Keywords

1 Introduction

Driven by the increasing demands on green mobility and structural safety, extensive research efforts have been devoted to developing advanced high-strength steels (AHSS), which play an essential role in lightweight automotive engineering. The quenching and partitioning (Q&P) steel is considered to be a very promising representative of the third generation of AHSS due to its excellent combination of strength and ductility. In the current development of Q&P steels, most studies are focused on the influence of processing parameters on the final tensile properties [1]. The mechanical performance of Q&P steels under other loading conditions needs to be systematically evaluated for the industrial application of this new grade of AHSS.

The local and global formability are both important factors that need to be considered for the application of AHSS [2,3,4,5]. It is observed that the global formability, represented by the forming limit curve as the necking resistance, of a commercial Q&P980 steel, is better than a dual-phase DP980 steel, which is attributed to the strong strain hardening rate [1]. There are few studies in the literature on the local formability (fracture resistance) of Q&P steels, which still requires more systematic investigations [6]. The effects of stress states, especially the Lode angle parameter, on ductile fracture have received intensive attention in the recent development of ductile fracture theories [2, 7,8,9]. To achieve better accuracy in describing the fracture properties of high-strength metallic materials, many uncoupled ductile fracture criteria, microscopic mechanisms motivated or purely phenomenological, have been proposed and applied in the past two decades [10,11,12]. A common feature of these fracture criteria is that the fracture strain is expressed as a function of stress state parameters. The stress triaxiality and Lode angle parameter are the most often applied parameters to describe the stress states, which is also adopted in this study. The general aim of this study is to investigate the effects of stress states on the fracture properties of a laboratory Q&P steel with excellent tensile properties. The damage mechanics approach is adopted in this study, which is based on a combination of experimental characterization and finite element simulations.

2 Materials and Experiments

The mechanical properties of a laboratory Q&P steel have been systematically investigated in this study. The uniaxial tensile tests at room temperature and quasi-static loading conditions were performed using a smooth dog bone (SDB) specimen to obtain the engineering stress and strain curves, as shown in Fig. 1a. It can be seen that very good uniaxial tensile properties were achieved in the laboratory Q&P steel in this study. To investigate the fracture properties of the material, tensile tests were performed along the rolling direction using different specimen geometries at room temperature. Various notch configurations were used to obtain different stress states in the critical positions of these flat specimens. The overview of all specimens is shown in Fig. 1b. The force and displacement (at a gauge length of 40 mm) results were measured during the tensile tests.

Fig. 1
figure 1

Engineering stress and strain curve of the investigated Q&P steel (a) and overview of all specimens used for tensile tests (b). Specimen geometries: shear (SH), central hole (CH-R3), notched dog bone (NDB-R10), and plane strain tension (PS-R15 and PS-R3)

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3 Models

The material is assumed to be isotropic in the mechanical properties, the elastoplastic deformation behavior is thus described by the Mises plasticity model. The Voce hardening law is calibrated based on experimental results of uniaxial tensile tests at room temperature. The fracture properties of the investigated Q&P steel are described using the uncoupled damage mechanics approach. The fracture behavior under different stress states is predicted using the unified fracture criterion that was recently proposed by Shen et al. [13, 14]. The stress state is described using a general expression of stress triaxiality \(\eta\) and Lode angle parameter \(\overline{\theta }\). These two variables are calculated from invariants of stress tensors (\({I}_{1}\), \({J}_{2}\), \({J}_{3}\)).

$$\eta ={I}_{1}/\sqrt{27\cdot {J}_{2}}$$
(1)
$$\overline{\theta }=1-\frac{6}{\pi }\theta =1-\frac{2}{\pi }{{\text{cos}}}^{-1}(\sqrt{27/4}\cdot {J}_{3}\cdot {{J}_{2}}^{-3/2})$$
(2)

For proportional loading conditions, the stress state influence on fracture initiation strain is quantified using the Bai-Wierzbicki [10] phenomenological function, where \({C}_{1\sim 4}\) are four material parameters that need to be calibrated. To compensate for the non-proportional loading effects during deformation, the overall stress state of the specimen is represented by the average values of \(\eta\) and \(\overline{\theta }\) (\({\eta }_{{\text{avg}}}\), \({\overline{\theta }}_{{\text{avg}}}\)). The fracture indicator \({I}_{{\text{f}}}\) is accumulated over the non-proportional loading paths, and fracture occurs when \({I}_{{\text{f}}}\) reaches the unity.

