Modelling discontinuous dynamic recrystallization using a physically based model for nucleation

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Acta Materialia 57 (2009) 5218–5228 www.elsevier.com/locate/actamat

Modelling discontinuous dynamic recrystallization using a physically based model for nucleation D.G. Cram a, H.S. Zurob b, Y.J.M. Brechet c, C.R. Hutchinson a,* b

a Department of Materials Engineering, Monash University, Clayton, 3800 Vic., Australia Department of Materials Science and Engineering, McMaster University, Hamilton, ON, Canada L8S 4L7 c SIMAP, Institut National Polytechnique de Grenoble, St. Martin D’He`res 38402, France

Received 1 June 2009; received in revised form 16 July 2009; accepted 16 July 2009 Available online 11 August 2009

Abstract A physically based model for nucleation during discontinuous dynamic recrystallization (DDRX) has been developed and is coupled with polyphase plasticity and grain growth models to predict the macroscopic stress and grain size evolution during straining. The nucleation model is based on a recent description for static recrystallization and considers the dynamically evolving substructure size. Model predictions are compared with literature results on DDRX in pure Cu as a function of initial grain size, deformation temperature and strain rate. The characteristic DRX features such as single to multiple peak stress transitions, convergence towards a steady-state stress and grain size, and a power-law relationship between the stress and grain size are quantitatively reproduced by the model. The critical conditions for the onset of nucleation in the model are shown to compare well with Gottstein et al.’s experimentally determined critical stress criteria. Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Dynamic recrystallization; Nucleation of recrystallization; Subgrain growth; Copper; Grain size

1. Introduction The plastic deformation of metals results in energy storage within the material in the form of dislocations. It is commonly accepted that only 10% of the mechanical energy of deformation is stored as dislocations and the rest is dissipated in the form of heat. These dislocations are out of equilibrium and therefore tend to disappear during annealing. When annealing is performed after the deformation step, the elimination of dislocations takes place either by recovery (a slow and continuous process) or by ‘‘static recrystallization” (a fast and discontinuous process). When plastic deformation is carried out at high enough temperatures, the two processes, dislocation storage and dislocation elimination, may coexist and the phenomenon is *

Corresponding author. E-mail address: [email protected] (C.R. Hutchinson).

known as ‘‘dynamic recrystallization” (DRX) [1–5]. Similarly, DRX can occur in a slow and continuous manner, and one then speaks of ‘‘continuous dynamic recrystallization” or in a fast and discontinuous manner, and the phenomenon is called ‘‘discontinuous dynamic recrystallization” (DDRX) [5]. In continuous dynamic recrystallization, the dislocation density is reduced by a continuous evolution of the subgrain structure, both in size and misorientation. In DDRX, nucleation of new grains occurs and regions depleted of dislocations grow at the expense of regions full of dislocations. A key difference between dynamic and static recrystallization processes is that in the former energy is continuously pumped into the system and the ongoing deformation loads the recrystallized grains with new dislocations. Quantitative modelling of DRX requires differentiation between continuous dynamic recrystallization, for which no nucleation step is required, and DDRX, for which the nucleation step of a new grain is a key issue. For materials

1359-6454/$36.00 Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2009.07.024

D.G. Cram et al. / Acta Materialia 57 (2009) 5218–5228

with moderate to low stacking fault energies, e.g. Cu, dynamic recovery is comparatively slow and DRX occurs in a discontinuous manner in which the nucleation and growth stages are clearly distinguished. For materials with high stacking fault energies, e.g. Al, dynamic recovery is fast and a continuous type of DRX is observed. In this contribution we limit our discussion to DDRX. Metals that undergo DDRX exhibit a number of common characteristics which are well-established by experiment and have been summarized several times [1,4,5]. These are schematically illustrated in Fig. 1. Any model for DDRX should be able to explain the following characteristics of DDRX:  A critical strain ec must be reached before DRX occurs [1–3].  Single peak or multiple peaks (stress oscillations) are observed in the stress–strain response depending on applied strain rate, deformation temperature and initial grain size of the material [1,4,6] (Fig. 1a).  The stress–strain response converges toward a saturation stress rs, (Fig. 1a).  At steady-state, the grain structure is equiaxed [1,6,7] and the grain size converges toward a saturation value Ds, (Fig. 1b).  A power-law relationship between the steady-state grain size and the steady-state saturation stress [6–8] (Fig. 1c) is observed. The minimum ingredients necessary to build a physically based model for DDRX are: (i) a nucleation criterion (both

