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Optimum processing parameters for equal channel angular pressing
Mechanics of Materials 100 (2016) 1–11
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Research paper
Optimum processing parameters for equal channel angular pressing M.A. Agwa∗, M.N. Ali, Amal E. Al-Shorbagy Department of Mechanical Design and Production Engineering, Faculty of Engineering, Zagazig University, 44519 Zagazig, Egypt
a r t i c l e
i n f o
Article history: Received 4 July 2015 Revised 24 May 2016 Available online 7 June 2016 Keywords: Equal channel angular pressing Finite element method Strain coefficient of variance Channel angle Corner angle Friction
a b s t r a c t This study aims to explore the optimum processing parameters for minimum strain coefficient of variance in equal channel angular pressing based on the analysis of whole steady zone. In order to design the die correctly, it is important to understand the effect of die geometry and processing parameters on the deformation behavior, strain distribution and punch load. The influences of channel angle, corner angle and friction coefficient are analyzed using finite element modeling. Analytical models are also used to validate the obtained computational solutions. Results showed that channel angle, corner angle and friction coefficient have significant influences on deformation behavior, strain coefficient of variance and punch load evolution. Among all the combinations of 7 corner angles, 6 channel angles and 6 friction coefficients (set of 252 study cases), it is found that the optimum parameters for maximum strain homogeneity correspond to channel angle φ = 90o , corner angle ψ = 15o and friction coefficient μ = 0.3. The influence of friction coefficient on punch load is more pronounced than the influence of die geometry. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Equal channel angular pressing (ECAP) is an effective severe plastic deformation (SPD) technique that increases the material strength by accumulating a very large plastic strain into the workpiece without changing its cross-sectional area (Srinivas et al., 2013). The deformation of the billet is achieved by a simple shear taking place in a thin layer at the crossing plane of the equal channels (Segal, 1995). The cross section of workpiece remains the same during ECAP so the process can be repeated until the accumulated deformation reaches a desired level. Therefore, high strain, improved mechanical and physical properties, extreme fine grains and varying textures can be obtained in the processed materials (Arab et al., 2014). Valiev and Langdon (2006) showed that the formation of ultrafine grains in metals and alloys underlies a very significant enhancement in their mechanical and functional properties. In addition, different microstructures and mechanical properties can also be produced by changing the strain path in the billet, which in turn is achieved by changing the orientation of the billet from one pass to the next (Jin et al., 2009). The use of finite element method in the simulation and analysis of technological processes is constantly increasing. It comes as a powerful tool to predict and study the plastic behavior of the billet during ECAP. Many authors devoted their research to finite element modeling, simulation and analysis of ECAP for better understating
∗
Corresponding author. Tel.:+201110015465; fax: +20552304987. E-mail address: [email protected] (M.A. Agwa).
http://dx.doi.org/10.1016/j.mechmat.2016.06.003 0167-6636/© 2016 Elsevier Ltd. All rights reserved.
of the deformation behavior and strain distribution (see, for example, (Nagasekhar and Yip Tick-Hon, 2004)). Li et al. (2004) formulated a comprehensive finite element model to analyze the formation of the plastic deformation zone (PDZ) and evolution of the working load with ram displacement during a single pass of ECAP with 90o intersection angle. They stated that the coefficient of friction has negligible effect on strain distribution. Fuqian Yang et al. (2005) studied the effect of ECAP process variables on the evolution of plastic deformation and simulated the deformation of materials subjected to multi-pass ECAP by using multi-bent extrusion process. Yi-Lang Yang and Shyong Lee (2003) have reported that the strain is independent of friction for a strain hardening material. Balasundar and Raghu (2010) studied the effect of Coulomb and shear friction models on the deformation pattern, strain distribution and load. Patil Basavaraj et al. (2010) found that the outer corner has a significant influence on the distribution and inhomogeneity of strain in the body of workpiece. Aluminum alloy 6082 is a medium strength alloy with excellent corrosion resistance, machinability and weldability (Cerri et al., 2009). Alloy 6082 is widely used in highly stressed application, trusses, bridges and transport applications. The addition of Zr to this alloy plays a critical role by providing particles which impede grain growth at elevated temperature to enhance superplasticity (Lee et al., 2002). In this paper we give continuity to the above-mentioned researches by investigating the influence of ECAP processing parameters on the optimum strain homogeneity of modified 6082 Alu-
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It is important to know the strain in these regions in order to optimize the processing parameters. Owing to the complexity involved in plastic deformation of ECAP process, the closed form solution for the governing equations usually do not exist. Certain simplifications must be made in order to find analytical methods to obtain partial information about the mechanics of ECAP process. Segal et al. (1997) derived a path independent strain relationship for a sharp corner (ψ = 0) configuration with die/channel angle φ according to which the effective strain (equivalent plastic strain) ε is given as:
ε
2 = √ cot 3
φ 2
.
(1)
Another formula for the equivalent plastic strain ε in the material is calculated by Iwahashi et al. (1996) as:
ε
1 = √ 2 cot 3
φ+ψ
+ ψ csc
2
φ+ψ
.
2
(2)
Goforth et al. (20 0 0) proposed the upper bound model for predicting the ECAP of strain-hardening materials as:
1
ε= √
3
Fig. 1. Schematic diagram of ECAP process.
2 cot
•
•
Head: the front of the workpiece; Tail: the region at the back of the workpiece; Steady state zone: the region between the non-uniformly deformed head and tail regions where the strain is relatively uniform along the billet axis.
+ ψ sec2
2
γ = 2 tan
•
φ−ψ
φ−ψ 2
.
(4)
Model B: which was found to yield better predictions for the strains as compared to the model of Iwahashi et al. (1996) for dies with a fillet radius under frictionless conditions. Model B states that:
Fig. 2. The different regions of the deformed workpiece.
•
(3)
Model A: which was found to be better in predicting the strain generated in dies with external arc of curvature. In this model, the shear strain (γ ) is given as:
Fig. 1 is a schematic illustration of ECAP process in which a billet is pressed through a die that consists of two equal crosssectional channels intersecting at an angle (φ ) and with an outer corner angle (ψ ). As shown in Fig. 2, the workpiece can be divided into three main regions:
2
+ψ .
Note that the last two models tend to Segal model (Eq. (1)) for
γ = 2 tan
2. Analytical solutions of the effective plastic strain
ψ = 0. Milind and Date (2012) derived two analytical models for the relation between shear strain (γ ) and the ECAP angles φ and ψ . These two models are: •
minium alloy billet. The influences of channel angle (φ ), corner angle (ψ ) and friction coefficient (μ) between the die and the billet on the deformation behavior, equivalent plastic strain, strain homogeneity and punch load are studied. The paper proceeds as follows. In the next section, the analytical solutions of the effective plastic strain are described. Section 3 sheds light on the finite element model used in the analysis. In Section 4, the results of the finite element simulation are presented and discussed. Finally, Section 5 concludes.
