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Helical Extrusion Process of General Polygonal Section Shapes through Curved Dies
Journal of Manufacturing Processes 38 (2019) 38–48
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Helical Extrusion Process of General Polygonal Section Shapes through Curved Dies
T
⁎
Ahmed Waleed Husseina, , Mushrek A. Mahdib, Raneen Sami Abidc a
Department of Energy Engineering, University of Babylon, Babylon, Iraq Department of Automobile Engineering, University of Babylon, Babylon, Iraq c Department of Aerospace Manufacture and Engineering, Nanjing University of Aeronautics & Astronautics, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Helical extrusion Twist extrusion Upper bound theory Finite element method Streamlined dies ANSYS
Practically, the helical products used as a mechanical part to satisfy the needs of mechanical strength and artistic appearance. In addition, during recent 5 years it used to produce an ultra-fine grain metal structure. However, there are very little works regarding this type of extrusion so far. This work proposed a new formulation for helical deformation zone to produce general helical polygonal shapes through a streamlined die that are usually made by hot extrusion through taper die. The general die surface was represented analytically. The velocity and strain rate fields are derived depending on the volume constancy and the velocity boundary conditions. The upper bound forming pressure was obtained for various frictional conditions, area reduction, helix angle, and die length. The results show that the axis of the product does not rotate through the helical extrusion. The peak value of the strain rate is located close to the die outlet and decreases as the helix angle increases. The optimum die length becomes high as the helix angle increases. The forming pressure increases with increasing helix angle, area reduction, factor of friction, while decreases when the number of sides increases. The theoretical results were verified with previous work of zero twist and showed completely compatible. A finite element solution was done using hardening material model to verify the analytical results and metal flow and to examine the strain and stress fields in the product.
1. Introduction and background In structure industry, the helical products used as a mechanical part to satisfy the need of mechanical strength besides the artistic appearance. In addition, they are used in the power devices such as screw pumps and superchargers. Nowadays, the helical sections are made using casting, hot forging operations or machining processes. These methods have many disadvantages, time consumes, besides their high cost. Furthermore, during recent 5 years, the material science engineering showed that the severe plastic deformation produced by torsional shear stress produces ultra-fine grain metal structures. The main advantages of that structure are free of porosity and internal oxidation [1–4]. Despite increasing demand for the application of helical sections in the industry, however, very little analytical methods have been attempted so far. In the literature, few papers talking about the profiles with helical shapes were found. Yang et al. [5–7] used conformal transformation method to create the die surface and to examine the internal metal flow by transformation of any intermediate cross section
⁎
into a unit circle. The upper bound theory was applied to find the required extrusion pressure for the extrusion of clover, elliptic, and trochoidal gear sections. They used a rigid -perfectly plastic material model. Actually, using transformation method was very difficult and contained wide coordinate transformations as well as transformation function and in most, it is impossible to apply this technique for extrusion of complicated shapes. Salehi et al. [8,9] utilizing a Fourier approximation to find die cavity and then the admissible velocity field. The upper bound theory was used to calculate the required energy for twist extrusion of square and elliptical shapes through squared and elliptical die cross-section. In this works, the cross sections of deforming material remain unchanged throughout the twist extrusion process. Khalifa and Tekkaya [10] used a multi-segment die for hot extrusion of helical screw shaft from round billet. They used the experiments and finite element method to examine the effect of the friction and material flow against the twisting angle. There is no analytical analysis presented in this work. It becomes clear that there is no general systematic work deals the
Corresponding author. E-mail address: [email protected] (A.W. Hussein).
https://doi.org/10.1016/j.jmapro.2018.12.032 Received 17 August 2018; Received in revised form 15 November 2018; Accepted 24 December 2018 1526-6125/ © 2018 Published by Elsevier Ltd on behalf of The Society of Manufacturing Engineers.
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Nomenclature C L x, y , z m N W f(z)
vx,vy,vz vo, vf Wi Wf Ro Wt σo γL γz
Geometry constant Die length Cartesian co-ordinates systems Friction factor A number of polygonal sides Half of the polygon side The function used to descript the die profile and the streamlines
Velocity components Punch and product velocities, respectively Power required for plastic deformation Power loses due to frictional resistance Billet radius Total powers consume The yield stress of the billet material Total helix angle at the die outlet helical angle function
0 ≤ n ≤ Ro π 0≤ϕ≤ N
extrusion of helical sections in deep, in which the velocity and strain of every single point in the helical deformation zone can be obtained analytically. Furthermore, in all previous works, there is shear power loses due to velocity breaks in the inlet and outlet of the die, which increase the forming power required. In addition, there is little information about the stress and strain distributions throughout the extruded product. Therefore, this work developed a new method for finding optimum die profile, which yields minimum forming pressure for the extrusion of helical section shapes from round bar stock using general die configuration. The velocity and strain rate fields were derived analytically based on the volume constancy and velocity boundary conditions. Upper bound theory was used to determine the forming pressure for different helical polygonal section shapes (the sides from 1 to ∞) for a given required helix angle, required area reduction, material properties, and frictional conditions. In addition, the validation of results, metal flow, and stress and strain distributions were also obtained using FEM code ANSYS based on the experimental material data for AL 2024 from the literature.
0≤z≤L
(1)
Where N represents the number of sides for polygonal section. n is the radial distance from the die center to random point E' at the die inlet, ϕ is the inlet sweep angle, and ψ is the equivalent sweep angle at die outlet. In this work, the twist angle of the die can be expressed as a general polygonal function from the third degree as follows,
γ (z ) = r1 z 3 + r2 z 2 + r3 z + r4
(2)
Where ri (i = 1 to 4) are constants to be determined from the angular boundary conditions of the deformation zone. In order to reduce the angular shear loses at the inlet and outlet of the die, the following angular boundary conditions can be used,
