*Article* **Electron Backscatter Diffraction (EBSD) Analysis of Machinable Lead-Free Brass Alloys: Connecting Texture with Fracture**

**Athanasios Vazdirvanidis, Andreas Rikos, Anagnostis I. Toulfatzis and George A. Pantazopoulos \***

ELKEME Hellenic Research Centre for Metals S.A., 61st km Athens—Lamia National Road, 32011 Oinofyta, Greece; avazdirvanidis@elkeme.vionet.gr (A.V.); arikos@elkeme.vionet.gr (A.R.); atoulfatzis@elkeme.vionet.gr (A.I.T.)

**\*** Correspondence: gpantaz@elkeme.vionet.gr; Tel.: +30-2262-60-4463

**Abstract:** The current paper is related to the study of the microstructure and texture of two machinable lead-free brass alloys, namely CuZn42 (CW510L) and CuZn38As (CW511L), which were evaluated in the as-drawn and post heat treated condition. Electron backscatter diffraction (EBSD) was employed for the examination of the brass rods' crystallographic properties in order to correlate the effect of post processing heat treatment on the evolution of phase structure and texture towards the interpretation of dynamic (impact) fracture properties. It is shown that α- and β-phase volume fractions, mean grain size, and grain boundary misorientation are the most influential factors altering the fracture resistance of single- and dual-phase brass alloy rods. The role of grain boundary engineering, through the formation of coincidence site lattice (CSL) boundaries and their evolution during thermomechanical processing, is of major importance for the design of the mechanical behaviour of new eco-friendly machinable brass alloys.

**Keywords:** brass; machinability; coincidence site lattice; grain boundary; EBSD; lead-free brass

#### **1. Introduction and Literature Review**

Brass alloys (Cu–Zn) are broadly used industrial metallic materials in domestic, mechanical, and electrical engineering as a result of their significant formability, corrosion and mechanical resistance, electrical conductivity, and, of course, high machinability [1–3]. Lead (Pb) is a basic alloying element of conventional brass due to its significant positive effect on machinability, resulting in high lubrication activity at the tool/workpiece interface and intense chip-fracturing, affecting the cutting tool service life and process [4–7]. However, the use of brass as a component in drinking water installations has increased the demands for the elimination of lead toxicity, inaugurating a new class of copper alloys, the eco-friendly or lead-free/low-lead brass alloys. A simple version of lead-free/low-lead machinable alloys include (almost) binary Cu–Zn systems, such as CuZn42 (CW510L) and CuZn38As (CW511L) (see [8–10]). New eco-friendly systems containing various (micro)alloying elements, such as Si, Ti, Bi, Sn, Mg, Sb, and Al aim, in their majority, to offer a significant advantage in chip breaking, replacing Pb with other intermetallic or second-phase particle constituents [11–17]. A recent review, summarizing the entire spectrum of such an extended family of copper alloys, highlighting the various technical approaches and machinability improvement strategies, outlines the necessity of the optimization of the manufacturing process towards the fabrication of eco-friendly components, satisfying the requirement of the relevant environmental and health and safety regulations [18].

Evidently, chip rupture constitutes a significant aspect of machinability performance and it is strongly related to the fracture properties of brass alloys under different loading conditions [7,19]. Therefore, the study of the fracture behaviour of leaded and leadfree/low-lead brasses constitutes a significant element, not only for the insightful understanding of the role of microstructure (e.g., lead particle size and distribution, phase structure, grain structure) on chip breaking, but also for the design and optimization of

**Citation:** Vazdirvanidis, A.; Rikos, A.; Toulfatzis, A.I.; Pantazopoulos, G.A. Electron Backscatter Diffraction (EBSD) Analysis of Machinable Lead-Free Brass Alloys: Connecting Texture with Fracture. *Metals* **2022**, *12*, 569. https://doi.org/10.3390/ met12040569

Academic Editor: Shi-Hoon Choi

Received: 1 March 2022 Accepted: 25 March 2022 Published: 28 March 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

the machinability performance, without compromising the mechanical properties of the fabricated components [20,21].

Failure and embrittlement mechanisms, such as intermediate temperature embrittlement (ITE) and transgranular-to-intergranular fracture, together with properties-processing relationships, have been extensively studied in brass alloys systems (see [22–26]). Among the most typical failure mechanisms, dezincification, stress corrosion cracking (SCC), and liquid metal embrittlement (LME) causing hot-shortness due to localized (intergranular) Pb fusion and grain boundary decohesion are revealed in characteristic case histories studied in the authors' Institute, and representative works are shown in [27–29].

#### **2. Background Information and Research Objectives**

The microstructure of common brass alloys typically consists of one (single-phase, α-brasses) or two (dual-phase, α + β brasses) phases, as is shown by the well-known binary Cu–Zn phase diagram [30]. The α-phase is rich in copper, contains less than 37 wt.% zinc, and has a face-centred-cubic (fcc) crystal structure. Single, α-phase alloys have moderate strength and excellent formability at room temperature due to their low stacking-fault energy (SFE < 10 mJ/m2) (see [31]). Therefore, they easily form twins and they can also form coincidence site lattice (CSL) boundaries.

At a higher zinc content, the β-phase is formed, which possesses a body-centredcubic (bcc) crystal structure. The copper alloys which include a β-phase have excellent hot workability and machinability. The β-phase is also referred to as an intermetallic compound due to the almost 50–50 atomic stoichiometry of Cu and Zn. It is stable at 800 ◦C from 39 to 55 wt.% Zn and at 500 ◦C between 45 and 49 wt.% Zn [30].

Above the temperature of 470 ◦C, Cu and Zn atoms occupy random locations of the lattice, forming a disordered structure which is known as the β-phase, but below this critical temperature thermal vibrations are decreased and atoms take preferential sites, forming a long-range ordered structure known as the β'-phase. This is constituted by smaller regions of ordered structure called domains. Their boundaries are called antiphase boundaries (APBs) and they can be only observed with transmission electron microscopy (TEM). Quenching from the ordered β-phase region leads to fine domain size and increased hardness. The CuZn or β'-phase belongs to B2 super lattice materials, having a structure similar to that of FeAl [30,32].

Extrusion products with variable volume fractions of α- and β-phases can be produced by altering extrusion temperature. Higher extrusion temperatures favour the formation of a β-phase and result in alloys with a lower yield strength if the α-phase is considerably coarsened [25].

It has been shown that grain boundary engineering (GBE) can be performed in fcc metals of low stacking-fault energy (SFE) through thermo-mechanical processing (TMP) in order to improve their mechanical properties. The proposed mechanism is the formation of coincident site lattice (CSL) boundaries, through the development of annealing twin boundaries with low values of Σ, such as Σ3 boundaries, and its variants, Σ9 and Σ27, which together with low-angle grain boundaries (LAGBs; considered to have 5–15 degrees of misorientation) are resistant to intergranular cracking in comparison to the more brittle behaviour shown by high-angle grain boundaries (HAGBs), which have misorientation values larger than 15 degrees [33].

It has been found that one reason for the more brittle behaviour of HAGBs could be their tendency to attract segregated elements, as opposed to LAGBs and the CSL boundaries. When a crack is formed, propagation is preferential through the network of HAGBs, while the fracture mode can be altered from intergranular to transgranular when the crack meets LAGBs and CSL junctions. The amount of CSL boundaries is also linearly increased with decreasing mean grain size, enabling the manufacturing of fracture-resistant alloy products [34].

In dual-phase brass alloys, the grain boundaries' "character" should be evaluated separately for the two distinct phases, since during deformation and post-processing annealing, the α- and β-phases show different tendencies in HAGBs formation, with the α-phase exhibiting a high volume fraction of CSL boundaries, depending on the thermo-mechanical process scheme and the β-phase random values of HAGBs, which are considered of minor importance in GBE.

The current research is a follow-up work, aiming to shed light, using texture analysis, on the alteration of fracture mechanics and failure mechanisms of environmentally friendly brass alloys under different processing conditions, i.e., as-drawn and heat treated [35].

As presented in [35], apart from the benefits in machinability of CW510L (CuZn42) and CW511L (CuZn38As) alloys provided by heat treatment through the increase and stabilization of β-phase content, controversial results were obtained in impact toughness testing:


The fracture surfaces of the respective Charpy specimens have also been studied in detail in [35]. The CuZn42 alloy sample in the as-received condition exhibited a typical, ductile fracture failure mechanism with multiple size dimples, while after the heat treatment it principally showed an intergranular fracture topography, with minute dimples on the grain facets. In the CuZn38As alloy sample, a fully ductile behaviour with multiple size dimples and large, deep voids was observed in the as-received condition as compared to the heat treated condition, where finer and shallower dimples were evident [35].

