two-phase titanium alloys during isothermal processing

Hongwei Li, Chuan Wu, He YangState Key Laboratory of Solidication Processing, SchoXian 710072, PR China

a r t i c l e i n f o

Article history:Received 26 December 2012Received in nal revised form 2 May 2013Available online 16 May 2013

Keywords:A. DislocationsB. Crystal plasticity

The (a + b)-type titanium alloys have been intensely studied due to their excellent mechanical properties, such assion resistance, low density, high strength at elevated temperatures and good formability, prompting the use of thesein a wide range of elds from aerospace to biomedical (Majorell et al., 2002; Zhang et al. 2007). Hot working, includingisothermal processing, is the most applicable forming process for these alloys. During this process, complex microstructural

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Corresponding author. Tel./fax: +86 29 8849 5632.E-mail address: yanghe@nwpu.edu.cn (H. Yang).

International Journal of Plasticity 51 (2013) 271291

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International Journal of Plasticityhttp://dx.doi.org/10.1016/j.ijplas.2013.05.0011. Introductioncorro-alloysthe latter is about 0.550.75 times in the a + b region of that in the b region. The effects thatintergranular non-uniform deformation due to grain orientations and interphase non-uni-form deformation due to different properties do on DRX are quantied. Nucleation rateand the recrystallized fraction at phase interface are about 4 and 10 times, respectively, ofthose within the b phase. Nucleation rate at b-grain boundary seems uniform, while, therecrystallized fraction becomes more and more non-uniform with increasing deformation.

2013 Elsevier Ltd. All rights reserved.B. Constitutive behaviourC. Finite elementDynamic recrystallizationol of Materials Science and Engineering, Northwestern Polytechnical University, 127 Youyi West Road,

a b s t r a c t

A new crystal plasticity model was proposed for synchronously responding dynamic recrys-tallization (DRX) and thermomechanical behavior of wrought two-phase titanium alloys.Within the crystal plasticity framework, the theories for dislocation density evolution andDRXwere introduced andmodied. The shear strain rate of slip system calculated via crystalplasticity was employed to determine the dislocation density of a grain. Dislocation annihi-lation caused by dynamic recovery was incorporated. The evolution of the dislocation densi-ties in the matrix grain (M-grain) and the recrystallized grain (R-grain) was consideredindividually, thus, repeated nucleation of the recrystallized grainswas permitted. Theywereconsidered to takeplace once thedislocation density of a grain (M-grain or R-grain) reached acritical value. The equivalent dislocation density of the grains aggregate (R-grains and M-grain) was calculated via a volume-averaged approach using the recrystallized fraction.The recrystallized fractionwas updated by accounting for the percentage of the grain bound-ary density of the unrecrystallized matrix. This model was coded as a VUMAT in ABAQUS/Explicit and embedded in the nite element method for the simulation of isothermal com-pression on a two-phase titanium alloy. The predictions of the model are in good agreementwith experimental data of the IMI834 alloy. The synchronous coupling effects of thermaldeformation responses and DRX are accounted for, which improves the accuracy of the sim-ulation results, e.g., the maximal error of the predicted steady stress to experiment is 3.1%.Nucleation rate and R-grains growth are quantied under different deformation conditions.The former is far larger (more than 20 times) in the a + b region than that in the b region, andCrystal plasticity modeling of the dynamic recrystallization of

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272 H. Li et al. / International Journal of Plasticity 51 (2013) 271291evolution takes place and, in turn, signicantly inuences the deformation behavior of the material as well as its mechanicalproperties after deformation (Fan andYang, 2011). Dynamic recrystallization (DRX) is a typical phenomenonof themicrostruc-tural evolution of (a + b)-type titanium alloys in the hot working process. DRXwas conrmed by the observation of new equi-axeda grains (Huanget al., 2009; Furuhara et al., 2007; Seshacharyulu et al., 2000; Zong et al., 2009; Poorganji et al., 2009) and bgrains (Vo et al., 2008; Ding andGuo, 2004) of two-phase titaniumalloys experimentally deformed in the a + b phase eld. Thistype ofmetallurgical phenomenon initiates upon reaching a critical condition, such as a critical strain (Fan and Yang, 2011) or acritical dislocation density (Ding and Guo, 2001). Additionally, the initiation of DRX is accompanied by the nucleation of newgrains, assumed to occur on grain boundaries, at a rate dependent on temperature and subsequent grain growth. DRX results inthe reduction of systemic energy by reducing the dislocation density and ultimately affects the hot working behavior.

During recent years, considerable experimental work has been conducted to investigate DRX during the hot deformationof titanium alloys (Khan et al., 2007; Sung et al. 2010; Liss et al., 2009; Meyers et al., 2000; Yang and Wang, 2006). Theseworks usually relate DRX to the forming temperature, strain-rate and strain when considering the effects of DRX on the owbehavior of alloys. Compared to the experimental methods, numerical simulation is an ideal alternative tool for the study ofDRX due to the inherent cost savings of simulations (Vo et al., 2007). Phenomenological constitutive formulas based onexperimental data were developed to compute the kinetics of DRX (Zhou, 1998; Brown and Bammann, 2012). This typeof constitutive model was also integrated into the nite element method (FEM) to predict the recrystallized volume fractionand the average grain size of titanium alloys in hot forming (Sun et al., 2010). Clearly, these phenomenological constitutiveformulas lack detailed information about the deformation process at the grain level. In addition, these formulas are unable tocapture the morphology and topological characteristics that directly affect the mechanical properties of metal materials.

Meso-scale simulation methods, such as cellular automaton (CA) and Monte Carlo (MC) models, have been proposed tosimulate the temporal evolution of the recrystallization microstructure. The MC model is essentially thermodynamic be-cause it extrapolates a trajectory towards lower energy situations. The MC model predicted features related to the staticrecrystallization (SRX) and the DRX (Rollett et al., 1989, 1992; Ling and Anderson, 1992; Caleyo et al., 2002). Although thismodel can give information about recrystallized microstructures, it is not straightforward to scale the simulation for realistictimes and lengths. The CAmodel is essentially a kinetic approach to the problem of recrystallization, where the existence of adriving force is tacitly assumed and the microstructural evolution is traced over time. Thus, CA models are easy to scale intime and length but depend on compliance with thermodynamics (Mukhopadhyay et al., 2007). Recent applications of CAmodels increasingly concern fundamental metallurgical principles for dislocation evolution (Ding and Guo, 2001), ow stressdue to dislocation interaction (Qian and Guo, 2004), driving forces and grain boundary mobility (Marx et al., 1999).

In addition, the aforementioned phenomenological constitutive formulas, MC and CA models treat the relationship be-tween DRX and deformation parameters, such as strain and strain rate, from a macroscopic perspective or with an assump-tion of homogeneity. Thus, they generally fail to capture the local deformation information at the grain level. Critically, theoccurrence of DRX is closely correlated with this grain-level deformation information. Crystal plasticity modeling serves asan effective tool to overcome this modeling deciency.

Two key aspects of crystal plasticity modeling have been discussed in recent years. One aspect is the numerical algorithmfor solution of high-order crystal plasticity equations. Kalidindi and Anand (1992) proposed a typical fully implicit iterationmodel that was solved by the NewtonRaphson algorithm. Li et al. (2008) adopted a homotopy auto-changing continuationmethod to enhance numerical stability. Additionally, explicit algorithms (Kuchnicki et al., 2006, 2008; Rossiter et al., 2010;Lee et al., 2010a,b; Li and Yang, 2012) were proposed to improve numerical efciency. The other aspect of crystal plasticitymodeling recently explored is how to account for the interaction between neighbors. To address the problem of a lack ofgrain interaction considered in the Taylor model, Van Houtte et al. (2002) proposed a bicrystal model the so-called LAMELmodel which calculated two crystals at the same time in order to consider grain interactions. Van Houtte et al. (2005) im-proved the LAMEL model to the ALAMEL model by considering two neighboring domains, each representing the subdivisionof a grain. Subsequently, Mahesh (2009, 2010) proposed a binary-tree based model to maintain traction continuity across thegrain interface within an aggregate by dividing the aggregate to subaggregates and then subdividing the subaggregates untilthe smallest sub-divisions contained only single grains. Similarly, Kumar et al. (2011) and Kumar and Mahesh (2012) pro-posed a stack model based on the ALAMEL model to account for intra- interactions by means of an arbitrary number ofco-deforming domains. In addition, self-consistent models (Neil and Agnew, 2009; Fan and Yang, 2011; Knezevic et al.,2013) and crystal plasticity nite element models (CPFEM) (Abdolvand et al., 2011; Brahme et al., 2011; Venkataramaniet al., 2008; Shenoy et al., 2008) are good alternatives when considering grain interactions.

