- Email: [email protected]
An investigation into the rolling process of copper tubes
Journal of Materials Processing Technology 95 (1999) 139±144
An investigation into the rolling process of copper tubes R. Jose Montecinos*, E. Saul Arauco Department of Mechanical and Metallurgical Engineering, P. Universidad CatoÂlica de Chile, Santiago, Chile Received 23 April 1998
Abstract Currently a large number of seamless copper tubes are manufactured based on a primary tube obtained by means of a piercing process normally performed using a skewed roll piercer. There may be bene®cial economic conditions in manufacturing a primary tube by continuous casting, but the walls of the tube obtained are thick and the material has a coarse-grained structure. This paper investigates the cross-rolling process with a view to applying it to reducing the thickness of the wall of the primary tube obtained by continuous casting. The research is based on analyzing a plane-strain model of the process of a rigid±viscoplastic material using the ®nite-element method. The zone subjected to plastic deformation is de®ned based on modeling the geometry of the process, and evidence obtained from preliminary rolling tests is presented in order to con®rm the material ¯ow suppositions in the modeling. # 1999 Published by Elsevier Science S.A. All rights reserved. Keywords: Finite elements simulation; Tube rolling; Copper tubes
1. Introduction This paper focuses on the following tube manufacturing process. Initially, a primary copper tube is obtained by the continuous-casting method, using an externally water cooled graphite die and with a mandrel made of the same material inside. The tubes are then reduced by hot rolling against a mandrel in an elongating roller, or by extrusion. At the end of the process, the structure of the material can be re®ned by annealing, and thus, the tube is in a suitable condition to be reduced to its ®nal dimensions by stretching. The tubes obtained by continuous casting are characterized by their heavy walls and because the material has a coarse granular structure, and consequently, they must be hot-reduced in order to re-crystallize the material. Due to the heterogeneous structure of the primary tube and the probable presence of defects, rolling equipment with three cross-rolls at 1208 is used instead of equipment with two rolls. This paper analyses the use of an Assel type three-roll elongating mill, which has been used widely in the steel tube industry for reducing the primary tubes obtained in a skewed roll
*Corresponding author. Fax: +56-2-686-5828 E-mail address: [email protected] (R.J. Montecinos)
piercer. The analysis made in this paper focuses on the rolling process using three rolls inclined to the axis of the tube against a mandrel inside. In order to predict stresses, torque, material ¯ow, and stress and strain distribution, the plastic deformation in this process needs to be examined. The solution to this problem implies making a three-dimensional analysis of the deformations, since each roll acts locally on the wall of the tube, generating normal and angular deformation components in both the tube's longitudinal and circumferential directions. The work by Blazynski [2] includes an analysis of the Assel process based on force balance and on the supposition of homogeneous deformation, which allow the average stresses and pressure distribution on the roll's presumed contact area to be predicted. The results of separation force and torque are calculated based on the pressure distribution obtained and on the roll's contact area, which is estimated using the method proposed in this publication. Furthermore, it is shown that the circumferential deformation component is very small compared to the other two deformation components contained on an axial plane, which re¯ect the elongation of the material during the process. The circumferential angular deformation component re¯ects the torque effect suffered by the tube during the process. Consequently, in order to study the material ¯ow corresponding to the elongation suffered by the tube, the analysis in this paper
0924-0136/99/$ ± see front matter # 1999 Published by Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 9 9 ) 0 0 2 8 2 - 4
140
R.J. Montecinos, E.S. Arauco / Journal of Materials Processing Technology 95 (1999) 139±144
will be based on a model of non-homogeneous plane deformations contained on an axial plane, in accordance with the description that will be made below on the geometry of the tube zone subjected to plastic deformation. The problem that will be analyzed will be for large, non-permanent, plastic deformations at high temperature, where the material's strength is sensitive to strain rates. In this paper, the ®nite-element method based on a rigid± viscoplastic material is applied in order to analyze the deformation problem. This problem considers prescribed conditions of velocity on the border, and friction is included in the area of contact with the tool, according to the method developed by Chen and Kobayashi [7]. For the numerical solution using the ®nite-element method, a computer program has been developed in order to analyze two-dimensional plastic deformation problems for a rigid±plastic material and a rigid±viscoplastic material, based on the plastic ¯ow formulation. This program includes the possibility of using 4- and 8-node quadrilateral elements and 6-node triangular elements. The numerical results to the problem under analysis are obtained by the program and presented as a graph, simulating the ¯ow of the material in an axial section of the zone below the roll shoulder. These results are compared with the results of tube rolling tests carried out on experimental equipment.
