Comparative Evaluation of Forming Limit Curve Models for AlMg Alloys ^†

Abstract

This publication deals with the comparative analysis of theoretical and empirical models suitable for predicting forming limit curves in the case of aluminum–magnesium alloys. The Stören–Rice and Hill–Bressan–Williams equations were evaluated and justified by inverse parameter identification. They showed good agreement with the measured strain values. Using the graphical Modified Maximum Force Criterion, it was demonstrated that the Voce hardening law is more suitable than the Swift one for characterizing this group of alloys, but the weighted combination of the two hardening laws is preferable.

1. Introduction

Sheet metal parts represent an increasing share of modern production. One of their main applications is forming car body panels, the characteristic materials of which are steel and aluminum sheets. From the point of view of construction and production optimization, it is of paramount importance to know the strength and formability properties of each sheet material as accurately as possible. In connection with this, the main goal of this publication is the critical evaluation and analysis of the forming limit curve (FLC) models for this group of sheets.

The history of the development of the FLC is presented in several publications; for example, a detailed overview is given in the books by Marciniak [1] and Banabic [2]. Several researchers dealt with the theoretical and empirical models of the forming limit curves, among which the theoretical work of Hill [3] and Swift [4] is most significant. From these theories, several researchers have developed relationships suitable for modeling FLCs, the best known of which are the Hill–Swift (HS) model, as well as the model developed by Hill–Bressan–Williams (HBW) [5] and Stören and Rice (StR) [6]. By further developing Hill’s Maximum Force Criterion (MFC), Hora et al. enriched the range of models with the new Modified Maximum Force Criterion (MMFC) [7]. Based on this theory, the research group named Pham developed a new, closed-form solution [8,9,10]. In addition to theoretical models, several empirical relationships have been developed to describe FLCs [11,12,13]. The richness of models inspired researchers to review the known theories in their summary works and to provide their evaluations [14,15,16,17].

Based on comparative analyses presented in the literature, the Stören–Rice and Hill–Bressan–Williams models have the most promising prediction capabilities for aluminum alloys among hardening exponent (n) based bifurcation models [14,17]. At the same time, it was advisable to consider the applicability of the graphical MMFC criterion developed by Pham et al. [8,9] because this calculation method allows for the application of hardening functions other than Hollomon’s σ = Kεⁿ equation, where K is strength coefficient and n is the hardening exponent [17]. Among the known hardening laws, the Voce approximation containing three parameters seemed to be the most favorable for aluminum alloys, its well-known formula is σ = A + B (1 − exp (−Cε)) where A, B, and C are material constants [8].

The purpose of the following review is to highlight those publications that present the calculation procedures of the models tested below. Hill first developed the “zero-extension” theory to characterize the occurrence of local instability. The known result of the theory for calculating FLC strains is given in Equation (1).

$${\epsilon }_1^{*}={\displaystyle \frac{n}{1+\beta }} and {\epsilon }_2^{*}=\beta {\epsilon }_1^{*}$$

(1)

where ε₁* and ε₂* are the major and minor strains of the FLC, and β is the strain ratio (β = ε₂/ε₁). It soon became clear that Hill’s theory correctly describes the behavior of sheets only in the β < 0 range, so Swift derived new relationships for the β > 0 range of biaxial stretching; the combination of two models is named the Hill–Swift model.

The Bressan–Williams criterion can also be derived from the “zero extension” condition [5]. The relationships suitable for calculation are shown in Equation (2), but it should be noted that Formula (2) is recommended only for the range β > 0. The left branch is calculated using the Hill Equation (1), so this combination is called the Hill–Bressan–Williams (HBW) model in the literature.

$${\epsilon }_1^{*}={\displaystyle \frac{n}{\sqrt{1+\beta +{\beta }^2}}}{\left[{\displaystyle \frac{\sqrt{1+\beta +{\beta }^2}}{\sqrt{1+\beta }}}\right]}^{1/n} and {\epsilon }_2^{*}=\beta {\epsilon }_1^{*}where{\epsilon }_2>0$$

(2)

Stören and Rice also started from the bifurcation model when describing the FLC [6]. The original model has been analyzed by many researches; the equations given here (3) and (4) are taken from Zhang’s [16] publication:

$${\epsilon }_1^{*}={\displaystyle \frac{n}{1+\beta }}{\left\{{\displaystyle \frac{1-n}{2}}+{\left[{\displaystyle \frac{{\left(1+n\right)}^2}{4}}-{\displaystyle \frac{\beta n}{{\left(1+\beta \right)}^2}}\right]}^{1/2}\right\}}^{-1} \beta <0,$$

(3)

$${\epsilon }_1^{*}={\displaystyle \frac{3{\beta }^2+n{\left(2+\beta \right)}^2}{2\left(2+\beta \right)\left(1+\beta +{\beta }^2\right)}} {\epsilon }_2^{*}=\beta {\epsilon }_1^{*} \beta \ge 0$$

(4)

Previous theories used Hollomon’s power function to give the flow curve, so the hardening exponent (n) was used as material parameter everywhere in the formulas.

