Introduction

Metal forming refers to a manufacturing process in which the definite shape of a solid piece is permanently changed without any change in the mass or composition of the material1. Among the different manufacturing methods, metal forming processes have a special statement due to the production of parts with excellent mechanical properties and the least waste of material, as well as the wide application of its products in the manufacturing industries2. In the last two decades, the production of materials with very fine grains based on their physical and mechanical properties has attracted a lot of attention. Several methods have been introduced to produce these materials in the form of solid rods, sheets and discs, without any porosity or impurity3. These materials, which are known as super metals, demonstrate the unique properties such as high strength at room temperature, superplastic property at high temperature, low strain rate, and high corrosion resistance4. The change of common sliding patterns in materials with nanometer grain size and the activation of grain boundary sliding mechanisms are among the characteristics of these materials, which lead to their unique mechanical properties5. One of the most important methods of producing nanometer materials is the top–bottom approach, in which coarse material grains are converted into fine grains. These materials are known as polycrystalline materials whose average grain size is less than one micrometer. To transform a material with coarse grains into a fine-grained material, a very high strain must be applied to the material in order to rearrange the dislocation. When a metal undergoes the plastic deformation at low temperatures, the internal structure of the metal begins to resist further deformation. Therefore, to continue in deformation process, more stress should be applied, and this state created in the metal is called work hardening or strain hardening. In fact, the hardening phenomenon increases the strength and hardness of the metal material due to the change in the plastic form. In contrast to this increase in the strength, the ability to deform in the material decreases. This constrain has caused the increase in the strength of the metal to be limited due to the possibility of breaking the material by applying mechanical work. One of the methods of producing nanostructured materials, which has received more attention in the last two decades, is the use of severe plastic deformation (SPD). In these processes, the microstructural changes occur in the material due to the application of severe plastic deformation, and the dimensions of the material grains reach to nanometer level. There are various methods for applying severe plastic deformation in metallic materials. The unique feature of these processes is the constancy of the dimensions and no drastic change in the appearance of the material during the SPD process, as a result of which, the limitation in applying strain is removed and it becomes possible to achieve high strains in the material. For this reason, as a result of applying strain, it is possible to modify the microstructure, reduce the grain size, and improve the mechanical properties of the metal without changing the shape of the specimen6,7,8. There are different methods to carry out the severe plastic deformation process in materials. In the current research, the constrained groove pressing (CGP) method has been used. Shin et al.9 presented a method of fine graining based on corrugated and repeated flattening of the sheet, which is known as CGP. In this method, a sheet metal is bent by a curved mold and then by turning the metal sheet twice and applying force again by the mold pressure, the sheet returns to its original state10,11. The schematic of CGP method is shown in Fig. 1. Repeatedly performing this operation creates a large effective strain in the blank and turns the coarse grains of the material into the fine grains. The advantage of this method is that it can induce more severe plastic deformation to a sheet metal.

Figure 1
figure 1

The schematic of constrained groove pressing method.