$${\overline{\varepsilon }}_{{\text{f}}}\left(\eta ,\overline{\theta }\right)=\left({C}_{1}{{\text{exp}}}^{-{C}_{2}\eta }-{C}_{3}{{\text{exp}}}^{-{C}_{4}\eta }\right){\overline{\theta }}^{2}+{C}_{3}{{\text{exp}}}^{-{C}_{4}\eta }$$
(3)
$${I}_{{\text{f}}}=\int\limits_{0}^{{\overline{\varepsilon }}^{{\text{p}}}}\frac{1}{{\overline{\varepsilon }}_{{\text{f}}}\left(\eta ,\overline{\theta }\right)}d{\overline{\varepsilon }}^{{\text{p}}}$$
(4)

4 Finite Element Simulations

Finite element simulations of deformation and fracture behavior of different specimens subjected to remote tensile tests have been carried out in this study using the ABAQUS/Explicit with a user-defined subroutine. The critical region of fracture specimen models with half thickness has been discretized using solid elements (C3D8R) with a fine mesh (0.1 × 0.1 × 0.1 mm3). The classical J2 Mises plasticity model is applied in this study to simulate the elastoplastic deformation behavior of the material. The strain hardening behavior of the material is described by the Voce hardening law. The distribution of equivalent plastic strain on different cross-section planes at the fracture displacement is shown in Fig. 2 for NDB-R10 and PS-R15 specimens. In both specimens, the crack initiates at the symmetry center, which is the peak position of equivalent plastic strain. In addition, the localization in the thickness direction is not very pronounced, as shown in Fig. 2. The strong strain hardening behavior of the investigated Q&P steel is beneficial for the global formability as the necking resistance is improved, which is consistent with the observations in a medium-Mn steel with superior tensile properties [2].

Fig. 2
figure 2

The equivalent plastic strain contour on the mid-thickness plane and two other cross-section planes at the fracture displacements in the a NDB-R10 and b PS-R15 specimens

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In order to calibrate the fracture criterion, the evolution of local stress state variables and equivalent plastic strain has been extracted from the critical positions of different specimens, as shown in Fig. 3. The solid curves represent the evolution history of stress state variables while the dashed lines correspond to the average values of stress triaxiality (black curves) and Lode angle parameter (blue curves) until the fracture point (red symbols). There is a slight increase of stress triaxiality with increasing strain in these specimens. In general, it can be seen that the stress state in the critical elements does not change significantly during plastic deformation. It is observed that the overall stress triaxiality is similar in NDB-R10 and PS-R15 specimens. However, the failure strain of the PS-R15 specimen is much smaller than the NDB-R10 specimen, which proves the significance of the Lode angle parameter on the failure strain of the investigated Q&P steel.

Fig. 3
figure 3

Evolution of local stress state variables in the critical elements of different tensile specimens during deformation. a NDB-R10 and b PS-R15

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After collecting the critical stress and strain variables from simulation results of different geometries, the four parameters in the fracture criterion have been optimized. In the final step, the fracture criterion has been implemented in the finite element simulations to predict the fracture behavior in different specimens. The numerical simulation results (dashed curves) of force and displacement curves are compared with experimental results (dotted curves) in Fig. 4 for the NDB-R10 and PS-R15 specimens. The fracture behavior of the investigated Q&P steel in tensile tests using different specimens can be accurately predicted by the calibrated failure criterion.

Fig. 4
figure 4

Comparison between experimental and simulated fracture behavior in different specimens during tensile tests. a NDB-R10 and b PS-R15

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5 Conclusions

Superior tensile properties are obtained in a laboratory Q&P steel, which provides excellent necking resistance. The fracture properties of the Q&P steel have been investigated by performing tensile tests using specimens with optimized sample geometries. The uncoupled damage mechanics approach is adopted to simulate the deformation and fracture behavior of the material. The failure strain of the investigated laboratory high-strength Q&P steel shows a clear dependence on the stress state, in particular, the Lode angle parameter.