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Correlation between grain size and flow stress during steady-state dynamic recrystallization
Materials Science & Engineering A 638 (2015) 357–362
Contents lists available at ScienceDirect
Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
Correlation between grain size and flow stress during steady-state dynamic recrystallization Houquan Liang a,n, Hongzhen Guo a, Kai Tan a, Y.Q. Ning a, Xin Luo a, Gang Cao a, Jiajun Wang a,b, Pengliang Zhen a a b
School of Materials Science and Engineering; Northwestern Polytechnical University, 127 Youyixi Street; Xi’an 710012, PR China China Electronics Technology Group Corporation No. 39 Research Institute, 30, 3nd Zhangba Road; Xi’an 710077, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 26 December 2014 Received in revised form 18 April 2015 Accepted 19 April 2015 Available online 29 April 2015
The mechanical behaviour and microstructure evolution during hot deformation remain a fundamental industrial topic for metallic materials. In the present paper, the correlation between grain size and flow stress during steady-state dynamic recrystallization (DRX) has been investigated from an internal-variable perspective. From the competition between work-hardening and recrystallization softening, a dynamic model of DRX evolution has been generalized by proposing an identification criterion on DRX continuity or periodicity. The dependence of grain size on deformation parameters under the DRX steady state has been obtained from the dynamic balance. An Arrhenius constitutive model for the DRX steady state has also been constructed from the Estrin-Mecking formula, and the stress exponent has been fixed as a constant equal to 6. The correlation between grain size and flow stress has been validated as an inverse proportion with the relevant exponent n¼1/2. In addition, the influence of deformation parameters on flow curve shapes and grain coarsening/refinement has been characterized correspondingly. & 2015 Elsevier B.V. All rights reserved.
Keywords: Grain size Steady state Dynamic recrystallization (DRX) Constitutive model Derby function
1. Introduction The thermomechanical processing of metallic materials has attracted considerable interest from the research community and industry due to the integrated influence of plastic deformation and microstructural evolution [1]. Dynamic recrystallization plays a significant role in the thermomechanical processing of metals owing to the improvement in component properties through stress relief and texture control, such as enhancing grain refinement and promoting matrix uniformity [2,3]. Both the mechanical behaviour and microstructure evolution are significantly influenced by DRX evolution during hot deformation, which directly determine the processing path and ultimate component performance. Therefore, the mechanical and microstructural investigation of DRX evolution is of fundamental technological importance for the hot deformation of metallic materials. The complexity of the mechanical behaviour and microstructure evolution is due to the concurrent effects of work-hardening (WH), dynamic recovery (DRV) and recrystallization during hot deformation [4]. The mechanical behaviour exhibits a corresponding response to different microstructure evolutions, with the flow stress clearly increasing during WH and decreasing with DRX softening. The DRX
n
Corresponding author. Fax: þ 86 29 88493744. E-mail address: [email protected] (H. Liang).
http://dx.doi.org/10.1016/j.msea.2015.04.046 0921-5093/& 2015 Elsevier B.V. All rights reserved.
steady state, a dynamic balance of work-hardening and DRX softening at larger strains of hot deformation, remains a metallurgical topic due to the establishment of an ultimate stress level and microstructure state. Once the DRX steady state is achieved during hot deformation, a constant average grain size and flow stress could be reached with some interdependence [5]. Several previous papers have investigated the steady-state mechanical behaviour and microstructural features during hot deformation undergoing DRX inform the experimental and theoretical perspectives. Frommert and Gottstein [6] demonstrated the dependence of flow stress and grain size on deformation parameters during steady-state DRX through the isothermal compression of austenitic steel 800 H. The single peak and multiple peaks in flow curves have been individually observed with low and high Zener-Hollomon parameters (Z ¼ ε_ U expðQ =RTÞ1). In addition, the reverse relationship between DRX grain size DS and steady-state stress has been obtained under steady state, and the relevant exponent between lnσ and lnDS has been calculated to be less than 1. Graetz et al. [7] characterized the DRX steady-state behaviour during hot deformation for pure copper and 800 H steel. The DRX grain size sensitivity to flow stress at steady state has been identified as inversely proportional, with the relevant exponent fixed at 1/2. Their research results also confirmed 1 ε_ and T denotes the strain rate and deformation temperature, respectively. R is the gas constant, and Q represents the deformation activation energy.
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the inheritance relationship between DRX grain size and the deformation-induced subgrain size. Furthermore, the grain size distribution during steady-state DRX has been evaluated and found to remain constant. Huang et al. [8] constructed a single irreversible thermodynamics-based formulation to discuss the correlation between steady-state stress and deformation parameters during hot deformation. The influence of DRX and DRV on dislocation annihilation has been comprehensively considered. The steady-state stress dependence on temperature and strain rate has been characterized by introducing Zener-Hollomon parameters under experimental verification by pure copper creep loading. Montheillet et al. [9] developed a grain-scale approach to demonstrate the mechanical and microstructural behaviour during steady-state discontinuous dynamic recrystallization (DDRX) by isothermal compression of 304 L stainless steel. The steady-state stress and grain size have been observed with less sensitivity to the initial microstructure conditions than to the deformation parameters. Lin et al. [10] investigated the DRX kinetics for the isothermal deformation of a Ni-based superalloy, and a corresponding physical constitutive model has been proposed to characterize the influence of deformation parameters on the flow stress. A stable stress variation has been observed at larger strains, and its relationship with the steady-state DRX grain size has been demonstrated by introducing Zener-Hollomon parameters. In the present paper, the dependence of the grain size on flow stress under the dynamic recrystallization steady state has been investigated for the hot deformation of metallic materials. Their correlation has been analysed from the dynamic balance achieved by DRX grain expansion and shrinkage. The dynamic model of DRX continuity or periodicity has been generalized by constructing an identification criterion from the deformation parameters. The interdependence between flow stress and grain size under steady-state DRX, characterized by the Derby function, has been interpreted from an internal-variable perspective. An Arrhenius constitutive model for the DRX steady state has been proposed using the Estrin-Mecking formula, and the stress exponents are discussed. The influence of the deformation parameters on the DRX steady-state stress and grain size has also been characterized correspondingly.