The present study is exclusively focused on electron backscatter diffraction (EBSD), in order to seek potential fracture–texture relationships that could justify the eminent evolution of fracture behaviour and crack propagation modes in the as-received and heat treated brass alloys. More specifically, further attention was placed on the salient interpretation of the observed "paradox" demonstrated by the CuZn42 sample, described as follows:

*"The CuZn42 alloy after heat treatment exhibited improved fracture toughness, even though (i) the fully β-phase microstructure, established by the heat treatment, is expected to induce to fracture toughness deterioration and (ii) the obtained impact fracture topography presents an almost complete intergranular pattern which microscopically implies to lower impact energy."*

On the contrary, in the CuZn38As sample, the anticipated decrease in impact energy after the heat treatment was expected considering the attained phase transformations and the observed fracture mechanism demonstrated by the obtained fracture morphology which consists of smaller and shallower dimples.

To the best of the authors' knowledge, there is no similar study concerning the investigation and interpretation of the fracture mechanisms of low-lead/lead-free brass alloys under various thermomechanical processing conditions (as-drawn and heat treated), using detailed texture and grain boundary analysis through electron backscatter diffraction (EBSD) analysis.

#### **3. Materials and Methods**

#### *3.1. Brass Alloy Samples*

Two distinct lead-free brass alloys were employed for the present study, namely CuZn42 (CW510L) and CuZn38As (CW511L). The chemical composition of the studied alloy samples is shown in Table 1 (see also [35]).

The samples are originated from 35 mm diameter extruded and drawn rods of two types of lead-free brass alloys, namely CuZn42 (CW510L) and CuZn38As (CW511L). The CuZn42 alloy rod was heat treated at 775 ◦C for 60 min and the CuZn38As at 850 ◦C for 120 min and they were both water quenched for maximization of the β-phase percentage and suppression of α-phase precipitation with the aim to improve machinability. A simple schematic showing the heat treatment processes for the different brass alloys is illustrated in Figure 1. The respective fracture surfaces after Charpy impact testing are shown in Figure 2.


**Table 1.** Chemical composition of the brass alloy rods under examination (analysis by optical emission spectroscopy, elemental contents are expressed in wt.%).

**Figure 1.** Simple schematic diagram showing the performed heat treatment processes (blue blocks for CuZn42 and orange blocks for CuZn38As alloy samples). Heat treatment conditions—CuZn42: 775 ◦C for 60 min followed by water quenching; CuZn38As: 850 ◦C for 120 min followed by water quenching.

**Figure 2.** Macroscopic fracture surfaces after impact fracture toughness tests (Charpy) before and after heat treatment of CW510L and CW511L brass alloys. The average values of the induced impact energies are referred too.

#### *3.2. Electron Backscatter Diffraction (EBSD)*

EBSD analysis was performed in suitably sectioned impact-tested samples using an EDAX Hikari XP camera (EDAX, Mahwah, NJ, USA) mounted on a JEOL IT-800 HL (JEOL Ltd.,Tokyo, Japan) Scanning Electron Microscope (SEM), under 20 kV of accelerating voltage. A tilt of 70 degrees, 4 × 4 binning, and various magnifications and step sizes, depending on the grain size of the samples, prior to and following the heat treatment, were applied to assist in microstructure (phase structure), texture, and grain boundary analysis towards the interpretation of the alterations of fracture mechanisms.

Visualization of the strains caused by the impact tests, near the fracture surfaces, and examination of their distribution on the α- and β-phases, as well as the characterization and calculation of misorientation between adjacent grains, were performed. In the relative statistics, the 15◦ angle value was deployed as a distinction criterion of LAGBs from HAGBs.

Examination of the microstructure (phase structure) and fracture surface profile was performed in cross-sections parallel to the extrusion direction. The automatic collection and analysis of EBSD patterns was performed. Data mapping and information regarding the orientation, phase distribution, grain size, morphology, grain boundary, and local deformation of crystallographic regions was realized by using Orientation Image Mapping Software (OIM Analysis, version, EDAX, Mahwah, NJ, USA).

Coincident site lattice (CSL) boundaries are special character boundaries. These boundaries are classified in terms of Σ-values. The CSL boundaries are considered as special due to the fact that they possess a given fraction of atoms in the grain boundary plane which are coincident to both lattices which are separated by the grain boundary. The Σ-value denotes the fraction of atoms in coincidence. The detection of CSL boundaries is automatically performed by the OIM Analysis software.

The various EBSD settings used for the presented images/maps are summarized in Table 2.

**Table 2.** List of settings (scanned area, magnification, step size) used for the collection of the relevant EBSD patterns.


Two representative EBSD scans per sample from the matrix in the as-drawn and heat treated conditions are shown concerning the phase structure examination. The following maps were acquired from each scan (the results are presented in Section 4.1):


One additional scan per sample of the fracture surface profile after Charpy impact testing in the as-drawn and heat treated conditions was performed and the retrieved results were as follows (the results are presented in Section 4.2):


#### **4. Results**

#### *4.1. Microstructure and Texture Characterization*

4.1.1. CuZn42—As-Drawn Condition

The texture and volume fraction percentages of the α- and β-phases in CuZn42 were determined by EBSD (see Figure 3 and Table 3). A (111) fibre structure was exhibited by the α-phase and a single (101) fibre with minor (335) preferred orientations structure was observed by the β-phase (see Figure 3a,b). The results showed a 46 vol.% fraction for the fcc (α-phase) and a 54 vol.% fraction for the bcc (β-phase) (see Figure 3d).

**Figure 3.** CuZn42 alloy in the as-drawn condition. (**a**) Inverse pole figure (IPF-Z) map and (**b**) texture plots. (**c**) Misorientation and (**d**) phase maps. The α-phase is precipitated in various morphologies from the β-grains. Note: CuZn (β-phase), Cu2Zn (α-phase). Unit: μm.

The α-phase appeared in various forms, mainly exhibiting an allotriomorphic morphology nucleated on the β-phase grain boundaries, but also as coarse intra-crystalline "islands", typical of a slow cooling rate from a hot working temperature (Figure 3a). The mean grain size of both the α-phase and β-phases was 25 μm. A large fraction (20%) of the α-phase grains exhibited thermal or annealing twins. The microstructure was characterized by the presence of almost exclusively high angle grain boundaries (see Figure 3c and Table 4). No sub-grains (2–5 degrees) were detected, while low angle boundaries (5–15 degrees) were almost null and were located exclusively on α–α interface boundaries (see the green lines in Figure 3c). The high angle boundaries included 19% CSL boundaries, consisting of 10% Σ3 and the remaining 9% distributed to other CSL categories. All CSL

high angle boundaries were related to the α-phase, while approximately half of them were located on α–β interphase boundaries, with the other half on α–α boundaries.

**Table 3.** Results of EBSD scans, % phase volume fractions, mean grain size, % twinned grains, and mechanical testing data.


**Table 4.** Results of EBSD scans, distribution of grain boundaries' angles (%).


4.1.2. CuZn42—Heat Treated Condition

After the heat treatment, the microstructure consisted merely of macroscopic size (several thousand μm, i.e., mm) β-phase grains, while any α-phase precipitation had been suppressed by the rapid water quenching (Figure 4 and Table 3). A double (101) and (113) fibre structure was exhibited by the single β-phase alloy (Figure 4b). Exclusively high angle boundaries were observed (Figure 4c). The β-phase exhibited poor quality in EBSD scans in most of the grains, which rendered difficulties in processing the respective maps. This could be attributed to (i) the disorder–order transformation which led to fine ordered domain size and multiple orientations within a single grain or (ii) due to the minute lattice parameters deviation between the β- and β'-phase, which could not be resolved by the EBSD technique [36]. Interestingly, the examination showed that approximately 20% of the grain boundaries were CSL, with the higher percentage being Σ3 and the remainder being Σ7, Σ15, and Σ21 (Table 4). The coarse grain size is an indication of the exaggerated grain growth which took place during the heat treatment of the brass rod.

The quality of the β-phase regions in the EBSD maps in the as-drawn condition was higher than in the heat treated conditions, since any quenching effects resulting in fine domain sizes were avoided.

**Figure 4.** CuZn42 alloy in the heat treated condition. (**a**) Inverse pole figure (IPF-Z) map and (**b**) texture plot. (**c**) Misorientation map. The poor quality of the map was attributed to fine domain size within the macroscopic β-phase grains. Unit: μm.

#### 4.1.3. CuZn38As—As-Drawn Condition

The texture and microstructure of CuZn38As in the as-drawn condition is shown in Figure 5 and the summary of the results are listed in Tables 3 and 4. The material was strongly textured with the (001) direction oriented parallel to the extrusion/drawing direction (Figure 5a,b). The microstructure consisted of partially recrystallized, equiaxed α-phase grains with a 14 μm mean grain size with a 98 vol.% fraction, while the β-phase was aligned parallel to the extrusion direction (longitudinal sections were performed). The volume fraction of the β-phase was 2% and its mean grain size was 3 μm (Figure 5d). An approximate 52% of α-phase grains exhibited thermal twins. A total of 1% of the grain boundaries constituted subgrain boundaries (2–5 degrees), 17% consisted of LAGBs, and the remaining 82% consisted of HAGBs (Figure 5c). The subgrain and LAGBs were detected within α-phase regions and could be considered as an indication of a fully recovered and partially recrystallized microstructure. The CSL fraction constituted 33% of the total boundaries, which were mainly encountered in the α-phase twins and on α–β interface boundaries in equal amounts. Approximately 22% of the total boundaries were Σ3, 3% were Σ5, and the remaining 8% was distributed to other categories (see Table 4).

