CC BY
84
УДК 539.319
Effect of martensite volume fraction on strain partitioning behavior
of dual phase steel
A.K. Rana1, S.K. Paul2, P.P. Dey1
1 Mechanical Engineering Department, Indian Institute of Engineering Science and Technology, Shibpur, Howrah, 711103, India 2 Materials Science and Technology Division, CSIR-National Metallurgical Laboratory, Jamshedpur, 831007, India
Monotonic deformation behavior of ferrite-martensite dual phase steels with martensite volume of 13-43 % have been analyzed in the current investigation using micromechanics based finite element simulation on representative volume elements. The effects of martensite volume fraction on the strain partitioning behavior between soft ferrite matrix and hard martensite islands in dual phase steels during tensile deformation have been investigated. As a consequence of strain incompatibility between hard martensite and soft ferrite phases, inhomogeneous deformation and finally deformation localization occur during tensile deformation. Restricted local deformation in ferrite phase caused by the adjacent martensite islands triggers the local stress triaxiality development. As the martensite volume fraction increases, the local deformation restrictions in ferrite phase also increases and which results in higher stress triaxiality development. Similarly the strain partitioning behavior between ferrite matrix and martensite island is also influenced by the volume fraction of martensite. The strain partitioning coefficient increases with increasing martensite volume fraction.
Keywords: dual phase steel, representative volume element, strain partitioning, deformation localization, stress triaxiality
Влияние объемной доли мартенсита на распределение деформаций
в двухфазной стали
A.K. Rana1, S.K. Paul2, P.P. Dey1
1 Индийский институт техники и технологий, Шибпур, Хаора, 711103, Индия 2 Национальная лаборатория металлургии, Джамшедпур, 831007, Индия
В статье с использованием метода конечных элементов анализируется монотонное деформирование представительных элементов объема ферритно-мартенситных двухфазных сталей с различной долей мартенсита 13, 23, 33 и 43 %. Исследовано влияние объемной доли мартенсита на распределение деформаций между мягкой ферритной матрицей и твердой фазой мартенсита при растяжении. Несовместность деформации твердого мартенсита и мягкого феррита при растяжении приводит к развитию неоднородной деформации с последующей локализацией. Стесненная локальная деформация ферритной фазы, вызванная прилежащими зонами мартенсита, обуславливает возникновение трехосного локального напряженного состояния. По мере увеличения объемной доли мартенсита стесненность локальной деформации в ферритной фазе возрастает. Распределение деформаций между ферритной матрицей и мартенситом также зависит от объемной доли мартенсита. Коэффициент распределения деформаций возрастает с увеличением объемной доли мартенсита.
Ключевые слова: двухфазная сталь, представительный элемент объема, распределение деформаций, локализация деформации, трехосность напряжений
1. Introduction
Dual phase (DP) steel microstructure consists of hard martensite islands dispersed in a soft and ductile ferrite matrix. In dual phase steel, matrix (ferrite) ensures good cold formability, whereas the island (martensite) acts as strengthening element. The demand for such type of steel is ever increasing particularly in the automotive industry, where high strength and ductility permit weight reduction
without sacrificing passenger safety. With the reduction of vehicle weight, the fuel consumption and emissions will also be reduced.
Mechanical properties of dual phase steel have been explored extensively by various researchers [1, 2]. Using in-situ scanning electron microscopy and digital correlation analysis the existence of the inhomogeneous deformation at the microstructure level and deformation concentra-
© RanaA.K., Paul S.K., Dey P.P., 2017
tion in the soft phase (ferrite) was reported [3, 4]. Some other researchers reported that the ductile failure has been shown to be caused by deformation localization due to microstructure level heterogeneity [5-7]. Sodjit and Uthaisan-gsuk found that several short interrupted shear bands developed with low volume fraction of martensite, but when volume fraction of martensite increased, long continuously pronounced localizing shear band appeared in the dual phase microstructure [8]. Paul showed that ultimate tensile strength increases while percentage of elongation decreases with increase in martensite volume fraction in dual phase steel [9]. Similar observation has also been reported by Lai and his coresearchers [10]. Apart from empirical mixture rules [11, 12] and Eshelby-based homogenization concept [13, 14], the finite element simulations on representative microstructures [6, 15-20] are also utilized to understand local deformation response and predict the stress-strain behavior. The inhomogeneous deformation behavior of dual phase steel arises from the strain partitioning between the softer ferrite phase and hard martensite phase. However, less number of literatures is available in the open domain on the strain partitioning behavior of dual phase steel. Effect of martensite volume fraction on the strain partitioning of dual phase steel is investigated in the present work. According to authors' knowledge, the literature on this topic is not present.
