The differences of damage initiation and accumulation of DP steels: a numerical and experimental analysis
Felix Pütz, Fuhui Shen, Markus Könemann, Sebastian Münstermann
mmanuscript No. (will be inserted by the editor)
The differences of damage initiation and accumulation of DP steels: anumerical and experimental analysis
Felix P ¨utz * · Fuhui Shen · Markus K¨onemann · Sebastian M ¨unstermann
Received: date / Accepted: date
Abstract
Many studies have examined the damage behaviourof dual-phase steels already. It is a topic of high interest,since understanding the mechanisms of damage during form-ing processes enables the production of steels with improvedproperties and damage tolerance. However, the focus wasrarely on the comparison between representatives of this steelclass, and the numerical simulation for the quantificationof damage states was not thoroughly used. Therefore, thisstudy compares the damage initiation and accumulation oftwo dual-phase steels (DP800 and DP1000), which are usedin the automotive industry. Additionally, parameter sets of aphenomenological damage mechanics model with coupleddamage evolution are calibrated for each material. The com-bined analysis reveals an earlier initiation of damage for theDP800, where the damage accumulation phase is prolonged.For DP1000 the damage nucleates only shortly before ma-terial failure. The material model is able to correctly predictthe behaviour, while experimental analysis confirms the pre-diction via light optical and SEM metallography.
Keywords steel · dual-phase · ductile damage · damagemodel · FEM simulation · damage tolerance The usage of dual-phase (DP) steels has been on the rise inrecent years. Especially the automotive industry shows highinterest in the development of these advanced high strengthsteels (AHSS), since DP steels show high strength valueswhile still maintaining good formability. Thus, a lightweight * Felix P¨utzIntegrity of Materials and Structures, Intzestraße 1 52072 AachenTel.: +49-241-8028415Fax: +49-241-8092253E-mail: [email protected] component design can be achieved by reducing componentthickness while still keeping the identical safety conditions(Davies and Magee, 1979). These specific properties resultfrom a distinct microstructure, that is composed of a soft fer-ritic phase with hard martensite islands on the grain bound-aries and triple points of ferrite grains. Due to the differ-ence in mechanical properties of the two phases, plastic be-haviour of DP steels shows a relatively low yield to ten-sile ratio, pronounced strain hardening and excellent globalformability. The reason for this extraordinary property pro-file lies in the partitioning of stress and strain between theinvolved phases, allowing for multiple degrees of freedomfor microstructural design (Bieler et al., 2009) .The strain partitioning between ferrite and martensitedepends vastly on the specific microstructure. Marteau et al.reported that the local microstructural neighbourhood is thecritical factor for strain heterogeneity (Marteau et al., 2013).Strain accumulates mostly in the ferrite forming localizedbands with an angle of 45-50 ◦ with respect to the loading di-rection (Ghadbeigi et al., 2010; Tasan et al., 2014a), whereasthe martensite carries the majority of the applied stress (Tasanet al., 2014b). Therefore, martensite is elastically deformedfor materials with low martensite content, while its defor-mation behaviour is plastic for high contents (Shen et al.,1986). The local microstructure especially is determiningthe strain distribution, e.g. average size of martensite islandsand global distribution (Park et al., 2014; Saai et al., 2014).Due to this inhomogeneity in the material constituents’behaviour, the microscopic damage modes of dual-phase steelsdiffer quite significantly to those of common structural steels.Where for structural steels the inclusions play the major rolefor void initiation, in DP steels damage incidents occur in re-lation to the two phases, martensite and ferrite (Tasan et al.,2010). Mechanisms for damage initiation in dual-phase steels a r X i v : . [ c ond - m a t . m t r l - s c i ] A p r Felix P¨utz * et al. are mostly decohesion of the martensite/ferrite interface, crack-ing of the martensite phase, or a localization of plastic strainin the ferrite phase, which results in debonding of the fer-rite grain boundaries (Ahmad et al., 2000). The mode forthe damage initiation depends on the microstructure and theresulting strain heterogeneity (Kadkhodapour et al., 2011).Therefore, grain size and martensite content do play an im-portant role (Maire et al., 2008; Ramazani et al., 2013; Tasanet al., 2015). Additionally, martensite morphology influencesthe early damage nucleation (Ghadbeigi et al., 2013; Heet al., 1984). Besides, observations have shown, that for bandedmartensite cracking is far more likely than a decohesion ofthe interface boundary of ferrite and martensite (Avramovic-Cingara et al., 2009).To assess the material’s properties and predict the loadbearing capabilities of structures, damage mechanics modelsare widely used for DP steels, e.g. in the automotive indus-try. In the field of damage mechanics, two different modeltypes exist: Coupled and uncoupled models (Besson, 2010).For the coupled damage mechanics models usually a dam-age variable is employed to reduce the yield potential ac-cording to the softening resulting from ductile damage inthe material during deformation. In case of the coupled mod-els, micromechanical