For modeling of recrystallization, Raabe and Becker (2000) made the rst attempt to couple the CPFEMwith a probabilisticCA for simulating theprimary SRXof aluminum. In this approach, theCPFEMaccounts for the crystallographic slip and the rota-tion of the crystal lattice during plastic deformation, while the CA algorithm accounts for the kinetics of recrystallization. Forcoupling these twomodels, the calculated state variables in the CPFEMwere translated into those used in the CA grid and pointlocations in the CPFEMweremapped onto the quadratic CAmeshes. Thus, as noted by Xiao et al. (2008), this approach does notreect the effects of a deformed topology on the dynamicmicrostructure evolution expected in DRX. To address this modelingfailure, Xiao et al. (2008) built a quasi-synchronous integration model by coupling the CA approach with a uniform topologydeformation technique for tracking the change in grain topology during plastic deformation.

While the deformed topology affects the DRX process, the DRX itself affects subsequent deformation due to newly gen-erated grains, new grain orientations, dislocation density reduction, slip resistance reduction, volume fraction evolution ofthe recrystallized grains (R-grains), as well as other effects. The respective inuence of the plastic deformation and the DRX

take place simultaneously and interactively. That is to say, the actual coupling model should account for these two aspectssynchronously. Unfortunately, studies accounting for this coupling are rare. Especially for heterogeneous materials such asa + b titanium alloys, different properties between the a phase and the b phase originate from the different crystallographicstructures of these two phases. Therefore, accounting for the signicantly differential properties of the two materials at theinterfaces between a and b is an essential challenge for modeling.

In the present work, a crystal plasticity model for synchronously responding thermomechanical behavior and DRX of a + btitanium alloys was established. The evolution of the dislocation density at the grain level, including the matrix grain (M-grain) and R-grains, addresses the actual interaction of hot deformation and DRX. In consideration of signicant develop-ments in modeling dislocation dynamics for plastic anisotropy (Benzerga, 2008; Wang et al., 2009; Akarapu et al., 2010;Lee et al., 2010a; Austin and McDowell, 2011; Wang and Beyerlein, 2011; Gao et al., 2011; Huang et al., 2012), creep behavior(Preuner et al., 2009; Basirat et al., 2012), thin-lm plasticity (Liu et al., 2009, 2011) and other behaviors, the thermomechan-ical process included dislocation generation, multiplication and annihilation to account for DRX and dynamic recovery. Acritical dislocation density at the grain level was adopted for the initiation of the DRX. The difference in dislocation densitiesinside and outside the R-grains provided a driving force for grain growth. The integrated dislocation density of the M-grainand the R-grains was calculated through the volume-averaged approach using the recrystallized fraction. Using this model,the thermomechanical behavior and DRX characteristics of the IMI834 alloy, a near-a titanium alloy, during isothermalworking was investigated.

2. Physically based constitutive modeling for DRX

2.1. Illustration of modeling approach

The target material is the IMI834 alloy whose actual microstructure prior to deformation at 1025 C is shown in Fig. 1(a).In this microstructure, the primary alpha, ap, the lamellar alpha, a0, and the matrix (b) can be observed. The a phase (includ-ing ap and a0) will decrease in volume with increasing temperature and will disappear when temperature exceeds the

H. Li et al. / International Journal of Plasticity 51 (2013) 271291 273Fig. 1. Illustration of modeling approach: (a) actual microstructure prior to deformation at 1025 C (Vo et al. 2008); (b) assumption of spherical a and bgrains and recrystallized nuclei taking place at grain boundary; (c) assumption of the constant volume of the M-grain and R-grain; (d) R-grains growth andthe repeated nucleation.

refersAn

assumnuclea

are tresity of

dislocWh

Thsideresitive

274 H. Li et al. / International Journal of Plasticity 51 (2013) 271291systems are represented in a continuum sense as a plastic shear strain c. Thus, the crystal plasticity theory is consideredto be a physically based theory. Many authors (Taylor, 1938; Hill, 1966; Rice, 1971; Asaro, 1983; Needleman and Tvergaard,1993) have made vital contributions to the development of crystal plasticity theory. In the rate-dependent crystal plasticitymodel, the shear strain rates of slip systems are directly linked to the instantaneous resolved shear stresses imposed on theslip systems. A representative relationship is the specic power law ow rule employed by many authors (Kalidindi et al.,1992; Zhang et al., 2007; Li et al., 2008), which reads,

_ca _c0 sa

sa

1v

sgnsa; 1

where _ca is the shear strain rate of the ath slip system, _c0 is a reference shear strain rate, v is the rate sensitivity coefcient ofmaterial, sa is the resolved shear stress acting on the slip system a and sa is the deformation resistance caused by the dis-location interaction.

Dislocations start to glide under a certain deformation gradient F which has a multiplicative decomposition property ex-pressed as,

F F Fp; 2where the plastic deformation gradient, Fp, represents the plastic shear deformation along the crystalline slip plane, and thenon-plastic component, F, represents the rotation and elastic stretch of the lattice.

The plastic part of the velocity gradient is dened as

Lp _FpFp1; and _Fp Lp Fp; 3and can also be calculated through the crystal plasticity theory as

Lp Xa

_caSa0; Sa0 ma0 na0: 4

Here, ma0 and na0 are time-independent orthonormal unit vectors dening the slip direction and the normal to the slip

plane of the ath slip system in the crystallographic conguration, Sa0 is the Schmid tensor and the subscript 0 denotes thevariable in the crystallographic system.

The Green strain is given by,

E 12FTF I; 5

where I is the second-order identity tensor. The elastic constitutive equation is given by,e main physical mechanism for plastic deformation of the two-phase titanium alloy at elevated temperatures is con-d to be controlled by the motions of dislocations. Rate-dependent crystal plasticity is a tool to account for the rate-sen-and temperature-sensitive deformation behavior. In crystal plasticity theory, massive dislocations gliding along slipgrain boundary density will be introduced, which was reported by Fan and Yang (2011). The detailed modeling approachfor DRX will be discussed in details in Section 2.4.

2.2. Rate dependent crystal plasticityation density of the R-grain and the new R-grain will be updated according to volume-averaged method.en DRX happens, the fraction of the R-grain will be accounted for. To this end, the concept of mobile and immobilecalculated individually in the same way. The volume-averaged dislocation densities of the R-grain and M-grain are taken toaccount for the slip resistance in the crystal plasticity framework. When the dislocation density of the R-grain reaches thecritical value for DRX, the repeated nucleation will happen (Fig. 1(d)). When the repeated nucleation happens, the equivalentated as one grain with the total volume and are assumed with the same grain orientation. The initial dislocation den-the new R-grain is given a near zero value (e.g. 100 m2). The dislocation densities of the R-grain and M-grain will beDeformation taking place at the M-grain and R-grain will be calculated by the crystal plasticity model. Since there willexist a mass of recrystallized nuclei during DRX, it is difcult to calculate each of them individually. So, all of the R-grainsption is made that the recrystallized grain only occupies the space of the specied matrix grain which invokes itstion so that the total volume of the R-grain and M-grain keeps unchanged (Fig. 1(c)).at grain boundary once the dislocation density of the matrix exceeds a critical value. The dislocation density of the matrixcan be calculated by the dislocation density evolution theory including the effects of hardening and dynamic recovery. A for-mula considering temperature and strain rate dependence together with nucleation site effect will be adopted to account forthe nucleation. When the R-grains grow, they will occupy the space of its neighbors. Thus, it brings forward difculty formodeling since the calculation of the current grain should consider the volume change of other grains. Here, anotherto the ap.assumption is made here that all grains are spheroid as shown in Fig. 1(b). The recrystallized nucleation will take placetransus. For the convenience of modeling, the lamellar alpha does not be considered in the work so that a grain in the context

T R : E; 6where T is the stress measure that is the elastic work conjugate to the strain measure E and R is a fourth-order elasticitytensor. The relationship between T and the Cauchy stress T is,

T det FF1TFT: 7In Eq. (1), the resolved shear stress may be approximated by

sa T : Sa0: 8The slip resistance (sa in Eq. (1)) is related to the dislocation structure and microstructure, expressed as

sa 1Nc1 exph=h0 k=

d

p c2lb

qaP

p: 9

The rst term on the right is the temperature-dependent initial slip resistance, c1 is a constant, h is the working temper-ature, h0 is the reference temperature. The second term on the right represents the size-effect on slip resistance. k is the HallPetch coefcient and d is the grain size. It should be mentioned here that the rst two terms is divided by the total number ofslip syearly aand kbelievhardeing, this a t

Fig. 2.

H. Li et al. / International Journal of Plasticity 51 (2013) 271291 275The spatial geometric relationship of dislocation gliding direction with the directions of slip systems: (a) edge dislocation and (b) screw dislocation.P

which will be talked about in next section.

2.3. Dislocation density evolution

Dislocation density serves as an intermediate variable relating DRX to microscopic deformations. Many researchers havemodeled dynamic microstructural evolution during metal processing based on the theory of dislocation density (Ma and Rot-ers, 2004; Lin et al., 2005). The evolution of dislocation density is considered a complex process governed by dislocation mul-tiplication and annihilation.