2. Microstructure characterization using finite element modeling
2.1. Micromechanical simulation of dual phase steel
Dual phase steel with 780MPa tensile strength is used for finite element simulation in the present work. Tensile specimens of 50mm gauge length and 12.5mm gauge width (ASTM E8) are machined parallel to the rolling direction from the as-received 1mm thick steel sheets. The specimen is tested at room temperature using an electromechanical tensile testing machine at a crosshead speed of 1 mm/min which roughly corresponds to a strain rate of 3.33 -10-4 s-1. The material properties of ferrite and martensite phases of DP780 steel are collected from the literature [6].
Von Mises elastic-plastic material law is assumed for each single phase. To define the isotropic stress-strain behavior of each individual phase in the calculations, a model based on dislocation theory [21, 22] was used. The stressstrain relation can be written as
a = ay + o.MG^I1 - , ^
where a is the stress at true strain of 8. The material constants of this model is collected from a previous study [22].
The first term in Eq. (1) is the yield stress which is the summation of friction stress, solid solution strengthening, precipitation strengthening with Nb, Ti and/or V, grain size [23], and it can be describe as
a y = 70 + 37Mn + 83Si + 2918Nsol + 33Ni -
- 30Cr + 680P + 38Cu +11Mo + 5000C + (2)
Vd
where friction stress is 70 MPa, d is the grain size [mm], alloy contents are in wt %.
The second term takes care of the dislocation strengthening as well as work softening due to recovery. The parameter a = 0.33 is constant, M = 3 is the Taylor factor, G = 80 GPa is the shear modulus, b = 2.5 • 1010 m is the Burger's vector, Kr is the recovery rate. For ferrite a value of 10-5/da is used, where da is the ferrite grain size in meters. For martensite the Kr value used is 41, L = = 3.8-10-8 m is the dislocation mean free path for martensite [22].
2D representative volume element (RVE) is considered in the present study. Each phase is modeled as a homogeneous isotropic part of the RVE. Detailed interaction between interphase boundaries is ignored, as it is considerably small (atomic sizes), compared to the RVE being modeled.
In the present investigation, the strain partitioning between the ferrite and martensite phases is determined from micromechanics based finite element simulation results. The strain partitioning between the soft and hard phases normally occurs due to stress-strain flow property difference between the phases. The difference in the amount of strain between the two phases (martensite and ferrite) for certain overall strain level under the uniaxial tensile loading of the DP780 steel is attributed to the strain partitioning among the phases.
Figure 1 shows schematic illustration of strain partitioning between the ferrite and martensite phases at different overall strain levels. The strain partitioning coefficient P can be written as
P = ^m, (3)
8f 8m
where for overall stress and strain level a and 8 of DP780 steel, the stress and strain level of ferrite are af and 8f,
Strain
Fig. 1. Schematic representation of stress and strain partitioning between ferrite and martensite phases of dual phase steel
and stress and strain level of martensite are om and sm, respectively.
2.2. RVE simulation of dual phase steel
In the current work, the effect of martensite volume fraction on strain partitioning, stress triaxiality and microstructural level local deformation response are modeled using RVE. In micromechanical modeling, both ferrite and martensite phases are considered as homogeneous individually. However, microstructurally dual phase steel is acting as inhomogeneous material since stress-strain behavior of ferrite and martensite phases are completely different.
