Microstructure Evolution in Cold-Rolled Pure Titanium: Modeling by the Three-Scale Crystal Plasticity Approach Accounting for Twinning

Microstructure Evolution in Cold-Rolled Pure Titanium: Modeling by the Three-Scale Crystal... Microstructure Evolution in Cold-Rolled Pure Titanium: Modeling by the Three-Scale Crystal Plasticity Approach Accounting for Twinning KAROL FRYDRYCH and KATARZYNA KOWALCZYK-GAJEWSKA A three-scale crystal plasticity model is applied to simulate microstructure evolution in hcp titanium subjected to cold rolling. Crystallographic texture and misorientation angle develop- ment, as an indicator of grain refinement, are studied. The impact of twinning activity on both phenomena is accounted for by combining the original three-scale formulation with the probabilistic twin-volume consistent (PTVC) reorientation scheme. The modeling results are compared with available experimental data. It is shown that the simulated textures are in accordance with the experimental measurements. The basic components of misorientation angle distribution, especially in the range of high angle boundaries, are also well reproduced. https://doi.org/10.1007/s11661-018-4676-2 The Author(s) 2018 I. INTRODUCTION The mechanisms governing the grain refinement in metals of hexagonal close packed (HCP) lattice experi- IT is well known that improvement of material encing large plastic strains, induced e.g., via severe properties can be achieved by tailoring the microstruc- plastic deformation (SPD) processes, are known to a ture. The microstructure can be refined by advanced much lesser extent. First of all, in those metals, the plastic deformation processes. The mechanisms leading number of easy slip systems is in general less than in to refinement are not yet fully described, in spite of the FCC metals, and twinning appears as an additional, huge effort that has been put into modeling. While this complementary mode. Second, in spite of the many statement is true in the case of face centered cubic (FCC) efforts to carry out cold SPD processes for commercially materials, it is even more relevant in the case of [9] pure titanium (CP Ti) and for AZ31b magnesium hexagonal close packed (HCP) materials. Mainly it is [10] alloy, they are usually performed at elevated temper- due to the fact that plastic deformation in HCP atures, which promote discontinuous dynamic recrys- [11] materials is usually more complicated than in FCC, tallization (DDRX). Beausir et al. assumed that the because of the insufficient number of easy slip systems DRX for magnesium occurs at a temperature equal to and the ease of twinning. 0:4T , where T is the melting point (T = 923 K for M M M Qualitative studies of the grain refinement phe- magnesium). Thus 0:4T ¼ 369 K ¼ 96 C, which is nomenon in FCC metals and alloys were reported in even less than the boiling point of water. Taking into References 1 through 7. Based on the indicated mech- account the simplicity of the 0:4T approximation and anisms of microstructure evolution, a three-scale crystal the fact that the temperature in the SPD process can [8] plasticity (3SCP) model was built. The model con- increase without additional heating, it is not surprising struction was motivated by the initially present defor- that Gu et al. observed the DRX in AZ31b subjected to mation-induced cell substructure, schematically shown equal channel angular pressing (ECAP) process at room [12] in Figure 1(a). Two types of grain boundaries can be temperature. Figueiredo and Langdon proposed the recognized, namely geometrically necessary boundaries model, in which DDRX is the mechanism responsible (GNBs) and incidental dislocation boundaries (IDBs). for grain refinement in the AZ31b alloy subjected to the The new grains are formed by gradual increase of ECAP process. New grains are formed near the initial misorientation angles across the boundaries. grain boundaries or twin boundaries. The reason for this is the presence of higher stresses in those areas, which promote the activity of non-basal slip systems—in the case of sufficiently small grain (with size below the KAROL FRYDRYCH and KATARZYNA KOWALCZYK- critical grain diameter), non-basal slip can activate GAJEWSKA are with the Institute of Fundamental Technological throughout the whole volume of the grain. The critical Research (IPPT), Polish Academy of Sciences, Pawin nskiego 5B, value of the grain diameter can be dependent upon the 02-106 Warsaw, Poland. Contact e-mail: kfryd@ippt.pan.pl Manuscript submitted January 11, 2018. given alloy, pressing temperature or the back-pressure used. Obtaining a homogeneous grain size distribution is METALLURGICAL AND MATERIALS TRANSACTIONS A [27–30] models. Apart from the approaches discussed in Reference 8, models applying the crystal plasticity finite element method (CPFEM) with many integration points [31–35] or elements per grain are also worth mentioning as a possible tool to predict the grain refinement. However, none of the models mentioned is able to fully predict the grain refinement in FCC materials. As the mechanisms leading to grain refinement in HCP metals and alloys are by far less known than the (a) (b) analogous mechanisms for FCC materials, there are accordingly fewer papers modeling this phenomenon. Fig. 1—(a) The schematic drawing of the dislocation-induced cell As already mentioned, the grain refinement in AZ31b substructure of a grain observed in FCC materials for small to [12,13] magnesium alloy is attributed to DDRX. The medium accumulated plastic strain according to Ref. [3], the picture simulation of grain refinement in an unconventional reprinted after Ref. [8]. (b) The schematic showing analogous substructure for commercially pure titanium, after twinning. process joining extrusion and ECAP was described in Reference 36. The process was performed at 350 C. The grain dimension after recrystallization was calculated thus feasible already in the first ECAP pass if the initial using a phenomenological equation, calibrated for grain size is lower than the critical diameter. If the initial compression tests. The authors claimed that they had grain size is greater than the critical value, the obtained an interesting agreement between the simula- microstructure obtained after the first pass is heteroge- tion and experimental data, but did not include direct neous and contains both small and large grains. The comparison. The grain refinement stemming from [12] model of Figueiredo and Langdon is consistent with DDRX can also be modeled using cellular automata [37] many experimental data for AZ31b magnesium alloy, (CA). Such an approach was applied by Gzyl et al. e.g., References 13 through 16. The strain, strain rate, and temperature at the integra- For pure titanium, the melting point T is equal to tion points were calculated using the finite element 1941 K; therefore, 0.4T = 776 K = 503 C, which is method. These data were then used to simulate the DRX a much greater value than in the case of magnesium. using the CA. The refinement in AZ31b subjected to According to the 0.4T criterion, the start of DRX is ECAP and I-ECAP was simulated and good agreement not expected in temperatures below 500 C, which between the results of modeling and experimental data [17,18] justifies the approach excercised by some authors, was achieved. who model the grain refinement of titanium assuming Grain refinement in CP Ti subjected to SPD processes similar mechanisms to those acting in FCC materials. at only slightly elevated temperatures was modeled These works deal mainly with processes performed at assuming mechanisms analogous to FCC materials. [17] [38] low temperature (below 0.4T ). Yang et al. analyzed Ding and Shin classified the orthogonal cutting as a the grain refinement mechanism in shear bands that severe plastic deformation process, because of the developed in cold-rolled CP Ti. They stated that the ultrafine grains developing inside chips. The grain mechanisms responsible for grain refinement are anal- dimensions inside chips were predicted using a model [39,40] ogous to those acting in FCC materials. On the other of dislocation evolution inside dislocation cells and hand, twinning plays also an important role in the walls between them. A similar approach was applied in fragmentation of original grains. The authors noticed Reference 41 for modeling the grain refinement in cold that the microstructure in shear bands is similar to the rolling of titanium. The maximum temperature was one observed in SPD processes. Taking this into determined to be 150 C. The average dislocation cell account, the supposition was made that the same dimensions were predicted in good agreement with mechanisms can lead to grain refinement in SPD experimental results. [18] processes. Zeng et al. analyzed the grain refinement In the present paper, the texture and grain refinement mechanisms acting during hot compression (tempera- of CP Ti deformed in cold rolling are studied. The ture range 673 to 973 K) of CP Ti extruded tube and evolution of microstructure of this material subjected to observed that twinning has strong impact upon the grain cold rolling was analyzed in References 42 through 46. refinement. At 723 K (53 K lower than 0.4T ), the high In References 42, 44 through 46, only experimental data [43] angle grain boundaries (HAGBs) developed by two were analyzed. Bozzolo et al. have modeled texture parallel processes: gradual increase of misorientation evolution during the process using the visco-plastic [47] angle across the low angle boundary (LAB)—analogous self-consistent (VPSC) model. The grain refinement mechanism to the one present in FCC metals, and by was not simulated in any of those papers. [42–46] deformation of the twin boundaries, cf. Figure 1(b). The authors of papers concluded that: There is quite a large number of papers reporting – Extension twinning systems of the family modeling of grain refinement in FCC metals. Our recent [8] fg 1012hi 1011 (E1) and contraction twinning systems paper shortly reviews the approaches used for mod- of the familyfg 1122hi 1123 (C1) were activated. eling the grain refinement using the macroscopic phe- [19,20] – Activation of E1 twinning requires a lower resolved nomenological, the mean-field crystal plasticity [8,21–26] shear stress (RSS) than C1, but the RSS for most models enhanced by some additional features grains is initially much higher for the contraction and the incremental energy minimization-based METALLURGICAL AND MATERIALS TRANSACTIONS A twinning mode due to the strong initial texture. As a of a number of subgrains, cf. Figure 2. The notion of result, the C1 twin modes are activated first and the subgrain denotes a single discrete orientation and is not E1 modes are initiated mainly as a secondary equal to the term subgrain in quantitative metallogra- twinning mode inside the C1 twins. phy, although it is conceptually related to