Rare Metals2018年第12期

A model for simulation of recrystallization microstructure in single-crystal superalloy

Run-Nan Wang Qing-Yan Xu Bai-Cheng Liu

Key Laboratory for Advanced Materials Processing Technology (MOE), School of Materials Science and Engineering, Tsinghua University

作者简介：*Qing-Yan Xu,e-mail:scjxqy@tsinghua.edu.cn;

收稿日期：18 August 2017

基金：financially supported by the National Key R&D Program of China (No.2017YFB0701503);the National Basic Research Program of China(No.2011CB706801);

A model for simulation of recrystallization microstructure in single-crystal superalloy

Run-Nan Wang Qing-Yan Xu Bai-Cheng Liu

Key Laboratory for Advanced Materials Processing Technology (MOE), School of Materials Science and Engineering, Tsinghua University

Abstract：

In the present investigation, a coupled crystal plasticity finite-element(CPFE) and cellular automaton(CA) model was developed to predict the microstructure of recrystallization in single-crystal(SX) Ni-based superalloy.The quasi-static compressive tests of [001] orientated SX DD6 superalloy were conducted on Gleeble3500 tester to calibrate the CPFE model based on crystal slip kinematics.The simulated stress-strain curve agrees well with the experimental results. Quantitative deformation amount was introduced in the deformed samples of simulation and experiment, and these samples were subsequently subjected to the standard solution heat treatment(SSHT).Results of CA simulation show that the recrystallization(RX) nucleation tends to occur at the third stage of SSHT process due to the high critical temperature of RX nucleation for the samples deformed at room temperature. The inhomogeneous RX grains gradually coarsen and compete to reach more stable status by reducing the system energy.Simulated RX grain density decreases from 7.500 to1.875 mm^-1,agreeing well with the value of 1.920 mm^-1from electron backscattered diffraction(EBSD) detection of the experimental sample.

Keyword：

Recrystallization; Single crystal; Crystal plasticity; Cellular automaton; Microstructure;

Received： 18 August 2017

1 Introduction

Single-crystal (SX) Ni-based superalloy has been widely used in the extreme conditions due to its superior mechanical properties and resistance to corrosion [ 1, 2] .However,the defects induced during directional solidification (DS) and heat treatment process,for example,the stray grains [ 3, 4] ,sliver [ 5] and recrystallization (RX) [ 6] ,seriously restrict the cost and application of SX castings.The high-angle boundaries induced by RX defects can degrade the creep and fatigue properties significantly.

In the past decades,the investigations of RX in SX superalloy mainly focused on the mechanical properties [ 6, 7, 8, 9, 10] ,microstructure [ 11, 12, 13, 14] ,annealing conditions [ 15] and RX nucleation [ 16, 17] .Mathur et al. [ 16] found that the formation of RX in SX superalloy was induced by the dislocations and micro y'in the surface eutectic particles formed during the DS process.The evolution of temperature field and the microstructure of castings during DS process have been studied by experiments and simulations,and corresponding works were reported in Refs. [ 18, 19, 20, 21, 22] .Lots of researches have been performed to simulate the recrystallization behavior for enhancing the material properties such as in steel and nonferrous alloy [ 23, 24, 25] ,while the microstructure simulation of RX defect in SX Nibased superalloy has rarely been reported.Li et al. [ 26] and Zambaldi et al. [ 27] have done the related work,but neither phenomenon-based model nor indentation test can give the accurately quantitative deformation amount in experiment and simulation at the same time.

In this study,the quasi-static compressive tests were conducted for SX DD6 superalloy by experiment and simulation based on crystal plasticity finite-element method (CPFEM).A cellular automaton (CA) model for microstructure simulation of RX was developed based on the calculated deformation.The samples deformed to 5%plasticity were subjected to the standard solution heat treatment.The evolution of RX microstructure was simulated to make comparison with the final grain morphology and density from electron backscattered diffraction(EBSD) detection.

2 Coupled CPFEM and CA model

The crystal plasticity theory developed by Taylor [ 28] ,Hill [ 29] and Hill and Rice [ 30] has gained a lot of popularity in describing the flow behavior of metals for a long time.It was coupled with finite-element method (FEM) to analyze the single-crystal deformation [ 31, 32, 33] .The driving force for RX nucleation and growth is the plastic deformation induced prior to the heat treatment,so an accurate calculation of plastic strain field during deformation is of crucial importance to the simulation of RX microstructure.In this study,the crystal plasticity theory was coded into a UMAT file (user-defined material),which acts as an interface between user and ABAQUS to be called during simulation.Simulated accumulated slip was imported to the CA model as the driving force for RX.