models are very popular, for instancethe Gurson-Tvergaard-Needleman (GTN) model (Gurson,1977; Tvergaard, 1981; Tvergaard and Needleman, 1984).Micromechanical models are characterized by the depictionof physical phenomena like void nucleation, growth and co-alescence through sets of parameters. Therefore, the param-eters are interdependent and thus, an extensive iteration pro-cess is necessary for the parameter calibration (West et al.,2012). Alternatively, phenomenological, coupled models areused to describe the damage in materials numerically. Incontrast to the micromechanical models, damage evolutionis treated in a macroscopic way, where a number of effectsare described by a mathematical formulation. A good exam-ple for this type of model is the Lemaitre model (Lemaitre,1985, 1992), which describes damage as an irreversible pro-cess.Contrary to that, uncoupled models describe the materialbehaviour including fracture without taking damage into ac-count. Both the Johnson – Cook (Johnson and Cook, 1985),as well as the Bai-Wierzbicki (BW) model are good ex-amples for this type of model (Bai and Wierzbicki, 2008).Further development has been applied by Lian et al., whocombined the advantages of uncoupled and coupled mod-els into a hybrid formulation, making it the modified Bai-Wierzbicki model (MBW) (Lian et al., 2013). The modeltherefore holds an easy formulation and combines it withthe influence of damage onto material behaviour. The modelhas been developed further since its inception. For the first version a locus for the damage initiation point, which wasdependent on both stress triaxiality and Lode angle was uti-lized. Additionally, a set of critical values for the damagevariable was applied, at which material fracture was assumedin the numerical simulation. Wu et al. changed that consider-ably by implementing a locus for the fracture, as well as con-sidering non proportional loading paths until the inceptionof ductile damage (Wu et al., 2017). A further developmentof the MBW model was made by Shen et al. to characterisethe influence of loading orientation, which was used to de-scribe the anisotropic ductile damage and fracture behaviourof pipeline steels (Shen et al., 2020). Since the MBW dam-age mechanics model is easy to use and calibrate, while alsodepicting the damage behaviour accurately, it is applied herefor the characterisation of damage behaviour in DP steels.While many studies focused on the damage in dual-phasesteels from an experimental standpoint, it is hard to experi-mentally determine the evolution of damage during the tests.Therefore, this study aims to enhance the experimental in-vestigation by performing finite element (FE) based numer-ical simulations that are utilized to quantify the damage inthe material during forming processes. Thus, in this study,two dual-phase steels, DP800 and DP1000 were compared.Their damage and fracture properties are distinctly different,while the strength is not very far apart. To compare the ma-terials behaviour, a damage mechanics model has been usedthat can describe both, damage initiation as well as ductilematerial fracture, while also taking the changes of the stressstate during deformation, due to non-proportional loadingeffects, into account . This allows a comparison of the dam-age initiation for different stress states between the mate-rials. Additionally, by means of a calibrated fracture locus,the damage accumulation phase can be analysed and com-pared. Thus, tensile tests were conducted on flat specimensof different geometries to gather information about materialsdeformations and damage properties under different stressstates. On that basis, the material parameters of the modifiedBai Wierzbicki model were calibrated. For the validation ofthe numerical results regarding the damage initiation and ac-cumulation of the investigated material, interrupted tensiletests were conducted and a metallography analysis was per-formed by using the light optical microscopy and scanningelectron microscopy (SEM).
In the present study, two dual-phase steels were evaluatedfor comparison purpose. Even though both materials are dual-phase steels, vastly different properties are observable. Thesevarying characteristics are obtained by distinct alloying con-cepts as well as heat treatment processes. he differences of damage initiation and accumulation of DP steels: a numerical and experimental analysis 3
Fig. 1
Microstructures of steels DP800 and DP100 revealed by HNO3 etching, in light optical metallography
Table 1
Chemical composition of dual-phase steels DP800 andDP1000, in mass-%C Si Mn Cr Mo Cu
DP800
DP1000
Fig. 1 shows a comparison of the respective microstruc-tures at a magnification of 1000. It is very well observablethat the average grain size of DP1000 is significantly smallerthan that of DP800. Additionally, DP1000 has increased marten-site contents of approximately 38% while DP800 containsabout 32%. For DP800 a pronounced banding of the marten-site in the microstructure is noticeable. Since the martensitebands run parallel to the rolling direction, there will be asignificant influence on the mechanical properties. A certainextent of failure anisotropy is expected due to the bandedmicrostructure, however, the anisotropic fracture propertiesare beyond the scope of this study and all tensile specimenswere manufactured perpendicular to the rolling direction ofboth DP steels. Both steels were delivered with a thicknessof 1.5mm; their respective chemical compositions are givenin
