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Forming of aluminum alloys at elevated temperatures – Part 1: Material characterization
International Journal of Plasticity 22 (2006) 314–341 www.elsevier.com/locate/ijplas
Forming of aluminum alloys at elevated temperatures – Part 1: Material characterization Nader Abedrabbo a, Farhang Pourboghrat a
b
a,*
, John Carsley
b
Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824-1226, United States General Motors Research & Development Center, Warren, MI 48090, United States Received 1 September 2004 Available online 2 June 2005
Abstract A temperature-dependent anisotropic material model for use in a coupled thermo-mechanical finite element analysis of the forming of aluminum sheets was developed. The anisotropic properties of the aluminum alloy sheet AA3003-H111 were characterized for a range of temperatures 25–260 °C (77–500 °F) and for different strain rates. Material hardening parameters (flow rule) and plastic anisotropy parameters (R0, R45 and R90) were calculated using standard ASTM uniaxial tensile tests. From this experimental data, the anisotropy coefficients for the Barlat YLD96 yield function [Barlat, F., Maeda, Y., Chung, K., Yanagawa, M., Brem, J.C., Hayashida, Y., Lege, D.J., Matsui, K., Murtha, S.J., Hattori, S., Becker, R.C., Makosey, S., 1997a. Yield function development for aluminum alloy sheets. J. Mech. Phys. Solids 45 (11/ 12), 1727–1763] in the plane stress condition were calculated for several elevated temperatures. Curve fitting was used to calculate the anisotropy coefficients of BarlatÕs YLD96 model and the hardening parameters as a function of temperature. An analytical study of the accuracy and usability of this curve fitting technique is presented through the calculation of plastic anisotropy R-parameters and yield function plots at different temperatures. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Thermo-mechanical; Temperature; Material anisotropy; Plastic anisotropy; Yield function
*
Corresponding author. Tel.: +1 517 432 0819; fax: +1 517 353 1750. E-mail address: [email protected] (F. Pourboghrat).
0749-6419/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijplas.2005.03.005
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1. Introduction Weight reduction has long been identified as a key priority for improving automotive fuel economy (Greene and DiCicco, 2000), and many studies often suggest substituting lightweight materials for typical steel applications. However, replacing steel in the structure and body of an automobile with lighter materials such as aluminum, magnesium, plastics, composites, etc., can be costly and is not simple or straightforward. Aluminum sheet, in particular, has lower formability at room temperature than typical sheet steel (Ayres and Wenner, 1979). High temperature forming methods based on superplastic behavior of Al–Mg alloys have been used to produce automotive closure panels that far exceed the conventional stamping formability of steel (Schroth, 2004). Superplastic formability however, requires finegrained microstructures and slower strain rates which can affect production cost. The formability of typical Ôoff-the-shelfÕ automotive aluminum sheet alloys (5182O and 5754-O) can be greatly improved by warm forming (Li and Ghosh, 2003) without the increased costs of refining the microstructure. The elevated temperature corresponds with decreased flow stress and increased ductility in the sheet, which can lead to deeper drawing and more stretching to form panels without design modifications to the stamped steel product. Furthermore, the serrated flow behavior of Al– Mg sheet alloys (dynamic strain aging/PLC effect) (Robinson and Shaw, 1994) and corresponding Lu¨derÕs line surface defects that result from the interactions of solutes with mobile dislocations can be avoided by deforming the sheet metal above a critical temperature. Finite element analysis (FEA) and simulations are vital tools in the automotive design and formability process to accurately predict deformation behavior during the stamping operation. Confidence in the numerical analysis of formability depends on the accuracy of the constitutive model describing the behavior of the material (Chung and Shah, 1992). This is especially important when the material exhibits anisotropic characteristics, as do most cold rolled sheet metals. Previously (Zampaloni et al., 2003; Abedrabbo et al., 2005b) the importance of using appropriate material models was demonstrated with respect to wrinkling and ironing during the sheet hydroforming process of an Al–Mg–Si alloy (AA6111-T4). The material model used was the anisotropic yield function proposed by Barlat et al. (1997a) (YLD96), which is based on a phenomenological description of the material. The use of anisotropic material models in FEA is further complicated by the material dependence on the anisotropy coefficients that require thorough material characterization under multiple loading conditions (Chung et al., 1996). This difficulty is further exacerbated when temperature effects are introduced. Because material hardening behavior and material response to loading conditions change at elevated temperatures, the anisotropy coefficients must be determined as a function of temperature so that the material model can account for these changes. Garrett et al. (in press) shows the effect of solution heat treatment on mechanical properties for AA6xxx alloys. Ayres (1979), Painter and Pearce (1980), Taleff et al. (1998, 2001), Takata et al. (2000), Naka et al. (2001), Li and Ghosh (2003) and Li and Ghosh (2004) studied
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the effects of elevated temperatures between 25 and 410 °C (77–770 °F) on the formability of aluminum alloys. Serrated flow behavior (dynamic strain aging) characteristic of AA5182 at room temperature (RT) was shown to diminish with elevated temperature deformation. Also, a larger total elongation in the post-uniform strain region was revealed along with a reduction in tensile strength. This indicates a shifting in the hardening behavior with temperature to a greater emphasis on strain-rate hardening relative to strain hardening. Several research papers characterized the tensile behavior of aluminum alloys at elevated temperatures (Li and Ghosh, 2003; Ayres, 1979; Painter and Pearce, 1980; Takata et al., 2000; Naka et al., 2001; van den Boogaard et al., 2001); the majority however, only studied the effect of elevated temperature on the evolution of the flow (hardening) stress. The effect of elevated temperature on the anisotropy of the material and how the yield surface of the aluminum alloy evolved as a function of temperature was not fully explored. In most cases, either HillÕs 1948 model (Hill, 1948) or the von Mises isotropic yield functions were used. van den Boogaard et al. (2001) characterized the behavior of AA5754-O in which two types of functions representing flow stress were used: the modified power law model and the Bergstro¨m model. The coefficients of the power law model were curve-fit exponentially as a function of temperature. The predictions of the material model, however, underestimated the values of the punch load in both models (Power-Law and Bergstro¨m models). Keum and Ghoo (2001, 2002) studied the effect of temperature on the anisotropic coefficients of the Barlat strain rate potential model (Barlat and Chung, 1993) for AA5052-H32. In this study, a third order polynomial function was used to fit the anisotropy coefficients, and a fifth order polynomial function was used to fit the coefficients for the power law model of flow stress. The material model, however, was assumed to reduce to the von Mises isotropic model as temperature increases. Furthermore, the plastic anisotropy parameters, R0, R45 and R90 almost remained constant with increasing temperature, which is opposite of what was found in the current research study. Other studies by Cleveland et al. (2003) and Masuda et al. (2003) also describe temperature and strain rate effects on flow stress and deformation behavior. Naka et al. (2003) conducted experimental tests to determine the yield locus of AL5083 at elevated temperatures. It was shown that the yield locus changes as a function of temperature and a higher order yield function (e.g., Barlat YLD96) is best for representing such a material. Vial-Edwards (1997) show methods for determining the yield loci for FCC and BCC sheet metals. In warm forming methods, the physics of dislocation movement suggests a thermally activated process. Zener and Hollomon (1944) studied the effects of strain rate and suggested a Zener–Hollomon parameter in which the relation between strain rate and temperature can be derived from statistical mechanics. However, as explained in van den Boogaard (2002), the Zener–Hollomon approach can only be used for small strain rates and temperature variations. When the strain rate itself is a function of temperature these types of models that incorporate the Zener–Hollomon parameter are inappropriate for the simulation of warm forming of aluminum. Gronostajski (2000) provides a list of different types of deformation dependent flow
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stresses for FEM analysis. Ha˚kansson et al. (2005) made a comparison of isotropic hardening and kinematic hardening in thermoplasticity. In the current research, material characterization of the anisotropy coefficients of the YLD96 (Barlat et al., 1997a) model and the modified power-law flow stress were evaluated as a function of temperature, as well as the effect of temperature on the evolution of the yield surface and on the plastic anisotropy parameters R0, R45 and R90. Several fitting functions were compared for both the anisotropy coefficients of the yield function and for the flow stress.
2. Theoretical analysis 2.1. Anisotropic constitutive model Accuracy of numerical analysis depends on the use of a constitutive model that precisely describes the behavior of the material. In a previous paper by the authors (Abedrabbo et al., 2005b), the anisotropic behavior of AA6111-T4 aluminum alloy sheets was studied and the importance of using an appropriate material model that captures the anisotropic behavior was shown. This prompted the use of BarlatÕs anisotropic material model (YLD96) for the current study. The YLD96 anisotropic yield function presented by Barlat et al. (1997a) is one of the most accurate yield functions for aluminum alloy sheets Barlat et al. (2003), because it simultaneously accounts for yield stress and r-value directionalities simultaneously. This yield function is based on a phenomenological description of the material, as an improvement to the YLD91 model (Barlat et al., 1991). To represent the behavior of these materials mathematically, a yield function was generalized to include the most general stress tensor with six components as shown in Eq. (1): a
a
a
U ¼ a1 jS 2 S 3 j þ a2 jS 3 S 1 j þ a3 jS 1 S 2 j ¼ 2 ra
ð1Þ
with a = 6 and a = 8 for BCC and FCC materials, respectively. S1, S2 and S3 are the is the flow stress. principal values of the stress tensor Sij, which is defined later. r The isotropic plasticity equivalent (IPE) stress is: S ¼ Lr;
ð2Þ
where for orthotropic symmetry, L is defined below with anisotropic coefficients ck. 3 2 ðc2 þc3 Þ c3 c2 0 0 0 3 3 3 7 6 c3 ðc3 þc1 Þ c1 6 0 0 07 3 3 7 6 3 7 6 c2 ðc1 þc2 Þ c1 7 6 3 0 0 0 3 3 ð3Þ Lij ¼ 6 7. 6 0 0 0 c4 0 0 7 7 6 7 6 4 0 0 0 0 c5 0 5 0
0
0
0
0
c6
For the plane stress case (rz = ryz = rzx = 0), Eq. (2) reduces to:
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3 2 ðc2 þc3 Þ Sx 3 6S 7 6 c3 6 y 6 7 6 3 S ij ¼ 6 7¼ 4 Sz 5 6 4 c2 2
S xy
3
c3 3 ðc3 þc1 Þ 3 c1 3
c2 3 c1 3 ðc1 þc2 Þ 3
0
0
0
32
3 rx 76 ry 7 07 7 76 . 6 74 0 7 5 5 0 rxy c6 0
The principal values of Sij, as defined in Eq. (1), are found as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Sx þ Sy Sx Sy S 1;2 ¼ þ S 2xy 2 2
ð4Þ
ð5Þ
and S3 = (S1 + S2) because of the deviatoric nature of Sij. The anisotropic coefficients ai of the yield function (1) are defined as: a1 ¼ ax cos2 b þ ay sin2 b; a2 ¼ ax sin2 b þ ay cos2 b; 2
ð6Þ
2
a3 ¼ az0 cos 2b þ az1 sin 2b. In the above equations, c1, c2, c3, c6, ax, ay, az0 and az1 are coefficients that describe the anisotropy of the material. The parameter 2b represents the angle between the line OA and the axis of the principal IPE stress, S1, as shown in Fig. 1, and is equal to: 2S xy 1 2b ¼ tan . ð7Þ Sx Sy With this yield function it is possible to increase the yield stress at pure shear without increasing the other plane strain yield stresses. It should be noted that by setting the anisotropy coefficients c1 = c2 = c3 = c6 = a1 = a2 = a3 = 1.0 and a = 2 (quadratic), the von Mises isotropic yield function is recovered. 2.2. Plastic anisotropy parameters The plastic anisotropy parameter Rh is defined as the ratio of width-to-thickness strain increments:
2
A Sxy
Sy
O
S2
S1
Sx Fig. 1. Determination of the anisotropy parameter 2b from MohrÕs circle.