φ+ψ
φ−ψ 2
+
sin ψ /2 × φ. cos(φ /2 − ψ /2 ) sin φ /2
(5)
According to Segal et al. (1997), the relationship between shear strain (γ ) and effective strain (ε ) in ECAP process is given as ε = √γ . Note that all expressions did not take into account the ef3
fects of friction, strain hardening, strain distribution and deformation gradient (Cerri et al., 2009). 3. Finite element analysis The finite element method is a numerical analysis procedure used to obtain approximate solutions to boundary value problems, which are found in every field of engineering. Most problems encountered in everyday practical applications do not have closedform solution. The accuracy of the simulations depends on the assumed boundary conditions and how well these conditions describe the real process. Many process parameters, such as contact, thermal and frictional conditions, are difficult to be determined correctly. The models developed in this study take into account the three sources of non-linearities present in ECAP problem, namely, material, geometry and boundary conditions. The ECAP die and the geometry of the workpiece are shown in Fig. 1. The mesh procedure is used to divide the model into small elements. The smaller these elements are, the more accurate the simulation will be. The disadvantage with smaller elements is
M.A. Agwa et al. / Mechanics of Materials 100 (2016) 1–11 Table 1 Properties of Aluminum alloy (6082Zr) (Cerri et al., 2009). Material property
Value
Young’s modulus Poisson ratio Density Strain hardening exponent (n) Strength coefficient (K)
69 GPa 0.33 2710 Kg/m3 0.23 291 MPa
3
components of its displacement vector u p (t ) = {un (t ) ut (t )}T , of p p its velocity vector u˙ p (t ) = {u˙ n (t ) u˙ t (t )}T and of its reaction vector p p p T r (t ) = {rn (t ) rt (t )} satisfy the Signorini unilateral contact conditions: p
unp (t ) − dnp ≤ 0,
rnp (t ) ≤ 0,
3.1. Contact, friction and material models The interaction between contacting bodies is defined by assigning a contact property model to the contact interaction. The contact property models available in Abaqus are mechanical, thermal, electrical and pure fluid contact models. The mechanical contact model which include friction model is used to define the force resisting the relative tangential motion of the surface. The possible friction models in Abaqus are frictionless, penalty, static-kinetic exponential decay, rough and Lagrange multiplier. It is currently widely accepted that Coulomb’s law is an appropriate way to describe the frictional phenomenon in many situations. Moreau (1986) stated that although in many technical situations for which Coulomb’s law may yield only moderately accurate results, it simply cannot be substituted since, quite often in practice, one does not have access to relevant properties of the contacting surfaces that would furnish a more refined knowledge of the frictional phenomenon. Many authors consider that it is adequate to use Coulomb’s model in many situations (Agwa, 2011). For any contact candidate particle, the normal (n) and tangential (t)
(unp (t ) − dnp )rnp (t ) = 0,
(6)
where dn is the initial normal gap between particle p and the obstacle (Fig. 3(a)). Coulomb’s friction law (Fig. 3(b)) may be written as a conjunction of an inequality and a nonûsmooth equation
|rtp (t )| + μrnp (t ) ≤ 0, the substantial increase in execution time. Different meshes were tested to find an optimum mesh density. The commercial finite element code Abaqus/Explicit was used to carry out all simulations. Three factors were varied in the simulations: channel angle (φ ), corner angle (φ ) and friction coefficient (μ). Channel angle (φ ) ranges from 75o to 150o , corner angle (ψ ) ranges from 0o to 90o with an increment of 15o for both angles and friction coefficient (μ) from 0 to 0.3. Billet with dimension of (10 mm × 10 mm × 60 mm) was modeled for 2D plane strain simulation with elastic-plastic material properties of Aluminum alloy with zirconium addition (6082Zr). Strain hardening material properties of the modified 6082 Aluminum alloy used for simulations are shown in Table 1 (Cerri et al., 2009). The billet material was modeled with four-node bilinear plane stain quadrilateral (CPE4R) element (Abaqus, 2009). The die and punch were modeled as analytical rigid elements. All simulations were performed with speed of 1 mm/s (Nagasekhar et al., 2005). Due to the strong non-linearities caused by friction, contact, large deformation and elastic-plastic behavior, one of the main challenges in this type of computational analysis is the convergence to a stable solution. For such complicated problems Abaqus/Standard may not be able to find a converging solution and will fail in that case (Oldfield et al., 2005). In this study, Abaqus/Explicit was used to model the ECAP problem. The load steps used in the ECAP analysis were chosen to be very small (Sheng et al., 2005). This allows such complicated problems, whose solutions cannot be found by Abaqus/Standard, to be solved by Abaqus/Explicit. The main issue with Abaqus/Explicit is its much greater computing load due to the very small load increments/time steps used for assuring convergence. Smooth fillet radius of 0.75 mm at the inner corner was chosen to avoid the convergence problems (Nagasekhar et al., 2007). Mass scaling was used for all simulation to prevent failure of the mesh during large deformation and to reduce the time of computation.
p
rtp (t )u˙ tp (t ) − μrnp (t )|u˙ tp (t )| = 0.
(7)
Abaqus plasticity models are usually formulated in terms of a yield surface, a flow rule and hardening law. The classical metal plasticity model allows to define the yield and inelastic flow of a metal at relatively low temperature, where loading is relatively monotonic and creep effects are unimportant. In the classical plastic behavior of a material, the von Mises yield surface is used. The von Mises yield criterion states that plastic yielding occurs, when √ the octahedral shearing stress reaches a critical value (σy / 3), where σ y is the yield strength. For von Mises plasticity it can be p shown that the equivalent plastic strain rate ε˙ is defined as
p
ε˙ =
2 p ε˙ : ε˙ p . 3
(8)
The nonlinear material response was modeled using isotropic hardening with a von Mises yield criterion. With isotropic hardening the metalÆs yield stress evolves as the metal accumulates plastic strains σy (ε p ). Nonlinear empirical idealization of the plastic hardening, in most cases, provides more accurate prediction of the material behavior. The most commonly used form for the nonlinear isotropic hardening rule is the power law given as:
n σy ( ε p ) = σy 0 + K ε p ,
(9)
where σ y0 is the initial yield stress. In this study, the penalty option is used that based on the Coulomb friction model. The friction coefficient was supposed to be isotropic. The value of friction coefficient between the billet and die was assumed to range from 0 to 0.3. The elastic-plastic behavior of the material is modeled by the linear elastic model and the classical metal plasticity model. 3.2. Model verification and mesh selection In order to verify that the ECAP problem is modeled correctly, a test case with the same parameters of Fig. 8 (a) in Cerri et al. (2009) was solved and compared in terms of contour plot, strains, stresses and deformation shape (see Fig. 4(a,b)). Good matching was obtained between the two results. The percentage error ranges from 2% for frictional cases to 5% for frictionless cases. This error may be attributed to factors such as (i) the difference in fillet radii at inside and outside corners of the die that was not specified by Cerri et al. (2009), (ii) the different number of load increments/time steps and (iii) the difference in the finite arithmetic precision of the computers used which resulted in different roundoff errors that may propagate due to the large number of time increments. Before performing the finite element simulation, a convergence test was carried out for channel angle (φ = 90o ), corner angle (ψ = 0o ) and friction coefficient (μ = 0) to assess the sensitivity of the results to mesh refinement. Five studies for different number of elements (840, 170 0, 320 0, 660 0 and 13,160 elements) were carried out to calculate the average and maximum equivalent plastic strain in the steady state region as shown in Fig. 5. The differences between the average plastic strain for both 840 and 1700 meshes
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Fig. 3. Graphs of (a) the unilateral contact law and (b) the friction law of Coulomb.
Fig. 4. Comparison between the equivalent plastic strain at integration points (PEEQ) (a) the present work and (b) Fig. 8 (a) of Cerri et al. (2009), both under the same conditions.
Fig. 5. Convergence test for different number of elements.
with 13,160 mesh are significant compared with the differences between the average plastic strain for both 3200 and 6600 meshes with 13,160 mesh (see Fig. 5 (a)). The same observation can be obtained in Fig. 5 (b) but for the maximum equivalent plastic strain. Accordingly, the mesh consists of 3200 elements was finally chosen to provide adequate precision in the simulations and related calculations. 4. Results and discussion 4.1. Deformation behavior The deformation behavior was examined in the whole steady state zone after one pass of ECAP. The influence of both channel
angle and corner angle under frictionless (μ = 0) and friction (μ = 0.2) conditions on deformation behavior is investigated. 4.1.1. Combined effect of channel angle and friction Fig. 6 shows the influence of channel angle (φ ) on the deformation behavior. During simulations there are two gaps appeared, namely, corner and channel gaps (see Fig. 2). These gaps are a normal consequence of billet bending during deformation. The head of billet bends toward the top of outlet channel, which leads to the formation of corner gap, while the contact between the side surface of head region and die channel leads to the formation of channel gap. Under frictionless condition, the increasing in the channel angle was accompanied by increasing in the corner gap found at the die corner (Fig. 6 (a)–(d)). The front end of the deformed sam-
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Fig. 6. Equivalent plastic strain distribution for ψ = 0o and different channel angles indicated below each curve. Left: frictionless (μ = 0). Right: friction (μ = 0.2).
