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Rare Metals2019年第10期

Constitutive modeling of a directionally solidified nickel-based superalloy DZ125 subjected to thermal mechanical creep fatigue loadings

Xiao-An Hu Xiao-Guang Yang Duo-Qi Shi Hui-Chen Yu Ting-Ting Ren

School of Energy and Power Engineering,Beihang University

Science and Technology on Advanced High Temperature Structural Materials Laboratory,Beijing Institute of Aeronautical Materials

Air Force Armament Department,China Aviation Museum

作者简介：*Xiao-Guang Yang e-mail:yxg@buaa.edu.cn;

收稿日期：17 March 2015

基金：financially supported by the National Basic Research Program of China (No.2015CB057400);

Constitutive modeling of a directionally solidified nickel-based superalloy DZ125 subjected to thermal mechanical creep fatigue loadings

Xiao-An Hu Xiao-Guang Yang Duo-Qi Shi Hui-Chen Yu Ting-Ting Ren

School of Energy and Power Engineering,Beihang University

Science and Technology on Advanced High Temperature Structural Materials Laboratory,Beijing Institute of Aeronautical Materials

Air Force Armament Department,China Aviation Museum

Abstract：

A transversely isotropic continuum elasto-viscoplasticity model,which was developed from Chaboche's unified constitutive model,was formulated to capture the thermal mechanical creep fatigue deformation behavior of a directionally solidified nickel-based superalloy.A fourthorder tensor was introduced to model material anisotropy.In order to model the tertiary creep behavior,the Kachanov damage evolution equation was coupled into the stress tensor.Based on the test results of uniaxial tensile,fatigue,and creep loadings at isothermal temperature conditions,the material parameters are obtained.Thermal mechanical fatigue(TMF) and creep-fatigue interaction test results were used to verify the robustness of the model.Additionally,strain-temperature-dependent stress-strain responses under TMF loadings were analyzed using the present model.Under strain-controlled conditions,both of the stress ranges and mean stresses are strongly influenced by the strain-temperature phases,a key parameter for TMF tests.

Keyword：

Constitutive; Thermal mechanical fatigue; Creep; Viscoplastic; Superalloy;

Received： 17 March 2015

1 Introduction

Components such as gas turbine blades in aircraft engines operating at high temperatures always experience heating and cooling transients during startup and shutdown,with a steady state introducing the dwell effect.Complex loadings,including low cycle fatigue,thermal mechanical fatigue (TMF),creep,and creep-fatigue interaction (CFI),will cause failures of these components.Since gas turbine blades are commonly made from directionally solidified(DS) or single-crystal (SC) superalloys,the material anisotropy could introduce additional complication in formulating the constitutive model,and further for the lifetime assessment.

For DS and SC superalloys,the macroscopic stress/strain response is anisotropic due to their micros truc ture.Constitutive modeling of these materials considering both anisothermal conditions and material anisotropy is a challenge to the engineering application.Up to now,a variety of constitutive models based on elastic viscoplasticity and crystal plasticity theory has been proposed and developed [ 1, 2, 3, 4] .Though much progress has been achieved during the past 20 years concerning multiscale approaches of the inelastic analysis of structure components,there is still the need of using more macroscopic approaches of the material constitutive modeling at the continuum scale of the components [ 5] .The unified viscoplastic constitutive models,as one kind of phenomenological constitutive models,have been proposed by Prager [ 6] ,Perzyna [ 7, 8] ,Frederick and Armstrong [ 9] ,and further developed by Walker [ 10] ,Bodner and Partom [ 11] ,Deseri and Mares [ 12] and Chaboche [ 2] ,etc.The general form of this constitutive model consists of flow rule,isotropic,and kinematic hardening evolution equations.More detailed discussion on constitutive models was recently reviewed by Chaboche [ 2] .

