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Microstructure-Based Multiscale Analysis of Hot Rolling of Duplex Stainless Steel Using Various Simulation Software

  • Sukeharu NomotoEmail author
  • Mototeru Oba
  • Kazuki Mori
  • Akinori Yamanaka
Open Access
Thematic Section: 2nd International Workshop on Software Solutions for ICME
Part of the following topical collections:
  1. 2nd International Workshop on Software Solutions for ICME

Abstract

We proposed a microstructure-based multiscale simulation framework using various commercial simulation software and applied it to analyze the hot rolling of duplex stainless steel. According to the Integrated Computational Material Engineering (ICME) concept, we established a procedure to bridge various simulation software from nano- to macroscopic length scales. Using our framework, first, microstructure evolutions by multiphase field (MPF) simulations coupled with the Calculation of Phase Diagrams (CALPHAD) database were performed. In our application, we simulated the columnar and equiaxed solidification during the continuous casting of duplex stainless steel. In the MPF simulations, the temperature field in the slab was calculated by heat conduction analysis using a finite element method (FEM). Then, the macroscopic elastic and plastic mechanical properties of the microstructure obtained by the MPF simulations were estimated by the virtual material test using a nonlinear FEM based on the homogenization method. Because the elastic constants of single δ-ferrite and γ phases in the microstructure are necessary for the virtual material test, they were calculated by molecular dynamics and first principle calculations. Furthermore, the plastic stress–strain properties of the single phases were estimated on the basis of the results of nano-indentation and uniaxial tensile tests. Subsequently, the hot rolling of the slab was simulated using an elastoplastic FEM with the mechanical properties obtained by the virtual material test. Finally, the static recrystallization in the rolled slab was again simulated by the MPF method.

Keywords

Integrated computational material engineering Multiphase field method First principles Molecular dynamics simulation Virtual material test Finite element method Solidification Hot rolling Duplex stainless steel 

Introduction

The multiphase field (MPF) method was first proposed approximately 20 years ago [1]. It enabled the prediction of microstructure evolution during recrystallization and phase transformations. Nowadays, the MPF method coupled with a Calculation of Phase Diagrams (CALPHAD) thermodynamic database, e.g., Thermo-Calc, is recognized as one of the most powerful tools in numerical simulations for engineering metal and alloy processes [2, 3, 4]. Thus, the MPF method coupled with the thermodynamic database is a crucial part of the Integrated Computational Materials Engineering (ICME) framework [5]. Several MPF software including commercial software and open sources have already been released. In particular, the Microstructure Evolution Simulation Software (MICRESS) is a versatile commercial MPF simulation software (http://web.micress.de/) and has been widely used to simulate microstructure evolution in practical multicomponent alloys and to design process conditions for obtaining the desired microstructure in alloys.

The successful simulation of microstructure evaluations using MICRESS has motivated the estimation of mechanical properties from the microstructure. By constructing a representative volume element (RVE) of a microstructure in the macroscopic mechanical structure on the basis of the MPF simulation results, the mechanical properties of the structure can be estimated by a virtual material test using the commercial finite element method (FEM) software ABAQUS and the HOMAT program, which is based on the mathematical homogenization theory [6, 7]. The estimated mechanical properties can be further used in the macroscale FEM analysis of the structure using ABAQUS.

Although we can estimate the mechanical properties of alloys, which depend on their underlying microstructures, by the virtual material test, the mechanical properties of each single phase in microstructures must be identified by other simulations and experiments. Nevertheless, recent advancements in computer performance have enabled performing first principles and molecular dynamic (MD) calculations with extremely large number of atoms [8, 9]. In fact, recent studies have reported that the elastic modulus of single phases in alloys, which depends on their chemical composition, can be estimated. However, the elastoplastic mechanical properties of alloys have not been estimated by nanoscale simulations with sufficient accuracy even using a fast computer [10, 11]. Experimental techniques such as the nano-indentation test should be effectively employed for elastoplastic mechanical properties that are difficult to be estimated by nanoscale simulations [12, 13].

Numerous studies have performed multiscale simulations in the field of alloys [14, 15]. However, most previous studies have performed multiscale simulations using the provided mechanical property value of the single phase. In this study, we propose a multiscale simulation framework using various commercial simulation software for nano-, micro-, and macroscopic length scales, in which the elastic constant of the single phase is obtained by a nanoscale simulation. The concept of our multiscale and software bridging simulation framework is illustrated in Fig. 1. In this study, as illustrated in Fig. 2, our simulation framework is applied to the multiscale simulation of hot rolling processes along with solidification during continuous casting of duplex stainless steel comprising two stable phases: δ-ferrite and γ phases (http://web.micress.de/ICMEg1/presentations_pdfs/Laschet_met_plast.pdf#search='HOMAT+MICRESS'.). The reliability of software bridging for multiscale analysis of the ICME concept was examined through this numerical simulation study and it is discussed herein.
Fig. 1