#### 4.1.4. CuZn38As—Heat Treated Condition

The texture and microstructure of CuZn38As in the heat treated condition are shown in Figure 6 and the results are summarized in Tables 3 and 4. The α-phase exhibited a (001) and (335) double fibre texture with equal amounts of both directions while the β-phase was strongly textured with prevailing (101) directions and minor (112) (see Figure 6a,b).

**Figure 5.** CuZn38As alloy in the as-drawn condition. (**a**) Inverse pole figure (IPF-Z) map and (**b**) texture plot. (**c**) Misorientation and (**d**) phase map. Note: CuZn (β-phase), Cu2Zn (α-phase). Unit: μm.

**Figure 6.** CuZn38As alloy in the heat treated condition. (**a**) Inverse pole figure (IPF-Z) map and (**b**) texture plot. (**c**) Misorientation and (**d**) phases map. Note: CuZn (β-phase), Cu2Zn (α-phase). Unit: μm.

After the heat treatment, the α-phase grains were coarsened, with a 23 μm mean grain size, and the fraction of the twinned grains was reduced to 21% (see Figure 6 and Tables 3 and 4). The β-phase percentage was increased to 14% and the banding morphology was altered since a more uniform distribution of the β-phase was created. The β-phase mean grain size was 6 μm, which was double the size of that prior to the heat treatment condition, and it exhibited a plate-like, elongated morphology (Figure 6d). The grain boundaries' character was also significantly altered by the heat treatment, i.e., 22% of the total boundaries were sub-grain boundaries, 7% were LAGBs, and the remaining 71% were HAGBs. Among them, 21% was characterized as CSL boundaries, with Σ3 and Σ49 occupying 8% of the total boundaries each, Σ5 and Σ9 occupying 3% in total, and the remainder being distributed to other categories (Table 4).

#### *4.2. Microstructure Characterization of the Impact Specimens' Fracture Surface Profile* 4.2.1. Sample CuZn42—As-Drawn and Heat Treated Conditions

The profile of the fracture surface of the impact test specimen in the as-drawn condition is shown in Figure 7a. The existence of micro-dimples was apparent as various size protrusions, not exceeding 2 μm. These protrusions did not exhibit the same amount of misorientation values in the KAM map, the latter being dependent on the phase where they were formed. The higher values were found by dimples formed within α-phase regions on the fracture surface or within the β-phase having a short distance from the α-phase, at maximum up to 10 μm. The α-phase network presented high misorientation values in the KAM map at even higher distances from the fracture surface.

**Figure 7.** Phase and KAM maps of the fracture surface profile of Charpy impact tested CuZn42 alloy samples: (**a**) phase map, (**b**) KAM map in the as-drawn condition, and (**c**) KAM map in the heat treated condition. Unit: μm.

In Figure 7c, the fracture surface profile of the CuZn42 alloy sample after the heat treatment is shown. It is typical of the occurrence of a brittle, intergranular fracture, with the presence of mechanical twins formed during the impact test. Mechanical twins can be produced in bcc metals under conditions of shock loading in decreased temperatures [31]. The "noise" in the map did not allow a definite recognition of deformed and non-deformed areas, but other than the angle boundaries which were high angle and lead to local high

misorientation values, a uniform distribution of lattice rotations was observed, close and far from the fracture surface.

In Figure 8, lens-shaped twins are shown extending across the width of the grain (deformation twins do not extend to adjacent grains). This is the main existing deformation mechanism exhibited by the coarse, single β-phase brass alloy, since the crack tip created a limited plastic zone size compared to the grain size, hindering the activation of multiple sets of slip systems in adjacent grains, and dislocation emission was impeded by the existing microstructural barriers, causing dislocation pile-up to the activated slip systems and grain boundaries [37]. Twinning behaviour in bcc alloys is considered important in orientation changes that enable the activation of new slip systems in order to allow deformation by slip. In the intergranular crack, shown in Figure 8, on both the fracture surface and across the secondary crack, no appreciable deformation has occurred, leading to the hypothesis that mechanical twinning produced by the shock loading was the predominant deformation mechanism, explaining the reason for the uniform misorientation values for the same grain close and far from the fracture surface.

**Figure 8.** Mechanical twins developed on the fracture surface of β-phase grains after Charpy impact test; CuZn42 alloy sample in heat treated condition, section transverse to fracture surface. Combined image quality and IPF map. Unit: μm.

#### 4.2.2. CuZn38As—As-Drawn and Heat Treated Conditions

The fracture surface profile of the sample in the as-drawn condition is shown in Figure 9a,b. It manifested appreciable ductility and the coarse formation of dimples reaching a size of 30 μm, which is multiple times the size of the dimples of the CuZn42 alloy in the same metallurgical condition. The equiaxed α-phase grains exhibited extensive strain markings close to the fracture surface as a contributing mechanism to the energy absorbance potential (Figure 10a). These constitute very fine deformation twins which are observed as a major deformation mechanism in low stacking-fault energy alloys such as the <sup>α</sup>-brass (20 mJ·m<sup>−</sup>2) [38]. The fine grain size of the <sup>α</sup>-phase grains contributed to improved ductility and the respective high energy values in the Charpy impact test (104 J).

In the heat treated condition, the fracture surface was also deformed, showing the presence of shallower dimples reaching a maximum size of 17 μm, since the more resistant to deformation β-phase did not allow the development of coarser dimples (Figure 9c,d). The α-phase grain structure, which was coarser (23 μm) than in the as-drawn condition, negatively affected the material's deformation capacity during the impact test, leading to a lower energy absorption (84 J). Strain markings in the α-phase were also developed close to the fracture surface and were more readily observed in the coarser α-phase grains, especially in the heat treated condition (see Figure 10).

**Figure 9.** Phase and KAM maps of the fracture surface profile of Charpy impact tested CuZn38As alloy samples in the (**a**,**b**) as-drawn and (**c**,**d**) heat treated conditions. Unit: μm.

**Figure 10.** Strain markings close to the fracture surface of (**a**) CuZn38As as-drawn and (**b**) CuZn38As heat treated alloy sample. EBSD image quality map. Unit: μm.

#### **5. Discussion**

The presented results, concerning the fracture behaviour of brass alloys, can be summarized and classified in two main categories: single-phase alloys and dual-phase (α + β) alloys.

#### *5.1. Single-Phase Alloys: CuZn42, Heat Treated and CuZn38As, As-Drawn Condition*

The CuZn38As sample, in the as-drawn condition, possessed a higher amount of αphase (98 vol.%: it can be considered for simplicity purposes as a "single-phase" alloy) and a finer grain size, rendering a high deformation and energy absorbance capacity. Moreover, it contained the higher fraction of LAGBs and CSL boundaries (~50% in total), which promote ductile, transgranular fracture modes. The high extent of strain markings formation was an additional indication of a strained material with a high energy absorbance capacity.

The opposite condition was observed by the CuZn42 alloy sample in the heat treated condition, which showed a single β-phase structure and macroscopic grain size. These attributes favoured the development of mainly brittle intergranular cracking.

However, the average impact energy value for CuZn42 at the heat treated condition presented a slight increasing tendency (from 47 to 52 J; i.e., approximately 10%), compared with the same alloy at the as-drawn condition. The fact that CSL boundaries were detected (~20%) also led to the appearance of a minor transgranular crack propagation mode in the heat treated CuZn42 alloy. In addition, the occurrence of mechanical twinning constituted the dominant deformation mechanism of this sample with appreciable strain energy absorption during the shock loading. The incremental tendency of impact energy could potentially be explained by the following phenomena:


In addition, the higher impact energy can be also supported by the higher hardness of the heat treated CuZn42 (fully β-phase) as compared to the as-drawn one, i.e., approximately 140 HV vs. 130 HV [35].

Grain boundaries are mostly influential on dynamic and static mechanical properties behaviour and can act as preferred locations for crack initiation and propagation. HAGBs hinder dislocation motion as a result of their high degree of misorientation, leading to dislocation pile-up and localized stress concentration that can induce failure [39]. In contrast, the slip systems of neighbour grains at LAGBs possess a higher alignment which favours dislocation mobility and retain slip homogeneity across the grain boundaries. Conclusively, LAGBs resist intergranular fracture, which tends to propagate along HAGBs [39].

#### *5.2. Dual-Phase (α + β) Alloys: CuZn38As, Heat Treated Condition and CuZn42, As-Drawn Condition*

Concerning the samples of CuZn42 in the as-drawn condition and CuZn38As in the heat treated condition, which constitute dual-phase alloys, the significant difference in energy absorbance capacity and dimple size, which are indicators of higher plasticity (for CuZn38As), seem to be related to the following factors (see also Tables 3 and 4):


#### **6. Conclusions and Further Research**

From the crystallographic examination of the single- and dual-phase brass rods, the following conclusions could be derived:


β-phase volume fraction inherits a negative effect on impact toughness for dual-phase brasses (α + β brasses).


