A non-associated constitutive model based on yld2004-18p yield criterion and its applications on sheet metal forming analysis

Abstract

For sheet metals, anisotropy is a significant property affecting sheet metal forming processes. The anisotropy of sheet metals is caused by the rolling process, and several anisotropic constitutive models have been proposed under the non-associated flow rule to describe the deformation and stress anisotropies of sheet metals independently. However, most of them are based on yield functions that are only identified by the experimental data of orthogonal axes, or yield functions that are applicable to only the plane stress state. In this study, the yld2004-18p yield function, which can be used to analyze the three-dimensional stress state in multiple axes with high accuracy and acceptable identification cost, is used to develop a non-associated constitutive model and subsequently applied to sheet metal forming analysis. Finite element analysis results of circular cup drawing and hole expansion demonstrate the capability of the yld2004-18p-based non-associated constitutive model in more accurately describing both the deformation and stress anisotropies of sheet metals.

Introduction

The anisotropy of metallic materials is affected by the manufacturing process. For sheet metals, the anisotropy is caused by the rolling process, where the developed rolling textures result in anisotropic R-values (deformation anisotropy) and anisotropic yield stresses (stress anisotropy) for rolled sheets. Anisotropy is known to have a significant effect on sheet metal forming processes. In the circular cup drawing process, ears appear on the cup after drawing due to the anisotropy¹. In the circular blanking process, because of the anisotropy of sheet metals, the product becomes oval-shaped even though the punch is circular². The effect of anisotropy is also demonstrated in the springback after unloading during the bending process³ and tension leveling^4,5. Therefore, it is significant to consider the anisotropy of metallic materials when studying sheet metal forming processes to better predict the forming results. More accurate predictions can significantly contribute to the optimization of forming processes.

Various anisotropic yield functions have been proposed to capture these phenomena in sheet metal forming. The best-known function is the Hill48 yield function⁶, which is an extension of the von Mises yield function for isotropic bodies. The Hill48 yield function is in a quadratic form, and it is still one of the most widely used constitutive models in the field of sheet metal forming due to its convenience and user-friendliness for engineers and researchers^7–11. The Hill79 yield function¹², which is an extension of the Hill48 yield function to a non-quadratic form, is proposed to control the curvature of the yield surface. Barlat et al. developed an anisotropic yield criterion by incorporating an additional shear component to the Hosford yield function, which is also an extension of the Hill48 yield function to a non-quadratic form to control the curvature of the yield surface, and presented it as the yld91 yield function¹³. The conventional Hill48 yield function comprises two sets of anisotropic parameters, i.e., one for deformation anisotropy identified from R-values, and the other for stress anisotropy identified from yield stresses. Whereas the anisotropic parameters of the conventional Hill48 yield function can be identified using the R-values or yield stresses in different directions to describe either the deformation anisotropy or stress anisotropy, the anisotropic parameters of the yld91 yield function can only be identified from the directional yield stresses to describe the stress anisotropy; hence, the deformation anisotropy cannot be captured by the yld91 yield criterion. By contrast, the yld96 yield function¹⁴ is a generalization of the yld91 yield function and can simultaneously capture both deformation and stress anisotropy. However, the yld96 yield function is imperfect. One of its major problems is that it cannot guarantee the convexity of the yield surface at all times; Hence, the yld2000-2d yield function¹⁵, which is a yield function for the plane stress state, was proposed based on linear transformations to guarantee the convexity of the yield surface. Subsequently, the yld2004-18p yield function¹⁶ was proposed as an extension of the yld2000-2d yield function to the three-dimensional stress state. Both the yld2000-2d and yld2004-18p models have gained widespread application in the field of sheet metal forming due to their relatively high accuracy at an acceptable identification cost.

The abovementioned Hill and yld family anisotropic yield functions, as well as some recent polynomial-type anisotropic yield functions (e.g., those proposed by S. Soare¹⁷ and F. Yoshida¹⁸), were all proposed based on the associated flow rule; consequently, they have evolved into a complex formulation to capture multiaxial stress and deformation anisotropy in a single complex function. The plasticity theory is composed of 1) yield criterion or function, 2) flow rule, and 3) hardening rule. The plastic potential function can be separated from the yield function in the flow rule to simplify the plasticity theory reasonably and is known as the non-associated flow rule. By developing a constitutive model using the non-associated flow rule that was then validated, Stoughton¹⁹ pioneered the use of the non-associated flow rule in sheet metal forming. et al. developed a non-associated constitutive model using the Hill48 yield function and the Karafilis–Boyce yield function to perform circular cup drawing analysis and obtained better results compared with using the associated flow rule²⁰. Park and Chung constructed a material model using the yld2000-2d yield function based on the non-associated flow rule and performed a circular cup drawing analysis, from which the shape of six ears was predicted successfully²¹. Paulino and Yoon adopted Hill48 as the yield function and yld2000-2d as the plastic potential function under the non-associated flow rule to predict the cup height profile after mini-die drawing, and the eight ears observed experimentally were predicted correctly with higher computational efficiency compared to the non-associated yld2000-2d and associated yld2004-18p models²². Some recent analyses using the non-associated flow rule have been applied to not only circular cup drawing²³, but also to hole expansion to predict the thinning behaviors of sheet metals²⁴, to forming limit deformation to improve the forming limit curve prediction²⁵, to deep drawing of channel sections to predict the springback behaviors²⁶, as well as to tension leveling to predict the residual curvatures of leveled strips²⁷. However, most of the non-associated constitutive models in sheet metal forming are based on yield functions such as Hill48, which are only identified by the experimental data of orthogonal axes, or yield functions such as yld2000-2d, which are applicable to only the plane stress state. In practice, under acceptable identification cost, choosing a constitutive model that describes the three-dimensional stress state can provide better prediction accuracy for sheet metal forming than models based on a plane stress state. For instance, many studies^1,28–30 have demonstrated that the forming anisotropy prediction accuracy of the yld2004-18p model in a three-dimensional stress state is higher than that of the yld2000-2d model in a plane stress state. However, few studies have been conducted to reproduce the anisotropic behavior more accurately with three-dimensional stress state and acceptable identification cost using constitutive models involving the non-associated flow rule.

