**1. Introduction**

The term cross-rolling is used for different special types of rolling processes [1]. Here, cross-rolling means a modified version of conventional or unidirectional sheet rolling, where the sheet—and consequently, the rolling direction (RD)—is rotated by 90◦ about its normal direction (ND) between subsequent passes [2–5]. The outcome is that a more random orientation distribution, that is, four-fold texture, is obtained, which is usually accompanied by a lower amount of plastic anisotropy. Li et al. applied cross-rolling on magnesium sheets and concluded that large basal plane scatter was achieved, which resulted in a more random texture [2]. Wronski et al. used cross-rolling on low carbon ferritic steel and copper and found that plastic anisotropy was decreased in copper; however, it increased in the low carbon steel [3]. Huh et al. applied cross-rolling to suppress cube texture formation in 5182 aluminum after annealing and observed a close to random structure and better formability [4]. Tang et al. obtained similar results on cross-rolled AZ31 aluminum, and besides negligible earing, they found that increased ductility was also achieved by cross-rolling [5]. Based on the literature, it can be safely stated that, in general, a more random texture and decreased plastic anisotropy is expected in aluminum alloys after cross-rolling.

Besides deep-drawing tests, several methods have been established to estimate earing. Fukui and Kudo found that earing can be predicted from the Lankford value Δr = (r0 + r90)/2 − r45 [6]. Since then, methods have been developed based on mechanical response [7], crystallography [8,9], and more complex theories were established [10]. Nowadays, prediction of earing is usually performed by finite element (FE) methods [11].

Recently, a new, simple method has been developed by the authors to predict earing based on pole figure data [12]. The advantage of the method over others is that qualitative information can be obtained quickly and directly from pole figures. If quantitative data is demanded, only one deep-drawing test is required. Furthermore, it can be combined with non-destructive (sample-cutting-free) texture measurement methods [13]. Thus, it is possible to predict earing in a truly non-destructive manner, which can have high potential in cases when unique, high value objects are to be examined. The aim of the present work is to extend the developed earing prediction method to the case of cross-rolling. For this, the method was applied on unidirectionally and cross-rolled 5056 type aluminum sheets and the results were validated with deep-drawing tests.

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

Sheets made of 5056 type aluminum having an initial thickness of 4 mm have been unidirectionally- (UD) and cross-rolled (CR) with a VonRoll experimental roll stand at the University of Miskolc. Other than the mode of rolling (UD or CR), all applied parameters were the same. Two sets of samples were produced with a di fferent number of passes. The first set of samples (three UD samples, marked as A1, A5, and A6; and three CR samples, marked as A3K, A4K, and A9K) were cold-rolled to a ~1 mm final thickness through 6 passes. Sample A4K was rotated around ND between passes clockwise, while samples A3K and A9K were rotated around ND alternately back and forth prior to passes. The reduction during the first pass was 0.6 mm, and 0.5 mm during the subsequent passes. The final thickness of the UD samples was 1.04, 1.03, and 1. 03 mm; and that of the CR samples was 1.14, 1.13, and 1.13 mm. The second set of samples, rolled with 12 passes (one UD, marked as A10; and one CR, marked as A11K) were cold-rolled to a ~0.99 and 1.10 mm final thickness, respectively. The reduction during the first pass was 0.3 mm and 0.25 during the following passes. Sample A11K was rotated around ND alternately back and forth prior to passes. Table 1 summarizes the properties of the investigated samples.


**Table 1.** The investigated samples and their properties. UD: unidirectionally; CR: cross-rolled.

Samples were cut along the longitudinal sections of the final sheets and prepared for optical microscope examinations with the use of Barker etchant. Optical microscope images were obtained with a Zeiss M1m microscope using polarized light.

Samples with a diameter of 30 mm were cut for texture measurements and samples with 50 mm were cut for deep-drawing tests. Texture examinations were carried out with a Bruker D8 Advance X-ray di ffractometer using CuK α radiation equipped with an Eulerian cradle operating with a 40 kV tube voltage and a 40 mA heating current. {111}, {200}, and {220} pole figures were measured up to CHI = 75◦ sample tilting. Complete recalculated {200} pole figures were obtained with the software of the equipment, TexEval. Orthotropic deformation was used during recalculation for both UD and CR samples.

The applied earing prediction method is described in detail elsewhere [12]. Briefly, recalculated and complete {200} pole figures were obtained by texture analysis from the measured {111}, {200}, and {220} incomplete pole figures. Then, the CHI-cuts, which show the recalculated intensity values versus CHI of the complete {200} pole figures were plotted for each PHI (sample rotation) angle. The method applies the principle according to which the CHI-cuts can be approximated with a sum of Gaussian curves [14]. Thus, the CHI-cuts of the recalculated {200} pole figures were fitted with Gaussian curves. The area of each individual Gaussian curve was determined and then the areas were weighted (multiplied) with the sin of the CHI values corresponding to the peaks of the Gaussian curves. Afterwards, the weighted areas of the Gaussian curves were summarized to the given PHI. The summarized, weighted {200} intensities were plotted versus PHI, which shows the predicted relative cup height.

Deep-drawing tests were performed at the John von Neumann University with Erichsen deep-drawing equipment. The diameter of the samples was 50 mm and that of the drawn cups was 33 mm, giving a drawing ratio of 1.51. The applied blank holder force was 1 kN. Average earing was calculated as the ratio of average ear height and average cup height [12]. Predicted average earing was calculated similarly from the weighted {200} intensity curves and divided by a scaling factor. Since predicted average earing is calculated from measured X-ray intensity and not cup height, the use of a scaling factor is required to obtain comparable values. The determination of the scaling factor is based on comparison to experimental data. The scaling factor was determined by dividing the predicted average earing calculated form weighted {200} intensities with average earing. It was determined to both UD and CR cases. The obtained values were 40 and 60, respectively.