The fact that CSL boundaries were also observed in the single β-phase CuZn42 heat treated alloy, constituted mostly by Σ3 boundaries, offers a promising ground for further research since the investigation of the influence of texture on single β-phase properties will provide valuable knowledge towards the development of advanced copper alloys with superior mechanical behaviour and machinability.

**Author Contributions:** Conceptualization, A.V. and G.A.P.; primary materials testing and methodology, A.V., A.R. and A.I.T.; sample preparation, A.R.; writing—original draft preparation, A.V.; writing—review and editing, A.I.T. and G.A.P.; project supervision and management, G.A.P. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data supporting the findings of this study are available upon request from the corresponding author.

**Acknowledgments:** The encouragement and support of ELKEME management is highly appreciated.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


**Shih-Kang Kuo 1, Yi-Liang Ou <sup>1</sup> and Dung-An Wang 2,3,\***


**Abstract:** An analytical model of a steel strip under alternate bending/reverse bending during a roller leveling process is developed. A combined isotropic/kinematic hardening model is implemented through a combined hardening parameter. A formulation of the change of the effective stress as a function of the change of the effective strain under cyclic loading is combined with the developed analytical model to predict the stress distributions and residual curvature of a steel strip under roller leveling efficiently and accurately. Dissimilar to the commonly used assumption of one contact point between the stripe and the rolls, an effective radius modelling the wrap-around contact characteristics is proposed. An arc contact of the strip around a roll is described by the contact model. An oscillatory behavior of the residual curvature is observed when a range of roll intermesh setting is considered. The contact model added to the analytical model may enhance the accuracy in predicting the oscillatory behavior of the residual curvatures. A range of the roll intermesh setting can be suggested by the developed model to obtain a flat strip after roller leveling.

**Keywords:** roller leveling; steel strip; combined hardening; analytical model

#### **1. Introduction**

Steel strips after rolling and annealing may possess defects such as wavy edges, center buckles, cambers, and twists. The flatness defects can be corrected by leveling in steel mills. Amor et al. [1] reported that shape defects of metal strips after rolling process or coiling operation can be removed by leveling in order to meet the quality requirement. Customers may cut the as-received flat stripes in order to achieve end functionalities. Unacceptable shape defects found after cutting the strips are serious issues of concern raised by the customers of steel mills. Li et al. [2] reported that these flatness/shape defects are mainly due to residual stresses generated from rolling, annealing, leveling, coiling, etc. Morris et al. [3] classified the shape defects into two categories—latent and manifest. Shape defects of the latent category appear flat prior to cutting operation, whereas manifest defects, such as wavy edges and center buckles, are visible in the coiled form. Conventionally, tension levelers are used upstream to remove manifest defects. Roller leveling with lower line tension and smaller rolls has been the essential step to flatten metal strips and attenuate the effects of inhomogeneous distribution of residual stress in strips. Roller leveling is a complex forming process involving multiple, alternate bending and reverse bending cycles. In order to achieve more effective leveling operation without resorting to on site trial and error approach, a lot of efforts have been devoted to the development of efficient, accurate simulation tools for roller leveling.

Finite element analyses and analytical modeling are two main approaches adopted to investigate the roller leveling process. Hira et al. [4] developed an analytical model for

**Citation:** Kuo, S.-K.; Ou, Y.-L.; Wang, D.-A. An Analytical Model for Stress and Curvature Prediction of a Strip Leveling Process. *Metals* **2022**, *12*, 757. https://doi.org/10.3390/ met12050757

Academic Editors: George A. Pantazopoulos and Zhengyi Jiang

Received: 14 March 2022 Accepted: 24 April 2022 Published: 28 April 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

calculation of curling and residual stress of a strip in longitudinal and width directions after tension leveling. Isotopic material hardening was considered in their model. Doege et al. [5] analyzed a levelling process using the Euler–Bernoulli beam theory and a combined isotropic and kinematic hardening model. Contact points between the strip and the rolls were computed iteratively assuming that only one contact point between the roll and the strip exists. They calculated the range of roll intermeshes for zero strip curvature at the exit roll for a seven-roll leveling system. Details of the verification of their analytical modeling were not described. Behrens et al. [6] developed an analytical model to find suitable settings of a leveler for flat strips. Contact points between the metal strip and the rolls were calculated by assuming that the curve of the strip was composed of arcs and common tangents between two rolls. Lengths of several segments dissected from the strip longitudinally were equalized with different plastic deformations to eliminate wave defects of the strip. Dratz et al. [7] also adopted this common tangent contact condition to develop their roller leveling model. Residual stress and curvature results of the strip after levelling were not reported. The common tangent assumption contradicts with the fact that the curvature between two rolls exhibits a nonlinear distribution. Liu et al. [8] developed an analytical model to analyze the residual stress and curvature of a plate subjected to roller leveling. Isotropic hardening was adopted in their model and a single contact between the plate and each roll was assumed. Chen et al. [9] developed an analytical model of a roller leveler for the strip with transverse and longitudinal wave defects. They divided the strip longitudinally in order to find a suitable bending amount of the rolls to attain equal length of each longitudinal section. Higo et al. [10] presented a theoretical model for a roller leveler based on Euler–Bernoulli beam theory and a single point contact model between a strip and each roll. They examined the influence of the exit roll intermesh on the residual curvature of the strip.

Morris et al. [11] built a finite element model to analyze the deformed shape and surface residual stress of a steel strip after a three-roll leveling process. They concluded that kinematic work-hardening can represent the loading condition experienced during tension levelling more closely than that of isotropic work-hardening by comparing model predictions of the deformed shapes with experiments. Schleinzer and Fischer [12] presented a finite element model for roller leveling of rails. Combined isotropic/kinematic hardening was assumed for the material. A friction coefficient of 0.2 was used between the rail and the rolls. Park and Hwang [13] conducted finite element analyses and experiments to examine the residual curvature and stress distribution of strips after roller leveling. They verified that the strips with various initial curvatures can attain residual curvatures with small deviation given an optimum entrance roll intermesh. Huh et al. [14] built a finite element model for a roller leveler in order to study the effects of process parameters on the curvature solution of roller leveling. They identified the parameters with significant effects on the residual curvature through a design of experiment approach. Roberts et al. [15] carried out a finite element study to compare modeled curvature distributions with experiments of a two-roll stretch-bending process. Their results showed that a reduction in the wrap-around contact length was caused by decreasing intermesh. Jin et al. [16] conducted a three-dimensional finite element analysis of a roller leveler. They presented residual stress distribution in the strip. Kim et al. [17] developed a finite element model for a roller leveling process. They implemented an implicit stress integration procedure in a constitutive material model which can capture the material yield behavior and the Bauschinger effect during reverse bending. Grüber et al. [18] employed a finite element analysis of roller levelling to determine several combinations of roll intermeshes to attain a flat strip after levelling. They identified that the roll intermesh of the center load triangle and the last load triangle of a seven-roll leveler is crucial to reach desired values of flatness and residual stress distribution.

Leveler operators have relied on parameter settings for strips with various yield strengths and through-gauges provided by leveler suppliers. Due to degradation of the leveler through years of usage, empirical methods were used to fine tune the settings for production. Although manifest defects appeared to be invisible after leveling, latent defects may still exist and cause the end products to fail to satisfy customer requirements. Analytical solutions of residual stress and curvature of the roller leveling process are necessary to provide rapid parameter settings to aid in its operation. In this investigation, an analytical model for a nine-roll roller leveler is developed. The nine-roll cold leveler was installed at the plate mill plant of the Chinese Steel Company for leveling heavy gauge steel plates. One of the motivations of this investigation is to provide operational parameters for the leveler to improve the quality of steel plates. Solutions of stress distributions after each bending/reverse bending and residual curvature of the strip are obtained based on the Euler–Bernoulli beam theory within a two-dimensional geometric framework. The material hardening is implemented through a combined isotropic/kinematic hardening parameter. Finite element analyses are carried out to verify the accuracy of the developed model. Results of the stress distributions and residual curvature of a steel strip are compared with the results based on the analyses. Finally, a roll intermesh range to produce a flatness condition of the strip is presented.

#### **2. Analytical Model**

Figure 1 is a schematic of a nine-roll leveler. The strip has the thickness *t*, the length *L*, and the width *W*. The rolls have a radius of *R* and a roll spacing of *D*. A strip enters the entry roll from the left, undergoes alternate bending and reverse bending, then leaves the exit roll. Flatness defects can be removed by the plastic deformation under the leveler rolls. Partial reduction in the flatness defect is ensued by insufficient plastic deformation. The intermesh settings, usually in a linear, declining trend, provide for the gradual flattening of the shape defects as the strip travels through the leveler. The roll intermeshes *p*2, *p*4, *p*6, and *p*<sup>8</sup> are indicated in Figure 1. The value of the roll intermesh is positive when the gap between the top roll and the bottom roll is larger than the strip thickness. In this investigation, the amount of the roll intermesh is defined by the inclination angle *θ* of the upper roll carriage and the roll intermesh at the next-to-last roll *p*<sup>8</sup> as indicated in Figure 1. The bottom rolls are fixed, and the top rolls are moved downward to specified roll intermeshes. The width of the strip is assumed to be sufficiently large. Therefore, the strain in the width direction is neglected, and the plane strain condition is considered. Since the leveling process can be viewed as a series of three-point bending/reverse bending actions with groups of three rolls, the analytical model of the roller leveling process is conducted based on the framework of Euler–Bernoulli beam theory.