In this study, with the aim of developing a constitutive model that offers relatively higher accuracy while not increasing the identification cost, we used the yld2004-18p yield function, which can be used to analyze the three-dimensional stress state with the results of multiaxial experimental values, to develop a non-associated constitutive model and subsequently applied to sheet metal forming analysis. To evaluate the developed constitutive model, we performed a finite element (FE) analysis of the circular cup drawing and hole expansion, since sheet metals are subjected to deformation and loads in different directions, and the capability of the developed constitutive model in describing deformation and stress anisotropy can be verified easily by comparing the analysis and experimental results. Based on the circular cup drawing, the performance of the constitutive model in capturing deformation anisotropy can be obtained by comparing the predicted cup height with the experimental results. For hole expansion, the performance of the constitutive model in capturing both deformation and stress anisotropy can be evaluated more precisely by verifying the thinning behaviors of the expanded sheet metal.

Non-associated constitutive model based on yld2004-18p yield criterion

Plastic potential function and yield function

In plasticity theory, the plastic strain increment is defined using the flow rule as follows:

1

where is the plastic strain increment, is the Cauchy stress tensor, is the plastic potential function, and is the plastic multiplier. When the plastic potential function is the same as the yield function , Eq. (1) is known as the associated flow rule; otherwise, it is known as the non-associated flow rule. The yld2004-18p yield criterion¹⁶ shown below was adopted as the basic expression of the yield function and plastic potential function in this study to develop the non-associated constitutive model, as it can express both normal anisotropy and in-plane anisotropy for both stress and deformation with sufficient accuracy.

2

Here, and are the principal values of and , respectively; and are linear transformations of the deviatoric stress tensor , expressed as

3

where and are the anisotropic parameter matrices defined as

4

Here, () and () are anisotropic parameters of the yld2004-18p yield criterion. The deviatoric stress tensor is defined as

5

where

6

Accordingly, the equivalent stress of the yld2004-18p yield criterion can be defined as

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Subsequently, the equivalent plastic strain can be calculated using the work conjugate rule as follows:

8

Substituting Eqs. (1)–(8) yields the increment form of the equivalent plastic strain, as follows:

9

Here, the plastic potential function and yield function are both in the form of the yld2004-18p yield function as Eq. (7). However, the anisotropic parameters (components of and in Eq. 4) are different, as discussed in the next section.

Parameter identification

The yld2004-18p function contains 18 anisotropy parameters (components of and in Eq. (4). The original yld2004-18p yield criterion was proposed under the associated flow rule; hence, the plastic potential function and the yield function are the same, and all the anisotropy parameters are identified by minimizing the following error function:

10

where the superscript pr represents the predicted values of the directional yield stresses or R-values, and the superscript ex presents the experimental values of the directional yield stresses or R-values. is the yield stress in the direction, and is the R-value in the direction. and are the weight coefficients for the fitting of the yield stresses and R-values, respectively. The experimental values of the directional yield stresses and R-values are measured from tensile tests at , , , , , , from the rolling direction , and biaxial tensile tests . The predicted yield stress in the direction is obtained by substituting the corresponding stress state to Eq. (7) to calculate the equivalent stress, and the predicted R-value in the direction is expressed as

11

The proof for Eq. (11) can be found in the first section of Methods. Because the stress and deformation anisotropies are described by a single function, which is the yield function under the associated flow rule, both the yield stresses and R-values are considered in this error function to identify the anisotropic parameters for the original yld2004-18p yield criterion under the associated flow rule. The recommended values of weight coefficients for the identification of stress anisotropy () and deformation anisotropy () are 1.0, and 0.1, respectively¹⁶.

In this study, a constitutive model was developed under the non-associated flow rule to capture the stress and deformation anisotropies more flexibly; therefore, the two sets of anisotropic parameters can be identified independently for the yld2004-18p function when it is adopted as the yield function or the plastic potential function. For the yld2004-18p yield function and plastic potential function, the anisotropic parameters are identified by minimizing the error functions defined in Eqs. (12) and (13), respectively.

12

13

Only the yield stresses were adopted to identify the anisotropic parameters of the yield function to capture the stress anisotropy, whereas only the R-values were adopted to identify the anisotropic parameters of the plastic potential function to capture the deformation anisotropy. The gradient descent method was adopted to minimize the error functions mentioned above for the parameter identification of the yld2004-18p function under the associated or non-associated flow rule (the initial values of all parameters were set to 1.0 in the parameter identification).

Prediction of anisotropies

To validate the developed non-associated constitutive model based on the yld2004-18p yield criterion, the experimental data of two anisotropic sheet metals, i.e., AA2090-T3 (shown in Table 1) and DP980 (shown in Table 2), were adopted to identify the anisotropic parameters of the yld2004-18p function under the associated and non-associated flow rules to predict the stress and deformation anisotropies. The identified parameters of the yld2004-18p function for AA2090-T3 and DP980 are listed in Tables 3 and 4, respectively (the yld2004-18p function identified by both the directional yield stresses and R-values using Eq. (10) is referred to as yld2004-sr, the yld2004-18p function identified by directional yield stresses using Eq. (12) is referred to as yld2004-s, and the yld2004-18p function identified by directional R-values using Eq. (13) is referred to as yld2004-r).