it. – Twinning is an important grain refinement mecha- The representative volume element (RVE) in 3SCP nism, especially in the initial stages of deformation. model consists of NG metagrains. In each metagrain, a Twin boundaries contribute to the fraction of high nominal orientation Q ð0Þ is initially assigned. Meta- angle boundaries and the texture evolves due to the grains are themselves aggregates of subgrains—each twin reorientation. consists of NS subgrains, whose orientations are – For higher strains, the twinning saturates and then obtained by slightly distorting the nominal metagrain most of the deformation is accommodated by slip on orientations. The initial orientation of subgrain i is thus the prismatic planesfg 1010 : The grain refinement obtained as follows: continues by means of mechanisms analogous to g g Q ð0Þ¼ R ð0ÞQ ð0Þ; ½1 those occurring in FCC metals. i i where The main goal of this paper is to verify to which extent the recently proposed computationally efficient three-s- [8,48] R ð0Þ¼ RnðÞ ; dw ð0Þ ½2 cale crystal plasticity (3SCP) model is applicable to simulate texture evolution and grain refinement in tita- is a rotation matrix around arbitrary axis n about the nium. To this aim, first, the 3SCP model is extended to small angle dw ð0Þ2hi 0; Dw : The axis n and the account for twinning and next, the simulations of cold distortion angle dw are assigned to each subgrain by rolling are performed for CP titanium. In the proposed random generation. extension of the 3SCP model, the probabilistic twin-vol- During the simulation of some plastic deformation ume consistent (PTVC) reorientation scheme is used in process, one can study the grain refinement occurring by order to properly account for the appearance of twin-re- evolution of subgrain orientations. The following equa- lated orientations at the metagrain level. Modeling results tion describes the current misorientation angle dw ðtÞ ij are analyzed and compared to experimental data avail- [49] between subgrains i and j of the metagrain g : able in the literature. The quality of model predictions is discussed and possible sources of discrepancies between g g 2 cosðdw Þ¼ trðR ðtÞÞ  1; ½3 ij ij simulations and experimental data are indicated. The paper is organized as follows: After this intro- where ductory section, the 3SCP model formulation is recalled g g g T in Section II. First, the construction of a polycrystalline R ðtÞ¼ Q ðtÞQ ðtÞ : ½4 ij i j representative volume is outlined, and then the consti- Note that in Reference 8, the misorientation angle was tutive descriptions used on the lowest level of calculated with respect to the mean orientation of the microstructure and rules of micro-macro transition are metagrain. In this work, we have not used this approach presented. The extension of the original framework due in order to directly observe misorientation angles to twinning activity is also discussed. In Section III, the between matrix and twinned domains. parameters of the crystal plasticity model are identified The original 3SCP model is extended to account for using the standard two-scale formulation, and in Sec- twinning by assuming relevant formulation of crystal tion IV the three-scale approach is applied to predict the plasticity theory at the lowest level of microstructure grain refinement and texture evolution in CP titanium (i.e., subgrain level) and adopting the PTVC scheme for subjected to cold rolling. The outcomes of simulations [50] twin-related reorientation. The PTVC scheme are compared with the experimental data available in [7] accounts for lattice reorientation due to twinning. It the literature. Modeling results are discussed in detail preserves consistency of the reorientation probability in Section V. The paper is concluded in Section VI. with the current twin volume fraction in the grain, which results from twinning activity. The scheme originates from a statistical concept described in Reference 51. II. MODEL OF GRAIN REFINEMENT Detailed discussion concerning this condition can be [8] found in Reference 52. Within the 3SCP model, the The three-scale model applied in this paper to PTVC scheme is applied at the level of each subgrain, so describe the microstructure evolution of HCP titanium if the reorientation condition is true a given subgrain was originally developed in order to enhance the texture [47] within the metagrain is reoriented according to the most predictions of the two-scale VPSC model and addi- active twinning system. The PTVC scheme ensures that tionally predict the grain refinement in FCC materials the volume fraction of twin-reoriented subgrains within by means of providing the misorientation angle distri- the metagrain is consistent with the accumulated twin butions. In this section, the formulation of the model volume fraction in that metagrain predicted by the will be reviewed and its extension to model twinning will crystal plasticity model. Since the whole subgrain is be presented. reoriented instantaneously the scheme does not account The main difference between the classical two-scale for a growth of a twin inside the subgrain and model and the 3SCP approach is the additional subgrain development of the layered substructure. Such level. Each grain is now called a metagrain and consists METALLURGICAL AND MATERIALS TRANSACTIONS A Lower micro-level (initial state) Upper micro-level gi Twin-reoriented subgrains gi a (t) Lower micro-level Macro-level (deformed state) Fig. 2—Schematic view of the three-scale model. morphological features are captured by models in which deformation according to the law summarized in additional level of microstructure is introduced: (i) the Table I. The physical background for the applied [53] total Lagrangian approach (ii) the composite grain formulation is extensively discussed in Reference 61. [54] [55] model based on the formulae valid for laminates (iii) Additionally, the fact that mechanical properties of [62] application of the ALAMEL model like in Reference 56, the twins are modified with respect to the matrix is or (iv) variational energy-based model proposed in taken into account. A scale factor l is introduced for Reference 57. Extensive discussion on their capabilities slip and twin systems, and the critical shear stresses for can be found in Reference 52. The crystal plasticity model the (sub)grain reoriented by PTVC scheme are changed 0 0 with twinning used in the present study at the subgrain to ls where s is the initial value of s . The quantities c c 0 ðabÞ level is formulated within the Eulerian large-strain s ; b; s ; f ; l; q are material parameters to be sat sat framework. The fundamentals of crystal plasticity theory determined by some identification procedure. can be found in References 58 and 59, and the details of Two micromechanical schemes, related to the partic- twinning incorporation in References 50 and 60. The ular mean-field formulation, have to be selected for the short summary of the approach is outlined below. scale transition between the subgrain-metagrain and In terms of kinematics the additive decomposition of metagrain-polycrystal levels. In Reference 8, the influ- the velocity gradient l into elastic and plastic parts is ence of the choice of mean-field models on the predic- exploited. Elastic stretches are neglected (the elastic part tions of misorientation evolution were studied. Here, e p l is equal to the lattice spin), while the plastic part l is taking into account the results of the mentioned study, [63] specified as the Taylor model is applied for the transition between grain aggregate and metagrain levels, and the tangent 2M N X X l [47,64] [47,65] p k k k l l l l TW variant of the VPSC scheme for the transition l ¼ c_ m  n þ c_ m  n ; where c_ ¼ c f ; between the metagrain and subgrain levels. The choice k¼1 l¼1 of the Taylor model for the higher micro-macro ½5 transition, and the VPSC for the lower can be supported kðlÞ as follows: Within the three-scale model, the subgrain is and c  0 is the rate of shearing on the k(l)th slip primarily interacting with its close vicinity, represented (twin) system; M and N denote the number of slip and in the self-consistent model by the averaged metagrain twin systems, respectively. Quantity f is the rate of behavior, and much less intensively with the whole volume fraction of twins that appeared due to activity of polycrystal. The averaged metagrain behavior calculated TW the twin system l, while c is the characteristic twin in the VPSC model is continuously modified with shear specified by the lattice geometry. Twinning is increasing misorientation angles between subgrains described as a unidirectional slip mode. and their twin-related reorientation. The iso-strain The classical viscoplastic power law Taylor model disables strain redistribution between metagrains, thus increases strain heterogeneity between c_ ¼ c_ ½6 subgrains, and the resulting misorientation angle, enabled by the VPSC scheme. Use of the VPSC scheme for both scale transitions would decrease strain relates the shear rate on the slip or twin system r with the r r r heterogeneity and the resulting misorientation between (non-negative) resolved shear stress s ¼hm  r  n i subgrains because the strain redistribution would take (where hi  ððÞ þ j  jÞ) and c_ is the reference shear place in two stages. Better accordance with experimental rate. The critical shear stresses s evolve during plastic METALLURGICAL AND MATERIALS TRANSACTIONS A data is observed for the first selection of scale-transition experimental ones presented in References 42 and 44. schemes. Additionally, the Taylor scheme allows for Since the stress–strain response is not on the focus of this easy parallelization of computation. Thus, in order for paper, therefore the value of s for prismatic slip was set c0 the 3SCP model to be effective, the proposed choice of to 1. For the detailed prediction of stress–strain response, mean-field models is used. the additional verification of parameters should be It is worth mentioning that the proposed three-scale performed. The parameters are shown in Table II.Itis formulation can be easily, at least conceptually, extended assumed that initially grains have spherical shape and the to the multi-scale framework. This could be done by shape evolves to ellipsoidal due to the deformation. introducing some threshold value for misorientation Figure 3 shows the slip and twin systems activities in the angle upon which the subgrain is subdivided into domains cold rolling simulation. The most active slip system is the with orientations slightly disturbed with respect to the prismatic one, which is justified by its lowest critical current orientation of parent subgrain. The same proce- resolved shear stress (CRSS) and texture favorable for the dure can be applied when the subgrain is reoriented due to activity of the prismatic slip family. At the beginning, the twinning. Nevertheless, such