Ma et al. (2006a,b,c) have established a set of equations to describe the dislocation multiplication and annihilation pro-cess for aluminum with a face-centered cubic (FCC) structure. It is accepted that edge dislocation plays a core role in thedeformation of FCC or BCC crystals. However, for a metal such as titanium or titanium alloys with the hexagonal close-packed (HCP) structure, screw dislocation may account for the plastic deformation (Castany et al., 2007). In this study, weassumed that both the edge and screw dislocation were straight. Thus, for the edge dislocation, the dislocation direction,or tangent vector, is perpendicular to the dislocations Burgers vector (slip direction m), while for the screw dislocation,the tangent vector is parallel to m, as illustrated in Fig. 2 where m and n denote the slip direction and normal of slip plane.Thus, the forest and parallel dislocation densities may be modied follows,

qaF XNb1

qbI j cosna;nb mbj; qaP XNb1

qbI j sinna;nb mbj Edge dislocation; 10a

qaF XNb1

qbI j cosna;mbj; qaP XNb1

qbI j sinna;mbj Screw dislocation: 10bstems, N, to apply resistance on each slip system by an assumption of the initial individual slip resistance would lin-dd up to the overall resistance. The error by this assumption would be compensated through the determination of c1by tting experimental data. The third term on the right, neglecting the mobile dislocation density which is usuallyed to be lower than the immobile one by at least one order of magnitude (Ma and Roters, 2004), represents the workning which is mostly considered to be a result of dislocation interaction from all slip systems. During dislocation glid-e interaction of mobile dislocations with immobile dislocations causes hardening, or a resistance to continuous slip. c2ting constant, l is the shear modulus, b is the Burgers vector, qa is the parallel dislocation density for slip system a

total nTh

immoloy, the immobilization is mainly caused by the interaction of the mobile dislocations with forest dislocations. In Ma and

be giv

Eqs. (1densit

wherethe re

a a

2.4.1. Initiation of DRX determined by a critical dislocation density

276 H. Li et al. / International Journal of Plasticity 51 (2013) 271291q < qc; No DRX;

where, qc is the critical dislocation density, which can be calculated by (Ding and Guo, 2001)

qc c5Q int _eblMQ2line

!1=3: 18

Here, Qint is the interface energy, _e is macroscopic strain rate of grains aggregate, l 1= qp is the average free distance ofdislocation, M is the boundary mobility, which is calculated in Eq. (21), Qline is the line energy, which is proportional tolb2, and c5 is a tting parameter.qP qc; DRX happens;

17DRX in titanium alloys takes place when a critical condition is met (Vo et al., 2007, 2008). Critical dislocation density isusually considered to be an effective index indicating the occurrence of DRX at the grain level (Humphreys and Hatherly,2004). That is to say, DRX occurs once the dislocation density of a grain, q PaqaI , reaches a critical value,here q1str is the dislocation density of the primary R-grain and q2ndr is the dislocation density of the new R-grains.

2.4. Dynamic recrystallizationqar qa1str1 fr qa2ndrfr; 16next section. At the case of repeated nucleation of the recrystallized grain, fm needs to shrink by fr accordingly since weassume the new R-grains occupy the space of the primary R-grain, as illustrated in Fig. 1(d). At this time, the equivalent dis-location density of the recrystallized aggregate can also be calculated by Eq. (15), which is rewritten asf = fm + fr is the total recrystallized fraction. Here, fm denotes the recrystallized fraction of the M-grain and fr denotescrystallized fraction of the recrystallized grain. fm and fr evolve in the same way which will be discussed in details inWhen DRX takes place, the equivalent dislocation density of the aggregate is the volume-weighted dislocation density ofthe M-grain and the R-grain, expressed as

qa qam1 f qar f : 150)(14) account for the evolution of dislocation density of the matrix prior to DRX. It is mentionable that dislocationies of the R-grain and M-grain, qam and qar , evolve in the same way according to Eqs. (10)(14) when DRX takes place.where c40 and f are tting constants, R is the ideal gas constant, and Qdeform is the deformation activated energy.Therefore, a complete dislocation density evolution equation for each slip system can be given by,

_qaI _qa _qa1: 14c4 c40 _ca exp QdeformRh ; 13where c3 is a constant and _qa is the increasing rate of the immobile dislocation density. The second process is dislocationannihilation, which is mainly caused by the interaction of mobile dislocations with immobile dislocations. At higher temper-atures, this process is mainly controlled by the cross slip of screw dislocations, which is given by,

_qa1 c4qaI _ca; 12

where c4 is a function depending on the strain rate and temperature that describes the thermally activated process per-formed by screw dislocation cross-slip. This term is given by (Fan and Yang, 2011),

1=fen by,

_qa c3qaF

p_ca; 11Roterss work (2004), the authors deduced that the dislocation density increase caused by the immobilization process couldumber of slip systems.ere are two processes contributing to the change in the dislocation density (Ma and Roters, 2004). The rst is thebilization of mobile dislocations, which increases the dislocation density. In each phase of the two-phase titanium al-Here, qaF represents the forest dislocation density of the ath slip system, respectively; m and n are the slip direction andthe normal of the slip plane, respectively. In these relationships, trigonometric functions (cos and sin) perform the projectionof qbI , the immobile dislocation density for slip system b, onto the planes forest dislocation and parallel dislocation. N is the

Thesearies oboth temperature and strain rate as follows,

Rh

Here,strain rate of the grain which has a relationship with _ca as _c _ca. Q is the activated energy for nucleation. This function

where

0 m

case (It is generally accepted that R-grains growth is achieved by the migration of grain boundaries. The moving velocity of

where

P P P : 22The driving force Pd is provided by the difference in dislocation densities between the M-grain and the R-grain. It is cal-

m I m r I r

Th

r

whereboundary energy of the high angle boundary, which can be directly calculated as follow (Chen et al., 2010),

H. Li et al. / International Journal of Plasticity 51 (2013) 271291 277Qm lbom

4p1 t : 26

Here, Om is the misorientation when the grain boundary becomes a high angle boundary (set to 15 in this research) and tis the Poissons ratio. Notably, this pressure is only signicant when the new grains are small.r is the equivalent curvature radius of the mobile grain boundaries, which will be discussed later, and Qm is the grainenergy decreases while the interfacial energy increases due to the increase of grain boundary density which will be discussedlater. Therefore, the increased boundary energy serves as a retarding pressure on the grain growth. The retarding pressure isgiven by

Pr 2Qm ; 25a a

e retarding pressure Pr is provided by changes of interfacial energy. During the process of R-grains growth, the storedculated by (Xiao et al., 2008),

Pd c8qm qrlb2; 23

where c8 is a constant, qm and qr are the dislocation densities of the M-grain and the R-grain, respectively. They are ac-counted for by the sum of the dislocation densities in all slip systems, i.e.,

q X

qa and q X

qa 24d rvation energy per mole.The net pressure P in Eq. (20) can be attributed to two aspects, described asKBh Rh

d is the thickness of characteristic boundary, D0b is the diffusion coefcient at 0 K and Qdiffu is the self-diffusion acti-where M is the boundary mobility and P is the net pressure on the boundary, which are dened by Eqs. (21) and (22)(Mclean, 1957), respectively,

M bdD0b exp Qdiffu

; 21grain boundaries is given by (Humphreys and Hatherly, 2004)

v MP: 20Eq. (19a)) to the actual case.boundary). The grain boundary area per unit volume with a grain radius of rm is 3/2r by the assumption of spheroidal grains.Eq. (19a) can be considered as the nucleation in the unit volume with the grain radius of r0. So, the ratio of grain boundarydensity of the actual case to that of the ideal case is r /r . This factor is used here to scale the nucleation rate from the idealRh rm

r0 is the radius of recrystallizing nuclei, rm is the M-grain radius. r0/rm reects the effect of nucleation site (grainact

describes the ideally uniform volume nucleation without consideration of nucleation site effect. However, in fact, nucleationusually occurs at grain boundary. So, a modied function is proposed here as

_n c6 _cc7 exp Qact

r0: 19bc6 and c7 are material constants which can be determined by tting experimental data (see Section 3.1), _c is the shearP_n c6 _cc7 exp Qact

: 19ainhomogeneities may refer to pre-existing microstructural features such as the second-phase particles, grain bound-r shear bands (Cram et al., 2009). Ding and Guo (2001) considered the nucleation rate of DRX, _n; to be a function of2.4.2. Nucleation and R-grains growthIt is well known that the newly recrystallized nuclei usually originate at inhomogeneities in the deformedmicrostructure.

DDRX(Cramthe matrix is gradually occupied by the R-grains of low dislocation density. Suggested by Fan et al. (2010) and Fan and Yang

ferentTh

After nucleation, the R-grains grow driven by the net pressure on the boundaries which is given in Eq. (22). Therefore, themobil

g m

_r v _n r nr _s ; 29

with t

impin

sm v

neighare p

278 H. Li et al. / International Journal of Plasticity 51 (2013) 271291(sm/sim)1/2 (v/r), may result in a discrepancy from the actual value. So, c9 also serves to compensate the errors producedby the two terms.

Therefore, the evolution of the mobile grain boundary density can be updated by

_sm _snm _sgm _sim: 31This equation indicates the mobile grain boundary density increases with the increasing recrystallized nuclei and growth

of R-grains, and decreases when different R-grains impinge.Meanwhile, nucleation, R-grains growth and impinging also cause the evolution of the immobile grain boundary density.