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Rh ¼
dew . det
319
ð8Þ
The thickness strain, however, is difficult to measure accurately in a thin sheet. Thickness strains are instead calculated from measurements of the longitudinal and width strains assuming constancy of volume as follows: det ¼ ðdel þ dew Þ.
ð9Þ
Therefore, two strain measurements are required, namely the longitudinal (del) and the width strains (dew), in order to calculate the plastic anisotropy parameters. For isotropic materials, R-values are equal to 1.0 for any direction h. R-values not equal to 1.0 indicate that plastic anisotropy exists in the material. A high R-value suggests that the material has a high resistance to thinning and thickening, which implies better formability of the material. If R-values depend on h, then the material is planar anisotropic, otherwise it is planar isotropic. 2.3. Anisotropy coefficients calculation In order for the constitutive model to accurately represent the aluminum alloy sheet at large strains, the anisotropic coefficients describing the behavior of the material, c1, c2, c3, c6, ax, ay, and az1 must be calculated (az0 can be set equal to 1.0) using seven test results. az0 can be set to 1.0 because increasing az0 will decrease ax and ay simultaneously as these parameters are related. Four stress states, uniaxial tension along the rolling, 45°, and transverse directions and balanced biaxial stress (Green et al., 2004), provide the required seven data points (Barlat et al., 1997b). A description of the data points from experiments with respect to the yield function is shown in Fig. 2. A summary of the experimental data needed to calculate the coefficients is given in Table 1. The yield stress could be used as input data instead of the flow stress. However, it may be difficult to accurately measure yield stress in the bulge test as well as in uniaxial tension because the slope of the stress–strain is steep and yielding is not always a discrete event in many aluminum alloys. Also, the yield stress is associated with very small plastic strain and might not reflect the anisotropy of the material over a larger strain range. For these reasons, flow stresses at equal amount of plastic work (Wb = Wu) were selected as input data rather than yield stress, as shown in Fig. 3. In this paper, a value of Wb = Wu = 20 MPa per unit volume was used to extract flow stressesr0, r45 andr90, as the material exhibited a sufficient amount of plastic strain Table 1 Experimental data needed to calculate yield function coefficients for plane stress states Test
Balanced biaxial
Uniaxial
Orientation Flow stress R-Value
N/A rb N/A
0° r0 R0
45° r45 R45
90° r90 R90
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Plane strain x=0 Uniaxial tension
Balanced biaxial tension
+Y
45˚
Plane strain y=0 1
1
-X
+X
Uniaxial tension
Pure shear -Y
2
Fig. 2. Characterization of required tests to be used in calculation of the anisotropic values in relation to the yield locus.
Fig. 3. Flow stresses at equal amounts of plastic work (Wb = Wu).
(18%) at that level of plastic work. The normalized values r0/r0, r45/r0 and r90/r0 were then used as input to calculate the anisotropy coefficients of the yield function. Flow stresses extracted at different values of W would not significantly change those stress ratios, and therefore consistent results can be obtained as long as sufficiently large values of W is used. The anisotropy coefficients were calculated as follows. For uniaxial tension in the rolling direction (0°), the principal IPE stresses become: ðc2 þ c3 Þ r0 ; 3 c3 S 2 ¼ r0 ; 3 c2 S 3 ¼ r0 . 3
S1 ¼
ð10Þ
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The yield function (1) can then be written as:
a 3 r ¼ 0. F 1 ¼ ax jc2 c3 j þ ay j2c2 þ c3 j þ az0 jc2 þ 2c3 j 2 r0 a
a
a
ð11Þ
Similar equations could be written for uniaxial tension in the 45°-direction, the transverse direction (90°) and the balanced biaxial tension as follows: a 3 r F 2 ¼ax j2c1 þ c3 ja þ ay jc3 c1 ja þ az0 j2c3 þ c1 ja 2 ¼ 0; ð12Þ r90 a a ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r r c c 2 c c 2 2 1 2 1 2 2 þ ð2c6 Þ þ a2 ðc1 þ c2 Þ þ þ ð2c6 Þ F 3 ¼a1 ðc1 þ c2 Þ 3 3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia a c c 2 2 r 2 1 þ a3 þ ð2c6 Þ2 2 ¼ 0; ð13Þ r45 3 where a1, a2 and a3 are as defined by Eq. (6). a
a
a
F 4 ¼ ax j2c1 þ c2 j þ ay j2c2 þ c1 j þ az0 jc2 c1 j 2
a 3 r ¼ 0. rb
ð14Þ
The other necessary sets of equations are derived from the plastic anisotropy definitions for R0, R45 and R90, with using Eq. (9) is defined as follows: Rh ¼
dew . ðdew þ del Þ
ð15Þ
From the normality rule: oU e_ xx ¼ k_ ; orxx oU ; e_ yy ¼ k_ oryy e_ xy þ e_ yx ¼ k_
ð16Þ oU . orxy
where k_ is a proportionality factor. After substitution, the plastic anisotropy is found as follows. For uniaxial tension performed along an arbitrary orientation h measured counterclockwise from the xaxis, the ratio of the strain rates Rh is defined by: Rh ¼
oU=or90þh ¼ oU=orzz
oU or90þh oU or90þh
oU þ or h
;
ð17Þ
where oU oU orxx oU oryy oU orxy ¼ þ þ2 ; orh orxx orh oryy orh orxy orh or
ð18Þ
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oU oU oU oU ¼ cos2 h þ sin2 h þ 2 cos h sin h orh orxx oryy orxy
ð19Þ
oU oU oU oU ¼ sin2 h þ cos2 h 2 cos h sin h; or90þh orxx oryy orxy
ð20Þ
and
which leads to Rh ¼
oU orxx
sin2 h þ oroUyy cos2 h 2 oroUxy cos h sin h ; oU oU þ orxx oryy
ð21Þ
where h = 0°, 45° and 90°, and U is the yield function. These systems of equations were solved for the values of the anisotropy coefficients c1, c2, c3, c6, ax, ay, and az1 using a non-linear solver (e.g., Newton–Raphson) with initial values corresponding to the isotropic situation (c1 = c2 = c3 = c6 = 1.0). 2.4. Constitutive equations (flow stress) Flow stress ( r) represents the size of the yield function during deformation. Metals undergoing plastic deformation at high temperatures and different strain rates should be modeled according to the physical behavior of the material (Gronostajski, 2000). An appropriate constitutive equation describing changes in the flow stress of the material depends on deformation conditions such as temperature and strain rate. Wagoner et al. (1988) proposed a flow rule that includes the strain-rate sensitivity: m e_ n p p ðe ; e_ Þ ¼ Kðe þ e0 Þ r ; ð22Þ esr0 where K (strength hardening coefficient), n (strain-hardening exponent) and m (strain-rate sensitivity index) are material constants. ep is the effective plastic strain and e_ is the strain rate. e0 is a constant representing the elastic strain to yield and esr0 is a base strain rate (a constant). This model was primarily selected over other types of hardening laws (e.g., Voce) because it represented the hardening behavior of the current material accurately, and it incorporates strain rate effects. Gronostajski (2000) describes other types of hardening laws that could be used (e.g., Backofen, Grosman) to represent other materials, including those with hardening saturation behavior. The coefficients K, n and e0 of Eq. (22) were solved for simultaneously using Levenberg–Marquardt algorithm. The strain-rate sensitivity index m was determined by the following equation using experimental data (Figs. 8 and 9): m¼
lnðr2= r1 Þ . lnð_e2 =_e1 Þ
ð23Þ
From the data acquired for e0, it was found that its value is very small (see Table 4). Another method was also used to solve for e0, which represents the elastic strain to
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323
yield by solving for the intersection of the linearly elastic loading equation with the strain hardening equation: r ¼ Ee; r ¼ Ken This provides the elastic strain at yield as: 1 ½n1 E e0 ¼ ; K
ð24Þ
ð25Þ
where E is the YoungÕs modulus. Comparing the values of e0 calculated using Eq. (25), to the ones determined using the Levenberg–Marquardt algorithm it was found that the differences between them are very small. Also the values of e0 are generally too small to have a great effect on the resulting stresses in forming processes. Furthermore, it was observed that e0 does not show significant temperature dependence. In this paper, e0 calculated from Eq. (25) was therefore used. Material properties change with increasing temperature. To include temperature effects in the flow rule, the material constants K, n and m are expressed as a function of temperature, and the flow stress equation becomes: mðT Þ e_ ðep ; e_ ; T Þ ¼ KðT Þðep þ e0 ÞnðT Þ r . ð26Þ esr0 The hardening model described by Eq. (26) was used assuming isotropic hardening behavior.
3. Experimental procedure 3.1. Materials Chemical composition and initial mechanical properties of the aluminum alloy (AA3003-H111) used in this study are shown in Tables 2 and 3. All samples were taken from a single lot of material. Because AA3003 is very soft in the annealed condition, one cold roll pass (H111) was applied to these sheets in order to provide some rigidity during handling. 3.2. Experimental procedure Uniaxial tests with standard ASTM-E8 rectangular dog-bone shaped samples shown in Fig. 4, were performed on an Instron Model 1127, screw-driven frame with a 4.5 kN load cell, 25 mm axial extensometer (50% strain max) and 12.7 mm transverse extensometer (30% strain max). The tensile samples were prepared from the aluminum sheet metal at 0° (rolling direction, RD), 45° and 90° (transverse direction, TD) from the rolling direction of the sheet. For the measurement of plastic anisotropy parameters, ASTM-E517 specifies that the test be performed
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Table 2 Chemical composition of AA3003-H111 (wt%) Al
Mg
Cu
Mn
Fe
Si
Cr
Zn
Ti
Ni
bal
0.02
0.07
1.10
0.50
0.21
0.005
0.01
0.02
0.005
Table 3 As-received mechanical properties of AA3003-H111 Thickness (mm)
UTS (MPa)
Yield strength (MPa)
Total elongation (%)
0.96
114.0
63.0
32.0
Fig. 4. Rectangular tension test specimen, ASTM-E8. All dimensions in mm.
at a 0.5/min (0.0083 s1) straining rate. Sample temperature was controlled with an Instron Model 3119 oven with a convection heating system. Tests were performed for several elevated temperatures in the range of 25–260 °C (77–500 °F), with the results of three tests averaged for each temperature. It should be mentioned that only a small variation in the stress–strain behavior was noticed between the three tests for each temperature. Tests in the experiments were performed at a fixed ambient temperature. To study the strain-rate sensitivity of the material, uniaxial tests under several strain rates (0.001–0.08 s1) were performed at each temperature. Bulge testing was performed on sheet samples from the same lot of material in order to determine rb at room temperature. At the moment, experimental calculation of the balanced biaxial data at elevated temperature is under development. Based on experimental observations from uniaxial tests at 0°, 45° and 90°, see Fig. 5, it was noticed that at room temperature the stress values at 45° and 90° coincide with the stress value from bulge test. Since stress values at 45° and 90° continued to coincide at elevated temperatures, it was assumed that a good estimate for the bugling stress at elevated temperature would be the average of the stress values at 45° and 90°. This assumption is unproven and will be studied and reported separately in a future publication. It should be noted that other methods could be used to predict rb. For example, bulge stress rb could be calculated from HillÕs 1948 model (Hill, 1948) or YLD91 model (Barlat et al., 1991) using only the available uniaxial tension tests data. In this paper however, it was not deemed appropriate to present this approach until after experimental bulge tests are completed, at which time a comparison of all these methods will be reported.
