Development of Recrystallization Texture Prediction Method ... - J-Stage

ISIJ International, Vol. 52 (2012), No. 4, pp. 592–600

Development of Recrystallization Texture Prediction Method Linking with Deformation Texture Prediction Model Toshiharu MORIMOTO,1) Fuyuki YOSHIDA,1) Yuji KUSUMOTO1) and Akira YANAGIDA2) 1) Nakayama Steel Works, Funamachi, Taisho-ku, Osaka, 551-8551 Japan. Hodogaya-ku, Yokohama, 240-8501 Japan.

2) Yokohama National University, Tokiwadai,

(Received on February 4, 2011; accepted on May 27, 2011)

T.M.C.P. (Thermo Mechanical Control Process) has been widely used in the steel industry. We have produced ultra low-carbon steel strip and ferritic stainless-steel strip through T.M.C.P. rolling method. Next, we analyze rolling texture and recrystallization texture of the ultra low-carbon strip and the ferritic stainless- steel strip using our developed texture prediction model. We can predict rolling texture accurately using the conventional Taylor model. Moreover, we precisely predict recrystallization texture classifying the total number of microscopic slips which are calculated using the Taylor model. We consider that these calculated results prove nucleation-oriented model and two types of recrystallization and grain growth mechanisms exit in our studies. One mechanism is that grains which had the lowest total number of microscopic slips are preferred orientation for the hot rolled and annealed ferritic stainless-steel strip. The other mechanism is that grains which had the highest total number of microscopic slips are preferred orientation for the cold rolled and annealed ultra low-carbon strip. KEY WORDS: T.M.C.P.; ultra low-carbon strip; ferritic stainless-steel strip; Taylor model; recraystallization texture; total number of microscopic slips.

are necessary.8) Thus, we aim to roll ultra low-carbon steels with a low reheating temperature, high-reduction rolling and rapid cooling immediately after the final rolling by using single roll-driven mills of different diameters and a laminar flow cooling device between stands in the finishing train. High-reduction rolling at a low temperature in the finishing train is known to improve the ridging property and drawability of ferritic stainless-steel.9) This is the high residual strains in ferritic stainless-steel hot strips rolled with high reduction in the non-recrystallized region promotes recrystallization in the hot strip annealing process. However, the rolling load at a low temperature in high-reduction rolling is too large to produce ferritic stainless-steel strips industrially in the conventional hot strip mill. Thus, we aim to roll ferritic stainless-steels with low-temperature, highreduction rolling by using single roll-driven mills of different diameters and a laminar flow cooling device in the finishing train. Since the texture of steel greatly affects its mechanical properties of steel, ultra low-carbon steels and ferritic stainless-steels are produced by the T.M.C.P. method, which enables the requirements of these steels to be met. There are three types of steel texture: the deformation texture formed in hot rolled or cold rolled steel, the transformation texture formed by transformation from the austenite structure to the ferrite structure and the recrystallization texture formed during recovery, recrystallization and grain growth.10) We previously reported a method of predicting the formation of deformation and transformation textures before.11) Many mechanisms for the formation of the recrystallization tex-

1. Introduction Thermo mechanical controlled processing (hereafter T.M.C.P.) has been widely used to produce fine-grained steels and optimize of the reheating temperature, the finishing temperature of rolling, the rolling reduction and the cooling rate.1,2) Recently, low-carbon fine-grained hot strips and high-carbon fine-grained strips have been rolled in tandem hot strip mills by the T.M.C.P. method.3,4) However, ultra low-carbon steels and ferritic stainless-steels have never been rolled in a tandem hot strip mill by this method. Three production techniques in the tandem hot strip mill have been reported to improve the formability of ultra lowcarbon steel. First, low-temperature reheating in a furnace causes carbide and sulfide precipitates unresolved in a steel slab.5) Second, high-temperature coiling causes the condensation of carbide and sulfide precipitates after coiling.6) Third, high-reduction hot rolling results the formation of fine-grained hot strips.7) When such ultra low-carbon hot strips are cold rolled and annealed, preferred {111} oriented grains preferentially grow and the final strips have high Lankford values. However, if an ultra low-carbon steel slab is reheated at 1 050°C in a furnace, it is difficult to finish rolling to obtain thin gauge strips above the Ar3 transformation temperature when a conventional schedule is used in the finishing train, because the Ar3 transformation temperature of ultra low carbon steel is more than 910°C. High-reduction final stand rolling to produce fine-grained steels and rapid cooling immediately after the final rolling to prevent grain growth © 2012 ISIJ

