Experimental–Numerical Investigation of the Ductile Damage of TRIP 780 Steel

Abstract

This study presents a combined experimental–numerical methodology to calibrate the mechanical behavior of an advanced high-strength steel (AHSS) with transformation-induced plasticity (TRIP) effects, incorporating both initial plastic anisotropy and ductile damage. The investigated TRIP 780 grade, widely used in the automotive industry for its exceptional strength–ductility balance, exhibits a complex deformation response that demands accurate constitutive modeling for reliable sheet metal forming simulations. The methodology minimizes the number of required specimen geometries without compromising accuracy. Three flat-sheet specimens were employed: standard uniaxial tension (UT) and two double-notched designs reproducing intermediate (ID) and plane strain (PS) modes. Experiments combined digital image correlation with finite element analysis. Hill’s 48 quadratic yield criterion captured the initial anisotropy of the TRIP 780 sheet, while the parameters of a phenomenological ductile damage model were calibrated from the experimental data. The TRIP effect under UT was quantified by X-ray diffraction, showing a decrease in retained austenite from 9.9% (as-received) to 3.2% at 21% equivalent plastic strain. Fractography revealed damage initiation dominated by void nucleation at phase boundaries. The proposed approach yielded stress–strain predictions with R² values exceeding 0.99. This simplified approach offers a cost-effective and experimentally feasible framework for constitutive modeling of AHSS grades, enabling practical applications in advanced sheet forming simulations.

1. Introduction

Advanced high-strength steels (AHSS) are a key solution in the automotive industry for improving energy efficiency and safety. Compared to conventional steels, AHSS exhibits a superior strength–ductility ratio, enabling the production of thinner components without compromising structural integrity. This reduction in thickness contributes to lighter vehicle designs, leading to lower fuel consumption and reduced emissions [1]. The exceptional performance of AHSS is attributed to its multiphase microstructure and the activation of microstructural transformation mechanisms during forming. These steels behave similarly to composite materials, leveraging the ductility of softer phases, such as ferrite, while benefiting from the high strength of more challenging phases, such as martensite. Additionally, microstructural transformation mechanisms, such as transformation-induced plasticity (TRIP) [2,3] and twinning-induced plasticity (TWIP), enhance the strength–ductility balance under mechanical loading [4]. Replacing conventional steel with AHSS enables significant mass reduction (up to 25%) while leading to immediate and continuous reductions in CO₂ emissions throughout the vehicle’s life cycle. Furthermore, AHSS recycling is more efficient and has a lower environmental impact than aluminum and other lightweight materials, such as carbon fiber-reinforced plastics (CFRP), reinforcing its sustainability advantage [5,6].

The broad applicability of AHSS stems from its high strength-to-weight ratio, wide range of strength levels, good weldability, relatively low manufacturing cost, and high recyclability [7]. This study focuses on AHSS TRIP 780 grades with a minimum tensile strength exceeding 780 MPa. This steel is widely used in automobile structural and safety components, such as cross members, longitudinal beams, B-pillar reinforcements, and sills [8]. However, while advancements in AHSS technology have enhanced strength and ductility, they have also introduced new challenges in accurately modeling mechanical behavior. Folle et al. [9] reviewed the literature on the behavior of high-strength steel during forming processes and their prediction through numerical simulations. The complex response of these steels depends on the specific application and manufacturing processes, particularly those involving multiple strain paths, high strain rates, and thermal effects. Accurately predicting the mechanical behavior of AHSS requires advanced constitutive and damage models, particularly for steels with TRIP effects, in which anisotropy and microstructural transformations play a significant role.

Bai and Wierzbicki [10] conducted a comparative study of sixteen damage models, categorizing them into three groups: physics-based, phenomenological, and empirical. These models were calibrated using experimental data from TRIP 780 steel. Among them—Modified Mohr–Coulomb (MMC), Cockcroft–Latham, Wilkins, Gurson–Tvergaard-Needleman (GTN), Modified GTN for shear, and Nielsen–Tvergaard—the MMC model exhibited the lowest mean error. To address ductile damage in metals, Bai and Wierzbicki [11] proposed modifying the Mohr–Coulomb model, initially developed for geomaterials, incorporating the effects of hydrostatic stress and the Lode parameter. The MMC model requires experimental evaluation of the stress triaxiality factor and the effective true strain at the onset of damage for different strain paths. Also, the number of specimen geometries needed depends on the sophistication of the methodology. Bai and Wierzbicki [11] used five specimen types—dog-bone (uniaxial tension), the flat specimen with cutouts (tension), the disk specimen (equibiaxial tension), and two butterfly specimens (tension and simple shear)—to identify the five material parameters of the MMC model for TRIP 780 steel.

While recent methodologies provide accurate damage modeling, they often require multiple complex specimen geometries and testing setups, increasing experimental costs and complexity. This study proposes a simplified numerical–experimental method that reduces the number of necessary specimen geometries while maintaining accuracy. By relying on standard testing equipment and commercial digital image correlation (DIC) and finite element analysis (FEA) software widely used in the automotive industry, this approach aims to provide a more accessible and cost-effective alternative for damage model calibration. For this purpose, three specimen geometries were selected: a dog-bone specimen for uniaxial tension (UT) and double-notched (DN) specimens with 1 mm and 5 mm radii to achieve plane-strain tension and intermediate strain modes, respectively. These specimens were chosen to capture key deformation modes relevant to AHSS forming processes, ensuring an effective balance between experimental feasibility and modeling accuracy. The ductile damage calibration was performed using Hill’s 48 plastic anisotropy yield criterion within Abaqus/CAE software (version 6.9). Additionally, microstructural analyses were conducted to investigate damage mechanisms and assess the dominant stage of the TRIP effect, highlighting its relevance for damage calibration.

