Multi-length scale modeling of martensitic transformations in stainless steels

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Acta Materialia 60 (2012) 6508–6517 www.elsevier.com/locate/actamat

Multi-length scale modeling of martensitic transformations in stainless steels Hemantha Kumar Yeddu ⇑, Vsevolod I. Razumovskiy, Annika Borgenstam, ˚ gren Pavel A. Korzhavyi, Andrei V. Ruban, John A Department of Materials Science and Engineering, KTH Royal Institute of Technology, Brinellva¨gen 23, 10044 Stockholm, Sweden Received 12 May 2012; received in revised form 10 July 2012; accepted 2 August 2012 Available online 1 October 2012

Abstract In the present work a multi-length scale model is developed to study both the athermal and stress-assisted martensitic transformations in a single crystal of 301 type stainless steel. The microstructure evolution is simulated using elastoplastic phase-field simulations in three dimensions. The input data for the simulations is acquired from a combination of computational techniques and experimental works. The driving force for the transformation is calculated by using the CALPHAD technique and the elastic constants of the body-centered cubic phase are calculated by using ab initio method. The other input data is acquired from experimental works. The simulated microstructures resemble a lath-type martensitic microstructure, which is in good agreement with the experimental results obtained for a stainless steel of similar composition. The martensite habit plane predicted by the model is in accordance with experimental results. The Magee effect, i.e. formation of favorable martensite variants depending on the loading conditions, is observed in the simulations. The results also indicate that anisotropic loading conditions give rise to a significant anisotropy in the martensitic microstructure. Ó 2012 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Phase-field models; Ab initio; Multi-length scale model; Martensitic phase transformation; Microstructure

1. Introduction Stainless steels have become important engineering materials, due to their high strength and corrosion resistance, and have made their presence inevitable in a wide range of applications. The high strength of stainless steels can be mainly attributed to the high-strength constituent, namely martensite. The high strength of martensite can in turn be attributed to the strong solid solution strengthening effect of carbon, as well as the complex microstructure. Because of its importance in determining the properties of steels, it is essential that the formation of martensite is thoroughly understood. Martensite forms under various thermomechanical conditions due to the rapid diffusionless phase transformation of austenite into martensite. Based on the mode of ⇑ Corresponding author. Tel.: +46 87906219; fax: +46 8100411.

E-mail address: [email protected] (H.K. Yeddu).

formation, martensitic transformation in steels can be classified as: athermal, i.e. by rapid quenching of steel; isothermal, i.e. by holding the steel at a constant temperature close to the martensite start temperature; stress-assisted, i.e. by application of stress that is below the yield limit of the steel; or strain-induced, i.e. by plastic deformation of steel. Several theoretical works have been performed to understand the formation of athermal [1,2], isothermal [3,4], and stress- and strain-induced martensite [5–8], and have contributed significantly to the understanding of the overall transformation. A number of studies have been undertaken to elucidate some of the complexities associated with the transformation, such as the nucleation [2,9], autocatalysis [10–12], morphology [13–15] and crystallography [14–20] of martensite. From the above studies, it has been well established that, from a morphological point of view, martensite can be formed as laths and plates, depending on the

1359-6454/$36.00 Ó 2012 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.actamat.2012.08.012

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alloy composition [13]; and, from a crystallographical point of view, it can be formed in 24 different crystallographic variants, which can be grouped into three main groups, known as the Bain groups [15,19,20]. Despite the extensive studies performed to understand martensitic transformation, some of the aspects of the transformation are still vague and need to be further explored, perhaps by performing in situ studies using synchrotron radiation. However, the rapidity of the transformation, approximately 1000 m s1 [21] in the case of plate martensite, makes it very difficult to be studied by in situ experiments. Moreover, the difficulties in attaining the experimental set-ups, with their complex loading conditions, makes the study of stress-assisted martensite formation even more complicated. The recent advances in modeling the phase transformations at the mesoscopic length scale by using the phase-field technique [22–24] seem promising and can be an aid in the study of martensitic transformation. Several works have been performed to study martensitic transformations using the phase-field technique [25–38]. Moreover, due to the growing interest in integrated computational materials engineering, i.e. materials design and optimization by means of a combination of multi-length scale modeling approaches, the possibilities of coupling the phase-field technique with other computational techniques need to be explored. In the present work, three-dimensional phase-field simulations of athermal and stress-assisted martensitic transformation occurring in a single crystal of a 301 type stainless steel with an alloy composition of Fe–17%Cr–7%Ni (expressed in weight percent) are performed. The input data for the simulations are acquired by using different computational techniques, viz. CALPHAD to calculate the thermodynamic data and the ab initio method to calculate electronic-level properties such as elastic constants. The other necessary input data is acquired from experimental works in order to ensure a physically based model. 2. Phase-field simulations 2.1. Phase-field model