subdivisions would lead to second most active system was 1122 1123 twinning the excessively increased computational cost and have been postponed in the present analysis. III. SELECTION OF MODEL PARAMETERS In order to perform the simulation using the 3SCP model, it is first necessary to identify a set of material parameters for the single crystal model. In the present study, the parameters were obtained using the tangent variant of the two-scale VPSC model. The parameters Fig. 3—Slip and twin systems activities obtained in the cold rolling were selected so as to predict correct texture in CP Ti sheet simulation using the VPSC model for the selected set of parameters (Color figure online). subjected to cold rolling, corresponding to the Table I. Summary of Hardening Description within the Applied Crystal Plasticity Model Hardening Law with Four Types of Interactions Slip–Slip (ss), Slip-Twin (st), Twin-Slip (ts), and Twin-Twin (tt) P P M ðssÞ 2MþN ðstÞ r rþM r q r q q q qþM _ _ _  _ _ For Slip (r  M) s_ ¼ s_ ¼ H h c þ H h c ; where c ¼ c þ c rq rq c c q¼1 q¼2Mþ1 ðssÞ ðstÞ P P M ðtsÞ 2MþN ðttÞ r r q r q For Twinning (r>2M) s_ ¼ H h c þ H h c_ rq rq c ðtsÞ q¼1 ðttÞ q¼2Mþ1 Hardening Moduli r as c Slip–Slip H ¼ h 1 ðasÞ 0 s sat Slip–Twin at TW r 0 Twin–Slip H ¼ r at TW ðatÞ s f f c sat Slip–Twin ðabÞ ðabÞ ðabÞ r q Latent Hardening Submatrices h ¼ q þð1  q Þjn  n j rq For the physical background and further details see Ref. [61]. Table II. Single Crystal Plasticity Parameters Used in the Paper Basal System Interaction s (MPa) h (MPa) bs =f (MPa)/– l (–) Prism. Twin I Pyr hc þ ai c0 0 sat sat Prism. slip-slip 1.00 1.79 1 1.14 1 1.0 1.0 1.0 slip-twin — 0.00 — 0.01 — 2.0 Basal slip-slip 2.94 0.01 1 14.29 2 1.8 1.8 1.8 slip-twin — 0.01 — 0.01 — 2.0 I Pyr hc þ ai slip-slip 2.94 0.01 1 14.29 2 1.8 1.8 1.8 slip-twin — 0.01 — 0.01 — 2.0 E1 twin-slip — 0.07 1 0.01 — 1.0 1.0 1.0 twin-twin 1.41 0.01 — 0.00 1 1.0 C1 twin-slip — 0.07 1 0.01 — 1.0 1.0 1.0 twin-twin 2.00 0.01 — 0.00 1 1.0 Abbreviations: prism.: prismatic, I pyr. hc þ ai: first-order pyramidal hc þ ai; E1: 1012 1011 extension twinning, C1: 1122 1123 contraction twinning. METALLURGICAL AND MATERIALS TRANSACTIONS A (C1), which was activated despite its relatively high CRSS, initial shape of subgrains is spherical and evolves to because of the favorable texture. Subsequently, the E1 ellipsoidal due to the deformation. Deformation in the twinning was activated, mainly as a secondary twinning in rolling process was approximated imposing the follow- primary C1 twins. After the saturation of twinning ing macroscopic velocity gradient: activity, the deformation proceeded further by prismatic 2 3 10 0 slip supported by basal and pyramidal ones. The predicted 4 5 scenario is in accordance with the conclusions of Refer- L ¼ 00 0 ; ½7 ences 42, 44 through 46 reported in the introduction. 00  1 According to References 42 and 44, in the present case where the directions of axes 1, 2, 3 are rolling direction twinning saturates at higher thickness reductions because (RD), transverse direction (TD), and normal direction the grain size has become so small that further twinning is (ND), respectively. impossible. In the simulations, since the model does not The additional 3SCP model parameters were set to explicitly account for the grain size, the volume fraction of TW n ¼ 10 and DW ¼ 0:3 deg. Figures 5 and 6 show the pole twinning f (see Table II) at which it saturates is an figures obtained in the simulation. Using the FC-Tg additional model parameter f . The function of harden- sat variant of the 3SCP model, it was possible to correctly ing modulus H for the critical shear stress for twinning is tt predict the final texture obtained experimentally, cf. formulated in a way which accounts for the geometrical Reference 42—Figures 6 and 9, Reference 43—Figure 1, effect of twin boundaries in reducing the mean free path [66] Reference 44—Figure 8 and Reference 45—Figure 5.In distance. This formulation follows earlier work. particular, it was possible to predict two ‘‘c’’ fibers Figure 4 showsfg 0001 ; 1010 and 1120 pole inclined from the normal to the sheet plane (fg 0001 pole figures of (a) initial and (b) final texture in the VPSC figure). It was also possible to predict the positions of simulation of cold rolling. The textures are demon- maxima on 1010 and 1120 pole figures. The strated using the convention in which y-axis of an orthogonal crystal frame is coaxial with a axes of temporary existence of the ‘‘c’’ fiber normal to the sheet hexagon, cf. Appendix. Such convention was chosen in plane observed in References 42 through 44 was, order to enable the direct comparison with experimental however, not predicted in the 3SCP simulation. In data presented in References 42 through 46. Reference 43, the presence of the fiber was attributed to multiple twinning reorientations. On the other hand, this fiber was not observed in Reference 45. As shown in Figure 7, as compared to the two-scale VPSC model (Figure 3), in the three-scale model an increased activity IV. SIMULATION OF MICROSTRUCTURE of the hard pyramidal and basal modes at the cost of EVOLUTION OF COLD-ROLLED PURE easy prismatic slip is observed in the twin-reoriented TITANIUM crystallites. This change is due to the use of the Taylor Then, the 3SCP model was applied in order to transition scheme at the upper level and seems to be the simulate the microstructure evolution in CP Ti subjected main source of difference in the texture image resulting to cold rolling. Equivalently with a two-scale model, an from two simulations. (a) (b) Fig. 4—fg 0001 ; 1010 and 1120 pole figures of (a) initial and (b) final texture in the VPSC simulation of cold rolling to a total thickness reduction of 90 pct. METALLURGICAL AND MATERIALS TRANSACTIONS A Fig. 5— 0001 ; 1010 and 1120 pole figures obtained in the 3SCP model simulation of cold rolling to a total thickness reduction of 10, 20, fg and 30 pct. Figure 8 shows plots of misorientation angle distri- misorientation axes are oriented along 1010 and [42] bution obtained experimentally by Chun et al. and 1120 directions. According to Reference 42,and to calculated using the results of 3SCP model simulation. our calculations, they correspond to the activity of Figure 9 presents the calculated misorientation-axis twinning modes C1 and E1, respectively. They represent distribution for the strain reduction 20 and 90 pct, the misorientation angle at the boundary between correspondingly. The latter results can be compared primary C1 twin and secondary E1 twin (64.4 deg) as with Figure 8 in Reference 67. The simulated distribu- well as between the matrix and primary C1 twin (85.0 tions were obtained by calculating the misorientation deg). The simulation predicts also additional maximum angles and axes between randomly selected pairs of around 48.4 deg. It can be verified that this value subgrains in each metagrain. The randomly chosen pairs represents misorientation angle between the secondary of subgrains are intended to mimic the neighboring E1 twin and the matrix. It seems that this peak is not areas present in real material. For any subgrain, a present in the experimental distributions, in which [67] number of neighbors can be randomly selected. This weighted length of the boundary is used, because number was assessed to be 12; therefore first, the the number and length of boundaries between secondary misorientation distributions were calculated for misori- twins and matrix are very small, cf. Figure 11. There- entation angles between each subgrain and its randomly fore, another misorientation angle distributions were selected 12 ‘‘neighbors.’’ In order to increase the calculated and are shown in the c column. This time the numerical efficiency of the analysis, in the second step misorientation angles between secondary twins and for each subgrain only one ‘‘neighbor’’ was randomly matrix were not taken into account. There is qualitative chosen. The misorientation angle distribution obtained agreement between the position and relative strength of using these two approaches was very similar (cf. the peaks obtained in the simulation and experiment. On Figure 10); therefore, the results presented in the paper the other hand in the misorientation-axis distribution were obtained for the case of one neighbor per each plots, poles corresponding to matrix-secondary twin subgrain. Column b presents distributions obtained for boundaries are seen in Reference 67 (see Figure 8 every pair of ‘‘neighbors.’’ Two maxima present in the therein), in which a discrete point corresponds to a experiment, namely for 64.4 and 85.0 deg, were quali- single boundary independently of its length. Thus, the tatively predicted in the simulation. The corresponding calculated misorientation-axis distributions shown in METALLURGICAL AND MATERIALS TRANSACTIONS A Fig. 6— 0001 ; 1010 and 1120 pole figures obtained in the 3SCP model simulation of cold rolling to a total thickness reduction of 40, 60, fg and 90 pct. that the orientation relationship between parent and twin grain deviates from the ideal one as deformation proceeds. Recently, some experimental studies of this [68] phenomena were performed for magnesium alloy and [69] for titanium. One of the conclusions in the first paper was that the dislocation-twin interaction tailors the orientation of twin boundary and leads to the actual twin boundary deviating from the theoretical one. Similarly, in the second paper, it was found that the deformation may result in either increasing or lowering Fig. 7—Slip and twin systems activities obtained in the cold rolling simulation using the 3SCP model (Color figure online). of the boundaries misorientation and different segments of the same grain boundary may develop different misorientation angles. Figure 9 are presented for the case when the boundaries between the pairs composed of subgrain subject to the secondary twinning and the untwinned one are not excluded. V. DISCUSSION It is useful to note that with increasing deformation the peaks become weaker, similarly as in the experiment. As shown in Figure 8, the model was unable to This, of course, should be attributed to decreasing quantitatively predict the fraction of low angle bound- activity of twinning as the deformation proceeds and aries. In every case, the number of low angles acquired increasing discrepancy between the orientation of twins in the simulation is lower than the measured one. This (or twinned subgrains in the model) created at early may be the result of simplifications applied in the stages of deformation. An increasing spread of orienta- twin-related reorientation procedure. The procedure tions around the theoretical peaks 85 and 64.4 deg in was formulated in such a way so as to properly predict misorientation angle distribution plots and the corre- the texture of the material and to ensure the consistency sponding increasing