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j o u r n a l o f m a t e r i a l s p r o c e s s i n g t e c h n o l o g y 2 0 9 ( 2 0 0 9 ) 3416–3425

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An innovative extrusion die layout design approach for single-hole dies Chao Lin ∗ , Rajesh S. Ransing School of Engineering, University of Wales Swansea, Singleton Park, Swansea, SA2 8PP, UK

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Article history:

To avoid distortion of complex extrudate profiles, it is essential that the exit velocity of the

Received 22 June 2007

metal is uniform. The most important design parameter that controls the exit velocity is

Received in revised form

bearing length. However, this factor is not sufficient to achieve optimal design settings. This

27 July 2008

paper proposes an innovative layout design approach using a geometry based bearing length

Accepted 29 July 2008

design methodology to minimise variations in exit velocity. This paper focuses on single hole extrusion dies and develops an optimisation algorithm suitable for any die profile. Since the optimised die design produces less bearing length difference, more uniform exit velocity


and higher design quality would be achieved using the proposed approach. © 2008 Elsevier B.V. All rights reserved.

Extrusion Optimisation Die layout design Geometrical reasoning methods





Extrusion is an important forming process. By choosing proper material, die shapes and extrusion processes, many useful extrudates with simple or complex shapes can be made. Different types of materials are widely used by industry depending on its extrudability. As one of the most important materials, Aluminium alloys can provide cheap, light, shaped and strength products.Therefore, technology for aluminium extrusion, such as extrusion die design, is one of the most important and popular areas for both research and industry. The basic parts of extrusion equipment include a container, ram and die (Fig. 1). During the extrusion process, the ram pushes a cylindrical billet in the container against a die. The billet material is forced to flow through the die opening so that the extrudate takes the shape of the die opening profile. Solid

dies can create shaped bars, simple section and semi-hollow section shapes. By using hollow dies, tubes and hollow section extrudates can be made. A schematic parts of a solid extrusion die is shown in Fig. 2.


Extrusion die design

A non-uniform material flow can lead to various defects. As discussed by Laue and Stenger (1981), the non-uniform flow caused by temperature variations in a billet, limited billet length and the friction between billet and container walls, as well as the friction between billet and die. Using a billet which marked with a grid, the flow pattern inside it can be checked. As an example given by Onuh et al. (2003), Fig. 3 illustrates two cross-sectional flow pattern inside an extruded billet. Undistorted grids in the billet indicate uniform flow and vice versa. It is clear that the flow remains quite uniform before the billet enters the dead zone, which marked as ‘Z’ in the figure. And

Corresponding author. E-mail addresses:, (C. Lin). 0924-0136/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2008.07.042

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to change the flow rate is the adjustment of frictional resistance in the extrusion die. Following are design factors which mentioned by Saha (2000) and used by designers for extrusion dies. • • • • •

Fig. 1 – Extrusion press (schematic).

Die layout design Bearing length design Pocket design Choke and relief design Bridge and porthole design

During the extrusion process, the friction between billet and bearing retards the flow. Longer bearing length generates greater friction and reduces the flow rate. A recess (also called as pocket) is often made in the front side of the die to allow easy entry of the material. Wider recesses lower the friction and boost the flow rate. Due to the friction between billet and container wall, the flow rate varies along radial direction inside a billet and hence different location and orientation can also influence the flow rate.


Fig. 2 – Solid extrusion die (schematic).

the flow becomes non-uniform after it enters the dead zone, which causes all kind of defects in the products. Therefore, all extrusion die designers strive to achieve uniform flow at die opening to improve their products. For a given extrudate profile, material and container, the extrudate section thickness, material properties and friction between billet and container are fixed. The only possible way

Traditional approach for extrusion die design

A trial-and-error approach is frequently used for die design but it is a slow and expensive method. Besides, the design quality depends on designers’ experience and skill. Since the whole optimisation process is undertaken by a designer, it is extremely difficult to transfer the design knowledge and skills from one person to another. It is also quite difficult to standardise the technique and maintain the design quality. Numerical methods, such as finite element method (FEM), are widely used for extrusion process. The simulation results can provide flow rate, stress, strain and temperature distri-

Fig. 3 – Cross-sectional flow pattern inside an extruded billet. The dead metal zones are marked as ‘Z’ (Onuh et al., 2003).