For the high-temperature superalloy application,some sophisticated constitutive models were developed by Manonukul et al. [ 13] ,M cke and Bernhardi [ 14] ,Becker and Hackenberg [ 15] ,and Shenoy et al. [ 16, 17] .The cyclic plasticity and creep behaviors of high-temperature superalloy are well modeled by these contributors.Complex loadings,i.e.,TMF cycle and biaxial creep loadings,are commonly used to verify the accuracy of the developed models.However,these models are not general enough for the anisotropic materials except for the Shenoy's model which covers rate dependency,anisotropy,wide range of temperature,creep,cyclic plasticity,etc.For the isotropic IN718 superalloy,a constitutive model [ 15] was developed for the unified description of rate-dependent and rate-independent material behavior.The main contribution was the introduction of a limit surface concept.With the proposed model,both rate-dependent and rate-independent inelasticity of a polycrystalline superalloy can be treated considering a wide range of temperature.However,the model was only suitable for the isotropic materials.Further modification may be extended to anisotropic conditions,i.e.,through coupling anisotropic tensor in the flow rule.Constitutive modeling of DS superalloy GTD-111 was extensively studied [ 16, 17] .Both crystal plasticity model and transversely isotropic viscoplasticity model were successfully applied to model isothermal fatigue (IF),creep,and TMF considering anisotropy of the material.Generally,crystal plasticity models require much more computational resources compared to viscoplasticity models,and thus the latter are applied more widely for the stress/strain of component level analyses.

In the present study,a modified Chaboche's model was developed to model tensile,cyclic,and creep deformation behaviors of a DS superalloy DZ125,considering both anisothermal conditions and material anisotropy.A fourth-order tensor was introduced to take anisotropic tensile properties of the DS superalloy into account.A damage evolution equation was coupled within stress tensor to capture the tertiary creep behavior.The isothermal tensile,creep,and fatigue test data were used to fit the parameters using an Levenberg-Marquardt (L-M) optimization strategy.Then,the creep-fatigue interaction and TMF test results were used to verify the accuracy of the model.

2 Experimental

DZ125 is a DS Ni-based superalloy used for turbine blades and vanes.The nominal chemical compositions of the material of the alloy are listed in Table 1.The solution heat treatment was as follows:1180℃/2 h+1230℃/3 hair cooling+1100℃/4 h air cooling+870℃/20 h air cooling.The microstructure of DZ125 DS superalloy mainly containsγ/γ'two phases.The size ofγ'phase is about 0.4-1.5μm,and the average size of fineγ'phase in the dendrite regions is about0.4μm.The experiments were performed in both of longitudinal and transverse orientations,including monotonic tensile at different strain rates,low cyclic fatigue,creep and creep fatigue,and TMF tests.

3 Development of constitutive model

The total strain rate tensor ( ) is assumed to be decomposed additively into elastic part ( ) and inelastic part( ),

The elastic part is expressed in terms of Hooke's law,

where is stress rate and C_ijkl represents the elastic tensor for anisotropic materials,and it is written as:

where E,G,andμare elastic modulus,shear modulus,and Poisson ratio,respectively.Subscript 3 represents the longitudinal direction,and subscripts 1 and 2 represent the two transverse directions,respectively.For the DS material,five independent constants are necessary to characterize the anisotropic elastic behavior.

In order to model the tertiary creep deformation,introducing a damage equation into the effective stress is a useful way.The damaged effective stress (σ_ij) is defined referring to strain equivalent principle [ 1] ,

whereσ_ij is stress tensor for virgin material;I is a fourthorder unit tensor;D is fourth-order damage tensor.Recently,based on the anisotropic continuum damage mechanics,a generic anisotropic continuum damage model has been proposed [ 18, 19] .However,since only longitudinal creep is concerned in the present study,for simplicity,an isotropic damage is selected.Thus,the damage in Eq.(4) is reduced to a scalar,and the expression of the effective stress is rewritten as:

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Table 1 Chemical compositions of DZ125 superalloy (wt%)

The Rabotnov-Kachanov law [ 20, 21, 22] is approved for the creep damage characterization [ 23, 24] .Such form of damage characterizing the tertiary creep behavior can be coupled into the effective stress in Chaboche’s model.The classical Rabotnov-Kachanov damage evolution is expressed as:

where is deviatoric components of effective stress,and J( ) refers to the second invariant of the effective stress.pa,pr and pk are three material parameters determined by the experimental creep deformation curves.A viscoplastic potential (Ψ) is adopted here using the same form as Chaboche's model [ 1] :

where the symbol<>is Heaviside function,defined as(u)=u if u>0 and (u)=0 if u=0;and K and n are material parameters.The yield function (F) is expressed as:

where are deviatoric components of back stress tensors,R(p) is isotropic hardening term depending on the plastic strain,and k₀ is temperature-dependent material constant,representing the initial yield stress.

