5.2.4. Twin Volume Fraction

The magnitude of shear strain alone does not present a complete quantitative estimation of twinning. A complete estimation requires the magnitude of the volume of twin in the overall plasticity. In view of this requirement, the twin volume fraction is estimated during the course of deformation, as represented through Figure 12. In conjunction with twin systems' activity, the twin volume fraction represents the ratio of active to total number of twin planes. A significant variation of twin volume is observed under tension and compression in all steels and crystallographic directions. As in the case of shear strain, the twin volume fraction becomes nearly steady for all crystallographic directions of TWIP 1 and 2, but not for TWIP 3, after a certain equivalent strain. The magnitude of twin volume fraction for TWIP steels is presented in Table 6. As is evident, the largest magnitude of twin volume is observed in the [100] crystallographic direction under tension and compression for all three TWIP steels. Overall, the twin volume fraction is not exceeded by 0.333 (33.33 %) in all directions and steels. This shows that the contribution of twinning in the plasticity of TWIP steels 1, 2, and 3 is limited to 33.33 % for a single crystal.


**Table 6.** Highest values of twin volume fraction for TWIP 1, 2, and 3 steels under tension and compression.

**Figure 12.** Twin volume fraction of a single crystal of TWIP steels subjected to tension and compression.

#### **6. Conclusions**

A numerical scheme is developed and implemented for modeling the elastic-plastic deformation behavior of twinning-induced plasticity steel, in which mechanical twinning contributes significantly to plasticity along with a crystallographic slip. Initially, a constitutive formulation of equations, reported in earlier work, is briefly discussed. Afterward, a numerical integration procedure is established by identifying primary variables, discretizing constitutive equations in the time domain, developing an iterative scheme, and introducing a time sub-stepping algorithm. A numerical integration scheme is then incorporated in finite element software ABAQUS through a user-defined material subroutine. Finite element models of single-crystal and polycrystal material points are developed and simulation results are compared with the published experimental observations. They are in close accord with the maximum error of 16.15% in TWIP steels for equivalent stress. However, the error becomes higher beyond 0.3 strain. This puts limitations on the current model to be implemented, with more accuracy, at high strain deformation. This must be further explored in future work. In addition, further simulations are executed to quantify slip and twin systems' activity, deformation behavior, shear strain pattern, and twin volume fraction of three TWIP steels subjected to uniaxial tension and compression. The slip and twin systems' activity shows that the slip contribution is higher for all crystallographic directions in all steels; however, the highest level of twin activity (number of active twin systems) is observed in the [100] direction. In addition, a higher magnitude of stress at 0.45 equivalent strain is observed in tension (1150 to 1300 MPa) than compression (780 to 850 MPa) in all polycrystal TWIP steels. It may indicate a more prominent role of slip and twin systems' reorientation and interactions in tension. Moreover, the twin shear strain becomes nearly constant for all crystallographic directions of TWIP 1 and 2 after 0.2 equivalent strain, but not for TWIP 3. It is also found that the fraction of twin volume is not surpassed by 33.33% for all directions and TWIP steels. The development and implementation of a fully implicit numerical integration scheme in modeling twinning-induced plasticity provide an effective platform to estimate the deformation behavior of TWIP steels. The current work can be enhanced to incorporate martensitic phase transformation and damage criterion in a coupled slip, twinning, and transformation-induced plasticity model.

**Author Contributions:** R.K.: conceptualization, methodology, investigation, and writing—original draft preparation; T.P.: supervision, and writing—review and editing; A.A.: validation and visualization; S.Z.Q.: supervision, formal analysis; S.M.: software and visualization. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** The authors acknowledge the support of Imam Mohammad Ibn Saud Islamic University for this research.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A. Slip and Twin Systems of Austenite Crystal**

Face-centered cubic (FCC) austenite crystal's slip and twin systems are represented in terms of Miller indices and unit vectors in Tables A1 and A2, respectively.


**Table A1.** Slip systems of FCC crystal.


**Table A2.** Twin systems of FCC crystal.

#### **References**

