Deformation and Recrystallization of Hexagona, Results for Zinc

Acta mater. 49 (2001) 3791–3801 www.elsevier.com/locate/actamat

DEFORMATION AND RECRYSTALLIZATION OF HEXAGONAL METALS: MODELING AND EXPERIMENTAL RESULTS FOR ZINC ´ 1†, O. ENGLER‡1 and H. R. WENK2 D. E. SOLAS1, C. N. TOME MST Division, Los Alamos National Laboratory, Los Alamos NM 87545, USA and Geology and Geophysics Department, University of California, Berkeley CA 94720, USA

1

2

( Received 11 September 2000; received in revised form 3 July 2001; accepted 3 July 2001 )

Abstract—A polycrystal approach that divides the grains into small cells and accounts for local interactions

in a self consistent way is used to calculate deformation and texture evolution of hexagonal zinc. As this model incorporates local effects, it predicts intragranular deformation and gives a description of the deformed microstructure in terms of misorientation between elements and variation in stored energy. This provides information which can be used as a basis for simulating recrystallization processes. The grains are composed of parall parallele elepip pipedi edicc cells, cells, and a Monte Monte Carlo Carlo algori algorithm thm is used used for simula simulatin ting g static static recrys recrystal talliz lizati ation. on. Nucleation and boundary mobility depend on the misorientation between cells and on the local variation in stored energy. The model is applied to simulate the kinetics of static recrystallization and the associated change in crystallographic texture in zinc polycrystals. Experimental results obtained by deforming zinc in plane strain compression compare well with the predictions and are consistent with a mechanism where nucleation nucleation occurring occurring in highly highly deformed deformed domains domains controls controls the recrystall recrystallizati ization on kinetic. kinetic.  2001 2001 Acta Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Recrystallization; Texture; Polycrystal model

1. INTRODUCTI INTRODUCTION ON

The past decade has witnessed witnessed great advance in sophisticated methods for simulating plastic deformation of aggregates. An obvious, and also necessary, extension of such line of research is to incorporate recrystallization effects into the simulations, since recrystall talliz izat atio ion n ofte often n acco accomp mpan anie iess plas plasti ticc form formin ing g proces processes ses.. The intere interest st in comput computer er simula simulatio tions ns of recrystallization is driven by the need to make quantitative predictions of the microstructure and properties of materials as affected by annealing treatments. Such processing-d processing-driven riven approach may yield a description description of recrystalli recrystallizatio zation n kinetics, kinetics, average average recrystall recrystallized ized grain grain size, size, and crysta crystallo llogra graphi phicc textur texture. e. Anothe Anotherr motivation motivation for simulation simulation of recrystall recrystallizati ization on is the need for improved understanding of the highly complex plex phenom phenomena ena of recrys recrystal talliz lizati ation, on, specifi specifical cally, ly, nuclea nucleatio tion n and growth growth.. The change changess in textur texturee and grai grain n size size that that occu occurr durin during g anne anneal alin ing g and and thei theirr dependence dependence on microstruct microstructural ural mechanisms mechanisms provides provides † To whom all correspondence should should be addressed. Tel.: +1-505-665-0892; fax: +1-505-667-8021. E-mail address: [email protected] (C. N. Tome´ ) ‡ Now with VAW aluminium aluminium AG, Research and Development, 53014 Bonn, Germany.

a logica logicall link link to develo develop p detail detailed ed recrys recrystal talliz lizati ation on models. The needs needs and future future direct direction ionss for simula simulatio tion n of recrys recrystal talliz lizat ation ion have have been been addres addressed sed by a group group of experts at the recent conference “Recrystallization ’96” in Monterey, CA [1]. Most of the simulation models used focus on predicting grain size and texture development during recrystallization. The initial spatial distribution of crystallographic orientations and stored plastic tic energy energy,, necess necesssar sary y for descri describin bing g nuclea nucleati tion on and priorii in thes growth growth,, is impose imposed d a prior thesee mode models ls.. As a consequence, deformation evolution is not an integral part part of the formal formalism ism and, and, oftent oftentime imes, s, these these model modelss cannot address dynamic recrystallization. The latter limitations limitations are overcome overcome by some deformation models based on crystal plasticity, which are coupled with probabilistic laws to simulate recovery and recr recrys ysta tall lliizati zation on.. Amon Among g them them,, a mode modell developed developed by Radhakrishna Radhakrishnan n et al. [2] couple coupless the finite element method (FEM) with the Monte Carlo technique so as to account for local effects in cubic aggre aggrega gate tes. s. A simp simple ler— r—bu butt also also more more limi limite ted— d— approach based on the one-site self-consistent viscoplastic (VPSC) model was successfully used to simulate static and dynamic recrystallization in geologic materials of various symmetry [3–5]. In this approach

1359-6454/01/$20.00  2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 5 4 ( 0 1 ) 0 0 2 6 1 - 0