ple is more uniform for the smaller channel angles 75o and 90o (Fig. 6 (a)–(b)) than the other two angles 105o and 120o (Fig. 6 (c)–(d)) accordingly, wastage of the sample during multiple ECAP passes for both 75o , 90o channel angle is less than the other angles. With the adding of friction, the corner gap begins to disappear which may be attributed to the increased constraint frictional drag at the billet surface. This drag force acts like a back pressure leading to the filling of the corner gap and makes the material flow fits better the die outer corner. The nonuniform ends of the billet at channel angle greater than 90o change slightly because friction reduces the upward bending of the sample (Fig. 5 (e)–(h)). From the above discussion, it can be recognized that (i) the acute and
right channel angles are preferred than the other obtuse angles; (ii) friction reduces the corner gap significantly. 4.1.2. Combined effect of corner angle and friction The influence of corner angle (ψ ) on the deformation behavior is shown in Fig. 7. Under frictionless condition, as corner angle increases, the corner gap decreases and the channel angle formed at the outlet channel increases (Fig. 7 (a)–(d)). With the increase of corner angle, the plastic deformation zone (PDZ) (see Fig. 2) begins to spread and covers the corner angle in which it was found to be narrow for sharp corner angles. It may be observed that the front end of the sample started to bend upward and generated nonuniform deformation at the end of the billet with the increase of corner angle. Minimizing of this nonuniform region (front end)
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M.A. Agwa et al. / Mechanics of Materials 100 (2016) 1–11
Fig. 7. Equivalent plastic strain distribution for φ = 90o and different corner angles indicated below each curve. Left: frictionless (μ = 0). Right: friction (μ = 0.2).
is of interest for maximizing the length of the useful homogeneous portion of the billet. Under friction condition, both corner gap and channel gap decrease and the tip of the billet begin to drop down compared with frictionless condition. From the above discussion it may be concluded that, the smaller corner angles are preferred than the other angles. 4.2. Average effective strain in steady zone The average effective plastic strain (ε ave ) from finite element analysis can be calculated by taking the average of the strain values p over the whole steady state area in plane strain conditions. ε ave is computed over the steady zone area IJKL ( see Fig. 2) as: p
εapve =
n 1 p εi , n
(10)
i=1
where n is the number of integration points in the steady zone and εip is the equivalent plastic strain at integration point i. Fig. 8 shows the relation between the average equivalent plastic strain and friction coefficient for channel angle ranges from 75o to
120o and corner angle from 0o to 90o , both with increment of 15o . It can be observed that the average equivalent plastic strain decreases significantly with the increase of both channel and corner angles. The flow of the material across the die becomes smoother and the deformation begins to be far from the severe plastic deformation as the channel and corner angles increase. The effect of corner angle on the average equivalent plastic strain becomes insignificant as the channel angle increases, this gives an indication that the corner angle affects only in the cases of small and sharp channel angles (75o , 90o ). With the increase of friction, the average equivalent plastic strain increases, but it can be observed from Fig. 8 that the effect of friction is relatively small compared with the effect of both channel and corner angles. The strain obtained by the analytical models (Eqs. (2)–(5)) and the average equivalent plastic strain from the simulation for different channel angles 75o , 90o , 105o and 120o at different corner angles under frictionless condition are shown in Fig. 9. The analytical and numerical strain results are matching well for φ = 90o which validates the results obtained by simulation. But, as we may notice, there is a small difference between them. This difference is due to
M.A. Agwa et al. / Mechanics of Materials 100 (2016) 1–11
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Fig. 8. Values of the average equivalent plastic strain with friction coefficient for different channel and corner angles.
the lacks that found in the analytical models in which it didn’t take into account the effect of strain hardening, strain distribution and deformation gradient assuming a homogeneous value of strain in the whole workpiece. For φ = 75o , the two models (A and B) of Milind and Date (2012) gave lower values than the other models (FEM, Iwahashi and Goforth) as shown in Fig. 9 (a). However, the two models (A and B) gave higher values than the other models for φ = 105o and 120o (Fig. 9 (c,d)). This gives indication that the models of Milind and Date (2012) are valid only for φ = 90o (Fig. 9 (b)). From the same figure we conclude that, as the channel and corner angles increase, the equivalent plastic strain decreases and this is the same observation that obtained from Fig. 8. 4.3. Strain coefficient of variance
p p ε max − ε min , ε apve
(11)
where ε max and ε min denote, respectively, the maximum and the minimum equivalent plastic strain selected from all strain values p computed at integration points over the steady state area IJKL. ε ave p
p
CVε p =
Std ε p
ε apve
,
(12)
where standard deviation Std ε p is a statistical term that measures the amount of variability or dispersion of equivalent plastic strain p at each integration point around the average effective strain (ε ave ). Dispersion is the difference between the actual value and the average value. Mathematically standard deviation is the square root of the average of the squared differences from the average equivalent p plastic strain ε ave ;
Although the analytical models give useful quantitative results, they just represent the average effective strain developed in the workpiece during ECAP. Neither the analytical models nor the experimental methods can show the distribution of the effective strain in the workpiece after ECAP or inhomogeneous deformation behavior. The degree of strain inhomogeneity can be estimated across the area IJKL of the steady zone using the equation defined by Li et al. (2004):
C=
denotes the average of the equivalent plastic strain computed from Eq. (10). Another expression for quantifying the strain inhomogeneity in a better way was adopted by Zaïri et al. (2006). It is defined by coefficient of variance of strain (CVε p ) given as:
Std ε = p
n 1 p (ε i − ε apve )2 n
12
.
(13)
i=1
The strain inhomogeneity defined by CVε p is better than that defined by C because it uses standard deviation, which is based on the distribution of strain taking into account the value of strain at all integration points in the section, while C based only on the difference between two extreme strain values. Consequently the most appropriate way of quantifying the strain inhomogeneity is given by the coefficient of variance CVε p . Fig. 10 illustrates the influence of channel angle, corner angle and friction on strain homogeneity by using coefficient of variance presented by Eq. (13). The lower values of coefficient of variance indicate better homogeneity. Strain homogeneity in ECAP is mainly dependent on the simple shear deformation. According to the the-
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Fig. 9. Values of the average equivalent plastic strain from numerical solution and analytical models under frictionless condition for different channel and corner angles.
ory of the ECAP (Segal, 1995), simple shear is confined to the thin layers at the crossing plane of the channels in which the contour lines of plastic strain are concentrated on the plane connecting the two corners of the channel. With the increase of corner angle, the plastic deformation zone widened and covered the corner angle (Fig. 7), the region of nonuniform deformation spreads out from the lower part of the billet toward the upper part and the deformation began to deviate from the simple shear resulting in decreasing the strain homogeneity. It can be observed from Fig. 10 that as corner angle increases the strain inhomogeneity increases in most of the cases except for ψ = 0o at channel angles of 75o , 90o and 105o (Fig. 10 (a)–(c)). From the same figure we conclude that, as the channel angle increases, the effect of corner angle on coefficient of variance becomes insignificant. Fig. 10 also illustrates that, friction coefficient has a significant effect on the strain homogeneity in case of smaller corner angles. The search for optimum parameters for maximum strain homogeneity is equivalent to solution of a set of problems. Among all the combinations of 7 corner angles, 6 channel angles and 6 friction coefficients (7 × 6 × 6 = 252 study cases), Table 2 presents the selected optimum parameters for minimum value of the coefficient of variance. The values in the table are arranged according to the coefficient of variance of strain (CVε p ); such that the smaller the coefficient of variance (CVε p ) the better the strain homogeneity. From Table 2, it can be drawn that small corner angles and acute and right channel angles are recommended to obtain optimum strain homogeneity. It may also be observed that moderate and high coefficients of friction help in improving the strain homogeneity. The global optimum parameters are those of case No. 1, which correspond to channel angle φ = 90o , corner angle ψ = 15o and coefficient of friction μ = 0.3.
