In this work, the damage behavior of cold-rolled zinc-coated dual-phase steel sheets 600 and 800 grades was evaluated by means of standard uniaxial tensile testing, microstructural characterization and finite element modeling. The void formation in both steels was investigated as a function of the plastic strain level from digital image processing of scanning electron micrographs to quantify the average void size and to determine the corresponding measures of void density, void area fraction and void aspect ratio. Based on the void analysis and the experimental uniaxial tensile test data, a four-step procedure is proposed to identify the parameters of the Gurson–Tvergaard–Needleman (GTN) damage model. Bearing in mind the non-uniqueness of the GTN model parameters, 3D finite element simulations using a single element and an 1/8 symmetry uniaxial tensile specimen model were performed in a systematic manner to investigate the role of both nucleation and failure parameters on the uniaxial behavior. At low straining levels, most of the void initiated by debonding of globular aluminum-oxide inclusions identified by EDS in both dual-phase steels. At higher straining levels, the void density evolution is found to be more pronounced in DP600 steel owing to the faster void nucleation rate at the ferrite grain boundaries. A good agreement between predictions and experimental load-elongation is achieved allowing for the identification of the full set of GTN damage model parameters of both DP600 and DP800 steels. The proposed damage calibration of dual-phase steels can be very helpful in simulations of practical sheet metal forming processes.

In the last few years, advanced high strength steels (AHSS), such as dual-phase (DP) steels, have been received great importance in the automotive industry. These steels are used to compose the structural parts of the vehicles aiming to the passengers’ safety. The DP steels, one of the most prominent AHSS, provide a good compromise between sheet metal formability (low initial yield stress) and enhanced mechanical properties (high ultimate tensile strength) thanks to the ferritic–martensitic structure which is usually obtained by a continuous annealing process [1]. AHSS sheets allow manufacturing of structural components with reduced thickness and, thus, producing lighter vehicles with the compromise of reducing the fuel consumption and greenhouse gas emissions. The structural automotive components manufactured from metallic sheets are typically formed using a punch-die tooling to obtain a desired geometry part. In industrial sheet metal forming process of conventional steel grades, localized necking usually governs the blank fracture. However, this is not the prevailing mechanism in the case of AHSS as, for instance DP steels, in which failure modes can be controlled by thickness instability and either ductile tensile fracture or ductile shear fracture [2]. Such failure mechanisms in DP steels are strongly related to the microstructural heterogeneity, namely, the fracture may start by decohesion of the harder martensite phase located at the boundary along with the softer ferrite matrix which, in turn, withstands the strain localization process. Microstructural damage analysis performed in a DP600 steel grade using digital image correlation to analyze the local strains during uniaxial tensile test, inside a scanning electron microscope, revealed that the martensite phase is also plastically deformable and that the damage evolution occurred owing to both void coalescence and micro-crack formation within the microstructure [3]. The ductile fracture behavior of DP steels has also been analyzed by quantitative analysis of voids as a function of the strain level in uniaxial tensile test using in situ X-ray tomography [4,5] and in terms of the strain-path [6] or the stress triaxiality [7,8].

Considering the complex intrinsic multiphase characteristic of AHSS, the investigation of microstructure role as well as the initiation and quantification of the damage process is crucial to describe and identify the material plastic behavior in sheet metal forming numerical simulations. In this experimental–numerical context, the current work evaluates the damage behavior of cold-rolled zinc-coated dual-phase steel sheets DP600 and DP800 grades by means of standard and interrupted uniaxial tensile tests, microstructural characterization for quantifying the phases and grain size measures, void analysis and numerical finite element simulations for identifying the material parameters of a phenomenological damage model. Microstructural measurements in the as-received conditions to quantify the average void area fraction allowed to define an effective stress measure to account for the damage behavior of both DP steels as a function of the plastic strain level imposed in uniaxial tensile deformation mode. Besides, the nucleation parameters of the Gurson–Tvergaard–Needleman damage model [9,10], hereafter, GTN model, were determined from the maximum uniform plastic strains values. Then, the corresponding yield loci parameters were determined from the minimization of an error function defined between the forecasted and experimental load–elongation uniaxial tensile testing curves.

2Experimental Procedure2.1Materials and mechanical testingDual-Phase steels 600 and 800 grades, produced by USIMINAS (Brazil) as cold rolled zinc-coated sheets with nominal thickness of 1.2mm were investigated by means of uniaxial tensile testing, microstructural characterization and finite element modeling. Chemical composition of both dual-phase steels is listed in Table 1. Uniaxial tensile testing was performed in a screw-driven universal testing machine with 20kN load capacity equipped with a 50mm gauge extensometer. The uniaxial specimens were manufactured by NC milling from sheets taken along the sheet rolling direction. Continuous uniaxial tests were performed at room temperature using a constant cross-head speed of 1mm/min (nominal strain-rate of 3×10−4s−1) up to specimen fracture. To measure the void area fraction as a function of the tensile axis specimen strain, interrupted tests were performed at four increasing straining levels. For DP600 steel, the strain levels were defined as 5%, 10%, 15% and 19.5%, while for DP800 steel the corresponding strain levels were taken as 5%, 10%, 15% and 18%. For this procedure, a total of 16 specimens were tested using a constant cross-head speed of 1mm/min. After testing, the specimens were carefully cut in the Buehler IsoMet 1000 precision cutter to analyze the micro-voids through the uniaxial specimen thickness direction. The analyzed regions are schematically shown in Fig. 1.

