1 Introduction

With increasing difficulty of extracting large onshore oil fields and the continuous decline in oil production, marine oil has become an important field of oil development [1,2,3,4]. Marine oil pipelines are essential lifelines for offshore oil production systems since they are the primary means of transporting offshore oil resources. They must serve in harsh marine environments including low temperature and high-pressure condition. They not only must tolerate variations of static stresses including internal and external pressures, but also have to handle dynamic loads, such as the changes of the seabed, wind, waves, and S-shaped laying as well as J-shaped laying during pipeline installation [5, 6]. High-deformability pipeline steel with polygonal ferrite and bainite (PF + B) is the optimum dual phase microstructure to guarantee the safety of oil pipeline transportation [1]. Studies have indicated that the coexistence of soft and hard phases improved strain-hardening behavior, which improved its strength and ensured its deformability [7]. The key factors affecting the strength and plasticity of dual phase steel included phase volume fraction, grain size, and phase distribution morphology [8,9,10,11,12]. However, there was a significant contradiction in the influence of phase content on strength–plasticity. Akbarpour and Ekrami [13] investigated the effect of bainite content on the strength and plasticity of PF + B dual phase steel. Results showed that the yield strength and tensile strength of the experimental steel gradually increased and plasticity gradually decreased with the bainite increasing. The work of Zhang et al. [14] showed that the yield strength and tensile strength of PF + B dual phase steel exhibited a two-stage variation rule with the bainite increasing. When the bainite content was less than 40%, the strength decreased with the increase in bainite volume fraction. When bainite was the matrix, the strength increased with the increase in bainite volume fraction. Whether bainite or ferrite as the matrix, its plasticity gradually decreased with the increase in bainite. Tang et al. [15] reported that as the volume fraction of bainite increased, the strain hardening exponent and uniform elongation of PF + B dual phase steel first increased and then decreased, and the yield strength ratio showed the opposite trend, reaching the maximum value when the bainite volume fraction was about 30%. The contradictory research results should be reduced by the inability to control other variables (like grain size and texture) during the research process.

The finite element simulation method based on the actual microstructure characteristics of microstructure can be used to analyze the influence of a single variable on strength and plasticity. Ishikawa et al. [16] studied the plastic deformation behavior of PF + B dual phase steel with bainite volume fraction of 16% and 46%, respectively. The digital image correlation (DIC) method and representative volume element (RVE) method were used to characterize and simulate the stress/strain during the deformation process. However, their results ignored the influence of crystallographic parameters of each phase. Liu et al. [17] established a crystal plasticity finite element model (CPFEM) based on the measured crystallographic parameters and grain size distribution characteristics of ferrite and martensite (F/M) dual phase steel, and the influence of a single texture feature on strength-plasticity in F/M dual phase steel was analyzed. The results indicated that the initial crystal orientation significantly influenced the localization of shear strain and damage behavior.

In this work, PF + B dual phase steel with different bainite contents (18–53 vol.%) was obtained by the thermal–mechanical controlling process. The 3D-CPFEM were established according to the actual microstructure characterization to analyze the effect of single bainite variable on the microstructural stress–strain distribution and strength-plasticity during deformation.

2 Experimental

2.1 Materials

The experimental steel used in this study was smelted in a vacuum induction melting furnace and cast into ingots. Subsequently, a billet with the size of 300 mm × 100 mm × 260 mm was obtained through free forging. The chemical composition of the experimental steel is shown in Table 1. Ar1 and Ar3 temperatures were determined as 660 and 740 °C, respectively.

Table 1 Chemical compositions of experimental steel (wt.%)
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The experimental steels with different phase contents were obtained through a hot rolling process. The original thickness of the steel billet was 80 mm. The billet was reheated to 1200 °C for 2 h, and then was rolled to the thickness of 32 mm through 8 passes. Controlling the relaxation termination temperature obtained PF + B dual phase steel with different bainite volume fractions. Setting the relaxation termination temperature (RTT) after rolling at the dual phase zone (between Ar1 and Ar3), the specific temperatures were 720, 700, 680, and 660 °C, respectively. Subsequently, the billets were cooled to 360 °C at a cooling rate of ~ 15 °C /s, and finally air cooled to room temperature. The specific rolling process route is shown in Fig. 1.

