Anisotropic Hardening Behaviour and Springback of Advanced High-Strength Steels

Abstract

Advanced high-strength steels (AHSSs) exhibit large, and sometimes anisotropic, springback recovery after forming. Accurate description of the anisotropic elasto-plastic behaviour of sheet metals is critical for predicting their anisotropic springback behaviour. For some materials, the initial anisotropy is maintained while hardening progresses. However, for other materials, anisotropy changes with hardening. In this work, to account for the evolution of anisotropy of a dual-phase steel, an elastoplastic material constitutive model is developed. In particular, the combined isotropic–kinematic hardening model was modified. Tensile loading–unloading, uniaxial and biaxial tension, and tension–compression tests were conducted along the rolling, diagonal, and transverse directions to measure the anisotropic properties, and the parameters of the proposed constitutive model were determined. For validation, the proposed model was applied to a U-bending process, and the measured springback angles were compared to the predicted ones.

1. Introduction

Weight reduction, in order to improve fuel efficiency and meet CO₂ regulations for addressing global warming while maintaining safety regulations, is an important issue in automotive manufacturing [1,2]. In this work, we investigate the application of advanced high-strength steels (AHSSs), with good strength and formability, in automotive parts. The demand for automotive parts made of AHSSs is based on their excellent impact resistance, which is an asset for the reinforcement of the car body structure, and which depends on their high strength [3,4,5,6,7]. However, along with their high strength and low thickness, the AHSSs exhibit anisotropic properties and large springback recovery after forming [8,9]. The higher their strength and the thinner the sheet metals, the greater the tendency for anisotropy and springback to occur [10,11]. For some metals, the anisotropy of a material follows a tendency determined at the initial yielding, which does not change as hardening progresses. However, in several cases of materials in which the anisotropic tendency changes according to the hardening progress, there is a restriction to express the phenomenon of anisotropic springback problem with a yield function and the conventional hardening rules. Therefore, the accurate description of the anisotropic elasto-plastic behaviour of sheet metals is critical for predicting their anisotropic springback behaviour.

To predict springback accurately, it is important to use sophisticated elastic material models [12,13,14,15]. A constant elastic modulus is widely used, but it is not suitable for representing anisotropic and nonlinear unloading behaviour. Lems [16] investigated a change in the elastic modulus following the plastic strain at low temperature and its recovery. Morestin and Boivin [17] proved that the variation in the elastic modulus following a plastic strain allows better numerical analysis of the elasto-plastic phenomenon in the case of the springback problem. Steels, for example, have a reduction of up to 20–30% of their initial elastic modulus ($E_0$), while for aluminium alloys, in general, the reduction is 20%. The chord modulus model [18] represents a changing elastic modulus behaviour according to the hardening progress by reducing the elastic modulus with the increase in equivalent plastic strain. This model can improve the accuracy in springback prediction, because it is effective in expressing the reduction phenomenon and is computationally efficient. Its disadvantage is that a nonlinear stress–strain response cannot be captured. As a result, the stress–strain description is accurate only when fully loaded. In the quasi-plastic-elastic (QPE) model proposed by Sun and Wagoner [19], the elastic modulus is maintained as the initial value until reaching a certain level of stress based on the QPE rules, and then, the elastic modulus, according to the strain amount, is nonlinearly reduced until reaching the next plastic behaviour.

The springback predictions are also sensitive to yield functions for materials having highly anisotropic properties [20,21,22]. The Yld2000-2d yield function [23] is widely used in sheet metal forming simulations. The Bauschinger effect is usually described by introducing back stresses, as in the Armstrong-Frederick hardening rule [24]. To account for the nonlinear hardening and changing anisotropic tendency, variable parameters and tensors have been introduced into the back stress evolution rule [25,26]. Recently, the homogeneous yield function-based anisotropic hardening (HAH) model [27] was proposed, wherein the plastic behaviour of a metallic material subjected to multiple or continuous strain path changes is described using a collapse of the yield function. A fluctuating term of the HAH model, along with phenomenological or dislocation density-based hardening equations, can depict an anisotropic hardening according to the change in various strain paths [28,29,30].