2.1.1. Athermal martensitic transformation The microstructure evolution is governed by the phasefield equation: q¼v X @gp dG ¼  Lpq dgq @t q¼1

where

dG dgq

phase-field variable gq, v is the total number of martensite domains and Lpq is a matrix of kinetic parameters. From a physical point of view, the Gibbs energy consists of three parts: Z  chem  G¼ GV þ Ggrad þ Gel ð2Þ V V dV V

corresponds to the chemical part of the Gibbs where Gchem V energy density of an unstressed system at the temperature under consideration. Ggrad is the extra Gibbs energy density V caused by the interfaces. Gel V is the elastic strain energy density that arises due to the elastic strain induced into the material by the transformation. The chemical part of the Gibbs energy density Gchem , v expressed as a Landau-type polynomial [28,35], is given by:   1   1 1  2 A g1 þ g22 þ g23  B g31 þ g32 þ g33 Gchem ðg ;g ;g Þ ¼ 1 2 3 v Vm 2 3   1  2 2 þ C g1 þ g22 þ g23 ð3Þ 4 where Vm is the molar volume and the coefficients A, B, C are expressed in terms of Gibbs energy barrier and the driving force [35]. The gradient energy density term, Ggrad as presented in V [28] can be expressed as: ¼ ð1=2Þ Ggrad V

ð1Þ

is a variational derivative that serves as a driving

force for the formation of martensite denoted by the

p¼v X @gp @gp bij ðpÞ @ri @rj p¼1

ð4Þ

where r(x, y, z) is the position vector expressed in Cartesian coordinates. bij is the gradient coefficient matrix expressed in terms of the interfacial energy, molar volume and Gibbs energy barrier. The elastic energy density term Gel V can be expressed as: Z ij ðrÞ Gel cijkl el ð5Þ kl ðrÞdij ðrÞ V ¼ 0ij ðrÞ

where cijkl is the tensor of elastic modulii, 0ij ðrÞ is the stressfree transformation strain, ij(r) is the total strain and el kl ðrÞ is the elastic strain. The stress-free transformation strain is given by: 0ij ðrÞ ¼

A brief overview of the phase-field model to simulate the athermal and stress-assisted martensitic transformations is presented below; the detailed derivations can be found in Refs. [35,38], respectively.

6509

p¼v X

gp ðrÞ00 ij ðpÞ

ð6Þ

p¼1

where 00 ij ðpÞ is the Bain strain, given in 2 3 2 3 0 0 1 6 7 6 00 0  ;  ð1Þ ¼ 0 ð2Þ ¼ 00 4 5 40 1 ij ij 0 0 1 0 2 3 1 0 0 6 7 00 ij ð3Þ ¼ 4 0 1 0 5 0 0 3

turn by: 3 0 0 7 3 0 5; 0

1 ð7Þ

where, 3 is a compressive transformation strain and 1 is a tensile transformation strain, defined based on the lattice constants of austenite (afcc) and martensite (abcc, cbcc)

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[35]. Thus, the martensite (Bain) variants 1, 2 and 3 are 00 00 governed by 00 ij ð1Þ, ij ð2Þ and ij ð3Þ, respectively. The elastic strain is given by: pl 0 el kl ðrÞ ¼ kl ðrÞ  kl ðrÞ  kl ðrÞ

ð8Þ

pl kl ðrÞ

where is the plastic strain. The plastic strain comes into existence only when the elastic stress exceeds the yield limit and the material undergoes plastic deformation, which acts as a relaxation of the elastic stress. The evolution of the plastic strain field is governed by Eq. (9) [33,35,39]: @pl dGshear ij ðrÞ ¼ k ijkl plv @t dkl ðrÞ

