A unique numerical iterative approach for modelling individual phase stress-strain curves in dual phase steel

Understanding the effects of martensite volume fractions (V m) in dual phase (DP) steel resulting from heat treatment is crucial for designing structures for mechanical impact resistance and optimizing manufacturing processes. DP steel’s material behaviour depends heavily on its microstructure properties. While stress-strain curves for individual phases in DP steels are often determined using empirical models, extensive experimental data is required to establish empirical model constants. This research aims to achieve two main objectives: firstly, to calibrate stress-strain curves for pure ferrite and pure martensite using limited experimental data using micromechanical adaptive iteration algorithm (MAIA). This calibration involves using stress-strain data from DP steels with varying V m during the calibration stage and additional data for verification. Secondly, to conduct a comprehensive sensitivity analysis of MAIA to assess its capabilities and limitations. Microstructure-based finite element (FE) models, simulated with ABAQUS/Standard, are employed to predict stress-strain curves under uniaxial tensile test conditions. The MAIA approach successfully calculated ferrite and martensite stress-strain curves that could predict plastic behaviour of DP steel with different V m, which agreed with experimental work. Key advantages of this approach include avoiding complex 3D microstructure geometries and requiring only two experimentally obtained stress-strain curves with different Vm for material constant calibration, along with another curve for validation. However, the experimental data selected for calibration must have a V m difference between 20%–50% and one of the DP steels must have a low martensite volume fraction. The FE micromechanical model could capture the effect of softening of martensite phase and strengthening of ferrite phase as compared to its bulk properties for DP steel. The effect of V m on strain hardening rate was also successfully captured. This technique comes with obvious shortcomings, such as excluding crystal plasticity behaviour, and change in chemical composition within the individual phase with varying martensite volume fraction.


Introduction
Incorporating the effects of microstructure into finite element (FE) models for manufacturing process requires the strain hardening constants in the constitutive models for each phase.Carbon steel commonly compromises of multiple microstructures, such as ferrite, martensite, bainite, and pearlite.Numerous research was carried out to produce constitutive equations to determine stress-strain curves of these single microstructures.Tomota et al [1] established equations to determine the single microstructure material constants for Swift's equation.These equations were a function of chemical compositions, grain size and lamellar spacing for pearlite.Pierman et al [2] developed empirical equations to calculate the stress-strain curve of martensite phase, primarily focus on the effect of carbon content within the phase.There were other material constants used in the equation, some were determined through curve fitting and others were assumed.Rodriguez and Gutierrez [3] presented a unified formulation to estimate the tensile stress-strain curves for the single microstructure.This formulation considers the chemical compositions, lattice friction and the evolution of dislocation density with strain.Several research papers [4][5][6][7] adapted the equations proposed by Amirmaleki et al however, the values required for the equations are challenging to obtain experimentally.This led to a situation where same values were used even for different materials with different chemical composition [4][5][6][7].
Several researchers [2,[8][9][10][11][12] have examined the impact of Vm on the strength of dual phase (DP) steel.Nevertheless, there has been limited research on investigating how varying V m affects the behaviour of ferrite and martensite phase as a mixture in the DP steel.Koo et al [13] and Tamura et al [14] conducted experimental tensile test on DP steel and observed that an increase in V m led to higher yield and ultimate tensile strength.Additionally, Koo et al [13] and Tamura et al [14] also observed that the martensite within DP steel exhibited lower hardness compared to pure martensite, while the ferrite in DP steel displayed greater harder compared to pure ferrite.
Accurate stress behaviour of ferrite and martensite phases were dependent on the appropriate values assigned for the parameters in the constitutive models.Allain et al [10] developed a generic mechanical mean field model capable of determining the stress behaviour of fully ferritic and fully martensitic steel.The calibrated models demonstrated remarkable success in capturing the effects of V m , grain size, carbon content and interactions between ferrite and martensite phases.The parameters required in the model were calibrated from at least 60 tensile curves of DP steel.Shrot and Bäker [11] faced similar challenges in terms of obtain parameters for material models using Johnson-Cook (JC) empirical equation.The FE cutting models built Figure 1.Flow chart showing the three stages in micromechanical adaptive Iteration algorithm to estimate σ f andσm.Reproduced with permission from [12].
were limited by the assumptions of input parameters of the JC equation.Due to the complexity of experimentally deriving the parameters of the material model, FE models on machining processes relied on a scarce pool of literature on these material parameters.In attempt to solve this problem, Shrot and Bäker [11] did an inverse identification of material parameters technique to calibrate the material models.
The concept of calibrating and conducting inverse identification of material parameters is not foreign.Halim and Ng [12] introduced the micromechanical adaptive iteration algorithm (MAIA), a method designed to estimate the stress-strain curves for single phase ferrite and martensite in a given DP steel alloy.The overall methodology of MAIA approach is shown in figure 1.This technique involves utilizing experimental stress-strain data from DP steels with two distinct martensite volume fractions (V m ) during the calibration stage, along with additional experimental stress-strain data from a DP steel with a different V m for results verification.The overall technique and nomenclature are detailed in section 3 of this paper.The MAIA technique assumed that for a given DP steel with a given geometry of martensite particles, there is a ferrite and a martensite phase stress-strain curves that could be used to model the stress-strain curve of given DP steel with varying V m .
MAIA technique is in no way suggesting that the physical phenomenon of the microstructure is unnecessary or mandating a particular way of calculating ferrite and martensite phase stress-strain curve.Rather, it is a unique approach to determine phase stress-strain curves without necessitating precise measurements of the physical characteristics and properties of the microstructure.
The objectives of this paper are (i) using MAIA approach together with limited experimentally obtained stress-strain curves to calibrate the strain hardening properties of pure ferrite and pure martensite and, (ii) to perform a comprehensive sensitivity analysis on MAIA to establish the extent of its capabilities and limitations.The calibration of pure martensite and pure ferrite phase stress-strain curves was established by using DP steels with two different range of V m .The DP steel ferrite grain size was between 1.4-2.3µm.One of the key research focuses was to investigate the effects on different range of V m values used during model calibration to accurately predict the stress-strain curve for pure ferrite and pure martensite.The uni-axial tensile test configuration was employed in this research.The key performance measure is based on using microstructure-based FE models to predict the stress-strain curves.The commercially available FE implicit code, ABAQUS/Standard, was used and no fracture was incorporated into the model.The range of martensite volume fractions used for the current investigation were between 20%-67%.Six combination groups of two different martensite volume fractions during calibration were investigated.Some groups were designed to have either a minimum or maximum range of V m .The phenomena of ferrite strengthening and martensite softening in DP steels with increasing V m was also extensively discussed from a solid mechanics perspective.Additionally, the ability of the calibrated ferrite and martensite phase stress-strain curves in capturing the effect of V m on strain hardening rate was also investigated.

