**4. Discussion**

Most analytical models used for plotting FLDs detect the forming limits at the necking point of materials, but calculating major and minor limit strains by numerical simulations is still required. In this study, these limiting strain values were detected from logarithmic strain tensor components, where the maximum value inside the tensor matrix represents the major strain, and minimum value is minor strain. Logarithmic strain is widely used in numerical simulations and is the best indicator for true strain, as its ability to count the strain values at every time interval and gives us an exact number when deformation occurs by a series of increments [31]. The strain tensor of this matrix represents an imaginary square unit, where minor strain is the minimum logarithmic change value in the unit square, and major strain is the maximum logarithmic change in this unit.

The increasing value of ultimate tensile strength by increasing phase fraction of the martensite is because of the enhanced reinforcement by uniformly distributed hard martensite particles [32]. On the other hand, the reduced capability of the material to undergo global strains with more martensite fraction is due to the brittle nature of martensite and

the comparatively less force shifted to the ferrite grains. Furthermore, the martensite's decreased grain size provokes the surrounding ferrite's plastic deformation; therefore, the DP steel sample with a small martensite grain size showed comparatively better formability [33].

The behavior of FLDs agrees with the results of Duancheng Ma [30], where it applies on the right side of FLD with and without periodic boundary conditions, as shown in Figure 5 for the M-K approach. On the other hand, in the K-B approach, the simulations, as shown in Figure 5b, agree with Basak [8], where results are presented for DP steels 600, 800, and 1000, and G Béres [12], who worked on aluminum alloy Al 2008-T4. In Figure 5a, the FLDs with a relative difference in trends influenced by martensite fraction compared to those in Figure 5b, where no significant difference was observed, establish the argument that the M-K approach corresponds better to the crystal plasticity-based modeling approach. Furthermore, this effect was observed to be more pronounced in the case of using the M-K approach to evaluate the influence of variation in the martensite grain size, as shown in Figure 8.

Keeler–Brazier's equations did not illustrate the difference of formability by changing martensite grain sizes. Furthermore, they did not show a significant necking band, as the damage values at plane strain loading conditions were not altered, as shown in Figures 8 and 10. On the other hand, the damage values' dependency on engineering stress–strain curves can clarify the 0.2% proof stress points, onset of necking, and fracture for the M-K approach.

In addition, n-values were calculated for both models using Equation (6).

$$
\sigma\_t = \mathbf{k} \,\,\varepsilon\_t^n \tag{6}
$$

where σ<sup>t</sup> is the true stress, k is a constant, E<sup>t</sup> is the true strain, and n is the strain hardening coefficient. These n-values can keep their higher strength values in the pre-necking zones, reducing the risk of local strain accumulations and uniformly distributing the strain over the whole domain, improving materials' formability. True stress–strain curves of DP steels and the plastic work performed by biaxial and uniaxial loading using crystal plasticity models show a good agreement with Equations (7) and (8) used during experimental work.

$$
\sigma\_t = \sigma\_a \text{ (}1 + \epsilon\_a\text{)}\tag{7}
$$

$$
\varepsilon\_t = \ln\left(1 + \varepsilon\_{\mathfrak{e}\_{\mathfrak{e}}}\right) \tag{8}
$$

where σ<sup>t</sup> is the true stress, E<sup>t</sup> is the true strain, σ<sup>e</sup> is the engineering stress, and E<sup>e</sup> is the true strain. These equations are based on the ISO 16842 standard for studying biaxial tensile testing on sheet metals [34].

The local damage behavior of the DP steel samples upon varying loading conditions showed different outcomes, which can help understand the corresponding effect of microstructure. This is also affected by the morphology of the martensite particles, their orientation, aggregation, and presence in small islands. The initiation and propagation of local damage at the martensite–ferrite grain interface are because of the decohesion of the comparatively weak point in the aggregate [35,36]. The direction of the damage propagation in the case of tensile loading in ductile ferrite matrix, i.e., at 45 degrees, is caused by the shear bands [29]. A further extensive study is needed to understand this effect by considering the contributing factors.

#### **5. Conclusions**

The dependence of microstructural attributes on the formability limit of DP steels was analyzed. Specifically, the averaged global behavior and the effect of different loading conditions on damage initiation were investigated. Several 3D RVEs with varying microstructural attributes were simulated using a crystal plasticity-based numerical simulation model called DAMASK. The global and local stress, strain, and damage evolution

of various RVEs revealed the internal phenomena during deformation. The global results were processed to obtain FLDs according to K-B and M-K criteria. Appealing outcomes were observed, which can be summarised by the following points.


