Prediction of the Maximum Dislocation Density in Lath Martensitic Steel by Elasto-Plastic Phase-Field Method

Zhenhua Cong1, Yoshinori Murata1,+, Yuhki Tsukada2 and Toshiyuki Koyama2

1Department of Materials, Physics and Energy Engineering, Graduate School of Engineering, Nagoya University, Nagoya 464-8603, Japan 2Department of Materials Science and Engineering, Graduate School of Engineering, Nagoya Institute of Technology, Nagoya 466-8555, Japan

On the basis of the two types of slip deformation (TTSD) model of lath martensite, the martensitic transformation was simulated in Fe 0.1 mass% C steel by an elasto-plastic phase-ﬁeld method. The TTSD model allowed us to predict the total dislocations for the necessity of the formation of lath martensite, which is taken as the upper limit of dislocation density in lath martensite. The calculated dislocation density by the simulation was reasonable to be higher than the observed dislocations in value but to be the same in order. This consistence indicates that the calculation method based on the TTSD model is credible, together with the calculation of the habit plane predicted by the TTSD model. [doi:10.2320/matertrans.M2012067]

(Received February 20, 2012; Accepted June 25, 2012; Published August 8, 2012) Keywords: dislocation density, slip deformation, phase-ﬁeld method, lath martensite

