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Abstract

The mechanical properties of metallic strips can be improved by producing an ultra-fine-grained microstructure. The equal-channel angular rolling process is a severe plastic deformation technique suitable for such a purpose. In this article, the effect of the equal-channel angular rolling parameters on the equivalent plastic strain is investigated and their optimum values are reported. The analysis is focused on the following parameters: thickness ratio (K), die channel angle (ϕ), die outer corner angle (ψ), fillet radius (r), friction coefficient between strip and rolls (µ), and friction factor between strip and die (m). Using a combination of the finite element modeling and the response surface methodology with central composite design, a simple and efficient quadratic model is developed to predict the equivalent plastic strain dependence on the equal-channel angular rolling process parameters. The analysis of variance is applied to assess the validity of the model and find the significant parameters. The results show that the die channel angle is the most significant parameter, while the friction coefficient between strip and rolls has no considerable effect on equivalent plastic strain. Moreover, the following optimum values of the parameters are obtained: K = 0.96, ϕ = 90.10°, ψ = 21.21°, r = 2.31 mm, µ = 0.20, and m = 0.19. An increase of about 14.86% in the equivalent plastic strain can be achieved by adopting these values for the process parameters.

Keywords

Ultra-fine-grained microstructure equal-channel angular rolling severe plastic deformation response surface methodology analysis of variance

Introduction

Extensive research has been conducted to investigate the structural evolution and the change in the mechanical response associated with severe plastic deformation (SPD). The studies proved that materials with ultra-fine-grained (UFG) structure can be obtained by inducing high plastic strains. Conventional metal forming techniques such as rolling, extrusion, and drawing are not able to develop sufficient strains for producing fine-grained components.¹ The SPD processing techniques have been developed to overcome this disadvantage.

Among the SPD techniques, equal-channel angular pressing (ECAP)² has gained a commercial significance in recent years, since it can be used as a procedure through which various bulk UFG materials can be produced without changing their cross section. ECAP is able to develop plastic strains of ∼1 at a single pass.^3,4 Since negligible change occurs in the cross-sectional shape of specimens after ECAP, a multi-pass operation is also possible.^4–7 Therefore, this method has been used to produce UFG bulk materials with enhanced mechanical properties.^3,4,8 According to the Hall–Petch relationship,⁹ reducing the grain size leads to an increase in the strength and hardness characteristics. Therefore, materials with uniform UFGs can be used as high-strength materials. However, the ECAP process has two major disadvantages. First, it cannot handle long and thin sheet metal, being restricted to bulk metals with rectangular and circular cross sections. Second, ECAP cannot be applied in a continuous manner, which results in low production efficiency and high cost.^10,11

Several attempts have been made to overcome these disadvantages. Techniques including equal-channel angular drawing (ECAD),¹² accumulative roll bonding (ARB),¹³ repetitive corrugation and straightening (RCS),¹⁴ continuous shear deformation process (conshearing),¹⁵ continuous confined strip shearing (C2S2),¹⁶ continuous frictional angular extrusion (CFAE),¹⁷ equal-channel angular sheet extrusion,¹⁸ ECAP-Conform,¹⁹ and so on have been recently developed as continuous ECAP methods usable for the production of UFG sheet materials.

Lee et al.²⁰ introduced the so-called C2S2 process based on ECAP, which enables the production of metallic strips in a continuous manner and can induce shear strains into a long and thin metallic sheet. The newly developed process was termed dissimilar channel angular pressing (DCAP) and equal-channel angular rolling (ECAR) to distinguish this technique from the conventional ECAP. During this process, a thin metal strip is fed through the feeding rolls into the inlet channel. In order to prevent metal strip escape from the rolls’ gap, the inlet cross section of the channel is slightly thinner than that of the strip. Then the material flows into the forming zone where the inlet and the outlet channels intersect at a specific angle. The strip then flows into the outlet channel, which has the original thickness of the strip, allowing the metallic strip to recover its initial thickness and leave the ECAR die with its original shape. Figure 1(a) illustrates the ECAR process with an initial strip thickness of 2 mm, inlet channel thickness of 1.8 mm, outlet channel thickness of 2 mm, die channel angle of 120°, and die outer corner angle of 0°. Figure 1(b) shows the ECAR die mounted on the rolling mills and the load cells by means of which the upper roller force was measured.

Figure 1.

(a) Schematic illustration of ECAR process and (b) ECAR die mounted on the rolling mills.

