**1. Introduction**

Metals are polycrystalline aggregates composed of numerous grains that can be considered perfect crystals if the materials are fully recrystallized. In order to predict the behavior of polycrystalline materials subjected to macroscopic load, the relationship between microstructures and properties should be known, and therefore, physically based modeling is of key importance. The understanding of microstructural changes is particularly important in setting up the thermomechanical processing (TMP) route. In TMP, the as-cast microstructure is subjected to continuous microstructural transformation. In the case of flat-rolled products, such as Al sheets, this transformation is conditioned by both deformation and recrystallization phenomena, since after casting, the material is subjected to hot rolling, which is followed by cold rolling, and finally the sheet is recrystallized during the final annealing process.

Various modeling approaches that employ diverse length scales (Figure 1) have been developed to reveal the behavior of materials during TMP [1]. It is a common practice to employ phenomenological models for the optimization of properties in groups of metals that do not reveal substantial varieties in chemical composition [1,2]. This strategy likewise works for tailoring a certain metal grade for a particular application. The main disadvantage of phenomenological approximations is that the model parameters need to be recalibrated in the case of a metal grade change or if the TMP quantitative indicators are varied in a wider range of processing windows. In more advanced numerical formulations, such as finite element modeling (FEM), materials are considered as continua, and thus, the mesoscopic

phenomena, occurring on the grain levels, are not accounted for in this engineering length scale (Figure 1). Nevertheless, FEM is successfully employed in many fields, especially when there is a need to simulate the response of a system subjected to mechanical load. On the other end of the length-scale spectrum, macroscopic properties are derived from interatomic potentials and by engaging basic principles of molecular thermodynamics, which ensures generic information about the relationships between atomic arrangements and macroscopic properties; however, this formulation is applicable only for ideal systems. The application of atomistic length-scale simulations to metals is restricted due to limitations in computational resources and also because of the fact that polycrystalline structures contain too many crystal defects even in a fully recrystallized state (typically 10<sup>8</sup> m<sup>−</sup>2).

**Figure 1.** Variety of length scales involved in modeling the microstructure–properties relationship.

Since a 1-mm-thick metallic sheet (A4 size (21 cm × 29.7 cm)) contains approximately 4 billion grains of various crystal orientations, assuming that the average grain size is 25 μm, it becomes clear that neither engineering (continuum) nor atomistic scales are capable of revealing the true nature of mesoscopic transformation involved in TMP. To capture the essence of microstructural changes in industrially produced materials, both mean-field and full-field crystal plasticity (CP) models are employed [1,3–9]. Either approach implements a certain homogenization scheme insofar as performing calculations for the above-mentioned 4 billion crystals, consisting of even smaller entities, would require enormous computational power. In CP, homogenization implies a characteristic length scale (mesoscopic) represented by grains of polycrystalline aggregate, and these entities are treated as perfect crystals (the in-grain heterogeneities are usually not taken into account). The mesoscopic scale tends to fall between two ends of the characteristic length spectrum, and this level of order is important from a practical perspective, as many of the known properties are controlled by the grain size and crystallographic texture. CP approaches treat the polycrystalline material as a continuum at the level of the crystal because the grain size is order of magnitudes larger compared to the size of molecular domains.

Full-field models [1,3,4] allow for analyzing not only the microscopic and macroscopic responses of material but also account for the evolution of heterogeneities at both levels. Although these methods give rise to a more comprehensive representation of a microstructure and provide detailed information on the deformation flow on meso- and macrolength scales, their practical implementation is limited because of extensive computational time. In contrast, the mean-field models are more computationally efficient, but they have fewer degrees of freedom compared to the full-field methods. For instance, in mean-field approximations [5–9], among other things, each grain interacts with a neighboring one in a way defined by the homogenization scheme, and the misorientation in the grain developed during deformation cannot be captured. Even though some microstructural features are ignored in Taylor-type homogenization CP formulations (such as "visco-plastic self-consistent (VPSC) [7], advanced Lamel (Alamel) [5,6], Cluster V [8], etc.), simulations on a mesoscopic scale are still capable of providing generic knowledge on the relationship between the structure, properties, and performance of a polycrystalline aggregate. Polycrystal models have been successfully employed in simulating the evolution of texture in metals (subjected to diverse strain modes), calculating the plastic strain ratio as well as predicting the cup-earing [5–11].

Among the many homogenization theories, Alamel [5,6] was employed in this study to simulate the texture evolution in an Al alloy. In this mean-field approach, the grain assembly is subdivided into a number of clusters consisting of two grains, whereas the interaction inside the cluster is governed by the randomly chosen grain boundary orientation. For a given pair of grains, the equilibrium of stresses is accomplished via the strain compatibility. Similarly to the above-listed Taylor-type models [5–9], the grain pair is subjected to a strain mode identical to the macroscopic one. In crystal plasticity formulations (including Alamel), the evolution of texture in face centered cubic (FCC) metals during cold deformation is related to the crystallographic slip on octahedral slip systems, whereas the macroscopic strain mode is approximated by various models. In order to make the application of crystal plasticity modeling practically attainable, the CP should be coupled with the computationally effective approach, which is capable of accurate prediction of deformation flow in a material. When it comes to rolling, the simplest approximation is the plane strain compression, which disregards many aspects of the process. Because of its simplicity, this strain mode is often used in simulations and tends to provide a reasonable estimate of overall (bulk) texture evolution [5,6,8,10]. In FEM, the e ffect of many technological parameters on the deformation flow can be taken into account, while the displacement history is accurately calculated for various materials at diverse temperatures; however, the model set-up and the calculation procedure are time-consuming. In view of this, deformation is often approximated by analytical solutions [12–18] such as flow-line models, which are capable of capturing many aspects of the process; however, the practical implementations of these computationally e ffective approaches are limited by fitting parameters, which have to be derived from the experimental data for each particular case.

