Study on Ductility Failure of Advanced High Strength Dual Phase Steel DP590 during Warm Forming Based on Extended GTN Model

Abstract

Under warm forming, the damage parameters of extended GTN model are easily affected by temperature, and the failure of materials under warm forming is less studied by using this model. In this paper, based on the extended GTN model, the damage parameters of DP590 at different temperatures are determined by experiment and numerical simulation, studying the trend of damage parameters changing with temperature. Through the analysis of shear specimens at different angles and temperatures, we studied the changes in shear damage and void damage under different conditions, discussing the influence of shear damage and void damage on competitive fracture failure under warm forming. We modify the damage by using a function based on stress triaxiality, and present a competitive failure equation considering temperature and stress triaxiality. It is found that the extended GTN model can be applied to the failure study of DP590 steel under warm forming.

1. Introduction

Continuum damage (CDM) and micromechanical damage are two representative methods to study the ductile damage of materials. For the CDM, Lemaitre [1] puts forward Lemaitre model by considering the influence of stress. Mashayekh [2] proposed an improved Lemaitre model and studied the forming process of materials through this model. This approach is derived on a thermodynamic framework and it takes no account of the mechanical behavior of micro defects in the material.

Gurson et al. [3,4,5] proposed that the failure of materials was mainly due to the nucleation, growth, and aggregation of microvoids in materials by studying the nucleation of spherical cavities. In order to more accurately explain the hardening of materials and the interaction between cavities, Tvergaard and Needleman imported [6] three extended parameters and the GTN model is obtained. GTN model has been widely used in the study of material failure and damage, but there are still some limitations. The ordinary GTN model is used to predict the triaxiality of low stress, that is, when the material with shear stress is destroyed, it cannot accurately describe the evolution law of void volume. Many experts and scholars have studied this. Nahshon and Hutchinson [7] introduce a new parameter to determine the development rate of shear damage without changing the mechanical behavior of materials under axisymmetric stress state, the model with this parameter can simulate the fracture form under shear stress under low stress triaxiality. Wei J et al. [8] extended the formula of shear deformation on damage evolution, so that the model can be applied to material failure under extensive stress triaxiality. Liang X [9] proposed a new shear damage mechanism based on McClintock considering the accumulation of void volumes in the shear band. Based on this model, Ying L [10] further calibrated the new parameters in the tensile process of 22MnB5, and established the corresponding numerical implementation method. He W [11] evaluated the evolution of void volume fraction and shear damage. The improved GTN model was used to predict the ductile fracture during piperotation, and good prediction results were obtained. Zao H A, et al. [12] proposed an improved shear propagation GTN model containing two independent damage mechanisms to predict the ductile fracture of materials under different stress states. Gatea S [13] and others used the extended GTN in the research of progressive forming and found that the extended GTN model improved the modeling accuracy of cracks compared with the original GTN model. Liu J [14] combined the GTN model with the maximum shear stress criterion and established a shear extended GTN model to predict the void volume fracture process of Q345. It is found that its prediction ability is better than the original GTN model. Through research, Sun Q [15] found that the identification parameters of shear modified GTN damage model can effectively characterize the mechanical properties, damage evolution and ductile failure of materials during small punch test. Li X [16] et al., established a coupling model by shear extended GTN damage model, and found that the shear size can promote the shear damage of materials. Wu P [17] proved that the main mode of failure changed from microvoid growth to shear damage under the condition of shear stress by superimposing hydrostatic pressure in numerical simulation. Zhou et al. [18] combined the damage mechanics concept of Lemaitre and Jean [19] with the GTN cavity growth model and proposed an extended GTN model. The model combines the concept of damage mechanics with the void growth model, and couples the void volume parameters and shear damage parameters into the yield function. It is proposed that the void volume damage and shear damage adopt a separate critical damage state. When the damage parameters are unified, the material will fail. In order to study the material failure and fracture more accurately, the finite element method has been widely used. Yao D [20] proposed a finite element aided test (fat) method to obtain the uniaxial full range constitutive relationship of A5083 steel and SS316L before failure. The fracture stress and strain were obtained by finite element analysis. Churyumova T A [21] using the experimental values of the relative crosssection reduction and fifinite element calculation of the stress triaxiality got the critical values of the modified Rice and Tracy fracture criteria. Aldakheel F [22] develop a new theoretical and computational framework for the phase fifield modeling of ductile fracture in conventional elastic–plastic solids under fifinite strain deformation, and verify its accuracy by numerical simulation.

