Modelling the evolution of recrystallization texture for a non-grain oriented electrical steel

Computational Materials Science 149 (2018) 57–64

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Modelling the evolution of recrystallization texture for a non-grain oriented electrical steel Hak Hyeon Leea, Jaimyun Junga, Jae Ik Yoona, Jae-Kyoum Kimb, Hyoung Seop Kima,c,

T

⁎

a

Department of Materials Science and Engineering, Pohang University of Science and Technology (POSTECH), Pohang 37673, Republic of Korea Pohang Research Lab. Steel Products Research Group 2, POSCO, Pohang 37859, Republic of Korea c Center for High Entropy Alloys, Pohang University of Science and Technology (POSTECH), Pohang 37673, Republic of Korea b

A R T I C LE I N FO

A B S T R A C T

Keywords: Crystal plasticity Recrystallization Finite element method Electrical steel

A new methodology based on the strain energy release maximization (SERM) theory and Avrami-type kinetics is introduced to predict the evolution of recrystallization texture in a non-grain oriented (NGO) electrical steel. The deformation orientation and the activated slip system of each orientation, which can be developed by cold rolling for a hot-rolled NGO electrical steel, were calculated using the finite element method and visco-plastic self-consistent model. Afterwards, the recrystallization orientations that can evolve from each deformation orientation were determined by the SERM theory, and their fraction over the annealing time was calculated based on the Avrami-type kinetic equation. As a result, this approach for the NGO electrical steel could successfully predict the formation of γ-fiber with strong {1 1 1}〈1 1 2〉 component during recrystallization, which was in good agreement with the experimental results.

1. Introduction As a soft magnetic material with a body-centered cubic (BCC) structure, Si steel exhibits easiest magnetization along the 〈1 0 0〉 crystal direction. For non-grain oriented (NGO) electrical steels, 〈1 0 0〉 directions parallel to sheet surface plane are preferred for superior magnetic properties [1–3]. Unfortunately, conventional rolling process is known to strengthen α-fiber (〈1 1 0〉//RD) and γ-fiber (〈1 1 1〉//ND), which are unfavorable for the magnetization. In order to transform the crystal orientation developed by cold rolling into texture favorable for the magnetization, recrystallization is essential, but the changes in microstructure and texture during recrystallization are vastly complex and not fully understood. Therefore, in an effort to nurture the easy magnetization direction, the evolution of recrystallization texture for NGO electrical steels is still an ongoing subject of interest. Because recrystallization texture of a polycrystalline material strongly depends on its processing history [1,2,4–18], diverse processing methods including cold rolling of columnar grains [7–8], α → γ → α phase transformations [9–10], and conventional [12–14] and unconventional rolling schemes [15–18] have been studied to understand the evolution of recrystallization texture for the NGO electrical steel. While these works laid valuable grounds for how specific orientations may nucleate or grow, very few efforts have been made in actually

⁎

predicting the evolution of recrystallization texture for the NGO electrical steel, which can significantly aid the development of processing methods. For a model to properly predict the evolution of recrystallization texture, the model should firstly be able to predict what orientations will be formed during recrystallization, and secondly be able to describe the kinetics of recrystallization. While the latter can be handled by employing Avrami-type equation, the former one is often problematic. Currently, most of the theories used to describe the evolution of recrystallization texture are based on oriented nucleation (ON) and oriented growth (OG) theories, but both theories are not quite clear as to which nuclei are preferred given a certain deformation history. This difficulty often becomes an obstacle in predicting the final recrystallization texture because recrystallization texture heavily depends on deformation history. A suitable theory to correlate deformation mode with stable recrystallization texture is the strain energy release maximization (SERM) theory [19,20]. The theory postulates that a stable recrystallized grain is developed if the grain is oriented in a way such that the strain energy by dislocations in the deformed grain is minimized, or the strain energy release upon recrystallization is maximized. This theory has been successfully utilized in a diverse class of materials such as Al containing high Mn austenitic steels [21], Al and Cu [22], Co thin film [23], low carbon steels [24], and grain-oriented electrical steels [25]. Also, in

Corresponding author at: Department of Materials Science and Engineering, Pohang University of Science and Technology (POSTECH), Pohang 37673, Republic of Korea. E-mail address: [email protected] (H.S. Kim).

https://doi.org/10.1016/j.commatsci.2018.03.013 Received 10 October 2017; Received in revised form 5 March 2018; Accepted 7 March 2018 0927-0256/ © 2018 Elsevier B.V. All rights reserved.