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bution in the billet. Devadas and Celliers (1992) use finite element method to simulate material flow for extrusion die design using a viscous non-Newtonian incompressible fluid model. Due to the limitation of computer speed at that time, plane-strain model instead of rigorous three-dimensional analysis was used. Comparison has been done between plainstrain and three-dimensional simulation results. At last, an empirical design formula has been introduced as an application of the simulation work. However, how to use the simulation result to determine the constants of the formula is not clear in the paper. Lee et al. (2002a) demonstrated three-dimensional material flow simulations for flat-die channel sections using finite element method. A non-steady thermo-rigid-viscoplastic model has been used for the threedimensional simulation. The research work gives strain, temperature, exit velocity and deformed shape of extrudates. Lof and Huétink (2000a,b) use Arbitrary Lagrangian Eulerian (ALE) finite element method to simulate extrusion process. Their model takes not only plastic but also elastic behaviour of materials to accurate the simulation results. The research work focuses on metal flow in bearing area. The simulation result gives pressure, stress, velocity distribution in the billet. As one of the popular design approach, numerical methods can be used to simulate the extrusion process and indicate the properties such as strain, temperature, pressure and velocity. But they are difficult to implement, time-consuming. Another issue is that numerical simulation cannot directly give die designs. Therefore, numerical simulation is not an easy and efficiency way for extrusion die design. Empirical design approach is another popular way for extrusion die design. During past years, a lot of design rules and formulae have been introduced and studied. Lotzenhiser (1977) discussed extrusion bearing length as well as layout design rules for single- and multi-hole extrusion dies. However, the mentioned design rules are very brief. For example, a design rule for single-hole die is to place the die opening at the centre of the die and keep the thinnest part of the profile towards the die centre. Castle et al. (1988) discussed the design and construction factors for extrusion dies. The research work mentioned that bearing length variation is the main method of controlling the flow. They also gave a set of bearing length design guidelines which are widely used in Europe. Miles et al. (1996, 1997) studied flat-faced aluminium extrusion die design factors. Since the bearing is the most common method to control the flow, as mentioned by them, the research work focused on bearing length design. They also introduced new bearing length design formulae which can be used to get reasonable die designs. Lee et al. (2002b) demonstrated an approach to determinate extrusion die design formula factors using numerical simulation results. To validate the approach, three-dimensional simulation results have been used to determine two factors of a bearing length design formula. The final result shows that reasonable bearing length design can be achieved using the determined design formula. Empirical design rules and formulae provide a easy and quick way to get reasonable extrusion die designs. But it is difficult to get good designs for complicated extrudates using this approach. And it is also difficult to get the design rules or formulae.

2. The proposed approach for extrusion die layout design Extrusion layout design is not as important as bearing length or pocket design. However, it is another vital design factor for extrusion dies. Research work for layout design are much less than for bearing length or pocket design during past years. And few research work which covered this topic only gave general design rules rather than design formulae or approaches. A novel extrusion die layout design approach is described and validated in the rest of this paper. The new approach shows how to quantify the design quality of a die using its bearing length estimation and how to optimise the layout design of the die. The research work also illustrates that an optimal layout design can be calculated automatically using the proposed approach.


An overview of the proposed approach

As mentioned, an optimal extrusion die design should give a uniform exit flow with a reasonable simple design. It is widely known that large value or dramatic changes of bearing length in a die lead to defects or failures. It is clear that an high quality design should have a low bearing length and bearing length variation. Therefore, bearing length is used to estimate the design quality for the new approach. To quantify the design quality, bearing length difference (BLD) is introduced and used in this paper. Since the flow inside a billet has different velocity in different place, different layout design leads to different bearing length design for the same die opening profile. An optimal layout design gives the location and orientation by which low bearing length and bearing length variation can be achieved. Since the die layout design quality can be quantified by corresponding BLD value, the optimisation work is to find out the best location and orientation by which minimum BLD value can be achieved. Details of this design approach are demonstrated in following sections.

2.2. Bearing length prediction using empirical design formulae As one of the most important design factor for extrusion dies, bearing length directly influence the friction between billets and dies. Though the flow behaviour is very complex, several useful guidelines are widely used by extrusion die designers to estimate the bearing lengths. Following are the widely used guidelines given by Castle et al. (1988) for bearing length design:

• Bearing length should be proportional to the width of die openings • The fact that material flow rate at the centre of the extrusion billet is faster than that in the peripheral areas should be considered • Bearing length should be correct for a particular geometry structures, such as tips, as they can influence flow of material inside the billet

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Fig. 4 – Layout design representation in global coordinate system (A) and in local coordinate system (B).