2. Materials and Methods

The mechanical behavior of a cold-rolled, zinc-coated TRIP 780 steel sheet with a nominal thickness of 1.6 mm was characterized, considering both anisotropy and ductile damage. The TRIP 780 steel sheet was supplied by ArcelorMittal Vega (São Francisco do Sul, Brazil), and its chemical composition in the as-received condition is given in

Table 1. Chemical composition of the TRIP 780 steel [%wt]. Reprinted from Ref. [12].

C	Si	Mn	P	S	Al	Cr	B
0.22	0.27	1.89	0.02	0.002	1.54	0.18	0.0003
Ti	Mo	Nb	Sn	Ca	Ni	V	Sb
0.004	0.005	0.018	0.003	0.0005	0.009	0.003	0.003

.

2.1. Uniaxial Tension Specimen Tests

Uniaxial tension (UT) specimens (dog-bone shape) were used to evaluate the mechanical behavior of the material and to quantify the transformation of retained austenite (RA) into the martensite phase due to plastic deformation. The UT specimens were manufactured according to ASTM E8 [13], and their geometry and dimensions are shown in Figure 1. To minimize phase transformation during specimen preparation, strips measuring 25 mm × 200 mm were machined using a milling process with a low feed rate and abundant coolant.

The uniaxial tensile testing campaign used a universal testing machine, EMIC-Instron model DL300000 (Instron, Norwood, MA, USA). The load was recorded using a 20 kN S-type load cell, and strain measurements were obtained using the VIC-3D Digital Image Correlation (DIC) system from Correlated Solutions. The DIC system requires a high-contrast stochastic pattern to be applied to the specimen surface within the gauge length. First, a matte white aerosol ink was used as a background layer. After approximately 5 to 10 min, a matte black ink was carefully sprayed over the white background. The drying time of the black ink was about 2 min. The tests were conducted immediately after the painting process, ensuring that the curing time did not exceed 20 min. Once fully cured, the ink loses elasticity and may detach from the surface during deformation. The DIC software (version 9) evaluated the speckle pattern quality before starting the tests.

Two tests were performed in two groups: (i) uniaxial tensile tests up to fracture and (ii) interrupted tensile tests. The first group described the material’s mechanical behavior, including elasticity, work-hardening, plastic anisotropy, and damage. UT specimens were manufactured in three angular orientations with respect to the rolling direction (RD) to evaluate anisotropy: 0°, 45°, and 90°. The second group was employed to quantify the TRIP effect as a function of the effective plastic strain. According to the von Mises yield criterion, the effective plastic strain was assumed to equal the plastic strain measured in uniaxial tensile tests within the uniform deformation region of the specimen gauge length. Based on the experimental results of the uniaxial tensile testing, a numerical simulation was performed using the material’s elastoplastic properties. The elongation over the 50 mm gauge length was then compared with the amount of effective plastic strain. To ensure this procedure, a contact extensometer was mounted on the tensile specimen alongside the DIC system (Figure 2), interrupting the tests at increasing plastic straining levels up to the maximum uniform elongation. Accordingly, five equidistant plastic strain levels were defined between the as-received undeformed and fractured conditions.

Table 2. Interrupted UT testing campaign.

Orientation	0° RD
Plastic strain [%]	3.8	7.5	12	17	21
Elongation [%]	3.98	7.56	11.28	15.14	19.12

resumes the plastic straining levels for each UT testing procedure performed on the specimen’s angular orientation at the RD. A 3 mm/min crosshead speed was used in the se uniaxial tensile testing campaign, corresponding to a nominal strain rate of 10⁻³ s⁻¹. Three specimens were tested under each condition to ensure the repeatability of the tests.

2.2. Double-Notched Specimen Tests

Two groups of double-notched specimen geometries were used: (i) Plane Strain (PS), which allows obtaining a strain path close to the plane strain tension state in the specimen center, and (ii) intermediate deformation (ID), which provides a strain path between those of the UT and PS specimens [14]. The detailed geometries and dimensions of these double-notched specimens are shown in Figure 3. Due to their geometric complexity, both specimens were manufactured using the wire electrical discharge machining (EDM) process. Furthermore, both specimen groups were subjected to uniaxial tensile loading up to fracture. The same universal testing machine and DIC system used for UT testing were employed to evaluate the mechanical behavior of the TRIP 780 steel under the PS and ID deformation modes. Three specimens were tested for each specimen geometry. To maintain a nominal strain rate of 1 $\times$ 10⁻³ s⁻¹, in the same order as the UT specimen tests, constant crosshead speeds of 0.114 mm/min and 0.594 mm/min were set for the PS and ID specimens, respectively.

2.3. X-Ray Diffraction Analysis

TRIP steels contain a significant retained austenite, which can be transformed into martensite under plastic straining. To estimate the initial retained austenite (RA) volume fraction and its transformation as a function of effective strain, X-ray diffraction (XRD) analysis was performed on specimens subjected to interrupted uniaxial tension (UT). After testing, 15 mm-long samples were extracted from the central region of the gauge length using a precision cutter (Isomet 1000, Buehler, Lake Bluff, IL, USA). The samples were then subjected to a chemical pickling process in 18% hydrochloric acid to remove the zinc coating, ensuring that the coating did not interfere with the XRD peak intensity measurements.

XRD analysis was performed according to the ASTM E975 standard [15] using a Shimadzu LabX XRD 6000 diffractometer (Shimadzu Corporation, Kyoto, Japan). This method applies to carbon and alloy steels with a near-random crystallographic orientation of both phases, provided that the RA content is at least 1% by volume.

De Meyer et al. [16] quantified the RA volume fraction ($V_{\gamma }$) in TRIP steel using XRD and different equations derived from the ASTM E975 standard, as proposed in the literature. Besides achieving accurate quantification, the results indicated a slight variation, with an error of less than 2%. Castanheira et al. [12] employed an XRD system with a Co-Kα radiation tube to analyze a TRIP 780 steel, while Guzmán and Monsalve [17] and Ashrafi et al. [18] used a Cu-Kα radiation tube. Both studies reported a retained austenite fraction of approximately 8% in the as-received condition. This study performed the XRD analysis using a Cr-Kα radiation tube, operated at 35 kV and 30 mA. The scanning range was set from 60° to 160°, with a 2°/min scan speed.