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smooth surface without surface roughness.

calculating the orientation of the recrystallized grain, the activities of slip systems are used, which means that orientation of the recrystallized grain depends on deformation modes. The fact that the SERM theory accounts for deformation mode is a huge advantage for predicting deformation history dependent recrystallization textures. In this work, we employed a combined finite element method (FEM) and visco-plastic self-consistent (VPSC) modeling approach to characterize the evolution of deformation texture developed by cold rolling for a hot-rolled NGO electrical steel. Afterwards, the recrystallization orientations that can evolve from each deformed grain were predicted based on the SERM theory, using the slip activities calculated in the combined FEM-VPSC. Finally, the evolution of the recrystallization texture at the specific annealing time and temperature was evaluated using the Avrami-type kinetic equation.

3. Numerical procedure 3.1. Finite element method A 2-D finite element simulation for a cold rolling process was conducted with isotropic von Mises yield criterion, using commercial ABAQUS 6.9 software. The roll diameter was designed to be 127 mm, and the initial dimension of a 2-D plane strain matrix representing the electrical steel sheet was set to the length of 80 mm and thickness of 1 mm. The friction coefficient between the roll and matrix was assumed to 0.08, and the lower boundary of the matrix was constrained along the RD due to the symmetric rolling condition. A reduction ratio of 0.35 was achieved so that the extracted velocity gradient history can be repeated four times to reach a final thickness of approximately 0.175. The isotropic hardening model was based on the experimental hardening curve of the hot-rolled NGO electrical steel, and Young’s modulus and Poisson’s ratio were 190 GPa and 0.3, respectively. The element type of the matrix was a fully-integrated quadrilateral element with 4 nodes (CPE4) and the number of elements was 4000. Because the VPSC code requires velocity gradient components (Lij) as the input deformation history, a user subroutine UMAT code for ABAQUS 6.9 was written to calculate and extract the components of each element [27,28]. Given a material point in the reference frame X and the material point in the deformed configuration x, the deformation gradient tensor F and velocity gradient tensor L are calculated as follows:

2. Experimental procedure The NGO electrical steel used in the present work was fabricated by POSCO, containing 2.9 wt% Si. The cold rolling of a hot-rolled steel sheet was subsequently conducted to a final thickness of 0.35 mm. The cold-rolled sheets were isothermally annealed at 750 °C and 830 °C for the various periods of time in Ar atmosphere using a tube-type furnace. The microstructure and crystallographic texture were measured on the plane perpendicular to the transverse direction (TD) by the electron backscatter diffraction (EBSD) using the field emission scanning electron microscopy (FESEM: XL-30S FEG, Philips Co., Netherlands). The acceleration voltage was 25 kV, and the working distance between the beam and the measured surface was 12 mm. The step sizes used for the measurements varied depending on the magnification and the fraction of the recrystallization. The measured EBSD data, including the orientation distribution functions (ODFs), were treated using the orientation imaging microscopy analysis software (TSL OIM Analysis 7.3, EDAX Inc., USA). Before the analyses, the EBSD data were cleaned up using the grain confidence index (CI) standardization, followed by the single iteration of grain dilation, and the minimum reliable confidence index was set to 0.1. The recrystallized grains were distinguished from the partially-recrystallized specimen using the grain orientation spread (GOS) parameter, and the grains with a GOS value less than 3° were regarded as recrystallized grains. In order to investigate the kinetics of the recrystallization, the recrystallized fraction for each annealed specimen was experimentally examined using Vickers microhardness tester. Before the hardness test, the specimens were mechanically ground and polished using SiC adhesive paper with the 1200 grit, and the hardness test was carried out under a load of 300 gf and holding time of 10 s. The equation calculating the recrystallized fraction (f) can be written as [26]

f=

HVini−HV (t ) , HVini−HVfin

F=

∂x , ∂X

(2)

L=

∂v ∂v ∂X = = FḞ −1, ∂x ∂X ∂x

(3)

F ̇ = LF .