Fig. 1. Geometry of the process: (a) action of two rolls; b) volume of displaced material.
2. Geometric modeling of the process The purpose of geometric modeling is to de®ne the zone subjected to plastic deformation, considering the way in which the rolls would make contact with the piece. In this way, the contact area can be estimated, which is important for determining the power and force required for the process. Fig. 1(a) shows the action of two rolls and their angle of slant in relation to the axis of the tube. Fig. 1(b) shows schematically the volume of the material displaced by the roll during the process. The rolls' advance depends on the angle of slant measured in relation to the roller's axis and is expressed by the following equation [2]. 2rd0 sin : (1) f n In Eq. (1), is a coef®cient that depends on the axial slide effect, and having a value of less than unity; n is the number of rolls of the roller; and rd0 is the diameter of the tube at the roll shoulder. In this process, the geometry of the transverse and longitudinal sections is shown in Figs. 2 and 3. The contact zone in this model can be estimated [4] by the conservation of the volume of a material element during its passage below the roll shoulder and until it exits the zone. The uniformity of the respective expressions for the element's volumes allows the contact area to be evaluated as a function of rolling height h, which is the difference in
Fig. 2. Geometry of the transverse section.
the radius of the tube between the initial instant before deformation and an intermediate instance subsequent to it. This function can be applied in order to calculate, for example, rolling couple T expressed according to the following equation : Z rl T dArr ; (2) 0
where the area element depends on variable h, which takes on values between zero at the beginning and h0 when it
Fig. 3. Geometry of the longitudinal section.
R.J. Montecinos, E.S. Arauco / Journal of Materials Processing Technology 95 (1999) 139±144
141
reaches the outlet. The detailed analysis can be reviewed in other works by various authors, such as, for example, [2].
where g is a value of 0.01 and there is a 0.03 volumetric change in the material [6].
3. Finite-element model for rigid±viscoplastic material
4. Discretization by means of finite elements
The formulation used for the plastic deformation problem is expressed by the variation equation [3]: Z Z Z T _ dV L_Tv _v dV ÿ fs us dSc v v Sc Z ÿ F T u dSf ; (3)
Adhering to the discretization procedure stipulated in the ®nite-element method, the velocity ®eld in the deformation zone is determined by the velocity at the nodes and the velocity at each point of an element by means of interpolation functions N(m), according to the following equation:
Sf
where L is a large constant that penalizes volumetric deformation, and fs is the force minimizing Eq. (4). A system of non-linear equations for the strain rate is obtained. The equation of a ®ctitious material with linear viscosity, given by Eq. (4) and which, in turn, allows Eq. (3) to be linearized, was used for the behavior of the material: _m
c ;
(4)
where c and m are constants of the material determined by means of experiment. In this problem, the border condition is imposed by the normal velocity of the roll shoulder over the tube's contact surface, expressed by U i ui ;
(5)
where ui is the normal prescribed velocity on the edge imposed by the roll. The ®ctitious rigid±viscoplastic material responds to nonNewtonian laws of ¯ow with a constant density, where the viscosity of the material varies as a function of the effective strain rate [5]. In the rolling problem, non-linearity effects basically occur in two ways: geometric non-linearity occurs as a result of large deformations and rotations of the tube; material non-linearity occurs when there are non-linear ratios in the stress±strain ratios. The following equation is used for evaluating the viscosity of the ®ctitious viscous material in each element of the discretized grid: _
1 ; 3 _
(6)
where is the effective strain rate evaluated in the center of each element, thus avoiding problems of divergence of the viscosity during the analysis. The criterion of Osakada and Nakano [6] is used in evaluating the effective strain rate of the material, using a volumetric compressibility coef®cient g, which is the ratio of the volumetric deformation and the hydrostatic stress: s 4 3 1 _x ÿ _y _x _y _ xy _v ; _ 9 4 g _v _x _y ;
(7)
U m N m u m :
(8)
Furthermore, the element must meet the conditions of continuity and compatibility throughout its domain, and continuity of the ®rst derivatives is required. The strain rate of the element is expressed by _ m B m U m :