ð9Þ

shear is the shear where pl ij ðrÞ is the local plastic strain and Gv energy density. kijkl is the plastic kinetic coefficient, expressed as:

k ijkl ¼ kc1 ijkl

ð10Þ

where c1 ijkl denotes the compliance tensor and k is a parameter called plastic relaxation rate, which controls the rate at which the elastic stresses are relaxed by means of plastic deformation. Finally, the total strain is calculated by solving the mechanical equilibrium equation: ! p¼v @kl X 00 @gp ðrÞ @pl ðrÞ cijkl  kl ðpÞ  kl ¼0 ð11Þ @rj @rj @rj p¼1 The anisotropic elastic properties of different phases are taken into account by considering different tensors of elastic modulii cijkl for different phases. In order to consider different cijkl, an expression as shown in Eq. (12) is employed such that, for a given phase, i.e. gp = 0 or gp = 1, the corresponding tensors of elastic modulii cfcc ijkl or cbcc ijkl are considered respectively, whereas in the interface a weighted cijkl that depends on the weight yielded by the phase-field variable is considered. bcc cijkl ¼ cfcc ijkl ð1  gp Þ þ cijkl gp

ð12Þ

As both the phases are of cubic structure, both the tensors are expressed by Eq. (13) [40], although the values of the elastic modulii in the two tensors are different. The elastic modulii of the face-centered cubic (fcc) phase are acquired from experimental data, whereas that of the body-centered cubic (bcc) phase are acquired from ab initio calculations, as explained in Section 3. 3 2 c11 c12 c12 0 0 0 6c 0 0 7 7 6 12 c11 c12 0 7 6 7 6 c c c 0 0 0 12 12 11 cubic 7 6 cijkl ¼ 6 ð13Þ 0 0 c44 0 0 7 7 6 0 7 6 4 0 0 0 0 c44 0 5 0

0

0

0

0

c44

2.1.2. Stress-assisted martensitic transformation In order to model the stress-assisted martensitic transformation, the extra Gibbs energy density caused by the externally applied stress needs to be included in the Gibbs energy of the system, as presented in Refs. [27,38]. Z  chem  appl G¼ dV ð14Þ Gv þ Ggrad þ Gel v v þ Gv V

where Gchem , Ggrad and Gel v v v are the chemical, gradient and elastic parts of the Gibbs energy density, expressed as presented in Section 2.1.1. Gappl is the extra Gibbs energy denv sity aroused due to the externally applied stress. In the case of stress-assisted martensitic transformation, where transformation occurs due to externally applied elastic stresses, the application of external stresses does not directly lead to extensive plastic deformation unless the martensitic transformation occurs. Thus the external stresses affect the transformation strains and thereby affect the martensitic transformation, which in turn gives rise to plastic deformation in the material. This is popularly known as the TRIP, or transformation-induced plasticity, effect. Thus Gappl can be expressed as: v 0 ¼ rappl Gappl ij ij ðrÞ v

ð15Þ

rappl ij

where is the externally applied stress tensor, expressed by the Cauchy stress tensor as: 2 3 rxx rxy rxz 6 7 ð16Þ ¼ 4 ryx ryy ryz 5 rappl ij rzx rzy rzz It can be seen from Eq. (15) that the applied stresses affect the transformation strains 0ij ðrÞ, which in turn affect the total strain ij(r) and plastic strain pl ij ðrÞ, contained in Gel v in Eq. (14). This ensures that the plastic deformation does not occur by mere application of external elastic stresses, unless the martensitic transformation occurs. In the absence of gradient and elastic energies, a schematic of the effect of uniaxial tensile stress, along the Xdirection, on the formation of martensite variants is shown in Fig. 1. In the case of a variant with a compressive

Fig. 1. Effect of applied stress on the thermodynamics of martensitic transformation.

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transformation strain along the loading direction, i.e. variant 1 according to Eq. (7), the external mechanical energy reduces the available driving force and also increases the height of the Gibbs energy barrier, DG*, implying that the formation of variant 1 is very difficult. In the case of a variant with a tensile transformation strain along the loading direction, i.e. variants 2 or 3 according to Eq. (7), it can be seen that the external mechanical energy adds up to the available chemical driving force as well as reduces DG*, indicating that the formation of variants 2 and 3 is favored under the corresponding stress state. Thus the externally applied mechanical energy can either contribute to or detract from the formation of some variants, as explained in the pioneering work of Patel and Cohen [41].