Although it is possible to produce severe plastic strains using SPD processes like ECAP, ECAR, and ARB, if the technological parameters are not set properly, the efficiency of these processes will be inappropriate. For this reason, many attempts have been made to optimize the parameters of the SPD processes. Srinivasan and Cherukuri²¹ optimized the ECAP parameters for strain homogeneity. In their work, the effect of parameters such as the outside corner angle, inner radius, and shear friction on the strain homogeneity was investigated. Ebrahimi et al.²² used the finite element method (FEM) to obtain higher values of the effective strain, more uniform strain distributions, and lower pressing forces in ECAP. Luri and Luis Pérez²³ performed a numerical investigation on the effect of the geometrical parameters of the dies and the part on the punch force in the equal-channel angular extrusion process. Xu et al.²⁴ investigated the influence of the geometrical and frictional parameters of the die on the deformation behavior of the pure aluminum sheets in the ECAR process. Wei et al.²⁵ showed that the distribution of the effective strain in most regions of the workpiece is uniform in ECAR. As the die channel angle increases, the effective strain decreases and becomes more uniform over the whole specimen. Kvačkaj et al.²⁶ used the FEM to investigate the effects of the die angle and friction parameters on the stress and strain distribution, strain homogeneity, shear deformation, and the required torque in ECAR.

The traditional optimization method considers one factor at a time and keeps the level of all the other variables fixed. Using this method in applications with multiple effective variables is expensive and time-consuming. Furthermore, the interaction between factors is ignored in this method. Therefore, in order to overcome these disadvantages, other methods like response surface methodology (RSM) can be used in order to evaluate the effect of several factors and responses with an acceptable number of experiments. RSM also provides simple relationships for the prediction of responses. Many successful optimizations and predictions have been performed using RSM in different fields. Some examples of RSM application in the field of metal forming are as follows: Venkateswarlu et al.²⁷ optimized the friction stir processing parameters using RSM. They obtained an improved microstructure for magnesium AZ31B alloy, which showed better mechanical properties and formability. Velmanirajan et al.²⁸ utilized RSM to predict the formability of aluminum alloys. Naceur et al.²⁹ optimized tool geometry in sheet metal forming in order to reduce the springback using RSM. Although ECAR processes are widely investigated in recently published articles, there is no work devoted to the optimization of the process parameters. To partially fill this gap, in this research, RSM and the analysis of variance (ANOVA) are employed to obtain the optimum values of the ECAR parameters. These parameters are the thickness ratio (K, the ratio of strip thickness at inlet channel to the strip thickness at outlet channel), die channel angle ( $ϕ$ ), die outer corner angle ( $ψ$ ), fillet radius ( $r$ ), friction coefficient between strip and rolls ( $μ$ ), and friction factor between strip and die ( $m$ ). The higher values of the strains would be obtained in ECARed specimens by setting the optimum values of these parameters.

Finite element analysis

Finite element analysis is an effective tool to predict the material flow properties and field variables, especially for the metal forming processes like ECAR. In this research, AA5083 aluminum alloy specimens with 300 mm length, 30 mm width, and 2 mm thickness were employed. Two-dimensional (2D) finite element simulations were carried out using the commercial ABAQUS/Explicit software to obtain the distribution of the equivalent plastic strain (PEEQ) in the specimens. The mechanical response of the aluminum alloy was described using an isotropic elastoplastic constitutive model. The upper and lower dies and rollers were represented as discrete rigid bodies. A constant angular speed of 1.15 rad/s was assumed for the rollers. The friction force between the rolls and the strip acts as a driving force to push the strip through the ECAR die. The Coulomb friction model with a constant coefficient of friction $μ$ was applied between strip and rolls. The friction conditions between the die and the strip were modeled using the constant shear model

τ = m \frac{\bar{σ}}{\sqrt{3}}

(1)

where $τ$ is the friction shear stress, $\bar{σ}$ is the effective stress, and m is the friction factor. The value of the friction factor depends on the characteristics of the contact surfaces. The as-received material was annealed at 345 °C. The mechanical properties required for calibrating the elastoplastic constitutive model were obtained from uniaxial tensile tests performed on annealed specimens. These properties are presented in Table 1. The four-node bilinear plane strain element (CPE4R) was employed to mesh the model. Figure 2 illustrates the FEM simulation of the ECAR process.

Table 1.

Material properties of the AA 5083 aluminum alloy.

Density, ρ (kg/m³)	Poisson’s ratio, υ	Young’s modulus, E (GPa)	Yield stress (MPa)	Strength coefficient, K (MPa)	Strain hardening exponent, n
2650	0.33	70.3	164	540	0.21

Figure 2.

Finite element model of the ECAR process.