(9)
By replacing Eqs. (9) and (10), and the Von Mises plastic ¯ow rule from Eq. (7) in the functional (Eq. (4)), and by minimizing this in relation to the velocity of the nodes, the following system of equations is determined: Z Z M1 Z X T B DBU dV LBT QBU dV ÿ N T F dS _ v v St 1 M2 Z X N T ffs @fs =@Uus g dS 0: (10) 1
Sc
In Eq. (10), friction was modeled using uni-dimensional isoparametric elements, applying the following equation: 2 jU0 j ; f ÿ mk tanÿ1 u0
(11)
where f is the force of friction per area unit, m is the friction factor, k is the shear stress, U0 is the relative velocity of the tube in relation to the roll, and u0 is a small positive constant compared to U0. The analysis was made using 4-node elements; a viscosity distributed evenly throughout the material with an arbitrary value of 105 MPa.s being applied in order to initiate the simulation on the computer. The convergence of the numeric solution to the problem was imposed with the condition that the relative viscosity in the element remained within a tolerance of 0.001, as expressed in Eq. (12): kk 0:001: kk
(12)
The grid for analysis was discretized using the ALGOR ®nite-element commercial program. 5. Conditions of the rolling process of copper tubes and experimental verification The aim of simulating the problem of plastic deformation in tube rolling using the ®nite-element method is to deter-
142
R.J. Montecinos, E.S. Arauco / Journal of Materials Processing Technology 95 (1999) 139±144
Fig. 5. Contact zone between the tube and the roll.
Fig. 4. Model of material for T 4008C [1].
mine the material ¯ow, velocity ®eld, and effective strain rate. The data of the problem analyzed are speci®ed below: (i) diameter of roll is 105 mm; (ii) velocity of roll is 76.1 rpm; (iii) normal velocity of roll imposed on the edge is 0.1 mm; (iv) speed of advance of the roll is 1.6 mm; (v) diameter of tube is 30 mm; (vi) rolling temperature: 4008C; (vii) c [1] 158.59 (MPa); (viii) m [1] 0.046; (ix) diameter of mandrel 17 mm; (x) reduction 0.2; (xi) helix angles and conicity 38; (xii) dimensions of sample; length 30 mm, wall thickness 6 mm. The model of the chosen material is shown by the stress± strain rate curve in Fig. 4 for a temperature of 4008C. Cold rolling tests were performed on the samples up to an area reduction of 52.4% in order to verify the process by experiment. The ®nal dimensions coincided with the ®nite-element model. The rolled samples were observed by means of photomacrographs, the grains of the material being observed to lengthen towards the less thick zone. Fig. 5 shows that the angle of the sample in the conical zone is less than 458, which is the angle of the roll shoulder. This con®rms that the hypothesis of the model is correct for establishing the border conditions for imposing velocities. This also con®rms that this process is viable in elongating copper tubes, with contact occurring mainly at the exit from the roll. A small amount of material was observed to tend to accumulate in the thicker section, and if this accumulation
increases, rolling will become dif®cult. Thus, the deformation of the material must be known in order to properly design the roll so as to avoid such problems. 6. Results and discussion The initial grid is shown in Fig. 6, which represents a longitudinal section of a sample of tube, where the thicker zone represents the initial condition of the tube before the process, and the less-thick zone represents the tube at the end of the process. 302 isoparametric 4-node elements were used in the rolling process simulation. Deformation occurred in incremental stages with a constant normal velocity of 0.1 mm/rev imposed by the roll; in this case, deformation and stress are evaluated at the end of each increase starting from the initial values. Figs. 7 and 8 show the evolution of the grid's deformation for a 10% and a 20% reduction in thickness, respectively, using a 1 : 1 scale. Friction occurs in the contact zone between the tube and the roll, and its initial length is equivalent to the roll's advance. The results in Figs. 7 and 8 show that the material in the less-thick zone has an even axial displacement in the grid, and that this increases as the deformation of the tube increases. Likewise, the material in the thicker zone does not show any displacement, which is maximum only next to the contact point. One can also observe that, according to the 458 angle used for the roll shoulder, the material does not touch the roll's conical zone, so there is no interference between the roll and
Fig. 6. Initial grid with 4-node elements and edge conditions.