Table 1 . While the alloying concepts show noticeablesimilarities, some minor differences are present.The carbon content for DP1000 is decreased comparedto DP800, thus leading to higher carbon concentration in themartensite phase for the DP800, since the phase fraction ofmartensite is higher for DP1000. Thus, it is to be expectedthat the strength of the martensite is reduced for DP1000due to the decreased carbon content, therefore leading toa bigger contrast of properties between ferrite and marten-site in the DP800. Furthermore, manganese and chromiumcontents are different, which leads to slight disparities dueto solid solution hardening. Additionally, Si as well as Mnand Cr reduce the critical cooling rate needed for forming martensite, thus influencing the respective time - tempera-ture - transformation graphs. On top of that the solubility ofcarbon in ferrite is reduced by silicon. Therefore, both ma-terials will have very distinct processing routes tailored tothe respective production process. For the characterizationof the resulting mechanical properties, isothermal, uniaxialquasistatic tensile tests were carried out on flat specimenswithout a notch. To ensure a proper depiction of the mate-rial’s properties, three tests were carried out per DP steel.A video extensometer was used to capture the elongation ofthe specimen during deformation, where the starting lengthof 40mm was used. The necking took place inside the areatracked by the extensometer for all 6 tensile tests, ensuringa good comparability.The results for both materials are shown in
Fig. 2 . Thescatter shown is the difference in fracture, resulting fromthe three tensile tests per material mentioned before. Fromthis figure it is obvious, that DP1000 shows higher strength,while the strain at fracture of DP800 is considerably higher.The higher strength is a result of both; the higher marten-site phase fraction as well as significantly refined grains inDP1000.In addition, there is a clear scatter in the elongation val-ues, with elongation at fracture between 18 and 23 percentfor DP800, while the scatter for the DP1000 is about onepercent. This variation in elongation at fracture can be ex-plained by the clearly pronounced band structures, whichcan lead to significant deviations depending on the posi-tion of the bands in the specimen. Since DP1000 fracturesshortly after uniform strain, the necking is far less pronouncedthan in DP800. During necking the stress state in the samplecan change quite significantly leading to non-proportionalloading paths during the deformation of the sample. Thus,to describe the materials behaviour after the uniform strain,it is necessary for the material model to consider the effectsof the changes of stress state during deformation. Therefore,
Felix P¨utz * et al.
Fig. 2
Engineering stress-strain curve of uniaxial tensile tests ofDP800 and DP1000 a development of the existing MBW model was required todetermine the material behaviour and damage accumulationmore realistically.
In the framework of continuum damage mechanics, the mod-ified Bai Wierzbicki (MBW) model has been proposed byLian et al. (2013) and widely applied to describe the dam-age and fracture behaviours of various grades of steels (Lianet al., 2013; M¨unstermann et al., 2017; Wu et al., 2017; No-vokshanov et al., 2015; Shen et al., 2020; Liu et al., 2020).Like in other damage mechanics models, the significant in-fluences of stress state on the ductile fracture are consideredthrough defining a strain based criterion which is usually aweighted function of two particularly important variables,the stress triaxiality η and the Lode-angle parameter θ thatare related to the three stress invariants. I = tr [ σ ] = ( σ + σ + σ ) (1) J = [ σ ] = [( σ − σ ) + ( σ − σ ) + ( σ − σ ) ] (2) J = (cid:16) σ − I (cid:17) · (cid:16) σ − I (cid:17) · (cid:16) σ − I (cid:17) (3) η = I √ · J = I σ = ( σ + σ + σ ) (cid:114) [( σ − σ ) + ( σ − σ ) + ( σ − σ ) ] (4) θ = cos − (cid:32) · √ · J · J / (cid:33) (5) θ = − θπ (6)where σ , σ and σ are principal stresses and σ is thevon Mises equivalent stress. For the material model, the Lode-angle parameter θ was used, which has a linear relationshipwith the Lode-angle θ .The effects of stress state on plasticity in some metallicmaterials have been reported, while steels typically show anegligible pressure sensitivity, therefore, only the effects ofLode-angle parameter are considered in the yield criterionof the MBW model. Φ = σ ( σ ) − ( − D ) · σ y ( ε p , θ ) ≤ σ y ( ε p , θ ) = σ y ( ε p ) · (cid:104) c s θ + ( c ax θ − c s θ ) · (cid:16) γ − γ m + m + (cid:17)(cid:105) (8) γ = √ − √ · (cid:34) sec (cid:0) θ · π (cid:1) − (cid:35) (9) c ax θ = (cid:40) c t θ , for θ ≥ . c c θ , for θ < . (10)Where D is a scalar variable to quantify the damage ef-fects, yield stress σ y is determined by the equivalent plasticstrain ε p and Lode-angle parameter θ , and σ y ( ε p ) corre-sponds to the flow stress at given equivalent plastic strain ε p under the reference stress state. c s θ , c t θ , c c θ are the nor-malised strength under shear, tension and compression stateand m is a material parameter with positive integral valuesthat describes the Lode-angle sensitivity. γ is another stressstate parameter with unique correlation to the Lode-angleparameter θ . According to the derivation of Lian et al., (Lianet al., 2013), the yield locus of MBW model is convex if thematerial parameters are located within the specific range of √ ≤ c s θ c ax θ ≤