(4)

If a time increment is given as Δt , the fully implicit time integration of Eq. (4) is

Fτ = exp(ΔtL τ ) Ft ≃ (I + ΔtL τ ) Ft ,

(5)

where t , τ , and L τ represent the time at the beginning, the time at the end of the time increment, and the velocity gradient at τ , respectively. Then, L τ can be expressed as follows:

L τ = (Fτ Ft−1−I )/Δt .

(6)

3.2. Visco-plastic self-consistent model In order to predict the texture evolution and calculate the slip activities of individual grains during the cold rolling process, the deformation history from FEM simulation was imported into the VPSC model. In the model, which was originally introduced by Lebensohn and Tomé [29,30], each grain within a polycrystalline aggregate is treated as an anisotropic ellipsoidal inclusion immersed in a homogeneous effective medium (HEM) with average property for the polycrystal. The behavior of a grain is described using the strain-rate sensitivity approach, in which shear rate is defined as

(1)

where the HVini , HVfin , and HV (t ) represent the recovered hardness value immediately before the recrystallization occurs, the fully-recrystallized hardness value, and the hardness value at annealing time t (min). The moment when the recrystallized fraction is equal to 0% (only recovery) or 100% (full recrystallization) was determined by observing the microstructure in each annealing time via optical microscopy (OM) after chemical etching with the 4% Nital solution. In order to obtain the hardening curve required for the finite element analysis, uniaxial tensile test on three specimens machined from the hot-rolled sheet was carried out at a strain rate of 1 × 10−3 s−1 using a universal testing machine (UTM, model 1361, Instron Co., USA). The specimens were dog-bone-shape plate specimens with 5.0 mm gauge length along the rolling direction (RD), 2.5 mm gauge width, and 2.0 mm thickness. In order to evaluate the highly accurate strain, a digital image correlation (DIC) method (ARAMIS 5 M, GOM mbH, Germany) was used in the tensile tests after patterning on a

γ ṡ = γ0̇ ⎛ ⎝ mijs =

|ms : σ| ⎞n sign (ms : σ ) τs ⎠

(7)

1 s s (bi nj + bjs nis ). 2

(8)

τs

represent the grain stress In the above expressions, σ , γ̇0 , n , and tensor, reference shear strain rate, the inverse of the strain rate sensitivity, and threshold stress for slip, respectively, and the colon indicates double dot product. Also, ms is the symmetric Schmid tensor of sth slip 58

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system, where b s and ns are the Burgers and normal vector of such slip system. The evolution of threshold stress as a function of strain is given by Voce hardening law as

θs ⎞ ⎛ τ s = τ0s + (τ1s + θ1s ) ⎜1−exp ⎛⎜−Γ 0s ⎞⎟ ⎟, τ1 ⎠ ⎝ ⎠ ⎝

crystals, resulting in one or more AMSDs [19,20,31,32]. Therefore, all the stress directions are used as long as the shear strain of each slip system is above a critical value, which is 0.1 in the present study. Finally, the orientation that can maximize strain energy release can be determined by applying a rotation about an 〈1 1 0〉 axis for AMSD to be parallel with MYMD [31,32].

(9)

nslip

Γ=

∑

3.4. Recrystallization kinetics

Δγ s (10)

s=1

In order to describe the recrystallization kinetics of the NGO electrical steel, a well-known form of the Avrami-type equation for sitesaturated nucleation is employed as below:

where τ0s , τ1s , θ0s , and θ1s are the hardening parameters. Crystal plasticity in a continuum framework incorporates slip as an additional kinematic degree of freedom for shear by decomposing plastic velocity gradient as

Lp =

∑

f = 1−exp [−FN (Gt )3],

nslip

nslip

msγ ṡ

+

s=1

∑

q sγ ṡ

= ε̇ + ω (11)

s=1

qs ,

ε̇, and ω represent skew-symmetric Schmid tensor, a strain where rate, and a rotation rate (spin), respectively. Then, Eqs. (7) and (11) yield the visco-plastic constitutive behavior in a given grain as nslip

εij̇ = γ0̇

∑ s=1

G = mp,

ms : σ n mijs ⎛ s ⎞ ⎝ τ ⎠

(12)

p=

dγ (i) dε, dε

4 fi = fi,def ⎧1−exp ⎡− πNi (mpt )3⎤ ⎫, ⎨ ⎣ 3 ⎦⎬ ⎩ ⎭

where the fi,def is the volume fraction of i deformed grain before recrystallization. The number of recrystallization nuclei per unit volume is calculated as follows:

Ni =

τ0{110}

245

150

95

1

< 111 >

: τ0{112}

< 111 >

γ0̇

n

0.001

25

D3

QN exp ⎜⎛− RX ⎟⎞, ⎝ kT ⎠

(19)

are a fitting constant, shear strain, critical where C0 , γd , γc , D, and shear strain, grain size of each deformed grain, and activation energy for nucleation, respectively [26,34,35]. Since shear strain for each deformed grain and the orientation of recrystallized grain are provided by VPSC-SERM calculations, the evolution of recrystallization texture can be estimated using Eqs. (18) and (19). In Table 2, the parameters are used in this work to fit the experimentally observed JMAK curve. In order to predict the evolution of recrystallization texture for the NGO electrical steel, various models, such as FEM, VPSC, the SERM theory, and Avrami-type kinetic equation, were integrated in this study. The relationship between each model was summarized in the flow chart (Fig. 1). At first, the velocity gradient components (Lij) for the cold rolling were calculated using FEM based on the experimental hardening curve of the hot-rolled NGO electrical steel. Then, the velocity gradient components and the initially hot-rolled texture were used as the input for the VPSC calculations. Afterwards, the recrystallized orientations that can evolve from each deformed grain were determined by the SERM theory, using the cold-rolled texture and slip activities for each deformed grain. Also, the recrystallized fraction (fi) of each grain at a

Table 1 Parameters used for VPSC simulation. C44 (GPa)

C0 (γd−γc ) × 1014

N QRX

where is the accumulated shear strain of the i slip system, which is calculated via crystal plasticity theory, and ε is the equivalent plastic strain. This calculation is rather straight forward in the case of face centered cubic crystals where all slip directions are related each other through associated slip planes. Only the vector sum of the accumulated shear strain multiplied to the vector of the respective slip direction is the AMSD [20]. On the other hand, the activation of different slip planes may give rise to more than one effective slip directions for BCC

C12 (GPa)

(18) th

(13)

C11 (GPa)

(17)

where Qb , k, and T represent activation energy for grain boundary movement, Boltzmann constant, and temperature, respectively. If recrystallized region of a deformed grain does not grow into another deformed grain, the constrained volume of recrystallized region is

th

γ (i)

(16)

Q m = m 0exp ⎛ b ⎞, ⎝ kT ⎠

Calculations on recrystallization texture rely on the theory of recrystallization via strain energy release maximization (SERM), which was originally proposed by Lee [19,20] and modified by Park et al. [31,32] for BCC crystals. According to the SERM theory, recrystallization takes place by releasing the maximum amount of strain energy accumulated during the deformation process, and the orientations of recrystallized grains are determined such that the maximum internal stress direction, or the absolute maximum stress direction (AMSD) characterized by dislocation stress fields, of the deformed grain is parallel to the minimum Young’s modulus direction (MYMD) of the recrystallized grain [19,20]. In the case of electrical steel with BCC structure, the MYMD corresponds to the 〈1 0 0〉 directions. The AMSD is calculated by taking into account the amount of shear strain in each slip system (Eq. (13)).

∫0

1 ρμb2, 2

where ρ , μ , and b are the stored dislocation density, shear modulus, and Burgers vector, respectively. A constant grain boundary mobility in the following form is used [33,34]

3.3. Strain energy release maximization theory

γ (i) =

(15)

where m is the time-independent grain boundary mobility. The driving force for the grain boundary motion is mainly provided by the stored dislocation energy as follows:

The stress and strain rate in each grain are calculated by the interaction with HEM until the condition is satisfied that the macroscopic stress and strain rate must be consistent with the average stress and strain rate of all grains. For this study, the deformation history, namely velocity gradient tensor, from FEM simulations and initially hot-rolled texture were used as inputs for VPSC model to simulate cold-rolled texture. The model parameters used for the VPSC calculations are given in Table 1.

e

(14)

where f, F, N, and G represent the recrystallized fraction, shape factor, the number of nuclei, and growth rate, respectively. This particular form will assume site-saturated nucleation with isotropic growth and a constant growth rate. The growth rate is in linear relation to the acting driving force p:

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Table 2 Parameters used for the recrystallization kinetics. b (10−10 m)

μ (GPa)

Qb (J)

QRXN (J)

C0

m0 (m3 s−1 N−1)

2.03

80

1.99 × 10−19

2.33 × 10−19

4 × 104

9 × 10−8

given T and t was calculated based on an Avrami-type kinetic equation. Finally, the evolution of recrystallization texture was expressed in terms of ODF.