2. Function and verification A dynamic balance between work-hardening and DRX softening is achieved under the DRX steady state of hot deformation, where the flow stress and DRX grain size are both stable with strain [11]. There exists a phenomenological function to characterize the dependence between these balanced variables, as illustrated by Fig. 1, proposed by Derby [12]: σ DS n ¼K ð1Þ μ b where DS denotes the DRX steady-state grain size. m is the material shear modulus. σ/m represents the normalized flow stress. b is the magnitude of the Burgers vector. K denotes a constant, usually within the range from 1 to 10, and n is a relevant exponent equal to 1/2 2/3. The Derby function has been validated with excellent application for the hot deformation of various alloy systems [12,13]. Stress-strain curves for the isothermal compression of austenitic steel alloy 800 H [6,7] are exhibited in Fig. 2 under different conditions, with the steady state achieved at a true strain of approximately 0.4. With increasing temperature T or decreasing strain rate ε_ , the flow curves were shifted to lower stress levels. An obvious stress decrease after the peaks in most curves could be observed as the result of DRX softening. A notable difference could also be identified for the DRX steady state in that stress fluctuations (multiple peaks) appeared in flow curves during hot deformation with high temperatures and low
Fig. 1. Relationship between flow stress normalized by the shear modulus and DRX grain size normalized by the Burgers vector for metallic materials, in verification of the Derby function.
strain rates, while curves with the opposite conditions exhibited a continuous softening trend after the stress peak (single peak). This phenomenon could be associated with the dependence of the DRX type on the deformation parameters. Fig. 3 demonstrates the microstructural and textural evolution under continuous deformation at strains of (a) 50%, (b) 70% and (c) 90% for alloy 800 H deformed with ε_ ¼ 0:001s 1 at 1100 1C. A true strain of 0.4 during hot deformation, with the DRX steady state achieved from flow curves in Fig. 1, corresponds to a 70% deformation. From comparison between strains of 50% and 70%, the enhanced deformation level could provide sufficient distortion storage for boundary combination and migration, correspondingly resulting in an obvious DRX-formed grain growth. A clear grain expansion could be identified in the o0 0 1 4 component owing to DRX directional growth under a fixed driving pressure gradient (from comparison between Fig. 3(a) and (b)). With the DRX steady state established, a relatively stable grain size could be maintained. Little grain size variation could be observed from 70% to 90% deformation, with a certain orientation distribution change (from comparison between Fig. 3(b) and (c)). Fig. 4 exhibits the microstructural and textural evolution for alloy 800 H [6] deformed at 1100 1C with (a) ε_ ¼ 0:01s 1 at 98%; (b) ε_ ¼ 0:001s 1 at 98%; (c) strain rate change from 0.001 s 1 to 0.01 s 1 at 56% and further deformation to a total strain of 96%; (d) strain rate change from 0.01 s 1 to 0.001 s 1 at 57%, total strain 96%. From comparison between Fig. 4(a) and (b), lower strain rates result in a prolonged deformation time for sufficient grain growth under high temperatures. A clear grain size sensitivity to strain rate could be observed as the DRX-formed grain expands from high to low rate. Furthermore, lower strain rates mean a lower deformation resistance under the same conditions. Therefore, the correlation between the grain size and flow stress under different strain rates keeps consistent with the Derby function, in which lower stress levels correspond to larger DRX grain sizes. Comparing Fig. 4(a, c) and (b, d), respectively, an interesting phenomenon could be observed in which the DRX steady-state grain size is insensitive to the processing route of deformed materials and is only significantly dependent on the deformation parameters at steady state. This phenomenon could also validate the Derby function, showing that the same stress levels under the DRX steady state correspond to similar grain sizes during hot deformation.
H. Liang et al. / Materials Science & Engineering A 638 (2015) 357–362
359
Fig. 2. Typical flow curves of austenitic steel alloy 800 H at different strain rates and temperatures obtained from isothermal compression.
Fig. 3. EBSD orientation images and inverse pole figures of the microstructure and texture for the respective grain size distributions of austenitic steel 800 H deformed with ε_ ¼ 0:001s 1 at T ¼ 1100 1C to strains of (a) 50%, (b) 70% and (c) 90% (This graph was taken from K. Graetz, C. Miessen, G. Gottstein, Acta Mater 67 (2014) 58–66.).
With the assumption that non-distortion DRX grains nucleate and bulge spherically from the deformed portion, the thermodynamic condition for DRX steady state could be demonstrated as the balance between the boundary surface energy increases and the volume free energy decreases as [13]:
ΔG ¼ γ GB U8π RdR E U 4π R2 dR ¼ 0
with
1 2 E ¼ μb ρSS 2
DS U σ 2SS ¼ 4α2 M 2 μ U γ GB
ð5Þ
The same form with the Derby function could be directly identified from Eq.(5), with the relevant exponent fixed as n ¼1/2.