The RA volume fraction, $V_{\gamma }$, was determined using Equation (1):

$$V_{\gamma }={\displaystyle \frac{\raisebox{1ex}{$I_{\gamma }$}\!\left/ \!\raisebox{-1ex}{$R_{\gamma }$}\right.}{\raisebox{1ex}{$I_{\alpha }$}\!\left/ \!\raisebox{-1ex}{$R_{\alpha }$}\right.+\raisebox{1ex}{$I_{\gamma }$}\!\left/ \!\raisebox{-1ex}{$R_{\gamma }$}\right.}}$$

(1)

In Equation (1), indices α and γ denote the ferrite (F) plus martensite (M) phase (α-iron: Body-Centered Cubic, BCC) and austenite phase (γ-iron: Face-Centered Cubic, FCC), respectively, whereas the parameter $I$ stands for the integrated intensity of each diffraction peak and $R$ is a factor depending on the interplanar spacing ($hkl$), the Bragg angle (θ), the crystal structure, and the phase composition. The values of $R_{\alpha }$ and $R_{\gamma }$ values were calculated for Cr-Kα radiation using the Powder Cell 2.4 software, following the ASTM E975 standard [15]. For the α phase (ferrite + martensite), the (110), (200), and (211) planes yielded 2θ angles of 68.82°, 106.40°, and 156.40°, with corresponding R values of 101.50, 20.73, and 190.80, respectively. For the γ phase (austenite), the (111), (200), and (220) planes were associated with 2θ angles of 66.88°, 79.04°, and 128.30°, and R values of 75.24, 34.78, and 47.88, respectively [15]. The XRD results were analyzed and plotted in OriginLab^® software (version 10.1), where the Lorentz equation was applied to integrate the intensity peaks.

2.4. Microstructural Characterization

The microstructural characterization of the TRIP 780 steel in the as-received condition was performed to identify the microconstituents of its multiphase structure. Samples with dimensions of 10 mm $\times$ 12.5 mm $\times$ 1.6 mm (thickness) were sectioned using a precision cutter (Isomet 1000, Buehler) under high cooling conditions. To ensure comprehensive analysis, the samples were mounted in acrylic cold-curing resin and prepared for examination in three different orientations: transverse, longitudinal, and planar, as schematically illustrated in Figure 4.

The samples were mechanically prepared using SiC abrasive grinding papers with grit sizes of 220, 320, 400, 600, 800, 1000, 1500, and 2000, employing a Struers Knuth-Rotor 2 machine (Struers, Copenhagen, Denmark). Polishing was performed with a water-based alumina fine polishing suspension (1 μm particle size), using distilled water as a lubricant. All samples were etched with a sodium metabisulfite solution (20 g of Na₂S₂O₃ dissolved in 100 mL of distilled H₂O) for 13 s to reveal the microstructure. Micrographs were obtained using an optical microscope (OM) and a scanning electron microscope (SEM). An Olympus BX51M microscope (Olympus Corporation, Tokyo, Japan) was employed for OM analysis, while SEM images were acquired using a Hitachi TM4000Plus II (Hitachi, Ltd., Tokyo, Japan). The ferrite phase volume fraction was estimated using OM micrographs and processed through threshold analysis in ImageJ software (version 1.52a). SEM micrographs were used to evaluate damage mechanisms and ferrite grain size analysis, as per ASTM E112 standard [19]. To estimate the ferrite average grain size ($G_S$) and the ASTM grain size number, longitudinal section samples with different strain levels were analyzed along the sheet RD: as received, 3.8%, 17%, and fractured conditions. A 2000$\times$ magnification was used for imaging, and measurements were taken in five regions of the sample surface. The Heyn linear intercept procedure was applied to the micrographs using ImageJ software.

2.5. Work-Hardening and Plastic Anisotropy

An experimental–numerical methodology identified the plastic and damage behaviors of the investigated AHSS TRIP 780 steel. First, the elastoplastic behavior was evaluated based on uniaxial tensile test results at the sheet RD. The Young’s modulus ($E$) and Poisson’s ratio (ν) were determined from the elastic domain data. The work-hardening behavior was fitted with the true stress, $\sigma$, and true plastic-strain data, ${\epsilon }^p$, using Swift’s equation:

$$\sigma =K{\left({\epsilon }_0+{\epsilon }^p\right)}^n$$

(2)

in which $K$ (MPa) is the strength coefficient, ${\epsilon }_0$ is the pre-strain, and $n$ is the strain-hardening exponent.

Hill’s quadratic yield criterion [20] was adopted to account for the initial plastic anisotropy, as it is widely used in commercial finite element software [21]. Hill’s 48 quadratic yield criterion has also been employed to model the anisotropic damage behavior of metallic sheets during the forming process [22]. In the general 3D case, Hill’s quadratic yield anisotropic criterion effective stress, ${\overline{\sigma }}_{Hq}$, is defined under the isotropic work-hardening assumption as a function of the Cauchy stress tensor ($\mathit{\sigma }$) components:

$${\overline{\sigma }}_{Hq}=\sqrt{F{\left({\sigma }_{22}-{\sigma }_{33}\right)}^2+G{\left({\sigma }_{33}-{\sigma }_{11}\right)}^2+H{\left({\sigma }_{11}-{\sigma }_{22}\right)}^2+2L{\sigma }_{23}^2+2M{\sigma }_{31}^2+2N{\sigma }_{12}^2}$$