Stress, MPa

600

4. Results 4.1. Characteristics of the initially hot-rolled NGO electrical steel Prior to predicting the texture evolution by cold rolling and recrystallization, the mechanical properties and microstructure of the initially hot-rolled sheet were preferentially evaluated. For the engineering stress-strain curve in Fig. 2, the yield strength, ultimate tensile strength, and total elongation of the hot-rolled specimen are 388.3 MPa, 518.5 MPa, and 51.6%, respectively. As shown in Fig. 3, the initial microstructure of the specimen is composed of the equiaxed grains with an average grain size of 173.0 μm and very low in-grain misorientation, which is a characteristic of hot-rolled and annealed single phase alloys. Also, the texture exhibits typical weak rolling texture components after the hot rolling and annealing process. These mechanical and microstructural properties are set as input data for the FEM simulations and VPSC calculations of the cold rolling.

450

300

True

150

Engineering 0

0

10

20

30

40

50

Strain, % Fig. 2. The stress-strain curve of the initially hot-rolled NGO electrical steel.

〈1 1 2〉 component. From the cold-rolled microstructure shown in Fig. 4c, one can observe that grains constitute a band-like structure with notable shear bands aligned roughly 30° to 45° with respect to the RD within γ-fiber grains, indicated by red dotted lines. These shear bands not only result in reduced image quality, but also introduce a strong ingrain misorientation within γ-fiber grains, which leads to the discrepancies between experimental and simulated textures.

4.2. Evolution of deformation texture After cold rolling, the final deformation textures from the EBSD measurement and VPSC calculation are represented in Fig. 4a and b, respectively. The experimental result indicates that deformation texture after 83% reduction is characterized by a strong α-fiber and relatively weak γ-fiber (Fig. 4a). This result is qualitatively in good agreement with the VPSC result shown in Fig. 4b. The α-fiber in the experiment is stronger than that of the simulated result while the opposite is true for the γ-fiber components. This discrepancy has likely emerged from intense shear bands within γ-fiber components, in particular the {1 1 1}

4.3. Prediction for stable recrystallized orientation The orientations capable of maximizing the strain energy release during recrystallization are predicted based on the SERM theory. The AMSD for each deformed grain is determined using the shear strain and activated slip systems calculated by VPSC. For instance, a deformed grain with (0 0 1)[1 −1 0] crystal orientation has (0 1 −1)[1 −1 −1], (1 0 −1)[1 −1 1], (1 −1 2)[−1 1 1], and (−1 1 2) [1 −1 1] active slip

Fig. 1. Flow chart for the methodology to predict the evolution of recrystallization texture.

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(a)

(b)

RD 300 m

300 m

ND Fig. 3. The microstructural characteristics of the hot-rolled specimen measured by EBSD: (a) Inverse pole figure map and image quality map, (b) ODFs plotted in the φ2 = 0°, 45° sections.

systems, which leads to the effective slip directions of [−1 1 1] and [1 −1 1]. The [−1 1 1] effective slip direction is attained by the activation of (0 1 −1)[1 −1 −1] and (1− 1 2)[−1 1 1] slips, i.e. by a reaction of 0.372 [1 −1 −1] + 1.468[−1 1 1]. The coefficient in front of each slip direction indicates the amount of shear on the slip system. The other effective slip system, [1 −1 1], is also calculated by following the procedure, i.e. by a reaction of 0.573[1 −1 1] + 1.149[1 −1 1]. Hence, the AMSDs of a deformed grain with (0 0 1)[1 −1 0] orientation are [−1 1 1] and [1 −1 1], and a 〈0 0 1〉 direction rotates about a common 〈1 1 0〉 axes for AMSDs to be parallel to the MYMD. Applying the transformation matrix of the rotation angles less than 90° among rotation angles calculated for each deformed orientation, the stable recrystallized orientation for each deformed grain is predicted. The recrystallized orientations were calculated for 519 deformed grains, and the overall texture is represented in ODF plot (Fig. 5). The result indicates that the intensity of the α-fiber components, which was strongly developed by cold rolling (Fig. 4b), is significantly weakened. On the other hand, the γ-fiber, especially the {1 1 1}〈1 1 2〉 component, mostly transforms into {1 1 1}〈1 1 2〉 . Thus, according to the SERM

Fig. 5. Stable recrystallized texture predicted by the SERM theory.