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10. Iso-surfaces of the phase-field variable (g = 0.5) are shown in all the figures. The red, blue and green colored iso-surfaces of the phase-field variables correspond to martensitic variants 1, 2 and 3, respectively. t* indicates the dimensionless time. 11. As the experimental data related to the mobility of the martensitic interface is ambiguous, the matrix of kinetic parameters Lpq in Eq. (1) that governs the mobility of the martensitic interface is considered to be the identity matrix. 12. The entire mathematical formulation used in the phase-field calculations is solved using tetrahedral finite elements and by using Dirichlet boundary conditions. Computations are performed on a mesh with 50  50  50 grid points.

2.2. Simulation data 2.3. Simulations The input data for the simulations is: The following simulations are performed: 1. A single grain of austenite undergoing martensitic transformation is considered to be the simulation domain and has a physical size of around 1 lm. 2. A spherical martensite nucleus, with a radius of 0.1 lm, is considered to pre-exist in the center of the simulation domain. 3. Thermodynamic parameters corresponding to an Fe–17% Cr–7% Ni alloy (expressed in weight percent) are considered. The martensite start temperature Ms(= 263 K) for the above alloy is acquired from experimental work [42]. The corresponding driving force (= 3600 J mol1) at Ms is calculated by using the Thermo-Calc software, which is based on the CALPHAD technique, with the TCFE6 database [43]. 4. The interface thickness, d, is considered to be 1 nm and the interfacial energy, c, is considered to be 0.01 J m2 [44]. 5. Bain strains, 1 = 0.1316 and 3 = 0.1998, are calculated based on the experimentally measured ˚ ) and lattice constants of austenite (afcc = 3.5918 A ˚ ) [45], corresponding martensite (abcc = cbcc = 2.874 A to an alloy with a similar composition as mentioned above. 6. The elastic modulii for austenite (fcc) are acquired from experimental measurements [46] as: C11 = 209 GPa, C12 = 133 GPa and C44 = 121 GPa. 7. The elastic modulii for martensite (bcc) are acquired from ab initio calculations, as explained in Section 4.1, as: C11 = 248 GPa, C12 = 110 GPa and C44 = 120 GPa. 8. The yield limits considered for austenite and martensite are raust: = 500 MPa [47] and rmart: = 800 MPa y y [48], respectively. 9. The externally applied stresses are chosen such that they are less than the yield stress of both austenite and martensite.

1. Athermal martensitic transformation 2. Stress-assisted martensitic transformation by application of uniaxial tensile stress of 150 MPa along the Xaxis, i.e. 2 3 150 0 0 6 7 rappl ¼4 0 0 05 ij 0 0 0 3. Stress-assisted martensitic transformation by application of biaxial compressive stresses of 150 MPa along the X- and Y-axes, i.e. 2 3 150 0 0 6 7 rappl ¼4 0 150 0 5 ij 0 0 0 4. Stress-assisted martensitic transformation under a triaxial loading condition with compressive stress of 150 MPa along the X-axis, tensile stresses of 150 MPa along the Y-axis and 150 MPa along the Z-axis, i.e. 2 3 150 0 0 6 7 rappl ¼4 0 150 0 5 ij 0 0 150

3. Ab initio calculations The present electronic structure calculations are based on the density functional theory (DFT) [49] and have been performed by means of the exact muffin-tin orbitals (EMTO) method [50–52]. Substitutional disorder in the bcc Fe–Cr alloys is treated within the coherent potential approximation (CPA) [53]. In the single-site DFT formalism, additional contributions to the one-electron potentials of alloy components and the total energy are