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Several researches have been carried out in the field of severe plastic deformation using the constrained groove pressing process. Lee and Park12 performed the simulation of the CGP process using the finite element method. The purpose of their research was to study the effective strain distribution in the workpiece. Based on above research, the strain distribution in the thickness of the sheet was not the same, and the effective strain in the center of the thickness was more than the points near the surface. Ghazani et al.13 performed the simulation of CGP process on an aluminum sheet. They considered the deformation distribution to be an inhomogenous distribution even after four passes and believed that the amount of this inhomogeneity in the process will not be lost by increasing the number of forming times. Peng et al.14 investigated on the refinement strengthening of annealed AZ31 magnesium alloy sheets under the CGP process at a temperature of 200 °C using Deform-3D software. They found that the yield strength and tensile strength of the specimens increased from 132 and 240 MPa to 155 MPa and 285 MPa, respectively. Nazari et al.15 focused on the influences of heat treatment on the microstructure and mechanical properties of copper sheets with thickness of 3 mm under the constrained groove pressing process. They found that the heat treatment made decreases the average grain size in the first cycle and increases it in the subsequent cycles. Moradpour et al.16 presented a new method to perform the CGP process on Al-5052 alloy. This method was obtained from the combination of the basic conventional and the square rotational methods. An acceptable agreement between the simulation results and their practical activities was reported. Gorbanhosseini et al.17 researched on the effects of elevated temperature CGP process on the microstructure, mechanical properties, anisotropy and texture of 2024 aluminum sheet. According to their experimental results, the maximum amount of texture intensity belongs to the one-pass sample, and on the other hand, the two-pass workpiece had the lowest amount of absolute plane anisotropy, which can be useful for secondary forming operations. In another research work, they investigate the anisotropy and texture of AZ91 Mg sheets subjected to various passes of elevated-temperature constrained groove pressing. A comprehensive research on development of new fiber textures for mechanical properties improvement of Al alloy has been carried out by Ghorbanhosseini et al.18. Based on their findings, the hardness, yield and ultimate strengths of 2024 Aluminum alloy in the first CGP cycle have been improved by, in turn, 68%, 45%, and 21%. Bhardwaj et al.19 experimentally studied the effect of high-temperature CGP process on the microstructure, mechanical properties and hardening behavior of Ti6A14V alloy. They found that no precipitates were observed in the SEM images, which confirmed the strain hardening effect of the CGP process and inherent voids as the main reason for the failure. Keyvani et al.20 analyzed the microstructural characteristics and electrochemical behavior of pure copper through the CGP process. According to the results of their research, increasing the CGP cycles reduces the corrosion current density and produces less defective passive films. Tavajjohi et al.21 focused on distribution of residual stress in the copper sheets deformed via CGP operation. According to their findings, this type of stress in the surface and central layers of the specimen are, in turn, compressive and tensile. Fereshteh-Saniee et al.22 conducted a comprehensive study on the influence of CGP operation on microstructure evolution, crystallographic texture and mechanical properties of Al 2024 and magnesium AZ91D sheets at elevated temperature. Based on the hardness test results, it was found that the annealed sample has the maximum homogeneity with the minimum inhomogeneity coefficient of 5.80 (for aluminum alloy) and 1.86 (for magnesium alloy). The effects of annealing on the microstructure evolution and mechanical properties of titanium sheet under CGP operation were investigated by Song et al.23. Their results revealed that the recrystallization occurred in the pure titanium deformed by the CGP process that was annealed at 650 °C, which leads to a decrease in tensile strength. Sharma et al.24 conducted a comprehensive study on the mechanical properties of AZ31 magnesium sheet and the effects of its deformation in CGP process. In their research, the influence of various experimental factors on strengthening the yield strength and elongation was investigated. Eftekhari Shahri et al.25 analyzed the effect of die geometry in the CGP process, experimentally and numerically. They found that non-uniformity in hardness distribution is correlated with non-uniformity in strain distribution. Hosseini Faregh et al.26 studied on the influence of the die angle on the deformation behavior of copper sheet in the constrained groove pressing operation. The results of their research showed that the first pass of the process causes the crystal size in both die sets to decrease to nanometer sizes and the dislocation density in the specimen increases, significantly. A comprehensive study on constrained groove pressing method has been carried out via Kumar27. In his research, the recent developments in the field of the above-mentioned method including cross CGP, pad CGP and refinements of die design have been reported. In another research work, Bhardwaj et al.28 investigated on changing the pressing speed on microstructure and mechanical properties of Ti6Al4V deformed via CGP process. Based on their results, the critical speed in order to achieve a suitable combination of strength, hardness and ductility was considered to 0.20 mm/sec. Wang et al.29 presented a CGP-FDEM hybrid analysis to evaluate fracture characteristic and support effect around deep lined tunnels. They reported that the greatest influence of reinforcement was observed in the excavation damaged zone. An in-depth study on CGP operation to achieve homogeneously improved properties has been carried out by Sawalkar and Field30. In this research, by providing a new die for creating combined bending and constant shear strain, the properties of copper sheet were improved, significantly.