Two-dimensional RVE models are considered in this work. The size of the RVE is 1000 x 1000 ¡xm. The average area of an elliptical martensite island is around 5 ¡m with aspect ratio of 2. The commercial finite element software ABAQUS is used in the present analyses. Two dimensional plane strain elements are adopted for meshing the complete RVE. In order to simulate uniaxial tensile loading condition, all the nodes along the top edge are given the same displacements in the y direction, while they can freely move in the x direction during the tensile loading. All the nodes along the bottom edge are constrained not to move in the y direction, but are allowed to freely move in the x
direction. Because the RVE is assumed to be the representative in-plane microstructure for the DP780 steel under examination, the macroscopic engineering stress is obtained by dividing the reaction force of the RVE in the y direction by the initial area, i.e., the initial thickness for the model. The engineering strain in the y direction is obtained by dividing the displacements of the top edge by initial length of the model.
3. Results and discussion
To study the effect of martensite volume fraction on tensile behavior of dual phase steel, various RVEs are constructed with martensite volume of 13, 23, 33 and 43 %. Generated RVEs are shown in Fig. 2. The stress-strain behavior of ferrite and martensite phases are shown in Fig. 3 (calculated from Eqs. (1) and (2)). In RVEs, martensite islands are randomly distributed in ferrite matrix. Kim et al. [18] pointed out that random distribution with no particular orientation of soft phase has no significant effect on the strain partitioning at two phase interface. Kadkhodapour et al. [17] considered interphase boundary during RVE simulation of dual phase steels. Mishnaevsky [24], and Qing and Mishnaevsky [25] also considered interface boundary during finite element simulation of composite material. In
Fig. 2. Representative volume element with various percentage of ferrite and martensite: 87 % ferrite and 13 % martensite (a), 77 % ferrite and 23 % martensite (b), 67 % ferrite and 33 % martensite (c), 57 % ferrite and 43 % martensite (d)
0 —■—i—■—i—■—i—■—i—■—i—■—i—■—i—■— 0.0 0.1 0.2 0.3 0.4 True strain
Fig. 3. Stress-strain response of ferrite and martensite phases
the present investigation interface of two phases is not considered. This present study is limited by the analysis of martensite volume fraction value only, however taking into account interface properties will lead to changing the numerical values of parameters slightly but will not change the revealed trends.
To validate the 2D RVE model, DP780 steel is selected for the current study. Approximately 77 % ferrite and 23 %
1000
0-1-■-,-•-,-•-
0.00 0.05 0.10 0.15
True strain
Fig. 4. Comparison of finite element RVE simulation (1) with experimental tensile test results (2) of DP780 steel (REV with 77 % ferrite and 23 % martensite)
martensite phases present in the original microstructure of DP780 steel. 2D RVE of DP780 steel is shown in Fig. 2, b. Figure 4 shows that the simulated stress-strain curve by 2D RVE model is matched well with experimental (tensile test) result. Figure 5 shows the equivalent plastic strain distribution of the dual phase steels with 23 % martensite vol-
Fig. 5. Distribution of equivalent plastic strain in dual phase steel with martensite volume faction of 23 % at the various overall engineering strain levels of 2.5 (a), 5.0 (b), 7.5 (c) and 10.0 % (d)
Fig. 6. Distribution of equivalent plastic strain in dual phase steels with martensite volume of 13 (a), 23 (b), 33 (c) and 43 % (d) at overall engineering strain levels of 5 %
ume fraction for overall engineering strain of 2.5, 5.0, 7.5 and 10.0 % respectively. Figure 5 shows that the deformation is inhomogeneous from the beginning of plastic deformation and it grows with further straining. The equivalent plastic strain distribution at overall engineering strain of 5 % for various volume fraction of martensite is shown in Fig. 6. Inhomogeneous plastic deformation takes place for all dual phase steels with different volume fraction of martensite. However, localized deformation is higher for higher volume fraction of martensite. This localized deformation is detected in the softer phase ferrite irrespective of martensite volume fraction. Similar observation is reported by other research groups [6, 15-20].
For detailed investigation dual phase steel with martensite volume fraction of 0.23 is selected. Simulated and experimental tensile stress-strain behavior of dual phase steel with martensite volume fraction of 0.23 is matched well and depicted in Fig. 4. Von Mises equivalent plastic strain and equivalent stress at integration points in the RVE are collected for further analysis for various overall engineering strain levels. Then those von Mises equivalent plastic
strain and equivalent stress are plotted as histograms in Fig. 7. The bin size of histograms for von Mises equivalent plastic strain is 0.005, while for von Mises equivalent stress is 20 MPa. It is observed that with increasing overall engineering strain, the distribution become wider and the peak of distribution shifts towards higher values for von Mises equivalent plastic strain (Fig. 7, a), whereas starting position of the distribution not altered. Similarly the distribution of von Mises equivalent stress fully shifts toward right and distribution become wider (Fig. 7, b) with increasing overall engineering strain.