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taken into consideration [54,55]. The screening contributions to the on-site electrostatic potential V iscr and electrostatic energy Escr are determined as explained in Ref. [54,55]: e 2 qi bX and Escr ¼ V scr ci qi V scr i ¼ a i 2 i S where e is the electron charge and S is the Wigner–Seitz (atomic sphere) radius. qi is the average net charge inside the atomic sphere and ci is the concentration of the ith alloy component. The screening parameters a and b are evaluated from the locally self-consistent Green’s function method within the EMTO formalism [56–58]. For the ferromagnetic bcc random alloy Fe–17%Cr–7%Ni, the values of the screening parameters are determined to be b = 1.14, aFe = 0.659, aNi = 0.809 and aCr = 0.74. The total energies are calculated using the generalized gradient approximation [59] within the full charge density formalism [60]. All the self-consistent EMTO–CPA calculations are performed using an orbital momentum cutoff of lmax = 3 for partial waves. The integration over the Brillouin zone is performed using a 37  37  37 grid of special k-points determined according to the Monkhorst–Pack scheme [61]. The core states are recalculated at each selfconsistency iteration. In order to include the effect of lattice expansion, the present calculations are performed with the experimental temperature-dependent lattice parameters [45,62]. The contribution from electronic excitations is included in the usual form of Fermi-function smearing (electronic temperature) [63,64]. 3.1. Elastic property calculations A uniform lattice distortion in the elastic property calculations is imposed on the lattice by transforming the set of primitive vectors, ai, to a set of new vectors, a0i , using a strain tensor e as: 0 01 0 1 a1 a1 B 0C B C ð17Þ @ a2 A ¼ ðI þ eÞ  @ a2 A 0 a3 a3 where I is a 3  3 identity matrix. Even though there are 21 independent elastic constants (Cij), the symmetry of the cubic lattice reduces this number to only three independent constants, i.e. C11, C12 and C44, as shown in Eq. (13). A uniform isotropic straining (compression or expansion) of the lattice yields the bulk modulus B, which is a linear combination of two elastic constants: B ¼ ðC 11 þ 2C 12 Þ=3

In order to find all three cubic elastic constants, two more types of lattice strains should be considered. Following the procedure proposed by Mehl et al. [65], a volume-conserving orthorhombic strain is considered as: 0 1 x 0 0 B C ð19Þ e ¼ @ 0 x 0 A 0

x2 1x2

0

for which the total energy is an even function of distortion x at constant volume V as: DEðxÞ ¼ DEðxÞ ¼ V ðC 11  C 12 Þx2 þ O½x4 

The C44 elastic constant is calculated using a monoclinic volume-conserving strain as: 0 1 0 2x 0 Bx C e¼@2 0 0 A ð21Þ 0

0

x2 4x2

for which the total energy dependence has the form: 1 DEðxÞ ¼ DEðxÞ ¼ VC 44 x2 þ O½x4  2

ð22Þ

The value of distortion x is varied from zero, for the equilibrium state, to 0.05 with a step of 0.01, in accordance with Mehl et al.’s prescription [65]. For each type of volume-conserving distortion, the calculated total energies are fitted by a parabola using x2 as the variable. The values of elastic constants are obtained from the derivatives of DE with respect to x2. Once the single-crystal elastic constants are known, related properties of polycrystalline alloys can also be evaluated [66,67]. A complication here is that modulii of a cubic single crystal may depend on direction, due to the elastic anisotropy. An elastic anisotropy parameter AG introduced by Zener [68] can be used as a measure of anisotropy. In the case of an isotropic cubic crystal AG = 1, the more it deviates from this value, the larger is the anisotropy. The shear modulus anisotropy is usually expressed as the ratio between two elastic constants, C44 and C0 = (C11  C12)/2, as: AG ¼

2C 44 C 11  C 12

ð23Þ

Table 1 Single-crystal elastic constants (GPa), bulk modulus B (GPa), Young’s modulus E (GPa) and anisotropy constants AG (dimensionless) of Fe– ˚ [45] (the 17%Cr–7%Ni alloy at 263 K and for a lattice constant of 2.87 A results are also compared to those of Ref. [69]).

ð18Þ

The bulk modulus is evaluated using the calculated total energies as a function of the Wigner–Seitz radius in the interval between 2.60 and 2.70 Bohr with a step of 0.01. The energy minimum is located using a third-order polynomial fit.

ð20Þ

This work (263 K) Ref. [69] (0 K)b a

a

C11

C12

C44

AG

B

2.87 2.84

248 273

110 142

121 126

1.75 1.83

156 186a

Calculated from the elastic constants using the relations in the previous section. b Results obtained using the EMTO method for Fe70Ni10Cr20 alloy [69].

