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外 文 資 料 翻 譯 Overview of adaptable die design for extrusionsW.A. Gordon.C.J. Van Tyne.Y.H. MoonABSTRACTThe term “adaptable die design” is used for the methodology inwhich the tooling shape is determined or modified to produce someoptimal property in either product or process. The adaptable die design method, used in conjunction with an upper bound model, allows the rapid evaluation of a large number of die shapes and the discovery of the one that produces the desired outcome. In order forthe adaptable die design method to be successful, it is necessary tohave a realistic velocity field for the deformation process through extrusion dies of any shape and the velocity f ield must allow f lexibility in material movement to achieve the required materialflow description. A variety of criteria can be used in the adaptable die design method. For example, dies which produce minimaldistortion in the product. A double optimization process is used todetermine the values for the f lexible variables in the velocity f ieldand secondly to determine the die shape that best meets the given criteria. The method has been extended to the design of dies for non-axisymmetric product shapes. ? 2006 Elsevier B. V. All rights reserved. Keywords: Extrusion; Die design; Upper bound approach; Minimum distortion criterion 1. IntroductionNew metal alloys and composites are being developed to meet demanding applications. Many of these new materials as well astraditional materials have limited workability. Extrusion is ametalworking process that can be used to deform these difficult materials into the shapes needed for specific applications. For asuccessful extrusion process, metalworking engineers and designersneed to know how the extrusion die shape can affect the final product. The present work focuses on the design of appropriate extrusion dieshapes. A methodology is presented to determine die shapes that meetspecific criteria: either shapes which pro-duce product with optimal set of specified properties, such as minimum distortion in theextrudate, or shapes which produce product by an optimized process, such as minimum extrusion pressure. The term “adaptable die design”is used for the method nology in which the die shape is determined or modified to produce some optimal property in either product orprocess. This adaptable die design method, used in conjunction withanupper bound model, allows the rapid evaluation of a large number of die shapes and the discovery of the one that can optimize the desired outcome. There are several conditions that need to be met forthe adaptable die design method to be viable. First, a generalized but realistic velocity field is needed for use in an upper bound model tomathematically describe the f low of the material during extrusionthrough dies of any shape. Second, a robust crite- rion needs to be established for the optimization of the die shape. The criterion must be useable within an upper bound model. The full details of the method are presented elsewhere [1–6]. In the present paper, following a review of previous models for extrusion, thef lexible velocity f ield for the deformation region in a direct extrusionwill be briefly presented. This velocity field is able to characterize the flow through a die of almost any configuration. The adaptable equation, which describes the die shape, is also presented. The constants in this die shape equation are optimized with respect to acriterion. The criterion, which can be used to minimize distortion, ispresented. Finally, the shape of an adaptable die, which produces ofan extruded product with minimal distortion, is presented. Theobjective of the present paper is to provide a brief overview of the adaptable die design method. 2. Background2.1. Axisymmetric extrusion Numerous studies have analyzed the axisymmetric extrusion of acylindrical product from a cylindrical billet. Avitzur[7–10] proposedupper bound models for axisymmetric extrusion through conical dies. Zimerman and Avitzur [11] modeled extrusion using the upper bound method, but with generalized shear boundaries. Finite elementmethods were used by Chen et al. [12] and Liu and Chung [13] to model axisymmetric extrusion through conical dies. Chen and Ling[14] and Nagpal [15] analyzed other die shapes. They developed velocity fields for axisymmetric extrusion through arbitrarily shapeddies. Richmond[16] was the first to propose the concept of astreamlined die shape as a die profile optimized for minimaldistortion. Yang et al. [17] as well as Yang and Han [18] developed upper bound models for streamlined dies. Srinivasan et al.[19] proposed a controlled strain rate die as a streamlined shape, whichimproved the extrusion process for materials with limited workability. Lu and Lo [20] proposed a die shape with an improved strain rate control. 