Introduction

As a high-performance alloy, Twinning-Induced Plasticity (TWIP) steel with strong work hardening, high strength and good ductility due to austenite twinning during plastic deformation has received considerable attention in auto body manufacturing (Ref 1-4).

Mechanical twinning plays a key role in the plastic deformation of TWIP steels, and the deformation mechanism is closely related to the stacking fault energy (SFE), which is very sensitive to the content of chemical elements in material. For example, the added Mn (Ref 5), C (Ref 6) and N (Ref 7) contents can increase SFE and promote the mechanical twinning while Si (Ref 8, 9), Al (Ref 10-12) and Cu (Ref 13) additions have the opposite effects. However, the mechanical twinning only occurs in the special SFE range from 18 to 45 mJ/m2 (Ref 14). Martensite transformation is the dominant mechanism when SFE is lower than 18 mJ/m2 while dislocation glide controls the strain hardening process of material when SFE exceeds 45 mJ/m2. In addition, the value of SFE is also sensitive to the deformation temperature of material, so are the deformation mechanism (Ref 14-19), flow stress and work hardening (Ref 20-22).

The deformation mechanisms of TWIP steels are strongly affected by the strain rate and stress state of material. Considering strain rate sensitivity, the strain hardening behaviors of TWIP steels were studied by conducting experiments under various strain rates (from 10−4 to 103 s−1) (Ref 23-34). Portevin-Le Châtelier (PLC) bands (Ref 23, 24) often appear during the deformation process. The flow stress and density of deformation twins and dislocations of material decrease with increasing the deformation temperature (Ref 25-27). TWIP steels show negative strain rate sensitivity of the strain hardening (Ref 25, 28-32) due to the twinning rate (Ref 26, 30) and show the suppression of dislocations and deformation twins at high strain rate (Ref 29). However, the TWIP steels with Al addition have positive strain rate sensitivity (Ref 28, 33). Considering the influence of the stress state, Jacques et al. (Ref 34) studied the work hardening and mechanical twinning of TWIP steel under uniaxial tension, simple shear and rolling processes, respectively. The results show that the twinning rate, the number of activated twinning systems in each grain, twin thickness and transmission of twins across grain boundaries are dependent on the imposed stress state.

Based on the shear-band theory, a mechanical twinning model was developed in this paper with the stress state of material considered. Furthermore, the effects of stress state on the strain hardening behaviors of TWIP steel (Fe-20Mn-1.2C) are analyzed using the dislocation-based model.

Theoretical Model

Twinning Model

With the assumption that the operative shear bands can be in the form of mechanical twins (Ref 35, 36), the relationship between the twinning volume fraction and the effective plastic strain can be described as:

$$dF_{\text{tw}} = \left( {F_{{{\text{tw}},\hbox{max} }} - F_{\text{tw}} } \right)A_{f} d\varepsilon \quad \left\{ {\begin{array}{*{20}l} {\varepsilon \le \varepsilon_{\text{ini}} } \hfill & {F_{\text{tw}} = 0} \hfill \\ {\varepsilon\, > \varepsilon_{\text{ini}} } \hfill & {F_{\text{tw}} \ne 0} \hfill \\ \end{array} } \right.$$
(1)

where \(F_{\text{tw}}\) is the volume fraction of twinning, \(F_{{{\text{tw}},\hbox{max} }}\) is the maximum volume fraction of twinning, \(\varepsilon_{\text{ini}}\) is the initial plastic strain when twinning is triggered (Ref 17), ɛ is the effective plastic strain, and A f is the material parameter with the following form:

$$A_{f} = \alpha \eta r\left( {1 - f_{\text{sb}} } \right)\left( {f_{\text{sb}} } \right)^{r - 1} P + \dot{g}\frac{{\eta \left( {f_{\text{sb}} } \right)^{r} }}{{\sqrt {2\pi } }}\exp \left[ { - \frac{1}{2}\left( {\frac{{g - \bar{g}}}{{\sigma_{g} }}} \right)^{2} } \right]$$
(2)

where η is geometric constant, and r is material constant.