ð2Þ
where R represents the DRX grain radius. γGB and E denote the unit surface energy and volume free energy, respectively. The DRX steady-state grain radius could be obtained from Eq.(2) as (ρSS denotes the steady-state dislocation density.): 2γ RS ¼ GB E
obtained as:
ð3Þ
The flow stress variation attributed to dislocation evolution and interaction could be quantified by the Taylor function as [14]: pffiffiffi σ ¼ α M μb ρ ð4Þ where α is a Taylor constant equal to 0.5, and M is the conversion factor between the shear and normal strains, 3.06 [14]. With Eq.(4) substituted into Eq.(3), a quantitative correlation between the DRX steady-state stress σ SS and grain size DS could be
3. Correlation interpretation 3.1. DRX type identification A dynamic model of the DRX process during hot deformation was proposed by Stüwe et al. [15] informs a dislocation evolution perspective. In a dynamic model, the DRX process could be activated by a critical dislocation density ρC achieved from work hardening. With DRX the activated and conducted, the distortion storage is consumed by substructure formation and boundary migration. The dislocation density in the deformed portion would decrease, and newly formed grains from the DRX could be considered with no distortion. The density decrease leads to the lack of a driving pressure for the progress of the DRX. The DRX process would be suppressed once the dislocation density decreases lower than a critical value. However, the DRX process occurs during the ongoing hot deformation,
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Fig. 4. EBSD orientation images and inverse pole figures of the microstructure and texture for the respective grain size distributions of austenitic steel 800 H deformed at 1100 1C: (a) with ε_ ¼ 0:01s 1 at 98%; (b) with ε_ ¼ 0:001s 1 at 98%; (c) strain rate change from 0.001 s 1 to 0.01 s 1 at 56% and further deformation to a total strain of 96%; (d) strain rate change from 0.01 s 1 to 0.001 s 1 at 57%, total strain 96% (This graph was taken from M. Frommert, G. Gottstein, Mater. Sci. Eng. A 506 (2009) 101–110.).
and uninterrupted dislocation supplements are produced from the continuous straining. Assumed that the dislocation density increases at a rate of ρ_ , whether the DRX critical density storage is maintained or re-achieved directly determines the progress of the DRX. With the DRX steady-state grain size DR established, if the dislocation density is always maintained beyond ρC during the hot deformation, the DRX softening will occur continuously, referred to as continuous dynamic recrystallization (CDRX). However, if the dislocation density could not be re-accumulate to ρC again because of the softening effect, the DRX process would temporarily cease, with the work-hardening behaviour accelerated in the distortion that is in preparation for the next DRX onset. This intermittent but regular occurrence of DRX is characterized as periodic dynamic recrystallization (PDRX). With t R representing the recrystallization time, the identification of DRX type during the steady state could be expressed as: 8 ρ_ U t R o ρC ) PDRX > < ρ_ U t R ¼ ρC ) Balance > : ρ_ U t 4 ρ ) CDRX R C
ð6Þ
Demonstrated from flow curves, the various curve shapes correspond to different DRX types with different dislocation evolution. Consistent with the periodic hardening and softening influence during PDRX, the flow stress correspondingly exhibits obvious fluctuations after a stress peak (multiple peaks). On the other hand, for deformation undergoing CDRX, a continuous decrease (single peak) could be identified in flow curves after a stress peak owing to an uninterrupted DRX softening effect. Fig. 5
Fig. 5. Schematic diagram for flow curve shapes attributed to different DRX types during hot deformation.
illustrates a schematic diagram for flow curve shapes incorporating different DRX types. The driving pressure for the directional migration of boundaries arises from the distortion difference between the deformed portion with the DRX critical dislocation density and the adjacent DRX nuclei with no distortion. According to dynamics theory, the migration rate for the grain boundaries newly formed during DRX could be given as [15,16]: v ¼ M BM U ΔP
with
1 2
ΔP ρC μb2
ð7Þ
H. Liang et al. / Materials Science & Engineering A 638 (2015) 357–362
where MBM represents the boundary mobility. The mobility capacity increases with the temperature owing to the diffusion dependence. The boundary mobility is usually assumed to be proportional to the self-diffusivity at the grain boundaries, which could be calculated by [13]: Q dif f u βδDGB V m M BM ¼ ð8Þ with D ¼ D Uexp GB 0 2 RT b R where DGB denotes the self-diffusion coefficient along the grain boundary, and Qdiffu represents its activation energy. D0 is the selfdiffusivity at absolute zero 0 K. δ is the grain boundary thickness, taken equal to 0.5 1 nm. Vm is the molar volume, and β is a fraction parameter usually estimated to be between 0.1 and 0.5. With Eqs.(7, 8) combined, the recrystallization time t R for DRX steady-state grain size formation could be approximately evaluated as: Q dif f u DS 2R U DS tR ¼ ð9Þ ¼ U ρC 1 Uexp ν βμδV m D0 RT According to the Orowan equation [17], the dislocation generation originates from the continuous trapping by existing dislocations and could be quantified by assuming the mean free slip distance, the dislocation that the Burgers vector b could migrate, equal to L: dρ ε_ ¼ dt bL
ð10Þ
