# Damage Analysis of Third-Generation Advanced High-Strength Steel Based on the Gurson–Tvergaard–Needleman (GTN) Model

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## Abstract

**:**

^{TM}) through microstructural and X-ray diffraction (XRD) analyses, uniaxial tensile and plane-strain tension testing, and numerical simulations. The damage behavior of this steel is described with the Gurson–Tvergaard–Needleman (GTN) model using an identification procedure based on the uniaxial tensile and initial microvoids data. The microstructure of the CR980XG3

^{TM}steel is composed of ferrite, martensite–austenite islands, and retained austenite with a volume fraction of 12.2%. The global formability of the CR980XG3

^{TM}steel, namely the product of the uniaxial tensile strength and total elongation values, is 24.3 GPa%. The Lankford coefficient shows a weak initial plastic anisotropy of the CR980XG3

^{TM}steel with the in-plane anisotropy close to zero (−0.079) and the normal anisotropy close to unity (0.917). The identified GTN parameters for the CR980XG3

^{TM}steel provided a good forecast for the limit strains defined according to ISO 12004-2 standard from the uniaxial tensile and plane-strain tension data.

## 1. Introduction

_{2}emissions, the high fossil fuel prices harm the global economy. Thus, reducing the dependence on fossil fuels by increasing the efficiency and sustainability of new passenger cars and light commercial vehicles became a priority [1]. The European Union agreed to set a reduction target of 37.5% CO

_{2}for vehicles by 2030 with the 2021 baseline [2]. In response to the European Commission proposal, the European Automobile Manufacturers Association (ACEA) advocates a realistic target to reduce 20% of CO

_{2}emissions [3]. In this context, the advanced high-strength steels (AHSS) have gained considerable importance in the automotive industry. This steel class has been developed to satisfy the requirements of vehicle performance and weight optimization. Several AHSS steel grades can be applied in the vehicle design, each strategically placed in the Body-in-White (BIW) to increase the crashworthiness performance. The excellent compromise between uniaxial tensile strength and elongation allows manufacturing of the BIW components with reduced thickness, producing lighter vehicles resulting in fuel economy and reducing greenhouse gas emissions.

^{TM}steel. The 980XG3

^{TM}steel exhibits an excellent combination of strength and ductility. According to the Japanese industrial standard (JIS), Hance and Link [13] found an ultimate tensile strength of 1017 MPa and a total elongation of 25% using the specimen Nº 5. Stamping tests were carried out with the 980XG3

^{TM}steel to manufacture components with reduced thickness for the Fiat Chrysler Automobiles [14].

^{TM}steel sheet plastic behavior between uniaxial tensile and plane-strain tension deformation modes.

## 2. Materials and Methods

^{TM}steel investigated in this work is a commercially available product being delivered to us as an uncoated cold-rolled sheet with a nominal thickness of 1.58 mm. The chemical composition and processing parameters are not given hereafter to respect the confidentiality requirements of the steel producer.

#### 2.1. Microstructural Analysis

^{TM}steel was performed following standard metallographic preparation procedures. The mechanical grinding, conventional polishing, followed by electropolishing using a Struers A2 electrolyte, were used to reveal the as-received microstructure. The sample preparation was completed by electrolytic polishing and etching using Struers LectroPol-5 automatic equipment operated at 35 V. The micrographs of the as-received CR980XG3

^{TM}steel were obtained with the scanning electron microscope Hitachi model SU-70.

#### 2.2. Mechanical Testing

^{TM}steel sheet were assessed from uniaxial tensile tests. The digital image correlation (DIC) system manufactured by GOM coupled with the ARAMIS-5M software (GOM GmbH, Braunshweig, Germany) measured the strain fields. The specimens were manufactured by CNC (Computer Numerical Control) milling machining MIKRON VCE 500 (Mikron Machining, Agno, Switzerland) from guillotined sheet strips. The dog-bone specimen dimensions are depicted in Figure 2.

^{−3}s

^{−1}. The strain measurements were recorded with an image acquisition rate of 1 frame/s.

#### 2.3. Microvoid Analysis

#### 2.4. GTN Damage Model Parameters Calibration

^{TM}steel was performed by finite element simulations of the uniaxial tensile test. Here, the idea is to vary the set of parameters to obtain a predicted nominal stress–strain curve close to the experimental uniaxial tensile test results, including post-necking behavior. Figure 5 shows the schematic geometrical model proposed to simulate the uniaxial tensile test. The simulation test was conducted by fixing one grip section and applying a prescribed constant speed at the other grip section. The quasi-static numerical simulations were performed with the ABAQUS/explicit commercial finite element code, wherein the GTN model is available as porous metal plasticity. In all numerical simulations, the specimen has meshed with the C3D8R element type: 8-nodes linear brick, reduced integration, and hourglass control [38]. The functionality Element Deletion was selected in the finite element ABAQUS mesh module. This option removes the finite element when it loses the strength capacity completely.