SOLAS et al.: HEXAGONAL METALS

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the nucleation probability increases with the stored plastic energy of the different orientations (grains). As for growth, the stored energy of the grain is compared with the average stored energy, and grains with low stored energy grow, while grains with high stored energy shrink. However, in the one-site model the grain domain is the basic spatial unit. As a consequence, local effects such as strain localization and misorientation, which frequently control recrystallization, are not accounted for in the model. In the present paper we overcome this limitation by extending the one-site approach and coupling an N - site VPSC model with a Monte Carlo model. Within the N -site scheme the microstructure is composed of a compact arrangement of grains and each grain is subdivided into parallelepipedic elements. The N -site VPSC model was originally developed to describe texture development associated with deformation in a two-phase “granite” (quartz and mica) [6]. Since this model accounts for topological effect, local variations in orientation and stored energy can be determined, which makes it a promising candidate for recrystallization simulations. An improved version of the N -site model of Canova et al. [6] is presented in a recent paper [7], where it is used to describe plastic deformation and localization in fcc and hexagonal closed-packed (hcp) aggregates. In the present paper, nucleation and growth criteria are incorporated into the N -site model for simulating static recrystallization using a Monte Carlo technique. The scheme is applied to simulate the deformation and recrystallization of polycrystalline zinc, and results are compared with experimental measurements. The material was deformed in plane strain at elevated temperature so as to suppress mechanical twinning during deformation. A hcp structure was chosen for several reasons. Due to the lower crystal symmetry and smaller number of available slip systems hcp materials are plastically more anisotropic than cubic structures, causing a stronger orientation dependence of accumulated strain energy. Furthermore, in most cubic metals like Al-alloys and steels, recrystallization is dominated by the existence of deformation heterogeneities that may act as nucleation sites—including transition bands, secondphase particles and shear bands [8] —which cannot be easily tackled by the present approach. Recrystallization of low symmetry materials, on the other hand, is usually simpler in that some of the components of the deformation texture grow by consuming other texture components. Finally, deformation and recrystallization of hcp structures is relevant for understanding anisotropy in geological systems, such as quartz and calcite [5, 9, 10], which deform on similar slip systems as hcp materials, and hcp ⑀-iron, which composes the inner core of the Earth [11, 12]. 2. THE

N -SITE

DEFORMATION MODEL

The N -site deformation model is based on a viscoplastic self-consistent scheme, where the polycrys-

tal is composed of a compact arrangement of grains and each grain is divided into ’brick ’ shaped cells. For a description of the polycrystal theory the reader is referred to Refs. [13 –15]. A discussion of the N site model as it applies to plastic deformation can be found in Ref. [7] and only its basic aspects will be described in what follows. The implementation of recrystallization mechanisms into the N -site model is done in detail in Section 3. In the classical one-site approach, each material element (cell) is embedded in and interacts with a homogeneous equivalent medium (HEM) which has the average stiffness of the aggregate. The properties of the HEM are not known a priori and have to be calculated self-consistently. Within the N -site approach, each (parallelepipedic) cell interacts with the 26 neighbor cells, and this cluster interacts, in turn, with the HEM. The cells are assigned a crystallographic orientation, which is initially the same for all cells belonging to the same grain domain. Deformation is imposed incrementally and cells deform plastically by slip on crystallographic defined systems. The shear rate in each system is a power of the resolved shear stress divided by a threshold value t s. The strain rate in the element is given by the sum over the shears contributed by all systems:

͸ ͩ ͪ

˙0 Dij ϭ g

msij

s

ms:S

n

t s

(1)

Here S is the deviatoric stress tensor for the element. During deformation the crystal associated with each cell reorients and hardens. The relation between the deviatoric stress and the plastic strain rate of the element can be expressed as a first order Taylor expansion with respect to a reference value as: S( x) ϭ Q( D( x0)): D( x) ϩ S0( D( x0))

(2)

This constitutive form is often called a “tangent” law, where Q is the local stiffness, and S0 is the stress intercept at zero strain rate. The response of the HEM (same as the aggregate) is also assumed to be described by a tangent law: ¯ (D ¯ ):D ¯ ϩ S¯ 0( D ¯ ) S¯ ϭ Q

(3)

A discretization is enforced within each cell domain “v”: the material properties (hardening), the deviatoric stress Sv, and the plastic rate Dv are assumed to be homogeneous. The hypothesis of discretization introduces discontinuities of stress and displacement across the element boundaries and, as a consequence, compatibility and equilibrium are only ful filled approximately. This discretization, however, allows us to solve the stress equilibrium equation and to


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derive an interaction equation which links the stress and strain rate of the cells with the overall stress and strain rate of the aggregate: ¯ ϩ Dv ϭ D ϩ

͸

¯ ϪQ ¯ :( DvϪ D ¯ )) ⌫vv:(SvϪS

(4)