One of the most analyzed bifurcation theory models is the modified maximum force (MMFC) criterion based on the publication by Hora [7]. This model, containing partial differential equations, cannot be solved (calculated) in a closed form, so the points of the forming limit curve can be obtained by iterative methods. In order to avoid complicated calculations, the authors of [8] formulated various simplifying conditions so that closed-form formulas could be derived. We can find relationships in the literature [8,9,10] that can be used for practical calculations, applying the yield criteria of Mises and Hill48 and Swift’s hardening law.

2. Materials and Methods

During this experimental work, the tensile test results and the forming limit curve of 12 aluminum–magnesium alloys were determined using the same testing technique in our laboratory. The measurement results were summarized in a homogeneous database. The following aluminum grades were investigated: one tested sheet was AA5049 (AlMg2), six were AA5754 (AlMg3), and five were AA5182 (AlMg4.5). The sheet thicknesses ranged from 1.2 mm to 3 mm. Most of the sheets had a hardness grade of H22, and some sheets were annealed (O).

The FLC measurements were carried out according to the Nakajima test by a punch with a diameter of 100 mm. The measurements and evaluation were controlled by the GOM ARAMIS^® 6M hardware–software system according to [18]. Seven types of standard-sized specimens were prepared for the tests; 3 parallel tests of each size were available. The tensile tests were carried out according to the [19] standard by Instron 5582-type equipment (Norwood, MA, USA) with a capacity of 100 kN on 3 parallel test specimens in each direction related to rolling by 0–45–90°.

3. Results

The tensile test results related to the rolling direction by 90° are given in

Table 1. Tensile test results.

Material	Rp0.2 (MPa)	Rm (MPa)	Ag (%)	Af (%)	n	r
AA5049-H22-t1,4	95	200	16.80	18.18	0.115	0.605
AA5754--H22-t1,35	150	224	14.01	14.63	0.173	0.743
AA5754-H22-t3	178	243	13.00	15.50	0.158	0.792
AA5754-1-H22-t2,5	147	234	15.57	20.70	0.216	0.600
AA5754-1-O-t2,5	80	217	21.06	25.26	0.380	0.678
AA5754-2-H22-t2,5	141	236	18.90	22.90	0.212	0.569
AA5754-2-O-t2,5	116	229	21.10	26.40	0.285	0.708
AA5182-H22-t1,2	144	262	14.50	15.00	0.276	0.667
AA5182-O-t3	132	260	20.50	22.00	0.264	0.713
AA5182-H22-t3	230	318	11.30	13.30	0.157	0.734
AA5182-O-t1,5	109	262	23.38	26.68	0.362	0.720
AA5182-H-t1,5	129	269	21.73	24.49	0.310	0.761

Rp0.2—flow stress at 0.2% strain; Rm—ultimate tensile strength; Ag—engineering strain at maximum load, Af—engineering strain at fracture; n—hardening exponent, calculated between 4 and 6% engineering strain using the Hollomon equation; r—planar anisotropy, evaluated between 8 and 12% engineering strain.

. According to the standard [18], the central axis of Nakajima specimens must be parallel with this direction for aluminum sheets.

The results of the Nakajima test were reported by GOM-ARAMIS^® 6M software with seven FLC points of ε*₂-ε*₁ minor–major strain pairs.

Table 2. Characteristic values of forming limit curves.

Material	ε*1 (0)	ε*1 (min)	ε*2 (min)	ε*1 (0.2)
AA5049-H22-t1,4	0.18	0.17	0.01	0.31
AA5754--H22-t1,35	0.17	0.12	0.01	0.28
AA5754-H22-t3	0.17	0.15	0.03	0.22
AA5754-1-H22-t2,5	0.16	0.15	0.04	0.24
AA5754-1-O-t2,5	0.22	0.21	0.03	0.27
AA5754-2-H22-t2,5	0.19	0.16	0.02	0.24
AA5754-2-O-t2,5	0.26	0.25	0.05	0.35
AA5182-H22-t1,2	0.19	0.17	0.03	0.27
AA5182-O-t3	0.25	0.24	0.05	0.32
A5182-H22-t3	0.12	0.11	0.02	0.23
AA5182-O-t1,5	0.28	0.26	0.06	0.28
AA5182-H-t1,5	0.26	0.24	0.05	0.27

shows the following interpolated strains from these seven points: ε*_{1 (0)}, ε*_{2 (min)}, and ε*_{1 (min)} plus ε*_{1 (0.2)} measured at ε₂ = 0.2.

The results of the table confirm the basic characteristic of aluminum alloys that the planar strain ε*_{1 (0)} and the minimum value of FLC ε*_{1 (min)} differ from each other; the vertices of the curves shifted slightly to the right and downwards. This is obviously caused by pre-forming, which is not known for the sheets in question but is presumably related to the final rolling phase and roller straightening.