Chemical composition of DP600 and DP800 steels (% weight).

Steel | C | Mn | P | S | Si | Cu | Ni | Cr |
---|---|---|---|---|---|---|---|---|

DP600 | 0.092 | 1.704 | 0.018 | 0.005 | 0.036 | 0.035 | 0.012 | 0.026 |

DP800 | 0.104 | 1.779 | 0.017 | 0.006 | 0.017 | 0.025 | 0.011 | 0.026 |

Steel | Mo | Sn | Al | Nb | V | Ti | B | N |

DP600 | 0.188 | 0.001 | 0.042 | 0.016 | 0.008 | 0.004 | 0.0001 | 0.0078 |

DP800 | 0.210 | 0.001 | 0.038 | 0.015 | 0.008 | 0.004 | 0.0001 | 0.0076 |

Average grain sizes and volume fraction measurements were conducted on as-received samples prepared using standard metallographic procedures. The samples were firstly polished and etched with Nital 10% solution during 4s and then analyzed by scanning electron microscopy (SEM) technique with the Carl Zeiss EVO MA10 microscope model. The grain sizes and volume fractions of both DP600 and DP800 constituents were determined from digital images of twenty regions by using magnifications equal to 5000 and 7000× for DP600 and DP800 steels, respectively. The corresponding volume fractions were calculated in accordance with ASTM E562 standard procedure [11]. Both ferrite and bainite grain sizes were measured in conformity with ASTM E112 standard [12], whereas, the martensite grain size was determined from martensite islands using ASTM E1245 procedure [13].

To analyze and quantify the micro-voids, the specimens were polished using firstly alumina suspension and secondly with diamond suspension. After that, each tested specimen was analyzed without etching the microstructure on the SEM Carl Zeiss EVO MA10 model. For this purpose, a zoom of 500× was used and eight images were captured at three regions of each deformed specimen which resulted in a total of twenty-four images for each uniaxial tensile specimen. The quantification of the micro-voids was performed using the software Image-J with the help of the “Analyze Particles” function. From the image analysis, the void density (number of voids per μm2 × 1000) is obtained as well as the void area fraction (Σ void area/total area × 100%), the void aspect ratio (mean void length/mean void width) and the average void size (Σ void area/void number).

To determine the initial porosity parameter or the initial void volume fraction, which is required to describe the material damage behavior in the adopted model, GTN model detailed in the next section, a representative volume element (RVE) is assumed to be equivalent to the binarized image obtained from the SEM analysis, as schematically depicted in Fig. 2. The RVE is composed of a spherical void embedded concentrically in a spherical metal matrix. Assuming the material matrix as a larger sphere of diameter D and a single void as a concentrically small sphere of diameter d, as shown in Fig. 2, to be equivalent to the actual binarized SEM areas of the material matrix and voids, respectively, the initial void volume fraction (f0) can thus be estimated from the experimentally measured initial fraction area of voids (fA) determined by digital image analysis performed in the as-received conditions, that is:

As some voided regions were related to inclusions in both investigated dual-phase steels, inclusion analysis was also performed using the ASTM E45-13 standard [14]. For this purpose, the as-received samples were polished and etched with Nital 3% solution during 3 s and then analyzed in the Nikon optical microscopy Eclipse LV150 using polarized light. Energy-dispersive X-ray spectroscopy (EDS) element analysis was also performed to identify globular inclusions content using the FEI Quanta 250 FEG model scanning electron microscopy.

2.3Damage analysis and finite element modelingThe GTN model is adopted in this work to describe the plastic damage behavior of both dual-phase steels. The GTN model has been widely used to describe the micromechanical effects of damage in ductile metals and is defined by the following yield function [9,10]:

withwhere fc is a critical value of void volume fraction and fu=1/q1 while ff are the void volume fraction at fracture. In Equation (2)q1, q2 and q3 are material parameters, whereas, σ¯ is the von Mises equivalent stress measure, f∗ is the damage parameter, Σ is the stress triaxiality factor defined by the ratio between the hydrostatic stress σh=σkk/3 and the yield or flow stress σy. In the GTN model, the damage arises partly from the growth of existing micro-voids and the nucleation of new voids by mechanisms of cracking or the interface decohesion of inclusions and or precipitates. The damage evolution equation in the GTN model is written in terms of an additive rate decomposition of void growth (VG) and void nucleation (VN) terms as [10]:in whichIn Eq. (4), ε˙P stands for the plastic strain-rate second-order tensor and I is the second-order identity tensor, while ε¯˙P is the equivalent plastic strain-rate. In Equation (5), AN is defined by a nucleation plastic-strain which follows a normal strain distribution with an average plastic strain value ɛN and a corresponding standard deviation SN. The parameter fN is the void volume fraction of nucleating particles. The damage material behavior using the GTN model is completely defined by three material parameters (q1, q2, q3), three void nucleation parameters (ɛN, SN, fN) and two failure parameters (fC, fF) along with the initial porosity r0=(1–f0), usually taken >0.9. In this work, the GTN model parameters were identified from finite element simulations of the experimental continuous uniaxial tensile testing performed in DP600 and DP800 sheets performed at a constant cross-head speed of 1mm/min. To identify the GTN damage model nucleation parameters, firstly a single 3D solid element with only one integration point was adopted along with 1/8 symmetry displacement boundary conditions. Afterwards, the uniaxial tensile tests were simulated using an 1/8 symmetry finite element model with 8-node linear 3D solid elements with reduced integration. The numerical simulations were performed with the ABAQUS/Explicit commercial finite element code wherein the GTN model is available. The proposed finite element mesh for the uniaxial tensile test specimen is depicted in Fig. 3 with the imposed displacement boundary conditions (symmetry along X=0 plane, Y=0 plane and Z=0 plane) and constant prescribed velocity along the X-axis in the corresponding region of the universal mechanical testing machine grip.