2. Configuration of the extrusion die
γ (0) = 0, and
∂γ (z ) ∂z z=0
=0
γ (L) = γL, and
∂γ (z ) ∂z z=L
=0
(3)
Where γL is the total helix angle at the die outlet (0 ≤ γL < 900) . Hence, using these angular boundary conditions, Eq. (1) becomes,
For a smooth cross-sectional change from the initial round billet to the final helical extruded product, the die surface must be designed without any rapid change of die surface along the die axis. The die geometry in three-dimensional extrusion process is shown in Fig. 1. Let, Ro the radius of the round billet, w the half side of polygon, and L the die length. As a result of the symmetry of the deformation zone, variations of n, ϕ and z are
z 2 z 3 γ (z ) = 3 γL ⎛ ⎞ − 2γL ⎛ ⎞ ⎝L⎠ ⎝L⎠
(4)
The billet is forced to flow through the die that created by a number of streamlines to the final helical section. Therefore, the points E and G on the die entrance are connected with the equivalent points B’’ and B’ at
Fig. 1. Proposed die geometry and the velocity field for general polygonal section shape. 39
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A.W. Hussein et al. N w2
the die outlet respectively. Assumed that the billet material passing through the sector area O’EG at the die inlet goes through the triangle area OB’B’’ at the die outlet conserving the extrusion ratio. The stream path defined by the points O’, E , B’’ and O becomes a three dimensional stream path. Taking the assumptions that the surface O’EB’’O consists of a number of streamline paths, hence, the particle at random point E' at the die inlet will go towards the equivalent point F' at the die exit preserving the proportionality of the position. The locations of the points E ′ and F ′ in the Cartesian coordinate are given by
E¯ (nsin(∅), ncos(∅), 0) F¯ (x1, y1 , L)
1
tan(ψ) =
c=
OB '=
w cos(γL)
(8)
BB '= w sin(γL)
x = f1 (z ) = a1 z 3 + a2 z 2 + a3 z + a4
(10)
The triangle (m′H ′F ′) has,
m'H '= H 'F 'tan(γL)
(12)
Therefore,
k (1 − tan(ψ) tan(γL) ) cos(γz )
x (0) = nsin(ϕ) ,
∂x ∂z z = 0
=0
y (0) = n cos (ϕ) ,
∂y ∂z z = 0
=0
(11)
And,
Om '= OH '− m'H '
x (L) = x1,
∂x ∂z z = L
=0
y (L) = y1 ,
∂y ∂z z = L
=0
(13)
x = n A (ϕ) + n B (ϕ, z , ) y = n C (ϕ) + n D (ϕ, z ) z=z
(26)
Where,
nw cos(γL) Ro
(14)
A (ϕ) = sin(ϕ);
Therefore,
nw (1 − tan(ψ) tan(γL) ) Ro
w w tan (ψ) B (ϕ, z ) = ⎡ ⎛ cos(γz )(1 − cϕtan(γz )) ⎞ tan (γL) + ⎢ ⎝ Ro cos(γL) Ro ⎠ ⎣
(15)
− sin(ϕ) ⎤ f (z ) ⎥ ⎦
But,
y1 = (Om')cos(γL)
(16)
C (ϕ) = cos(ϕ)
Therefore, (17)
w D (ϕ, z ) = ⎡ ⎛ cos(γz )(1 − cϕtan(γz )) ⎞ − cos(ϕ) ⎤*f (z ) ⎠ ⎣ ⎝ Ro ⎦
x1 = mm ’+ m’F ’
(18)
z 3 z 2 f (z ) = −2 ⎛ ⎞ + 3 ⎛ ⎞ ⎝L⎠ ⎝L⎠
mm ’= y1 tan (γL)
(19)
nw cos(γL)[1 − tan(ψ) tan(γL) ] Ro
Since,
m’F =
(25)
Using Eqs. (24) and (25), with Eq. (4), the coordinate of any point along the stream path is given by,
From the proportionality of the position between the die inlet and die outlet,
y1 =
(24)
Where ai and bi (i = 1–4) are constants can be determined using specific velocity boundary conditions. In order to develop a streamline path that does not produce any rapid change of the metal direction, the following boundary conditions must be satisfied,
(9)
H 'F '= OH 'tan(ψ)
(23)
N π cot ⎛ ⎞ π ⎝N ⎠
y = f2 (z ) = b1 z 3 + b2 z 2 + b3 z + b4 z=z (7)
Om '=
N π cot ⎛ ⎞ ϕ = c ϕ π ⎝N ⎠
The stream paths in Fig. 1 can be characterized by polynomial curves f (z ) satisfying a smooth metal flow. The previous works have been reported that the forming load is not very sensitive to higher order polynomial curves that represent the die profile [11,12]. So, any point through the stream path E'F' is given by
(6)
k cos(γL)
k=
(22)
Where,
(5)
OH '=
Om '=
tan( π N ) w 2tan(ψ) = 1 R 2ϕ π Ro2 2 o
Hence,
Where x1 and y1 are the coordinates of a particle at the exit section of the die, which can be determined as follows:
HH '= k tan(γL)
2
k tan(ψ) m'H ' = sin(γL) (cos(γL))2
Eq. (26) give the relationship between the n, ϕ and z coordinate system and Cartesian coordinate system Fig. 2 shows the overall die surface for the extrusion of helical square section (N = 4) from round billet produced by numerical application of Eqs. (26) in MATLAB.
(20)
Hence,
x1 = y1tan(γL) +
n w tan(ψ) Ro cos(γL)
3. Velocity and strain rates fields (21)
In order to evaluate the velocity field in the Cartesian coordinate system, the coordinate transformation must be made. Therefore, the Jacobian of Eqs. (26) is given by,
Since the extrusion ratio must be constant throughout the extrusion process, therefore the following equation can be written, 40
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A.W. Hussein et al.
P12 = −n (A ϕ + Bϕ)
P13 = n2 [Dz (A ϕ + Bϕ) − Bz (Cϕ + Dϕ)]
P21 = −(C + D) P22 = (A + B )
P = n [Bz (C + D) − Dz (A + B )] P31 = P32 = 0
P33 = det J Therefore, from coordinate transformation it yields, ∂vx
∂vx
∂vy
∂vy
∂vy
∂vy
⎡ ∂n ⎤ ⎡ ∂x ⎤ ⎢ ∂vx ⎥ ⎢ ∂vx ⎥ − 1 ⎢ ∂y ⎥ = [J ] * ⎢ ∂ϕ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ∂vx ⎥ ⎢ ∂vx ⎥ ⎣ ∂z ⎦ ⎣ ∂z ⎦ ⎡ ∂n ⎤ ⎡ ∂x ⎤ ⎢ ∂vy ⎥ ⎢ ∂vy ⎥ − 1 ⎢ ∂y ⎥ = [J ] * ⎢ ∂ϕ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ∂vy ⎥ ⎢ ∂vy ⎥ ⎣ ∂z ⎦ ⎣ ∂z ⎦
Fig. 2. Overall die surface of the twisted squared section (L = 20 mm, π Ro = 12.5 mm, w = 8 mm, γL = 4 ). ∂x
⎡ ∂n ⎢ ∂y J= ⎢ ∂n ⎢ ⎢ ∂z ⎣ ∂n
∂x ∂ϕ
∂x ⎤ ∂z
∂y ∂ϕ
∂y ⎥ ∂z ⎥
∂z ∂ϕ
⎡ ∂n ⎤ ⎡ ∂x ⎤ ⎢ ∂vy ⎥ ⎢ ∂vy ⎥ − 1 ⎢ ∂y ⎥ = [J ] * ⎢ ∂ϕ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ∂vy ⎥ ⎢ ∂vy ⎥ ⎣ ∂z ⎦ ⎣ ∂z ⎦
⎥
∂z ⎥ ∂z ⎦
Now, the Strain rate components ε˙ is given by the following tensor [2],
(27)
The determinant of Eq. (27) is given by
det J = − n g (ϕ, z )
ε˙ij = (28)
∂Vj ⎞ 1 ⎛ ∂Vi + ⎜ ⎟ ∂x i ⎠ 2 ⎝ ∂x j
g (ϕ, z ) = (A ϕ + Bϕ ) (C + D) − (A + B ) (Cϕ + Dϕ) The velocity of the entrance must be billet velocity vo, while at the exit of the die it should be vf, that’s due to the volume constancy. Therefore,
εxx = εyy =
π Ro2
εzz =
N w2 tan( π N )
(29)
1
1
[det J ]z = 0 1 . Vo = . vo det J g (ϕ, z )
n Bz ∂x *vz = vo ∂z g (ϕ, z )
(31)
vy =
n Dz ∂y *vz = vo ∂z g (ϕ, z )
(32)
=
∂vy
=
H1 , det J H2 , det J H3 , det J
∂vy ⎤ ∂x ⎦
∂vz ∂x ⎤ ⎦
εyz = 2 ⎡ ∂z + ⎣
=
∂vz ⎤ ∂y ⎦
=
H4 , 2 det J
H5 ,and 2 det J
=
H6 . 2 det J
(36)
Where
(Bzϕ*g − Bz *gϕ)*(−C − D) ⎤ Bz (Cϕ + Dϕ ) H1 = (n vo) ⎡ + ⎥ ⎢ g g2 ⎦ ⎣ (Dzϕ*g − Dz *gϕ )(A + B ) ⎤ Dz (−A ϕ − Bϕ ) H2 = (n vo) ⎡ + ⎥ ⎢ g g2 ⎦ ⎣
Where Ai, Bi, Ci, and Di, (i = z or ϕ ) represent the partial differential functions with respect to z or ϕ. Hence, the inverse of Eq. (27) is given by,
P11 P12 P13 P21 P22 P23 P31 P32 P33
∂vz ∂z
∂vx
∂vx
1
(30)
vx =
1 det J
=
εzx = 2 ⎡ ∂z + ⎣
Using direction cosines, other velocity components Vx and Vy can be obtained as follows,
[J ]−1 =
∂vx ∂x ∂vy ∂y
εxy = 2 ⎡ ∂y + ⎣
The form of Eq. (29) can be satisfied by the following procedure
vz =
(35)
Where Vi and Vj are the velocity components, and xi and xj are the coordinate variables. Therefore, the strain rate components become,
Where,
vf =
(34)
H3 = (n vo) ⎡ ⎢ ⎣
−gϕ (Bz (C + D) − Dz (A + B )) g2
+
gz ⎤ g⎥ ⎦
⎡ Bz (−A ϕ − Bϕ) ⎞ ⎛ (Bzϕ g − Bz gϕ )(A + B ) ⎞ H4 = (n vo) ⎢ ⎜⎛ ⎟ + ⎜ ⎟ g g2 ⎠ ⎝ ⎠ ⎣⎝ ( D g D g ) ( C D ) − − − D ( C + D ) zϕ z z ϕ ϕ ϕ ⎞⎤ ⎞⎟ + ⎛⎜ + ⎛⎜ ⎟⎥ g g2 ⎝ ⎠ ⎝ ⎠⎦
(33)
Where:
P11 = n (Cϕ + Dϕ) 41
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Table 1 Numerical and physical properties used in FEM.
