility of describing the planar r-value anisotropy, the flow stress directionality and the shape of yield locus. Specifically for the description of (V x , V y ) yield locus, it offers a freedom to give two plane-strain stress points, V a and V c , besides two uniaxial stresses V 0 , V 90 and the biaxial stress V b . A user-friendly scheme of material parameter identification has been presented. It includes an idea of using a simple yield function, such as Hill48 or Yld89, for the estimations of V a and V c as well as V D and rD . Consequently, for the parameter identification, we need only limited experimental data of r-values and yield stresses in three directions ( r0 , r45 , r90 and V 0 , V 45 , V 90 ) plus equi-biaxial yield stress ( V b ).
3. 4.
Once all the material parameters for plane stress (2D) yield function are determined, no additional parameters are needed for 3D version of the model. The validity of the model has been confirmed by comparing the numerical simulations of the planar r-value anisotropy, rD , and the flow stress directionality, V D , with the experimental data of several types of steel sheets (a high r-value IF steel and high strength steel sheets of 440-980MPa TS levels, totally more than twenty types of sheets). The yield loci calculated by the model well capture the shape of several types of steel sheets. Furthermore, FE simulation result of hole expansion showed that it is useful for accurate numerical simulation of sheet metal forming.
REFERENCES 1. F. Yoshida, T. Uemori, and K. Fujiwara, Int. J. of Plasticity 18, 2002, pp. 633-659. 2. F. Yoshida and T. Uemori, Int. J. of Plasticity 18, 2002, pp. 661-686. 3. F. Yoshida and T. Uemori, Int. J. of Mechanical Sciences 45, 2003, pp. 1687-1702. 4. P. A. Eggertsen and K. Mattiasson, Int. J. of Mechanical Sciences 52, 2010, pp. 804-812. 5. A. Ghaei and D. E. Green, Computational Material Science, 48, 2010, pp. 195-205. 6. R. Hill, Proc. Royal Soc. London A 193, 1948, pp.281-297 7. R. Hill, J. Mech. Phys. Solids 38, 1990, pp. 405-417. 8. R. Hill, J. Mech. Phys. Solids 42, 1994, pp. 1803-1816. 9. M. Gotoh, Int. J. Mech. Sci. 19, 1977, pp.505-512. 10. F. Barlat and J. Lian, Int. J. Plasticity 5, 1989, pp. 51-66. 11. F. Barlat, J. C. Brem, J. W. Yoon, K. Chung, R. E. Dick, D. J. Lege, F. Pourboghrat, S.-H.. Choi and E. Chu, Int. J. Plasticity 19, 2003, pp.1297-1319. 12. F. Barlat, J. W. Yoon and O. Cazacu, Int. J. Plasticity 23, 2007, pp.876-896. 13. D. Banabic, H. Aretz, D. S. Comsa and L. Paraianu, Int. J. Plasticity 21, 2005, pp.493-512. 14. H. Vegter and A. H. van den Boogaard, Int. J. Plasticity 22(2006), pp.557-580. 15. W. Hu, Int. J. Plasticity 23 (2007), pp.620-639. 16. S. Soare, Int. J. Plasticity 24(2008), pp.915-944. 17. K. Hashimoto, T. Kuwabara, E. Iizuka and J. W. Yoon, Tetsu-to-Hagane 96, 2010, pp.27-33. 18. T. Kuwabara, A. Van Bael and E. Iizuka, Acta Materialia 50, 2002, pp.3717-3729. 19. T. Kuwabara, S. Ikeda and K. Kuroda, J. Mat. Process. Technology 80-81, 1998, pp.517-523. 20. W. Hu, Int. J. Plasticity 21 (2005), pp.1771-1796. 21. K. Kitayama, T. Kobayashi, T. Uemori and F. Yoshida, Tetsu-to-Hagane 97, 2011, pp.221-229.
