Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 51, 1, pp. 105-116, Warsaw 2013 A GENERALIZED UPPER BOUND SOLUTION FOR EXTRUSION OF BI-METALLIC RECTANGULAR CROSS-SECTION BARS THROUGH DIES OF ANY SHAPE Heshmatollah Haghighat, Parvaneh Amjadian Razi University, Mechanical Engineering Department, Kermanshah, Iran e-mail: hhaghighat@razi.ac.ir In this paper, a generalized upper bound solution is developed for extrusion of bi-metallic rectangular cross-section bars through dies of any shape. The internal, shearing and fric- tional power terms are derived and they are used in the upper bound model. By using the developed upper bound model, the extrusion pressures for two types of die shapes, an optimum wedge shaped die as a linear die profile and an optimum streamlined die shape as a curved die profile, are determined. The corresponding results for those two die shapes are also determined by using the finite element code and compared with the upper bound results. These comparisons show a good agreement. Key words: extrusion, bimetal, plane strain, upper boundmethod, FEM Nomenclature bc,bs – widths of core and sleeve materials, respectively dQ – differential volume flow P – extrusion pressure J∗ – externally supplied power of deformation L,Lo – length of die and initial bi-metallic billet length m1,m2 – friction factor between sleeve and die, and between core and sleeve r,θ,z – cylindrical coordinate system rf,ro – radial position of surfaces S2 and S4, and of surfaces S1 and S3 x – punch displacement S – surface of integration S1-S4 – cylindrical surfaces of velocity discontinuity t1f, t2f – half thickness of sleeve and core in final product t1o, t2o – half thickness of sleeve and core in initial bar U̇r, U̇θ, U̇z – radial, angular and third components of velocity V – volume of integration Vf,Vo – velocity of final product and initial billet, respectively Ẇf,Ẇi,Ẇs – friction power losses along the wall, internal power of deformation and shear power losses along surface of velocity discontinuity, respectively α – angle of line connecting initial to final point of die γ – arbitrary angle on cylindrical surfaces S1 and S3 β – angle of interface in deformation zone ∆V – velocity difference ε̇rr, ε̇θθ, ε̇zz – normal strain rate components in radial, angular and third directions ε̇rθ, ε̇rz, ε̇zθ – shear strain rate components η – local angle of die surface with respect to local radial velocity component 106 H. Haghighat, P. Amjadian σc,σs – mean flow stress of core and sleeve material ψ(r),ψi(r) – angular position of die and interface as function of radial position 1. Introduction Bimetals are components made up of two separate metallic units, each occupying a distinct position in the component. Bimetal components make possibility of combining properties of dissimilar metals. The compressive state of stress in extrusion and the possibility of producing metallurgical bonds between the two metals make this process a suitable choice for producing bimetal bars (Berski et al., 2004). In this process, alike othermetal formingprocesses, estimation and minimization of the extrusion pressure is important. The upper bound technique as an analytical method and the finite element method have been widely used for the analysis of the extrusion of barsmade of mono-metal and bi-metallic materials. One of the limitations of most of the current FE solution schemes for metal forming is that they do not provide parametric analysis. Hence, any parametric investigation is usually done manually by changing one FE model to another until a feasible solution is obtained. Analytical solutions facilitate parametric study and can aid an FE modeler in seeking out optimal extrusion process conditions. After the upper boundmodel has determined the optimal die shapes, a finite element model has then been used to study the extrusion through dies with these optimal shapes. A number of people have used the upper bound method and FEM to analyze the mono- metal and bimetal extrusion process through a variety of die shapes. Zimmerman and Avitzur (1970)modeled