⎡ n2Dz (Dz (A ϕ + Bϕ) − Bz (Cϕ + Dϕ )) ⎞ H5 = vo ⎢ ⎜⎛ ⎟ g ⎠ ⎣⎝ 2 (D g − D g )(B (C + D ) − D (A + B )) n zϕ z ϕ z z ⎛ ⎞ + ⎜ ⎟ 2 g ⎝ ⎠ 2 (D g − D g ) g ( A + B ) n zz z ⎤ ϕ z ⎛ ⎞ ⎞ − ⎜⎛ ⎟ − ⎜ ⎟⎥ g g2 ⎝ ⎠ ⎝ ⎠⎦ 2 ⎡ gϕ (C + D) ⎞ ⎛ n Bz (Dz (A ϕ + Bϕ) − Bz (Cϕ + Dϕ)) ⎞⎟ H6 = vo ⎢ ⎜⎛ ⎟ + ⎜ 2 g g ⎠ ⎝ ⎠ ⎣⎝ 2 − + − + n ( B g B g )( B ( C D ) D ( A B )) zϕ z z z ϕ ⎛ ⎞ + ⎜ ⎟ g2 ⎝ ⎠
n2 (Bzz g − Bz gz ) ⎞ ⎤ − ⎛⎜ ⎟⎥ g ⎝ ⎠⎦ Where gi (i = z or ϕ ) represent the partial differential functions with respect to z or ϕ It was proved analytically that the calculated strain rates satisfy the volume constancy condition i.e. εxx + εyy + εzz = 0 .
(37)
Where Wi is the power required for the plastic deformation, Wf is the frictional power loses at the billet-die interface. These powers can be obtained using the following equations, 2 3
L
∫0 ∫0
π
N
(ε˙xx R ∫0 o ⎡
2 + ε˙ 2 + ε˙ 2) yy zz
2
⎣
1
+ ε˙ xy 2 + ε˙ xz 2 + ε˙ yz 2⎤ ⎦
2
|detJ | dn
aϕ dz Wf = m
σo 3
L
∫0 ∫0
π
N
1
[Vx 2 + Vy 2 + Vz 2]n = Ro cos(α )
∂ (x , z ) ∂ (ϕ, z )
tanξ =
dϕ dz
=
andtanλ =
∂x ∂y
(38)
2π Ro Lc Vo mc σo 3
0.12
Eqs. (30)–(32) and Eq. (36) were numerically solved using MATLAB 2017b in order to find the velocity and strain rate fields inside the deformation zone for the streamlined die under the consideration. Figs. 4 shows the velocity field along x, y, and z directions for specific conditions. For helical angle 45°, the metal velocities vx and vy are vanished along with the die axis, while the axial velocity vz gradually increases to its highest value at the die outlet. This means that material along the die axis moves in a straight line without any angular rotation. Figs. 5 and 6 show velocity contours for velocity components vx and vy. It can be noted that, at zero sweep angle φ and random radial position (n), the velocity components in the x-and y directions increase in value as the helical angle increases. When the velocity components have negative values, this indicated that the metal flow is along the negative direction of the axis. Figs. 7 and 8 show the velocity components at 30° sweep angle. It is noted that the velocity component vx shifted toward the die inlet as the helical angle increases, while the velocity component vy increases in value and begins in zero value at the center and increases to approach it peak value at the surface of the die. Fig. 9 shows the velocity field along the z-direction. As the helix angle increases, the axial velocity increases gradually toward the die exit. In addition, it is observed from this figure that the axial velocity does not depend on the angular position.
Where α is the inclination angle of any element of the die surface with respect to the projected surface of the element on the x–z plane [12]. The frictional loses due to billet-container interface can be found from the next equation,
wfc =
75.8 MPa 73.1 GPa 0.33 0.1
6.1. The velocities and strain rates distributions
2 s
(cosξ )2 + (cosλ )2 ∂y , ∂z
25 mm 25 mm 30 mm 50 mm 25 mm Tetrahedral Solid186 Square 60%,80% CONTA174 & TARGE170 0.12, 0.1 45°
6. Results and discussion
= n (A ϕ + Bϕ )
1 cos(α )
S=
∂ (x , z ) ∂ (ϕ, z )
Billet diameter Billet length Container length /steel Container outside diameter Container inside diameter Element used Product Shape Area Reduction Contact elements Friction factor Twist angle AL 2024-O Yield stress Elastic modulus Poisson's Ratio Friction factor AL 2024 Ref. [13]
metal flow, and to examine the strain and stress fields in the deformation zone, numerical investigations were accomplished using the finite element code ANSYS (V.15). In the literature, most previous papers used a simple rigid-plastic material model and neglected the strain-hardening phenomenon. For that reason, here, two FE models were constructed. The first model used an elastic-rigid plastic material model (AL 2024-O) to make a validation for the upper bound solution results. The second FE model used an elastoplastic material model (AL 2024) from the literature to show the effect of the strain-hardening phenomenon [13]. The mechanical properties of the materials and the dimensions of the extrusion tools are given in Table 1. Using rings compression test, the friction factor m = 0.12 was taken [13]. The load was applied in the form of axial displacement of 25 mm. The geometry of the extrusion setup including the die surface was created using SolidWorks. The complete FE model is shown in Fig. 3.
Since the stream paths of the die surface do not produce any velocity discontinuities from the inlet to the outlet of the die, then the upper bound of the total extrusion power (J) required to extrude helical polygonal sections reduced to:
Wi =
AL 2024-T4, AL 2024-O
σ¯ = 118.7 + 229.4 ε¯ 0.247 MPa Friction factor
4. Upper bound theory
J = 2N (Wi + Wf ) + wfc
Billet material
(39)
The calculation of Eq. (37) can be accomplished numerically for given yield stress σo and friction factor (m) using 10 points Gaussian quadrature method. A program in MATLAB was developed to find the velocity components, strain rates, and the extrusion pressure for a given yield stress, billet dimensions, area reduction (AR), relative die length (L/Ro), and the number of polygonal sides (N). 5. Finite element simulation In order to make a validation for the results of the analytical model, 42
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Fig. 5. velocity contour along the x-direction.
Fig. 3. Finite element model.
Fig. 6. velocity contour along the y-direction.
Fig. 4. velocity field along the die axis.
Figs. 10–12 shows the strain rate distribution for specific metal conditions. In general, it can be noted that as the twist angle increases, the strains increase components increase in the deformation zone. In addition, these figures clearly show that the sum ε˙ x , ε˙ y, and ε˙ z strains at any point inside the die is zero and does not depend on the twist angle γL and that verified the volume constancy condition. Fig. 13 shows the effective strain rate throughout the die length. It increases gradually to reach its peak value close to the die outlet and
Fig. 7. velocity contour along the x-direction.
43
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then decreases. As the twist angle increases, the position of the peak point moves away from the die exit and that is a very important factor in extrusion die design. 6.2. Extrusion pressure Regarding the production of a helical section shapes, the extrusion pressure was found for a twisted polygonal section whose helix angle is given by γL in Eq. (4). The plots of the total relative forming pressure (Pavg/σo) and its components with the relative axial die length (L/Ro) are shown in Fig. 14. It can be seen from the figure that the internal power of deformation (Wt) gradually decreases with increasing relative die length. The frictional load (Wf) has always an increasing tend with relative die length due to the high surface area of the die. The total relative forming pressure (J) decreases with increasing relative die length up to optimum value and then it increases. The extrusion pressure for the helical shape with N = 4 is shown in Fig. 15 for different helical angles. Noted in the figure that the optimum relative die length comes to be higher as the helical angle increases. In the case of no twist, the relative extrusion pressure decreased to the Gunasekera’s case [12]. Fig. 16 showed the variation of the relative extrusion pressure against various helical angle and frictional conditions. The relative extrusion pressure noted to be high at the highest value of twist angle. While Fig. 17 shows the influence of the area reduction on the relative forming pressure for various frictional conations. It showed that the relative forming pressure increases with the increase of the extrusion area reduction. The effect of the number of polygonal sides on the relative forming pressure is shown in Fig. 18. It decreases as the number of the polygonal sides increase. As the number of sides approaches to infinite, the case reduced to become a twisted circular shape. Table 2 shows the comparison between the upper bound average forming pressure and that obtained using the FE method. The extrusion conditions were L/Ro = 2, m = 0.1, AL2024-O with zero hardening, area reduction 60%, N = 4, γL = 45o , and billet velocity 1 mm/sec. It could be seen that the values of upper bound forming pressure and that obtained using FEM were in good agreement at the steady state conditions.