The volume fraction of the shear-band \(f_{\text{sb}}\) can be described as:

$$df_{\text{sb}} = (1 - f_{\text{sb}} )\alpha d\varepsilon$$
(3)

where α is a stress state-dependent parameter reflecting the rate of shear-band formation and has the following form:

$$\alpha = \alpha_{0} + \alpha_{1} \sum$$
(4)

where α 0 and α 1 are material parameters. \(\Sigma = \frac{{\sigma_{1} + \sigma_{2} + \sigma_{3} }}{{3\bar{\sigma }}}\) is stress state (Ref 36).

The probability parameter P is assumed to be a Gaussian distribution of twinning probability and has the form of cumulative probability distribution function:

$$P = \frac{1}{{\sqrt {2\pi } \sigma_{g} }}\int_{ - \infty }^{g} {\exp \left( { - \frac{1}{2}\left( {\frac{{g^{{\prime }} - \bar{g}}}{{\sigma_{g} }}} \right)^{2} } \right)} dg^{\prime}$$
(5)

where \(\bar{g}\) and σ g are dimensionless mean value and standard deviation of a given probability distribution function, respectively.

The twinning driving force g relating to the deformation temperature and stress state is defined as:

$$g = g_{0} + g_{1} \sum$$
(6)

where g 0 and g 1 are material parameters.

Dislocation-Based Model

To take into account the contribution of twinning to the total plastic shear strain, a mixture law was adopted (Ref 17):

$$d\gamma = \left( {1 - F_{\text{tw}} } \right)d\gamma_{g} + \gamma_{t} dF_{\text{tw}}$$
(7)

where γ is the effective shear strain of material, γ g is the shear strain of dislocation gliding, and γ t is the twinning shear strain.

The Taylor factor M links the effective plastic strain ɛ to the effective shear strain γ as follows:

$$d\gamma = Md\varepsilon$$
(8)

Combining Eq 1, 8, and 9, the relationship between the shear strain of dislocation gliding γ g and the effective plastic strain ɛ can be deduced as:

$$\frac{{d\gamma_{g} }}{d\varepsilon } = \frac{M}{{F_{tw,\hbox{max} } - F_{tw} }} - \gamma_{t} A_{f}$$
(9)

The relation between the flow stress and the dislocation density of twinning or glide area is assumed to obey the classical form (Ref 3, 4):

$$\sigma_{i} = \sigma_{0i} + \chi MGb\sqrt {\rho_{i} }$$
(10)

where σ 0i is the initial flow stress of twinning or glide area, χ is material constant, and M is Taylor factor. G and b are shear modulus and Burgers vector, respectively. ρ i is the dislocation density of twinning or glide area.

The flow stress σ of TWIP steels is assumed to be the weighted sum of the stress of twinning and glide:

$$\sigma = F_{\text{tw}} \sigma_{\text{tw}} + \left( {1 - F_{\text{tw}} } \right)\sigma_{g}$$
(11)

According to Mecking-Kocks theory (Ref 37, 38), the evolution of dislocation density for glide or twinning area is the competition result of the generation and annihilation of dislocation:

$$\left( {\frac{{d\rho_{i} }}{{d\bar{\gamma }_{g} }}} \right) = \left( {\frac{{d\rho_{i} }}{{d\bar{\gamma }_{g} }}} \right)^{ + } + \left( {\frac{{d\rho_{i} }}{{d\bar{\gamma }_{g} }}} \right)^{ - }$$
(12)

The generation of dislocation density can be expressed as:

$$\left( {\frac{{d\rho_{i} }}{{d\bar{\gamma }_{g} }}} \right)^{ + } = \frac{1}{b}\left( {\frac{1}{{d_{i} }} + k_{i} \sqrt {\rho_{i} } } \right)$$
(13)

where d i is the grain size or the average twins spacing, and k i is the corresponding dislocation generation parameter.