As it is clear from the previous studies, the most of conducted researches are related to checking the variation of mechanical and metallurgical properties in the specimens deformed via constrained groove pressing process in the experimental manner. Moreover, lower studies concentrate on the performing of CGP process on the special light weight alloys. Therefore, in this study, investigation of the damage evolutions for Al 2024 – T3 alloy is considered. One of the methods of simulating the CGP process is the use of material models, which has received less attention in the aforementioned operation. In the present research work, the Johnson–Cook model is considered for numerical simulation, and the additional explanations of the aforementioned method are provided in the “Conclusions”. The geometrical variables studied in the current research include teeth number, teeth angle and sheet thickness, and the purpose of their evaluation is to reduce the equivalent strain and equivalent force target factors, simultaneously. In order to determine the weight’s coefficients of these target factors and select the superior numerical test, the Shannon’s Entropy and Simple Additive Weighting method has been used. Also, the damage parameter, stress triaxiality, and equivalent plastic strain of Al 2024 worksheet have been investigated during the best practical condition of CGP process.

Experimental procedures

In the current section, the practical experiments of constrained groove pressing process are discussed. A view of the experimental set and specimens produced at different temperatures are shown in Fig. 2. The upper and lower grooving dies were made by wire cutting method with the groove angle of 45 °C and the width of each groove equal to 2 mm (equivalent to the thickness of the used workpiece). Two upper and lower flat dies were also used to smooth the congressed blank. In order to manufacture the introduced dies, MO40 hot-work steel (1.7225 steel) has been used. It should be noted that the practical tests have been performed at 25 °C (room temperature) and 300 °C. The production of CGPed specimens at elevated temperature was done by placing three 500 W elements in the dies. Santam press machine with a capacity of 40 tons and a ram velocity of 1 mm/min has been employed to carry out the CGP process of Al sheet (see Fig. 2a). The experimental procedures of the constrained groove pressing process has been done in four stages, which include grooving (step 1), flattening (step 2), grooving after 180° rotation about the central axis (step 3), and finally flattening (step 4). In the present study, 2024 aluminum alloy sheet has been used as a workpiece. High strength-to-weight ratio is the most important feature of this alloy. The main applications of 2024 Al alloy are in the aerospace industry (aircraft body), engine and car rims, heat transfer devices (radiator fins), light and heat reflectors, heat exchangers and decorative items.

Figure 2
figure 2

A view of (a) practical setup of CGP process to deform the specimens at process temperatures of (b) 25 °C and (c) 300 °C.

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Numerical simulation

In order to numerically study the severe plastic deformation process via constrained groove pressing method, Abaqus software has been employed. The die set is modeled in the software as two groove dies, two flat dies and initial blank. All dies are introduced as a rigid three-dimensional part and a square workpiece of Al-2024 with a length of 100 mm and a thickness of 2 mm is defined as a 3D deformable part. The mechanical properties of the Al sheet were obtained using the stress–strain diagram extracted from the standard tensile test and imported to the software16. Considering that the most of sheet metal forming processes at low speed are known as quasi-static, in the present FE-simulation, the dynamic explicit step is chosen as the solution to the problem. In the current simulation, the amounts of time for each step and mass scale parameters are considered to, in turn, 20 s and 106. In the simulated die set, the die-workpiece contact surface is considered to be frictional with tangential behavior. To define the frictional field, the penalty method is selected, in which the relative movement is allowed to the surfaces in contact with each other. The amount of slippage between mentioned surfaces is limited and controlled by the software so that the shear stress caused by this slippage does not exceed from its critical level. According to Ref.31, the value of friction coefficient is considered to 0.1. In order to fill the teeth of the congress dies with the sheet under the CGP process, with the lower die fixed, the upper die must pull down to the thickness of the sheet after contacting the surface of the workpiece. For this reason, the simulation of CGP process is considered as displacement control type. C3D8R element, which is a three-dimensional, 8-node and reduced-linear element has been used for meshing the deformable blank. To perform the finite element simulation in elevated temperature, C3D8T element has been employed. In order to investigate more precisely the strain distribution in the sheet thickness, the number of four elements along the thickness is defined. This number of elements in the thickness direction has been selected based on the convergence of the maximum force required to perform the first step of CGP process. The mesh dependency diagram is given in Fig. 3. Based on the convergence test, the number of sheet elements in the process simulation has been considered to 4800.

Figure 3
figure 3

The maximum force required to perform the first step of CGP process versus the number of elements.