Schematic illustration of the multiscale and software bridging simulation proposed in this study (http://web.micress.de/, http://www.thermocalc.com/, http://www.engineering-eye.com/FINAS_TPS/, https://www.vasp.at/, http://www.3ds.com/simulia/, http://web.micress.de/ICMEg1/presentations_pdfs/Laschet_met_plast.pdf#search='HOMAT+MICRESS') [16] (Color figure online)

Fig. 2

Schematic illustration of continuous casting and hot rolling of duplex stainless steel simulated in this study (http://www.nisshin-steel.co.jp/saiyo/process/) (Color figure online)

Methodology

MPF Simulation of Solidification During Continuous Casting

The chemical composition of duplex stainless steel is assumed to be the same as that of SUS304 steel: Fe, 18 %; Cr, 8 %; and Ni, 0.08 % C (wt%). The dimensions of the slab used in the hot rolling experiment are shown in Fig. 3. The size of the small slab was obtained by communication with Nisshin Steel Corporation. The experimental observation of the solidified slab (Fig. 3) revealed that the microstructure in the surface region of the slab is columnar, while the microstructure in the inner region is equiaxed. Therefore, we performed MPF simulations of columnar and equiaxed solidifications in these two regions using MICRESS coupled with the Thermo-Calc software. We employed the TCFE7 database for the chemical-free energy of steels and the MOBFE3 database for the diffusion mobility of alloying elements (http://www.thermocalc.com/solutions/by-material/iron-based-and-steels/, http://www.thermocalc.com/media/6014/mobfe3_flyer_bh.pdf).
Fig. 3

Model of the slab used in continuous casting and hot rolling (Color figure online)

The temperature field in the slab during continuous casting must be provided for the MPF simulations. The initial temperature and cooling rate at the slab surface were defined as 1733 K and 25 K/s, respectively. These values were obtained by communication with Nisshin Steel Corporation. To obtain the temperature field in the slab, we performed thermal conduction analysis using FINAS/STAR TPS (http://www.engineering-eye.com/FINAS_TPS/), which is FEM-based heat treatment software. A snapshot of the calculated temperature field in the slab after cooling for 36 s is shown in Fig. 4. Variation in temperature at the center of the slab is shown in Fig. 5. According to this thermal conduction analysis, the temperature gradient in the surface region of the slab was approximated as 400 K/cm.
Fig. 4

Distribution of temperature in the slab after continuous cooling for 36 s (Color figure online)

Fig. 5

Variation in temperature at the slab center

The columnar solidification was simulated using the defined cooling rate, 25 K/s, and the obtained temperature gradient, 400 K/cm. The size of the computational domain was defined as 64 × 64 × 128 μm3. The domain was divided into 128 × 128 × 256 finite difference grids. The grid spacing was 0.5 μm. The initial δ-ferrite of the body-centered cubic (bcc) structure nucleus was placed in the four bottom corners, as shown in Fig. 6. The distance between the δ grains was approximately defined as the order of the primary dendrite arm spacing (http://lammps.sandia.gov).
Fig. 6

Schematic illustration of initial position of the δ-ferrite nucleus, cooling rate, and temperature gradient used for the MPF simulation of columnar solidification (Color figure online)

The dimensions of the computational domain were set to 64 × 64 × 64 μm3. The grid spacing was the same as that in the columnar solidification simulation. The initial δ-ferrite nucleus was randomly placed. The mean distance between the nucleus was assumed to be 30 μm.

After the formation of the δ-ferrite phase, the γ phase formed on the interface between the δ-ferrite phase and liquid phase. We assumed random nucleation of the γ phase below 1720 K according to the equilibrium calculation using Thermo-Calc, as shown in Fig. 7. In the MPF simulation using MICRESS, the time interval for the nucleation was set to 0.01 s, and the mean distance between the nuclei was set to 3 μm [3].
Fig. 7

Equilibrium phase fractions versus temperature (Color figure online)

The interfacial energies for each interface shown in Table 1 were referenced in this study, in which the microstructure evolution, including σ phase formation of a stainless steel system, was simulated using MICRESS [3]. The interfacial mobility values for the liquid interface were additionally referenced. On the other hand, the interfacial mobility values for the δ–γ interface were determined to be the temperature dependency, as shown in Table 2. The procedure for determining these values is explained as follows. Firstly, the interfacial mobility at 1733 K for maintaining the balance between interfacial movement and solute diffusion in the interface was found by gradually decreasing by one from the very large value, thereby resulting in numerical stability. Next, an approximately average activation energy was estimated by referencing Arrhenius plot data of Cr and Ni diffusivities, which were listed in the MICRESS output file. Then, the temperature dependency rate compared with the unit defined at 1733 K was exponentially calculated at each temperature shown in Table 2 by using the activation energy. Finally, the interfacial mobility at 1733 K was multiplied by the rate at each temperature.
Table 1

Interfacial energy and mobility

Interface

Energy

Mobility

J/m2

m4/J/s

δ–liquid

0.2

1 × 10−10

δ–γ

0.7

(Table 2)

