Hill48 criterion is one of the simplest and most common yield functions to describe the anisotropic material behaviors, and its constant parameters are determined with explicit formulas. The formulation of the Hill48 yield function is expressed as [22]:

σ ̅2 = F(σy − σz)2 + G(σz − σx)2 + H(σx − σy)2 + 2(Lσyz2 + Mσzx2 + Nσxy2). (1)

For plane stress conditions, Hill48 is as follows:

σ ̅2 = (G + H)σx2 − 2Hσxσy + (F + H)σy2 + 2Nσxy2, (2)

where F, G, H, and N are material coefficients for the Hill48 yield function that are determined by using the r-value in different directions. The relations to compute the Hill48's constant parameters are given in the appendix, and these values are listed in Table 2.

Because of the Hill48 equation limitations discussed in the current section, the polynomial-type yield function was proposed by Gotoh. This fourth-order polynomial yield function is applicable to predict the yield behavior of the materials under plane stress conditions. Gotoh's yield function is expressed as [23]:

σ ̅4 = A1σx4 + A2σx3σy + A3σx2σy2 + A4σxσy3 + A5σy4 + (A6σx2 + A7σxσy + A8σy2)σxy2

4 (3)

+ A9σxy .

A1 to A9 are material parameters for the Gotoh yield function that are obtained by using the direct approach as stated in the appendix. The constant parameters of the Gotoh yield function are listed in Table 3.

It is widely reported that the YLD2000-2d is a proper yield criterion to describe the anisotropic yield behavior of the aluminum sheets [6]. For this yield function, the formulation is expressed as:

ϕ = ϕ′ + ϕ′′ = 2σ ̅m, (4)

where σ ̅ is equivalent stress and m is the constant value that for FCC metals, m=8. Functions ϕ′ and ϕ′′are obtained as below [24]:

ϕ′ = |X1 ′ − X2 ′ |m,

ϕ′′ = |2X2 ′′ + X1 ′′|m + |2X1 ′′ + X2 ′′|m.

Xi ′ and Xj′′ (i, j=1,2) are the principal values of the matrix X′ and X′′, and are expressed as:

′ 1 ′ ′ ′ ′ 2 ′2

Xi = 2 (X11 + X22 ± √(X11 − X22) + 4X12 ), (6) ′′ 1 ′′ ′′ ′′ ′′ 2 ′′2 Xi = (X11 + X22 ± √(X11 − X22) + 4X12 ).

Components of X′ and X′′ are obtained as:

X′ = L′σ,

X′′ = L′′σ,

where L′ and L″ are linear transformation tensors that are calculated by eight constant parameters as follows:

L′11 2/3 0 0 L′12 −1/3 0 0 α1

L′21 = 0 −1/3 0 [α2]. (8)

L′22 0 2/3 0 α7 [L′ ] [ 0 0 1] L′′ 11 −2 2 8 − 2 0 α3 L′′ 12 1 − 4 − 4 4 0 α4 ′′ 1

L21 = 4 − 4 − 4 1 0 α5 . (9)

′′ 9 α L22 −2 8 2 − 2 0 6 [L′′ ] [ 0 0 0 0 9] [α8]

α1 to α8 are eight anisotropic parameters which could be computed by eight mechanical properties of sheet metals measured experimentally [25]. The six data are r values and flow stresses in 0°, 45°, and 90° direction from rolling (r0, r45, r90, σ0, σ45, σ90), and the other two input data are the balanced biaxial r value and flow stress(, rb, σb). Yld2000-2d yield function coefficients are presented in Table 5, and the relations to calculate α1 to α8 are given in the appendix.

In order to examine the accuracy of the computed coefficients for the above yield functions, the yield surface of each criterion is plotted.

Fig. 1 shows the yield surface predicted by different yield criteria and available experimental data in [5] plane stress conditions. However, Gotoh and Yld2000-2d functions effectively matched the test data; a slight deviation is seen in the yield surface based on the Hill 48 function in Fig. 1.