themono-metal extrusion using the upper boundmethodwith generalized shear boundaries. Osakada et al. (1973) described the hydrostatic extrusion of bi-metallic bars with hard cores through conical dies by the upper boundmethod. Nagpal (1974) developed velocity fields for axisymmetricmono-metal extrusions througharbitrarily shapeddies.Chen et al. (1979) usedfinite element analysis to examine the axisymmetric extrusion through conical dies.Avitzur (1983) summarized the factors that affect simultaneous flow of layers in the extrusion of a bimetal bar through conical dies. Tokuno and Ikeda (1991) verified the deformation in extrusion of composite bars by experimental and upper bound methods. Yang et al. (1999) studied the axisymmetric extrusion of composite bars through curveddies by experimental andupperbound methods. Sliwa (1997) described the plastic zones in the forward extrusion of metal composites by experimental and upper boundmethods.Chitkara andAleem (2001a,b) theoretically studied the mechanics of extrusion of axisymmetric bi-metallic tubes from solid circular bars using fixed mandrel with application of generalized upper bound and slab method analyses. They investigated the effect of different parameters such as the extrusion ratio, frictional conditions, and shape of the dies and themandrels on the extrusion pressures.Kang et al. (2002) designed a die for the hot forward andbackward extrusion process ofAl-Cu clad composite by experimental investigations and FEM simulations. Hwang and Hwang (2002) studied the plastic deformation behavior within a conical die during composite bar extrusion by experimental and upper bound methods. Haghighat and Asgari (2011) proposed a generalized spherical velocity field for the bi-metallic tube extrusion process through dies of any shape. Haghighat and Amjadian (2011) developed velocity fields for the mono-metal plane strain extrusion through arbitrarily curved dies. The purpose of this paper is to develop a generalized upper bound solution for flow of bi- metallic rectangular cross-section bars during extrusion through dies of any shape. Based on thismodel, for a given die shape and process parameters, the optimumdie length and extrusion pressure are derived. The FEM simulation on the extrusion of a bi-metallic rectangular cross- section bar composed of a copper sleeve layer and an aluminum core layer is also conducted. A generalized upper bound solution for extrusion... 107 2. Upper bound analysis Based on the upper bound theory, for a rigid-plastic Von-Misses material and amongst all the kinematically admissible velocity fields, the actual one that minimizes the power required for material deformation is expressed as J∗ = 2√ 3 σ0 ∫ v √ 1 2 ε̇ijε̇ij dV + σ0√ 3 ∫ SV |∆V | dS +m σ0√ 3 ∫ Sf |∆V | dS − ∫ St TiVi dS (2.1) where σ0 is the mean flow stress of the material, ε̇ij the strain rate tensor, m the constant friction factor, V the volume of the plastic deformation zone, SV and Sf the area of velocity discontinuity and frictional surfaces, respectively, St theareawhere tractionsmayoccur, ∆V the amount of velocity discontinuity on the frictional and discontinuity surfaces and Vi and Ti are the velocity and tractions applied on St, respectively. The extrusion of the bi-metallic rectangular cross-section bar through a die of arbitrary curved shape is modelled as shown in Figs. 1 and 2. An initially billet, made up two different ductile materials with the mean flow stresses σc and σs, is considered. The subscripts c and s denote the inner material, core, and outer material, sleeve, respectively. The material starts as a bar with the sleeve thickness 2t1o and core thickness 2t2o and it extrudes into a product of thicknesses 2t2f and 2t1f for the sleeve and core through a curved die, respectively. Because of the width of the core bc and width of the sleeve bs are long, bc ∼= bs ≫ 2t1o, the process can be assumed as a plane strain extrusion. Fig. 1. The extrusion process of a bi-metallic rectangular cross-section billet through an arbitrarily curved die To analyze the process, thematerial