Liquid–γ

0.3

1 × 10−10

Table 2

Interfacial mobility for the δ–γ interface as a function of temperature

Temperature/K

Interfacial energy/m4/J/s

1733.0

2.0 × 10−12

1700.0

1.51 × 10−12

1600.0

5.94 × 10−13

1500.0

2.07 × 10−14

1400.0

6.21 × 10−15

1300.0

3.05 × 10−15

MD and First Principles Calculations for Estimation of Elastic Constants

To estimate the elastic constants of the γ phase, the MD calculation using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) open-source MD software was used [16]. On the other hand, the first principles calculation using Vienna Ab initio Simulation Package (VASP) commercial software was used to estimate the elastic constants of the δ phase (https://www.vasp.at/) [17, 18]. This is because there was an insufficient number of accurate interatomic potential parameters of the δ phases in the Fe–Cr–Ni ternary and Fe–Cr–Ni–C quaternary alloys for MD calculation in the open data resources. However, VASP required more calculation time than LAMMPS.

We first calculated the elastic constants of a face-centered cubic (fcc) structure in an Fe–Cr–Ni ternary alloy at 0 K using LAMMPS with the embedded atom potential for an Fe–Ni–Cr alloy proposed by Bonny et al. [19]. To construct the initial atomic configuration, the lattice parameter was calculated through energy minimization with the conjugate gradient algorithm. The total number of Fe, Cr, and Ni atoms was 4000. These atoms were randomly distributed in the Fe–18 %Cr–8 %Ni (wt%) composition. A typical distribution of atoms is shown in Fig. 8. To obtain statistically meaningful results, the elastic constants were estimated for 11 different atom distribution cases because the calculated internal energy depended on the initial structure. The elastic constants were calculated by referencing the method proposed by Nishimatsu [20].
Fig. 8

One case of distribution of Fe, Cr, and Ni atoms in an fcc structure used for MD calculation (Color figure online)

Next, to estimate the elastic constants of the fcc structure in the Fe–Cr–Ni–C quaternary alloy at 0 K, we calculated the elastic constants of the fcc structure in the Fe–Cr–Ni alloy doped with carbon atoms. The carbon atoms were inserted into an fcc structure constructed for the above simulations so that the average concentration of carbon atoms was 0.08 wt %. To consider the inserted carbon atoms in the MD simulation, we employed the Lennard–Jones potential for metal and carbon atoms [21].

In the first principles calculation for the δ phase using VASP, the electron wave functions were described by the Blochl method in the implementation of Kresse and Joubert [22, 23]. The generalized gradient approximated functional of Perdew, Burke, and Ernzerhof was used for treating the exchange and correlation for electrons [24]. Plane waves were included up to a cutoff energy level of 300 eV. For integration within the Brillouin zone, specific k points were selected using 4 × 4 × 4 Monkhorst–Pack grids for a bcc structure. The Kosugi algorithm was applied in this calculation. Convergence of the calculation was investigated by the decreasing energy difference of less than 10−6 eV in repeating the self-consistent loop. The lattice parameters were obtained from the corresponding energy minimization at constant volumes and by fitting a Birch–Murnaghan equation of state to the resulting energy–volume data [25, 26].

The total number of atoms for the first principles calculation was set to 54 because the calculation was much more time consuming than the MD simulation. In this calculation, carbon was not considered on account of its composition of 0.08 wt %, which was too small for the 54-atom model. An example of the initial distribution of atoms is shown in Fig. 9. Calculations of 11 different atom distribution cases were additionally performed. The elastic constant value was obtained by taking the average of 11 cases.
Fig. 9

One case of distribution of Fe, Cr, and Ni atoms with a bcc structure used for VASP calculation (Color figure online)

Nano-Indentation Test for Estimation of Stress–Strain Curve for Single Phases

The nano-indentation test was used for estimating the stress–strain curve of a single δ-ferrite phase. In the nano-indentation test, a Berkovich triangular pyramid indenter was used. The maximum load was set to 100 mN, and the displacement velocity was 13.324 m/s. In this study, a reverse analysis technique proposed by Dao et al. [27] was used for obtaining the stress–strain curves. They proposed the approximate function between the force–depth curve obtained by the nano-indentation test and the parameters on the basis of their FEM analysis. They assumed the following simple power law:

$$ \sigma =\left\{\begin{array}{cc} E\varepsilon & \sigma \le {\sigma}_{\mathrm{y}}\\ {} R{\varepsilon}^n& \sigma \ge {\sigma}_{\mathrm{y}}\end{array}\right.,\kern0.5em {\sigma}_{\mathrm{y}}= E{\varepsilon}_{\mathrm{y}}= R{\varepsilon}_{\mathrm{y}}^n $$
(1)

where E is Young’s modulus, ε is the strain, σ denotes the stress, σ y represents the yield stress, and ε y is the corresponding yield strain, which is defined in Eq. (1). In addition, R is the strength index and n is the strain hardening exponent. All parameters in Eq. (1) can be calculated by the reverse analysis technique. Nonetheless, Young’s modulus E was obtained from the nanosimulation result. The representative of the uniaxial strain–stress relationship of anisotropic elastic constants C 11 was used as Young’s modulus E.