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whereas the phase field method required many parameters.

ture in steels have been proposed, including the oriented nucleation model,12) oriented grain growth model,13) dislocation pass model14) and dislocation build- up model.15) In other models, such as the crystal rotation model,16,17) dislocation twist area model18,19) and encroachment model, recrystallized grains encroach on non-recrystallized deformed grains.20) However, no definite conclusions have been reached regarding the validity of each model. In the dislocation twist area model, it is predicted that the recrystallization texture of ultra low-carbon steels is inherited from the cold rolling texture. In this model, the deformation history of cold rolling is taken into account and the residual dislocation is used as a parameter to predict the recrystallization texture. Dillamore proposed a prediction model using the Taylor factor of the pre cold rolling texture.21) However, the dislocation twist area model is a different type of model because crystal slips and rotations that occur during cold rolling are taken into account. In this study, we develop a new prediction method based on the dislocation twist theory. First, we predict the rolling texture using the crystal plasticity by applying Lee’s elastic-plastic decomposition rule.22) Second, we predict the recrystallization texture from the predicted rolling texture. Furthermore, we explain the mechanism of recrystallization texture formation used in our prediction. Since nucleation during the recrystallization of polycrystals is generally not only in the transition bands but also in the grain boundaries,23) within which the range of orientations mostly exists. The combination analysis with the finite element crystal plasticity method and the phase field method has recently been carried out to calculate recrystallization and grain growth structures.24,25) Our method of using the total numbers of each crystal slip to predict the recrystallization and grain growth textures has not been previously reported. Our method has many advantageous features compared with the phase field method. For example, even though it is based on simple assumptions, it enables us to predict the recrystallization texture quantitatively. Moreover, it can predict the recrystallization texture of polycrystals with few parameters and in a short time,

Fig. 1.

2. Ultra Low Carbon Steel and Ferritic Stainless Steel Rolled by the T.M.C.P. Method The total length of the Nakayama tandem hot strip mill is 192 m, and both the exit temperature and coiling temperature are high even when the reheating temperature in the furnace is low. Figure 1 shows an illustration of the Nakayama tandem hot strip mill. At the final three stands of the finishing train, there are single roll-driven rolling stands with different diameters and laminar flow cooling devices. The chemical composition of the ultra low carbon steel we used is 0.002C–0.01Si–0.15Mn–0.008P–0.005S wt% with trace amounts of niobium and titanium. Table 1 shows the rolling pass schedules of rough rolling and finish rolling. The reheating temperature is 1 050°C. Using a slab of 230 mm thickness, five passes of rough rolling are carried out. In the finishing train, six passes of the finishing rolls are carried out to give an exit thickness of 2.6 mm. The reduction in the final three stands in the finishing train is about 35%. The exit temperature of the finishing roll is 880°C because this is the temperature after cooling immediately after the final rolling. Figure 2 shows the optical microstructures of hot strips and TEM images of precipitates. The ferrite grain size is 28 microns and the precipitate size is large, confirming the effect of a low reheating temperature, high reduction rolling with immediate cooling and high coiling temperature. Table 2 shows the mechanical properties of hot strips. Large precipitates result in a low yield stress strip, and a purer steel results strip with high total elongation. We investigate the properties of the cold rolled and annealed strips using hot strips rolled the T.M.C.P. method. The cold rolling reduction was 80% and the annealing temperature was 800°C, which was applied for 60 s. A description of Fig. 2 is given above shows the microstructures of strips after cold rolling and annealing. The ferrite grain size is large. This is precipitates condense during hot rolling, the ultra low-carbon steel becomes purer and grains with {111} preferential

Schematic illustration of Nakayama tandem hot strip mill.

Table 1.

Tandem hot rolling condition of ultra low-carbon strip.

Rheating Temp.

Slab Thick.

R1

R2

R3

R4

R5

1 050°C

230 mm

172 mm

129 mm

94 mm

64 mm

48 mm

F1

F2

F3

F4

F5

F6

Finishing Temp.

Coiling Temp.

27.4 mm

16.5 mm

10.1 mm

6.2 mm

4.1 mm

2.6 mm

*880°C

750°C

* Finishing temperature means temperature just after F6 cooling.