814
Determination of the Forming Limit Diagram of an UltraThin Ferritic Stainless Steel Sheet Hyuk Jong Bonga, Frédéric Barlata, and Myoung-Gyu Leea a
Materials Mechanics Laboratory, Graduate Institute of Ferrous Technology, Pohang University of Science and Technology, Pohang, Gyeongbuk, 790-784, Korea Abstract. The forming limit diagram (FLD) of a 98μm-thick ferritic stainless steel sheet sample was determined using a modified Marciniak test and the conventional Nakazima test (ASTM E2218-2). Wrinkling and undesired fracture were not observed in the modified Marciniak test. However, for some strain paths, wrinkling of the sheet specimens and undesired fracture at the upper die radius and the draw bead locations occurred in the Nakazima test. In spite of the problems observed with the Nakazima test, similar FLDs were obtained with the two methods. Keywords: Forming limit diagram, modified Marciniak test, Nakazima test, ultra-thin ferritic stainless steel sheets PACS: 60.20.F-
INTRODUCTION During recent decades, there has been a world-wide surge of efforts to replace internal combustion engines with more fuel-efficient and environmentally friendly technologies. Particularly, fuel cell technology has received great interest because of its characteristic high energy conversion efficiency and zero pollutant emissions. In the development of commercial fuel cells using metallic bipolar plates, a key obstacle has been the mass of the bipolar plates, which represents roughly 60% of the total mass cells1. Therefore, it is necessary to reduce the mass of the bipolar plates by using metallic sheet thinner than 0.1 mm. While reducing mass, fuel cell efficiency must be maintained and requires forming deep micro-channels in the bipolar plate. However, it has been an engineering challenge to form ultra-thin ( R0 then the deformation is elastic in the entire cross section and the moment will be
t0 / 2 E y2 E 1 t03 M max 2 dy 2 2 R0 12 0 1 R0 1
(6)
The load Q (/ unit width) was estimated according to [4]
Q (1 4 t 0 / w) t 02 Rm / 2 w
(7)
where Rm is the ultimate tensile strength of the selected sheet. This value was used by the author in the first iteration to get the bending angle then was modified after that if the desired bending angle not obtained. Using this value to control the algorithm iterations was modified. The desired bending angle was used directly to control the interactions and a significant improvement was noticed, see below. The procedure used to calculate the variation of R0 along x axis of the sheet (toward the die edge) was used to judge whether the deformation is elastic in the entire section (R0 < R0*) or elastic/plastic. According to [1], the variation took the pattern of the two equations in two different regions. Upon replication of that work a discontinuity was detected as appearing in Fig. 3: M ate rial 904L, t0 = 20.9 m m , σ=327+1873ε E= 200,000 M pa, w = 150 m m Q (pe r unit w idth = 1394.5) N/m m
1/R0i
0< x < 45 mm
1/R0i 75 > x > 45 mm
0.0008 0.0007
y = -1E-05x + 0.0007 1/R0i (mm-1)
0.0006 0.0005 0.0004 0.0003 0.0002
y = -8E-06x + 0.0006
0.0001
x (m m )
0 0
20
40
60
80
FIG. 3. The variation of curvature 1/R0 with x axis as reproduced in the current investigation using the presented integration of Eqs (4), Material 904L, thickness = 20.9 mm with die width w = 150 mm. Q is calcu. by Eq. (7)
Revision was performed in order to determine the difference source. An integration of Eq. (4) revealed that some terms are missing in the presented integration. The correct result of the integral must be
M max R0 * x
2
2 * E 8 2 K e t 03 2 x Ke 0 e 0 .2887 0 e t 0 2 9 9 R0 3 3 1
(8)
which yields exact results found in [1] as in Fig. 4. Rf which represents the radius after unloading was calculated recognizing that the unloading is pure elastic recovery.
1154
FIG. 4. The variation of curvature 1/R0 with x as reproduced in the current investigation using Eqs (8), Material 904L, thickness = 20.9 mm with die width w = 150 mm. Q is calcu. by Eq. (7). (Linear hardening)
Determination of springback angle Eqs. relating radii before and after unloading can only be solved numerically [1] to find deflection y(x) in order to get the final profile of the sheet metal before/after unloading. This profile will directly give the springback angle in the form of αf-α0 (Fig. 1). Steps of the numerical solution are inspired by [5], this numerical integration yielded much better correlation of springback angles than the correlation presented in [1] The Euler-Bernoulli equation is presented by 2 d 2 y dy 2 / 1 dx dx
3/ 2
M EI
(9)
where M=moment, E=Young’s modulus, I = moment of inertia. In the author’s study and this investigation the right hand side is in the form of F(x). In Eq. (16) set dy/dx = tanθ leads to
d (tan ) 2 / 1 tan dx
3/ 2
F ( x) , simplifying leads to, (sec ) 2 d
dx
/(sec )
2 3/ 2
F ( x)
which can be written as
cos d = F(x)dx
(10)
eq. (10) was then numerically integrated to get the angle θ which represent =Ф0/2 and Фf/2 in Fig. 1 for the before/after unloading cases respectively. The two angles used in the calculations of springback are related to Ф0/2 and Фf/2 by the following two Eqs.
0 180 2 0 / 2 2 arctan(dy / dx) (before unloading)
(11)
f 180 2 f / 2 2 arctan( dy / dx ) (after unloading)
(12)
and
In order to validate comparison with the experimental data, the author in [1] had to apply an iterative procedure in the following manner[1]: 1.
A certain load was selected and used to calculate dy/dx prior to unloading (dy/dx = tanΦ0/2, see Fig. 2)
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2.
Assume that load was insufficient to bend this sheet in the experimental part ( 0 180 2 0 / 2, Fig .3 ).
3. 4.
Then the load was increased until the selected bending angle or dy/dx was obtained. The iteration over the moment equation was then used to finally get dy/dx, y(x) and αf. The used load was then compared to the load calculated by Eq. (7)
Noticing points 1-4 above and how α0 was considered fixed between the theoretical and experimental parts, this is actually more like feeding the procedure with a specific α0 instead of relying on the one that corresponds to the force Q calculated by Eq. (7). Alternatively in this investigation the program was fed with the specific α0 then by the way of iterating over α0 itself and changing Q value (another level of iteration was then introduced over that described in [1] over the x axis).
Non-Linear hardening Assuming that the material behaviour is described best by
0e Ke e n , in the plastic region
(13)
where n =strain-hardening exponent. Same line of analysis as above come up to
M
max
x* R 0 E 2 0 1
2
n t 0 / 2 2 2Ke 2 y y2 dy * R 3 0 e R0 3 3 R0 x 0
n
ydy
(14)
Integration of Eq. (14) yields n
n
3 2 2 E 1 * 2 t02 * 4 Ke 2 1 Mmax R R x e x 0 0 0 3 n 2 3 R 3 12 R0 3 4 0
t0 n2 * n2 x R0 2
(15)
The radius at which the yielding R0 is given as a function of e by *
*
R0* 3 t 0 E / 4 1 2 e* where
(16)
e* = the effective stress at the elastic-plastic interface and was calculated by
0e * e
* K e e E
n
(17)
RESULTS In [7] springback behavior of cold rolled transformation induced plasticity (TRIP) steels in air v-bending process was reported. The improved algorithm presented here was able to reproduce the reported results for the TRIP steel with accuracy over 97%. Only data for a constant modulus was used for the comparison. However, the extension of algorithm presented here can be performed and will include the variation in modulus into account. One example for T1.45 as mentioned in [7] is to use the equation E 201 105 GPa (18) Such an equations needs to be introduced to the moment equation like Eq. (8) above and included into the iterative algorithm to converge to specific bending angle, hence calculate the springback more accurately.