spread around 1120 and 1010 between the volume fraction of reoriented grains in each directions for the misorientation axis illustrate the fact metagrain and the twin volume fraction stemming from METALLURGICAL AND MATERIALS TRANSACTIONS A (a) (b) (c) Fig. 8—The misorientation angle distribution for a sheet of CP Ti cold rolled to a total thickness reduction of 10, 20, 30, and 40 pct obtained in [42] (a) experiment and in the simulation with the 3SCP model, (b) for any randomly selected pair of subgrains, (c) excluding the pairs composed of subgrain subject to the secondary twinning and the untwinned one. LAB stands for low angle boundary fraction. its activity as a pseudo-slip mode. Within the procedure, Yet another option would be to apply the reorienta- the whole subgrain is reoriented when some condition is tion scheme similar to the one proposed in Reference 53, fulfilled, so that the number of subgrains within the where the orientations of twins are predefined and metagrain does not increase during the simulation. In during the simulations the volume fractions of different the real material, the dislocation substructure can twin variants are measured. In such a variation, the develop inside the twin. Its development is supposed simulation of rolling using the two-scale model with the to be responsible for the high fraction of LABs present Taylor transition could be first performed, wherefrom in experimental data. The 3SCP model is unable to the evolution of volume fractions of each twin variant predict the grain refinement inside the twins. In order to for every metagrain could be obtained. Then, for each do so, the fourth level of microstructure could be added, metagrain, composed of subgrains of the respective [54] for example assuming the composite grain model at variant taken in relevant evolving proportions, the the level of subgrain. This, however, would make the simulation using the VPSC model could be performed model much more numerically costly and is beyond the (with a disabled primary twin mode in subgrains related scope of the present paper. to the respective variant). On this lower level, the METALLURGICAL AND MATERIALS TRANSACTIONS A 94 (a) (b) Fig. 9—Misorientation-axis distributions calculated for the randomly selected pairs of subgrains in the CP Ti rolled to reductions of (a) 20 pct and (b) 90 pct. (a) (b) Fig. 10—Comparison of misorientation distributions calculated for (a) 12 and (b) 1 randomly selected neighbors. secondary twins and intra-twin refinement could be approach, the secondary twins could be also modeled accounted for. Such procedure would be again much during the first simulation using the Taylor model on the more computationally demanding than the one pro- upper micro-macro scale, but in such a case the model posed in the present study, but it should be at least more would probably be more or less as costly as the efficient than the four-scale model. On the other hand, it four-scale model, because the number of metagrains still could not describe the possible refinement inside the times primary twin variants times secondary twin secondary twins. To include such effects in this variants had to be accounted for. METALLURGICAL AND MATERIALS TRANSACTIONS A It is maybe important to note that in the model we between subgrains belonging to the same phase (matrix do not have any information about the relative or a specific twin variant). In such a case, it appears frequencies of boundaries between subgrains belonging that the fraction of low angle grain boundaries to matrix and twins. Such information may be supplied (misorientation angle below 15 deg) is about 100 pct by experiment and used in the model to improve the for reductions up to 60 and about 90 pct for 90 pct procedure of selecting pairs of neighbors for calculat- reduction. To conclude, one can state that the fraction ing misorientation angle distribution. Following this of LABs can be correctly predicted using the 3SCP idea, in order to have greater insight, we have also model if the distributions obtained for matrix-matrix, calculated distributions of misorientation angles matrix-twin and twin-twin boundaries are combined in appropriate proportions. It could be interesting to apply experimental data concerning the relative num- ber of every type of boundary and see to what extent the results of simulation are consistent with experi- mental misorientation distribution. This way one could also assess the error stemming from not including the intra-twin refinement. Figure 12 shows the misorientation distributions calculated for sets of misorientation angles as shown in Figure 8 appended by adding also the sets of misori- entation angles between subgrains belonging to the same phase mentioned in the previous paragraph. It can be seen that the fraction of LABs for 10 and 20 pct thickness reductions is more or less similar to the one observed experimentally. Unfortunately, for reductions of 30 and 40 pct, the fraction of LABs is again lower than the experiment. This can be explained by rising proportion of the intra-phase boundaries to the inter-phase boundaries stemming from ongoing intra- Fig. 11—The schematic view of the primary (in red) and secondary (in green) twins in single titanium grain showing that the number of phase refinement occurring by mechanisms analogous boundaries between secondary twins and matrix is very low. The to the ones existing in FCC materials with higher schematic can be compared with experimental microstructures, as thickness reduction. shown, e.g., in Fig. 2 in Ref. [46] (Color figure online). Fig. 12—The misorientation angle distribution for a sheet of CP Ti cold rolled to a total thickness reduction of 10, 20, 30, and 40 pct obtained in the simulation with the 3SCP model. The distributions are plotted by combining (in equal proportions) the sets of misorientation angles between randomly selected subgrains and angles between subgrains belonging to the same phase (matrix or a given twin variant). METALLURGICAL AND MATERIALS TRANSACTIONS A VI. SUMMARY AND CONCLUSIONS which permits unrestricted use, distribution, and reproduction in any medium, provided you give The aim of the paper was to examine the applicability appropriate credit to the original author(s) and the [8] of the recently proposed 3SCP model to the case of source, provide a link to the Creative Commons grain refinement in HCP material. Within the approach, license, and indicate if changes were made. [50] the probabilistic twin volume consistent scheme was used to account for appearance of twin-related orienta- tions. The simulations of texture evolution and grain APPENDIX: CONVENTIONS USED IN PLOTTING refinement in CP Ti subjected to cold rolling were THE TEXTURES OF HCP METALS AND ALLOYS performed. The results were compared with the exper- imental data available in References 42 through 45.It The stereographic projection of HCP crystals depends was found that the simulated textures were in accor- on the convention used for the denomination of dance with the experimental ones. Concerning the orthogonal coordinate system. The hexagonal system predictions of the misorientation angle distribution, is very convenient for describing the HCP crystals, due the basic components of the distributions observed in to its ability to naturally take into account the crystal experiments in the range of high angle boundaries were symmetries. As this system is not orthogonal, it has to well identified. The peaks are associated with twinning be transformed to the orthogonal one in order to enable activity. However, the application of the 3SCP model to the description of the lattice orientation. Two conven- the case of the HCP material also had difficulties, tions for this transformation are used in the pole namely the fraction of low angle boundaries was figure plotting software: underestimated by the model. Possible solutions to the problem were discussed. To this aim, some kind of a 1. x axis is parallel to a (i ¼ 1; 2; 3, the choice of i is four-scale model could be developed, a different reori- arbitrary because of the crystal symmetry), cf. entation procedure can be applied, or the postprocessing Figure A1(a), of the data can be enhanced using experimental data. 2. y axis is parallel to a , cf. Figure A1(b). To the best of the authors’ knowledge, there is no model that is able to predict misorientation angle z axis is always parallel to the c axis of the crystal, and distributions in the microstructure of HCP metals, and the unspecified axis (y in convention 1 and x in therefore, the results presented in the paper can be convention 2.) is perpendicular to both defined axes. considered as novel. The capabilities of the 3SCP model In order to show the significance of the choice of as compared to other grain refinement models have been convention Figure A2 shows theðÞ 0001 ; 1010 and summed up in Reference 8. First, contrary to the model [21,22] of Leffers, the orientations within the microstruc- 1120 pole figures plotted for a single orientation ture do not have to be predefined. This enables described by Euler angles (0, 0, 0). It is readily seen, that application of the model to study any deformation the choice of convention does not have any influence on process. For example, it has been applied to study theðÞ 0001 pole figure, because the ‘‘c’’ direction is the ECAP in References 8 and 48 and rolling in Reference same in both conventions. The choice of the convention 48 and the present paper. Second, the intra-granular is however very important in case of 1010 and 1120 strain heterogeneity is considered, contrary to models pole figures, namely it leads to their mutual interchange. presented in References 23 through 25. Third, contrary As the choice of the convention is crucial for the pole to References 23 through 26 the number of orientations figures obtained it should be made consciously and during the simulation does not change, which increases explicitly stated in the publication. Such an approach the computational efficiency of the model. In the present could save many misunderstanding across different analysis, it is demonstrated that those advantages are groups of researchers. not lost when the 3SCP model is combined with PTVC reorientation scheme to account for twinning. The proposed approach provides results of statistical char- acter. In the model, there is no information concerning the spatial distribution of subgrains in the metagrain or metagrains in the polycrystalline representative volume. This can be treated as a drawback but also as an asset because the outcomes, when a sufficiently large number of orientations are taken for the analysis, are expected to be statistically representative and not related to the specific microstructure realization. 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Microstructure Evolution in Cold-Rolled Pure Titanium: Modeling by the Three-Scale Crystal Plasticity Approach Accounting for Twinning