(3)

wherein the indices (1, 2, 3) correspond to the material orthotropic symmetry axes (x, y, z) assumed aligned with the rolling (RD), transverse (TD), and in-plane normal (ND) sheet directions, respectively. In Equation (3), $F$, $G$, $H$, $L$, $M$, and $N$ are material parameters describing the initial plastic anisotropy. For thin sheets, the material parameters ($F$, $G$, $H$, and $N$) were determined from Lankford $r$-values obtained from uniaxial tension tests in three angular orientations at 0°, 45°, and 90° to the rolling direction ($r_0$, $r_{45}$, and $r_{90}$) with the uniaxial tensile yield (or flow) stress at the rolling direction (${\sigma }_0$). In the commercial finite element (FE) code ABAQUS, these parameters are defined by the anisotropic yield stress ratios, $R_{ij}$, as [23]

$$\begin{gathered} F={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{R_{22}^2}}+{\displaystyle \frac{1}{R_{33}^2}}-{\displaystyle \frac{1}{R_{11}^2}}\right) \\ G={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{R_{33}^2}}+{\displaystyle \frac{1}{R_{11}^2}}-{\displaystyle \frac{1}{R_{22}^2}}\right) \\ H={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{R_{11}^2}}+{\displaystyle \frac{1}{R_{22}^2}}-{\displaystyle \frac{1}{R_{33}^2}}\right) \\ L={\displaystyle \frac{3}{2R_{23}^2}} \\ M={\displaystyle \frac{3}{2R_{31}^2}} \\ N={\displaystyle \frac{3}{2R_{12}^2}} \end{gathered}$$

(4)

stress ratios to unity to provide an effective stress measure equal to the uniaxial tensile yield (or flow) stress at the rolling direction (RD); that is, ${\overline{\sigma }}_{Hq}={\sigma }_0$. Moreover, out-of-plane anisotropic stress ratios related to the through-thickness shear effects are suitably assumed as isotropic, as in the von Mises yield criterion; namely, $R_{23}=R_{31}=1$. The other anisotropic stress-ratios are defined from Lankford $r$-values [24,25]:

$$\begin{gathered} R_{22}=\sqrt{{\displaystyle \frac{r_{90}\left({1+r}_0\right)}{r_0\left(1+r_{90}\right)}}} \\ {R}_{33}=\sqrt{{\displaystyle \frac{r_{90}\left(1+r_0\right)}{\left(r_0+r_{90}\right)}}} \\ {R}_{12}=\sqrt{{\displaystyle \frac{{3r}_{90}\left(1+r_0\right)}{\left({1+2r}_{45}\right)\left(r_0+r_{90}\right)}}} \end{gathered}$$

(5)

The $r$-values were determined from the DIC measurements of the length and width of the specimen during the uniaxial tensile tests. The $r$-value was calculated as the slope of the trend line of the relationship between the width strain (${\epsilon }_w$) and the thickness strain (${\epsilon }_t$). Assuming plastic incompressibility and neglecting small elastic-strains, the $r$-value can be defined as a function of the width strain and length strain (${\epsilon }_l$), as [26]

$$r=-{\displaystyle \frac{{\epsilon }_w}{\left({\epsilon }_w+{\epsilon }_l\right)}}$$

(6)

2.6. Damage Modeling

In the ABAQUS FE code, the ductile fracture of metals can be predicted with an element removal technique using a classical continuum damage mechanics (CDM) approach based on an evolution law describing the rate of the material stiffness degradation, which is activated once an initiation damage criterion is verified. Then, a scalar damage variable, $D$, ranging from 0 (undamaged) to 1 (wholly fractured), accounts for the progressive material degradation, namely,

$$\mathit{\sigma }=\left(1-D\right)\tilde{\mathit{\sigma }}$$

(7)

where $\mathit{\sigma }$ is the stress tensor whereas $\tilde{\mathit{\sigma }}$ is the current effective stress tensor; that is, corresponding to the undamaged material. The effective plastic strain to the onset of damage, ${\overline{\epsilon }}_D^p$, depends on the stress triaxiality factor, $\eta$, which is defined as the ratio between the hydrostatic stress (${\sigma }_h={\sigma }_{kk}/3$) and the von Mises effective stress (${\overline{\sigma }}_{vMises}$), i.e., $\eta ={\sigma }_h/{\overline{\sigma }}_{vMises}$, and the effective plastic strain rate (${\dot{\overline{\epsilon }}}^p$):

$${\overline{\epsilon }}_D^p=f\left(\eta ,{\dot{\overline{\epsilon }}}^p\right)$$

(8)

Also, a damage indicator increasing monotonically with the plastic straining process is introduced as

$${\omega }_D=\int {\displaystyle \frac{d{\overline{\epsilon }}^p}{{\overline{\epsilon }}_D^p\left(\eta ,{\dot{\overline{\epsilon }}}^p\right)}}$$

(9)

Then, once the damage onset criterion is met, ${\omega }_D=1$, the material degradation is activated based on the scalar damage variable $D$ (see Equation (7)). For this purpose, a degradation rate of material stiffness is also needed to describe stress softening. In the ABAQUS code, a linear degradation is defined to account for the damage evolution law as a function of the characteristic length element, $L_e$; the effective plastic strain rate, ${\dot{\overline{\epsilon }}}^p$; and an effective plastic displacement, ${\overline{u}}_f^p$; that is,

$$\dot{D}={\displaystyle \frac{L_e{\dot{\overline{\epsilon }}}^p}{{\overline{u}}_f^p}}={\displaystyle \frac{{\dot{\overline{u}}}^p}{{\overline{u}}_f^p}}$$

(10)

To complete the ductile fracture using the available damage model in the ABAQUS FE code, one needs to define the material parameters ${\overline{\epsilon }}_D^p$ and ${\overline{u}}_f^p$. In advanced high-strength steels, the damage process that may cause catastrophic failure is usually triggered by the mechanisms of microvoid formation: nucleation, growth, and coalescence. In dual-phase steels, for instance, the initial stage of damage is commonly attributed to the void nucleation from debonding between hard inclusions and grain boundaries. Still, the nucleation of voids may occur from the decohesion at the interfaces of the soft ferrite matrix and hard martensite phases or otherwise from the fracture of particles [27]. In this study, the damage mechanism considered is controlled solely by void growth based on the Rice and Tracey (R-T) void expansion model, wherein it is supposed that the stress triaxiality factor, $\eta$, influences the effective plastic strain at fracture, ${\overline{\epsilon }}_f^p$, as [28,29]

$${\overline{\epsilon }}_f^p=\alpha exp\left(-\beta \eta \right)$$

(11)