The values of the parameters in the finite element model were selected according to the experimental condition. The geometrical parameters, the thickness ratio ( $K = 0.9$ ), die channel angle ( $ϕ = 120 \circ$ ), die outer corner angle ( $ψ = 0 \circ$ ), and fillet radius ( $r = 0.5 mm$ ) were set based on the geometries of the dies and the strip used in the experimental tests. But the value of friction coefficient between strip and rolls ( $μ$ ) was calibrated based on its effect on the rolling load. The upper roller force was obtained from the simulation for different values of $μ$ and was compared with experimental data. This comparison for the value of $μ = 0.13$ was depicted in Figure 3. One may note that the finite element model predicts the rolling force with an acceptable accuracy. The maximum values of the experimental and computed rolling forces are 6.53 and 6.65 ton (about 2% error), respectively. This indicates that the value of $μ$ has been calibrated with good accuracy. Furthermore, this value is in agreement with the results presented in Jeswiet.³⁰ Finally, the value of the friction factor between strip and die was selected as $m = 0.12$ according to the simulation conditions and the results published in the literatures.^24,25

Figure 3.

Comparison between the experimental and computed forces for the first ECAR pass.

Figure 4 shows some specimens after single-pass ECAR ( $ϕ = 120 \circ$ ). One may note that the specimens have two parts. The first part is the rolled region of the strip before entering the shear deformation zone, while the second part is the region of the strip with the imposed shear deformation.

Figure 4.

Single-pass ECARed samples ( $ϕ = 120 \circ$ ).

Figure 5(a) shows PEEQs obtained by numerical simulations in the middle path of the strip with respect to distance (longitudinal direction of the sample). The sharp increase in the PEEQ value visible on every curve in Figure 5(a) corresponds to the transition from the rolled to the severely deformed regions of the specimens. As one may note in Figure 5(a) that increased maxima of the PEEQ can be obtained by decreasing the die channel angle.

Figure 5.

(a) Effect of the die angle on the equivalent plastic strain (PEEQ) along the middle path of strip and (b) shear deformation after single ECAR (channel angles of 120°).

During ECAR process, plastic strains with three major sources are imposed to the strip. The first one is the normal plastic strain that is imposed to the strip during rolling and reduction of the thickness. The second and the most important part of the plastic strain is the shear strain that is imposed to the strip during passing through the intersection of the die channels and the last one is the shear plastic strain caused by the frictional sliding of the strip during ECAR process. Figure 5(b) illustrates the distortion of the elements due to shear plastic deformations, which are imposed at the intersection of the die channels and frictional contact surfaces (frictional contact between top and bottom surfaces of the strip with die surface). The first and the second plastic strains can be calculated by analytical formulations. Iwahashi et al.³¹ developed an analytical model for calculation of the shear strain in ECAP process. This model that was modified by Lee et al.¹⁶ for calculation of the shear strain imposed to the strip at the shear deformation zone in the ECAR process. This shear strain can be obtained using

γ = K^{2} [2 \cot (\frac{ϕ + ψ}{2}) + ψ cosec (\frac{ϕ + ψ}{2})]

(2)

where $γ$ is the shear strain. On the other hand, the normal strain can be calculated using

ε = \ln (\frac{t_{0}}{t})

(3)

where $t_{0}$ is the initial thickness of the strip and $t$ is the thickness of the strip after rolling process. By knowing the values of the normal and shear strains, it is possible to evaluate the PEEQ in the ECAR process utilizing the following equation³²

ε_{eq} = N {[\frac{2}{3} {4 ε^{2} + \frac{γ^{2}}{2}}]}^{\frac{1}{2}}

(4)

where $N$ is the number of passes. By placing the values of $ϕ = 120 \circ$ , $ψ = 0 \circ$ , $K = 0.9$ , $t_{0} = 2 mm$ , and $t = 1.8 mm$ (which are the same values used in the finite element model) in equations (2) and (3), the normal and shear strains for single-pass ECAR were calculated as $ε = 0.105$ and $γ = 0.935$ . By placing these values in equation (4), the PEEQ value for a single ECAR pass, $N = 1$ , was obtained as $ε_{eq} = 0.566$ . This value is less than the value obtained by numerical strain ( $ε_{eq} = 0.74$ , Figure 5(a) for $ϕ = 120 \circ$ ). This difference is due to uniform deformation assumption and elimination of the friction effects in derivation of equations (2) and (4). Also, it is worth to mention that the shear strain value can be estimated by measuring the distortion angle of the elements after passing the intersection of the die channels. From Figure 5(b), this inclination angle was obtained as ∼43.5°, which gives the value of shear strain as $γ = \tan (43.5) = 0.949$ . This value is very close to the analytical value of shear strain ( $γ = 0.935$ ). The comparison of the numerical and analytical results shows that the numerical model is able to predict the PEEQ with a reasonable accuracy and can be used with a high confidence.

Central composite design

Box–Behnken and central composite designs (CCDs) are the most recommended types of RSM designs. In this research, CCD was applied for experimental design. Generally, CCD can be classified into three types, namely, circumscribed, inscribed, and face-centered CCD. Among these CCDs, the central composite inscribed (CCI) design is suitable for situations in which the limits of the factors are truly limits.³³ In this study, since the specified upper and lower limits of the parameters (factors) satisfy these conditions, the CCI design was used. The CCI design considers the upper and lower limits as the start points and creates a factorial or fractional factorial design within these limits.