(a)

(b)

RD

(c) ND

60 m

Fig. 4. Deformation texture upon 83% reduction: ODFs plotted in the φ2 = 0°, 45° evaluated by (a) EBSD and (b) VPSC. (c) Inverse pole figure map for the cold-rolled specimen.

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(b)

350 o

750 C o 830 C

Recrystallized fraction

Vickers hardness, HV

(a)

300

250

200 1

1.0

0.5

Exp_830 ˚C Fitting_830 ˚C Exp_750 ˚C Fitting_750 ˚C

0.0 0

10

Annealing time, min

5

10

15

20

Annealing time, min

Fig. 6. (a) Variation of Vickers hardness over the annealing time at 750 °C and 830 °C, (b) experimental recrystallized fractions (symbols) and JMAK curves calculated by Avrami-type kinetic equation (lines).

theory, it is considered that both α- and γ-fiber components mainly release maximum strain energy via recrystallizing into {1 1 1}〈1 1 2〉, which is in a good agreement with previous literature dealing with recrystallization texture of cold-rolled steels [7,8].

{1 1 1}〈1 1 2〉 and {1 1 2}〈0 1 2〉 components intensify. In general, the calculated results for both 750 °C and 830 °C indicate that initially formed {1 1 1}〈1 1 2〉 simply strengthens over annealing time. In order to experimentally verify the results predicted by this model, the recrystallization textures of the partially-recrystallized specimens are evaluated by EBSD, as shown in Fig. 8. At the early stage of the recrystallization, the recrystallized grains are characterized by a γ-fiber and a fair amount of Goss ({1 1 0}〈0 0 1〉). Upon further annealing, the {1 1 1}〈1 1 2〉 component strengthens while Goss gradually weakens. After full recrystallization, both annealed specimens at 750 °C and 830 °C exhibit strong {1 1 1}〈1 1 2〉 component with a trace amount of Goss and {0 0 1}〈1 2 0〉 . Generally, the recrystallization texture remains almost unchanged over annealing time. In other words, recrystallization texture seems to be largely attributed to oriented nucleation. The experimental results in which the initially formed {1 1 1} 〈1 1 2〉 component strengthens over time are in good agreement with the simulated results. Nonetheless, the model failed to predict Goss, which nucleates at the early stage of recrystallization, and {0 0 1} 〈1 2 0〉, which is mildly observable from the experimental result at the final stage of recrystallization. The limitation of the model is mainly due to the fact that the present construct of the model does not take into account microstructural variables and normal grain growth. The detailed explanation about the discrepancy is treated in the following section.

4.4. Evolution of the recrystallization texture The isothermal recrystallization kinetics is generally characterized based on the Johnson-Mehl-Avrami-Kolmogorov (JMAK) equation:

f = 1−exp (−kt n ),

(20)

where n is the Avrami exponent indicating the nucleation behavior during the recrystallization. The fraction of recrystallization is determined from Vickers hardness results (Fig. 6a). The fully-recrystallized specimens shared an identical hardness value of approximately 196 HV. The experimental recrystallization fractions after isothermal annealing at 750 °C and 830 °C are represented as symbols in Fig. 6b, and it is confirmed that the Avrami exponent (n) is 2.83 by calculating the slope of Avrami plots from the recrystallized fraction. Since the Avrami exponent is approximately close to 3, a site-saturated nucleation is assumed. Finally, the recrystallized fraction in each deformed grain is individually determined by the Avrami-type kinetic model based on the site-saturated nucleation, and the recrystallized fraction for whole grain is plotted as line in Fig. 6b. Consequentially, the recrystallized fractions calculated by the model are in good agreement with the experimental results in both 750 °C and 830 °C. The changes in recrystallization texture over annealing time at 750 °C and 830 °C are calculated by combining the recrystallization kinetics with stable recrystallized orientations predicted by the SERM theory (Fig. 7). At the early stage of the recrystallization, γ-fiber with notable {1 1 1}〈1 1 2〉 is formed. As the recrystallization progresses, the

5. Discussion 5.1. Discrepancies between the predicted and experimental recrystallization texture As shown in Fig. 4c, a small amount of the deformed (1 1 0)//ND,

Fig. 7. Evolution of the predicted recrystallization texture over the annealing time at 750 °C and 830 °C.