The dual-phase (DP) steels studied in this work are generally subjected to intercritical annealing of cold rolled strips followed by quenching and as a result, they have a microstructure consisting of a ferritic matrix and a martensitic islands [31]. The quenching converts the ferrite-mixed austenite phase to martensite, leading to a ferrite–martensite two-phase system. The complex microstructure of DP steels may cause complicated behaviour upon changes in loading paths.

In this work, to account for the evolution of anisotropy in a DP steel, an elastoplastic material constitutive model is developed. In particular, the combined isotropic–kinematic hardening model was modified. Tensile loading–unloading, uniaxial and biaxial tension, and tension–compression tests were conducted along the rolling (RD), diagonal (DD), and transverse (TD) directions to measure the anisotropic properties, and the parameters of the proposed constitutive model were determined. For validation, the proposed model was applied to a U-bending process, and the measured and predicted springback angles were compared.

2. Constitutive Equations

2.1. Elasticity

A stress-strain relationship of a material under a non-yielding condition may be expressed by the generalised Hooke’s law [32]:

$$\mathsf{\sigma }=\mathbf{C}{\mathsf{\epsilon }}^e$$

(1)

where ${\mathsf{\epsilon }}^e$ is elastic strain, $\mathsf{\sigma }$ is Cauchy stress, and C is the stiffness matrix. Under the plane stress condition and with the isotropic elasticity, Equation (1) may be rewritten in the matrix form as [32]:

$$\left[\begin{array}{c}{\sigma }_{11}\\ {\sigma }_{22}\\ {\sigma }_{12}\end{array}\right]=\frac{1}{1-{\nu }^2}\left[\begin{array}{ccc}E& \nu E& 0\\ \nu E& E& 0\\ 0& 0& G\left(1-{\nu }^2\right)\end{array}\right]\left[\begin{array}{c}{\epsilon }_{11}^e\\ {\epsilon }_{22}^e\\ {\epsilon }_{12}^e\end{array}\right]$$

(2)

where $E$ is the Young’s modulus, $G$ is the shear modulus, and $\nu$ is the Poisson’s ratio. The Young’s modulus can be express by various elastic material models, as shown in Figure 1. The constant modulus model underestimates the elastic recovery, whereas the chord modulus model overestimates it. The QPE model [19] can depict the elastic unloading behaviour by introducing a QPE function and its nonlinear evolution. In the QPE model, the elastic modulus is given by:

$$E=E_0-E_1\left[1-\exp (-b{\displaystyle \int }||d\mathsf{\epsilon }-d{\mathsf{\epsilon }}^p||)\right]$$

(3)

where $E_0$ is the initial elastic modulus and $E_1$ and $b$ are material parameters determined by loading–unloading tests. $d\mathsf{\epsilon }$ and $d{\mathsf{\epsilon }}^p$ are, respectively, increment of total strain and plastic strain. In the QPE model, the QPE surface is defined where the constant elastic modulus is used:

$$f_1={\varphi }_1\left(\mathsf{\sigma }-{\mathsf{\alpha }}^{\ast }\right)-R=0$$

(4)

where $f_1$, ${\mathsf{\alpha }}^{\ast },\mathrm{and}$ $R$ are the QPE function, its center, and the size of the QPE function, respectively. The size of the QPE function is assumed constant in this work. In the QPE mode, the stress is located outside of $f_1$ and inside of f.

2.2. Yield Criterion

The Yld2000-2d yield function [23], f, composed of two functions, ${\varphi }^{\prime }$ and ${\varphi }^{″}$, was used to account for the anisotropic yielding

$$f\left(\mathsf{\sigma }\right)=\frac{{\varphi }^{\prime }+{\varphi }^{″}}{2}={\overline{\sigma }}^m$$

(5)

where $\mathsf{\sigma }$ is the stress, $\overline{\sigma }$ is the effective stress, and m is the yield function exponent. The details of the yield function are given in Appendix A.