Table 2 The selected optimum processing parameters (φ , ψ , μ) for maximum strain homogeneity conditions in the steady state zone IJKL. No. 1 2 3 4 5 6
φ
ψ o
90 75o 105o 120o 135o 150o
o
15 15o 15o 0o 0o 0o
μ
εapve
C
Std ε p
CVε p
0.3 0.1 0.3 0.15 0.3 0.3
1.0563 1.3531 0.8175 0.6490 0.4660 0.2962
0.6575 0.5142 0.7903 0.9924 1.0922 1.3031
0.0839 0.1186 0.0931 0.0854 0.0770 0.0599
0.0794 0.0877 0.1139 0.1316 0.1652 0.2021
4.4. Punch load evolution In this subsection, we investigate how the value of the punch load varies with corner angle, channel angle and friction coefficient. The load required to extrude the billet for different channel angles 75o , 90o , 105o and 120o with respect to corner angle under frictionless and frictional (μ = 0.2) conditions are shown in Fig. 11. The punch load-displacement curves also explain the deformation steps in which the load required to deform the billet increase initially until reaching a peak value, it then drops and reaches a steady value as the deformation progresses. Under frictionless and friction conditions, with the increase of channel and corner angles, the load required to extrude the billet by the punch decreases due to the free flow of the material across the die. From the same figure we conclude that, as the coefficient of friction increases from μ = 0 to μ = 0.2, the punch load increases significantly (more than two times) because the friction operates in the reverse direction of the moving surface. Therefore, in order to reduce the punch load and raise the die life, high friction coefficients should be avoided during ECAP process. The in-
M.A. Agwa et al. / Mechanics of Materials 100 (2016) 1–11
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Fig. 10. Values of the coefficient of variance of strain with coefficient of friction for different channel and corner angles.
fluence of friction on punch load is greater than the influence of channel and corner angles.
The following conclusions can be summarized from the study: •
5. Conclusion Finite element simulations were carried out with Abaqus/Explicit to study the influences of channel angle, corner angle and friction coefficient on the deformation behavior, strain homogeneity and punch load. The simulations were carried out for φ ∈ [75o , 150o ], ψ ∈ [0o , 90o ] and μ ∈ [0, 0.3]. Among all the combinations of 252 study cases, we presented the selected optimum parameters for minimum strain coefficient of variance.
•
•
From the analysis of deformation behavior, it was found that (i) the acute and right channel angles are preferred than the other obtuse angles; (ii) smaller corner angles maximize the homogeneous portion of the billet; (iii) friction reduces the corner gap significantly. The average equivalent plastic strain generated in the sample is greatly affected by channel and corner angles. With the increase of channel and corner angles, the material will flow smoothly across the die inducing lower strain across the billet. There is a good matching between numerical and analytical results but still a difference between them due to the limitation of the assumption in the analytical models.
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M.A. Agwa et al. / Mechanics of Materials 100 (2016) 1–11
Fig. 11. Punch load-displacement curves. Left: μ = 0. Right: μ = 0.2.
•
•
With the increase of the corner angle, the strain homogeneity decreases. As the channel angle increases, the effect of corner angle on the strain homogeneity becomes insignificant. This study also proved that, the friction has a significant effect on the strain homogeneity in case of smaller corner angles.
•
Small corner angles and acute/right channel angles are recommended to obtain optimum strain homogeneity. Moderate and high coefficients of friction help in improving the strain homogeneity. The optimum parameters for maximum strain homogeneity can be achieved with φ = 90o , ψ = 15o and μ = 0.3.
M.A. Agwa et al. / Mechanics of Materials 100 (2016) 1–11 •
The required punch load to extrude the billet is greatly affected by friction coefficient. For example, punch load is doubled if we increase friction coefficient μ from 0 to 0.2. The influence of friction on punch load is more pronounced than the influence of die geometry. Therefore, in order to reduce the punch load and raise the die life, high friction coefficient should be avoided during ECAP process.
Acknowledgments A word of recognition is due to Professor António Pinto da Costa (Departamento de Engenharia Civil, Arquitetura e Georrecursos and ICIST, Instituto Superior Técnico, Lisbon, Portugal) for all the support and guidance that he has been providing to the first author for many years. References Abaqus, 2009. Abaqus 6.9 Theory Manual. Hibbitt, Karlsson & Sorensen, Inc. Agwa, M.A., 2011. Friction-induced dynamic instabilities and solution (non-)uniqueness in contact mechanics. Technical University of Lisbon, Instituto Superior Técnico, Lisbon, Portugal Ph.D. thesis. Arab, M.S., El Mahallawy, N., Shehata, F., Agwa, M.A., 2014. Refining SiCp in reinforced Al-SiC composites using equal-channel angular pressing. Mater. Des. 64, 280–286. Balasundar, I., Raghu, T., 2010. Effect of friction model in numerical analysis of equal channel angular pressing process. Mater. Des. 31, 449–457. Cerri, E., De Marco, P.P., Leo, P., 2009. FEM metallurgical analysis of modified 6082 Aluminum alloys processed by multipass ECAP: Influence of material properities and different process settings on induced plastic strain. J. Mater. Process. Technol. 209, 1550–1564. Fuqian Yang, Aditi Saran, Kenji Okazaki, 2005. Finite element simulation of equal channel angular extrusion. J. Mater. Process. Technol. 166, 71–78. Goforth, R.E., Hartwig, K.T., Cornwell, R., 20 0 0. chapter 1. Kluwer Academic Publishers, Dordrecht (Netherlands), pp. 3–12. Iwahashi, Y., Wang, W., Nemoto, M., Langdon, T., 1996. Principle of equal-channel angular pressing for the processing of ultra-fine grained materials. Scripta. Mater. 35, 143–146.
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Jin, Y.G., Son, I.H., Kang, S.H., Im, Y.T., 2009. Three-dimensional finite element analysis of multi-pass equal-channel angular extrusion of Aluminum AA1050 with split dies. Mater. Sci. Eng. A 503, 152–155. Lee, S., Utsunomiya, A., Akamatsu, H., Neishi, K., Furukawa, M., Horita, Z., Langdon, T., 2002. Influence of scandium and zirconium on grain stability and superplastic ductilities in ultrafine-grained Al-Mg alloys. Acta. Mater. 50, 553–564. Li, S., Bourke, M.A.M., Beyerlein, I.J., Alexander, D.J., Clausen, B., 2004. Finite element analysis of the plastic deformation zone and working load in equal channel angular extrusion. Mater. Sci. Eng. A 382, 217–236. Milind, T.R., Date, P.P., 2012. Analytical and finite element modeling of strain generated in equal channel angular extrusion. Int. J. Mech. Sci. 56, 26–34. Moreau, J.J., 1986. Dynamique des systèmes à lyaisons unilatérales avec frottement sec éventuel: éssais numériques. Technical Report 85-1. Laboratoire de Mécanique Générale des Milieux Continus, Université des Sciences et Téchiques du Languedoc. Nagasekhar, A.V., Yip Tick-Hon, 2004. Optimal tool angles for equal channel angular extrusion of strain hardening materials by finite element analysis. Comput. Mater. Sci. 30, 489–495. Nagasekhar, A.V., Yip Tick-Hon, Li, S., Seow, H.P., 2005. Effect of acute tool-angles on equal channel angular extrusion/pressing. Mater. Sci. Eng. A 410–411, 269–272. Nagasekhar, A.V., Yip Tick-Hon, Seow, H.P., 2007. Deformation behavior and strain homogeneity in equal channel angular extrusion/pressing. Mater. Sci. Eng. A 192–193, 449–452. Oldfield, M., Ouyang, H., Mottershead, J.E., 2005. Simplified models of bolted joints under harmonic loading. Comput. Struct. 84, 25–33. Patil Basavaraj, V., Uday Chakkingal, Prasanna Kumar, T.S., 2010. Influence of outer corner radius in equal channel angular pressing. World Acad. Sci. Eng. Technol. 38, 714–720. Segal, V.M., 1995. Materials processing by simple shear. Mater. Sci. Eng. A 197, 157–164. Segal, V.M., Hartwig, K.T., Goforth, R.E., 1997. In situ composites processed by simple shear. Mater. Sci. Eng. A 224, 107–115. Sheng, D., Eigenbrod, K.D., Wriggers, P., 2005. Finite element analysis of pile installation using large-slip frictional contact. Comput. Geotechnics 32, 17–26. Srinivas, B., Ch. Srinivasu, Banda Mahesh, Md. Aqheel, 2013. A review on severe plastic deformation. Adv. Mater. Manuf. Char. 3 (1), 291–296. Valiev, R.Z., Langdon, T.G., 2006. Principles of ECAE as a processing tool for grain refinement. Prog. Mater. Sci. 51, 881–981. Yi-Lang Yang, Shyong Lee, 2003. Finite element analysis of strain conditions after equal channel angular pressing. J. Mater. Process. Technol. 140, 583–587. Zaïri, F., Aour, B., Gloaguen, J.M., Naït-Abdelaziz, M., Lefebvre, J.M., 2006. Numerical modelling of elastic-viscoplastic equal channel angular extrusion process of a polymer. Comput. Mater. Sci. 38, 202–216.