1. Introduction study, the total dislocations for the necessity of the formation of lath martensite steel is counted by simulation using an The martensite phase in steels exhibits several morphol- elasto-plastic phase-field model based on the TTSD model, ogies such as lath, plate and butterfly, depending on the and the result is compared with the experimental results alloying elements.1,2) Among them, lath martensite exhibits reported until date. high strength, wear resistance, and toughness.37) The martensitic microstructures must be characterized accurately 2. Evaluation of the Maximum Dislocation Density in terms of orientation, morphology, transformation disloca- Based on the TTSD Model tion density, and retained austenite. In recent years, Morito et al. observed the martensitic orientation and microstruc- According to Iwashita et al., the martensitic transformation tures by means of TEM, SEM and EBSD.811) Spanos et al. is accomplished by coupling lattice deformation and plastic adopted EBSD and serial sectioning to establish 3-D mor- deformation.14) The lattice deformation (Bain deformation) phology of martensite lath,7) which provided further detailed realizes the transformation from the austenite phase with a insights into lath orientation, distributions and shapes. face-centered cubic (fcc) lattice to a body-centered tetragonal High dislocation density is inevitable in lath martensite, (bct) lattice. After that, the length of the c-axis is adjusted to which accommodates the large strain induced by martensitic accommodated the strain induced by Bain deformation. Due transformation and subsequent interface gliding. Wayman to the strain induced by Bain deformation is so large that classified the dislocations in the martensite phase into two plastic deformation is inevitable. In the present study, the types: transformation dislocations and interface disloca- plastic deformation is realized by dislocation slip along two tions.5) Morito et al. used a TEM method to measure the independent slip systems as shown in Fig. 1, which is called dislocation densities in nickel steels and carbon steels, and as the TTSD model. The crossed planes shown in Fig. 1(a) they reported that the dislocation density for lath martensite are the two types of slip systems, ½101ð101 Þ¡0 and 15 ¹2 14) is approximately 1.11 © 10 m in a Fe0.18C steel and ½101 ð101Þ¡0 . Through TTSD model, the habit plane 14 ¹2 12) 3.8 © 10 m in a Fe11Ni steel. In addition, Cong et al. {557}£ and lattice correspondence between the martensite used the X-ray diffraction (XRD) method to detect the dislocation density of lath martensite in low carbon steels (0.020.09 mass% C) and the dislocation density is 4.87 © (a) (b) 1014 m¹2 in a Fe10Cr5W0.02C steel.13) However, all these studies focus purely on experimental results, which [101] (101) cannot relate the dislocation with the formation mechanism α' or of lath martensite. [101] (101) Recently, Iwashita et al. developed a two types of slip α' deformation (TTSD) model to explain the formation mechanism of lath martensite.14) In this model, high dislocation density introduced by martensitic transformation is realized by two inevitable independent slip systems. In this Fig. 1 Skeleton of plastic deformations along two slip systems, i.e., ½101ð101 Þ¡0 and ½101 ð101Þ¡0 . b1 and b2 are the Burgers vector for the +Corresponding author, [email protected] two slip systems. Prediction of the Maximum Dislocation Density in Lath Martensitic Steel by Elasto-Plastic Phase-Field Method 1599 phase and the austenite phase are successfully explained without any rotation matrix. Figure 1(b) shows that each slip system can be taken as a combination of two a=2h111i¡0 dislocation slips with the Burgers vectors of b1 and b2, which can usually be observed in practical steels. Compared to directly performing the slip deformation along h111i slip system, the TTSD model can represent the plastic deformation simply. Moreover, the TTSD model can well explain the {557}£ habit plane in the formation process of lath Fig. 2 Slip deformation skeleton for (a) a random state and (b) an assumed martensite. state, where the intervals between the neighboring slip planes are the Assuming that the plastic deformation is accommodated same. throughoutly by these dislocation slips, the total dislocations for the necessity of the formation of the lath martensite can be evaluated. The idea for using the phase-field method two adjacent slip planes for each slip system is m, then the to model a dislocation is establishes by Nabarro15) that value of D can be given by eq. (3): dislocations can be taken as a set of coherent misfitting D ¼ m dhkl: ð3Þ platelet inclusions. For simplisity, a dislocation loop is described as a sheared pletelet with thickness and the region The distance between the (hkl)¡A planes can be obtained from inside the platelet is sheared by a Burgers vector b.16) By eq. (4). extending this discription to a spatial region with a 1 dhkl ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; ð4Þ population of dislocations, the average plastic strain pave, h2 k2 l2 þ þ caused by dislocation slip is given by 2 2 2 a¡0 a¡0 c¡0 jbj p ¼ ; ð Þ where h, k and l are the Miller indices of the slip planes, and ave D 1 a¡A and c¡A are the lattice parameters of the martensite phase. where «b« is the magnitude of the Burgers vector and D is By inserting the values of D(101) estimated from eq. (1) and the average distance between the neighboring slip planes, d(101) in eq. (3), we can evaluate the number of lattice planes that is, dislocation planes. In the formation process of lath slipping along the h101i¡0 system, m(101). As a result, the martensite, a lot of dislocations are necessary for the plastic number of lattice planes slipping actually along the h111i¡0 accommodation. After the martensitic transformation, some direction m(111) should be twice that of m(101). By substituting dislocations are resided in the martensite crystal, which can the values of m(111) and d(111) into eq. (3), we obtain D(111). be observed by experiments, whereas some dislocations pass Now with the help of D(111) and eq. (2), the total amount through out of the martensite crystal using for the formation of dislocations for the necessity of the formation of lath of lath boundaries, which cannot be observed directly by martensite in practical steels can be evaluated. experiments. In the present study, we focuses on the total dislocations contributing on the formation of lath martensite, 3. Elasto-Plastic Phase-Field Method which is taken as the upper limit of dislocation density of lath martensite, μlim. In the martensitic transformation, μlim For the martensitic transformation, the field variable ºiðrÞ should contribute to the plastic deformation for moderating (i = 1, 2, 3) is introduced to describe the Bain deformation the strain by Bain lattice deformation. Here, we give the and i = 1, 2, 3 is used to distinguish the three coordinate distance between neighboring dislocations by a rough coincidences; that is, the c-axis of the bct phase is along the estimation as three equivalent h100i directions in the austenite matrix. Here pffiffiffiffiffiffiffiffi r is the positional