The annihilation of dislocation density can be written as:

$$\left( {\frac{{d\rho_{i} }}{{d\bar{\gamma }_{g} }}} \right)^{ - } = - f_{i} \rho_{i}$$
(14)

where f i is the dislocation annihilation parameter.

Combining Eq 9, 12-14, the relationship between the effective plastic strain ε and the dislocation density ρ can be deduced as:

$$\frac{{d\rho_{i} }}{d\varepsilon } = \frac{1}{b}\left( {\frac{1}{{d_{i} }} + k_{i} \sqrt {\rho_{i} } - f_{i} \rho_{i} } \right)\left( {\frac{M}{{1 - f_{\text{tw}} }} - \gamma_{t} A_{f} } \right)$$
(15)

Material

The steel used in this paper to verify the proposed model is the austenitic Fe-20Mn-1.2C steel (Ref 34, 39, 40). It was hot-rolled from the cast ingot down to 3.5 mm in thickness, then was cold rolled to 1.5-mm-thick sheet and finally was annealed for recrystallization prior to mechanical testing and microstructure characterization. For comparison, the measured loads and elongations were converted into equivalent Von Mises stresses and strains. The rolling stress-strain relationship was established by estimating the equivalent strain evolution from the thickness reduction in the pre-strain samples and the equivalent stress of these samples from their yield strength measured in tension or their Vickers hardness (Ref 34). The tested equivalent stress-equivalent strain curves are illustrated in Fig. 1. The mechanical tests including simple shear, uniaxial tension and rolling are applied, and the corresponding mechanical properties are listed in Table 1. The model parameters have been fitted to describe the experimental results as accurately as possible, and the corresponding values are summarized in Tables 2 and 3 for the twinning model and the strain hardening model, respectively.

Fig. 1
figure 1

Equivalent stress-equivalent strain curves

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Table 1 Mechanical properties of Fe-20Mn-1.2C TWIP steel with different loading (Ref 34, 39, 40)
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Table 2 Material parameters for twinning model
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Table 3 Material parameters for strain hardening model
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Results and Discussion

Model Verification

The twinning model is developed based on the shear-band framework, in which the shear-band volume fraction and twinning driving force are controlled by stress state, Σ (Here we defined Σ = −1 for rolling, Σ = 0 for shear and Σ = 0.33 for tension).

The results from the experimental and calculated data are shown in Fig. 2, which indicate that twins appear earlier in shear than that in rolling and tension (in Fig. 3), while the twinning rate is slower than that of tension and seems to saturate at a smaller value (Ref 34), and the twinning volume fraction increases with Σ at large plastic strain (Ref 41).

Fig. 2
figure 2

Verification of twinning volume fraction with different stress state

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Fig. 3
figure 3

Twinning distribution at 0.1 strain (TEM): (a) Tension, (b) Shear

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The strain hardening behaviors including stress-strain curve, hardening rate and strain hardening exponent are presented in Fig. 4(a)-(c) (Ref 34). The results show that the strength and failure strain in tension are higher than those in rolling and shear (see Fig. 4a), which is due to the twinning volume fraction as shown in Fig. 2. However, the multi-axial compression can improve the strength of material and lead to a higher strength and hardening rate in rolling than that in shear (in Fig. 4a and b). The trend of the strain hardening exponent curve under different stress states is similar to the trend of the stress-strain curve as shown in Fig. 4(c).

Fig. 4
figure 4

Verification of strain hardening behaviors under different stress states. (a) Stress-strain curve, (b) hardening rate and (c) strain hardening exponent

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Twinning and Glide Effects on Strain Hardening

The strain-induced deformation twins are preferentially formed in a local deformation region (Ref 1), the thin twin lamella divides the original grains as strain increases (Ref 2, 3), and the dislocation piles up at the twin boundaries (Ref 42). With the help of Eq 15, the dislocation density of twinning and glide and the general dislocation density of material are calculated with results shown in Fig. 5. The results show that the general dislocation mainly depends on the dislocation glide at initial deformation stage, and then, the dislocation density of twinning area increases with austenite twining (Ref 43).