¯ ϪQ ¯ :( DvЈϪ D ¯ )) ⌫vvЈ:(SvЈϪS

vЈv

where ⌫vvЈ is a tensor that mechanically couples the volume elements v and vЈ. The first term of equation (4) corresponds to the Taylor full constraints approach. The second term accounts for the interaction of element v with the HEM, and leads to the onesite approach. The third term adds the interactions between v and other elements vЈ, which leads to the N -site approach. The coupling tensor ⌫vvЈ depends on ¯ , the shape of the volume elements v and vЈ, and on Q their relative position. While within the one-site Fig. 1. Grain size distribution and initial microstructure approach strain and orientation are homogeneous (30×30×30 elements with periodic boundary conditions). within the grain domain, the N -site coupling induces strain and orientation heterogeneity within the grain domain, specially in the vicinity of grain boundaries each numerical orientation is assigned a real crystal[7], as will be shown in what follows. The numerical lographic orientation from a set of randomly generprocedure for solving the integral equation has been ated Euler angles. developed for parallelepipedic domains in order to be able to update the element shape with deformation [7]. Although the procedure keeps track of the strain 3. RECRYSTALLIZATION MODEL history associated with each element, the shape updating is the same for all elements and equal to the averAfter deformation, the structure consists of a regage distortion. This approach accounts for element ular mesh composed of parallelepipedic cells (as interaction in a first order approximation, and is a opposed to the cubic cells of the initial state) with the limitation compared to Finite Element approaches, average shape of the aggregate. As a consequence, where the individual distortion of the elements is while it is not necessary to remesh the structure in explicitly accounted for. order to link the N -site VPSC scheme with the Monte To generate the initial microstructure, in this appliCarlo procedure, we need to account for the noncation we use a three-dimensional Monte Carlo model cubic shape of the cells in the Monte Carlo procedure. similar to the one presented by Anderson et al. [16] An energy, E vtot, is assigned to each cell as the sum to model grain growth. A detailed description is given of the stored energy, E vstored, and the grain boundary in Ref. [7]. Each cube is assigned a number, which energy, E vgb. The stored energy is assumed to be prorepresents a fictitious numerical orientation, so that a portional to the dislocation density or, equivalently, grain is defined by a set of connected cells with the to the square of the yield stress: same number. Grain boundaries are assumed to exist between elements with different orientations. It must E vstored ϭ E 0 (t sϪt s0)2. (5) be emphasized that the aim of this Monte Carlo model s is to generate the grain microstructure for the N -site model and not to simulate the physical process of grain growth. In our simulations the grain microstrucHere, t s0 and t s are the threshold shear stresses in systure is composed by 30×30×30 elements, to which tem s before and after deformation, respectively, and periodic boundary conditions are imposed. This rep E 0 is a normalization factor which gives units of resents a reasonable compromise between compu- energy. The grain boundary energy depends on the tational demands on one hand, and a realistic rep- misorientation of a given site v with respect to its 26 resentation of the aggregate for simulating neighbors [17] and we express it as: deformation on the other hand. The microstructure generated with this Monte d 30 v vvЈ Carlo approach (Fig. 1) is used as input of the N -site g vvЈ (6a) E gb ϭ d model. It comprises 475 grains (there is an average vЈ of 56.8 elements per grain), and the grain size distribution mimics the one observed in a real microstructure. Once that the initial microstructure is generated, with

͸

͸


3794

vvЈ

g

ϭ

Ά

ͩ ͪ

qvvЈ qvvЈ g m [1Ϫln ] when qvvЈрq q q ∗

∗

∗

when qvvЈϾq

g m

(6b)

∗

where g m is the specific energy of high-angle grain boundaries, q is the misorientation limit for lowangle boundaries (taken as 15° in the present simulations), and d 0 is the size of the elements in the undeformed cubic structure. d vvЈ is the distance between the centers of the cell v and its neighbor vЈ and qvvЈ is the crystallographic misorientation between v and vЈ. The grain boundary energy as a function of the misorientation expressed in equation (6b) corresponds to the well-known Read–Shockley equation Ј is proportional to the surface [18]. The term d 30 / dvv between neighboring elements and is particularly relevant in the case of a non-cubic mesh. The mobility M of the boundary between the two sites is calculated according to [19] ∗

ͩ ͪ

M vvЈ ϭ M m[1Ϫexp(Ϫ

qvvЈ 3 )] 10

(7)

where M m is the average mobility of high-angle grain boundaries, and the misorientation angle q is expressed in degrees. This expression saturates when qϾ15°. 3.1. Nucleation