N. Abedrabbo et al. / International Journal of Plasticity 22 (2006) 314–341
325
160
140
True Stress (MPa)
120
100
80
60
40
20 0
45
90
b
0 0
50
100
150 Temperature (˚C)
200
250
300
Fig. 5. Stress values at equal values of plastic work (20 MPa/unit-vol.) as a function of temperature. Data shown for tests at 0°, 45° and 90° directions at multiple elevated temperatures and for balanced biaxial tension (bulge) data at room temperature.
4. Results and discussion 4.1. Hardening model Fig. 6 shows the true stress–true strain behavior at room temperature for rolling direction (RD), 45° and TD uniaxial tests as well as the balanced biaxial (bulge) test. As seen from the bulge test, much larger strains can be obtained compared to uniaxial strains. Fig. 7 shows the engineering stress–strain behavior at several temperatures for the rolling direction. As temperature increased, flow stress of the material decreased with a corresponding increase in the elongation to failure. Two methods were used to measure strain-rate sensitivity of the material. In the first, several tensile tests were performed on identical samples at different strain-rates (from 0.001 to 0.08 s1) and at several elevated temperatures. For example, Fig. 8 shows the true stress–true strain curves at 204 °C (400 °F) in the rolling direction. Considering that the sample-to-sample variation (thickness, width, etc. . .) may cause scatter in the results, it was desirable to perform the strain-rate sensitivity analysis using a single sample with the ‘‘Jump-rate Test’’ method (Wagoner and Chenot, 1996). In this method, the crosshead speed is increased to produce a ‘‘jump’’ in the strain rate at some predetermined level of strain. Fig. 9 shows results of this method at several elevated temperatures in which crosshead speed jumps were 10–50–150 mm/mm/min. As seen in Figs. 8 and 9, AA3003-H111 exhibits very small strain-rate sensitivity at room temperature, but with increasing temperature, the material becomes more strain rate sensitive.
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Balanced Biaxial
True Stress (MPa)
140 120 100 80 60 40
Rolling Direction
20
45˚-Direction Transverse Direction
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
True strain
Fig. 6. True stress strain data of AA3003-H111 at room temperature for uniaxial tests at 0°, 45° and 90° directions and for balanced biaxial tension (bulge) data.
120 25˚C 100 66˚C
Tensile Stress (MPa)
121˚C 80 177˚C 204˚C
60
232˚C 40
20
0 0
10
20
30
40
50
Tensile Strain (%)
Fig. 7. Engineering stress strain curves of AA3003-H111 at several elevated temperatures for the rolling direction.
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327
-1
0.08s
100
-1
0.05s
-1
0.01s
80
True Stress (MPa)
0.001s
-1
60
40
20
0 0.00
0.05
0.10
0.15 0.20 True Strain
0.25
0.30
0.35
Fig. 8. True stress strain data of AA3003-H111 from uniaxial tests at 204 °C (400 °F) at several strain rates for the rolling direction.
160 150mm/mm/min
25˚C
140 93.3˚C
50mm/mm/min 120
True Stress (MPa)
10mm/mm/min 100 204.4˚C 80 260˚C 60 40 20 0 0
0.05
0.1
0.15
0.2 True Strain
0.25
0.3
0.35
0.4
Fig. 9. True stress strain data of AA3003-H111 from uniaxial tests using the jump-rate strain method. The jumps in crosshead speed were 10–50–150 mm/mm/min (at temperature of 260 °C, at cross head speed of 150 mm/mm/min the specimen failed, therefore there is no information available).
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Table 4 Material properties of AA3003-H111 at elevated temperatures Temperature °C
K MPa
n
e0
m
R0
R45
R90
25 38 66 93 121 149 177 204 232 260
199.82 186.41 175.78 168.41 146.98 139.18 119.65 106.32 93.820 77.320
0.215 0.200 0.187 0.179 0.175 0.163 0.157 0.137 0.132 0.116
8.30E 04 5.00E 04 3.16E 04 6.64E 04 3.20E 04 7.40E 04 6.20E 04 2.21E 04 3.71E 04 5.05E 04
0.003 0.004 0.004 0.005 0.010 0.015 0.030 0.045 0.065 0.080
0.827 1.074 1.092 1.247 1.262 1.273 1.413 1.614 1.688 2.035
1.126 1.424 1.457 1.690 1.742 1.787 1.975 1.977 2.281 2.485
0.773 0.892 0.978 0.879 0.982 0.943 1.007 1.105 1.236 1.328
The values shown below represent an average of three tensile tests. The hardening values are for the rolling direction.
From the results of uniaxial tension tests performed at different constant strain rates, and at several elevated temperatures, values for the Holloman hardening rule (K, n, e0 and m) and the plastic anisotropy parameters (Rh) were calculated as a function of temperature. For example, Table 4 shows values of K, n, e0 obtained with Levenberg–Marquardt algorithm, and m for the rolling direction. Similar results were also found for 45° and the transverse directions. For the finite element simulation, values of K, n and m only from the rolling direction were used to represent isotropic hardening. As shown in Figs. 10 and 11, the values of K and n decrease linearly with temperature. This is in line with the decreasing flow stress and flattening of the hardening curve shown in Fig. 7. Fig. 12 shows the variation of the strain-rate sensitivity index, m, as a function of temperature for AA3003-H111. At lower temperatures, m values are very small, indicating the material is strain-rate insensitive; but at higher temperatures, the material exhibits a significant sensitivity to strain-rate. An exponential function, as shown on the graph, was used to represent the behavior of m as a function of temperature. Fig. 13 shows the variation of the plastic anisotropy parameters R0, R45 and R90, measured at a straining rate of 0.5/min, with respect to temperature. As mentioned earlier, R-values higher than 1.0 indicate good formability and resistance to thinning. As can be seen from Fig. 13, the values of R0, R45 and R90 increase with temperature, which suggests that the formability of the aluminum sheet also enhances at elevated temperature. Table 5 shows a summary of the equations used to fit the hardening parameters for the flow rule, along the rolling direction, as a function of temperature. Curve fitting at 45° and the transverse directions are not required, since only the values of hardening in the rolling direction are used in the finite element simulation. 4.2. Barlat’s yield 96 anisotropy coefficients Results from the uniaxial and bulge tests were implemented into Eqs. (11)–(21), and BarlatÕs YLD96 model coefficients were calculated at several temperatures with
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329
250
K-Values (MPa)
200
150 K(T) = -0.5058*T + 210.4 100
50
Experimental Data
Curve Fit
0 0
50
100
150 Temperature (˚C)
200
250
300
Fig. 10. Strength hardening coefficient (K) of AA3003-H111 along the rolling direction as a function of temperature at 0.5 mm/mm strain rate. Solid line is a linear curve fit.
0.25
0.20
n(T) = - 0.0004*T + 0.2185
n-Value
0.15
0.10
0.05
Experimental Data
Curve Fit
0.00 0
50
100
150
200
250
300
Temperature (˚C)
Fig. 11. Strain-hardening exponent (n) of AA3003-H111 along the rolling direction as a function of temperature at 0.5/min strain rate. Solid line is a linear curve fit.
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m-value
m(T) = 0.0018*e 0.05 0.04 0.03 0.02 0.01
M-value Expon. (M-value)
0.00 0
50
100
150
200
250
300
Temperature (˚C)
Fig. 12. Strain-rate sensitivity index (m) of AA3003-H111 along the rolling direction as a function of temperature. Solid line is an exponential curve fit.
3.0
2.5
R - Value
2.0
1.5
1.0
0.5
R0
R45
R90
0.0 0
50
100
150
200
250
300
Temperature ˚C
Fig. 13. Plastic anisotropy parameters (Rh) of AA3003-H111 as a function of temperature, calculated at 0.5/min strain rate (ASTM-E517).
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Table 5 Summary of equations used to fit hardening parameters for the power law flow rule Hardening parameter
Rolling direction, 0°
Unit
K(T) n(T) m(T)
=0.5058*T + 210.40 =0.0004*T + 0.2185 =0.0018*exp(0.0147*T)
MPa
Temperatures in °C.
the results shown in Table 6. It should be noted that in BarlatÕs yield function az0 = 1.0, and the non-quadratic parameter for aluminum alloys (FCC) is a = 8. Fig. 14 is a plot of the yield function for AA3003-H111, comparing BarlatÕs YLD96 model at 25 °C (77 °F) with the isotropic von Mises yield function (normal). Fig. 15 is a similar plot comparing BarlatÕs ized stress values with respect to r YLD96 function at several elevated temperatures (normalized stress values with re), indicating that temperature has an effect on the shape of the yield surface spect to r (change in the local slopes at different temperatures). This change in the shape is particularly noticed near the balanced biaxial region of the yield surface. This is to be expected, since any change in the plastic anisotropy parameters (R-values) would affect the shape of the yield surface. Fig. 16 shows the effect of ax, ay and az0 on the yield surface shape (Barlat et al., 1997a,b). The az1 parameter primarily affects the yield surface for the stress state with rxy and therefore its effect cannot be seen in a 2D plot (Fig. 16) where a constant shear stress is assumed. Due to the significant effect that temperature has on material properties, the application of a temperature-dependent constitutive model for accurate analysis of warm forming becomes imperative. It is also necessary that this temperature-dependent constitutive model be used in a coupled thermo-mechanical finite element analysis of warm forming process where the thermal analysis provides temperature as input ˇ anadija and Brnic´, 2004). It is from such a coupled analto the mechanical model (C ysis that deformation stresses corresponding to both thermal and mechanical deformation could be accurately calculated. This issue is discussed in more detail in Part 2 of this paper (Abedrabbo et al., 2005a). Table 6 BarlatÕs YLD96 model anisotropy coefficients calculated at several elevated temperatures Temperature, °C
c1
c2
c3
c6
ax
ay
az1
25 38 66 93 121 149 177 204 232 260
1.11698 1.16219 1.22994 1.28713 1.08469 1.18035 1.12801 1.17424 1.25255 1.23298
0.95454 1.04235 1.02323 1.04776 1.01709 1.03178 1.06198 1.06811 1.20946 1.09940
1.0030 0.9877 0.9968 0.9953 1.0116 1.0049 1.0003 1.0058 0.9564 0.9939
1.0429 1.0985 1.0888 1.0721 0.9639 1.0185 0.9782 0.9932 0.9938 0.9904
0.9130 0.6516 0.5028 0.4810 0.8060 0.6182 0.6650 0.5382 0.3160 0.3920
1.3960 0.7866 0.8513 0.7034 0.7864 0.7380 0.5930 0.5294 0.2427 0.4515
1.2500 1.0400 1.1560 1.5080 2.1100 1.8057 2.1460 1.9010 1.9690 2.1230
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Fig. 14. 2D-Plot of von Mises yield function vs. BarlatÕs YLD96 yield function at 25 °C (77 °F) for AA3003-H111.