## 3. Results

#### 3.1. As-Received Microstructure

^{TM}steel for the as-received condition is shown in Figure 7, in which the peaks are identified as $\alpha $-iron or $\gamma $-iron.

^{TM}steel sheet is 12.2%, which was very close to the value of ~12% reported for the Q&P 980 steel [19,20].

#### 3.2. Mechanical Properties

^{TM}steel determined in the rolling direction are the Young modulus, $E$ = 195 ± 5 GPa, and Poisson’s ratio $\nu $ = 0.289 ± 0.003. The plastic behavior of the CR980XG3

^{TM}sheet was evaluated from the yield stress defined at 0.2% of the plastic-strain offset (${S}_{y}$), ultimate tensile strength (${S}_{u}$), uniform elongation (${e}_{u}$), total elongation (${e}_{t}$), and Lankford’s anisotropy coefficient ($r$-value). The $r$-value was defined as the slope between the width and the thickness strains between yield strength and the maximum uniform plastic strain values. Table 2 resumes the average mechanical properties of the CR980XG3

^{TM}steel as a function of the angular orientation and the corresponding standard deviation values.

^{TM}steel is classed as a current third-generation AHSS, as shown in Figure 8. Its ultimate tensile strength (1040 MPa) and total elongation (23.4%) allow reaching 24.3 GPa% as performance in the global formability diagram, which is close to the Q&P results reported in the literature [11,19,20,39]. The quenching and partitioning steels are still under the desired performance target, namely 30 and 35 GPa% (Figure 1). However, they significantly outperform the well-established first-generation AHSS, such as DP and TRIP.

^{TM}steel sheet can be regarded as isotropic given that the values found for $\Delta r$ and $\overline{r}$ are close to zero and unity, respectively. However, in Figure 9a, one can observe deviations in the uniaxial tensile engineering stress–strain behavior at 0°, 45°, and 90° in-plane angular orientations. Figure 9b further emphasizes this observed anisotropic behavior wherein the CR980XG3

^{TM}steel would present small amplitudes of earing formation in the cup-drawing test. The formability of the CR980XG3

^{TM}steel is close to that of DP600, with higher strength. Figure 9c shows the yield stress and strength values normalized by the corresponding data at the rolling direction. The yield stress behavior presents a monotonic increase of about 11% between 0° and 90° orientations, whereas the ultimate tensile strength varies by less than 5%.

^{TM}steel sheet was fitted according to the Hollomon, Ludwik, Swift, and Voce work-hardening equations. Table 3 presents the average values of the fitted work-hardening parameters along with the corresponding goodness-of-fit (R

^{2}) values. The Ludwik work-hardening equation corresponds to the first term of the Johnson–Cook plasticity model, which is widely adopted in finite element simulations [42]. Alternatively, the pre-strain term ${\epsilon}_{0}$ in Swift’s equation can be used to describe the initial plastic yielding or a prior deformation process as, e.g., from a cold-rolling skin pass. The best fits were obtained with the Swift and Voce work-hardening equations with R

^{2}equal to 0.996 and 0.999, respectively.

#### 3.3. Limit Strains

#### 3.4. Void Results

#### 3.5. Full Set of GTN Damage Model Parameters

#### 3.5.1. Effective Work Hardening and Initial Porosity

^{TM}steel sheet. The corresponding $A$ and $B$ values in Equation (12) are also indicated in Figure 13a. The predicted stress–strain curves of Swift and Voce work-hardening equations obtained from Equation (11) are compared with the experimental true-stress and true plastic strain in Figure 13b. Both work-hardening equations show good agreement with the experimental uniaxial tensile behavior observed in the uniform plastic-strain domain. Despite presenting a better fit, Voce’s equation provides saturation of the work hardening for larger strains. Using the GTN damage model, such plastic behavior will rapidly lose the load-bearing capacity at the necking onset. In this way, the Swift work-hardening equation was selected to describe the behavior of the fully dense matrix since it provides a better fit to the experimental stress–strain data when considering the softening effect. From the result presented in Figure 13a and Equations (12)–(14), the initial porosity was estimated as: $r{d}_{0}=\left(1-{0.00232}^{3/2}\right)=0.99988$.

#### 3.5.2. Yield Locus Parameters (${q}_{1}$, ${q}_{2}$, ${q}_{3}$)

^{TM}steel sheet are ${q}_{1}=1.74$, ${q}_{2}=0.83$, and ${q}_{3}={q}_{1}^{2}=3.03$.