vector. º ðrÞ (i = 1, 2, 3) ranges from 0 to 1 D 1= μ : ð2Þ i lim and 0 represents austenite phase, where 1 represents the full Assuming that all of the dislocations contriute to the plastic martensite phase at a certain i. In the present simulation, deformation, the value of D can be estimated from the the lath martensite phase is formed only when ºiðrÞ ² 0.7. ¡ average plastic strain pave, which is available from the Another field variable pi ðrÞ (i = 1, 2, 3) is considered to ¡ simulation results by using phase-field model. As mentioned describe the plastic deformation and the value of pi ðrÞ above, TTSD model is based on ½101ð101 Þ¡0 and represents the local plastic strain produced by dislocations. ½101 ð101Þ¡0 slip systems in bct crystals as shown in ¡ represents the number of slip systems, i.e., ½101ð101 Þ¡0 or ¡ Fig. 1(a). Therefore, the value of D evaluated from eq. (1) ½101 ð101Þ¡0 . In our simulation, the value of pi ðrÞ ranges is the distance between the neighboring slip planes along from 0, which means no plastic deformation, to 1.21, which ½101ð101 Þ¡0 or ½101 ð101Þ¡0. To obtain the total dislocations is the maximum of plastic strain determined by eqs. (1) and for the necessity of the formation of martensite phase in a real (3). The plastic deformation will choose the slip system, ¡ case, the value of D along h101i¡0 should be transformed to which has the bigger value of pi ðrÞ, to accommodate the the value along h111i¡0 , as shown in Fig. 1(b). strain caused by Bain deformation. For a real case, the slip planes should arrange randomly as The martensitic transformation is a minimization process shown in Fig. 2(a). For simplicity, it is assumed that the of the total free energy for the phase-field simulation. Here intervals between neighboring slip planes are the same, as the total free energy is defined by the GinzburgLandau-type shown in Fig. 2(b). If the number of lattice planes between Gibbs free energy functional, which is a sum of chemical free 1600 Z. Cong, Y. Murata, Y. Tsukada and T. Koyama energy Echem, gradient energy Egrad, and elastic strain energy 1 17) 0 E : ¤¾klðrÞ¼ fmkðrÞnl þ mlnkðrÞg· ðrÞnn; ð11Þ el 2 mn E ¼ E ðfº ðrÞgÞ þ E ðfp¡gÞ þ E ðfº ðrÞg; fp¡gÞ: total chem i grad el i ·0 C ¾0 ³ r where mn klmn kl. km( ) is the Green function tensor ð5Þ and is defined as below21) The chemical term is taken as the driving force for ¤mk nmnk martensitic transformation, which can be approximated mkðrÞ¼ : ð12Þ ® 2®ð1 ¯Þ by the conventional GinzburgLandau phenomenological fi ¾0 r coarse-grained functional of eld variables. It contains the klð Þ is the total eigen strain and is given by local specific free energy and non-local gradient terms, X3 i.e.:18) ¾0 ¼ ð¾B ðiÞº ðrÞÞ Z "#kl kl i ¬ X3 i¼1 º 2 X ¡ ¡ ¡ ¡ Echem ¼ f0ðfºiðrÞgÞ þ ðrºiðrÞÞ dr; ð6Þ b n þ n b 2 þ i i i i p¡ðrÞ ; ð Þ r i¼1 b¡ i 13 ¡ 2j i j where f is the specific free energy and is defined as 0 8 9 ! where the first term describes the eigen strain caused by the < X3 X3 X3 2= a b c Bain deformation and the second term is the eigen strain f ¼ f º2 º3 þ º2 : ð7Þ 0 : i i i ; attributed to the plastic deformation. By inserting all the 2 i¼1 3 i¼1 4 i¼1 terms to eq. (9), the elastic strain energy can be evaluated. Here, a, b and c are the coefficients of the Landau polynomial As a result, the total free energy Estr, for the martensitic expansion. In this study, they are chosen as a = 0.1, transformation is determined. b = 3a + 12 and c = 2a + 12.17) ¦f is the driving force The dynamics of martensitic transformation is controlled for the martensitic transformation, which is calculated by by the AllenCahn equation:22) Thermo-Calc with CALPHAD method. The second term in @Mðr;tÞ ¤Etotal ¼LM ; ð14Þ eq. (6) is the gradient part due to the inhomogeneity of the @t ¤Mðr;tÞ field variable ºiðrÞ. ¬º is a coefficient positively defined ¡ second-rank tecsor and r@=@ri is a differential operator. where M(r,t) ðM ¼ ºi;pi Þ are the field variables of the The gradient energy Egrad describes the contribution of the coordinate vector r and evolution time t, and LM is the kinetic core energy of the dislocations to plastic accommodation an parameter of each field variable. it is represented by the following equation:16) Z 4. Numerical Simulation ¬ X3 X E ¼ p ½n¡ rp¡ ½n¡ rp¡ dr; ð8Þ grad 2 ðiÞ ðiÞ ðiÞ ðiÞ r i¼1 ¡ The evolution of lath martensite in Fe0.1 mass% C steel where ¬p is the gradient energy coefficient to guarantee a was simulated at 300 K by the elasto-plastic phase-field smooth transition of the deformation strain field profile on the model in 3-D space. The simulation was performed in a 3 austenite/martensite interface and ni is the unit vector of the cubic with N (N = 64) meshes and the mesh size is 4 nm. slip plane normal. Therefore, the computational domain is 256 © 256 © According to Khachaturyan,19) the elastic strain energy is 256 nm. For the initial state, a dislocation loop with a radius given by of 12 nm is set in the center of the austenite cubic and the Z growth of lath martensite with time evolution is simulated 1 0 0 Eel ¼ Cklmnf¾klðrÞ¾ ðrÞgf¾mnðrÞ¾ ðrÞgdr; ð9Þ around the dislocation loop. The shape of the martensite 2 kl mn r lath is taken as a thin plate with thickness. The time step where Cklmn is the elastic coefficient matrix. For simplisity, ¦t* is set to be 0.001 and the symbol of asterisk represents the material is assumed to be isotropic due to that the elastic a dimensionless simulation time. The lattice parameters constants of lath martensite are not available up to date. both of the austenite phase and martensite phase are ¹10 Therefore, the tensor Cklmn can be expressed as Cklmn ¼ estimated to be a£ = 3.599 © 10 m, a¡A = 2.867 © ¹10 ¹10 ¤kl¤mn þ ®ð¤km¤ln þ ¤kn¤lmÞ in terms of the Lamé constants 10 m and c¡A = 2.880 © 10 m, respectively, in Fe and ®, which are estimated from Young’s modulus and the 0.1 mass% C steel. Assuming the calculation system is Bulk modulus for an isotropic cubic crystal.20) Here, ¤(x)is isotropic, the Lamé constants and ® are estimated to be the Dirac delta function. ¾klðrÞ is the total strain, which is 123 and 72 GPa from both the Young’s modulus and the 23) defined as the sum of the homogeneous strain ¾kl and the Bulk modulus of pure iron. The driving force for the heterogeneous strain ¤¾kl: martensitic transformation, ¦f is calculated to be 5085 J/mol in a Fe0.1 mass% C steel at 300 K based on Thermo-Calc ¾ ðrÞ¼¾ þ ¤¾ ðrÞ: ð10Þ kl kl kl data base. The gradient coefficients with respect to the ¡ ¾kl describes the macroscopic shape deformation of the field variables ºiðrÞ and pi ðrÞ are fitted to be 1.6 © system. When the macroscopic shape of the system is fixed 10¹14 J·m2/mol and 30 © 10¹14 J·m2/mol, respectively. The 16) during the transformation, the homogeneous strain is zero. kinetic parameter LM for each field variable is set to be 1. ¤¾ fi TheR heterogeneous strain kl,isdened to satisfy With the minimization of the total free energy controlled 19) V ¤¾kl ¼ 0. According to the theory of elasticity, ¤¾kl is by the kinetic equation, the martensitic transformation was given as performed. Prediction of the Maximum Dislocation Density in Lath Martensitic Steel by Elasto-Plastic Phase-Field Method 1601