The elastic behavior of both dual-phase steels is assumed to be isotropic and linear and is defined by the Young modulus (207,000MPa) and the Poisson's ratio (0.29). The experimental uniaxial tensile true-stress and plastic true-strain data were fitted to the Swift's hardening equation. Then, the corresponding plastic behavior is defined in a tabulated form as a function of the plastic strain. For this purpose, it is assumed an isotropic work-hardening behavior described by the associated flow rule together with the von Mises isotropic yield criterion. In effect, both DP600 and DP800 steels present an anisotropic behavior in both uniaxial flow stress and Lankford R-values. The average normal plastic anisotropy values of DP600 and DP800 steels are equal to 0.93 and 0.86, respectively [15]. Nevertheless, the plastic anisotropy effects are neglected in the present work as a first approximation to describe phenomenologically the main ductile fracture mechanisms, specifically, the nucleation and growth of voids, of both DP600 and DP800 as a function of the plastic straining level resulting from the uniaxial tensile test.

3Results and discussion3.1Microstructures and mechanical propertiesThe microstructures of DP600 and DP800 steels obtained by SEM from the as-received conditions along the rolling direction are shown in Fig. 4. The corresponding average grain sizes (d0) and volume fractions (Vv) are listed in Table 2. Both steels have a ferritic–martensitic (F–M) structure together with bainite (B) phase the volume fractions of which do not exceed 1%. The average martensite volume fractions determined from SEM digital image analysis of DP600 and DP800 microstructures are equal to 27.3% and 30.1%, respectively. The average ferrite and martensite grain sizes determined for DP600 steel are equal to 5.1μm (5.8μm for DP800 steel) and 1.1μm (1.3μm for DP800 steel), respectively. For the martensite volume fraction range from 11% and 37%, similar average grain sizes were determined by Lai et al. [16], namely, ferrite grain size of 5.3μm and martensite grain size of 2μm (DP600) and ferrite grain size of 4.8μm and martensite grain size of 2.6μm (DP800). The average mechanical properties of DP600 and DP800 steels obtained from the uniaxial tensile true-stress and true-total strain measures at the rolling direction are summarized in Table 3. These properties include the yield stress (σy), ultimate yield strength (σu), uniform strain (ɛu), total strain (ɛT), hardening exponent (n) and plastic anisotropy coefficient (r). Owing to the higher martensite volume fraction, DP800 grade provided higher yield stress and ultimate yield strength values accompanied by a reduction of the uniform strain and the total strain. Both dual-phase steels presented a small plastic anisotropy value (r<1) at the rolling direction. Improved performance of both strength and formability of dual-phase steels, in comparison to similar yield strength of High Strength Low Alloy (HSLA)-steels, can be effectively achieved by low yield stress to ultimate yield strength ratio along with a high uniform to total strain ratio. The yield stress to ultimate yield strength ratio obtained from both DP600 and DP800 grades is close to 0.52 and the corresponding total to uniform strain-ratio is equal to 1.49 for DP600 (respectively, 1.46 for DP800). The uniaxial true-stress and true-strain curves at the sheet rolling direction of both dual-phase steels are compared in Fig. 5. Despite the small difference of the martensite volume fraction obtained for DP600 steel (VM = 27.3%) and DP800 steel (VM = 30.1%), improved mechanical properties were achieved with the later grade. From thermodynamic calculations, performed with the help of Thermo-Calc software, using the chemical composition of some main elements together with the measured volume fractions of both ferrite and martensite, the forecasted C contents of ferrite and martensite of DP600 (respectively, DP800) are equal to 0.0066wt% (0.0067wt%) and 0.32wt% (0.33wt%). From the microstructural characterization of dual-phase steels and corresponding micromechanical modeling simulations performed by Lai et al. [16], it is shown when the martensite volume fraction is higher than 19% that the tensile strength increases with the martensite C content in detriment to the uniform elongation. Also, the numerical investigations obtained for VM≥ 19% revealed that the local plastic strain in the martensite phase increases with martensite C content and, therefore, suggests an improved co-deformation with the ferrite phase which can be achieved with the presence of the harder second phase.

Scanning electron microscopy (SEM) pictures obtained with Nital 10% solution from the as-received conditions along the sheet rolling direction: (a) DP600 (5000×) and (b) DP800 (7000×). The letters F, M and B indicated on the grains stand for ferrite, martensite and bainite, respectively.

Uniaxial mechanical properties of DP600 and DP800 steels [15].