2.3. Hardening Law

The back stress, $\mathsf{\alpha }$, is introduced to account for the Bauschinger effect in the Armstrong-Frederick hardening model [24]:

$$f\left(\mathsf{\sigma }-\mathsf{\alpha }\right)-{\overline{\sigma }}_{iso}^m=0$$

(6)

where the yield function size of the isotropic hardening law of Voce type [33], ${\overline{\sigma }}_{iso}$, is given by:

$${\overline{\sigma }}_{iso}={\overline{\sigma }}^0+Q\left(1-\exp \left(-b\overline{\epsilon }\right)\right)$$

(7)

where ${\overline{\sigma }}^0$, $Q$, and $b$ are material parameters. The yield function is an m-th order homogeneous function, whose size is decided by the equivalent plastic strain $\overline{\epsilon }$. Yielding occurs when the yield function equals its size. In the Armstrong–Frederick hardening model, the increment in the back stress is composed of two terms:

$$d\mathsf{\alpha }=d{\mathsf{\alpha }}_1-d{\mathsf{\alpha }}_2=C\frac{\left(\mathsf{\sigma }-\mathsf{\alpha }\right)}{{\overline{\sigma }}_{iso}}d\overline{\epsilon }-\gamma \mathsf{\alpha }d\overline{\epsilon }$$

(8)

where $C$ and $\gamma$ are material parameters. The second term of the furthest right side of Equation (8) is called a recall term, which is introduced by expressing the transient behaviour of the gradually disappearing memory of the material when subjected to a rapid change in the stress. The hardening rate increases if $C$ of the first term increases and non-linear hardening appears when $\gamma$ increases under a reverse loading condition.

Chung et al. [25] proposed a back stress evolution rule, where $C$ and $\gamma$ are functions of the equivalent plastic strain, for expressing general hardening behaviour. In this paper, $C$ and $\gamma$ were expressed as:

$$C\left(\overline{\epsilon }\right)=C_1+C_2\exp \left(-C_3\overline{\epsilon }\right)$$

(9)

$$\gamma \left(\overline{\epsilon }\right)={\gamma }_1+{\gamma }_2\exp \left(-{\gamma }_3\overline{\epsilon }\right)$$

(10)

where $C_1$, $C_2$, $C_3$, ${\gamma }_1$, ${\gamma }_2$, and ${\gamma }_3$ are material parameters.

To account for the anisotropic evolution of back stress [26], the back stress evolution rule was modified as:

$$d\mathsf{\alpha }=d{\mathsf{\alpha }}_1-d{\mathsf{\alpha }}_2={\mathbf{\Gamma }}_{1}C\frac{\left(\mathsf{\sigma }-\mathsf{\alpha }\right)}{{\overline{\sigma }}_{iso}}d\overline{\epsilon }-{\mathbf{\Gamma }}_2\gamma \mathsf{\alpha }d\overline{\epsilon }$$

(11)

where ${\mathbf{\Gamma }}_1$ and ${\mathbf{\Gamma }}_2$ are 3 × 3 diagonal tensors given by:

$${\mathbf{\Gamma }}_1=\left[\begin{array}{ccc}1& 0& 0\\ 0& {\Gamma }_{1,22}& 0\\ 0& 0& {\Gamma }_{1,12}\end{array}\right]$$

(12)

$${\mathbf{\Gamma }}_2=\left[\begin{array}{ccc}1& 0& 0\\ 0& {\Gamma }_{2,22}& 0\\ 0& 0& {\Gamma }_{2,12}\end{array}\right]$$

(13)

This model is denoted as ‘the hardening model with anisotropic evolution’. If the back stress evolves isotropically (${\Gamma }_{1,22}={\Gamma }_{1,12}={\Gamma }_{2,22}={\Gamma }_{2,12}=1$), it is denoted as ‘the hardening model with the isotropic evolution’.

3. Experiments

A dual-phase steel with tensile strength of 980 MPa and thickness of 1.1 mm (DP980) was used for the mechanical and U-bending tests. The uniaxial tension, uniaxial tensile loading–unloading, and biaxial tension tests were conducted to determine the material parameters of the QPE model and the Yld2000-2d yield function. The hardening parameters of the isotropic and anisotropic evolution models were obtained from the tension–compression tests. The U-bending test was conducted to evaluate the effect of the anisotropic evolution on springback.