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Research paper
Optimum processing parameters for equal channel angular pressing M.A. Agwa∗, M.N. Ali, Amal E. Al-Shorbagy Department of Mechanical Design and Production Engineering, Faculty of Engineering, Zagazig University, 44519 Zagazig, Egypt
a r t i c l e
i n f o
Article history: Received 4 July 2015 Revised 24 May 2016 Available online 7 June 2016 Keywords: Equal channel angular pressing Finite element method Strain coefficient of variance Channel angle Corner angle Friction
a b s t r a c t This study aims to explore the optimum processing parameters for minimum strain coefficient of variance in equal channel angular pressing based on the analysis of whole steady zone. In order to design the die correctly, it is important to understand the effect of die geometry and processing parameters on the deformation behavior, strain distribution and punch load. The influences of channel angle, corner angle and friction coefficient are analyzed using finite element modeling. Analytical models are also used to validate the obtained computational solutions. Results showed that channel angle, corner angle and friction coefficient have significant influences on deformation behavior, strain coefficient of variance and punch load evolution. Among all the combinations of 7 corner angles, 6 channel angles and 6 friction coefficients (set of 252 study cases), it is found that the optimum parameters for maximum strain homogeneity correspond to channel angle φ = 90o , corner angle ψ = 15o and friction coefficient μ = 0.3. The influence of friction coefficient on punch load is more pronounced than the influence of die geometry. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Equal channel angular pressing (ECAP) is an effective severe plastic deformation (SPD) technique that increases the material strength by accumulating a very large plastic strain into the workpiece without changing its cross-sectional area (Srinivas et al., 2013). The deformation of the billet is achieved by a simple shear taking place in a thin layer at the crossing plane of the equal channels (Segal, 1995). The cross section of workpiece remains the same during ECAP so the process can be repeated until the accumulated deformation reaches a desired level. Therefore, high strain, improved mechanical and physical properties, extreme fine grains and varying textures can be obtained in the processed materials (Arab et al., 2014). Valiev and Langdon (2006) showed that the formation of ultrafine grains in metals and alloys underlies a very significant enhancement in their mechanical and functional properties. In addition, different microstructures and mechanical properties can also be produced by changing the strain path in the billet, which in turn is achieved by changing the orientation of the billet from one pass to the next (Jin et al., 2009). The use of finite element method in the simulation and analysis of technological processes is constantly increasing. It comes as a powerful tool to predict and study the plastic behavior of the billet during ECAP. Many authors devoted their research to finite element modeling, simulation and analysis of ECAP for better understating
∗
Corresponding author. Tel.:+201110015465; fax: +20552304987. E-mail address: [email protected] (M.A. Agwa).
http://dx.doi.org/10.1016/j.mechmat.2016.06.003 0167-6636/© 2016 Elsevier Ltd. All rights reserved.
of the deformation behavior and strain distribution (see, for example, (Nagasekhar and Yip Tick-Hon, 2004)). Li et al. (2004) formulated a comprehensive finite element model to analyze the formation of the plastic deformation zone (PDZ) and evolution of the working load with ram displacement during a single pass of ECAP with 90o intersection angle. They stated that the coefficient of friction has negligible effect on strain distribution. Fuqian Yang et al. (2005) studied the effect of ECAP process variables on the evolution of plastic deformation and simulated the deformation of materials subjected to multi-pass ECAP by using multi-bent extrusion process. Yi-Lang Yang and Shyong Lee (2003) have reported that the strain is independent of friction for a strain hardening material. Balasundar and Raghu (2010) studied the effect of Coulomb and shear friction models on the deformation pattern, strain distribution and load. Patil Basavaraj et al. (2010) found that the outer corner has a significant influence on the distribution and inhomogeneity of strain in the body of workpiece. Aluminum alloy 6082 is a medium strength alloy with excellent corrosion resistance, machinability and weldability (Cerri et al., 2009). Alloy 6082 is widely used in highly stressed application, trusses, bridges and transport applications. The addition of Zr to this alloy plays a critical role by providing particles which impede grain growth at elevated temperature to enhance superplasticity (Lee et al., 2002). In this paper we give continuity to the above-mentioned researches by investigating the influence of ECAP processing parameters on the optimum strain homogeneity of modified 6082 Alu-
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It is important to know the strain in these regions in order to optimize the processing parameters. Owing to the complexity involved in plastic deformation of ECAP process, the closed form solution for the governing equations usually do not exist. Certain simplifications must be made in order to find analytical methods to obtain partial information about the mechanics of ECAP process. Segal et al. (1997) derived a path independent strain relationship for a sharp corner (ψ = 0) configuration with die/channel angle φ according to which the effective strain (equivalent plastic strain) ε is given as:
ε
2 = √ cot 3
φ 2
.
(1)
Another formula for the equivalent plastic strain ε in the material is calculated by Iwahashi et al. (1996) as:
ε
1 = √ 2 cot 3
φ+ψ
+ ψ csc
2
φ+ψ
.
2
(2)
Goforth et al. (20 0 0) proposed the upper bound model for predicting the ECAP of strain-hardening materials as:
1
ε= √
3
Fig. 1. Schematic diagram of ECAP process.
2 cot
•
•
Head: the front of the workpiece; Tail: the region at the back of the workpiece; Steady state zone: the region between the non-uniformly deformed head and tail regions where the strain is relatively uniform along the billet axis.
+ ψ sec2
2
γ = 2 tan
•
φ−ψ
φ−ψ 2
.
(4)
Model B: which was found to yield better predictions for the strains as compared to the model of Iwahashi et al. (1996) for dies with a fillet radius under frictionless conditions. Model B states that:
Fig. 2. The different regions of the deformed workpiece.
•
(3)
Model A: which was found to be better in predicting the strain generated in dies with external arc of curvature. In this model, the shear strain (γ ) is given as:
Fig. 1 is a schematic illustration of ECAP process in which a billet is pressed through a die that consists of two equal crosssectional channels intersecting at an angle (φ ) and with an outer corner angle (ψ ). As shown in Fig. 2, the workpiece can be divided into three main regions:
2
+ψ .
Note that the last two models tend to Segal model (Eq. (1)) for
γ = 2 tan
2. Analytical solutions of the effective plastic strain
ψ = 0. Milind and Date (2012) derived two analytical models for the relation between shear strain (γ ) and the ECAP angles φ and ψ . These two models are: •
minium alloy billet. The influences of channel angle (φ ), corner angle (ψ ) and friction coefficient (μ) between the die and the billet on the deformation behavior, equivalent plastic strain, strain homogeneity and punch load are studied. The paper proceeds as follows. In the next section, the analytical solutions of the effective plastic strain are described. Section 3 sheds light on the finite element model used in the analysis. In Section 4, the results of the finite element simulation are presented and discussed. Finally, Section 5 concludes.