3D micromechanical FE model
The micromechanical FE model used was a 3D discrete microstructure as shown in figure 2(a).Equations ( 1)-(3) were applied on xy-plane, yz-plane and xz-plane respectively, where u, v, and w denotes displacement in the x, y and z directions respectively.As shown in figure 2(a), normal loading condition was created by applying displacement in the y-direction on surface S3.User defined multi point constrain using equations ( 4) and ( 5) was applied to surfaces S1 and S2 respectively.The length L used in this model is 1 mm, w s1 and w A represent displacement in the z-direction for all nodes on surface S1 and node A respectively.u s2 and u A represent displacement in the x-direction for all nodes on surface S2 and node A respectively.The deformed shape of the model is shown in figure 2(b).Figure 3 shows the micromechanical models with various martensite volume fraction and the martensite elements were randomly assigned.

MAIA
MAIA used in this paper was initially developed by Halim and Ng [12].The overall workflow of MAIA is shown in figure 1, which consists of three main stages.The first stage involves the initial estimation of ferrite and martensite phase stress-strain curves.This process includes obtaining material constants from experimental tensile test data published by Scott et al [15], as described in section 3.1.This is followed by the initialization process to acquire the material constants of ferrite and martensite phase stress-strain curves for the first iteration of FE models as detailed in section 3.2.The second stage employs adaptive iteration algorithm to calibrate the ferrite and martensite phase stress-strain curves.In this stage, results are extracted from FE models, the difference between the experimental and FE modelled results were calculated, and this difference was utilized to calculate the ferrite and martensite phase stress-strain curves for the subsequent iteration.This process is detailed in sections 3.3-3.5.The final stage is known as the validation stage.

Tensile test experimental data
The DP steel used in this current investigation had a 0.182% C, 0.19% Si, 1.5% Mn, 0.14% V, 0.008% N, 0.1% Mo, 0.03% Cr, 0.019% Al and Fe balance (all weight percentage).The DP steel had different martensite volumetric fraction ranging from 20% to 67%.This was achieved by annealing the DP steel at different temperatures and isothermal holding durations, which was carried out by Scott et al [15].The stress-strain curves of DP steels with various V m were digitized using WebPlotDigitizer [16].The data points were then curve fitted using least square regression method to obtain the material parameters of the plasticity model.The plasticity model used for this paper was modified Voce law as shown in equation ( 6), where σ 0 , R ∞ , b and R 0 are the material parameters to be determined.The notation and material parameters used to represent the experimental uniaxial tensile test data are shown in table 1, These stress-strain curves were used in MAIA to estimate ferrite and martensite phase stress-strain.DP steels with two different V m were used in the calibration stage to obtain ferrite and martensite stress-strain curves.With the experimental data shown in table 1, there are several combinations of pairing V m to calibrate ferrite and martensite phase stress-strain curves.In the initial development of MAIA [12], stress-strain curves of DP steel with minimum and maximum V m were used to calibrate the stress-strain curves of ferrite and martensite phase.Results were validated using DP steels with V m within the range of minimum and maximum V m used during calibration.One of the primary objectives for this current research is to investigate the effects of using different V m combination to determine 100% ferrite and 100% martensite phase stress-strain curves during calibration stage.
Table 2 shows the experimental data used in the calibration process for the various groups.Group A used DP with minimum and maximum V m ; Group B utilized DP steel with minimum and mid-value V m ; Group C employed DP steel with mid-value and maximum V m .To ensure consistency, only two iterations were carried out for all the groups shown in table 2.

Initial estimates of ferrite and martensite phase stress-strain curves
The first stage is for initial estimation of stress-strain curves for the individual phase of ferrite and martensite.This requires experimentally obtained stress-strain curves of two DP steels of the same alloy with different martensite volume fraction V m .Experimentally obtained stressstrain curves exp σ A% and exp σ B% were applied in the constant strain rule of mixture (ROM) shown in equations ( 7) and ( 8) respectively.The initial estimations of martensite (σ mi ) and ferrite (σ fi ) stress-strain curves were solved simultaneously using equations ( 9) and ( 10) respectively.Figure 4 illustrates the process by which σ mi and σ fi were calculated at one strain value.This procedure was repeated at various strain values multiple times.In current study, σ mi and σ fi were collected from 100 different strain values, The MAIA process detailed in sections 3.2-3.5 utilizes experimental tensile test data of DP steel with V m of 20% and 67% for both initialization and iteration stage.Table 3 showed an example of using equations ( 9) and (10) to derive the initial estimates of ferrite and martensite phase stress-strain data.These values were subsequently curve-fitted using least square regression to determine the material parameters of the plasticity models used.In this case, the modified Voce law was utilized.
The initial estimates of the ferrite and martensite phase stress-strain curves served as material parameters in the micromechanical FE models.The result data from FE models will constitute the first iteration results in the second stage of MAIA.

