Finite element simulation of strain induced austenite–martensite transformation and fine grain production in stainless steel

Computational Materials Science 40 (2007) 90–100 www.elsevier.com/locate/commatsci

Finite element simulation of strain induced austenite–martensite transformation and fine grain production in stainless steel N. Bontcheva a

a,*

, P. Petrov b, G. Petzov b, L. Parashkevova

a

Institute of Mechanics, Bulgarian Academy of Sciences, Department of Solid Mechanics, Acad. G. Bontchev St., Bl. 4, BG-1113 Sofia, Bulgaria b Technical University Varna, 1 Studentska St., BG-9010 Varna, Bulgaria Received 20 March 2006; received in revised form 8 November 2006; accepted 9 November 2006

Abstract Grain refinement due to phase transformation is an effective method for improving the mechanical properties of steel. An approach is proposed in the present work based on the FEM, for numerical simulation of the microstructure evolution as a result of hot rolling and subsequent cold torsion, with evaluating of the aggregate flow stress, depending on the austenite and martensite volume fractions changes developing during strain induced phase transformation. Grain refinement in 304 stainless steel at four different technological schedules is considered. Results of numerical simulation are compared with experimental data. Coupling of the thermoplastic deformation with microstructure evolution is realized.  2006 Elsevier B.V. All rights reserved. PACS: 62.20.Fe; 81.05.Bx; 81.20.Hy; 81.30.Kf; 81.40.Ef Keywords: Grain refinement; FEM; Recrystallization; Strain-induced phase transformation

1. Introduction Grain refinement due to phase transformation is an effective method for improving the mechanical properties of steel. The strain-induced austenite–martensite transformation in stainless steel with metastable austenite leads to forming of new smaller grains, enriched with dislocations [1]. Such a microstructure appears at a temperature, rather lower than that of hot forging. Martensite, as well as dislocations are unstable at thermal changes and reverse diffusionally into finer-grained austenite. The entire cycle of austenite–martensite–austenite transformation provides considerable increase of the plasticity of the austenitic steel

*

Corresponding author. Tel.: +359 2 979 6437; fax: +359 2 870 7498. E-mail address: [email protected] (N. Bontcheva).

0927-0256/$ - see front matter  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2006.11.003

type 304 at hot forming at strain rates in the range of 102– 104 s1 and temperature 850 C [2]. The grain refinement can be achieved by preliminary controlled rolling of the steel. Thermo-mechanical parameters contributing to a favorable microstructure can be predicted by finite element simulations [3]. Further development of metastable microstructure of stainless steel may be provoked by severe plastic deformation. Such intensive straining can be realized by axial torsion. The aim of the present work is to propose an approach based on the FEM, for numerical simulation of the microstructure evolution as a result of hot rolling and subsequent cold torsion, with evaluating of the aggregate flow stress, depending on the austenite and martensite volume fractions changes developing during strain induced phase transformation.

N. Bontcheva et al. / Computational Materials Science 40 (2007) 90–100

91

2.2. Modeling of the thermo-mechanical behaviour of the material

2. Simulation of the rolling process 2.1. Modeling of the thermo-mechanical process

e_ ij ¼

3 e_ sij ; 2 rd

ð1Þ

where e_ ij is the strain-rate tensor, sij is the stress deviator of the Cauchy stress tensor rij, e_ is the effective strain rate and rd is the strain resistance. The large plastic strains and intensive friction lead to significant heat generation and the process is characterized with considerable temperature changes, depending on the technological parameters. Material properties are temperature sensitive. This leads to the necessity of considering a coupled thermo-plastic process. The temperature T inside the deformed body is determined by the equation 1 Q_ T T_ ¼ kT T ;ii þ ; qcT qcT

Q_ T ¼ k T rde_ ;

ð2Þ

where kT denotes conductivity coefficient, cT is specific heat supply, q is material density, and kT is the Taylor–Qinney coefficient, determining the fraction of the strain-rate energy, which turns into heat. The fraction of the remainder of the strain-rate energy is expected to cause changes in dislocation density, grain boundaries and precipitation of second phase. Coulomb friction law at contact surfaces is used. The initial temperature is prescribed. Convection heat exchange between the body and the tools and between the body and the surrounding area is considered. The examined process takes place under conditions of rather small velocities, hence dynamic effects are neglected and the process is considered as quasi-static. Temperature changes in the metal, due to cooling during interpass times, are also calculated in time. The angular velocities of the rolls are prescribed.