R.J. Montecinos, E.S. Arauco / Journal of Materials Processing Technology 95 (1999) 139±144
143
Fig. 7. Deformed grid with 4-node elements after six increments.
Fig. 8. Deformed grid with 4-node elements after 12 increments.
Fig. 9. Velocity field (mm/s) after 12 increments.
the tube, with contact occurring only at the end of the reduction. The velocity ®eld in the material is shown in Fig. 9, where the velocity gradients vary from zero at the restricted nodes to a maximum value in the zone next to the point of contact with the roll. The ¯ow is also observed to move in the direction of the less-thick section, where the velocity is noted to be uniform throughout the section. The effective strain rate after 12 increments is shown in Fig. 10. The maximum velocity is achieved in the zone next to the contact point and decreases upon moving away from that point, becoming very small in the material's rigid zone. The result of the effective stress analysis is shown in Fig. 11. A threshold effective stress value of 120 MPa was de®ned in order to de®ne the rigid zone and the plastic zone,
Fig. 10. Effective strain rate (sÿ1) in the material after 12 increments.
so that there is plastic ¯ow when the stress exceeds this value, the material being considered to be rigid when it is below this value. As was to be expected, the stresses are observed to be maximum in the zone next to the point of contact and decrease as upon moving away from that point.
Fig. 11. Effective stress (MPa) after 12 increments.
144
R.J. Montecinos, E.S. Arauco / Journal of Materials Processing Technology 95 (1999) 139±144
7. Conclusions This paper analyzed the ¯ow, effective strain rate, and velocity ®eld in a simulation of a hot copper tube elongation process, considering a model of non-homogeneous plane deformation and a rigid±viscoplastic material using the ®nite-element method. The results of the analysis con®rm the experiment's observations that the maximum ¯ow of material occurs in the zone next to the point of contact and that such ¯ow is towards the less-thick zone or outlet. Additionally, for the 458 angle conical zone of the roll shoulder, it was con®rmed that the tube material does not come into contact with the roll in this zone; thus, the analysis allows one to predict that the contact between both occurs mainly at the exit from the roll in the cylindrical zone, which was observed in the rolling experiments made using samples of copper tubes obtained by continuous casting. This analysis has established that the main effect of rolling with cross-rolls is to reduce the tube, making the material ¯ow lengthwise, this helping towards understanding the process being investigated. 8. Nomenclature f n a rd 0 rd rm rr qrl h0 _ij
advance of roll (mm) number of rolls angle of advance (radians) outer radius of outlet (mm) radius of tube (mm) radius of mandrel (mm) radius of roll angle of contact of roll rolling depth (mm) strain-rate tightener
_v Ui D fs L U(m) effective stress N(m) n,m |U0| u0 k
volumetric strain rate velocity in material contact zone matrix of rigid±viscoplastic material force of friction penalization constant element velocity interpolation matrix effective strain rate coefficients of the material relative velocity between the roll and the tube small constant compared with relative velocity shear±stress (MPa)
References [1] T. Altan, F.W. Boulger. Flow stress of metals and its application in metal forming analysis, J. Eng. Ind., Trans. ASME. (1973) 447± 457. [2] T.Z. Blazynski, Plasticity and Modern Metal-Forming Technology, Elsevier Applied Science Publishers, UK, 1989. [3] C.H. Lee, S. Kabayashi, New solutions to rigid plastic deformation problems using matrix method, J. Eng. Ind., Trans. ASME. (1973) 865±873. [4] J. Montecinos, S. Arauco, ModelacioÂn del Proceso de LaminacioÂn Rotatoria de Tubos de Cobre Obtenidos por Colada Continua. Anales VII Congreso Nacional de MecaÂnica y Ramas Afines. Universidad Austral, Valdivia, Chile, 1996, pp. 365±369. [5] S.I. Oh, Finite element analysis of metal forming process with arbitrarily shaped dies, Int. J. Mech. Sci. 24(8) (1982) 479±493. [6] K.Y. Osakada, J. Nakano, Finite element method for rigid plastic analysis of metal forming ± formulation for finite deformation, Int. J. Mech. Sci. 24(8) (1982) 459±468. [7] C.C. Chen, S. Kobayashi, Rigid-plastic finite element analysis of ring compression, Appl. Num. Methods of Forming Processes, ASME, AMD, 28 (1978) 163±174.