Fig. 8. velocity contour along the y-direction.
6.3. Metal flow in the deformation zone In this paper, in Section 2, it was assumed that the billet material passing through the sector area at the die inlet would go through the triangular area at the die outlet conserving the area reduction of the extrusion. That leads to the assumption that the starting point at the die inlet and the ending point at the die outlet of the streamline path preserved the proportionality of the position. Using the FE solution at the end of 706 substep (time 0.7), these assumptions can be justified. Therefore, four nodes were taken for this consideration at different position in the deformation zone (A, B, C, and D). The initial position, the displacement after the deformation (Fig. 19), and the final position of these nodes are given in Table 3. In this analysis, the final position of these nodes were obtained using its initial position and the displacement obtained from the FE solution. The new position of any point (i) in the deformation zone can be found from the following equations,
Fig. 9. velocity contour along the z-direction.
X 'i = Xi + UX i Y 'i = Yi + UY i
(40)
Where X, Y are initial position of the node, X ’, Y ′ are the final position of the node, and UX i, UY i are the displacement components of the node (i). As shown in Fig. 20, the sector area (OAB) at the die inlet transfers to the triangular area (OA’B’) at the die exit. The area of sector is 10.226 mm2, while the area of the exit triangle is 4.77 mm2, which makes the area reduction to be 57% while the theoretical value was 60% with error about 5%. In same manner, the proportionality of the position for the nodes C
Fig. 10. strain rate contour along the x-direction.
44
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Fig. 14. Forming pressure for various die length.
Fig. 11. strain rate contour along the y-direction.
Fig. 15. Forming pressure for various die length and helix angle.
Fig. 12. strain rate contour along the z-direction.
Fig. 16. Forming pressure for various helix angle and frictional conditions. Fig. 13. Variation of the effective strain rate along the die length.
45
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Fig. 17. Forming pressure for various helix angle and area reductions.
Fig. 18. Effect of the polygonal shape on the extrusion stress and die length.
Table 2 Comparison between upper bound and FEM results. Method
Forming pressure
Upper bound FEM
158.36 150.3
and D,
OD 9.34 OD' 6.08 = = 0.74, while = = 0.81 7.42 OC 12.5 OC '
Fig. 19. Displacement solution at the end of 706 Substep: (A) Ux, (B) Uy, and (C) Uz.
(41)
Hence, the error is about 8%. These errors because the plastic deformation is highly inhomogeneous, since the metal flow is faster at the center of the die and its motion decreases at the die-billet interface.
Table 3 Node sample for study of the position.
6.4. Stresses and strains Fig. 21 shows the Von Mises plastic strain and corresponding stress contours in the deformation zone in the case of extrusion of AL 2014-O with zero hardening condition. The maximum values exist at the die surface and be minimum at the die axis. Fig. 22 shows the forming load against its displacement. It can be noted that the forming load increases 46
Points
Node
X
Y
UX
UY
X’
Y’
A B C D
16888 16889 16902 17052
10.83 9.92 9.04 −6.76
6.25 7.61 8.62 6.45
−2.10 −2.33 −12.02 4.32
−5.04 −5.48 −2.22 −0.88
8.72 7.59 −2.98 −2.44
1.21 2.13 6.82 5.57
Journal of Manufacturing Processes 38 (2019) 38–48
A.W. Hussein et al.
Fig. 22. forming load for AL 2024-O billet.
Fig. 20. the proportionality position of the sample nodes.
Fig. 23. Von Mises for AL 2024 [13]: (A) stress field (MPa), and (B) plastic strain field.
not linearly up to steady state value. Fig. 23 shows the Von Mises plastic strain and corresponding stress contours in the deformation zone in the case of extrusion of AL 2024 with hardening condition [13]. The extrusion condition were L/Ro = 2, m = 0.12, area reduction 60%, N = 4, γL = 45o and billet velocity
Fig. 21. Von Mises for AL 2024-O: (A) stress field (MPa), and (B) plastic strain field.
47
Journal of Manufacturing Processes 38 (2019) 38–48
A.W. Hussein et al.
• As the twist angle increases, the position of the peak point of the • • •
effective strain rate moves away from the die exit toward the die entrance and that is a very important factor in the design of the extrusion die. The optimal relative die length comes to be larger as the twist angle increases. In the case of no twist, the forming pressure reduced to the Gunasekera’s case. The forming pressure becomes high as the helix angle, area reduction, and the factor of friction increase, while it decreases as the number of sides of the polygonal shape increases. The finite element solution shows that the maximum strain and corresponding stress located closed to die outlet at the surface and become minimum at the die axis in which the velocity components along x and y directions vanish.
References [1] Valiev RZ, Islamgaliev RK, Alexandrov IV. Bulk nanostructured materials from severe plastic deformation. Prog Mater Sci 2000;45:103–89. [2] Beygelzimer Y, Varyukhin V, Synkov S, Orlov D. Useful properties of twist extrusion. Mater Sci Eng A 2009;503:14–7. [3] Zendehdel H, Hassani A. Influence of twist extrusion process on microstructure and mechanical properties of 6063 aluminum alloy. Mater Des 2012;37:13–8. [4] Orlov Dmitry, Beygelzimer Yan, Synkov Sergey, Varyukhin Viktor, Tsuji Nobuhiro, Horita Zenji. Plastic flow, structure and mechanical properties in pure Al deformed by twist extrusion. Mater Sci Eng A 2009;519:105–11. [5] Yang DY, Altan T. Analytical and experimental investigation into lubricated three dimensional extrusion of general helical sections. CIRP Ann-Manuf Technol 1986;35(1):169–72. [6] Park YB, Yoon JH, Yang DY. Finite element analysis of steady state three dimensional helical extrusion of twisted sections using recurrent boundary conditions. Int J Mech Sei 1994;36(2):137–48. [7] Yang DY, Lange K. Investigation into non-.sTeady-state three dimensional extrusion next term of a Trochoidal previous term helical next term gear by the rigid. Plastic finite element method. CIRP Ann-Manuf Technol AM I 1994:229–33. [8] Seyed Salehi M, Anjabin N, Kim HS. An upper bound solution for twist extrusion process. Met Mater Int 2014;20(5):825–34. [9] Seyed Salehi M, Serajzadeh S. A new upper bound solution for the analysis of the twist extrusion process with an elliptical die cross-section. Proc IMechE 2009;223. Part C: J. Mechanical Engineering Science. [10] Khalifa NB, Tekkaya AE. Newest developments on the manufacture of helical profiles by hot extrusion. J Manuf Sci Eng 2011;13:061010. [11] Hussein AW, Kadhim AJ. Mathematical analyses and numerical simulations for forward extrusion of circular, square, and rhomboidal sections from round billets through streamlined dies. J Manuf Sci Eng 2017;139(6):064501. [12] Gunasekera JS, Hoshino Sl. Analysis of extrusion of polygonal sections through streamlined dies. J Eng Ind 1985;107 / 229. [13] Yang DY, Lee CM, Cho JR. Analysis of axisymmetric extrusion of rods by the method of weighted residuals using body-fitted coordinate transformation. Int J Mech Sci 1990;32(2):101–14.
Fig. 24. forming load for AL 2024 billet [13].
1 mm/sec. It can be noted that the maximum values of the strains and equivalent stress occurs close to the exit region of the die at the die surface. This result was seen in Fig. 13 in which the effective strain rate reaches its peak value near the die exit. Another note can be showed in Fig. 23 that the strains and corresponding stresses is minimum along the die axis in which the velocity components vx and vy vanished as seen in Fig. 4. Fig. 24 shows the pressure of the extrusion punch against its displacement. It can be noted that the punch load increases gradually until it reached steady-state value, which represents the required forming pressure. 7. Conclusions An analytical representation of the die surface for helical section shapes in three-dimensional process was proposed. The design includes a smooth shape change of the metal flow in the die, which produced less forming pressure. The velocity field has been obtained based on the volume constancy and velocity boundary conditions. The main conclusions of this work were:
• The metal at the die axis moves in a straight line from the inlet to the outlet of the die and there is no angular motion.