φ+ψ
φ−ψ 2
+
sin ψ /2 × φ. cos(φ /2 − ψ /2 ) sin φ /2
(5)
According to Segal et al. (1997), the relationship between shear strain (γ ) and effective strain (ε ) in ECAP process is given as ε = √γ . Note that all expressions did not take into account the ef3
fects of friction, strain hardening, strain distribution and deformation gradient (Cerri et al., 2009). 3. Finite element analysis The finite element method is a numerical analysis procedure used to obtain approximate solutions to boundary value problems, which are found in every field of engineering. Most problems encountered in everyday practical applications do not have closedform solution. The accuracy of the simulations depends on the assumed boundary conditions and how well these conditions describe the real process. Many process parameters, such as contact, thermal and frictional conditions, are difficult to be determined correctly. The models developed in this study take into account the three sources of non-linearities present in ECAP problem, namely, material, geometry and boundary conditions. The ECAP die and the geometry of the workpiece are shown in Fig. 1. The mesh procedure is used to divide the model into small elements. The smaller these elements are, the more accurate the simulation will be. The disadvantage with smaller elements is
M.A. Agwa et al. / Mechanics of Materials 100 (2016) 1–11 Table 1 Properties of Aluminum alloy (6082Zr) (Cerri et al., 2009). Material property
Value
Young’s modulus Poisson ratio Density Strain hardening exponent (n) Strength coefficient (K)
69 GPa 0.33 2710 Kg/m3 0.23 291 MPa
3
components of its displacement vector u p (t ) = {un (t ) ut (t )}T , of p p its velocity vector u˙ p (t ) = {u˙ n (t ) u˙ t (t )}T and of its reaction vector p p p T r (t ) = {rn (t ) rt (t )} satisfy the Signorini unilateral contact conditions: p
unp (t ) − dnp ≤ 0,
rnp (t ) ≤ 0,
3.1. Contact, friction and material models The interaction between contacting bodies is defined by assigning a contact property model to the contact interaction. The contact property models available in Abaqus are mechanical, thermal, electrical and pure fluid contact models. The mechanical contact model which include friction model is used to define the force resisting the relative tangential motion of the surface. The possible friction models in Abaqus are frictionless, penalty, static-kinetic exponential decay, rough and Lagrange multiplier. It is currently widely accepted that Coulomb’s law is an appropriate way to describe the frictional phenomenon in many situations. Moreau (1986) stated that although in many technical situations for which Coulomb’s law may yield only moderately accurate results, it simply cannot be substituted since, quite often in practice, one does not have access to relevant properties of the contacting surfaces that would furnish a more refined knowledge of the frictional phenomenon. Many authors consider that it is adequate to use Coulomb’s model in many situations (Agwa, 2011). For any contact candidate particle, the normal (n) and tangential (t)
(unp (t ) − dnp )rnp (t ) = 0,
(6)
where dn is the initial normal gap between particle p and the obstacle (Fig. 3(a)). Coulomb’s friction law (Fig. 3(b)) may be written as a conjunction of an inequality and a nonûsmooth equation
|rtp (t )| + μrnp (t ) ≤ 0, the substantial increase in execution time. Different meshes were tested to find an optimum mesh density. The commercial finite element code Abaqus/Explicit was used to carry out all simulations. Three factors were varied in the simulations: channel angle (φ ), corner angle (φ ) and friction coefficient (μ). Channel angle (φ ) ranges from 75o to 150o , corner angle (ψ ) ranges from 0o to 90o with an increment of 15o for both angles and friction coefficient (μ) from 0 to 0.3. Billet with dimension of (10 mm × 10 mm × 60 mm) was modeled for 2D plane strain simulation with elastic-plastic material properties of Aluminum alloy with zirconium addition (6082Zr). Strain hardening material properties of the modified 6082 Aluminum alloy used for simulations are shown in Table 1 (Cerri et al., 2009). The billet material was modeled with four-node bilinear plane stain quadrilateral (CPE4R) element (Abaqus, 2009). The die and punch were modeled as analytical rigid elements. All simulations were performed with speed of 1 mm/s (Nagasekhar et al., 2005). Due to the strong non-linearities caused by friction, contact, large deformation and elastic-plastic behavior, one of the main challenges in this type of computational analysis is the convergence to a stable solution. For such complicated problems Abaqus/Standard may not be able to find a converging solution and will fail in that case (Oldfield et al., 2005). In this study, Abaqus/Explicit was used to model the ECAP problem. The load steps used in the ECAP analysis were chosen to be very small (Sheng et al., 2005). This allows such complicated problems, whose solutions cannot be found by Abaqus/Standard, to be solved by Abaqus/Explicit. The main issue with Abaqus/Explicit is its much greater computing load due to the very small load increments/time steps used for assuring convergence. Smooth fillet radius of 0.75 mm at the inner corner was chosen to avoid the convergence problems (Nagasekhar et al., 2007). Mass scaling was used for all simulation to prevent failure of the mesh during large deformation and to reduce the time of computation.
p
rtp (t )u˙ tp (t ) − μrnp (t )|u˙ tp (t )| = 0.
(7)
Abaqus plasticity models are usually formulated in terms of a yield surface, a flow rule and hardening law. The classical metal plasticity model allows to define the yield and inelastic flow of a metal at relatively low temperature, where loading is relatively monotonic and creep effects are unimportant. In the classical plastic behavior of a material, the von Mises yield surface is used. The von Mises yield criterion states that plastic yielding occurs, when √ the octahedral shearing stress reaches a critical value (σy / 3), where σ y is the yield strength. For von Mises plasticity it can be p shown that the equivalent plastic strain rate ε˙ is defined as
p
ε˙ =
2 p ε˙ : ε˙ p . 3
(8)
The nonlinear material response was modeled using isotropic hardening with a von Mises yield criterion. With isotropic hardening the metalÆs yield stress evolves as the metal accumulates plastic strains σy (ε p ). Nonlinear empirical idealization of the plastic hardening, in most cases, provides more accurate prediction of the material behavior. The most commonly used form for the nonlinear isotropic hardening rule is the power law given as:
n σy ( ε p ) = σy 0 + K ε p ,
(9)
where σ y0 is the initial yield stress. In this study, the penalty option is used that based on the Coulomb friction model. The friction coefficient was supposed to be isotropic. The value of friction coefficient between the billet and die was assumed to range from 0 to 0.3. The elastic-plastic behavior of the material is modeled by the linear elastic model and the classical metal plasticity model. 3.2. Model verification and mesh selection In order to verify that the ECAP problem is modeled correctly, a test case with the same parameters of Fig. 8 (a) in Cerri et al. (2009) was solved and compared in terms of contour plot, strains, stresses and deformation shape (see Fig. 4(a,b)). Good matching was obtained between the two results. The percentage error ranges from 2% for frictional cases to 5% for frictionless cases. This error may be attributed to factors such as (i) the difference in fillet radii at inside and outside corners of the die that was not specified by Cerri et al. (2009), (ii) the different number of load increments/time steps and (iii) the difference in the finite arithmetic precision of the computers used which resulted in different roundoff errors that may propagate due to the large number of time increments. Before performing the finite element simulation, a convergence test was carried out for channel angle (φ = 90o ), corner angle (ψ = 0o ) and friction coefficient (μ = 0) to assess the sensitivity of the results to mesh refinement. Five studies for different number of elements (840, 170 0, 320 0, 660 0 and 13,160 elements) were carried out to calculate the average and maximum equivalent plastic strain in the steady state region as shown in Fig. 5. The differences between the average plastic strain for both 840 and 1700 meshes
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M.A. Agwa et al. / Mechanics of Materials 100 (2016) 1–11
Fig. 3. Graphs of (a) the unilateral contact law and (b) the friction law of Coulomb.
Fig. 4. Comparison between the equivalent plastic strain at integration points (PEEQ) (a) the present work and (b) Fig. 8 (a) of Cerri et al. (2009), both under the same conditions.
Fig. 5. Convergence test for different number of elements.
with 13,160 mesh are significant compared with the differences between the average plastic strain for both 3200 and 6600 meshes with 13,160 mesh (see Fig. 5 (a)). The same observation can be obtained in Fig. 5 (b) but for the maximum equivalent plastic strain. Accordingly, the mesh consists of 3200 elements was finally chosen to provide adequate precision in the simulations and related calculations. 4. Results and discussion 4.1. Deformation behavior The deformation behavior was examined in the whole steady state zone after one pass of ECAP. The influence of both channel
angle and corner angle under frictionless (μ = 0) and friction (μ = 0.2) conditions on deformation behavior is investigated. 4.1.1. Combined effect of channel angle and friction Fig. 6 shows the influence of channel angle (φ ) on the deformation behavior. During simulations there are two gaps appeared, namely, corner and channel gaps (see Fig. 2). These gaps are a normal consequence of billet bending during deformation. The head of billet bends toward the top of outlet channel, which leads to the formation of corner gap, while the contact between the side surface of head region and die channel leads to the formation of channel gap. Under frictionless condition, the increasing in the channel angle was accompanied by increasing in the corner gap found at the die corner (Fig. 6 (a)–(d)). The front end of the deformed sam-
M.A. Agwa et al. / Mechanics of Materials 100 (2016) 1–11
5
Fig. 6. Equivalent plastic strain distribution for ψ = 0o and different channel angles indicated below each curve. Left: frictionless (μ = 0). Right: friction (μ = 0.2).