The second deviatoric stress tensor invariant,J,is in the form of modified Von Mises type,which is:

where M_ijkl is a fourth-order tensor considering the yield surface anisotropy,which is used to describe the anisotropic characterization of the DS super alloy.Therefore,the modified inelastic strain rate is then obtained from Drucker's normality hypothesis and can be written as the following expression:

where p is the accumulated inelastic strain rate,and the expression is:

The Ohno-Wang modification is applied in the evolution equation for the back stress describing nonlinear kinematic hardening:

and

whereΦ(p)=Φ_s+(1-Φ_s)e^-ωp and C^(k)=c^(k)a^(k).Φ_s,ω,c,a,β,m and r are material parameters.Φ(p) is the function of accumulated inelastic strain considering the change of elastic stiffness during the cyclic loading condition.T and T are temperature and temperature rate,respectively.The first item in Eq.(13) refers to Prager’s linear hardening;the second item is dynamic recovery which is known as Ohno-Wang modification;the third item is describing static recovery;the forth additional item was used to consider the temperature effect.It was also introduced in the unified viscoplastic constitutive equations proposed by Chaboche [ 25] .Walker [ 10] ,Ohno and Wang [ 26] also discussed this subject.Ohno and Wang [ 26] pointed out that the hysteresis loops may shift unreasonably in stress when this temperature rate term is absent in the linear or nearly linear kinematic hardening.

To consider the temperature dependence of the isotropic hardening rule,Benallal and Cheikh [ 27] have proposed a temperature rate term.Thus,the isotropic hardening (R)evolution equation can be written in the following expression:

where b,Q,γ,and m are material parameters used to describe deformation induced by isotropic hardening,and Q_r is asymptotic value for the yield surface.The first term in Eq.(14) refers to the cyclic hardening or softening phenomenon.The second term describes time-dependent recovery effect.If the recovery term is neglect,the isotropic hardening reached steady state when R approaches its asymptotic value.In the present study,the second term is neglected because the cyclic hardening of DZ125 seems to be steady in a few cycles,i.e.,20cycles.The third term,suggested by Benallal and Cheikh [ 27] ,is added to consider temperature rate effect.Frenz et al. [ 28] also verified its necessity for the anisothermal conditions.

In order to consider material anisotropy,Tsai and Wu [ 29] proposed a general theory for the strength of anisotropic materials.A generalized forth tensor can be used to capture the yield surface of the anisotropic materials.If the interaction between the normal stress and shear stress is neglected,the anisotropic material tensor ([M]) can be written as:

Furthermore,the interaction between shear stress can be neglected,and Eq.(15) is simplified as:

For the symmetry of the orthotropic material,there are nine independent parameters in Eq.(16),which are M₁₁,M₂₂,M₃₃,M₁₂,M₁₃,M₂₃,M₄₄,M₅₅,and M_66.For the transversely isotropic materials,M₂₂=M₁₁,M₅₅=M₆₆,and M₁₃₌M_23.The number of the parameters is reduced to 6.Furthermore,it can be assumed that inelastic strain is incompressible,and the modified flow rule in the principal stress space should be reduced to uniaxial stress state in both of transverse and longitudinal orientations,the number of independent parameters in Eq.(16) is further reduced to 3 (i.e.,M₁₁,M₃₃ and M₅₅) for DS superalloys.Hence,they must satisfy the following equations:

To sum up,there are a total of 28 material parameters in the modified Chaboche model.E₁₁,E₃₃,G₂₃,μ₁₂,andμ₂₃are employed to describe the elastic behavior of DS superalloy;M₁₁,M₃₃,and M₅₅ are used to describe the material anisotropy;K,n,k₀,a¹,a²,c¹,and c² are employed to characterize the inelastic behavior of the material;b and Q are employed to describe the cyclic hardening and softening,corresponding to isotropic hardening;m¹,m²,Φ_s,ω,r¹,r²,β¹,andβ² are employed to describe mean stress relaxation corresponding to kinematic hardening;pa,pr,and pk are used to describe the tertiary creep behavior of the materials.