**Figure 1.** A schematic of a nine-roll leveler.

#### *2.1. Material Model*

In development of the analytical model, friction forces at the roll/strip interface, tension forces in the strip, and gravity are neglected. Figure 2 schematically shows a differential element of the strip. The strip has a thickness of *t* and a width of *W*. A Cartesian coordinate system is also shown in the figure. *x* represents the longitudinal direction, and *y* is along the thickness direction. The neutral axis is assumed to coincide with the mid-surface of the strip. The origin of the *y* coordinate is at the middle of the beam as

shown in the figure. *ρ* is the radius of the curvature of the differential element under the moment loading *M*. Longitudinal strain *ε<sup>x</sup>* is expressed as

$$
\varepsilon\_x = -\kappa y \tag{1}
$$

where *ε<sup>x</sup>* and *κ* [1/m] are the longitudinal strain and the curvature, respectively. In the elastic regime, the longitudinal stress *σ<sup>x</sup>* [N/m2] is related to the longitudinal strain *ε<sup>x</sup>* by

$$
\sigma\_{\mathbf{x}} = E \varepsilon\_{\mathbf{x}} \tag{2}
$$

where *E* [N/m2] is the Young's modulus. The material hardening behavior is given by an exponential law

$$
\overline{\sigma} = \sigma\_s + Q\_i \left( 1 - e^{-b\_i \overline{\varepsilon}} \right) \tag{3}
$$

where *σ* [N/m2] and *ε* are the effective stress and the effective plastic strain, respectively. *σ<sup>s</sup>* [N/m2] is the initial yield stress. *Qi* [N/m2] and *bi* are the isotropic hardening parameters, where *Qi* specifies the maximum change in the size of the yield surface, and *bi* defines the rate at which the size of the yield surface changes as plastic straining develops. For uniaxial loadings as the strip bending case, *σ* and *ε* can be taken as *σ<sup>x</sup>* and *εx*, respectively. Relatively large plastic deformation is considered for the roller levelling. This simplification of *ε* ≈ *ε<sup>x</sup>* contributes to a very small fraction of error to the results. Values of (*Q*1, *b*1) and (*Q*2, *b*2) are obtained by fitting experimental stress–strain curve with Equation (3) for the first half cycle and the second half cycle of the loading, respectively. Therefore, *Q*<sup>1</sup> and *b*<sup>1</sup> are used to model the material hardening behavior in the first half loading cycle. *Q*<sup>2</sup> and *b*<sup>2</sup> are adopted for the subsequent loadings. The bending moment *M* [N·m] of the cross section of the strip is given as

**Figure 2.** A schematic of a differential element of the strip.

The sign convention for the moment *M* and the curvature *κ* is related to the orientation of the coordinate axes.

During the first bend loading, the material of the strip exhibits isotropic hardening behavior. In the subsequent bend loadings, the material hardening of the strip is taken as a combined isotropic/kinematic type. Zhang et al. [19] formulated the change of the effective stress Δ*σ* [N/m2] as a function of the change of the effective strain Δ*ε* under cyclic bend loading as

$$|\Delta\overline{\sigma}| = \left\{ \begin{array}{c} E|\Delta\overline{\varepsilon}| \\ \sigma\_{\varepsilon} + Q\_{i} \left( 1 - e^{-b\_{i} \left( |\Delta\overline{\varepsilon}| - \frac{|\Delta\overline{\sigma}|\_{\text{lim}}}{E} \right)} \right) \\ \end{array} , |\Delta\overline{\varepsilon}| \ge \frac{|\Delta\overline{\sigma}|\_{\text{lim}}}{E} \end{array} \tag{5}$$

where <sup>|</sup>Δ*σlim*<sup>|</sup> [N/m2] is the elastic limit of the effective stress change at the stress reversal point and can be written as

$$|\Delta \overline{\sigma}\_{lim}| = (1+m)|\overline{\sigma}\_r| + (1-m)(2\sigma\_s - |\overline{\sigma}\_r|) \tag{6}$$

where <sup>|</sup>*σr*<sup>|</sup> [N/m2] is the magnitude of the effective stress at the point of stress reversal. *m* is the combined hardening coefficient. *m* = 1 and *m* = 0 correspond to the isotropic hardening case and kinematic hardening case, respectively, and 0 < *m* < 1 is for the case of combined hardening. The value of *m* for each bending during the roller leveling is calibrated by matching the stress–strain curves based on the model of Equations (1)–(6) to the experimental measurements in the corresponding cycles over a strain range expected in the roller leveling process. Kotov et al. [20] experimentally demonstrated the validity of a kinematic hardening model of a steel strip under roller leveling.

#### *2.2. Contact Model*

Contact points between the strip and the rolls can be computed iteratively assuming that only one contact point between the roll and the strip exists [5,10]. Behrens et al. [6] and Dratz et al. [7] adopted the common tangent contact condition to develop their roller leveling model. The single point contact and common tangent contact assumptions render a simple means to model the contact between the strip and rolls. A line contact between the strip and each roll is assumed in the initial modeling work here. In the two-dimensional model considered in this investigation, the strip contacts with each roll tangentially as shown in Figure 3. The expanded radius of the roll *Re* [m] is given as

**Figure 3.** Tangential contact condition of the strip with the rolls.

The tangent of the contact angle *λ<sup>i</sup>* [rad] is the gradient of the longitudinal axis of the strip

$$
tan\lambda\_{\bar{i}} = \left.\frac{dy}{d\mathfrak{x}}\right|\_{x\_{\bar{i}}}\tag{8}$$

where *y*(*x*) is the deflected curve of the longitudinal axis of the strip, (*xi*,*yi*) is the *i*th contact point, and *λ<sup>i</sup>* is the *i*th contact angle. The contact point (*xi*, *yi*) [m] is

$$\begin{aligned} x\_i &= x\_\varepsilon + R\_\varepsilon \sin \lambda\_i \\ y\_i &= y\_\varepsilon - R\_\varepsilon \cos \lambda\_i \end{aligned} \tag{9}$$

where (*xc*, *yc*) [m] is the coordinates of the center of the roll. Müller et al. [21] also used the expanded radius of the roll to calculate the contact point between a strip and a roll in a leveler.

A recursive scheme is adopted to calculate the curvature, bending moment, deflected curve of the strip, and the contact points. Initially, the location of the contact points and the curvature *κ<sup>i</sup>* of the strip curve at the *i*th contact point are assumed. The bending moment *Mi* at the *i*th contact point is computed using Equations (1)–(6). Based on the assumption of a linear distribution of the moment between the contact points, the curvature of the deflected strip curve is estimated. Using the curvature distribution *κ*(*x*), the deflected strip curve *y* [m] is calculated by

$$y(\mathbf{x}) = \kappa(\mathbf{x})d\mathbf{x} + \mathbf{C}\_1\mathbf{x} + \mathbf{C}\_2 \tag{10}$$

where *C*<sup>1</sup> and *C*<sup>2</sup> are the integration constants, and can be determined by the known contact points. A Newton–Raphson algorithm is used to obtain converged solutions of the deflected strip curve, moment distribution, curvature distribution, and contact points that satisfy both the Euler–Bernoulli beam theory and the geometrical constraints. Guan et al. [22] adopted a similar curvature integration approach to find the contact points for a roller leveler. They verified their model by comparing the deflection curves calculated by their model with those from experiments.

#### *2.3. Residual Curvature*

The flatness of the strip after exiting the leveler is related to the residual curvature. The bending and reverse bending cycle is repeated until the material reaches the last roll in the leveler. The strip is free to rotate at the last roll where the bending moment applied to the strip can be considered as zero. The internal stress distribution of the cross section of the strip at the last roll should result in a zero-bending moment. This condition can be enforced by imposing an artificial bending moment of the same magnitude but opposite sign to the moment at the last roll before unloading. Assuming no reverse yielding during unloading, the residual curvature *κ* [1/m] of the strip at the last roll can be expressed as

$$
\kappa' = \kappa - \frac{M}{EI} \tag{11}
$$

where *κ* and *M* are the curvature and moment, respectively, before unloading, and *I* is the second moment of inertia of the cross section of the strip. Hosford and Caddell [23] and Guan et al. [24] also adopted this approach to calculate the residual curvature for beam bending problems.