2.1 Crystal plasticity model

2.1.1 Kinematics and constitutive laws

The plastic deformation is assumed to arise from the crystallographic dislocation slip.The Schmid stress,or resolved shear stress on a slip system,is assumed here to be the driving force for slip.The total deformation gradient(F) can be expressed as:

where F^p and F^e represent the plastic shear and stretching and rotation of the lattice,respectively.The rate of change of F^p is related to the slipping rate ( )of theαslip system by:

where the sum ranges over all activated slip systems,unit vectors s^αand m^αare the slip direction and normal to slip plane in the reference configuration,respectively.

s^e(α)=F^es^αis the vector lying along the slip direction of systemαin the deformed configuration.A normal to the slip plane which is the reciprocal base vector to all such vectors in the slip plane follows:

The velocity gradient (L) in the current state is:

where F is the time derivative of F,and the symmetric rate of stretching (D) and the antisymmetric spin tensor (Ω) can be decomposed into lattice parts (D^e andΩ^e) and plastic parts (D^p andΩ^p) as:D=D^e+D^p,Ω=Ω^e+Ω^p,D^e+Ω^e=E^e(F^e)^-1 and .

The relationship between Jaumann rate ( ) and Cauchystress (σ)follows:

where I is the second-order unit tensor,σthe material derivative ofσand is the corotational stress rate on axes spinning with the material.The crystalline slip depends on the current stress (σ) through the Schmid stress,τ^α=m^e(α)· ·S^e(α),whereρ₀ andρare the reference and current mass density.Rate of change of Schmid stress(τ^α) follows:

2.1.2 Hardening of rate-dependent crystalline materials

According to the rate-dependent model,the shearing ratefollows: ,where ,τ^αand gαare reference shearing rate,resolved shear stress and slip resistance of theαslip system. is the shearing rate of theβslip system.h_αβis the matrix of hardening modulus.

Self-hardening:

Latent hardening:

where h₀ is the initial hardening modulus,σ₀ is the yield stress,andτ_s is the saturated stress and q the constant.Verification of model was performed by the uniaxial compression of[001]oriented SX CMSX-4 superalloy.The parameters used in the CPFEM simulation are presented in Table 1,where RT,AHT and AC are short for room temperature,as-heat treated and as-cast conditions,respectively.Figure 1 shows stress-strain curve for the quasi-static compressive test of[001]oriented SX CMSX-4superalloy.The experimental data are from Ref. [ 27] .

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Table 1 Parameters used in CPFEM simulations (RT)

Fig.1 Stress-strain curve for quasi-static compressive test of[001]oriented SX CMSX-4 superalloy

2.2 CA model for recrystallization simulation

The driving force of RX is plastic deformation which is essentially the dislocation slip produced prior to heat treatment.The stored deformation energy for CA modeling is expressed in terms of a reference dislocation density,which is related to the amount of accumulated shear.The effect of as-cast dendritic morphology on RX behavior was taken into consideration through different activation energy in dendritic arms (DAs) and interdendritic regions (IDRs).

2.2.1 Model of RX nucleation

The RX nucleus usually occurred in the region with large deformation,so the continuous nucleation model is suitable for describing this process.Considering the temperature of heat treatment and the heterogeneity of as-cast sample,the nucleation rate (N) follows:

where C₀ is constant coefficient (1.0×10⁹ s^-1·J^-1);R,T and P denote universal gas constant,degree kelvin and driving force for RX,respectively;P is related to the reference dislocation density of 2×10⁵ J·m^-2,which was multiplied by the local amount of accumulated shear (γ);and Q_a and P^c are activation energy and critical stored energy of RX nucleation,respectively.

whereγ_c is critical shear set 2%in this study,and E_lagb is the low-angle grain boundary energy set 0.6 J·m^-2.According to our previous research,the onset temperature of RX formation varies with temperature at which the samples are deformed [ 34] .For the samples with 5%plastic strain deformed at RT,RX grains only form at the annealing temperature above 1310℃.Hence,the critical temperature of RX nucleation is set 1311℃.

2.2.2 Model of RX coarsening and growing

The velocity of interface migration (v) depends on the grain boundary mobility for the static recrystallization (M) and driving pressure for the grain boundary movement (P),as shown in Eq.(13).M can be described by the Arrhenius formula:

where M₀ is grain boundary mobility,Qb is the activation energy for grain boundary motion,D₀ is the diffusion constant,k is the Boltzmann constant and b is the Burger's vector.

When RX process is completed,the grains will still coarsen under the control of grain boundary energy (E) in Eq.(15) and curvature (κ) in Eq.(16).The misorientation angle (θ) is calculated from Euler angle(Φ₁,Φ,Φ₂) of adjacent grains,and the corresponding algorithm can be found from Ref. [ 26] .

where E_m is the large grain boundary energy,θ_m is the critical misorientation of large grain boundary,αis the constant,c_s is the CA element size,c_i is the number of elements which have the same state of central element,N'=25 and Kink=15 for 2D simulation.In view of the inconformity of meshes used in CA and FEM,the simulated results of FEM calculation ( ) must be mapped to CA meshes by a conversion algorithm,and the mapped result in CA cell (f_CA) follows:

where l is the number of FEM nodes within R distance from the CA cell (i,j,k),and d_n is the real distance between the CA cell and FEM node.The key parameters used in the simulation are listed in Table 2.Different symbol values were used to distinguish the cells of DAs and IDRs in the computational domain.