On the other hand, the estimation of the stress–strain curve for a single γ phase from the nano-indentation test was very complicated owing to the deformation-induced martensitic transformation in the preparation of the sample and the nano-indentation. Therefore, we calculated the flow stress of a single γ phase by assuming the following linear function:

$$ \sigma ={f}_{\upgamma}\kern0.1em {\sigma}_{\upgamma}+{f}_{\updelta}\kern0.1em {\sigma}_{\updelta},\kern0.75em {f}_{\upgamma}+{f}_{\updelta}=1\kern0.75em \Rightarrow \kern0.75em {\sigma}_{\upgamma}=\frac{\sigma -\left(1-{f}_{\upgamma}\right){\sigma}_{\updelta}}{f_{\upgamma}\kern0.1em } $$
(2)
where σs is the stress–strain curve of SUS304 stainless steel, and fγ and f δ are the volume fractions of the δ and γ phases, respectively, which were obtained from the calculated equiaxed microstructure. The stress–strain curves of SUS304 stainless steel at various temperatures could be obtained from the experimental data reported by JAEA [28]. The stress–strain curves of the δ phase, γ phase, and SUS304 stainless steel at room temperature are shown in Fig. 10.
Fig. 10

Stress–strain curves of δ phase, γ phase, and SUS304 stainless steel at room temperature (Color figure online)

The stress–strain curves of the single phases at high temperature were calculated by assuming logarithmic scaling of the yield stress and the strain hardening exponent between room temperature and high temperature (900 °C). The following linear relationship in the plastic regime was obtained from Eq. (1) by taking the natural log:
$$ \log \sigma = \log R+ \log {\varepsilon}^n,\kern0.5em \log {\sigma}_{\mathrm{y}}= \log R+ \log {\varepsilon}_{\mathrm{y}}^n $$
(3)

Then, logR could be eliminated as:

$$ \log \sigma = n\left( \log \varepsilon - \log {\varepsilon}_{\mathrm{y}}\right)+ \log {\sigma}_{\mathrm{y}} $$
(4)

The equation parameters are slope n and intercept logσ y. Comparing these parameters for SUS304 at room temperature, \( {n}^{\left(\mathrm{M},\mathrm{RT}\right)}, \log {\sigma}_{\mathrm{y}}^{\left(\mathrm{M},\mathrm{RT}\right)} \), with those at 900 °C, \( {n}^{\left(\mathrm{M},900\right)}, \log {\sigma}_y^{\left(\mathrm{M},900\right)} \), the temperature scaling factor α RT → 900 was obtained as:

$$ {\upalpha}_{\mathrm{RT}\to 900}=\frac{n^{\left(\mathrm{M},900\right)}}{n^{\left(\mathrm{M}, RT\right)}}\cdot \frac{ \log {\sigma}_{\mathrm{y}}^{\left(\mathrm{M},900\right)}}{ \log {\sigma}_{\mathrm{y}}^{\left(\mathrm{M},\mathrm{RT}\right)}} $$
(5)

Applying the factor to the difference of the δ phase and SUS304, we could obtain the stress–strain curve of the δ phase at 900 °C as:

$$ \frac{ \log {\sigma}^{\left(\mathrm{M},900\right)}- \log {\sigma}^{\left(\updelta, 900\right)}}{ \log {\sigma}^{\left(\mathrm{M},\mathrm{RT}\right)}- \log {\sigma}^{\left(\updelta, \mathrm{RT}\right)}}={\alpha}_{\mathrm{RT}\to 900} $$
(6)

and the stress–strain curve of the γ phase at 900 °C as:

$$ \frac{ \log {\sigma}^{\left(\mathrm{M},900\right)}- \log {\sigma}^{\left(\upgamma, 900\right)}}{ \log {\sigma}^{\left(\mathrm{M}, RT\right)}- \log {\sigma}^{\left(\upgamma, \mathrm{RT}\right)}}={\alpha}_{\mathrm{RT}\to 900} $$
(7)

Note that this logarithmic scaling was applied only in the condition in which the stress–strain curve of SUS304 was provided by the power law, Eq. (1).

Homogenization

The elastic constants of the columnar and equiaxed microstructures were estimated using HOMAT software, which is based on the mathematical homogenization method [14]. HOMAT is used as a seamless interface to estimate the elastic constants of the microstructure simulated by MICRESS (http://web.micress.de/ICMEg1/presentations_pdfs/Laschet_met_plast.pdf#search='HOMAT+MICRESS'). Furthermore, it generates finite element meshes of the RVE according to the distributions of phases, crystal grains, and crystallographic orientations provided by MICRESS under the periodic boundary condition. To calculate the elastic constants of RVEs by substituting the global unit strain component in the asymptotic expansion equation, HOMAT executes the calculation by its own FEM code with an iterative BiCGstab solver. By using the elastic constants of δ and γ phases at 900 °C calculated by MD and first principles calculations, the homogenized elastic constants of the columnar and equiaxed microstructures at 900 °C were calculated.