Table 1.

Material properties of AA2090-T3 sheet.

Material	AA2090-T3*
Sheet thickness (mm)	1.60
Young’s modulus (MPa)	61,000
Poisson’s ratio	0.33
Stress–strain curve (MPa)

Material properties of AA2090-T3 are reported in Yoon et al.³¹

Table 2.

Material properties of DP980 sheet.

Material	DP980*
Sheet thickness (mm)	1.20
Young’s modulus (MPa)	210,000
Poisson’s ratio	0.33
Stress–strain curve (MPa)
Normalized yield stresses
	1.000	0.9906	0.9813	0.9915	1.0034	1.0178	1.0312	0.9860
R-values
	0.6910	0.7600	0.8880	1.0500	1.0800	1.0200	0.9590	0.6976

Material properties of DP980 are reported in Benchmark1 of NUMISHEET2018³²

Table 3.

Anisotropic parameters of yld2004-18p for AA2090-T3 sheet.

	yld2004-18p-sr	yld2004-18p-s	yld2004-18p-r
		0.0569
	1.2086	0.9860	0.8729
		0.1986	0.0788
	0.4198	0.9314	0.9801
	0.6058	0.5871	0.6396
	1.2442	1.3309	1.4167
	1.0560	0.9447	1.0768
	1.0891	1.0907	1.0690
	1.0938	1.0925	1.0237
	0.8930	1.0763	0.9362
	0.4520	0.4324	0.7828
	0.4569	0.6137	0.6837
	0.9206	0.9659	0.7782
	1.1227	1.1519	1.1211
	1.3167	1.4567	1.1683
	1.1660	1.1752	1.1471
	1.1210	1.1164	1.0516
a	8	8	8

Table 4.

Anisotropic parameters of yld2004-18p for DP980 sheet.

	yld2004-18p-sr	yld2004-18p-s	yld2004-18p-r
	0.9787	0.9822	0.9631
	1.0156	1.0162	1.0405
	0.9563	0.9729	0.9672
	0.9923	0.9937	0.9666
	1.0380	1.0254	1.0364
	0.9700	0.9846	0.9707
	1.0230	1.0119	1.0279
	1.0000	1.0000	1.0000
	1.0000	1.0000	1.0000
	0.9787	0.9821	0.9631
	1.0156	1.0163	1.0405
	0.9563	0.9731	0.9672
	0.9923	0.9937	0.9667
	1.0379	1.0255	1.0364
	0.9700	0.9845	0.9707
	1.0229	1.0117	1.0279
	1.0000	1.0000	1.0000
	1.0000	1.0000	1.0000
a	6	6	6

Using the identified anisotropic parameters of the yld2004-18p yield function under the associated or non-associated flow rule, the yield stresses and R-values of AA2090-T3 and DP980 at each angle from the rolling direction were calculated and compared with the experimental values (as shown in Figs. 1 and 2). For both AA2090-T3 and DP980, the material characterization data required to identify the anisotropic parameters of the associated and non-associated constitutive models based on the yld2004-18p yield criterion were the same. However, the constitutive model based on the non-associated flow rule fitted more accurately to the experimental data compared with that based on the associated flow rule. This indicates that the developed non-associated yld2004-18p constitutive model can improve the prediction accuracy for the deformation and stress anisotropies without increasing experimental cost compared with the associated yld2004-18p constitutive model. However, the number of parameters of the non-associated yld2004-18p constitutive model is 36, which is much more than the number of input experimental data used for parameter identification (totally 16 sets of experimental data as shown in Eqs. 12 and 13). The redundant parameters will raise an issue of uniqueness for the parameter identification (i.e. various sets of constitutive parameters that can fit the input experimental data may be obtained during the parameter identification).

Figure 1.

Experimental and predicted (a) yield stresses and (b) R-values of AA2090-T3.

Figure 2.

Experimental and predicted (a) yield stresses and (b) R-values of DP980.

However, as mentioned in the previous section, in the parameter identification, the initial values of all parameters were set to 1.0, and the gradient descent method was adopted to minimize the error functions. In this case, for both the AA2090-T3 and DP980, the identification procedure was robust, and the parameters listed in Tables 3 and 4 were the only ones can be obtained. For other initial values, the identified values of parameters may be different but all the input experimental data can be well-captured by the identified yield and plastic potential surfaces. This means that the orthotropic anisotropy, as well as the in-plane anisotropy at increments can be well-captured by the calibrated non-associated yld2004-18p model even though the identified parameters may not be unique. Different sets of identified parameters do not cause different predictions of orthotropic anisotropy. They only cause slightly different in-plane anisotropy predictions at degree increments less than , which do not affect the prediction accuracy of in-plane anisotropy at increments. Some previous relatively simple yield functions do not have enough parameters to capture all the available anisotropies in stresses and R-values accurately, while this can be avoided by utilizing our proposed method. It is the redundant parameters of the developed non-associated yld2004-18p model that improve the prediction accuracy of the orthotropic anisotropy and in-plane anisotropy at increments because the parameter identification becomes more flexible considering the redundancy compared with the associated yld2004-18p model. Note that only 16 of the 18 parameters of the yld2004-18p function are independent as indicated by van den Boogaard et al.³³, which makes the non-uniqueness problem during the parameter identification of the developed non-associated yld2004-18p model even less serious.