An investigation into the rolling process of copper tubes R. Jose Montecinos*, E. Saul Arauco Department of Mechanical and Metallurgical Engineering, P. Universidad CatoÂlica de Chile, Santiago, Chile Received 23 April 1998
Abstract Currently a large number of seamless copper tubes are manufactured based on a primary tube obtained by means of a piercing process normally performed using a skewed roll piercer. There may be bene®cial economic conditions in manufacturing a primary tube by continuous casting, but the walls of the tube obtained are thick and the material has a coarse-grained structure. This paper investigates the cross-rolling process with a view to applying it to reducing the thickness of the wall of the primary tube obtained by continuous casting. The research is based on analyzing a plane-strain model of the process of a rigid±viscoplastic material using the ®nite-element method. The zone subjected to plastic deformation is de®ned based on modeling the geometry of the process, and evidence obtained from preliminary rolling tests is presented in order to con®rm the material ¯ow suppositions in the modeling. # 1999 Published by Elsevier Science S.A. All rights reserved. Keywords: Finite elements simulation; Tube rolling; Copper tubes
1. Introduction This paper focuses on the following tube manufacturing process. Initially, a primary copper tube is obtained by the continuous-casting method, using an externally water cooled graphite die and with a mandrel made of the same material inside. The tubes are then reduced by hot rolling against a mandrel in an elongating roller, or by extrusion. At the end of the process, the structure of the material can be re®ned by annealing, and thus, the tube is in a suitable condition to be reduced to its ®nal dimensions by stretching. The tubes obtained by continuous casting are characterized by their heavy walls and because the material has a coarse granular structure, and consequently, they must be hot-reduced in order to re-crystallize the material. Due to the heterogeneous structure of the primary tube and the probable presence of defects, rolling equipment with three cross-rolls at 1208 is used instead of equipment with two rolls. This paper analyses the use of an Assel type three-roll elongating mill, which has been used widely in the steel tube industry for reducing the primary tubes obtained in a skewed roll
*Corresponding author. Fax: +56-2-686-5828 E-mail address: [email protected] (R.J. Montecinos)
piercer. The analysis made in this paper focuses on the rolling process using three rolls inclined to the axis of the tube against a mandrel inside. In order to predict stresses, torque, material ¯ow, and stress and strain distribution, the plastic deformation in this process needs to be examined. The solution to this problem implies making a three-dimensional analysis of the deformations, since each roll acts locally on the wall of the tube, generating normal and angular deformation components in both the tube's longitudinal and circumferential directions. The work by Blazynski [2] includes an analysis of the Assel process based on force balance and on the supposition of homogeneous deformation, which allow the average stresses and pressure distribution on the roll's presumed contact area to be predicted. The results of separation force and torque are calculated based on the pressure distribution obtained and on the roll's contact area, which is estimated using the method proposed in this publication. Furthermore, it is shown that the circumferential deformation component is very small compared to the other two deformation components contained on an axial plane, which re¯ect the elongation of the material during the process. The circumferential angular deformation component re¯ects the torque effect suffered by the tube during the process. Consequently, in order to study the material ¯ow corresponding to the elongation suffered by the tube, the analysis in this paper
0924-0136/99/$ ± see front matter # 1999 Published by Elsevier Science S.A. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 9 9 ) 0 0 2 8 2 - 4
140
R.J. Montecinos, E.S. Arauco / Journal of Materials Processing Technology 95 (1999) 139±144
will be based on a model of non-homogeneous plane deformations contained on an axial plane, in accordance with the description that will be made below on the geometry of the tube zone subjected to plastic deformation. The problem that will be analyzed will be for large, non-permanent, plastic deformations at high temperature, where the material's strength is sensitive to strain rates. In this paper, the ®nite-element method based on a rigid± viscoplastic material is applied in order to analyze the deformation problem. This problem considers prescribed conditions of velocity on the border, and friction is included in the area of contact with the tool, according to the method developed by Chen and Kobayashi [7]. For the numerical solution using the ®nite-element method, a computer program has been developed in order to analyze two-dimensional plastic deformation problems for a rigid±plastic material and a rigid±viscoplastic material, based on the plastic ¯ow formulation. This program includes the possibility of using 4- and 8-node quadrilateral elements and 6-node triangular elements. The numerical results to the problem under analysis are obtained by the program and presented as a graph, simulating the ¯ow of the material in an axial section of the zone below the roll shoulder. These results are compared with the results of tube rolling tests carried out on experimental equipment.