The virtual tensile tests were performed by the ABAQUS/standard solver with the HOMAT interface to calculate the stress–strain curves of the columnar and equiaxed microstructures. In the virtual tensile test with ABAQUS and HOMAT, the global strain was applied to RVEs under periodic boundary conditions, and the homogenized stress–strain relationship was calculated. In this study, the von Mises yield criterion was supposed, and the strain–stress relationship was calculated by the uniaxial virtual tensile test. Using the elastoplastic stress–strain curves of the columnar and equiaxed RVEs at 900 °C, the virtual tensile test up to a total global strain of 1 % was performed. The calculated strain–stress curves were linearly extrapolated to 50 % strain.

Hot Rolling Simulation Using ABAQUS

The hot rolling of duplex stainless steel comprising columnar and equiaxed solidified microstructures was simulated using the ABAQUS/Explicit solver. The conditions of the hot rolling simulation were determined based on communication with Nisshin Steel Corporation. The conditions of the hot rolling simulation were set as follows. The diameter of the roll was 160 mm. The roller surface velocity was set to 18 m/min. The hot rolling reduction was set to 40 %.

Under these conditions, the hot rolling finish time was 0.4 s. A single pass of hot rolling was simulated. By considering the symmetry of the slab, a quarter of the slab was modeled. The slab was discretized by an eight-node linear brick element with reduced integration (C3D8R element). The elastic constants and the stress–strain properties of the columnar and equiaxed microstructures calculated above were assigned to the elements in the inner and surface regions of the slab, respectively. The friction coefficient between the roll and metal was 1.0. To accelerate the simulation, the element stable time increment was set to 1.0 × 10−7 s by variable mass scaling. As kinetic energy (ALLKE) is less than 0.001 % of the total strain energy (ALLIE), the simulation could still be assumed to be quasi-static after applying mass scaling.

Recrystallization

The static recrystallization in the rolled slab was qualitatively simulated by the MPF method using MICRESS. We performed two-dimensional MPF simulations of the static recrystallization at the cross sections passing through the center of the RVEs, as shown in Fig. 11. The initial distribution of the crystal grains in the deformed microstructure in the RVEs was selected the solidified microstructures obtained at the final step of the MPF simulations. We supposed that the static recrystallization was driven by the plastic energy density stored by the hot rolling process for 0.4 s. The residual δ phase was assumed to be transformed into the γ phase before the recrystallization. The threshold energy density of the recrystallization calculations was assumed to be 100 MPa under the maximum plastic energy density of the equiaxed region.
Fig. 11

Cross sections passing through the center of the RVEs where the two-dimensional MPF simulations of the static recrystallization were performed. a Columnar case. b Equiaxed case (Color figure online)

The parameters in the simulations of the static recrystallization were obtained from the example data implemented in MICRESS (http://web.micress.de/). The interfacial energy and interfacial mobility between the recrystallized and matrix grains were 5 × 10−5 J/cm2 and 7.5 × 10−3 cm4/J/s, respectively. The anisotropy of the interfacial energy was described using the Read–Shockley equation for the low-angle grain boundary. On the other hand, the anisotropy of the interfacial mobility was expressed by Humphreys’ model. The threshold misorientation between the low- and high-angle grain boundaries was set to 15°. We assumed random nucleation of the recrystallized grain of the γ phase only at the interface between the γ phases. The crystal orientation of the recrystallized grain was randomly determined so that the misorientation between the recrystallized grain and the substrate grain was ±15°. The time interval of the nucleation and the minimum distance between the recrystallized grains were set to 5 × 10−5 s and 2.5 μm, respectively.

Results

MPF Simulation of Solidification During Continuous Casting

The MPF simulations of the microstructure evolution are shown in Fig. 12. As expected, the δ phase first forms and then δ–γ transformation occurs at the δ–liquid interface. The carbon concentration distributions during the columnar and equiaxed solidifications are shown in Fig. 13. During the formation of the δ phase, the carbon atoms diffuse into the liquid phase and the carbon concentration is subsequently increased in the γ phase. The interstitial solute atom concentration distributions of the columnar and equiaxed solidification structures at 14.0 and 22.0 s, respectively, are shown in Fig. 14. By comparing them with the phase distributions in Fig. 12, slight concentration partitions of Cr and Ni can been seen at the same positions of δ and γ interfaces. Thus, it is confirmed that these microstructure evolution calculations were successfully performed. The phase distributions of the columnar microstructure at 1383 K and the equiaxed microstructure at 1352 K were used to construct the RVEs. The volume fractions of the untransformed δ phase in the columnar and equiaxed solidified microstructure were 12.3 and 11.9 %, respectively.
Fig. 12

Evolution of microstructures during the columnar and equiaxed solidifications. a Columnar. b Equiaxed (Color figure online)

Fig. 13

Distributions of carbon concentration during the columnar and equiaxed solidifications. a Columnar. b Equiaxed (Color figure online)

Fig. 14

Distributions of chromium and nickel concentrations of the columnar and equiaxed solidifications at 14 and 22 s, respectively. a Columnar. b Equiaxed (Color figure online)

MD and First Principles Calculations for Estimation of Elastic Constants

The averaged values of the elastic constants C 11, C 12, and C 44 of the fcc structure at 0 K by using LAMMPS were obtained as 229.7 GPa, 154.3 GPa, and 126.9 GPa, respectively. The deviation of the calculated elastic constant distribution in 11 cases was narrow. The maximum and minimum C 11 of the fcc structure were obtained as 231.5 and 227.6 GPa, respectively. The maximum and minimum C12 were 154.8 and 153.4 GPa, respectively. The maximum and minimum C44 were 127.7 and 126.4 GPa, respectively. Therefore, the elastic constants of the fcc structure were defined by arithmetic averaging of ones of 11 cases. The calculated elastic constants C 11, C 12, and C 44 of Fe–18 %Cr–12 %Ni (wt %) composition were determined as 215.9 GPa, 144.6 GPa, and 128.9 GPa, respectively. These constant values were in good agreement with the experimental ones [21].