The microstructural changes during recrystallization are based on the two fundamental mechanisms of recrystallization, the formation of the new grains at specific sites in the as-deformed microstructure and their subsequent growth by consumption of the deformed neighborhood. Both, nucleation and growth of the recrystallized grains, are thermally activated processes, the driving force of which is provided by the energy stored during the deformation. However, this driving force is too low to enable homogeneous nucleation through thermal fluctuations, as is the case in phase transformations. Rather, the “nuclei” (i.e. subgrains) are present in the microstructure that formed during the preceding deformation, mostly in the vicinity of local heterogenities. Accordingly, the term “nucleation” is not quite appropriate, yet it is commonly used because of the analogy of recrystallization processes with solidification reactions, or phase transformations where nucleation does take place. In the present model, two possible nucleation mechanisms have been implemented, and the one of choice has to be pre-selected before the simulation. Our intention is to determine which one is the active recrystallization mechanism in zinc, by comparing the experimental texture with the recrystallization textures predicted with each mechanism. The first mech-

anism, illustrated in Fig. 2(a) corresponds to a case where nucleation occurs in highly deformed regions, that is, in elements with a high stored energy. One site is selected at random and —provided it is unrecrystallized, i.e. its stored energy is larger than zero—a nucleation probability is calculated according to

ͩ

ͪ

E vstoredϪ E 1 Pvnucl ϭ 1Ϫexp Ϫ E 2Ϫ E 1

(8)

E 1 is a threshold (nucleation may only take place if the stored energy is larger than this value); the value of E 2 determines the nucleation dynamics: when E 2 is close to E 1 (as is the case in this work) all nucleation events occur at the beginning of the simulation (site saturated nucleation) while, when E 2 is much larger than E 1, continuous nucleation can be simulated. When the value of the nucleation probability, Pnucl, exceeds a randomly generated number between 0 and 1, nucleation takes place. This means the stored energy in the element is reset to zero, that is, the critical resolved shear stresses adopt their original value, yet the crystallographic orientation of the element is preserved. The second mechanism corresponds to the strain induced boundary migration (SIBM), sometimes referred to as bulging mechanism [Fig. 2(b)]. SIBM occurs between two regions with a large difference in stored energy [20]. In the Monte Carlo algorithm one element (site 2) and one of its neighbors (site 1) are selected at random. If these two sites are unrecrystallized (i.e. their stored energy is larger than zero), nucleation by SIBM may take place in site 1. In that case, the orientation of site 1 is changed to that of site 2, and the stored energy of site 1 is reset to zero. For this mechanism to happen the following two additional conditions must be ful filled. 1 2 Ϫ E vstored Ͼ E 3 E vstored

1 ⌬ E ϭ E vtot 1 Ϫ E vtot

(initial configuration)

(9a)

(9b)

(final configuration) Ͼ0

Equation (9a) requires the stored energy in the nucleating element to exceed the one of the neighbor site by at least a threshold E 3. Equation (9b) requires that the driving force for the process, given by the difference in energy between the initial and the nucleated configuration, has to be positive. The nucleation probability is derived from:

Pnucl ϭ

⌬ EM 1Ϫ2t inc

d 1Ϫ2

(10)


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Fig. 2. (a) Nucleation in regions with high stored energy; (b) nucleation by SIBM; (c) grain growth (dark gray, site with high stored energy; light gray, site with low stored energy; white, recrystallized site).

where M 1Ϫ2: is the directional mobility of the grain boundary, given by equation (7), ⌬ EM 1Ϫ2 corresponds to the velocity of the boundary, and t inc is the time increment. The migration distance of the grain boundary, that is, ⌬ EM 1Ϫ2t inc, is compared to the distance d 1Ϫ2 between the centers of sites 1 and 2. Nucleation is allowed if this value is larger than a random number between 0 and 1. A method to estimate t inc, as well as the justi fication of the probability law, will be given in the following subsection, in connection with the growth mechanism which it resembles. 3.2. Growth

The growth mechanism incorporated in the model is very similar to the SIBM bulging mechanism described above, except that condition (9a) is not enforced. One site (site 2) and one of its six first neighbors (site 1) are selected at random [Fig. 2(c)]. Growth is only possible if site 2 is already recrystallized. Note, however, that growth may take place even when both, site 1 and site 2, are recrystallized, depending on the minimization of the total grain boundary energy [equation (9b)]. The probability for growth is: Pgrow ϭ

⌬ EM 1Ϫ2t inc

d 1Ϫ2

.

(11)

Here ⌬ E is given by equation (9b), M is the mobility, t inc the time increment, d 1Ϫ2 the distance between the centers of sites 1 and 2. Grain boundary migration is allowed if this probability is larger than a random number generated in the interval [0, 1]. Recovery can be accounted for as follows: when one site that has been selected at random does not satisfy the conditions for either nucleation or growth, its stored energy may be reduced by a fixed factor. This procedure would lead to an exponential decrease