To our knowledge, it is common to use the extended GTN model to study the failure of materials at room temperature. However, under the condition of warm forming, the damage parameters will be affected by temperature, it is necessary to recalibrate and understand the change trend of damage parameters with temperature. On the other hand, the evolution of shear damage and void damage under different temperatures and different stress states needs to be further studied. If we can establish influence of shear damage and void damage on competitive fracture failure, it will be very helpful to verify whether the extended GTN model can be applied to the failure study of DP590 steel under warm forming.

In this paper, the damage parameters of materials at different temperatures can be determined by employing experiments and numerical simulation. Through the analysis of shear specimens at different angles and temperatures, we will study the changes of shear damage and void damage under different conditions. The effects of shear damage and void damage on material failure under warm forming conditions will be also discussed. The damage is corrected by using a function based on stress triaxiality, and a failure equation considering temperature and stress triaxiality is given.

2. Experiments

2.1. Material

Advanced high strength dual phase steel is mainly composed of ferrite and martensite. In this study, we selected DP590 dual phase steel which produced by Shanghai Baosteel company. Its chemical composition shows in

Table 1. The chemical composition of DP5900 and its mass fraction%.

C	Si	Mn	P	S	Al
0.08	0.46	1.75	0.014	0.004	0.032

.

According to the Chinese GB/T 4338–2006 high temperature tensile test method for metallic materials. Uniaxial tensile tests were carried out on DP590 specimens with a thickness of 1 mm at 293K, 373K, 473K, 573K, 673K, and 773K. The tensile specimen is cut from the DP590 plate in the rolling direction (RD), 45 direction and transverse direction (TD). The test was carried out on WDW-20D material performance testing machine with the tensile speed 0.2 mm/min at different temperatures. The dimensions of uniaxial tensile specimens are shown in Figure 1.

Before the tensile test, heating the DP590 test specimen to the test temperature, and the temperature of the test specimen remains stable until the test specimen is stretched to fracture. Uniaxial tensile tests were carried out on the same three groups of samples at different temperatures. In order to ensure the accuracy of the results, the average value of each group of tests was taken. Test results are shown in Figure 2.

According to experimental results, when the temperature is higher than 673 K, the plasticity of DP590 will obviously improve. Because when the temperature is high, the warm forming of DP590 can reduce the obstruction of grain boundary to the internal dislocation movement of the material, improve the ability of dynamic recovery and recrystallization of the material, and promote the elimination of material defects and internal stress, which is reflected in the improvement of plasticity and the reduction of strength. When the temperature is 473 K–573 K, the strength of DP590 is high and the phenomenon of “blue brittleness” will appear. This is because C and N atoms anchor the dislocation rapidly, which makes it difficult for the dislocation to slip, forming the blue embrittlement phenomenon.

We use Equations (1) and (2) to convert the engineering stress–strain curve obtained from the test into the real stress–strain curve.

$$\epsilon =\ln (1+{\epsilon }_{\mathrm{nom}})$$

(1)

$$\sigma ={\sigma }_{\mathrm{nom}}(1+{\epsilon }_{\mathrm{nom}})$$

(2)

$\epsilon$, ${\epsilon }_{\mathrm{nom}}$ stand for real strain and engineering strain respectively, $\sigma$, ${\sigma }_{\mathrm{nom}}$ stand for real stress and engineering stress respectively.