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LS-Dyna commercial software was used to achieve independent comparison basis for the thin sheets. Shell elements with full integration formulation were used for the blank. Symmetry was utilized and plane strain conditions were imposed, Fig (5). Implicit solver was used for springback analysis using Dynain file approach. The dynein file approach was preferred over the seamless springback approach, see [8] because convergence was some time hard to achive and better stabilization control was generally required. Large springback and flexible parts are considered difficult springback problems even for simple geometry as in this case. The contact stiffness was reduced below the recommended values for metal forming by LS-Dyna, [6]. By selection of suitable contact stiffness scale and reasonable mass scaling, error down to 5% was obtained as compared to the improved algorithm presented here.
FIG. 5. LS-Dyna analysis for Material is 904L, with die width = 300 mm.
The implicit analysis by LS-Dyna deals with the stresses in formed material which is the same starting point of calculations as presented in this paper, however the convergence by the algorithm is guaranteed and proved accurate. Fig. 6 displays the normalized springback obtained theoretically, (α0/αf)theoretical vs. the normalized springback found experimentally (α0/αf)experimental as noticed by the author in a specific range of bending angle α0 (87o≤α0≤86o) for material 253 MA . The author states that the pattern shown may initiate the idea of having a correction factor that is able to bring all the theoretical values to coincide with the experimental values.
FIG. 6. Normalized springback obtained theoretically, (α0/αf)theoretical, vs. normalized springback found experimentally (α0/αf)experimental as noticed by the study author Material is 253 MA. The thick line represents (α0/αf)theoretical = (α0/αf)experimental.
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Fig 7 displays the corresponding normalized data; for a different material however; associated with the trend lines as obtained in this investigation. This figure displays the results obtained by both linear and non-linear models, they show very good correlation.
FIG. 7. Normalized springback obtained theoretically, (α0/αf)theoretical, vs. normalized springback found experimentally (α0/αf)experimental as per this investigation, Material is 904L MA. The thick line represents (α0/αf)theoretical = (α0/αf)experimental.
CONCLUSIONS The following can be concluded:
A partially analytical model (including numerical integration) is correctly reproduced from [1], and by employing improved accurate numerical solution, the model is able to predict the springback values of the sheet metals for both thin and thick sheets. The model gives results that are comparable to those reported by implicit solvers, the convergence in the presented model is guaranteed and easy to implement. Non-linear hardening (Ludwik model) does not add a significant value to the springback results. Linear model does then maintain simpler analysis and does achieve enough accuracy in such applications.
REFERENCES 1. N. Asnafi, “Springback and fracture in v-die air bending of thick stainless steel sheets”, Materials and Design 21(2000) 217236. 2. Johnson W, Mellor PB. Engineering plasticity. London, Great Britain: van Nostrand Reinhold Company Ltd, 1973:124-128. 3. William F. Hosford, Robert M. Caddell Metal Forming.Cambridge University Press, ISBN: 9780521881210. 4. Weinmann KJ, Shippel RJ. Effect of tool and workpiece geometries upon bending forces and springback in 908 v-die bending of HSLA steel plate. Proceedings of the Sixth North American Manufacturing Research Conference (NAMRC)., held in Gainsville-Fl. Dearborn Michigan: SME, 1978:220-227. 5. Bisshopp, K. E., and Drucker, D. C., “Large Deflections of Cantilever Beams,” Quarterly of Applied Mathematics, Vol. 3, 1945, pp. 272-275. 6. H. V. Gajjar, A. H. Gandhi and H. K. Raval, “Finite Element Analysis of Sheet Metal Air-bending Using Hyperform LSDYNA,” Int. J. Mech.Sys. Sci. Eng., vol. 1, no. 2, pp. 117-122, 2008. 7. Fei, D., Hodgson, P., 2006. Experimental and numerical studies of springback in air v-bending process for cold rolled TRIP steels. Nuclear Engineering and Design, 236(18):1847-1851. 8. Maker, B.N. and Zhu, X, “Input Parameters for Metal Forming Simulation Using LS-DYNA,” April, 2000, available at www.feainformation.com/forming_parameters2.pdf or by email from
[email protected].
