Trans. Nonferrous Met. Soc. China 24(2014) 3621-3631

Extrusion force analysis of aluminum pipe fabricated by CASTEX using expansion combination die

Fu-rong CAO¹, Jing-lin WEN¹, Hua DING¹, Zhao-dong WANG², Chuan-ping YU³, Fei XIA¹

1. School of Materials and Metallurgy, Northeastern University, Shenyang 110819, China;

2. State Key Laboratory of Rolling and Automation, Northeastern University, Shenyang 110819, China;

3. Dandong Feituo Aluminum Products Co., Ltd., Dandong 118000, China

Received 24 November 2013; accepted 14 April 2014

Abstract: To determine the extrusion force of pipe fabricated by continuous casting and extrusion (CASTEX) using an expansion combination die, the metallic expansion combination die was divided into diversion zone, expansion zone, flow dividing zone, welding chamber, and sizing zone, and the corresponding stress formulae in various zones were established using the slab method. The deformation zones of CASTEX groove were divided into liquid and semisolid zone, solid primary gripping zone, and solid gripping zone, and the formulae of pipe extrusion forces were established. Experiments were carried out on the self-designed CASTEX machine to obtain the aluminum pipe and measure its extrusion force using the expansion combination die. The experimental results of radial extrusion force for aluminum pipe are in good agreement with the calculated ones.

Key words: continuous casting and extrusion; continuous extrusion; expansion combination die; aluminum pipe; stress analysis; extrusion force

1 Introduction

Unlike conventional extrusion [1,2] and other types of extrusion [3-5], CONFORM, continuous extrusion forming, turns the friction resistance between the billet and the container into the driving force of billet deformation. As long as various billets or feedstock such as rod, particles, and powders of aluminum, copper, and polymeric material are fed continuously into the entrance of CONFORM machine, a wide range of products such as wire, rod, profile, and tube that meet the requirements can be produced continuously at the exit of extrusion die. Since the appearance of CONFORM, its application in major industrial and scientific fields has been increasing. MITCHELL [6] pointed out that CONFORM was a commercially viable method of producing hollow sections and tubing for air conditioning, refrigeration and cable television applications. CHURCH [7] introduced the application of CONFORM to the production of sheathe cable and to the manufacture of various solid and hollow sections in Europe and put forward the development of an expansion chamber, leading to the die through which profile whose cross section was larger than that of the feedstock could be made. HAWKES and MORGAN [8] reported the CONFORM extrusion of copper or aluminum solid sections and the sheathing or cladding operations. LANGERWEGER and MADDOCK [9] and MADDOCK [10] changed the solid feedstock of CONFORM into molten metal and presented an innovative continuous casting and extrusion (CASTEX) machine with the expansion chamber. Although the idea of the expansion chamber was put forward, no report was available to produce real pipes in practical CASTEX process using an expansion combination die except Northeastern University in China. In recent years, RAAB et al [11] and XU et al [12] have explored the application of ECAP (equal channel angular pressing)-CONFORM to the fabrication of ultrafine-grained (UFG) aluminum wire and Al-6061 rod. Recently, FENG et al [13], YUN et al [14], and HE et al [15] have reported the continuous extrusion of Cu-0.16Cr-0.12Zr alloy, have investigated the continuous extrusion and rolling forming of copper strip, and achieved 200% elongation to failure at 473 K and 1×10^-4s^-1 in AZ31 magnesium alloy processed by ECAP-CONFORM, respectively.