• Finally, a minimum value of about 2–3 mm should be maintained for bearing length in order to maintain die strength. Following all kind of guidelines, different bearing length design formulae were introduced. Eq. (1) is the most commonly used design formula which given by Miles et al. (1996). where b is the bearing length at boundary point p in mm, w is the width of section at p in mm, r is the distance of section at p from die centre in mm, R is the distance of furthest part of orifice from die centre in mm and C is a constant, typically equals to 1/80. According to the formula, the section width, radial distance to die centre and maximum radial distance, w, r and R, are design variables for bearing length design. b = w(2 + C(R − r))


As a widely used formula, Eq. (1) is used to demonstrate the proposed approach. However, it should be noticed that the computational framework of this approach is applicable for any other bearing length design methods.


Bearing length and bearing length difference (BLD)

For a given die layout, die opening section width, radial distance and maximum radial distance are fixed. And the values of constants, e.g. constant C can be determined using experiments or numerical simulation results. Therefore, all the variables required by Eq. (1) can be pre-determinated and hence the bearing length can be calculated. In the proposed approach, medial axis transform (MAT), which generated using CADfix (2005), is used to calculate the die opening width w. Coordinates of points which lie on die opening boundaries are used to calculate radial distance r. R is the maximum value of r (Fig. 5). By using Eq. (1), bearing length distribution of any die opening profile can be easily calculated. And the corresponding BLD can be calculated using Eq. (2), where bmax and bmin are the maximum and minimum bearing lengths for the given layout design. BLD = bmax − bmin



Bearing length difference (BLD) distribution

In Eq. (1), variable r and R are related to die opening location and orientation. The die opening locations and orientations can be described in two ways. The first method uses a global coordinate system, and assumes die centre as the original point. Three variables, x, y, and rotation angle ˛ are needed for describing location and orientation as shown in Fig. 4(A). The other method is based on a local coordinate system. As shown in Fig. 4(B), a reference point of die opening is used as the original point and the centre of a die is then defined by local coordinates x and y in the coordinate system. As a result, die opening locations and orientations can be described by only two design variables. The proposed approach uses the second method to get r and R values. It is trivial that for any given x and y , w, r and R are fixed and therefore BLD can be calculate using Eq. (2). The layout design problem is thus simplified to find out the optimal values of x and y to achieve the minimum BLD. BLD distribution is used to find the minimum BLD. To get the BLD distribution, a grid of points would be used for the calculation. Each point in the grid corresponds to a possible position of die centre. The procedure for calculating BLD distribution for any given die opening profile is described as follows: (1) Generate the medial axis transform for a given die opening profile. (2) Choose a reference points grid. The size of the grid should be larger than extrusion die. The density of points in the grid is kept high to maintain precision and accuracy. (3) Choose a reference point in the grid. (4) Calculate bearing length distribution and store the maximum and minimum values of bearing length as bmax and bmin , respectively. (5) Calculate BLD using Eq. (2). Save the result with the corresponding reference point. (6) Repeat steps 3–5 for all other unused reference points. To demonstrate the proposed approach, a ‘L’-shape die profile is used. The shape and medial axis transform of the profile are shown in Fig. 5. The BLD distribution result is shown


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Fig. 5 – ‘L’-shape profile with its medial axes.

Fig. 6 – Bearing length difference (BLD) distribution of ‘L’-shape die opening.

in Fig. 6. As shown in the legend of this figure, the darker area of the distribution result indicates higher bearing length difference value and vice versa. Since the objective of the optimisation process is to achieve a lower value for BLD, reference point locations in the light area of the figure become potential choices for the die centre. For the optimisation process, all possible BLD values are possible choice for the input variable, and is called the ‘design space’ of the corresponding die opening profile. It is trivial that the BLD values are distributed in the whole X–Y plan. Hence the ‘design space’ for the L-shape profile is the whole X–Y plan. It is obvious that not all reference point in the plan are suitable for the die layout design. Therefore, design space refinement is an essential process before the optimisation algorithm can be implemented.