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Numerical Modeling for Springback Predictions by Considering the Variations of Elastic Modulus in Stamping Advanced High-Strength Steels (AHSS) Hyunok Kim1* and Menachem Kimchi1 1
Edison Welding Institute, 1250 Arthur E. Adams Drive, Columbus, Ohio 43221-3585, U.S. *Corresponding author: 1-614-688-5239, 1-614-688-5001 and
[email protected] Abstract. This paper presents a numerical modeling approach for predicting springback by considering the variations of elastic modulus on springback in stamping AHSS. Various stamping tests and finite-element method (FEM) simulation codes were used in this study. The cyclic loading-unloading tensile tests were conducted to determine the variations of elastic modulus for dual-phase (DP) 780 sheet steel. The biaxial bulge test was used to obtain plastic flow stress data. The non-linear reduction of elastic modulus for increasing the plastic strain was formulated by using the Yoshida model that was implemented in FEM simulations for springback. To understand the effects of material properties on springback, experiments were conducted with a simple geometry such as U-shape bending and the more complex geometry such as the curved flanging and S-rail stamping. Different measurement methods were used to confirm the final part geometry. Two different commercial FEM codes, LS-DYNA and DEFORM, were used to compare the experiments. The variable elastic modulus improved springback predictions in U-shape bending and curved flanging tests compared to FEM with the constant elastic modulus. However, in S-rail stamping tests, both FEM models with the isotropic hardening model showed limitations in predicting the sidewall curl of the S-rail part after springback. To consider the kinematic hardening and Bauschinger effects that result from material bending-unbending in S-rail stamping, the Yoshida model was used for FEM simulation of S-rail stamping and springback. The FEM predictions showed good improvement in correlating with experiments. Keywords: Springback, Stamping, Advanced high strength steel. PACS: NUMI_ABS214
INTRODUCTION Auto industry experiences the increase of AHSS applications for automotive structure to improve the crashworthiness without corresponding weight increases. AHSS is known to have unique elastic and plastic material behaviors. This becomes more difficult for stamping engineers and tooling designers to reduce or compensate for springback in their production and tooling design. Therefore, the reliable prediction and practical compensation of springback are important to allow production of the desirable stamped parts. Authors conducted the springback analyses using a simple V-die bend test and FEM [1]. S-rail stamping study was conducted and the springback of Srail parts was conducted using FEM with an isotropic hardening model and a variable elastic modulus [2]. In this paper, the new results of the S-rail stamping and springback are presented by considering a kinematic hardening model and a variable elastic modulus. The objective of this study was to improve the prediction of springback in FE simulation of stamping AHSS parts by considering the variation of elastic modulus, kinematic hardening, and the Bauschinger effect.
APPROACH The following three different testing methods were used in this study to evaluate springback (Fig. 1):
U-Bend Test (UBT) Curved-Flanging Test (CFT) S-Rail Stamping Test (SST)
The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 1159-1166 (2011); doi: 10.1063/1.3623734 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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c) Final part shapes predicted by FEM
Figure 1. Schematic of various forming tests with S-shaped tooling and the final shape predicted by FEM.
Two different commercial FEM codes, LS-DYNA and DEFORM-3D, were used to analyze the experimental results and predict springback. As shown in Fig. 1, various stamping tests were conducted with an S-rail die at a 160-ton hydraulic press with the CNC-controlled cushion system. Final blank shapes and their locations at a S-rail die for different testing methods were determined to avoid any cracking and wrinkling using preliminary stamping trials and FEA.
RESULTS Characterization of material properties A dual phase (DP) 780 material was used in this study. The elastic and plastic material properties of DP780 were determined by using the ASTM standard tensile test and biaxial bulge test. Three samples were tested for each condition. Table 1 shows the sheet material properties that were obtained from the tensile test in both rolling and transverse directions. Table 1. Summary of ASTM tensile test results. Sheet Material DP 780 (t0=1 mm)
Tensile Testing Direction RD TD
0.2% Yield (MPa) 538.6 531.7
UTS (MPa) 865.5 872.4
Total Elongation (%) 19.3 15.1
A larger range of true stress-strain data for DP 780 was obtained by conducting the viscous pressure bulge (VPB) test, because the tensile test gave a relatively small range of data. As compared in Fig. 2, while the tensile test reaches a maximum strain of 0.11 for DP 780, the VPB test reaches higher maximum strains, up to 0.37, which are often observed in stamping processes.
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Figure 2. True stress-strain data obtained from the VPB test and tensile test for DP 780.
The cyclic loading-unloading tensile test was conducted to quantify the change of elastic modulus of DP 780 materials as the strain increased. The elastic loading and unloading modulus were obtained by calculating the slopes of apparent elastic loading and unloading curves, as shown in Fig. 3.
Loading
Unloading
Figure 3. True stress-strain curves in the loading and unloading tensile test for DP 780 material.
The apparent elastic unloading modulus is more relevant to springback than the apparent elastic loading modulus because springback takes place when the workpiece is released from the punch or die. The apparent elastic modulus of DP 780 was found to decrease about 28% from the initial elastic properties to the elastic modulus at 0.11 strain (Fig. 4). The variation of elastic modulus during unloading was used to determine the mathematical forms of the variable elastic modulus model, the “Yoshida-Uemori model”, as shown in Fig. 4. Three coefficients (E0, EA, and ζ) in the Yoshida model were determined by fitting the outputs of the model with the measured data at a 97% confidence level.
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Figure 4. Yoshida model vs. experimental data for variable elastic modulus of DP 780.