The theoretical mechanics analysis and simulation on CONFORM or CASTEX process using finite element (FEM) simulation, upper bound method, slab method, and slip line method have been increasing. First, FEM simulation was carried out. PENG et al [16] predicted the processes of defect initiation and development during CONFORM process. KIM et al [17] investigated the effects of several process parameters, such as material flow, defect occurrence, temperature and effective strain distribution, on the process characteristics. CHO and JEONG [18] addressed a parametric investigation on the occurrence of the surface defect in CONFORM process using FEM simulation. YUN et al [19] and WEI et al [20] investigated the metallic flow, strain field and temperature field of UFG copper rod using continuous ECAP, simulated the deformation behavior of continuous ECAP and investigated the effect of die angle and contact friction on the strain and stress distribution, flow homogeneity, shear deformation and torque, respectively, by FEM method. ZHAO et al [21] investigated the process of aluminum sheath during continuous extrusion using FEM simulation. Second, upper bound analyses were conducted. CAO et al [22] and KIM et al [23] derived the upper bound driving power equation for the CASTEX and the CONFORM processes, respectively. Third, slab method was used to determine the extrusion force. TIROSH et al [24] established a unit pressure distribution equation of CONFORM extrusion for wire. SHI et al [25] got a formula of CASTEX force for aluminum-steel cladding wire. SONG et al [26] established a formula of CONFORM extrusion force for copper bar. CAO et al [27] derived a formula of CONFORM expanding extrusion force for Al-5Ti-1B alloy and pure aluminum solid sections. Fourth, SEGAL [28] utilized the slip line method to study the mechanics of continuous ECAP of rectangular billets.

In terms of aforementioned application and theoretical analysis, the authors developed a CASTEX process with the expansion combination die so that the pipe with cross section larger than that of the feedstock was produced, and the corresponding equipment and relevant tools were designed. According to literature survey, there was little information available reporting on the mechanics analysis of aluminum pipe processed by CASTEX extrusion using the slab method. In this work, by using the expansion combination die designed by ourselves, the CASTEX process was divided into several deformation zones and the extrusion stress formulae were derived zone by zone. On the basis of wheel groove force analysis, the tangential force and radial force of driving the wheel groove for the CASTEX process were determined. Experiments were carried out on the self-designed CASTEX machine to obtain aluminum pipes and to compare the experimental force with the calculated one.

2 Mechanics analysis of CASTEX process using expansion combination die

The schematic diagram of the CASTEX equipment using the expansion combination die is shown in Fig. 1. Molten metal flows from the entrance of extruding wheel into the extrusion cavity formed by the wheel groove and the grip segment. Under the action of friction forces imposed by the two sides of wheel groove and the bottom surface, the molten metal solidifies, is deformed in the extrusion cavity and extruded out of the extrusion die to obtain the pipe product in required shape and properties, as shown in Fig. 2.

Fig. 1 Schematic diagram of CASTEX equipment using expansion combination die

Fig. 2 Aluminum pipes produced by CASTEX using expansion combination die

According to the deformation characteristics of feedstock, the solidification and deformation processes in the wheel groove are divided into three deformation zones: liquid and semisolid zone l₀, solid primary gripping zone l₁ and solid gripping zone l₂; the deformation process in the expansion combination die is divided into five deformation zones: 1) diversion zone, 2) expansion zone, 3) flow dividing zone, 4) welding chamber, and 5) sizing zone, as shown in Fig. 3. Mechanics analysis was carried out zone by zone from the beginning of the sizing zone of the die to the wheel entrance.

Fig. 3 Schematic diagrams of expansion combination die and its structure dimensions

2.1 Sizing zone

Fig. 4 Stress acting on infinitesimal element of sizing zone

The stress analysis in the sizing zone is shown in Fig. 4.  and  are normal pressures acting on the internal and external surfaces of the sizing zone of the die, respectively.  is the normal pressure of the rigid die acting on the metal. σ_n is the normal stress caused by the recovery of elastic deformation force.  and  are assumed to be approximately equal to σ_s, the yield stress, i.e., Because of relative movement between the billet and the sizing zone of the die, friction stress occurs. τ′ and τ are friction stresses acting on the internal and external surfaces of the sizing zone of the die, respectively, and abide by Coulomb’s law of friction. The coefficients of friction on the internal and external surfaces in the sizing zone of the die are assumed to be f₇:

                         (1)

The differential equation of static force equilibrium for the infinitesimal element along horizontal x axis direction is

   (2)

Equation (2) reduces to

                      (3)

It is assumed that the stress acting on the boundary between the sizing zone and the welding chamber is σ₇. Based on the boundary condition, when x=0, σ_x=0; when x=l₇, σ_x=σ₇.

Definite integral of Eq. (3) becomes

Above equality reduces to

                              (4)

where f₇ is the coefficient of friction between the billet and the sizing zone of die, σ_s is the flow stress of the material, l₇ is the length of the sizing zone of the die, d₈ is the diameter of the sizing zone of the die, and d₉ is the diameter of mandril.