The optimization procedure was designed based on the six-factor CCI design consisting of the thickness ratio ( $K$ ), die channel angle ( $ϕ$ ), die outer corner angle ( $ψ$ ), fillet radius ( $r$ ), friction coefficient between strip and rolls ( $μ$ ), and friction factor between strip and die ( $m$ ). Table 2 shows the levels of these factors and their coded values. The α value in this table is equal to 2.828, which was recommended by Design-Expert 7.0 software.³⁴ According to this table, it is necessary to define five levels for each factor in the CCI design. A total number of 86 samples with different combinations of factor levels were prepared in random order, as shown in Table 3.

Table 2.

Input factors and their coded and actual values used for optimization.

Input parameters	Symbol	Coded levels
		Axial (−α)	−1	0	1	Axial (+α)
Thickness ratio	A	0.9	0.92	0.935	0.95	0.975
Die channel angle (°)	B	90	102.93	110	117.07	130
Die outer corner angle (°)	C	0	24.24	37.5	50.76	75
Fillet radius (mm)	D	0.5	1.15	1.5	1.85	2.5
Friction coefficient between strip and rolls	E	0.08	0.12	0.14	0.16	0.2
Friction factor between strip and die	F	0.09	0.14	0.17	0.21	0.26

Table 3.

Six-factor CCI design with numerical and predicted values of PEEQ. Fact: Factorial points.

Standard order	Type	A	B	C	D	E	F	Actual plastic strain value	Predicted plastic strain value
1	Fact	−1	−1	−1	−1	−1	−1	1.048	1.07
2		1	−1	−1	−1	−1	−1	1.043	1.05
3		−1	1	−1	−1	−1	−1	0.7799	0.78
4		1	1	−1	−1	−1	−1	0.7487	0.76
5		−1	−1	1	−1	−1	−1	0.9657	0.98
6		1	−1	1	−1	−1	−1	0.9818	0.96
7		−1	1	1	−1	−1	−1	0.7434	0.74
8		1	1	1	−1	−1	−1	0.715	0.72
9		−1	−1	−1	1	−1	−1	1.153	1.13
10		1	−1	−1	1	−1	−1	1.119	1.11
11		−1	1	−1	1	−1	−1	0.8429	0.84
12		1	1	−1	1	−1	−1	0.8355	0.82
13		−1	−1	1	1	−1	−1	1.046	1.05
14		1	−1	1	1	−1	−1	1.038	1.03
15		−1	1	1	1	−1	−1	0.798	0.8
16		1	1	1	1	−1	−1	0.7893	0.79
17		−1	−1	−1	−1	1	−1	1.049	1.06
18		1	−1	−1	−1	1	−1	1.054	1.04
19		−1	1	−1	−1	1	−1	0.7804	0.77
20		1	1	−1	−1	1	−1	0.7488	0.75
21		−1	−1	1	−1	1	−1	0.961	0.97
22		1	−1	1	−1	1	−1	0.9918	0.95
23		−1	1	1	−1	1	−1	0.7443	0.73
24		1	1	1	−1	1	−1	0.7155	0.71
25		−1	−1	−1	1	1	−1	1.15	1.12
26		1	−1	−1	1	1	−1	1.11	1.1
27		−1	1	−1	1	1	−1	0.843	0.83
28		1	1	−1	1	1	−1	0.8336	0.81
29		−1	−1	1	1	1	−1	1.051	1.04
30		1	−1	1	1	1	−1	1.033	1.02
31		−1	1	1	1	1	−1	0.7996	0.8
32		1	1	1	1	1	−1	0.7619	0.78
33		−1	−1	−1	−1	−1	1	1.055	1.06
34		1	−1	−1	−1	−1	1	1.268	1.11
35		−1	1	−1	−1	−1	1	0.7818	0.77
36		1	1	−1	−1	−1	1	0.8023	0.82
37		−1	−1	1	−1	−1	1	0.9592	0.98
38		1	−1	1	−1	−1	1	1.049	1.02
39		−1	1	1	−1	−1	1	0.7448	0.74
40		1	1	1	−1	−1	1	0.8644	0.78
41		−1	−1	−1	1	−1	1	1.133	1.13
42		1	−1	−1	1	−1	1	1.164	1.17
43		−1	1	−1	1	−1	1	0.846	0.84
44		1	1	−1	1	−1	1	0.9091	0.88
45		−1	−1	1	1	−1	1	1.043	1.04
46		1	−1	1	1	−1	1	1.055	1.09
47		−1	1	1	1	−1	1	0.7969	0.8
48		1	1	1	1	−1	1	0.8677	0.85
49		−1	−1	−1	−1	1	1	1.064	1.05
50		1	−1	−1	−1	1	1	1.065	1.1
51		−1	1	−1	−1	1	1	0.7825	0.76
52		1	1	−1	−1	1	1	0.8181	0.81
53		−1	−1	1	−1	1	1	0.964	0.97
54		1	−1	1	−1	1	1	1.032	1.02
55		−1	1	1	−1	1	1	0.7455	0.73
56		1	1	1	−1	1	1	0.7565	0.77
57		−1	−1	−1	1	1	1	1.13	1.12
58		1	−1	−1	1	1	1	1.136	1.16
59		−1	1	−1	1	1	1	0.8545	0.83
60		1	1	−1	1	1	1	0.8757	0.87
61		−1	−1	1	1	1	1	1.052	1.03
62		1	−1	1	1	1	1	1.079	1.08
63		−1	1	1	1	1	1	0.7972	0.79
64		1	1	1	1	1	1	0.8163	0.84
65	Axial	−2.828	0	0	0	0	0	0.9229	0.9
66		2.828	0	0	0	0	0	0.9182	0.93
67		0	−2.828	0	0	0	0	1.332	1.36
68		0	2.828	0	0	0	0	0.578	0.61
69		0	0	−2.828	0	0	0	0.9701	1
70		0	0	2.828	0	0	0	0.8494	0.83
71		0	0	0	−2.828	0	0	0.8077	0.83
72		0	0	0	2.828	0	0	1.016	1
73		0	0	0	0	−2.828	0	0.8731	0.93
74		0	0	0	0	2.828	0	0.8819	0.9
75		0	0	0	0	0	−2.828	0.8804	0.87
76		0	0	0	0	0	2.828	0.9172	0.96
77	Center	0	0	0	0	0	0	0.8861	0.92
78		0	0	0	0	0	0	0.8861	0.92
79		0	0	0	0	0	0	0.8861	0.92
80		0	0	0	0	0	0	0.8861	0.92
81		0	0	0	0	0	0	0.8861	0.92
82		0	0	0	0	0	0	0.8861	0.92
83		0	0	0	0	0	0	0.8861	0.92
84		0	0	0	0	0	0	0.8861	0.92
85		0	0	0	0	0	0	0.8861	0.92
86		0	0	0	0	0	0	0.8861	0.92