48
Contents lists available at ScienceDirect
Journal of Manufacturing Processes journal homepage: www.elsevier.com/locate/manpro
Helical Extrusion Process of General Polygonal Section Shapes through Curved Dies
T
⁎
Ahmed Waleed Husseina, , Mushrek A. Mahdib, Raneen Sami Abidc a
Department of Energy Engineering, University of Babylon, Babylon, Iraq Department of Automobile Engineering, University of Babylon, Babylon, Iraq c Department of Aerospace Manufacture and Engineering, Nanjing University of Aeronautics & Astronautics, China b
A R T I C LE I N FO
A B S T R A C T
Keywords: Helical extrusion Twist extrusion Upper bound theory Finite element method Streamlined dies ANSYS
Practically, the helical products used as a mechanical part to satisfy the needs of mechanical strength and artistic appearance. In addition, during recent 5 years it used to produce an ultra-fine grain metal structure. However, there are very little works regarding this type of extrusion so far. This work proposed a new formulation for helical deformation zone to produce general helical polygonal shapes through a streamlined die that are usually made by hot extrusion through taper die. The general die surface was represented analytically. The velocity and strain rate fields are derived depending on the volume constancy and the velocity boundary conditions. The upper bound forming pressure was obtained for various frictional conditions, area reduction, helix angle, and die length. The results show that the axis of the product does not rotate through the helical extrusion. The peak value of the strain rate is located close to the die outlet and decreases as the helix angle increases. The optimum die length becomes high as the helix angle increases. The forming pressure increases with increasing helix angle, area reduction, factor of friction, while decreases when the number of sides increases. The theoretical results were verified with previous work of zero twist and showed completely compatible. A finite element solution was done using hardening material model to verify the analytical results and metal flow and to examine the strain and stress fields in the product.
1. Introduction and background In structure industry, the helical products used as a mechanical part to satisfy the need of mechanical strength besides the artistic appearance. In addition, they are used in the power devices such as screw pumps and superchargers. Nowadays, the helical sections are made using casting, hot forging operations or machining processes. These methods have many disadvantages, time consumes, besides their high cost. Furthermore, during recent 5 years, the material science engineering showed that the severe plastic deformation produced by torsional shear stress produces ultra-fine grain metal structures. The main advantages of that structure are free of porosity and internal oxidation [1–4]. Despite increasing demand for the application of helical sections in the industry, however, very little analytical methods have been attempted so far. In the literature, few papers talking about the profiles with helical shapes were found. Yang et al. [5–7] used conformal transformation method to create the die surface and to examine the internal metal flow by transformation of any intermediate cross section
⁎
into a unit circle. The upper bound theory was applied to find the required extrusion pressure for the extrusion of clover, elliptic, and trochoidal gear sections. They used a rigid -perfectly plastic material model. Actually, using transformation method was very difficult and contained wide coordinate transformations as well as transformation function and in most, it is impossible to apply this technique for extrusion of complicated shapes. Salehi et al. [8,9] utilizing a Fourier approximation to find die cavity and then the admissible velocity field. The upper bound theory was used to calculate the required energy for twist extrusion of square and elliptical shapes through squared and elliptical die cross-section. In this works, the cross sections of deforming material remain unchanged throughout the twist extrusion process. Khalifa and Tekkaya [10] used a multi-segment die for hot extrusion of helical screw shaft from round billet. They used the experiments and finite element method to examine the effect of the friction and material flow against the twisting angle. There is no analytical analysis presented in this work. It becomes clear that there is no general systematic work deals the
Corresponding author. E-mail address: [email protected] (A.W. Hussein).
https://doi.org/10.1016/j.jmapro.2018.12.032 Received 17 August 2018; Received in revised form 15 November 2018; Accepted 24 December 2018 1526-6125/ © 2018 Published by Elsevier Ltd on behalf of The Society of Manufacturing Engineers.
Journal of Manufacturing Processes 38 (2019) 38–48
A.W. Hussein et al.
Nomenclature C L x, y , z m N W f(z)
vx,vy,vz vo, vf Wi Wf Ro Wt σo γL γz
Geometry constant Die length Cartesian co-ordinates systems Friction factor A number of polygonal sides Half of the polygon side The function used to descript the die profile and the streamlines
Velocity components Punch and product velocities, respectively Power required for plastic deformation Power loses due to frictional resistance Billet radius Total powers consume The yield stress of the billet material Total helix angle at the die outlet helical angle function
0 ≤ n ≤ Ro π 0≤ϕ≤ N
extrusion of helical sections in deep, in which the velocity and strain of every single point in the helical deformation zone can be obtained analytically. Furthermore, in all previous works, there is shear power loses due to velocity breaks in the inlet and outlet of the die, which increase the forming power required. In addition, there is little information about the stress and strain distributions throughout the extruded product. Therefore, this work developed a new method for finding optimum die profile, which yields minimum forming pressure for the extrusion of helical section shapes from round bar stock using general die configuration. The velocity and strain rate fields were derived analytically based on the volume constancy and velocity boundary conditions. Upper bound theory was used to determine the forming pressure for different helical polygonal section shapes (the sides from 1 to ∞) for a given required helix angle, required area reduction, material properties, and frictional conditions. In addition, the validation of results, metal flow, and stress and strain distributions were also obtained using FEM code ANSYS based on the experimental material data for AL 2024 from the literature.
0≤z≤L
(1)
Where N represents the number of sides for polygonal section. n is the radial distance from the die center to random point E' at the die inlet, ϕ is the inlet sweep angle, and ψ is the equivalent sweep angle at die outlet. In this work, the twist angle of the die can be expressed as a general polygonal function from the third degree as follows,
γ (z ) = r1 z 3 + r2 z 2 + r3 z + r4
(2)
Where ri (i = 1 to 4) are constants to be determined from the angular boundary conditions of the deformation zone. In order to reduce the angular shear loses at the inlet and outlet of the die, the following angular boundary conditions can be used,