3.1. Uniaxial Tensile and Loading–Unloading Tests

The uniaxial tensile and loading–unloading tests were carried out using a universal testing machine. The ASTM E8 standard size specimens were manufactured by wire electrodischarge machining. The effect of the coupon manufacturing methods on tensile properties were discussed elsewhere [34]. The uniaxial tensile tests were conducted for the RD, DD, and TD, as shown in Figure 2a. The loading–unloading tests were conducted for pre-strains of 3, 5, and 7% in the RD, as shown in Figure 2b. The material properties were obtained following the standard test methods ASTM E8 (Standard Test Method for Tension Testing of Metallic Materials) and E517 (Standard Test Method for Plastic Strain Ratio r for Sheet Metal), as listed in

Table 1. Uniaxial tensile properties of DP980.

Direction	Young’s Modulus [GPa]	Yield Strength [MPa]	r-Value	Tensile Strength [MPa]	Elonagtion [%]
RD	202.2	618.1	0.757	1014.1	15.6
DD	206.1	610.7	0.851	1015.6	16.4
TD	208.8	626.7	0.856	1021.9	14.0

.

The parameters of the QPE model were derived from the loading–unloading data, as listed in

Table 2. Parameters of the QPE model.

${\mathit{E}}_0$ [GPa]	${\mathit{E}}_1$ [GPa]	$\mathit{b}$	$\mathit{R}$ [MPa]
205.8	105.7	345	300

.

3.2. Biaxial Tensile Test

The balanced biaxial (BB) tensile test was carried out using a biaxial tensile tester [35] with a cruciform specimen, as shown in Figure 3. A constant loading rate of 9.8 kN/min was used. The plastic strain ratio in balanced biaxial tension $r_b$ was obtained. The results of the balanced biaxial tension are shown in Figure 4 and listed in

Table 3. Balanced biaxial tensile properties of DP980.

Elastic Modulus [GPa]	Yield Strength [MPa]	${\mathit{r}}_{\mathit{b}}$
297.0	601.9	0.915

.

The Yld2000-2d parameters were obtained using the data in Table 1 and Table 3, as listed in

Table 4. Parameters of the anisotropic Yld2000-2d yield function.

$\mathbf{m}$	${\mathsf{\alpha }}_1$	${\mathsf{\alpha }}_2$	${\mathsf{\alpha }}_3$	${\mathsf{\alpha }}_4$	${\mathsf{\alpha }}_5$	${\mathsf{\alpha }}_6$	${\mathsf{\alpha }}_7$	${\mathsf{\alpha }}_8$
6	0.978	0.978	1.049	1.008	1.026	1.044	0.998	1.023

. The variation of the yield strength and the comparison of the initial yield criteria of the von-Mises and Yld2000-2d yield functions are shown in Figure 5.

3.3. Tension-Compression Test

In order to investigate the anisotropic evolution of hardening, tension–compression tests were carried out in the three material directions using a tension–compression tester, as shown in Figure 6. In the tension-compression test, the sheet specimen is placed between the comb-shaped anti-buckling fixtures. Buckling of the sheet metals, which may occur during compression, is suppressed by the hydraulic force acting perpendicular to the sheet plane through the comb-shaped fixtures. Details of the tension-compression test can be found elsewhere [36]. In the tension-compression tests, the tension by a strain of 7% was followed by the compression to a strain of −7%. The true stress-accumulated plastic strain curves can be divided into three stages, as shown in Figure 7. In stage A, where hardening hardly progresses, the anisotropic tendency of yielding matches the initial yield strength trend in Figure 5a. The curves of RD and TD become similar in stage B, and the flow stress of RD becomes larger than that of TD in stage C where the initial anisotropic tendency is reversed. The change in anisotropic tendency is illustrated in Figure 8.

The hardening model parameters were obtained from the tension–compression data by optimisation using the fminsearch function of the MATLAB program, as listed in

Table 5. Parameters of hardening models with isotropic and anisotropic evolution.