Using the famous swift model to describe the mechanical properties of DP590 steel at different temperatures, taking 293 K as an example. As shown in Figure 3.

$$\sigma =K({\epsilon }_0+{\overline{\epsilon }}_{\mathrm{p}}{)}^n$$

(3)

where K is the hardening coefficient and n is the material hardening index.

The determined elastoplastic parameters are shown in

Table 2. Elastoplastic mechanical parameters.

Temperature (K)	E (Mpa)	v	K (Mpa)	n
293	215,000	0.3	1145	0.198
373	212,000	0.3	971	0.137
473	198,000	0.3	1064	0.129
573	182,000	0.3	1112	0.149
673	176,000	0.3	654	0.06
773	165,000	0.3	366	0.063

.

2.2. Tensile Shear Test

In order to study whether the extended GTN model is applicable to the failure of DP590 dual phase steel at different temperatures and low stress triaxiality, we designed a series of shear specimens (as shown in Figure 4), and the shear tests at different temperatures were carried out by tensile testing machine.

Before the tensile shear test, heating the DP590 shear specimen to the test temperature, and the temperature of the specimen remains stable until the specimen is stretched to fracture. Figure 5 shows the partial shear specimen. Figure 6 shows the load–displacement curve of different angle shear specimens at different temperatures.

3. Constitutive Model

3.1. GTN Damage Model

Gurson model considers the factors of void volume damage. On the basis of Gurson model, tvegard and niedman modified it into GTN damage model by introducing void volume interaction parameters. GTN damage model is a commonly used constitutive model in meso damage analysis of materials. The yield function of GTN damage model is expressed as:

$$\mathsf{\Phi }={\left(\frac{{\sigma }_{eq}}{{\sigma }_m}\right)}^2+2q_1{f}^{*}\cos \mathrm{h}\left(-\frac{3q_2{\sigma }_h}{2{\sigma }_m}\right)-\left[1+q_3{f}^{*}{}^2\right]=0$$

(4)

where ${\sigma }_{eq}$ is the Mises equivalent stress, ${\sigma }_h$ is the hydrostatic pressure, ${\sigma }_m$ is the equivalent stress of the material, $q_1,q_2,q_3$ is the strengthening parameter, $f^{*}$ is the void volume fraction function:

$$f^{*}=\{\begin{array}{c}f\\ f_c+\delta \left(f-f_c\right)\end{array}\begin{array}{cc}& \begin{array}{c}f\le f_c\\ f>f_c\end{array}\end{array}$$

(5)

$$\delta =\frac{f_u^{*}-f_c}{f_f-f_c}$$

(6)

where $f_c$ is the critical void volume fraction when the voids begin to polymerize, $f_f$ is the void volume fraction when the material ruptures, $\delta$ is the void growth acceleration factor, $f_u^{*}$ is the $f^{*}$ value when the stress in the yield equation is 0.

3.2. Extended GTN Model

GTN model is more accurate in predicting the necking fracture phenomenon of materials, but the effect of the model is not accurate in predicting the material failure dominated by shear damage. In the case of low stress triaxiality, the growth of void volumes is restrained, only a small number of void volumes can nucleate, the void volume fraction changes little, and the ordinary GTN model cannot accurately predict the failure of materials. In order to overcome these defects, Zhou et al. [18] introduced the concept of continuous damage into the GTN model, the shear damage variable is defined as D_S. Shear damage only affects the deviatoric stress state, and the change of plastic volume is affected by porosity. The yield function of the extended GTN model can be expressed as:

$$\mathsf{\Phi }={\left(\frac{{\sigma }_{eq}}{{\sigma }_m}\right)}^2+2q_1{f}^{*}\cos \mathrm{h}\left(-\frac{3q_2{\sigma }_h}{2{\sigma }_m}\right)-\left[1+{\left(q_1{f}^{*}+D_s\right)}^2-2D_s\right]=0$$

(7)

$$D=q_1{f}^{*}+D_S$$

(8)

where D is the total damage, D_S is shear damage.