Fig. 1. Geometry of the process: (a) action of two rolls; b) volume of displaced material.
2. Geometric modeling of the process The purpose of geometric modeling is to de®ne the zone subjected to plastic deformation, considering the way in which the rolls would make contact with the piece. In this way, the contact area can be estimated, which is important for determining the power and force required for the process. Fig. 1(a) shows the action of two rolls and their angle of slant in relation to the axis of the tube. Fig. 1(b) shows schematically the volume of the material displaced by the roll during the process. The rolls' advance depends on the angle of slant measured in relation to the roller's axis and is expressed by the following equation [2]. 2rd0 sin : (1) f n In Eq. (1), is a coef®cient that depends on the axial slide effect, and having a value of less than unity; n is the number of rolls of the roller; and rd0 is the diameter of the tube at the roll shoulder. In this process, the geometry of the transverse and longitudinal sections is shown in Figs. 2 and 3. The contact zone in this model can be estimated [4] by the conservation of the volume of a material element during its passage below the roll shoulder and until it exits the zone. The uniformity of the respective expressions for the element's volumes allows the contact area to be evaluated as a function of rolling height h, which is the difference in
Fig. 2. Geometry of the transverse section.
the radius of the tube between the initial instant before deformation and an intermediate instance subsequent to it. This function can be applied in order to calculate, for example, rolling couple T expressed according to the following equation : Z rl T dArr ; (2) 0
where the area element depends on variable h, which takes on values between zero at the beginning and h0 when it
Fig. 3. Geometry of the longitudinal section.
R.J. Montecinos, E.S. Arauco / Journal of Materials Processing Technology 95 (1999) 139±144
141
reaches the outlet. The detailed analysis can be reviewed in other works by various authors, such as, for example, [2].
where g is a value of 0.01 and there is a 0.03 volumetric change in the material [6].
3. Finite-element model for rigid±viscoplastic material
4. Discretization by means of finite elements
The formulation used for the plastic deformation problem is expressed by the variation equation [3]: Z Z Z T _ dV L_Tv _v dV ÿ fs us dSc v v Sc Z ÿ F T u dSf ; (3)
Adhering to the discretization procedure stipulated in the ®nite-element method, the velocity ®eld in the deformation zone is determined by the velocity at the nodes and the velocity at each point of an element by means of interpolation functions N(m), according to the following equation:
Sf
where L is a large constant that penalizes volumetric deformation, and fs is the force minimizing Eq. (4). A system of non-linear equations for the strain rate is obtained. The equation of a ®ctitious material with linear viscosity, given by Eq. (4) and which, in turn, allows Eq. (3) to be linearized, was used for the behavior of the material: _m
c ;
(4)
where c and m are constants of the material determined by means of experiment. In this problem, the border condition is imposed by the normal velocity of the roll shoulder over the tube's contact surface, expressed by U i ui ;
(5)
where ui is the normal prescribed velocity on the edge imposed by the roll. The ®ctitious rigid±viscoplastic material responds to nonNewtonian laws of ¯ow with a constant density, where the viscosity of the material varies as a function of the effective strain rate [5]. In the rolling problem, non-linearity effects basically occur in two ways: geometric non-linearity occurs as a result of large deformations and rotations of the tube; material non-linearity occurs when there are non-linear ratios in the stress±strain ratios. The following equation is used for evaluating the viscosity of the ®ctitious viscous material in each element of the discretized grid: _
1 ; 3 _
(6)
where is the effective strain rate evaluated in the center of each element, thus avoiding problems of divergence of the viscosity during the analysis. The criterion of Osakada and Nakano [6] is used in evaluating the effective strain rate of the material, using a volumetric compressibility coef®cient g, which is the ratio of the volumetric deformation and the hydrostatic stress: s 4 3 1 _x ÿ _y _x _y _ xy _v ; _ 9 4 g _v _x _y ;
(7)
U m N m u m :
(8)
Furthermore, the element must meet the conditions of continuity and compatibility throughout its domain, and continuity of the ®rst derivatives is required. The strain rate of the element is expressed by _ m B m U m :