#### **3. Finite Element Analysis**

In order to examine the accuracy of the developed analytical model, a two-dimensional finite element analysis of a roller leveler is carried out. A quasi-static condition is assumed in the finite element analyses. Due to the sufficiently large width, the strain in the width direction can be neglected, and the plane strain condition is considered in the finite element analysis. Both the upper and lower rolls are modeled as rigid bodies. A strip and a nine-roll leveler are shown in Figure 1. A Cartesian coordinate system is also shown in the figure. The direction of the *z* axis is given by the right-hand rule. In this investigation, the strip is moved forward by a displacement boundary condition and the bottom rolls are assumed to be fixed in the *x* and *y* directions and free to rotate with respect to the *z* axis. The displacements in the −*y* direction of the upper rolls are specified to represent the amount of roll intermeshes, while their displacements in the *x* direction are constrained. As shown in Figure 1, a uniform displacement is applied in the +*x* direction to the right edge surface of the strip, and the displacement in the *y* direction for the right edge surface is constrained to represent the strip travel horizontally along the +*x* direction.

The commercial software, Abaqus (6.14, Dassault Systemes, Walthem, MA, USA), is adopted to compute the stress and deformation of the strip during the leveling process. Mises yield surface is used with the nonlinear isotropic/kinematic hardening model in Abaqus. The nonlinear kinematic hardening component is modeled through the back stress to describe the translation of the yield surface. The isotropic hardening component is modeled through the equivalent stress as a function of the equivalent plastic strain to define the size of the yield surface. The hardening law for the back stress *α* [N/m2] is

$$
\dot{\alpha} = \frac{\mathbb{C}}{\sigma\_t} (\sigma\_\text{x} - \alpha) \dot{\varepsilon}^{pl} - \gamma a \dot{\varepsilon}^{pl} \tag{12}
$$

where *C* [N/m2] and *γ* are kinematic hardening parameters that are calibrated from symmetric strain, cyclic test data. *σ<sup>t</sup>* and · *ε pl* are the current size of the yield surface and the plastic strain rate, respectively. Figure 4 schematically shows a stabilized cycle. The plastic strain *εpl* is determined by

$$
\varepsilon^{pl} = \varepsilon\_x - \frac{\sigma\_x}{E} - \varepsilon\_0 \tag{13}
$$

where *ε*<sup>0</sup> is the strain value of the intercept of the left half of the stabilized cycle with the strain axis as shown in Figure 4. The value of *α* [N/m2] is given as

$$
\alpha = \sigma\_{\overline{x}} - \sigma\_{\overline{t}} \tag{14}
$$

where *σ<sup>t</sup>* = (*σ*<sup>1</sup> + *σn*)/2 is the current size of the yield surface. The stresses *σ*<sup>1</sup> and *σ<sup>n</sup>* are marked in Figure 4. To solve the back stress hardening law of Equation (12) over this stabilized cycle, with the first data point of *σ<sup>x</sup>* = *σ*<sup>1</sup> and *εpi* = 0, the expression of *α* [N/m2] is

$$\alpha = \frac{\mathbb{C}}{\gamma} \left( 1 - e^{-\gamma \varepsilon^{pl}} \right) + (\sigma\_1 - \sigma\_t) \varepsilon^{-\gamma pl} \tag{15}$$

**Figure 4.** A schematic of a stabilized cycle.

Data pairs of *α*, *εpl* extracted from the stabilized stress–strain curve are used to calibrate values of the kinematic hardening parameters *C* and *γ*.

The isotropic hardening model of the Abaqus is defined as

$$
\sigma\_t = \sigma\_s + Q\_\infty \left( 1 - e^{-b\tilde{\mathbf{v}}^{pl}} \right) \tag{16}
$$

where *σ<sup>t</sup>* [N/m2] and *σ<sup>s</sup>* [N/m2] are the current size and the initial size of the yield surface, respectively, and *εpl* is the equivalent plastic strain. *Q*<sup>∞</sup> [N/m2] and *bi* are the isotropic hardening parameters of the finite element model. The isotropic hardening parameters are calibrated from the data of the current size of the yield surface *σt*-equivalent plastic strain *εpl* relation based on a symmetric strain-controlled cyclic experiment with strain range Δ*ε* as shown schematically in Figure 4. For the *i*th cycle, the current size of the yield surface *σ<sup>t</sup>* [N/m2] is calculated by

$$
\sigma\_{\mathbf{i}} = \sigma\_{\mathbf{i}}^p - \mathfrak{a}\_{\mathbf{i}} \tag{17}
$$

where *σ<sup>p</sup> <sup>i</sup>* [N/m2] and *<sup>α</sup><sup>i</sup>* [N/m2] are the peak tensile stress and the back stress, respectively, in the *i*th cycle. *α<sup>i</sup>* [N/m2] is calculated by

$$\alpha\_i = \frac{\sigma\_i^p + \sigma\_i^n}{2} \tag{18}$$

where *σ<sup>n</sup> <sup>i</sup>* [N/m2] is the compressive stress with the same *<sup>ε</sup>pl* value as the peak tensile stress *σp <sup>i</sup>* in the *i*th cycle. Since the value of the back stress *α<sup>i</sup>* in each cycle at a particular strain level is nearly the same based on the model, *<sup>α</sup><sup>i</sup>* is approximated by the values of *<sup>σ</sup><sup>p</sup>* <sup>1</sup> and *<sup>σ</sup><sup>n</sup>* 1 . The equivalent plastic strain *εpl* corresponding to the *i*th cycle is

$$\mathbb{E}^{pl} = \frac{4i - 3}{2} \Delta \varepsilon^{pl} \tag{19}$$

where <sup>Δ</sup>*εpl* can be approximated by <sup>Δ</sup>*<sup>ε</sup>* <sup>−</sup> <sup>2</sup>*σ<sup>p</sup>* <sup>1</sup> /*E*.

Figure 5a is a mesh of the finite element model. Two-dimensional plane strain 4-noded CPE4R element is employed in the model. The total number of elements is 8188. Figure 5b is a close-up view of the mesh near a work roll. Ten elements are used in the thickness direction of the strip in order to obtain an accurate solution of the stress–strain distribution. The "analytical rigid surface" in Abaqus is used to model the rolls. Frictionless contact is assumed at the interface between the strip and the rolls.

**Figure 5.** (**a**) A finite element model. (**b**) A close-up view of the mesh near a roll.

#### **4. Analyses, Results, and Discussions**

*4.1. Analysis and Results of the Analytical Model*

Figure 6 shows the stress–strain curves of a mild steel under a uniaxial, symmetric strain-controlled, cyclic test. The strain range Δ*ε* is 0.015. Based on the first half of the stress–strain curve of the first cycle, the calibrated values of the Young's modulus *E* and the initial yield stress *σ<sup>s</sup>* are 219.8 GPa and 329.7 MPa, respectively, and *Q*<sup>1</sup> and *b*<sup>1</sup> have values of 66.3 MPa and 742.7, respectively. The values of *Q*<sup>2</sup> and *b*<sup>2</sup> are calibrated as 128.3 MPa and 604.7, respectively, based on the second half of the stress–strain curve of the first cycle.

**Figure 6.** Stress–strain curves of a uniaxial, symmetric strain-controlled, cyclic test.

By specifying the combined hardening coefficient *m*, computational stress–strain curves based on the analytical model of Equations (1)–(11) with the strain range Δ*ε* of 0.0015 can be obtained. For a nine-roll roller leveler, the total number of bends equals seven. The values of the combined hardening coefficient *m* for the seven bends can be calibrated by comparing the computational stress–strain curves with the stress–strain curves of the mild steel shown in Figure 6 for the first seven bend loadings. Figure 7 shows the computational stress–strain curves and the stress–strain curves of Figure 6. The computational curves agree with those of the mild steel. The calibrated *m* values for the second to the seventh bending are 0.0784, 0.2, 0.3, 0.4, 0.45, and 0.5, respectively. The hardening behavior of the material for the first bending is taken as isotropic.

**Figure 7.** Stress–strain curves for calibration of the combined hardening coefficients.

Consider the case where the strip has the thickness *t* (=20 mm), the length *L* (=2480 mm), and the width *W* (=100 mm). The radius *R* of the rolls is 85 mm. The roll spacing *D* is 150 mm. The strip is taken as flat and free of residual stress. The upper roll carriage has an inclination angle *θ* of 0.1◦ and the roll intermesh at the next to the last roll *p*<sup>8</sup> ranges from −1.2 mm to 0.8 mm. Note that the value of the roll intermesh is positive when the gap between the top

roll and the bottom roll is larger than the strip thickness. Therefore, a negative value of the intermesh means the work roll plunges into the strip.