Applications

Numerical implementation

The developed non-associated constitutive model based on the yld2004-18p function was implemented in the Abaqus/Standard via a user-defined material (UMAT) subroutine using the convex cutting-plane algorithm³⁴ (implementation details in the second section of Methods), which is a semi-implicit stress integration algorithm that can guarantee the work conjugate numerically. The Hill48 function, which remains one of the most widely used constitutive models in the field of sheet metal forming to date, was also implemented in the Abaqus/Standard via a UMAT subroutine as the yield or plastic potential function for later accuracy comparisons in this study. Note that the derivatives of Hill48 function were calculated analytically considering its relatively simple form, and the derivatives of yld2004-18p function were calculated numerically considering its relatively complex form. Usually R-value is often overestimated with a semi-implicit stress integration algorithm³⁵. In this study, we carefully controlled the step increment size within 0.001 with the total time period of 1.0 in simulation to ensure the prediction accuracy of R-values. A one-element analysis was conducted to compare the predicted R-values with the analytical results for both the associated and non-associated yld2004-18p models (details in the third section of Methods, showing the prediction accuracy of R-values using convex cutting-plane algorithm). The static implicit FE analysis of the circular cup drawing for AA2090-T3 and hole expansion for DP980 was conducted using the UMAT to further evaluate the simulation accuracy of the developed yld2004-18p-based non-associated constitutive model in sheet metal forming analysis. For the circular cup drawing analysis, the simulation accuracy is evaluated by the cup height. For the hole expansion analysis, the simulation accuracy is evaluated by the thickness plastic strain. For comparison, the Hill48 function was adopted as the yield function or plastic potential function (Hill48 yield criterion with associated and non-associated flow rules, details in the fourth section of Methods) to simulate the circular cup drawing and hole expansion. Hereinafter, the yld2004-18p function and Hill48 function identified by directional yield stresses are referred to as yld2004-18p-s and Hill48-s, respectively; the yld2004-18p function and Hill48 function identified by directional R-values are referred to as yld2004-18p-r and Hill48-r, respectively; and the yld2004-18p function identified by both the directional yield stresses and R-values is referred to as yld2004-18p-sr.

Circular cup drawing test

The FE analysis of the circular cup drawing was performed under the experimental conditions reported by Yoon et al.³¹. A schematic view of the circular cup drawing test is shown in Fig. 3, and the experimental conditions are summarized in Table 5. One quarter of the cup was simulated considering the symmetry. The punch, holder, and die were modeled as analytical rigid bodies, and the blank was discretized using 7440 incompatible eight-node solid elements (62, 40, and 3 in the circumferential, radial, and thickness directions, respectively). Note that a case study was conducted to decide this mesh size considering both the calculation efficiency and accuracy.

Figure 3.

Schematic view of circular cup drawing.

Table 5.

Experimental conditions of circular cup drawing of AA2090-T3.

	AA2090-T3
Punch diameter (mm)	97.46
Punch profile radius (mm)	12.70
Die opening diameter (mm)	101.48
Die profile radius (mm)	12.70
Blank diameter (mm)	158.76
Blank-holder force (KN)	22.2
Friction coefficient	0.1

The friction coefficient was set to 0.1 following the suggestions of Yoon et al³¹

The properties of the AA2090-T3 blank are listed in Table 1, and the Swift law was used to fit the stress–strain curve of the hardening law. The developed non-associated constitutive model based on the yld2004-18p yield criterion, as well as the original yld2004-18p yield criterion under the associated flow rule, were adopted to conduct the FE analysis of the circular cup drawing for the AA2090-T3 sheet using the UMAT subroutine in Abaqus/Standard (parameters listed in Table 3). For comparison, the Hill48 function was also adopted as the yield function or plastic potential function to simulate the circular cup drawing for the AA2090-T3 sheet (parameters listed in Table 8).

Table 8.

Anisotropic parameters of Hill48 for AA2090-T3 sheet.

	H	F	G	L, M, N
Hill48-s	0.6368	0.5703	0.3632	2.5711
Hill48-r	0.1746	0.2522	0.8254	2.2381

Figure 4 summarizes the cup height distributions obtained via experiment and FE analysis using the associated and non-associated Hill48 and yld2004-18p constitutive models, and the computation time of each model is listed in Table 6. It can be seen that for both the Hill48 and yld2004-18p yield criteria, the non-associated models yield higher computation time of circular cup drawing than the associated models. The yld2004-18p models do not take much more computation time than the Hill48 models, because the derivatives are calculated numerically for yld2004-18p function but analytically for Hill48 function during the numerical implementation as mentioned in the section Numerical implementation. As for the results of the associated and non-associated constitutive models based on the Hill48 function, none of the models based on the associated flow rule (associated Hill48-r and associated Hill48-s) can reproduce the ear around that occurred in the experiment, while the constitutive model based on the non-associated flow rule (non-associated Hill48) reproduced a small amount of ear around . As for the constitutive models using the yld2004-18p function under the associated or non-associated flow rule, both models can reproduce the ear around , while the overall predicted cup height, particularly the ear peak, obtained by the non-associated flow rule (non-associated yld2004-18p) was closer to the experimental values than obtained by the associated flow rule (associated yld2004-18p-sr). Figure 5 shows the average relative error of the predicted cup heights of each model for all measured positions. The average relative error of the predicted cup heights for all measured positions is defined as

14

where is the cup height at the measured position i, and N is the total number of all measured positions. The superscript pr and ex represent the predicted and experimental cup heights, respectively. It can be seen that the non-associated constitutive models can yield better predictions than the associated constitutive models, and the developed yld2004-18p-based non-associated constitutive model yields the best prediction. However, the peak position of the ear obtained by the yld2004-18p-based non-associated constitutive model was , whereas the experimental value was at . The yld2004-18p function used the results of uniaxial tensile tests at increments from the rolling direction to the transverse direction to identify the anisotropic parameters. Therefore, it cannot yield accurate predictions for the directions appearing between the and . This characteristic of the yld2004-18p function leads to the difference between the predicted and experimental peak positions. Note that although the associated Hill48-s and associated yld2004-18p-sr constitutive models yield the same average relative errors, the associated Hill48-s constitutive model cannot predict the small ear around while the associated yld2004-18p-sr constitutive model can do. The higher prediction accuracy of associated yld2004-18p-sr constitutive model is benefited from its more characterization data.