Processing data from micromechanical FE model
The average FE modelled in-situ true stress and true strain values in the y-direction of ferrite and martensite phase were calculated by volume averaging using equations ( 11)- (14).ε and σ  11) 13) σm (MPa) equation ( 12) 14) FE σ 20% (MPa) equation ( 16) The strain and stress values of the homogenized stress-strain curve of the DP steel were calculated using equations ( 15) and ( 16) respectively, Examples of calculated in-situ ferrite, in-situ martensite, and homogenized modelled stressstress curves of DP steel with V m of 20% are shown in table 4. The same process was also conducted for FE micromechanical model with V m of 67% to generate the identical table of values.
Figure 5. Illustration of the process to calculate differences between FE modelled results with experimental results for DP steel with V mA .(a) calculate local phases and homogenized stress-strain curves.(b) Determine stress and strain inhomogeneity between phases at specific global strain.(c) Relate the stress and strain inhomogneity between phases with the experimental stress-strain curves.(d) the differences between FE modelled and experimental stress-strain curves were calculated as vector, represented with d A%i .Reproduced with permission from [12].

Calculation on the difference between experiment and FE model
The second stage is the adaptive iteration stage.The main concept in this stage is to calibrate ferrite and martensite stress-strain curves by using the differences between the FE modelled results and experimental results to compute the next estimates in the iteration process.In this stage, i and i + 1 represents current and next estimates respectively.The procedure to determine the differences between experiment stress-strain curve and FE modelled stress-strain curve for DP steel with V mA is illustrated in figure 5.In figure 5(a), FE models the in-situ stress-strain curves of ferrite (σ f ) and martensite (σ m ) phase using equations ( 11)-( 14).The FE modelled homogenized stress-strain curve (FE σ A%i ) was obtained from equations ( 15) and ( 16).Next, the stress and strain inhomogeneity between ferrite and martensite were determined at multiple points.The relationships between ferrite, martensite, and homogenous stress-strain curves at a given global strain were connected by straight line, as illustrated in figure 5(b).In figure 5(c), the points where the straight lines intersect with the experiment stress-strain curves (exp σ A% ) were calculated.The differences between experiment (exp σ A% ) and modelled (FEσ A%i ) stressstrain curves were computed as vectors as shown in figure 5 Step εm (−) eq. ( 11) 13) σm (MPa) eq. ( 12) σ f (MPa) eq. ( 14) FE ε 20% (−) eq. ( 15) In figure 5(b), the lines drawn show the stress and strain inhomogeneity between the phases.The lines were represented with the equation a 0 + a 1 ε, and were obtained by performing linear regression on the three points at a given global strain.Table 5 presents examples of the linear equations formed at two different time steps for the FE model with V m of 20%.
Table 6 displays examples of values representing the points of intersections between the experimental stress-strain curve and the lines that represent stress and strain inhomogeneity between the phases, as illustrated in figure 5(c).To determine the point of intersection, the strain values were calculated numerically using (17).The material parameters of the experimental data are shown in table 1.Since there could be a 100 points of intersection to calculated, an Excel macro was created to automate the calculation, The difference between the experimental and the FE modelled stress-strain curves was represented as a vector, as illustrated in figure 5(d).At a given global strain, the vector d A%i was calculated by taking the point where experimental stress-strain curve intersects with the line a 0 + a 1 ε and subtracting it from the point of the homogenized stress strain curves.An example of calculated difference vector for Vm of 20% (d 20%i ) is shown in table 7. The same process was carried out for DP steel with a V m of 67%, and subsequently, the difference between experimental and FE modelled stress strain curves (d 67%i ) was obtained.

Subsequent estimation of ferrite and martensite phase stress-strain curves
The differences of the stress-strain curves between FE modelled results and experimental results for DP steel with V mA and V mB are expressed using ROM shown in equations ( 18) and ( 19) respectively, Table 6.Examples of values of points of intersection between the experimental stressstrain curves with the lines that relates with the stress and strain inhomogeneity between phases, as shown in figure 5(c).

Micromechanical FE model with
Linear equations that represent lines on figure 5(b) Points where experimental stress-strain curve intersects with the lines a 0 + a 1 ε.Shown in figure 5(c) Step FE σ 20% (MPa) eq. ( 16) a 0 a 1 ε (−) Obtained by solving eq. ( 17) FE σ 20% (MPa) eq. ( 16) Obtained by solving eq. ( 17) where d fi and d mi are the differences between current and next estimates of ferrite and martensite respectively.Both d mi and d fi were solved simultaneously using equations ( 20) and ( 21) respectively.
Figure 6.Relationship between current estimates and new estimates of phase stressstrain curves as specified by equations ( 22) and ( 23).Reproduced with permission from [12].
Finally, the next estimates of ferrite (σ f i +1 ) and martensite (σ m i +1 ) stress-strain curves could be calculated using equations ( 22) and ( 23) respectively.Figure 6 illustrates the relationship between current and new estimates of the phase stress-strain curves.The calculated data points σ m i +1 and σ f i +1 were curve fitted to obtain the new sets of material parameters to be used for the next iteration of calculation, Table 8 shows examples of calculated vector representing the differences between current and next estimates of ferrite and martensite stress-strain curves.The calculated next estimate of martensite (σ m i +1 ) and ferrite (σ f i +1 ) curves were subsequently utilized as material input for micromechanical FE models, which generates the FE modelled homogenized DP steel stress-strain curves for the second iteration.
The iteration process continued until the estimated ferrite and martensite phase stress-strain curves produced modelled homogeneous stress-strain curves with an error that satisfies the user convergence condition.The error is calculated using equation (24), where A exp represent area under the stress-strain curve of the experiment result, and A error is the area enclosed by the experimental stress-strain curve and the FE modelled stress-strain curve until the uniform elongation of the experimental result.Figures 7(a) and (b) illustrate the area of A exp and A error respectively.
In the validation stage, the final estimated ferrite and martensite stress-strain curves obtained from the second stage are compared and validated against experimental tensile test data.The validation process used experimental tensile test data of DP steel with different values of V m .This process determines how well the estimated ferrite and martensite stress-strain curves could represent the overall mechanical behaviour of DP steel with different V m .20) Vector d f% i to represent difference between σ fi and σ fi +1 .Eq. ( 21) Step ∆ε ∆σ(MPa) ∆ε ∆σ(MPa) ∆ε ∆σ(MPa) ∆ε ∆σ(MPa)