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Johnson–Cook material model

One of the practical methods of numerical simulation in forming processes is to use a material model to introduce the properties of the desired material. In severe plastic deformation processes (such as CGP), the complexities of material behavior during the deformation process have caused the material model to have a significant influence on finite element simulation outcomes. The Johnson–Cook material model, regarding the effect of strain hardening, the strain rate of the material and the effect of its softening, is considered as one of the most widely used models to determine the plastic behavior of the material during the numerical simulation process. The Johnson–Cook material model for aluminum sheets in the different manufacturing processes has also been utilized by previous researchers as well32,33,34,35,36,37,38. This applied model has constants whose accurate determination, considering the type of process investigated, plays an important role in the behavior of the material. In general, Johnson–Cook material model equation is expressed as follows (Eq. (1))39:

$$ \overline{\sigma } = [A + B\left( {\overline{\varepsilon }^{pl} } \right)^{n} ][1 + C\ln (\frac{{\dot{\varepsilon }^{pl} }}{{\dot{\varepsilon }_{o} }})](1 - (T^{*} )^{m} ) $$
(1)

In the above equation, \(\overline{\sigma }\) and \(\overline{\varepsilon }^{pl}\) represent the rate dependent yield stress and equivalent plastic strain, respectively. Also, \(C\) parameter indicates the strain rate constant and \(\dot{\varepsilon }_{o}\) is the reference strain rate. It should be noted that \(A\),\(B\) and \(n\) are material parameters and these constants are extracted at \(T^{*}\)- temperature. In order to obtain the equivalent plastic strain at damage initiation, Eq. (2) is employed39. With this regard, d1 to d4 represent damage parameters.

$$ \overline{\varepsilon }_{D}^{pl} = [d_{1} + d_{2} \exp (\frac{{d_{3} p}}{q})][1 + d_{4} \ln (\frac{{\dot{\overline{\varepsilon }}_{pl} }}{{\dot{\varepsilon }_{o} }})] $$
(2)

In Table 1, the physical properties, Johnson–Cook and damage parameters related to Al-2024-T3 are given. The aforementioned properties have been used as input in the numerical simulation of the constrained groove pressing process.

Table 1 General properties related to Al-2024-T339.
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Results and discussion

Due to the fact that in the present research, three-CGP passes simulation has been done, therefore, the process simulation will include twelve steps, totally. It should be noted that the numerical simulation of the All CGP passes was performed in 240 s. Also, both room (25 °C) and elevated (300 °C) temperatures were considered in these simulations. In the following sub-sections, the numerical simulation results are presented in detail.

Validation of the numerical simulation and material model

In the current sub-section, the validation of FE- simulation findings and the accuracy of the applied Johnson–Cook model are discussed. Considering that the problem consisting of twelve steps and each step takes 20 s, the amount of run time to solve the aforementioned process in three passes is equal to 240 s. Figure 4 shows the damage evolution versus process time for the specimen deformed at room (part a) and elevated (part b) temperatures. The Johnson–Cook damage model assumes that damage accumulates in material elements when plastic strains are generated. When this accumulated damage reaches its critical value, it suddenly accelerates and causes the collapse of those elements. The D-damage parameter varies from zero (soundness part) to one (complete rupture). The amount of this variable is obtained from the ratio of the growth of the equivalent strain to the plastic strain. The critical value of the damage parameter is usually considered to be around 0.3 based on existing standards. Regarding that in the present study, the value of this variable reached to 1 in the 17th second for deformation at room temperature, it is expected that the workpiece ruptured at this time (see Fig. 4a). The damage parameter at room temperature has a faster growth rate than at elevated temperatures. The reason for this is that the elongation at elevated temperatures increases for a material under forming process, and this factor causes defects (such as dislocations) to behave in such a way that the material fails later. Accordingly, the failure values at elevated temperatures are lower than room temperature.

Figure 4
figure 4

Damage evolution diagram for CGPed-specimen at (a) room and (b) 300 °C process temperatures.

