Investigation on the ductile fracture of a high-strength dual-phase steel with anisotropic damage mechanics model

.. In this study, a hybrid experimental and numerical investigation is implemented to characterize the plasticity and ductile fracture behavior of a high-strength dual-phase steel. Uniaxial tensile tests are conducted along the three typical directions of rolled sheet metals for the anisotropic plastic behavior, while the hydraulic bulge test is applied for the flow behavior under equiaxial biaxial tension. Further tensile tests are conducted on various featured dog-bone specimens to study the fracture behavior of the material from the uniaxial to plane-strain tension. On the numerical side, the evolving non-associated Hill48 (enHill48) plasticity model considering anisotropic hardening and plastic strain ratio evolution is employed to describe the anisotropic plastic deformation. The extended enHill48 model with damage and fracture formulation is further calibrated and validated in the study to describe the ductile fracture behavior of the steel under various stress states. Through a comparison of the results based on the evolving anisotropic model with the isotropic Mises model, it is concluded that even for materials that show only minor initial plastic anisotropy, it could develop a non-negligible influence on the large plastic deformation and the prediction of both deformation and fracture shows profound improvement with the evolving anisotropic plasticity model.


Intr Introduction oduction
Dual-phase (DP) steels are usually low-carbon low-alloy materials, which have dispersed martensitic islands embedded in the ductile ferrite matrix as the microstructure [1].Accompanying the development of advanced high strength steels (AHSS), DP steels are widely adopted in the automotive industries, due to their attractive combination of high strength with balanced ductility and good formability at low production costs [2].However, the complexity of the damage and fracture behavior of these materials increases consequently.In order to understand and predict the failure of the material, damage mechanics model development becomes a common interest among researchers.Ductile materials tend to have strain hardening and considerably large plastic deformation as fracture response.Therefore, the ductile crack initiation and propagation in DP steels is a crucial issue in fracture analysis.Accurate numerical prediction of ductile fracture initiation requires fracture models that reflect the physical mechanisms of the fracture process as reasonable as possible, yet do not require extensive material testing for calibration.
There is a wild range of ductile fracture models [3] that have been proposed based on different mechanical backgrounds, from the microscopic level in terms of voids nucleation to macroscopic point of view with continuum mechanics [4][5][6][7][8][9][10].
Among them, a hybrid damage mechanics model, modified Bai-Wierzbicki (MBW) has been proposed by Lian et al. [11].It is based on the original uncoupled Bai-Wierzbicki (BW) [4] fracture model, which emphasizes the Lode angle ESAFORM 2021.MS14 (Formability Metals), 10.25518/esaform21.4322effect on plastic deformation and ductile fracture.Its unique feature is that it differentiates damage from fracture, with an explicit damage initiation criterion and a continuum damage mechanics (CDM) based damage evolution for progressive damage accumulation until the final fracture.Due to the hybrid characteristics of the MBW model, it has the potential and flexibility in describing the ductile failure of a variety of materials featured with distinguished damage mechanisms, either with a pronounced damage evolution stage before the final rupture [11,12] or in other cases where the failure is triggered shortly after damage initiation with rapid damage evolution [13,14].
The above-mentioned models have been only developed and validated for isotropic materials.However, most of the metallic materials show anisotropic deformation behavior due to the nature of the structure as well as the processing history.Various anisotropic plasticity models in conjunction with advanced testing methods have been developed in the past decades especially for metallic material [15][16][17][18][19].To improve the accuracy of the prediction on plasticity and ductile fracture, an evolving non-associated Hill48 (enHill48) model [20], describing the anisotropic hardening and Lankford coefficient (r-value) evolution at different strain levels simultaneously, has been developed.The accuracy and flexibility of the enHill48 model have been approved by the accurate cold formability prediction of the ferritic stainless steel in [21].The extension of the enHill48 anisotropic model to predict fracture by combing the MBW fracture model has been introduced and validated in the previous study [22].
In this study, the fracture model based on the enHill48 anisotropic plasticity model is applied to a DP steel, DP1000, for describing the deformation and fracture behavior under various stress states.The steel shows only minor initial plastic anisotropy and has been described by the isotropic MBW model with satisfactory accuracy in the previous study [14].
By the current work, we aim to demonstrate that non-negligible plastic anisotropy will develop during the deformation for the material with minor initial anisotropy, and through a comparison with the isotropic MBW model, a higher level of accuracy in the description of both deformation and fracture could be achieved with the evolving anisotropic plasticity and fracture model.DP1000 sheet steel with a thickness of 1.5 mm has been investigated in this study.The material is composed of ferrite and martensite phases, with an approximate martensite phase fraction of 45%.The detailed microstructural characterization can be found in [23].To characterize the material anisotropic plasticity behavior, uniaxial tensile tests of the smooth dog-bone (SDB) specimens along three different directions with respect to the rolling direction (RD) from thin sheets have been performed at room temperature (RT, 25°C) and quasi-static (QS, 0.0001 s −1 ).The specimens have been manufactured according to DIN EN ISO 68921 [24].The geometry design and the loading directions are shown in Fig. 1 (a).Detailed experimental setup and procedures on tensile tests can be found in our previous study [14].The experimental engineering stress vs. strain curves along RD, transverse direction (TD), and diagonal direction (DD) from uniaxial tensile tests are shown in Fig. 1 (b).The experimental result shows that the material plasticity properties, from yield strength, ultimate tensile strength, to fracture elongation are somewhat scattered among directions.Furthermore, with an in-depth investigation on the anisotropy evolution, not only the flow stress, i.e. the hardening behavior, but also the plastic strain ratio, i.e. r-value, are evolving during the plastic deformation.The characteristic stress-strain curves and r-value evolution curves with scattered deviation among three parallel tests are shown in Fig. 2 (a) and (b), respectively.In addition, hydraulic bulge tests as described in [12], have been performed for the equibiaxial tension condition.
Investigation on the ductile fracture of a high-strength dual-phase steel with anisotro...