In addition, Fig. 2 shows the comparison between the computed normalized yield stresses and r-value with the test results. The variation of these values with respect to the angle from rolling direction (θ) is computed by using the following formulas [26]:

σx = σθ cos2θ, σy = σθ sin2θ, σxy = σθ cosθ sinθ. (10)

σθ 1

= . (11)

Y σ ̅(1, θ) 2 ∂σ ̅ 1 ∂σ ̅ 2 ∂σ ̅ sin θ − sin2θ + cos θ ∂σx 2 ∂σxy ∂σy

rθ = − . (12)

∂σ ̅ ∂σ ̅ ∂σx ∂σy As depicted in Fig. 2a, all yield functions successfully estimated the r-value of the aluminum AA5182-0. Although Hill48 calculates the r-value with good accuracy, the prediction of the normalized yield stress is significantly distorted, as shown in Fig. 2a. According to the results observed in Fig. 1 and Fig. 2, Gotoh and Yld2000-2d provided good accuracy in predicting the yield condition.

In contrast, the values predicted by the Hill48 function have a large deviation from experimental results, and this means Hill48 cannot be close to the real yield condition. Therefore, in the next sections of this paper, the Gotoh and Yld2000-2d yield functions are used to plot the forming limit curve of the AA5182-0 sheet.

σ ̅ = Ks(ε ̅ + ε0)ns, (13)

where Ks and ns are material constants for Swift work hardening. ε ̅ is the effective strain in the plastic region, and ε0 is a constant representing the elastic strain to the yield point. The Swift coefficients for the AA5182-0 alloy are given in Table 6.

Swift and Voce formulas. Kim-Tuan model is defined as follows [8]:

σ ̅ = σ0 + Kk[1 − exp(−tkε)](ε + ε0)hk], (14)

where Kk, tk, and hk are the material constants, and σ0 is the initial yield strength. The coefficients of the Kim-Tuan hardening model are computed according to the appendix relations and listed in Table 6.

This advanced strain hardening model, which describes the isotropic-distortion behavior of aluminum and titanium sheet metal, is expressed as [9]:

h amk

σ ̅ = +Kmk(ε + ε0) mk (1 + ∗ ), (15)

1 + exp(tmk(ε ̅ − ε )) where S, Kmk, hmk, amk, tmk, and ε∗ are modified Kim-Tuan equation coefficients, which are presented in

Fig. 3 indicates the comparison between Swift, Kim-Tuan, and modified Kim-Tuan stress-strain curves and available experimental data to validate the strain hardening relationships and their constant parameters. The curve fitting technique is the most popular method for identifying the coefficients of the hardening model to decrease the difference between the predicted flow stresses and experimental results. As shown in Fig. 3, by fitting the hardening equations to the test data under uniaxial tensile test, it can be observed that almost all models can estimate the experimental flow stress and give satisfactory accuracy to the test result. However, it will be seen that these hardening models show different results for forming limits in the next sections.

During the last decades, a lot of studies reported the effect of various constitutive models on the accuracy of the predicted FLC. In this section, to investigate the effects of material models on theoretical forming limits, Gotoh and Yld2000-2d yield functions are employed, and also the influences of the Swift, Kim-Tuan, and modified Kim-Tuan stress-strain relationships are examined. The M-K instability theory, by considering the initial groove angle and imperfection ratio f0 = 0.999, is utilized to estimate the forming limit diagram in the linear loading path condition.

Fig. 5 indicates the comparison between the FLC predicted by different hardening models based on the Yld2000-2d yield function and experimental data for AA5182-0 alloy. As can be seen in Fig. 5, the predicted FLC according to all hardening models is in good agreement with experimental results. On the left side of the Diagram, forming limits based on the Swift and Modified Kim-Tuan hardening models are almost at a similar level of accuracy, and their results are in better agreement with the experimental data than those predicted by the Kim-Tuan. The Kim-Tuan model underestimates the forming limit strains for AA5182-0 alloy when the strain ratio is negative. The reasons for this behavior of the hardening laws are explained in detail in [21].

On the right side of the Diagram, the Swift hardening model overestimates the limit strains. At the same time, the calculated FLCs according to both the Kim-Tuan and modified Kim-Tuan relationships demonstrate an acceptable accuracy for the formability of AA5182-0. As illustrated in Fig. 5, Swift and modified Kim-Tuan can predict the FLC0, FLC in the plane strain condition, with better accuracy than the Kim-Tuan model. Results indicate that the modified Kim-Tuan model can predict almost all parts of the DiagramDiagram better than the Swift and Kim-Tuan models. Therefore, the modified Kim-Tuan strain hardening law that describes the isotropic-distortion behavior of sheet metal [9] is the proper model to predict the formability of AA5182- 0 sheets by using the Yld2000-2d. To prove the above results, the absolute error for predicted limit strains is measured by using the following equation:

cal exp

|ε1 − ε1 |

err(%) = exp × 100, (21)

where ε1 exp and ε1 cal are experimental and predicted major strain, respectively. The calculated errors are presented in Table 7. As indicated in Table 7, the modified Kim-Tuan hardening model can predict the most accurate limit strains by applying the Yld2000-2d yield function for AA5182-0 alloy.