Fig. 7 – Clearance zone between opening and container wall.

remaining part of the die forms a design space within which die openings can be located. White circular area in Fig. 8 is the corresponding design space and die openings marked as 1–5 are five possible layout designs. Although the original definition of clearance zone is simple, automatic computation of this zone using computers is not straight forward. In order to overcome this difficulty, a concept of clearance zone around die opening along with the corresponding ‘valid design space’ is introduced. As mentioned, a proper clearance can be maintained by reserving a ring-shape zone within a die (Fig. 7). In the other way, the proper clearance can be kept using a ‘clearance zone’ around the die opening shape. An example of die opening layout and its corresponding clearance zone are shown in Fig. 9(A). Since the shape of an extrusion die is always circular, all concave parts of the clearance zone, hatched area between clearance zone and the edge in this case, do not remain in contact with the edge of a die. Therefore, a modified clearance zone, which has a curvilinear convex shape, is given as the hatched area in Fig. 9(B). Due to the complexity of the calculations, simple convex polygon is used instead of the curvilinear shape to describe the clearance zone. An example of simplified clearance zone for the ‘L’-shape die opening is shown in Fig. 9(C).

2.5. Refinement of the design space to include a clearance zone For single-hole die layout design, a proper clearance between die opening and the side wall has to be maintained. For a given profile of die opening, this clearance corresponds to the minimum distance between any container wall and die opening boundary points. The clearance can be described as a ring-shape zone close to the inner side of the die, with a thickness equal to the permissible minimum clearance value. An example is shown in Fig. 7. The hatched ‘ring’-shaped area is the clearance zone for the ‘L’-shape die opening profiles. The

Fig. 8 – Possible location and orientation of die openings.

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Fig. 9 – Concave clearance zone (A), convex clearance zone (B) and simplified clearance zone (C).

It is clear now that the clearance zone for a given die profile can be calculated in two steps: (1) Create an enlarged polygon following the shape of die opening profile with a uniform thickness, which is equal to the minimum clearance distance, between the polygon and the profile. (2) Convert concave polygon into convex shape.


Automatic computation of the valid design space

If a proper clearance zone is available, possible location and orientation of die opening can be easily chosen. This is accomplished by moving or rotating along die opening with its clearance zone making sure that both of them are within the die plate. The clearance zone around the die opening thus works like a car ‘bumper’ and maintains a proper clearance for the layout design. Possible location and orientation of ‘L’shape die opening is shown in Fig. 10(A). Although a convex clearance zone method is generally used by designers, it is not suitable for developing a computer aided technique. A concept of the ‘valid design space’ which is related with the local coordinate system is introduced to overcome this complexity.

The different location and orientation of die opening is linked to the movement of a die opening within the die plate. For three possible die opening location and orientation that are marked as 1–3 in Fig. 10(A), the die centre positions relative to the corresponding die opening are shown in Fig. 10(B). For the first position, left-bottom corner of the clearance zone touches the die plate. The contact point and die centre are indicated as P1 and O1 in Fig. 10(B). Please note that the vector n1 which starts from P1 and ends at O1 is a radius of the circle (die plate) and it is perpendicular to the curvilinear boundary line of clearance zone at point P1 . Relative positions of die centre for the second and third cases are also shown in this figure. This property enables the formulation of the following algorithm which then provides the valid design space. In Fig. 11, the inner white ‘L’-shape area indicates the die opening. Grey area is the simplified convex-hull indicating the clearance zone of the profile as mentioned before. For each point on the boundary of the clearance zone (such as P1 , P2 , . . ., P13 in the figure), a corresponding vector of the same length as the die radius is drawn on that point. The direction of the vector is perpendicular to the tangent line at that point and it points to the ‘inner’ part of the profile. The end point of the vector gives a relative position of the die centre (O1 , O2 , . . ., O13 in Fig. 11). The locus of end points of vectors define a closed

Fig. 10 – Die opening location and orientation relative to die plate (A), and die plate position in relation to the location and orientation of die opening (B).


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zone cannot be obtained in the area close to the left-bottom corner of the profile. Similarly, for die centre located outside the curve n3 , v1 , n4 , the distance between the profile’s rightbottom corner and container wall is less than the minimum required. Therefore, curve n1 , v1 and v1 , n4 ought be the right boundary of valid design zone. The final boundary of valid design space is obtained by removing the invalid overlapping parts of the original results (Fig. 12(A)). The final step for single-hole die layout design is to find the optimal position for die centre. Since the design objective is to achieve minimum bearing length difference between maximum and minimum values, this step is implemented by combining the BLD distribution with the valid design space. By using the combination of BLD distribution result of the ‘L’shape die opening (Fig. 6) and its valid design space (Fig. 12(A)), the BLD distribution for all possible layout design is calculated (Fig. 12(B)). Based on this, the optimal position for the die centre is found out and is marked as point O in Fig. 12(B). Its corresponding layout design is shown in Fig. 12(C).