2.2 Welding chamber

The stresses acting on the infinitesimal element of the welding chamber are shown in Fig. 5.  and  are normal pressures acting on the internal and external surfaces of the welding chamber of the die, respectively.  and where the semicone anglecan be obtained according to the force equilibrium along vertical direction. For simplification, it is assumed that the normal pressures acting on the internal and external surfaces are approximately equal,, and their coefficients of friction are also the same. The shear stresses acting on the boundary between the movable metal and the “dead” metal (D.M.) zone attain the maximum values. The coefficients of friction obey

Fig. 5 Stresses acting on infinitesimal element of welding chamber

The differential equation of force equilibrium for the infinitesimal element along horizontal x axis direction is

                       (5)

Neglecting differential quantities of the second order, in consideration of , Eq. (5) reduces to

     (6)

Substituting Tresca’s yield criterion, σ_n-σ_x=σ_s, into Eq. (6) leads to

                   (7)

Let , Eq. (7) becomes

                       (8)

Rearranging Eq. (8) yields

     (9)

Let

,

Therefore,

       (10)

where C is a constant of integration.

Solving the above formula gives

                  (11)

It is assumed that the stress acting on the boundary between the welding chamber and the flow dividing zone is σ₆. Based on the boundary condition, when D_x=d₈, σ_x=σ₇; when D_x=d₆, σ_x=σ₆.

Equation (11) reduces to

       (12)

In Eq. (12), numerical integration for the second item shall be implemented to gain the calculated results.

2.3 Flow dividing zone

Flow dividing zone consists of four holes in kidney shape, as shown in Fig. 6. The flow dividing hole in kidney shape is assumed to be equivalent to a rectangle whose side lengths are h₅ and b₅. The depth of the hole is equal to l₅. Friction stress is determined according to the maximum shear stress. The coefficient of friction is assumed to be It is assumed that where  and  are the normal pressures.

Fig. 6 Four flow dividing holes in male die of combination die

As shown in Fig. 7, the differential equation of force equilibrium for the infinitesimal element along horizontal x axis direction is

    (13)

                          (14)

It is assumed that the stress acting on the boundary between the flow dividing zone and the expansion zone is σ₅ and the length of flow dividing zone is l₅. Based on the boundary condition, when x=0, σ_x=σ₆; when x=l₅, σ_x=σ₅.

Fig. 7 Stress acting on infinitesimal element of flow dividing zone

Definite integral of Eq. (12) becomes

                    (15)

Therefore,

                        (16)

2.4 Expansion zone

The infinitesimal element to be chosen is shown in Fig. 8. The shear stress, τ, acting on the boundary between the metal and the die reaches the maximum value,and the coefficient of friction . The differential equation of force equilibrium for the infinitesimal element along horizontal x axis direction is

  (17)

Fig. 8 Stress acting on infinitesimal element of expansion zone

Neglecting differential quantities of the second order, in consideration of , Eq. (17) reduces to

         (18)

Hence,

Substituting Tresca’s yield criterion, σ_n-σ_x=σ_s, into above formula leads to

Hence,

                (19)

It is assumed that the stress acting on the boundary between the expansion zone and the diversion zone is σ₄. Based on the boundary condition, when D_x=d₄, σ_x=σ₅; when D_x=d₃, σ_x=σ₄.

Definite integral of Eq. (19) becomes

           (20)

Because

;

The limit of integration on the right of equality sign is reversed and becomes

           (21)

Therefore,

         (22)

where the semicone angle .

2.5 Diversion zone

The infinitesimal element to be chosen is shown in Fig. 9. The shear stress, τ, acting on the boundary between the metal and the die reaches the maximum value, . The differential equation of force equilibrium for the infinitesimal element along horizontal x axis direction is

     (23)

Fig. 9 Stress acting on infinitesimal element of diversion zone

Equation (23) reduces to

                          (24)

Hence,

                          (25)

It is assumed that the stress acting on the boundary between the diversion zone and the solid gripping zone is σ₃ and the length of diversion zone is l₃. Based on the boundary condition, when x=0, σ_x=σ₄; when x=l₃, σ_x=σ₃.