It can be described by

_sim 14 _snm simsmv c10 _sim: 32

The three terms on the right of Eq. (32) represent the reduction of the immobile grain boundary density by nucleation andR-grains growth and its increase by R-grains impinging, respectively. c10 is a tting constant.boring R-grains; v/r represents the increasing rate of the radius of the mobile grain boundaries. These two termsroportional to the impinging rate of mobile grain boundaries. Theoretical estimation of impinging probability,_sim c9sm sim r ; 30

where, c9 is a material parameter reecting the sensitivity of the impinging rate to the number of neighboring R-grains of acertain R-grain; (sm/sim)1/2 is approximately proportional to the ratio of the R-grains radius and the average distance betweenging can be calculated by

1=2the deduction on this equation.The R-grains impinge during their growth. When different R-grains impinge, new grain boundaries are formed at the

touching surface of two impinging R-grains. Meanwhile, the former mobile grain boundaries of each R-grain disappear.The newly forming grain boundaries are immobile as dislocation structures. Thus, mobile grain boundaries convert to immo-bile grain boundaries when different R-grains impinge. The decreasing rate of mobile grain boundaries caused by R-grainsr sm0 m

he initial condition r np r0. n _nt is the nuclei in unit volume at time t. Please refer to Appendix A for the details of_sm r v ; 28

where v is the velocity of the mobile grain boundary which is proportional to the boundary mobility and the net pressure onthe boundaries as given in Eq. (21), and r is the equivalent curvature radius of the mobile grain boundaries. The radius in-creases with nucleation and R-grains growth, since they increase the mobile grain boundaries, and decreases as a result of R-grains impinging, when the mobile grain boundaries convert to the immobile one. The radius can be calculated by

r0 2 4pr2 p ne grain boundary density increases while the R-grains grow according to

2slution. The initial immobile grain boundary density has the relationship with the initial grain size as s0im 3=d by an assump-tion of spheroidal grains. The immobile grain boundary serves as the nucleation site. Therefore, the evolution of mobile grainboundary density caused by nucleation is related to

_snm _n s0: 27Here, s0 4pr20 is the immobile grain boundary area of a nucleus, _ns0 represents the incremental rate of mobile grain

boundary density.R-grains impinge or the driving force reaches zero (Fan and Yang, 2011).e deformation information (e.g. shear strain rate) of the grain can be taken to calculate the grain boundary density evo-(2011), mobile grain boundaries and immobile grain boundaries are introduced to describe the grain boundary migrationduring nuclei growth. All initial grain boundaries are immobile before recrystallization, which serve as nucleation sites. Mo-bile grain boundaries initialized by nucleation, sweep through the deformedmatrix, leading to the annihilation of dislocationstructures as well as immobile grain boundaries. Mobile grain boundaries convert to immobile grain boundaries when dif-is phenomenologically characterized by repeated nucleation and boundary migration of low dislocation density grainset al., 2009; Fan and Yang, 2011). Driven by the stored energy due to dislocations accumulation, the nuclei grow and2.4.3. Evolution of grain boundaries density in DRXDiscontinuous dynamic recrystallization (DDRX) can be depicted by the traditional nucleation and growth theory, i.e.

What is worth mentioning, the same theory is suitable to account for the repeated recrystallization of the recrystallizedgrains. When the repeated DRX takes place, the grain size of the recrystallized grain at this time is used to determine theinitial immobile grain boundary density.

H. Li et al. / International Journal of Plasticity 51 (2013) 271291 279Fig. 3. Incremental procedure for coding the model presented in this work.

2.4.4. Evolution of recrystallized fraction and grain sizeThe recrystallized fraction is measured by the percentage of unrecrystallized matrix, and can be expressed as

_f 1 f smv : 33The average recrystallized grain size rg can be related to the equivalent curvature radius of the mobile grain boundaries r.

The migration of the R-grains boundaries is equivalent to the movement of the equivalent curvature. Thus, there exists4pr2gvn: 4pr2v , where n _nDt is the number of nuclei per unit volume at the end of the period. The average recrystallizedgrain size is then calculated by

rg r=n

p: 34

2.5. Numerical procedure

2.5.1. Crystallographic system for stress updateDuring the constitutive updating, all tensors and vectors should be in the same system. In this model, tensors provided by

nite element method, such as F, and T, are in the global system (C), while tensors related to grains, e.g. Sa and R, are in the

The ployed

h0 Reference temperature (K) 180 (9)R Ideal gas constant (J K1 mol1) 8.31 (13) (19) and (21)mKB

QactQdefo

280 H. Li et al. / International Journal of Plasticity 51 (2013) 271291dDob Product of boundary thickness and diffusion coefcient (m s ) 5.7 10 (21)r0 Radius of a nucleus (m) 0.5 109 (29)Activated energy for nucleation (J mol ) 150 10 (19)rm Activated energy for deformation (J mol1) 150 103 (13)

3 1 14b Burgers vector (m) 2.95 1010 (a)2.86 1010 (b)

(23)

Qm Boundary energy of high angle (J m2) 0.43 (25) and (26)Qdiffu Diffusion activation energy (J mol1) 97 103 (a)

153 103 (b)(21)

1 3Poissons ratio 0.39 (26)Boltzman constant (J K1) 1.38 1023 (21)imental data obtained in the a + b region deformation on the basis of the determined parameters of b grain. Of course, onlysome of the data are used for parameter tting and the others are taken for model verication.

Table 1Model constants with certain physical meanings.

Symbol Meaning (unit) Value Equations

_c0 Reference shear strain rate 0.001 (1)tting procedure involves two steps. At the rst step, the experimental data obtained in b region deformation is em-to determine the parameters of b grain. Subsequently, the parameters for a grain are determined by tting the exper-0

crystallographic system (C0). So, one side of them needs to be transferred to the other system in order to ensure all of them inthe same system before constitutive calculation. The rotation tensor R performs the transformation as

D R D0 RT and D0 R1 D RT 35In Eq. (35), D stands for a tensor in the global system and D0 in the crystallographic system. According the analysis in Li

and Yang (2012), in the current work, all tensors in C were transferred to C0 at rst. Then, the constitutive update was per-formed with Sa0 and R unchanged for all crystals in C0. Then, the calculated stress T and orientation rotation tensor R weretransferred back to C to do the volume-averaged calculation for stress.

2.5.2. Explicit incremental algorithmIn the present work, an Euler forward integration scheme was developed to determine the evolution of state variables

during loading. The detailed numerical procedure for this model is illustrated by the owchart in Fig. 3. Following this ow-chart, a user material subroutine VUMAT is developed under the environment of ABAQUS/Explicit.

3. Model calibration

3.1. Parameter determination

A near-a titanium alloy, IMI834 alloy (Ti5.8Al4Sn3.5Zr0.7Nb0.5Mo0.35Si0.06C) is taken as the material for mod-el calibration. This alloy has a b transus of 1318 K. Recrystallization is observed in the b phase but not in the primary a phase.Therefore, only b recrystallization is considered in this model. Constants of IMI834 alloy that are physically meaningful andindependent of strain, strain rate or temperature are listed in Table 1. Fitting parameters are determined by tting the exper-imental data of IMI834 alloy on ow stress, recrystallized fraction and recrystallized grain size reported by Vo et al. (2008).

The temperature dependent shear modulus is taken to be (Fan and Yang 2011)

l=GPa 49:02 5:821exp181 K=h 1 : 36

Although the shear modulus of the a phase is generally higher than that of the b phase, the discrepancy is not signicant.Thus, we assume that both the a and b grains have the same shear modulus. Based on the assumption of spheroidal grain, therecrystallized fraction is approximately equal to (Ding and Guo, 2001)

f _n e_e43pr3g : 37

By combing with Eq. (19b), this equation can be rewritten as f 4=3 c6 pr2r0 e _cc71 expQ act=Rh with the approxi-mation of _e _c. Thus, two groups of experimental data containing the recrystallized fraction, f, and the average recrystallizedgrain size, rg, at 1373 K and 1 s1 and 0.1 s1 can determine the parameters for nucleation, i.e., parameters c6 and c7 in Eq.(19b). Parameters for the evolution of mobile and immobile grain boundary density are taken from the literature (Fan andYang, 2011).

Vo et al. (2008) pointed out the peak strain of this metal was small, as measured as 0.01, and varied slightly with tem-perature and strain rate. It indicates a small value of the critical dislocation density. Therefore, the parameter c5 in Eq. (18) isdetermined by tting the peak strain of this metal deformed at 1373 K and 1 s1.