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4. Results and discussion 4.1. Ab initio calculations

Fig. 2. Calculated characteristic surface showing the Young’s modulus as a function of crystallographic direction in a random FM Fe–17%Cr–7%Ni alloy. The values on the color scale and on the axes are in GPa. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

The calculated single-crystal elastic constants, the elastic anisotropy constant and the estimated Young’s modulii are shown in Table 1. The present results are compared with the theoretical values of Zhang et al. [69] for an alloy with a similar composition. It can be seen that there is a significant difference between the two results, especially in the values of C11, C12 and B. The difference can be attributed to a much lower lattice parameter being used in the work of Zhang et al. [69] compared to the present study [62]. The similarities in the values of C44 obtained in both works can be attributed to the negligible effect of lattice expansion on the corresponding elastic modulus [62]. The elastic anisotropy constant of the alloy is different from unity, which means that the system is elastically anisotropic. Hence an elastic constant such as Young’s modulus will acquire different absolute values in different crystallographic directions. Fig. 2 shows the anisotropy of the modulus in Fe–17%Cr–7%Ni alloy, obtained from the present calculations. Comparing the present results with those of

Fig. 3. Martensitic microstructure evolution during athermal martensitic transformation. Snapshots taken at (a) t* = 0, (b) t* = 5, (c) t* = 10 and (d) t* = 75.

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The top view of the microstructure obtained at t* = 75 shown in Fig. 3d is presented in Fig. 4. It can be seen that the three different colored martensite (Bain) variants form in groups, as marked by the dashed circles. This result is different from that observed in the case of carbon steels [35], where only two out of the three Bain variants are formed in a group.

Fig. 4. Top view of the microstructure obtained during athermal martensitic transformation at t* = 75, i.e. Fig. 3d. Dotted circles indicate the groups of the three Bain variants.

the earlier work by Razumovskiy et al. [62], it can be seen that Ni slightly increases the anisotropy of the elastic constants of Fe–Cr binary alloys in a ferromagnetic state. 4.2. Phase-field simulations 4.2.1. Athermal martensitic transformation The microstructure evolution due to athermal martensitic transformation is shown in Fig. 3. It can be observed that, as the transformation progresses, the pre-existing spherical martensitic embryo transforms into an ellipsoidshaped martensite unit. The material tends to minimize the stresses caused by the growing martensite unit by means of autocatalysis, i.e. by nucleation of martensite variants governed by other Bain strain tensors. As the stresses are reduced, the rate of formation of new variants reduces and thereafter the energy is reduced by the coarsening of the martensite variants.

4.2.2. Stress-assisted martensitic transformation The microstructure obtained under a uniaxial tensile loading condition mentioned in case-2 of Section 2.3 is shown in Fig. 5. The resultant microstructure is mainly composed of two variants, i.e. variants 2 and 3, in blue and green, respectively, with a very small volume fraction of variant 1 (in red). The variants that minimize the net Gibbs energy by minimizing the externally applied energy, i.e. minimize Gappl in Eq. (15), are favored. As variant 1 is v governed by the Bain strain tensor 00 ij ð1Þ in Eq. (7), it is compressed along the X-axis and expanded along the Yand Z-axes, and hence maximizes Gappl under uniaxial tenv sile loading along the X-axis. Hence variant 1 is not favored under uniaxial tensile loading along the X-axis. However, in the case of variants 2 and 3, governed by 00 ij ð2Þ and 00 ð3Þ in Eq. (7) respectively, the tensile transformation ij strains are along the X-axis, which minimize Gappl and v hence their formation is favored. The results obtained are in good agreement with the experimental results reported by Tsu et al. [70] and Oliver [71] that the effect of applied stress is to promote the formation of those martensite variants which cause the greatest elongation along the tensile axis such that the mechanical potential energy is minimized. The small volume fraction of the “unfavorable” variant 1 is due to autocatalysis, i.e. minimization of internal stresses through the formation of martensitic variants [35]. The microstructure obtained under a biaxial compressive loading condition mentioned in case-3 of Section 2.3 is shown in Fig. 6. It can be seen that only variants 1 and 2 are favored, with a small volume fraction of variant 3 forming at the grain boundaries, i.e. the boundaries of the simulation domain, which could be due to autocatalysis. In contrast to the tensile stress, a compressive stress favors the

Fig. 5. Stress-assisted martensitic microstructure evolution under uniaxial tensile loading. Snapshots taken at (a) t* = 10, (b) t* = 20 and (c) t* = 30.