F Fr ractur acture char e charact acterization erization
To experimentally characterize the ductile fracture properties of the investigated material, tensile tests on various specimen geometries under the QS loading conditions have been performed.A category of all flat samples at laboratory level in the current study is shown in Fig. 3 (a), including one central-hole (CH) with a diameter of 6 mm (CHD6), one notched dog-bone (NDB) specimen with radii of 50 mm (NDBR50), and two plane-strain (PS) specimens with notch radii of 2 mm (PSR2) and 16 mm (PSR16).The detailed geometry design is explained in [25].A sketch demonstrating the characteristic stress states in the space of stress triaxiality and Lode angle parameter is depicted in Fig. 3 [11].er [11].3).The flow curve along the reference direction, i.e.RD, is expressed by Y(̅ P ).The flow potential  controls the evolving strain components according to the non-associated flow rule (NAFR) with the equivalent plastic strain rate ̅ ̇P, which is not equal to the conventional nonnegative plastic multiplier , as summarized in Eq. ( 4) and Eq. ( 5).Detailed derivation can be found in Lian et al. [20].
Investigation on the ductile fracture of a high-strength dual-phase steel with anisotro...

4322/4
The equivalent stress () is adopted with the corresponding parameters listed in Eq. ( 6), where  and  are the flow stress and r-value along the corresponding loading direction , and  is the flow stress under the equibiaxial tension condition obtained from the hydraulic bulge test.Since the material is a flat rolling sheet, only in-plane anisotropy is considered.Therefore, the parameters of  and  are typically set as 3 according to Aretz [26].

Hy Hybrid damage mechanics model brid damage mechanics model
In the case of proportional loading, the damage initiation locus (DIL) is formulated as a critical plastic strain ̅ ddi with the dependence of stress states as shown in Eq. ( 7), where 1, 2, 3, 4 are four damage initiation parameters in the symmetric DIL.To characterize the stress states, the stress triaxiality  and Lode angle parameter ̅ are two independent variables that have been widely applied.As shown in Eq. ( 8) and Eq. ( 9), since the stress state is evolving during plastic deformation, the average stress triaxiality avg and the average Lode angle parameter ̅ avg have been adopted.For proportional loading, once the DIL is reached during plastic deformation, the damage initiation is triggered.The damage initiation indicator dd in Eq. (10), which is integrated during the accumulation of plastic strain, is applied in the non-proportional loading conditions.The ductile damage evolution is based on the energy dissipation theory and simplified as linear law.The damage evolution rate as shown in Eq. ( 11), is controlled by the equivalent stress at the damage initiation point ddi and the dissipated energy f between damage initiation and fracture.The critical damage value cr determines the ductile fracture locus (DFL) as Eq. ( 12).The crack propagation in finite element (FE) simulation is accomplished via element deletion, and it starts at the point when the ductile fracture indicator df in Eq. ( 13), reaches unity.The hybrid damage mechanics model, in conjunction with the enHill48 plasticity model, is implemented as a VUMAT user subroutine in ABAQUS/Explicit.To ensure the convergence and enhance computational efficiency, the tangent cutting plane algorithm has been applied [27].From the test results, the localized necking happens at a relatively small strain.Therefore, an appropriate extrapolation based on the experimental data is applied to obtain the strain hardening behavior of the material at large deformation.
The flow curves of plastic deformation are derived in the formulation of the hardening function according to the combined Swift and Voce law in Eq. ( 14), with hardening parameters (, ̅ 0, , 0, , , ).The calibration procedure is similar to the one shown in [28] with an additional modification by introducing a piecewise combination at discrete strain levels.The evolving anisotropic parameters (σ, σ, σ, σ) as shown in Eq. ( 6), are calibrated using the flow stress obtained from uniaxial tensile tests along three distinguished loading directions as well as the equibiaxial tension condition.
The evolution of r-values, obtained from uniaxial tensile tests, is fitted by the Voce exponential law with constant 1C 3 as shown in Eq. ( 15).Based on the r-value evolution, the evolving anisotropic parameters (r, r, r, r) can be calibrated as discussed in Eq. ( 6).
The normalized stress distribution and r-value evolution at three true strain levels 0.002, 0.02, and 0.2, i.e. from the onset of yielding until the extrapolation at large strain level along three loading directions, are depicted in Fig. 5  Investigation on the ductile fracture of a high-strength dual-phase steel with anisotro... surfaces in our previous study [25], damage initiation in DP1000 steel is found to take place at a very late stage of plastic deformation.Therefore, the critical instant, at which a sudden load drop occurs on the force-displacement curve, is determined as the damage initiation point for all specimens.The local variables ̅ ddi, avg, ̅ avg at the critical position and instant in these four specimens are used to calibrate the four damage initiation parameters 1~4 according to Eq.( 7).(blue dashed lines) and enHill48 based anisotropic MBW hybrid ductile model (red solid lines), are depicted in Fig. 6 (a), (b), and (c), respectively.In addition, the previous results with the isotropic extended MBW (eMBW) model [14] indicated by black dotted lines, are also shown as a reference.From the comparison, it is obvious that the evolving plasticity model gives precise predictions on the plastic deformation behavior.The most remarkable improvement is in describing the softening behavior of the NDBR50, for which the isotropic model overestimates the force considerably.
The overestimation in terms of the force also limits the fracture prediction of the isotropic eMBW model, while the enHill48 based anisotropic MBW shows a great match on force and the final fracture point.More tests with a larger range of stress states will be incorporated into the study to further exam the predictive capability of the developed model.Furthermore, a validation test focusing on the edge crack will be also developed in an upcoming study.