In the continuation of this section, the theoretical FLC of AA5182-0 aluminum alloy under linear loading path is calculated according to the Gotoh yield function by using the Swift, Kim-Tuan, and modified Kim-Tuan work hardening models. As illustrated in Fig. 6, the Swift and modified Kim-Tuan models verified the experimental data only in the vicinity of the plane strain state, and the FLC predicted by the Kim-Tuan model almost does not match the experimental data. A Comparison of results between Fig. 5 and Fig. 6 proves that, however, both Gotoh and Yld2000-2d yield functions describe the yield behavior of the AA5182-0 accurately according to Fig. 1 and Fig. 2, the forming limit diagram predicted by Gotoh is not appropriate, and the computed FLCs by it are poor.

According to the above explanation, the Yld2000-2d criterion is a suitable yield function to determine the limit strains for aluminum alloys. Also, according to Fig. 5, the modified Kim-Tuan strain hardening law shows the best prediction based on the Yld2000-2d for AA5182-0 sheets. For this reason, in the next section, the modified Kim-Tuan hardening law and the Yl2000-2d non-quadratic anisotropic yield function are utilized together to compute the forming limit diagrams.

The M-K criterion code was developed to calculate the theoretical forming limit diagram of AA5182-0 under combined loading paths. This modified approach was used to analyze FLCs of AA5182-0 aluminum alloy sheet for a variety of nonlinear strain paths that consist of two loading stages. In multi-stage loading processes, the first stage has different magnitudes of pre-strain along the following modes: uniaxial tension (ρ = −0.5), plane-strain tension (ρ = 0), and equi-biaxial tension (ρ = 1). In the second stage, FLC was determined following the previous pre-strain by considering a series of linear paths in the range between uniaxial tension and equi-biaxial tension (−0.5 ≤ ρ ≤ 1).

Fig. 7 shows the forming limit diagrams obtained by combining the modified Kim-Tuan work hardening law and Yld2000-2d yield function in the bilinear strain path condition. To study the influences of different strain paths on the limit strains, the FLCs were plotted under various pre-strains along the uniaxial tension, plane strain, and equi-biaxial tension paths. In Fig. 7, the curves with square markers show the effect of 0.1 and 0.2 pre- effective strains in the uniaxial tension direction on the FLC. Compared to the forming limits under the linear path, this pre-strain enhances and improves the formability of the positive minor strains, but moves the left side of the curve downward; also, the pre-strain shifts the Diagram towards the negative minor strains.

The curves with a triangle marker in Fig. 7 indicate the calculated forming limit diagrams after ε1 =0.1 and 0.2 pre-strain in the plane-strain direction. It can be observed that after a pre-strain along the plane-strain condition, the forming limit curve moves up for all deformation modes except plane-strain. Therefore, however, pre-strain along the (ρ = 0) improves the forming limits, FLC0 remains almost unchanged. The curves with pentagonal markers in Fig. 7 illustrate the predicted FLC of AA5182-0 sheet specimens pre-loaded to ε1 = 0.1 and 0.2 in equi-biaxial tension direction. It was found that the formability of the limit strains that are on the left side of the intersection increases remarkably, but the forming limits of the other part of the FLC decrease and shift toward the right side of the curve.

In Fig. 7, the path dependency of the FLC can be observed by considering various pre-strains. According to the above discussions, the forming limit strains are strongly path dependent, and the loading path plays an important role in the prediction of them, which means the pre-strain can shift the Forming limit strain curve in the strain space and improve the formability of the metal sheet. Also, it was seen that the strain path dependence of limit strains under bilinear loading paths with higher pre-strains is more obvious. Therefore, the larger the pre-strain, the more influence it has on the forming limits.

To investigate the path dependence of the forming limit curve in stress space, the Kim-Tuan and modified Kim-Tuan strain hardening laws are chosen. Also, the Yld2000-2d yield function is utilized to study the sensitivity of limit stresses to the strain path. Fig. 8 indicates the computed FLSC under various quantities of pre-strain along uniaxial tension for two different hardening models.