3 Experimental

The second-generation SX superalloy DD6,whose nominal chemical composition is presented in Table 3 [ 35] ,was used in this study.DD6 superalloy ingot was melted,and directional solidification (DS) was performed in an industrial vacuum Bridgman furnace to obtain the SX cylinder bars which are 180 mm in length and 15 mm in diameter.The small cylinder specimens,whose diameter is 6 mm and the length is 10 mm,were cut from as-cast cylinder bars using electrical discharge machining.Only the samples within 10°misorientation angle from[001]direction were employed.The compression flow curves were used to calibrate the crystal plasticity model.Compressive test was performed on Gleeble3500 tester at room temperature at a low strain rate of 1×10^-3 S^-1 to simulate the quasi-static compressive process.For the samples of quasi-static compressive test,the relationship between true and engineering strain (ε_t,s_e)/stress (σ_t,σ_e) follows:ε_t=-ln(1-ε_e) andσ_t=σ_e*(1-ε_e),respectively.

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Table 2 Key parameters used in CA simulation

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Table 3 Nominal chemical composition of DD6 superalloy (wt%)

The as-cast samples with a quantitative deformation of5%were chosen to undergo the standard solution heat treatment (SSHT 1290℃,1 h+1300℃,2 h+1315℃,4 h,air cooling).The annealed sample was cut from the middle cross section of cylinder to perform polishing.The as-polished samples were etched using the Marble's reagent and then observed by optical microscope (OM,Zeiss AM10OM).The samples for electron backscattered diffraction(EBSD) detection were electrochemical polished using HClO₄ (10%) and C₂H₅OH (90%).The data were collected from a MIRA3 LMH FSEM equipped with an Oxford EBSD detector at an accelerating voltage of 20 kV and step size of2-10μm range.High-angle grain boundaries (>15°) were shown by black lines in inverse pole figure (IPF).The critical value of area to define a true grain was set to be 20μm²because of the existence of EBSD calibration error.The density of RX grains in the viewing zone can be calculated by:N/A,where N and A are the number of RX grains and the area of viewing zone,respectively.For the RX grains located at the side and corner of the viewing zone,the number should be multiplied by 1/2 and 1/4,respectively.

4 Results and discussion

4.1 Simulated and experimental quasi-static compression

The compression flow curves were used to calibrate the crystal plasticity model.Two cylinder samples were compressed,and the true stress-strain curves for quasi-static compressive test of[001]oriented as-cast DD6 superalloy are shown in Fig.2.The material parameters for CPFEM simulation are demonstrated in Table 1.The simulated curve of the same process is shown as the red line in Fig.2.It conforms well to the experimental results,revealing that the simulation results based on crystal plasticity can accurately describe the deformation scale and status in SX samples.By employing the deformation of 5%plasticity,the total cumulative shear strain y on the middle section of sample is shown in Fig.3.Accumulated shear strain gradually decreases from the center of the circle to the periphery.

In addition,during compression process,the inhomogeneous strain distribution can be induced by the anisotropy of SX material with fcv structure as well as the friction between the head face of sample and the pressure head of Gleeble tester.For the Ni-based superalloy deformed at RT,the slip face and direction are{111}and<110>,respectively.Three end-fixed samples,whose axial direction is along[001],[110]and[111],were simulated to test the anisotropy.The black dashed lines in Fig.4a,c,e represent that the axial directions of the cylinder samples coincide with[001],[110]and[111]orientation of fcc lattice,respectively.The shear strain on the head faces of deformed samples expresses fourfold,threefold and double symmetric distributions(Fig.4b,d,f),which conform to the periodicity and symmetry of{111}<110>slip system for fcv structure.Owing to the long distance between the head face and the middle cross section,the anisotropy does not influence the distribution of shear strain on the middle cross section (Fig.3).