under deformation is divided into six zones, as shown in Fig. 2. The die surface, which is labeled as ψ(r) in Fig. 2, is given in the cylindrical coordinate system (r,θ,z). The origin of the cylindrical coordinate system is located at the point O which is defined by the intersection of the midline with a line at the angle α that goes through the point where the die begins and the exit point of the die. In zones I and II, the incoming materials are assumed to flow horizontally as a rigid body with velocity Vo. In zones V and VI, the extruded materials are assumed to flow horizontally as a rigid body with velocity Vf. Zones III and IV are the deformation regions, where the velocity is complex. These zones are surrounded by four velocity discontinuity surfaces S1, S2, S3 and S4. In addition to these surfaces, there are two frictional surfaces between the sleeve and container and the die surface and sleeve S5. The surfaces S1 and S3 are located at a distance ro from the origin, and the surfaces S2 and S4 are located at a distance rf from the origin. The 108 H. Haghighat, P. Amjadian Fig. 2. Schematic diagram of a plane strain extrusion of a bi-metallic strip through a curved die and the cylindrical coordinate system mathematical equations for radial positions of four velocity discontinuity surfaces S1, S3 and S2, S4 are given by ro = t1o − t2o sinα rf = t1f − t2f sinα (2.2) where α is the angle of the line connecting the initial point of the curved die to the final point of the die and tanα = t1o − t1f L (2.3) where L denotes the die length. The interface surface between the inner and the outermaterials is defined by ψi(r), which is the angular position of the interface surface as a function of the radial distance from the origin. The angle β, shown in Fig. 2, is given by sinβ = t2o t1o sinα (2.4) 2.1. Velocity field in the deformation zone The first step in the upper bound analysis is to choose an admissible velocity field for the material undergoing plastic deformation. The velocity field that derives from incompressibility condition and satisfies the velocity boundary conditions is a kinematically admissible velocity field. The velocity component in the radial direction within the deformation zone U̇r can be obtained by assuming volume flow balance. In Fig. 2, the volume flow of thematerial across the S1 and S3 surface at the point (ro,γ,z) in the radial direction is dQ =−Vocos(γ)bro dγ (2.5) The volume flow of the material in the radial direction at the point (r,θ,z) in the deformation zone is dQ = U̇rbrdθ (2.6) Equating Eqs. (2.5) and (2.6), the radial velocity in the deformation zone is U̇r =−Vo ro r cosγ dγ dθ (2.7) A generalized upper bound solution for extrusion... 109 Regarding the relationship between the angular positions γ, on the surfaces S1 and S3, and the angular position θ in the deformation zone and assuming proportional distances from the midline in the deformation zone (Gordon et al., 2007), one obtains sinγ sinα = sinθ sinψ (2.8) where the angle ψ is the angular position of a point on the die profile. Differentiating Eq. (2.8) yields cosγ dγ dθ =sinα cosθ sinψ (2.9) Replacing Eqs. (2.9) into Eq. (2.7) gives the radial velocity component in the deformation zone as U̇r =−Vo ro r sinα sinψ cosθ (2.10) When the die shape reduces to a linear die profile, it has a single value (i.e. ψ(r)= α) and Eq. (2.10) reduces to the velocity field proposed by Avitzur (1968) for the mono-metal plane strain extrusion through a wedge shaped die. The full velocity field for the flow of the material in the deformation zones is obtained by invoking volume constancy. The volume constancy in cylindrical coordinate system is defined as ε̇rr + ε̇θθ + ε̇zz =0 (2.11) where ε̇ii is the normal strain rate component in the i-th direction. The strain rates components in cylindrical coordinates are defined as ε̇rr = ∂U̇r ∂r ε̇θθ = 1 r ∂U̇θ ∂θ + U̇r r ε̇zz = ∂U̇z ∂z ε̇rθ = 1 2 (∂U̇θ ∂r + 1 r ∂U̇r ∂θ − U̇θ r ) ε̇θz = 1 2 ( ∂U̇θ ∂z + 1 r ∂U̇z ∂θ ) ε̇zr = 1 2 (∂U̇r ∂z + ∂U̇z ∂r ) (2.12) For the plane strain extrusion (i.e. U̇z = 0), the full velocity field is obtained by placing U̇r, from Eq. (2.10) into Eqs. (2.11) and (2.12), solving for U̇θ and applying appropriate boundary conditions. The boundary conditions are as U̇θ|θ=0 = 0 for the midline and for the die surface (U̇θ/U̇r)|θ=ψ = r∂ψ/∂r. Then, the velocity components can be given by U̇r =−Vo ro r sinα sinψ cosθ U̇θ =−Voro sinα sinψ ∂ψ ∂r cotψ sinθ U̇z =0 (2.13) Based on the established velocity field, the strain rate field for the deformation zone can be obtained by Eqs. (2.12) as ε̇rr =−ε̇θθ = Vo ro r2 sinα sinψ (1+r ∂ψ ∂r cotψ)cosθ ε̇rθ = 1 2 Vo ro r2 sinα sinψ [ 1−r2cotψ∂ 2ψ ∂r2 +r2(1+cot2ψ) (∂ψ ∂r )2 +cot2ψ (∂ψ ∂r )2 +rcotψ ∂ψ ∂r ] sinθ ε̇zz = ε̇θz = ε̇zr =0 (2.14) 110 H. Haghighat, P. Amjadian With the strain rate field and the velocity field, the standard upper bound method can be implemented. This upper bound model involves calculating the internal power of deformation over the deformation zone volume, calculating the shear power losses over the four surfaces of velocity discontinuity, and the frictional power losses between the sleeve and the die andbetween the sleeve and the container. 2.2. Internal power of deformation The internal power of deformation in the upper boundmodel is Ẇi = 2√ 3 σ0 ∫ v √ 1 2 ε̇ijε̇ij dV (2.15) The internal power of zones I, II, V and VI are zero, and the equation to calculate the internal power of deformation in zone III is ẆiIII = 2σs√ 3 b ro ∫ rf ψ(r) ∫ ψi(r) √ 1 2 ε̇2rr + 1 2 ε̇2 θθ + ε̇2 rθ rdθ dr (2.16) where σs is the mean flow stress of the sleeve material, and is determined by σs = 1 ε ε ∫ 0 σ dε ε = ln t1o − t2o t1f − t2f (2.17) and ψi(r) is the angular position of the interface surface as a function of the radial distance from the origin O, and is given by ψi(r)= sin −1 [sinβ sinα sinψ(r) ] (2.18) The general equation to calculate the internal power of deformation in zone IV is determined as ẆiIV = 2σc√ 3 b ro ∫ rf ψi(r) ∫ 0 √ 1 2 ε̇2rr + 1 2 ε̇2 θθ + ε̇2 rθ rdθ dr (2.19) where σc is the mean flow stress of core material and is given by σc = 1 ε ε ∫ 0 σ dε ε = ln t2o t2f (2.20) 2.3. Shear power losses The equation for the power losses along the shear surface of velocity discontinuity is ẆS = σ0√ 3 ∫ Sv |∆V |dS (2.21) A generalized upper bound solution for extrusion... 111 The shear power losses along the velocity discontinuity surfaces S1, S2, S3 and S4 can be given by ẆS1 = σs√ 3 Vorob α ∫ β ( 1+ro ∂ψ ∂r ∣ ∣ ∣ r=ro cotα ) sinθ dθ ẆS2 = σs√ 3 Vorob α ∫ β ( 1+ro to tf ∂ψ ∂r ∣ ∣ ∣ r=rf cotα ) sinθ dθ ẆS3 = σc√ 3 Vorob β ∫ 0 ( 1+ro ∂ψ ∂r ∣ ∣ ∣ r=ro cotα ) sinθ dθ ẆS4 = σc√ 3 Vorob β ∫ 0 ( 1+ro to tf ∂ψ ∂r ∣ ∣ ∣ r=rf cotα ) sinθ dθ (2.22) 2.4. Friction power losses The general equation for the friction power losses for the surface with a constant friction factor m1 is Ẇf = m1 σ0√ 3 ∫ Sf |∆V |dS (2.23) For the die surface S5 dS5 = √ 1+ ( r ∂ψ ∂r )2 dr |∆V |= |U̇r cosη+ U̇θ sinη|θ=ψ (2.24) where η is the local angle of the die surface with respect to the local radial velocity component and cosη = 1 √ 1+ ( r∂ψ ∂r )2 sinη = r ∂ψ ∂r √ 1+ ( r∂ψ ∂r )2 (2.25) The frictional power losses along the die surface are calculated as Ẇf5 = σs√ 3 m1V0br1o ro ∫ rf sinα rtanψ [ 1+ ( r ∂ψ ∂r )2] dr (2.26) where m1 is the constant friction factor between the sleeve and die. The frictional power losses along the container surface S7 can be given by Ẇf7 = σs√ 3 m1Vob(Lo −x) (2.27) where x is the displacement of the punch and Lo is the initial bi-metallic billet length. Based on the upper bound model, the total power needed for a bi-metallic plane strain extrusion process can obtained by summing the internal power and the power dissipated on all frictional and velocity discontinuity surfaces. Then, the total upper bound solution for the extrusion pressure is given by Pe = ẆiIII +ẆiIV +Ẇs1+Ẇs2+Ẇs3+Ẇs4+Ẇf5+Ẇf7 bt1oVo (2.28) 112 H. Haghighat, P. Amjadian AMATLAB program has been implemented for the previously derived equations and has been used to study the plastic deformation for different die shapes and friction conditions. It includes the parameter L, the die length that should be optimized. 