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for the critical conditions for the onset of nucleation and the nucleation rate); (ii) a growth law for the recrystallized grains growing into an unrecrystallized matrix; and (iii) in order to feed the growth law, the stored energy of a given grain must be described, and therefore the deformation of a plastically heterogeneous material (some grains being freshly recrystallized and others still fully deformed) must be calculated. For each step, different levels of approximations are possible. For example, a purely phenomenological criterion for nucleation of DDRX might be adopted, N_ ð_e; T Þ (e.g. [9]), or the stored energy might be assumed proportional to the local plastic work. The deformation of the plastically heterogeneous mixture could be described either using the classical bound approximations (Taylor or static) or, at the other extreme, using the more refined self-consistent models [10,11]. In the present contribution, we propose a simple, physically based model with two internal variables (dislocation density and grain size) in which the nucleation step is described by the instability of an evolving subgrain structure, the heterogeneous plastic deformation by an approximation to the self-consistent models, the relation between plastic flow and stored energy is obtained from a physically based model for work-hardening [12], and the growth rate by a simple linear relation between the grain boundary velocity and the difference in stored energy [4,9]. The overall framework is a mean field theory where each grain is embedded into an average medium [13]. The objectives are to describe quantitatively the effect of temperature, applied strain rate and initial grain size, and to recover the characteristic features of DDRX described above. The material we consider for the application of the model is pure Cu, which has been extensively studied [6,7,14]. We feel that the key innovation of the present paper with respect to the existing literature is the proposition of a physically based nucleation criterion instead of the usually proposed phenomenological criteria [9,13,15,16]. 2. DRX model 2.1. Model structure

Fig. 1. Schematic illustration of the typically observed experimental characteristics for DRX with changes in deformation conditions (_e;T): (a) stress–strain response showing the transition from single to multi-peak; (b) evolution of an initial grain size D0 to a steady-state size Ds; (c) stress dependence on the steady-state grain size.

The polycrystal metal is represented by a set of spherical grains embedded in a medium with properties representative of the aggregate [13] (Fig. 2). Each grain is defined by its diameter (Di), orientation/Taylor factor (Mi), dislocation density (qi) and subgrain size distribution (P(vi)). The advantage of the mean field description is its simplicity. However, it is not able to capture spatially heterogeneous microstructural effects such as necklacing [17]. Initialization of the model requires definition of the deformation temperature (T) and applied strain rate ð_emac Þ, and the initial grain size and grain orientation distributions. The model numerically iterates through the macroscopic strain steps (demac) and at each step calculations of the polyphase plasticity, grain nucleation and grain

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Grain 1

Grain 2

D1 M 1 ρ1

D2 M2 ρ2

Grain i

Di M i ρi

Polycrystal matrix ρ D

Fig. 2. Schematic illustration of grains immersed within an average polycrystal matrix aggregate (average medium). Di is the grain size, Mi is the orientation/Taylor factor and qi is the dislocation density.

growth (Fig. 3) are made. Each of the modules are coupled as discussed in Sections 2.2–2.4. 2.2. Modelling polyphase plasticity The driving force for both nucleation of new grains and grain growth is the stored energy. The essence of DRX is the coexistence of grains that have nucleated at different times and therefore have undergone different straining histories and have different stored energies. It is therefore of key importance to describe with reasonable accuracy the plastic deformation of the heterogeneous material (the recently recrystallized grains being the softest grains) and the relationship between the strain history and the stored energy. It is also desirable that such models remain simple enough to retain a transparent picture of their physical content. The plastic deformation of heterogeneous materials, and especially of polycrystals, has been extensively studied (for a review, see [18]). The simplest approach is to consider

either a homogeneous strain (Taylor) or a homogeneous stress (static). Both are approximations: the first does not ensure equilibrium across the grain boundaries while the second does not ensure strain compatibility between grains. Furthermore, there is an intuitive feature which should be reproduced by the plasticity model we chose: the softer grains deform more than the hard ones (which contradicts Taylor’s hypothesis), and the stored energy, related to the dislocation density and therefore to the flow stress, varies from grain to grain (which is incompatible with the static approximation). A self-consistent model could be a solution for this dilemma, but such a model is rather computationally demanding and a simpler approach is welcome. Indeed, a simple approximation has recently been proposed by Bouaziz and Buessler [19] in which each grain behaves so that the mechanical work done in each is equal. Although this approach does not seem to rely on any physical basis, and it is obviously not true for extreme contrasts such as that between void and material, it has been shown for a range of two-phase materials to provide a good approximation to the more rigorous and sophisticated self-consistent approach [19]. This simple assumption has the dual advantage of capturing the intuitive features outlined above and can be easily implemented. The ‘‘iso-work increment assumption” [19] is used to partition the macroscopically applied strain, demac, between the various grains (i). Each grain undergoes the same work done, K, where: dei ¼

