Strain hardening exponent and strain rate sensitivity exponent of dual-phase steels at quasi-static strain rates during tensile testing

Abstract The effects of quasi-static strain rates on the tensile properties of two commercial ferrite-martensite dual-phase DP 600 and DP 800 steels were investigated using strip-shaped samples. The investigation was done by uniaxial tensile tests, covering applicable quasi-static strain rates. The two dual-phase steels show positive strain rate sensitivity. It is found that, as the flow stress increases, the strain-rate sensitivity exponent m decreases. The drop in the strain-rate sensitivity exponent m with strain is largely attributed to the decreased true strain rate caused by the increased instantaneous length of the specimen as the deformation progresses. To better describe the flow behavior of DP steels, a relationship combining the effect of both strain hardening exponent n and strain-rate sensitivity exponent m on the slope of the stress-strain is correlated. A good agreement between the extended Hollomon model and experimental tensile test data from stress-strain measurements is found.


ABOUT THE AUTHOR
Walid Khraisat is an associate professor in industrial and mechanical engineering.Dr. Khraisat has a major in materials science engineering.He earned his PhD's degrees from Chalmers University/ Sweden in 2004.He published more than 22 research articles in the fields of powder metallurgy, microstructural analysis and mechanical testing of steel.He currently works at the University of Jordan at the department of Industrial Engineering.This research work is a part of an ongoing research activity to include the aspect of sustainability.Emphasis on sustainability continues to expand for several reasons: increasing energy and material costs, increasing public pressure for improved environmental, health, and safety performance, and shifting consumer preference for green products.

Introduction
The dual-phase steel (DP) is a distinct steel belonging to the family of advanced high strength steels (AHSS) with increased formability (Amirthalingam et al., 2010).The AHSS steels are characterized as steels with yield strength values greater than 300 MPa and a tensile strength greater than 600 MPa (Bhadeshia & Honeycombe, 2006;Cooman, 2008).Because of its increased strength and better formability, AHSS can be used to make lighter parts while maintaining superior strength and safety performance.Making them the material of choice for automakers and infrastructure to support renewable energy such as solar power racks and wind turbine in places where winds are strong and the likelihood of earthquakes are high (Lesch et al., 2017;Loulijat et al., 2023;Ansari et al., 2023).
Dual phase (DP) steels are produced as both hot rolled and cold rolled steels having a microstructure consisting of two phases, ferrite and martensite.The strain rate sensitivity for AHSS decreases as the nominal strength of the steel increases (Movahed et al., 2009;Sarwar & Priestner, 1996).In addition, both the work-hardening and the strength of the steel increase as the strain-rate increases (Beynon et al., 2005;Tarigopula et al., 2006).
Strain hardening and strain rate sensitivity can be determined by uniaxial tensile tests at different strain rates, from 0.001S À 1 to 10 S À 1 .In the testing machine, the tensile test is carried out at a constant crosshead velocity (v).To accurately predict the behavior of dual phase steels, the choice of constitutive equations using the strain hardening n and strain rate hardening m exponents based on the experimental σ-ε data, should consider that n and m are not constant throughout the uniform deformation.This is because m is strain rate dependent and the true strain rate during the tensile test is not constant.
The prediction of the flow curve is mainly based on empirical equations, and the most popular formulation is the Ludwik-Hollomon equation and its variants.The limitation of these models, mathematically speaking, is the assumption that the indices used in the equations are most of the time considered constants (K, n, and m) which reduces the applicability of these equations when conditions are changed (Shin et al., 2022).In some cases, the use of different n values is required in order to fit the flow curve over the entire deformation range (Strinfellow & Parks, 1991).
The true train rate, ε • , is related to the gradient of ε with respect to time (t) is given by: Where, v is the crosshead speed of the tensile testing machine.
The total derivative of ε • is given by: According to equation 2 the true strain rate decreases as the instantaneous length L of the specimen increases.The second term in the right-hand side of eq.2 indicates that the strain rate is not constant during tensile testing.The true strain ɛ which is the rate of instantaneous increase in the instantaneous gauge length L is: Considering the uniaxial case, the true strain is defined by letting dL be the incremental change in gauge length and L 0 the gauge length at the beginning of that increment.Then, the corresponding true plastic strain increment (dε) becomes: Combining both equations 1 and 3 the following equation is obtained From eq.5 it is clear that the actual strain rate is dependent on strain.The strain rate exponentially decreases very rapidly at first, and then becomes slower as time passes.The total derivative at constant testing speed v of eq. 2 can be written as: The strain rate sensitivity m decreases with increasing plastic strain level, which is a consequence of the increase in the sample's length.This means that to run tensile tests at constant ε • requires that the crosshead velocity v steadily increases with the increasing length to maintain a constant ε • .Otherwise, during tensile testing, strain rate changes continuously affect the plastic flow behavior of strain rate-sensitive steels.Also, during deformation of DP steel grain refinement occurs which can effectively increase the strain rate sensitivity exponent and reduce the strain hardening exponent (Kang et al., 2009;Wei et al., 2004).From the literature review concerning the cold deformation of DP steel it can be concluded that characterizing the tensile behavior of DP steel at quasi static strain rates still considers the n value as constant and the m is given a zero value.This means that the n exponent is not a function of strain rate.
In this work, the variations in deformation behavior of DP 600 and DP 800 steels with strain rate will be demonstrated by considering that the true strain rate is not constant during tensile testing.A relationship is derived from the extended Hollomon equation to describe the material's hardening response based on the combined strain hardening and strain rate hardening effect.This relationship can be obtained by a linear regression of experimental σ-ε data where the slope of the curve lnσ vs. lnε is the value (n-mε).