Ferrite and martensite stress-strain curves in Group A
Group A used DP steels with minimum and maximum V m of 20% and 67% to calibrate pure ferrite and pure martensite stress-strain curves.Results were validated using DP steels with V m within the range of 20%-67%.Results obtained in the calibration stage are shown in figure 8. Figure 8(a) shows the calculated pure ferrite and pure martensite stress-strain curves used as material input for iteration 1 and 2. Figures 8(b) and (c) shows the corresponding modelled stress-strain curves of DP steel using calculated phase stress-strain curves from iteration 1 and iteration 2 respectively.In the first iteration, FE models underpredict the stress-strain curves of DP steels when compared with the experimental results.The differences in the FE modelled and experimental results were used in the calibration algorithm to calculate the phase stressstrain curves used for iteration 2. By the second iteration, stress-strain curves of DP steels produced by FE models agree well with the experimental result.
In the verification process, the calculated ferrite and martensite phase stress-strain curves from iteration 2 were used in the FE models to predict stress-strain curves of DP steel with V m of 27%, 42%, 52% and 62%.Figures 9(a)-(d) compared FE modelled stress-strain curves of DP steels with experimental results.Figure 9(e) shows the corresponding calculated error between FE results and experimental data.The calculated ferrite and martensite phase stressstrain curves could predict stress-strain curves of DP steel with V m of 27% and 42% that agreed well with experiment, however it over predicts DP steel with Vm of 52% and 62% by 5.3% and 4% respectively.There may be several reasons to the overprediction of modelled stress flow curves for DP steels with V m of 52% and 62%.One of the more obvious ones would be the over simplified FE micromechanical models used.The models did not include fracture mechanism, effects of grain size, grain boundaries and morphology.
In DP steels, softer ferrite goes through larger strain and lower stress than the harder martensite [17], thus causing strain incompatibility between the phases.Figure 10 shows the effect of V m on FE modelled strain ratio between ferrite and martensite phases . The modelled ε f εm curves show that ferrite phase goes through larger deformation than the martensite phase.The strain ratio ε f εm decreases with increasing V m , which indicates that there is lower strain incompatibility between the phases with increasing amount of martensite in DP steels.Similar behaviour was observed experimentally by Shen et al [18] and Su and Garland [19].The strain ratio ε f εm showed an initially increase during the early stage of deformation, followed by a subsequent decrease after reaching global strain values of around 0.03.This behaviour aligns with the three stages of stress flow relationship between ferrite and martensite in the DP steel published by [20,21].During the first stage, both ferrite and martensite deform elastically, resulting in a nearly equal strain ratio close to 1.In the second stage, ferrite deform plastically while martensite continues to deform elastically, causing an increase in the strain ratio between the two phases.In the third stage, both ferrite and martensite deform plastically.The onset of the third stage of deformation is the cause of the decrease in strain ratio after global strain values of 0.03.Various research had been dedicated to measure the strain incompatibility between ferrite and martensite phases in DP steels.Tasan et al [22] reported strain ratio ε f εm of DP800 steel with values varying from 5 to 17, and concluded that morphology of martensite affects the strain distribution in DP steels.Ososkov et al [23] measured local strain partitioning of DP600 and reported ε f εm values between 3.8-6 for global strain up to 0.17, and found that the strain partitioning behaviour of DP steels is significantly affected by the spatial distribution of ferrite and martensite phase.Ghadbeigi et al [24] experimentally measured strain ratio ε f εm of DP1000 with values 1.1-1.2.The modelled strain ratio ε f εm values shown in figure 10 varied between 1.15-1.29,which indicated that even though ferrite phase strained preferably than martensite phase, the degree of strain partitioning between ferrite and martensite phase is substantially lower than the findings from Tasan et al [22] and Ososkov et al [23].The experimental data used here had vanadium added to the DP steels as one of the alloys (refer to table 1).The alloy vanadium significantly reduced the ferrite grain size in DP steels, enhancing the strength in ferrite phase [15,25,26].Vanadium alloy added to DP steels also caused martensite softening [15,25].The combined effects of vanadium on strengthening of ferrite and softening of martensite caused the substantial reduction in strain incompatibility between martensite and ferrite phases.This resulted in lower strain ratio value in the simulation.Figure 11(a) compares the FE modelled stress-strain curves of unconstrained single ferrite phase condition (referred to hereafter as bulk ferrite) with the average in-situ ferrite phase stress-strain curves of DP steel with V m of 20% and 67%.Bulk ferrite phase stress-strain curve is the calculated material property as an input material property in the FE models for iteration 2. The modelled average in-situ ferrite stress-strain curves were obtained from the FE models using equations ( 13) and ( 14).The FE modelled strength of ferrite in the mixture of DP steel is higher than the bulk ferrite.The modelled in-situ strength of ferrite in the mixture of DP steel increases its strength with increasing V m .Fonstein [27] from experimental observation concluded that the properties of ferrite and martensite as a mixture in the DP steel can be different from its single phase condition.Su and Gurland [19] observed that ferrite phase in the mixture of DP steel behaves differently from its individual phase condition, and assumed that the ferrite phase in the mixture of DP steel strain hardened more than bulk ferrite, this trend was similar to figure 11(a).Upon quenching in the final stage of intercritical annealing, the transformation of austenite to martensite phase formed high dislocation density in the ferrite phase near the martensite island due to the transformation strain [27,28].This dislocation density increased strength of the ferrite phase in the DP steel.Koo et al [13] also mentioned that the enhancement of ferrite strength in DP steel is contributed by the effects of dislocation hardening and the free mean path of ferrite as according to the Hall-Petch equation.However, the ferrite stress-strain curve used in the FE model for the current research does not account the effect of both ferrite grain size and dislocation density.Perhaps, the differences in the in-situ and bulk phase properties were due to the interactions between the two phases, which was successfully captured in the FE models.Figure 11(b) compares the stress-strain curve of unconstrained single martensite phase (referred to from hereafter as bulk martensite) with the average in-situ martensite phase stress-strain curves of DP steel with Vm of 20% and 67%.For martensite phase, the modelled strength of average in-situ martensite phase is lowered than bulk martensite phase.Davies [29], Koo et al [13] and Tamura et al [14] observed experimentally that martensite in the mixture of DP steel have lower strength when compared to martensite formed from a fully austenitic steel (bulk martensite), which agree with the graph shown in figure 11(b).The modelled in-situ strength of martensite in the mixture of DP steel increases as V m increases.However, it is important not to conclude that a lower in-situ martensite stress-strain curves indicate a softer and more deformable martensite elements.Figure 11(b) must be interpreted and analysed in conjunction with figure 10, which detailed the effect of V m on strain ratio between ferrite and martensite in DP steel.Connecting these two graphs suggests that as V m increases, the martensite in the DP steel bears more load and deforms earlier.This behaviour has been experimentally observed by [27].Results shown in figures 11(a) and (b) demonstrate that with one set of ferrite and martensite phase stress-strain curves, the FE models could simulate the strengthening of ferrite phase and the softening of martensite phase in the mixture of DP steel.One of the main explanations for the softening of martensite phase and strengthening of ferrite phase in DP steel, as V m increase, primarily revolves around the change in carbon content within the martensite and ferrite phase.This is clearly specified in the constitutive equations developed by Tomota et al [1], Rodriguez et al [3] and Pierman et al [2].Therefore, it is believed that the stress-strain curves for martensite and ferrite phases should vary with V m .Perhaps, within a certain range of V m , there could be a mechanical phenomenon to elucidate the changes in martensite and ferrite behaviour with V m .