1. The conventional normality rule is applied in he differences of damage initiation and accumulation of DP steels: a numerical and experimental analysis 5 the MBW model and the plastic strain components are up-dated according to the following equation and d λ is a non-negative plastic multiplier. d ε p = d λ · δ Φδ σ (11)In the coupled damage mechanics model, two individualcriteria have been defined to identify the ductile damage ini-tiation (DDI) and ductile fracture (DF), which correspondsto the initiation of degradation on microscopic scale in thematerial and the loss of load carrying capacity on the macro-scopic scale. Numerically, damage initiation, in this study, isdefined as the onset of macroscopic softening due to dam-age, which must be taken into account by the numerical rep-resentation of the material behaviour. In order to considerthe change of stress state during plastic deformation, the av-erage values of the stress triaxiality η avg and the Lode-angleparameter θ avg have been used to describe the stress state fornon-proportional loading paths (Wu et al., 2017; Mu et al.,2020). η avg = ε p (cid:90) ε p η ( ε p ) d ε p (12) θ avg = ε p (cid:90) ε p θ ( ε p ) d ε p (13)Since the damage is dependent on stress state it is neces-sary to define equations for the initiation of damage, as wellas the fracture, that represent this dependency. Therefore,the damage initiation locus (DIL) and ductile fracture lo-cus (DFL) have been defined as two individual equations f di and f d f with the stress triaxiality and the Lode-angle pa-rameter as independent variables. The instantaneous and av-erage values of the independent stress state variables havebeen used in the damage and fracture criteria under propor-tional and non-proportional loading conditions, respectively.Under non-proportional loading conditions, these two equa-tions describe the critical equivalent plastic strains at the mo-ment of damage initiation and ductile fracture, respectively. f di ( η avg , θ avg ) = (cid:104) ( D e − D η avg + D e − D η avg ) − D e − D η avg (cid:105) θ avg + ( D e − D η avg − D e − D η avg ) θ avg + D e − D η avg (14) f d f ( η avg , θ avg ) = (cid:104) ( F e − F η avg + F e − F η avg ) − F e − F η avg (cid:105) θ avg + ( F e − F η avg − F e − F η avg ) θ avg + F e − F η avg (15) where D - D and F - F are material parameters usedto define the damage initiation locus and ductile fracture lo-cus. Under the condition that D = D , D = D and F = F , F = F , the DIL and DFL are symmetric with respect tothe Lode-angle parameter and four independent parametersare enough to define the corresponding loci. Based on previ-ous experimental observations, a cut-off value of the stresstriaxiality η c exists, below which the initiation and evolu-tion of ductile damage cannot be triggered due to pressureeffects. η c = − as a reasonable estimation has been adoptedin the MBW model (Wu et al., 2017). Therefore, when thestress triaxiality is lower than η c , the equations f di and f d f are set to be infinite. The damage initiation specified by thismodel is unrelated to the materials mechanisms of damageinitiation, e.g. micro crack formation, void formation. In-stead, it aims to describe the aggregative accumulation of thedefects and their influence on the load bearing capabilities.For this step a plasticity model is no longer able to describethe materials mechanical behaviour (Keim et al., 2019). Forthe non-proportional loading, two indicators have been ap-plied to describe the ductile damage initiation I dd and ductilefracture I d f respectively to consider the effects of stress stateevolution. I dd = (cid:90) ε p d ε p ε pdi ( η avg , θ avg ) w ith ε pdi ( η avg , θ avg ) = (cid:40) + ∞ , η avg ≤ η c f di ( η avg , θ avg ) , η avg > η c . (16) I d f = (cid:90) ε p ε p , cdi d ε p ε pd f ( η avg , θ avg ) w ith ε pd f ( η avg , θ avg ) = (cid:40) + ∞ , η avg ≤ η c f d f ( η avg , θ avg ) , η avg > η c . (17)The values of equivalent plastic strain and equivalentstress at the moment of damage initiation ( I dd =
1) are de-fined as two characteristic variables ε p , cdi and σ cdi , respec-tively: ε p , cdi = ε p ( I dd = ) (18) σ cdi = σ ( I dd = ) (19)After the damage initiation criterion is fulfilled, damageevolution is controlled according to the energy dissipationtheory. Depending on the shape of damage initiation locusand ductile fracture locus, when the indicator of the duc-tile fracture I d f reaches unity, the damage variable D doesnot necessarily reach unity. Therefore, a critical value of the Felix P¨utz * et al. damage variable D cr exists, at which the material point willfail regardless of the value of the D variable: D cr = σ cdi G f (cid:0) ε pd f − ε pdi (cid:1) (20)Where G f is a material parameter which controls thedamage evolution rate. Linear damage evolution is assumedin the MBW model, which is expressed as: D = , I dd < D cr · I d f , I dd ≥ ∧ I d f < , I dd ≥ ∧ I d f ≥ I d f reachesunity, the final crack propagation is triggered and failure oc-curs. Therefore, the model, hereafter called npMBW-19, iscapable of representing the influence of the necking, andthus the change of stress state, during deformation. The calibration approach for the material models for bothsteels follows roughly the approach of Lian et al. (Lian et al.,2013, 2014). Since the calibrated npMBW-19 model needsto be able to account for various stress states, the calibrationof the material model is carried out on a variety of sam-ple geometries. By varying the sample geometries in ten-sile tests, different stress states can be accomplished. In thisstudy three differently notched specimen types were appliedfor the calibration of the material model in addition to theuniaxial tensile test. Used specimen types were: Notcheddogbone samples (varying notch geometries at the edge ofthe sample), central hole samples (round, as well as ellipticalholes in the center of the specimen) and plane strain samples(notch with different radii over the thickness of the sample).The applied specimens for each material can be seen in