The local deformation of soft ferrite phase is constrained by nearby martensite islands and leads to inhomogeneous deformation and strain partitioning. The level of deformation constrained in the ferrite phase solely depends upon the amount and distribution of surrounded martensite. As a consequence of deformation constrained, the local stress state alters in different location of the ferrite. Therefore, inhomogeneous microstructure leads to build up of local stress triaxiality. Stress triaxiality can be defined as the ratio of mean stress and von Mises equivalent stress. For
80000-
60000
& 40000-
20000-
Engineering strain 0.5«
300 500 700
Equivalent stress, MPa
900
Fig. 7. Distributions of von Mises equivalent plastic strain (a) and von Mises equivalent stress (b) at various overall engineering strain levels
overall engineering strain of 5.0 %, distributions of local stress triaxiality in ferrite and martensite phases are shown in Fig. 8 for different martensite volume fractions. Stress triaxiality values in the histograms are plotted by taking a common bin size of 0.02 and plotted with respect to normalized frequency. Frequency of the stress triaxiality is normalized by the total number of integration points. Histogram of stress triaxiality for each phases show a bell-shaped distributions. It is definite from Fig. 8 that the bandwidth of the distribution of stress triaxiality in both ferrite and martensite phases increase with increasing martensite volume fraction in dual phase steel. Similarly, the peak of distribution also decreases for both ferrite and martensite phases with increasing martensite volume fraction in dual phase steel. However, the peak of distribution slightly shifts towards the positive side in ferrite phase and towards the negative side in martensite phase. For comparison purpose, the stress triaxiality distribution in ferrite and martensite phases of dual phase steel with 23 % volume fractions of martensite at overall engineering strain of 5.0 % is illustrated in the Fig. 9. The distribution of stress triaxiality is visible different in ferrite and martensite phases. The peak
of the distribution for martensite is higher than peak of the distribution for ferrite.
Furthermore, in order to clearly understand the stress levels (i.e. stress partitioning) of each phases, the von Mises equivalent stress distributions are computed for RVEs of dual phase steels with martensite volume of 13, 23, 33 and 43 %. Von Mises equivalent stress distributions in ferrite and martensite phase are shown in Fig. 10 for overall engineering strain of 5.0 % and various volume fraction of martensite. With increasing volume fraction of the martensite in dual phase steel the width of the distribution in ferrite phase increases, however peak of the distribution slightly shifts towards positive von Mises equivalent stress. Exactly similar observation is noticed for the distribution of martensite phase, but secondary peak is observed after 1544 MPa von Mises equivalent stress. The secondary peak is visible due to yielding of martensite. Strain hardening of martensite is relatively low, so after yielding increment of stress is also small. Secondary peak in martensite phase increases with increasing volume fraction of the martensite in dual phase steel. For dual phase steel with martensite volume fraction of 13 %, the secondary peak is small which
0.12i
§ 0.08-
^ 0.04 "
o £
0.00 -0.4
0.12
0.0 0.4
Stress triaxiality
0.00 -0.4
0.4 0.8
Stress triaxiality
Fig. 8. Distributions of stress triaxiality for martensite (a) and ferrite (b) phase in dual phase steels with martensite volume of 13, 23, 33 and 43 % at overall engineering strain levels of 5 %
Fig. 9. Stress triaxiality distribution at ferrite (1) and martensite (2) phases in dual phase steel with martensite volume of 23 % at overall engineering strain levels of 5 %
indicates little amount of martensite yielded at the overall engineering strain of 5 %.