3. Results and discussion Tomake a comparison with the developedmodel, a bimetal bar composed of aluminium as the core layer and copper as the sleeve layer is used. The flow stresses for copper and aluminium in room temperature are obtained as (Hwang and Hwang, 2002) σAl =189.2ε 0.239MPa σCu =335.2ε 0.113MPa (3.1) Friction factors m1 = 0.2 and m2 = 0.9, t1o = 10mm, t2o = 6mm, t1f = 7mm are adopted during the analytical solution and FEM simulation. The developed upper bound model can be used for the bi-metallic plane strain extrusion through dies of any shape if the die profile is expressed as an equation of ψ(r). Two types of die shapes are examined in the present investigation. The first die shape is wedge shaped die as a linear die profile. This profile has a single constant value, i.e. ψ(r) = α. The second die shape is taken from thework byYang andHan (1987). They created a streamlined die shape as a fourth-order polynomial whose slope is parallel to the axis at both entrance and exit. The die shape of Yang and Han (1987) was expressed in the cylindrical coordinate system. In Fig. 3, the extrusion pressure of the wedge shaped die and the Yang and Han die shape obtained from theupperboundmethod are comparedwith each other.The extrusion pressure of Yang andHan die shape is lower than that of conical die. Because this curved die has a smooth transition at the die entrance and exit, the shearing in the velocity discontinuity surfaces is zero. At a die length called the optimum die length, the extrusion pressure is minimized. As shown in Fig. 3, the optimumdie length in the case of the wedge shaped die is 7.9mm, and in the case of Yang and Han die shape is of 10.7mm. Fig. 3. Comparison between the analytical extrusion pressure values versus die length for the wedge shaped die and Yang andHan die shape The extrusion process is simulated by using the finite element code ABAQUS. Two die shapes are used in simulations: (a) optimum wedge shaped die with 7.9mm die length and (b) optimumYang andHandiewith 10.7mmdie length.Considering the symmetry in geometry, plane strain models are used for FEM analyses. In each case, the whole model is meshed with CAX4R elements. The punch and the die are modeled as rigid materials. The die model is fixed by applying displacement constraint on its nodes, while the punch model is loaded by specifying displacement in the axial direction. Figure 4a illustrates themesh used to analyze the A generalized upper bound solution for extrusion... 113 deformation of the optimum Yang and Han die shape. Deformed models of the sleeve and core are shown in Fig. 4b. This figure shows that aluminum leaves the deformation zone sooner than copper. Since the flow stress of aluminum is lower than that of copper, the former extrudes first. As the applied stress increases the flow stress of copper, simultaneous flow of the two metals continues. Fig. 4. (a) Finite element mesh and (b) deformedmesh in the bi-metallic plane strain extrusion process In Figs. 5a and 5b, the extrusion pressure variation during the whole extrusion process obtained from the upper bound solution are compared with the FEM for the wedge shaped die and Yang and Han die shape having 8mm and 11mm die lengths, respectively. The results show a good agreement between the upper bound data and the FEM results. As shown in these figures, at the early stage of extrusion, an unsteady state deformation occurs, and thematerials have not yet filled up the cavity of the die completely. Thus, the extrusion pressure increases as the extrusion process proceeds. After thematerials have filled up the cavity of the die completely, the extrusion pressure decreases gradually. The gradual decrease in the load-displacement curves is due to decreasing of the frictional surface area in the container as the punch is advanced. The results show a good agreement between the analysis and FEM results. As shown in Figs. 5a,b, the analytically predicted pressure is about 14% higher than the FEM result, which is due to the nature of the upper bound