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Materials Science; Metallic Materials; Characterization and Evaluation of Materials; Structural Materials; Surfaces and Interfaces, Thin Films; Nanotechnology
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Abstract

Microstructure Evolution in Cold-Rolled Pure Titanium: Modeling by the Three-Scale Crystal Plasticity Approach Accounting for Twinning KAROL FRYDRYCH and KATARZYNA KOWALCZYK-GAJEWSKA A three-scale crystal plasticity model is applied to simulate microstructure evolution in hcp titanium subjected to cold rolling. Crystallographic texture and misorientation angle develop- ment, as an indicator of grain refinement, are studied. The impact of twinning activity on both phenomena is accounted for by combining the original three-scale formulation with the probabilistic twin-volume consistent (PTVC) reorientation scheme. The modeling results are compared with available experimental data. It is shown that the simulated textures are in accordance with the experimental measurements. The basic components of misorientation angle distribution, especially in the range of high angle boundaries, are also well reproduced. https://doi.org/10.1007/s11661-018-4676-2 The Author(s) 2018 I. INTRODUCTION The mechanisms governing the grain refinement in metals of hexagonal close packed (HCP) lattice experi- IT is well known that improvement of material encing large plastic strains, induced e.g., via severe properties can be achieved by tailoring the microstruc- plastic deformation (SPD) processes, are known to a ture. The microstructure can be refined by advanced much lesser extent. First of all, in those metals, the plastic deformation processes. The mechanisms leading number of easy slip systems is in general less than in to refinement are not yet fully described, in spite of the FCC metals, and twinning appears as an additional, huge effort that has been put into modeling. While this complementary mode. Second, in spite of the many statement is true in the case of face centered cubic (FCC) efforts to carry out cold SPD processes for commercially materials, it is even more relevant in the case of [9] pure titanium (CP Ti) and for AZ31b magnesium hexagonal close packed (HCP) materials. Mainly it is [10] alloy, they are usually performed at elevated temper- due to the fact that plastic deformation in HCP atures, which promote discontinuous dynamic recrys- [11] materials is usually more complicated than in FCC, tallization (DDRX). Beausir et al. assumed that the because of the insufficient number of easy slip systems DRX for magnesium occurs at a temperature equal to and the ease of twinning. 0:4T , where T is the melting point (T = 923 K for M M M Qualitative studies of the grain refinement phe- magnesium). Thus 0:4T ¼ 369 K ¼ 96 C, which is nomenon in FCC metals and alloys were reported in even less than the boiling point of water. Taking into References 1 through 7. Based on the indicated mech- account the simplicity of the 0:4T approximation and anisms of microstructure evolution, a three-scale crystal the fact that the temperature in the SPD process can [8] plasticity (3SCP) model was built. The model con- increase without additional heating, it is not surprising struction was motivated by the initially present defor- that Gu et al. observed the DRX in AZ31b subjected to mation-induced cell substructure, schematically shown equal channel angular pressing (ECAP) process at room [12] in Figure 1(a). Two types of grain boundaries can be temperature. Figueiredo and Langdon proposed the recognized, namely geometrically necessary boundaries model, in which DDRX is the mechanism responsible (GNBs) and incidental dislocation boundaries (IDBs). for grain refinement in the AZ31b alloy subjected to the The new grains are formed by gradual increase of ECAP process. New grains are formed near the initial misorientation angles across the boundaries. grain boundaries or twin boundaries. The reason for this is the presence of higher stresses in those areas, which promote the activity of non-basal slip systems—in the case of sufficiently small grain (with size below the KAROL FRYDRYCH and KATARZYNA KOWALCZYK- critical grain diameter), non-basal slip can activate GAJEWSKA are with the Institute of Fundamental Technological throughout the whole volume of the grain. The critical Research (IPPT), Polish Academy of Sciences, Pawin nskiego 5B, value of the grain diameter can be dependent upon the 02-106 Warsaw, Poland. Contact e-mail: kfryd@ippt.pan.pl Manuscript submitted January 11, 2018. given alloy, pressing temperature or the back-pressure used. Obtaining a homogeneous grain size distribution is METALLURGICAL AND MATERIALS TRANSACTIONS A [27–30] models. Apart from the approaches discussed in Reference 8, models applying the crystal plasticity finite element method (CPFEM) with many integration points [31–35] or elements per grain are also worth mentioning as a possible tool to predict the grain refinement. However, none of the models mentioned is able to fully predict the grain refinement in FCC materials. As the mechanisms leading to grain refinement in HCP metals and alloys are by far less known than the (a) (b) analogous mechanisms for FCC materials, there are accordingly fewer papers modeling this phenomenon. Fig. 1—(a) The schematic drawing of the dislocation-induced cell As already mentioned, the grain refinement in AZ31b substructure of a grain observed in FCC materials for small to [12,13] magnesium alloy is attributed to DDRX. The medium accumulated plastic strain according to Ref. [3], the picture simulation of grain refinement in an unconventional reprinted after Ref. [8]. (b) The schematic showing analogous substructure for commercially pure titanium, after twinning. process joining extrusion and ECAP was described in Reference 36. The process was performed at 350 C. The grain dimension after recrystallization was calculated thus feasible already in the first ECAP pass if the initial using a phenomenological equation, calibrated for grain size is lower than the critical diameter. If the initial compression tests. The authors claimed that they had grain size is greater than the critical value, the obtained an interesting agreement between the simula- microstructure obtained after the first pass is heteroge- tion and experimental data, but did not include direct neous and contains both small and large grains. The comparison. The grain refinement stemming from [12] model of Figueiredo and Langdon is consistent with DDRX can also be modeled using cellular automata [37] many experimental data for AZ31b magnesium alloy, (CA). Such an approach was applied by Gzyl et al. e.g., References 13 through 16. The strain, strain rate, and temperature at the integra- For pure titanium, the melting point T is equal to tion points were calculated using the finite element 1941 K; therefore, 0.4T = 776 K = 503 C, which is method. These data were then used to simulate the DRX a much greater value than in the case of magnesium. using the CA. The refinement in AZ31b subjected to According to the 0.4T criterion, the start of DRX is ECAP and I-ECAP was simulated and good agreement not expected in temperatures below 500 C, which between the results of modeling and experimental data [17,18] justifies the approach excercised by some authors, was achieved. who model the grain refinement of titanium assuming Grain refinement in CP Ti subjected to SPD processes similar mechanisms to those acting in FCC materials. at only slightly elevated temperatures was modeled These works deal mainly with processes performed at assuming mechanisms analogous to FCC materials. [17] [38] low temperature (below 0.4T ). Yang et al. analyzed Ding and Shin classified the orthogonal cutting as a the grain refinement mechanism in shear bands that severe plastic deformation process, because of the developed in cold-rolled CP Ti. They stated that the ultrafine grains developing inside chips. The grain mechanisms responsible for grain refinement are anal- dimensions inside chips were predicted using a model [39,40] ogous to those acting in FCC materials. On the other of dislocation evolution inside dislocation cells and hand, twinning plays also an important role in the walls between them. A similar approach was applied in fragmentation of original grains. The authors noticed Reference 41 for modeling the grain refinement in cold that the microstructure in shear bands is similar to the rolling of titanium. The maximum temperature was one observed in SPD processes. Taking this into determined to be 150 C. The average dislocation cell account, the supposition was made that the same dimensions were predicted in good agreement with mechanisms can lead to grain refinement in SPD experimental results. [18] processes. Zeng et al. analyzed the grain refinement In the present paper, the texture and grain refinement mechanisms acting during hot compression (tempera- of CP Ti deformed in cold rolling are studied. The ture range 673 to 973 K) of CP Ti extruded tube and evolution of microstructure of this material subjected to observed that twinning has strong impact upon the grain cold rolling was analyzed in References 42 through 46. refinement. At 723 K (53 K lower than 0.4T ), the high In References 42, 44 through 46, only experimental data [43] angle grain boundaries (HAGBs) developed by two were analyzed. Bozzolo et al. have modeled texture parallel processes: gradual increase of misorientation evolution during the process using the visco-plastic [47] angle across the low angle boundary (LAB)—analogous self-consistent (VPSC) model. The grain refinement mechanism to the one present in FCC metals, and by was not simulated in any of those papers. [42–46] deformation of the twin boundaries, cf. Figure 1(b). The authors of papers concluded that: There is quite a large number of papers reporting – Extension twinning systems of the family modeling of grain refinement in FCC metals. Our recent [8] fg 1012hi 1011 (E1) and contraction twinning systems paper shortly reviews the approaches used for mod- of the familyfg 1122hi 1123 (C1) were activated. eling the grain refinement using the macroscopic phe- [19,20] – Activation of E1 twinning requires a lower resolved nomenological, the mean-field crystal plasticity [8,21–26] shear stress (RSS) than C1, but the RSS for most models enhanced by some additional features grains is initially much higher for the contraction and the incremental energy minimization-based METALLURGICAL AND MATERIALS TRANSACTIONS A twinning mode due to the strong initial texture. As a of a number of subgrains, cf. Figure 2. The notion of result, the C1 twin modes are activated first and the subgrain denotes a single discrete orientation and is not E1 modes are initiated mainly as a secondary equal to the term subgrain in quantitative