#### 3.5.3. Void Nucleation Parameters (${\epsilon}_{N}$, ${S}_{N}$, ${f}_{N}$)

#### 3.5.4. Failure Parameters (${f}_{C}$, ${f}_{F}$)

^{TM}steel under the uniaxial tension deformation mode was ${f}_{F}$ = 0.095. Thus, the identified parameters of the GTN damage model are presented in Table 7.

#### 3.5.5. Mesh Sensitivity

#### 3.6. Numerical Predictions of the In-Plane Limit Strains

^{TM}steel sheet behavior under sheet metal forming processes conditions with ${\epsilon}_{2}\le 0$.

## 4. Discussion

^{TM}steel was evaluated using mechanical tests and microstructural analysis to calibrate the parameters of the GTN damage model. The numerical predictions of limit strains showed that the selected GTN model parameters provided a similar behavior to the experimentally observed results, representing the left side of the forming limit curve. This good agreement can be ascribed to the low planar anisotropy behavior of the CR980XG3

^{TM}steel. In this context, the methodology proposed in this work proved to be capable of predicting the macro-mechanical behavior of a recent third-generation AHSS. Many researchers in the literature have already introduced important extensions to the GTN damage model, some more widespread and adopted than others. According to Bergo et al. [44], contributions aimed at solving the GTN model’s limitation of correctly predicting material failure under low stress triaxiality, as well as extensions related to incorporating different microvoid features, such as the shape, orientation, and rotation, have been added to the GTN model by several authors in the last 20 years.

^{TM}steel between the uniaxial and plane-strain tension deformation modes.

## 5. Conclusions

^{TM}steel sheet from mechanical tests and finite element modeling. The material presented a global formability ${S}_{u}\times {e}_{t}$ = 24.3 GPa% within the region known as the current third-generation advanced high-strength steels. This excellent formability results from its multiphase microstructure composed of martensite, ferrite, and retained austenite. From the performed testing procedures and proposed finite element modeling, the main conclusions are summarized as follows:

- The standard mechanical testing provided the average mechanical properties, namely the yield stress (604 MPa), ultimate tensile strength (1040 MPa), and total elongation (23.4%). The tested material presents a weak initial plastic anisotropy, with a planar anisotropy close to zero (−0.079) along with the normal anisotropy coefficient close to unity (0.917).
- In the as-received state, the XRD analysis provided a retained austenite volume fraction of 12.2%, which, in turn, is prone to transform into martensite during the early stages of plastic straining.
- The CR980XG3
^{TM}steel provided an experimental Lankford r-value close to the unity, and thus isotropic plasticity could be assumed as a first approximation in the modeling. The identified damage parameters of the GTN model were able to reproduce the experimental load-elongation obtained from the uniaxial tensile test. The mesh sensitivity analysis also showed that the mesh size does not influence the finite element predictions in the uniform elongation domain. However, as the necking appears, the smaller the mesh size, and hence the deformation is more localized. A mesh size of 0.4 mm in the gauge length zone was enough to fit the experimental data. - A simple methodology for calibrating the parameters of the GTN model was performed based on the adopted mechanical testing and finite element simulations. The calibration method provided the complete set of the GTN damage model parameters for the CR980XG3
^{TM}steel, namely $r{d}_{0}$ = 0.99988, ${q}_{1}$ = 1.74, ${q}_{2}$ = 0.83, ${\epsilon}_{N}$ = 0.18, ${S}_{N}$ = 0.07, ${f}_{N}$ = 0.035, ${f}_{C}$ = 0.05, and ${f}_{F}$ = 0.095. Moreover, the calibrated GTN parameters provided an excellent forecast for the experimental limit strains located on the left-hand side of the forming limit curve.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Council of European Union–European Parliament. Regulation (EU) Nº 333/2014 of European Parliament of the Council of 11 March 2014 amending Regulation (EC) Nº 443/2009 to define the modalities for reaching the 2020 target to reduce CO
_{2}emissions from new passenger cars. Off. J. Eur. Union**2014**, 103, 15–21. [Google Scholar] - Mock, P. CO
_{2}Emission Standards for Passenger Cars and Light Commercial Vehicles in European Union. ICCT—The International Council on Clean Transportation. 2019. Available online: https://theicct.org/publication/co2-emission-standards-for-passenger-cars-and-light-commercial-vehicles-in-the-european-union/ (accessed on 9 December 2021). - ACEA—European Automobile Manufacturers Association. The European Commission’s Proposal on Post-2021 CO
_{2}Targets for Cars and Vans. ACEA European Automobile Manufacturers Association, March 2018. Available online: https://www.acea.auto/publication/position-paper-european-commission-proposal-on-post-2021-co2-targets-for-cars-and-vans/ (accessed on 9 December 2021). - Matlock, D.K.; Speer, J.G. Chapter 11—Third Generation of AHSS: Microstructure Design Concepts. In Proceedings of the International Conference on Microstructure and Texture in Steel and Other Materials, Jamshedpur, India, 5–7 February 2008; Haldar, A., Suwas, S., Bhattacharjee, D., Eds.; Springer: Berlin/Heidelberg, Germany, 2008; pp. 185–205. [Google Scholar]
- Grajcar, A.; Kuziak, R.; Zalecki, W. Third generation of AHSS with increased fraction of retained austenite for the automotive industry. Arch. Civ. Mech. Eng.
**2012**, 12, 334–341. [Google Scholar] [CrossRef] - Aydin, H.; Essadiqi, E.; Jung, I.-H.; Yue, S. Development of 3rd generation AHSS with medium Mn content alloying compositions. Mater. Sci. Eng. A
**2013**, 564, 501–508. [Google Scholar] [CrossRef] - Speer, J.; Matlock, D.K.; De Cooman, B.C.; Schroth, J.G. Carbon partitioning into austenite after martensite transformation. Acta Mater.
**2003**, 51, 2611–2622. [Google Scholar] [CrossRef] - Yi, H.L.; Sun, L.; Xiong, X.C. Challenges in the formability of the next generation of automotive steel sheets. Mater. Sci. Technol.
**2018**, 34, 1112–1117. [Google Scholar] [CrossRef] - Sugimoto, K.-I.; Hojo, T.; Kobayashi, J. Critical assessment 29: TRIP-aided bainitic ferrite steels. Mater. Sci. Technol.
**2017**, 33, 2005–2009. [Google Scholar] [CrossRef] - Branagan, D.; Parsons, C.; Machrowicz, T.; Cischke, J.; Frerichs, A.; Meacham, B.; Cheng, S.; Justice, G.; Sergueeva, A. Effect of Deformation during Stamping on Structure and Property Evolution in 3rd Generation AHSS. Open J. Met.
**2018**, 8, 15–33. [Google Scholar] [CrossRef] [Green Version] - Wang, L.; Speer, J.G. Quenching and Partitioning Steel Heat Treatment. Met. Microstruct. Anal.
**2013**, 2, 268–281. [Google Scholar] [CrossRef] [Green Version] - Shen, I. General Motors Applies Third-Generation Advanced High-Strength Steel in New Vehicles for China—Open Innovation Enables Fast Adoption of Lightweight Technology. GM Corporate Newsroom, Shanghai, 1 January 2015. Available online: https://media.gm.com/media/cn/en/gm/news.detail.html/content/Pages/news/cn/en/2015/dec/1201_advanced-steel.html (accessed on 12 October 2021).
- Hance, B.M.; Link, T.M. Effects of fracture area measurement method and tension test specimen type on fracture strain values of 980 class AHSS. In Proceedings of the IOP Conference Series: Materials Science and Engineering, 38th International Deep Drawing Research Group Annual Conference, Enschede, The Netherlands, 3–7 June 2019; Volume 651, p. 012061. [Google Scholar] [CrossRef]
- Macek, B.; Lutz, J. Virtual and Physical Testing of Third-Generation High Strength Steel—Evaluating a New High-Strength Steel’s Ability to Improve on an Existing Stamped-Steel Production Part. SAE International—Automotive Engineering. 18 August 2020. Available online: https://www.sae.org/news/2020/08/stamping-ahss-tech-paper (accessed on 12 October 2021).
- Kempken, J. Chapter 16—Technological challenges and solutions for the production of state-of-the-art second-and third-generation AHSS grades. In Advanced High Strength Steel, Lecture Notes in Mechanical Engineering; Roy, T.K., Ed.; Springer: Dusseldorf, Germany, 2018; pp. 143–148. [Google Scholar]
- Paykani, M.A.; Shahverdi, H.R.; Merismaeilli, R. First and third generation of advanced high strength steel in FeCrNiBSi system. J. Mater. Process. Technol.