By substituting Eqs.(9,10) into Eq.(6), the dynamic balance condition between the DRX grain expansion and shrinkage could be described as: Q dif f u 2R UDS ¼ ρ2C U ε_ exp ð11Þ RT βμδV M D0 bL Jonas et al. [18] analysed twenty-six flow curves determined under DRX conditions on different steels to characterize the recrystallization kinetic behaviour. The stress levels from the DRX criticality and steady state are considered similar because the critical condition exactly corresponds to the balanced state between work-hardening and DRX softening as: σ C σ SS . With the dislocation density substituted by Eq.(4), Eq.(11) could be re-written as:
κ R U DS U ε_ expðQ dif f u =RTÞ ¼ σ 4SS with κ R ¼
2α4 μ3 M 4 b R βδV m D0 bL
361
pffiffiffi Owing to the similar magnitudes of RC and 1= ρ, an equivalent result could be derived from equivalent infinitesimal mathematical theory as: R4C ρ2 -0 ) 1 expð 0:7R4C ρ2 Þ 0:7R4C ρ2
ð15Þ
Upon substituting Eq.(15) into Eq.(14), the expression for the annihilation coefficient Ω could be estimated as:
ε_ n Ω Ω0 _ U 0:7R4C ρ2 Uexp ε
Q dif f u RT
ð16Þ
The dynamic balance at the DRX steady state is characterized by the dislocation variation rate dropping to zero as dρ ¼ h Ω U ρ ¼ 0 from the Estrin-Mecking model. dε SS
With the combination of Eqs.(4,16), the dependence of the steady-state stress on the deformation parameters could be obtained in an Arrhenius power type equation:
σ 6SS ¼ κ σ U ε_ U expðQ dif f u =RTÞ with κ σ ¼
1:43α5 μ5 M 5 b h Ω0 R4C ε_ n 5
ð17Þ
The stress exponent, dðln ε_ Þ=dðln σ Þ, could be directly obtained by taking the natural logarithm on Eq.(17) as a constant equal to 6. Fig. 6 demonstrates the sensitivity of the steady-state stress to the strain rate by a linear regression analysis between ln σ SS and ln ε_ according to the experimental data from [22]. It could be validated for the hot deformation of pure copper that the stress exponent remains as a constant close to 6, insensitive to the processing parameters. In addition, Table 1 lists the stress exponent, calculated by peer studies, for hot deformation on various metallic materials, all consistent with the fixed stress exponent of 6, as given in Eq.(17). With Eq.(17) substituted into Eq.(12) to characterize the dependence of DRX grain size on the deformation parameters, the identification criterion for the DRX balanced state, Eq.(6),
3
ð12Þ
3.2. Steady-state constitutive With the assumption of the dislocation distributing uniformly in a deformed matrix, the Estrin-Mecking relation [19] has been validated with excellent consistency for the dislocation density variation during hot deformation undergoing DRX: dρ ¼ h Ωρ dε
ð13Þ
where h denotes the dislocation multiplication coefficient, related to the lattice structure and mechanical parameters. Ω represents the annihilation coefficient, sensitive to deformation conditions [20]. With the assumption that the probability of two opposite-sign dislocation recombinations satisfies the requirement of p ¼ 1 expð ρ=ρÞ2 , Picu and Majorell [21] proposed a thermal model to describe the annihilation coefficient Ω as: Q ε_ n ð14Þ Ω ¼ Ω0 _ ð1 expð 0:7R4C ρ2 ÞÞ U exp dif f u ε RT where Ω0 is a proportionality constant And RC denotes the critical radius beyond which dislocations cannot cross-slip and recombine.
Fig. 6. Stress sensitivity to strain rate obtained from the linear regression between ln σ SS and ln ε_ , with pure copper deformation as a precedent.
Table 1 Typical stress exponent n for the hot processing of different alloys.
Stress Exponent
TC18 Ti [23]
Fe-Mn- Ni55Fe19Ga26[26] Superalloy AA7075 ZK 60 Mg Al steel [27] Al[28] [25] [24]
5.6
6
5.5
5.2
5.4
5.7
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The reverse relationship between the DRX steady-state stress
and the Zener-Hollomon parameters has been obtained, with the correlation exponent fixed at 1/3. Multiple peaks in flow curves and grain coarsening during hot deformation have been identified as occurring under high temperatures and low strain rates, as characterized by PDRX. Single peak and grain refinement occurs with the opposite deformation conditions for the CDRX effect.
Acknowledgement This work was supported by the High-end CNC Machine Tools and Basic Manufacturing Equipment project (2012 ZX 04010-081). Dr. Y.Q. Ning is supported by the Hong Kong Scholar Program (Grant No. XJ2014047). Fig. 7. Dependence of flow curve shape and microstructure evolution on deformation parameters with the identification of the DRX type.
could be illustrated as: DS U Z 1=3 ¼ κ σ =κ R ¼ Const 2=3
with
Z ¼ ε_ U expðQ =RTÞ
ð18Þ
The reverse relationship between the DRX steady-state grain size and the Zener-Hollomon parameters could be directly obtained from Eq.(18), and the correlation exponent 1/3 gives a good agreement with the experimental result by [29]. Fig. 7 exhibits a deformation mechanism schematic map to demonstrate the dependence of the flow curve shape and microstructure evolution on the deformation parameters. Characterized by Eq.(18) and Eq.(6), multiple fluctuations in the PDRX and grain coarsening occur with low Zener-Hollomon parameters under high temperatures and low strain rates, and vice versa for single peak and grain refinement during CDRX. By combining Eqs.(12, 17), the ultimate formula relating the flow stress and grain size under the DRX steady state during hot deformation could be obtained as: DS U σ 2SS ¼ κ σ =κ R
ð19Þ
with the Derby function validated and interpreted.
4. Conclusion
The dependence of grain size on flow stress under DRX steady
state during the hot deformation of metallic materials has been investigated with validation and interpretation of the Derby function. An Arrhenius constitutive model has been constructed to characterize the dependence of the DRX steady-state stress on the deformation parameters, with the stress exponent fixed as a constant equal to 6.