4 Simulations under isothermal and anisothermal loadings

The experimental data were used to fit model parameters and simulate model correlations by ABAQUS/UMAT.The fits are observed to agree with the experimental results in both of the longitudinal (L) and transverse (T) orientations over the entire temperature range for the baseline conditions.Furthermore,complex loadings,i.e.,creep-fatigue interaction and thermo-mechanical fatigue cycling,are also predicted in a reasonable way.

4.1 Tensile stress/strain response

Figure 1 shows the experimental and simulated uniaxial tensile stress-strain response of DZ125 superalloy at various temperatures.The results show that the simulations agree very well with the test results for all of the considered temperatures.The rate dependence in the longitudinal orientation was characterized by Chaboche's constitutive model.Owing to the induction of anisotropic material tensor,stress and strain responses in longitudinal and transverse orientations are both well modeled,and thus material anisotropy is well modeled through the modification of Chaboche's model.

Fig.1 Experimental (symbols) and simulated (lines) uniaxial tensile stress/strain responses of DZ125 superalloy at various temperatures in longitudinal (L) and transverse (T) orientations with strain rates of 1×10^-3 and 5×10^-3:a 20℃,b 650℃,c 760℃,d 850℃,and e 980℃

4.2 Cyclic stress/strain response

The strain-controlled fatigue tests with strain ratio of R=-1 and R=0 were used to characterize the cyclic hardening/softening and mean stress relaxation behavior,respectively.Meanwhile,the finite element method (FEM)simulations considering four temperatures (i.e.,20,760,850,and 980℃) and two crystallographic orientations(i.e.,longitudinal and transverse directions) were conducted by the modified Chaboche's constitutive model.The simulations at 650℃are not demonstrated for their similarity.

Figures 2 and 3 demonstrate the experimental and simulated results of the first hysteresis loop of DZ125superalloy with strain ratio of R=-1 at 20,760,850,and980℃,for longitudinal and transverse orientations,respectively.Figure 4 shows the predicted loops and experimental data with strain ratio of R=0,which is known as asymmetric cycle,at 760 and 980℃.The fittings are in good agreement with the experimental results in longitudinal and transverse orientations.As it can be seen from the test data,the anisotropic characterization of this material is significant.Thus,choosing the tensor (M) to modify the viscoplastic flow rule is a reasonable way for this anisotropic material.

The results for simulating the cyclic hardening/softening behavior of DZ125 superalloy in longitudinal and transverse orientations at various temperatures were compared with the experimental data,as shown in Fig.5.The experimental results reveal that this superalloy demonstrates cyclic hardening below 850℃and cyclic softening at higher temperatures,i.e.,980℃.The cyclic hardening or softening,which corresponds to an increase or decrease in the size of the elastic domain or yield surface,is usually described by introducing the isotropic hardening variable (R) [ 30] .It can be seen from Eq.(14)that Q>0 (Table 2,when temperatures below 850℃)will make R increase,which corresponds to the cyclic hardening as shown in Fig.5a-c,while Q<0 will decrease R,which corresponds to the cyclic softening as shown in Fig.5d.The results for simulated and experimental mean stress relaxation behavior of DZ125 superalloy in longitudinal and transverse orientations are presented in Fig.6.The test results reveal that mean stress relaxation in longitudinal orientation is much more significant than in transverse orientation at temperatures of below 850℃,while at higher temperatures,i.e.,980℃,the difference is small.The simulations are in agreement with the test data for most of the conditions except the case at 850℃where the mean stress is underestimated.It is appreciated that the trend is reasonable.It should be noted that the linear kinematic hardening is proportionate to the plastic strain in the classical Chaboche theory,which leads to excessive inelastic strain [ 31, 32] .When Ohno-Wang modification is introduced in the classical Chaboche theory,mean stress relaxation behavior can be described much better,as shown in Fig.6.

