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外文資料翻譯 Overview of adaptable die design for extrusions W.A. Gordon.C.J. Van Tyne.Y.H. Moon ABSTRACT The term “adaptable die design” is used for the methodology in which the tooling shape is determined or modified to produce some optimal 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 for the adaptable die design method to be successful, it is necessary to have a realistic velocity field for the deformation process through extrusion dies of any shape and the velocity field must allow flexibility in material movement to achieve the required material flow description. A variety of criteria can be used in the adaptable die design method. For example, dies which produce minimal distortion in the product. A double optimization process is used to determine the values for the flexible variables in the velocity field and 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. Introduction New metal alloys and composites are being developed to meet demanding applications. Many of these new materials as well as traditional materials have limited workability. Extrusion is a metalworking process that can be used to deform these difficult materials into the shapes needed for specific applications. For a successful extrusion process, metalworking engineers and designers need to know how the extrusion die shape can affect the final product. The present work focuses on the design of appropriate extrusion die shapes. A methodology is presented to determine die shapes that meet specific criteria: either shapes which pro-duce product with optimal set of specified properties, such as minimum distortion in the extrudate, 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 or process. This adaptable die design method, used in conjunction with anupper 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 for the adaptable die design method to be viable. First, a generalized but realistic velocity field is needed for use in an upper bound model to mathematically describe the flow of the material during extrusion through 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, the flexible velocity field for the deformation region in a direct extrusion will 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 a criterion. The criterion, which can be used to minimize distortion, is presented. Finally, the shape of an adaptable die, which produces of an extruded product with minimal distortion, is presented. The objective of the present paper is to provide a brief overview of the adaptable die design method. 2. Background 2.1. Axisymmetric extrusion Numerous studies have analyzed the axisymmetric extrusion of a cylindrical product from a cylindrical billet. Avitzur[7–10] proposed upper 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 element methods 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 shaped dies. Richmond[16] was the first to propose the concept of a streamlined die shape as a die profile optimized for minimal distortion. 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, which improved 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 increasing reduction 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 velocity field to allow the distorted grid to change with friction. They found that increasing friction causes more distortion in the extruded product. Chen et al.[12] con-firmed that distortion increases with increasing reduction, die angle, and friction. Other research work has focused on non-conical die shapes. Nagpal [15] refined the upper bound 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 a die shape, which minimized force and redundant strain. Richmond and Devenpeck [16,22,23], instead of assuming a particular type of die shape, decided to design a die based upon some feature of the extruded product. Using slip line analysis and assuming ideal and frictionless conditions, Richmond [16] proposed a stream-lined sigmoidal 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. The control 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 a large region of the deformation zone. Lu and Lo [20] used a refined slab method to account for friction and material property changes in the deformation zone. Kim et al. [26] used FEM to design an axisymmetric controlled strain rate die. They used Bezier curves to describe the die shape and minimized the volumetric effective strain rate 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 models were developed Basily and Sansome[28] , Boer et al.[29] , and Boer and Webster [30] . Kiuchi [31] studied non-axisymmetric extrusions through 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 field 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 a general upper bound model to study extrusion of elliptic and rectangular sections. Han and Yang [38] modeled the extrusion of trocoidal gears. Yang et al. [39] also used finite element analysis to con-firm the experimental and upper bound analysis of the clover sections. Non-axisymmetric three-dimensional extrusions have been studied further by using upper bound elemental technique [40] and spatial elementary rigid zones [41,42] . Streamlined dies have been the 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 of local 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-entrant geometries. Ponalagusamy et al. [45] proposed using Bezier curves for designing streamlined extrusion dies. Kang and Yang[46] used finite element models to predict the optimal bearing length for an “ L” shape extrusion. Studies on the design of three-dimensional extrusion dies have been limited. The controlled strain rate concept has only been applied to axisymmetric extrusions and not to three-dimensional extrusions [19,20,26]. 