σy (MPa) | σu (MPa) | ɛu (%) | ɛT (%) | n | r | |
---|---|---|---|---|---|---|

DP600 | 392±4 | 749±7 | 15.2±0.3 | 22.6±0.3 | 0.20 | 0.52±0.3 |

DP800 | 451±3 | 867±8 | 12.7±0.2 | 18.5±0.6 | 0.17 | 0.58±0.2 |

For finite element purposes, the true plastic-strain and corresponding true stress values were fitted according to the Swift's hardening equation defined as:

in which K, ɛ0 and n are the strength coefficient, pre-strain and hardening exponent, respectively. The Swift's hardening equation parameters obtained for both steels are listed in Table 4 together with the R-squared fitting values.3.2Void analysisTo quantify the void formation in both dual-phase steels, the as-received and deformed uniaxial tensile specimens were cut perpendicular to the rolling direction, see Fig. 1. Both DP600 and DP800 steels presented small voids and globular inclusions in the as-received conditions. A globular inclusion in the ferrite matrix of the as-received DP600 steel sheet can be seen in Fig. 6(a). Such hard particles will serve as nucleation sites for the formation of microvoids as the plastic strain increases, as shown in the SEM image of Fig. 6(b), wherein a void region is located near two inclusions in DP800 steel deformed in uniaxial tension at a level of 18% total strain. According to the ASTM E45-13 method, the identified inclusions in the as-received conditions of both dual-phase steels are mainly composed by fine globular oxides with severity levels of 0.5 and 1. This standard procedure does not assign a severity level for the inclusions with a width lesser than 2μm. However, the image analysis revealed such inclusions in both DP600 and DP800 steels. Sulphides and silicates inclusions with 0.5 severity level were also identified using the optical microscopy technique along with polarized light. The element analysis obtained from the energy-dispersive X-ray spectroscopy (EDS) of the inclusions observed in the specimens deformed in uniaxial tension at 10% and 18% total strain for DP600 and DP800 steels, EDS results shown in Fig. 7(a) and (b), respectively, revealed the principal concentrations of oxygen (38.6 and 42.1% normalized wt% for DP600 and DP800, respectively) and Aluminum (26.4 and 33.7% normalized wt% for DP600 and DP800, respectively) and, therefore, indicate the presence of aluminum-oxide inclusion type in both investigated dual-phase steels.

Scanning electron microscopy images of the analyzed dual-phase steels: (a) globular inclusion in the ferrite matrix of DP600 in the as-received condition (8000×) and (b) elongated voided areas formed around two inclusions in the DP800 deformed in uniaxial tension at a level of 18% (7000×).

With increasing level of straining, microvoids might nucleate at the ferrite–martensite interfaces, between martensite particles and near the inclusions. In the present work, most of the microvoids formed at different straining levels are associated to the harder globular inclusions observed in both dual-phase steels. Conversely, at higher straining levels, which are close to the uniform elongation values of DP600 and DP800 steels, the voids were found near to the F–M interfaces, at the ferrite matrix and near to martensite islands revealing, thus, the strong dependence of the void formation mechanisms on the plastic straining level. Such nucleation of voids, growth and quasi-coalescence characteristics can be observed in Figs. 8 and 9 for two different uniaxial strain levels imposed to DP600 (10% and 19.5%) and DP800 (10% and 18%) steels, respectively. These micrographs were taken parallel to the major strain axis, that is, along the uniaxial specimen length which is the sheet rolling direction. Firstly, one can observe the formation of two kinds of microvoids in the DP600 steel deformed at 10% of total strain, see Fig. 8(a), namely, microvoid V1 which is associated to an inclusion I and a second microvoid V2 formed at the ferrite matrix surrounded by martensite islands. Besides, it is worth to envisage a possible coalescence of these two voided regions if the straining localization process would have further continued in the softer ferrite matrix. At a higher level of uniaxial strain, micrograph shown in Fig. 8(b), a larger microvoid is formed in DP600 microstructure matrix which is elongated along the major strain axis. This microvoid appeared to initiate at the inclusion vicinity and is assisted on the opposite extremity by the decohesion at the martensite–ferrite interface. At 10% of total strain, DP800 steel microstructure showed a microvoid formation which is very similar to that observed for the same level of uniaxial straining in DP600 steel, see Figures 8(a) and 9(a). For a higher strain level of 18%, a microvoid propagation in DP800 steel shown in Fig. 9(b) is partially associated to the inclusion enhanced by a decohesion along the interfaces between the ferrite matrix and martensite islands, which act as a preferred propagation path. Also, it is possible to observe some particles resulting from the matrix degradation. For the same level of straining in DP800 steel, a plausible formation sequence of microvoids which are not related to the inclusions is presented in Fig. 10. A narrow void can be observed in the ferrite matrix, Fig. 10(a), which can be viewed as representative of the initial aspect of microvoids nucleated in the ferrite. Conversely, more advanced void formation stages can be observed in Fig. 10(b) and (c) which, in turn, may have originated in the ferrite matrix and then were propagated as the deformation increased in ferrite grains surrounded by the martensite islands. The plastic localization process endured by the edges of the ferrite grains results in higher stress concentration levels in these regions leading, therefore, to the nucleation and growth of voids in the ferrite matrix and the interface decohesion of the ferrite and martensite islands.