Parameter c3 in Eq. (11) represents the increasing rate of the immobile dislocation density due to immobilization of themobilthe sa

H. Li et al. / International Journal of Plasticity 51 (2013) 271291 281Fig. 4. The determination of rate sensitivity coefcient of the IMI834 alloy by tting experimental data.recovery may gradually balance the dislocation generation after peak strain, which results in a steady state during deforma-tion. Therefore, the order of magnitude of immobile dislocations in the matrix during the steady state may be estimatedaccording to Eq. (9), and the ratio of c4 to c3 can be determined according to the equivalence of Eqs. (11) and (12). In thisstudy, the order of magnitude of the ratio of c40 to c3 is approximately 107. Parameter c8 in Eq. (23) is a constant between0 and 1, which reects the magnitude of the driving force on grain boundary migration, thus the velocity of grain boundarymigration. Namely, the parameter reects the changing rate of grain boundary density and then the changing rate of therecrystallized fraction. The changing rate of recrystallized fraction determines the changing rate of ow stress under the con-dition of certain dynamic recovery. Therefore, c8 can be determined by tting the ow stress at 1373 K and 1 s1. It is set to0.5 in this study.

The initial slip resistance of the b grain is temperature dependent, the value of which was determined by tting the exper-imental yield stress. The measured yield stress at 1373 K under the strain rate of 1 s1 is used. Thus, parameter c1 in Eq. (9),reecting initial resistance, is determined. Parameter h0 is set to 180 K according to Fan and Yang (2011). The average grainsizes of the M-grain and R-grain change slightly during DRX. The HallPetch effect inuences the slip resistance slightly. TheHallPetch coefcient k is related to shear modulus and Burgers vector as k Hl

b

p, and the constant H is set to 0.031

according to Fan and Yang (2011). The parameter c2 in Eq. (9) is determined by tting the measured ow stresses at1373 K and 1298 K under the strain rate of 1 s1. Similarly, the measured yield stress at 1298 K under the strain rate of1 s1 is used to determine c1 for the a grain. At this temperature, the volume fraction of the a phase is 0.12 (Fan and Yang,2011).

Rate sensitivity coefcients, v, in Eq. (1) is obtained from the experimental ow stress according to the relation of

lnr v ln _e const: 38e dislocation. Parameter c40 in Eq. (13) represents the dislocation annihilation produced by dislocation interactions inme system and dislocation climb or cross slip between different systems. Dislocation annihilation during dynamic

The lnr ln _e relationship of the IMI834 alloy deformed in the b phase eld and the a + b phase eld is shown in Fig. 4,illustrating that v is slightly temperature-dependent, which is larger in the b region (0.269) than that in the a + b region(0.243), in a limited strain rate range of 0.011 s1. Although a remarkable discrepancy of v of the a phase from that ofthe b phase for Ti6Al4V alloy was noted, the same value of vwas identied for the two phases of the near-a titanium alloy,IMI834 (Fan and Yang, 2011).

Since the a phase is harder than the b phase, it is almost undeformable. So, the other parameters of the grain except forthose aforementioned are set to the same with those of the b phase. All tting parameters are listed in Table 2.

3.2. Model verication

Beyond the experimental data used for parameter determination, the rest including ow stress, recrystallized fraction andaverage recrystallized grain size were employed for model verication. The deformation process is an ideal isothermal com-pression, i.e., no friction between the material and dies and no temperature change during the process were considered. Dueto the isothermal process, no phase transformation takes place. This deformation process was modeled through a single cubeelement in ABAQUS/Explicit. 100 grains with random orientations were taken into account in the simulation. The phase vol-ume fraction for modeling was taken from the literature (Fan and Yang, 2011). The initial b grain diameter was 350 lm in bregion and 50 lm in the a + b region, and the initial a grain size was 30 lm. For the a phase, the most common deformationmechanism depends on the basal, prismatic and pyramidal glide of screw dislocations with the same glide direction h11 20i(hai-type slip). Moreover, at high temperature, the activation of the rst-order pyramidal planes with slip direction h11 23i(ha + ci-type slip) becomes possible (Zhang et al., 2007). Twinning is not considered in this study due to the high aluminum

Table 2Parameters determined by tting experimental data.

c1 (MPa) c2 c3 (m1) c40 c5 c6 c7 c8 c9 c10 f

5.1 105 (a)2.2 105 (b)

0.4 5.0 107 (a)4.7 107 (b)

200 (a)97 (b)

3.0 106 2.5 1016 2.3 0.5 8.0 0.3 6.0

Table 3Active slip systems in the a and b grains.

282 H. Li et al. / International Journal of Plasticity 51 (2013) 271291Fig. 5. Comparison of predicted ow stress with experimental measurements at the b region (1373 K) and the a + b region (1298 K).

content (Williams et al., 2002) and high deformation temperature used. For the b phase, the deformation mode follows thedescriptions of Mayeur and McDowell (2007). Gliding of the slip systems of the {110} planes with h111i directions accountsfor the deformation. The detailed information of active slip systems is listed in Table 3. The calculated results on ow stress,recrystallized fraction, and recrystallized grain size under the conditions of 1373 K, 1333 K, 1303 K, 1273 K and 1 s1, 0.1 s1,and 0.01 s1 were compared with the experimental data. They are shown in Figs. 57, respectively. In Fig. 7, the ZenerHoll-omon parameter is Z _e expQ=RT (Q = 180 kJ mol1). Consistency between the calculated results and the experimentaldata is good, which indicates the reliability of this model.

Fig. 6. Comparison of the predicted recrystallized fraction with experimental measurements.

H. Li et al. / International Journal of Plasticity 51 (2013) 271291 283Fig. 7. Comparisons of the predicted average diameter of the recrystallized grains with experimental measurements.

4. Results and discussions

4.1. Stressstrain responses

As shown in Fig. 5, the ow stresses of the IMI834 alloy are measured and predicted at given temperatures and strainrates. The temperature and strain rate dependent stress responses are easily found both in the b region (1373 K) and inthe a + b region (1298 K). The almost same small peak strain (0.02) is found in these two regions. It may indicate thatDRX takes place soon after deformation starting, which will be conrmed by discussing the evolution of recrystallizationkinetics later. However, the material shows a distinct peak stress at all of the strain rates in the a + b region, while no peakstress in the b region. In the b region, the ow stress rapidly increases to a steady value and then keeps almost unchanged. Inthe a + b region, the ow stress reduces dramatically after rapidly reaching the peak stress and then arrives at a steady stage.Moreover, the peak stress increases with the increasing strain rate. The different features in ow stress response indicatedifferent softening mechanisms in these two deformation regions. Dynamic recovery and DRX are the two important soft-ening mechanisms for titanium alloy. Dynamic recovery, annihilating dislocation structures, competes with work hardening,inducing dislocation multiplication, and results in a balance in dislocation density. Therefore, dynamic recovery makes theow stress keep a steady level. During DRX, a mass of new R-grains are generated. They occupy a portion of the matrix andannihilate all of the dislocation structure in the occupied volume. So, DRX may cause a reduction of the equivalent disloca-tion density and then the peak stress. However, the fraction of R-grains at the small strain (0.02) is still very small, which canbe found in Fig. 6. At this time, DRX may be unable to cause this peak stress. Semiatin and Bieler (2001a,b) attributed thepeak stress to the dynamic spheroidization of the colony alpha. Unfortunately, this mechanism has not yet been consideredin the model, which results in the discrepancy in the predicted peak stress to the experimental one.

4.2. DRX kinetics

As shown in Fig. 6, the predicted recrystallized fraction agrees with the experimental data of the IMI834 alloy in the b

284 H. Li et al. / International Journal of Plasticity 51 (2013) 271291region and the a + b region. Small recrystallized fractions are predicted and measured in the b region, while relatively largerfractions in the a + b region. Therefore, there are a major effect of DRX in the a + b region and a minor effect in the b region.These kinetics exhibit an S-type curve indicating a rapid increase at the beginning of DRX and then a slight increase to asteady state. In the b region, the recrystallized fraction increases slightly with decreasing strain rate, while almost does notchange with temperature. However, it increases remarkably with increasing strain rate and decreasing temperature in thea + b region. These features can be interpreted from the theory of nucleation and growth.

At the rst aspect, the nucleation rate is proportion to deformation temperature, strain rate since c7 = 2.3, and the nucle-ation sites (i.e., inversely proportional to grain size) (Eq. (19b)). However, whether the nucleation happens or not is depen-dent on whether the dislocation density of the M-grain reaches the critical value or not. Therefore, the higher dislocationdensity of the M-grain helps gain the higher probability of nucleation. Here, the nuclei per unit volume generated with

Fig. 8. Nuclei generated in unit volume (1 mm3) with strain under different temperatures and strain rates.

increasing strain to 1.0 under different strain rates and temperatures are shown in Fig. 8. It can be found the nuclei are muchmore in the a + b region than those in the b region. It can be interpreted as followings. One is the larger grain boundary den-sity in the a + b region since there is a small grain size in this region. The other is the larger dislocation density of the M-grainsince the dynamic recovery is weaker due to the lower temperature. It also can be found the maximal ratio of the nucleinumber in the a + b region to that in the b region reaches up to about 50. However, the ratio of the grain boundary density(nucleation site), which can be approximate to 3/d, in the a + b region to that in the b region is only 7. It may be explained as

Fig. 9. Grain boundary migration velocity under different temperatures and strain rates.