3. The adaptable die design method The 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 round bar to a rectangular shape [6]. The major criterion used in developing the method was to minimize the distortion in the product. The present paper 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 shape 3.1. Velocity field An upper bound analysis of a metal forming problem requires a kin matically admissible velocity field. Fig. 1 shows the process parameters 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 plastic material with flow strength, 0 ? .he friction, which exists between the deformation zone in he work piece and the die, is characterized by a frictional shear tress, 0 /3 mf ?? ? , 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 f R . Rigid body flow occurs in zones I and III, with velocities of 0 v and f v , respectively. Zone II is the deformation region, where the velocity is fairly complex. Two spherical surfaces of velocity discontinuity Γ 1 and Γ 2 separate the three velocity zones. The surface Γ 1 is located a distance 0 r from the origin and the surface Γ2 is located a distance f r from the origin. The coordinate system is centered at the convergence point of the die. The convergence point is defined by the intersection of the axis 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 angular position 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 flow in the deformation region is the sine-1 velocity field [1,2] . This velocity field uses a base radial velocity, r v , which is modified by an additional term comprised to two functions with each function containing pseudo-independent parameters to determine the radial velocity component in zone II: rr Uv ? ? ?? (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 r v is determined by assuming proportional distances in a cylindrical sense from the centreline: 2 0 0 2 sin cos sin r r vv r ? ? ? ?? ?? ?? ?? (2) This velocity field was found to be the best representation of the flow in the deformation region of an extrusion process for an arbitrarily shaped die. The ε function is represented as a series of Legendre polynomials that are orthogonal over deformation zone. The representation of ε is: ?? 0 a n ii i AP x ? ? ? ? (3) Where ?? 0 00 2/1 1/ f f RR r xw i t h RRr ? ? ?? ?? ? i a being the coefficients of the Lengendre polynomials Pi(x) and a n being the order of the representation. There is a restriction that: ?? 1 3 a n odd i i A A ? ?? ? ， ? ? 0 2 a ne v e n i i A A ? ?? ? (4) The remaining higher order coefficients (A2 to A a n ) are the pseudo-independent parameters, with values determined by minimization of the total power. Legendre polynomials are used so that 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. The γ function that satisfies the boundary conditions and allows the best description of the flow is: 0 1 1c o s 1c o s b i n i i BB ? ? ? ? ?? ? ?? ?? ? ?? ? (5) where 0 1 1 b n i i B B i ? ?? ? ? and the high order coefficients B1 to B b n are pseudo-independent parameters with values determined by minimization of the total power. The order of the representation is b n . It has been shown [3] that a n = 6 and b n = 2 are usually sufficient to provide reasonable flexibility for the flow field in the deformation region. 3.2. Die shape The die shape is described by the function ψ (r). The adaptable die shape is described by a set of Legendre polynomials: ?? 0 c n ii i cp x ? ? ? ? (6) where ?? ?? 0 0 0 2/1 1/ f f RR r x with r RR ? ? ?? ?? ? and i c being the coefficients of the Legendre polynomials Pi(x). The order of the Legendre polynomial representation is c n . The boundary conditions at the entrance and exit of the deformation region require that: At r = 0 r , ψ= α At r = f r , ψ= α (7) If a streamlined die is used then this function must meet two additional boundary conditions: At r = 0 r , 0 tan r r ? ? ? ?? ? ? At r = f r , 0 0 tan f R r rR ? ? ? ?? ? ? (8) 3.3. Distortion criteria The criterion that was found to minimize the distortion in the extrusion product involves minimizing the volumetric effective strain rate deviation [4,5] . The volumetric effective strain rate deviation in the deformation zone is: Where with: (10) and ij ? ? are the components of the strain rate field. 3.4. Determining the adaptable die shape The search for the optimal coefficients for the Legendre polynomials representing the die shape is not constrained. A nested optimization 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, andμ is the height to width ratio of the rectangular product. (For interpretation of the references to colour in this figure legend, the reader is referred 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 flow to non-axisymmetric three-dimensional flow in the deformation region 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 flexibility in the rotational direction (i.e. ψ(r, φ ). This flexibility allows die shape adaptation with respect to the rotational coordinate, φ . 4. Die shape to minimize distortion To illustrate the adaptable die design method a specific three-dimensional example is presented. An extrusion upper bound model was used to determine adaptable die surface shapes, which minimize distortion through minimizing the volumetric effective strain rate deviation in the deformation zone for the extrusion of a cylindrical billet into a round cornered rectangular product. Two schemes were used. In the first method there was no flexibility allowed in the rotational direction, ψ (r), whereas in the second method 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 flexibility 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 flexibility is different from the die shape obtained without 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 die shape 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 in the 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 the rotational direction—plotted along rectangular symmetry planes. Fig. 5. Extrudate distorted grid along rectangular symmetry planes for extrusion through a streamlined die shape with no adaptation as the rotational direction compared to a streamlined die shape with adaptation as a function of 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. Summary This paper presented an overview of the “adaptable die design” methodology. The full details of the method are given elsewhere [1–6]. In order to use the adaptable die design method in conjunction with an upper bound analysis, it is necessary to have a velocity field in the deformation region with sufficient flexibility so that the model can be closer to the real flow. The specific criterion of producing a product with minimal distortion involves minimizing the volumetric strain rate deviation. A double optimization process is used (1) to de