K ; ri

ð1Þ

dei is the strain increment experienced by grain i, and ri is the flow stress of grain i. In this description, softer grains deform more than harder grains in the same demac step. The work done in each step is chosen arbitrarily to control the demac step. The macroscopic strain is calculated over all grains: P V i dei demac ¼ P ; ð2Þ Vi where Vi is the volume of grain i. In order to proceed, a constitutive equation between the stress increment and the strain increment is required, and such an expression should be readily interpreted in terms of dislocation densities which, in turn, give access to the stored energies. A Voce hardening law is used to describe the strain-hardening of each grain [20]:   dri ri ; ð3Þ ¼ HII 1  dei rs;i

Fig. 3. Summary of the model structure and its operation.

where rs,i is the steady-state stress of grain i and HII is the stage II work-hardening rate. The advantage of this law is that, in pure metals, it can be derived from dislocation/dislocation interactions as proposed by Kocks et al. [20]. The steady-state stress rs,i for grain i is a function of the strain rate (_ei ) in grain i and the temperature [21]: log(rs,i/l) / (kT/lb3)  ln(107/_ei )

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(where k is the Boltzmann constant and l is the shear modulus) and can be easily obtained from the available experimental data [20]. The relationship between the flow stress of grain i and the dislocation density is classically written as [21]:  m pffiffiffiffi e_ i ; ð4Þ ri ¼ aM i lb qi e_ ref where a is a numerical constant of the order 0.1–0.5, e_ ref is a reference strain rate and m is the strain rate sensitivity. The macroscopic stress is given by a volume average of the stress over all grains: P V i ri : ð5Þ rmac ¼ P Vi This procedure allows us to monitor the current flow stress and stored energy of each grain as a function of deformation. It also allows a calculation of the macroscopic stress– strain curve provided the state of recrystallization and its evolutions are known. In order to obtain the evolution of the state of recrystallization, a model for the nucleation and growth rate of new grains is necessary.

critical strain for nucleation and a nucleation rate depending on strain, strain rate and temperature. In order to apply this approach to the nucleation events during DDRX modifications are necessary to capture the continuous evolution of the subgrain structure and stored energy due to the ongoing plastic deformation, but the ‘‘instability criterion” leading to nucleation remains the Bailey–Hirsch criterion. As a first approximation, in this work we consider grain boundary bulging as the only mechanism for nucleation of DRX grains. As a result, at large strains, if annealing twins do play a pivotal role in the nucleation of new grains, we may expect to see some discrepancy between the experimental observations and the predictions of our approach. It is assumed that a nucleation event can occur during DRX when a subgrain located on a grain boundary reaches a critical radius, rc. Nucleation assisted from twin boundaries or deformation bands is not considered at this stage. The critical subgrain size (Eq. (6)) corresponds to the condition when the stored energy of grain i, Gi = ½qilb2, is large enough to overcome the capillary force of a subgrain of radius ri, lying on the grain boundary of grain i. Once this condition is met, the subgrain can bulge into the matrix as a new grain, as shown schematically in Fig. 4 [29]: rc;i ¼