1.5 1
y
/
0.5 0 -0.5 25˚C 93˚C 149˚C 204˚C 260˚C
-1 -1.5 -1.5
-1
-0.5
0 x
0.5
1
1.5
/
Fig. 15. 2D-Plot of BarlatÕs YLD96 yield function for AA3003-H111 at several elevated temperatures. Normalized stress values.
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333
1.5 x
1 x
y
y
0
y
/
0.5
z0
-0.5 z0
-1 -1.5 -1.5
-1
-0.5
0
0.5
1
1.5
/
x
Fig. 16. Effect of ax, ay and az0 on the yield function (Barlat et al., 1997).
At a first glance, it seems that the coefficients of Barlat YLD96 in Table 6 are fluctuating randomly without any apparent trend. This is attributed to the fact that the experimental data needed to calculate the anisotropic coefficients of YLD96 yield function (Table 1) do not linearly increase with temperature. Looking at the stress values in Fig. 5 it is noticed that they decrease with temperature, while plastic anisotropy parameters (R0, R45 and R90) in Fig. 13 increase with temperature. This opposite behavior causes anisotropy coefficients of the yield function to fluctuate. Therefore, it is important to use higher order fitting functions to capture these variations. To show the importance of varying the anisotropy coefficients with temperature, the plastic anisotropy parameters R0, R45 and R90 were first calculated directly from the yield function using Eq. (21) and the values of the yield function coefficients at the discrete temperatures listed in Table 6. Then, the plastic anisotropy parameters (R0, R45 and R90) were calculated at those same temperatures but this time values of the yield function anisotropy coefficients at 25 °C were used for all temperatures. In both cases, however, the temperature dependent hardening values from Table 5 were used, as flow stress in general only affects the size of the yield function and not the plastic anisotropy parameters. Fig. 17 shows the result of the two tests for the R0 case. From Fig. 17, it could be seen that the plastic anisotropy parameter R0 closely followed the experimental data when the values of anisotropy coefficients at each temperature from Table 6 were used. However, the resulting plastic anisotropy parameters remained constant when the yield function coefficients were assumed to remain constant for all temperatures. This indicates the importance of using temperature dependent anisotropy coefficients, when accurate modeling results are expected. In order to develop an anisotropic material model for use in a coupled thermomechanical finite element analysis of sheet metal forming processes, the anisotropy
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2.0
R0
1.5
1.0
0.5
Experimental-R0 Variable anisotropy coefficients Constant values @25˚C 0.0 0
50
100
150 Temperature (˚C)
200
250
300
Fig. 17. Plot of experimental values of plastic anisotropy parameter (R0) of AA3003-H111 compared with predictions from the Barlat YLD96 material model. Calculations were done first using variable values of the yield function with each temperature and then by using a fixed value for all temperatures.
coefficients of a yield function must be represented as a function of temperature. Thus, curve-fitting methods were used to fit the anisotropy coefficients shown in Table 6. Figs. 18 and 19 show a sample plot of some of the YLD96 coefficients, e.g., c6 and az1, as a function of temperature. In these plots, two levels of polynomial curve fitting, 3rd and 5th order were used to represent variation with respect to temperature. The fit functions used for all seven anisotropy coefficients are shown in Table 7. To determine the validity and accuracy of the 3rd and 5th order polynomial curve fit functions, the plastic anisotropy parameters R0, R45 and R90 were calculated directly from the yield function using Eq. (21) and compared to experimentally measured values. All material parameters used in these calculations, i.e., hardening parameters and BarlatÕs YLD96 function anisotropy coefficients, were set as temperature-dependent equations from Tables 5 and 7. The results at discrete points are shown in Figs. 20–22. While both results of 3rd and 5th order polynomial curve fit functions of the plastic anisotropy parameters R0 and R45 closely agree with the experimental data, the 5th order fit function appears to match the data slightly better than the 3rd order function. On the other hand, in the case of plastic anisotropy in the transverse direction, R90, neither of the two functions agrees very well with experimental data at temperatures lower than 150 °C. This difference could be attributed to possible inaccuracies associated with both experimental tests as well as with the fitting of the yield function anisotropy coefficients. This issue is discussed in more detail in Part 2 of this paper (Abedrabbo et al., 2005a).
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1.2
1.0
C6(T)
0.8
0.6
0.4
0.2 Experimental
3th Order Fit
5th Order Fit
0.0 0
50
100
150
200
250
300
Temperature (˚C)
Fig. 18. Plot of (c6) as a function of temperature for AA3003-H111. 3rd and 5th order polynomial curve fits are shown also.
2.5
2.0
z1(T)
1.5
1.0
0.5
Experimental
3th Order Fit
5th Order Fit
0.0 0
50
100
150 Temperature (˚C)
200
250
300
Fig. 19. Plot of (az1) as a function of temperature for AA3003-H111. 3rd and 5th order polynomial curve fits are shown also.
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Table 7 Barlat YLD96 material model anisotropy coefficients for AA3003-H111 as a function of temperature using two different order polynomial fit functions YLD96 coefficient
3rd order fit
5th order fit
c1
=1.0446 + 0.0045834T 3.9615E 5T2 + 9.7409E 8*T3 =0.9867 + 0.000373T + 1.1717E 7T2 + 3.0052E 9T3 =0.9860 + 0.0003334T 1.7857E 6T2 + 1.593E 9T3 =1.0592 + 0.000855T 1.3728E 5T2 + 3.696E 8T3 =1.0146 0.010636T + 8.3876E 5T2 2.0664E 7T3 =1.4253 0.012009T + 6.4874E 5T2 1.3432E 7T3 =0.86396 + 0.007906T + 3.0245E 6T2 6.2854E 8T3
=0.90902 + 0.0097045T 6.2517E 5T2 3.3687E 7T3 + 3.7316E 9T4 7.6631E 12T5 =0.90308 + 0.0023977T + 3.5398E 5T2 8.9808E 7T3 + 5.4066E 9T4 9.8763E 12T5 =0.98895 + 0.0010519T 3.7432E 5T2 + 4.7418E 7T3 2.3519E 9T4 + 3.9325E 12T5 =0.76161 + 0.017737T 0.0003099T2 + 2.2357E 6T3 7.2652E 9T4 + 8.8146E 12T5 =1.8838 0.05496T + 0.00073166T2 3.8433E 6T3 + 7.5607E 9T4 3.0894E 12T5 =2.683 0.079188T + 0.0011348T2 7.0955E 6T3 + 1.9271E 8T4 1.8287E 11T5 =2.7451 0.096914T + 0.0017925T2 1.2899E 5T3 + 4.0712E 8T4 4.7071E 11T5
c2 c3 c6 ax ay az1
Temperatures in °C.
2.5
2.0
R0
1.5
1.0
0.5
Experimental-R0
3rd-Order-Fit
5th-Order-Fit
0.0 0
50
100
150 Temperature (˚C)
200
250
300
Fig. 20. Plot of experimental values of plastic anisotropy parameter (R0) for AA3003-H111 compared with predictions from BarlatÕs YLD96 material model where the yield function parameters are a function of temperature. Calculations were done using 3rd and 5th order curve fits for the yield function anisotropy coefficients.
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337
3.0
2.5
R45
2.0
1.5
1.0
0.5 Experimental-R45
3rd-Order-Fit
5th-Order-Fit
0.0 0
50
100
150
200
250
300
Temperature (˚C)
Fig. 21. Plot of experimental values of plastic anisotropy parameter (R45) for AA3003-H111 compared with predictions from BarlatÕs YLD96 material model where the yield function parameters are a function of temperature. Calculations were done using 3rd and 5th order curve fits for the yield function anisotropy coefficients.
1.4
1.2
1.0
R90
0.8
0.6
0.4
0.2
Experimental-R90
3rd-Order-Fit
5th-Order-Fit
0.0 0
50
100
150
200
250
300
Temperature (˚C)
Fig. 22. Plot of experimental values of plastic anisotropy parameter (R90) for AA3003-H111 compared with predictions of BarlatÕs YLD96 material model where the yield function parameters are a function of temperature. Calculations were done using 3rd and 5th order curve fits for the yield function anisotropy coefficients.
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Fig. 23. Plot of BarlatÕs YLD96 yield function for AA3003-H111 using anisotropy coefficients extracted from experimental data compared to anisotropy coefficients from both the 3rd and 5th order curve fits shown in Table 7. Temperature was 25 °C (77 °F).
Fig. 24. Plot of BarlatÕs YLD96 yield function for AA3003-H111 using anisotropy coefficients extracted from experimental data compared to anisotropy coefficients from both the 3rd and 5th order curve fits shown in Table 7. Temperature was 204 °C (400 °F).
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The experimentally determined anisotropy coefficients were compared to those calculated from the 3rd and 5th order polynomial curve fit functions from Table 7 for a temperature of 25 °C (77 °F) in the plot of BarlatÕs YLD96 yield function shown in Fig. 23. There is very little difference between these two yield surface curves. Fig. 24 shows the yield function plot for a temperature of 204 °C (400 °F). Again in both cases, the yield functions coincide with very small differences. (All ). yield function plots show normalized values with respect to r
5. Conclusions In Part 1 of this paper, a temperature-dependent anisotropic material model to be used in the finite element analysis model has been developed for the aluminum alloy sheet AA3003-H111. Using experimental data from uniaxial and bulge tests in different directions, the anisotropy coefficients of the Barlat YLD96 model for several elevated temperatures in the range of 25–260 °C (77–500 °F) have been calculated. Using two polynomial curve fitting functions (3rd and 5th order), the anisotropy coefficients of the yield function were determined as a function of temperature. From experimental tests, a strain-rate dependent hardening flow rule was also determined as a function of temperature. The curve fitting functions were verified by a comparison of the plastic anisotropy parameters (R0, R45 and R90) that were extracted from experimental tests to the values predicted from the yield function in which the anisotropy coefficients vary with temperature. Also, the normalized yield functions at several temperatures compared favorably to experimental results. The curve fit functions representing the anisotropy of the yield function could be used in a coupled thermo-mechanical finite element analysis to provide the correct temperature-dependent yield function at each step in the simulation of forming processes for aluminum sheet alloys. In Part 2 of this paper (Abedrabbo et al., 2005a), modeling issues related to the implementation of this temperature dependent yield function for the analysis of warm forming and prediction of failure of aluminum sheets will be discussed. Future plans include extending the current approach and applying it to Barlat YLD2000 yield function (Barlat et al., 2003; Yoon et al., 2004) for several aluminum alloys of interest to industry. From the results obtained in this research, it is expected that both the anisotropy coefficients and the hardening behavior of other aluminum alloys would also change as a function of temperature. Therefore, a comprehensive characterization of the material is needed if accurate finite element analysis of sheet forming processes at elevated temperatures is expected.
Acknowledgements The authors thank General Motors for the support of this research project. The authors sincerely thank Drs. Paul Krajewski and Anil Sachdev from the
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GM Research and Development Center and Dr. Frederic Barlat from ALCOA for their assistance and helpful discussions in support of this research.