metallogra- twinning mode inside the C1 twins. phy, although it is conceptually related to it. – Twinning is an important grain refinement mecha- The representative volume element (RVE) in 3SCP nism, especially in the initial stages of deformation. model consists of NG metagrains. In each metagrain, a Twin boundaries contribute to the fraction of high nominal orientation Q ð0Þ is initially assigned. Meta- angle boundaries and the texture evolves due to the grains are themselves aggregates of subgrains—each twin reorientation. consists of NS subgrains, whose orientations are – For higher strains, the twinning saturates and then obtained by slightly distorting the nominal metagrain most of the deformation is accommodated by slip on orientations. The initial orientation of subgrain i is thus the prismatic planesfg 1010 : The grain refinement obtained as follows: continues by means of mechanisms analogous to g g Q ð0Þ¼ R ð0ÞQ ð0Þ; ½1 those occurring in FCC metals. i i where The main goal of this paper is to verify to which extent the recently proposed computationally efficient three-s- [8,48] R ð0Þ¼ RnðÞ ; dw ð0Þ ½2 cale crystal plasticity (3SCP) model is applicable to simulate texture evolution and grain refinement in tita- is a rotation matrix around arbitrary axis n about the nium. To this aim, first, the 3SCP model is extended to small angle dw ð0Þ2hi 0; Dw : The axis n and the account for twinning and next, the simulations of cold distortion angle dw are assigned to each subgrain by rolling are performed for CP titanium. In the proposed random generation. extension of the 3SCP model, the probabilistic twin-vol- During the simulation of some plastic deformation ume consistent (PTVC) reorientation scheme is used in process, one can study the grain refinement occurring by order to properly account for the appearance of twin-re- evolution of subgrain orientations. The following equa- lated orientations at the metagrain level. Modeling results tion describes the current misorientation angle dw ðtÞ ij are analyzed and compared to experimental data avail- [49] between subgrains i and j of the metagrain g : able in the literature. The quality of model predictions is discussed and possible sources of discrepancies between g g 2 cosðdw Þ¼ trðR ðtÞÞ  1; ½3 ij ij simulations and experimental data are indicated. The paper is organized as follows: After this intro- where ductory section, the 3SCP model formulation is recalled g g g T in Section II. First, the construction of a polycrystalline R ðtÞ¼ Q ðtÞQ ðtÞ : ½4 ij i j representative volume is outlined, and then the consti- Note that in Reference 8, the misorientation angle was tutive descriptions used on the lowest level of calculated with respect to the mean orientation of the microstructure and rules of micro-macro transition are metagrain. In this work, we have not used this approach presented. The extension of the original framework due in order to directly observe misorientation angles to twinning activity is also discussed. In Section III, the between matrix and twinned domains. parameters of the crystal plasticity model are identified The original 3SCP model is extended to account for using the standard two-scale formulation, and in Sec- twinning by assuming relevant formulation of crystal tion IV the three-scale approach is applied to predict the plasticity theory at the lowest level of microstructure grain refinement and texture evolution in CP titanium (i.e., subgrain level) and adopting the PTVC scheme for subjected to cold rolling. The outcomes of simulations [50] twin-related reorientation. The PTVC scheme are compared with the experimental data available in [7] accounts for lattice reorientation due to twinning. It the literature. Modeling results are discussed in detail preserves consistency of the reorientation probability in Section V. The paper is concluded in Section VI. with the current twin volume fraction in the grain, which results from twinning activity. The scheme originates from a statistical concept described in Reference 51. II. MODEL OF GRAIN REFINEMENT Detailed discussion concerning this condition can be [8] found in Reference 52. Within the 3SCP model, the The three-scale model applied in this paper to PTVC scheme is applied at the level of each subgrain, so describe the microstructure evolution of HCP titanium if the reorientation condition is true a given subgrain was originally developed in order to enhance the texture [47] within the metagrain is reoriented according to the most predictions of the two-scale VPSC model and addi- active twinning system. The PTVC scheme ensures that tionally predict the grain refinement in FCC materials the volume fraction of twin-reoriented subgrains within by means of providing the misorientation angle distri- the metagrain is consistent with the accumulated twin butions. In this section, the formulation of the model volume fraction in that metagrain predicted by the will be reviewed and its extension to model twinning will crystal plasticity model. Since the whole subgrain is be presented. reoriented instantaneously the scheme does not account The main difference between the classical two-scale for a growth of a twin inside the subgrain and model and the 3SCP approach is the additional subgrain development of the layered substructure. Such level. Each grain is now called a metagrain and consists METALLURGICAL AND MATERIALS TRANSACTIONS A Lower micro-level (initial state) Upper micro-level gi Twin-reoriented subgrains gi a (t) Lower micro-level Macro-level (deformed state) Fig. 2—Schematic view of the three-scale model. morphological features are captured by models in which deformation according to the law summarized in additional level of microstructure is introduced: (i) the Table I. The physical background for the applied [53] total Lagrangian approach (ii) the composite grain formulation is extensively discussed in Reference 61. [54] [55] model based on the formulae valid for laminates (iii) Additionally, the fact that mechanical properties of [62] application of the ALAMEL model like in Reference 56, the twins are modified with respect to the matrix is or (iv) variational energy-based model proposed in taken into account. A scale factor l is introduced for Reference 57. Extensive discussion on their capabilities slip and twin systems, and the critical shear stresses for can be found in Reference 52. The crystal plasticity model the (sub)grain reoriented by PTVC scheme are changed 0 0 with twinning used in the present study at the subgrain to ls where s is the initial value of s . The quantities c c 0 ðabÞ level is formulated within the Eulerian large-strain s ; b; s ; f ; l; q are material parameters to be sat sat framework. The fundamentals of crystal plasticity theory determined by some identification procedure. can be found in References 58 and 59, and the details of Two micromechanical schemes, related to the partic- twinning incorporation in References 50 and 60. The ular mean-field formulation, have to be selected for the short summary of the approach is outlined below. scale transition between the subgrain-metagrain and In terms of kinematics the additive decomposition of metagrain-polycrystal levels. In Reference 8, the influ- the velocity gradient l into elastic and plastic parts is ence of the choice of mean-field models on the predic- exploited. Elastic stretches are neglected (the elastic part tions of misorientation evolution were studied. Here, e p l is equal to the lattice spin), while the plastic part l is taking into account the results of the mentioned study, [63] specified as the Taylor model is applied for the transition between grain aggregate and metagrain levels, and the tangent 2M N X X l [47,64] [47,65] p k k k l l l l TW variant of the VPSC scheme for the transition l ¼ c_ m  n þ c_ m  n ; where c_ ¼ c f ; between the metagrain and subgrain levels. The choice k¼1 l¼1 of the Taylor model for the higher micro-macro ½5 transition, and the VPSC for the lower can be supported kðlÞ as follows: Within the three-scale model, the subgrain is and c  0 is the rate of shearing on the k(l)th slip primarily interacting with its close vicinity, represented (twin) system; M and N denote the number of slip and in the self-consistent model by the averaged metagrain twin systems, respectively. Quantity f is the rate of behavior, and much less intensively with the whole volume fraction of twins that appeared due to activity of polycrystal. The averaged metagrain behavior calculated TW the twin system l, while c is the characteristic twin in the VPSC model is continuously modified with shear specified by the lattice geometry. Twinning is increasing misorientation angles between subgrains described as a unidirectional slip mode. and their twin-related reorientation. The iso-strain The classical viscoplastic power law Taylor model disables strain redistribution between metagrains, thus increases strain heterogeneity between c_ ¼ c_ ½6 subgrains, and the resulting misorientation angle, enabled by the VPSC scheme. Use of the VPSC scheme for both scale transitions would decrease strain relates the shear rate on the slip or twin system r with the r r r heterogeneity and the resulting misorientation between (non-negative) resolved shear stress s ¼hm  r  n i subgrains because the strain redistribution would take (where hi  ððÞ þ j  jÞ) and c_ is the reference shear place in two stages. Better accordance with experimental rate. The critical shear stresses s evolve during plastic METALLURGICAL AND MATERIALS TRANSACTIONS A data is observed for the first selection of scale-transition experimental ones presented in References 42 and 44. schemes. Additionally, the Taylor scheme allows for Since the stress–strain response is not on the focus of this easy parallelization of computation. Thus, in order for paper, therefore the value of s for prismatic slip was set c0 the 3SCP model to be effective, the proposed choice of to 1. For the detailed prediction of stress–strain response, mean-field models is used. the additional verification of parameters should be It is worth mentioning that the proposed three-scale performed. The parameters are shown in Table II.Itis formulation can be easily, at least conceptually, extended assumed that initially grains have spherical shape and the to the multi-scale framework. This could be done by shape evolves to ellipsoidal due to the deformation. introducing some threshold value for misorientation Figure 3 shows the slip and twin systems activities in the angle upon which the subgrain is subdivided into domains cold rolling simulation. The most active slip system is the with orientations slightly disturbed with respect to the prismatic one, which is justified by its lowest critical current orientation of parent subgrain. The same proce- resolved shear stress (CRSS) and texture favorable for the dure can be applied when the subgrain is reoriented