In this way, the material damage onset parameter, ${\overline{\epsilon }}_D^p$, can be identified as the effective plastic strain at fracture, ${\overline{\epsilon }}_f^p$, which, in turn, is measured from the experimental testing [28]. Only proportional loading conditions are considered to account for the initial material anisotropy, and the principal strain and stress axes are coincident with the material orthotropic symmetry axis. In this case, the effective plastic-strain measure in the sense of Hill’s quadratic anisotropic yield criterion is [30,31]

$${\overline{\epsilon }}^p=\sqrt{{\displaystyle \frac{\left(F+H\right){\epsilon }_1^2+\left(G+H\right){\epsilon }_2^2+2H{\epsilon }_1{\epsilon }_2}{FG+GH+HF}}}$$

(12)

Equation (12) estimates the effective plastic strain at the onset of damage using the total principal true in-plane strains (${\epsilon }_1$, ${\epsilon }_2$) determined from the DIC measurements of the tests performed on the adopted specimen geometries (see Section 3.3).

On the other hand, the effective plastic strain at fracture measure, ${\overline{\epsilon }}_f^p$, is mesh dependent; that is, it will be influenced by the mesh size of characteristic element length, $L_e$, in Equation (10), and the element formulation. This question is solved using the energy required to open unit area, $G_f$ [23]:

$$G_f={\int }_{{\overline{\epsilon }}_0^p}^{{\overline{\epsilon }}_f^p}{L}_e{\sigma }_{y}d{\overline{\epsilon }}^p={\int }_0^{{\overline{u}}_f^p}{\sigma }_{y}d{\overline{u}}^p$$

(13)

where ${\sigma }_y$ represents the flow stress after the damage onset. By observing that ${\dot{\overline{u}}}^p=0$ before damage onset, while ${\dot{\overline{u}}}^p=L_e{\dot{\overline{\epsilon }}}^p,$ and from the linear damage evolution law, given in Equation (10), the effective plastic displacement at fracture is obtained as [23]

$${\overline{u}}_f^p={\displaystyle \frac{2G_f}{{\sigma }_{y0}}}$$

(14)

Equation (14) implies that the energy dissipation during the damage evolution is equal to $G_f$ if the material response has a perfectly plastic behavior with a constant flow stress, ${\sigma }_{y0}$, when the damage onset criterion is achieved. The critical $G_f$ value (or equivalently ${\overline{u}}_f^p$) was determined until the predicted numerical post-necking behavior matched the experimental stress–strain curves. A critical damage value, $D_{critical}$, set to activate the element removal technique in the ABAQUS/Explicit, was set by a critical damage value. This corresponds to the maximum degradation that the material can withstand before fracture. This value can be determined from Equation (7) as

$$D_{critical}=1-\left({\displaystyle \frac{{\sigma }_f}{\overline{\sigma }}}\right)$$

(15)

where ${\sigma }_f$ is the true-stress measure at the failure and $\overline{\sigma }$ is the effective stress measure.

2.7. Finite Element Modeling

Figure 5 shows the models proposed in the finite element (FE) simulations of the uniaxial and double-notched tensile specimens. A one-fourth symmetry FE model was adopted with symmetric boundary conditions at the half-thickness, wherein the normal displacements were set to zero. The experimental tests were reproduced by imposing a prescribed speed at the nodes on the right grip region, as shown by the red boxes in Figure 4. In contrast, an encastre boundary condition was defined for the nodes at the left grip region. The quasi-static simulations were performed using the commercial finite element (FE) code Abaqus/Explicit. For all specimens, the same mesh size (0.2 mm $\times$ 0.2 mm $\times$ 0.2 mm) was used in the region of interest, as shown in Figure 5. Santos et al. [32] conducted a sensitivity analysis using a mesh with an element size of 0.2 mm, which resulted in a refined mesh yielding numerical predictions close to the experimental results for the UT and PS tests. Moreover, the characteristic element length has a direct influence on the definition of the $G_f$ parameter, as shown in Equation (13) and Figure 16. The specimens were meshed with C3D8R elements (8-node linear brick elements with reduced integration and hourglass control) [23]. The UT specimen model is meshed with 33,952 elements, while ID and PS are meshed with 35,124 and 24,600 elements, respectively. A mass scaling value of 10,000 was adopted to speed up all simulations. To verify the quasi-static loading condition, the ALLKE (kinetic energy)/ALLIE (internal energy) ratio remained less than 1% in all FE testing models, which indicates that the mass scaling factor was not excessively selected.

The experimental uniaxial and double-notched tensile tests were simulated using the material parameters calibrated from the proposed numerical–experimental procedure. The numerical predictions of the true stress and total true strain measures were compared to validate the material modeling with the experimental data obtained from load and DIC measurements. The R-squared goodness-of-fit measure (R²) was adopted to evaluate the ability of the calibrated material parameters to predict the experimental curves.

3. Results

3.1. Retained Austenite and TRIP Effect

Figure 6 presents the X-ray diffraction (XRD) results from the interrupted uniaxial tensile tests at increasing equivalent plastic strain levels (3.8 up to 21%) with the as-received (0%) condition. The characteristic diffraction peaks of ferrite/martensite (α/α’) and retained austenite (γ) were identified at the expected angles, exhibiting variations in intensity. Due to the intrinsic nature of the TRIP (transformation-induced plasticity) effect, a fraction of the retained austenite (RA) in the microstructure underwent strain-induced transformation into martensite during plastic deformation. This phase transformation was confirmed by the progressive reduction in the intensity of the (111)γ, (200)γ, and (220)γ peaks, with increasing equivalent plastic strain. Such observed behavior substantiates the occurrence of the TRIP effect in the investigated TRIP780 steel sheet.