Evaluation of Springback in the UBT and FEA The UBT was conducted for DP 780 material with the S-rail die and the blank holder. The specimen size of 40 × 185 mm was determined by considering the straight-side edges of punch as well as the easiness to measure the bending angle of specimen side-wings. Two different blank holder forces (BHF), 2 and 8 tons, were used to investigate the stretching effect on springback. Most of the tests were conducted with no lubricant. All the measurement of springback angle of tested samples was done by using an X-Y table and a high-resolution camera. Measurements were made for all the tested samples (e.g., three samples for each condition). As the BHF increased, the springback was found to decrease. FEM analyses for the UBT were conducted by using two different commercial FEM codes, DEFORM-3D and LS-DYNA. The same input parameters of material properties, friction, and the same element size were used in both FEM codes. The coefficient of friction was defined to be 0.25 that is reasonable input for a dry friction condition used in experiments. LS-DYNA springback simulations were run using both a variable (see Fig. 4) and constant value (207 GPa) for the elastic modulus. The isotropic hardning model was used in simulation. Two LS-DYNA simulations considered both the variable elastic modulus and the constant elastic modulus. Two different springback angles in the channel opening angle (Springback 1) and the flange angle (Springback 2) were compared between FEM predictions and experiments. The comparison of springback in UBT at a BHF of 2-tons is given in Fig. 5. In the BHF 2-ton, LS-DYNA with variable elastic modulus showed the best correlations with experiments, while DEFORM-3D and LS-DYNA with the constant elastic modulus showed similar predictions and both results gave about 4 to 5 degrees error with respect to experiments.
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Figure 5. Comparison of springback in U-shaped stretch bending of DP 780 at a BHF of 2 tons.
Evaluation of Springback in the CFT and FEA The springback was investigated with the CFT. In the CFT, non-uniform stress and strain conditions cause more severe dimensional errors such as variations of channel opening and side-wall curls, as well as twisting of the drawn channel. The effects of CFT were investigated with tests conducted at two different BHF levels. The CFTs were conducted for DP 780 with an S-rail die by placing the rectangular sample (60 × 185 mm) with respect to the S-rail punch. Most of the experiments were conducted without lubricant. The six different angles on the tested specimens which were measured are shown in Fig. 6.
Figure 6. Locations of the measurements for various angles after springback.
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The mid-section profile was difficult to accurately measure using the camera system or the manual-type protractor. Therefore, the side-edge profiles of tested samples were measured and the average value of measured angles was used to quantify springback. Final bending angles of FEM predictions and experiments at 8-ton BHF are compared in Fig. 7. The variable elastic modulus gave better predictions of the flange angles, βin and βout. However, for the channel angles (өin and өout) and the shoulder angles (αin and αout), it is not clear which elastic modulus model gave better predictions compared to experiments. This may be caused by possible measurement errors for these angles. It was more difficult to accurately measure the channel angles and the shoulder angles compared to the flange angles, because of the sidewall curl and twisting.
Figure 7. Comparison of the final bending angles of curved flanging between FEM predictions and experiments at a BHF of 8 tons.
Evaluation of Springback in the SST and FEA In the SST, the distributions of stress and strain on the part are more complicated than for UBT and CFT. Because of complex geometry and stress conditions, three different types of springback, defined twisting of the channel, channel opening, and sidewall curl, occur in S-rail stamping. Experiments were conducted with DP 780 (t0=1.0 mm) materials. Fig. 8 shows the stamping test setup and the CAD models of the initial blank geometry and the designed stamping geometry. FEM simulations for S-rail stamping were conducted by using a commercial FEM code, LS-DYNA. Two different elastic modulus models; a constant (207 GPa) and the variable (as given in Fig. 4); two different hardening models; the isotropic hardening and kinematic hardening models; were used for DP 780 in the FEM simulations. Shell elements with a 2-mm element size were used for the sheet blank model. The coefficient of friction was input as 0.125 to consider the lubrication at the tool-workpiece interface. As shown in Fig. 8, the FEM model was carefully set up by considering the relative positions of the initial sheet blank to other punches and dies, which were measured in the experiments.
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Figure 8. S-rail tooling and the initial blank and final stamping geometries, and FEM model
In the test, the final depth of stamping was determined to be 20 mm and BHF was limited to 2 tons to avoid any cracking of sheet material. A pressure pad force on the top of the hat surface was measured to be increased from 10 to 44 kN as the stroke increase up to 20 mm during stamping. The initial blanks were lubricated with a stamping lubricant before the test. The stamped part was scanned by the 3-D white light scanning method to obtain a 3-D CAD model from the actual stamped part. The isotropic hardening model did not provide accurate results of springback regardless of elastic modulus models; a constant or variable. The detailed results has been published in [2]. Therefore, in this study, the YoshidaUemori model [3] was used to consider the effects of the kinematic hardening and the Bauschinger effect on springback. This model was implemented by using an advanced material model (*MAT_125) that is available in LS-DYNA [4]. The detailed parameters for DP 780 were provided by a steel supplier [5], as shown in Table 2. Two different sections on the stamping were compared for springback between the actual and FEM predicted final part geometry as shown in Fig. 9. Two reference points at the top flat sectional profile were fixed to compare the multiple profiles together. TABLE 2. Parameters for the Yoshida-Uemori model for DP 780 Variable CB Y SC K Value 453 MPa 291 MPa 513 MPa 62 Variable EA COE IOP C1 Value 149 GPa 54 0.0 0.052
RSAT 700 MPa C2 0.955
SB 449 MPa
H 0.95
Figure 9: Two selected sections for springback comparison
Fig. 10 compares springback results for two different sections. New simulation results (A and B) with the Yoshida-Uemori model showed better correlations with experiments (Exp) in comparison to the isotropic hardening model results (C and D). The variable elastic modulus (A) gave better predictions of the channel opening (θ) and the flange angle (β), while the constant elastic modulus showed slightly better correlation with experiment for the dieshoulder angle (α).