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In Fig. 5a, the equivalent plastic strain (PEEQ) contour of CGPed-sample is shown in the 17th sec. It can be seen that at this moment, the maximum PEEQ is equal to 0.47, which at room temperature, this amount of strain indicates the failure of aluminum part. Therefore, it can be concluded that there is a good correlation between the damage material model and the failure onset. The experimental specimen formed in practical conditions similar to the FE- simulation is shown in Fig. 2b. Based on the observations made, the experimental sample also failed at 17th sec. It can be claimed that the finite element simulation outcomes have an acceptable accuracy. In Fig. 5b, the 10th step of FE- simulation at room temperature is presented, which shows the fully rupture of the deformed sheet. The trend of damage parameter changes for the deformed specimen at 300 °C is shown in Fig. 4b. As can be seen, in the first step, the amount of this parameter reached to 0.2 with an almost steep slope, which is quite reasonable based on the operation mechanism of constrained groove pressing process. In the following, it can also be stated that the amount of this variable is increasing with a slight slope due to the increase of process passes; so that at the end of the third passes (12th step), its value is less than 0.3.

Figure 5
figure 5

Equivalent plastic strain of deformed part for CGPed-sample in (a) 17th sec (step 1) and (b) 200th sec (step 10) at room temperature.

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Based on this, it can be acknowledged that the CGPed- sample does not fail at 300 °C process temperature until the end of the third pass. As mentioned earlier, due to the increase in elongation in the forming process at elevated temperatures, the dislocations are placed in relation to each other in such a way that their connection happens later when the sample is subjected to load. This factor has caused that at elevated temperatures, the damage parameter has a much lower growth rate and the failure occurs later in comparison with CGPed sample manufactured in the room temperature. Therefore, it can be seen that no failure was observed in the three-pass specimen CGPed at 300 °C process temperature. The three-pass experimental part shown in Fig. 2c is proof of this issue. According to the change trend of damage variable for workpiece CGPed at elevated temperature, it can be said that if the process is performed in the fourth pass, it is possible to increase the amount of damage parameter to more than 0.3 and as a result the damage of the specimen can be expected.

Choosing the superior geometrical condition

As mentioned earlier, several factors influence on the quality of the fine-grained specimens manufactured by constrained groove pressing process. In the present research work, the process variables including the number of teeth (N), the thickness of the sheet (Thickness) and the angle of the teeth (Theta) have been considered. Several criteria can be presented to demonstrate the improvement of the quality of the CGPed products. In this study, the effects of the aforementioned variables on both factors, namely equivalent strain (Target Factor 1 (Eq. 3)) and equivalent force (Target Factor 2 (Eq. 4)) have been investigated, simultaneously40. Decreasing the amount of equivalent strain leads to a more uniform distribution of strain in the CGPed specimen, and by reducing the amount of forming force, the amount of energy consumption is reduced. Accordingly, it is very important to choose the best numerical test for the simultaneous control of both response variables.

$$ T.F.\,\,1 = \frac{{\overline{\varepsilon }_{\max } - \overline{\varepsilon }_{ave} }}{{\overline{\varepsilon }_{ave} }} $$
(3)
$$ T.F.\,\,2 = \frac{{F_{\max } }}{{10^{7} \times W}} $$
(4)

where, \(\overline{\varepsilon }_{\max }\) and \(\overline{\varepsilon }_{ave}\), in turn, denote the maximum and average effective plastic strains induced into the blank, and \(F_{\max }\) and \(W\) represent the maximum force in the load–displacement curve and sheet weight, respectively. In Table 2, the numerical simulation results assisted by Johnson–Cook material model in CGP operation are presented. As it is clear, the process variables, namely N (in four level), Thickness (in two level), and Angle (in three level) are considered and the numerical simulations have been carried out in full factorial mode.

Table 2 The numerical simulation results obtained from CGP process.
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Considering that the maximum reduction of both objective functions is desirable, one should look for the superior numerical test. Investigating the effects of variables on objective functions in multi-objective problems are of particular importance41,42,43. Accordingly, in the current research, the Shannon’s Entropy method has been employed in order to determine the weight’s coefficients of target factors. Shannon’s Entropy is one of the multi-criteria decision-making (MADM) methods for calculating the weights of the criteria. This method requires a criterion-option matrix. In this method, which is presented by Shannon and Weaver in 1974, Entropy expresses the amount of uncertainty in a continuous probability distribution. The main idea of the aforementioned method is that the greater the dispersion in the values of an index, the more important that index is. Entropy method assigns a weight to each criterion. In general, the total weight of the criteria should be equal to one44,45. The calculation steps for determining the weight of the target factors via Shannon’s Entropy method are described in Fig. 6. Based on the Shannon’s Entropy method, the weight’s coefficients for equivalent strain (Target Factor 1) and equivalent force (Target Factor 2) were calculated to, in turn, 0.38 and 0.62.