Definite integral of Eq. (25) becomes

                    (26)

Therefore,

                         (27)

2.6 Liquid and semisolid zone, solid primary gripping zone and solid gripping zone

2.6.1 Lengths and stress distribution of liquid and semisolid zone, solid primary gripping zone and solid gripping zone

The deformation zones along the wheel groove are divided into liquid and semisolid zone l₀, the solid primary gripping zone l₁ and solid gripping zone l₂, as shown in Fig. 10(a). The distribution of friction stress τ and tangential stress σ_n in the wheel groove is shown in Fig. 10(b).

Fig. 10 Division of metallic solidification and deformation zones along wheel groove and stress distribution diagram

1) Length of zone l₀

Liquid metal is poured into the groove and dynamic solidification occurs when friction stress τ=0 and tangential stress σ_n=0. The length of l₀ obeys

                 (28)

where D₀ is the wheel diameter, W is the groove height or width, l₁ is the length of solid primary gripping zone, l₂ is the length of solid gripping zone, D_t=d₃, and γ is the wrapping angle in rad.

2) Length of l₁ zone

Metal fills the groove gradually. τ starts from zero and increases till fσ_s; σ_n, the tangential stress, starts from zero and increases till σ_s, as shown in Fig. 10(b). The metal in section B-C is taken as the stress body. The metal in this section is subjected to the friction forces on both sides of the wheel groove and the tangential force is transferred from the front metal. The friction force of the groove bottom acting on the metal counteracts that of the grip segment acting on the metal in section B-C.

It is assumed that the friction stress in the section B-C is τ₁, so , which satisfies the boundary condition of l=0, τ₁=0; l=l₁, τ₁=fσ_s. Due to the simultaneous action of friction forces on both sides of the wheel groove, the friction force of the wheel groove acting on the stress body becomes

   (29)

The tangential force is

                            (30)

Static force equilibrium of section B-C gives

                      (31)

Therefore,

                                (32)

where f is the coefficient of friction and f=0.5.

3) Length of zone l₂

τ is equal to fσ_s. Tangential stress σ_n in front of the abutment increases, causing the metal to change the flow direction and the extrusion stress produced in the extrusion direction to reach σ₃ which is required by extruding the metal in the expansion combination die out. Let n_s=σ₃/σ_s, the metal in section C-D is taken as the stress body. The metal in this section is subjected to the friction forces on both sides of the wheel groove and the tangential force σ_n is transferred from the abutment. The friction force of the groove bottom acting on the metal counteracts that of the grip segment acting on the metal in section C-D.

The tangential force equilibrium in section C-D gives

                 (33)

According to Tresca’s yield criterion of maximum shear stress,

                              (34)

Substituting Eq. (34), and  into Eq. (33) gives

                              (35)

                            (36)

2.6.2 Determination of CASTEX extrusion force

The CASTEX extrusion force means the tangential force, P_t, and the radial force, P_r, sustained by the wheel groove of CASTEX wheel. The wheel is taken as the force body. For the convenience of presentation, the force diagram of the wheel groove is shown in Fig. 11. The size of cross section of the groove is W×W.

Fig. 11 Force diagrams of wheel groove for CASTEX extrusion

The forces acting on the side faces and the bottom surface of l₁ zone (acting on the middle point of l₁) are

,           (37)

The forces acting on the side faces and the bottom surface of l₂ zone (acting on the middle point of l₂) are

,                    (38)

The flash forces caused by the clearance or gap (acting on the middle point of l₂) are

,                    (39)

The flash forces on the three sides of the abutment (acting on the middle point of Z) are

,                    (40)

The forces of diversion zone acting on the wheel groove are

,                     (41)

where b is the length of single side flash and Z is the length of flash caused by the leakage between the abutment and the wheel groove.

Thus, the tangential force of deformation material acting on the wheel groove is

            (42)

The radial force of deformation material acting on the wheel center, P_r, is determined by the following relation:

                            (43)

                             (44)

where P_x is the horizontal component, P_y is the vertical component, and θ is the direction angle.

Resolution of forces is carried out for Fig. 11(b) and leads to

, i=1, 2, 3, 4, 5       (45)

Combining the above formulae for i, horizontal component, P_x, and vertical component P_y, of radial force, P_r, are obtained.