Figure 4.

Cup heights obtained from experiment and Hill48-based and yld2004-18p-based associated and non-associated constitutive models.

Table 6.

Computation time of each model for circular cup drawing simulation of AA2090-T3.

Model	Computation time
associated Hill48-r	6,254 s
associated Hill48-s	6,439 s
non-associated Hill48	11,519 s
associated yld2004-18p-sr	10,283 s
non-associated yld2004-18p	13,009 s

(CPU clock speed: 2.90 GHz; Number of cores per CPU: 4; Number of CPUs: 1; Main memory: 8 GB; Operating system: Windows 10)

Figure 5.

Average relative error of predicted cup heights of associated and non-associated Hill48 and yld2004-18p constitutive models for all measured positions.

Both the deformation and stress anisotropies can affect the earing in drawing test³⁶. To evaluate the effect of deformation anisotropy and stress anisotropy in this circular cup drawing test, the Hill48 function and yld2004-18p function were adopted as the yield function or plastic potential function under the non-associated flow rule to conduct the FE analysis. Figure 6 summarizes the cup height distributions predicted by the non-associated constitutive models based on the Hill48 function and yld2004-18p function. As shown, the distributions of cup heights predicted by the same plastic potential function were similar. In other words, it can be concluded that the selection of the plastic potential function contributed more to the predicted cup height in the circular cup drawing than the yield function under the non-associated flow rule, and that the deformation anisotropy is the dominant factor affecting the cup height.

Figure 6.

Cup heights obtained from experiment and non-associated constitutive models based on Hill48 function and yld2004-18p function.

Hole expansion test

An FE analysis of the hole expansion test for the DP980 sheet was conducted. The hole expansion test was performed as the Benchmark 1 at NUMISHEET 2018³². A schematic view of the hole expansion test and tool dimensions is shown in Fig. 7, and the initial diameter of the blank hole was 30 mm. One-quarter of the blank was simulated considering the symmetry. The punch and upper and lower dies were modeled as analytical rigid bodies, and the blank was discretized by 6480 incompatible eight-node solid elements (40, 43, and 3 in the circumferential, radial, and thickness directions, respectively). Note that a case study was conducted to decide this mesh size considering both the calculation efficiency and accuracy. The friction coefficient between the blank and the punch was set to 0.1. For the other contacts, the friction coefficient was set to 0.15.

Figure 7.

Schematic view of hole expansion test and tool dimensions.

The material properties of the DP980 blank are summarized in Table 2, and the Kim–Tuan equation³⁷ was used to fit the stress–strain curve of the hardening law. The developed non-associated constitutive model based on the yld2004-18p yield criterion, as well as the original yld2004-18p yield criterion under the associated flow rule, were adopted to conduct the FE analysis of hole expansion for the DP980 sheet using the UMAT subroutine in Abaqus/Standard (parameters listed in Table 4). As in the circular cup drawing analysis, the Hill48 function was adopted as the yield function or plastic potential function to conduct the FE analysis such that a comparison can be performed (parameters listed in Table 9). Note that the bead constraint shown in Fig. 7 was substituted by the full constraint at the outer edge of the blank in the FE analysis. However, in the experiment, it was discovered that the thickness reduction of the expanded blank was suppressed by the inflow of material from the bead. Therefore, the thickness reductions obtained from the FE analysis were generally larger than the experimental results³⁸. If additional materials can flow through the bead, the overall thickness reductions can be affected. When comparing the present analysis with the experimental data to evaluate the capability of each constitutive model, it is preferable to compare the shapes of the thickness plastic strain distributions instead of the exact values, because the errors of exact values can be brought by the FE modeling that cannot consider the inflow of material from the bead. In both the experiment and simulation, the thickness plastic strain at each specified location is calculated using the measured thickness before and after deformation.

Table 9.

Anisotropic parameters of Hill48 for DP980 sheet.

	H	F	G	L, M, N
Hill48-s	0.4559	0.4845	0.5441	1.5201
Hill48-r	0.4086	0.4261	0.5914	1.5771

Figure 8 summarizes the distribution of the thickness plastic strain in three radial directions: (rolling direction), (diagonal direction), and (transverse direction) from the rolling direction, when the punch stroke reached 15 mm for the Hill48-based and yld-2004-18p-based constitutive models under the associated or non-associated flow rule. It can be seen that all of the Hill48-based and yld2004-18p-based associated and non-associated models shown in Fig. 8 can predict the radial thinning behavior of the rolling, diagonal, and transverse directions in the hole expansion test correctly, because the tensile tests in these three directions were used to calibrate the Hill48 and yld2004-18p functions. While the associated yld2004-18p constitutive model yields the best predicted shape of the thickness plastic strain distribution in the rolling direction, the non-associated yld2004-18p constitutive model yields the best predictions in the diagonal and transverse directions. The computation time of each model is listed in Table 7, and it can be seen that for both the Hill48 and yld2004-18p yield criteria, the non-associated models yield higher computation time of hole expansion than the associated models. Similar to the circular cup drawing simulations, the yld2004-18p models do not take much more computation time than the Hill48 models, because the derivatives are calculated numerically for yld2004-18p function, but analytically for Hill48 function during the numerical implementation as mentioned in the section Numerical implementation.