ple is more uniform for the smaller channel angles 75o and 90o (Fig. 6 (a)–(b)) than the other two angles 105o and 120o (Fig. 6 (c)–(d)) accordingly, wastage of the sample during multiple ECAP passes for both 75o , 90o channel angle is less than the other angles. With the adding of friction, the corner gap begins to disappear which may be attributed to the increased constraint frictional drag at the billet surface. This drag force acts like a back pressure leading to the filling of the corner gap and makes the material flow fits better the die outer corner. The nonuniform ends of the billet at channel angle greater than 90o change slightly because friction reduces the upward bending of the sample (Fig. 5 (e)–(h)). From the above discussion, it can be recognized that (i) the acute and
right channel angles are preferred than the other obtuse angles; (ii) friction reduces the corner gap significantly. 4.1.2. Combined effect of corner angle and friction The influence of corner angle (ψ ) on the deformation behavior is shown in Fig. 7. Under frictionless condition, as corner angle increases, the corner gap decreases and the channel angle formed at the outlet channel increases (Fig. 7 (a)–(d)). With the increase of corner angle, the plastic deformation zone (PDZ) (see Fig. 2) begins to spread and covers the corner angle in which it was found to be narrow for sharp corner angles. It may be observed that the front end of the sample started to bend upward and generated nonuniform deformation at the end of the billet with the increase of corner angle. Minimizing of this nonuniform region (front end)
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M.A. Agwa et al. / Mechanics of Materials 100 (2016) 1–11
Fig. 7. Equivalent plastic strain distribution for φ = 90o and different corner angles indicated below each curve. Left: frictionless (μ = 0). Right: friction (μ = 0.2).
is of interest for maximizing the length of the useful homogeneous portion of the billet. Under friction condition, both corner gap and channel gap decrease and the tip of the billet begin to drop down compared with frictionless condition. From the above discussion it may be concluded that, the smaller corner angles are preferred than the other angles. 4.2. Average effective strain in steady zone The average effective plastic strain (ε ave ) from finite element analysis can be calculated by taking the average of the strain values p over the whole steady state area in plane strain conditions. ε ave is computed over the steady zone area IJKL ( see Fig. 2) as: p
εapve =
n 1 p εi , n
(10)
i=1
where n is the number of integration points in the steady zone and εip is the equivalent plastic strain at integration point i. Fig. 8 shows the relation between the average equivalent plastic strain and friction coefficient for channel angle ranges from 75o to
120o and corner angle from 0o to 90o , both with increment of 15o . It can be observed that the average equivalent plastic strain decreases significantly with the increase of both channel and corner angles. The flow of the material across the die becomes smoother and the deformation begins to be far from the severe plastic deformation as the channel and corner angles increase. The effect of corner angle on the average equivalent plastic strain becomes insignificant as the channel angle increases, this gives an indication that the corner angle affects only in the cases of small and sharp channel angles (75o , 90o ). With the increase of friction, the average equivalent plastic strain increases, but it can be observed from Fig. 8 that the effect of friction is relatively small compared with the effect of both channel and corner angles. The strain obtained by the analytical models (Eqs. (2)–(5)) and the average equivalent plastic strain from the simulation for different channel angles 75o , 90o , 105o and 120o at different corner angles under frictionless condition are shown in Fig. 9. The analytical and numerical strain results are matching well for φ = 90o which validates the results obtained by simulation. But, as we may notice, there is a small difference between them. This difference is due to
M.A. Agwa et al. / Mechanics of Materials 100 (2016) 1–11
7
Fig. 8. Values of the average equivalent plastic strain with friction coefficient for different channel and corner angles.
the lacks that found in the analytical models in which it didn’t take into account the effect of strain hardening, strain distribution and deformation gradient assuming a homogeneous value of strain in the whole workpiece. For φ = 75o , the two models (A and B) of Milind and Date (2012) gave lower values than the other models (FEM, Iwahashi and Goforth) as shown in Fig. 9 (a). However, the two models (A and B) gave higher values than the other models for φ = 105o and 120o (Fig. 9 (c,d)). This gives indication that the models of Milind and Date (2012) are valid only for φ = 90o (Fig. 9 (b)). From the same figure we conclude that, as the channel and corner angles increase, the equivalent plastic strain decreases and this is the same observation that obtained from Fig. 8. 4.3. Strain coefficient of variance
p p ε max − ε min , ε apve
(11)
where ε max and ε min denote, respectively, the maximum and the minimum equivalent plastic strain selected from all strain values p computed at integration points over the steady state area IJKL. ε ave p
p
CVε p =
Std ε p
ε apve
,
(12)
where standard deviation Std ε p is a statistical term that measures the amount of variability or dispersion of equivalent plastic strain p at each integration point around the average effective strain (ε ave ). Dispersion is the difference between the actual value and the average value. Mathematically standard deviation is the square root of the average of the squared differences from the average equivalent p plastic strain ε ave ;
Although the analytical models give useful quantitative results, they just represent the average effective strain developed in the workpiece during ECAP. Neither the analytical models nor the experimental methods can show the distribution of the effective strain in the workpiece after ECAP or inhomogeneous deformation behavior. The degree of strain inhomogeneity can be estimated across the area IJKL of the steady zone using the equation defined by Li et al. (2004):
C=
denotes the average of the equivalent plastic strain computed from Eq. (10). Another expression for quantifying the strain inhomogeneity in a better way was adopted by Zaïri et al. (2006). It is defined by coefficient of variance of strain (CVε p ) given as:
Std ε = p
n 1 p (ε i − ε apve )2 n
12
.
(13)
i=1
The strain inhomogeneity defined by CVε p is better than that defined by C because it uses standard deviation, which is based on the distribution of strain taking into account the value of strain at all integration points in the section, while C based only on the difference between two extreme strain values. Consequently the most appropriate way of quantifying the strain inhomogeneity is given by the coefficient of variance CVε p . Fig. 10 illustrates the influence of channel angle, corner angle and friction on strain homogeneity by using coefficient of variance presented by Eq. (13). The lower values of coefficient of variance indicate better homogeneity. Strain homogeneity in ECAP is mainly dependent on the simple shear deformation. According to the the-
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M.A. Agwa et al. / Mechanics of Materials 100 (2016) 1–11
Fig. 9. Values of the average equivalent plastic strain from numerical solution and analytical models under frictionless condition for different channel and corner angles.
ory of the ECAP (Segal, 1995), simple shear is confined to the thin layers at the crossing plane of the channels in which the contour lines of plastic strain are concentrated on the plane connecting the two corners of the channel. With the increase of corner angle, the plastic deformation zone widened and covered the corner angle (Fig. 7), the region of nonuniform deformation spreads out from the lower part of the billet toward the upper part and the deformation began to deviate from the simple shear resulting in decreasing the strain homogeneity. It can be observed from Fig. 10 that as corner angle increases the strain inhomogeneity increases in most of the cases except for ψ = 0o at channel angles of 75o , 90o and 105o (Fig. 10 (a)–(c)). From the same figure we conclude that, as the channel angle increases, the effect of corner angle on coefficient of variance becomes insignificant. Fig. 10 also illustrates that, friction coefficient has a significant effect on the strain homogeneity in case of smaller corner angles. The search for optimum parameters for maximum strain homogeneity is equivalent to solution of a set of problems. Among all the combinations of 7 corner angles, 6 channel angles and 6 friction coefficients (7 × 6 × 6 = 252 study cases), Table 2 presents the selected optimum parameters for minimum value of the coefficient of variance. The values in the table are arranged according to the coefficient of variance of strain (CVε p ); such that the smaller the coefficient of variance (CVε p ) the better the strain homogeneity. From Table 2, it can be drawn that small corner angles and acute and right channel angles are recommended to obtain optimum strain homogeneity. It may also be observed that moderate and high coefficients of friction help in improving the strain homogeneity. The global optimum parameters are those of case No. 1, which correspond to channel angle φ = 90o , corner angle ψ = 15o and coefficient of friction μ = 0.3.