(9)
By replacing Eqs. (9) and (10), and the Von Mises plastic ¯ow rule from Eq. (7) in the functional (Eq. (4)), and by minimizing this in relation to the velocity of the nodes, the following system of equations is determined: Z Z M1 Z X T B DBU dV LBT QBU dV ÿ N T F dS _ v v St 1 M2 Z X N T ffs @fs =@Uus g dS 0: (10) 1
Sc
In Eq. (10), friction was modeled using uni-dimensional isoparametric elements, applying the following equation: 2 jU0 j ; f ÿ mk tanÿ1 u0
(11)
where f is the force of friction per area unit, m is the friction factor, k is the shear stress, U0 is the relative velocity of the tube in relation to the roll, and u0 is a small positive constant compared to U0. The analysis was made using 4-node elements; a viscosity distributed evenly throughout the material with an arbitrary value of 105 MPa.s being applied in order to initiate the simulation on the computer. The convergence of the numeric solution to the problem was imposed with the condition that the relative viscosity in the element remained within a tolerance of 0.001, as expressed in Eq. (12): kk 0:001: kk
(12)
The grid for analysis was discretized using the ALGOR ®nite-element commercial program. 5. Conditions of the rolling process of copper tubes and experimental verification The aim of simulating the problem of plastic deformation in tube rolling using the ®nite-element method is to deter-
142
R.J. Montecinos, E.S. Arauco / Journal of Materials Processing Technology 95 (1999) 139±144
Fig. 5. Contact zone between the tube and the roll.
Fig. 4. Model of material for T 4008C [1].
mine the material ¯ow, velocity ®eld, and effective strain rate. The data of the problem analyzed are speci®ed below: (i) diameter of roll is 105 mm; (ii) velocity of roll is 76.1 rpm; (iii) normal velocity of roll imposed on the edge is 0.1 mm; (iv) speed of advance of the roll is 1.6 mm; (v) diameter of tube is 30 mm; (vi) rolling temperature: 4008C; (vii) c [1] 158.59 (MPa); (viii) m [1] 0.046; (ix) diameter of mandrel 17 mm; (x) reduction 0.2; (xi) helix angles and conicity 38; (xii) dimensions of sample; length 30 mm, wall thickness 6 mm. The model of the chosen material is shown by the stress± strain rate curve in Fig. 4 for a temperature of 4008C. Cold rolling tests were performed on the samples up to an area reduction of 52.4% in order to verify the process by experiment. The ®nal dimensions coincided with the ®nite-element model. The rolled samples were observed by means of photomacrographs, the grains of the material being observed to lengthen towards the less thick zone. Fig. 5 shows that the angle of the sample in the conical zone is less than 458, which is the angle of the roll shoulder. This con®rms that the hypothesis of the model is correct for establishing the border conditions for imposing velocities. This also con®rms that this process is viable in elongating copper tubes, with contact occurring mainly at the exit from the roll. A small amount of material was observed to tend to accumulate in the thicker section, and if this accumulation
increases, rolling will become dif®cult. Thus, the deformation of the material must be known in order to properly design the roll so as to avoid such problems. 6. Results and discussion The initial grid is shown in Fig. 6, which represents a longitudinal section of a sample of tube, where the thicker zone represents the initial condition of the tube before the process, and the less-thick zone represents the tube at the end of the process. 302 isoparametric 4-node elements were used in the rolling process simulation. Deformation occurred in incremental stages with a constant normal velocity of 0.1 mm/rev imposed by the roll; in this case, deformation and stress are evaluated at the end of each increase starting from the initial values. Figs. 7 and 8 show the evolution of the grid's deformation for a 10% and a 20% reduction in thickness, respectively, using a 1 : 1 scale. Friction occurs in the contact zone between the tube and the roll, and its initial length is equivalent to the roll's advance. The results in Figs. 7 and 8 show that the material in the less-thick zone has an even axial displacement in the grid, and that this increases as the deformation of the tube increases. Likewise, the material in the thicker zone does not show any displacement, which is maximum only next to the contact point. One can also observe that, according to the 458 angle used for the roll shoulder, the material does not touch the roll's conical zone, so there is no interference between the roll and
Fig. 6. Initial grid with 4-node elements and edge conditions.