The volume fraction of RA ($V_{\gamma }$) in the as-received condition was estimated at ~9.9%, as shown in the plot of Figure 7. The integration of intensity peaks from the XRD analysis of specimens subjected to uniaxial tensile loading revealed that the transformation of RA into martensite takes place at a higher rate during the initial stages of plastic deformation. Near 10% true plastic strain, almost 50% of the initial RA had already been transformed. However, the TRIP effect gradually stabilizes, preventing complete phase transformation. At 21% plastic strain, the remaining volume fraction of RA is ~3.2%.

3.2. Microstructural Analysis

The microstructure of the TRIP 780 steel in the as-received condition is shown in Figure 8a, wherein it is possible to identify a ferritic (F) matrix (brown) with dispersed islands of martensite/retained austenite (M/RA) (white). Due to the absence of topographical contrast, differentiation between martensite and retained austenite was not feasible. At the periphery of these islands, dark brown grains suggest the presence of bainite (B). The identified microconstituents agree with the microstructural characterizations reported by Castanheira et al. [12] and Soleimani et al. [3] for TRIP steel grades.

The ferrite volume fraction in the investigated TRIP780 steel sheet was determined as 52.5 $\pm$ 0.5%, which is within the expected range for this microconstituent in TRIP steels (50–55%) as reported by Kuziak et al. [1] and Soleimani et al. [3]. Figure 8b reveals the presence of martensite/retained austenite bands (MB) aligned to the rolling direction (RD). The formation of these bands has been documented in previous studies [12,33]. According to Ramazani et al. [33], the presence of such bands enhances the steel’s formability by reducing the dispersion of martensite/retained austenite islands, which promotes localized plastic deformation of the more ductile ferrite phase. However, their findings also indicate that TRIP steels containing these bands exhibit reduced mechanical strength due to the localized nature of strain accommodation.

The grain size ($G_S$) values obtained from longitudinal section micrographs at different deformation stages are listed in

Table 3. Average grain size of ferrite matrix with increasing strain level.

Strain Level	$\mathbf{Grain}\mathbf{Size},{\mathit{G}}_{\mathit{S}}$ [μm]	ASTM Grain Size
As received	$2.47\pm$ 0.18	14.0
${\overline{\epsilon }}_p$ = 3.8%	$2.60\pm$ 0.25	13.9
${\overline{\epsilon }}_p$ = 17%	$2.92\pm$ 0.20	13.6
Fractured	$3.15\pm$ 0.18	13.3

. For reference, Table 3 also presents the standardized ASTM grain size number. The micrographs of the present TRIP780 steel from the as-received condition qualitatively indicate a high degree of grain refinement. This microstructural observation is quantitatively supported by an estimated average grainsize of 2.47 μm. In the subsequent stages, the calculated $G_S$ values revealed a progressive elongation trend in the ferrite grains, which can be attributed to the plastic deformation imposed during the uniaxial tensile test. This behavior is associated with the preferential reorientation of grains and the development of deformation-induced texture, characteristic of materials with pronounced plastic anisotropy [34].

The SEM micrographs confirm a significant elongation of the ferrite matrix grains along the direction of the applied stress. This morphological evolution indicates deformation-induced grain reorientation, a typical response of ferritic steels under tensile loading. Figure 9 shows two conditions: the as-received and the post-fractured stage, highlighting the progressive grain elongation during the deformation process.

SEM analysis of the as-received samples revealed the presence of microvoids and cavities, likely induced during the metallographic preparation process. The sequential sanding and polishing steps may have selectively removed harder particles, such as inclusions, leading to the formation of these undesired features. The void nucleation and coalescence mechanisms characteristic of ductile damage were observed in the fractured specimens. The SEM micrograph in Figure 10a shows multiple voids near the fracture edge. Notably, two predominant nucleation sites were identified: (i) decohesion at the interface between soft and hard phases and (ii) void nucleation via fracture of harder second-phase constituents due to localized deformation. Figure 10b illustrates a sequence of aligned microvoids merging into a large void, initiating the void coalescence process. This microstructural evolution ultimately compromises the material’s load-bearing capacity, thus leading to ultimate failure.

3.3. Material Parameters Calibration

From the uniaxial tensile (UT) test using a constant crosshead speed of 3 mm/min and digital image correlation, the mechanical properties of the TRIP 780 steel with a nominal thickness of 1.60 mm were obtained. Figure 11 shows the experimental result of the engineering stress–strain curve obtained from the uniaxial tensile test.

Table 4. Mechanical properties of TRIP 780 steel determined from UT testing at the RD.

$\mathit{E}$ [GPa]	ν	${\mathit{S}}_{\mathit{y}}$ [MPa]	${\mathit{S}}_{\mathit{u}}$ [MPa]	${\mathit{e}}_{\mathit{u}}$ [%]	${\mathit{e}}_{\mathit{t}}$ [%]	$\mathit{G}\mathit{F}\mathit{I}$ [GPa%]
202 $\pm$ 6	0.35 $\pm$ 0.02	456 $\pm$ 1	802 $\pm$ 2	20.5 $\pm$ 0.4	24.7 $\pm$ 0.5	19.8 $\pm$ 0.4

lists the values of Young’s modulus ($E$), Poisson’s ratio ($\nu$), yield strength ($S_y$), ultimate tensile strength ($S_u$), uniform elongation ($e_u$), total elongation ($e_t$), and global formability index ($GFI$).

The plastic behavior of the TRIP 780 steel in the rolling direction was described using Swift work-hardening (see Equation (2)). This accounts for initial plastic yielding and prior deformation history, such as that induced by a cold-rolling skin-pass, through the strain term (${\epsilon }_0$). The true-stress predictions obtained with the fitted Swift’s parameters, summarized in

Table 5. Fitted Swift work-hardening parameters for the TRIP 780 steel.