**2016**, 238, 256–265. [Google Scholar] - Hance, B. Advanced High Strength Steel (AHSS) performance level definitions and targets. SAE Int. J. Mater. Manf.
**2018**, 11, 505–516. [Google Scholar] [CrossRef] - Noder, J.; Gutierrez, J.E.; Zhumagulov, A.; Dykeman, J.; Ezzat, H.; Butcher, C. A Comparative Evaluation of Third-Generation Advanced High-Strength Steels for Automotive Forming and Crash Applications. Materails
**2021**, 14, 4970. [Google Scholar] [CrossRef] [PubMed] - Li, W.; Ma, L.; Peng, P.; Jia, Q.; Wan, Z.; Zhu, Y.; Guo, W. Microstructural evolution and deformation behavior of fiber laser welded QP980 steel joint. Mater. Sci. Eng. A
**2018**, 717, 124–133. [Google Scholar] [CrossRef] - Guo, W.; Wan, Z.; Peng, P.; Jia, Q.; Zou, G.; Peng, Y. Microstructure and mechanical properties of fiber laser welded QP980 steel. J. Mater. Process. Technol.
**2018**, 256, 229–238. [Google Scholar] [CrossRef] - Guo, W.; Wan, Z.; Jia, Q.; Ma, L.; Zhang, H.; Tan, C.; Peng, P. Laser weldability of TWIP980 with DP980/B1500HS/QP980 steels: Microstructure and mechanical properties. Opt. Laser Technol.
**2020**, 124, 105961. [Google Scholar] [CrossRef] - Pereira, A.B.; Santos, R.O.; Carvalho, B.S.; Butuc, M.C.; Vincze, G.; Moreira, L.P. The Evaluation of Laser Weldability of the Third-Generation Advanced High Strength Steel. Metals
**2019**, 9, 1051. [Google Scholar] [CrossRef] [Green Version] - Zhao, H.; Huang, R.; Sun, Y.; Tan, C.; Wu, L.; Chen, B.; Song, X.; Li, G. Microstructure and mechanical properties of fiber laser welded QP980/press-hardened 22MnB5 steel joint. J. Mater. Res. Technol.
**2020**, 9, 10079–10090. [Google Scholar] [CrossRef] - Sun, Q.; Lu, Y.; Chen, J. Identification of material parameters of a shear modified GTN damage model by small punch test. Int. J. Fract.
**2020**, 222, 25–35. [Google Scholar] [CrossRef] - Gurson, A. Continuum theory of ductile rupture by void nucleation and growth. Part I. Yield criteria and flow rules for porous ductile media. J. Eng. Mater. Technol.
**1977**, 99, 2–15. [Google Scholar] [CrossRef] - Tvergaard, V.; Needleman, A. Analysis of the cup-cone fracture in a round tensile bar. Acta Metall.
**1984**, 32, 157–169. [Google Scholar] [CrossRef] - Li, G.; Cui, S. A review on theory and application of plastic meso-damage mechanics. Theor. Appl. Fract. Mech.
**2020**, 109, 102686. [Google Scholar] [CrossRef] - Gholipour, H.; Biglari, F.; Nikbin, K. Experimental and numerical investigation of ductile fracture using GTN damage model on in-situ tensile tests. Int. J. Mech. Sci.
**2019**, 164, 105170. [Google Scholar] [CrossRef] - Abbasi, M.; Ketabchi, M.; Izadkhah, H.; Fatmehsaria, D.; Aghbash, A. Identification of GTN model parameters by application of response surface methodology. Procedia Eng.
**2011**, 10, 415–420. [Google Scholar] [CrossRef] [Green Version] - Kami, A.; Dariani, B.M.; Vanini, A.S.; Comsa, D.S.; Banabic, D. Numerical determination of the forming limit curves of anisotropic sheet metals using GTN damage model. J. Mater. Process. Technol.
**2015**, 216, 472–483. [Google Scholar] [CrossRef] - ASTM. Standard E975-13; Standard Practice for X-ray determination of retained austenite in steel with Near-Random Crystallographic Orientation. American Society for Testing and Materials; ASTM: West Conshohocken, PA, USA, 2013.
- ISO 12004-2:2007; Metallic Materials—Sheet and Strip—Determination of Forming Limit Curves—Part 2: Determination of Forming-Limit Curves in Laboratory. International Organization for Standardization: Geneva, Switzerland, 2007.
- Schwindt, C.D.; Stout, M.; Iurman, L.; Signorelli, J.W. Forming Limit Curve Determination of a DP-780 Steel Sheet. Procedia Mater. Sci.
**2015**, 8, 978–985. [Google Scholar] [CrossRef] [Green Version] - Samei, J.; Green, D.E.; Cheng, J.; Lima, M.S.D.C. Influence of strain path on nucleation and growth of voids in dual phase steel sheets. Mater. Des.
**2016**, 92, 1028–1037. [Google Scholar] [CrossRef] - Chaboche, J. Continuum damage mechanics: Present state and future trends. Nucl. Eng. Des.
**1987**, 105, 19–33. [Google Scholar] [CrossRef] - Santos, R.O.; da Silveira, L.B.; Moreira, L.P.; Cardoso, M.C.; da Silva, F.R.F.; Paula, A.D.S.; Albertacci, D.A. Damage identification parameters of dual-phase 600–800 steels based on experimental void analysis and finite element simulations. J. Mater. Res. Technol.
**2019**, 8, 644–659. [Google Scholar] [CrossRef] - Faleskog, J.; Gao, X.; Shih, C.F. Cell model for nonlinear fracture analysis—I. Micromechanics calibration. Int. J. Fract.
**1998**, 89, 355–373. [Google Scholar] [CrossRef] - Abaqus User’S Manual; Version 6.9; Dassault Systemes Simulia Corp.: Providence, RI, USA, 2009.
- Finfrock, C.B.; Clarke, A.J.; Thomas, G.A.; Clarke, K.D. Austenite Stability and Strain Hardening in C-Mn-Si Quenching and Partitioning Steels. Met. Mater. Trans. A
**2020**, 51, 2025–2034. [Google Scholar] [CrossRef] - Navarro-López, A.; Hidalgo, J.; Sietsma, J.; Santofimia, M.J. Characterization of bainitic/martensitic structures formed in isothermal treatments below the Ms temperature. Mater. Charact.
**2017**, 128, 111–127. [Google Scholar] [CrossRef] - Jatczak, C.F. Retained Austenite and Its Measurement by X-ray Diffraction. In SAE Technical Paper Series; SAE: Warrendale, PA, USA, 1980. [Google Scholar]
- Pantalé, O.; Gueye, B. Influence of the constitutive flow law in FEM simulation of the radial forming process. J. Eng.
**2013**, 2013, 231847. [Google Scholar] [CrossRef] [Green Version] - Slimane, A.; Bouchouicha, B.; Benguediab, M.; Slimane, S.-A. Parametric study of the ductile damage by the Gurson–Tvergaard–Needleman model of structures in carbon steel A48-AP. J. Mater. Res. Technol.
**2015**, 4, 217–223. [Google Scholar] [CrossRef] [Green Version] - Bergo, S.; Morin, D.; Hopperstad, O.S. Numerical implementation of a non-local GTN model for explicit FE simulation of ductile damage and fracture. Int. J. Solids Struct.
**2021**, 219–220, 134–150. [Google Scholar] [CrossRef] - Cao, T.-S.; Maire, E.; Verdu, C.; Bobadilla, C.; Lasne, P.; Montmitonnet, P.; Bouchard, P.-O. Characterization of ductile damage for a high carbon steel using 3D X-ray micro-tomography and mechanical tests—Application to the identification of a shear modified GTN model. Comput. Mater. Sci.
**2014**, 84, 175–187. [Google Scholar] [CrossRef] - Requena, G.; Maire, E.; Leguen, C.; Thuillier, S. Separation of nucleation and growth of voids during tensile deformation of a dual phase steel using synchrotron microtomography. Mater. Sci. Eng. A
**2014**, 589, 242–251. [Google Scholar] [CrossRef] - Maire, E.; Bouaziz, O.; Di Michiel, M.; Verdu, C. Initiation and growth of damage in a dual-phase steel observed by X-ray microtomography. Acta Mater.
**2008**, 56, 4954–4964. [Google Scholar] [CrossRef] - Chen, D.; Li, Y.; Yang, X.; Jiang, W.; Guan, L. Efficient parameters identification of a modified GTN model of ductile fracture using machine learning. Eng. Fract. Mech.
**2021**, 245, 107535. [Google Scholar] [CrossRef] - Saeidi, N.; Ashrafizadeh, F.; Niroumand, B.; Forouzan, M.; Barlat, F. Damage mechanism and modeling of void nucleation process in a ferrite–martensite dual phase steel. Eng. Fract. Mech.
**2014**, 127, 97–103. [Google Scholar] [CrossRef] - Morin, L.; Leblond, J.-B.; Kondo, D. A Gurson-type criterion for plastically anisotropic solids containing arbitrary ellipsoidal voids. Int. J. Solids Struct.
**2015**, 77, 86–101. [Google Scholar] [CrossRef]