   (46)

      (47)

As actual directions of stress vectors are opposite to those shown in Fig. 11(b), it is necessary to add a minus sign before P_x and P_y and final P_x and P_y are obtained:

      (48)

     (49)

where

,

, ,

, .

These angles are determined according to the geometrical relation.

Equations (42), (43), (44), (48) and (49) are the derived formulae for radial CASTEX extrusion.

3 Experimental process and calculated results

3.1 Experimental process and results

CASTEX tests of aluminum pipes were conducted on the self-designed CASTEX machine with the expansion combination die. The wheel diameter was 300 mm. Liquid metal at 710 °C was poured into the rotary extrusion wheel groove and a pipe with a dimension of d50 mm×5 mm was obtained by extruding deformation. In the shoe, a sensor was mounted at the outlet side of the extrusion die to measure the radial force. Prior to the measurement of the radial force, the sensor was calibrated by a model DY-15 stabilized voltage power supply, a model YD-15 dynamic electrical resistance strain gauge and a model SC-16 light ray recording oscillograph on the universal tensile test machine. Thermocouples were inserted into the holes drilled on the grip segments and an X-Y four point temperature measuring device was used to measure the deformation temperature. The temperature that metal entered the die was measured to be 400 °C. According to WEN [29], the flow stress, σ_s, of aluminum at deformation temperature of 400 °C and strain rate of 3.07 s^-1 was 23.4 MPa and the coefficient of Coulomb’s friction, f₇, was 0.275.

3.2 Calculated results

The technical parameters of CASTEX machine and extrusion process are shown in Table 1.

Table 1 Technical parameters of CASTEX machine and extrusion process

The structure dimensions of the expansion combination die corresponding to Fig. 3(b) are shown in Table 2.

Table 2 Structure dimensions of expansion combination die (mm)

The calculated stresses in various deformation zones and the comparison of the calculated extrusion radial force with the experimental one are shown in Table 3. The experimental radial force is in good agreement with the calculated one with an error of less than 10% under the present experimental condition.

Table 3 Calculated stresses in various deformation zones and comparison of the calculated extrusion radial force with the experimental one

4 Conclusions

1) An expansion combination die for the CASTEX process is designed. The metallic expansion combination die is divided into diversion zone, expansion zone, flow dividing zone, welding chamber, and sizing zone. The corresponding stress formulae in various zones are established using the slab method.

2) The deformation zones of CASTEX groove are divided into liquid and semisolid zone, solid primary gripping zone, and solid gripping zone. The formulae of pipe extrusion forces are established.

3) Experiments were carried out on the self- designed CASTEX machine to obtain the aluminum pipe and measure its extrusion force using the expansion combination die. The experimental results of radial extrusion force for aluminum pipe are in good agreement with the calculated ones.

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扩展组合模连续铸挤铝管材的挤压力分析

曹富荣¹，温景林¹，丁 桦¹，王昭东²，于传平³, 夏 飞¹

1. 东北大学 材料与冶金学院，沈阳110819；

2. 东北大学 轧制技术与连轧自动化国家重点实验室，沈阳110819；

3. 丹东飞拓铝品有限公司，丹东 118000

摘  要：为了确定管材扩展组合模连续铸挤过程的变形力，将金属扩展组合模的模腔划分为导流区、扩展区、分流区、焊合区和定径区，分析各区金属的受力状态；用切块法建立各区应力计算公式。将金属连续铸挤型腔划分为液相区与半固态区、固态初始夹紧区和固态夹紧区，建立管材连续铸挤挤压力计算公式。在自行设计的连续铸挤机上进行铝管扩展组合模连续铸挤实验并测量其挤压力，获得的径向挤压力实验结果与理论计算结果吻合。

关键词：连续铸挤；连续挤压；扩展组合模；铝管；应力分析；挤压力

(Edited by Wei-ping CHEN)

Foundation item: Projects (51334006, 50274020) supported by the National Natural Science Foundation of China

Corresponding author: Fu-rong CAO; Tel: +86-24-83670183; E-mail: cfr-lff@163.com

DOI: 10.1016/S1003-6326(14)63507-X