Fig. 1
figure 1

TMCP process for preparation of PF + B dual phase steel with different phase contents

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2.2 Microstructural characterization and mechanical property test

The specimens were obtained from the rolling billet by cut machine, and then subjected to grinding, polishing and etching (4 vol.% Nital solution for about 15 s). Microstructure was observed along the rolling direction (RD)–normal direction (ND) plane by optical microscopy (OM) and scanning electron microscopy (SEM). 10 OM images were selected for calculating the volume fraction of the bainite through ImageJ software. The electron backscatter diffraction (EBSD) analysis tests were conducted by Zeiss Gemini SEM 300 field emission scanning microscopy (FE-SEM). The specimens were electropolished at − 20 °C in an electrolyte solution containing 10 vol.% perchloric acid and 90 vol.% alcohol with a TenuPol-5 electrolytic polishing machine, and the voltage and scanning step used for EBSD data collection were 20 kV and 0.15 μm, respectively.

The dog-bone-shaped tensile specimens with gauge length of 20 mm and diameter of 3 mm were prepared along the rolling direction. The room temperature tensile tests were conducted on the SANS-CMT 5205 tensile machine at a strain rate of 1.0 × 10–3 s−1. The tensile fracture morphologies were observed by SEM.

Nanoindentation experiments were conducted on bainite and polygonal ferrite phases to study the micromechanical properties. The load–displacement curve to describe the changes in the material during compression under external forces, and the loading–unloading curve was used to analyze the experimental results. Firstly, the nanoindentation test specimens was electropolished and positioned using a Vickers hardness tester. Subsequently, nanoindentation experiments were conducted on the polygonal ferrite and bainite corresponding to the positioning area. The test was conducted on the Triboundeter nanoindentation mechanical tester, using a Berkovich triangular conical indenter with a curvature radius of approximately 100 nm. The maximum loading force during the experiment was 4000 µN, with loading time of 5 s and a constant loading rate of 0.5 mN/s.

2.3 Crystal plasticity model

The crystal plasticity (CP) method based on single crystals has been successfully applied to predict the mechanical response of polycrystalline materials. The deformation amount F in single crystal deformation can be expressed as Eq. (1) [18]:

$$ F = \frac{\partial x}{{\partial X}} $$
(1)

where X and x represent the initial and current positions of particles in a single crystal, respectively. F can be decomposed into plastic shear deformation (Fp) and elastic deformation (Fe). Fp is mainly related to crystal rotation, slip, and twinning, while Fe is mainly caused by lattice stretching and lattice distortion, and it is assumed that Fe is not affected by crystal slip. However, Fp cannot be represented by simple parameters, but rather by plastic deformation rate gradients [19]:

$$ {\dot{F}}_{{\text{p}}}=L{F}_{{\text{p}}}$$
(2)

where L is the rate gradient [20], and it is related to the slip systems and twinning initiated during internal deformation of grains, which can be expressed as:

$$ L = \mathop \sum\limits_{\alpha = 1}^{N} {\mathop \gamma \limits^{ \cdot } }^{(\alpha )} m^{(\alpha )} \otimes n^{(\alpha )} $$
(3)

where α represents slip systems; \({\dot{\gamma }}^{(\alpha )}\) represents the shear rate of α slip systems; \({m}^{(\alpha )}\) and \({n}^{(\alpha )}\) represent the united vectors of slip direction and the normal direction of slip plane; and N is the number of the activated slip systems. \({\dot{\gamma }}^{(\alpha )}\) is mainly determined by the initial shear rate (\({\dot{\gamma }_{0}}\)), and the relationship of the shear stress on this slip system (\({\tau }_{0}^{\left(\alpha \right)}\)) and the critical shear stress of the slip system to initiate movement (\({\tau }_{{\text{c}}}^{\left(\alpha \right)}\)) can be determined as:

$$ {\dot{\gamma }}^{(\alpha )} = {\dot{\gamma }} \left( {\frac{{\tau_{0}^{(\alpha )} }}{{\tau_{{\text{c}}}^{(\alpha )} }}} \right)^{\frac{1}{m}} {\text{sign}}\left( {\tau_{0}^{(\alpha )} } \right) $$
(4)

where m is the strain rate sensitivity index and is determined as 0.5 for this material. \({\dot{\gamma }_{0}}\) is related to the type of phase in the microstructure. In each slip system, \({\tau }_{{\text{c}}}^{\left(\alpha \right)}\) is defined by the Piola–Kirchhoff stress, and it can be calculated using Eq. (5):

$$ \dot{\tau }_{\text{c}}^{(\alpha )} = \mathop \sum \limits_{\beta }^{N} {h_{\alpha \beta } } \dot{\gamma }^{(\beta) } $$
(5)

where β represents slip systems; \({h}_{\alpha \beta }\) is the strain hardening rate; and \({\dot{\gamma }}^{(\beta) }\) is the shear rate of β slip systems. Accumulated shear strain of all slip systems (γ) can be expressed as:

$$ \gamma = \sum\limits_{\alpha = 1}^{N} {\int_{0}^{t} {\left| {{\dot{\gamma }} ^{(\alpha )} } \right|} } {\text{d}}t $$
(6)

where t is the loading time. During the plastic deformation process of polycrystals, there is deformation heterogeneity within and among grains. The orientation inside the grains will gradually change with the deformation process, and coordinated deformation was formed among grains through the interaction between slip and grain boundaries. This led to the transformation of the grain orientation that was originally easy to deform into an orientation not conducive to deformation, and the unfavorable orientation became a favorable orientation. In addition, while dislocations were formed inside the grains, the sharpening of the dislocation envelope will gradually increase the orientation difference at the sub-grain boundaries. As the density of dislocations increased, entanglement occurred among dislocations, leading to an increase in material strength and a decrease in plasticity, and resulting in strain hardening of the material. This caused the critical shear stress initiated by the slip system in the crystal to increase with the increase in strain, and the plastic deformation process of the material was closely related to its strain hardening rate. hαβ of the material can be expressed by the following equation:

$$ h_{\alpha \beta } = q_{\alpha \beta } \left[ {h_{{0}} \left( {1 - \frac{{\tau_{0}^{(\beta )} }}{{\tau_{{\text{c}}}^{(\beta )} }}} \right)^{a} } \right] $$
(7)

Thus, the critical shear stress of the slip system can be expressed as:

$$ \tau_{{\text{c}}}^{(\alpha )} = \tau_{0}^{(\alpha )} + \int_{0}^{t} {q_{\alpha \beta } } \left[ {h_{0} \left( {1 - \frac{{\tau_{0}^{(\beta) } }}{{\tau_{{\text{c}}}^{(\beta) } }}} \right)^{\alpha } } \right]\dot{\gamma }^{{^{(\beta )} }} {\text{d}}t $$
(8)

where \({\tau }_{0}^{(\beta )}\) and \({\tau }_{{\text{c}}}^{(\beta) }\) refer to the shear stress of the β slip system and the critical shear stress of the slip system initiation, MPa, respectively; \({h}_{0}\) and a are the initial hardening modulus parameter between among slip systems; and \({q}_{\alpha \beta }\) is the hardening coefficient among different slip systems, which is self-hardening in this model and always has a value of 1. In the present study, the slip systems considered in the simulation were mainly {110}<111> and {112}<111> , with a total of 24 slip systems. In summary, the material parameters that need to be determined during simulation are summarized in Table 2.

Table 2 Material parameters in phenomenological power-law model
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3 Results and discussion

3.1 Microstructures

The microstructures with different RTTs are presented in Fig. 2. The volume fraction of bainite decreased as RTT decreased. The long relaxation time contributed to enhancing the driving force of austenite to ferrite transformation. Thus, ferrite experienced a long nucleation and growth time, leading to a decrease of bainite.