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of the finishing rolls was 827°C. After the hot strip was annealed at 850°C for 8 hours, the cold rolling reduction was 83% and the annealing temperature was 920°C, which was applied for 120 s. Figure 3 shows optical microstructures of the hot strip, annealed hot strip and annealed cold strip of ferritic stainless steel. Table 4 shows the mechanical properties of the hot strip, annealed hot strip and annealed cold strip of ferritic stainless steel. The annealed cold strip has both a good anti-ridging property and good drawability. Thus, we predict that strains of the ferritic stainless-steel hot strips rolled by the T.M.C.P. method promoted strongly recovery and recrystallization at the hot strip annealing process. After cold rolling and annealing, the steel has grains with a strong {111} preferential orientation grains and few colonies of grains inherited from the casting structure. Figure 4 shows E.B.S.D. orientation maps of the hot band, hot and annealed band, cold band and cold and annealed band at the center of thickness about ultra low-carbon steel and ferritic stainless-steel. The hot band orientation of ultra low-carbon steel was random because of transformation from austenite to ferrite. But in the hot band of ferritic stainless-steel, you can see elongated microstructure and hot rolling texture as like as cold rolling texture in the cold band of ultra low-carbon steel because it does not transform from austenite to ferrite. An ultra low-carbon steel had many {1,1,1} preferred grains when it’s cold rolled and annealed. A ferritic stainless-steel was annealed twice and

orientation grow more quickly during annealing. Table 2 also gives the mechanical properties of the cold rolled and annealed strips. Both tensile strength and yield strength are low. Moreover, ultra low-carbon steels have high Lankford values. Thus, ultra low carbon steels rolled by the T.M.C.P. method are suitable for use as drawing products. The chemical composition of the ferritic stainless-steel we used is 0.07C–0.35Si–0.30Mn–0.019P–0.004S –16.32Cr wt%. Table 3 shows the rolling pass schedules of rough rolling and finish rolling. The reheating temperature was 1 160°C. Using a slab of 250 mm thickness, seven passes of rough rolling are carried out. In the finishing train, six passes of the finishing rolls were carried out to give an exit thickness of 3.0 mm. The reduction in the final three stands in the finishing train was about 30%. The exit temperature Table 2. Mechanical properties of ultra low-carbon strip rolled by T.M.C.P. method. 0.2% Proof stress (MPa)

Tensile strength (MPa)

Total elongation (%)*

153

299

54.3

91

281

50.7

Hot rolled band Cold rolled and annealed band

Average Lankford value

2.3

(*Gauge length 50 mm). Table 3.

Tandem hot rolling condition of ferritic stainless-steel strip.

Rheating Temp.

Slab Thick.

R1

R2

R3

R4

R5

R6

R7

1 160°C

250 mm

211 mm

172 mm

136 mm

102 mm

74 mm

51 mm

36 mm

F1

F2

F3

F4

F5

F6

Finishing Temp.

Coiling Temp.

19.9 mm

12.9 mm

8.2 mm

5.6 mm

4.1 mm

3.0 mm

827°C

732°C

Fig. 2.

Optical microstructure and precipitation TEM image of ultra low carbon steel rolled by T.M.C.P. method.

Fig. 3.

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Optical microstructure of ferritic stainless-steel rolled by T.M.C.P. method.

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slipping direction, respectively. Thus, the plastic strain rate DP and plastic spin WP can be defined using Eqs. (2) and (3), respectively.

{1,1,1} preferred grains was few at the cold rolled and annealed process. 3. Texture Analysis of Ultra Low-carbon Steel and Ferritic Stainless-steel

1 ⎛ ∂u j ∂ui Dij p = ⎜ + 2 ⎜⎝ ∂xi ∂x j

11)

3.1. Rolling Texture Analysis Model Figure 5 shows a flow chart of rolling texture analysis and recrystallization texture analysis. Number of b.c.c. crystals was ten thousands. Here, we suggested the assumptions in rolling texture analysis. The critical shear stress of all slip systems was the same and calculated from the equivalent flow stress using the Taylor Factor. And the Taylor Factor was constant. The equivalent flow stress was analyzed from the measured rolling load by the Orowan calculations model.26) Hereafter, Xi (i = 1,2,3) is vector component of X in orthogonal coordinate system. A displacement velocity vector u can be expressed as

⎞ 1 ⎟⎟ = γ ai b j + a j bi = Pij γ. ..... (2) ⎠ 2

(

)

u = γ ( x ⋅ a ) b . .............................. (1) Here, γ, x, a and b are the slip rate, the position, the unit vector normal to the slipping plane and the unit vector in the Table 4. Mechanical properties of ferritic stainless-steel strip rolled by T.M.C.P. method. 0.2% Proof Tensile Total Average stress strength elongation Lankford (MPa) (MPa) (%)* value Hot rolled band

451

583

20.1

Hot rolled and annealed band

251

448

34.5

Cold rolled and annealed band

285

489

30.5

1.3

Ridging value (micron)

12.1 Fig. 5.

(*Gauge length 50 mm).