Scanning electron microscopy images from the DP600 uniaxial specimens deformed at two levels of total strain: (a) 10% (5000×) and (b) 19.5% (5000×). These micrographs were taken at regions cut parallel to the sheet rolling direction, i.e., along the major strain axis. The letters F, M, V and I designate ferrite, martensite, void and inclusion, respectively.

Scanning electron microscopy images from the DP800 uniaxial specimens deformed at two levels of total strain: (a) 10% (5000×) and (b) 18% (10,000×). These micrographs were taken at regions cut parallel to the sheet rolling direction, i.e., along the major strain axis. The letters F, M, V and I designate ferrite, martensite, void and inclusion, respectively.

Scanning electron microscopy images from the DP800 uniaxial specimens deformed at 18% of total strain with a conceivable sequence of microvoids formation: (a) nucleated void in the ferrite matrix (20,000×), (b) void propagation in ferrite grains (10,000×) and (c) void surrounded by martensite islands (5000×). These micrographs were taken at regions cut parallel to the sheet rolling direction, i.e., along the major strain axis. The letters B, F, M, V and I designate bainite, ferrite, martensite, void and inclusion, respectively.

The quantitative void analysis was performed from the binarized SEM images obtained by assembling several micrographs as, for instance, the binarized images in Fig. 11, taken from the uniaxial tensile specimens deformed at levels of 19.5% and 18% total strain for DP600 and DP800 steels, respectively. For the as-received conditions and for each imposed strain level, the corresponding digital image analysis provided the quantification of the void area fraction (Σ Void area/Total area × 100%), void density (Void number per μm2×1000), mean void size (Σ Void area/Void number) and void aspect ratio (Mean void length/Mean void width). These results are shown in Figs. 12 and 13 as a function of the imposed longitudinal total strain in the uniaxial testing. The data at zero strain correspond to the void measurements on as-received conditions. For the sake of conciseness, the void measurements data are grouped in two graphs, namely, (1) void density and void area and (2) void aspect ratio and mean void size, Figs. 12 and 13, respectively. From the void results, one can firstly observe that the void density, void area and mean void size increased with the imposed uniaxial strain. The present void measures are qualitatively consistent with the voids data determined by Samei et al. [6] and Saedi et al. [7] for DP600 and DP800 steels, respectively.

However, the current void data showed dispersed values at some straining levels. Mostly of the measured voided regions in both steels are related to the inclusions which influenced the void measurements values due to the average inclusion size. Likewise, the heterogeneity with respect to the number of voids in some of analyzed regions may have contributed to the higher standard deviation values of the average void results. Despite these deviations, at larger strains one can observe a higher void density in DP600 which can be ascribed to the faster void nucleation rate, as shown in Fig. 12(a). Conversely, the average values of void area and mean void size obtained for both steels are very close with a slight increase in the mean void size with the strain level. In both dual-phase steels, the average values of the measured void aspect ratio are greater than 1 and, therefore, indicate that the voids assumed an ellipsoidal shape along the major stress/strain direction.

3.3Finite element analysisIn this section, the effective work-hardening parameters are firstly obtained from the measured fraction area of voids and the true-stress and true-strain hardening curve fitted to the experimental uniaxial results determined at the rolling direction. Secondly, the nucleation parameters of the GTN damage model (ɛN, SN, fN) are evaluated from the simulations of a single 3D finite element model. The initial porosity parameter, r0=1−f0, is determined from the initial average volume fraction of voids, f0, which, in turn, is calculated from the initial fraction area of voids fA using Equation (1). Afterwards, the yield loci parameters (q1, q2, q3) are defined from the micromechanical predictions determined in the numerical study performed in [17]. Lastly, the two failure parameters (fC, fF) are obtained from the experimental values of the maximum yield strength and the ultimate fracture strain. From this three-step procedure, six sets for the GTN model parameters are proposed bearing in mind that the identification of the full set is not unique, namely, different parameters sets may result in the same damage behavior. To overcome this, the full parameters set can be obtained from the minimum standard error response (SE):

defined between the experimental uniaxial force–elongation curves of both DP600 and DP800 steels and the corresponding numerical predictions obtained with the proposed sets of the GTN model parameters using a 3D finite element model of the uniaxial tensile specimen.3.4Effective work-hardeningThe first step of the proposed procedure consisted in defining an effective plastic behavior for which the material matrix is completely dense, that is, with no initial voids. For this purpose, the work-hardening equation describing the experimental plastic behavior of both dual-phase steels, Equation (6), is corrected using the concept of a damage effective stress which is defined considering the area which effectively supports the uniaxial tensile load F[18]. The effective stress concept is schematically depicted in Fig. 14 with a representative volume element (RVE) in which F is the load, whereas, S and Sv stand for the overall section and the voided areas of the RVE, respectively. Thus, the effective stress measure can be defined as:

In Eq. (8), fA=S/SV denotes the fraction area of voids and σy is the stress defined from Equation (6) which parameters were fitted to the experimental uniaxial tensile curves, see Table 4. Assuming the hypothesis of isotropy meaning that the voids are equally distributed in all directions as well as independent of the mechanical loading path, tension or compression, the voided fraction area fA can thus be viewed as a scalar damage variable.