The initial material subjected to hot deformation is austenitic stainless steel. Its chemical composition defined by spectral analysis is shown in Table 1. The mechanical properties of the material are time-temperature dependent. The material is strain hardening, strain rate sensitive and temperature softening. The strain resistance is given by the relation [4–6]:    e rd ¼ nt r02e_ nr 0 þ Dre exp  ex11e_ nr 1     TT   b e  r02e_ nr 0  rs2e_ nr s 1  exp  ; ð3Þ n 2 ex21e_ r where



r01 rs1 e_ 1 e_ 1 ln ; ns ¼ ln ln ; r02 rs2 e_ 2 e_ 2



ex11 ex21 e_ 1 e_ 1 n1 ¼ ln ln ; n2 ¼ ln ln ; ex12 ex22 e_ 2 e_ 2   T nt ¼ exp nts 1  ; Tb     T1 T1 ln r22 1 ; nts ¼ ln r11  Tb Tb  e_ r ¼ e_ e_ b : n0 ¼ ln

ð4Þ

where ex11, ex21, ex22, ex12, r01, r02, rs1, rs2, Dr, Tb and e_ b are material constants experimentally obtained in [6] and given in Table 2. The behaviour of the material, according to Eq. (3) is seen in Fig. 1. The material constants, appearing in Eq. (2) are given in Table 3.

400 350 300 T=900 °C 250

σd, MPa

As plastic strains during hot metal forming processes are much larger than the elastic ones, the rigid-plastic model is applied. The von Mises yield condition and the associated flow rule are applied, leading to the relation

strain rate 5 1/s strain rate 10 1/s strain rate 50 1/s strain rate 100 1/s

200 150

Table 1 Chemical composition of 304 stainless steel

T=1200 °C

100

%C

%Si

%Mn

%P

%S

%Cr

%Mo

%Ni

0.049

0.757

1.71

0.023

0.018

18.55

0.349

%Co

%Cu

%Nb

%Ti

%V

%W

%N

%Fe

0.088

0.404

0.01

0.032

0.064

0.084

69.51

8.29

%Al

50

0.0017 0

0.029

0

0.4

0.8

εp

1.2

1.6

2

Fig. 1. Material behaviour.

Table 2 Experimentally obtained parameters Dr (MPa)

Tb (C)

r01 (MPa)

r02 (MPa)

rs1 (MPa)

rs2 (MPa)

ex11

ex21

ex12

ex22 (s1)

e_ b

236

1100

93.9

80.8

152.1

128.8

0.264

0.260

0.261

0.251

1

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N. Bontcheva et al. / Computational Materials Science 40 (2007) 90–100

  Q Dst ¼ a2em3 Dm0 4 exp  ds ; RT

Table 3 Material constants for the thermal process 2

Mass density, q (kg/m )

Conductivity, kT (W/m/K)

Specific heat, cT (kJ/kg/K)

kT

7800

23

0.574

0.9

ð9Þ

where a2, m3, m4 and Qds are material constants and D0 is initial grain size. The average grain size in the neighbourhood of a material point is [9–13]:

2.3. Modeling of the microstructure development

D ¼ Rst Dst þ ð1  Rst Þ D0 :

The austenite grain size changes simultaneously during the deformation and recrystallization processes. We consider the values of critical reduction and holding time as factors influencing the grain size evolution [7]. Static (st), dynamic (d) and metadynamic (md) recrystallizations take place during the rolling process. The recrystallized fraction Rj (j = st, d, md) depends on temperature, effective strain and effective strain rate and is time dependent:   Rj ¼ Rj T ; e; e_ ; t ; j ¼ st; d; md: ð5Þ

As soon as the limit Rst = Rlimit = 0.99 is reached, grain growth takes place according to the relation [12]    gr m5  m Qsg Dst ¼ Dprev 5 þ a3 tst exp  ; ð11Þ RT