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Contents lists available at ScienceDirect
Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
Correlation between grain size and flow stress during steady-state dynamic recrystallization Houquan Liang a,n, Hongzhen Guo a, Kai Tan a, Y.Q. Ning a, Xin Luo a, Gang Cao a, Jiajun Wang a,b, Pengliang Zhen a a b
School of Materials Science and Engineering; Northwestern Polytechnical University, 127 Youyixi Street; Xi’an 710012, PR China China Electronics Technology Group Corporation No. 39 Research Institute, 30, 3nd Zhangba Road; Xi’an 710077, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 26 December 2014 Received in revised form 18 April 2015 Accepted 19 April 2015 Available online 29 April 2015
The mechanical behaviour and microstructure evolution during hot deformation remain a fundamental industrial topic for metallic materials. In the present paper, the correlation between grain size and flow stress during steady-state dynamic recrystallization (DRX) has been investigated from an internal-variable perspective. From the competition between work-hardening and recrystallization softening, a dynamic model of DRX evolution has been generalized by proposing an identification criterion on DRX continuity or periodicity. The dependence of grain size on deformation parameters under the DRX steady state has been obtained from the dynamic balance. An Arrhenius constitutive model for the DRX steady state has also been constructed from the Estrin-Mecking formula, and the stress exponent has been fixed as a constant equal to 6. The correlation between grain size and flow stress has been validated as an inverse proportion with the relevant exponent n¼1/2. In addition, the influence of deformation parameters on flow curve shapes and grain coarsening/refinement has been characterized correspondingly. & 2015 Elsevier B.V. All rights reserved.
Keywords: Grain size Steady state Dynamic recrystallization (DRX) Constitutive model Derby function
1. Introduction The thermomechanical processing of metallic materials has attracted considerable interest from the research community and industry due to the integrated influence of plastic deformation and microstructural evolution [1]. Dynamic recrystallization plays a significant role in the thermomechanical processing of metals owing to the improvement in component properties through stress relief and texture control, such as enhancing grain refinement and promoting matrix uniformity [2,3]. Both the mechanical behaviour and microstructure evolution are significantly influenced by DRX evolution during hot deformation, which directly determine the processing path and ultimate component performance. Therefore, the mechanical and microstructural investigation of DRX evolution is of fundamental technological importance for the hot deformation of metallic materials. The complexity of the mechanical behaviour and microstructure evolution is due to the concurrent effects of work-hardening (WH), dynamic recovery (DRV) and recrystallization during hot deformation [4]. The mechanical behaviour exhibits a corresponding response to different microstructure evolutions, with the flow stress clearly increasing during WH and decreasing with DRX softening. The DRX
n
Corresponding author. Fax: þ 86 29 88493744. E-mail address: [email protected] (H. Liang).
http://dx.doi.org/10.1016/j.msea.2015.04.046 0921-5093/& 2015 Elsevier B.V. All rights reserved.
steady state, a dynamic balance of work-hardening and DRX softening at larger strains of hot deformation, remains a metallurgical topic due to the establishment of an ultimate stress level and microstructure state. Once the DRX steady state is achieved during hot deformation, a constant average grain size and flow stress could be reached with some interdependence [5]. Several previous papers have investigated the steady-state mechanical behaviour and microstructural features during hot deformation undergoing DRX inform the experimental and theoretical perspectives. Frommert and Gottstein [6] demonstrated the dependence of flow stress and grain size on deformation parameters during steady-state DRX through the isothermal compression of austenitic steel 800 H. The single peak and multiple peaks in flow curves have been individually observed with low and high Zener-Hollomon parameters (Z ¼ ε_ U expðQ =RTÞ1). In addition, the reverse relationship between DRX grain size DS and steady-state stress has been obtained under steady state, and the relevant exponent between lnσ and lnDS has been calculated to be less than 1. Graetz et al. [7] characterized the DRX steady-state behaviour during hot deformation for pure copper and 800 H steel. The DRX grain size sensitivity to flow stress at steady state has been identified as inversely proportional, with the relevant exponent fixed at 1/2. Their research results also confirmed 1 ε_ and T denotes the strain rate and deformation temperature, respectively. R is the gas constant, and Q represents the deformation activation energy.
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the inheritance relationship between DRX grain size and the deformation-induced subgrain size. Furthermore, the grain size distribution during steady-state DRX has been evaluated and found to remain constant. Huang et al. [8] constructed a single irreversible thermodynamics-based formulation to discuss the correlation between steady-state stress and deformation parameters during hot deformation. The influence of DRX and DRV on dislocation annihilation has been comprehensively considered. The steady-state stress dependence on temperature and strain rate has been characterized by introducing Zener-Hollomon parameters under experimental verification by pure copper creep loading. Montheillet et al. [9] developed a grain-scale approach to demonstrate the mechanical and microstructural behaviour during steady-state discontinuous dynamic recrystallization (DDRX) by isothermal compression of 304 L stainless steel. The steady-state stress and grain size have been observed with less sensitivity to the initial microstructure conditions than to the deformation parameters. Lin et al. [10] investigated the DRX kinetics for the isothermal deformation of a Ni-based superalloy, and a corresponding physical constitutive model has been proposed to characterize the influence of deformation parameters on the flow stress. A stable stress variation has been observed at larger strains, and its relationship with the steady-state DRX grain size has been demonstrated by introducing Zener-Hollomon parameters. In the present paper, the dependence of the grain size on flow stress under the dynamic recrystallization steady state has been investigated for the hot deformation of metallic materials. Their correlation has been analysed from the dynamic balance achieved by DRX grain expansion and shrinkage. The dynamic model of DRX continuity or periodicity has been generalized by constructing an identification criterion from the deformation parameters. The interdependence between flow stress and grain size under steady-state DRX, characterized by the Derby function, has been interpreted from an internal-variable perspective. An Arrhenius constitutive model for the DRX steady state has been proposed using the Estrin-Mecking formula, and the stress exponents are discussed. The influence of the deformation parameters on the DRX steady-state stress and grain size has also been characterized correspondingly.