The FLSC under pre-effective strains ε ̅ = 0.1, 0.15, 0.18, 0.2, and 0.25 by using the Kim-Tuan work hardening law are illustrated in Fig. 8a. It can be seen that there is no significant difference between diagrams under pre-strains ε ̅ = 0.1 and 0.15 and the linear condition. However, the FLSC obtained after ε ̅ = 0.18, 0.2, and 0.25 pre-strain in uniaxial tension is significantly different.

Then, the FLSC was calculated by the modified Kim-Tuan model after ε ̅ = 0.1, 0.2, 0.22, and 0.25 pre-effective strains in the uniaxial tension direction. As indicated in Fig. 8.b, no strain path dependency was observed for pre-strains ε ̅ = 0.1 and 0.2. Also, plotted diagrams show FLSC after ε ̅ = 0.1 and 0.2 pre-strains almost coincide with the original FLSC, but pre-strains ε ̅ = 0.22 and 0.25 created significant curves.

The criterion for this behavior (dependence or independence of FLSC) is the magnitude of pre-effective strain compared to the value of the effective forming limit strain in a plane strain state and linear loading condition (effective FLC0). As indicated in Fig. 5, different work hardening models create different limit strains and FLC0, and also an effective FLC0. Briefly, if effective pre-strain in the multistep loading path process is less than this critical value, the final limit stresses will coincide with the FLSC in the linear condition. However, if the strain path changed after this critical value, the limit stresses will be affected by the pre-strain.

According to the above, for nonlinear loading paths with pre-effective strain less than the limit effective strain in a plane strain state, FLSC is path independent and appropriate for industrial applications, and for multi-stage processes with pre-strain larger than the critical effective strain, FLSC will be path dependent. For a detailed examination, the effective plastic forming limit curve of the AA5182-0 according to the Kim-Tuan and modified Kim-Tuan models is plotted in Fig. 9. As illustrated, in most regions of the Diagram, the effective limit strains based on the modified Kim-Tuan are higher than those of the Kim-Tuan model. Also, the effective FLC0 for the modified Kim-Tuan model is larger than that of the Kim-Tuan model.

Results indicate that the chosen strain hardening model has a great influence on the critical effective strain, and also on the path dependence of FLSC. Hence, this critical value is very important, and the magnitude of that for different hardening laws must be calculated. In fact, in this section, the effect of various stress-strain relationships on the path dependence of the FLSC by computing the critical strains is investigated. The effective FLC0 for the Kim-Tuan and modified Kim-Tuan strain hardening laws were determined, and they are equal to 0.175 and 0.212, respectively. Accordingly, FLSC is path independent with a larger pre-strain value based on the modified Kim-Tuan law than the Kim-Tuan hardening model.

The reason for this behavior is a difference in the magnitude of the effective FLC0 for these various hardening models. Therefore, the predicted FLSC by using the modified Kim-Tuan can be independent in the larger pre-strain than the Kim-Tuan model. Also, since the path dependence of FLSC is related to the value of the limit effective strain in plane strain, the accuracy of the FLC0 is very important. As shown in Fig. 5, the modified Kim-Tuan model can predict the FLC0 with good accuracy, and the Kim-Tuan model underestimates the FLC0. Therefore, the modified Kim-Tuan hardening model is a better hardening law to estimate the path independence of the FLSC than the Kim-Tuan model.

To survey the influence of yield criteria on the sensitivity of the forming limit stress curves to the strain path, Gotoh and Yld2000-2d functions using the modified Kim-Tuan strain hardening laws are applied. Fig. 10 presents the forming limit stresses for various pre-effective strains in the uniaxial tension direction, considering different yield functions.

Therefore, the limit stresses by using the Gotoh function for pre- effective strain 0.22 are path independent, and by Yld2000-2d are path dependent.

As discussed earlier, for small values of pre-strain, the FLSC is path independent, but if the pre-major strain increases to the effective FLC0 value, the strain path affects the FLSC. As indicated in Fig. 8 and Fig. 10, the sensitivity of the forming limit stress diagram to the strain path leads to an upward shift in the limit stresses in the vicinity of the plane strain condition. The results demonstrated that selected yield criteria similar to the hardening model can affect the path dependency of the FLSC, and all of these results were concluded from the shape of the forming limit diagram of each yield function.