The calculated elastic constants C 11, C 12, and C 44 of the fcc structure in the Fe–Cr–Ni–C alloy at 0 K were 261.9, 165.4, and 133.3 GPa, respectively. Comparison of the calculated elastic constants with those for the Fe–Cr–Ni alloy revealed that the elastic constants were increased by doping with carbon atoms. The elastic constants C 11, C 12, and C 44 of the bcc structure at 0 K by using VASP were calculated as 239.2, 121.1, and 116.1 GPa, respectively.

The elastic constants at 0 K were calculated from the abovementioned MD and first principles simulations. However, to simulate hot rolling at 900 °C, the elastic constants had to be estimated at a higher temperature. Nevertheless, no accurate interatomic potential exists at a finite temperature for MD simulations. Therefore, the temperature dependency of the elastic constants was assumed to be linear on the basis of the experimental data for the Fe–Ni binary alloy and pure Fe, in which the measured elastic constants were linearly explained with temperature [29, 30]. Figure 15 shows the elastic constants of the fcc and bcc structures as a function of temperature. The elastic constants of single phases shown in Fig. 15 were used for the calculation of the elastic constants of the solidified microstructures using HOMAT and the virtual material test.
Fig. 15

Temperature dependency of the elastic constants of a the fcc structure and b bcc structure (Color figure online)

Nano-Indentation Test for Estimation of Stress–Strain Curve for Single Phases

Figure 16 shows the force–depth curve obtained from the nano-indentation test. In Fig. 15, the C 11 value at room temperature (293.15 K) is 224.5 GPa for the δ-ferrite. In the reverse analysis, Young’s modulus, E = 224.5 GPa, was used, and the other parameters were calculated by the reverse analysis technique. The results at room temperature were σ y = 114.1  MPa, R = 1.723 GPa, and n = 0.358. From Eq. (2), the identified parameters for the single γ phase at room temperature are shown in Table 3. From Eqs. (6) and (7), the parameters at high temperature could be obtained, as given in Table 4, and these stress–strain relationships are shown in Fig. 17.
Fig. 16

Force–depth curve obtained by the nano-indentation test for the δ-ferrite phase

Table 3

Parameters for the stress–strain curve of a single γ phase at room temperature

Parameter

(SUS304)

δ

γ

E [GPa]

237.5

224.5

239.3

σ y [MPa]

140.3

114.1

145.7

R [GPa]

0.589

1.723

0.487

n

0.193

0.358

0.162

Table 4

Parameters for the stress–strain curves of δ phase, γ phase, and SUS304 at 900 °C

Parameter

(SUS304)

δ

γ

E [GPa]*

172.5

180.3

171.4

σ y [MPa]

60.8

58.6

61.2

R [GPa]

0.0771

0.0885

0.0752

n

0.030

0.051

0.026

Fig. 17

Stress–strain curves of the δ phase, γ phase, and SUS304 at 900 °C (Color figure online)

Homogenization

The anisotropic elastic constants obtained by HOMAT are described by the following constitutive equation:
$$ \left\{\begin{array}{c}\hfill {\sigma}_{11}\hfill \\ {}\hfill {\sigma}_{22}\hfill \\ {}\hfill {\sigma}_{33}\hfill \\ {}\hfill {\sigma}_{12}\hfill \\ {}\hfill {\sigma}_{13}\hfill \\ {}\hfill {\sigma}_{23}\hfill \end{array}\right\}=\left[\begin{array}{cccccc}\hfill {C}_{11}\hfill & \hfill {C}_{12}\hfill & \hfill {C}_{13}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill {C}_{12}\hfill & \hfill {C}_{22}\hfill & \hfill {C}_{23}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill {C}_{13}\hfill & \hfill {C}_{23}\hfill & \hfill {C}_{33}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {C}_{44}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {C}_{55}\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {C}_{66}\hfill \end{array}\right]\kern0.5em \left\{\begin{array}{c}\hfill {\varepsilon}_{11}\hfill \\ {}\hfill {\varepsilon}_{22}\hfill \\ {}\hfill {\varepsilon}_{33}\hfill \\ {}\hfill {\gamma}_{12}\hfill \\ {}\hfill {\gamma}_{13}\hfill \\ {}\hfill {\gamma}_{23}\hfill \end{array}\right\} $$
(8)