2. Function and verification A dynamic balance between work-hardening and DRX softening is achieved under the DRX steady state of hot deformation, where the flow stress and DRX grain size are both stable with strain [11]. There exists a phenomenological function to characterize the dependence between these balanced variables, as illustrated by Fig. 1, proposed by Derby [12]: σ DS n ¼K ð1Þ μ b where DS denotes the DRX steady-state grain size. m is the material shear modulus. σ/m represents the normalized flow stress. b is the magnitude of the Burgers vector. K denotes a constant, usually within the range from 1 to 10, and n is a relevant exponent equal to 1/2 2/3. The Derby function has been validated with excellent application for the hot deformation of various alloy systems [12,13]. Stress-strain curves for the isothermal compression of austenitic steel alloy 800 H [6,7] are exhibited in Fig. 2 under different conditions, with the steady state achieved at a true strain of approximately 0.4. With increasing temperature T or decreasing strain rate ε_ , the flow curves were shifted to lower stress levels. An obvious stress decrease after the peaks in most curves could be observed as the result of DRX softening. A notable difference could also be identified for the DRX steady state in that stress fluctuations (multiple peaks) appeared in flow curves during hot deformation with high temperatures and low
Fig. 1. Relationship between flow stress normalized by the shear modulus and DRX grain size normalized by the Burgers vector for metallic materials, in verification of the Derby function.
strain rates, while curves with the opposite conditions exhibited a continuous softening trend after the stress peak (single peak). This phenomenon could be associated with the dependence of the DRX type on the deformation parameters. Fig. 3 demonstrates the microstructural and textural evolution under continuous deformation at strains of (a) 50%, (b) 70% and (c) 90% for alloy 800 H deformed with ε_ ¼ 0:001s 1 at 1100 1C. A true strain of 0.4 during hot deformation, with the DRX steady state achieved from flow curves in Fig. 1, corresponds to a 70% deformation. From comparison between strains of 50% and 70%, the enhanced deformation level could provide sufficient distortion storage for boundary combination and migration, correspondingly resulting in an obvious DRX-formed grain growth. A clear grain expansion could be identified in the o0 0 1 4 component owing to DRX directional growth under a fixed driving pressure gradient (from comparison between Fig. 3(a) and (b)). With the DRX steady state established, a relatively stable grain size could be maintained. Little grain size variation could be observed from 70% to 90% deformation, with a certain orientation distribution change (from comparison between Fig. 3(b) and (c)). Fig. 4 exhibits the microstructural and textural evolution for alloy 800 H [6] deformed at 1100 1C with (a) ε_ ¼ 0:01s 1 at 98%; (b) ε_ ¼ 0:001s 1 at 98%; (c) strain rate change from 0.001 s 1 to 0.01 s 1 at 56% and further deformation to a total strain of 96%; (d) strain rate change from 0.01 s 1 to 0.001 s 1 at 57%, total strain 96%. From comparison between Fig. 4(a) and (b), lower strain rates result in a prolonged deformation time for sufficient grain growth under high temperatures. A clear grain size sensitivity to strain rate could be observed as the DRX-formed grain expands from high to low rate. Furthermore, lower strain rates mean a lower deformation resistance under the same conditions. Therefore, the correlation between the grain size and flow stress under different strain rates keeps consistent with the Derby function, in which lower stress levels correspond to larger DRX grain sizes. Comparing Fig. 4(a, c) and (b, d), respectively, an interesting phenomenon could be observed in which the DRX steady-state grain size is insensitive to the processing route of deformed materials and is only significantly dependent on the deformation parameters at steady state. This phenomenon could also validate the Derby function, showing that the same stress levels under the DRX steady state correspond to similar grain sizes during hot deformation.
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Fig. 2. Typical flow curves of austenitic steel alloy 800 H at different strain rates and temperatures obtained from isothermal compression.
Fig. 3. EBSD orientation images and inverse pole figures of the microstructure and texture for the respective grain size distributions of austenitic steel 800 H deformed with ε_ ¼ 0:001s 1 at T ¼ 1100 1C to strains of (a) 50%, (b) 70% and (c) 90% (This graph was taken from K. Graetz, C. Miessen, G. Gottstein, Acta Mater 67 (2014) 58–66.).
With the assumption that non-distortion DRX grains nucleate and bulge spherically from the deformed portion, the thermodynamic condition for DRX steady state could be demonstrated as the balance between the boundary surface energy increases and the volume free energy decreases as [13]:
ΔG ¼ γ GB U8π RdR E U 4π R2 dR ¼ 0
with
1 2 E ¼ μb ρSS 2
DS U σ 2SS ¼ 4α2 M 2 μ U γ GB
ð5Þ
The same form with the Derby function could be directly identified from Eq.(5), with the relevant exponent fixed as n ¼1/2.