Table 2 The selected optimum processing parameters (φ , ψ , μ) for maximum strain homogeneity conditions in the steady state zone IJKL. No. 1 2 3 4 5 6
φ
ψ o
90 75o 105o 120o 135o 150o
o
15 15o 15o 0o 0o 0o
μ
εapve
C
Std ε p
CVε p
0.3 0.1 0.3 0.15 0.3 0.3
1.0563 1.3531 0.8175 0.6490 0.4660 0.2962
0.6575 0.5142 0.7903 0.9924 1.0922 1.3031
0.0839 0.1186 0.0931 0.0854 0.0770 0.0599
0.0794 0.0877 0.1139 0.1316 0.1652 0.2021
4.4. Punch load evolution In this subsection, we investigate how the value of the punch load varies with corner angle, channel angle and friction coefficient. The load required to extrude the billet for different channel angles 75o , 90o , 105o and 120o with respect to corner angle under frictionless and frictional (μ = 0.2) conditions are shown in Fig. 11. The punch load-displacement curves also explain the deformation steps in which the load required to deform the billet increase initially until reaching a peak value, it then drops and reaches a steady value as the deformation progresses. Under frictionless and friction conditions, with the increase of channel and corner angles, the load required to extrude the billet by the punch decreases due to the free flow of the material across the die. From the same figure we conclude that, as the coefficient of friction increases from μ = 0 to μ = 0.2, the punch load increases significantly (more than two times) because the friction operates in the reverse direction of the moving surface. Therefore, in order to reduce the punch load and raise the die life, high friction coefficients should be avoided during ECAP process. The in-
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Fig. 10. Values of the coefficient of variance of strain with coefficient of friction for different channel and corner angles.
fluence of friction on punch load is greater than the influence of channel and corner angles.
The following conclusions can be summarized from the study: •
5. Conclusion Finite element simulations were carried out with Abaqus/Explicit to study the influences of channel angle, corner angle and friction coefficient on the deformation behavior, strain homogeneity and punch load. The simulations were carried out for φ ∈ [75o , 150o ], ψ ∈ [0o , 90o ] and μ ∈ [0, 0.3]. Among all the combinations of 252 study cases, we presented the selected optimum parameters for minimum strain coefficient of variance.
•
•
From the analysis of deformation behavior, it was found that (i) the acute and right channel angles are preferred than the other obtuse angles; (ii) smaller corner angles maximize the homogeneous portion of the billet; (iii) friction reduces the corner gap significantly. The average equivalent plastic strain generated in the sample is greatly affected by channel and corner angles. With the increase of channel and corner angles, the material will flow smoothly across the die inducing lower strain across the billet. There is a good matching between numerical and analytical results but still a difference between them due to the limitation of the assumption in the analytical models.
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Fig. 11. Punch load-displacement curves. Left: μ = 0. Right: μ = 0.2.
•
•
With the increase of the corner angle, the strain homogeneity decreases. As the channel angle increases, the effect of corner angle on the strain homogeneity becomes insignificant. This study also proved that, the friction has a significant effect on the strain homogeneity in case of smaller corner angles.
•
Small corner angles and acute/right channel angles are recommended to obtain optimum strain homogeneity. Moderate and high coefficients of friction help in improving the strain homogeneity. The optimum parameters for maximum strain homogeneity can be achieved with φ = 90o , ψ = 15o and μ = 0.3.
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The required punch load to extrude the billet is greatly affected by friction coefficient. For example, punch load is doubled if we increase friction coefficient μ from 0 to 0.2. The influence of friction on punch load is more pronounced than the influence of die geometry. Therefore, in order to reduce the punch load and raise the die life, high friction coefficient should be avoided during ECAP process.
Acknowledgments A word of recognition is due to Professor António Pinto da Costa (Departamento de Engenharia Civil, Arquitetura e Georrecursos and ICIST, Instituto Superior Técnico, Lisbon, Portugal) for all the support and guidance that he has been providing to the first author for many years. References Abaqus, 2009. Abaqus 6.9 Theory Manual. Hibbitt, Karlsson & Sorensen, Inc. Agwa, M.A., 2011. Friction-induced dynamic instabilities and solution (non-)uniqueness in contact mechanics. Technical University of Lisbon, Instituto Superior Técnico, Lisbon, Portugal Ph.D. thesis. Arab, M.S., El Mahallawy, N., Shehata, F., Agwa, M.A., 2014. Refining SiCp in reinforced Al-SiC composites using equal-channel angular pressing. Mater. Des. 64, 280–286. Balasundar, I., Raghu, T., 2010. Effect of friction model in numerical analysis of equal channel angular pressing process. Mater. Des. 31, 449–457. Cerri, E., De Marco, P.P., Leo, P., 2009. FEM metallurgical analysis of modified 6082 Aluminum alloys processed by multipass ECAP: Influence of material properities and different process settings on induced plastic strain. J. Mater. Process. Technol. 209, 1550–1564. Fuqian Yang, Aditi Saran, Kenji Okazaki, 2005. Finite element simulation of equal channel angular extrusion. J. Mater. Process. Technol. 166, 71–78. Goforth, R.E., Hartwig, K.T., Cornwell, R., 20 0 0. chapter 1. Kluwer Academic Publishers, Dordrecht (Netherlands), pp. 3–12. Iwahashi, Y., Wang, W., Nemoto, M., Langdon, T., 1996. Principle of equal-channel angular pressing for the processing of ultra-fine grained materials. Scripta. Mater. 35, 143–146.
11
Jin, Y.G., Son, I.H., Kang, S.H., Im, Y.T., 2009. Three-dimensional finite element analysis of multi-pass equal-channel angular extrusion of Aluminum AA1050 with split dies. Mater. Sci. Eng. A 503, 152–155. Lee, S., Utsunomiya, A., Akamatsu, H., Neishi, K., Furukawa, M., Horita, Z., Langdon, T., 2002. Influence of scandium and zirconium on grain stability and superplastic ductilities in ultrafine-grained Al-Mg alloys. Acta. Mater. 50, 553–564. Li, S., Bourke, M.A.M., Beyerlein, I.J., Alexander, D.J., Clausen, B., 2004. Finite element analysis of the plastic deformation zone and working load in equal channel angular extrusion. Mater. Sci. Eng. A 382, 217–236. Milind, T.R., Date, P.P., 2012. Analytical and finite element modeling of strain generated in equal channel angular extrusion. Int. J. Mech. Sci. 56, 26–34. Moreau, J.J., 1986. Dynamique des systèmes à lyaisons unilatérales avec frottement sec éventuel: éssais numériques. Technical Report 85-1. Laboratoire de Mécanique Générale des Milieux Continus, Université des Sciences et Téchiques du Languedoc. Nagasekhar, A.V., Yip Tick-Hon, 2004. Optimal tool angles for equal channel angular extrusion of strain hardening materials by finite element analysis. Comput. Mater. Sci. 30, 489–495. Nagasekhar, A.V., Yip Tick-Hon, Li, S., Seow, H.P., 2005. Effect of acute tool-angles on equal channel angular extrusion/pressing. Mater. Sci. Eng. A 410–411, 269–272. Nagasekhar, A.V., Yip Tick-Hon, Seow, H.P., 2007. Deformation behavior and strain homogeneity in equal channel angular extrusion/pressing. Mater. Sci. Eng. A 192–193, 449–452. Oldfield, M., Ouyang, H., Mottershead, J.E., 2005. Simplified models of bolted joints under harmonic loading. Comput. Struct. 84, 25–33. Patil Basavaraj, V., Uday Chakkingal, Prasanna Kumar, T.S., 2010. Influence of outer corner radius in equal channel angular pressing. World Acad. Sci. Eng. Technol. 38, 714–720. Segal, V.M., 1995. Materials processing by simple shear. Mater. Sci. Eng. A 197, 157–164. Segal, V.M., Hartwig, K.T., Goforth, R.E., 1997. In situ composites processed by simple shear. Mater. Sci. Eng. A 224, 107–115. Sheng, D., Eigenbrod, K.D., Wriggers, P., 2005. Finite element analysis of pile installation using large-slip frictional contact. Comput. Geotechnics 32, 17–26. Srinivas, B., Ch. Srinivasu, Banda Mahesh, Md. Aqheel, 2013. A review on severe plastic deformation. Adv. Mater. Manuf. Char. 3 (1), 291–296. Valiev, R.Z., Langdon, T.G., 2006. Principles of ECAE as a processing tool for grain refinement. Prog. Mater. Sci. 51, 881–981. Yi-Lang Yang, Shyong Lee, 2003. Finite element analysis of strain conditions after equal channel angular pressing. J. Mater. Process. Technol. 140, 583–587. Zaïri, F., Aour, B., Gloaguen, J.M., Naït-Abdelaziz, M., Lefebvre, J.M., 2006. Numerical modelling of elastic-viscoplastic equal channel angular extrusion process of a polymer. Comput. Mater. Sci. 38, 202–216.