Secondly, from the average measured values of the voided area fraction fA as a function of the total longitudinal strain in uniaxial tension, results taken from Fig. 12, and neglecting the elastic strains, a linear equation is proposed to describe the evolution of fA as a function of the longitudinal plastic strain, namely:

Fig. 15 shows the linear fitting results determined from the experimental average void fraction area of DP600 and DP800 steels. The values of the α and β coefficients of Equation (8) are indicated in these figures. A rather good agreement is found with the proposed linear approximation. The effective true stress curves obtained from Eq. (8) along with Eq. (9) are compared with the experimental true-stress and true plastic-strain data in Fig. 16 and to the Swift fitting work-hardening predictions obtained with Eq. (6). Henceforward, the effective stress curves are used in all finite element simulations.

Comparison between the experimental true-stress and true plastic-strain obtained from uniaxial tensile testing, the corresponding curves fitted with swift work-hardening equation and the corresponding effective true-stress measure determined from the experimental fraction area of voids: (a) DP600 and (b) DP800.

The nucleation of new voids in the GTN model is controlled by the equivalent plastic strain-rate along with a probability density function, which follows a normal distribution, see Eqs. (4) and (5), and depends on the calibration of three parameters (ɛN, SN and fN). ɛN is a mean value of characteristic nucleation plastic-strain and SN its corresponding standard deviation. The parameter fN stands for the volume fraction of the nucleated voids. In the GTN model, the nucleation of new voids description is completed with the initial porosity value which is determined as a function of the initial void volume fraction as r0=1−f0. This can be performed by assuming that the initial void area fraction, obtained from the digital image analysis of SEM images, is equivalent to an RVE matrix composed by a sphere of diameter D with a single concentrically small sphere of diameter d accounting for all initial voids of the as-received conditions of both dual-phase steels. This RVE correspondence, detailed in Section 2, see Fig. 2, allows one to determine the initial average void volume fraction (f0) from the measured initial average fraction area of voids (fA) by means of Eq. (1) and, hence, to determine the initial average porosity as r0=1−f0. The initial average values of the obtained porosity parameters, along with the average initial fraction area and calculated initial fraction volume of voids of DP600 and DP800 steels are listed in Table 5.

To identify the nucleation set parameters (ɛN, SN, fN), it is firstly assumed that the nucleation of new voids fN reaches a maximum value at the uniform plastic strain εuP obtained in the uniaxial tensile test. Thus, the value of the characteristic nucleation plastic strain AN can be estimated as εN=εuP/2. Then, to select an appropriate value of the standard deviation, SN, the probability density function to nucleate new voids, see Eq. (5), must be evaluated in the range of the plastic straining of the material which, in all cases is limited up to the εuP-value. Plotting both the probability density function for void nucleation, AN defined by Eq. (5), and the resulting nucleation void volume fraction, NVVF, as a function of the equivalent plastic-strain, is an appropriated way to better explain the choice of the GTN damage model nucleation parameters SN and fN. The nucleation void volume fraction is calculated by integration of Eq. (5):

These plots are shown in Figs. 17 and 18 based on the experimental data obtained from the uniaxial tensile tests of both dual-phase steels. Two sets of (SN, fN) parameters were firstly chosen to virtually reproduce the same uniaxial tension plastic behavior for DP600 and DP800 steels. The first set with a very small value of the standard deviation (SN=0.030, fN=0.010) and (SN=0.028, fN=0.014) for DP600 and DP800 steels, respectively. Conversely, the second set defined by choosing a larger SN value, which is in the range of dual-phase steels (SN=0.200, fN=0.028) for DP600 steel and (SN=0.200, fN=0.042) for DP800. A small standard deviation SN increases the probability of nucleating new voids near to the values of the characteristic nucleation strain, ɛN and, consequently, resulting in a more heterogeneous void nucleation mechanism during the plastic straining process, as shown in Fig. 17(b). Likewise, increasing the standard deviation SN value to 0.2 offers a small variation of the probability void nucleation density function in the range of interest and a constant void nucleation rate. Thus, the initial choice for the remaining nucleation parameters is defined by the two sets bounded between small and large values of the standard deviation, namely, (SN=0.030, fN=0.010) and (SN=0.200, fN=0.028) for DP600 and (SN=0.028, fN=0.014) and (SN=0.200, fN=0.042) for DP800.

For steels, the typical values of the GTN yield locus parameters qii=1,2,3 in Eq. (2) are q1=1.5, q2=1.0 and q3=q12=2.25[10]. The parameter q1 affects the load-bearing capacity of the material, namely, q1>0 provide a decrease of the yield strength leading to a material softening due to void growth in detriment of the work-hardening of the matrix material. The second GTN yield locus parameter q2 is associated to the stress triaxiality factor in the GTN yield function, Σ=σn/σy in Eq. (2), which depends on the hydrostatic stress and the material yield or flow stress σy. Choosing high σ2-values (>1.0) will produce a strong softening of the matrix material.

West et al. [19] proposed an experimental–numerical procedure to determine the GTN damage parameters of DP600 as a function of the effect of the stress triaxiality factor Σ. From the work of West et al. [19], the yield loci parameters calibrated to describe the behavior of DP600 steel are q1=1.2, q2=0.9 and q3=1.44 for 0.45≤Σ≤0.67 which, in turn, is the range of stress triaxiality between a uniaxial tensile specimen with a central hole (Σ=0.45) and a wide plane-strain tension specimen (Σ=0.67).