Fig. 5
figure 5

Dislocation evolution of material, twinning and glide

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The effects of twinning and dislocation glide on the strain hardening behaviors of material under different stress states are shown in Fig. 6. The results indicate that the strain hardening of TWIP steel mainly depends on glide at the beginning. The effects of twinning on strain hardening become dominant when the twinning volume fraction reaches about 5% under different stress states (in Fig. 2), and its influence increases evidently with increasing strain.

Fig. 6
figure 6

Strain hardening of material, twinning and glide. (a) Stress-strain curve, (b) hardening rate and (c) strain hardening exponent

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Stress State Effects on Strain Hardening Behaviors

The effects of stress state on the strain hardening behaviors of TWIP steel are studied. Four kinds of stress state are given, i.e., Σ =  0.33 (uniaxial compress), Σ = 0.33 (uniaxial tension), Σ = 0.58 (plane stretching) and Σ = 0.67 (biaxial stretching). The calculated parameters under different stress states are similar to those of uniaxial tension so that the influence of stress state on the strain hardening behaviors of TWIP steel can be determined. The twining volume fraction, twining parameters (α, g and P) and strain hardening behaviors are shown in Fig. 7, 8, 9, and 10, respectively.

Fig. 7
figure 7

Twinning volume fraction under different stress states (Σ = −0.33, 0.33, 0.58 and 0.67)

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Fig. 8
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Parameters α and g as a function of stress state

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Fig. 9
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Parameter P as a function of stress state

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Fig. 10
figure 10

Strain hardening prediction of TWIP steels under different stress states. (a) Stress-strain curve, (b) hardening rate and (c) strain hardening exponent

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The results indicate that: (1) the increased Σ benefits austenite twinning (in Fig. 7). (2) Possible explanation is that the increased Σ can improve the twinning probability P by influencing the shear-band intersection and twinning drive force (in Fig. 8 and 9).

More austenite twinning due to the larger Σ can lead to higher dislocation density in twinning area, which can promote the strength (in Fig. 10a), the hardening rate (in Fig. 10b) and the strain hardening exponent (in Fig. 10c) of material. In general, the stress state can improve the strain hardening behaviors of TWIP steel.

As a nonlinear function of strain, the twinning rate increases quickly with increasing strain and then decreases after reaching the maximal value. The change in twinning rate is controlled by stress state and a larger value of stress state Σ results in higher twinning rate. Finally, twinning rates of different stress state reach a similar value at large strain (in Fig. 11).

Fig. 11
figure 11

Twinning rate under different stress states

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Conclusions

The effects of stress state on the strain hardening behaviors of Fe-20Mn-1.2C TWIP steel are studied in this paper using a developed dislocation-based model. Based on the analysis above, it can be concluded that:

  1. 1.

    The twinning model is presented by considering the influence of stress state on the shear-band intersection, the volume fraction and driving force of twinning, respectively. Meanwhile, the strain hardening model is established by taking the effects of austenite twinning on the flow stress of twinning and glide area into account. The proposed models are verified with the experimental results of Fe-20Mn-1.2C TWIP steel obtained from shear, tension and rolling tests, respectively.

  2. 2.

    The strain hardening of twinning and glide area is evaluated utilizing the modeled results. The results indicate that the strain hardening of TWIP steel mainly depends on glide at the beginning. The effects of twinning on strain hardening are dominant when the twinning volume fraction of different stress state is about 5% (in Fig. 2), and its influence increases evidently with increasing strain.

  3. 3.

    The effects of stress state (Σ = −0.33, 0.33, 0.58 and 0.67) on the strain hardening of TWIP steel are predicted. The results indicate that the stress state benefits austenite twinning, and higher austenite twinning due to the increased stress state can promote the strain hardening behaviors of TWIP steel.