4. Discussion

4.1. Analysis of the Stören–Rice and Hill–Bressan–Williams Models

Using the hardening exponents obtained from the tensile test (n-values in Table 1) and the measured seven FLC minor and major strains valid for each material, the models according to Stören–Rice and Hill–Bressan–Williams were calculated using Equations (1)–(4), and then the mean absolute percentage errors (MAPEs) of deviations from the measured values were evaluated using the following expression: MAPE = 1/N Σ^N_{i = 1}|(ε_meas − ε_calc)/ε_meas|100%.

Assuming that, according to the literature, the measured n values can be replaced by the logarithmic strain measured at the end of uniform elongation, we calculated this parameter with the following formula: ε_u = LN (1 + A_g/100). After that, we evaluated the MAPE again according to the condition n ≈ ε_u. Subsequently, inverse parameter identification was performed to determine the n_c (calculated n) parameters in the formulae from the measured data. To accomplish this, an initial n_ci = n value was used to calculate the squared deviations between the measured and calculated points, then their sum was minimized to obtain the n_c value that best approximates the given StR or HBW function. This brought the average error of approximations below 8%. Figure 1a shows graphs of models and measured values calculated for the AA5754-1-H22-t2.5 sheet.

Comparing the empirical FLCs of the dataset, it can be stated that annealed sheets (O) show higher curves related to cold-rolled (H22) versions, as the formability of annealed sheets is better than the cold-formed ones. In the case of all measured AlMg alloys, the equi-biaxial strain limits are significantly higher than the uniaxial major strains; the difference is about 20%. This is in accordance with several measured results; for example, in [14,16]. The average of the MAPEs determined for all 12 sheets is shown in Figure 1b. This confirms the earlier statement that the worst approximation is given by the n value determined from the tensile test, and the most favorable approximation is given by the n_c obtained by the inverse parameter identification. It can also be seen that the two models approximate the measured values with practically identical errors for any n parameters.

The relationship between n and ε_u and the calculated values n_c obtained by the inverse parameter identification was also determined. It is possible to conclude that the n_c values of the StR and HBW models differ little from each other. A comparison of the regression coefficients showed that uniform elongation (ε_u) gives a better estimation for n_c than the hardening exponent (n). The relationships obtained offer an estimation for the calculation of n_c from n or ε_u.

4.2. Analysis of the MMFC Model

Using the relationships in publications [8,9,10], we determined the approximate functions according to the simplified calculation of the MMFC model for six selected FLCs. The calculations used the Mises and Hill48 yield criteria combined with the Swift and Voce hardening laws. These variables resulted in four combinations denoted by the letters S-M, S-H, V-M, and V-H on the charts (the first letter refers to hardening law and the second refers to yield criteria). The model materials selected for analysis are briefly marked as A1-A6; four of them are AA5754 (AA5754-1 O and H22; AA5754-2 O and H22 t = 2.5), and two are AA5182 (AA5182-O and H22 t = 1.5).

To compare the measured and predicted values, FLC points were assigned according to Table 2. For each strain, the percentage errors per four model variants were calculated with the following formula: ERR% = (ε_meas − ε_calc)/ε_meas100%, as shown in

Table 3. Percentage error of different models (ERR%).

Strain	S-M	V-M	S-H	V-H
ε1(0) − c	37.16	–30.36	38.10	–27.76
ε1(min) − c	49.62	–24.02	50.74	–21.18
ε1(0,2) − c	49.08	–27.29	54.90	–16.73

. Each calculated strain approximates the measured values with a minor error in the case of the Voce function. The errors of the Hill48 and Mises yield criteria are nearly identical.

Figure 2a shows the location of the measured and estimated values for the minimum major strain ε_1(min). It can be clearly seen that the values estimated by the Swift hardening law are above the measured values for both yield criteria, and the data of the Voce hardening law are below it. It can also be proved that the Mises and Hill yield criteria give practically identical results, so the position of the curves is basically determined by the hardening law.

The ERR%s calculated for the six sheets are illustrated in Figure 2b, which shows that the use of the Voce function is preferable. These results also show that the measured FLC points are between the two limits of hardening laws, so the best approximation can be given by the combination of the Swift and Voce hardening laws with a suitably chosen weight factor w between 0.3 and 0.6.

The analysis of the effectiveness of each model highlights two problems. One is the question of the minimum point of the FLC, which, according to each model, is at ε₂ = 0 (plain strain condition). In contrast, the values of ε_{2 (min)} and ε_{1 (min)} in Table 2 show that the measured minimum points are in the range of ε₂ = 0.02 − 0.05, so there is an error between the measured and calculated values around the minimum values of the FLC. Another difference is that the StR and HBW models contain a single parameter that determines both the estimated position of the FLC and the slope of the two curve branches. A comparison of measured and calculated results shows that in the ε₂ < 0 range, the calculated curves approximate the measurement points relatively well, but the slope of the curves fitted to ε₂ > 0 values differ from the model calculations, which increases the percentage error. This statement is valid not only to the n-based StR and HBW models but also to the applied MMFC model.

Based on these findings, the refinement of models is possible only within certain limits by optimizing material parameters. To further improve the models, it is necessary to consider the deviation between ε_{1 (min)} and ε_{1 (0)} for this group of alloys.