2.3. Modelling grain nucleation A common nucleation mechanism for a recrystallized grain in pure metals involves grain boundary bulging as described by Bailey and Hirsch [22] (strain-induced boundary migration), in which a subgrain can bulge and grow into the deformed matrix as a new grain once it reaches a critical size, becoming a viable nucleus. This condition is met when the stored energy due to dislocations can overcome the energy associated with grain boundary curvature. Prior to the bulging process, Sakai [4] suggests that potential subgrains for nucleation can be enhanced through grain boundary corrugation and localized shearing. Wusatowska-Sarnek et al. [7] proposed that the localized shearing causes the gradual accumulation of subgrain misorientation within a grain. In the case of Cu and other low stacking fault energy metals deformed to relatively low strains at elevated temperature, there appears to be general acceptance that the nucleation event is a Bailey–Hirsch-like process [7,17,23,24]. However, at larger strains, approaching the steady-state, there is some indication that annealing twins formed during grain boundary migration [25,26] may play an important role in subsequent grain nucleation [7,17,27,28]. Due to the heterogeneous nature of the DRX nucleation event, it has been an ongoing challenge to model the process as a macroscopic response. Recently, a physically based nucleation description for static recrystallization was successfully developed by Zurob et al. [29] based on the Bailey–Hirsch subgrain bulging mechanism and a subgrain size distribution. In this approach, the subgrain size evolves due to recovery and subgrain growth processes. The output of this model is a

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2c ; Gi

ð6Þ

where c is the grain boundary energy. The evolution of the average subgrain size (ri ) of grain i is assumed to have two linearly additive competing terms: a term which increases  r þ the average subgrain size due to subi , and a term which decreases the average grain growth d dt size due to the increasing stress applied on the dislocation  ri  . structure d dt Subgrain growth is driven by capillary effects (Eq. (7)):  þ dri 2c ¼ M LAG  sub ; ð7Þ ri dt where MLAG is the subgrain boundary mobility and csub is the subgrain boundary energy The stress-dependent subgrain size effect (Eq. (8)) is derived from Raj and Pharr [30] (di/b  Ksub(l/ri)) and the Voce hardening law (Section 2.2):

Fig. 4. Schematic illustration of a subgrain growing (left) and once it reaches the critical size (right) bulging into the deformed matrix as a new strain-free grain [29].

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      dri dri dri dei ¼ dt dri dei dt     1 l ri  e_ i ; ¼  K sub b 2  HII 1  rs;i 2 ri

ð8Þ

where di is the average subgrain diameter and Ksub is a constant. It is assumed that a subgrain structure is established after very small strains, which in the case of Cu is experimentally well supported [31–33]. The subgrain size distribution in each grain can be approximated by a Rayleigh distribution [29,34] and this is represented schematically in Fig. 5. For the purpose of calculation it is convenient to normalize the subgrain size by its average size, to obtain a subgrain size probability density function P(v) (vi = ri/ri ). From this distribution, there will exist a fraction of subgrains (Fsub,i) within each grain that have a size greater than the critical size, vc,i, (vc,i = rc,i/ri ) and hence are large enough to nucleate. This fraction is shown as the shaded area in Fig. 5. This fraction of subgrains Fsub,i has an analytical solution given by ! Z 1 pv2c;i F sub;i ¼ P ðvi Þdvi ¼ exp : ð9Þ 4 vc;i At each point during straining, both the stored energy driving nucleation and the subgrain size distribution in each grain change. Each grain has a specific ‘‘potential” for nucleation. The number (or fraction) of subgrains able to nucleate from each grain (Nnuc,i) is given by the product of the fraction of subgrains able to nucleate (Fsub,i) and the number of subgrains lying on the grain boundary (Nsub,i): ! pv2c;i 64R2 N nuc;i ¼ F sub;i  N sub;i ¼ exp ð10Þ  2 2i : 4 p ri

The number of subgrains lying on the grain boundary  64R2 N sub;i ¼ p2 r2i is estimated by dividing the grain boundary i   surface area 4pR2i by the average area of the circle formed by the random  2 intersection of a plane and a sphere of rap pr4 i . Summing Nnuc,i over the total number of dius ri, P grains, Nnuc,i, during a strain step, demac, provides a criterion for the number of subgrains that should nucleate in the system as new grains. For model implementation, each new grain nucleates from a randomly selected grain and has an initial size equal to that of its instantaneous critical radius rc,i. The equivalent volume is subtracted from the parent grain. The orientation/Taylor factor Mi assigned to the new grain is random and hence independent of the parent grain. A low initial dislocation density of q = 109 m2 is assigned to each new grain. After the nucleation events have occurred during the strain step demac, it is assumed the fraction of the largest subgrains in every grain’s distribution, Fsub, have been consumed. These subgrains are no longer available for further nucleation, which correctly takes into account of the depletion of the nucleation potential of each grain following a nucleation event. This approach to nucleation does not require the explicit definition of a critical strain, stress or dislocation density for nucleation to occur nor does it require assumptions about the functional form of the nucleation rate. Both the critical conditions for nucleation and the nucleation rate (and its evolution) are outputs of the nucleation framework adopted. 2.4. Modelling nucleus growth The growth of each grain is driven by the stored energy (dislocation density) difference between the grain (qi) and the effective medium ( q) in which it resides. The grain boundary velocity is assumed proportional to the stored energy difference across the boundary: dDi ¼ M HAG lb2 ð q  qi Þ; dt