Shear damage is regarded as a function of plastic strain and stress state. The definition formula is as follows:

$$D_s={\left(\frac{{\epsilon }_m^{pl}}{{\epsilon }_f^s}\right)}^n$$

(9)

where ${\epsilon }_f^s$ is the fracture strain under pure shear state; n is the weakening index of the material and the ${\epsilon }_m^{pl}$ plastic strain of the matrix. In the above formula, stress triaxiality and Lode parameter are introduced and extended to any stress state.

$$\stackrel{·}{D_s}=\mathsf{\psi }\left(\theta ,\mathrm{T}\right)\frac{n{D}_s^{\frac{n-1}{n}}}{{\epsilon }_f^s}\stackrel{·}{{\epsilon }_m^{pl}}$$

(10)

where $\mathsf{\psi }\left(\theta ,\mathrm{T}\right)$ is the weight function [23].

$$\mathsf{\psi }\left(\theta ,\mathrm{T}\right)=\{\begin{array}{c}g\left(\theta \right)\\ g\left(\theta \right)\left(1-k\right)+k\end{array}\begin{array}{cc}& \begin{array}{c}T>0\\ T\le 0\end{array}\end{array}$$

(11)

where k is the weight factor when the triaxiality of stress is negative, which can be calibrated by the data of tensile test.

$$g\left(\theta \right)=1-\frac{6\left|\theta \right|}{\pi }$$

(12)

where $\theta$ is Lode angle. At this time, the evolution of plastic strain is expressed material equivalent plastic power and macro equivalent power:

$$\stackrel{·}{{\epsilon }_m^{pl}}=\frac{\sigma :d{\epsilon }^p}{\left(1-D/q_1\right){\sigma }_m}$$

(13)

where $\left(1-D/q_1\right)$ can ensure that here is no shear damage.

4. Parameters Calibration

4.1. The Yield Surface Coefficients

Before applying the extended GTN damage model to predict the damage of dual phase steel at different temperatures, 11 uncertain parameters in the model need to be calibrated. Among them, the extended GTN model is used to calibrate n and ${\epsilon }_f$ parameter, and the GTN model is used to calibrate other parameters. By consulting a number of literature [24,25,26,27,28,29,30,31,32], most researchers use the strengthening coefficient q₁ = 1.5, q₂ = 1, q₃ = 2.25, proposed by Tvergaard when using GTN model to study the failure of different materials. In this paper, the coefficients are selected according to the results of literature review.

4.2. Nucleation Rate Function Parameters and Void Nucleation Parameters

For the standard deviation of nucleation strain S_n, according to the above literature research, it can be found that the standard deviation of nucleation strain is generally taken as S_n = 0.1.

For the normal temperature forming of DP590, the average ${\epsilon }_N$ strain void volume nucleation is generally taken as 0.3. According to Liu Wenquan’s research [33], when the forming temperature of highstrength steel is considered, the average strain of void volume nucleation should consider the influence of temperature. Therefore, ${\epsilon }_N$ is the strain value when strain local deformation occurs in the tensile sample at different temperatures, as shown in

Table 3. Values of ${\epsilon }_N$ at different temperatures.

Temperature (K)	${\mathit{\epsilon }}_{\mathit{N}}$
293	0.3
373	0.289
473	0.275
573	0.271
673	0.36
773	0.45

.

In this paper, the parameter inverse method is used to identify the damage parameters at different temperatures. In the process of parameter calibration, central composite design and response surface analysis method are adopted [34,35,36], and the peak point and fracture point of the material tensile curve are used as the key positions of damage parameter calibration. The load R₁ at the peak point, the displacement R₂ at the peak point, the load R₃ at the fracture point, and the displacement R₄ at the fracture point are taken as the optimization objectives. When the error between the prediction curve and the test curve is small, the appropriate damage parameter value can be found taking the damage parameter calibration at 573 K as an example.