Fig.3 and
Fig. 4
The type of notch of the sample is abbreviatedwith an r continuing with the radius, for the notched dogbone samples.The corresponding stress states, characterised by the Lode-angle parameter and the stress triaxiality in the applied sam-ples are delineated in
Table 2 . To achieve multiple stressstates, notches were modified with various radii to gain ge-ometries of different stress states within one sample type.Per specimen geometry, three tensile tests were performed in
Table 2
Stress states of utilised sample geometries
Sample geometry Lode-angle Stress triaxiality η parameter θ Uniaxial tensile (UT) 1 Notched dog bone (DB) 0.3 - 0.8 0.4 - 0.6Central hole (CH) ˜1 0.3 - 0.4Plane strain (PS) 0 0.5 - 0.7
Table 3
Hollomon-Voce fitting parameters for steels DP800 andDP1000 α K n A B C
DP800 0.5138 1843 0.44 1167 820.4 100DP1000 0.5879 2000 0.1127 725.5 300 57.2 accordance with the procedure described earlier for the uni-axial tensile test. Afterwards, simulations of the experimentswere conducted, using ABAQUS, to achieve a comparisonbetween the force - displacement curves of experimentallydetermined values and simulated ones.For the determination of the base flow curve, the uniax-ial tensile tests (T), presented in the previous chapter, wereutilised. From the determined engineering stress-strain curve,the true stress-true strain curve was calculated until the uni-form elongation point. This data was then used to fit the Hol-lomon - Voce hardening model to the material’s flow curvevia the Matlab curve fitting tool. σ = α · ( K ε np ) + ( − α ) · ( A − B · e − C ε p ) (22)This specific hardening model was chosen, since it showsa good compromise between accurate representation at lowplastic strains and realistic hardening behaviour for higherstrains. In Table 3 the parameters for the Hollomon-Vocemodels are given for both, DP800 and DP1000.After the calibration of the flow curve the basic param-eters of the MBW model were determined ( c s θ , c t θ , c c θ , m).This was done by iterating over multiple simulations usinga range of different sample geometries.Subsequently the damage and fracture parameters of thenpMBW-19 model were determined. Damage and fracturecriteria in this material model are described by equations 14and 15. Therefore, the specified locus needs to be calibratedfor both events, damage initiation and fracture (Lian et al.,2013; Wu et al., 2017). For the damage initiation locus, acomparison of force and displacement curve between simu-lation and experimental results was used. Since the damagedescribed in this model is related to the accumulated dam-age incidents, a threshold method has been utilised to findthe numerical damage initiation. For that reason, the numer-ical onset of damage was determined as the point where the he differences of damage initiation and accumulation of DP steels: a numerical and experimental analysis 7 Fig. 3
Applied tensile specimens for steel DP800
Fig. 4
Applied tensile specimens for steel DP1000 deviation between simulated and experimental force and dis-placement curves was apparent. Similar methods have beenused by other authors within the damage mechanics field(Børvik et al., 2001; Bouchard et al., 2011).At this step, the Lode-angle parameter and stress tri-axiality as well as the equivalent plastic strain (PEEQ) aretaken from the simulation. Since these parameters may varylocally, the element is chosen that shows the most criticalstate of stress and thus is most likely to encounter damagefirst. By extracting the Lode-angle parameter, stress triaxial-ity and equivalent plastic strain for a multitude of differenttensile geometries, data points are gathered in the space de-fined by these three variables. Applying the curve fitting toolof
Matlab , a function can be defined that describes the de-sired surface while using the obtained results as supportingpoints. For non-proportional loading paths it is necessary to average the stress state of the critical element, where dam-age happens first, over the simulated steps (Wu et al., 2017).After determining the locus for the onset of damage (DIL),the effect that damage has on the component needs to be ad-justed. In the npMBW-19 model parameter G f is calibratedto adjust the speed at which damage accumulates in the sim-ulated material. G f is defined as the energy dissipation be-tween damage initiation and complete fracture. When thesoftening is specified the fracture locus can be determined.The approach used for this determination follows the onefrom the damage initiation locus closely. This time the pointfor the experimental fracture is compared to the simulation.The step where the fracture should occur is identified andLode-angle parameter, stress triaxiality and equivalent plas-tic strain are extracted for the critical elements. Again, thestress states are averaged from the point of damage initiationto the presumed fracture of the sample. After gathering the Felix P¨utz * et al.