The average PEEQs of the strips were considered as the desired response in our study. The values of the average PEEQs (actual value of the response, Table 3) were obtained by least-square fitting a straight line to the upper chunks of the PEEQ curves. The predicted values of the response were obtained using RSM. According to Table 3, 10 center points have been selected in our RSM design. The actual plastic strain values for these center points are identical because they were calculated in the absence of external noises.

ANOVA

A custom quadratic model was developed for the prediction of plastic strains. The ANOVA method was used to assess the fitness of the model to the numerical data. The results of the ANOVA analysis are listed in Table 4. In this table, the F-value is the ratio of the model mean square to the appropriate error mean square. Larger F-values indicate the likelihood that the variance contributed by the model is significantly larger than the random error.³⁵ The probability of a larger F-value (p-value or prob > F) is the probability of seeing the observed F-value if the null hypothesis is true (there is no factor effect). Small probability values call for rejection of the null hypothesis. In other words, if the p-value is very small (less than 0.05), then the terms in the model have a significant effect on the response. The p-value equals the proportion of the area under the curve of the F-distribution that lies beyond the observed F-value.³⁴

Table 4.

ANOVA table for the equivalent plastic strain.

Source	SS	DOF	Mean square	F-value	p-value (prob > F)
Model	1.62	9	0.18	210.53	<0.0001
A	3.91E−03	1	3.91E−03	4.56	0.036
B	1.41	1	1.41	1649.74	<0.0001
C	0.072	1	0.072	84.52	<0.0001
D	0.08	1	0.08	93.05	<0.0001
E	1.68E−03	1	1.68E−03	1.96	0.1659
F	0.016	1	0.016	18.87	<0.0001
AF	0.017	1	0.017	19.91	<0.0001
BC	0.01	1	0.01	12.05	0.0009
B²	8.69E−03	1	8.69E−03	10.14	0.0021
Residual	0.065	76	8.57E−04
Lack of fit	0.065	67	9.72E−04
Pure error	0.000	9	0.000
Cor total	1.69	85

SS: sum of squares; DOF: degree of freedom; Cor total: corrected total sum of squares.

In Table 4, we included only six independent factors (Table 2) and the significant quadratic terms, that is, AF, BC, and B². The F-value of the custom quadratic model is equal to 210.53, which implies that the model is robust. According to Table 4, the probability of occurrence for such a large F-value is less than 0.01% (p < 0.0001), mainly due to the noises that appear in the data acquisition.

The p-values less than 0.05 identify the significant terms. According to this criterion, A, B, C, D, F, AF, BC, and B² are significant terms. All the other terms (characterized by p-values greater than 0.05) were removed from the analysis.