2.2. Distortion and die shape analysis Numerous analytical and experimental axisymmetric extrusion investigations have examined the die shape and resulting distortion. Avitzur [9] showed that distortion increases with increasingreduction and die angle for axisymmetric extrusion through conical dies. Zimerman and Avitzur [11] and Pan et al. [21] proposed further upper bound models, including ones with flexibility in the velocityfield to allow the distorted grid to change with friction. They found that increasing friction causes more distortion in the extrudedproduct. Chen et al.[12] con-firmed that distortion increases with increasing reduction, die angle, and friction. Other research work hasfocused on non-conical die shapes. Nagpal [15] refined the upperbound approach to study alter- native axisymmetric die shapes. Chen and Ling [14] used the upper bound approach to study the flow through cosine, elliptic, and hyperbolic dies in an attempt to find adie shape, which minimized force and redundant strain. Richmondand Devenpeck [16,22,23], instead of assuming a particular type ofdie shape, decided to design a die based upon some feature of the extruded product. Using slip line analysis and assuming ideal andfrictionless conditions, Richmond [16] proposed a stream-linedsigmoidal die, which has smooth transitions at the die entrance and exit. The streamlined die shape is the basis for many efforts in axisymmetric extrusion die design. Yang et al. [17] , Yang and Han [18] , and Ghulman et al. [24] developed upper bound models using streamlined dies. Certain materials, such as metal matrix composites,can be successfully extruded only in a narrow effective strain rate range, leading to the development of controlled strain rate dies. Thecontrol of the strain rate in the deformation zone came from studies that showed fiber breakage during the extrusion of whisker reinforced composites decreases when peak strain rate was minimized[25] . Initially developed by Srinivasan et al. [19] , the streamlined die shape attempts to produce a constant strain rate throughout alarge region of the deformation zone. Lu and Lo [20] used a refined slab method to account for friction and material property changes inthe deformation zone. Kim et al. [26] used FEM to design an axisymmetric controlled strain rate die. They used Bezier curves todescribe the die shape and minimized the volumetric effective strainrate deviation in the deformation zone. 2.3. Three-dimensional non-axisymmetric extrusion analysis Both the upper bound and finite element techniques have been used to analyze three-dimensional non-axisymmetric extrusions.Nagpal [27] proposed one of the earliest upper bound analyses for non-axisymmetric extrusion. Upper bound and finite element modelswere developed Basily and Sansome[ 28] , Boer et al.[29] , and Boer and Webster [30] . Kiuchi [31] studied non-axisymmetric extrusionsthrough straight converging dies. Gunasekera and Hoshino [32–34] used an upper bound model to study the extrusion of polygonal shapes through converging dies as well as through streamlined dies.Wu and Hsu [35] proposed a flexible velocity field to extrude polygonal shapes through straight converging dies. Han et al. [36]created a velocity f ield from their previous axisymmetric upper bound model [37] in order to study extrusion through streamlined dies that produced clover- shaped sections. Yang et al. [37] applied ageneral upper bound model to study extrusion of elliptic and rectangular sections. Han and Yang [38] modeled the extrusion oftrocoidal gears. Yang et al. [39] also used finite element analysis tocon-firm the experimental and upper bound analysis of the cloversections. Non-axisymmetric three-dimensional extrusions have beenstudied further by using upper bound elemental technique [40] and spatial elementary rigid zones [41,42] . Streamlined dies have beenthe proposed die shape for most three-dimensional extrusion. The shape of the die between the entrance and exit has been selected by experience and feel rather than rigorous engineering principles. Nagpal et al. [43] assumed that the final position of a point that was initially on the billet is determined by ensuring that area reduction oflocal segments was the same as the overall area reduction. Once the final position of a material point was assumed, a third order polynomial was fit between the die entrance and exit points. Gunasekera et al.[44] refined this method to allow for re-entrantgeometries. Ponalagusamy et al. [ 45] proposed using Bezier curves for designing streamlined extrusion dies. Kang and Yang[46] used f inite element models to predict the optimal bearing length for a n“ L” shape extrusion. Studies on the design of three-dimensional extrusiondies have been limited. The controlled strain rate concept has only been applied to axisymmetric extrusions and not to three-dimensionalextrusions [19,20,26]. 