Grain size D is changing according to the recrystallization factors and to the currently reached grain size:   DD ¼ DD D; T ; e; e_ ; t : ð6Þ Static recrystallization takes place in volumes where the critical strain ec = bep is not yet reached; ep is the peak strain in the flow curve at the current strain rate and temperature (Fig. 2); b is a coefficient, usually taking values between 0.8 and 0.95 [8]. The recrystallized fraction at time t in the case of static recrystallization is given with [9–11] " !# trecst Rst ðtÞ ¼ 1  exp 0:693 st ; ð7Þ t0;5 where tst0;5 is the time for 50% recrystallization, defined with   Qs tst0;5 ¼ a1em1 Dm0 2 exp : ð8Þ RT where D0 is the initial grain size, a1, m1, m2 and Qs are material constants, R is the gas constant, and T is the current Kelvin temperature. trecst is the time elapsed since the static recrystallization started at the point under consideration. The following relations describe the grain size development during static recrystallization [9,10,12]:

4=3

2

ð10Þ

where Dprev is the value of Dst at the time tstgrowth when Rst = Rlimit was reached. The time tst is measured from that moment, i.e. tst ¼ t  tstgrowth ;

ð12Þ

where t is the real processing time. a3, m5 and Qsg are material constants. Dynamic recrystallization starts as soon as the critical strain ec is reached. In the case of hot rolling process and deformation degree less than 20% it is possible to ignore dynamic recrystallization [14]. But in the case of large deformation, dynamic recrystallization should be taken into account. The recrystallized fraction is given with an expression, similar to (7) [6]: "   # k tdyn d Rd ðtÞ ¼ 1  exp A ; tf A ¼  ln ð1  Rf Þ; tf ¼ ts  tidyn ; kd ¼

log ðln ð1  Ri Þ= ln ð1  Rf ÞÞ ; t log idyn tf

Ri ¼ 0:05; Rf ¼ 0:95: ð13Þ

where ts is the time when steady state is reached (Fig. 2). We assume that es ð14Þ ts ¼ ; e_ where e_ is the current effective strain rate at the point under consideration. Grain size evolution is defined with: Dd ¼ a4 Z m6 ;

σd

where a4 and m6 are material constants,   Qd Z ¼ e_ exp RT

σs

εs ε

εp Fig. 2. Peak strain.

ð15Þ

ð16Þ

is the Zener–Hollomon parameter and Qd is the activation energy. As soon as the limit Rd = Rlimit is reached, metadynamic recrystallization starts. Metadynamic recrystallization takes place during technological pause (interpass time), provided that static recrystallization is completed, or during a deformation process if the dynamic recrystallization fraction Rd (13)

N. Bontcheva et al. / Computational Materials Science 40 (2007) 90–100

reaches 99%. At the beginning of a deformation pass, following an interpass time, metadynamic recrystallization could continue at certain points if the strain rate there is still too low or even zero. This depends usually on the geometry of the applied tools. Dynamic recrystallization at such points will start after reaching the value of the Zener–Hollomon parameter corresponding to the end of the previous dynamic recrystallization. Metadynamic recrystallization fraction is described with [10,11,15,16] 2 !k m 3 trecmd 5; Rmd ðtÞ ¼ 1  exp 40:693 md ð17Þ t0;5 where km is material constant, trecmd is measured from the moment tmd when metadynamic recrystallization started at the point under consideration, i.e. trecmd ¼ t  tmd

ð18Þ

and the time tmd 0;5 for 50% recrystallization is given with [10,11,15]:   Qmd m7 ¼ a Z exp tmd ; ð19Þ 5 0;5 RT where a5, m7 and Qmd are material constants. Grain size during metadynamic recrystallization is given with [10]: Dmd ¼ a6 Z mmd8 ;

ð20Þ

where a6 and m8 are material constants and Zmd is given with:   Qmd1 _ Z md ¼ e exp : ð21Þ RT

93

If metadynamic recrystallization takes place during interpass time, where in fact e_  0, the value of e_ in (19) and (21) is equal to the one reached when metadynamic recrystallization started. The average grain size in the neighbourhood of the point under consideration is given with [10,11,15]: 4=3

2

D ¼ Rmd Dmd þ ð1  Rmd Þ Dprev ;