**Figure 2.**Geometry, dimensions (mm) of the uniaxial tensile test specimen, and slices cut from the gauge length region for the void formation analysis.

**Figure 3.**Plane-strain tension specimen geometry and dimensions (mm) [33].

**Figure 4.**(

**a**) Representative volume element (RVE): effective stress measure and (

**b**) definition of the initial void volume fraction from the measured void area fraction [36].

**Figure 6.**SEM micrographs of the as-received CR980XG3

^{TM}steel sheet with (

**a**) 1000× and (

**b**) 10,000× highlighting the ferrite (F), martensite–austenite (MA), and retained austenite (RA).

**Figure 9.**Uniaxial tensile test results of the CR980XG3

^{TM}steel sheet: (

**a**) engineering stress–strain curves at 0°, 45°, and 90°, and angular evolutions of the (

**b**) Lankford r-values, (

**c**) normalized yield stress and ultimate tensile strength, and (

**d**) uniform and total elongation.

**Figure 10.**DIC measurements during the in-plane tests: (

**a**) major strain and (

**b**) minor strain fields obtained from the uniaxial tension specimen, and the selected (

**c**) major strain and (

**d**) minor strain fields from the double-notched plane strain specimen.

**Figure 11.**SEM micrographs (1500×) and corresponding binarized image of CR980XG3

^{TM}steel sheet in (

**a**,

**b**) as received, (

**c**,

**d**) 13% true strain, and (

**e**,

**f**) fractured condition.

**Figure 12.**Void analysis results: (

**a**) void density and void area fraction, (

**b**) void aspect ratio and mean void size.

**Figure 13.**(

**a**) Experimental void area fraction as a function of the true longitudinal plastic strain obtained for the CR980XG3

^{TM}steel sheet under uniaxial tension concerning the rolling direction and (

**b**) comparison between the experimental true-stress and true plastic-strain data and the predicted work-hardening curves in the rolling direction.