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Forming of aluminum alloys at elevated temperatures – Part 1: Material characterization Nader Abedrabbo a, Farhang Pourboghrat a
b
a,*
, John Carsley
b
Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824-1226, United States General Motors Research & Development Center, Warren, MI 48090, United States Received 1 September 2004 Available online 2 June 2005
Abstract A temperature-dependent anisotropic material model for use in a coupled thermo-mechanical finite element analysis of the forming of aluminum sheets was developed. The anisotropic properties of the aluminum alloy sheet AA3003-H111 were characterized for a range of temperatures 25–260 °C (77–500 °F) and for different strain rates. Material hardening parameters (flow rule) and plastic anisotropy parameters (R0, R45 and R90) were calculated using standard ASTM uniaxial tensile tests. From this experimental data, the anisotropy coefficients for the Barlat YLD96 yield function [Barlat, F., Maeda, Y., Chung, K., Yanagawa, M., Brem, J.C., Hayashida, Y., Lege, D.J., Matsui, K., Murtha, S.J., Hattori, S., Becker, R.C., Makosey, S., 1997a. Yield function development for aluminum alloy sheets. J. Mech. Phys. Solids 45 (11/ 12), 1727–1763] in the plane stress condition were calculated for several elevated temperatures. Curve fitting was used to calculate the anisotropy coefficients of BarlatÕs YLD96 model and the hardening parameters as a function of temperature. An analytical study of the accuracy and usability of this curve fitting technique is presented through the calculation of plastic anisotropy R-parameters and yield function plots at different temperatures. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Thermo-mechanical; Temperature; Material anisotropy; Plastic anisotropy; Yield function
*
Corresponding author. Tel.: +1 517 432 0819; fax: +1 517 353 1750. E-mail address: [email protected] (F. Pourboghrat).
0749-6419/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijplas.2005.03.005
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1. Introduction Weight reduction has long been identified as a key priority for improving automotive fuel economy (Greene and DiCicco, 2000), and many studies often suggest substituting lightweight materials for typical steel applications. However, replacing steel in the structure and body of an automobile with lighter materials such as aluminum, magnesium, plastics, composites, etc., can be costly and is not simple or straightforward. Aluminum sheet, in particular, has lower formability at room temperature than typical sheet steel (Ayres and Wenner, 1979). High temperature forming methods based on superplastic behavior of Al–Mg alloys have been used to produce automotive closure panels that far exceed the conventional stamping formability of steel (Schroth, 2004). Superplastic formability however, requires finegrained microstructures and slower strain rates which can affect production cost. The formability of typical Ôoff-the-shelfÕ automotive aluminum sheet alloys (5182O and 5754-O) can be greatly improved by warm forming (Li and Ghosh, 2003) without the increased costs of refining the microstructure. The elevated temperature corresponds with decreased flow stress and increased ductility in the sheet, which can lead to deeper drawing and more stretching to form panels without design modifications to the stamped steel product. Furthermore, the serrated flow behavior of Al– Mg sheet alloys (dynamic strain aging/PLC effect) (Robinson and Shaw, 1994) and corresponding Lu¨derÕs line surface defects that result from the interactions of solutes with mobile dislocations can be avoided by deforming the sheet metal above a critical temperature. Finite element analysis (FEA) and simulations are vital tools in the automotive design and formability process to accurately predict deformation behavior during the stamping operation. Confidence in the numerical analysis of formability depends on the accuracy of the constitutive model describing the behavior of the material (Chung and Shah, 1992). This is especially important when the material exhibits anisotropic characteristics, as do most cold rolled sheet metals. Previously (Zampaloni et al., 2003; Abedrabbo et al., 2005b) the importance of using appropriate material models was demonstrated with respect to wrinkling and ironing during the sheet hydroforming process of an Al–Mg–Si alloy (AA6111-T4). The material model used was the anisotropic yield function proposed by Barlat et al. (1997a) (YLD96), which is based on a phenomenological description of the material. The use of anisotropic material models in FEA is further complicated by the material dependence on the anisotropy coefficients that require thorough material characterization under multiple loading conditions (Chung et al., 1996). This difficulty is further exacerbated when temperature effects are introduced. Because material hardening behavior and material response to loading conditions change at elevated temperatures, the anisotropy coefficients must be determined as a function of temperature so that the material model can account for these changes. Garrett et al. (in press) shows the effect of solution heat treatment on mechanical properties for AA6xxx alloys. Ayres (1979), Painter and Pearce (1980), Taleff et al. (1998, 2001), Takata et al. (2000), Naka et al. (2001), Li and Ghosh (2003) and Li and Ghosh (2004) studied
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the effects of elevated temperatures between 25 and 410 °C (77–770 °F) on the formability of aluminum alloys. Serrated flow behavior (dynamic strain aging) characteristic of AA5182 at room temperature (RT) was shown to diminish with elevated temperature deformation. Also, a larger total elongation in the post-uniform strain region was revealed along with a reduction in tensile strength. This indicates a shifting in the hardening behavior with temperature to a greater emphasis on strain-rate hardening relative to strain hardening. Several research papers characterized the tensile behavior of aluminum alloys at elevated temperatures (Li and Ghosh, 2003; Ayres, 1979; Painter and Pearce, 1980; Takata et al., 2000; Naka et al., 2001; van den Boogaard et al., 2001); the majority however, only studied the effect of elevated temperature on the evolution of the flow (hardening) stress. The effect of elevated temperature on the anisotropy of the material and how the yield surface of the aluminum alloy evolved as a function of temperature was not fully explored. In most cases, either HillÕs 1948 model (Hill, 1948) or the von Mises isotropic yield functions were used. van den Boogaard et al. (2001) characterized the behavior of AA5754-O in which two types of functions representing flow stress were used: the modified power law model and the Bergstro¨m model. The coefficients of the power law model were curve-fit exponentially as a function of temperature. The predictions of the material model, however, underestimated the values of the punch load in both models (Power-Law and Bergstro¨m models). Keum and Ghoo (2001, 2002) studied the effect of temperature on the anisotropic coefficients of the Barlat strain rate potential model (Barlat and Chung, 1993) for AA5052-H32. In this study, a third order polynomial function was used to fit the anisotropy coefficients, and a fifth order polynomial function was used to fit the coefficients for the power law model of flow stress. The material model, however, was assumed to reduce to the von Mises isotropic model as temperature increases. Furthermore, the plastic anisotropy parameters, R0, R45 and R90 almost remained constant with increasing temperature, which is opposite of what was found in the current research study. Other studies by Cleveland et al. (2003) and Masuda et al. (2003) also describe temperature and strain rate effects on flow stress and deformation behavior. Naka et al. (2003) conducted experimental tests to determine the yield locus of AL5083 at elevated temperatures. It was shown that the yield locus changes as a function of temperature and a higher order yield function (e.g., Barlat YLD96) is best for representing such a material. Vial-Edwards (1997) show methods for determining the yield loci for FCC and BCC sheet metals. In warm forming methods, the physics of dislocation movement suggests a thermally activated process. Zener and Hollomon (1944) studied the effects of strain rate and suggested a Zener–Hollomon parameter in which the relation between strain rate and temperature can be derived from statistical mechanics. However, as explained in van den Boogaard (2002), the Zener–Hollomon approach can only be used for small strain rates and temperature variations. When the strain rate itself is a function of temperature these types of models that incorporate the Zener–Hollomon parameter are inappropriate for the simulation of warm forming of aluminum. Gronostajski (2000) provides a list of different types of deformation dependent flow
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stresses for FEM analysis. Ha˚kansson et al. (2005) made a comparison of isotropic hardening and kinematic hardening in thermoplasticity. In the current research, material characterization of the anisotropy coefficients of the YLD96 (Barlat et al., 1997a) model and the modified power-law flow stress were evaluated as a function of temperature, as well as the effect of temperature on the evolution of the yield surface and on the plastic anisotropy parameters R0, R45 and R90. Several fitting functions were compared for both the anisotropy coefficients of the yield function and for the flow stress.
2. Theoretical analysis 2.1. Anisotropic constitutive model Accuracy of numerical analysis depends on the use of a constitutive model that precisely describes the behavior of the material. In a previous paper by the authors (Abedrabbo et al., 2005b), the anisotropic behavior of AA6111-T4 aluminum alloy sheets was studied and the importance of using an appropriate material model that captures the anisotropic behavior was shown. This prompted the use of BarlatÕs anisotropic material model (YLD96) for the current study. The YLD96 anisotropic yield function presented by Barlat et al. (1997a) is one of the most accurate yield functions for aluminum alloy sheets Barlat et al. (2003), because it simultaneously accounts for yield stress and r-value directionalities simultaneously. This yield function is based on a phenomenological description of the material, as an improvement to the YLD91 model (Barlat et al., 1991). To represent the behavior of these materials mathematically, a yield function was generalized to include the most general stress tensor with six components as shown in Eq. (1): a
a
a
U ¼ a1 jS 2 S 3 j þ a2 jS 3 S 1 j þ a3 jS 1 S 2 j ¼ 2 ra
ð1Þ
with a = 6 and a = 8 for BCC and FCC materials, respectively. S1, S2 and S3 are the is the flow stress. principal values of the stress tensor Sij, which is defined later. r The isotropic plasticity equivalent (IPE) stress is: S ¼ Lr;
ð2Þ
where for orthotropic symmetry, L is defined below with anisotropic coefficients ck. 3 2 ðc2 þc3 Þ c3 c2 0 0 0 3 3 3 7 6 c3 ðc3 þc1 Þ c1 6 0 0 07 3 3 7 6 3 7 6 c2 ðc1 þc2 Þ c1 7 6 3 0 0 0 3 3 ð3Þ Lij ¼ 6 7. 6 0 0 0 c4 0 0 7 7 6 7 6 4 0 0 0 0 c5 0 5 0
0
0
0
0
c6
For the plane stress case (rz = ryz = rzx = 0), Eq. (2) reduces to:
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3 2 ðc2 þc3 Þ Sx 3 6S 7 6 c3 6 y 6 7 6 3 S ij ¼ 6 7¼ 4 Sz 5 6 4 c2 2
S xy
3
c3 3 ðc3 þc1 Þ 3 c1 3
c2 3 c1 3 ðc1 þc2 Þ 3
0
0
0
32
3 rx 76 ry 7 07 7 76 . 6 74 0 7 5 5 0 rxy c6 0
The principal values of Sij, as defined in Eq. (1), are found as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Sx þ Sy Sx Sy S 1;2 ¼ þ S 2xy 2 2
ð4Þ
ð5Þ
and S3 = (S1 + S2) because of the deviatoric nature of Sij. The anisotropic coefficients ai of the yield function (1) are defined as: a1 ¼ ax cos2 b þ ay sin2 b; a2 ¼ ax sin2 b þ ay cos2 b; 2
ð6Þ
2
a3 ¼ az0 cos 2b þ az1 sin 2b. In the above equations, c1, c2, c3, c6, ax, ay, az0 and az1 are coefficients that describe the anisotropy of the material. The parameter 2b represents the angle between the line OA and the axis of the principal IPE stress, S1, as shown in Fig. 1, and is equal to: 2S xy 1 2b ¼ tan . ð7Þ Sx Sy With this yield function it is possible to increase the yield stress at pure shear without increasing the other plane strain yield stresses. It should be noted that by setting the anisotropy coefficients c1 = c2 = c3 = c6 = a1 = a2 = a3 = 1.0 and a = 2 (quadratic), the von Mises isotropic yield function is recovered. 2.2. Plastic anisotropy parameters The plastic anisotropy parameter Rh is defined as the ratio of width-to-thickness strain increments:
2
A Sxy
Sy
O
S2
S1
Sx Fig. 1. Determination of the anisotropy parameter 2b from MohrÕs circle.