Hardening Model with Isotropic Evolution
${\overline{\sigma }}^0$ [MPa]	$Q$ [MPa]	$b$	$C_1$ [MPa]	$C_1$ [MPa]	$C_1$	${\gamma }_1$	${\gamma }_2$	${\gamma }_3$
583.9	409.7	3.861	17,896	26,144	9.537	34.74	80.27	6.551
Hardening Model with Anisotropic Evolution
${\overline{\sigma }}^0$ [MPa]	$Q$ [MPa]	$b$	$C_1$ [MPa]	$C_1$ [MPa]	$C_1$	${\gamma }_1$	${\gamma }_2$	${\gamma }_3$
583.9	409.7	3.861	17,896	26,144	9.537	34.74	80.27	6.551
${\Gamma }_{1,22}$	${\Gamma }_{1,12}$	${\Gamma }_{2,22}$	${\Gamma }_{2,12}$
0.948	0.975	1.014	0.940

. The MATLAB function finds parameters that minimise the error function value using the Nelder–Mead method [37]. After calculating a residual by performing a finite element (FE) analysis using an estimated value, the fminsearch function calculates the updated parameter values and repeats the FE analysis. This process is repeated until no further reduction in residual is achieved. In the case of the hardening model with isotropic hardening, it was optimised only for the rolling direction data.

In Figure 9, the tension-compression curves in the three material directions were compared. In the case of the isotropic evolution, it follows the initial trend of anisotropy determined by the yield function. However, in the case of the anisotropic evolution, the anisotropic tendency is changed during reverse loading. The errors of fit of the tension–compression curves predicted by the isotropic and anisotropic evolution models are compared in Figure 10. The error of fit in the RD value is similar because the parameters of the isotropic model were obtained from the RD data. However, the errors in the DD and TD are smaller for the anisotropic models.

4. Application

U-bending tests were conducted using specimens with size of 300 $\times$ 30 $\times$ 1.1 mm³ for the three material directions. The experiments were performed using a servo-press. The dimensions of the test are shown in Figure 11a. The holding force was linearly increased from 5.65 kN to 7.92 kN from the beginning to the end of the punch stroke. The specimens before and after forming and springback are shown in Figure 12a.

Simulations of the U-bending process were carried out using the commercial finite element software Abaqus/Standard, as shown in Figure 11b. Four-node shell elements with reduced integration (S4R) were used with an element size of 0.5 mm. The number of integration points through the thickness was seven. The coefficient of friction was assumed as 0.12. The deformation of the tools may affect the U-bending test results significantly [38]. However, for simplicity, the punch, die, and holder were assumed as rigid bodies in this work.

The measured and calculated shapes of the specimens after springback are compared for the three material directions in Figure 13. The predicted shapes showed good agreements with the measurements in RD, while there are some differences in DD and TD. The predictions with the isotropic and anisotropic evolution showed little differences for RD because the material parameters of the isotropic evolution model were fitted for RD. However, the predictions with the isotropic and anisotropic evolution showed differences in TD, although the amount of the difference is small.

In order to evaluate the springback quantitatively, three measures of springback ($\rho$, ${\theta }_1$, and ${\theta }_2$), defined in Figure 12b, were taken from the specimen geometry, as shown in Figure 14. The predictions with the isotropic and anisotropic evolution showed little differences in the U-bending shape of the RD and DD specimens. For the TD specimens, the predictions with the anisotropic evolution showed better agreements with the measurements. This is because the anisotropic hardening evolution rule reduces the error when representing the stress-strain behaviour in directions other than RD. In the case of the isotropic evolution model, the initial yielding anisotropic tendency of the yield function is maintained in the springback result of U-bending. However, the initial anisotropic tendency was changed during deformation, as shown in Figure 8. In the case of the anisotropic evolution model, the evolving anisotropy tendency is represented, which is more similar to the measurements.

5. Conclusions

To account for the evolution of anisotropy of a dual-phase steel, an elastoplastic material constitutive model is developed by modifying the combined isotropic-kinematic hardening model. The tensile loading–unloading, uniaxial and biaxial tension, and tension–compression tests were conducted along the rolling, diagonal, and transverse directions to measure the anisotropic properties, and the parameters of the proposed constitutive model were determined. For validation, the proposed model was applied to a U-bending process, and the measured and predicted springback angles were compared. The following conclusions were drawn:

• The dual-phase steel studied in this work exhibited a change in the anisotropic properties during deformation.
• The hardening model with the anisotropic evolution successfully captured the evolution of the angular variation of anisotropic properties.
• Using the hardening model with anisotropic evolution, it was possible to accurately predict the springback in different directions.