Then, the columnar and equiaxed microstructures are respectively obtained as:

$$ \begin{array}{lll}\bullet \kern0.5em {C}_{11}\kern0.5em =\kern0.5em 205.31,\kern0.5em \hfill & {C}_{22}\kern0.5em =\kern0.5em 205.32,\hfill & {C}_{33}\kern0.5em =\kern0.5em 172.36\hfill \\ {}\bullet \kern0.5em {C}_{12}\kern0.5em =74.13,\hfill & {C}_{13}\kern0.5em =107.08,\hfill & {C}_{23}\kern0.5em =107.09\hfill \\ {}\bullet \kern0.5em {C}_{44}=53.63,\hfill & {C}_{55}\kern0.5em =95.05,\hfill & {C}_{66}=95.05\hfill \end{array}\kern0.5em \left(\mathrm{GPa}\right) $$
and
$$ \begin{array}{lll}\bullet \kern0.5em {C}_{11}\kern0.5em =\kern0.5em 219.64,\kern1em \hfill & {C}_{22}\kern0.5em =\kern0.5em 211.20,\hfill & {C}_{33}\kern0.5em =\kern0.5em 205.55\hfill \\ {}\bullet \kern0.5em {C}_{12}\kern0.5em =80.59,\hfill & {C}_{13}\kern0.5em =86.29,\hfill & {C}_{23}\kern0.5em =94.71\hfill \\ {}\bullet \kern0.5em {C}_{44}=60.33,\hfill & {C}_{55}\kern0.5em =66.14,\hfill & {C}_{66}=70.83\hfill \end{array}\kern0.5em \left(\mathrm{GPa}\right) $$

It is evident that the elastic anisotropy of the columnar microstructure is stronger than that of the equiaxed case according to the anisotropy parameter A = 2C 44 / (C 11 − C 12).

The obtained elastoplastic stress–strain curves by the virtual tensile test are shown in Fig. 18. No significant difference exists between the two cases, except for the yield stress, because the volume fractions of the constituent phases were almost the same in both cases.
Fig. 18

Stress–strain curves of columnar and equiaxed microstructures at 900 °C (Color figure online)

Hot Rolling Simulation Using ABAQUS

The distributions of the plastic strain energy density and the von Mises equivalent stress on the surface and the symmetric cross section of the slab at 0.16 s obtained by ABAQUS are shown in Fig. 19. Figure 20 shows the distribution of the plastic energy density at the finish time of 0.4 s. The maximum plastic strain energy density in the surface region of the slab, where the columnar microstructures are formed, is 195.5 MPa, while that in the internal region, including the equiaxed microstructures, is 116.7 MPa. In the following section, we discuss the MPF simulations of the static recrystallization in these regions, which were driven by the plastic strain energy stored in the microstructure during hot rolling.
Fig. 19

Distributions of plastic strain energy density and the von Mises equivalent stress at 0.16 s during hot rolling. a Strain energy density. b von-Mises equivalent stress (Color figure online)

Fig. 20

Distribution of plastic strain energy density after hot rolling (Color figure online)

Recrystallization

The evolution of the recrystallized grains is shown in Fig. 21. It is evident that the nucleation of the recrystallized grain is concentrated in the early stage of the recrystallization until 0.003 s in both cases. After that, the grain growth is the dominant phenomena.
Fig. 21

Evolution of the recrystallized grains with time. a Columnar case. b Equiaxed case (Color figure online)

Discussion

In this study, the solidification microstructure evolution calculations in a continuous casting process were performed by dividing the slab model space into only two regions—equiaxed and columnar. Thus, the homogenized mechanical properties of the slab were also defined in two regions. To more precisely simulate the hot rolling process, solidification microstructures had to first be obtained at many positions in the slab space. Furthermore, to perform these solidification calculations, the temporal temperature distribution had to be estimated by considering not only the complex temperature boundary condition but also the melt flow effect. To date, the multiphase field method coupling the fluid flow equation has been well studied [31]. However, a method of solving the highly nonlinear problem in which melt flow must be solved in the conditions of microscale and macroscale coupling remains undetermined.

In this microstructure simulation, the number of RVE grids was 125 × 125 × 125 and 125 × 125 × 256 for equiaxed and columnar solidification cases, respectively. These grid numbers were decided by not only referencing a hominization study using MICRESS and HOMAT—in which 100 × 100 × 100 grids were applied to equiaxed microstructure evolution—but also by considering a reasonable calculation time consumption, even if executing parallel computing of the MICRESS Open-MP feature [32]. Of course, a larger number of grids in RVE leads to a larger number of grains in the solidified microstructure, which will result in more accurate homogenization. Thus, a higher performance computation technique is required by adapting the simulation software to highly parallelizing programming, e.g., Open-MPI [33].

The grain size just before the hot rolling process must be larger than the solidified one because the grain growth advances in the period between continuous casting and hot rolling processes. This calculation can be performed by MICRESS. However, the number of grains in RVE will become very small in accordance with the large grain size after coarsening. This will affect the homogenization accuracy. This was assumed for the solidified microstructure to be applied to the hot rolling process calculation.