Fig. 4.

Flow chart of calculation for rolling texture and recrystallization texture using total amount of microscopic slips.

Ultra low-carbon and ferritic stainless-steel orientation map of hot rolled, annealed, cold rolled and final annealed respectively.

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1 ⎛ ∂u j ∂ui Wij P = ⎜ − 2 ⎜⎝ ∂xi ∂x j Pij =

where

I is the identity tensor.

⎞ 1 ⎟⎟ = γ ai b j − a j bi = Qij γ, .... (3) ⎠ 2

(

)

(

)

(

)

⋅ R = Ω ⋅R.................................. (15) R = exp Ω dt ≈ I + dtΩ + 1/2dt 2Ω 2............ (16)

1 ai b j + a j bi .......................... (4) 2

3.2. Analysis of Rolling Texture Obtained the Taylor model with the Asaro rate-dependent rule (hereafter referred to as the Taylor model), Figs. 6 and 7 respectively show the predicted and observed cold rolling texture of ultra low-carbon steel and hot rolling texture of ferritic stainless-steel. When we predict cold rolling texture of ultra-low carbon steel, an initial texture assumed to be random. Furthermore, when we predict hot rolling texture of ferritic stainless-steel, an initial texture also assumed to be random because recrystallization occurs in the rougher rolling mill. Parameters of Asaro rate-dependent rule were same as our previous report.11) Here, γ0 is 100 and m is 0.05. We measured cold rolling texture of ultra low- carbon steel and hot rolling texture of ferritic stainless-steel by electron backscattering diffraction at the center in the thickness direction. Observed area was shown as Fig. 4. The prediction accuracy for ultra low-carbon cold rolling texture and ferritic stainless-steel hot rolled texture is high. Ferritic stainless steels hardly recrystallize during hot rolling in the finisher. Thus, we can accurately simulate hot rolling texture of ferritic stainless-steel strip considered that six finishing passes were in non recrystallized region mentioned as Table 3. To predict rolling texture more precisely, we guess three modifications below. First, measured orientations will be used as initial orientations. Second, the critical shear stresses of all slip systems will not be the same. Finally, the Taylor Factor will not be constant because the Taylor Factor of b.c.c is 2.75 in a case of random orientations.

1 Qij = ai b j − a j bi ........................... (5) 2 Here, the plastic strain rate DP ≅ D is calculated using the Orowan calculation model.26) A b.c.c. crystal lattice has 12 slip systems of the {110} type, 12 slip systems of the {112} type and 24 slip systems of the {123} type. Strain equilibrium, the shear stress of each slip system, the rate-dependent rule and boundary condition are expressed using Eqs. (6)–(9), respectively. Equation (6) implies that the sum of the slip values γ for each slip system in a crystal is equal to the sum of the micro strains ε . Equation (7) gives the relationship between micro stress σ and the resolved shear stress τ acting on the slip plane. Equation (8) is the Asaro rate-dependent rule and m is a material parameter.27) For the boundary conditions, the macro strain E is assumed to be equal to the micro strain ε in accordance with Taylor theory, as shown in Eq. (9). dε i = ∑ Pij dγ j ............................. (6)

τ j = ∑ Pijσ i ............................... (7) ⎛τ j dγ j = γ0 * dt * ⎜ ⎜τy ⎝

1/ m

⎞ ⎟ ⎟ ⎠

* sign(τ j ) ............... (8)

ε = E...................................... (9) Here, γ0 , τ y and dt are the reference slip strain rate, the critical shear stress and the time increment, respectively. The critical shear stress τ y can be expressed as