2. Configuration of the extrusion die
γ (0) = 0, and
∂γ (z ) ∂z z=0
=0
γ (L) = γL, and
∂γ (z ) ∂z z=L
=0
(3)
Where γL is the total helix angle at the die outlet (0 ≤ γL < 900) . Hence, using these angular boundary conditions, Eq. (1) becomes,
For a smooth cross-sectional change from the initial round billet to the final helical extruded product, the die surface must be designed without any rapid change of die surface along the die axis. The die geometry in three-dimensional extrusion process is shown in Fig. 1. Let, Ro the radius of the round billet, w the half side of polygon, and L the die length. As a result of the symmetry of the deformation zone, variations of n, ϕ and z are
z 2 z 3 γ (z ) = 3 γL ⎛ ⎞ − 2γL ⎛ ⎞ ⎝L⎠ ⎝L⎠
(4)
The billet is forced to flow through the die that created by a number of streamlines to the final helical section. Therefore, the points E and G on the die entrance are connected with the equivalent points B’’ and B’ at
Fig. 1. Proposed die geometry and the velocity field for general polygonal section shape. 39
Journal of Manufacturing Processes 38 (2019) 38–48
A.W. Hussein et al. N w2
the die outlet respectively. Assumed that the billet material passing through the sector area O’EG at the die inlet goes through the triangle area OB’B’’ at the die outlet conserving the extrusion ratio. The stream path defined by the points O’, E , B’’ and O becomes a three dimensional stream path. Taking the assumptions that the surface O’EB’’O consists of a number of streamline paths, hence, the particle at random point E' at the die inlet will go towards the equivalent point F' at the die exit preserving the proportionality of the position. The locations of the points E ′ and F ′ in the Cartesian coordinate are given by
E¯ (nsin(∅), ncos(∅), 0) F¯ (x1, y1 , L)
1
tan(ψ) =
c=
OB '=
w cos(γL)
(8)
BB '= w sin(γL)
x = f1 (z ) = a1 z 3 + a2 z 2 + a3 z + a4
(10)
The triangle (m′H ′F ′) has,
m'H '= H 'F 'tan(γL)
(12)
Therefore,
k (1 − tan(ψ) tan(γL) ) cos(γz )
x (0) = nsin(ϕ) ,
∂x ∂z z = 0
=0
y (0) = n cos (ϕ) ,
∂y ∂z z = 0
=0
(11)
And,
Om '= OH '− m'H '
x (L) = x1,
∂x ∂z z = L
=0
y (L) = y1 ,
∂y ∂z z = L
=0
(13)
x = n A (ϕ) + n B (ϕ, z , ) y = n C (ϕ) + n D (ϕ, z ) z=z
(26)
Where,
nw cos(γL) Ro
(14)
A (ϕ) = sin(ϕ);
Therefore,
nw (1 − tan(ψ) tan(γL) ) Ro
w w tan (ψ) B (ϕ, z ) = ⎡ ⎛ cos(γz )(1 − cϕtan(γz )) ⎞ tan (γL) + ⎢ ⎝ Ro cos(γL) Ro ⎠ ⎣
(15)
− sin(ϕ) ⎤ f (z ) ⎥ ⎦
But,
y1 = (Om')cos(γL)
(16)
C (ϕ) = cos(ϕ)
Therefore, (17)
w D (ϕ, z ) = ⎡ ⎛ cos(γz )(1 − cϕtan(γz )) ⎞ − cos(ϕ) ⎤*f (z ) ⎠ ⎣ ⎝ Ro ⎦
x1 = mm ’+ m’F ’
(18)
z 3 z 2 f (z ) = −2 ⎛ ⎞ + 3 ⎛ ⎞ ⎝L⎠ ⎝L⎠
mm ’= y1 tan (γL)
(19)
nw cos(γL)[1 − tan(ψ) tan(γL) ] Ro
Since,
m’F =
(25)
Using Eqs. (24) and (25), with Eq. (4), the coordinate of any point along the stream path is given by,
From the proportionality of the position between the die inlet and die outlet,
y1 =
(24)
Where ai and bi (i = 1–4) are constants can be determined using specific velocity boundary conditions. In order to develop a streamline path that does not produce any rapid change of the metal direction, the following boundary conditions must be satisfied,
(9)
H 'F '= OH 'tan(ψ)
(23)
N π cot ⎛ ⎞ π ⎝N ⎠
y = f2 (z ) = b1 z 3 + b2 z 2 + b3 z + b4 z=z (7)
Om '=
N π cot ⎛ ⎞ ϕ = c ϕ π ⎝N ⎠
The stream paths in Fig. 1 can be characterized by polynomial curves f (z ) satisfying a smooth metal flow. The previous works have been reported that the forming load is not very sensitive to higher order polynomial curves that represent the die profile [11,12]. So, any point through the stream path E'F' is given by
(6)
k cos(γL)
k=
(22)
Where,
(5)
OH '=
Om '=
tan( π N ) w 2tan(ψ) = 1 R 2ϕ π Ro2 2 o
Hence,
Where x1 and y1 are the coordinates of a particle at the exit section of the die, which can be determined as follows:
HH '= k tan(γL)
2
k tan(ψ) m'H ' = sin(γL) (cos(γL))2
Eq. (26) give the relationship between the n, ϕ and z coordinate system and Cartesian coordinate system Fig. 2 shows the overall die surface for the extrusion of helical square section (N = 4) from round billet produced by numerical application of Eqs. (26) in MATLAB.
(20)
Hence,
x1 = y1tan(γL) +
n w tan(ψ) Ro cos(γL)
3. Velocity and strain rates fields (21)
In order to evaluate the velocity field in the Cartesian coordinate system, the coordinate transformation must be made. Therefore, the Jacobian of Eqs. (26) is given by,
Since the extrusion ratio must be constant throughout the extrusion process, therefore the following equation can be written, 40
Journal of Manufacturing Processes 38 (2019) 38–48
A.W. Hussein et al.
P12 = −n (A ϕ + Bϕ)
P13 = n2 [Dz (A ϕ + Bϕ) − Bz (Cϕ + Dϕ)]
P21 = −(C + D) P22 = (A + B )
P = n [Bz (C + D) − Dz (A + B )] P31 = P32 = 0
P33 = det J Therefore, from coordinate transformation it yields, ∂vx
∂vx
∂vy
∂vy
∂vy
∂vy
⎡ ∂n ⎤ ⎡ ∂x ⎤ ⎢ ∂vx ⎥ ⎢ ∂vx ⎥ − 1 ⎢ ∂y ⎥ = [J ] * ⎢ ∂ϕ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ∂vx ⎥ ⎢ ∂vx ⎥ ⎣ ∂z ⎦ ⎣ ∂z ⎦ ⎡ ∂n ⎤ ⎡ ∂x ⎤ ⎢ ∂vy ⎥ ⎢ ∂vy ⎥ − 1 ⎢ ∂y ⎥ = [J ] * ⎢ ∂ϕ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ∂vy ⎥ ⎢ ∂vy ⎥ ⎣ ∂z ⎦ ⎣ ∂z ⎦
Fig. 2. Overall die surface of the twisted squared section (L = 20 mm, π Ro = 12.5 mm, w = 8 mm, γL = 4 ). ∂x
⎡ ∂n ⎢ ∂y J= ⎢ ∂n ⎢ ⎢ ∂z ⎣ ∂n
∂x ∂ϕ
∂x ⎤ ∂z
∂y ∂ϕ
∂y ⎥ ∂z ⎥
∂z ∂ϕ
⎡ ∂n ⎤ ⎡ ∂x ⎤ ⎢ ∂vy ⎥ ⎢ ∂vy ⎥ − 1 ⎢ ∂y ⎥ = [J ] * ⎢ ∂ϕ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ∂vy ⎥ ⎢ ∂vy ⎥ ⎣ ∂z ⎦ ⎣ ∂z ⎦
⎥
∂z ⎥ ∂z ⎦
Now, the Strain rate components ε˙ is given by the following tensor [2],
(27)
The determinant of Eq. (27) is given by
det J = − n g (ϕ, z )
ε˙ij = (28)
∂Vj ⎞ 1 ⎛ ∂Vi + ⎜ ⎟ ∂x i ⎠ 2 ⎝ ∂x j
g (ϕ, z ) = (A ϕ + Bϕ ) (C + D) − (A + B ) (Cϕ + Dϕ) The velocity of the entrance must be billet velocity vo, while at the exit of the die it should be vf, that’s due to the volume constancy. Therefore,
εxx = εyy =
π Ro2
εzz =
N w2 tan( π N )
(29)
1
1
[det J ]z = 0 1 . Vo = . vo det J g (ϕ, z )
n Bz ∂x *vz = vo ∂z g (ϕ, z )
(31)
vy =
n Dz ∂y *vz = vo ∂z g (ϕ, z )
(32)
=
∂vy
=
H1 , det J H2 , det J H3 , det J
∂vy ⎤ ∂x ⎦
∂vz ∂x ⎤ ⎦
εyz = 2 ⎡ ∂z + ⎣
=
∂vz ⎤ ∂y ⎦
=
H4 , 2 det J
H5 ,and 2 det J
=
H6 . 2 det J
(36)
Where
(Bzϕ*g − Bz *gϕ)*(−C − D) ⎤ Bz (Cϕ + Dϕ ) H1 = (n vo) ⎡ + ⎥ ⎢ g g2 ⎦ ⎣ (Dzϕ*g − Dz *gϕ )(A + B ) ⎤ Dz (−A ϕ − Bϕ ) H2 = (n vo) ⎡ + ⎥ ⎢ g g2 ⎦ ⎣
Where Ai, Bi, Ci, and Di, (i = z or ϕ ) represent the partial differential functions with respect to z or ϕ. Hence, the inverse of Eq. (27) is given by,
P11 P12 P13 P21 P22 P23 P31 P32 P33
∂vz ∂z
∂vx
∂vx
1
(30)
vx =
1 det J
=
εzx = 2 ⎡ ∂z + ⎣
Using direction cosines, other velocity components Vx and Vy can be obtained as follows,
[J ]−1 =
∂vx ∂x ∂vy ∂y
εxy = 2 ⎡ ∂y + ⎣
The form of Eq. (29) can be satisfied by the following procedure
vz =
(35)
Where Vi and Vj are the velocity components, and xi and xj are the coordinate variables. Therefore, the strain rate components become,
Where,
vf =
(34)
H3 = (n vo) ⎡ ⎢ ⎣
−gϕ (Bz (C + D) − Dz (A + B )) g2
+
gz ⎤ g⎥ ⎦
⎡ Bz (−A ϕ − Bϕ) ⎞ ⎛ (Bzϕ g − Bz gϕ )(A + B ) ⎞ H4 = (n vo) ⎢ ⎜⎛ ⎟ + ⎜ ⎟ g g2 ⎠ ⎝ ⎠ ⎣⎝ ( D g D g ) ( C D ) − − − D ( C + D ) zϕ z z ϕ ϕ ϕ ⎞⎤ ⎞⎟ + ⎛⎜ + ⎛⎜ ⎟⎥ g g2 ⎝ ⎠ ⎝ ⎠⎦
(33)
Where:
P11 = n (Cϕ + Dϕ) 41
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A.W. Hussein et al.