Figure 6
figure 6

Determination of weights coefficient for response factors assisted by Shannon’s Entropy method.

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After calculating the weight’s coefficients of the target factors, now the superior test should be selected. There are different methods to choose the best numerical test. In the present research, the Simple Additive Weighting (SAW) method has been used. In this method, the overall score of a candidate solution is determined by the weighted sum of all feature values. In this method, which is also known as the weighted linear combination method, after de-scaling the decision matrix, using the weight coefficients of the criteria, a weighted unscaled decision matrix is obtained and the score of each option is calculated according to this matrix. The results of the Entropy-SAW hybrid technique for the equivalent strain and equivalent force (separately and simultaneously) are given in Fig. 7. The amount of the objective functions for single target factors are presented in Fig. 7a,b. Based on the observations made, it can be seen that the lowest amount of equivalent strain (Fig. 7a) and equivalent force (Fig. 7b) are corresponded to, in turn, Tests No. 7 and No. 23 (See Table 2).

Figure 7
figure 7

The heat map plots obtained from Entropy-SAW hybrid technique for choosing the superior test based on (a) equivalent strain, (b) equivalent force, and (c) combination of two target factor.

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In Fig. 7a,b, the amount of the objective function is determined based on the values of the process variables. It should be noted that the amount of thickness can be seen on the horizontal axis and the number and angle of the teeth can be observed on the vertical axis. Based on the guide bar on the right side of the graph, the amount of the objective functions is determined based on the color scheme. For example, the lowest amount of equivalent strain has a thick blue surface and belongs to the experiment in which the values of N, Theta, and thickness are considered to 11 and 45°, and 1 mm, respectively (Test No. 7 in Table 2). Also, in the case that the amount of N, Theta, and thickness are, in turn, 11, 60° and 2 mm (See Test No. 23 in Table 2), the amount of equivalent force is minimized. In Fig. 7c, the numerical tests are ranked based on the weight of the objective functions and their values using Entropy-SAW hybrid technique. Accordingly, the maximum score belongs to Test No. 23, where the values of N, Theta, and thickness are equal to 11, 60° and 2 mm, respectively. Considering the importance of both objective functions, the superior numerical test has been selected by considering the weight of these two functions, simultaneously. The study of the stress triaxiality and plastic strain distribution in the continuation of the research has been carried out based on the superior test conditions.

Stress triaxiality

In this sub-section, the aim is to investigate the amount of triaxiality stress in the specimens deformed at 25 °C and 300 oC process temperatures. Stress triaxiality is a quantitative parameter to evaluate the stress limit. This parameter is an important factor for the failure mode of the material, which is introduced to the material in the loading operation46,47. One of the proposed methods to assess the effect of triaxial stress on the fracture strain of the material is presented by Bridgman48. Based on this definition, the amount of this parameter is calculated from the ratio of hydrostatic stress to von Mises stress. Stress triaxiality has a significant influence on the amount of tensile strain of a material before ductile failure. The value of this parameter is calculated as bellows (Eq. (5)):

$$ \eta = \frac{{\sigma_{m} }}{{\sigma_{eq} }} $$
(5)

where, \(\sigma_{m}\) and \(\sigma_{eq}\) are defined as follows (Eqs. (6) and (7)):

$$ \sigma_{m} = \frac{{\sigma_{1} + \sigma_{2} + \sigma_{3} }}{3} $$
(6)
$$ \sigma_{eq} = \sqrt {\frac{1}{2}[(\sigma_{1} - \sigma_{2} )^{2} + (\sigma_{2} - \sigma_{3} )^{2} + (\sigma_{3} - \sigma_{1} )^{2} ]} $$
(7)