**Figure 15.**Major strain predictions of the uniaxial tensile test without the failure parameters: (

**a**) regions at the necking and far from the necking and (

**b**) corresponding major strain history.

**Figure 16.**Proposed identification procedure for the critical void volume fraction ${f}_{c}$-value: contours plots of the (

**a**) major strain and (

**b**) void volume fraction, and (

**c**) void volume fraction measures as a function of the major strain in the necking region.

**Figure 17.**Images of the uniaxial tensile specimen during the testing at (

**a**) the unloaded state, (

**b**) immediately before the fracture, and (

**c**) after the fracture.

**Figure 18.**(

**a**) Finite element simulation of the uniaxial tensile test: experimental digital image and predicted deformed specimen, and (

**b**) experimental uniaxial tensile test load elongation and predicted results obtained with the full set parameters of the GTN damage model for the CR980XG3

^{TM}steel.

**Figure 19.**Different mesh sizes in the gauge length region of the uniaxial tensile specimen: (

**a**) 1.6 mm, (

**b**) 0.8 mm, (

**c**) 0.4 mm, and (

**d**) 0.2 mm.

**Figure 21.**(

**a**) Experimental and predicted major and minor strains at the centerline section of the uniaxial tensile specimen, and (

**b**) limit strains from the uniaxial tensile test simulations using the GTN model.

**Figure 22.**(

**a**) Mesh refinement adopted in the flat double-notched specimen, and (

**b**) experimental true stress–strain curve of the plane strain test and predicted results obtained with the complete set of parameters of the GTN model.

**Figure 23.**(

**a**) Experimental and predicted major and minor strain in the centerline of the double-notched plane-strain sample and (

**b**) limit strains from the plane-strain tension test simulations using the GTN model.

**Figure 24.**Comparison between the experimental left-hand side of the forming limit curve and the numerical prediction determined for the CR980XG3

^{TM}steel sheet using flat specimens and the GTN model.

**Table 1.**Theoretical ${R}_{\alpha}$ and ${R}_{\gamma}$ using Cu radiation (data from [41]) and integrated intensity per angular diffraction.

Phase Index | hkl | 2θ | R_{index} | I_{index} |
---|---|---|---|---|

α | 110 | 44.6 | 233.8 | 6100.6 |

200 | 65.0 | 31.9 | 1615.1 | |

211 | 82.2 | 60.9 | 2643.9 | |

220 | 98.9 | 20.6 | 573.0 | |

γ | 111 | 43.6 | 182.8 | 2079.5 |

200 | 50.8 | 81.6 | 105.8 | |

220 | 74.6 | 44.4 | 214.1 | |

311 | 90.6 | 51.3 | 159.2 |

Orientation (θ) | ${\mathit{S}}_{\mathit{y}}\left(\mathbf{MPa}\right)$ | ${\mathit{S}}_{\mathit{u}}\left(\mathbf{MPa}\right)$ | ${\mathit{e}}_{\mathit{u}}(\%)$ | ${\mathit{e}}_{\mathit{t}}(\%)$ | r-Value |
---|---|---|---|---|---|

0° | 604 ± 7 | 1040 ± 9 | 18.0 ± 0.5 | 23.4 ± 0.2 | 0.861 ± 0.003 |

45° | 643 ± 4 | 1015 ± 8 | 17.9 ± 0.2 | 23.0 ± 0.6 | 0.957 ± 0.004 |

90° | 668 ± 7 | 1023 ± 8 | 16.9 ± 0.6 | 21.9 ± 1.3 | 0.895 ± 0.005 |

**Table 3.**Parameters of the work-hardening equations of the CR980XG3

^{TM}steel sheet determined from the uniaxial tensile tests performed at the rolling direction.