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Rh ¼
dew . det
319
ð8Þ
The thickness strain, however, is difficult to measure accurately in a thin sheet. Thickness strains are instead calculated from measurements of the longitudinal and width strains assuming constancy of volume as follows: det ¼ ðdel þ dew Þ.
ð9Þ
Therefore, two strain measurements are required, namely the longitudinal (del) and the width strains (dew), in order to calculate the plastic anisotropy parameters. For isotropic materials, R-values are equal to 1.0 for any direction h. R-values not equal to 1.0 indicate that plastic anisotropy exists in the material. A high R-value suggests that the material has a high resistance to thinning and thickening, which implies better formability of the material. If R-values depend on h, then the material is planar anisotropic, otherwise it is planar isotropic. 2.3. Anisotropy coefficients calculation In order for the constitutive model to accurately represent the aluminum alloy sheet at large strains, the anisotropic coefficients describing the behavior of the material, c1, c2, c3, c6, ax, ay, and az1 must be calculated (az0 can be set equal to 1.0) using seven test results. az0 can be set to 1.0 because increasing az0 will decrease ax and ay simultaneously as these parameters are related. Four stress states, uniaxial tension along the rolling, 45°, and transverse directions and balanced biaxial stress (Green et al., 2004), provide the required seven data points (Barlat et al., 1997b). A description of the data points from experiments with respect to the yield function is shown in Fig. 2. A summary of the experimental data needed to calculate the coefficients is given in Table 1. The yield stress could be used as input data instead of the flow stress. However, it may be difficult to accurately measure yield stress in the bulge test as well as in uniaxial tension because the slope of the stress–strain is steep and yielding is not always a discrete event in many aluminum alloys. Also, the yield stress is associated with very small plastic strain and might not reflect the anisotropy of the material over a larger strain range. For these reasons, flow stresses at equal amount of plastic work (Wb = Wu) were selected as input data rather than yield stress, as shown in Fig. 3. In this paper, a value of Wb = Wu = 20 MPa per unit volume was used to extract flow stressesr0, r45 andr90, as the material exhibited a sufficient amount of plastic strain Table 1 Experimental data needed to calculate yield function coefficients for plane stress states Test
Balanced biaxial
Uniaxial
Orientation Flow stress R-Value
N/A rb N/A
0° r0 R0
45° r45 R45
90° r90 R90
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Plane strain x=0 Uniaxial tension
Balanced biaxial tension
+Y
45˚
Plane strain y=0 1
1
-X
+X
Uniaxial tension
Pure shear -Y
2
Fig. 2. Characterization of required tests to be used in calculation of the anisotropic values in relation to the yield locus.
Fig. 3. Flow stresses at equal amounts of plastic work (Wb = Wu).
(18%) at that level of plastic work. The normalized values r0/r0, r45/r0 and r90/r0 were then used as input to calculate the anisotropy coefficients of the yield function. Flow stresses extracted at different values of W would not significantly change those stress ratios, and therefore consistent results can be obtained as long as sufficiently large values of W is used. The anisotropy coefficients were calculated as follows. For uniaxial tension in the rolling direction (0°), the principal IPE stresses become: ðc2 þ c3 Þ r0 ; 3 c3 S 2 ¼ r0 ; 3 c2 S 3 ¼ r0 . 3
S1 ¼
ð10Þ
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The yield function (1) can then be written as:
a 3 r ¼ 0. F 1 ¼ ax jc2 c3 j þ ay j2c2 þ c3 j þ az0 jc2 þ 2c3 j 2 r0 a
a
a
ð11Þ
Similar equations could be written for uniaxial tension in the 45°-direction, the transverse direction (90°) and the balanced biaxial tension as follows: a 3 r F 2 ¼ax j2c1 þ c3 ja þ ay jc3 c1 ja þ az0 j2c3 þ c1 ja 2 ¼ 0; ð12Þ r90 a a ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r r c c 2 c c 2 2 1 2 1 2 2 þ ð2c6 Þ þ a2 ðc1 þ c2 Þ þ þ ð2c6 Þ F 3 ¼a1 ðc1 þ c2 Þ 3 3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia a c c 2 2 r 2 1 þ a3 þ ð2c6 Þ2 2 ¼ 0; ð13Þ r45 3 where a1, a2 and a3 are as defined by Eq. (6). a
a
a
F 4 ¼ ax j2c1 þ c2 j þ ay j2c2 þ c1 j þ az0 jc2 c1 j 2
a 3 r ¼ 0. rb
ð14Þ
The other necessary sets of equations are derived from the plastic anisotropy definitions for R0, R45 and R90, with using Eq. (9) is defined as follows: Rh ¼
dew . ðdew þ del Þ
ð15Þ
From the normality rule: oU e_ xx ¼ k_ ; orxx oU ; e_ yy ¼ k_ oryy e_ xy þ e_ yx ¼ k_
ð16Þ oU . orxy
where k_ is a proportionality factor. After substitution, the plastic anisotropy is found as follows. For uniaxial tension performed along an arbitrary orientation h measured counterclockwise from the xaxis, the ratio of the strain rates Rh is defined by: Rh ¼
oU=or90þh ¼ oU=orzz
oU or90þh oU or90þh
oU þ or h
;
ð17Þ
where oU oU orxx oU oryy oU orxy ¼ þ þ2 ; orh orxx orh oryy orh orxy orh or
ð18Þ
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oU oU oU oU ¼ cos2 h þ sin2 h þ 2 cos h sin h orh orxx oryy orxy
ð19Þ
oU oU oU oU ¼ sin2 h þ cos2 h 2 cos h sin h; or90þh orxx oryy orxy
ð20Þ
and
which leads to Rh ¼
oU orxx
sin2 h þ oroUyy cos2 h 2 oroUxy cos h sin h ; oU oU þ orxx oryy
ð21Þ
where h = 0°, 45° and 90°, and U is the yield function. These systems of equations were solved for the values of the anisotropy coefficients c1, c2, c3, c6, ax, ay, and az1 using a non-linear solver (e.g., Newton–Raphson) with initial values corresponding to the isotropic situation (c1 = c2 = c3 = c6 = 1.0). 2.4. Constitutive equations (flow stress) Flow stress ( r) represents the size of the yield function during deformation. Metals undergoing plastic deformation at high temperatures and different strain rates should be modeled according to the physical behavior of the material (Gronostajski, 2000). An appropriate constitutive equation describing changes in the flow stress of the material depends on deformation conditions such as temperature and strain rate. Wagoner et al. (1988) proposed a flow rule that includes the strain-rate sensitivity: m e_ n p p ðe ; e_ Þ ¼ Kðe þ e0 Þ r ; ð22Þ esr0 where K (strength hardening coefficient), n (strain-hardening exponent) and m (strain-rate sensitivity index) are material constants. ep is the effective plastic strain and e_ is the strain rate. e0 is a constant representing the elastic strain to yield and esr0 is a base strain rate (a constant). This model was primarily selected over other types of hardening laws (e.g., Voce) because it represented the hardening behavior of the current material accurately, and it incorporates strain rate effects. Gronostajski (2000) describes other types of hardening laws that could be used (e.g., Backofen, Grosman) to represent other materials, including those with hardening saturation behavior. The coefficients K, n and e0 of Eq. (22) were solved for simultaneously using Levenberg–Marquardt algorithm. The strain-rate sensitivity index m was determined by the following equation using experimental data (Figs. 8 and 9): m¼
lnðr2= r1 Þ . lnð_e2 =_e1 Þ
ð23Þ
From the data acquired for e0, it was found that its value is very small (see Table 4). Another method was also used to solve for e0, which represents the elastic strain to
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323
yield by solving for the intersection of the linearly elastic loading equation with the strain hardening equation: r ¼ Ee; r ¼ Ken This provides the elastic strain at yield as: 1 ½n1 E e0 ¼ ; K
ð24Þ
ð25Þ
where E is the YoungÕs modulus. Comparing the values of e0 calculated using Eq. (25), to the ones determined using the Levenberg–Marquardt algorithm it was found that the differences between them are very small. Also the values of e0 are generally too small to have a great effect on the resulting stresses in forming processes. Furthermore, it was observed that e0 does not show significant temperature dependence. In this paper, e0 calculated from Eq. (25) was therefore used. Material properties change with increasing temperature. To include temperature effects in the flow rule, the material constants K, n and m are expressed as a function of temperature, and the flow stress equation becomes: mðT Þ e_ ðep ; e_ ; T Þ ¼ KðT Þðep þ e0 ÞnðT Þ r . ð26Þ esr0 The hardening model described by Eq. (26) was used assuming isotropic hardening behavior.
3. Experimental procedure 3.1. Materials Chemical composition and initial mechanical properties of the aluminum alloy (AA3003-H111) used in this study are shown in Tables 2 and 3. All samples were taken from a single lot of material. Because AA3003 is very soft in the annealed condition, one cold roll pass (H111) was applied to these sheets in order to provide some rigidity during handling. 3.2. Experimental procedure Uniaxial tests with standard ASTM-E8 rectangular dog-bone shaped samples shown in Fig. 4, were performed on an Instron Model 1127, screw-driven frame with a 4.5 kN load cell, 25 mm axial extensometer (50% strain max) and 12.7 mm transverse extensometer (30% strain max). The tensile samples were prepared from the aluminum sheet metal at 0° (rolling direction, RD), 45° and 90° (transverse direction, TD) from the rolling direction of the sheet. For the measurement of plastic anisotropy parameters, ASTM-E517 specifies that the test be performed
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Table 2 Chemical composition of AA3003-H111 (wt%) Al
Mg
Cu
Mn
Fe
Si
Cr
Zn
Ti
Ni
bal
0.02
0.07
1.10
0.50
0.21
0.005
0.01
0.02
0.005
Table 3 As-received mechanical properties of AA3003-H111 Thickness (mm)
UTS (MPa)
Yield strength (MPa)
Total elongation (%)
0.96
114.0
63.0
32.0
Fig. 4. Rectangular tension test specimen, ASTM-E8. All dimensions in mm.
at a 0.5/min (0.0083 s1) straining rate. Sample temperature was controlled with an Instron Model 3119 oven with a convection heating system. Tests were performed for several elevated temperatures in the range of 25–260 °C (77–500 °F), with the results of three tests averaged for each temperature. It should be mentioned that only a small variation in the stress–strain behavior was noticed between the three tests for each temperature. Tests in the experiments were performed at a fixed ambient temperature. To study the strain-rate sensitivity of the material, uniaxial tests under several strain rates (0.001–0.08 s1) were performed at each temperature. Bulge testing was performed on sheet samples from the same lot of material in order to determine rb at room temperature. At the moment, experimental calculation of the balanced biaxial data at elevated temperature is under development. Based on experimental observations from uniaxial tests at 0°, 45° and 90°, see Fig. 5, it was noticed that at room temperature the stress values at 45° and 90° coincide with the stress value from bulge test. Since stress values at 45° and 90° continued to coincide at elevated temperatures, it was assumed that a good estimate for the bugling stress at elevated temperature would be the average of the stress values at 45° and 90°. This assumption is unproven and will be studied and reported separately in a future publication. It should be noted that other methods could be used to predict rb. For example, bulge stress rb could be calculated from HillÕs 1948 model (Hill, 1948) or YLD91 model (Barlat et al., 1991) using only the available uniaxial tension tests data. In this paper however, it was not deemed appropriate to present this approach until after experimental bulge tests are completed, at which time a comparison of all these methods will be reported.