of the stored energy with time, just as it has been observed experimentally during recovery [21]. However, this option has not been used for the simulations presented in this paper. In most Monte Carlo simulations, the unit of time is the Monte Carlo Step (MCS), which is proportional to number N of attempted transitions, where N is the number of lattice sites. In our simulation we introduce a time increment t inc which is adjusted after each MCS using the maximum value of ( M ⌬ E ·t inc / d) corresponding to the MCS. If this value was larger than 1 the corresponding boundary would move by a distance larger than the cell dimension. Vice versa, if ( M ⌬ E ·t inc / d) is much smaller than 1, an unnecessarily large number of MCS are required to complete the recrystallization process. In the simulation, an initial guess of t inc is introduced for the first MCS. After that, the time is automatically adjusted so that there are enough successful attempts during each MCS. This procedure is particularly important at the end of the recrystallization when only the elements with lower stored energy remain and the driving force decreases. The probability law for growth must take into account the shape of the mesh, such that the motion of a boundary is not arti ficially faster in one direction than in the others. The probability law used in the present model [equation (11)] was validated by studying how a spherical recrystallized grain evolves in a deformed polycrystal with 30×30×30 elements (Fig. 3). The simulations were run on a deformed mesh with shape factors of 1.284, 1.000 and 0.779 in the rolling direction (RD), normal direction (ND) and transverse direction (TD), respectively, which corresponds to plane strain deformation with a thickness reduction of 25%. The microstructure was generated with a Monte Carlo procedure and an approximately spherical grain composed of 680 elements was defined in the center of this structure (main axes span 9, 11 and 15 elements along the RD, TD and ND,

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Fig. 3. Growth sequence of a recrystallized grain in a deformed matrix with paralellepipedic elements with ratios 1.284, 1.000 and 0.779 in the rolling, transverse and normal directions, respectively, corresponding to a thickness reduction of 25% by plane strain deformation. (a) 0 mcs; (b) 60 mcs; (c) 120 mcs; (d) 180 mcs.

respectively). The stored energy is de fined to be zero inside the central grain, and equal to 1 for the outside elements. Then, the grain shape is allowed to evolve, controlled by the growth mechanism described earlier. Figure 3 shows a sequence of simulation results with increasing MCS. The results are presented as a two-dimensional section through the middle of the three-dimensional structure along the RD/ND plane. In this plane, which corresponds to a longitudinal section of a deformed specimen, the aspect ratio of the elements is maximum. Since the grain boundary energy g enters the Monte Carlo simulations, different ratios E stored / g of stored energy and specific grain boundary energy (ranging from 10 to 0.1) were used so as to study the influence of this parameter on the grain shape evolution. In all simulations, the grain shape remained almost spherical. One example, with E stored / g = 0.5, is presented in Fig. 3(b–d). These simulations confirm that with the present probability law [equation (11)] grain growth remains isotropic even for deformed meshes. 4. EXPERIMENTAL RESULTS

In order to assess the possibilities of the present approach to simulate recrystallization, the model was applied to the deformation and recrystallization of polycrystalline high purity zinc. The as-received material had an average grain size of 150 µm [Fig. 4(a)] and an almost random texture. EDS analysis done on a ground surface showed only zinc peaks, indicating that impurities, if present, are below the 0.1% limit. Samples for metallographical investigations and texture measurement were polished with alumina (5 and 1 µm) and silica colloidal and finally

etched for 10 s in dilute nitric acid (25% HNO 3 in water). Micrographs were obtained with an optical microscope and polarized light. The spatial arrangement of the deformation and recrystallization texture orientations was studied by electron back-scatter diffraction (EBSD) [22]. The electron beam in the SEM (Philips XL30) was controlled to scan a sampling area of 1500×1290 µm in steps of 7.5 µm, and for each point the crystallographic orientation was determined by EBSD. This technique, commonly referred to as orientation imaging microscopy (OIM ) [23], allows reproduction of the microstructure of a given sample from the crystallographic orientations of the microstructural constituents, for example, grains or subgrains. Furthermore, from the orientation data collected with this technique {0002} and {101¯ 0} pole figures were produced and will be presented as contoured pole densities in stereographic projection (note that the pole figures are projected in the RD/ND plane, and the TD coincides with the center of the pole figure). Samples with initial size of 20 ×9.9×8 mm3 (RD, TD, and ND, respectively) were deformed by channel-die compression at a temperature of 125 °C. This sample geometry allowed for a fairly homogeneous deformation by avoiding the formation of shear bands. The samples were wrapped in a Te flon sheet and a molybdenum sulfide lubricant was used to reduce friction. The samples were held for 10 min at test temperature before the deformation started, to ensure a uniform sample temperature. In order to verify that the microstructure of the sample was not modified by the heating process, one sample was sub jected to annealing for as much as 1 h at 150°C. No noticeable grain growth was detected, which implies that the initial microstructure remains stable during


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Fig. 4. Microstructure of the polycrystalline zinc: (a) undeformed; (b) deformed by 25% at 100 °C; (c) deformed by 25% at 125°C; (d) deformed by 25% at 125 °C and recrystallized for 30 min at 225 °C.