theory. Fig. 5. Comparison of analytical and FEM extrusion pressure-displacement curves for the wedge shaped die (a) and for the Yang andHan die shape (b) 114 H. Haghighat, P. Amjadian The effect of die length on the extrusion pressure for different values of the friction factor is shown in Fig. 6. As expected, for a given value of the friction factor, the extrusion pressure is minimized in the optimum die length. It is observed that the optimum die length decreases when the shearing friction factor increases.Thisfigure, also, shows that an increase in the friction factor tends to increase the extrusion pressure. Fig. 6. The effect of die length on the extrusion pressure for different values of the friction factor for the Yang andHan die shape Since the developed upper bound solution is faster than FEM analysis, it can be very bene- ficial in studying the influence of multiple variables on the bi-metallic rectangular cross-section bar extrusion process and for a given die shape, it can be used for finding the optimum die length which minimizes the extrusion pressure. 4. Conclusions In this research, a generalized velocity field for use in the upper bound model of the extrusion process of the bi-metallic rectangular cross-section bar through dies of any shapewas developed. By using the developed upper boundmodel, optimum die lengths for the wedge shaped die and also for theYang andHan die shapehave been determined.The extrusion pressure for these two dies has been found by using the finite element code, ABAQUS, and compared with analytical results. These comparisons show a good agreement, and the extrusion pressure of the Yang and Han die shape was lower than in the wedge shaped die. The developed upper boundmodel can be used for fast estimation of the extrusion pressure of bi-metallic rectangular cross-section bars through dies of any shape and for a given die shape and process parameters. It can be used for finding the optimumdie length whichminimizes the extrusion pressure. References 1. Avitzur B., 1968,Metal Forming: Processes and Analysis, McGraw-Hill, NewYork 2. Avitzur B., 1983,Handbook of Metal-Forming Processes, Wiley, NewYork 3. Berski S.,DyjaH.,BanaszekG., JanikM., 2004,Theoretical analysis ofbimetalbar extrusion process in double reduction dies, Journal of Materials Processing Technology, 154, 153-183 A generalized upper bound solution for extrusion... 115 4. 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YangD.Y.,HanC.H., 1987,Anew formulation of generalizedvelocity field for axisymmetric for- ward extrusion through arbitrarily curved dies,Transactions of the ASME, Journal of Engineering for Industry, 109, 161-168 18. Yang D.Y., Kim Y.G., Lee C.M., 1999, An upper-bound solution for axisymmetric extrusion of composite bars through curved dies, International Journal of Mechanical Sciences, 31, 565-575 19. ZimmermanZ.,AvitzurB., 1970,Metalflowthroughconical convergingdiesa lowerupperbound approach using generalized boundaries of the plastic zone,Transactions of the ASME, Journal of Engineering for Industry, 92, 119-129 116 H. Haghighat, P. Amjadian Uogólnione rozwiązanie ograniczenia górnego problemu wytłaczania bimetalowych belek o przekroju prostokątnym za pomocą tłoczników o dowolnym kształcie Streszczenie W pracy wyprowadzono uogólnione rozwiązanie ograniczenia górnego problemu wytłaczania bime- talowych belek o przekroju prostokątnym za pomocą tłoczników o dowolnym kształcie. Wyznaczono wyrażenia potęgowe opisujące wewnętrzne, ścinające oraz tarciowe obciążenia i zastosowano je w mo- delu matematycznym omawianego zagadnienia. Zaproponowana metoda badań pozwoliła na określenie nacisku podczas wytłaczania elementów przy użyciu dwóch typów tłocznika – optymalnego klinowego o profilu liniowym oraz tłocznika o zarysie krzywoliniowym, opływowym. W obydwu przypadkach ob- liczenia porównano z wynikami uzyskanymi z metody elementów skończonych. Porównanie to wykazało dobrą zgodność rozwiązania ograniczenia górnego z rezultatamiMES. Manuscript received January 5, 2012; accepted for print March 26, 2012