The strain partitioning coefficient ß is calculated from Eq. (3) for different overall engineering strain levels and for dual phase steels with various volume fraction of martensite. Only mean value of stress and strain states are considered for computation of strain partitioning coefficient ß. Variation of strain partitioning coefficient ß for dual phase steels with various volume fraction of martensite are shown
Martensite volume fraction
— 13 23 -a- 33 °/ -o- 43 %
о д
<D
O4 <D
<D N
О
0.05
0.00
200 400 600 800 Equivalent stress, MPa
1000
0.00
200 600 1000 1400 Equivalent stress, MPa
1800
Fig. 10. Distributions of von Mises equivalent stress for ferrite (a) and martensite (b) phase in dual phase steels with martensite volume of 13, 23, 33 and 43 % at overall engineering strain levels of 5 %
Fig. 11. Variation of strain partitioning coefficient в at different overall engineering strain levels with various volume fraction of martensite
in Fig. 11. Log of strain partitioning coefficient в is used in Fig. 11 as variation is huge for dual phase steels with various volume fraction of martensite. Irrespective of martensite volume fraction, the strain partitioning coefficient в decreases with increase in overall engineering strain. However at all overall engineering strain levels, the strain partitioning coefficient в increases with increasing volume fraction of martensite. Therefore, it can be concluded that the strain partitioning between ferrite and martensite phases depends upon the volume fraction of martensite.
4. Conclusions
The strain partitioning behavior between softer ferrite phase and harder martensite phase has been modeled using the representative volume elements of dual phase steels with various martensite volume fractions. Deformation response of each phase in the microstructural level is also investigated in this study.
From the beginning of the plastic deformation the deformation in microstructural level is inhomogeneous and the deformation inhomogeneity increases with increasing straining.
For same level of overall engineering strain, the deformation inhomogeneity increases with increasing martensite volume fraction of dual phase steel.
Local stress triaxiality builds up in the ferrite phase due to local deformation restriction by the existence of adjacent hard martensite phase. Stress triaxiality development in ferrite and martensite phases alter with change in martensite volume fraction of dual phase steel.
Strain partitioning between soft ferrite and hard martensite phases occurs due to strength difference between them. The strain partitioning coefficient в increases with increasing volume fraction of martensite.
References
1. Taylor M.D., Choi K.S., Sun X.Matlock D.K., Packard C.E., Xu L Barlat F. Correlations between nanoindentation hardness and macro-
scopic mechanical properties in DP980 steels // Mat. Sci. Eng. A. -Struct. - 2014. - V. 597. - P. 431-439.
2. Sirinakorn T., Wongwises S., Uthaisangsuk V. A study of local deformation and damage of dual phase steel // Mater. Design. - 2014. -V. 64. - P. 729-742.
3. Kang J., Ososkov Y., Embury J.D., Wilkinson D.S. Digital image correlation studies for microscopic strain distribution and damage in dual phase steels // Scripta Mater. - 2007. - V. 11. - No. 56. - P. 9991002.
4. Ghadbeigi H., Pinna C., Celotto S., Yates J.R. Local plastic strain evolution in a high strength dual-phase steel // Mat. Sci. Eng. A. Struct. - 2010. - V. 527. - No. 18. - P. 5026-5032.
5. Sun X., Choi K.S., Liu W.N., Khaleel M.A. Predicting failure modes and ductility of dual phase steels using plastic strain localization // Int. J. Plasticity. - 2009. - V. 25. - No. 10. - P. 1888-1909.
6. Paul S.K., Kumar A. Micromechanics based modeling to predict flow behavior and plastic strain localization of dual phase steels // Comp. Mater. Sci. - 2012. - V. 63. - P. 66-74.
7. Paul S.K. Micromechanics based modeling of dual phase steels: Prediction of ductility and failure modes // Comp. Mater. Sci. - 2012. -V. 56. - P. 34-42.
8. Sodjit S., Uthaisangsuk V. Microstructure based prediction of strain hardening behavior of dual phase steels // Mater. Design. - 2012. -V. 41. - P. 370-379.
9. Paul S.K. Effect of martensite volume fraction on stress triaxiality and
deformation behavior of dual phase steel // Mater. Design. - 2013. -V. 50. - P. 782-789.
10. Lai Q., Brassart L., Bouaziz O., Goune M., Verdier M., Parry G., Perlade A., Brechet Y., Pardoen T. Influence of martensite volume fraction and hardness on the plastic behavior of dual-phase steels: Experiments and micromechanical modeling // Int. J. Plasticity. -2016.- V. 80. - P. 187-203.