Theoretical analysis
The flow curve, in the region of uniform plastic deformation, can be expressed using the extended Hollomon equation which is used to represent the flow curve after yielding (Gang et al., 2022;Hosford, 2013;Schindler et al., 2003;Shi & Meuleman, 1995) where σ is the flow stress ε � is the strain rate K is the strength coefficient, n is the strain hardening exponent and m is the strain rate sensitivity exponent.Then, by applying the chain rule of differentiation to compute the total derivative of σ, we obtain: where the subscripts denote the quantity being held constant when calculating derivatives.
Combining eqs. 2 and 8, the following equation is obtained: Now assuming that the extended Hollman equation can be used to describe the stress-strain response, eq.7, and using eq.9 then dσ, will be given by: Which can be expressed as: At constant v eq.11 becomes: The right side of eq. 12 is the differential of lnσ with respect to lnε: The strain hardening exponent n can be obtained by linear regression of experimental data, where the slope of the curve lnσ vs. lnε is the value (n-mε).
Then, using the criterion for the onset of diffuse necking known as Considére criterion expressed as (Bian et al., 2017): Using the Considére criterion in eq.14 the condition for necking will be: So eq. 15 describes the material's strain hardening response based on the combined strain hardening and strain rate hardening.The strain rate sensitivity is dependent on the strain rate, which is dependent on the plastic strain (eq.5).As a result, the m-value decreases with increasing plastic strain level during tensile testing.So according to eq. 15 the n-value will increase to maintain the constant slope value of the line segment of the curve lnσ vs. lnε.

Experimental procedure
The stress-strain data obtained from tensile tests were analyzed.Mechanical properties and material flow curve parameters proposed by Hollomon were evaluated and examined to determine the strain rate effect on the strain hardening characteristics of DP 600 and DP 800.

Tensile testing
Uniaxial quasi-static tests were performed using a Shimadzu Universal-Materials-Testing-Machine in quasi-static strain rates range 10 −4 − 1 s −1 .Due to the slow nature of the quasi-static experiments isothermal deformation is assumed, thus no material softening takes place.Yielding is continuous in both the dual-phase steels as seen in Figures 1 and 2.

Specimen geometry
Strips were used for the uniaxial tension test.The strips were cut from the plates in a direction parallel to the rolling direction and then machined into tensile specimens with dimensions of 100 mm gauge length, 1 mm thick and 10 mm in width.The materials under investigation were two cold drawn dual-phase grade steel plates DP600 and DP 800.The chemical compositions of these materials are given in Table 1.The mechanical properties were tested at room temperature, and the tensile velocity was 5 mm/ min.The work-hardening parametrs n and m fitted from the tensile testing are shown in table 2. To measure the strain rate sensitivity exponents for DP600 and DP800 steels strain rate jump tests were conducted by changing the tensile velocity from 5 to 10 mm/min corresponding to strain rates 0.0008 s −1 and 0.00167 s −1 , respectively.The strain rate sensitivity exponent m was calculated based on the change of the flow stress obtained from the strain rate jump tests as given by (Dieter, 1986;Straffelini, 2023): Where F 1 is the tension force at strain rate 1 and F 2 is the tension force at strain rate 2 during the jump test.The strain rate sensitivity parameter m was measured several times at different strains for both DP 600 and DP 800 steels.Some of these measured m values are shown in Table 3.These measured m values will be used to investigate whether m values for the DP 600 and DP 800 steels may vary with the plastic strain or not and also to determine if the stress-strain response of DP 600 and DP 800 steels is sensitive to strain rate or not.