ð22Þ

where Dprev is the grain size at the beginning of metadynamic recrystallization. As soon as the limit Rmd = Rlimit is reached. metadynamic grain growth starts, according to the relation [10,11,16]:    gr m9 Qmg m9  Dmd ¼ Dmd þ a7 tmd exp  ; ð23Þ RT where m9, a7 and Qmg are material constants and tmd is the time elapsed since metadynamic grain growth started (since tmdgrowth): tmd ¼ t  tmdgrowth :

ð24Þ

All material constants, concerning the microstructure model are given in Tables 4–6. The strains es and ep (Fig. 2) depend on temperature and strain rate. Their values were experimentally obtained in [6] for different values of T and e_ . An approximation is made in the form     ep T ; e_ ¼ A1 þ A2 T þ A3 T 2 þ e_ A4 þ A5 T þ A6 T 2 ð25Þ     es T ; e_ ¼ B1 þ B2 T þ B3 T 2 þ e_ B4 þ B5 T þ B6 T 2 using the experimental data. The values of the coefficients Ai and Bi, i = 1, 2, . . ., 6 are given in Table 7. It can be seen from the relations above that recrystallization depends on effective strain, effective strain rate and

Table 4 Material constants ai i

1

2

3 tst

ai

15

2.3 · 10

4 tst

<1s 7

4.0 · 10

343

5

6

7 tmd < 1 s

>1s

1.5 · 10

27

2000

4

2.6 · 10

1.1

7

1.2 · 10

tmd < 1 s 8.2 · 1025

Table 5 Material constants mi i

mi

1

2

2.5

3

0.5

2

4

0.4

5

6

tst < 1 s

tst > 1 s

2

7

0.15

7

0.8

8

0.23

9 tmd < 1 s

tmd < 1 s

2

7

Table 6 Material constants R (J/mol/K)

Qs, Qmd (kJ/mol)

Qds (kJ/mol)

8.314

230

45

Qsg (kJ/mol)

Qd (kJ/mol)

tst < 1 s

tst > 1 s

113

400

405

Qmd1 (kJ/mol) 300

Qmg (kJ/mol) tmd < 1 s

tmd < 1 s

113

400

94

N. Bontcheva et al. / Computational Materials Science 40 (2007) 90–100

Table 7 Coefficients of the functions ep and es i

1

2

3

4

5

6

Ai Bi

0.683 14.415

9.157E04 2.049E02

1.00E06 8.070E06

3.368E03 0.133

3.157E06 2.482E04

1.493E14 1.140E07

no

ε < bε p

In pause?

yes

calculate Rst Rst < 0. 99

no

static

dynamic

no

calculate D = Dd

yes

calculate D = Dst

Fig. 3 shows the flow chart of the numerical process that was implemented in the FE-program MARC for numerical simulation of the microstructure evolution as a result of hot rolling. At any time the effective strain e and the peak strain ep are different at different material points. This means that the numerical calculation will develop along different branches of the flow chart. Thus the coupling of the microstructure and the thermo-mechanical models is ensured and appearance of different microstructure zones in the workpiece, as well as their development in time is possible.

yes

calculate Rd yes

Rd < 0. 99 no

calculate D = Dstgr

calculate Rmd gr calculate D = Dmd

Return

metadynamic

Fig. 3. Flow chart of the calculation of the recrystallization process.

temperature. Microstructure evolution depends on the kind of recrystallization that occurs in the deformable body during deformation. Hence the thermo-mechanical and the microstructure models must be coupled for obtaining realistic microstructure development in the body during hot deformation.

2.4. Numerical simulation of the rolling process The initial body subjected to hot deformation is austenitic stainless rod. Shape rolls with three different gauges are used for the hot rolling process. Two types of gauges were considered. Their geometry and dimensions are given in Fig. 4. Rods with diameter 20 mm are used for gauges type 1 and rods with 23 mm diameter for gauges type 2. After the first and the second passes the workpiece is rotated at

Gauge 1

Gauge 2

14.3

16

Gauge 3

5.2

R3

11.4

5.6 11.7

10.5 5 13.47

8.5

24 Gauge 1

Gauge 2

Gauge 3

15.5

18

15

5.2 15.4

14

3

14

25.2

3

9

10

Fig. 4. Geometry of the gauges; (a) type 1; (b) type 2. Table 8 Schedule of the rolling process Variant