The equivalent von-Mises stress, or equivalent tensile stress shown in equation ( 25) is used to calculate the equivalent stress of the ductile material under multiaxial loading from the results of uniaxial tensile tests.In equation (25), σ v is the equivalent von Mises ensile stress, σ 1 , σ 2 and σ 3 are the principal stresses.This equation forms an open circular cylinder surface in a principal stress coordinate system, In a 3D DP uniaxial tensile model, it could be simplified into a constant strain or a constant stress model.Figure 12(a) provides the diagram representation of von Mises yield surfaces for ferrite and martensite.At constant strain or constant stress condition, σ 1 = σ 3 which reduced the cylindrical surface into a straight line in positive σ 2 axis as shown in figures 12(b) and (c) respectively.In figure 12, σ m is the equivalent stress for martensite and σ f is the equivalent stress for ferrite.σ 1f , σ 2f and σ 3f are the principal stresses for ferrite, σ 1m , σ 2m and σ 3m are the principal stresses for martensite.Under constant strain condition, both ferrite and martensite elements undergo equal strain but different stress values.Figure 13(a) shows the configuration of a 3D DP microstructure model that undergoes constant strain in a uniaxial tensile stress condition, with V m of 50%. Figure 13(b) shows the corresponding modelled stresses with respect to nodal displacement.The graph shows that at any given pulling displacement, the stress values of ferrite and martensite in σ 1 and σ 3 equal to zero, the stress values of ferrite and martensite in the σ 2 are the same as its equivalent stresses.It is worthwhile to note that the equivalent stresses are the bulk phase stress-strain curves, while the stress values in the pulling direction correspond to the in-situ phase stress-strain curves.Figure 13(c) illustrates the relationship between principal stresses and equivalent stresses for the ferrite and martensite phases in the principal stress coordinate system under constant strain condition or in this case, U2 = 0.019 mm.The stress values of ferrite and martensite in the 2-direction are the same as its equivalent stress values (σ 2 m = σ m and σ 2 f = σ f ).The overall stress value σ 2 follows the rules of mixture calculated using equation ( 16).This model and the graph shown in figure 13(b) reveals that DP microstructure, arranged under constant strain configuration, the in-situ ferrite and martensite phase stress-strain curves are behaving the same as the bulk ferrite and martensite phase stress-strain curves.
Under constant stress condition, both ferrite and martensite elements have the same stress in the 2-direction (σ 2 f = σ 2 m = σ 2 ). Figure 14(a) shows the configuration of a 3D DP microstructure model that undergoes constant stress in a uniaxial tensile stress condition, with V m of 50%. Figure 14(b) shows the corresponding modelled stresses as a function of displacement in the pulling direction.In this scenario, although the overall model is under uniaxial tensile stress condition, but individually as a phase, both ferrite and martensite elements are not in uniaxial tensile state.The principal stresses of ferrite and martensite in the 1-and 3-directions are not equal to zero, but the combined stresses for each individual phase will lead to its phase equivalent stress value by using equation (25).Despite of both ferrite and martensite elements deforming in similar manner, the principal stresses in the 1-and 3-directions are positive for ferrite phase and negative for martensite phase.These phenomena could be explained by referring to figure 14(c).Figure 14(c) shows the stress states of ferrite and martensite under constant stress at a given strain in the principal coordinate system.At constant stress condition, stresses  of ferrite and martensite phase in the pulling direction must be the same (i.e.σ 2 f = σ 2 m = σ 2 ), causing σ 2 f > σ f and σ 2 m < σ m , which explain the higher stresses for ferrite phase and lower stresses for martensite phase as compared to its bulk phase stress-strain curves.
The 3D microstructure model of DP steel as shown in figure 15 above, it contained random distribution of martensite elements in the ferrite matrix.The overall model contains combination of constant strain and constant stress conditions.In regions where the microstructure has constant strain configuration, the in-situ stresses will align with the corresponding values of their bulk stresses.In areas where the microstructure has constant stress configuration, it provides a physical interpretation that contributes the overall strengthening of ferrite phase and softening martensite phase.Therefore, with a given range of V m and grain size, one could reasonably assume a constant bulk ferrite and martensite stress-strain curves in DP steel and still could capture changes in the in-situ behaviour of ferrite and martensite phase stress-strain curves.
The diagrams presented in figure 16 illustrate the effect of V m on the in-situ phase stressstrain curves in comparison to its bulk stress-strain curves under constant stress configuration.Figure 16(a) shows the configuration of DP microstructure with a V m of 20% under constant stress in a uniaxial tensile condition.Figure 16(b) displays the corresponding principal stresses in relation to the pulling distance.Figure 16(c) shows the relationship between the constant pulling stress σ 2 with the phase principal stresses.It is imperative to be mindful of the fact that that overall uniaxial tensile stress condition pulling in the 2-direction means σ 11 = σ 33 = 0.In this case, since V m is 20%, σ 11 = σ 33 = 0.2 (−1504) + 0.8 (373) ≈ 0. Therefore, just like the constant stress model with V m of 50%, as shown in figure 14, overall model is under uniaxial condition, even though individual phase is not under uniaxial tensile condition.Irrespective of the V m content, when under constant stress configuration, the in-situ stresses of the ferrite phase and martensite phase will be higher and lower than its bulk stress-strain curves respectively.Notably, the stresses of the in-situ ferrite phase increase less with low V m , and the stresses of martensite phase decrease more with low V m .This explains the variation in the in-situ stresses of the ferrite and martensite phases in DP steels with varying V m , as shown in figure 11.Figures 16(e) and (f) illustrates how reducing V m influences the in-situ martensite stress values and its strain behaviour respectively.While the stress-strain curves of the martensite phase decrease with lower V m , it does not indicate that the martensite phase becomes more deformable with decreasing V m .Instead, it simply suggests that, under constant stress configuration, as V m increases, martensite phase in the DP steels bears more load and deformation begins earlier.
The ability of the FE micromechanical models and the calculated ferrite and martensite phase to capture strain hardening behaviour of DP steels was also examined.Figure 17 shows the effect of V m on the FE modelled strain hardening rate of the DP steel with strain.The DP steel displays a high initial strain hardening rate and decreases continuously as the strain increases, which aligns with earlier experimental findings [30].The high initial strain hardening rate is contributed by the back stresses formed due to the high dislocation density between the ferrite and martensite interface, as discussed by [31].
At the initial stage of deformation (up to true strain value of approximately 0.035), the FE modelled strain hardening rate of DP steel increases with higher V m .Beyond the true strain value of 0.035, the strain hardening rate of DP steel decreases with higher V m .The reverse role of Vm on strain hardening rate occurs at a strain value of approximately 0.035 aligns purposefully with the strain ratio behaviour illustrated in figure 10.At the initial stage of strain hardening rate, ferrite deformed plastically while martensite is still in its elastic region.During this phase, the effect of back stress caused by the elastic deformation of martensite is linear with V m [9], thereby amplifying the strain hardening rate with increasing V m .After the strain value of approximate 0.035, both ferrite and martensite phase are deforming plastically.The back stress induced by elastic deformation of martensite no longer increases with strain but remain constant.As V m increases, the strain hardening rate experience greater decline, attributed to the increase in plastic deformation of martensite with increasing V m .The strain hardening rate decreases more substantially with increasing V m due to the increased in plastic deformation of martensite phase at high V m .This trend on the effect of increasing V m on the strain hardening rate at higher strain were also observed experimentally by Scott et al [15] and Lai et al [9].Similar study was also carried out Sarosiek and Owen [32], which they carried out experimental work to observe the work hardening behaviour of DP steel at small strain.They observed that the work hardening index in Ludwik equation increases with increasing V m , and the order reversed at higher strain.This is an exciting finding, as it indicated that using one set of ferrite and martensite stress-strain curves, the non-linear effect of strain hardening behaviour with varying V m was also effectively captured in the FE simulations.