Fig. 2
figure 2

Microstructure of PF + B dual-phase steel with different relaxation termination temperatures. a 660 °C; b 680 °C; c 700 °C; d 720 °C

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The volume fraction of bainite was obtained by counting 10 metallographic microstructure images. The evolution law of bainite content with RTT is presented in Fig. 3a, and the variety rule of bainite content was consistent with the results observed in Fig. 2. The bainite content was determined as 18% (660 °C), 27% (680 °C), 35% (700 °C), and 53% (720 °C) in different RTTs, respectively. In order to minimize the effect of grain size, areas on each hot-rolled steel plate with small difference in average grain size were selected for subsequent research. The grain size distribution at the selected location is shown in Fig. 3b. The results indicated that the average grain size of polygonal ferrite in the selected area was within 5–7 μm. For convenience, the specimens with different bainite contents were named as PF + 18%B, PF + 27%B, PF + 35%B, and PF + 53%B, respectively.

Fig. 3
figure 3

Evolution rule of bainite volume fraction (a) and grain size (b) of PF after different relaxation termination temperatures

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3.2 Mechanical properties

Figure 4a shows the engineering stress–strain curves of four specimens. There are no obvious yield plateau and it presents a continuous yield characteristic. The variation rule of yield strength, tensile strength, total elongation, and uniform elongation of each specimen are shown in Fig. 4b. The specific mechanical properties are shown in Table 3. As the volume fraction of bainite increased, the yield strength gradually increased, but when the bainite content was below 35%, the tensile strength did not change much. The total elongation and uniform elongation showed a trend of increasing first and then decreasing with the increase in bainite content, reaching the maximum value when the bainite was 35%. The strain hardenability of steel was usually represented by the strain hardening index (n). The higher the value of n, the higher the strain hardenability of the steel. The strain hardening index can be fitted to lnσ–lnε of the uniform plastic deformation stage based on the Hollomon equation (Eq. (9)). The curve was obtained as shown in Eq. (10) [21]:

$$ \sigma = K\varepsilon^{n} $$
(9)
$$ \ln \sigma = \ln K + n\ln \varepsilon $$
(10)

where σ is true stress, MPa; ε is true strain; and K is the strength coefficient, MPa.

Fig. 4
figure 4

Engineering stress–strain curves (a) and tensile properties (b) of experimental steels

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Table 3 Mechanical properties of four specimens
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As shown in Fig. 4b, the variation rule of the strain hardening index was consistent with that of elongation. When the bainite content was 35%, the strain hardening index also reached its maximum value, indicating that the strain hardenability of the steel was optimal at this time. According to the variation law of comprehensive mechanical properties, the steel has the optimal strength–plasticity matching when the bainite reached 35%.

In ferrite + martensite dual-phase steel, the elongation usually showed a gradual decrease as the martensite increased [22]. High martensite content could induce martensite to bear greater strain during deformation, thereby increasing the rate of micro-void initiation and making it more prone to unstable propagation, and finally resulting in fracture. In general, the strain will preferentially transfer to the bainite as the bainite increases in PF + B dual phase steel, leading to a decrease in its plasticity. However, in this study, the elongation did not show a monotonic decrease trend, but reached its peak at a bainite volume fraction of 35%. This result was similar to the research results of Tang et al. [15]. Nonetheless, the existing studies have not focused on the influence of volume fraction on the microstructure deformation characteristics and plastic damage behavior during the deformation process. In order to clarify the reason for achieving the optimal strength plastic matching when the bainite content was 35%, the present study investigated the plastic deformation damage mechanism of PF + B dual phase steel by 3D-CPFEM analysis.

3.3 Establishment of 3D-CPFEM

Based on EBSD data, a 3D-CPFEM was established. The grain in bainite with high angle grain boundary (> 15°) was defined as an effective grain that affected deformation behavior [23, 24]. The grain size distribution of polygonal ferrite and bainite is shown in Fig. 5a. The 3D-CPFE models with bainite volume fractions of 18%, 35%, and 53% were constructed using DREAM 3D software [25], as shown in Fig. 5b and c, with a model size of 100 μm × 100 μm × 5 μm. Figure 5b presents the phase distribution map, and Fig. 5c shows the crystal orientation distribution (COD) map. The orientation distribution map (Fig. 5e) of bainite and polygonal ferrite in the model was established, which had similar characteristics to the pole map (Fig. 5d) measured in PF + 18%B specimen. During simulation, the material was loaded along x direction at a loading rate of 1 × 10–3 s−1, with loading time of 250 s.

