refinement, Mechanical properties.

UFG materials would definitely play an important role in the future of industrialization on Earth due to their unique mechanical and physical properties. There are several techniques and methods through which these kinds of materials were produced [1]. The SPD process, widely regarded as an effective technique for producing Nanostructured/UFG materials, has enabled their production. In the SPD process, the grain size of materials is reduced to the nanoscale, thereby improving mechanical properties. According to the Hall-Petch relation, a reduction in grain size normally increases material strength.

The fabricated materials produced through the bottom-up approach are rarely used in industry due to their porous nature. Conversely, in the case of a top-down process, nanostructured materials are fabricated using plastic deformation processes. The fabricated materials produced by such severe plastic deformations play an important role owing to their non-porous nature, high strength and toughness, and suitable dimensions for performing mechanical/physical tests. To date, several techniques have been introduced for fabricating ultrafine-grained materials. The most suitable technique depends on the type of materials used for the fabrication process. However, some of these techniques, such as equal channel angular pressing [2], [3] and high-pressure torsion [4], [5] can be used only for bulk metals. In contrast, other techniques, including repetitive corrugation and straightening [6], [7], accumulative roll bonding [8], [9] constrained Groove rolling [10], and Constrained Groove Pressing (CGP), are suitable for producing nanocrystalline sheets. Among all these techniques, the CGP method was first introduced by Shin et al. [11] and is exclusively designed for sheet metals, with several advantages compared to other SPD techniques.

In the CGP process, the sheets undergo considerable shear deformation using grooves and flat dies alternately [12]. The imposition of this amount of shear plastic deformation on the sheet results in enhanced mechanical properties of the sheet material, including strength, mean hardness, and grain refinement, as reported in recent studies [13–15].

According to recent experimental studies, there have been notable grain refinement and enhanced mechanical properties of the material [12], [13], [15]. In recent times, the CGP process has been studied using different materials, including aluminum [14], copper sheets [16], and brass [17]. It is necessary to note that finite element analyses and experimental studies have been conducted to investigate the performance of various new processes, such as the semi-CGP process [18] and the rubber pad CGP process [19]. Moreover, comparative studies have been conducted to assess the performance of the RCS and CGP processes, and the results clearly indicate that the CGP process is superior to the repetitive corrugation and straightening process [20].

Several experiments have demonstrated that CGP is a reliable technique for optimizing the microstructure and improving the mechanical properties of low-carbon steel sheets. According to Alihosseini and Dehghani [21], CGP could efficiently refine the coarse-grained structure to ultra-fine grain (260–270 nm), yield strength up to 563 MPa after three passes, and enhance bake hardenability by 170 percent compared with their coarse-grained counterparts. Moreover, the study by Khodabakhshi and Kazeminezhad [22] showed that the CGP process considerably increases dislocation density and electrical resistivity by up to 100%. In the same vein, the work by Khodabakhshi et al. [23] demonstrated that CGP-processed UFG low-carbon steel exhibits a reduced work-hardening rate but enhanced strain-rate sensitivity.

Regarding new process modification techniques, Kumar [24] proposed cross-route-Constrained Groove Pressing (cross-CGP). Cross-CGP is a process optimization approach to enhance strain homogeneity and improve the mechanical properties of low-carbon steel sheets. The results showed that the tensile strength and hardness of low-carbon steel sheets processed using the cross-CGP technique are greater than those of conventional CGP, particularly at higher pass numbers.

Kumar [25], in an extensive review of developments in CGP, discussed recent advancements in CGP modification techniques, including improved die designs, annealing techniques, cross-CGP, ring CGP, hybrid SPD, covered-sheet case CGP, and rubber pad CGP, among others.

Regarding the numerical simulations, Khodabakhshi et al. [26] carried out three-dimensional finite element modeling of the CGP-cross route procedure. They demonstrated that the values and uniformity of strain and hardness distributions obtained by FEM simulations agreed with experimental findings for low-carbon steel sheets. Using DEFORM-3D software, Singh et al. [27] conducted numerical simulations of the CGP of 316 austenitic stainless steel (and hence extended to low-carbon steel) and confirmed that the yield strength values calculated using analytical models agreed well with experimental findings.