Conclusions Conclusions
In this study, the plasticity and ductile fracture behavior of DP1000 steel has been investigated with a hybrid experimental and numerical method.The following conclusions can be drawn: • From the experimental characterization, DP1000 steel is recognized with only minor plastic anisotropy at the initial deformation stage, but since the anisotropic hardening and r-value keep developing along deformation, the accumulated influence of anisotropy is noticeable and crucial to an accurate numerical prediction of ductile fracture.
• By considering the anisotropic hardening and plastic strain ratio evolution at different strain levels simultaneously, the enHill48 plasticity model gives a precise prediction on the anisotropic plastic behavior.
• The extended enHill48 model in conjunction with the MBW damage model improves the already satisfactory prediction of deformation and fracture under various stress states by the damage model based on isotropic plasticity to a great level of accuracy.It shows that even for materials with minor or moderate plastic anisotropy, considering anisotropy is critical to precisely describe the large deformation and ductile fracture behavior.

A Ackno cknow wledgments ledgments
This contribution shows the current research work of the research project 'Digital solution for microstructure and process design of DP and Q&P steels with tailored mechanical property profiles'.The financial support by the Technology Industries of Finland Centennial Foundation is gratefully acknowledged.The authors wish to acknowledge the CSC -IT Center for Science, Finland, for computational resources under project 2001353.

Fig. 1 .
Fig. 1.Uniaxial t Fig. 1.Uniaxial tensile t ensile test specimens and r est specimens and results.esults.(a) Geometry details and loading dir (a) Geometry details and loading direction sk ection sket etches of the SDB specimen, (b) engineering str ches of the SDB specimen, (b) engineering stress-str ess-strain curv ain curves along es along thr three loading dir ee loading directions.ections.

Fig. 3 .
Fig. 3. Plasticity and fr Fig. 3. Plasticity and fractur acture t e tests o ests ov ver v er various str arious stress stat ess states.es.(a) Ov (a) Overview of the sample geometry erview of the sample geometry, (b) e , (b) experimental t xperimental tests f ests for char or charact acteristic str eristic stress stat ess states in the space of str es in the space of stress ess triaxiality and Lode ang triaxiality and Lode angle par le paramet ameter[11].er[11].
(a) and (b), respectively.It is evident that the enHill48 model can well capture the anisotropic plasticity of the investigated materials with high accuracy in terms of both stress and r-value distribution, especially under the consideration of anisotropy evolution.

Fig. 5 .
Fig. 5.The e Fig. 5.The ev volution of anisotr olution of anisotrop opy in e y in experiments and calibr xperiments and calibrat ated enHill48 model.ed enHill48 model.(a) Distribution of normalized str (a) Distribution of normalized stress at diff ess at differ erent str ent strain le ain lev vels, (b) r-v els, (b) r-value dir alue directionality at diff ectionality at differ erent str ent strain le ain lev vels.els.

Fig. 6 .
Fig. 6.Experimental and numerical simulation comparison on f Fig. 6.Experimental and numerical simulation comparison on for orce-displacement r ce-displacement responses of fr esponses of fractur acture t e tensile t ensile tests, ests, with the gr with the green shaded ar een shaded area as the scatt ea as the scatter among par er among parallel t allel tests.ests.(a) CHD6, (b) PSR2, (c) NDBR50.(a) CHD6, (b) PSR2, (c) NDBR50.The predicted force-displacement responses for CHD6, PSR2, and NDBR50 specimens, with enHill48 plasticity model