Fig. 5
figure 5

Parameters for CPFE model and corresponding 3D-CPFE model. a Effective grain size distribution; b phase distribution map; c crystal orientation distribution; d reality pole figures of PF and bainite; e pole figures of established models

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3.4 Determination of material parameters

To determine the mechanical properties of polygonal ferrite and bainite phase in multiphase materials, the nanoindentation method was used. The stress–strain curve of the corresponding phase can be obtained through reverse inference. In the present study, the combination of 3D-CPFE and nanoindentation inverse method was used to obtain the material constitutive parameters required for the crystal plasticity finite element simulation of the studied PF + B dual phase steel. The entire process is shown in Fig. 6.

Fig. 6
figure 6

Material parameters calibration process

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The plastic behavior of many pure and alloy metals can be approximated by the power law, and the true stress–strain relationship can be represented by Eq. (11) [26]:

$$ \sigma = \left\{ {\begin{array}{*{20}c} {E\varepsilon } & {\sigma \le \sigma_{{\text{y}}} } \\ {\sigma_{{\text{y}}} (1 + E\varepsilon_{{\text{p}}} /\sigma_{{\text{y}}} )^{n} } & {\sigma > \sigma_{{\text{y}}} } \\ \end{array} } \right. $$
(11)

where εp is the strain during the plastic deformation stage; σy is the yield strength, MPa; and E is elastic modulus, GPa.

The load–displacement curves of PF and bainite are shown in Fig. 7a. Figure 7b shows the typical indentation response curve (P–h curve), where P is the load, and h is displacement parameter, in which the relevant parameters that determine mechanical properties and stress/strain characteristics are described. The load–displacement curve is considered to follow Kick's power law, P = Ch2, where C depends on the elastic–plastic material properties and the geometry of the indenter [27]. At the maximum depth hm or maximum load Pm, the average contact pressure Pave = Pm/A (A is the true contact area measured at hm) can be considered as the hardness (H) of the indentation material. After unloading, the slope of P–h was initially linear, where Pu was the unloading force. When fully unloaded (zero load), the remaining depth was hr. The area below the loading and unloading curves represented the total work of loading Wt and elastic unloading We, respectively. Therefore, the area enclosed by the loading and unloading curves was the plastic work (Wp) during the indentation process, Wp = Wt − We.

Fig. 7
figure 7

Mechanical properties and model of bainite and polygonal ferrite single-phase. a Load–displacement curves; b typical nanoindentation curve; c reverse calculation of true stress–strain curve; d CPFE model

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During the nanoindentation experiment, n value of a material was easily obtained from the relationship between Wp and Wt (or hr, hm), and the degree of elastic rebound will depend on the degree of hardening. The yield strength of materials was generally obtained by analyzing the curvature C during the loading process. The research [28] indicated that σ0.033 was independent of strain hardening, where Er is contact modulus, and σ0.033 can be presented in Eq. (12):

$$ \frac{C}{{\sigma_{{{0}{\text{.033}}}} }} = - 1.131\left[ {\ln \left( {\frac{{E_{{\text{r}}} }}{{\sigma_{{{0}{\text{.033}}}} }}} \right)} \right]^{3} + 13.635\left[ {\ln \left( {\frac{{E_{{\text{r}}} }}{{\sigma_{{{0}{\text{.033}}}} }}} \right)} \right]^{2} - 30.594\left[ {\ln \left( {\frac{{E_{{\text{r}}} }}{{\sigma_{{{0}{\text{.033}}}} }}} \right)} \right] + 29.267 $$
(12)