11. Kuang S., Kang Y.L., Yu H., Liu R.D. Stress-strain partitioning analysis of constituent phases in dual phase steel based on the modified law of mixture // Int. J. Miner. Metall. Mater. - 2009. - V. 16. - P. 393-398.
12. Paul S.K., Mukherjee M. Determination of bulk flow properties of a material from the flow properties of its constituent phases // Comp. Mater. Sci. - 2014. - V. 84. - P. 1-12.
13. Mazinani M., Poole W.J. Effect of martensite plasticity on the deformation behavior of a low-carbon dual-phase steel // Metall. Mater. Trans. A. - 2007. - V. 38. - No. 2. - P. 328-339.
14. Brassart L., Doghri I., Delannay L. Self-consistent modeling of DP steel incorporating short range interactions // Int. J. Mater. Form. -
2009. - V. 2. - No. 1. - P. 447-450.
15. Choi K.S., Liu W.N., Sun X., Khaleel M.A. Influence of martensite mechanical properties on failure mode and ductility of dual-phase steels // Metall. Mater. Trans. A. - 2009. - V 40. - No. 4. - P. 796-809.
16. Uthaisangsuk V., Prahl U., Bleck W. Modelling of damage and failure in multiphase high strength DP and TRIP steels // Eng. Fract. Mech. - 2011. - V. 78. - No. 3. - P. 469-486.
17. Kadkhodapour J., Butz A., Ziaei-Rad S., Schmauder S. A micro mechanical study on failure initiation of dual phase steels under tension using single crystal plasticity model // Int. J. Plasticity. - 2011. -V. 27. - No. 7. - P. 1103-1125.
18. Kim E.Y., Yang H.S., Han S.H., Kwak J.H., Choi S.H. Effect of initial microstructure on strain-stress partitioning and void formation in DP980 steel during uniaxial tension // Met. Mater. Int. - 2012. -V. 18.- No. 4. - P. 573-582.
19. Marvi-Mashhadi M., Mazinani M., Rezaee-Bazzaz A. FEM modeling of the flow curves and failure modes of dual phase steels with different martensite volume fractions using actual microstructure as the representative volume // Comp. Mater. Sci. - 2012. - V. 65. -P. 197-202.
20. Chen P., Ghassemi-Armaki H., Kumar S., Bower A., Bhat S., Sada-gopan S. Microscale-calibrated modeling of the deformation response of dual-phase steels // Acta Mater. - 2014. - V. 65. - P. 133-149.
21. Rodriguez RM., Gutierrez I. Unified formulation to predict the tensile curves of steels with different microstructures // Mater. Sci. Forum. -2003. - V. 426-432. - P. 4525-4530.
22. Uthaisangsuk V., Prahl U., Bleck W. Modelling of damage and failure in multiphase high strength DP and TRIP steels // Eng. Fract. Mech. - 2011. - V. 78. - P. 469-486.
23. Pickering F.B. The Effect of Composition and Microstructure on Ductility and Toughness // Towards Improved Ductility and Toughness. - Tokyo: Climax Molybdenum Company, 1971. - P. 9-32.
24. Mishnaevsky L., Jr. Damage mechanisms of hierarchical composites: Computational modelling // Phys. Mesomech. - 2015. - V. 18. -No. 4.- P. 416-423.
25. Qing H., Mishnaevsky L., Jr. 3D multiscale micromechanical model of wood: From annual rings to microfibrils // Int. J. Solids Struct. -
2010. - V. 47. - No. 9. - P. 1253-1267.
Поступила в редакцию 21.10.2016 г.
Сведения об авторах
Rana Amit Kumar, PhD student, Indian Institute of Engineering Science and Technology, India, amit.mme@gmail.com
Paul Surajit Kumar, Senior Scientist, MST Division, CSIR-National Metallurgical Laboratory, India, paulsurajit@yahoo.co.in, surajit@nmlindia.org Dey Partha Pratim, Associate Professor, Indian Institute of Engineering Science and Technology, India, ppdey2000@yahoo.com