ð11Þ

where MHAG is the grain boundary mobility and is estimated as a fraction of Turnbull’s estimate [35] (which has previously been shown to be a good approximation [36,37]). M HAG 

Fig. 5. The subgrain size distribution within each grain of average radius r. The shaded area represents the fraction of subgrains larger than the critical size rc.

1 d  Dgb ðT Þ  V m ; 5 b2 RT

ð12Þ

where d is the grain boundary width, Vm is the molar volume, R is the gas constant, and Dgb(T) is the grain boundary diffusivity. ) With this description a highly deformed grain (qi > q ) will shrink, whereas a relatively strain-free grain (qi < q will grow. In the case of a growing grain of dislocation density qi, the boundary migration leaves behind a new region free of dislocations. The dislocation density of grain i after an increment of boundary migration is recalculated assuming instantaneous homogenization of the density throughout the grain.

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3. Results

Fig. 6. Schematic illustration of how the surface area shared by neighbouring grains and the resultant grain boundary velocities can be represented as an average velocity for each grain i.

, is The average dislocation density of the medium, q defined by a surface-area-weighted average over all grains: Pn 2 Di qi  ¼ Pi¼1 : ð13Þ q n 2 i¼1 Di It may seem surprising that a surface-weighted average rather than a volume-weighted average is used. The grain boundary of grain i will see an average dislocation density based on the dislocation densities of the neighbouring grains as shown schematically in Fig. 6. This average is scaled by the relative magnitude of surface area exposed to grain i. Eq. (13) has been derived in Ref. [13] on the basis of conservation of volume of the system during the simulation. 2.5. Model parameters Pure Cu has been chosen as the model system due to its relatively isotropic plastic properties (i.e. polyphase plasticity is expected to be described reasonably accurately by the ‘‘Iso-Work” approach) and its suitability to a mean field model. The parameters used for model simulations are summarized in Table 1. The equations were integrated using a strain increment typically of the order of demac = 1  104. This increment is small enough to ensure no effect of the step size on the integration routine.

Model simulations were performed over a range of different hot-working parameters (temperatures: 725–975 K, strain rates: 1  103–2  101 s1) and initial grain sizes (D0 = 25–155 lm). The initial grain size distribution consists of grains that are normally distributed with a standard deviation of D0/3. As a result of the initial grain size allocation, a very small number of grains were sometimes assigned negative sizes. These were reassigned new initial sizes by randomly selecting a positive grain size within the normal distribution. A grain orientation distribution represented by a Taylor factor assigned to each grain is normally distributed about 3.06 (standard deviation of 0.2) with lower and upper limits of 2.3 and 4.2, respectively. A low initial dislocation density was assigned to all grains. The stress–strain response and microstructural evolution of the model is compared with the experimental data for pure Cu during DRX reported by Gao et al. [6] and Blaz et al. [14]. Blaz et al. has the advantage that it reports simultaneous measurements of the effect of temperature and initial grain size on the DRX behaviour. Gao et al. work shows the effect of strain rate on DRX behaviour. 3.1. Effect of deformation temperature The effect of deformation temperature on the stress– strain curves (experimental [14] and calculated) is shown in Fig. 7. The agreement between experiment and model is considered quite reasonable. The experimental observations (Fig. 7a) indicate that with increasing deformation temperature, the steady-state stress rs and the peak stress rp occur at earlier strains. This feature is well reproduced by the model calculations and is due to the combined effects of accelerated nucleation kinetics and increased grain boundary mobility with increasing temperature. The experimental observations also indicate an increase in the steady-state grain size with increasing deformation temperature. This feature is quantitatively captured by the model (Fig. 7c) with the calculated and experimental steady-state grain sizes corresponding to within ±5 lm

Table 1 Parameters used for the model simulations of DRX in pure Cu. Symbol

Parameter

Value

Used in Eq. no.

Ref.