The data provided by ANOVA were used to obtain a quadratic relationship between the response (PEEQ) and the significant terms A, B, C, D, F, AF, BC, and B² as follows

\begin{matrix} Equivalent plastic strain = 0.92 + 0.00699 A \\ - 0.13 B - 0.03 C + 0.032 D \\ + 0.014 F + 0.016 AF + 0.013 BC + 0.0086 B^{2} \end{matrix}

(5)

Equation (5) was used to predict the PEEQ values. A comparison between the predictions of equation (5) and the finite element results is shown in Table 3. One may note the high accuracy of the model. The coefficient of correlation (R²) for the model is equal to 0.9614, which is quite reasonable for the response surface. Another approach is to employ the sum of squares (SS) to test the statistical significance of the terms and the model. For this purpose, we can use

P % = \frac{S S_{Model}}{S S_{Total}} \times 100 or P % = \frac{S S_{Model terms}}{S S_{Total}} \times 100

(6)

so that P% is the percentage contribution of the model or model terms. The P% for the model is about 96.14%, which indicates that the conducted model has a high confidence level. In this regard, P% for “lack of fit” is about 3.86% meaning that the model is well reproducing the actual data. The same conclusion can be formulated with respect to the high values of R² (adjusted R² is 95.69%). Moreover, the p-values of the second-order terms, that is, AF, BC, and B² in equation (5), are <0.0001, 0.0009, and 0.0021, respectively. This indicates that they have a significant effect on the model.

Graphical interpretation of the models

Response contour and surface plots are useful tools to investigate the interaction and the individual effects of the process parameters on PEEQ. Figure 6 shows the combined effects of the thickness ratio (A) and friction factor between strip and die (F) on PEEQ. The die channel angle, die outer corner angle, fillet radius, and friction coefficient between strip and rolls were hold at 110°, 37.5°, 1.5 mm, and 0.14 (center point values), respectively. According to Figure 6, the highest value of PEEQ (1.09) was obtained at the highest values of the thickness ratio (0.97) and friction factor between strip and die (0.26). The PEEQ usually increases with the increase in the thickness ratio and the friction between strip and die. But Figure 6 unexpectedly shows that the PEEQ decreases at lower thickness ratios with an increase in the friction between strip and die. Also, this figure shows that the PEEQ decreases at the lower friction factors with an increase in the thickness ratio. This indicates that there is an interaction between the friction factor and the thickness ratio in the ECAR process. With an increase in the thickness ratio, two events will happen. First, the shear strain and consequently the PEEQ will increase due to the increase in the difference between the inner and the outer arc lengths at the shear deformation region. Second, the flow of the material through shear deformation region will be facilitated due to the decrease in the required thickness recovery, which results in a lower value of the PEEQ. So it seems that at a lower value of the friction factor between the strip and the die, the PEEQ is more influenced by the second circumstance, and hence, its value decreases with the increase in the thickness ratio. As mentioned above, Figure 6 shows that at the lower values of the thickness ratio, the PEEQ decreases slightly with the increase in the friction factor. Such a behavior could be the result of some changes in shear strain with respect to different process parameters.³⁶ The explicit discussion about the other possible reasons requires more investigations.

Figure 6.

(a) Response surface and (b) contour plots of the equivalent plastic strain with A and F factors.

In Figure 6, the blue and red colors are related to the minimum (0.578) and maximum (1.332) values of PEEQ. The green color, which is dominant in Figure 6, shows the average value of PEEQ so that in these ranges of thickness ratio and friction factor between strip and die (other parameters are fixed at the center point), an average PEEQ can be achieved.

Figure 7 shows the combined effects of the die channel angle (B) and die outer corner angle (C) on PEEQ. The thickness ratio, fillet radius, friction coefficient between strip and rolls, and friction factor between strip and die were hold at 0.94, 1.5 mm, 0.14, and 0.17 (center point values), respectively. As one may note that the highest value of PEEQ (1.55) corresponds to the lowest values of the die channel angle (90°) and the die outer corner angle (0°). However, there are some limitations for the selection of these angles; inappropriate choice of these angles (beside the values of other parameters) increases the forming forces extensively and can prevent the strip from feeding into the die channel in some cases. This causes the strip dragging at the intersection of the die channels and the failure of the ECAR process. Therefore, these limitations must be considered when selecting the die channel and corner angles. Figure 7 also shows that the PEEQs change in a wider range as compared to Figure 6 and depicts that the die channel angle is the most significant factor in the ECAR process.

Figure 7.

(a) Response surface and (b) contour plots of the equivalent plastic strain with B and C factors.