the heating. The samples were deformed to 25% thickness reduction with a constant strain rate of ⑀ ˙ = 10Ϫ3 sϪ1. The deformation temperature, 125 °C, was chosen to ascertain that deformation proceeded merely by dislocation slip, that is, without twinning and without dynamic recrystallization. According to the deformation maps compiled by Frost and Ashby [24], for low strain-rates the prevalent deformation mechanism in zinc changes from crystallographic slip and mechanical twinning at room temperature to creep mechanisms (without twinning) at temperatures in excess of 100 °C. At higher temperatures—above 270°C—dynamic recrystallization is initiated. Frost and Ashby [24] obtained these data from different experimental results [25, 26], and it is likely that mechanism will vary depending on the precise composition and impurity content of the zinc. Therefore, several deformation trials at different deformation temperatures were performed to determine the optimum deformation temperature for the zinc used in our study. A sample deformed at 100°C still revealed traces of twins in some grains [Fig. 4(b)] while twinning was not observed at 125 °C [Fig. 4(c)]. Finally, the samples were annealed for 1 h at 225°C in an air furnace. The resulting microstructure shows a characteristic coarse grain recrystallized microstructure [Fig. 4(d)]. Pertaining to the texture changes accompanying deformation at 125 °C, the {0002} pole figure shows that the c-axis tends to be oriented along the compression direction (i.e. the ND), but there are also some grains with c-axis close to the TD [Fig. 5(a)]. Note that the basal component close to the ND is rotated by about 20° towards the RD. This is consistent with cold rolling textures of zinc alloys reported by Philippe et al. [27]. The prismatic planes {101¯ 0} are fairly uniformly oriented in the RD/TD plane [Fig. 5(b)]. After recrystallization, the basal component near the TD disappears, as seen in the (0002) pole

Fig. 5. Experimental OIM pole figures of the deformed and recrystallized zinc.

figure [Fig. 5(c)], while there is not much change in the prismatic poles [Fig. 5(d)]. This means that grains with the c-axis close to the TD tend to disappear during the recrystallization process. This observation is confirmed by the OIM maps of Fig. 6, where increasingly dark shadings stand for orientations with the caxis closer to the TD. It is evident that the volume of grains with the c-axis close to the TD decreases during recrystallization. 5. MODELING RESULTS

5.1. Deformation

In this section we present plane strain deformation simulations for a zinc aggregate at medium temperature. At room temperature basal slip is the easier to


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Fig. 6. Orientation imaging maps of the deformed and recrystallized zinc (EBSD); (a) deformed by 25% at 125°C; (b) deformed by 25% at 125 °C and recrystallized for 30 min at 225°C. Darker areas correspond to grain with c-axis closer to the transverse direction.

activate system in zinc, while pyramidal slip, prismatic slip, and compressive twinning have associated much higher critical stresses of activation [27]. We do not observe twinning at 125°C, and we assume that pyramidal and prism slip will be relatively easier to activate than at room temperature. The latter observation reflects in the systems and relative critical shear stresses used in the present simulation (see Table 1). Simulation results are not very sensitive to the precise values of CRSS. Hardening of the slip systems, which is required to generate energy gradients between cells, is assumed to increase linearly with the accumulated shear strain ⌫ in the grain: ⌬t s ϭ

h⌬⌫ with ⌬⌫ ϭ

͸

⌬g s

tivity, the resulting deformation textures are not very sensitive to the precise value of n used in the simulations. As a consequence, the stress exponent n which enters in the rate sensitive kinetic law [equation (1)] is set to n = 5 in order to speed up the numerical convergence. An equivalent strain increment ⌬⑀VM = 0.01 is imposed at each deformation step. Figure 7 shows the {0002} and {101¯ 0} pole figures predicted for plane strain deformation after 25% thickness reduction. Similarly as in the experimental pole figures [Figs 5(a) and (b)], a component corresponding to the c-axis tilted about 20° from the ND towards the RD, and a weaker component close to the TD can be observed. As local intragranular deformation takes place, we observe the development of misorientations between cells which originally belonged to the same grain and, thus, had the same orientation. This localization of deformation and formation of misorientation starts near the grain boundaries and propagates inside the grain as deformation proceeds. The low-angle boundary distribution in the deformed structure is similar to the one observed in fcc materials [28, 29] except that for hcp aggregates the average misorientation for low-angle boundaries is about 5 ° rather than the 3 ° characteristic of fcc materials [7]. Orientation maps, misorientation maps and stored energy maps are depicted in Fig. 8 for one section through the threedimensional deformed structure. In the orientation map [Fig. 8(a)], the darker areas correspond to grains with the c-axis closer to the TD. In Fig. 8(b) the misorientation angle w is proportional to the thickness of the lines, which identify the intervals 2 –5°, 5–10°, 10–15° and 15–90°. Note that the original grain boundaries all exceed 15 ° misorientation, that is, they were all of the high-angle type. Stored energy is calculated using equation (5) and is represented in Fig. 8. We observe that all grains with the c-axis close to the ND have associated low stored energy.