R.J. Montecinos, E.S. Arauco / Journal of Materials Processing Technology 95 (1999) 139±144
143
Fig. 7. Deformed grid with 4-node elements after six increments.
Fig. 8. Deformed grid with 4-node elements after 12 increments.
Fig. 9. Velocity field (mm/s) after 12 increments.
the tube, with contact occurring only at the end of the reduction. The velocity ®eld in the material is shown in Fig. 9, where the velocity gradients vary from zero at the restricted nodes to a maximum value in the zone next to the point of contact with the roll. The ¯ow is also observed to move in the direction of the less-thick section, where the velocity is noted to be uniform throughout the section. The effective strain rate after 12 increments is shown in Fig. 10. The maximum velocity is achieved in the zone next to the contact point and decreases upon moving away from that point, becoming very small in the material's rigid zone. The result of the effective stress analysis is shown in Fig. 11. A threshold effective stress value of 120 MPa was de®ned in order to de®ne the rigid zone and the plastic zone,
Fig. 10. Effective strain rate (sÿ1) in the material after 12 increments.
so that there is plastic ¯ow when the stress exceeds this value, the material being considered to be rigid when it is below this value. As was to be expected, the stresses are observed to be maximum in the zone next to the point of contact and decrease as upon moving away from that point.
Fig. 11. Effective stress (MPa) after 12 increments.
144
R.J. Montecinos, E.S. Arauco / Journal of Materials Processing Technology 95 (1999) 139±144
7. Conclusions This paper analyzed the ¯ow, effective strain rate, and velocity ®eld in a simulation of a hot copper tube elongation process, considering a model of non-homogeneous plane deformation and a rigid±viscoplastic material using the ®nite-element method. The results of the analysis con®rm the experiment's observations that the maximum ¯ow of material occurs in the zone next to the point of contact and that such ¯ow is towards the less-thick zone or outlet. Additionally, for the 458 angle conical zone of the roll shoulder, it was con®rmed that the tube material does not come into contact with the roll in this zone; thus, the analysis allows one to predict that the contact between both occurs mainly at the exit from the roll in the cylindrical zone, which was observed in the rolling experiments made using samples of copper tubes obtained by continuous casting. This analysis has established that the main effect of rolling with cross-rolls is to reduce the tube, making the material ¯ow lengthwise, this helping towards understanding the process being investigated. 8. Nomenclature f n a rd 0 rd rm rr qrl h0 _ij
advance of roll (mm) number of rolls angle of advance (radians) outer radius of outlet (mm) radius of tube (mm) radius of mandrel (mm) radius of roll angle of contact of roll rolling depth (mm) strain-rate tightener
_v Ui D fs L U(m) effective stress N(m) n,m |U0| u0 k
volumetric strain rate velocity in material contact zone matrix of rigid±viscoplastic material force of friction penalization constant element velocity interpolation matrix effective strain rate coefficients of the material relative velocity between the roll and the tube small constant compared with relative velocity shear±stress (MPa)
References [1] T. Altan, F.W. Boulger. Flow stress of metals and its application in metal forming analysis, J. Eng. Ind., Trans. ASME. (1973) 447± 457. [2] T.Z. Blazynski, Plasticity and Modern Metal-Forming Technology, Elsevier Applied Science Publishers, UK, 1989. [3] C.H. Lee, S. Kabayashi, New solutions to rigid plastic deformation problems using matrix method, J. Eng. Ind., Trans. ASME. (1973) 865±873. [4] J. Montecinos, S. Arauco, ModelacioÂn del Proceso de LaminacioÂn Rotatoria de Tubos de Cobre Obtenidos por Colada Continua. Anales VII Congreso Nacional de MecaÂnica y Ramas Afines. Universidad Austral, Valdivia, Chile, 1996, pp. 365±369. [5] S.I. Oh, Finite element analysis of metal forming process with arbitrarily shaped dies, Int. J. Mech. Sci. 24(8) (1982) 479±493. [6] K.Y. Osakada, J. Nakano, Finite element method for rigid plastic analysis of metal forming ± formulation for finite deformation, Int. J. Mech. Sci. 24(8) (1982) 459±468. [7] C.C. Chen, S. Kobayashi, Rigid-plastic finite element analysis of ring compression, Appl. Num. Methods of Forming Processes, ASME, AMD, 28 (1978) 163±174.