Tm l HII a e_ ref m Ksub c MLAG csub d D0gb Qgb

Melting temperature Shear modulus Stage II strain-hardening rate Dislocation junction strength Strain rate reference quantity Strain rate sensitivity Subgrain size constant Grain boundary energy Subgrain boundary mobility Subgrain boundary energy Grain boundary thickness Pre-exponential grain boundary diffusion Activation energy grain boundary diffusion

1356 K h  i Pa 35:4  109 1  0:5 T 300 Tm 0.05 lPa 0.3 4.5  107 s1 0.0222 50 0.625 J m2 0.075MHAG m4 J1 s1 0.25 J m–2 1  109 m 2.35  105 m2 s1 107 kJ/mol1

(4) and (11) (3) (4) (4) (4) (8) (6) (7) (6) (12) (12) (12)

[29] [29] [20,21] [38] Calculation from [20] Calculation from [20] [30] [29] [29] Estimate [39] [39] [39]

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Fig. 7. Stress–strain curves and grain size evolution for pure Cu over different temperatures (T = 725–975 K) for e_ = 2  103 s1 and D0  78 lm. (a) Experimental r–e [14]; (b) model r–e; (c) model D–e.

(at 725 K steady-state is not reached, for both experiment and model). It can be seen that the strain at which the steady-state grain size and stress is reached for each temperature are in close agreement (e.g. 875 K at e  0.4). Importantly, the model also describes the experimentally well-established transition in stress–strain response from single to multi-peak stress (Fig. 7a) with an increase in deformation temperature. As described by Sakai and Jonas [1], if D0 > 2Ds, then a single stress peak is observed experimentally, whereas if D0 < 2Ds multi-stress peaks are observed. At 725 and 775 K, the calculated Ds is 13.1 and 18.5 lm respectively (D0  6.0Ds and D0  4.2Ds), and hence only a single stress peak was observed. Increasing the deformation temperature to 875 and 975 K, the Ds is 37.1 and 61.2 lm, respectively (D0  1.3Ds and D0  2.1Ds), and multi-stress peaks are observed. This is reasonably consistent with the empirical criteria: D0 < 2Ds. 3.2. Effect of applied strain rate The effect of applied strain rate on the model calculations is compared with the experimental observations at 775 K in Fig. 8. At relatively low strain rates (<2.7  102 s1) good agreement between the calculated and experimentally observed stress–strain response is observed. However, at higher rates (e.g. 2.7  102 and 2  101 s1) the experimental observations indicate softening at large strains which is not captured by the model (Fig. 8a and b). Under these conditions, the model calculations of the average grain size evolution converge to the steady-state much slower than the experimental observations. Both of these points suggest there is a deficiency in the rate of nucleation at high applied strain rates, and this feature will be further explored in the discussion section of this contribution. At low applied strain rates, the steadystate stress and grain size are reached at similar strains

(e.g. 1  103 s1 at e  0.5). At high strain rates, the lack of DRX nucleation gives the impression that a steady-state stress is reached well before the steady-state grain size (e.g. 2  101 s1 at e  0.6). The stress level under these conditions is dominated by the work-hardening behaviour of the material. It is not a steady-state stress with respect to a balance between hardening due to work-hardening and softening through DRX. The transition from single to multi-peak in the stress–strain response, as a function of applied strain rate, is well captured by the model. 3.3. Effect of initial grain size The effect of initial grain size on the stress–strain response and the steady-state grain size is compared with experiment in Fig. 9. In agreement with experimental observations the model’s final grain size (Ds = 18.5 lm) is independent of the initial grain size D0. Regardless of the initial microstructure both the stress and the grain size converge towards a steady-state defined only by the processing conditions. Furthermore, a decrease in the peak stress is observed experimentally with a decrease in the initial grain size (Fig. 9a). This feature is properly captured by the model and is directly related to the enhanced nucleation associated with an increased grain boundary area per unit volume. For each of the different initial grain sizes considered, the grain size and stress evolution reach their steadystate at similar strains. A decrease in initial grain size D0 from 78 to 50 lm is also able to predict the single to multi-peak transition in the stress–strain response. 3.4. Steady-state grain size evolution Experimentally it is has been observed that the steadystate grain size Ds shows a power-law relationship with the steady-state stress rs [8,40]. This is typically represented

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Fig. 8. Stress–strain curves and grain size evolution for pure Cu over different applied constant strain rates (_e = 2.9  104–2  101 s1). (a) Experimental r–e [6] for T = 773 K and D0  340 lm. 4NCu represents a Cu purity of 99.99%. (b) Model r–e for T = 775 K and D0  78 lm. (c) Model D–e for T = 775 K and D0  78 lm.