The strain hardening index can be obtained from Eq. (13):

$$ \begin{aligned} \frac{{1{\text{d}}P_{u} }}{{E_{{\text{r}}} h_{{\text{m}}} {\text{d}}h}}\left| {h_{{\text{m}}} } \right. & = ( - {1}{\text{.40557}}n^{{3}} { + 0}{\text{.77526}}n^{{2}} { + 0}{\text{.15830}}n - {0}{\text{.06831}})\left[ {\ln \left( {\frac{{E_{{\text{r}}} }}{{\sigma_{{{0}{\text{.033}}}} }}} \right)} \right]^{3} + \\ & ({17}{\text{.93006}}n^{{3}} - {9}{\text{.22091}}n^{{2}} - {2}{\text{.37733}}n{ + 0}{\text{.86295}})\left[ {\ln \left( {\frac{{E_{{\text{r}}} }}{{\sigma_{{{0}{\text{.033}}}} }}} \right)} \right]^{2} + \\ & ({ - 79}{\text{.99715}}n^{{3}} { + 40}{\text{.55620}}n^{{2}} { + 9}{\text{.00157}}n - {2}{\text{.54543}})\left[ {\ln \left( {\frac{{E_{{\text{r}}} }}{{\sigma_{{{0}{\text{.033}}}} }}} \right)} \right] + \\ & ({122}{\text{.65069}}n^{{3}} - {63}{\text{.88418}}n^{{2}} - {9}{\text{.5893}}n{ + 6}{\text{.20045}}) \\ \end{aligned} $$
(13)

The calculated strength and strain hardening index of bainite and PF are shown in Table 4.

Table 4 Key parameters for reverse analysis of true strain–stress curves of PF and bainite
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The calculated true stress–strain curves of bainite and polygonal ferrite are shown in Fig. 7c, and CPFEM of bainite and polygonal ferrite used for simulation is shown in Fig. 7d. By adjusting the constitutive parameters, it can be determined that when the material constitutive parameters were set values as shown in Table 5, and the simulation results can be well matched with the calculated results (Fig. 8).

Table 5 Material parameters in phenomenological power-law model for bainite and ferrite
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Fig. 8
figure 8

Experiment and simulation stress–strain curves of polygonal ferrite (a) and bainite single phase (b)

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3.5 Stress/strain distribution during deformation

Figure 9 shows the stress/strain distribution behavior of PF + B dual-phase model with different bainite contents at tensile strains of 5% and 25% conditions. As the tensile strain increased, a significant stress/strain gradient appeared in the model. In the initial stage of deformation, the strain was mainly distributed in polygonal ferrite, and as the strain continued to increase, there would also be a certain strain distribution in bainite. During the tensile deformation process, obvious strain concentration characteristics appeared. The strain concentration at B/PF interface was defined as Cint, and the strain concentration in polygonal ferrite was defined as CP.

Fig. 9
figure 9

Stress/strain distribution behavior of PF + B dual-phase model with different bainite contents at tensile strains of 5% and 25% of PF + 18%B model (a, b, e, f, i, j), PF + 35%B model (c, d, g, h, l) and PF + 53%B model (m–r). a–d, m, n Phase distribution maps; e–h, o, p strain distribution maps

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When the strain increased to 25%, both Cint and Cp points were observed (Fig. 9c–h, o–p). The content of bainite has a significant impact on the strain concentration characteristics. When the volume fraction of bainite was 18%, its strain concentration was mainly located in the polygonal ferrite, and a small amount of Cint points were observed (Fig. 9e, f). When the volume fraction of bainite increased to 35% (Fig. 9g, h), the strain concentration points at the phase interface significantly increased. When bainite reached 53% (Fig. 9o, p), only a few strain concentration points in polygonal ferrite can be observed, while more linear strain concentration features appeared near the phase interface. As the volume fraction of bainite increased, the strain concentration in polygonal ferrite decreased, while the strain concentration at the phase interface increased. This indicated that the phase content was an important factor affecting the form of strain concentration behavior [29, 30].