Optimization

The optimum values of the thickness ratio, die channel angle, die outer corner angle, fillet radius, friction coefficient between strip and rolls, and friction factor between strip and die were obtained using the data generated by the statistical analysis. Design-Expert searches for a combination of factors that simultaneously satisfy the constraints set for each of the response and factors. These conditions are stated in Table 5. The optimum values of the factors (in their specified ranges) led to the maximum value of PEEQ. These optimum values are listed in the last column of Table 5. In this condition, the predicted PEEQ (1.53) shows an improvement of about 14.86% as compared to the highest value of PEEQ (1.332). It is also worthwhile mentioning that the possibility of accomplishing the ECAR process must be considered in the selection of the optimum values of the parameters. For this purpose, finite element simulation of the ECAR process was performed using the optimum values of the parameters. Figure 8(a) shows the PEEQ curve obtained from this numerical simulation. Furthermore, Figure 8(b) illustrates the PEEQ distribution at the strip after the optimization. The accomplishment of the finite element simulation and the reasonable agreement between the predictions of the optimization procedure and the finite element simulation (∼4% relative error) prove the validity of the obtained optimum values. On the other hand, the values of the parameters, which lead to the maximum PEEQ in Figure 7 (1.55), are not acceptable. Because in this case, the finite element simulation of the ECAR process aborts due to the failure in feeding the strip into the ECAR die.

Table 5.

Optimization goal and constraints for all factors and the response.

Input parameters	Goal	Lower	Upper	Optimum
Thickness ratio	In range	0.9	0.975	0.96
Die channel angle (°)	In range	90	130	90.10
Die outer corner angle (°)	In range	0	75	21.21
Fillet radius (mm)	In range	0.5	2.5	2.31
Friction coefficient between strip and rolls	In range	0.08	0.2	0.20
Friction factor between strip and die	In range	0.09	0.26	0.19
Equivalent plastic strain	Maximize	0.578	1.332	1.53

Figure 8.

(a) PEEQ along the middle path of strip and (b) the distribution of the PEEQ after the optimization.

Conclusion

The numerical simulation of the ECAR process was conducted utilizing 2D finite element analysis. The RSM and the ANOVA were employed to study the effects of the ECAR parameters on the induced PEEQ. On the basis of this analysis, the following conclusions can be formulated:

A quadratic model is able to predict the PEEQ by considering the significant terms provided by ANOVA. This model can be used as an efficient tool to evaluate the PEEQ with high accuracy.

ANOVA showed that the thickness ratio, die channel angle, die outer corner angle, fillet radius, and friction factor between strip and die have significant effects on the PEEQ induced during the ECAR processes. Furthermore, the contour and surface plots of PEEQ show that the die channel angle and die outer corner angle have the most important influence on PEEQ (die channel angle is the most significant parameter).

Optimum values of ECAR parameters, which result in maximum PEEQ, were derived. The optimum values of the parameters are $K = 0.96$ , $ϕ = 90.10$ , $ψ = 21.21$ , $r = 2.31 mm$ , $μ = 0.20$ , and $m = 0.19$ , which lead to an increase of about 14.86% in the maximum level of PEEQ.

Footnotes

Declaration of conflicting interests

The authors declare that there is no conflict of interest.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

References

Valiev

Langdon

. Principles of equal-channel angular pressing as a processing tool for grain refinement. Prog Mater Sci 2006; 51: 881–981.

Segal

. Materials processing by simple shear. Mat Sci Eng A: Struct 1995; 197: 157–164.

Furukawa

Utsunomiya

Matsubara

. Influence of magnesium on grain refinement and ductility in a dilute Al–Sc alloy. Acta Mater 2001; 49: 3829–3838.

Nakashima

Horita

Nemoto

. Development of a multi-pass facility for equal-channel angular pressing to high total strains. Mat Sci Eng A: Struct 2000; 281: 82–87.

Valiev

Islamgaliev

Alexandrov

. Bulk nanostructured materials from severe plastic deformation. Prog Mater Sci 2000; 45: 103–189.

Semiatin

DeLo

Shell

. The effect of material properties and tooling design on deformation and fracture during equal channel angular extrusion. Acta Mater 2000; 48: 1841–1851.

Djavanroodi

Sedighi

Hashemi

. Equal channel angular pressing of copper wire. Proc IMechE, Part B: J Engineering Manufacture 2014; 228(3): 1–7.

Shin

Kim

. Microstructural evolution in a commercial low carbon steel by equal channel angular pressing. Acta Mater 2000; 48: 2247–2255.

Meyers

Mishra

Benson

. Mechanical properties of nanocrystalline materials. Prog Mater Sci 2006; 51: 427–556.

10.

Tamimi

Ketabchi

Parvin

. Microstructural evolution and mechanical properties of accumulative roll bonded interstitial free steel. Mater Design 2009; 30: 2556–2562.

11.

Cheng

Chen

Xia

. Effect of channel clearance on crystal orientation development in AZ31 magnesium alloy sheet produced by equal channel angular rolling. J Mater Process Tech 2007; 184: 97–101.