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Fig. 6. Stress-assisted martensitic microstructure evolution under biaxial compressive loading. Snapshots taken at (a) t* = 5, (b) t* = 10 and (c) t* = 30.

Fig. 7. Stress-assisted martensitic microstructure evolution under triaxial loading. Snapshots taken at (a) t* = 5, (b) t* = 10 and (c) t* = 30.

Fig. 8. Illustration of the prediction of habit plane of one of the martensite variants.

formation of those martensite variants that cause the greatest compression along the direction of compressive loading. Hence, formation of variants 1 and 2 is observed in the compressive loading case, as their compressive transformation strains are aligned along the compressive

loading axes, i.e. X and Y. Moreover, it can be observed that all the variants are oriented along the (0 0 1)c plane. The microstructure obtained under the triaxial loading condition mentioned in case-4 of Section 2.3 is shown in Fig. 7. It can be seen that the anisotropy in the loading conditions gives rise to an anisotropic microstructure evolution. It can also be seen that the anisotropic loading conditions cause the variants to grow along different habit planes. Moreover, all the variants seem to be equally favored under this loading condition. It can also be observed that the variants tend to form in two segments on either side of the central red-colored martensite variant. Based on the above results, it can be summarized that the application of external elastic stresses favor the formation of only those martensite variants that are favorably oriented with respect to the applied stress, which is popularly known as the Magee effect [5,8]. 4.2.3. Habit plane prediction The habit plane of the simulated microstructure shown in Fig. 3 can be predicted, as shown in Fig. 8. The model predicts the habit plane to be (2 1 1)c, which is in agreement with the experimental observations in stainless steels [72].

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5. Conclusions The simulated microstructures resemble a lath-type martensitic microstructure, which is in good agreement with the experimental results on a stainless steel of similar composition [42,72]. It can be seen that the martensite variants tend to form in groups of three Bain variants. Autocatalysis play a major role in the martensitic microstructure evolution in stainless steels. The habit plane predictions seem to be in accordance with the experimental results of Kelly [72]. The results clearly show the Magee effect, i.e. for a given stress state only the variants with the best orientation with respect to the applied stress are formed. The results also show that the microstructure can be significantly different, depending on the loading conditions. Moreover, anisotropic loading conditions give rise to a significant anisotropy in the martensitic microstructure. Based on the results, it seems feasible to build multilength scale models to study practically important materials, e.g. stainless steels. Moreover, acquiring the input parameters for the simulations from experimental data imparts physical validity to such multi-length scale models. However, the model could be improved further by incorporating the strain hardening in the plasticity equations. The possibilities to couple other computational techniques, e.g. molecular dynamics, crystal plasticity or phase-field crystal, with the present model also need to be explored further. Acknowledgements This work was performed within the VINN Excellence Center Hero-m, financed by Vinnova, the Swedish Governmental Agency for Innovation Systems, Swedish Industry and KTH Royal Institute of Technology. Computer resources for this study were provided by the Swedish National Infrastructure for Computing (SNIC) and MATTER Network at the National Supercomputer Center (NSC), Linko¨ping. The authors would like to thank Prof. Staffan Hertzman, Assoc. Prof. Malin Selleby, Dr. Peter Hedstro¨m and Dr. Wei Xiong for the ideas and the discussions. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/ j.actamat.2012.08.012. References [1] Easterling KE, Tho¨len AR. Acta Metall 1980;28:1229. [2] Ghosh G, Olson GB. Acta Metall Mater 1994;42:3361. [3] Raghavan V. In: Olson GB, Owen WS, editors. Martensite. Materials Park, OH: ASM International; 1992. [4] Levitas VI, Idesman AV, Olson GB, Stein E. Philos Mag A 2002;82:429. [5] Cherkaoui M, Berveiller M, Lemoine X. Int J Plast 2000;16:1215.

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