Results discussions
The tensile stress and strain data are presented in Figure 1 of DP600 and DP800 steels, respectively, at quasi static strain rates.The curves show typical characteristics of the tensile behavior of DP steels with low yield strength, continuous yielding, and high strain hardening at the early stages of plastic deformation.For the two investigated DP steels, the higher martensite fractioned sample DP800, possesses higher strength (YS and UTS) and lower elongation values compared with DP600.
The strain rate sensitivity parameter m for DP 600, and DP 800 shown in Table 3 indicate a positive strain-rate sensitivity.DP600 steel shows higher m-values than DP800 steel.The m-values are quite low compared to most metals which typically fall in the range of 0.02 to 0.2.
Plotting the true stress-true strain curves for both steels DP 600 and DP 800 on a logarithmic scale, the strain hardening curves are obtained in Figure 2. The least square regression factor (R 2 ) is used to obtain the best linear fit of ln(ε)-ln(σ).The (R 2 ) value for the best linear fit is kept to a minimum of 0.995.The linear segments for the different strain hardening stages for DP 600 and DP 800 are fitted with equations as shown in Table 4.The strength coefficient K, the slope of the linear segments, and the R 2 value used for the best linear fit are all shown in Table 4.
Increasing the martensite content from DP 600 to DP 800 raises the number of strain hardening stages.This is in agreement with many authors (Davies, 1978;Fonstein, 2017;Soliman & Palkowski, 2020;Zhao et al., 2014).The variation of strain hardening of DP 600 obeyed the two-stage work hardening mechanism as seen in Figure 2.However, the variations of strain hardening of DP 800 obeyed the three-stage strain hardening mechanism, characterized by a rapid decrease at low stresses (stage I) followed by a gradual decrease at higher stresses (stage-II) a third stage (stage III) with the lowest strain hardening ability as seen in Figure 2.
The number of strain hardening stages in DP 600 compared to DP 800 steels decreases from 3 to 2 stages.This is consistent with literature (Bag et al., 1999;Byun & Kim, 1993;Soliman & Palkowski, 2021) findings that the number of strain hardening stages is related to the martensite amount in the microstructure (V m ).The higher the V m is the higher the constraining effect of martensite for ferrite deformation and the higher the strain hardening exponent n in the first stage.The strain hardening behavior of DP steels is directly influenced by V m and strength of martensite, which increases with carbon content.Therefore, all stages of deformation of DP 800 have higher strain hardening compared to all stages of deformation of DP 600 this is evident by examining all the equations in Table 4 it is clearly evident that the Holloman equation parameters (work hardening exponent and strength coefficient) in stage I and II for DP 800 steel were increased compared to DP 600 steel which is a result of the increase in Vm.
To reproduce the strain hardening rate (θ=dσ/dε) of DP 600 and DP 800 steels eq. 13 is used.The value of dlnσ/dlnε is assumed to be equal to the slope of the linear segments in Table 3 thus eq. 13 becomes: The work hardening behavior of DP 600 and DP 800 is analyzed by investigating the variation of the work hardening rate (θ= dσ/dε) with the true stress.Figure 3 shows the θ-σ curve for both DP 600 and DP 800 steels.The presence of three stages of work hardening in DP 800 steel can be related to the activation of different work hardening mechanisms (Colla et al., 2009;Tomita & Okabayashi, 1985;Tomota et al., 2008;Zuo et al., 2012).Stage I which is characterized by higher work hardening is attributed to the ferrite deformation, while Stage II is related to restrained ferrite deformation due to the presence of the hard martensite phase (Mazaheri et al., 2015;Nam & Bae, 1999).Martensite is not observed to deform plastically to a significant extent before necking, so as the martensite content increases in the dual-phase microstructure, the strength ratio decreases (Khraisat & Abu Al Jadayil, 2018), i.e. the material becomes less sensitive to strain rate changes.Most of the plastic deformation is concentrated in ferrite and since DP 800 has the highest martensite content it is least sensitive to strain rate increase.The decrease in θ remains up to the plastic instability that, in turn, defines the ultimate strength.The strain hardening rate θ where instability was initiated was higher than the theoretical values of Considere and Hart's theoretical criterion (dθ/dε=σ), probably due to the fact that the flow curve is deviating from the extended Hollomon equation, which is caused by the fact that ferrite and martensite yield at different stress states and due to changes in stress and strain partitioning.Non-linearity can be seen in Figure 3 for both DP 600 and DP 800 steels.DP 800 shows nonlinearity at higher strain levels in stage II, whereas DP600 at lower strain levels in stage I.The nonlinearity in Figure 3 indicates that the flow curve is deviating from the extended Hollomon equation.
To correctly represent the flow curve after yielding using the extended Hollomon equation different values of m-values were used at different plastic strains.Table 5 shows some of the m-values used for different strain ranges.Table 5 gives an overview of the m-values for DP 600 and DP 800 steels.An increase of the strain rate sensitivity exponent m is observed from 0,006  down to 0.0003 for DP 600 and from 0.0132 down to 0.00001.Evaluation of the stress-strain behavior for DP steels is done by comparing the experimental stress-strain curves with the calculated stress from the extended Hollomon equation.
Comparing the m values from Table 5 with the values in Table 3 it is clear that they differ significantly.The m values from strain-rate jump tests are strain rate dependent and increase with the increasing strain rate, also the m values are obtained from the slope of the ln (σ)/ln (ε) curve, assuming that the microstructure remains constant and is not affected by any thermal influences.However, under low strain rate conditions (quasi-static conditions), the test period is long, thus isothermal conditions take place and a negligible increase in temperature is observed.This is not the case during high strain rates where adiabatic conditions take place during testing.During higher deformation rates work hardening occurs in the ferrite phase and softening occurs in the martensite phase due to adiabatic conditions leading to an increase in the martensite ductility (Wang et al., 2013).
Both the tensile test and the extended Hollomon flow curves have been plotted together and match each other quite well for both steels DP 600 and DP 800.The true stress-true plastic strain flow curves acquired from the uniaxial tensile test are plotted in Figures 4 and 5 for both steels, accompanied by the modeling curves.As deformation advances, gradual decrease in strain rate leads to a decrease in the strain rate sensitivity exponent m which contributes to destabilizing the plastic flow which decreases resistance to necking (Ghosh, 1977;Shi & Meuleman, 1995;Wagoner & Wang, 1983;Zhu et al., 2022).That is, the flow stress of the metal decreases, with the decrease in the strain rate at the same deformation temperature.This is because, as the strain rate decreases, the time required for the DP steel to produce the same amount of deformation becomes longer and longer, so that the DP steel does have enough time to complete the softening effect and the work hardening effect is less and less significant causing the decrease in flow stress Analysis of the strain hardening behavior of DP 600 steel shows a double strain hardening stage with an n value significantly lower than that of DP 800 steel.The decrease in n-value from DP 600 to DP 800 is a result of increasing the volume fraction of martensite (Kumar et al., 2008).From