due to activity of the prismatic slip family. At the beginning, the twinning. Nevertheless, such subdivisions would lead to second most active system was 1122 1123 twinning the excessively increased computational cost and have been postponed in the present analysis. III. SELECTION OF MODEL PARAMETERS In order to perform the simulation using the 3SCP model, it is first necessary to identify a set of material parameters for the single crystal model. In the present study, the parameters were obtained using the tangent variant of the two-scale VPSC model. The parameters Fig. 3—Slip and twin systems activities obtained in the cold rolling were selected so as to predict correct texture in CP Ti sheet simulation using the VPSC model for the selected set of parameters (Color figure online). subjected to cold rolling, corresponding to the Table I. Summary of Hardening Description within the Applied Crystal Plasticity Model Hardening Law with Four Types of Interactions Slip–Slip (ss), Slip-Twin (st), Twin-Slip (ts), and Twin-Twin (tt) P P M ðssÞ 2MþN ðstÞ r rþM r q r q q q qþM _ _ _  _ _ For Slip (r  M) s_ ¼ s_ ¼ H h c þ H h c ; where c ¼ c þ c rq rq c c q¼1 q¼2Mþ1 ðssÞ ðstÞ P P M ðtsÞ 2MþN ðttÞ r r q r q For Twinning (r>2M) s_ ¼ H h c þ H h c_ rq rq c ðtsÞ q¼1 ðttÞ q¼2Mþ1 Hardening Moduli r as c Slip–Slip H ¼ h 1 ðasÞ 0 s sat Slip–Twin at TW r 0 Twin–Slip H ¼ r at TW ðatÞ s f f c sat Slip–Twin ðabÞ ðabÞ ðabÞ r q Latent Hardening Submatrices h ¼ q þð1  q Þjn  n j rq For the physical background and further details see Ref. [61]. Table II. Single Crystal Plasticity Parameters Used in the Paper Basal System Interaction s (MPa) h (MPa) bs =f (MPa)/– l (–) Prism. Twin I Pyr hc þ ai c0 0 sat sat Prism. slip-slip 1.00 1.79 1 1.14 1 1.0 1.0 1.0 slip-twin — 0.00 — 0.01 — 2.0 Basal slip-slip 2.94 0.01 1 14.29 2 1.8 1.8 1.8 slip-twin — 0.01 — 0.01 — 2.0 I Pyr hc þ ai slip-slip 2.94 0.01 1 14.29 2 1.8 1.8 1.8 slip-twin — 0.01 — 0.01 — 2.0 E1 twin-slip — 0.07 1 0.01 — 1.0 1.0 1.0 twin-twin 1.41 0.01 — 0.00 1 1.0 C1 twin-slip — 0.07 1 0.01 — 1.0 1.0 1.0 twin-twin 2.00 0.01 — 0.00 1 1.0 Abbreviations: prism.: prismatic, I pyr. hc þ ai: first-order pyramidal hc þ ai; E1: 1012 1011 extension twinning, C1: 1122 1123 contraction twinning. METALLURGICAL AND MATERIALS TRANSACTIONS A (C1), which was activated despite its relatively high CRSS, initial shape of subgrains is spherical and evolves to because of the favorable texture. Subsequently, the E1 ellipsoidal due to the deformation. Deformation in the twinning was activated, mainly as a secondary twinning in rolling process was approximated imposing the follow- primary C1 twins. After the saturation of twinning ing macroscopic velocity gradient: activity, the deformation proceeded further by prismatic 2 3 10 0 slip supported by basal and pyramidal ones. The predicted 4 5 scenario is in accordance with the conclusions of Refer- L ¼ 00 0 ; ½7 ences 42, 44 through 46 reported in the introduction. 00  1 According to References 42 and 44, in the present case where the directions of axes 1, 2, 3 are rolling direction twinning saturates at higher thickness reductions because (RD), transverse direction (TD), and normal direction the grain size has become so small that further twinning is (ND), respectively. impossible. In the simulations, since the model does not The additional 3SCP model parameters were set to explicitly account for the grain size, the volume fraction of TW n ¼ 10 and DW ¼ 0:3 deg. Figures 5 and 6 show the pole twinning f (see Table II) at which it saturates is an figures obtained in the simulation. Using the FC-Tg additional model parameter f . The function of harden- sat variant of the 3SCP model, it was possible to correctly ing modulus H for the critical shear stress for twinning is tt predict the final texture obtained experimentally, cf. formulated in a way which accounts for the geometrical Reference 42—Figures 6 and 9, Reference 43—Figure 1, effect of twin boundaries in reducing the mean free path [66] Reference 44—Figure 8 and Reference 45—Figure 5.In distance. This formulation follows earlier work. particular, it was possible to predict two ‘‘c’’ fibers Figure 4 showsfg 0001 ; 1010 and 1120 pole inclined from the normal to the sheet plane (fg 0001 pole figures of (a) initial and (b) final texture in the VPSC figure). It was also possible to predict the positions of simulation of cold rolling. The textures are demon- maxima on 1010 and 1120 pole figures. The strated using the convention in which y-axis of an orthogonal crystal frame is coaxial with a axes of temporary existence of the ‘‘c’’ fiber normal to the sheet hexagon, cf. Appendix. Such convention was chosen in plane observed in References 42 through 44 was, order to enable the direct comparison with experimental however, not predicted in the 3SCP simulation. In data presented in References 42 through 46. Reference 43, the presence of the fiber was attributed to multiple twinning reorientations. On the other hand, this fiber was not observed in Reference 45. As shown in Figure 7, as compared to the two-scale VPSC model (Figure 3), in the three-scale model an increased activity IV. SIMULATION OF MICROSTRUCTURE of the hard pyramidal and basal modes at the cost of EVOLUTION OF COLD-ROLLED PURE easy prismatic slip is observed in the twin-reoriented TITANIUM crystallites. This change is due to the use of the Taylor Then, the 3SCP model was applied in order to transition scheme at the upper level and seems to be the simulate the microstructure evolution in CP Ti subjected main source of difference in the texture image resulting to cold rolling. Equivalently with a two-scale model, an from two simulations. (a) (b) Fig. 4—fg 0001 ; 1010 and 1120 pole figures of (a) initial and (b) final texture in the VPSC simulation of cold rolling to a total thickness reduction of 90 pct. METALLURGICAL AND MATERIALS TRANSACTIONS A Fig. 5— 0001 ; 1010 and 1120 pole figures obtained in the 3SCP model simulation of cold rolling to a total thickness reduction of 10, 20, fg and 30 pct. Figure 8 shows plots of misorientation angle distri- misorientation axes are oriented along 1010 and [42] bution obtained experimentally by Chun et al. and 1120 directions. According to Reference 42,and to calculated using the results of 3SCP model simulation. our calculations, they correspond to the activity of Figure 9 presents the calculated misorientation-axis twinning modes C1 and E1, respectively. They represent distribution for the strain reduction 20 and 90 pct, the misorientation angle at the boundary between correspondingly. The latter results can be compared primary C1 twin and secondary E1 twin (64.4 deg) as with Figure 8 in Reference 67. The simulated distribu- well as between the matrix and primary C1 twin (85.0 tions were obtained by calculating the misorientation deg). The simulation predicts also additional maximum angles and axes between randomly selected pairs of around 48.4 deg. It can be verified that this value subgrains in each metagrain. The randomly chosen pairs represents misorientation angle between the secondary of subgrains are intended to mimic the neighboring E1 twin and the matrix. It seems that this peak is not areas present in real material. For any subgrain, a present in the experimental distributions, in which [67] number of neighbors can be randomly selected. This weighted length of the boundary is used, because number was assessed to be 12; therefore first, the the number and length of boundaries between secondary misorientation distributions were calculated for misori- twins and matrix are very small, cf. Figure 11. There- entation angles between each subgrain and its randomly fore, another misorientation angle distributions were selected 12 ‘‘neighbors.’’ In order to increase the calculated and are shown in the c column. This time the numerical efficiency of the analysis, in the second step misorientation angles between secondary twins and for each subgrain only one ‘‘neighbor’’ was randomly matrix were not taken into account. There is qualitative chosen. The misorientation angle distribution obtained agreement between the position and relative strength of using these two approaches was very similar (cf. the peaks obtained in the simulation and experiment. On Figure 10); therefore, the results presented in the paper the other hand in the misorientation-axis distribution were obtained for the case of one neighbor per each plots, poles corresponding to matrix-secondary twin subgrain. Column b presents distributions obtained for boundaries are seen in Reference 67 (see Figure 8 every pair of ‘‘neighbors.’’ Two maxima present in the therein), in which a discrete point corresponds to a experiment, namely for 64.4 and 85.0 deg, were quali- single boundary independently of its length. Thus, the tatively predicted in the simulation. The corresponding calculated misorientation-axis distributions shown in METALLURGICAL AND MATERIALS TRANSACTIONS A Fig. 6— 0001 ; 1010 and 1120 pole figures obtained in the 3SCP model simulation of cold rolling to a total thickness reduction of 40, 60, fg and 90 pct. that the orientation relationship between parent and twin grain deviates from the ideal one as deformation proceeds. Recently, some experimental studies of this [68] phenomena were performed for magnesium alloy and [69] for titanium. One of the conclusions in the first paper was that the dislocation-twin interaction tailors the orientation of twin boundary and leads to the actual twin boundary deviating from the theoretical one. Similarly, in the second paper, it was found that the deformation may result in either increasing or lowering Fig. 7—Slip and twin systems activities obtained in the cold rolling simulation using the 3SCP model (Color figure online). of the boundaries misorientation and different segments of the same grain boundary may develop different misorientation angles. Figure 9 are presented for the case when the boundaries between the pairs composed of subgrain subject to the secondary twinning and the untwinned one are not excluded. V. DISCUSSION It is useful to note that with increasing deformation the peaks become weaker, similarly as in the experiment. As shown in Figure 8, the model was unable to This, of course, should be attributed to decreasing quantitatively predict the fraction of low angle bound- activity of twinning as the deformation proceeds and aries. In every case, the number of low angles acquired increasing discrepancy between the orientation of twins in the simulation is lower than the measured one. This (or twinned subgrains in the model) created at early may be the result of simplifications applied in the stages of deformation. An increasing spread of orienta- twin-related reorientation procedure. The procedure tions around the theoretical peaks 85 and 64.4 deg in was formulated in such a way so as to properly predict misorientation angle distribution plots and