Work-Hardening Equation | $\mathit{K}$ (MPa) | $\mathit{n}$ | ${\mathit{\sigma}}_{\mathit{y}}\left(\mathbf{MPa}\right)$ | ${\mathit{\epsilon}}_{0}(\%)$ | R^{2} | |
---|---|---|---|---|---|---|

Hollomon | ${\sigma}_{hard\_eqn}=K{\left({\epsilon}^{p}\right)}^{n}$ | 1723 ± 9 | 0.187 ± 0.002 | - | - | 0.977 |

Ludwik | ${\sigma}_{hard\_eqn}={\sigma}_{y}+K{\left({\epsilon}^{p}\right)}^{n}$ | 1569 ± 11 | 0.386 ± 0.009 | 474 ± 11 | - | 0.993 |

Swift | ${\sigma}_{hard\_eqn}=K{\left({\epsilon}_{0}+{\epsilon}^{p}\right)}^{n}$ | 1880 ± 7 | 0.231 ± 0.002 | - | 0.69 ± 0.03 | 0.997 |

Voce | ${\sigma}_{hard\_eqn}={\sigma}_{y}+K\left(1-{\mathrm{e}}^{-n{\epsilon}^{p}}\right)$ | 643 ± 1 | 14.914 ± 0.065 | 634 ± 1 | - | 0.999 |

$\mathit{\theta}$ (deg) | Uniaxial Tension | Plane Strain | ||
---|---|---|---|---|

${\mathit{\epsilon}}_{2}$ | ${\mathit{\epsilon}}_{1}$ | ${\mathit{\epsilon}}_{2}$ | ${\mathit{\epsilon}}_{1}$ | |

0 | −0.1480 ± 0.0070 | 0.3240 ± 0.0200 | 0.0109 ± 0.0026 | 0.1525 ± 0.0062 |

90 | −0.1490 ± 0.0110 | 0.3200 ± 0.0280 | 0.0098 ± 0.0016 | 0.1345 ± 0.0234 |

Elastic Properties | Isotropic Effective Work Hardening (Swift) | |||||
---|---|---|---|---|---|---|

$\mathit{E}$ (MPa) | ν | $\mathit{K}$ (MPa) | ${\mathit{\epsilon}}_{0}$ | $\mathit{n}$ | A | B |

192,000 | 0.289 | 1880 | 0.0069 | 0.231 | 0.00762 | 0.00232 |

$\mathit{r}{\mathit{d}}_{0}$ | ${\mathit{q}}_{1}$ | ${\mathit{q}}_{2}$ | ${\mathit{q}}_{3}$ | ${\mathit{\epsilon}}_{\mathit{N}}$ | ${\mathit{S}}_{\mathit{N}}$ | ${\mathit{f}}_{\mathit{N}}$ |
---|---|---|---|---|---|---|

0.99988 | 1.74 | 0.83 | 3.03 | 0.18 | 0.07 | 0.035 |

$\mathit{r}{\mathit{d}}_{0}$ | ${\mathit{q}}_{1}$ | ${\mathit{q}}_{2}$ | ${\mathit{q}}_{3}$ | ${\mathit{\epsilon}}_{\mathit{N}}$ | ${\mathit{S}}_{\mathit{N}}$ | ${\mathit{f}}_{\mathit{N}}$ | ${\mathit{f}}_{\mathit{C}}$ | ${\mathit{f}}_{\mathit{F}}$ | Element Deletion |
---|---|---|---|---|---|---|---|---|---|

0.99988 | 1.74 | 0.83 | 3.03 | 0.18 | 0.07 | 0.035 | 0.050 | 0.095 | yes |

**Table 8.**Mesh sensitivity—minimum standard error of load–elongation curve, CPU time, and the number of elements.

Mesh Size: | 1.6 mm | 0.8 mm | 0.4 mm | 0.2 mm |
---|---|---|---|---|

$\mathrm{Minimum}\mathrm{standard}\mathrm{error}(SE$) | 262.2 N | 261.9 N | 260.8 N | 265.4 N |

Normalized CPU time | 1.0× | 2.6× | 9.4× | 50.7× |

Normalized Nº of elements | 1.0× | 2.5× | 16.8× | 126.7× |

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## Share and Cite

**MDPI and ACS Style**

Santos, R.O.; Moreira, L.P.; Butuc, M.C.; Vincze, G.; Pereira, A.B.
Damage Analysis of Third-Generation Advanced High-Strength Steel Based on the Gurson–Tvergaard–Needleman (GTN) Model. *Metals* **2022**, *12*, 214.
https://doi.org/10.3390/met12020214

**AMA Style**

Santos RO, Moreira LP, Butuc MC, Vincze G, Pereira AB.
Damage Analysis of Third-Generation Advanced High-Strength Steel Based on the Gurson–Tvergaard–Needleman (GTN) Model. *Metals*. 2022; 12(2):214.
https://doi.org/10.3390/met12020214

**Chicago/Turabian Style**

Santos, Rafael O., Luciano P. Moreira, Marilena C. Butuc, Gabriela Vincze, and António B. Pereira.
2022. "Damage Analysis of Third-Generation Advanced High-Strength Steel Based on the Gurson–Tvergaard–Needleman (GTN) Model" *Metals* 12, no. 2: 214.
https://doi.org/10.3390/met12020214