H. Li et al. / International Journal of Plasticity 51 (2013) 271291 285Fig. 10. Evolution of dislocation density of the recrystallized grain (R-grain) and the matrix (M-grain) under different temperatures and strain rates.

that the dislocation density of the M-grain in the a + b region is much higher than that in the b region, which can be found inFig. 10 in Section 4.3. That is to say it is with more probability to nucleate in the a + b region than that in the b region so thatthe accumulation of nuclei with strain in the a + b region is much higher than that in the b region.

In addition, the nuclei increase with increasing strain rate in the a + b region, while decrease in the b region (Fig. 8),though the nucleation rate is proportion to strain rate according to Eq. (19). This difference may also be attributed to thedislocation density of the matrix, which will be discussed in Section 4.3. This may be the reason why the recrystallized frac-tion increases with increasing strain rate in the a + b region, while decrease in the b region.

At the other aspect, R-grains growth relies upon grain boundary migration velocity and time. Fig. 9 shows the grainboundary migration velocity (Eq. (20)) under different temperatures and strain rates. The curves exhibit a continuous de-crease from a peak value. At this time, the dislocation density of the nucleus is very small (near to zero), and then the dif-ference of the dislocation density between the matrix and the R-grain reaches the maximum. The maximal differenceprovides the largest driving force on nucleus for its growth. As strain increasing, the hardening raises the dislocation densityof the R-grain, which decreases the growing speed of the R-grain. Although the retarding force decreases due to the contin-ually growing average radius of R-grain, its inuence on the growing speed of the R-grain is insignicant to mention. Thisfeature contributes to the S-type evolution of recrystallized fraction. Additionally, this gure indicates a higher grain

286 H. Li et al. / International Journal of Plasticity 51 (2013) 271291Fig. 11. Effects of the a phase on the b recrystallized fraction: (a) and (b) the initial phase conguration; (c) and (d) the distribution of the b recrystallizedfraction mainly caused by nucleation at the strain of 0.01; (e)(j) the distribution of the b recrystallized fraction at the strain of 0.1, 0.2 and 0.25; (e), (g) and(i) are the results with a volume fraction of a phase of 16% (VF|a = 16%); (f), (h) and (j) are the results with VF|a = 26%. The calculation conditions are 1273 Kand 1 s1.

boundary migration velocity at a higher temperature and larger strain rate. Although the higher grain boundary migrationvelocity acts on the recrystallized grain in the b region, much less nuclei generated in this region (Fig. 8) lead to the smallrecrystallized fraction. Since the nuclei generated in the b region decrease slightly with increasing strain rate, while thegrowth time is larger under the lower strain rate by reaching the same strain. So, it results in a smaller recrystallized fractionunder a larger strain rate, though the grain boundary migration velocity is larger. However, this rule reverses in the a + bregion since the rule of nucleation reverses in this region.

4.3. Evolution of dislocation density

The grain boundary migration characteristics are attributed to the dislocation density evolution according to Eqs. (20)(24). The difference in dislocation density between the M-grain and R-grain supplies the driving force for grain boundarymigration. So, the evolution of dislocation density is studied here. Fig. 10 shows the dislocation density evolution of theM-grain and R-grain under different temperatures and strain rates. For the M-grain, the dislocation density monotonicallyincreases to near the saturation value under a small strain due to the stronger strain hardening and dynamic recovery effect,while the dislocation density for the R-grains increases gradually at rst before accumulating to the saturation value due tothe very low level initial dislocation density. Lower level dislocation density is found in the b region, while the higher level inthe a + b region. This is attributed to the stronger dynamic recovery in the b region than that in the a + b region due to thehigher temperature (relatively rapid climb or cross slip of mobile dislocation). However, nearly identical dislocation densitycan be found at two different temperatures both in the b region and in the a + b region. It indicates DRX plays a major role ondislocation density evolution in the a + b region than that in the b region. It also can be found the difference of dislocationdensities under different strain rates is smaller in the b region than that in the a + b region. This feature may be the reason forthe rule that nuclei number varies slightly with increasing strain rate in the b region, while varies remarkably in the a + bregion (see Fig. 8).

4.4. The effect of the a phase

Although there is no recrystallization taking place with the a grain, the a grain inuences the DRX of the b grain. Thiseffect can be attributed to two aspects: (1) phase interface serves as nucleation site; (2) a large shear strain occurs at thephase interface due to the harder a particle among the softer b matrix. Fig. 11 shows the evolution of the recrystallized frac-tion of two near-a titanium alloys with different area fractions of the a phase in the 2D model during the isothermal

H. Li et al. / International Journal of Plasticity 51 (2013) 271291 287Fig. 12. Shear strain experience of the conguration VF|a = 26% in Fig. 11.

and (ered elof 0.01 were provided, as shown in Fig. 11(c) and (d). These two pictures have the same legend as given in the middle of

the ph(j) shows the recrystallized fraction evolution with increasing strain. Almost identical recrystallized fractions are found from

this titribut

5. Conclusions

Appendix A. On the evolution of the equivalent curvature radius of mobile grain boundary

288 H. Li et al. / International Journal of Plasticity 51 (2013) 271291As mentioned in the context, the mobile grain boundary area increases with new grains appearance and R-grains growth,but decreases when R-grains impinging. Eq. (20) give the grain migration velocity of an R-grain, v. During a time incrementDt, there are n _nDt R-grains per volume. We assume all the R-grains grow with the velocity of v in the period. The volumeswept over by the mobile grain boundary of the R-grains in the period is

V 4pr20 _nDtv : A:1By setting v 0 dr=dt _r, the volume swept over by the equivalent mobile grain boundary is

V 0 4pr2v 0Dt: A:2The two volumes are approximately identical, thus there exists

v 0 v r0r

2_n: A:3

On the other hand, R-grains will impinge with R-grains growth and new R-grains appearance. At this time, the mobilegrain boundary area decreases since the mobile grain boundary converts to the immobile one. Therefore, the radius ofthe equivalent mobile grain boundary decreases.(1) A new crystal plasticity model was proposed to synchronously account for the DRX and thermomechanical behavior oftwo-phase titanium alloys. This model was coded as a VUMAT in ABAQUS/Explicit and embedded in FEM for simula-tion of isothermal processing on a two-phase titanium alloy.

(2) The correlation of predictions with experiments proved that the model correctly reects the following: no peak stressin the b region; large stress drop after peak stress under large strain rate in the a + b region; the recrystallized fractionincreases with decreasing temperature and increasing strain rate in the a + b region, while decreases with increasingstrain rate in the b region; signicant nuclei growth with decreasing strain rate in the b region.

(3) The results also show that the model captures many features of the isothermal process of two-phase titanium alloys asfollows: signicant nucleation in the a + b region; more nuclei, following larger R-grain growth speed and thus thehigher recrystallized fraction with increasing strain rate in the a + b region, while grain growth contributes more tothe recrystallized fraction than nucleation in the b region; DRX is signicant under a large strain rate in the a + bregion, resulting in signicant decrease of equivalent dislocation density and thus stress; the a-b interface improvethe nucleation and the R-grains growth due to a large shear strain at the phase interface.

Acknowledgments

The authors would like to thank the National Natural Science Foundation of China (51175428), the National Basic Re-search Program of China (2010CB731701), National Natural Science Foundation of China for Key Program (50935007),and the 111 Project (B08040) for supports given to this research.fraction becomes more and more signicant (Fig. 11(g)(j)). Especially, the b grain surrounded by a grains has the greatestrecrystallized fraction. This can be attributed to the large shear strain along the interface (see Fig. 12(c) and (d)). By compar-ing the results in Fig. 11(c), (e), (g) and (i) with those in Fig. 11(d), (f), (h) and (j), no signicant difference can be found underthe conditions of different alpha fractions. In addition, the recrystallization on the a grain would be considered in the case oftoo large alpha fraction since the a grain would also deform remarkably at that time. This is an issue for further investigation.me by comparing with those in Fig. 11(g)(j), as illustrated in Fig. 12(b)(d). At this time, phase interface does not con-e more to the DRX than b grain boundary. With increasing strain, the contribution of phase interface to recrystallizedFig. 11(e) and (f) at the b grain boundary and the ab interface. It can be attributed to the relatively smaller shear strain atase interface. It can be interpreted by the large shear strain rate at the interface as illustrated in Fig. 12. Fig. 11(e)

them. It can be found that the recrystallized fraction distributes non-uniform; the large value appears on the b grain at) give the initial phase congurations of the samples. The fractions of the a phase are 16% and 26%, respectively. Theements stand for the a grain. In order to study the effects on nucleation, the results on recrystallized fraction at a straincompression, and Fig. 12 shows shear strain history during the process. In order to compare the effects under different a areafractions, the simulated deformation processes were under the same conditions of 1273 K and 1 s1. The initial grain sizes ofthe a and the b phase were 30 lm and 50 lm, respectively. All parameters are the same as listed in Tables 1 and 2. Fig. 11(a)

nucChen, F

cellDing, R

Fan, X.Gmo

Furuha

equHuang,

meKalidin

537Khan, A

H. Li et al. / International Journal of Plasticity 51 (2013) 271291 289ranges of strain rates and temperatures. Int. J. Plasticity 23, 931950.Knezevic, M., McCabe, R.J., Tom, C.N., Lebensohn, R.A., Chen, S.R., Cady, C.M., Gray III, G.T., Mihaila, B., 2013. Modeling mechanical response and texture