Ferrite and martensite stress-strain curves in various groups
The calculated ferrite and martensite stress-strain curves using different pairs of V m to calibrate are shown in figures 18(a) and (b) respectively.Group A used DP steels with minimum and maximum V m of 20% and 67%; Group B used DP steel with minimum and mid-value V m of 20% and 42%; Group C used DP steel with mid-value and maximum V m of 42% and 67%.The ferrite stress-strain curves obtained from Group A and Group B are almost identical, whereas from Group C is lower.The calculated martensite stress-strain curves showed more differences between the groups.The differences in the stress flow curves became more substantial at higher strain values as detailed in figure 18(b).The consistency of results could perhaps be due to the comparable ferrite grain size of the DP used in this paper.The ferrite grain size of the DP steels used for this research vary from 1.4 to 2.3 µm [15].Similar findings were noted by Tamura et al [14], indication that the strength of DP steel is dependent on volume fraction, assuming that the alloys have similar grain size.Davies [29] did extensive experimental work on DP steel with various V m , ferrite grain size and chemical composition.Davies analysis showed that the strength of DP is solely dependent on ferrite grain size and V m .
The calculated error made during calibration and validation stage for Group A, B and C are shown in figures 19(a), (b) and (c) respectively.The error made by the FE models during calibration stage is expected to be low, as the ferrite and martensite phase stress-strain curves are calibrated to reduce the differences between modelled and experimental data.The error in the validation stage ranged from 1% to 6%.Regardless of the groups, the error is consistently higher for DP steel with V m of 52% and 62%.The estimated ferrite and martensite phase stress-strain curves for Groups A, B and C performed relatively well in the validation stage.The calculated phase stress-strain curves for the three groups did not differ much.This indicated the importance of using two DP steels with a difference of 20%-50% in the martensite content during calibration.
Additional groups of pairing V m to calibrate ferrite and martensite stress-strain curves were tested to confirm the suggestions made for the operating boundaries for MAIA.The parings for the additional groups are shown on table 9.There were main purposes for calibrating ferrite and phase stress-strain curves using the two steels shown in D. is to determine ferrite and martensite stress-strain curves that could produce FE modelled DP steel with V m of 52% that agrees well with experiment, and secondly to test the usability of ferrite and martensite stress-strain curves that were calibrated using both DP steels with high V m .The objective of testing Group E was to determine the importance of using low V m as one of the DP steels used in calibration stage.Finally, the pair in Group F was to find out the quality of ferrite and martensite phase stress-strain curves calibrated by using both DP steels with low V m , this is to confirm the requirement of using DP steels with a minimum difference of 20% in martensite volume fraction.
The calculated error made by Group D, E and F are shown in figures 20(a), (b) and (c) respectively.The error in Group D and F are substantially higher than the rest of the groups.This suggests that using experimental data of DP steel with both low V m , or both high V m in the  calibration stage may introduce mathematical bias in the calculation of ferrite and martensite phase stress-strain curves.The substantial increase in error for Group D and F confirmed that the pair of DP steels used in calibration stage must have a difference of 20%-50% in martensite volume fraction.The results obtained by Group E showed similar performance when compared with Group A, B and C.However, the error made for DP steel with V m of 67% is higher in Group E compared to Group A, B and C.There could be a possibility that the experimental data for DP steel with V m of 52% is an anomaly.Group A, B and E are consistent with the condition of being calibrated using two DP steels with a difference of 20%-50% in the martensite content and one of the DP steels used in calibration stage has a low V m of 20%.The calculated ferrite and martensite phase stressstrain curves obtained by Group A, B and E are shown in figure 21(a).The results show that with the above both conditions met, the ferrite stress-strain curves obtained were similar.This indicated the importance of using a DP steel with low V m for calibrating the ferrite stress-strain curves.As mentioned earlier, the consistency in the ferrite stress-strain curves might also be due to the ferrite grain size of the DP steels used did not change much with varying V m .The martensite phase stress-strain curves showed more variation in results, especially for Group E. The corresponding error in the FE model results are shown in figure 21(b).The difference in the martensite stress-strain curve for Group E affected FE model result the most at high V m .As it was mentioned earlier, one possibility would be that the experimental data for DP steel with V m of 52% is an anomaly.It could also perhaps be the result of an over simplified models that exclude several conditions such as grain boundaries, and fracture behaviour etc.