Table 1 Numerical and physical properties used in FEM.
⎡ n2Dz (Dz (A ϕ + Bϕ) − Bz (Cϕ + Dϕ )) ⎞ H5 = vo ⎢ ⎜⎛ ⎟ g ⎠ ⎣⎝ 2 (D g − D g )(B (C + D ) − D (A + B )) n zϕ z ϕ z z ⎛ ⎞ + ⎜ ⎟ 2 g ⎝ ⎠ 2 (D g − D g ) g ( A + B ) n zz z ⎤ ϕ z ⎛ ⎞ ⎞ − ⎜⎛ ⎟ − ⎜ ⎟⎥ g g2 ⎝ ⎠ ⎝ ⎠⎦ 2 ⎡ gϕ (C + D) ⎞ ⎛ n Bz (Dz (A ϕ + Bϕ) − Bz (Cϕ + Dϕ)) ⎞⎟ H6 = vo ⎢ ⎜⎛ ⎟ + ⎜ 2 g g ⎠ ⎝ ⎠ ⎣⎝ 2 − + − + n ( B g B g )( B ( C D ) D ( A B )) zϕ z z z ϕ ⎛ ⎞ + ⎜ ⎟ g2 ⎝ ⎠
n2 (Bzz g − Bz gz ) ⎞ ⎤ − ⎛⎜ ⎟⎥ g ⎝ ⎠⎦ Where gi (i = z or ϕ ) represent the partial differential functions with respect to z or ϕ It was proved analytically that the calculated strain rates satisfy the volume constancy condition i.e. εxx + εyy + εzz = 0 .
(37)
Where Wi is the power required for the plastic deformation, Wf is the frictional power loses at the billet-die interface. These powers can be obtained using the following equations, 2 3
L
∫0 ∫0
π
N
(ε˙xx R ∫0 o ⎡
2 + ε˙ 2 + ε˙ 2) yy zz
2
⎣
1
+ ε˙ xy 2 + ε˙ xz 2 + ε˙ yz 2⎤ ⎦
2
|detJ | dn
aϕ dz Wf = m
σo 3
L
∫0 ∫0
π
N
1
[Vx 2 + Vy 2 + Vz 2]n = Ro cos(α )
∂ (x , z ) ∂ (ϕ, z )
tanξ =
dϕ dz
=
andtanλ =
∂x ∂y
(38)
2π Ro Lc Vo mc σo 3
0.12
Eqs. (30)–(32) and Eq. (36) were numerically solved using MATLAB 2017b in order to find the velocity and strain rate fields inside the deformation zone for the streamlined die under the consideration. Figs. 4 shows the velocity field along x, y, and z directions for specific conditions. For helical angle 45°, the metal velocities vx and vy are vanished along with the die axis, while the axial velocity vz gradually increases to its highest value at the die outlet. This means that material along the die axis moves in a straight line without any angular rotation. Figs. 5 and 6 show velocity contours for velocity components vx and vy. It can be noted that, at zero sweep angle φ and random radial position (n), the velocity components in the x-and y directions increase in value as the helical angle increases. When the velocity components have negative values, this indicated that the metal flow is along the negative direction of the axis. Figs. 7 and 8 show the velocity components at 30° sweep angle. It is noted that the velocity component vx shifted toward the die inlet as the helical angle increases, while the velocity component vy increases in value and begins in zero value at the center and increases to approach it peak value at the surface of the die. Fig. 9 shows the velocity field along the z-direction. As the helix angle increases, the axial velocity increases gradually toward the die exit. In addition, it is observed from this figure that the axial velocity does not depend on the angular position.
Where α is the inclination angle of any element of the die surface with respect to the projected surface of the element on the x–z plane [12]. The frictional loses due to billet-container interface can be found from the next equation,
wfc =
75.8 MPa 73.1 GPa 0.33 0.1
6.1. The velocities and strain rates distributions
2 s
(cosξ )2 + (cosλ )2 ∂y , ∂z
25 mm 25 mm 30 mm 50 mm 25 mm Tetrahedral Solid186 Square 60%,80% CONTA174 & TARGE170 0.12, 0.1 45°
6. Results and discussion
= n (A ϕ + Bϕ )
1 cos(α )
S=
∂ (x , z ) ∂ (ϕ, z )
Billet diameter Billet length Container length /steel Container outside diameter Container inside diameter Element used Product Shape Area Reduction Contact elements Friction factor Twist angle AL 2024-O Yield stress Elastic modulus Poisson's Ratio Friction factor AL 2024 Ref. [13]
metal flow, and to examine the strain and stress fields in the deformation zone, numerical investigations were accomplished using the finite element code ANSYS (V.15). In the literature, most previous papers used a simple rigid-plastic material model and neglected the strain-hardening phenomenon. For that reason, here, two FE models were constructed. The first model used an elastic-rigid plastic material model (AL 2024-O) to make a validation for the upper bound solution results. The second FE model used an elastoplastic material model (AL 2024) from the literature to show the effect of the strain-hardening phenomenon [13]. The mechanical properties of the materials and the dimensions of the extrusion tools are given in Table 1. Using rings compression test, the friction factor m = 0.12 was taken [13]. The load was applied in the form of axial displacement of 25 mm. The geometry of the extrusion setup including the die surface was created using SolidWorks. The complete FE model is shown in Fig. 3.
Since the stream paths of the die surface do not produce any velocity discontinuities from the inlet to the outlet of the die, then the upper bound of the total extrusion power (J) required to extrude helical polygonal sections reduced to:
Wi =
AL 2024-T4, AL 2024-O
σ¯ = 118.7 + 229.4 ε¯ 0.247 MPa Friction factor
4. Upper bound theory
J = 2N (Wi + Wf ) + wfc
Billet material
(39)
The calculation of Eq. (37) can be accomplished numerically for given yield stress σo and friction factor (m) using 10 points Gaussian quadrature method. A program in MATLAB was developed to find the velocity components, strain rates, and the extrusion pressure for a given yield stress, billet dimensions, area reduction (AR), relative die length (L/Ro), and the number of polygonal sides (N). 5. Finite element simulation In order to make a validation for the results of the analytical model, 42
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Fig. 5. velocity contour along the x-direction.
Fig. 3. Finite element model.
Fig. 6. velocity contour along the y-direction.
Fig. 4. velocity field along the die axis.
Figs. 10–12 shows the strain rate distribution for specific metal conditions. In general, it can be noted that as the twist angle increases, the strains increase components increase in the deformation zone. In addition, these figures clearly show that the sum ε˙ x , ε˙ y, and ε˙ z strains at any point inside the die is zero and does not depend on the twist angle γL and that verified the volume constancy condition. Fig. 13 shows the effective strain rate throughout the die length. It increases gradually to reach its peak value close to the die outlet and
Fig. 7. velocity contour along the x-direction.
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then decreases. As the twist angle increases, the position of the peak point moves away from the die exit and that is a very important factor in extrusion die design. 6.2. Extrusion pressure Regarding the production of a helical section shapes, the extrusion pressure was found for a twisted polygonal section whose helix angle is given by γL in Eq. (4). The plots of the total relative forming pressure (Pavg/σo) and its components with the relative axial die length (L/Ro) are shown in Fig. 14. It can be seen from the figure that the internal power of deformation (Wt) gradually decreases with increasing relative die length. The frictional load (Wf) has always an increasing tend with relative die length due to the high surface area of the die. The total relative forming pressure (J) decreases with increasing relative die length up to optimum value and then it increases. The extrusion pressure for the helical shape with N = 4 is shown in Fig. 15 for different helical angles. Noted in the figure that the optimum relative die length comes to be higher as the helical angle increases. In the case of no twist, the relative extrusion pressure decreased to the Gunasekera’s case [12]. Fig. 16 showed the variation of the relative extrusion pressure against various helical angle and frictional conditions. The relative extrusion pressure noted to be high at the highest value of twist angle. While Fig. 17 shows the influence of the area reduction on the relative forming pressure for various frictional conations. It showed that the relative forming pressure increases with the increase of the extrusion area reduction. The effect of the number of polygonal sides on the relative forming pressure is shown in Fig. 18. It decreases as the number of the polygonal sides increase. As the number of sides approaches to infinite, the case reduced to become a twisted circular shape. Table 2 shows the comparison between the upper bound average forming pressure and that obtained using the FE method. The extrusion conditions were L/Ro = 2, m = 0.1, AL2024-O with zero hardening, area reduction 60%, N = 4, γL = 45o , and billet velocity 1 mm/sec. It could be seen that the values of upper bound forming pressure and that obtained using FEM were in good agreement at the steady state conditions.