More recently, Abdian and Kazeminezhad [28] reported the influence of inter-pass and first inter-pass annealing on the properties of low carbon steels after the CGP procedure, revealing that performing inter-pass annealing in the temperature range of 300–500 °C keeps hardness (160–170 Vickers) constant with the increment of CGP passes, and the elongation loss rate gets reduced. Moreover, Poojitha et al. [29] revealed that a combination of CGP followed by cold rolling reduced grain size to ~0.3 μm and increased yield strength to ~483 MPa for annealed ARMCO Fe low-carbon steel, starting from 90 MPa.

Satjabut et al. [30] conducted a study on the use of CGP prior to intercritical annealing in dual-phase steels. They observed that martensitic and tempered martensitic starting structures yield DP steels with an appropriate strength-elongation relationship.

Hence, even after several studies, there is still a need for experiments, along with finite element analysis, specifically conducted on low-carbon steel sheets after repeated applications of CGP, with attention to strain distribution, microstructural development, and mechanical behavior. The current work is dedicated to fulfilling this requirement by designing suitable CGP dies, performing two-dimensional finite element analysis using ABAQUS, and conducting tensile and hardness tests.

In the CGP method, two sets of dies are used: grooved dies to impose severe plastic deformation, and flattening dies to straighten the sheet. As illustrated in Fig. 1, according to the principles of the CGP process, the width and height of the die grooves must equal the sheet thickness, and the groove angle must be 45° to achieve pure shear deformation. These design dimensions result in a pure shear strain of γ=1.15 (equivalent von Mises strain of ε=0.58) in the deformed regions.

Referring to Eqs. (1)-(5), it is proved that the applied Von Mises effective strain in the deformed areas in each step of the pressing process is equal to 0.58. It must be noted that the process is considered to be performed under plane-strain conditions.

Referring to Eqs. (1)-(5), it can be shown that the von Mises effective strain induced in the deformed areas during each pressing step is equal to 0.58. It should be noted that the process is assumed to occur under plane strain conditions. The shear strain γ is related to the groove angle θ.

γxy = = = 1. (1)

γxy 1

εxy = = . (2)

εx = εy = εz = εyz = εzx = 0. (3)

εeff = √2 [(εx − εy)2 + (εy − εz)2 + (εz − εx)2] + 4 [εxy2 + εzy2 + εzx2]. (4)

ε = √4εxy2 = √4(γxy/2) = 1 = 0.58. (5)

eff 3 3 √3

Since any change in the specimen's dimensions can hinder the imposition of plastic strain and the attainment of high cumulative strain, most SPD methods are designed to keep the specimen's overall dimensions unchanged. As shown in Fig. 2, one complete pass of the CGP process consists of four pressing stages, each inducing a strain of 0.58 in the deformed regions. Upon completion of these four stages, a relatively uniform strain is introduced throughout the entire sheet, thereby finishing one CGP pass.

As proven above, pressing a plate-shaped workpiece using the grooved die imposes an effective strain of 0.58 in the deformed regions. In the second stage (the flattening step), the workpiece is flattened using a set of flattening dies; consequently, the inclined regions are subjected to a reverse effective strain of 0.58 again. At the end of this step, the sheet contains two distinct types of regions: 1) regions with an effective strain of 1.16, and 2) unstrained regions (zero strain). These regions alternate with a spacing equal to the sheet thickness. In Fig. 2, the white regions represent zero-strain areas, while the black regions indicate areas with an induced strain of 1.16.

The specimen is then rotated 180° about the axis perpendicular to the sheet plane and subjected to Eqs. (1)- (5) again. Finally, during the fourth stage (flattening of the corrugated sheet), a uniform strain of 1.16 is induced throughout the entire sheet.