Fig. 9. Stress–strain curves and grain size evolution for pure Cu for different initial grain sizes (D0  11–155 lm) for T = 775 K and e_ = 2  103 s1. (a) Experimental r–e [14]; (b) model r–e; (c) model D–e.

in the form shown in Eq. (14), where N is a constant ranging from 0.4–0.8 [4,40]: r / DN s :

ð14Þ

Derby and Ashby [40] provide a derivation for this relationship under steady-state conditions for which the nucleation rate and average grain size remain constant. The model calculations also predict a power-law relationship (Fig. 10) with a value of N = 0.528, which lies within the range experimentally observed [4,40]. The model calculations require selection of the initial grain size distribution. A normal distribution is assumed in the current work. However, once the steady-state conditions are achieved in DRX it is experimentally observed that an equiaxed grain structure is obtained. The model calculations of the evolution of the grain size distributions (fitted by Weibull probability density functions) as a func-

tion of strain for conditions T = 775 K, e_ = 2  103 s1 and D0 = 78 lm is shown in Fig. 11. During straining the distribution narrows considerably from the initially normally distributed distribution. 4. Discussion A physically based model for DDRX should be able to reproduce the principal experimentally observed characteristics of DDRX outlined in the Introduction. The model presented here is able to do this over a wide range of processing conditions. One example is the ability of the model to predict the transition from single to multi-peak stress– strain behaviour according to the criteria D0 = 2Ds [1] as a function of deformation temperature and applied strain rate. Luton et al. [2] first explained that the physical origin for this criterion is the relative synchronization in DRX

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D.G. Cram et al. / Acta Materialia 57 (2009) 5218–5228 7.75

7.7

log σs

7.65 N ~ -0.528 7.6

7.55

7.5 -5

-4.8

-4.6

-4.4

-4.2

-4

log Ds

Fig. 10. A log–log plot showing the model’s calculated power-law relationship (r / DN s ) between the steady-state grain size and steadystate stress. The model’s results have been linearly fitted with a dashed line where N  0.528.

Probability Density Function

0.04 0.035

ε=1

0.03 0.025

ε = 0.2

0.02 0.015

ε=0

0.01 0.005 0 0

50

100

150

200

Grain Size (μm) Fig. 11. The model’s grain size distribution evolution at different strains at an initial grain size of D0 = 78 lm for T = 775 K and e_ = 2  103 s1. As nucleation of DRX grains continues with strain, larger grains become consumed and smaller DRX grains grow into matrix, leading to the narrowing of the grain size distribution.

fraction cycles. Separation of DRX cycles can arise from high initial rates of increasing DRX volume fraction (high nucleation and growth rates). Here a macroscopic softening response will occur, but as the rate of DRX decreases approaching saturation, the work-hardening will dominate, and a stress oscillation will result. For this to be predicted in the model, it is clear that the nucleation and growth kinetics must be in tune with the work-hardening rate. The initial grain size dependence of the transition in stress oscillation behaviour comes from the increased nucleation with smaller grain sizes and from the fact that quick DRX kinetics which cause multi-peak stress oscillations inevitably lead to increased grain growth and Ds. The model was also able to converge naturally to a steady-state stress and grain size as observed experimentally. The physical origin of the dynamic steady-state is the balance between the rate of grain boundary migration