evolution of a-uranium as a function of strain rate and temperature using polycrystal plasticity. Int. J. Plasticity 43, 7084.tals. Int. J. Mech. Sci. 34, 309329.di, S.R., Bronkhorst, C.A., Anand, L., 1992. Crystallographic texture evolution in bulk deformation processing of fcc metals. J. Mech. Phys. Solids 40,569..S., Kazmi, R., Farrokh, B., 2007. Multiaxial and non-proportional loading responses, anisotropy and modeling of Ti6Al4V titanium alloy over widePlasticity 28, 141158.Humphreys, F.J., Hatherly, M., 2004. Recrystallization and Related Annealing Phenomena, second ed. Pergamon Press, Oxford.Kalidindi, S.R., Anand, L., 1992. An approximate procedure for predicting the evolution of crystallographic texture in bulk deformation processing of fcciaxed microstructure. Mater. Sci. Eng. A 505, 136143.M., Zhao, L., Tong, J., 2012. Discrete dislocation dynamics modelling of mechanical deformation of nickel-based single crystal superalloys. Int. J.59, 6469.Gao, Y., Zhuang, Z., Liu, Z.L., You, X.C., Zhao, X.C., Zhang, Z.H., 2011. Investigations of pipe-diffusion-based dislocation climb by discrete dislocation dynamics.

Int. J. Plasticity 27, 10551071.Hill, R., 1966. Generalized constitutive relations for incremental deformation of metal crystals by multi-slips. J. Mech. Phys. Solids 14, 95102.Huang, L.J., Geng, L., Li, A.B., Cui, X.P., Li, H.Z., Wang, G.S., 2009. Characteristics of hot compression behavior of Ti6.5Al3.5Mo1.5Zr0.3Si alloy with an., Yang, H., Sun, Z.C., Zhang, D.W., 2010. Quantative analysis of dynamic recrystallization behavior using a grain boundary evolution based kineticdel. Mater. Sci. Eng. A 527, 53685377.ra, T., Poorganji, B., Abe, H., Maki, T., 2007. Dynamic recovery and recrystallization in titanium alloys by hot deformation. J. Miner. Metals Mater. Soc.31633175.Ding, R., Guo, Z.X., 2004. Microstructural evolution of a Ti6Al4V alloy during beta phase processing: experimental and simulative investigations. Mater.

Sci. Eng. A 365, 172179.Fan, X.G., Yang, H., 2011. Internal-state-variable based self-consistent constitutive modeling for hot working of two-phase titanium alloys coupling

microstructure evolution. Int. J. Plasticity 27, 18331852.leation. Acta Mater. 57, 52185228.., Cui, Z., Liu, J., Chen, W., Chen, S., 2010. Mesoscale simulation of the high-temperature austenitizing and dynamic recrystallization by coupling aular automation with a topology deformation technique. Mater. Sci. Eng. A 527, 55395549.., Guo, Z.X., 2001. Coupled quantitative simulation of microstructural evolution and plastic ow during dynamic recrystallization. Acta Mater. 49,The probability for impinging is inversely proportional to the fraction of mobile grain boundary since impinging will eas-ily occur due to the less space at the case of large fraction of immobile grain boundary. So, there exists

pimpinging /4pr2

sm: A:4

Also, impinging probability is proportional to nucleation rate. Here, we use _snm instead of _n to keep the same physicalmeanings with Eq. (A.4). So, it can be expressed as

pimpinging / _snm: A:5In addition, the initial condition should be taken into account. We assume there is no impinging when all the R-grains are

recrystallized nuclei with the initial radius of r0. At this time, the equivalent curvature radius r n

pr0. Therefore, we formu-

late the decrease rate of the equivalent curvature radius as

_r 4pr2

smr np r0_snm: A:6

The initial condition is r np r0. Here, the fact of r np r0 not only accounts for the initial condition but also acts as a scalemaintaining the same unit with Eq. (A.3).

With the joint consideration of Eqs. (A.3) and (A.6), the evolution of the equivalent curvature radius can be written as

_r v r0r

2_n 4pr

2

smr np r0_snm: A:7

This is a modied version from that proposed by Fan et al. (2010).

References

Abdolvand, H., Daymond, M.R., Mareau, C., 2011. Incorporation of twinning into a crystal plasticity nite element model: evolution of lattice strains andtexture in Zircaloy-2. Int. J. Plasticity 27, 17211738.

Akarapu, S., Zbib, H.M., Bahr, D.F., 2010. Analysis of heterogeneous deformation and dislocation dynamics in single crystal micropillars under compression.Int. J. Plasticity 26, 239257.

Asaro, R., 1983. Micromechanics of crystals and polycrystals. Adv. Appl. Mech. 23, 1115.Austin, R.A., McDowell, D.L., 2011. A dislocation-based constitutive model for viscoplastic deformation of fcc metals at very high strain rates. Int. J. Plasticity

27, 124.Basirat, M., Shrestha, T., Potirniche, G.P., Charit, I., Rink, K., 2012. A study of the creep behavior of modied 9Cr1Mo steel using continuum damage

modeling. Int. J. Plasticity 37, 96107.Benzerga, A.A., 2008. An analysis of exhaustion hardening in micron-scale plasticity. Int. J. Plasticity 24, 11281157.Brahme, A.P., Inal, K., Mishra, R.K., Saimoto, S., 2011. The backstress effect of evolving deformation boundaries in FCC polycrystals. Int. J. Plasticity 27, 1252

1266.Brown, A.A., Bammann, D.J., 2012. Validation of a model for static and dynamic recrystallization in metals. Int. J. Plasticity 3233, 1735.Caleyo, F., Baudin, T., Penelle, R., 2002. Monto Carlo simulation of recrystallization in Fe50%Ni starting from EBSD and bulk texture measurements. Scr.

Mater. 46, 829835.Castany, P., Pettinari-Sturmel, F., Crestou, J., 2007. Experimental study of dislocation mobility in a Ti6Al4V alloy. Acta Mater. 55, 62846291.Cram, D.G., Zurob, H.S., Brechet, Y.J.M., Hutchinson, C.R., 2009. Modelling discontinuous dynamic recrystallization using a physically based model for

290 H. Li et al. / International Journal of Plasticity 51 (2013) 271291Kuchnicki, S.N., Cuitio, A.M., Radovitzky, R.A., 2006. Efcient and robust constitutive integrators for single-crystal plasticity modeling. Int. J. Plasticity 22,19882011.

Kuchnicki, S.N., Radovitzky, R.A., Cuitio, A.M., 2008. An explicit formulation for multiscale modeling of bcc metals. Int. J. Plasticity 24, 21732191.Kumar, M.A., Mahesh, S., 2012. Banding in single crystals during plastic deformation. Int. J. Plasticity 36, 1533.Kumar, M.A., Mahesh, S., Parameswaran, V., 2011. A stack model of rate-independent polycrystals. Int. J. Plasticity 27, 962981.Lee, M.G., Lim, H., Adams, B.L., Hirth, J.P., Wagoner, R.H., 2010a. A dislocation density-based single crystal constitutive equation. Int. J. Plasticity 26, 925938.Lee, M.G., Kim, S.J., Han, H.N., 2010b. Crystal plasticity nite element modeling of mechanically induced martensitic transformation (MIMT) in metastable

austenite. Int. J. Plasticity 26, 688710.Li, H.W., Yang, H., Sun, Z.C., 2008. A robust integration algorithm for implementing rate dependent crystal plasticity into explicit nite element method. Int.

J. Plasticity 24, 267288.Li, H., Yang, H., 2012. An efcient parallel-operational explicit algorithm for Taylor-type model of rate dependent crystal plasticity. Comput. Mater. Sci. 54,

255265.Lin, J., Liu, Y., Farrugia, D.C.J., Zhou, M., 2005. Development of dislocation-based unied material model for simulating microstructure evolution in multipass

hot rolling. Philos. Mag. 85, 19671987.Ling, S., Anderson, M.P., 1992. Monte Carlo simulation of grain growth and recrystallization in polycrystalline materials. JOM J. Miner. Metals Mater. Soc. 44,

3036.Liss, K.D., Schmoelzer, T., Yan, K., Mark, R., Matthew, P., Dippenaar, R., Helmut, C., 2009. In situ study of dynamic recrystallization and hot deformation

behavior of a multiphase titanium aluminide alloy. J. Appl. Phys. 106, 113526.Liu, Z.L., Liu, X.M., Zhuang, Z., You, X.C., 2009. A multi-scale computational model of crystal plasticity at submicron-to-nanometer scales. Int. J. Plasticity 25,

14361455.Liu, Z.L., Zhuang, Z., Liu, X.M., Zhao, X.C., Zhang, Z.H., 2011. A dislocation dynamics based higher-order crystal plasticity model and applications on conned

thin-lm plasticity. Int. J. Plasticity 27, 201216.Ma, A., Roters, F., 2004. A constitutive model for fcc single crystals based on dislocation densities and its application to uniaxial compression of aluminum

single crystals. Acta Mater. 52, 36033612.Ma, A., Roters, F., Raabe, D., 2006a. A dislocation density based constitutive model for crystal plasticity FEM including geometrically necessary dislocations.