ð2Þ
where R represents the DRX grain radius. γGB and E denote the unit surface energy and volume free energy, respectively. The DRX steady-state grain radius could be obtained from Eq.(2) as (ρSS denotes the steady-state dislocation density.): 2γ RS ¼ GB E
obtained as:
ð3Þ
The flow stress variation attributed to dislocation evolution and interaction could be quantified by the Taylor function as [14]: pffiffiffi σ ¼ α M μb ρ ð4Þ where α is a Taylor constant equal to 0.5, and M is the conversion factor between the shear and normal strains, 3.06 [14]. With Eq.(4) substituted into Eq.(3), a quantitative correlation between the DRX steady-state stress σ SS and grain size DS could be
3. Correlation interpretation 3.1. DRX type identification A dynamic model of the DRX process during hot deformation was proposed by Stüwe et al. [15] informs a dislocation evolution perspective. In a dynamic model, the DRX process could be activated by a critical dislocation density ρC achieved from work hardening. With DRX the activated and conducted, the distortion storage is consumed by substructure formation and boundary migration. The dislocation density in the deformed portion would decrease, and newly formed grains from the DRX could be considered with no distortion. The density decrease leads to the lack of a driving pressure for the progress of the DRX. The DRX process would be suppressed once the dislocation density decreases lower than a critical value. However, the DRX process occurs during the ongoing hot deformation,
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Fig. 4. EBSD orientation images and inverse pole figures of the microstructure and texture for the respective grain size distributions of austenitic steel 800 H deformed at 1100 1C: (a) with ε_ ¼ 0:01s 1 at 98%; (b) with ε_ ¼ 0:001s 1 at 98%; (c) strain rate change from 0.001 s 1 to 0.01 s 1 at 56% and further deformation to a total strain of 96%; (d) strain rate change from 0.01 s 1 to 0.001 s 1 at 57%, total strain 96% (This graph was taken from M. Frommert, G. Gottstein, Mater. Sci. Eng. A 506 (2009) 101–110.).
and uninterrupted dislocation supplements are produced from the continuous straining. Assumed that the dislocation density increases at a rate of ρ_ , whether the DRX critical density storage is maintained or re-achieved directly determines the progress of the DRX. With the DRX steady-state grain size DR established, if the dislocation density is always maintained beyond ρC during the hot deformation, the DRX softening will occur continuously, referred to as continuous dynamic recrystallization (CDRX). However, if the dislocation density could not be re-accumulate to ρC again because of the softening effect, the DRX process would temporarily cease, with the work-hardening behaviour accelerated in the distortion that is in preparation for the next DRX onset. This intermittent but regular occurrence of DRX is characterized as periodic dynamic recrystallization (PDRX). With t R representing the recrystallization time, the identification of DRX type during the steady state could be expressed as: 8 ρ_ U t R o ρC ) PDRX > < ρ_ U t R ¼ ρC ) Balance > : ρ_ U t 4 ρ ) CDRX R C
ð6Þ
Demonstrated from flow curves, the various curve shapes correspond to different DRX types with different dislocation evolution. Consistent with the periodic hardening and softening influence during PDRX, the flow stress correspondingly exhibits obvious fluctuations after a stress peak (multiple peaks). On the other hand, for deformation undergoing CDRX, a continuous decrease (single peak) could be identified in flow curves after a stress peak owing to an uninterrupted DRX softening effect. Fig. 5
Fig. 5. Schematic diagram for flow curve shapes attributed to different DRX types during hot deformation.
illustrates a schematic diagram for flow curve shapes incorporating different DRX types. The driving pressure for the directional migration of boundaries arises from the distortion difference between the deformed portion with the DRX critical dislocation density and the adjacent DRX nuclei with no distortion. According to dynamics theory, the migration rate for the grain boundaries newly formed during DRX could be given as [15,16]: v ¼ M BM U ΔP
with
1 2
ΔP ρC μb2
ð7Þ
H. Liang et al. / Materials Science & Engineering A 638 (2015) 357–362
where MBM represents the boundary mobility. The mobility capacity increases with the temperature owing to the diffusion dependence. The boundary mobility is usually assumed to be proportional to the self-diffusivity at the grain boundaries, which could be calculated by [13]: Q dif f u βδDGB V m M BM ¼ ð8Þ with D ¼ D Uexp GB 0 2 RT b R where DGB denotes the self-diffusion coefficient along the grain boundary, and Qdiffu represents its activation energy. D0 is the selfdiffusivity at absolute zero 0 K. δ is the grain boundary thickness, taken equal to 0.5 1 nm. Vm is the molar volume, and β is a fraction parameter usually estimated to be between 0.1 and 0.5. With Eqs.(7, 8) combined, the recrystallization time t R for DRX steady-state grain size formation could be approximately evaluated as: Q dif f u DS 2R U DS tR ¼ ð9Þ ¼ U ρC 1 Uexp ν βμδV m D0 RT According to the Orowan equation [17], the dislocation generation originates from the continuous trapping by existing dislocations and could be quantified by assuming the mean free slip distance, the dislocation that the Burgers vector b could migrate, equal to L: dρ ε_ ¼ dt bL
ð10Þ