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Figure 10: Comparison of FEM predictions and experiment
CONCLUSIONS The following conclusions can be drawn from this project: As the plastic strain increases, the elastic modulus significantly decreased up to 28% for DP 780 from the initial values. The springback decreased with increasing BHF from 2 to 8-tons, as confirmed in the CFT and the UBT results. However, the side-wall curl was increased for increasing the BHF. The isotropic hardening models with constant and variable elastic modulus showed limitations in SST to predict the channel opening and the sidewall curl compared with experiments. The Yoshida-Uemori model with the consideration of kinematic hardening and the Bauschinger effects showed better correlations with springback measurements in S-rail stamping compared to the isotropic hardening models regardless of the elastic modulus models. To predict the springback of the relative simple part geometry; V-die or U-die bending; the constant elastic modulus and the isotropic hardening model can be reasonable input data of FEM. However, to predict the spingback of the complicate stamping geometry that encounters the repeated bending and unbending, it is recommended to define the non-linear hardening behaviors and elastic modulus change in the material input data for reliable FEM simulations.
ACKNOWLEDGEMENTS This project was conducted with support from the State of Ohio, Department of Development, and Thomas Edison Program. The authors would like to thank Dr. Taylan Altan, Director of Engineering Research Center for Net-Shape Manufacturing (ERC/NSM), the Ohio State University and Dr. Boel Wadman, Project Manager of Swerea-IVF, for their technical supports and advices on this research program.
REFERENCES 1. Kim, H., Kimchi, M., Altan, T. Control of springback in bending and flanging advanced high-strength steels (AHSS), International Automotive Body Congress (IABC), 4-5 November 2009, Troy, Michigan, USA. 2. Kim, H., Kimchi, M., Kardes, N., Demiralp, Y., Mete, O., Altan, T., Predictions of springback in the S-rail stamping of AHSS using FEM with the variable elastic modulus, IABC, 3-4 November 2010, Troy, Michigan, USA. 3. Yoshida, F., Uemori, T., A model of large-strain cyclic plasticity describing the Bauschinger effect and workhardening stagnation, International Journal of Plasticity 18, 2002, pp. 661-686. 4. LSTC, LS-DYNA Keywords User’s Manual, Vol. 2, Material Models, Version 971,Rev 5.0, 2010, pp. 479-482. 5. Chen, X., Shi, M., Zhu, X., Du, C., Xia, Z.C., Xu, S., Wang, C.T., Springback prediction improvement using new simulation technologies, SAE Paper 2009-01-0981, 2009.
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Free Surface Roughness Prediction in Bending Under Tension Sergei Alexandrova, Ken-ichi Manabeb and Tsuyoshi Furushimab a
A.Yu.Ishlinskii Institute for Problems in Mechanics of the Russian Academy of Sciences, 101-1 Prospect Vernadskogo, 119526 Moscow, Russia b Department of Mechanical Engineering, Tokyo Metropolitan University, 1-1 Minami-osawa Hachioji, 192-0397 Tokyo, Japan
Abstract. The paper presents the first step of research aimed at determining the specific form of a new empirical equation for free surface roughness evolution. In contrast to conventional empirical equations for free surface roughness evolution, the new equation introduces a dependence of the increment of roughness parameters on two kinematic variables. Nevertheless, it can be conveniently represented by a curve in appropriate space. The experimental method chosen is the process of plane strain bending of a sheet stressed by tensions applied to the ends. When the convex surface of the sheet is considered, one point of the aforementioned curve can be obtained. The theoretical part of the approach proposed consists of an analytic solution for an idealized process and simple geometric considerations. Keywords: Free Surface Roughness, Empirical Equation, Bending Under Tension, Semi-Analytic Method. PACS: 46
INTRODUCTION The roughness has a great impact on the quality of surface after metal forming processes. Yet, it may have an effect on the overall deformation in course of such processes. The effect depends on the ratio of roughness parameters to a characteristic length of the process and, therefore, is more pronounced in microforming processes. A typical micro-forming process where the free surface roughness plays a crucial role is die-less drawing of microtubes. Theoretical methods for predicting the evolution of free surface roughness include empirical equations, for example [1, 2] and numerical methods, for example [3, 4]. The empirical equation proposed in [1] has been successfully used in applications [5, 6]. The empirical equation given in [2] generalizes the equation from [1] and is based on an analysis of several independent experimental results [7-13]. In contrast to the original equation, the new equation introduces a dependence of the increment of roughness parameters on two kinematic variables. Nevertheless, it can be conveniently represented by a curve in appropriate space. A general numerical method for predicting the free surface roughness evolution in the case of plane strain deformation has been proposed in [14]. This method relates the solution of a boundary value problem of a phenomenological theory of plasticity and the finite element solution for a representative element of material in the vicinity of the free surface. Both experimental and numerical methods can be adopted to find the specific shape of the aforementioned curve. When any plane strain process in which infinitesimal fibers normal to the free surface are subject to compression is considered, just one point of the curve can be found. In the present paper, the convex surface of the sheet subject to bending under tension is chosen for this purpose. An advantage of this process over the pure bending process is that much larger strain can be achieved. On the other hand, the theoretical analysis based on conventional assumptions is practically the same for both processes. In particular, a unified approach for the analysis of pure bending proposed in [15] has been successfully used for several material models in [16-19] and then extended to bending under tension in [20]. These solutions are semi-analytical. In particular, a numerical treatment is usually necessary for integrating and solving transcendental equations. However, in the case under consideration the experimental technique is based on displacement control. Therefore, one additional relation to connect kinematic variables is known in advance. As a
The 8th International Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, 1167-1174 (2011); doi: 10.1063/1.3623735 © 2011 American Institute of Physics 978-0-7354-0949-1/$30.00
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result, for any isotropic incompressible material the complete system of equations used to relate the measurable input variables and roughness parameters is obtained from simple geometric considerations without solving the stress equations. This means that the analysis is independent on the particular hardening law.