Table 4. Range of values of damage parameters.

Damage Parameters	Low Level	High Level	Intermediate Level
$f_0$	0.001	0.0028	0.0019
$f_c$	0.018	0.039	0.00285
$f_N$	0.023	0.041	0.032
$f_f$	0.12	0.22	0.17

shows the damage parameter coefficient levels of the central composite design.

Table 5. Design scheme of damage parameters and response values.

${\mathit{f}}_{\mathit{c}}$	${\mathit{f}}_{\mathbf{0}}$	${\mathit{f}}_{\mathit{N}}$	${\mathit{f}}_{\mathit{f}}$	${\mathit{R}}_{\mathbf{1}}$	${\mathit{R}}_{\mathbf{2}}$	${\mathit{R}}_{\mathbf{3}}$	${\mathit{R}}_{\mathbf{4}}$
0.018	0.001	0.041	0.12	5.85	7015.38	10.53	4853.26
0.018	0.0028	0.041	0.12	5.85	6999.51	10.26	4500.2
0.0075	0.0019	0.032	0.17	5.40	7003.7	11.43	4561
0.039	0.001	0.023	0.12	5.386	7013.54	8.19	5957.3
0.039	0.0028	0.023	0.22	5.85	6994.29	8.1	5956.5
0.039	0.001	0.041	0.22	5.490	7008.04	8.729	5692
0.039	0.001	0.023	0.22	5.2198	7007.12	8.189	5928.15
0.0285	0.0019	0.032	0.17	5.670	7001.69	8.81	5817
0.018	0.001	0.023	0.12	5.7598	7011.8	9.71	5089.2
0.0495	0.0019	0.032	0.17	5.220	6998.54	8.11	5762.78
0.018	0.001	0.023	0.22	5.7598	7011.8	9.8	4472.6
0.018	0.0028	0.023	0.22	5.5801	6996.77	9.18	5595.14
0.0285	0.0019	0.032	0.17	5.670	7001.69	8.81	5817
0.018	0.0028	0.023	0.12	5.5801	6996.77	9.26	5387.09
0.039	0.0028	0.023	0.12	5.490	6992.41	8.1	5895.69
0.0285	0.0019	0.032	0.07	5.670	7001.69	8.999	5374.74
0.0285	0.0001	0.032	0.17	5.400	7017.38	9.18	5303.39
0.0285	0.0019	0.032	0.17	5.670	7001.69	8.9	5652.21
0.0285	0.0019	0.014	0.17	5.822	7001.22	8.18	5869.5
0.039	0.0028	0.041	0.22	5.826	6993	8.37	5885.26
0.0285	0.0019	0.032	0.17	5.670	7001.69	9.025	5152.32
0.0285	0.0019	0.032	0.27	5.736	7001.25	9.222	4836.23
0.0285	0.0019	0.05	0.17	6.321	6998.23	9.72	4936.23
0.0285	0.0037	0.032	0.17	5.825	6989.23	8.92	5423.62
0.018	0.0028	0.041	0.22	5.8802	6996.77	10.34	4422.26
0.039	0.001	0.041	0.12	5.490	7008.04	8.4598	6125
0.039	0.0028	0.041	0.12	5.7762	6992.41	8.64	5296.56
0.018	0.001	0.041	0.22	5.825	7011.8	10.169	4255.11

lists 28 groups of damage parameter combinations and response values obtained from numerical simulation.

Analysis of ANVOA is an important method to evaluate the significance of response surface model. According to the analysis of ANOVA, we can find out whether the regression model is applicable.

Through ANVOA analysis (as shown in

Table 6. ANVOA analysis.