Modeling the evolution of deformation texture in Al alloys by combining the basic principles of CP theories and models that are capable of predicting deformation flow is vital, since many crystallographic features evolved in deformation are directly linked to the evolution of recrystallization texture [19–21]. The latter determines plastic anisotropy and limits the forming characteristics of a material.

Even though many e fforts have been made to model texture development during cold rolling, there are nevertheless many aspects of texture evolution that are still not explained by existing models. The goal of this contribution was to employ computationally e fficient and accurate numerical approaches that could ensure quantitatively reasonable texture simulations for the cold rolling process. For this, a well-established CP model (Alamel) [5,6] was coupled with the recently developed FLM [14]. To make the CP simulations practically applicable, a correlation was defined between the FLM model parameters, the roll gap geometry, and the friction coe fficient. This implies that combined Alamel–FLM-based modeling can be performed without a fitting procedure, allowing for analyses of the e ffect of technological quantitative indicators on texture evolution in cold rolling. It is shown that the Alamel–FLM texture simulations revealed both qualitative and quantitative resemblance to the counterparts calculated with the deformation history obtained by means of finite element modeling. The quality of the texture simulations was estimated by comparing the simulated textures to the experimentally measured ones.

#### **2. Materials and Computational Methods**

In the current investigation, the evolution of crystallographic texture was studied in an Al–Mg–Si Al alloy from 6016 series. The sheet (1.125 mm thick) was subjected to a 29.6% thickness reduction in a single pass with a roll diameter of 128.9 mm. Prior to cold rolling, the material was annealed at 550 ◦C to ensure a fully recrystallized microstructure. Both recrystallized (initial) and deformed (rolled) samples were subjected to through-thickness texture measurements by means of the electron backscattering di ffraction (EBSD) technique. The orientation data were collected and analyzed by commercial OIM-TSL-8 ® software (EDAX Inc., Mahwah, NJ, USA).

Sample preparation for orientation imaging microscopy (OIM) was performed according to the standard procedure, which comprises mechanical grinding and polishing as well as electrolytic polishing. The mechanical polishing procedure was finished with DiaDuo-2 Struers ®-type 1-μm diamond paste. Electrolytic polishing, as a final step of sample preparation for EBSD, was conducted for ~1 min at a voltage of 18 V with A2 Struers® electrolyte and was cooled to temperatures ranging between −5 and 0 ◦C.

A Hikari-type EBSD detector was attached to a scanning electron microscope (SEM) FEI®. The OIM data of the investigated material with a fully recrystallized microstructure were collected at an acceleration voltage of 20 kV, whereas in the cold-rolled sample, to avoid overlap of the acquired pattern with ones originating from the deeper layers, the acquisition was performed at a comparatively low acceleration voltage of SEM. An application of 15 kV guaranteed appropriate OIM data acquisition. During OIM measurements, the sample was 70◦ tilted with respect to the EBSD detector. EBSD mapping was performed on a hexagonal scan grid in the plane perpendicular to the sample transverse direction (TD-plane) extending over the entire thickness of the investigated sample. For accurate texture evaluation, the deformed sample was scanned at a step size of 1.5 μm, while the recrystallized material was investigated at a 5-μm step. In order to ensure a meaningful comparison between the experimentally measured and simulated orientation distribution functions (ODFs), the data were postprocessed with MTM-FHM software [22], and the textures were displayed in ϕ2 = const. sections.

The evolution of the cold-rolling texture was analyzed by a well-established Taylor-type homogenization approach called Alamel, the comprehensive formulation of which can be found elsewhere [5,6]. The simulations were performed by taking into account {111}<110> octahedral slip systems, which are typically activated in FCC metals during cold-forming processes. Prior to texture simulations, a homogenization of experimentally measured EBSD data, which involves the determination of the representative volume element, had to be performed. In the current case, the measured OIM data were converted to a continuous ODF and afterwards discretized to a set of 8000 equally weighted orientations.

To enable the calculation of texture evolution through the Taylor-type homogenization scheme employed, the deformation history in the form of strain velocity gradients should be provided to the given CP model. Both a two-dimensional finite element method (Deform 2D®) and a recently developed flow-line model (FLM) [14] were employed in the calculation of deformation flow. These approximations are capable of capturing strain heterogeneities across the thickness of a cold-rolled sheet. In the FLM and FEM simulations, the material subjected to rolling is considered to be plastic and isotropic. In FEM, the strain hardening phenomenon is described with a stress–strain curve fitted by piecewise linear segments, whereas the rolls are considered to be fully rigid objects. Contrary to FEM, the FLM employed neglects the strain hardening effect. The following material parameters were used in the FEM simulations for the isotropic aluminum matrix: *E* = 68.9 GPa (Young's modulus), ν = 0.33 (Poisson's ratio), and <sup>σ</sup>y = 80 MPa (yield stress). Since rolling is a symmetric deformation process, the deformation flow was simulated for the following layers: surface, 1/5, 2/5, 3/5, and 4/5 of the half-thickness and midthickness plane. The extracted outputs were used as an input for CP calculations. The results obtained for the FEM-CP and FLM-CP simulations were compared to each other and likewise to their experimentally measured counterparts.