By means of micromechanical simulations considering unit cell finite element models with a homogeneous material and a single spherical void, the former using GTN model, Faleskog et al. [17] proposed a procedure for the yield locus parameters calibration revealing that q1,q2 parameters exhibited a dependence on the strain-hardening exponent n and on the ratio between the yield stress σy and Young's modulus E. This procedure is adopted in the present work to define the yield loci parameters avoiding extensive experimental and or combined numerical investigations given that the non-uniqueness of the GTN damage model parameters. From the calibration results of the procedure developed by Faleskog et al. [17], reproduced in Fig. 19 by the plots of the q1,q2 parameters as a function of the strain-hardening exponent n and the ratio σy/E, the yield locus parameters found for DP600 steel are q1=1.86, q2=0.80 and q3=q12=3.46 whereas, the corresponding parameters for DP800 steel are q11.76, q2=0.83 and q3=q12=3.09. The usual values of the yield loci parameters for steels are also considered to describe the yield loci behavior of both dual-phase steels (q1=1.5, q2=1.0 and q3=q12=2.25). Thus, two sets of the GTN yield loci parameters are evaluated for each dual-phase steel.

Yield loci parameters q1, q2 determined from micromechanical predictions of Faleskog et al. [17] together with the experimental data of DP600 and DP800 steels: (a) parameter q1 and (b) parameter q2.

By choosing the possible values of both nucleation (ɛN, SN, fN) and yield loci q1,q2,q3 parameters together with the identification of the initial average porosity value r0, which is obtained from the measured fraction area of voids in the as-received state, the full set of GTN damage model parameters is completed by selecting two failure parameters (fc, fF). The critical void volume fraction, which is the failure parameter fc, is obtained from the plastic strain corresponding to the ultimate yield strength, that is, the uniform plastic strain. This can be justified by assuming that the damage increase will be faster when this critical value is reached owing to the coalescence of the voids.

In the GTN model, the coalescence of voids is accounted for in the void growth rate term, see Equation (4), which, in fact, depends on the volumetric plastic-strain rate variation. In the same way, the void volume fraction at fracture, the second failure parameter fF, is determined from the plastic-strain value at fracture. The failure parameters (fc, fF) are then assessed from the single 3D finite element model by plotting the numerical predictions of the total void volume fraction f, the void growth volume fraction fVG and the void nucleation volume fraction fVN as a function of the equivalent plastic-strain. For this purpose, an element removal procedure is performed wherein the single element is deleted once the condition f>fF is met at the integration point. This damage failure criterion by element removal results in zero force and, therefore, produces an element unloading process by means of material stiffness degradation. In Fig. 20, the numerical predictions obtained from a single 3D finite element model are compared to the experimental true-stress and true-strain data of the uniaxial tensile tests of dual-phase steels DP600 and DP800. These predictions were obtained from six different parameters sets of the GTN damage model for each dual-phase steel. Tables 6 and 7 summarize the evaluated sets of the GTN damage model parameters proposed for DP600 and DP800 steels, respectively. From these results it is worthwhile to observe that different combinations of the GTN damage parameters lead to equivalent stress–strain behavior for both dual-phase steels. However, small variations on the failure parameter fc, fF which describes the onset of the coalescence of existing voids, produce large variations on the unloading behavior of both DP600 and DP800 steels. The proposed procedure to identify the failure parameters (fc, fF) is depicted in Fig. 21 where the finite element predictions calculated with GTN model parameters set # 1 for the total void volume fraction fVN, void growth volume fraction fVG and void nucleation volume fraction fVN are plotted as a function of the equivalent plastic strain. Firstly, one can observe the saturation in the volume fraction of nucleated voids which, in fact, corresponds to the nucleation parameter fN and is equal to 0.010 for DP600 steel (0.014 for DP800). Secondly, the failure parameter fc is defined as 0.0168 for DP600 (and 0.0239 for DP800) from the abrupt change in the total void volume fraction resulting from the variation of the void growth volume fraction, as indicated in Fig. 21. Lastly, the failure parameter fc is determined as the total void volume fraction value f at the fracture condition and is equal to 0.018 for DP600 (and 0.026 for DP800).

Evaluated sets of the GTN damage model parameters: DP600 steel.

Set | r0 | ɛN | SN | fN | q1 | q2 | q3 | fC | fF |
---|---|---|---|---|---|---|---|---|---|

# 1 | 0.995 | 0.090 | 0.030 | 0.0100 | 1.5 | 1 | 2.25 | 0.01684 | 0.018 |

# 2 | 0.030 | 0.0100 | 1.86 | 0.80 | 3.46 | 0.01632 | |||

# 3 | 0.200 | 0.0280 | 1.5 | 1 | 2.25 | 0.01684 | |||

# 4 | 0.200 | 0.0280 | 1.86 | 0.80 | 3.46 | 0.01632 | |||

# 5 | 0.030 | 0.0104 | 1.86 | 0.80 | 3.46 | 0.01684 | |||

# 6 | 0.200 | 0.0293 | 1.86 | 0.80 | 3.46 | 0.01684 |

Evaluated sets of the GTN damage model parameters: DP800 steel.