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Fig. 8. Evolution of the experimental recrystallization texture over the annealing time at 750 °C and 830 °C.

Fig. 9. Inverse pole figure map for the recrystallized grains near the shear band in the partially-recrystallized specimen.

(1 0 0)//ND orientations is formed around the shear bands inclined at angles of 30–45° with respect to RD. From Fig. 9 and published literature [7,32], it is well established that the shear band in the deformed γ-fiber plays a major role in the formation of Goss oriented nuclei at the early stage of recrystallization. In particular, the intense shear deformation within shear bands enables fast nucleation of Goss, which explains the reason why Goss is observed at the early stage of recrystallization. Nonetheless, the precise morphology and deformation history of shear bands are not taken into account in the present work, explaining why the model failed to predict the early stage recrystallization of Goss component. NGO electrical steels develop strong Cube and near Cube ({0 0 1} 〈1 2 0〉) after annealing at elevated temperature, indicating that normal grain growth plays an important role. From SERM calculations, deformed {1 1 1}〈1 1 2〉 is able to release maximum strain energy if the recrystallization orientation is Cube or {1 1 1}〈1 1 2〉, but the fraction of Cube is too small to be captured in ODFs. Cube recrystallizing from {1 1 1}〈1 1 2〉 should retain a size advantage over other recrystallization orientations because deformed grains with {1 1 1}〈1 1 2〉 orientations have the most deformation energy. Nonetheless, because normal grain growth is not defined in the current model, Cube texture did not increase in its fraction due to its size advantage.

VPSC, the recrystallized texture originated near the shear band can be approximated by the SERM theory. Additionally, the general form of the constant grain boundary mobility (Eq. (17)) was employed in this model, but the grain boundary mobility of each grain can vary during recrystallization since the orientation relationship between adjacent grains influences the grain boundary migration. As the relationship between the misorientation and grain boundary mobility has been actively studied in the preceding literature [37–39], the prediction for the recrystallization texture can be more reliable if the misorientation between each recrystallized nuclei is taken into account in recrystallization kinetics. Finally, the effect of grain growth on the texture evolution could be reflected in the current model. The phase field model is applicable to the grain growth modeling as a mathematical method for interfacial dynamics. Using the phase field model, Jamshidian and Rabczuk successfully simulated the normal or abnormal grain growth according to microstructural length scale and shape, and they quantified texture evolution during grain growth depending on the relative magnitude between two types of driving force (curvature driving force and strain energy driving force) [40,41]. Therefore, the evolution of recrystallization texture will be predicted more precisely if the grain boundary kinetics based on the phase field approach is applied to the present model.

5.2. Improvement scheme in the present model

6. Conclusion

In order to enhance the accuracy of the current model, the deformed texture near the shear band should be separately calculated according to the direction of the shear deformation because the evolution of the recrystallization texture in the vicinity of the shear band relies on the inclined angle of the shear band with respect to RD [36]. After calculating the deformed texture in the characteristic shear direction by

In this work, the evolution of the recrystallization texture for 2.9 wt % Si non-grain oriented electrical steel was modeled. The deformation texture by cold rolling was calculated combining the finite element method (FEM) and visco-plastic self-consistent (VPSC) model. Afterwards, the possible recrystallization orientations were predicted based on the strain energy release maximization (SERM) theory. 63

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Finally, the evolution of the recrystallization texture at 750 °C and 830 °C was predicted using the Avrami-type kinetic equation. The model successfully predicted that the γ-fiber, especially {1 1 1} 〈1 1 2〉 component, was formed at early stage recrystallization and gradually strengthened over the annealing time because both α- and γfiber components developed by cold rolling were mostly converted to {1 1 1}〈1 1 2〉 by releasing maximum strain energy upon the recrystallization. The model was not successful in predicting formation of Goss and {0 0 1}〈1 2 0〉, which would require microstructural variables and normal grain growth to be taken into account.

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