Fig.2 Stress/strain response with strain ratio of R=-1 and strain range of 1.6%in longitudinal direction at various temperatures:a 20℃,b 760℃,c 850℃,and d 980℃

4.3 Creep deformation simulation

The simulated creep strains of DZ125 DS superalloy in longitudinal orientation at different temperatures were compared with the experimental results,as shown in Fig.7.Since the creep-related parameters are optimized using the lowest and highest stresses (i.e.,430 and520 MPa,respectively) at 850℃,it is appreciated that the present constitutive model can predict creep curves at511 and 460 MPa very well.This situation is the same as that at 760℃.It should be noted that the original Chaboche's model is enough to model the primary and steady creep.However,as the Kachanov damage is coupled,the model is extended to obtain the ability of modeling the tertiary creep deformation.This is especially important at the higher temperature under relative low stress where the tertiary creep mechanism dominates the failure.Though introduction of Kachanove damage law in Chaboche's model characterizes tertiary creep deformation,the modification will result in a slightly faster damage accumulation compared to experimental results as shown in Fig.7.

Fig.3 Stress/strain responses with strain ratio of R=-1 and strain range of 1.6%in transverse orientation at various temperatures:a 20℃,b 760℃,c 850℃,and d 980℃

4.4 Simulation under complex loadings

4.4.1 Creep fatigue loadings

Strain-controlled low cycle fatigue with tensile dwell tests was conducted.For 850 and 980℃conditions,the dwell time is 120 and 60 s,respectively.The experimental results are shown in Fig.8.Owing to the creep deformation during the dwell time,the stresses are relaxed.The simulated results are also presented in Fig.8.The results reveal that the short-term stress relaxation response during strain holds is well captured.Since they were not used to determine model parameters,the dwell fatigue simulations can be considered as true predictions.The modeling results are reasonable.

4.4.2 Thermal mechanical loadings

TMF tests are considered a good experimental way to simulate the true loadings for the high-temperature components.Take the highly cooled turbine blade as an example,the relationship between strain and temperature is very complex.There are several standard profiles simplified as an approximation,among which the most important ones are out of phase (OP) and in phase (IP).The predicted and experimental IP and OP TMF hysteresis loops of DZ125 are presented in Fig.9.Since the material data at different temperatures are gained from the baseline isothermal tests,a Calmat subroutine is employed to linearly interpolate the model parameters at the non-experimental temperatures.The overall prediction is reasonable.The stress asymmetry is well modeled.Some researchers explained that the meaning stress discrepancy among different TMF cycles (i.e.,IP,OP) is an important factor affecting the fatigue life,and such accurate modeling method is of critical importance.However,there are some discrepancies between the prediction and test result.Since maximum temperature of the isothermal tests is 980℃,which is 20℃lower than the upper temperature of the predicted cycle,the parameter extrapolation will bring error.Additionally,limited by the temperature rate of the test setup,i.e.,10℃·s^-1,the TMF test was performed at the relative low strain rate,and this may be another factor influencing the accuracy.Because of the limited TMF response of DZ125 DS superalloy,another DS superalloy which is named GTD-111 is considered.

Fig.4 Stress/strain response at strain ratio of R=0 and strain range of 1.6%in longitudinal (L) and transverse (T) directions:a 760℃,L;b 760℃,T;c 980℃,L;d 980℃.T

Using the test results taken from Shenoy and Gordon's researches [ 16, 33] ,the constitutive model parameters can be identified for GTD-111 DS superalloy.The parameters for GTD-111 are listed in Table 3.It should be noted that the creep effect is neglected for the present study,and thus the corresponding parameters are absent.Figure 10 shows the temperature-dependent stress-strain responses in longitudinal direction which are predicted using linear interpolation of the constitutive parameters for GTD-111 superalloy.The results indicate that the predictions are changed monotonously with temperature increasing.The linear interpolation is a useful way for the anisothermal conditions for both of DZ125 and GTD-111 superalloys.Figure 11shows the experimental and modeling results of GTD-111 DS superalloy under OP TMF loadings.The model appears to characterize the overall trends of the test results.Some discrepancies may be due to parameters interpolation.Therefore,the present model is capable for the TMF hysteresis loop simulations.