3. The adaptable die design methodThe adaptable die design method has been developed and is described in detail in a series of papers [1–5 ]. The method has been extended to non-axisymmetric three- dimensional extrusion of a roundbar to a rectangular shape [6]. The major criterion used in developing the method was to minimize the distortion in the product. The presentpaper provides a brief overview of the method and results from these previous studies. Fig. 1. Schematic diagram of axisymmetric extrusion using spherical coordinate system through a die of arbitrary shape3.1. Velocity field 3An upper bound analysis of a metal forming problem requires akin matically admissible velocity field. Fig. 1 shows the processparameters in a schematic diagram with a spherical coordinate system (r, θ , φ ) and the three velocity zones that are used in the upper bound analysis of axisymmetric extrusion through a die with an arbitrary die shape. The material is assumed to be a perfectly plasticmaterial with flow strength, ??0 .he friction, which exists between thedeformation zone in he work piece and the die, is characterized by africtional shear tress, ????mf ??0 / , where the constant friction factor,mf, can take values from 0 to 1. The material starts as a cylinder of radius Ro and is extruded into a cylindrical product of radius Rf . Rigid body flow occurs in zones I and III, with velocities of v0 and v f , respectively. Zone II is the deformation region, where the velocity is fairly complex. Twospherical surfaces of velocity discontinuity Γ 1 and Γ 2 separate the three velocity zones. The surface Γ 1 is located a distance r0 from the origin and the surface Γ 2 is located a distance rf from the origin. The coordinate system is centered at the convergence point ofthe die. The convergence point is defined by the intersection of theaxis of symmetry with a line at angle α that goes through the point where the die begins its deviation from a cylindrical shape and the exit point of the die. Fig. 1 shows the position of the coordinate system origin. The die surface, which is labeled ψ (r) in Fig. 1, is given in the spherical coordinate system. ψ(r) is the angularposition of the die surface as a function of the radial distance from the origin. The die length for the deformation region is given by the parameter L. The best velocity field to describe the f low in the deformation region is the sine-1 velocity f ield [1,2] . This velocity field uses abase radial velocity, vr , which is modified by an additional termcomprised to two functions with each function containing 0 2pseudo-independent parameters to determine the radial velocity component in zone II: Ur ??vr ???? ( 1 )The ε function permits flexibility of flow in the radial, r, direction,and the γ function permits flexibility of flow in the angular, θ ,direction. The value of vr is determined by assuming proportional distances in a cylindrical sense from the centreline: ??r ??sin2 ?v ???v ??0 ? cos? ( 2 )r??r ??sin ?This velocity field was found to be the best representation of the flow in the deformation region of an extrusion process for an arbitrarilyshaped die. The ε function is represented as a series of Legendre polynomials that are orthogonal over deformation zone. Therepresentation of ε is:na??????Ai Pi ??x?i?0( 3 )Where 2?????Rfx ?1??Rf/ R0 ???1/ R0with????rr0ai being the coefficients of the Lengendre polynomials Pi(x) and that: na being the order of the representation. There is a restriction na ?odd ?A1 ????i?3Ai ，na ?even?A0 ???????Aii?2( 4 )The remaining higher order coefficients (A2 to A na ) are the pseudo-independent parameters, with values determined byminimization of the total power. Legendre polynomials are used sothat higher order terms can be added to the function without causing significant changes in the coefficients of the lower order polynomials.This feature of the Legendre polynomials occurs because they are orthogonal over a finite distance. ?iThe γ function that satisfies the boundary conditions and allows the best description of the flow is: nb ??1??cos???????B0 ????Bi ? ? ( 5 )i?1 ??1??cos???where nb B B0 ????i?1 i ?1and the high order coefficients B1 to B nb are pseudo-independent parameters with values determined by minimization of the totalpower. The order of the representation is nb . It has been shown [3] that na = 6 and nb = 2 are usually sufficient to provide reasonable f lexibility for the f low f ield in the deformation region. 