It should be mentioned that \(\sigma_{i}\) are principal stresses. In Fig. 8, the stress triaxiality distribution for specimen CGPed at room and elevated temperatures are shown. The fluctuations in the aforementioned diagram are due to the corrugation-flattening behavior in the constrained groove pressing process. The closer the value of this type of stress is to one, the more likely the workpiece will failure. As it is known, in the first step (about 17th second), the value of this parameter in the negative state is close to 1 (at room temperature), and it is expected that this is the starting point of specimen failure. Comparing the amount of stress triaxiality at first pass with the amount of plastic strain (Fig. 5a) as well as the experimental sample (Fig. 2b) at this process temperature proves the accuracy of the findings. In the further passes, the intensity of the stress triaxiality fluctuations increases and even crosses one, which is expect to cause complete failure at these time points49. In this case, the specimen in Fig. 5b confirms the issue. As it is shown in the stress triaxiality distribution diagram for the CGPed-part at 300 °C process temperature, the fluctuating behavior occurred in a lower range; in such a way that the amount of this parameter will be less than 0.5 in all three passes of the CGP process. Based on this, it can be claimed that the mechanical work done on Al-2024 specimen at 300 °C will not cause it to fail. The maximum amount of stress triaxiality for specimen CGPed at elevated temperature (about 0.5) occurred in the first step, and in this same step, the maximum value of the damage parameter was also observed in Fig. 4b (about 0.2). Therefore, there is an acceptable correlation between the findings.

Figure 8
figure 8

Stress triaxiality distribution for specimens CGPed at (a) room and (b) 300 °C temperatures.

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Plastic strain distribution

Another effective parameter in the constrained groove pressing process is the equivalent strain distribution, which is discussed in this sub-section. By performing CGP process, more plastic strain was induced to the worksheet and, consequently, the 3-passed sample has the maximum plastic strain. Figure 9 demonstrates the equivalent plastic strain distribution for the specimens CGPed at 25 °C and 300 °C. In the deformed sample at room temperature, the steep increase of this parameter is evident in the first step. The maximum amount of equivalent plastic strain in this initial stage has reached about 1, which indicates the failure of the part. The values of damage parameter and triaxiality stress in the current step are also indicative of this issue. As can be seen in the distribution diagram of the equivalent plastic strain for the specimen CGPed at elevated temperature, the increasing trend of this variable is quite limited. The maximum amount of plastic strain after three passes of the CGP process reached less than 0.1, which indicates the sample without defect.

Figure 9
figure 9

Equivalent plastic strain distribution for specimens CGPed at room and elevated temperatures.

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Conclusions

In the current research, the constrained groove pressing process has been considered, experimentally and numerically. The material used in this study is 2024-T3 aluminum alloy and its plastic deformation behavior has been studied at room and elevated temperatures. In order to simulate the process, the Johnson–Cook material model has been employed in Abaqus software. The process variables included teeth number, teeth angle, and sheet thickness and their effects on equivalent plastic strain and equivalent force were studied, simultaneously. In this research, the Shannon’s Entropy method is used to weight the objective functions and the Simple Additive Weighting method is employed to select the superior practical test. A summary of the research findings are given bellows:

  1. 1.

    There was good agreement between numerical and experimental results of CGP process on Al 2024-T3 alloy at room and elevated process temperatures.

  2. 2.

    Based on the Entropy-SAW hybrid technique, the weight’s coefficients of equivalent strain and equivalent force were obtained as 0.38 and 0.62, respectively.

  3. 3.

    The CGP process for Al 2024-T3 alloy could not be performed at room temperature. The stress triaxiality and damage parameter exceed than the critical limit, and the worksheet was collapsed.

  4. 4.

    Three CGP passes could be performed at 300˚C for Al 2024–T3 alloy. The damage parameter and stress triaxiality were controlled by performing CGP process at this temperature.

  5. 5.

    Equivalent plastic strain (PEEQ) distribution was investigated in FE simulations. After three CGP passes, the PEEQ was reached to 0.1 at 300˚C.

  6. 6.

    The defects initiation for the worksheet which are produced and propagated until complete failure in CGP operation at room temperature, are coincident in experiment and simulation.

The specimens produced by severe plastic deformation methods have more favorable mechanical properties such as high strength, good fatigue life, high toughness and low temperature superplasticity. Due to the characteristics mentioned above, these materials have several applications in the aerospace, automotive, transportation, chemical processes, and defense industries. In the continuation of the present research work, it is possible to carry out the constrained groove pressing process at different velocities and process temperatures in the presence of different lubricants and study the effects of these factors on the mechanical properties of the manufactured parts.