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325
160
140
True Stress (MPa)
120
100
80
60
40
20 0
45
90
b
0 0
50
100
150 Temperature (˚C)
200
250
300
Fig. 5. Stress values at equal values of plastic work (20 MPa/unit-vol.) as a function of temperature. Data shown for tests at 0°, 45° and 90° directions at multiple elevated temperatures and for balanced biaxial tension (bulge) data at room temperature.
4. Results and discussion 4.1. Hardening model Fig. 6 shows the true stress–true strain behavior at room temperature for rolling direction (RD), 45° and TD uniaxial tests as well as the balanced biaxial (bulge) test. As seen from the bulge test, much larger strains can be obtained compared to uniaxial strains. Fig. 7 shows the engineering stress–strain behavior at several temperatures for the rolling direction. As temperature increased, flow stress of the material decreased with a corresponding increase in the elongation to failure. Two methods were used to measure strain-rate sensitivity of the material. In the first, several tensile tests were performed on identical samples at different strain-rates (from 0.001 to 0.08 s1) and at several elevated temperatures. For example, Fig. 8 shows the true stress–true strain curves at 204 °C (400 °F) in the rolling direction. Considering that the sample-to-sample variation (thickness, width, etc. . .) may cause scatter in the results, it was desirable to perform the strain-rate sensitivity analysis using a single sample with the ‘‘Jump-rate Test’’ method (Wagoner and Chenot, 1996). In this method, the crosshead speed is increased to produce a ‘‘jump’’ in the strain rate at some predetermined level of strain. Fig. 9 shows results of this method at several elevated temperatures in which crosshead speed jumps were 10–50–150 mm/mm/min. As seen in Figs. 8 and 9, AA3003-H111 exhibits very small strain-rate sensitivity at room temperature, but with increasing temperature, the material becomes more strain rate sensitive.
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Balanced Biaxial
True Stress (MPa)
140 120 100 80 60 40
Rolling Direction
20
45˚-Direction Transverse Direction
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
True strain
Fig. 6. True stress strain data of AA3003-H111 at room temperature for uniaxial tests at 0°, 45° and 90° directions and for balanced biaxial tension (bulge) data.
120 25˚C 100 66˚C
Tensile Stress (MPa)
121˚C 80 177˚C 204˚C
60
232˚C 40
20
0 0
10
20
30
40
50
Tensile Strain (%)
Fig. 7. Engineering stress strain curves of AA3003-H111 at several elevated temperatures for the rolling direction.
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327
-1
0.08s
100
-1
0.05s
-1
0.01s
80
True Stress (MPa)
0.001s
-1
60
40
20
0 0.00
0.05
0.10
0.15 0.20 True Strain
0.25
0.30
0.35
Fig. 8. True stress strain data of AA3003-H111 from uniaxial tests at 204 °C (400 °F) at several strain rates for the rolling direction.
160 150mm/mm/min
25˚C
140 93.3˚C
50mm/mm/min 120
True Stress (MPa)
10mm/mm/min 100 204.4˚C 80 260˚C 60 40 20 0 0
0.05
0.1
0.15
0.2 True Strain
0.25
0.3
0.35
0.4
Fig. 9. True stress strain data of AA3003-H111 from uniaxial tests using the jump-rate strain method. The jumps in crosshead speed were 10–50–150 mm/mm/min (at temperature of 260 °C, at cross head speed of 150 mm/mm/min the specimen failed, therefore there is no information available).
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Table 4 Material properties of AA3003-H111 at elevated temperatures Temperature °C
K MPa
n
e0
m
R0
R45
R90
25 38 66 93 121 149 177 204 232 260
199.82 186.41 175.78 168.41 146.98 139.18 119.65 106.32 93.820 77.320
0.215 0.200 0.187 0.179 0.175 0.163 0.157 0.137 0.132 0.116
8.30E 04 5.00E 04 3.16E 04 6.64E 04 3.20E 04 7.40E 04 6.20E 04 2.21E 04 3.71E 04 5.05E 04
0.003 0.004 0.004 0.005 0.010 0.015 0.030 0.045 0.065 0.080
0.827 1.074 1.092 1.247 1.262 1.273 1.413 1.614 1.688 2.035
1.126 1.424 1.457 1.690 1.742 1.787 1.975 1.977 2.281 2.485
0.773 0.892 0.978 0.879 0.982 0.943 1.007 1.105 1.236 1.328
The values shown below represent an average of three tensile tests. The hardening values are for the rolling direction.
From the results of uniaxial tension tests performed at different constant strain rates, and at several elevated temperatures, values for the Holloman hardening rule (K, n, e0 and m) and the plastic anisotropy parameters (Rh) were calculated as a function of temperature. For example, Table 4 shows values of K, n, e0 obtained with Levenberg–Marquardt algorithm, and m for the rolling direction. Similar results were also found for 45° and the transverse directions. For the finite element simulation, values of K, n and m only from the rolling direction were used to represent isotropic hardening. As shown in Figs. 10 and 11, the values of K and n decrease linearly with temperature. This is in line with the decreasing flow stress and flattening of the hardening curve shown in Fig. 7. Fig. 12 shows the variation of the strain-rate sensitivity index, m, as a function of temperature for AA3003-H111. At lower temperatures, m values are very small, indicating the material is strain-rate insensitive; but at higher temperatures, the material exhibits a significant sensitivity to strain-rate. An exponential function, as shown on the graph, was used to represent the behavior of m as a function of temperature. Fig. 13 shows the variation of the plastic anisotropy parameters R0, R45 and R90, measured at a straining rate of 0.5/min, with respect to temperature. As mentioned earlier, R-values higher than 1.0 indicate good formability and resistance to thinning. As can be seen from Fig. 13, the values of R0, R45 and R90 increase with temperature, which suggests that the formability of the aluminum sheet also enhances at elevated temperature. Table 5 shows a summary of the equations used to fit the hardening parameters for the flow rule, along the rolling direction, as a function of temperature. Curve fitting at 45° and the transverse directions are not required, since only the values of hardening in the rolling direction are used in the finite element simulation. 4.2. Barlat’s yield 96 anisotropy coefficients Results from the uniaxial and bulge tests were implemented into Eqs. (11)–(21), and BarlatÕs YLD96 model coefficients were calculated at several temperatures with
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329
250
K-Values (MPa)
200
150 K(T) = -0.5058*T + 210.4 100
50
Experimental Data
Curve Fit
0 0
50
100
150 Temperature (˚C)
200
250
300
Fig. 10. Strength hardening coefficient (K) of AA3003-H111 along the rolling direction as a function of temperature at 0.5 mm/mm strain rate. Solid line is a linear curve fit.
0.25
0.20
n(T) = - 0.0004*T + 0.2185
n-Value
0.15
0.10
0.05
Experimental Data
Curve Fit
0.00 0
50
100
150
200
250
300
Temperature (˚C)
Fig. 11. Strain-hardening exponent (n) of AA3003-H111 along the rolling direction as a function of temperature at 0.5/min strain rate. Solid line is a linear curve fit.
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m-value
m(T) = 0.0018*e 0.05 0.04 0.03 0.02 0.01
M-value Expon. (M-value)
0.00 0
50
100
150
200
250
300
Temperature (˚C)
Fig. 12. Strain-rate sensitivity index (m) of AA3003-H111 along the rolling direction as a function of temperature. Solid line is an exponential curve fit.
3.0
2.5
R - Value
2.0
1.5
1.0
0.5
R0
R45
R90
0.0 0
50
100
150
200
250
300
Temperature ˚C
Fig. 13. Plastic anisotropy parameters (Rh) of AA3003-H111 as a function of temperature, calculated at 0.5/min strain rate (ASTM-E517).
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Table 5 Summary of equations used to fit hardening parameters for the power law flow rule Hardening parameter
Rolling direction, 0°
Unit
K(T) n(T) m(T)
=0.5058*T + 210.40 =0.0004*T + 0.2185 =0.0018*exp(0.0147*T)
MPa
Temperatures in °C.
the results shown in Table 6. It should be noted that in BarlatÕs yield function az0 = 1.0, and the non-quadratic parameter for aluminum alloys (FCC) is a = 8. Fig. 14 is a plot of the yield function for AA3003-H111, comparing BarlatÕs YLD96 model at 25 °C (77 °F) with the isotropic von Mises yield function (normal). Fig. 15 is a similar plot comparing BarlatÕs ized stress values with respect to r YLD96 function at several elevated temperatures (normalized stress values with re), indicating that temperature has an effect on the shape of the yield surface spect to r (change in the local slopes at different temperatures). This change in the shape is particularly noticed near the balanced biaxial region of the yield surface. This is to be expected, since any change in the plastic anisotropy parameters (R-values) would affect the shape of the yield surface. Fig. 16 shows the effect of ax, ay and az0 on the yield surface shape (Barlat et al., 1997a,b). The az1 parameter primarily affects the yield surface for the stress state with rxy and therefore its effect cannot be seen in a 2D plot (Fig. 16) where a constant shear stress is assumed. Due to the significant effect that temperature has on material properties, the application of a temperature-dependent constitutive model for accurate analysis of warm forming becomes imperative. It is also necessary that this temperature-dependent constitutive model be used in a coupled thermo-mechanical finite element analysis of warm forming process where the thermal analysis provides temperature as input ˇ anadija and Brnic´, 2004). It is from such a coupled analto the mechanical model (C ysis that deformation stresses corresponding to both thermal and mechanical deformation could be accurately calculated. This issue is discussed in more detail in Part 2 of this paper (Abedrabbo et al., 2005a). Table 6 BarlatÕs YLD96 model anisotropy coefficients calculated at several elevated temperatures Temperature, °C
c1
c2
c3
c6
ax
ay
az1
25 38 66 93 121 149 177 204 232 260
1.11698 1.16219 1.22994 1.28713 1.08469 1.18035 1.12801 1.17424 1.25255 1.23298
0.95454 1.04235 1.02323 1.04776 1.01709 1.03178 1.06198 1.06811 1.20946 1.09940
1.0030 0.9877 0.9968 0.9953 1.0116 1.0049 1.0003 1.0058 0.9564 0.9939
1.0429 1.0985 1.0888 1.0721 0.9639 1.0185 0.9782 0.9932 0.9938 0.9904
0.9130 0.6516 0.5028 0.4810 0.8060 0.6182 0.6650 0.5382 0.3160 0.3920
1.3960 0.7866 0.8513 0.7034 0.7864 0.7380 0.5930 0.5294 0.2427 0.4515
1.2500 1.0400 1.1560 1.5080 2.1100 1.8057 2.1460 1.9010 1.9690 2.1230
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Fig. 14. 2D-Plot of von Mises yield function vs. BarlatÕs YLD96 yield function at 25 °C (77 °F) for AA3003-H111.