and nucleation of new grains which is influenced by the work-hardening rate during deformation. The present model can describe with reasonable accuracy the transitions between the different regimes in the stress–strain curve, the evolution of grain size, and their dependences on temperature and initial grain size (Fig. 1). A key innovation of the present paper is the proposition of a physically based description for DDRX nucleation by a grain boundary bowing mechanism. This has proven successful in capturing the influence of applied strain rate, temperature and grain size on the nucleation of recrystallization during DDRX. Recently, Gottstein et al. [41] have reported a critical stress condition for the initiation of DRX on the basis of experimental observations (for single and polycrystals) as a function of deformation temperature and strain rate. This condition suggests that the nucleation of DRX occurs when the stress approaches 85% of the saturation stress in the absence of DRX (i.e. the saturation stress in the Voce hardening law). The stress at which nucleation is initiated in our proposed model, as a function of deformation temperature, strain rate and initial grain size, is compared with Gottstein et al.’s criterion in Fig. 12. The agreement is quite reasonable. The model proposed does not reproduce the experimentally observed softening effects at high strain rates as shown in Fig. 8. Clearly the nucleation model under these conditions is underestimating the nucleation rate. With this respect, it is worth reconsidering the nucleation condition we have proposed, in the light of the interpretations put forward in the literature. Many DRX models in the literature assume that no DRX nucleation events can occur until the system reaches a critical strain ec or dislocation density qc. The most widely used criteria for the initiation of DRX is that proposed by Roberts and Ahlblom [42]. This description is based on the migration of a grain boundary into the deformed matrix and thermodynamically assesses whether the change in stored energy is large enough for the occurrence of a nucleation event. Polak and Jonas [43] suggest that the critical point for DRX initiation is a point of inflection in the work-hardening vs. stress curve. Gottstein et al. [41] propose a physical interpretation and suggests that this critical DRX nucleation point corresponds to the onset of stage IV work-hardening, associated with an instability in the subgrain structure. These interpretations shed some new light on what is missing in the present model. The nucleation conditions are here derived from a critical subgrain size. This critical subgrain size may not be reached fast enough in a high strain rate regime, and while the stress is constant the only mechanism to grow the subgrains is capillary forces, which are always small. However, in the high strain rate regime, relatively large strains can be obtained and it is the strain which governs the evolution in subgrain misorientation (the flow stress contribution dominates the subgrain size). It may well be that in the high strain rate experiments, sufficiently large strains are reached with rel-

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atively small subgrain sizes such that the misorientation increases to the point where subgrain mobility transitions from a small fraction of the high-angle grain boundary mobility to a much larger value. In that case, subgrain growth would be much faster than predicted in the present model and therefore nucleation would occur even under the higher strain rate conditions. A simple way to account for this is to incorporate a sub-boundary mobility which would be strain dependent as shown in Fig. 13. At low strain rates, subgrains are able reach a critical nucleation conditions before the critical misorientation strain (e) for a transition in boundary mobility is reached. For higher strain rates, e may occur before ec and, as shown in Fig. 13, a transition from low to high mobility occurs, and subgrain growth is significantly accelerated, aiding the nucleation of DRX. At the larger strains approaching steady-state, numerous authors suggest that annealing twins, formed by the migration of grain boundaries, play a pivotal role in the nucleation of new DRX grains [7,17,27,28], especially at relatively low strain rates. This effect has not been incorporated in the present approach and may play a role in explaining the lack of softening observed in the model at large strains in the high strain rate regime. The development of a quantitative description of annealing twin-

Fig. 13. Schematic representing how the critical strain ec for nucleation varies with strain rate. If at high-strain rates the ec has not been reached before the critical misorientation strain e (where the subgrain effectively represents a high-angle grain boundary), the subgrain can then exhibit a high-angle mobility which enhances nucleation.

induced DRX nucleation would be a useful area for future research. 5. Conclusions A physically based description for nucleation in static recrystallization has been extended and successfully applied to nucleation during DRX. The new physically based nucleation model for DDRX was coupled to grain growth and polyphase plasticity descriptions to describe the DRX of pure Cu. Comparisons between experiment and model calculations show good agreement over a wide range of deformation temperature, initial grain size and applied strain rate. The model has been able to successfully predict:  The transition from single to multi-peak stress response in agreement with the experimental criteria: (D0 = 2Ds).  The natural progression towards a steady-state stress and steady-state grain size with strain.  Steady-state grain sizes and stresses which are independent of the initial grain size and dependent only on the hot-working conditions.  A power-law relationship between the steady-state grain size and steady-state stress.  Critical strains ec for the initiation of DRX which are consistent with Gottstein et al.’s experimentally found critical stress condition, rc/rs = 0.85. The current model was unable to predict the significant stress softening that is experimentally observed at high strain rates. A reason for this may be that the currently applied nucleation description emphasizes subgrain size but neglects the evolution of subgrain boundary misorientation with strain. In reality, subgrain misorientation evolves with strain, and under high strain rate conditions, the misorientation may evolve sufficiently to take on a higher mobility and hence subgrain growth may be accelerated. Extension of the current model to include the effects of solid solution and precipitation is currently underway. It is hoped that optimal solutions for the hot-working process

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