Table 4
Calibrated npMBW-19 parameter sets for both materials c s θ c t θ c c θ m D D D DP800 0.95 1 0.9 6 0.5 2 0.365DP1000 0.95 1 0.97 6 0.4 1 0.1 D G f (cid:2) Jmm (cid:3) F F F F DP800 3 1.2 0.7 1 0.366 2DP1000 1.5 6.5 0.58 0.76 0.443 1.57 data for all sample geometries the locus is fitted in regardsto the obtained points using the
Matlab curve fitting tool.In
Fig. 5 the final results of this calibration process aredepicted for steel DP800. From this figure it is obvious, thata good match between experimental data and simulationswas been obtained. The scatter for the experimental testingcan be seen in the shaded areas.
Fig. 5
Comparison between experimentally obtained data (back-ground and lighter color) and simulation results for DP800
Likewise, the material model for DP1000 was calibrated.The applied flow curve can be seen in
Table 3 . Addition-ally, a damage initiation, as well as a ductile fracture lo-cus were calibrated using the same approach as describedabove for the DP800. By duplicating the approach statedabove, a good agreement with the experimental data couldbe reached (
Fig. 6 ). Contrary to the DP800 almost no scat-ter could be found during the tests of the DP1000 material,which also shows no significant banding in its microstruc-ture. The applied set of parameters can be found in
Table4 . Interestingly, the calibrated G f parameter for DP1000 ishigher which results in a slower development of the dam-age variable. This results in a fairly slow accumulation ofdamage after the initiation. For the scope of this study, it is important to differentiate be-tween failure and damage of a material or component. Be-
Fig. 6
Comparison between experimentally obtained data(backgroundand lighter color) and simulation results for DP1000 cause damage is the deterioration of materials properties be-fore failure, especially the load bearing capacity (Lemaitre,1992), it is not to be equated with component failure. Dam-age occurs on a microscale and is usually described as thedevelopment of voids inside the microstructure, while on amacroscale damage usually equates to cracks in the compo-nent and therefore can be seen as component failure. It istherefore highly relevant to differentiate between micro andmacroscale (Tekkaya et al., 2017). For numerical analysis,damage is defined as the macroscopic reduction of the stressduring loading, that cannot be described by basic plastic-ity modelling. Thus, Lemaitre introduced a factor for dam-age in a microstructure, which results in a reduction of theflow potential by the term (1 − D ), where D is the damagevariable (Lemaitre, 1985). The damage variable adopted byLian et al. shows some differences to the one postulated byLemaitre. While Lemaitre’s damage variable is calculatedbased on the area fraction of defects, Lian et al. refer tothe stress at damage initiation, divided by the energy re-quired to create new surfaces in a volume of the material(see G f ), an adaptation of the damage evolution law usedby ABAQUS finite element code (Lian et al., 2013). Ac-cordingly, both damage variables are scalar, but there arequite pronounced differences between both numerical dam-age rules. These differences between the damage modelsmust be distinguished, as well as the differences betweenmicro- and macroscopic damage phenomena.Because the damage law used in the MBW model doesnot refer to a physical material characteristic, like the areafraction of voids, except for the energy for cracks, it is arather more phenomenological approach to model the influ-ence of damage on the materials flow potential. The damageevolution, as discussed before, starts when a specific equiv-alent plastic strain (PEEQ) locally exceeds a certain thresh-old, which changes with stress states. The respective valuefor PEEQ is determined by the damage initiation locus. Af- he differences of damage initiation and accumulation of DP steels: a numerical and experimental analysis 9 Fig. 7
Comparison of the ductile damage initiation locus and the ductile fracture locus for DP800 (left) and DP1000 (right) ter this point, softening occurs in the simulation, which leadsto a direct reduction of the resulting stress compared to vir-gin materials. The length of this following phase where dam-age accumulates depends on the stress state which is con-sidered in the ductile fracture locus. The comparisons of theductile damage initiation locus and the ductile fracture lo-cus for each respective material are shown in
Fig. 7 . Theshape of the loci for DP800 and DP1000 are different, aswas to be expected. The distance between the plots is higherfor the DP 800 material which leads to a longer damage ac-cumulation phase. Merely for higher triaxialities and Lode-angle parameters around zero, the differences between theloci of DP800 and DP1000 is minimal. Some research sug-gests a different shape for the ductile damage initiation locusand the ductile fracture loci, especially for the area around astress triaxiality of 0, namely shear stress state (Papasideroet al., 2015). Nevertheless, based on the experimental andnumerical results in this study, the loci in
Fig. 7 constructedfor both steels using the corresponding calibrated damageand fracture parameters is validated within the range of in-vestigated stress states. In the case of an application of thecalibrated material model for even lower or higher stress tri-axialities, the loci would have to be revisited to confirm oradapt their shape.Due to the differences in the distance of the DIL and theDFL the damage accumulation phase is significantly differ-ent between the two steels.