Response	${\mathit{R}}_{\mathbf{1}}$	${\mathit{R}}_{\mathbf{2}}$	${\mathit{R}}_{\mathbf{3}}$	${\mathit{R}}_{\mathbf{4}}$
Model p value	<0.0001	<0.0001	<0.0001	0.0002
$R^2$ %	93.27	96.85	98.27	90.52

), it can be seen that the model is significant and effective. Therefore, the regression model of damage parameters at 573 K temperature is:

$$\begin{array}{cc}\hfill R_1=& 6.791+17.96f_c-374.03f_0-68.49f_n-1.348f_f+10986.04f_c{f}_0+876.54f_0{f}_f\hfill \\ & -814.77f_c^2+1240.94f_n^2\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill R_2=& 7024.08+91.04f_c-13767.94f_0+397.68f_n-60.98f_f-8062.16f_c{f}_n+13513.88f_0{f}_f\hfill \\ & +802566f_0^2-3023.72f_n^2+76.531f_f^2\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill R_3=& 11.628-152.948f_c-487.46f_0+55.61f_n-4.66f_f+5365.07f_c{f}_0-1143.65f_c{f}_n\hfill \\ & +7493f_0{f}_f+1816.03f_c^2+14.13f_f^2\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill R_4=& 4472.325+48981.1f_c+546198f_0-24768.43f_n+1690.19f_f-1258571f_c{f}_0+117353f_c{f}_n\hfill \\ & -1703910f_0{f}_n+341183f_0{f}_f-81471f_c^2-48665100f_0^2-41569.49f_f^2\hfill \end{array}$$

(17)

Finally, the damage parameter values under different temperatures are obtained by genetic algorithm (

Table 7. Damage parameter values at different temperatures.

Temperature	${\mathit{f}}_{\mathit{N}}$	${\mathit{f}}_{\mathbf{0}}$	${\mathit{f}}_{\mathit{c}}$	${\mathit{f}}_{\mathit{f}}$
293	0.016	0.0002	0.045	0.136
373	0.0195	0.0004	0.0395	0.128
473	0.021	0.00092	0.037	0.126
573	0.032	0.0012	0.03	0.123
673	0.029	0.00132	0.043	0.158
773	0.012	0.00138	0.056	0.172

). The flow chart of genetic algorithm is shown in the Figure 7.

According to the above damage parameters, we write VUMAT subroutine and use ABAQUS software to simulate the damage parameters at different temperatures. We compare the load displacement curve obtained by numerical simulation with the test load displacement curve, and the results are shown in Figure 8.

By comparing the curves, we found that the maximum error of the curve is 4.72%. Therefore, it can be considered that the GTN damage parameters obtained by numerical simulation combined with the test and RSM experimental design are accurate and can be used to predict the failure of unidirectional tensile specimens at different temperatures. Figure 9 shows the variation law of void volumerelated damage parameters with temperature.

It can be found that the initial void volume fraction increases slightly with the increase of temperature. When the temperature is high, the material will recover and recrystallize, so as to reduce the internal defects, reduce the volume fraction $f_N$ of cavity nucleation, and increase the critical void volume fraction $f_c$ and fracture void volume fraction $f_f$.

4.3. Shear Damage Parameters

The shear damage introduced into the extended model is an important state variable. The evolution of shear damage is affected by the material shear damage softening parameter n and effective failure strain. The shear parameters were calibrated by shear test and numerical simulation. Taking 293K as an example, the load–displacement curve under two parameter sets (Figure 10) and other model parameters take the above given values, and finally determine the value of n and ${\epsilon }_f$ at different temperatures.

The load–displacement curve test and numerical simulation curve of 0°shear parts at different temperatures are compared as shown in Figure 10. It can be seen that the calibrated shear damage parameters are appropriate.

5. Verification and Discussion

In this section, the damage parameters of the above test and calibration are numerically simulated by using VUMAT subroutine in the previous section and using ABAQUS software to study the ductile failure of DP590 at different temperatures.