5. Results and Discussion we calculate the distance between neighboring slip planes ¹8 D(101) to be 1.13 © 10 m. Substituting the values of D(101) d = © ¹10 Figure 3 shows the time evolution of the average value of and ð101Þ¡0 2.03 10 m into eq. (3), the value of m(101) plastic strain pave. It considers all the local values of plastic is estimated to be 56, and thus m(111) should be 112 for the strain in lath martensite along the two slip systems, actual slip systems in the martensite phase. By substituting ﬁ d = © ¹10 ½101ð101Þ¡0 and ½101ð101Þ¡0 . The gure reveals that the the values of m(111) and ð111Þ¡0 1.66 10 m into eq. (3) ¹8 average plastic strain increases with the progression of again, we calculated D(111) to be 1.86 © 10 m. Once the martensitic transformation and saturated in 30 time steps at value of D(111) is known, the maximum dislocation density a value of 0.036. This means that the martensitic trans- in a practical steel is evaluated to be 2.89 © 1015 m¹2 from formation is accomplished at t* = 30. Therefore, we use the eq. (2). The simulation result is almost twice that of the saturated value of pave to estimate the maximum dislocation experimental result in a Fe0.1 mass% C steel (1.55 © density in a full lath martensite. By inserting the values of 1015 m¹2) measured by Kehoe et al. using a TEM method.24) ¹10 pave and «b(101)« = 4.06 © 10 m in pure iron into eq. (1), The simulation result is deﬁnitely higher than the experimental result with respect to the value, but the orders of the dislocation density are the same. As mentioned in Section 2, the maximum dislocation density μlim considers the total dislocations for the necessity of the formation of lath martensite. In this sense, the calculation result is natural and right higher than the observed dislocation density. In our calculation, only the dislocations in the martensite phase are considered. In fact, the surrounding austenite phase should also contain some dislocations because of the strain originating from the martensite phase and they may be inherited into the lath martensite phase during martensitic transformation.5) However, it is argued that if the surrounding austenite phase is deformed during martensitic transformation, it will help to accommodate part of the strain in the