3.2. Die shape The die shape is described by the functionψ (r). The adaptable die shape is described by a set of Legendre polynomials: nc?????ci pi ??x?i?0( 6 )where 2?????Rfx ?1???Rf/ R0 ???1/ R0 ?with????rr0and ci being the coefficients of theLegendre polynomials Pi(x). The order of the Legendre polynomial representation is nc . The boundary conditions at the entrance and exit of the deformation region require that: At r =At r =r0 , ψ= αrf , ψ= α ( 7 )If a streamlined die is used then this function must meet two additional boundary conditions: iAt r = r0 ,??r0 ????r??????tan ?R0At r = rf , r0 ? ?r??? tan ?Rf( 8 )3.3. Distortion criteria The criterion that was found to minimize the distortion in the extrusion product involves minimizing the volumetric effective strainrate deviation [4,5] . The volumetric effective strain rate deviation inthe deformation zone is: Wherewith:(10 )and ? are the components of the strain rate f ield. ??ij3.4. Determining the adaptable die shape The search for the optimal coefficients for the Legendre polynomials representing the die shape is not constrained. A nestedoptimization routine is used with the velocity field (inside loop) being minimized with respect to the externally supplied power for the process, and the die shape (outside loop) being adapted to minimize the distortion criterion. The final shape is called an adaptable die shape, since the shape has adapted to meet the specified criterion. Fig. 2. Streamlined adaptable die shape with no adaptation in the rotational directiowith red = 0 . 60 , L/ Ro= 1. 0,mf= 0.1 , Rr/ Ro= 0. 1 and μ = 1.5. The area reduction ratio is red, Rr is the cornered radius of the rectangular product,and μ i s the height to width ratio of the rectangular product. (For interpretation of the references to colour in this figure legend, the reader isreferred to the web version of the article.) 3.5. Extension to three-dimensional non-axisymmetric shapes In extending the adaptable die design method from axisymmetric f low to non- axisymmetric three-dimensional flow in the deformationregion requires several special considerations [ 6 ]. First, the velocity field needs to be modified to allow for rotational movement in the deformation region. Second, the bearing region on the exit side of the die needs to be analyzed properly. Third, the functions used to describe the die shape need to have some f lexibility in the rotational direction (i.e. ψ(r, φ ). This f lexibility allows die shape adaptation with respect to the rotational coordinate, φ .4. Die shape to minimize distortionTo illustrate the adaptable die design method a specific three-dimensional example is presented. An extrusion upper boundmodel was used to determine adaptable die surface shapes, which minimize distortion through minimizing the volumetric effectivestrain rate deviation in the deformation zone for the extrusion of acylindrical billet into a round cornered rectangular product. Twoschemes were used. In the first method there was no flexibility allowed in the rotational direction, ψ (r), whereas in the secondmethod the die shape was able to adapt in the rotational direction, ψ(r, φ ). In both methods a streamlined condition was used at the entrance and exit regions of the die. The quarter sections ( φ =0 to π/2) of both die shapes are given in Figs. 2 and 3. Fig. 2is the die shape with no rotational f lexibility and in Fig. 3 the die was allowed to adapt its shape in the rotational direction to reduce distortion. For both of these examples the area reduction was 60% and the rectangular product had a width to height ratio, μ, of 1.5. In Fig. 4, the extrusion die surface shape on the two rectangular symmetry planes of the product is presented. The adaptable die shape with rotational f lexibility is different from the die shape obtainedwithout adaptation in the rotational direction especially along the φ =π /2 symmetry plane. The adaptable die shape geometry along the φ =π /2 symmetry plane increases the speed of the material in the deformation zone near the exit. Fig. 5 shows the resulting distorted grid in the extrudate on the two symmetry planes. The adaptable dieshape shows a smaller difference in distortion as compared to the die shape without the rotational flexibility.Fig. 3. Streamlined adaptable die shape with adaptation of the die shape inthe rotational direction with red = 0. 60 , L/Ro= 1. 0,mf= 0. 1,Rr/ Ro= 0 . 1 and μ= 1. 5. Fig. 4. Streamlined die shape with no adaptation as the rotational direction compared to streamlined die shape with adaptation as a function of therotational direction—plotted along rectangular symmetry planes. Fig. 5. Extrudate distorted grid along rectangular symmetry planes forextrusion through a streamlined die shape with no adaptation as the rotational direction compared to a streamlined die shape with adaptation as a functionof the rota-tional direction— plotted along rectangular symmetry planes. Ar is the width of he extrudate and Br is the height of the extrudate. 5. SummaryThis paper presented an overview of the “adaptable die design” methodology. The full details of the met