12.

Chakkingal

Suriadi

Thomson

. Microstructure development during equal channel angular drawing of Al at room temperature. Scripta Mater 1998; 39: 677–684.

13.

Saito

Tsuji

Utsunomiya

. Ultra-fine grained bulk aluminum produced by accumulative roll-bonding (ARB) process. Scripta Mater 1998; 39: 1221–1227.

14.

Huang

Zhu

Jiang

. Microstructures and dislocation configurations in nanostructured Cu processed by repetitive corrugation and straightening. Acta Mater 2001; 49: 1497–1505.

15.

Saito

Utsunomiya

Suzuki

. Improvement in the r-value of aluminum strip by a continuous shear deformation process. Scripta Mater 2000; 42: 1139–1144.

16.

Lee

Seok

Suh

. Microstructural evolutions of the Al strip prepared by cold rolling and continuous equal channel angular pressing. Acta Mater 2002; 50: 4005–4019.

17.

Huang

Prangnell

. Continuous frictional angular extrusion and its application in the production of ultrafine-grained sheet metals. Scripta Mater 2007; 56: 333–336.

18.

Saray

Purcek

Karaman

. Equal-channel angular sheet extrusion of interstitial-free (IF) steel: microstructural evolution and mechanical properties. Mat Sci Eng A: Struct 2011; 528: 6573–6583.

19.

Raab

Valiev

Lowe

. Continuous processing of ultrafine grained Al by ECAP–conform. Mat Sci Eng A: Struct 2004; 382: 30–34.

20.

Lee

Seok

Han

. Controlling the textures of the metal strips via the continuous confined strip shearing (C2S2) process. Mater Res Bull 2001; 36: 997–1004.

21.

Srinivasan

Cherukuri

. Optimization of the equal channel angular pressing (ECAP) process for strain homogeneity. Mater Sci Forum 2007; 539: 3655–3660.

22.

Ebrahimi

Rajabifar

Djavanroodi

. New approaches to optimize strain behavior of Al6082 during equal channel angular pressing. J Strain Anal Eng 2013; 48: 395–404.

23.

Luri

Luis Pérez

. Analysis and modelling by finite element method of the equal channel angular extrusion pressure. Proc IMechE, Part B: J Engineering Manufacture 2010; 224: 925–935.

24.

Zhao

Ren

. Numerical investigation of aluminum deformation behavior in three-dimensional continuous confined strip shearing process. Mat Sci Eng A: Struct 2008; 476: 281–289.

25.

Wei

Zhang

Wei

. Finite element analysis of deformation behavior in continuous ECAP process. Mat Sci Eng A: Struct 2009; 516: 111–118.

26.

Kvačkaj

Kováčová

. Finite element analysis of deformation behaviour in equal channel angular rolling process. Acta Metall Slov 2010; 16: 84–90.

27.

Venkateswarlu

Davidson

Tagore

GRN

. Analysis of sheet metal formability studies of friction stir processed Mg AZ31B alloy using response surface methodology. Proced Eng 2012; 38: 2228–2236.

28.

Velmanirajan

Thaheer

ASA

Narayanasamy

. Numerical modelling of aluminium sheets formability using response surface methodology. Mater Design 2012; 41: 239–254.

29.

Naceur

Guo

Ben-Elechi

. Response surface methodology for design of sheet forming parameters to control springback effects. Comput Struct 2006; 84: 1651–1663.

30.

Jeswiet

. A comparison of friction coefficients in cold rolling. J Mater Process Tech 1998; 80–81: 239–244.

31.

Iwahashi

Wang

Horita

. Principle of equal-channel angular pressing for the processing of ultra-fine grained materials. Scripta Mater 1996; 35: 143–146.

32.

Chung

Park

Lee

. An analysis of accumulated deformation in the equal channel angular rolling (ECAR) process. Met Mater Int 2006; 12: 289–292.

33.

Prasad

Hassan

Yang

. Response surface optimisation for the extraction of phenolic compounds and antioxidant capacities of underutilized Mangifera pajang Kosterm. Peels. Food Chem 2011; 128: 1121–1127.

34.

Design-Expert^® Software. Trial version 7.0, user’s guide. Stat-Ease, Inc: Minneapolis, MN, 2005.

35.

Anderson

Whitcomb

. RSM simplified: optimizing processes using response surface methods for design of experiments. New York: Productivity Press, 2005.

36.

Oruganti

Subramanian

Marte

. Effect of friction, backpressure and strain rate sensitivity on material flow during equal channel angular extrusion. Mat Sci Eng A: Struct 2005; 406: 102–109.

Modeling and optimization of equivalent plastic strain in equal-channel angular rolling using response surface methodology

Abstract

Keywords

Introduction

Finite element analysis

Central composite design

ANOVA

Graphical interpretation of the models

Optimization

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References