GENERAL APPROACH The empirical equation for predicting the evolution of free surface roughness proposed in [2] can be written in the form
'R ag where
§ Hn · ¨ H eq ¸¸ © ¹
H eq : ¨
(1)
'R is the increment of any roughness parameter, Ra or Rz , ag is an average grain size, H eq is the
equivalent strain,
Hn
is the logarithmic strain normal to a free surface, and : is a function of its argument. This
function should be determined either from experiment or from numerical simulation for several ratios
H n H eq . The
equivalent strain is defined by t
³[
H eq
eq
dt , [ eq
0
where
[eq
is the equivalent strain rate,
[ij
2 [ij[ij 3
(2)
are the components of the strain rate tensor, and t is the time. The
integral in (2) is taken over the actual strain path. Some general features on the function :
H
n
H eq
found in [2]
with the use of experimental data are illustrated in Figure 1. It has been hypothesized that the function has two local maximums and one local minimum. The local maximums are located in the vicinity of the points and
H n H eq
3 2 . The local minimum should be near the point H n H eq
shape of the curve are from (2) that addition,
H n H eq
Hn 0
E1 ! E
3 2
0 . The restrictions imposed on the
0 d H n H eq d 1 and J 1 ! J in the range 0 t H n H eq t 1 . It is seen
3 2 in the case of plane strain deformation, if the material is incompressible. If, in
then equation (1) can be rewritten in the form
'R ag Thus
in the range
H n H eq
§
H eq : ¨¨ ©
3· ¸ H eq : ps , : ps 2 ¸¹
§ 3· : ¨¨ ¸¸ © 2 ¹
(3)
'R is proportional to the equivalent strain and the coefficient of proportionality is ag : ps . Since the
average grain size is known, it is sufficient to make one experiment from which the roughness coefficients should be found to determine the value of : ps and thus one point of the curve shown in Figure 1. An alternative approach is to determine the evolution of roughness parameters numerically, for example with the use of the method proposed in [14]. It is seen from Figure 1 that results of three or four different experiments (or numerical solutions) should provide sufficient information to propose an analytic representation on the function of variation of its argument, 1 t H n surfaces under the condition
: H n H eq in the entire range
H eq t 1 . The present paper deals only with plane strain deformation of free
H n 0 . One of possible tests in this case is plain strain bending under tension. The
roughness parameters should be measured on the convex free surface of the sheet.
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FIGURE 1. Typical shape of the curve representing the function :.
BENDING UNDER TENSION An idealized process of bending under tension is illustrated in Figure 2. The initial shape is rectangular A1B1C1D1. Any intermediate shape is sector ABCD with AB and CD being circular arcs whereas AD and BC are straight lines. Such transformation of the rectangular into the sector is possible if a force F and a bending moment M are applied at BC and AD and, in addition, a uniform pressure p is applied over CD. It is obvious that F and p are not independent. However, this relation is not important for the purpose of the present paper. The pure bending process is obtained as a particular case when F 0 and p 0 . In this particular case, a unified approach for analysis of the process for quite an arbitrary material model has been proposed in [15]. This approach has been used in [16-19] and then extended to the process of bending under tension in [20]. A similar approach has been proposed in [21]. The starting point of the approach proposed and developed in [15, 20] is the following transformation equations between Eulerian Cartesian coordinates xy and Lagrangian coordinates ]K (see Figure 2)
]
x H
a
s s cos 2aK , 2 a a
a)
K= L/H C1
y
]
y H
a
b)
K= L/H
=0
y B
=0
=1
T H=h
p
2L
T O
(4)
M r F C
B1
=1
s sin 2aK a2
x
O1
O
rCD
x
h
rAB
D F A1
D1
M
A
K= L/H
K= L/H
FIGURE 2. Geometry of the problem and coordinate systems (a) initial shape, (b) intermediate shape.
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where H is the initial thickness of the sheet, s is a function of a, a is a function of the time, and a
0 at t
0 . The
dependence of a on t is immaterial for rate-independent materials. The function s a should be found with the use
of constitutive equations and the equilibrium equations. It is possible to verify by inspection that equations (4) correspond to the transformation of rectangular A1B1C1D1 into sector ABCD if
1 4
s at a
(5)
0 . It has been shown in [15] that the radii of curves AB and CD are given by
rAB H
s , a
sa a
rCD H
(6)
respectively. Moreover, the principal components of the strain tensor are determined from the equations
H]
1 ln ¬ª 4 ] a s ¼º 2
HK
(7)
The equivalent strain can be found from these equations in the form
H eq Since
]
1 ln ª 4 ] a s º¼ 3 ¬
(8)
0 on surface AB throughout the process of deformation, it follows from (8) that
H eq ]
0
H AB
ln 4 s 3
(9)
It is obvious that this value of the equivalent strain is involved in (3). When F is prescribed, for example F 0 in the pure bending process, numerical integration and solution to
transcendental equations are required to find the function s a . Once this function has been determined, the values of rAB , rCD and
H AB
can be found from (6) and (9) with no difficulty. However, the process of bending under
tension can be performed under displacement control. In this case the function s a can be determined from geometric relations independently of the constitutive model. In particular, the experimental set-up used in the present study is shown in Figure 3. In order to relate it to the idealized process illustrated in Figure 2, it is necessary to assume that the concave surface of the sheet is a circular arc. Then, the geometric scheme of the real process can be represented by a diagram shown in Figure 4. Here L and l do not change with the time. Therefore, points C1 and C2 are fixed in space. On the other hand, Z is a prescribed function of the time. The value of Z can be measured at any stage of the process. U is a function of the time and U o f as t o 0 . It is obvious that U rCD . Therefore, it follows from (6) that
U H
sa a
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(10)
Using elementary geometric relations (Figure 4) it is possible to arrive at
2l 2L
FIGURE 3. Experimental set-up.