Fig. 8. velocity contour along the y-direction.
6.3. Metal flow in the deformation zone In this paper, in Section 2, it was assumed that the billet material passing through the sector area at the die inlet would go through the triangular area at the die outlet conserving the area reduction of the extrusion. That leads to the assumption that the starting point at the die inlet and the ending point at the die outlet of the streamline path preserved the proportionality of the position. Using the FE solution at the end of 706 substep (time 0.7), these assumptions can be justified. Therefore, four nodes were taken for this consideration at different position in the deformation zone (A, B, C, and D). The initial position, the displacement after the deformation (Fig. 19), and the final position of these nodes are given in Table 3. In this analysis, the final position of these nodes were obtained using its initial position and the displacement obtained from the FE solution. The new position of any point (i) in the deformation zone can be found from the following equations,
Fig. 9. velocity contour along the z-direction.
X 'i = Xi + UX i Y 'i = Yi + UY i
(40)
Where X, Y are initial position of the node, X ’, Y ′ are the final position of the node, and UX i, UY i are the displacement components of the node (i). As shown in Fig. 20, the sector area (OAB) at the die inlet transfers to the triangular area (OA’B’) at the die exit. The area of sector is 10.226 mm2, while the area of the exit triangle is 4.77 mm2, which makes the area reduction to be 57% while the theoretical value was 60% with error about 5%. In same manner, the proportionality of the position for the nodes C
Fig. 10. strain rate contour along the x-direction.
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Fig. 14. Forming pressure for various die length.
Fig. 11. strain rate contour along the y-direction.
Fig. 15. Forming pressure for various die length and helix angle.
Fig. 12. strain rate contour along the z-direction.
Fig. 16. Forming pressure for various helix angle and frictional conditions. Fig. 13. Variation of the effective strain rate along the die length.
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Fig. 17. Forming pressure for various helix angle and area reductions.
Fig. 18. Effect of the polygonal shape on the extrusion stress and die length.
Table 2 Comparison between upper bound and FEM results. Method
Forming pressure
Upper bound FEM
158.36 150.3
and D,
OD 9.34 OD' 6.08 = = 0.74, while = = 0.81 7.42 OC 12.5 OC '
Fig. 19. Displacement solution at the end of 706 Substep: (A) Ux, (B) Uy, and (C) Uz.
(41)
Hence, the error is about 8%. These errors because the plastic deformation is highly inhomogeneous, since the metal flow is faster at the center of the die and its motion decreases at the die-billet interface.
Table 3 Node sample for study of the position.
6.4. Stresses and strains Fig. 21 shows the Von Mises plastic strain and corresponding stress contours in the deformation zone in the case of extrusion of AL 2014-O with zero hardening condition. The maximum values exist at the die surface and be minimum at the die axis. Fig. 22 shows the forming load against its displacement. It can be noted that the forming load increases 46
Points
Node
X
Y
UX
UY
X’
Y’
A B C D
16888 16889 16902 17052
10.83 9.92 9.04 −6.76
6.25 7.61 8.62 6.45
−2.10 −2.33 −12.02 4.32
−5.04 −5.48 −2.22 −0.88
8.72 7.59 −2.98 −2.44
1.21 2.13 6.82 5.57
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Fig. 22. forming load for AL 2024-O billet.
Fig. 20. the proportionality position of the sample nodes.
Fig. 23. Von Mises for AL 2024 [13]: (A) stress field (MPa), and (B) plastic strain field.
not linearly up to steady state value. Fig. 23 shows the Von Mises plastic strain and corresponding stress contours in the deformation zone in the case of extrusion of AL 2024 with hardening condition [13]. The extrusion condition were L/Ro = 2, m = 0.12, area reduction 60%, N = 4, γL = 45o and billet velocity
Fig. 21. Von Mises for AL 2024-O: (A) stress field (MPa), and (B) plastic strain field.
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• As the twist angle increases, the position of the peak point of the • • •
effective strain rate moves away from the die exit toward the die entrance and that is a very important factor in the design of the extrusion die. The optimal relative die length comes to be larger as the twist angle increases. In the case of no twist, the forming pressure reduced to the Gunasekera’s case. The forming pressure becomes high as the helix angle, area reduction, and the factor of friction increase, while it decreases as the number of sides of the polygonal shape increases. The finite element solution shows that the maximum strain and corresponding stress located closed to die outlet at the surface and become minimum at the die axis in which the velocity components along x and y directions vanish.
References [1] Valiev RZ, Islamgaliev RK, Alexandrov IV. Bulk nanostructured materials from severe plastic deformation. Prog Mater Sci 2000;45:103–89. [2] Beygelzimer Y, Varyukhin V, Synkov S, Orlov D. Useful properties of twist extrusion. Mater Sci Eng A 2009;503:14–7. [3] Zendehdel H, Hassani A. Influence of twist extrusion process on microstructure and mechanical properties of 6063 aluminum alloy. Mater Des 2012;37:13–8. [4] Orlov Dmitry, Beygelzimer Yan, Synkov Sergey, Varyukhin Viktor, Tsuji Nobuhiro, Horita Zenji. Plastic flow, structure and mechanical properties in pure Al deformed by twist extrusion. Mater Sci Eng A 2009;519:105–11. [5] Yang DY, Altan T. Analytical and experimental investigation into lubricated three dimensional extrusion of general helical sections. CIRP Ann-Manuf Technol 1986;35(1):169–72. [6] Park YB, Yoon JH, Yang DY. Finite element analysis of steady state three dimensional helical extrusion of twisted sections using recurrent boundary conditions. Int J Mech Sei 1994;36(2):137–48. [7] Yang DY, Lange K. Investigation into non-.sTeady-state three dimensional extrusion next term of a Trochoidal previous term helical next term gear by the rigid. Plastic finite element method. CIRP Ann-Manuf Technol AM I 1994:229–33. [8] Seyed Salehi M, Anjabin N, Kim HS. An upper bound solution for twist extrusion process. Met Mater Int 2014;20(5):825–34. [9] Seyed Salehi M, Serajzadeh S. A new upper bound solution for the analysis of the twist extrusion process with an elliptical die cross-section. Proc IMechE 2009;223. Part C: J. Mechanical Engineering Science. [10] Khalifa NB, Tekkaya AE. Newest developments on the manufacture of helical profiles by hot extrusion. J Manuf Sci Eng 2011;13:061010. [11] Hussein AW, Kadhim AJ. Mathematical analyses and numerical simulations for forward extrusion of circular, square, and rhomboidal sections from round billets through streamlined dies. J Manuf Sci Eng 2017;139(6):064501. [12] Gunasekera JS, Hoshino Sl. Analysis of extrusion of polygonal sections through streamlined dies. J Eng Ind 1985;107 / 229. [13] Yang DY, Lee CM, Cho JR. Analysis of axisymmetric extrusion of rods by the method of weighted residuals using body-fitted coordinate transformation. Int J Mech Sci 1990;32(2):101–14.
Fig. 24. forming load for AL 2024 billet [13].
1 mm/sec. It can be noted that the maximum values of the strains and equivalent stress occurs close to the exit region of the die at the die surface. This result was seen in Fig. 13 in which the effective strain rate reaches its peak value near the die exit. Another note can be showed in Fig. 23 that the strains and corresponding stresses is minimum along the die axis in which the velocity components vx and vy vanished as seen in Fig. 4. Fig. 24 shows the pressure of the extrusion punch against its displacement. It can be noted that the punch load increases gradually until it reached steady-state value, which represents the required forming pressure. 7. Conclusions An analytical representation of the die surface for helical section shapes in three-dimensional process was proposed. The design includes a smooth shape change of the metal flow in the die, which produced less forming pressure. The velocity field has been obtained based on the volume constancy and velocity boundary conditions. The main conclusions of this work were:
• The metal at the die axis moves in a straight line from the inlet to the outlet of the die and there is no angular motion.
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