With *p*<sup>8</sup> = −0.70 mm and the upper roll carriage inclined at the angle *θ* of 0.1◦, *p*2, *p*4, and *p*<sup>6</sup> have values of −2.32 mm, −1.78 mm, and −1.24 mm, respectively. Sixty divisions are taken along the thickness direction of the strip for calculation of the stress and strain distributions. Figure 8 shows the deformed strip between roll 1 and roll 9 based on the analytical model. Figure 9a–c shows the distribution of the bending moment per unit width, the curvature, and the deformed center line of the strip, respectively, between roll 1 and roll 9. The contact points between the strip and the rolls are marked by circles in the figure. One hundred nodes are used between the contact points in the computations. Figure 9a shows the linear distribution of the bending moment between the contact points. At the entry roll and the exit roll, the values of the bending moment are zero. Figure 9b shows the curvature distribution. Values of the curvatures are nearly two orders smaller than the roll curvature 1.176 × <sup>10</sup>−<sup>2</sup> mm<sup>−</sup>1. The curvature of the strip at the exit roll has a value of 3.054 × <sup>10</sup>−<sup>5</sup> mm<sup>−</sup>1. The positive value of the curvature indicates that the strip is bent upward. Higo et al. [10] pointed out that the abrupt increase in magnitude of the curvature near all contact points except the final one is due to the nonlinear material hardening behavior. This is evidenced by the nonlinear sections of the moment–curvature curves during the leveling process plotted in Figure 10. The moment and the curvature are normalized by the maximum bending moment *M*<sup>0</sup> and the maximum curvature *C*<sup>0</sup> for which elastic conditions hold, respectively.

**Figure 8.** The deformed strip between roll 1 and roll 9.

**Figure 9.** Distributions of (**a**) bending moment per unit width, (**b**) curvature, and (**c**) deformed center line of the strip between roll 1 and roll 9.

**Figure 10.** Normalized moment as a function of normalized curvature during leveling.

Figure 11a–c shows the stress distributions in the strip thickness direction when the strip travels through the leveler. The results computed by the analytical model are displayed by lines in the figure. Significant plastic deformations appear in the first three bends as seen in Figure 11a. The fractions of the plastic deformation are nearly 40%, 50%, and 60% for the first, the second, and the third bend, respectively. As the strip traverses through the fifth roll and the sixth roll, the region of plastic deformation stays at 60% as seen in Figure 11b. Figure 11c shows the stress distributions in the strip thickness direction at the eighth roll and the exit roll. The stress distribution at the exit roll can be taken as the residual stress distribution since the total moment applied to the strip at the exit roll is nearly zero. The stress at the *i*th roll is calculated based on a linear superposition assumption. The remnant stress of the cross section of the strip at the (*i* − 1)th roll and the loading stress at the *i*th roll are superimposed to obtain the stress distribution at the *i*th roll. Guan et al. [25] also adopted this stress inheritance law in their roller leveling model. Yonetani [26] reported that the stress of a microscopic segment at the cross section in a uniaxial stress state satisfies the linear superposition assumption. The uniaxial stress loading condition is also assumed in the model considered in this investigation.

#### *4.2. Analysis and Results of the Finite Element Model*

Finite element analyses are carried out to verify the accuracy of the analytical model. The material properties for the mild steel used in the finite element analyses are listed in Table 1. The isotropic hardening parameters, *Q*∞ and *b*, and the kinematic hardening parameters, *C* and *γ*, are calibrated from the uniaxial, symmetric strain-controlled, cyclic test with the strain range Δ*ε* of 0.015 as shown in Figure 6. The calibration procedure is described in Section 3.

**Table 1.** Material properties employed in the finite element analyses.


The strip is taken as flat and free of residual stress in the beginning of the leveling process. Smith [27] reported that the initially flat condition is useful in the leveling analysis since the incoming flatness defects of a strip may vary from one location to another, and a strip generally has some initially flat area. Figure 12a shows the initial configuration of the finite element model. A Cartesian coordinate system is also shown in the figure. A displacement of 1200 mm in the +*x* direction is given to the right edge surface of the

strip. The displacement equals the distance from the center of roll 1 to the center of roll 9. Therefore, the cross-section A (marked in Figure 12a) travels from roll 1 at the beginning of the analysis to roll 9 at the end of the analysis. The *y*-displacement of the nodes at the right edge surface is constrained to represent the horizontal motion of the strip. Figure 12b shows the deformed shape of the mesh when the cross-section A reaches roll 9. Then, the constraints at the right edge surface of the strip are released for the strip to spring back. Figure 12c shows the deformed shape of the mesh after spring-back.

**Figure 11.** Stress distributions of the strip at (**a**) rolls 2, 3, and 4; (**b**) rolls 5, 6, and 7; (**c**) rolls 8 and 9.

**Figure 12.** (**a**) The initial configuration, (**b**) a deformed configuration before spring-back, and (**c**) a deformed configuration after spring-back of the finite element model.

The stress distributions of the cross-section A at rolls 2–4, rolls 5–7, and rolls 8–9 based on the finite element analyses are shown in Figure 11a–c, respectively. The results based on the finite element analyses (FEA) are plotted with the markers in the figure. As shown in the figure, the analytical results agree with the results of finite element analyses. The discrepancy can be attributed to different contact conditions between the analytical model and the finite element model. Multiple contact points between the strip and each roll are observed in the finite element model. However, a single contact point between the strip and each roll is assumed in the analytical model.

#### *4.3. Residual Curvature*

Flatness is an important factor to evaluate the strip quality after roller leveling processes. The residual curvature of the strip at the exit roll can be used as a metric to evaluate the strip flatness after leveling. Figure 13 shows the residual curvatures *κ* of the strip as a function of the averaged intermesh based on the analytical model and the finite element analyses. The averaged intermesh is the averaged value of the roll intermeshes, *p*2, *p*4, *p*6, and *p*8, where *p*<sup>8</sup> ranges from −1.2 mm to 0.8 mm. The residual curvature *κ* is an indicator of the deviation of the strip from an initially flat surface at the entry roll. The curve with the dash-dot line and the markers represent the results based on the analytical model and the finite element analyses, respectively. The analytical predictions generally agree with the finite element analyses for the averaged intermeshes ranging from −1.71 mm to 0. As the averaged intermesh decreases from zero, the values of *κ* decreases. *κ* appears to exhibit an oscillatory behavior when the averaged intermesh is less than −0.11 mm based on the analytical model. As the averaged intermesh decreases further, the amplitude of the oscillation of *κ* grows. When the averaged intermesh is less than −1.41 mm (analytical predictions), *κ* has positive values of increasing magnitude. Five crossover points are observed at the averaged intermesh of −0.36 mm, −0.61 mm, −0.86 mm, −1.11 mm, and −1.41 mm (analytical predictions). Smith [27] reported that the several crossover points with zero residual curvature underlie the reason why successful leveling can be achieved by the series roll leveling process in practice.

In the analytical model, a point contact is assumed between the strip and each roll, which means the strip does not wind around the work rolls. This contradicts with the fact that multi-point contact between the strip and the roll predicted in the finite element analyses. Morris et al. [3] reported that the wrap angle near the exit roll has a significant influence on the flatness of the strip. Wrap-around contact length between the strip and the roll may depend on the intermesh and roll spacing. In describing the arc of contact of the strip around a roll, an effective radius can be assumed to model the wrap-around contact characteristics. The concept of the effective radius is illustrated schematically in Figure 14. Figure 14a shows the original contact model, where the strip contacts with roll *i*

tangentially. *Re* is the expanded radius of the roll, which is defined by Equation (7). (*xi*,*yi*) is the *i*th contact point, and *λ<sup>i</sup>* is the *i*th contact angle. Figure 14b shows that the strip makes contact elastically with roll *i* when subjected to an external force *Fi*. The circumferences of the deformed roll and the original roll are represented by the solid line and the dashed line, respectively, in Figure 14b. (*x i*,*y i* ) is the *i*th contact point, and *λ <sup>i</sup>* is the *i*th contact angle for the Hertz contact model. A local deformation is ensued to cause a reduction in the local radius of roll *i*. The effective radius of the roll *Rs* is given as

$$R\_{\mathbb{S}} = R\_{\mathbb{S}} - sF\_{\mathbb{I}} \tag{20}$$

where *s* is the contact compliance of roll *i*. The external force *Fi* is computed from the moment distribution in the analytical model. Based on the Hertz contact model, the contact point (*x <sup>i</sup>*, *y i* ) is

$$\begin{aligned} \mathbf{x'}\_i &= \mathbf{x}\_c + \mathbf{R}\_s \sin \lambda'\_i \\ \mathbf{y'}\_i &= \mathbf{y}\_c - \mathbf{R}\_s \cos \lambda'\_i \end{aligned} \tag{21}$$

where (*xc*, *yc*) is the coordinates of the center of the roll. As shown in Figure 14b, the deformed center line of the strip based on the analytical model with the Hertz contact compliance may be thought to have an arc segment bounded by the two virtual contact points, *U* and *V*, wrapped around the circumference of the original roll *i*, shown as a dashed circle in Figure 14b. When the contact angle *λ <sup>i</sup>* is very small which is the case in the roller leveler, the line segment and the arc segment bounded by the two end points *U* and *V* are approximately equal. The contact compliance of the rolls is taken as a fitting parameter in the analytical model with the Hertz contact compliance to fit the analytical predictions with the finite element computations. Yi et al. [28] considered the wrap angle on a roll during a roller leveling process by fitting an arc curve around a roll. A parameter determined by experiments is needed for curve fitting.