Figure 10 shows the strain distribution rule in bainite and polygonal ferrite during tensile deformation. The variations of average strain and interphase strain difference during tensile deformation are shown in Fig. 11. As the tensile strain increased, the strain distribution of polygonal ferrite and bainite showed a normal distribution curve that shifted to the right as a whole, indicating that the strain distribution in two phases gradually increased. The average strain curves showed a linear increase pattern, and the average strain of polygonal ferrite was higher than that of bainite. However, there was a certain strain distribution in both phases, indicating a coordinated deformation behavior during tensile test. As the bainite content increased, the average strain in the bainite and polygonal ferrite phases gradually increased. During the tensile deformation, when the volume fraction of bainite was below 35%, increasing the content of bainite did not affect the strain difference between two phases. However, when the content of bainite reached 53%, the strain difference increased significantly.

Fig. 10
figure 10

Strain distribution behavior in polygonal ferrite (a–c) and bainite (d–f). a, d PF + 18%B model; b, e PF + 35%B model; c, f PF + 53%B model

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Fig. 11
figure 11

Variations of average strain (a) and interphase strain difference (b) under different tensile strains

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The distribution and variation of stress are different. Figure 12 shows the stress distribution rule in polygonal ferrite and bainite during tensile deformation. Figure 13 shows the variations of average stress and interphase stress difference under different tensile strains. According to Figs. 9, 12, and 13, there was a significant stress gradient in polygonal ferrite and bainite, and the stress distribution in bainite was much higher than that in polygonal ferrite. As the tensile strain increased, the stress in polygonal ferrite gradually increased, but the maximum stress was still located in bainite. As the tensile strain increased, in addition to the bainite bearing stress, the polygonal ferrite also burdened a certain amount of stress due to strain hardening. Under the same strain conditions, as the bainite increased, the stress in bainite increased, and the peak stress distribution also increased. The stress difference between phases gradually increased with the increase in bainite content, but when the bainite content reached 53% and the tensile strain reached 25%, the increased amplitude was significantly increased.

Fig. 12
figure 12

Mises stress distribution behavior in polygonal ferrite (a–c) and bainite (d–f). a, d PF + 18%B model; b, e PF + 35%B model; c, f PF + 53%B model

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Fig. 13
figure 13

Variations of average Mises stress (a) and interphase stress difference (b) under different tensile strains

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The average strain of bainite and polygonal ferrite increased simultaneously, which contributed to the decrease of the strain difference among phases. In this case, the stress of both phases gradually increased under different tensile strains, indicating that a coordinated deformation behavior and strong strain hardening existed simultaneously. Therefore, when polygonal ferrite was the matrix (bainite content < 50%) in PF + B dual phase steel, moderately increasing the bainite content will enhance the coordinated deformability, and this excellently coordinated deformability increased the average stress/strain of both phases simultaneously, thereby maximizing the plasticity of the material, which was beneficial to improving the strength and plasticity of the material.

However, when the bainite content reached 53%, the average strain in polygonal ferrite further increased. In contrast, the average strain increment in bainite significantly decreased, which induced uneven deformation of bainite and polygonal ferrite. Meanwhile, the average stress increase in polygonal ferrite was minimal, while the average stress increase in bainite was significant, further exacerbating the strength difference between the two phases. Strain and stress were distributed separately in polygonal ferrite and bainite, reducing the coordinated deformation ability of two phases, and easily causing interphase plastic damage.

4 Conclusions

  1. 1.

    When the volume fraction of bainite was 35%, PF + B dual phase steel showed the optimal strain hardenability and strength plasticity matching.

  2. 2.

    The 3D-CPFE analysis results indicated that with the increase in bainite content from 18% to 35%, the interphase coordinated strain hardening did not continuously increase the strain difference, and the stress distribution in polygonal and bainite phases increased simultaneously, improving the coordinated deformability, which was beneficial to improving the strength and plasticity.

  3. 3.

    When the bainite content reached 53%, the deformability of ferrite was limited, exacerbating the non-uniformity of deformation between phases, increasing the interphase strain/stress difference, reducing the coordinated deformability between polygonal ferrite and bainite, and easily causing interphase plastic damage.