3.3. Analysis of the Recrystallization Texture Here we suggested together the conventional assumptions in the above model. The recrystallized grains have a range of orientations in the deformed state that satisfy the preferential orientation model. Nucleation occurs from piled-up dislocations at grain boundaries. The total number of microscopic slips is estimated as the total number of displaced dislocations. The magnitude of slips of crystals is calculated from the Schmid tensor expressed as Eq. (4). In other words, crystals with a low Schmid factor slip a longer distance than crystals with a high Schmid factor. Needless to say, the Taylor factor is approximately inversed to the Schmid factor. We introduce one further assumption here. The total number of microscopic slips is estimated as the total number of dislocations piled up at grain boundaries.28) This might be a self-evident assumption. We can accurately predict both the rolling texture of ultra low-carbon strip and ferritic stainless-steel strip using the Taylor model as mentioned in section 3.2. After applying crystals with random orientations by same strains and total spins, crystals slip in proportion to Schmid factors of each crystal. Crystals have orientation distribution, that is to say, rolling texture. Here, we selected 10 percents in order of a crystal with more total number of microscopic slips and 10 percents in order of a crystal with little total number of microscopic slips. Investigating actual

τ y = S/Taylor Factor........................ (10) S is equivalent flow stress, which was determined from the measured rolling force using Orowan rolling theory.26) The Taylor factor of b.c.c. crystals is 2.75. On the basis of Lee’s elastic-plastic decomposition of a deforming material,22) the deformation gradient tensor F and velocity gradient tensor L of a crystal can be expressed using the following equations. F = R⋅U ................................. (11) L = D + W ............................... (12) Here, R, U, D and W are the rotation tensor, stretch tensor, strain rate tensor, and total spin tensor. D and W are calculated using the Orowan calculation model.11,26) As a result, the lattice rotation spin tensor Ω is indicated by Eqs. (13) and (14).

Ω = W – W P (W = 0) Ω = W – WP

at center............ (13) at surface .................. (14)

The rotation tensor R is obtained using Eqs. (15) and (16). © 2012 ISIJ

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Fig. 6.

Fig. 7.

Texture prediction accuracy of ultra low-carbon cold strip and annealed strip.

Texture prediction accuracy of ferritic stainless-steel hot strip and annealed strip.

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Fig. 8.

Texture prediction accuracy of ferritic stainless-steel cold strip and annealed strip by continuous analysis.

the Taylor model, the orientation of the recrystallization grains is the same as that of the rolled grains. In other words, this result proves that the nucleation preferred grain model is correct. However, why is the rolled texture classified by the total number of microscopic slips using the Taylor model the same as the observed recrystallization texture of ultra low-carbon steel and ferritic stainless-steel? Generally speaking, recrystallized grains originate at grain boundaries or in the deformed bands of grains rather than at the triple junctions of grains.29,30) Thus, according to the Taylor model, this analysis probably corresponds to the condition of grain boundaries or the deformed bands in the grains. The orientation of ultra low -carbon steel cold rolled at reduction ratio of 80% or of ferritic stainless-steel hot rolled at reduction ratio of 85% in the nonrecrystallized region is considered to be almost stable. Next we briefly explain the stability of the crystal orientation. If the slip system is {h,k,l}, then the stable crystal orientation is {h,k,l} or {u,v,w} because the Schmid factor of these orientations is zero. However, {u,v,w} satisfies the Taylor rotation rules. The major slip systems of a b.c.c. crystal are {1,1,0} and {1,1,2}. Thus, upon replacing the slip plane with the slip direction, {1,1,1} and {1,1,1} become stable crystal orientations, which are known as the alpha fiber and gamma fiber orientations, respectively. Two recrystallization mechanisms of ultra low carbon steel have been proposed. In mechanism A, the gamma fiber grains with dislocations undergo self-recrystallization. In mechanism B, the gamma fiber grains with fewer dislocations make encroach on alpha fiber grains, which have more dislo-

recrystallization textures of ultra low-carbon strip and ferritic stainless-steel strip, we determined critical values of 10 percents total number of microscopic slips. Figure 6 shows the case when we selected crystals with the highest total amount of microscopic slips as those having the preferential orientation. We can predict the recrystallization texture of cold rolled and annealed ultra low-carbon steel strip precisely because this analysis simulated the gamma fiber structure encroach the alpha fiber structure. Figure 7 shows the case when we selected crystals with the lowest total amount of microscopic slips as those having preferential orientation. We can predict the recrystallization texture of hot rolled and annealed ferritic stainless-steel strips precisely because this analysis simulated the existing bow out into the grains which have the higher stored energy, otherwise nuclei develop from subgrains close to that grain boundaries. Next, as the predicted orientations of hot rolled and annealed ferritic stainless-steel strip in Fig. 7 assumed to be initial orientations, we calculated cold rolling texture and selected grains in order of 10 percents with high number of microscopic slips. Figure 8 shows the accuracy of predicting the texture of cold rolled and annealed ferritic stainless-steel strip. We believe that it is possible to analyze the continuous change in the texture from hot rolled strips to cold rolled and annealed strips by considering both hot rolling and cold rolling conditions. 3.4. Discussion According to the recrystallization textures of ultra lowcarbon steel and ferritic stainless- steel whose orientations were classified by total number of microscopic slips using © 2012 ISIJ