the corre- the texture of the material and to ensure the consistency sponding increasing spread around 1120 and 1010 between the volume fraction of reoriented grains in each directions for the misorientation axis illustrate the fact metagrain and the twin volume fraction stemming from METALLURGICAL AND MATERIALS TRANSACTIONS A (a) (b) (c) Fig. 8—The misorientation angle distribution for a sheet of CP Ti cold rolled to a total thickness reduction of 10, 20, 30, and 40 pct obtained in [42] (a) experiment and in the simulation with the 3SCP model, (b) for any randomly selected pair of subgrains, (c) excluding the pairs composed of subgrain subject to the secondary twinning and the untwinned one. LAB stands for low angle boundary fraction. its activity as a pseudo-slip mode. Within the procedure, Yet another option would be to apply the reorienta- the whole subgrain is reoriented when some condition is tion scheme similar to the one proposed in Reference 53, fulfilled, so that the number of subgrains within the where the orientations of twins are predefined and metagrain does not increase during the simulation. In during the simulations the volume fractions of different the real material, the dislocation substructure can twin variants are measured. In such a variation, the develop inside the twin. Its development is supposed simulation of rolling using the two-scale model with the to be responsible for the high fraction of LABs present Taylor transition could be first performed, wherefrom in experimental data. The 3SCP model is unable to the evolution of volume fractions of each twin variant predict the grain refinement inside the twins. In order to for every metagrain could be obtained. Then, for each do so, the fourth level of microstructure could be added, metagrain, composed of subgrains of the respective [54] for example assuming the composite grain model at variant taken in relevant evolving proportions, the the level of subgrain. This, however, would make the simulation using the VPSC model could be performed model much more numerically costly and is beyond the (with a disabled primary twin mode in subgrains related scope of the present paper. to the respective variant). On this lower level, the METALLURGICAL AND MATERIALS TRANSACTIONS A 94 (a) (b) Fig. 9—Misorientation-axis distributions calculated for the randomly selected pairs of subgrains in the CP Ti rolled to reductions of (a) 20 pct and (b) 90 pct. (a) (b) Fig. 10—Comparison of misorientation distributions calculated for (a) 12 and (b) 1 randomly selected neighbors. secondary twins and intra-twin refinement could be approach, the secondary twins could be also modeled accounted for. Such procedure would be again much during the first simulation using the Taylor model on the more computationally demanding than the one pro- upper micro-macro scale, but in such a case the model posed in the present study, but it should be at least more would probably be more or less as costly as the efficient than the four-scale model. On the other hand, it four-scale model, because the number of metagrains still could not describe the possible refinement inside the times primary twin variants times secondary twin secondary twins. To include such effects in this variants had to be accounted for. METALLURGICAL AND MATERIALS TRANSACTIONS A It is maybe important to note that in the model we between subgrains belonging to the same phase (matrix do not have any information about the relative or a specific twin variant). In such a case, it appears frequencies of boundaries between subgrains belonging that the fraction of low angle grain boundaries to matrix and twins. Such information may be supplied (misorientation angle below 15 deg) is about 100 pct by experiment and used in the model to improve the for reductions up to 60 and about 90 pct for 90 pct procedure of selecting pairs of neighbors for calculat- reduction. To conclude, one can state that the fraction ing misorientation angle distribution. Following this of LABs can be correctly predicted using the 3SCP idea, in order to have greater insight, we have also model if the distributions obtained for matrix-matrix, calculated distributions of misorientation angles matrix-twin and twin-twin boundaries are combined in appropriate proportions. It could be interesting to apply experimental data concerning the relative num- ber of every type of boundary and see to what extent the results of simulation are consistent with experi- mental misorientation distribution. This way one could also assess the error stemming from not including the intra-twin refinement. Figure 12 shows the misorientation distributions calculated for sets of misorientation angles as shown in Figure 8 appended by adding also the sets of misori- entation angles between subgrains belonging to the same phase mentioned in the previous paragraph. It can be seen that the fraction of LABs for 10 and 20 pct thickness reductions is more or less similar to the one observed experimentally. Unfortunately, for reductions of 30 and 40 pct, the fraction of LABs is again lower than the experiment. This can be explained by rising proportion of the intra-phase boundaries to the inter-phase boundaries stemming from ongoing intra- Fig. 11—The schematic view of the primary (in red) and secondary (in green) twins in single titanium grain showing that the number of phase refinement occurring by mechanisms analogous boundaries between secondary twins and matrix is very low. The to the ones existing in FCC materials with higher schematic can be compared with experimental microstructures, as thickness reduction. shown, e.g., in Fig. 2 in Ref. [46] (Color figure online). Fig. 12—The misorientation angle distribution for a sheet of CP Ti cold rolled to a total thickness reduction of 10, 20, 30, and 40 pct obtained in the simulation with the 3SCP model. The distributions are plotted by combining (in equal proportions) the sets of misorientation angles between randomly selected subgrains and angles between subgrains belonging to the same phase (matrix or a given twin variant). METALLURGICAL AND MATERIALS TRANSACTIONS A VI. SUMMARY AND CONCLUSIONS which permits unrestricted use, distribution, and reproduction in any medium, provided you give The aim of the paper was to examine the applicability appropriate credit to the original author(s) and the [8] of the recently proposed 3SCP model to the case of source, provide a link to the Creative Commons grain refinement in HCP material. Within the approach, license, and indicate if changes were made. [50] the probabilistic twin volume consistent scheme was used to account for appearance of twin-related orienta- tions. The simulations of texture evolution and grain APPENDIX: CONVENTIONS USED IN PLOTTING refinement in CP Ti subjected to cold rolling were THE TEXTURES OF HCP METALS AND ALLOYS performed. The results were compared with the exper- imental data available in References 42 through 45.It The stereographic projection of HCP crystals depends was found that the simulated textures were in accor- on the convention used for the denomination of dance with the experimental ones. Concerning the orthogonal coordinate system. The hexagonal system predictions of the misorientation angle distribution, is very convenient for describing the HCP crystals, due the basic components of the distributions observed in to its ability to naturally take into account the crystal experiments in the range of high angle boundaries were symmetries. As this system is not orthogonal, it has to well identified. The peaks are associated with twinning be transformed to the orthogonal one in order to enable activity. However, the application of the 3SCP model to the description of the lattice orientation. Two conven- the case of the HCP material also had difficulties, tions for this transformation are used in the pole namely the fraction of low angle boundaries was figure plotting software: underestimated by the model. Possible solutions to the problem were discussed. To this aim, some kind of a 1. x axis is parallel to a (i ¼ 1; 2; 3, the choice of i is four-scale model could be developed, a different reori- arbitrary because of the crystal symmetry), cf. entation procedure can be applied, or the postprocessing Figure A1(a), of the data can be enhanced using experimental data. 2. y axis is parallel to a , cf. Figure A1(b). To the best of the authors’ knowledge, there is no model that is able to predict misorientation angle z axis is always parallel to the c axis of the crystal, and distributions in the microstructure of HCP metals, and the unspecified axis (y in convention 1 and x in therefore, the results presented in the paper can be convention 2.) is perpendicular to both defined axes. considered as novel. The capabilities of the 3SCP model In order to show the significance of the choice of as compared to other grain refinement models have been convention Figure A2 shows theðÞ 0001 ; 1010 and summed up in Reference 8. First, contrary to the model [21,22] of Leffers, the orientations within the microstruc- 1120 pole figures plotted for a single orientation ture do not have to be predefined. This enables described by Euler angles (0, 0, 0). It is readily seen, that application of the model to study any deformation the choice of convention does not have any influence on process. For example, it has been applied to study theðÞ 0001 pole figure, because the ‘‘c’’ direction is the ECAP in References 8 and 48 and rolling in Reference same in both conventions. The choice of the convention 48 and the present paper. Second, the intra-granular is however very important in case of 1010 and 1120 strain heterogeneity is considered, contrary to models pole figures, namely it leads to their mutual interchange. presented in References 23 through 25. Third, contrary As the choice of the convention is crucial for the pole to References 23 through 26 the number of orientations figures obtained it should be made consciously and during the simulation does not change, which increases explicitly stated in the publication. Such an approach the computational efficiency of the model. In the present could save many misunderstanding across different analysis, it is demonstrated that those advantages are groups of researchers. not lost when the 3SCP model is combined with PTVC reorientation scheme to account for twinning. The proposed approach provides results of statistical char- acter. In the model, there is no information concerning the spatial distribution of subgrains in the metagrain or metagrains in the polycrystalline representative volume. This can be treated as a drawback but also as an asset because the outcomes, when a sufficiently large number of orientations are taken for the analysis, are expected to be statistically representative and not related to the specific microstructure realization. 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