Fig.2 Stress-strain curve for quasi-static compressive test of[001]oriented as-cast DD6 superalloy

Fig.3 Total cumulative shear strain y on middle section of sample with 5%plastic strain

4.2 Evolution of RX microstructure during heat treatment

The evolution of the optical metallographic micros tructure during SSHT process is shown in Fig.5.For the samples in as-cast deformed condition (Fig.5a) and first stage of SSHT (1290℃,0.5 h Fig.5b),the DA and IDRs can be clearly distinguished.RX grain has not nucleated at this stage.The degree of element segregation gradually decreases with the SSHT proceeding.Consequently,the DA and IDRs become almost uniform at the max SSHT temperature (1315℃),but slight IDRs and eutectic particles can still be identified in Fig.5c (1290℃,1 h+1300℃,2 h+1315℃,0.5 h).Figure 5d shows the final microstructure of samples subjected to complete SSHT process.As shown in Fig.5c,d,RX grains can be clearly observed when the annealing temperature reaches1315℃.The morphologies of RX grains in Fig.5d are more homogeneous than those in Fig.5c,and the sizes of grains are bigger.

The simulated evolution of RX microstructure during SSHT is shown in Fig.6a-d,as well as the experimental results of EBSD measurement (Fig.6e,f).Before heat treatment of 1315℃,no RX grain appears in the deformed region.At the beginning of solvus heat treatment (3.05 h),many misoriented RX grains begin to nucleate,and RX grains of different sizes inhomogeneously coexist in the simulated area.Subsequently,these grains gradually coarsen and compete to reach more stable status by reducing the system energy.Finally,the number of RX grains becomes less and less,and the large grains get the dominant position,as shown in the simulated (Fig.6d) and experimental (Fig.6f) results.

The variation of simulated and experimentally measured grain density during the SSHT process is presented in Fig.7.RX grains begin to nucleate at the third stage of SSHT,and the density of RX grains rapidly increases to7.500 mm^-1.With the increase of annealing time,the value gradually decreases.The simulated grain density decreases to 5.5 mm^-1 at 3.50 h and finally to1.875 mm^-1.From the EBSD observation,the grain densities on the middle sections of samples at 3.50 and 7.00 h are 4.950 and 1.920 mm^-1,respectively.The grain size and density usually depend on the activation energy for boundary migration.The velocity of interface migration under the solvus condition is faster than that under subsolvus condition.Hence,this leads to big grain size and small density.

It is well known that many small RX nuclei tend to appear in the dendritic arms from the onset of heat treatment [ 26, 36] .Subsequently,by undergoing long-time SSHT at high temperature (solvus and sub-solvus),the small grains will overgrow the dendritic arms and enter the interdendritic regions.RX behavior of deformed matrix at this stage is caused by the release of deformation stored energy.However,the behavior of RX nucleation strongly depends on the deformation temperature of samples.According to previous researches [ 34, 37] ,critical temperature of RX nucleation reaches as high as 1310℃for the samples with 5%plastic strain induced at room temperature.Hence,there is no RX nucleation until the solution temperature comes to the solvus 1315℃.At this time,the interdendritic regions and dendritic arms have become homogenized,and then the RX nucleation can rapidly grow and coarsen to decrease the interfacial energy (IFE).The process of RX grains growing and coarsening is controlled by the difference of IFE between small and large RX grains.The amount of grain boundaries and the scale of IFE can be reduced during this process spontaneously.Thus,large RX grains get the dominant position instead of fine grains after SSHT process.

Fig.4 Anisotropy of shear strain distribution on head face of deformed samples with axial directions of samples lying along different orientations of fcc lattice:a,b[001],c,d[110]and e,f[111]

Fig.5 OM images for microstructural evolution of deformed samples during SSHT:a as-cast deformed;b 1290℃,0.5 h;c 1290℃,1 h+1300℃,2 h+1315℃,0.5 h;d 1290℃,1 h+1300℃,2 h+1315℃,4 h

Fig.6 Simulated (a-d) and EBSD detected (e,f) RX microstructures of samples subjected to SSHT:a 0 h (inset showing dendritic morphology of as-cast sample for simulation),b 3.05 h,c 4.00 h;d 7.00 h,e 3.50 h and f 7.00 h (misoriented grains with different Euler angles marked by different colors)

Fig.7 Density of RX grains varying with annealing time and solution temperature

5 Conclusion

A coupled CPFEM and CA model was developed to simulate the RX microstructure of SX Ni-based superalloy.Quasi-static compressive tests were conducted to calibrate the CPFEM model.The simulated stress-strain curves for as-heat-treated CMSX-4 and as-cast DD6 SX superalloy agree well with the experimental results.As the driving force for RX,the plasticity which is represented by local dislocation density was quantitatively introduced in simulation and experiment.Then,the samples with 5%plastic strain were subjected to SSHT process.Simulated results show that high critical temperature of RX nucleation for the samples deformed at room temperature results in different RX behaviors in three stages of SSHT.RX nucleation tends to occur at the third stage (solvus) of SSHT process,and the inhomogeneous RX grains gradually coarsen and compete to reach stabilization by reducing the system energy.Finally,the large grains get the dominant position.Simulated RX grain density decreases from 7.500to 1.875 mm^-1,which agrees well with the value of1.920 mm^-1 from experimental sample by EBSD detection.

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