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gamma fiber grains.18) Moreover, dislocations build up at grain boundaries. In other words, subgrains are formed at grain boundaries and gamma fiber grains initially undergo self-recrystallization. Next, alpha fiber grains that are elongated in the rolling direction and located next to gamma

cations. The final texture of cold rolled and annealed texture of ultra low-carbon steel is determined by mechanism B. As gamma fiber grains have a less stable orientation than alpha fiber grains, gamma fiber grains rotate more than alpha fiber grains. Thus, a twisted area appears between the

Fig. 9.

Fig. 10.

Analyzed cold rolled and annealed texture and Lankford value of ultra low-carbon strip.

Analyzed final texture of ferritic stainless-steel if we idealized recrystallization texture of hot annealing.

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the total number of microscopic slips using the Taylor model is also high. The reason for this is that ultra low-carbon steels and ferritic stainless-steels recrystallize in accordance with the preference nucleation model. The hot rolled and annealed texture is classified by the lowest total number of microscopic slips because nuclei develop from subgrains close to the grain boundaries. In contrast, the cold rolled and annealed texture satisfies the highest total number of microscopic slips because gamma fiber grains encroach on alpha fiber grains in the recrystallization period of the latter. In the prediction of recrystallization texture, the total number of microscopic slips using the Taylor model has the same meaning as the conventional Taylor factors. However, our method of predicting the recrystallization texture is very simple and practical to use and accuracy. This will be useful tool to determine production condition from hot rolling to cold rolling and annealing. Determining critical total amount of microscopic slips of recrystallization is a problem to predict recrystallization texture more accurately.

fiber grains are encroached upon by the gamma fiber grains. For the above-mentioned reasons, a texture whose orientations classified by the total number of slips with the highest order using the Taylor model become a final recrystallization texture of cold rolled and annealed ultra low-carbon steel strip. In contrast, as hot rolled and annealed ferritic stainless-steel strip recrystallizes by mechanism A, the orientation classified by the total number of slips with the lowest order using the Taylor model is the same as that in the observed recrystallization texture. We guess that as dislocations of hot strip are unstable and these dislocations are digested at bulging boundaries of grains with few dislocations obeying the mechanism A. However, we guess that dislocations of cold strip are stable, entangled each another. Thus, in those dislocations, recrystallized grains with high angle boundary appear and eat dislocations of unrecrystallized grains obeying the mechanism B. Mechanism A has been argued to contradict to mechanism B by many researchers, but we consider the both mechanisms are valid. Thus, mechanism A is should be referred to as recrystallization and mechanism B should be referred to as grain growth, which means that coalescence and shrinkage occurs between recrystallized grains and nonrecrystallized grains.

Acknowledgments The authors are deeply grateful to Dr. Osamu Akisue for his suggestions.

4. Application of Texture Prediction Method to Steel Production Process

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Using both this texture prediction method and the Lankford prediction method,32) we analyze the relationship between the cold reduction ratio and Lankford values for cold rolled and annealed ultra low-carbon steel. Figure 9 shows that the higher cold reduction ratio became, the greater Lankford value became. But the predicted Lankford value was less than observed Lankford value because every grain covers a same area on our texture prediction method. However, in the actual ultra low-carbon cold rolled and annealed strip grains with {1,1,1} orientations are larger than ones with other orientations. Figure 10 shows the texture of cold rolled and annealed of ferritic stainless-steel if we idealized the hot rolled and annealed texture. Metallurgy of producing ferritic stainlesssteel with good drawability is that residual strains of hot rolled strip make recrystallization at the hot rolled annealing process. Thus, a ferritic stainless-steel hot annealed strip with random texture will be best. If a ferritic stainless-steel strip had random texture after hot rolling and annealing, a final cold rolled and annealed strip would have more superior properties than a conventional cold rolled and annealed strip. From Figs. 9 and 10, we can clearly understand relationship between texture and its mechanical properties. Our texture prediction method is very practical for use to determine conditions of steel production process. 5. Conclusions The prediction accuracy for the textures of cold rolled ultra low-carbon strip and hot rolled ferritic stainless-steel strip using the Taylor model is high. Moreover, the prediction accuracy of the recrystallization texture classified by

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