Conclusion
The hardening response of DP steel is based on the combined strain hardening and strain rate hardening effects.This combined effect can be obtained by a linear regression of experimental σ-ε data where the slope of the curve lnσ vs. lnε is the value (n-mε).
The drop in rate sensitivity exponent m with strain is largely attributed to the decreased strain rate caused by the increased instantaneous sample length as the deformation progresses.The present study has confirmed that the quasi static strain rate deformation behavior of DP 600 and 800 is significantly influenced by the strain rate and the strain rate sensitivity exponent m of DP 600 is higher than the strain sensitivity exponent of DP 800.The results show that the m-value is not constant and is highly dependent on the applied strain rate, strain level and testing method.The decrease in the strain hardening exponent n from DP 600 to DP 800 is a result of decreasing the strain rate sensitivity due to the decrease in the strain rate-sensitive ferrite.
Figure 3.The variation of the work hardening θ (MPa) with the true stress σ (MPa)for DP 600 and DP 800 steel showing the different stages of hardening at room temperature.

Figure
Figure 4. Stress-strain curves obtained from tensile test data of DP600 and from using the extended Hollman model (dashed).
Figure 5. Stress-strain curves obtained from tensile test data of DP800 and from using the extended Hollman model (dashed).

Table 5 . Summary of the extended Hollomon equations for the different strain hardening stages of deformed DP 600 and DP 800 steels. Where the true strain rate between brackets is taken from eq.5
Table 3 the lowest slope is for DP 800 stage 3, and from Table 4 the highest m values are found for DP 800 stage III.