Acta Mater. 54, 21692179.Ma, A., Roters, F., Raabe, D., 2006b. Studying the effect of grain boundaries in dislocation density based crystal-plasticity nite element simulations. Int. J.

Solids Struct. 43, 72877303.Ma, A., Roters, F., Raabe, D., 2006c. On the consideration of interactions between dislocations and grain boundaries in crystal plasticity nite element

modeling: theory, experiments, and simulation. Acta Mater. 54, 21812194.Mahesh, S., 2010. A binary-tree based model for rate-independent polycrystals. Int. J. Plasticity 26, 4264.Mahesh, S., 2009. A hierarchical model for rate-dependent polycrystals. Int. J. Plasticity 25, 752767.Majorell, A., Srivatsa, S., Picu, R.C., 2002. Mechanical behavior of Ti6Al4V at high and moderate temperatures. Part I: Experimental results. Mater. Sci. Eng.

A 326, 297305.Marx, V., Reher, F.R., Gottstein, G., 1999. Simulation of primary recrystallization using a modied three-dimensional cellular automaton. Acta Mater. 47,

12191230.Mayeur, J.R., McDowell, D.L., 2007. A three-dimensional crystal plasticity model for duplex Ti6Al4V. Int. J. Plasticity 23, 14571485.Mclean, D., 1957. Grain Boundaries in Metals. Oxford University Press, Oxford.Meyers, M.A., Nesterenko, V.F., LaSalvia, J.C., Xu, Y.B., Xue, Q., 2000. Observation and modeling of dynamic recrystallization in high-strain, high-strain rate

deformation of metals. J. Phys. IV France 10, 5156.Mukhopadhyay, P., Loeck, M., Gottstein, G., 2007. A cellular operator model for the simulation of static recrystallization. Acta Mater. 55, 551564.Needleman, A., Tvergaard, V., 1993. Comparison of crystal plasticity and isotropic hardening predictions for metalmatrix composites. J. Appl. Mech. 60, 70

76.Neil, C.J., Agnew, S.R., 2009. Crystal plasticity-based forming limit prediction for non-cubic metals: application to Mg alloy AZ31B. Int. J. Plasticity 25, 379

398.Poorganji, B., Yamaguchi, M., Itsumi, Y., Matsumoto, K., Tanaka, T., Asa, Y., Miyamoto, G., Furuhara, T., 2009. Microstructure evolution during deformation of

a near-a titanium alloy with different initial structures in the two-phase region. Scr. Mater. 61, 419422.Preuner, J., Rudnik, Y., Brehm, H., Vlkl, R., Glatzel, U., 2009. A dislocation density based material model to simulate the anisotropic creep behavior of single-

phase and two-phase single crystals. Int. J. Plasticity 25, 973994.Qian, M., Guo, Z.X., 2004. Cellular automata simulation of microstructural evolution during dynamic recrystallization of an HY-100 steel. Mater. Sci. Eng. A

365, 180185.Raabe, D., Becker, R.C., 2000. Coupling of a crystal plasticity nite-element model with a probabilistic cellular automaton for simulating primary static

recrystallization in aluminum. Model. Simul. Mater. Sci. Eng. 8, 445462.Rice, J.R., 1971. Inelastic constitutive relations for solids: an internal-variable theory and its application to metal plasticity. J. Mech. Phys. Solids 19, 433455.Rollett, A.D., Luton, M.J., Srolovitz, D.J., 1992. Microstructural simulation of dynamic recrystallization. Acta Metall. Mater. 40, 4355.Rollett, A.D., Srolovitz, D.J., Doherty, R.D., Anderson, M.P., 1989. Computer simulation of recrystallization in non-uniformly deformed metals. Acta Metall. 37,

627639.Rossiter, J., Brahme, A., Simha, M.H., Inal, K., Mishra, R., 2010. A new crystal plasticity scheme for explicit time integration codes to simulate deformation in

3D microstructures: effects of strain path, strain rate and thermal softening on localized deformation in the aluminum alloy 5754 during simple shear.Int. J. Plasticity 26, 17021725.

Semiatin, S.L., Bieler, T.R., 2001a. The effect of alpha platelet thickness on plastic ow during hot working of Ti6Al4V with a transformed microstructure.Acta Mater. 49, 35653573.

Semiatin, S.L., Bieler, T.R., 2001b. Effect of texture and slip mode on the anisotropy of plastic ow and ow softening during hot working of Ti6Al4V.Metall. Mater. Trans. A 32A, 17871799.

Seshacharyulu, T., Medeiros, S.C., Frazier, W.G., Prasad, Y.V.R.K., 2000. Hot working of commercial Ti6Al4V with an equiaxed ab microstructure:materials modeling considerations. Mater. Sci. Eng. A 284, 184194.

Shenoy, M., Tjiptowidjojo, Y., McDowell, D.L., 2008. Microstructure-sensitive modeling of polycrystalline IN 100. Int. J. Plasticity 24, 16941730.Sun, Z.C., Yang, H., Han, G.J., Fan, X.G., 2010. A numerical model based on internal-state-variable method for the microstructure evolution during hot-

working process of TA15 titanium alloy. Mater. Sci. Eng. A 527, 34643471.Sung, J.H., Kim, J.H., Wagoner, R.H., 2010. A plastic constitutive equation incorporating strain, strain-rate, and temperature. Int. J. Plasticity 26, 17461771.Taylor, G.I., 1938. Plastic strain in metals. J. Inst. Met. 62, 307324.Van Houtte, P., Delannary, L., Kalidindi, S.R., 2002. Comparison of two grain interaction models for polycrystal plasticity and deformation texture prediction.

Int. J. Plasticity 18, 359377.Van Houtte, P., Li, S., Seefeldt, M., Delannay, L., 2005. Deformation texture prediction: from the Taylor model to the advanced lamel model. Int. J. Plasticity

21, 589624.Venkataramani, G., Kirane, K., Ghosh, S., 2008. Microstructural parameters affecting creep induced load shedding in Ti-6242 by a size dependent crystal

plasticity FE model. Int. J. Plasticity 24, 428454.Vo, P., Jahazi, M., Yue, S., 2008. Recrystallization during thermomechanical processing of IMI834. Metall. Mater. Trans. 39A, 29652980.Vo, P., Jahazi, M., Yue, S., Bocher, P., 2007. Flow stress prediction during hot working of near-alpha titanium alloys. Mater. Sci. Eng. A 447, 99110.

Wang, Z.Q., Beyerlein, I.J., 2011. An atomistically-informed dislocation dynamics model for the plastic anisotropy and tension-compression asymmetry ofBCC metals. Int. J. Plasticity 27, 14711484.

Wang, Z.Q., Beyerlein, I.J., LeSar, R., 2009. Plastic anisotropy in fcc single crystals in high rate deformation. Int. J. Plasticity 25, 2648.Williams, J.C., Baggerly, R.G., Paton, N.E., 2002. Deformation behavior of HCP TiAl alloy single crystals. Metall. Mater. Trans. A33, 837850.Xiao, N., Zheng, C., Li, D., Li, Y., 2008. A simulation of dynamic recrystallization by coupling a cellular automaton method with a topology deformation

technique. Comput. Mater. Sci. 41, 366374.Yang, Y., Wang, B.F., 2006. Dynamic recrystallization in adiabatic shear band in a-titanium. Mater. Lett. 60, 21982202.Zhang, M., Zhang, J., McDowell, D.L., 2007. Microstructure-based crystal plasticity modeling of cyclic deformation of Ti6Al4V. Int. J. Plasticity 23, 1328

1348.Zhou, M., 1998. Constitutive modeling of the viscoplastic deformation in high temperature forging of titanium alloy IMI834. Mater. Sci. Eng. A A245, 2938.Zong, Y.Y., Shan, D.B., Xu, M., Lv, Y., 2009. Flow softening and microstructural evolution of TC11 titanium alloy during hot deformation. J. Mater. Process.

Technol. 209, 19881994.

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Crystal plasticity modeling of the dynamic recrystallization of two-phase titanium alloys during isothermal processing1 Introduction2 Physically based constitutive modeling for DRX2.1 Illustration of modeling approach2.2 Rate dependent crystal plasticity2.3 Dislocation density evolution2.4 Dynamic recrystallization2.4.1 Initiation of DRX determined by a critical dislocation density2.4.2 Nucleation and R-grains growth2.4.3 Evolution of grain boundaries density in DRX2.4.4 Evolution of recrystallized fraction and grain size

2.5 Numerical procedure2.5.1 Crystallographic system for stress update2.5.2 Explicit incremental algorithm

3 Model calibration3.1 Parameter determination3.2 Model verification

4 Results and discussions4.1 Stressstrain responses4.2 DRX kinetics4.3 Evolution of dislocation density4.4 The effect of the phase

5 ConclusionsAcknowledgmentsAppendix A On the evolution of the equivalent curvature radius of mobile grain boundaryReferences