Figure 8.

Distribution of thickness plastic strain in (a) , (b) , and (c) from rolling direction for Hill48-based and yld-2004-18p-based associated and non-associated constitutive models when punch stroke reached 15 mm.

Table 7.

Computation time of each model for hole expansion simulation of DP980.

Model	Computation time (s)
Associated Hill48-r	7214
Associated Hill48-s	6985
Non-associated Hill48	8027
Associated yld2004-18p-sr	7963
Non-associated yld2004-18p	8315

(CPU clock speed: 2.90 GHz; Number of cores per CPU: 4; Number of CPUs: 1; Main memory: 8 GB; Operating system: Windows 10)

Figure 9 shows the thickness plastic strain in the circumference direction at 2 mm from the hole edge when the punch stroke reached 15 mm for the Hill48-based and yld-2004-18p-based constitutive models under the associated or non-associated flow rule. As shown, the positions of the peaks and troughs obtained by all the Hill48-based and yld2004-18p-based associated constitutive models were opposite to those obtained from the experimental values, indicating that the circumferential thinning behaviors in the hole expansion test could not be predicted correctly by the constitutive model under the associated flow rule. The developed non-associated yld2004-18p constitutive model overcame this shortcoming of the associated yld2004-18p constitutive model and yielded the best prediction for the circumferential thinning behaviors (shape of the thickness plastic strain distribution in the circumference direction). It is worth mentioning that by adjusting the weight coefficient of the error function (Eq. 10) for a certain direction (e.g. ), the predictive stress and R-value results of the associated yld2004-18p model in corresponding direction may be improved while results in other directions deteriorate. However, we cannot know which direction has the most significant effect on the desired sheet metal forming result before we conduct the experiment and relevant numerical case studies, which is another issue deviating from the main purpose of this work. In this study, we adopted the widely used parameter identification method for the associated yld2004-18p model to evaluate its prediction ability for sheet metal forming.

Figure 9.

Distribution of thickness plastic strain in circumference direction at 2 mm from hole edge obtained Hill48-based and yld-2004-18p-based associated and non-associated constitutive models when punch stroke reached 15 mm.

To evaluate the effect of deformation anisotropy and stress anisotropy in this hole expansion test, the Hill48 function and yld2004-18p function were adopted as the yield function or plastic potential function under the non-associated flow rule to conduct the FE analysis. Figure 10 summarizes the thickness strain distributions predicted by different non-associated constitutive models based on the Hill48 function and yld2004-18p function. As shown, stress anisotropy (selection of yield function) dominated because the thickness plastic strain distributions obtained by the constitutive models with the same yield function exhibited similar shapes, whereas the constitutive models with different yield functions exhibited different shapes.

Figure 10.

Distribution of thickness plastic strain in circumference direction at 2 mm from hole edge obtained from experiment and non-associated constitutive models based on Hill48 function and yld2004-18p function when punch stroke reached 15 mm.

However, different plastic potential functions can also yield different results when the same function is selected as the yield function, as exemplified in Fig. 11. As shown, even though yld2004-18p-s was selected as the yield function for all models, the thickness strain distribution in the circumferential direction varied depending on the plastic potential function, indicating that the effect of the plastic potential cannot be disregarded. When yld2004-18p-s is selected as the plastic potential function to describe the deformation anisotropy, the thickness strain distribution in the circumferential direction is incorrect because the R-values are not considered in yld2004-18p-s. For the other models shown in Fig. 11, because the R-values are considered in the plastic potential function, the positions of the peaks and troughs of the thickness strain distribution in the circumferential direction are similar to the experimental results. Therefore, by comparing the shapes of the thickness plastic strain distributions instead of the exact values in Figs. 10 and 11, it can be concluded that the correct strain distribution trends in the circumferential direction cannot be obtained unless both the stress anisotropy (selection of yield function) and deformation anisotropy (selection of plastic potential function) are correctly described.

Figure 11.

Distribution of thickness plastic strain in circumference direction at 2 mm from hole edge obtained using yld2004-18p-s yield function and different plastic potential functions when punch stroke reached 15 mm.

Figure 12 visualizes the thickness plastic strain distribution when the punch stroke reached 20 mm. In the experiment, fracture occurred at 8 mm from the edge of the hole in the rolling direction when the punch stroke reached mm. Because the fracture law is not applied in this FE analysis, if we assume that the fracture occurs at the location where the plastic strain in the thickness direction is the lowest when the punch stroke reaches 20 mm, then it can be assumed that the fracture occurs at from the rolling direction in all the Hill48-based and yld2004-18p-based associated constitutive models. By contrast, both the Hill48-based and yld2004-18p-based non-associated constitutive models successfully predicted the fracture occurs at the rolling direction. While the non-associated Hill48 constitutive model predicted the location of the fracture at approximately 5 mm from the hole edge, the developed non-associated yld2004-18p constitutive model predicted the location of the fracture at approximately 6 mm from the hole edge, which was closer to the experimental result.

Figure 12.

Distribution of thickness plastic strain when punch stroke reached 20 mm for (a) associated Hill48-r, (b) associated Hill48-s, (c) associated yld2004-18p-sr, (d) non-associated Hill48, and (e) non-associated yld2004-18p constitutive models.