$\mathit{K}$ [MPa]	${\mathit{\epsilon }}_0$	$\mathit{n}$	R2
1464.3 $\pm$ 1.4	0.0020 $\pm$ 0.0001	0.2342 $\pm$ 0.0005	0.999

, are compared in Figure 12 to the experimental data, showing an excellent R²-value of 0.999. The resulting work-hardening curve was implemented in the Abaqus/CAE code in a tabular form for numerical simulation.

The anisotropic behavior of the TRIP 780 sheet was evaluated by determining the Lankford coefficients ($r_0$, $r_{45}$, $r_{90}$) at 0°, 45°, and 90° with respect to the rolling direction, calculated using Equation (6). Figure 13 shows the trend lines to determine the $r$-values as a function of width and thickness strains. Literature reports TRIP 780 steel grades with $r$-values comparable to those obtained experimentally in this study [35,36,37].

The normal anisotropy ($\overline{r}$) and planar anisotropy ($∆r$) values were calculated from these coefficients. The Hill’s 48 quadratic anisotropic yield function coefficients and anisotropic yield stress ratios were also determined for implementation in the Abaqus/CAE code as input parameters for the numerical simulations. The manifested material small planar anisotropy, that is, $∆r\ne 0$, indicates non-uniform deformation behavior in different in-plane sheet directions, which can result in low-amplitude earing defects in deep-drawing operations. Conversely, normal anisotropy ($\overline{r}$) is related to the sheet’s resistance to thinning, with higher ($\overline{r}$ > 1) values promoting improved formability and deeper drawability while minimizing wrinkling risks.

Table 6. Anisotropic parameters of TRIP 780 steel.

Lankford Coefficients and Normal and Planar Anisotropy
${\mathit{r}}_0$	${\mathit{r}}_{45}$		${\mathit{r}}_{90}$	$\overline{\mathit{r}}$	$∆\mathit{r}$
0.654	0.660		0.592	0.642	−0.037
Hill’s 48 Yield Function Coefficients:
$\mathit{F}$	$\mathit{G}$	$\mathit{H}$	$\mathit{L}$	$\mathit{M}$	$\mathit{N}$
0.668	0.605	0.395	1.476	1.476	1.476
Anisotropic Yield Stress Ratios:
${\mathit{R}}_{11}$	${\mathit{R}}_{22}$	${\mathit{R}}_{33}$	${\mathit{R}}_{12}$	${\mathit{R}}_{13}$	${\mathit{R}}_{23}$
1	0.970	0.886	1.008	1	1

summarizes the results of the anisotropic parameters for the TRIP 780 steel.

Table 7. Stress triaxiality factor and equivalent plastic-strain values at the onset of damage obtained from the proposed experimental–numerical procedure.

UT		ID		PS
Principal Strains
${\mathit{\epsilon }}_1$	${\mathit{\epsilon }}_2$	${\mathit{\epsilon }}_1$	${\mathit{\epsilon }}_2$	${\mathit{\epsilon }}_1$	${\mathit{\epsilon }}_2$
0.259	−0.091	0.132	−0.024	0.083	−0.003
Stress Triaxiality Factor–Equivalent Plastic Strain
$\mathit{\eta }$	${\overline{\mathit{\epsilon }}}^{\mathit{p}}$	$\mathit{\eta }$	${\overline{\mathit{\epsilon }}}^{\mathit{p}}$	$\mathit{\eta }$	${\overline{\mathit{\epsilon }}}^{\mathit{p}}$
0.333	0.259	0.489	0.135	0.570	0.089

presents the experimentally determined principal strain values (${\epsilon }_1$, ${\epsilon }_2$) at the onset of damage. The maximum recorded load for each evaluated deformation mode (UT, ID, and PS) was adopted to define the values of the in-plane principal strains, which, in turn, were calculated in the central region of the specimens from the DIC measurements, as depicted in Figure 14. Equation (12) was employed to calculate the equivalent plastic strain (${\overline{\epsilon }}^p$) at the onset of damage. Using the computed ${\overline{\epsilon }}^p$ values, the corresponding stress triaxiality ($\eta$) was obtained from the numerical simulations of all specimens. Consequently, the ($\eta$, ${\overline{\epsilon }}^p$) pairs obtained for UT, ID, and PS specimens were used to characterize the damage initiation criterion between the uniaxial tensile and plane-strain deformation modes, as described by Equation (11).

Figure 15 presents the equivalent plastic strain at the onset of damage as a function of the stress triaxiality factor, fitted using an exponential law according to Equation (11). The resulting fitted data were tabulated and implemented in the Abaqus/CAE code to define the ductile damage initiation criterion.

Table 8. Parameters of fitted exponential curve—Equation (11).

$\mathit{A}$	$\mathit{B}$	R2
1.081 $\pm$ 0.150	4.266 $\pm$ 0.352	0.989

provides the parameters of the exponential equation obtained with R² = 0.989, thus providing a reasonable estimate of the plastic-strain at the damage onset. On the other hand, unlike the uniaxial tension (UT) deformation mode, which exhibits a relatively uniform strain path until the damage onset, the ID and PS specimen deformation modes experienced a non-uniform strain state from the beginning of the straining process. In the PS specimen, the plane strain condition occurs at the central region, while strain variations emerge near the notch, as shown in Figure 14e. Since damage predominantly initiates in the notch regions, precisely determining damage onset in the specimen’s central area is more challenging. Consequently, the estimated equivalent plastic strain values at damage onset are conservative due to inherent experimental limitations in capturing the exact commencement of the damage.