(9)

s

Here, the same value of h = 0.1 was used for all systems, which corresponds to low hardening. When the expression for the stored energy [equation (5)] is combined with the simple hardening law [equation (9)], the stored energy becomes proportional to h2. Since all the systems are assigned the same rate sensiTable 1. Slip systems and relative critical shear stresses for Zn at 125°C Basal Pyramidal Prismatic

{0002}͗1¯ 21¯ 0͘ {101¯ 1}͗1¯ 1¯ 23͘ {101¯ 0}͗1¯ 21¯ 0͘

t = 1 t = 5 t = 10

Fig. 7. Pole figures from simulated deformation texture after 25% plane strain compression.


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Table 2. Recrystallization parameters E 1, E 2 (threshold and maximum stored energy for induced nucleation model), E 3 (threshold for SIBM nucleation model) and g m (misorientation energy parameter) used in the simulationsa E 1 E 2 E 3 g m Ͻ E storedϾ max ( E stored)

27.0 E 0 h2 27.1 E 0 h2 10.0 E 0 h2 5.0 E 0 h2 8.56 E 0 h2 45.1 E 0 h2

a The values of the stored energy E stored (average and maximum) derived from the deformation model are given for comparison with the recrystallization parameters ( E 0, normalization factor to give energy units; h, linear hardening coef ficient, taken equal to 0.1 in the simulations).

Fig. 8. Microstructure in a layer of the aggregate after deformation (simulation results): (a) orientation map (lines correspond to boundaries exceeding 15°); (b) misorientation map (the thickness of the boundaries is proportional to the misorientation angle between elements); (c) stored energy in units of E o (darker area corresponds to higher stored energy).

5.2. Recrystallization

In this section we apply the recrystallization model described in Section 3 to determine which nucleation mechanism may explain the texture evolution. Subsequently, we present the results concerning the kinetics of recrystallization and the microstructure evolution during recrystallization. The deformation model gives us the value of the stored energy E V stored and the crystallographic orientation for each element. The value of the speci fic grain boundary energy g m in equation (6) was adjusted so that the average grain boundary energy and the average stored energy are of the same order. Whereas the value of g m (more precisely, the ratio between g m and the stored energy) influences the kinetics of recrystallization [the Johnson –Mehl–Avrami–Kolmogorov (JMAK) exponent], it does not modify the simulated recrystallization textures. The reason is that for static recrystallization the stored energy and the orientation in each element remain constant until they are reset upon nucleation or growth. As a consequence, if the ratio changes, the nucleation and growth probabilities [equations (8), (10) and (11)] will change. The overall effect will be to affect the kinetics of the process, that is, the speed at which texture evolves, but not the final texture. This argument does not apply when dynamic recrystallization or recovery take place, because the stored energy keeps evolving differently in each element. Both mechanisms described in Section 3.2,

nucleation in high-energy regions (mechanism 1) and nucleation by SIBM (mechanism 2), were tested. The values of the respective parameters are listed in Table 2. E 1 and E 2 control initiation and saturation for the first mechanism, and E 3 is a threshold value for the second mechanism to take place. These parameters were adjusted to be consistent with the experimental evidence [Figs 4(a and d)] of site saturated nucleation, and with the grain size ratio of approximately 3 between the microstructure after and before recrystallization. Using these values, between 1500 and 2000 elements nucleate in the system within the first two Monte Carlo steps. This number is large enough to have a suf ficient number of recrystallized grains for good statistics of the final recrystallization texture, but still small enough so that the growth of the nuclei may play an important role. The recrystallization textures predicted by these simulations are presented in Figs 9(a and b) for mechanism 1, and in Figs 9(c and d) for mechanism 2. The recrystallization textures are totally different, depending on the nucleation mechanism being assumed. With the SIBM mechanism the intensity of the basal component at the center of the pole figure

Fig. 9. Results of the recrystallization texture simulations; (a), (b) results for high stored energy induced nucleation; (c), (d) results for SIBM nucleation.

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(i.e., the TD) increases substantially while, on the other hand, it tends to disappear when nucleation is controlled by the high-stored-energy mechanism. Since zinc deforms mainly by basal slip, and since this system cannot be activated in grains with the caxis perpendicular to the compression axis, such grains deform and reorient little upon plane strain deformation. As a consequence, these grains store little energy during deformation, which favors them for nucleation by SIBM [Figs 9(c and d)]. On the other hand, grains where basal slip is activated tend to reorient the c-axis towards the compression axis and to accumulate high strains. Thus, those grains will prevail if the recrystallization textures are controlled by nucleation of elements with high stored energy [Figs 9(a and b)]. Evidently, the experimental results shown in Fig. 5 can be explained if nucleation takes place in the highly deformed grains, followed by a growth of the nuclei into the less deformed grains which eventually get consumed. Figure 10 shows simulation results in a two-dimensional section through the recrystallized microstructure obtained with the high stored energy nucleation mechanism. In the orientation map [Fig. 10(a)] grains with light shadings, that is, large angular deviations of the c-axis from the TD, prevail, whereas grains with the c-axis parallel to the TD were not observed. The corresponding misorientation map [Fig. 10(b)] mainly displays high-angle grain boundaries, which is consistent with recrystallization being completed. The present recrystallization model can also be used to study the kinetics of recrystallization. Figure 11 shows the evolution of the recrystallized volume with time (Monte Carlo steps). At the beginning of recrystallization the JMAK exponent is slightly larger than 2. For nucleation at planar grain boundaries and site saturated nucleation an exponent of 2 would be expected. At the later stages of recrystallization the JMAK exponent decreases to values below 1. This