5.1. Finite Element Model

According to the shear tests and the damage parameters calibrated above, the numerical simulation of uniaxial tensile shear test is carried out for shear specimens from other angles. In order to better capture local deformation and damage, refined meshes and coarse meshes are used inside and outside the central region respectively. The size of the refined mesh is 0.2 mm × 0.2 mm, and the size of the coarsened mesh is 0.8 mm × 0.8 mm. Updated Lagrange formulations were used to handle the geometrical and material nonlinearities, and mass scaling factor was set to 20 so as to reduce the total simulation time. Fixed constraint was imposed at the bottom of the samples and the displacement loading was implemented on the top boundary. The finite element model of the test piece is shown in Figure 11.

5.2. Numerical Simulation Results

The numerical simulation results such as equivalent plastic strain, vvf, etc., are shown in Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16 below (take the shear specimen at 673 K as an example).

It can be seen from the above Figure 12, Figure 13, Figure 14, Figure 15, Figure 16 and Figure 17 that in the 0° tensile test, the shear damage starts at both ends of the sample notch and propagates to the center, and the crack propagation direction is parallel to the loading direction. With the increasing of specimen, the influence of shear damage becomes smaller and smaller, the initial position of shear damage gradually shifts to the center, and the crack propagation direction also develops vertical to the loading direction. When any variable of shear damage or void volume damage reaches the critical value, the material will fail.

Through numerical simulation, changes in void volume damage and shear damage at different temperatures are obtained. It can be seen from the above Figure 18 and Figure 19 that for samples with the same temperature and different angles, the influence of void volume damage on material damage increases with the increase of shear angle. This is consistent with the gradual development of fracture form from shear fracture to tensile fracture. For the shear specimen with the same angle, when the temperature changes from 293 K to 573 K, the void volume damage value is small under the same equivalent plastic strain. When the temperature changes from 573 K to 773 K, the void volume damage value will gradually increase. This shows that when the formability of DP590 becomes better, the void volume damage has a great influence on the failure of DP590.

In order to more accurately measure the influence of shear damage and void damage on competitive fracture failure, introduce a weight coefficient to describe the evolution process of shear damage and void damage.

$$D={\left(\frac{T}{T_L}\right)}^2\times q_1{f}^{*}+\sqrt{1-\left({\left(\frac{T}{T_L}\right)}^2\right)}\times D_S\hspace{1em}\hspace{1em}(T<T_L)$$

(18)

where $T_L$ is the stress triaxiality in pure tension, which can be calibrated by uniaxial tensile test specimen and $T$ is the stress triaxiality. With the continuous increase of stress triaxiality, the influence of shear damage decreases gradually. When the stress triaxiality of the specimen exceeds the critical stress triaxiality, the total damage is caused by void damage. In this way, the shear damage and void damage in the case of low stress triaxiality can be corrected, which can better conform to the evolution of total damage.

In order to find the influence of shear damage and void damage on competitive fracture failure and triaxiality of temperature and stress, taking 673 K as an example to extract the shear damage and stress triaxiality before the critical damage of materials as shown in

Table 8. Stress triaxiality and shear damage values at 673 K.

Angle	Stress Triaxiality	Shear Damage
0	0.0025	0.947
15	0.133	0.586
30	0.234	0.492
45	0.282	0.350
60	0.306	0.0786

and we used a multiple linear regression equation to describe the shear damage and stress triaxiality at different temperatures.

$$D_S=0.937-2.46\ast T-3.85{\mathrm{e}}^{-7}\times \mathrm{t}$$

(19)

It can be seen that temperature has little effect on shear damage, so we choose quadratic polynomial to more accurately describe the relationship between shear damage and stress triaxiality.

$$D_s=4.49\times T^2-4.17\times T+1.02$$

(20)

Because $f_c$ has little change under low stress triaxiality, only look for the relationship between $f_c$ and temperature. By fitting the relationship between $f_c$ and temperature, we get the expression of critical void volumes at different temperatures.