Set | r0 | ɛN | SN | fN | q1 | q2 | q3 | fC | fF |
---|---|---|---|---|---|---|---|---|---|

# 1 | 0.993 | 0.085 | 0.028 | 0.0140 | 1.5 | 1 | 2.25 | 0.02390 | 0.026 |

# 2 | 0.028 | 0.0140 | 1.76 | 0.83 | 3.09 | 0.02335 | |||

# 3 | 0.200 | 0.0420 | 1.5 | 1 | 2.25 | 0.02390 | |||

# 4 | 0.200 | 0.0420 | 1.76 | 0.83 | 3.09 | 0.02335 | |||

# 5 | 0.028 | 0.0146 | 1.76 | 0.83 | 3.09 | 0.02390 | |||

# 6 | 0.200 | 0.0445 | 1.76 | 0.83 | 3.09 | 0.02390 |

Next, the choice within the evaluated sets of the GTN model parameters will be based upon the minimum standard error (SE), defined by Equation (7), by comparing the experimental load–elongation curves of both dual-phase steels with the numerical results obtained from the 3D finite element model of the uniaxial tensile specimen geometry. For this, Fig. 22 compares the forecasted load–elongation curves with the experimental data of DP600 and DP800. The standard error (SE) values, also shown in these figures, designate the choice of the GTN model parameters, namely, set #5 for DP600 steel SE=166.9N and set # 6 for DP800 (SE=279.5N). The full sets of the GTN model full parameters determined for both dual-phase steels are summarized in Table 8. The nucleation parameters obtained for DP600 with a small standard deviation value SN=0.030 suggest a faster void nucleation rate in comparison to the DP800 steel SN=0.200. From the average void density measured as a function of the longitudinal true-strain, see Fig. 12, it is possible to observe a rapid increase in the void density of DP600 steel (from ∼ 10% true-strain) while DP800 showed a more homogeneous void density evolution. From the experimental average void aspect ratio, shown in Fig. 13, it is also possible to observe a small increase in the void aspect ratio of DP600 steel from ∼ 1.4 up to ∼1.6 indicating a slightly heterogeneous void nucleation.

The proposed four-step procedure to identify the full set of the parameters of the GTN damage model can be summarized as follows. In the first step, an effective work-hardening measure is defined by considering the evolution of the measured void area fraction as a function of the plastic strain. In the second step, the initial porosity and the GTN nucleation parameters are obtained from the measured values of the as-received average fraction area of voids, transformed to the void volume fraction, and the experimental uniform plastic strain, respectively. In the third step, the yield loci parameters of the GTN damage model were defined from the numerical predictions available in the literature determined using a unit cell micromechanical model. Lastly, the choice of the full set of damage parameters is completed based on the failure criterion and the minimum standard error between the forecasted and experimental uniaxial tensile testing load-elongation curves. Conversely, it should be observed that the present identification procedure of the GTN damage model parameters must be further validated by considering both the effects of the strain-path (or the stress triaxiality) and the initial sheet material plastic anisotropy. These effects are not addressed in the present work. In this direction, the proposed procedure could be supported using data from interrupted Forming Limit Curve tests together with digital image analysis of voids formation, as detailed in [6].

4ConclusionsIn this work, the mechanical behavior and microstructural characterization evaluated in the as-received conditions of dual-phase steels DP600 and DP800 are reported. The damage behavior of both dual-phase steels grades is investigated by performing interrupted uniaxial tensile testing to analyze and quantify the formation of voids as a function of the strain level. The voids observed at earlier straining levels are mostly associated to the globular aluminum-oxide inclusions identified by EDS in both dual-phase steels. Conversely, at straining levels close to the uniaxial uniform plastic strain values, the voids were formed near to the ferrite-martensite interfaces as well as in the ferrite matrix and close to martensite islands. The void analysis performed on the SEM images allowed to obtain the void density (number of voids per μm2), void area fraction (Σ void area/total area), void aspect ratio (mean void length/mean void width) and average void size (Σ void area/void number). The void density evolution in DP600 is more pronounced in comparison to DP800 steel. This can be attributed to the faster nucleation rate of voids in the ferrite matrix of DP600 at higher straining levels (>10%). In both dual-phase steels the average void aspect ratios are greater than 1 indicating, thus, that the microvoids assumed an ellipsoidal shape along the major strain direction. The damage behavior of DP600 and DP800 steels was also analyzed by means of finite element simulations of the uniaxial tensile testing. Good agreement between numerical predictions and load-elongation experimental data is achieved allowing for the identification of the full set of the GTN damage model parameters of both DP600 and DP800 steels.

Conflict of interestThe authors declare no conflicts of interest.

The authors express their sincere thanks to USIMINAS Technology Center (Brazil) which supplied the dual-phase steel sheets and to Prof. A. L. V. Costa e Silva (UFF) for helping with the thermodynamic calculations. L.B. Silveira acknowledges CAPES/DS program for granting the M.Sc. scholarships. D.A. Albertacci (PIBIC/UFF), A.S. Paula and L.P. Moreira (PQ research grant) acknowledge CNPq Brazilian agency.

*et al*.

*et al*.