Fig. 8 shows this difference util-ising the flow curves obtained from the uniaxial tensile testof both materials, as well as calculating the points for dam-age initiation and fracture under uniaxial tension condition ( η = , θ = ) based on calibrated material parameters. Thepoint for the damage initiation takes place at roughly thesame strain for both materials, while fracture is delayed sig-nificantly for DP800.To further examine precision of the numerical results, in-terrupted tensile tests were conducted for both materials. Foreach material a sample was therefore first tested until fail-ure and subsequent specimens of identical geometry werestopped after a distinct strain was reached. The lowest elon-gation used in this investigation was the uniform elongation, Fig. 8
Flow curves from uniaxial tensile test of DP800 and DP1000with numerically determined points of damage initiation and fractureunder uniaxial tension condition. as no or little damage is expected below this. This way ametallographic damage analysis could be carried out to in-vestigate the average amount of damage that could be ob-served in a sample. For both materials unnotched dog bonespecimens were utilised to ensure a good comparability. Forthe analysis of the damage in the material, light optical mi-croscopy was chosen, since a bigger area can be investi-gated by light optical analysis, where scanning electro mi-croscopy (SEM) pictures resolve only smaller areas of thesamples. However, it is not easily possible to differentiatebetween voids and inclusions in the material. Thus the areafraction that is detected is not quantitatively representativeof the actual void fraction. To find out about the area frac-tion for each sample, multiple pictures were taken to gatherinformation about the scatter band where the actual valueslie. For this analysis, the light optical pictures were con-verted to greyscale images, which were subsequently eval-uated by a threshold method, with which a differentiationbetween matrix material and voids/inclusion could be made.For these steps
Fiji was used as image analysis software(Rueden et al., 2017; Schindelin et al., 2012).The results of this analysis are depicted in
Fig. 9 . Tobetter compare the values for both materials, a normalisa-
Fig. 9
Comparison of light optically detected area fraction for DP800 and DP1000 for different strains tion was carried out, where the current strain was dividedby the respective fracture strain. A comparison of the val-ues for the detected area fraction reveals a gradual increasefor DP800, while for DP1000 no significant rise in fractioncan be observed until just before fracture of the sample. Thelarge scatter, especially at the beginning, can be explainedby the lack of necking, which means that the region of inter-est cannot be identified accurately.Thus, the damage accumulation phase for DP800 startsat lower strains relative to the fracture strain of the material.By contrast, the damage accumulation phase for DP1000starts very late and just before fracture. Therefore, the dam-age in the material behaves exactly as predicted using thenpMBW-19 model. To assess the damage state in the mi-crostructure, pictures were taken in the SEM. Especially forDP1000 an analysis for higher magnifications was necessaryto reveal if damage forms earlier than shortly before frac-ture. For 70% of the fracture strain, only very few eventsof damage initiation could be found under high magnifica-tion (
Fig. 10 ). While the amount of these initiation locationsincreases with the strain, growth is very limited (
Fig. 11 ).A comparison of the samples that are at 95% of fracturestrain reveals, that the voids in DP1000 (
Fig. 11 ) are signif-icantly smaller than in DP800 (
Fig. 12 ). It is therefore con-cluded, that the damage accumulation phase for DP1000 isindeed significantly shortened. In particular, it is noticeable,that no void in DP1000 exceeds a length of 1 µ m , while the Fig. 10
Evolution of damage in DP1000. Single martensite crack withmagnification of 5000 for 0.7 of fracture strain
DP800 features multiple larger voids. Additionally, voids forDP800 are more circular, while they are shaped like cracksfor DP1000 again leading to the conclusion, that there hasbeen no time for growth after initiation. This is in line withthe results demonstrated in
Fig. 8 , where a shorter damageaccumulation phase is present in DP1000 and thus a lowerdecrease of load bearing capabilities is to be expected.Besides, the damage initiation modes were investigated.For both materials, the prevalent modus for damage initia- he differences of damage initiation and accumulation of DP steels: a numerical and experimental analysis 11
Fig. 11
Evolution of damage in DP1000. Many voids have formed andgrown, magnification of 2000, 0.95 of fracture strain
Fig. 12
Damage shortly before fracture in DP800 tion was the cracking of martensite islands. For DP800 themartensite bands especially were sites for damage initiation.Furthermore, decohesion of ferrite and martensite islandswas found in the DP800 after about 80% of fracture strain.