$$f_c=91.36-91\times \exp \left(\frac{-0.5\times {\left(t-495\right)}^2}{12816}\right)$$

(21)

When the stress triaxiality is less than the critical stress triaxiality, shear damage is the dominant factor causing material damage. With the continuous increase of stress triaxiality, void damage gradually dominates. Therefore, establish a failure expression based on temperature and stress triaxiality with $T_L$ as the threshold value.

$$\{\begin{array}{c}{D}_s>4.49\times T^2-4.17\times T+1.02\\ f_c>91.36-91\times \exp \left(\frac{-0.5\times {\left(t-495\right)}^2}{12816}\right)\end{array}\begin{array}{cc}& \end{array}\begin{array}{c}T\le T_L\\ T>T_L\end{array}$$

(22)

when the stress triaxiality is greater than $T_L$, the specimen will undergo tensile fracture. When the stress triaxiality is less than $T_L$, the specimen will shear. At this time, the critical value can be determined by $D_S$ and $f_c$ respectively.

5.3. Result Analysis

According to the damage parameters determined above and the introduced correction function, we numerically simulate the shear specimens with different temperatures and angles, and extract the load–displacement curve (Figure 20).

Comparing the load–displacement curve in the Figure 20, the maximum error is within 5%. It can be considered that the extended GTN damage parameters obtained by combining test and numerical simulation are accurate, the introduced correction function can better predict the damage and the extended GTN model can be applicable to the shear damage prediction of DP590 dual phase steel under warm forming conditions. In order to verify the accuracy of the competitive failure of the extended GTN model, the tensile bending test and numerical simulation of DP590 are carried out. The results are shown in Figure 21.

By analyzing the numerical simulation results of different fillet radii, it is found that the fillet radius of 1 mm is shear fracture, the fillet radius of 5 mm is mixed fracture, and the fillet radius of 15 mm is tensile fracture. The fracture form and fracture position are consistent with the test results. It is verified that the competitive failure can predict the fracture of DP590 under different stress states.

6. Conclusions

In this paper, the damage parameters of materials at different temperatures are determined by experiment and numerical simulation. Through the analysis of shear specimens at different angles and temperatures, the changes of shear damage and void damage under different conditions are studied, and the effects of shear damage and void damage on material failure under warm forming conditions are discussed. A function based on stress triaxiality is used to modify the total damage D, and a competitive failure equation considering temperature and stress triaxiality is given.

• Uniaxial tensile tests are carried out on plane specimens of DP590 dual phase steel at different temperatures and the central composite design (CCD)—response surface method (RSM)—genetic algorithm (GA) method was used to back correct the elasticplastic mechanical parameters and extended GTN damage parameters at different temperatures. By comparing the load–displacement curves of test and numerical simulation, the calibrated damage parameters are found to be accurate, and was found that the damage parameter f₀ of the extended GTN model changes little with temperature at different temperatures. At higher temperature, the critical void volume integral f_c and fracture void volume fraction f_f increase due to the good recovery and recrystallization ability of the material. At the same time, due to the reduction of internal defects, the nucleation volume fraction f_n of the cavity reduced.
• Combined with test results, we simulated and analyzed the shear parts with different angles at different temperatures. For specimens with the same temperature and different angles, with the increase of stress triaxiality, the influence of void volume damage on material damage is greater. Moreover, under the same stress triaxiality, when the formability of DP590 increases, the void volume damage will gradually dominate.
• Using a function based on stress triaxiality to modify the total damage D, by extracting the load–displacement curves of tensile shear numerical simulation at different temperatures and angles, the maximum error is 4.75%, which shows that the correction of this function is accurate. A competitive failure equation was presented to judge the critical damage of materials. Through tensile bending test and numerical simulation, we find that the fracture form and fracture position are consistent, which shows that the competitive failure equation is effective. Experiment and simulation show that the extended GTN model can be applied to the study of ductile fracture of DP590 under warm forming, and can obtain accurate ductile fracture prediction.