Discussions

The validity of using the yld2004-18p yield criterion based on the non-associated flow rule is discussed in this section. Stoughton and Yoon derived the four stability conditions of the material model for sheet metal forming³⁹, and the readers can refer to their paper for details. It is well understood that the associated flow rule models can guarantee the stability conditions. However, a non-associated flow rule model may lead to negative plastic work under some conditions, which cannot meet the stability requirements. Hence, we first discuss whether the plastic work is positive. This is the condition in which the amount of work generated by plastic deformation is always greater than zero. The work generated by plastic deformation can be expressed as follows:

15

Substituting Eq. (1) into Eq. (15) yields

16

If the plastic potential function is a linear homogeneous function with respect to the Cauchy stress tensor , then from Euler’s homogeneous function theorem, the plastic work can be expressed as

17

Because is greater than or equal to zero, , and the requirement of positive plastic work is satisfied. The original yld2004-18p yield function under the associated flow rule is a linear homogeneous function with respect to the Cauchy stress tensor , and we applied it to the non-associated flow rule using independent anisotropic parameters for the yield function and plastic potential function, which does not affect the property of the yld2004-18p function; therefore, we can confirm that the positive plastic work is guaranteed when the yld2004-18p function is used as the plastic potential function under the non-associated flow rule.

Next, we discuss the incompressibility of plastic deformation. Hydrostatic stress contributes to volume change, whereas deviatoric stress, which is the Cauchy stress minus the hydrostatic stress, contributes to plastic deformation. Therefore, to satisfy volume invariance due to plastic deformation, the deviatoric stress must be used to define the plastic potential function. The derivative of the yld2004-18p function when it is adopted as the plastic potential function with respect to the Cauchy stress tensor, can be expressed by the chain rule as follows:

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \frac{\partial g(\varvec{\sigma })}{\partial \varvec{\sigma }}=&\left( \frac{\partial g(\varvec{\sigma })}{\partial \phi (\tilde{\varvec{s}}^{'}, \tilde{\varvec{s}}^{''})} \cdot \frac{\partial \phi (\tilde{\varvec{s}}^{'}, \tilde{\varvec{s}}^{''})}{\partial \tilde{\varvec{s}}^{'}}\cdot \frac{\partial \tilde{\varvec{s}}^{'}}{\partial \tilde{\varvec{s}}} \right. + \left. \frac{\partial g(\varvec{\sigma })}{\partial \phi (\tilde{\varvec{s}}^{'}, \tilde{\varvec{s}}^{''})} \cdot \frac{\partial \phi (\tilde{\varvec{s}}^{'}, \tilde{\varvec{s}}^{''})}{\partial \tilde{\varvec{s}}^{''}}\cdot \frac{\partial \tilde{\varvec{s}}^{''}}{\partial \tilde{\varvec{s}}} \right) \cdot \frac{\partial \tilde{\varvec{s}}}{\partial \varvec{\sigma }}. \end{aligned} \end{aligned}$$\end{document} 18

From the definition of the deviatoric stress tensor in Eq. (5), the following equation is satisfied:

19

Substituting Eq. (19) into Eq. (18) yields

20

This proves that the sum of the plastic strain increments in the x, y, and z directions is zero; hence, the incompressibility of plastic deformation is satisfied when the yld2004-18p function is used as the plastic potential function regardless of the flow rule.

Regarding the convexity of the non-associated yld2004-18p yield criterion, a function is convex if its Hessian matrix is positive semi-definite with respect to its independent variables, and the convexity can be preserved from the principal reference frame to any other transformed reference frame. For both the yield function and plastic potential function of the non-associated yld2004-18p yield criterion, they are all in the yld2004-18p function form, and the convexity of the yld2004-18p function has been proven by Barlat et al.¹⁶. Hence, the convexity of the non-associated yld2004-18p yield criterion can be guaranteed.

Although the validity of using yld2004-18p as a plastic potential function based on the non-associated flow law is discussed above, we cannot assert that the non-associated flow rule is strictly theoretically correct in sheet metal forming. However, the FE analysis results obtained using the developed yld2004-18p-based non-associated constitutive model in this study for the circular cup drawing and hole expansion tests were similar to the experimental values, and some experimental observations also indicated that the plastic potential was non-associated under certain conditions⁴⁰. Moreover, the circumferential thinning behaviors in the hole expansion test were successfully predicted, unlike using the associated constitutive models. This indicates the high analytical accuracy of the developed non-associated constitutive model based on the yld2004-18p yield criterion. However, compared with the associated constitutive model, adopting the non-associated constitutive model in FE analysis will increase the computation time, as shown in Tables 6 and 7. This is because the material stiffness matrix of the non-associated constitutive model is unsymmetric (the option “Use unsymmetric material stiffness matrix” should be checked when defining the material behaviors in Abaqus). Unlike the associated constitutive model, whose material stiffness matrix is symmetric, the material stiffness matrix of non-associated constitutive model should be fully saved during the FE analysis, which leads to lower calculation efficiency.

Another crucial issue worthy of discussion is the balance between the accuracy of the constitutive model and the model identification cost. The main improvement in accuracy of the non-associated yld2004-18p model presented in this study is compared to the original associated yld2004-18p model. This accuracy improvement is achieved without increasing the identification cost, which is one of the key significances of this work. In recent years, numerous constitutive models based on neural networks have been proposed, which offer further improvements in accuracy compared to traditional advanced constitutive models that rely on complex constitutive equations⁴¹. However, these neural network-based constitutive models require extensive identification data for training, and the high identification cost currently hinders their practical application despite their superior accuracy. Therefore, balancing model accuracy with identification cost, which is an important aspect of user-friendliness, has become a critical topic in current research on constitutive models. Both academia and industry place significant importance on the user-friendliness of constitutive models and have undertaken corresponding research and efforts (e.g., those proposed by Yoshida¹⁸ and Abspoel⁴²). This study aims to find the balance between accuracy and cost, offering a non-associated yld2004-18p model that is convenient to implement widely and achieves improvement in accuracy compared to the associated yld2004-18p.