Initial temperature (C)

Pause 1

Pause 2

Pause 3

Duration (s)

T (C)

Duration (s)

T (C)

Duration (s)

T (C)

1 2 3 4

1150 1000 1000 1000

20 20 5 5

1050 950 950 20

40 40 5 10

1000 900 900 900

20 20 20 10

20 20 20 20

N. Bontcheva et al. / Computational Materials Science 40 (2007) 90–100

95

and 3 (gauges type 1) and 100 rot/s for variant 4 (gauges type 2). The finite element code MARC is used for the numerical simulation. Due to the axial symmetry the quarter of the cross-section is considered. Generalized plain strain quadrilateral finite elements are applied. Tools are assumed rigid. The initial FE mesh is shown in Fig. 5. The microstructure model is implemented in the FE code by means of user subroutines. 2.5. Results

Fig. 5. Initial FE mesh for the rolling process.

90. After the third pass it remains at room temperature. Four variants of the rolling process are simulated. The schedule of the forming processes is given in Table 8. The angular velocity of the rolls is 25 rot/s for variants 1, 2

Fig. 6 shows grain size development during the rolling process for all four variants. It is seen that during the action of the gauges grains are diminishing. During interpass times metadynamic recrystallization takes place and grains are growing. Grain size distribution in the cross-section at the end of the rolling process is shown in Fig. 7. In the case of gauges type 1 (variants 1, 2 and 3) the grain size distribution in the final product is approximately homogeneous, while in the case of gauges type 2 (variant 4) the difference in the maximal and the minimal grain size in the cross-section is rather large and the grain size distribution is nonhomogeneous. This is due to the nonhomogeneous and insufficient straining of the metal in the case of variant

180

180 pause 1

160

pause 2

pause 3

140 grain size, μm

120

pause 2

160

Variant 1 core surface

140 grain size, μm

pause 1

pause 3

100 80

100 80

60

60

40

40

20

20

0

Variant 2 core surface

120

0 0

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 time, s

0

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 time, s

180 180

pause 1 pause 1

160

pause 2

pause 2

pause 3

160

pause 3

140 120

120 100

grain size, μm

grain size, μm

140

Variant 3 core surface

80

80

60

60

40

40

20

20

0 0

5

10

15 time, s

20

25

30

Variant 4 core surface

100

0

0

5

10

15 time, s

20

25

Fig. 6. Grain size development during the rolling process; (a) variant 1; (b) variant 2; (c) variant 3; and (d) variant 4.

30

96

N. Bontcheva et al. / Computational Materials Science 40 (2007) 90–100

Fig. 7. Grain size distribution in lm; (a) variant 1; (b) variant 2; (c) variant 3; and (d) variant 4.

Table 9 Grain size at the end of the rolling process Variant

Grain size Numerical simulation (lm)

1 2 3 4

Experiment (lm)

Maximum

Minimum

Average

38.6 26.9 26.6 27.1

39.4 27.5 27.2 50.7

39.09 27.26 26.98 39.17

– 25.9 – 28.50

4. The calculated average grain size in the cross-section for all four variants is given in Table 9. Experiments, corresponding to the regime of variants 2 and 4 were performed [2]. Table 9 shows that calculated results are in good agreement with the experimental ones. The unsufficient coincidence in variant 4 can be explained with the inhomogeneous grain size distribution in the cross-section. Experimental results depend in fact on the point at which measurements were taken. Fig. 8 shows distribution of recrystallization zones in the cross-section during the rolling process. It is notable that different zones are available

– static recrystallization, static grain growth, dynamic recrystallization, metadynamic recrystallization and metadynamic grain growth. Zones are developing and changing during the rolling process. 3. Simulation of the torsion process 3.1. Modeling of the mechanical process Cylinders, cut out from the rolled workpiece are subjected to torsion with prescribed angular velocity. Sticking between the cylinder and the tools is assumed. 3.2. Modeling of the material Torsion takes place at room temperature. Heat generation due to plastic strains does not increase the temperature considerably. That is why no thermal influence is taken into account. The material is considered elastic–plastic, as elastic strains during cold deformation are significant. Tools are assumed rigid.