1.5 1
y
/
0.5 0 -0.5 25˚C 93˚C 149˚C 204˚C 260˚C
-1 -1.5 -1.5
-1
-0.5
0 x
0.5
1
1.5
/
Fig. 15. 2D-Plot of BarlatÕs YLD96 yield function for AA3003-H111 at several elevated temperatures. Normalized stress values.
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333
1.5 x
1 x
y
y
0
y
/
0.5
z0
-0.5 z0
-1 -1.5 -1.5
-1
-0.5
0
0.5
1
1.5
/
x
Fig. 16. Effect of ax, ay and az0 on the yield function (Barlat et al., 1997).
At a first glance, it seems that the coefficients of Barlat YLD96 in Table 6 are fluctuating randomly without any apparent trend. This is attributed to the fact that the experimental data needed to calculate the anisotropic coefficients of YLD96 yield function (Table 1) do not linearly increase with temperature. Looking at the stress values in Fig. 5 it is noticed that they decrease with temperature, while plastic anisotropy parameters (R0, R45 and R90) in Fig. 13 increase with temperature. This opposite behavior causes anisotropy coefficients of the yield function to fluctuate. Therefore, it is important to use higher order fitting functions to capture these variations. To show the importance of varying the anisotropy coefficients with temperature, the plastic anisotropy parameters R0, R45 and R90 were first calculated directly from the yield function using Eq. (21) and the values of the yield function coefficients at the discrete temperatures listed in Table 6. Then, the plastic anisotropy parameters (R0, R45 and R90) were calculated at those same temperatures but this time values of the yield function anisotropy coefficients at 25 °C were used for all temperatures. In both cases, however, the temperature dependent hardening values from Table 5 were used, as flow stress in general only affects the size of the yield function and not the plastic anisotropy parameters. Fig. 17 shows the result of the two tests for the R0 case. From Fig. 17, it could be seen that the plastic anisotropy parameter R0 closely followed the experimental data when the values of anisotropy coefficients at each temperature from Table 6 were used. However, the resulting plastic anisotropy parameters remained constant when the yield function coefficients were assumed to remain constant for all temperatures. This indicates the importance of using temperature dependent anisotropy coefficients, when accurate modeling results are expected. In order to develop an anisotropic material model for use in a coupled thermomechanical finite element analysis of sheet metal forming processes, the anisotropy
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2.0
R0
1.5
1.0
0.5
Experimental-R0 Variable anisotropy coefficients Constant values @25˚C 0.0 0
50
100
150 Temperature (˚C)
200
250
300
Fig. 17. Plot of experimental values of plastic anisotropy parameter (R0) of AA3003-H111 compared with predictions from the Barlat YLD96 material model. Calculations were done first using variable values of the yield function with each temperature and then by using a fixed value for all temperatures.
coefficients of a yield function must be represented as a function of temperature. Thus, curve-fitting methods were used to fit the anisotropy coefficients shown in Table 6. Figs. 18 and 19 show a sample plot of some of the YLD96 coefficients, e.g., c6 and az1, as a function of temperature. In these plots, two levels of polynomial curve fitting, 3rd and 5th order were used to represent variation with respect to temperature. The fit functions used for all seven anisotropy coefficients are shown in Table 7. To determine the validity and accuracy of the 3rd and 5th order polynomial curve fit functions, the plastic anisotropy parameters R0, R45 and R90 were calculated directly from the yield function using Eq. (21) and compared to experimentally measured values. All material parameters used in these calculations, i.e., hardening parameters and BarlatÕs YLD96 function anisotropy coefficients, were set as temperature-dependent equations from Tables 5 and 7. The results at discrete points are shown in Figs. 20–22. While both results of 3rd and 5th order polynomial curve fit functions of the plastic anisotropy parameters R0 and R45 closely agree with the experimental data, the 5th order fit function appears to match the data slightly better than the 3rd order function. On the other hand, in the case of plastic anisotropy in the transverse direction, R90, neither of the two functions agrees very well with experimental data at temperatures lower than 150 °C. This difference could be attributed to possible inaccuracies associated with both experimental tests as well as with the fitting of the yield function anisotropy coefficients. This issue is discussed in more detail in Part 2 of this paper (Abedrabbo et al., 2005a).
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1.2
1.0
C6(T)
0.8
0.6
0.4
0.2 Experimental
3th Order Fit
5th Order Fit
0.0 0
50
100
150
200
250
300
Temperature (˚C)
Fig. 18. Plot of (c6) as a function of temperature for AA3003-H111. 3rd and 5th order polynomial curve fits are shown also.
2.5
2.0
z1(T)
1.5
1.0
0.5
Experimental
3th Order Fit
5th Order Fit
0.0 0
50
100
150 Temperature (˚C)
200
250
300
Fig. 19. Plot of (az1) as a function of temperature for AA3003-H111. 3rd and 5th order polynomial curve fits are shown also.
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Table 7 Barlat YLD96 material model anisotropy coefficients for AA3003-H111 as a function of temperature using two different order polynomial fit functions YLD96 coefficient
3rd order fit
5th order fit
c1
=1.0446 + 0.0045834T 3.9615E 5T2 + 9.7409E 8*T3 =0.9867 + 0.000373T + 1.1717E 7T2 + 3.0052E 9T3 =0.9860 + 0.0003334T 1.7857E 6T2 + 1.593E 9T3 =1.0592 + 0.000855T 1.3728E 5T2 + 3.696E 8T3 =1.0146 0.010636T + 8.3876E 5T2 2.0664E 7T3 =1.4253 0.012009T + 6.4874E 5T2 1.3432E 7T3 =0.86396 + 0.007906T + 3.0245E 6T2 6.2854E 8T3
=0.90902 + 0.0097045T 6.2517E 5T2 3.3687E 7T3 + 3.7316E 9T4 7.6631E 12T5 =0.90308 + 0.0023977T + 3.5398E 5T2 8.9808E 7T3 + 5.4066E 9T4 9.8763E 12T5 =0.98895 + 0.0010519T 3.7432E 5T2 + 4.7418E 7T3 2.3519E 9T4 + 3.9325E 12T5 =0.76161 + 0.017737T 0.0003099T2 + 2.2357E 6T3 7.2652E 9T4 + 8.8146E 12T5 =1.8838 0.05496T + 0.00073166T2 3.8433E 6T3 + 7.5607E 9T4 3.0894E 12T5 =2.683 0.079188T + 0.0011348T2 7.0955E 6T3 + 1.9271E 8T4 1.8287E 11T5 =2.7451 0.096914T + 0.0017925T2 1.2899E 5T3 + 4.0712E 8T4 4.7071E 11T5
c2 c3 c6 ax ay az1
Temperatures in °C.
2.5
2.0
R0
1.5
1.0
0.5
Experimental-R0
3rd-Order-Fit
5th-Order-Fit
0.0 0
50
100
150 Temperature (˚C)
200
250
300
Fig. 20. Plot of experimental values of plastic anisotropy parameter (R0) for AA3003-H111 compared with predictions from BarlatÕs YLD96 material model where the yield function parameters are a function of temperature. Calculations were done using 3rd and 5th order curve fits for the yield function anisotropy coefficients.
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337
3.0
2.5
R45
2.0
1.5
1.0
0.5 Experimental-R45
3rd-Order-Fit
5th-Order-Fit
0.0 0
50
100
150
200
250
300
Temperature (˚C)
Fig. 21. Plot of experimental values of plastic anisotropy parameter (R45) for AA3003-H111 compared with predictions from BarlatÕs YLD96 material model where the yield function parameters are a function of temperature. Calculations were done using 3rd and 5th order curve fits for the yield function anisotropy coefficients.
1.4
1.2
1.0
R90
0.8
0.6
0.4
0.2
Experimental-R90
3rd-Order-Fit
5th-Order-Fit
0.0 0
50
100
150
200
250
300
Temperature (˚C)
Fig. 22. Plot of experimental values of plastic anisotropy parameter (R90) for AA3003-H111 compared with predictions of BarlatÕs YLD96 material model where the yield function parameters are a function of temperature. Calculations were done using 3rd and 5th order curve fits for the yield function anisotropy coefficients.
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Fig. 23. Plot of BarlatÕs YLD96 yield function for AA3003-H111 using anisotropy coefficients extracted from experimental data compared to anisotropy coefficients from both the 3rd and 5th order curve fits shown in Table 7. Temperature was 25 °C (77 °F).
Fig. 24. Plot of BarlatÕs YLD96 yield function for AA3003-H111 using anisotropy coefficients extracted from experimental data compared to anisotropy coefficients from both the 3rd and 5th order curve fits shown in Table 7. Temperature was 204 °C (400 °F).
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The experimentally determined anisotropy coefficients were compared to those calculated from the 3rd and 5th order polynomial curve fit functions from Table 7 for a temperature of 25 °C (77 °F) in the plot of BarlatÕs YLD96 yield function shown in Fig. 23. There is very little difference between these two yield surface curves. Fig. 24 shows the yield function plot for a temperature of 204 °C (400 °F). Again in both cases, the yield functions coincide with very small differences. (All ). yield function plots show normalized values with respect to r
5. Conclusions In Part 1 of this paper, a temperature-dependent anisotropic material model to be used in the finite element analysis model has been developed for the aluminum alloy sheet AA3003-H111. Using experimental data from uniaxial and bulge tests in different directions, the anisotropy coefficients of the Barlat YLD96 model for several elevated temperatures in the range of 25–260 °C (77–500 °F) have been calculated. Using two polynomial curve fitting functions (3rd and 5th order), the anisotropy coefficients of the yield function were determined as a function of temperature. From experimental tests, a strain-rate dependent hardening flow rule was also determined as a function of temperature. The curve fitting functions were verified by a comparison of the plastic anisotropy parameters (R0, R45 and R90) that were extracted from experimental tests to the values predicted from the yield function in which the anisotropy coefficients vary with temperature. Also, the normalized yield functions at several temperatures compared favorably to experimental results. The curve fit functions representing the anisotropy of the yield function could be used in a coupled thermo-mechanical finite element analysis to provide the correct temperature-dependent yield function at each step in the simulation of forming processes for aluminum sheet alloys. In Part 2 of this paper (Abedrabbo et al., 2005a), modeling issues related to the implementation of this temperature dependent yield function for the analysis of warm forming and prediction of failure of aluminum sheets will be discussed. Future plans include extending the current approach and applying it to Barlat YLD2000 yield function (Barlat et al., 2003; Yoon et al., 2004) for several aluminum alloys of interest to industry. From the results obtained in this research, it is expected that both the anisotropy coefficients and the hardening behavior of other aluminum alloys would also change as a function of temperature. Therefore, a comprehensive characterization of the material is needed if accurate finite element analysis of sheet forming processes at elevated temperatures is expected.
Acknowledgements The authors thank General Motors for the support of this research project. The authors sincerely thank Drs. Paul Krajewski and Anil Sachdev from the
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GM Research and Development Center and Dr. Frederic Barlat from ALCOA for their assistance and helpful discussions in support of this research.
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