To study the impact of the CGP process on sheet deformation and to calculate plastic strains, FE analysis was performed under plane-strain conditions using ABAQUS. To reduce the time required for simulation and calculation and to obtain accurate results, all dies were modeled as analytical rigid bodies. Meshing of the sheet was done using CPE4R elements, i.e., 4-node bilinear plane-strain quadrilateral elements with reduced integration. The total number of two-dimensional elements used to discretize the sheet is 984, and the total number of nodes is 1017. A friction coefficient of 0.1 was considered, which falls within the acceptable range of 0.05-0.1.

All dies used in the CGP processing of low-carbon steel sheets have been designed and manufactured in accordance with the guidelines for CGP processing. The dimensions of the cavity in which the specimens are inserted are identical to those of sheet specimens. Sheets of low-carbon steel, measuring 102 mm x 50 mm x 2 mm, have been selected as the raw material.

To examine the influence of CGP processing on low-carbon steel sheets subjected to different pass-number processes, tensile and hardness tests were performed under identical conditions. Tensile test specimens in accordance with ASTM E8M were prepared from sheets subjected to varying numbers of passes. Hardness testing was conducted in the thickness direction of both deformed and as-received sheets.

Grain size in the specimens was determined before and after the CGP technique across various numbers of passes. For this reason, the specimens were prepared and polished in accordance with the standards ASTM E03 1 and ASTM E407 99, respectively. Grain size measurements were performed in accordance with ASTM E112 using images taken with an optical microscope at a magnification of 400×. The validation of the FEA model was reported in a previously published paper [18].

The results are divided into three subsections: Effective strain distribution from finite element simulation, microstructural evolution and grain size, and hardness measurements.

CGP was successfully used on low-carbon steel sheets; finite element simulations and experiments were conducted to examine strain, microstructure, grain refinement, and hardening. The following findings can be made:

I. Design of dies and strain examination: CGP dies with an inclination angle of 45° and groove depths equal to the sheet thickness (2 mm) have been fabricated. The FEA results in ABAQUS showed that the effective strain imposed by each press is roughly 0.58. After four presses (one cycle), the average cumulative effective strain is roughly 1.15, which is close to the theoretical value (1.16). Some strain inhomogeneity (1.0-2.5) was noted at groove corners, and it decreases significantly with increasing number of presses.

II. Microstructural refinement: By the use of an optical microscope, it was found that initial coarse equiaxial ferrite grains (average diameter 22.5 µm, ASTM Grain Size No. 8) become finer progressively through each cycle, and four passes give rise to an ultrafine-grained structure with an average grain diameter of 4.5 µm (ASTM Grain Size No. 13.5), which indicates a grain size reduction of 80%. Most pronounced refinement takes place in the first pass.

III. Improvement in hardness: Hardness was measured using the Vickers hardness test on the material, and the results indicated a significant increase in hardness from 52 HV in the as-received condition to 148 HV, 195 HV, 228 HV, and 252 HV after the 1st, 2nd, 3rd, and 4th passes, respectively. This result implies an overall increase of 385% in hardness after four passes. The increase in hardness decreased after additional passes due to saturation.

IV. Hall-Petch relationship: There exists a direct linear relationship between Vickers hardness values and the inverse square root of the average grain size (R² > 0.98). It was therefore concluded that grain refinement is the major hardening process for low-carbon steel samples subjected to the CGP treatment. The Hall-Petch constants were obtained as HV₀ ≈ 20 HV and k ≈ 18 HV·µm¹/².

V. Efficiency of the method: Simulation coupled with experimental results demonstrates that cyclic groove processing is an efficient technique for manufacturing ultrafine-grained steel sheets. Finite element analysis has demonstrated that this severe plastic deformation technique effectively improves both mechanical properties and microstructure while maintaining the sample sheet's dimensions.

The authors report no conflicts of interest related to this work.

All relevant data are presented in this manuscript.

No financial support was received for the conduct of this study.