Conclusions

Based on the non-associated flow rule, a constitutive model was developed using the yld2004-18p yield function, whose anisotropic parameters were identified from directional yield stresses, and the yld2004-18p plastic potential function, whose anisotropic parameters were identified from directional R-values. The developed non-associated constitutive model was implemented into the Abaqus/standard via the UMAT subroutine to conduct FE analyses of circular cup drawing and hole expansion to evaluate its capability in capturing stress and deformation anisotropies. In the circular cup drawing analysis, the cup height predicted by the developed non-associated yld2004-18p constitutive model was more similar to the experimental result, as compared with those by the constitutive models based on the Hill48 function and yld2004-18p function under the associated flow rule, as well as that by the constitutive model based on the Hill48 function under the non-associated flow rule. In the hole expansion test analysis, the thickness plastic strain distribution in the circumferential direction could not be predicted correctly by the constitutive models based on the associated flow rule; however, the developed yld2004-18p-based non-associated constitutive model predicted the trend successfully. The positive plastic work, the incompressibility of plastic deformation, and the convexity of the yield criterion were guaranteed in the developed non-associated constitutive model based on the yld2004-18p function. The FE analysis results of the circular cup drawing and hole expansion tests demonstrated the practicability of the non-associated yld2004-18p constitutive model developed in this study without increasing the identification cost compared to the associated yld2004-18p constitutive model. In future studies, the developed model can be further improved by considering more sheet metal behaviors (e.g., the anisotropic hardening, Bauschinger effect, fracture criterion), and more identification data (e.g., stresses in normal and shear planes). Considering more sheet metal behaviors with more identification data may further improve the simulation accuracy⁴³, but it is also crucial to strike a balance between model accuracy and identification cost when developing constitutive models for practical applications. An optimal approach ensures the developed constitutive model is both sufficiently accurate and user-friendly.

Methods

A proof for Eq. (11)

As per R. Hill’s monograph “The Mathematical Theory of Plasticity” (Chapter 12)⁴⁴, we have the following equation to calculate the R-values in the direction:

21

Considering the non-associated flow rule we used and the plastic incompressibility, we have

22

23

24

25

Substituting Eqs. (22), (23), (24), and (25) to Eq. (21), then Eq. (11) in the manuscript can be obtained.

Convex cutting-plane algorithm for implementation of constitutive model

The convex cutting-plane algorithm is one of the elastic-plastic operator split methods that contains two steps. In the first step, the total strain increment is assumed to be fully elastic and the trial stress is calculated using Eq. (26). The yield criterion F is given by Eq. (27).

26

27

where the superscript n denotes the nth step of the stress integration, the subscript k denotes the iteration number of the nth step of the stress integration (k=0 denotes initial state), represents the elastic stiffness tensor, represents the flow stress calculated by hardening rule. If , the trial stress is located inside or on the yield surface (the total strain increment is fully elastic), and the th step of stress integration can be conducted. If , the trial stress is located outside the yield surface (plastic strain occurs), and the second step called plastic corrector is conducted to bring back the stress onto the yield surface using Newton–Raphson iteration scheme.

Transform the yield criterion F using the Newton–Raphson method with respect to the plastic multiplier gives

28

According to the chain rule,

29

Substituting Eq. (29)–(28) gives

30

where , , , and .

Finally, the incremental change of the plastic multiplier in the kth iteration of the th stress integration is given by

31

The stress and strain are updated as follows during the iterative calculation until convergence.

32

33

One-element analysis for R-value prediction using convex cutting-plane algorithm

The static implicit FE analysis using one-element (1 mm1 mm1 mm, incompatible eight-node solid element) was conducted to evaluate the R-value prediction ability of the convex cutting-plane algorithm for the associated and non-associated yld2004-18p models. The material parameters of AA2090-T3 were used for the analysis (listed in Table 3), and the step increment size in the simulation was controlled within 0.001 with the total time period of 1.0. The predicted and analytical R-values are compared in Fig. 13, showing the R-value prediction accuracy of the stress integration algorithm in this study.

Figure 13.

Comparison of predicted and analytical R-values for AA2090-T3 in one-element analysis of (a) associated yld2004-18p and (b) non-associated yld2004-18p models.

Hill48 yield criterion under associated or non-associated flow rule

The Hill48 yield criterion is expressed as

34

Under the associated flow rule, the yield function and plastic potential function based on the Hill48 yield criterion are the same, as shown below:

35

The anisotropic parameters can then be identified using either directional R-values obtained from Eq. (36) to capture the deformation anisotropy (referred to as Hill48-r), or directional yield stresses obtained by Eq. (37) to capture the stress anisotropy (referred to as Hill48-s).

36a

36b

36c

36d

37a

37b

37c

37d

For the Hill48 yield criterion under the non-associated flow rule, the anisotropic parameters of the yield function and plastic potential function are independent. The parameters of the Hill48 yield function are identified using Eq. (36) with the directional yield stresses, and the parameters of the Hill48 plastic potential function are identified using Eq. (37) with directional R values.

Author contributions

Conceptualization, H.W.; methodology, H.W.; software, H.W. and A.Y.; investigation, H.W.; writing—original draft preparation, H.W.; writing—review and editing, H.W., C.F., S.X., G.Q., X.L. and J.Y.; methodology-advising, C.F.; supervision, J.Y.; funding acquisition, S.X. All authors have read and agreed to the published version of the manuscript.

Data Availability

The datasets used and analyzed during the current study are available from the corresponding author upon reasonable request.

Declarations

Conflict of interest

The authors declare no conflict of interest.