Fig. 11 – Valid design zone calculation based on the clearance zone of the profile.

curvilinear boundary as indicated by dashed line in the figure. The curve interlace at some position such as the cross-point between n2 and n3 . Each segment of the curve means the die plate touches a segment on clearance zone boundary. It is clear that if the clearance between container wall and die opening is maintained, none of die opening segments would be located in the clearance zone. This indicates that for the interlace part of the valid design zone boundary, the inner most curve should be chosen as the final result. For example, v1 is the cross point of curve n1 ,n2 and n3 ,n4 . If the die centre is located outside this curve n1 , v1 , n2 , some part of the extrusion die container must cross boundary P1 ,P2 . This means that the proper clearance


Validation of the proposed approach


Case study 1

To test the proposed layout design approach for single-hole dies, a simple channel-shape profile is used (Fig. 13(A)). The BLD distribution result is shown in Fig. 13(B). As shown in the figure, the BLD achieves its minimum value at the centre of the channel. According to the design approach, the optimal layout design for this channel shape is to put it at the centre of the die so that the centre of the channel is over the die centre. A schematic optimisation result is shown in Fig. 13(C) and has been confirmed as a good layout design by our industrial partner.


Case study 2

In this case, a general polygon die opening profile and a designer’s layout design which given by Lotzenhiser (1977) are

Fig. 12 – Valid design zone for L-shape die opening profile (A) (schematic), its BLD distribution inside the valid design zone (B), and the optimal layout design (C).

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Fig. 13 – Die opening profile with MAT (A), its BLD distribution (B), and the optimal layout design (C).

Fig. 14 – Die opening profile with MAT (A) (Lotzenhiser, 1977) and its BLD distribution (B).

used. Fig. 14(A) shows the profile shape and its medial axis transformation result. Fig. 14(B) shows the BLD distribution result of the profile. As can be seen, the lowest BLD value is located closely to the left tip of the profile. The designer’s design is shown in Fig. 15(A) and the optimal design obtained using the new approach is shown in Fig. 15(B). The optimisation result agreed well with the designer’s design. This result compares favourably with the established design guidelines and common practise for the single-hole die design where

designer tends to locate the thinnest part of the die opening at the centre of the die.


Case study 3

A realistic solid extrusion die design studies by Li et al. (2002) is used for this case study. The die opening profile and designer’s design for layout and bearing length are shown in Fig. 16(A). In the figure, DIM.A and DIM.B indicate the thickness of the

Fig. 15 – Die layout design given by designer (A) (Lotzenhiser, 1977), and the optimisation result (B).


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Fig. 16 – Die layout design by designer (Li et al., 2002) (A), and the optimisation result of die layout design (B).

profile leg and base. All other numbers around the profile are designed bearing lengths. All dimensions are in millimetres. The die centre is marked as O in the figure. The maximum and minimum bearing length are 4.0 mm and 1.5 mm and it is easy to get BLD value, which is 2.5 mm in this case. The optimisation result given by the new approach is illustrated in Fig. 16(B). As illustrated in the figure, the BLD value changes from 2.4 mm to 4.0 mm for different location within the chosen area. The result indicates that BLD values are quite close to the minimum in three areas which marked as A1, A2 and O. However, to hold die opening at a location corresponding to area A1 or A2, larger die diameter is required compare to the diameter required at O. Therefore, the location in area O where the minimum BLD achieved is the optimal location of die centre. This case study shows that for a complex die opening profile, the proposed approach gives the optimal result successfully. The optimal location and corresponding BLD value fit designer’s design quite well.



A novel approach has been presented in this paper that makes use of the medial axis transform and empirical bearing length design formula to obtain an optimal extrusion die layout design quickly and efficiently. A new design quality estimation method is demonstrated. This method uses BLD within a local coordinate system rather than a commonly used global coordinate system to quantify the quality of die layout designs. The method combines the benefits of geometrical reasoning technology and empirical bearing length design formulae to provide a innovative way to quantify the design quality automatically, easily and quickly. In this paper, ‘clearance zone’ and ‘valid design space’ have also been introduced. By using these two new concepts, proper extrusion die clearance can be maintained and an optimal layout design can be calculated automatically and efficiently. Since the proposed approach could calculate the optimal layout design automatically, there are no difficulties to extend its function for multi-hole die layout design.

Acknowledgements This work was carried out with the support from EPSRC-Grant No.: GR/R15610/01. The authors would also like to acknowledge The IRC in Material Processing of Birmingham university, ITI TranscenData Europe Ltd. (Cambridge, U.K.) and Hydro Aluminium Extrusion Ltd.


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