N. Bontcheva et al. / Computational Materials Science 40 (2007) 90–100

97

Fig. 8. Distribution of the recrystallization zones in the cross-section. 0–1 static recrystallization; 1–2 static grain growth; 2–3 dynamic recrystallization; 3–4 metadynamic recrystallization; 4–5 metadynamic grain growth; (a) pass 1; (b) pass 2.

3.3. Modeling of the austenite–martensite transformation

which corresponds to the austenite state, and f ð nM Þ > 1

if nM > 0

ð32Þ

According to [17], we consider two volume fractions: austenite nA and martensite nM, which develop under the restriction

during austenite–martensite transformation. We assume further the following expression for f(nM):

nA þ nM ¼ 1:

f ðnM Þ ¼ 1 þ C M nMM ;

ð26Þ

Q

ð33Þ

The kinetics of austenitic-martensite transformation is considered. The volume of strain-induced transformation is a function of equivalent plastic strain ep and temperature T: " #B )1 (  1 ep  D nM ¼ 1 þ ¼ 1 þ C C ðep ÞB ; ð27Þ C exp  T

where CM and QM are material constants. Relation (33) obviously obeys conditions (31) and (32).

where

Dt0 ffi; DA ¼ k A pffiffiffiffiffiffiffiffiffiffiffiffi 1 þ c2



 B D C C ¼ C exp  : T

ð28Þ

The values of CC and B are to be obtained on the base of experiments. The strain hardening of stainless steel in austenite state is given with [17]:  m e_ p n  rA ¼ K ð e Þ ; K ¼ K ; ð29Þ 0 d e_ 0 where K0, m and n are material constants and e_ 0 is reference strain rate. We assume that the strain resistance of the stainless steel rd during austenite–martensite transformation depends on the strain resistance of the steel in austenite state rA d and on the martensite volume fraction nM: rd ¼ rA d f ðnM Þ:

ð30Þ

The function f(nM) leads to a correction to the initial strain resistance rA d so, that f ðnM Þ ¼ 1 if nM ¼ 0;

ð31Þ

3.4. Microstructure model According to [18] the size of austenite grains is given with ð34Þ

where Dt0 is the initial grain size at the beginning of torsion, c is the shear strain and kA is experimentally obtained parameter. The initial grain size Dt0 is taken from the results at the end of the simulation of the rolling process – the average grain size in the cross-section at the end of rolling. The size of martensite grains is DM ¼

DA : 2:5

ð35Þ

The average grain size during torsion is then given with: D ¼ DA ð1  nM Þ þ DM nM :

ð36Þ

3.5. Material constants for the material under consideration The initial grain size Dt0 for the torsion process is taken from Table 9. The values of CC and B, obtained on the base of experiments [2] are CC = 0.5579, B = 2.11782. The approximation, according to relation (27) is shown in Fig. 9. Experimental data, given in Table 10 are taken from

98

N. Bontcheva et al. / Computational Materials Science 40 (2007) 90–100 1

1.35

700

1.3

0.8

600

1.25

500 σd, MPa

0.6

0.4

ξM = 1/(1+(CC/εB))) CC = 0.55779 B = 2.11782

1.2 400

experiments 1.15

σd

300

σdA 200

1.1

f(ξM)

0.2

1.05

100 0

0 0

0.5

1

1.5

2

2.5

3

f(ξM)

Martensite volume fraction

T=20°

800

0

0.2

0.4

0.6

0.8

1 1

strain

Plastic strain

Fig. 10. Influence of martensite and austenite on strain resistance.

Fig. 9. Martensite content vs. plastic strain at 20 C.

Table 10 Experimental data for matrenisite content Plastic strain Martensite fraction

0.34 0.15

0.42 0.23

0.49 0.30

0.56 0.34

[2] and are denoted by triangles in the figure. The value of kA in Eq. (34) is taken 0.8. The values of CM and QM, appearing in (33) are obtained from experimental data given in Table 11 [2] and have the values CM = 0.3355, QM = 0.027845. The values of the constants describing strain hardening of the austenite steel are [17]: _ K0 = 566.75 MPa, n = 0.18, m = 0.013 and e0 ¼ 0:0005. Young’s module is E = 210,000 MPa and Poisson ratio is m = 0.4. Fig. 10 shows the influence of martensite on strain resistance. 3.6. Numerical simulation of the torsion process Torsion with angular velocity 0.005 s1 is realized. The cylinders of 9 mm diameter and 45 mm length are subjected to torsion at 720. The initial FE mesh for the torsion process is shown in Fig. 11. Three dimensional, isoparametric, 4 + 1-node, low-order, tetrahedral finite elements are applied with additional pressure degree of freedom at each of the four corner nodes. Due to the severe element distortion during the process, autoremeshing procedure is applied. Large strains formulation is considered.