There are many reports that the elastic constant of the binary compound system is usually calculated by first principles [34, 35, 36]. However, a simulation report for the elastic constant of the ternary compound system has not been apparent. Thus, we newly attempted application of the first principles and MD simulation to obtain the elastic constant over the ternary compound. At first, we calculated the elastic constant of the bcc structure using first principles of the 54-atom model because the EAM potential parameter for the bcc structure of the ternary system could not be found for the MD calculation.

The first principles calculation results showed that the arrangement of Cr and Ni atom positions did not have an evident relation with the obtained elastic constant values because these calculations were performed using the very small model of 54 atoms. In other words, the number of Cr and Ni atoms in the first principles model was too small. Therefore, many more atoms, e.g., 108 or 250, at least, were required for the first principles calculation. On the other hand, we calculated the elastic constant of the fcc structure using the MD simulation using the 4000-atom model because the EAM potential parameter for the fcc structure of the Fe–Cr–Ni ternary system was referenced from the published article.

The temperature dependency of the elastic constants was approximately estimated in accordance with the experimental data for the Fe–Ni binary alloy and pure Fe, in which the elastic constants were linear with temperature. However, it was reported that the temperature dependency of the elastic constants in pure germanium is measured as a nonlinear relation [37]. Therefore, it is desirable for the temperature dependency of the elastic constant in the Fe–Cr–Ni–C system to be expected in the future by the MD simulation.

Only 1 % of the global strain was applied in the virtual tensile test, and the results were extrapolated. As the hot rolling reduction was set to 40 %, the metal was compressed by approximately 40 % on average. This means the extrapolation was very important in the hot rolling simulation and significantly affected the results. As plastic behavior of metal can be explained by strain hardening, the extrapolation will be more accurate with the appropriate hardening effect. If the effect is considered, stress decreases in the hot rolling simulation and the resultant plastic strain energy density also decreases.

The obtained homogenized elastic constants for the equiaxed microstructure are the orthotropic material properties shown in Eq. (8). However, if the microstructure was truly equiaxed, the property had to be isotropic. This means the components in Eq. (9) had to have had the following relationships:
$$ {\sigma}_{22}={\sigma}_{33}={\sigma}_{11},\kern0.5em {\sigma}_{13}={\sigma}_{23}={\sigma}_{12},\kern0.5em {\sigma}_{44}={\sigma}_{55}={\sigma}_{66}=\frac{1}{2}\left({\sigma}_{11}-{\sigma}_{12}\right) $$
(9)

However, the obtained components did not satisfy the above relationships. This anisotropicity of elastic constants may have affected the results of the hot rolling simulation. The same anisotropicity could be observed in elastic constants for columnar microstructure. Because 1- and 2-axis directions are symmetry, it must be observed in elastic constants. However, for example, C 11 and C 22 slightly differed. The method of obtaining isotropic elastic constants must be determined in the future.

The recrystallization calculations were performed in two-dimensional space even if the solidification simulations were executed in three dimensions. This is because the three-dimensional recrystallization by the multiphase field method still has some limitations or problems, such as uncertain interfacial mobility and anisotropic models in multiphase junction energy [38]. Thus, the recrystallization calculation model was set simply by reducing dimensions to avoid the complexity of many parameters.

The classical mean recrystallization energy model of MICRESS was applied to the calculations, in which the threshold for recrystallization was assumed to be slightly lower than the maximum strain energy density of the equiaxed microstructure case of 100 MPa. On the other hand, the local recrystallization model based on a dislocation density field is available in MICRESS. A dislocation model is necessary for estimation of recrystallization energy of the present model. A crystal plasticity FEM, of which the elastoplastic model can be directly related to the dislocation density, is better for coupling with the multiphase field method [39].

Summary

In this paper, we proposed a microstructure-based multiscale simulation framework using various commercial simulation software applications to simulate the microstructure evolution and elastoplastic deformation behavior of the materials. The proposed framework was then used to investigate the solidification in the continuous casting and static recrystallization after the hot rolling of duplex stainless steel. We demonstrated that our methodology enables simulating the microstructure evolution using the MPF method and estimating the mechanical properties of the microstructures by the MD, first principles calculations, and finite element simulations. The reliability of multiscale software bridging was confirmed by this study. However, we must improve all simulations to obtain quantitative results, especially for the estimation of the mechanical properties of the single phase.

Notes

Acknowledgements

The authors wish to thank Mr. Takafumi Kawagoe at Nisshin Steel Co., Ltd. for engineering suggestions. We would like to thank Editage (www.editage.jp) for English language editing.

Compliance with Ethical Standards

Competing Interests

The authors declare that they have no competing interests.

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Copyright information

© The Minerals, Metals & Materials Society 2017

Authors and Affiliations

  • Sukeharu Nomoto
    • 1
    Email author
  • Mototeru Oba
    • 1
  • Kazuki Mori
    • 1
  • Akinori Yamanaka
    • 2
  1. 1.ITOCHU Techno-Solutions Corp.TokyoJapan
  2. 2.Division of Advanced Mechanical Systems Engineering, Institute of EngineeringTokyo University of Agriculture and TechnologyTokyoJapan

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