4.5 TMF strain-stress analysis with different phase angles

In the actual turbine blade components,the strain and temperature vary at different locations as well as the service conditions of the gas turbine engine.This is much different from the fatigue test using standard specimens.In the present section,eight temperaturephase angles of TMF loadings are considered,which are-45°,-90°,and-135°CCD;45°,-90°,and 135°CD;IP and OP,as shown in Fig.12.In Sect.4.4,simulations under IP and OP TMF loadings are shown to be coincided with the experimental data.However,since test results under other phase angles are absent for DZ125,135°CD TMF test results of a single-crystal superalloy [ 34] were used for comparison.As shown in Fig.13,the shapes of TMF cycle under simulation and test data are almost the same.TMF simulations under eight different phase angles are demonstrated in Fig.14.From the simulations,it can be concluded:(1) the stress-strain response shapes are closely related with the phase angle,due to different temperature-strain histories;(2) the mean stresses as well as the stress ranges are also strongly dependent on the TMF phases.In Fig.15,the mean stress and stress range at stable cycle were plotted against phases.It should be noted that under CCD cycles,positive mean stresses were developed.Owing to relative low temperature under tensile stresses,the brittle oxides are easy to crack,resulting in early failures of the test specimens as well as the real components.

Fig.5 Experimental (symbols) and simulated (lines) stress ranges (Δσ) versus cycles (N) in longitudinal (L) and transverse (T) directions with strain ratio of R=-1 at various temperatures:a 650℃,b 760℃,c 850℃,and d 980℃

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Table 2 Constitutive model parameters of DZ125 DS superalloy at various temperatures

Fig.6 Experimental (symbols) and simulated (lines) mean stress (σm) versus cycles (N) in longitudinal (L) and transverse (T) directions with strain ratio of R=0 at various temperatures:a 650℃,b 760℃,c 850℃,and d 980℃

Fig.7 Experimental (symbols) and simulated (lines) creep strain in longitudinal direction of DZ125 DS superalloy at various temperatures:a 760℃,b 980℃,and c 850℃

Fig.8 Stress-strain response under creep fatigue loadings in longitudinal orientation with strain range of 1.6%:a 850℃(1st cycle),tensile hold time of 120 s,strain ratio of R=-0.875,and strain range of 1.5%;b 980℃(2nd cycle),tensile hold time of 60 s,strain ratio of R=0,and strain range of 1.2%

Fig.9 Stress-strain response for DZ125 DS superalloy under OP TMF 500-1000℃in longitudinal orientation,comparison of experimental data with model prediction:a IP and b OP

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Table 3 Constitutive model parameters of GTD-11 1 DS superalloy at various temperatures

Fig.10 Predicted stress and strain responses of GTD-111 DS superalloy in longitudinal orientation at various temperatures using linear interpolation of constitutive parameters

Fig.11 Stress-strain response of GTD-111 DS superalloy under OP TMF 538-927℃in longitudinal orientation (1st cycle),comparison of experimental data with model predictions:a predicted by present model and b predicted in Refs. [16,17] .

Δεand R_εbeing strain rate,strain range,and strain ratio

Fig.12 Various temperature-phase angles for TMF loadings:a-45°,-90°and-135°CCD;b 45°,-90°and 135°CD;c IP and OP

Fig.13 Stress-strain response simulation in longitudinal orientation of DZ125 under 135°CD TMF loadings a and its comparison with AM1superalloy at<001>direction b.Where Aε_m andε_m being strain range and strain

Fig.14 Stress-strain responses of DZ125 in longitudinal orientation with cycle time of 100 s and strain amplitude (ε_a) of 0.8%at 500-1000℃for various phase angles:a IP,b-45°,c-90°,d-135°,e 180°,f 135°,g 90°,and h 45°

Fig.15 Stress evolution versus phase angle in stable TMF cycle (10th cycle) at different strain ranges:a stress range and b mean stress

5 Conclusion

In the present study,the viscoplastic deformation of DS superalloy was experimentally carried out.The effect of temperature,crystallographic orientation,and strain rate on the tensile,cyclic and creep behaviors were all considered.A modified Chaboche's constitutive model was developed to capture the stress-strain response in the longitudinal and transverse orientations.The creep fatigue and TMF tests were used to validate the accuracy of the model.Since most of the simulations and predictions are reasonable,this model is a useful way for the component level analysis considering complex loading histories.Furthermore,the phase angle-dependent stress-strain responses under TMF loading were also discussed.Owing to different temperature-strain histories,the stress ranges and mean stresses are both strongly related with the phase angle.Thus,it is very important to capture this information when analyzing the engineer components.