Fig. 11. Initial FE mesh for the torsion process.

3.7. Results The results obtained by the finite element simulation are presented in Table 12. Comparison of the results corresponding to variant 2 with the experimental results shows good agreement. The table shows that the duration of interpass times has no significant influence on the final grain size (variant 2 and variant 3). The influence of the initial

Table 11 Experimental data for strain resistance Plastic strain Strain resistance

0.01 191

0.02 425

0.03 436

0.06 475

0.07 482

0.08 490

0.09 498

0.10 501

0.12 517

Plastic strain Strain resistance

0.15 540

0.20 574

0.25 593

0.30 620

0.35 643

0.41 662

0.46 677

0.51 693

0.56 704

Plastic strain Strain resistance

0.61 712

0.66 719

0.72 731

0.78 739

0.83 746

0.88 750

0.91 758

N. Bontcheva et al. / Computational Materials Science 40 (2007) 90–100 Table 12 Grain size at the end of the torsion process Variant

4. Conclusions

Grain size Numerical simulation (lm)

1 2 3 4

Experiment (lm)

Maximum

Minimum

Average

18.52 12.91 12.78 18.56

14.81 10.22 10.22 14.84

16.60 11.58 11.46 16.63

– 10.82 – 18.41

19 18 17 16

D, μm

variant 1 D0t=39.09 μm variant 2 D0t=27.26 μm

15

variant 3 D0t=26.98 μm

14

variant 4 D0t=39.17 μm

13 12 11 10 0

1

2

99

3

4

r, mm

Fig. 12. Grain size distribution in the cross-section at the end of torsion.

19 18

An approach is presented, based of the FE-system MARC for numerical simulation of the microstructure evolution of stainless 304 steel in hot rolling and consequent cold torsion with evaluating the aggregate austenite–martensite flow stress. The thermomechanical influence on microstructure formation during static, dynamic and metadynamic recrystallizations is considered. The microstructure evolution of 304 stainless steel in torsion is numerically simulated using physically based relations for strain-induced austenite–martensite transformation. The simulation determines the most appropriate regime for achieving adequate refining of microstructurte under given conditions. Numerical results are in good agreement with experimental observations. The numerical simulation shows that the grain size at the end of hot rolling and subsequent air-cooling depends mainly on the start- and less on of the end-rolling temperature. In cases of well-worked cross-section of the rods (variants 1–3) the higher temperature leads to larger grain size. The duration of interpass times during rolling has no significant influence on the grain size. As favorable thermomechanical schedule for refinement of microstructure at the present computational investigations is that of hot rolling according to variant 2, followed by cold torsion. Such partially transformed to martensite structure after reversing again into austenite has decreased the mean size up to less than 8 lm, as was shown in [2]. The represented FE-simulation, determining the influence of the hot shape-rolling on the microstructure development of austenitic steel type 304 is a successful example for realistic choice of favorable thermomechanical conditions, reducing the expensive industrial tests.

17

D, μm

16

References

var. 1 D0t=39.09 μm

15

var. 2 D0t=27.26 μm

14

var. 3 D0t=26.98 μm

13

var. 4 D0t=39.17 μm

12 11 10 9 8 0.52

0.56

0.6

0.64

0.68

0.72

equivalent plastic strain

Fig. 13. Grain size vs. plastic strain in the cross-section at the end of torsion.

temperature is but significant (variant 1 and variant 2). Fig. 12 shows the grain size distribution in the cross-section at the end of the torsion process. Fig. 13 represents grain size distribution vs. plastic strain for the same case. It is seen that grain size follows the value of plastic strain, which is approximately linearly distributed along the radius.

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