**Author Contributions:** Conceptualization, T.H. and F.Q.; methodology, T.H.; software, T.H., F.Q. and M.U.; validation, T.H., F.Q. and S.G.; formal analysis, M.U.; investigation, T.H. and F.Q.; resources, F.Q. and S.G.; data curation, T.H.; writing—original draft preparation, T.H., M.U. and F.Q.; writing review and editing, F.Q., M.U. and S.G.; visualization, T.H.; supervision, F.Q. and S.G.; project administration, S.G. and U.P.; funding acquisition, T.H. and U.P. All authors have read and agreed to the published version of the manuscript.

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

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The simulation data are not publicly available but can be shared upon request.

**Acknowledgments:** The authors acknowledge the DAAD Faculty Development for Candidates (Balochistan), 2016 (57245990)-HRDI-UESTP's/UET's funding scheme in cooperation with the Higher Education Commission of Pakistan (HEC) for sponsoring the stay of Faisal Qayyum at IMF TU Freiberg. This work was conducted with the DFG-funded collaborative research group TRIP Matrix Composites (SFB 799). The authors gratefully acknowledge the German Research Foundation (DFG) for the financial support of SFB 799. Furthermore, Freunde und Förderer der TU Bergakademie Freiberg e.V. is acknowledged for providing financial assistance to Muhammad Umar. The authors also acknowledge the support of Martin Diehl and Franz Roters (MPIE, Düsseldorf) for their help regarding the functionality of DAMASK. Finally, the competent authorities at Khwaja Fareed University of Engineering and Information Technology, (KFUEIT) Rahim Yar Khan, Pakistan, and TU BAF Germany are greatly acknowledged for providing research exchange opportunities to Muhammad Umar at the Institute of Metal Forming TU BAF, Germany, under a memorandum of understanding (MoU).

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

## **Nomenclature**


#### **Appendix A.**

#### *Appendix A.1. Running Simulation via CP Spectral Solver*

The numerical simulations were processed by incorporating ductile damage and recording more increments after the damage initiation than those without damage in each RVE. During numerical processing by the spectral method, each grid point inside the mesh of the RVE acts as a computation point and represents individually defined deformation mechanisms, phase fractions, grain orientation, and homogenization schemes. Each increment after damage initiation records the material degradation behavior in detail. Once the damage is initiated, the numerical processing slows down and becomes intensively computed, and crashes after specific material degradation occurs for the RVE. The simulations and the load increments in each loading condition in this work were processed till converging.

#### *Appendix A.2. Postprocessing Stage*

The completed numerical simulations were post-processed using customized subroutines on DAMASK [37]. FLDs were plotted by adopting major and minor strains. The following two approaches are generally accepted, and commonly employed techniques based on some equations and engineering stress and strain curves to adopt the major and minor forming limit strains as forming limit criteria.

#### Appendix A.2.1. M-K Approach

For the M-K approach, after plotting engineering stress and strain curves, the maximum stress (localized necking point) was detected, and the strain tensors were recorded along with the increments. Major and minor values were detected from the strain tensor matrix at an increment, corresponding to this maximum stress in the stress–strain curve, as shown in the following process chart (see Figure A1).

#### Appendix A.2.2. K-B Approach

The Keeler-Brazier model depends on the major true strain value FLD0, true on the thickness of the virtual samples, and the strain hardening coefficient when the minor true strain equals zero [13]. It presents Equations (A1)–(A3) to predict the FLDs following the activities, as shown in Figure A2, where major true strain E<sup>1</sup> is dependent on values of thickness t and strain hardening coefficient n.

For E<sup>2</sup> < 0 (uniaxial case)

$$
\varepsilon\_1 = \text{FLD}\_{0, \text{ true}} - \varepsilon\_2 \tag{A1}
$$

For E<sup>2</sup> > 0 (biaxial case)

$$e\_1 = \ln[0.6 \times (\exp(\varepsilon\_2) - 1) + \exp(\text{FLD}\_{0,\text{true}})] \tag{A2}$$

**Figure A2.** Process chart of plotting FLD via Keeler-Brazier (K-B) approach.

values

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values (ᡅ1)

2. Umar, M.; Qayyum, F.; Farooq, M.U.; Khan, L.A.; Guk, S.; Prahl, U. Analyzing the cementite particle size and distribution in heterogeneous microstructure of C45EC steel using crystal plasticity based DAMASK code. In Proceedings of the 2021 International Bhurban Conference on Applied Sciences and Technologies (IBCAST), Islamabad, Pakistan, 12–16 January 2021; p. 6. [CrossRef]