By substituting Eqs.(9,10) into Eq.(6), the dynamic balance condition between the DRX grain expansion and shrinkage could be described as: Q dif f u 2R UDS ¼ ρ2C U ε_ exp ð11Þ RT βμδV M D0 bL Jonas et al. [18] analysed twenty-six flow curves determined under DRX conditions on different steels to characterize the recrystallization kinetic behaviour. The stress levels from the DRX criticality and steady state are considered similar because the critical condition exactly corresponds to the balanced state between work-hardening and DRX softening as: σ C σ SS . With the dislocation density substituted by Eq.(4), Eq.(11) could be re-written as:
κ R U DS U ε_ expðQ dif f u =RTÞ ¼ σ 4SS with κ R ¼
2α4 μ3 M 4 b R βδV m D0 bL
361
pffiffiffi Owing to the similar magnitudes of RC and 1= ρ, an equivalent result could be derived from equivalent infinitesimal mathematical theory as: R4C ρ2 -0 ) 1 expð 0:7R4C ρ2 Þ 0:7R4C ρ2
ð15Þ
Upon substituting Eq.(15) into Eq.(14), the expression for the annihilation coefficient Ω could be estimated as:
ε_ n Ω Ω0 _ U 0:7R4C ρ2 Uexp ε
Q dif f u RT
ð16Þ
The dynamic balance at the DRX steady state is characterized by the dislocation variation rate dropping to zero as dρ ¼ h Ω U ρ ¼ 0 from the Estrin-Mecking model. dε SS
With the combination of Eqs.(4,16), the dependence of the steady-state stress on the deformation parameters could be obtained in an Arrhenius power type equation:
σ 6SS ¼ κ σ U ε_ U expðQ dif f u =RTÞ with κ σ ¼
1:43α5 μ5 M 5 b h Ω0 R4C ε_ n 5
ð17Þ
The stress exponent, dðln ε_ Þ=dðln σ Þ, could be directly obtained by taking the natural logarithm on Eq.(17) as a constant equal to 6. Fig. 6 demonstrates the sensitivity of the steady-state stress to the strain rate by a linear regression analysis between ln σ SS and ln ε_ according to the experimental data from [22]. It could be validated for the hot deformation of pure copper that the stress exponent remains as a constant close to 6, insensitive to the processing parameters. In addition, Table 1 lists the stress exponent, calculated by peer studies, for hot deformation on various metallic materials, all consistent with the fixed stress exponent of 6, as given in Eq.(17). With Eq.(17) substituted into Eq.(12) to characterize the dependence of DRX grain size on the deformation parameters, the identification criterion for the DRX balanced state, Eq.(6),
3
ð12Þ
3.2. Steady-state constitutive With the assumption of the dislocation distributing uniformly in a deformed matrix, the Estrin-Mecking relation [19] has been validated with excellent consistency for the dislocation density variation during hot deformation undergoing DRX: dρ ¼ h Ωρ dε
ð13Þ
where h denotes the dislocation multiplication coefficient, related to the lattice structure and mechanical parameters. Ω represents the annihilation coefficient, sensitive to deformation conditions [20]. With the assumption that the probability of two opposite-sign dislocation recombinations satisfies the requirement of p ¼ 1 expð ρ=ρÞ2 , Picu and Majorell [21] proposed a thermal model to describe the annihilation coefficient Ω as: Q ε_ n ð14Þ Ω ¼ Ω0 _ ð1 expð 0:7R4C ρ2 ÞÞ U exp dif f u ε RT where Ω0 is a proportionality constant And RC denotes the critical radius beyond which dislocations cannot cross-slip and recombine.
Fig. 6. Stress sensitivity to strain rate obtained from the linear regression between ln σ SS and ln ε_ , with pure copper deformation as a precedent.
Table 1 Typical stress exponent n for the hot processing of different alloys.
Stress Exponent
TC18 Ti [23]
Fe-Mn- Ni55Fe19Ga26[26] Superalloy AA7075 ZK 60 Mg Al steel [27] Al[28] [25] [24]
5.6
6
5.5
5.2
5.4
5.7
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The reverse relationship between the DRX steady-state stress
and the Zener-Hollomon parameters has been obtained, with the correlation exponent fixed at 1/3. Multiple peaks in flow curves and grain coarsening during hot deformation have been identified as occurring under high temperatures and low strain rates, as characterized by PDRX. Single peak and grain refinement occurs with the opposite deformation conditions for the CDRX effect.
Acknowledgement This work was supported by the High-end CNC Machine Tools and Basic Manufacturing Equipment project (2012 ZX 04010-081). Dr. Y.Q. Ning is supported by the Hong Kong Scholar Program (Grant No. XJ2014047). Fig. 7. Dependence of flow curve shape and microstructure evolution on deformation parameters with the identification of the DRX type.
could be illustrated as: DS U Z 1=3 ¼ κ σ =κ R ¼ Const 2=3
with
Z ¼ ε_ U expðQ =RTÞ
ð18Þ
The reverse relationship between the DRX steady-state grain size and the Zener-Hollomon parameters could be directly obtained from Eq.(18), and the correlation exponent 1/3 gives a good agreement with the experimental result by [29]. Fig. 7 exhibits a deformation mechanism schematic map to demonstrate the dependence of the flow curve shape and microstructure evolution on the deformation parameters. Characterized by Eq.(18) and Eq.(6), multiple fluctuations in the PDRX and grain coarsening occur with low Zener-Hollomon parameters under high temperatures and low strain rates, and vice versa for single peak and grain refinement during CDRX. By combining Eqs.(12, 17), the ultimate formula relating the flow stress and grain size under the DRX steady state during hot deformation could be obtained as: DS U σ 2SS ¼ κ σ =κ R
ð19Þ
with the Derby function validated and interpreted.
4. Conclusion
The dependence of grain size on flow stress under DRX steady
state during the hot deformation of metallic materials has been investigated with validation and interpretation of the Derby function. An Arrhenius constitutive model has been constructed to characterize the dependence of the DRX steady-state stress on the deformation parameters, with the stress exponent fixed as a constant equal to 6.
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