The parameter $G_f$, used to determine the effective plastic displacement at fracture, see Equations (13) and (14), was calculated through an iterative simulation process to achieve the best fit for the UT case. The $G_f$ value is dependent on the element’s characteristic length. For the adopted mesh (3D continuum solid elements) with an approximate characteristic length of 0.2 mm, the optimal value of $G_f$ was 400 mJ. Figure 16 illustrates the procedure for obtaining the appropriate $G_f$ value for the UT deformation mode. Next, the maximum degradation parameter was computed using Equation (15), yielding $D_{critical}$ = 0.0849. This indicates that the damage variable evolves once the damage initiation criterion is met, leading to a progressive reduction in the element’s load-bearing capacity. Suppose the element deletion option is enabled in the ABAQUS/CAE code. In that case, elements will be removed once the damage variable $D_{critical}$ reaches 8.49%, hence reproducing the fracture propagation.

3.4. Validations

After calibrating the material parameters, numerical simulations were conducted to validate the prediction of the mechanical behavior, considering both anisotropy and ductile damage in the TRIP 780 steel. Figure 17 compares the experimental results with the numerical predictions of the uniaxial tensile testing. The FEM predictions demonstrate that the anisotropic Hill’s parameters accurately captured the TRIP780 steel plastic anisotropy behavior, achieving R²-values of 0.999, 0.999, and 0.999 for the specimens’ orientation of 0°, 45°, and 90° to the rolling direction, respectively.

The calibrated parameters also demonstrated high accuracy in predicting the true stress response for all three specimen types (UT, ID, and PS), as seen in Figure 18. The numerical predictions yielded R²-values close to 1, thus indicating an excellent agreement between the predicted and experimental true stress–true strain behavior under the investigated deformation modes.

The principal total true strain values (${\epsilon }_1$ and ${\epsilon }_2$) at the damage onset for the UT (RD) deformation mode, obtained from FEM numerical predictions and DIC measurements, are compared in Figure 19. Figure 19a,b illustrate good agreement between the principal strain values obtained in the same region of interest. The predicted and experimental values of ${\epsilon }_1$ were 0.548 and 0.542, respectively. However, for the minor total principal strain, ${\epsilon }_2$, the FEM of the UT forecasted −0.199, whereas the DIC value was −0.141. The equivalent plastic strain at the fracture site of the UT specimen, calculated using the experimental ${\epsilon }_1$ and ${\epsilon }_2$ values, was 0.547. Figure 19c,d show the equivalent plastic strain distribution in the central region of the specimen at the onset of fracture and post-fracture, as predicted by the proposed FE model. The predicted values were 0.543 at fracture initiation and 0.547 after fracture, closely matching the experimentally resulting value. This agreement demonstrates that the proposed methodology provides a simplified but reliable description of the ductile damage behavior of the TRIP 780 steel.

The post-fracture predicted shapes of all investigated specimen geometries (UT, ID, and PS) are compared in Figure 20 to the corresponding experimental DIC instant frames. For comparison purposes, the finite element model geometries have all been mirrored, from which one can verify that the proposed experimental–numerical procedure reasonably describes the final fracture profiles to calibrate the material parameters of the TRIP780 steel. Figure 20 also presents a perpendicular image of the fractured region for each of the deformation modes.

4. Discussion

The findings of this study demonstrate the feasibility of a simplified experimental–numerical approach for calibrating the plastic and anisotropic behavior of a TRIP 780 steel. The adopted methodology reduced the number of required specimen geometries compared to conventional calibration approaches while maintaining high prediction accuracy. The Hill’s 48 quadratic anisotropic yield criterion effectively captured the initial plastic anisotropy behavior of this material, as indicated by the agreement between experimental and numerical true stress–strain curves. For this, the three Lankford r-values determined in uniaxial tensile tests (0°, 45°, and 90° with respect to RD) with the fitted uniaxial tensile work-hardening curve and their suitable integration in the Abaqus/CAE code ensured accurate description of the plastic behavior, which is essential for sheet metal forming simulations. The ductile damage model was successfully calibrated using a phenomenological approach, with the stress triaxiality and equivalent plastic strain at damage onset determined through experimental–numerical correlation. The resulting damage initiation criterion agreed with experimental observations, capturing the transition from plastic deformation to fracture. XRD analysis confirmed the deformation-induced transformation of retained austenite into martensite, significantly impacting the mechanical response of TRIP-aided steels. While the TRIP effect enhances ductility, it also introduces complexities in constitutive modeling due to the kinetics of phase transformation. Further investigations incorporating micromechanical phase transformation modeling would improve modeling accuracy, particularly by using XRD analysis to assess residual retained austenite under deformation modes beyond simple uniaxial tension. Such a micro-macro approach would help refine the model for different triaxiality states and explore conceivable strategies to exploit the TRIP effects in sheet metal forming processes. Finally, microstructural analysis provided valuable insights into the damage mechanisms governing failure in TRIP 780 steel. Void nucleation at phase boundaries and subsequent coalescence into larger cavities were consistent with classical ductile fracture models. These findings highlight the importance of considering microstructural evolution in damage plasticity modeling of metallic materials for engineering applications.

5. Conclusions

This study proposed and validated a simplified experimental–numerical methodology for calibrating the mechanical behavior of TRIP 780 steel, considering the initial plastic anisotropy and ductile damage. The key findings are summarized as follows:

• Hill’s 48 quadratic yield criterion effectively described the initial plastic anisotropy of the TRIP 780 steel sheet based on the experimental Lankford r-values to describe the behavior of UT, ID, and PS deformation modes.
• The adopted damage modeling accurately captured the experimentally observed fracture behavior, despite minor limitations in damage onset determination for complex strain paths (ID and PS).
• The strain-induced transformation of retained austenite into martensite was experimentally quantified, revealing a significant reduction in the retained austenite fraction with increasing plastic strain. Most of this phase transformation occurred before the onset of damage under uniaxial tension, suggesting that the TRIP effect has a greater influence on parameter calibration before the damage initiation criterion.
• The proposed methodology provides a cost-effective and experimentally feasible alternative for constitutive modeling and parameter calibration of AHSS grades, with practical applications in sheet metal forming process simulations. Future research should explore the integration of micromechanical modeling to further refine damage predictions and phase transformation effects in TRIP steels.