**Figure 13.** Residual curvature as a function of the averaged intermesh.

The residual curvatures *κ* of the strip calculated by the analytical model with the Hertz contact compliance are plotted in Figure 13. The curves with the compliance *s* = 0 and 4 × <sup>10</sup>−<sup>7</sup> mm/N appear to envelop the results based on the finite element analyses. The curve with the compliance *<sup>s</sup>* = 2 × <sup>10</sup>−<sup>7</sup> mm/N may be able to predict the general trend of the results based on the finite element analyses for the averaged intermesh ranging from 0 mm to −2 mm. The wrap-around contact condition between the strip and the rolls can be manifested by the analytical model with the Hertz contact compliance within an acceptable accuracy, compared with the results of the finite element analyses.

**Figure 14.** Illustration of (**a**) the original contact model; (**b**) the Hertz contact model.

A series of simulations was run to calculate the residual curvatures *κ* of the strip with an initial curvature *κ*<sup>0</sup> based on the analytical model with the Hertz contact compliance *<sup>s</sup>* = 2 × <sup>10</sup>−<sup>7</sup> mm/N. The initial stress in the strip is neglected without losing generality of the residual curvature predictions. Mathieu et al. [29] considered initial flatness defects in their finite element analyses of a leveling process. They introduced the flatness defects in the strip which was free of stress. Figure 15 shows the residual curvature as a function of the averaged intermesh for the strip with the initial curvature *κ*<sup>0</sup> varying between −<sup>1</sup> × <sup>10</sup>−<sup>4</sup> mm−<sup>1</sup> to 1 × <sup>10</sup>−<sup>4</sup> mm<sup>−</sup>1. As shown in Figure 15, at low levels of roll intermesh (averaged intermeshes greater than −0.1), the residual curvatures *κ* for the three cases of *<sup>κ</sup>*<sup>0</sup> <sup>=</sup> −<sup>1</sup> × <sup>10</sup>−<sup>4</sup> mm<sup>−</sup>1, 0, and 1 × <sup>10</sup>−<sup>4</sup> mm−<sup>1</sup> deviate from each other significantly. For the averaged intermesh in this level, the curve for the case of *<sup>κ</sup>*<sup>0</sup> <sup>=</sup> −<sup>1</sup> × <sup>10</sup>−<sup>4</sup> mm−<sup>1</sup> oscillates mostly in the positive-curvature region (*κ* > 0), in contrast to the cases of *κ*<sup>0</sup> = 0 and 1 × <sup>10</sup>−<sup>4</sup> mm<sup>−</sup>1, which oscillate between the positive *<sup>κ</sup>* region and the negative *<sup>κ</sup>* region. The initially bowed-down defect for the strip with *<sup>κ</sup>*<sup>0</sup> <sup>=</sup>−<sup>1</sup> × <sup>10</sup>−<sup>4</sup> mm−<sup>1</sup> exits the leveler with the residual curvature in the same direction for the averaged intermesh greater than −0.71. As the extent of the averaged intermesh increases, the residual curvatures for the three cases of *<sup>κ</sup>*<sup>0</sup> <sup>=</sup> −<sup>1</sup> × <sup>10</sup>−<sup>4</sup> mm<sup>−</sup>1, 0, and 1 × <sup>10</sup>−<sup>4</sup> mm−<sup>1</sup> gradually converge to the same values. For the values of the averaged intermesh less than −1.1, the curves for the three cases appear identical, where two cross over points with the interpolated averaged intermesh values of −1.46 and −1.13 were found. For the values of the averaged intermesh less than −1.46, positive residual curvatures were produced. For the averaged intermesh values within the interval of −1.46, −1.13, the minimum of the residual curvature is −0.27 × <sup>10</sup><sup>4</sup> mm<sup>−</sup>1. In this region of the leveler settings, the residual curvatures of the strip seem to be insensitive to its initial curvatures. This result can serve to the advantage of leveler operators to obtain nearly zero residual curvature for strips with various initial curvatures. Grüber et al. [18] also demonstrated robustness of the roll intermesh settings for a roller leveler regarding a change in the initial curvature. In this investigation, the plane strain condition is considered in the finite element analyses. The width-to-thickness ratio of the strip considered in the analyses is 5. Carvalho et al. [30] reported that, in order to develop near plain strain conditions, it is important to maintain a ratio of width-to-thickness greater than 5.

A combined isotropic/kinematic hardening is implemented to describe the material hardening of the strip in this investigation. Doege et al. [5] also adopted combined isotropic/kinematic hardening for their leveling model, where mathematical formulations of their hardening model were not presented. Detailed formulations of the hardening model are provided in this investigation. Doege et al. [5] presented analysis results of their model. Results of the stress distributions and residual curvature of a steel strip based on our analytical model are verified by the finite element analyses. Doege et al. [5] computed the contact points between the strip and the rolls by assuming only one contact point

between the strip and each roll. An effective radius modelling the wrap-around contact characteristics by the Hertz contact compliance is proposed to describe the arc of contact of the strip around a roll. A roll inter-mesh range to produce a flatness condition of the strip is presented based on the analytical model with the Hertz contact compliance.

**Figure 15.** Residual curvature as a function of the averaged intermesh for the strip with an initial curvature.

Indeed, the initial curvature considered in the model is the longitudinal wave defect of a strip. Behrens et al. [6] sectioned a strip longitudinally and showed that the length of all sections after leveling should be the same in order to remove a transverse wave defect. Therefore, bendable rolls, as practiced in the industry, can be applied into a leveler to cause various degree of plastic deformation in each longitudinal section to achieve equal length. Chen et al. [9] developed an analytical model of a roller leveler with consideration to the bending of the rollers to eliminate transverse wave defects. This approach can be implemented in the analytical model to extend its applicability. Park and Hwang [13] slit a strip longitudinally to calculate the initial curvature of each longitudinal section. Given proper roll intermesh settings, the longitudinal sections with various initial curvatures can reach similar values of residual curvature after leveling based on finite element analyses and experiments. The results shown in Figure 15 based on the developed model also provide evidence that the residual curvatures of the strips with different initial curvatures can converge to the same value given enough amount of roll intermesh.

Given the multiple forming processes and complex machine settings involved in the roller leveling, tradeoffs between a simple, efficient model and an elaborate, detailed model should be balanced. Baumgart et al. [31] described that the effects of parts of the leveler, such as support rolls, frames, posts, and adjustment screws, should also be considered in order to obtain a more accurate leveler model. Wang and Li [32] reported that stiffness of roll cassettes and leveler housing are important factors of the leveling process. In this study, a relatively simple analytical model was developed based on the assumptions of two-dimensional geometry, pure bending of the strip, and the uniaxial loading condition. Compared to previously reported models, the Hertz contact compliance implemented in the model relaxed the single point contact condition between the strip and the roll, and a relatively accurate prediction of the residual curvature can be attained. The model could serve as a guide in the development of strategies for effective adjustment of roller levelers.

#### **5. Conclusions**

An analytical model for analyzing the residual stress distribution and residual curvatures of a strip during roller leveling is developed. The model verified by the finite element analyses is based on a simplified, two-dimensional geometric framework. The combined

isotropic and kinematic material hardening model is implemented through a combined hardening parameter. The stress distributions after each bending/reverse bending and the residual curvature at the exit roll subjected to various leveler settings can be obtained by the developed model. The model was further enhanced by considering a Hertz contact model to eliminate the discrepancies between the model predictions and the finite element analyses. Based on the analytical model with the Hertz contact compliance, a range of the leveler settings can be determined to robustly obtain a strip with a nearly zero curvature at the exit roll.

Analytical models with combined isotropic/kinematic material hardening for roller leveling have been reported previously. A formulation of the change of the effective stress as a function of the change of the effective strain under cyclic bend loading is adopted in the developed analytical model with the combined hardening. The model is efficient and accurate in predicting the stress distributions and residual curvature of a steel strip. Unlike the assumption of a single contact point between the strip and the rolls, an effective radius modelling the wrap-around contact characteristics by the Hertz contact compliance is proposed in this investigation. An arc contact of the strip around a roll is described by the Hertz contact model. A roll intermesh range to produce a flatness condition of the strip can be obtained by the analytical model with the Hertz contact compliance.

**Author Contributions:** Conceptualization, S.-K.K. and D.-A.W.; methodology, D.-A.W.; software, Y.-L.O.; validation, D.-A.W.; formal analysis, D.-A.W.; resources, S.-K.K.; data curation, D.-A.W.; writing—original draft preparation, D.-A.W.; writing—review and editing, D.-A.W.; funding acquisition, S.-K.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by China Steel Corporation, grant number 106T7D-RE18 and the Ministry of Science and Technology, Taiwan, grant number MOST 106-2221-E-005-038.

**Data Availability Statement:** Data presented in this article are available at request from the corresponding author.

**Acknowledgments:** The authors would like to express their appreciation to the National Center for High-Performance Computing (NCHC), Taiwan for their assistance.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