O
2G
U
2J C1
C2
Z
2l 2L FIGURE 4. Idealized scheme of the process.
U
L , tan G sin J
l 2 § J G · 2 , 2 L sin ¨ ¸ sin J Z L l Z L cot J © 2 ¹
(11)
It follows from these equations that
tan J tan J G tan J tan J G 1 where
J G
tan J l L ª¬1 Z L tan J º¼
(12)
º » » ¼
(13)
should be excluded by means of
J G
ª sin J 2arcsin « « 2 ¬
2
l· §Z· § ¨ ¸ ¨1 ¸ © L¹ © L¹
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2
Then, equation (12) can be solved numerically for
J
sa a Since the function
J Z
. Combining (10) and (11)1 gives
L H sin J
(14)
has been determined, this equation provides a relation between s and a at any value of Z.
One more relation is required to find the functions s Z and a Z . It follows from (7) that
HK ] HK
1
where
HKCD
HK
ln UJ L (Figure 4). Using (11)
CD
is the value of the strain
HKCD
1 ln ª 4 s a º¼ 2 ¬
(15)
on surface CD (Figure 3). On the other hand, this strain can be calculated as
§ J · ln ¨ ¸ © sin J ¹
HKCD
(16)
Comparing (15) and (16) gives the second relation between s and a in the form
§ J · 4 s a ¨ ¸ © sin J ¹
2
(17)
Excluding s a between (14) and (17) leads to
a
J H
(18)
2 L
Then, using (17) 2
s
1§ J · J H ¨ ¸ 4 © sin J ¹ 2 L
It is seen from this solution that (5) is satisfied as
H AB
Since the function
J Z
(19)
J o 0 . Substituting (19) into (9) gives
2 1 ª§ J · Hº ln «¨ 2 J » ¸ L ¼» 3 ¬«© sin J ¹
has been determined, equation (20) provides the dependence of
(20)
H AB
on Z L .
ROUGHNESS EVOLUTION A bending under tension test was performed with the use of the experimental set-up shown in Figure 3 on an A1050 aluminum alloy sheet 0.7mm thick. The length of the specimen used for the current work was 120mm and
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the width was 20mm. The spans between the inner and outer supports had lengths of 10mm and 100mm, respectively. An average grain size in the specimens after this treatment was of the order of 25Pm. The initial arithmetic mean roughness was Ra | 0.35P m . Using the geometric parameters of the test the dependence of H AB on Z L has been found by means of (20). It is illustrated in Figure 5. Roughness parameters have been measured at several stages of the process (several Z L - values) with the use of a standard technique. Then, using the dependence of
H AB
on Z L (see Figure 5) the variation of the roughness parameters with
In particular, the variation of Ra with
H AB
H AB
has been obtained.
is depicted in Figure 6. It is seen from this figure that this dependence is
linear, which is in agreement with (3). Using this dependence and (3) it is possible to find that ag : ps | 9.5P m .
0.38 and one point of the curve shown in Figure 1 has been determined.
0.14 0.12
HAB
0.1 0.08 0.06 0.04 0.02 0 0
0.05
0.1
0.15
0.2
0.25
0.3
Z/L FIGURE 5. Variation of HAB with Z/L.
2
Arithmetical mean roughness Ra [Pm]
Thus : ps
1.5
1
0.5
0 0
0.05
0.1
0.15
Equivalent strain H$% FIGURE 6. Variation of the arithmetic mean roughness with strain.
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CONCLUSIONS A theoretical/experimental procedure based on the plane strain bending under tension test and its analytical treatment has been proposed and completed to determine one point of the curve representing the empirical equation for free surface roughness evolution proposed in [2]. The theoretical treatment is quite simple and reduces to numerical solution of equation (12). Another point of the curve can be in general found from the same test considering the concave surface of the sheet because it is also traction free. In this case H n ! 0 and therefore the argument of the function :
H
n
H eq
is
H n H eq
3 2 . Still another point can be obtained by measuring the
evolution of roughness parameters on the sides of the specimen where made (see Figure 1) the function
H n H eq | 0 . According to the assumptions
: H n H eq attains local maximums at H n H eq
r 3 2 and a local minimum
H n H eq | 0 . Therefore, the values of the function at three aforementioned points allow one to propose a rather accurate specific dependence of : on H n H eq in the entire range of H n H eq values, 1 d H n H eq d 1 . Other at
possible tests for determining the function
: H n H eq have been proposed in [2].
ACKNOWLEDGMENTS The first author acknowledges support from the Russian Foundation for Basic Research (Project 10-08-00083).
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