This study showed significant differences between two in-dustrially produced dual-phase steels. Starting with the ex-perimental results, the difference in fracture strain was foundto be significant with large scatter for the DP800. This scat-ter was attributed to the pronounced banding found withinthe material. For the numerical analysis it was found, thatthe change in the stress state during necking needs to beconsidered by the material model for proper simulation re-sults. Therefore, the MBW model was extended to accountfor non-proportional loading paths. The parameters of the model were then fitted for both materials to reveal the dis-parities in the material behaviour numerically. Subsequently,the damage initiation and fracture loci were calibrated. Thecomparison of simulation results to the experimentally ob-tained force-displacement curves reveals a high agreementfor both materials. Especially the differences in damage be-haviour were modelled precisely.The found differences during the experimental testingand analysis can be attributed to the differences in the mi-crostructure. Especially grain size and martensite content,but also the pronounced banding in the DP800 play an im-portant role for mechanical properties, as well as damageinitiation and accumulation. In this study, it was shown, thatthe damage in both dual-phase steels initiates at similar equiv-alent plastic strains. Oppositely, the fracture happens at vastlydifferent equivalent plastic, as well as global strains. Thisleads to completely different damage accumulation phasesin the material. The numerical simulations showed an ex-ceedingly different length of the damage accumulation phasefor the two steels. This difference was subsequently verifiedby experimental tests, where the amount of damage in thematerial after an interrupted tensile test was examined. Forthese tests it could be shown, that DP800 exhibits a pro-nounced damage accumulation phase, while DP1000 frac-tures shortly after a critical amount of voids forms in the ma-terial. Thus fracture occurs with almost no damage accumu-lation and the void growth phase is nearly skipped. There-fore, the different microstructures lead to specific damagecharacteristics, which in turn influence and change the spe-cific properties of the material. Additionally, the contrast inthe mechanical properties between the two phases for bothmaterials reinforce this effect. Since the carbon content inmartensite is relatively higher in DP800 than in DP1000,the martensite fractures earlier leading to a relatively earlydamage initiation and longer damage accumulation phase.For the DP1000 the the contrasts are not so distinct, whichleads to a comparatively late initiation of damage and failureshortly afterwards.During the comparison of experimental and numericalresults, it was quite obvious, that the presented material modelis able to accurately represent the experimental tensile testcurves, both uniaxial and notched specimens. The stress statesdo, however show only minor variance. Thus, for higher de-viations, an adjustment of the averaging scheme for deter-mining stress triaxiality and Lode angle parameter for theductile damage and fracture loci might be appropriate. Fur-thermore, the model shows a discontinuity around the valueof η c = − , since a fixed value at which no damage is ap-plied in the model, will be difficult to deal with, when get-ting close to it (e.g. η = − . paths in this study are exclusively above this value, this is apromising and important concern for future development.The comparison of numerical and experimental ductiledamage showed, that the presented material model is able toaccurately predict the damage initiation, damage accumula-tion and fracture of both materials. Nevertheless, the accu-racy of the damage initiation point in the material model isstill an important topic for further investigation. Since thelocation of the fracture locus strongly depends on the dam-age initiation locus, a high precision for the DIL is desirable.However, the commonly used method of direct current po-tential drop (DCPD) is not feasible for DP steel, since itsvoid volume is comparably low. Therefore, an improvementof the method to determine aforementioned damage initia-tion locus is necessary and currently examined. One possi-bility is to measure the density of the material to determinethe time of damage initiation (Hering et al., 2019; Schowt-jak et al., 2019; Meya et al., 2019).The analysis of damage initiation point and damage ac-cumulation by light optical microscopy is rather qualitativethan quantitative. Since statistical representativeness and ac-curacy have to be balanced for this type of examination, mi-cro voids are not detected in the pictures. Thus, the valuesreceived are not the void area fraction. On top of that, lightoptical pictures show inclusions in a similar colour to voids,isolation of voids for analysis purpose is rather difficult forlight optical microscopy. For a more quantitative result ofvoid area fraction SEM pictures with high resolution over abig area seem to be more promising.The presented results suggest that the damage, and there-fore the materials mechanical properties depend on the mi-crostructure of the respective steel. It is therefore of high in-terest to investigate the influence of each microstructural pa-rameter on the damage characteristics as well as the mechan-ical properties of the material. Three-dimensional represen-tative volume elements seem to be a promising approach toinvestigate the influence of different microstructural char-acteristics, like martensite volume content, martensite mor-phology and grain size. The funding of this research by the Deutsche Forschungs-gemeinschaft (DFG, German Research Foundation) – Pro-jektnummer 278868966 – TRR 188, is gratefully acknowl-edged.
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