Conclusions
(a) The MAIA method estimates ferrite and martensite phase stress-strain curves by using two DP steel with same chemical composition and different V m , reducing the need for complex characterization.However, the experimental data selected for calibration must have a martensite volume (V m ) fraction difference between 20% to 50% and one of the DP steels must have a low martensite volume fraction.(b) The algorithm captures the impact of vanadium on ferrite and martensite stress-strain curves in DP steel, showing vanadium-induced strengthening of ferrite and softening of martensite.The reduced in strain partition between the two phases due to the added vanadium was captured in the FE models.
(c) FE micromechanical models, based on calculated stress-strain curves, effectively capture softening of in-situ martensite and strengthening of in-situ ferrite in DP steel.d) In the initial stage of deformation, strain hardening rate increases with increasing V m , beyond strain value of approximately 0.035, the strain hardening rate decreases with increasing V m .This behaviour relates to the onset of martensite plastic deformation.(e) The changes in the mechanical behaviour of DP steel associated with increasing V m were successfully incorporated into the FE model.This achievement was attained using a single set of stress-strain curves for both the ferrite and martensite phases, without accounting for alterations in the chemical composition within each phase with increasing V m .(f) It is imperative to take note that the ferrite grain size of the DP steels tested in this paper vary from 1.4 to 2.3 µm.The effect of using wider range of ferrite grain size MAIA has not been investigated.The MAIA was successful in estimating pure ferrite and pure martensite stress-strain curves.This technique comes with obvious shortcomings, such as excluding crystal plasticity behaviour, effect of grain size and grain boundary, effect of change in chemical composition with varying martensite volume fraction, morphology etc.Though this technique fails to include individual physical effects of microstructure into its process, but collectively it was able to estimate pure ferrite and pure martensite stressstrain curves that could be used in FE Models to predict DP steel with different percentage of V m stress-strain curves with good accuracy.(g) Future work could involve embedding micromechanical FE model into the geometry of macroscale tensile test specimen.Additional properties, such as the effects of morphology and fracture properties, could be incorporated to better predict UTS and stress-strain curves beyond the necking stage.