Fig. 3 The average value of the plastic strain pave along the two slip martensite phase, thus resulting in the loss of dislocation systems. density in the martensite phase itself. This loss and the

¡ Fig. 4 The time evolution of the plastic strain pi ðrÞ (i = 1, 2, 3) along ½101 ð101Þ¡0 slip system at {111} plane for (a) t* = 2, (b) t* = 4, (c) t* = 10 and (d) t* = 30 by phase-ﬁeld simulation. 1602 Z. Cong, Y. Murata, Y. Tsukada and T. Koyama

¡ Fig. 5 The time evolution of the plastic strain pi ðrÞ (i = 1, 2, 3) along ½101ð101 Þ¡0 slip system at {111} plane for (a) t* = 2, (b) t* = 4, (c) t* = 10 and (d) t* = 30 by phase-field simulation. dislocations stored in the surrounding austenite phase cancel deformation along the two slip systems are complementary each other out. Therefore, the maximum dislocation density and they cooperated with each other to assist the plastic in a full martensite should be almost equal to our result. accommodation. In other words, during martensitic transformation, the total strain containing the surrounding austenite phase is consid- 6. Conclusions ered to be represented by the dislocations in this study, although the quantitative evaluation should be done in the The maximum dislocation density of lath martensite in a future. Fe0.1 mass% C steel was evaluated on the basis of a TTSD Figures 4 and 5 show the time evolution of the local plastic model. By employing an elasto-plastic phase-field method ¡ strain pi ðrÞ (i = 1, 2, 3) along the ½101ð101 Þ¡0 slip system based on the TTSD model, the average value of plastic strain and the ½101 ð101Þ¡0 slip system on the {111} plane by was evaluated to be approximately 0.036 for 30 time steps. phase-field simulation, respectively. In Figs. 4 and 5, the The evaluated maximum dislocation density was 2.89 © deep blue areas indicate that there are no slip deformation, 1015 m¹2. This result was reasonable to be higher than the while the red areas represent the most dramatic slip observed dislocation density in value but to be the same in deformation. For a specific value shown in Figs. 4 and 5, order. it may come from an arbitrary lattice corresponding in Bain deformation, where the value of i can be equal to 1, 2 or 3. REFERENCES But all the values distributed in a packet contain the local plastic strain for all the three cases of lattice corresponding, 1) G. Olson and W. Owen: Martensite, (ASM International, Materials i.e., i = 1, 2 and 3. Because of a dislocation loop set in the Park, Ohio, 1992). center of the austenite phase as the initial state, the slip 2) K. Otsuka and C. M. Wayman: Shape Memory Materials, (Cambridge University Press, Cambridge, 1999). deformation also originated from the center of the austenite 3) K. Wakasa and C. M. Wayman: Acta Metall. 29 (1981) 973990. phase and the range of the slip deformation extends with the 4) K. Wakasa and C. M. Wayman: Metallography 14 (1981) 4960. evolution of the martensitic transformation. It is to be noted 5) B. P. J. Sandvik and C. M. Wayman: Metall. Trans. A 14 (1983) 809 that the plastic strain of area “A” marked in Fig. 4(d) is very 822. 33 large, while in the same area “B” marked in Fig. 5(d), there 6) P. M. Kelly: Mater. Trans., JIM (1992) 235 242. 7) D. J. Rowenhorst, A. Gupta, C. R. Feng and G. Spanos: Scr. Mater. 55 is almostly no plastic strain along the other slip system. (2006) 1116. This result can be observed at all times and places by 8) S. Morito, X. Huang, T. Furuhara, T. Maki and N. Hansen: Acta Mater. comparing Figs. 4 and 5. So it is concluded that the slip 54 (2006) 53235331. Prediction of the Maximum Dislocation Density in Lath Martensitic Steel by Elasto-Plastic Phase-Field Method 1603

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