Fig. 10. Simulated microstructure after recrystallization with the high stored energy nucleation condition; (a) orientation of c-axis with respect to the TD; (b) misorientation map.

Fig. 11. Evolution of the recrystallized volume fraction×as a function of annealing time (units of Monte Carlo steps).

strong decrease is attributed to the sharp decrease in nucleation rate (i.e., constant number of nuclei) with progressing recrystallization. Furthermore, in contrast to the assumptions of the JMAK model, the total stored energy of the system decreases faster than linearly with recrystallized volume fraction, X , since the sites with higher stored energy recrystallize faster than the ones with lower stored energy. This would decrease grain boundary velocity and, consequently, further reduce the JMAK exponent. 6. DISCUSSION AND CONCLUSIONS

The N -site self consistent model predicts texture evolution and gives a description of the deformed microstructure, more specifically, the misorientation between elements and the local variation of stored energy. It also provides a simpler alternative to finite element implementations of deformation and recrystallization models [2], although it is subject to the limitation that the distortion of the elements is accounted for only in average. A positive feature is that the parameters of the model have a physical meaning (stored energy, grain boundary energy…) and experiments can be carried out to determine their exact values. The combined N -site deformation and recrystallization models permit us to account for recovery, plus nucleation and grain growth associated with static and dynamic recrystallization. Both texture and microstructure associated with recrystallization can be described with this method, and recrystallization kinetics as well. This N -site deformation approach is linked to a Monte Carlo model where nucleation and grain boundary velocity depend on the misorientation and the variation of stored energy. The Monte Carlo procedure takes into account the parallelepipedic mesh obtained after deformation. The structure does not have to be remeshed when making the transition between the deformation and the recrystallization model. A time scale (and thus a space scale) is also introduced. In zinc, basal slip is easy to activate, while pyramidal slip is much harder, and prism slip does not reorient the c-axis. As a consequence, during deformation


the c-axis tends to align with the compression direction except for those grains for which the c-axis is nearly perpendicular to the compression axis (where basal slip cannot be activated). Grains having the c axis aligned with the TD (“hard” grains) can be expected to deform less and to store little energy. The self-consistent model accounts for the directional rigidity of the grains and captures the latter behavior. The experimental evidence indicates that during static recrystallization the less deformed grains disappear, which means that the more deformed grains nucleate and consume the less deformed ones. This seems to favor a recrystallization model based on nucleation of high stored energy regions, rather than a SIBM mechanism. In this case, a model in which no new orientations are generated explains texture development during recrystallization. Interestingly, the zinc investigated in this study appears to have similar recrystallization behavior as several mineral systems, in which it was observed that the most highly deformed orientations ( “soft grains”) dominate the recrystallization texture, while low deformed grains disappear. This is the case for halite [3], olivine [5, 9, 31, 32], quartz [10], ice [30], and calcite [33] for most deformation conditions. It may be typical for materials with a high plastic anisotropy, that is, a large variation in Taylor factor for differently oriented grains. Zinc is of interest for geophysics because it can be considered as an analog for the high pressure polymorph of iron (⑀-iron), an hcp phase that constitutes the Earth’s solid inner core. Seismic evidence indicates that the inner core is anisotropic and anisotropy may be produced by deformation [11, 12]. At the conditions of the inner core, close to the melting point, recrystallization is likely of importance and the results for zinc will be of help for more realistic modeling of texture development. Among the drawbacks of this model we should mention that dislocation mechanics is not accounted for explicitly. The dislocation density can be calculated indirectly from the hardening law introduced in the model: the density is a function of the yield stress, which in turn depends on the accumulated shear in the element. However, when an element deforms we have no way to decide whether dislocations are being stored at the boundary between contiguous elements, or if they are stored at the grain boundary. Another limitation comes from the fact that some strain localization such as shear bands are not predicted by the model. These bands correspond to narrow regions of intense shear and they are major sites for recrystallized grains in cubic materials. Acknowledgements—The authors would like to acknowledge Carl Necker for providing the zinc used in this study. They appreciate stimulating discussions with Dave Embury, Fred

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Kocks and Carl Necker. HRW and CNT have been partially supported by IGPP-LANL.

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Deformation and Recrystallization of Hexagona, Results for Zinc

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