Table 5 .
(d), represented with d A%i .The process shown in figure 5 were repeated using the FE model and experimental result of DP steel with V mB to obtain d B%i .Examples of obtaining the lines shown in figure 5(b) to represent the stress and strain inhomogeneity between phases at specific global strain.Micromechanical FE model with Vm = 20% iteration 1. Linear equations that represent lines on figure 5(b).a 0 + a 1 ε

Figure 7 .
Figure 7. (a) Aexp is the area under the stress-strain curve obtained experimentally.(b)|Aerror| is the area enclosed by experimental stress-strain curve and FE modelled stressstrain curve up till uniform elongation of the experiment result.Reproduced with permission from[12].

Figure 8 .
Figure 8.Calibration results for Group A data set.(a) Calculated bulk ferrite and bulk martensite phase stress-strain curves used as material input in FE models for iteration 1 and iteration 2. (b) Stress-strain curves for DP steel with Vm = 67% and Vm = 20% using martensite and ferrite stress-strain curves from iteration 1. (c) Stress-strain curves for DP steel with Vm = 67% and Vm = 20% using martensite and ferrite stress-strain curves from iteration 2.

Figure 10 .
Figure 10.Effect of Vm on strain incompatibility between ferrite and martensite.

Figure 11 .
Figure 11.(a) Comparison FE modelled stress-strain curves of bulk ferrite and in-situ ferrite in DP steels with Vm of 20% and 67%.(b) Comparison of FE modelled stressstrain curves of bulk martensite and in-situ martensite in DP steels with Vm of 20% and 67%.

Figure 12 .
Figure 12.(a) Equivalent von Mises surface of ferrite and martensite phases in the principal stress coordinate system, (b) equivalent von Mises surface reduce to linear line under constant strain, and (c) under constant stress condition.

Figure 13 .
Figure 13.(a) Configuration of dual phase microstructure under constant strain in a uniaxial tensile condition.Diagram illustrates a model with Vm = 50%.(b) Corresponding modelled principal stresses and equivalent stresses of ferrite and martensite with pulling displacement.(c) Stresses of ferrite and martensite at a given strain on the principal stress coordinate system, observe that when σ 1 = σ 3 = 0, σ 2m and σ 2 f equal to its equivalent stresses.This graph shows the relationship at pulling displacement U2 = 0.0195 mm.

Figure 14 .
Figure 14.(a) Configuration of dual phase microstructure under constant stress in a uniaxial tensile condition.Diagram illustrate a model with Vm = 50%.(b) Corresponding modelled principal stresses and equivalent stresses of ferrite and martensite phases with pulling displacement.(c) The relationship between principal stresses and resultant stresses of ferrite and martensite phases at a given homogenized strain under constant stress condition for Vm = 50%.This graph shows the relationship at pulling displacement U2 = 0.0195 mm.

Figure 15 .
Figure 15.Example of 3D micromechanical model with random distribution of microstructure.The random distribution cause certain parts of the model to experience constant strain condition, while other parts experience a constant stress condition.

Figure 16 .
Figure 16.(a) Configuration of dual phase microstructure under constant stress in a uniaxial tensile test condition.Diagram illustrates a model with Vm = 20%.(b) Corresponding modelled principal stresses and equivalent stresses of ferrite and martensite phases with pulling displacement.(c) The relationship between principal stresses and resultant stresses of ferrite and martensite phases at a given homogenized strain under constant stress for Vm = 20%.This graph shows the relationship at pulling displacement of U2 = 0.0195 mm.(e) Effect of Vm on in-situ martensite phase stressstrain curves under constant stress configuration.(f) Effect of Vm on the strain behaviour of martensite phase elements.

Figure 17 .
Figure 17.Effect of Vm on strain hardening rate with strain.

Figure 18 .
Figure 18.Stress-strain curves and its corresponding modified Voce law material constants for (a) ferrite phase and (b) martensite phase.Calibrated for 2 iterations.

Figure 19 .
Figure 19.Calculated error between simulated and experimental data in iteration 2 and validation process for (a) Group A, (b) Group B, and (c) Group C.

Figure 20 .
Figure 20.Calculated error between simulated and experimental data in iteration 2 and validation process for (a) Group D, (b) Group E, and (c) Group F.

Figure 21 .
Figure 21.Calculated ferrite and martensite phase stress-strain curves, (b) corresponding error in FE modeled results in Group A, B and E.

Table 1 .
Experimental data and its corresponding notation and material parameters.

Table 2 .
[12]ous groups of DP steels with different pairs of Vm to calibrate ferrite and martensite stress-strain curves.Illustration of calculating initial σm i and σ fi using exp σ A% and exp σ B% at a single strain value.Reproduced with permission from[12].

Table 3 .
Example of initial estimates of martensite (σm i ) and ferrite (σ fi ) phase stressstrain curves values using V mA as 20% and V mB as 67% during initialization and calibration stage.

Table 4 .
Data taken from FE micromechanical models with Vm of 20%.Results from this FE models used ferrite and martensite stress-strain curves that was calculated in the initialization process.

Table 8 .
Examples of calculated d m% i and d f% i to be used to estimate the next martensite and ferrite stress strain curves respectively.