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Chinese Journal ofAeronautics 22(2009) 81-86

Chinese Journal of Aeronautics

www.elsevier.com/locate/cj a

A Quasi-equipotential Field Simulation for Preform Design of

P/M Superalloy Disk

Wang Xiaona, Li Fuguo*

School of Materials Science and Engineering, Northwestern Polytechnical University, Xi'an 710072, China Received 11 January 2008; accepted 24 April 2008

Abstract

On the basis of the minimum energy principle and the minimum resistance law, this article proposes a new method, termed equipotential field method, to design the proper preform for producing isothermo forged P/M superalloy disks. Using this new method, six variant preform contours are acquired with software ANSYS. The isothermal forging process of the P/M superalloy disk is simulated by using the industrial software MSC/Superform with thus obtained preforms so as to achieve the equivalent strain distribution in the final forging and the deformation degree distribution in preforming and final forming. By comparing the equivalent strains, the deformation degrees and other field variables, an optimized preform is acquired.

Keywords: superalloys; equipotential field; preforming; finite element method (FEM)

1. Introduction

P/M superalloy has long become the best candidate material to manufacture turbo-disks in the new generations of high thrust-weight ratio aero-engines working under high temperatures and stresses. As one of the most important techniques to produce P/M superalloy disk, isothermal forging includes three stages: upsetting, preforming and final forming.

Experience-dependent designing of conventional isothermal forging process is already outdated because it always results in low efficiency and high costs owing to expensive manpower, material and prolonged production cycle. Furthermore, the quality of traditional isothermal forgings turns to be usually unstable and unreliable. All these factors make it meaningless to study the classical method by identifying the effects of die geometric parameters on product quality.

This article tries to adopt the equipotential field method to design the felicitous preform for new-type P/M superalloy disk iso-forgings. Numerical simulation is used to make exact analysis of the isothermal forging process for producing a P/M superalloy disk, characterized by very sensitive metal flow and com-

Corresponding author. Tel.: +86-29-83070387. E-mail address: fuguolx@nwpu.edu.cn

Foundation item: Aeronautical Science Foundation of China (03H53048)

1000-9361/$ - see front matter © 2009 Elsevier Ltd. All rights reserved. doi: 10.1016/S1000-9361(08)60072-2

plicated deformation. Besides, the isothermal forging of the P/M superalloy disk is simulated by using software MSC/Superform in order to obtain the equivalent strain distribution in the final shape and the deformation degree distribution in preforming and final forming.

2. Theoretical Foundation of Electric-field Simulation

2.1. Basic principles

According to the basic principle of electrical analogy, if the physical object under study, such as a force field, magnetic field, or thermal flow field, has the same field equation in form as the electrostatic field equation, it can be simulated in very much the same way as the electrostatic field.

Comparability and similarity, two basic features of natural existence, are the objective foundations of the simulation theory and the principle of simulation technology. According to the principle of comparability, the velocity field for inner particles of material in the metal plastic-deformation process has the same field equation as the electrostatic field after suitable transformation, and thus, the flow characters can be described by a comparable electrostatic field[1].

The movement and strain of a continuous medium can be depicted by the relationship between the initial coordinates of a particle at the time t, xi, and the in-

stantaneous coordinates of the particle, xi + Aui (i = 1, 2, 3) at t + At. Thus, the displacement of the particle is Au after At, and the velocity field of the deformation zone can be expressed by the Euler variable:

... Au.

v(x,t) = !im(—)

At^0 At

which displays the instantaneous velocity of all particles in the deformation zone. When the velocity field, v is expressed by the stream function y/, or the potential function (¡), the velocity components (v1, v2, and v3 in three dimensions) turn to be an unknown velocity potential function ^,or velocity stream function yz.

Given that the material volume remains constant and the velocity field is non-spinning, the following equations hold true. div v = 0 rot v = 0,

then div grad ^ = V • V^ = 0

a V d V d V

which indicates that the velocity field potential function <fi(x1, x2, x3) has the form of a Laplace equation.

When ^ is a constant c1, the equipotential lines of a scalar field are obtained. Eq.(3) can be derived in a similar manner:

V V = —^ + —^ + —= 0

dx1 dx2

which indicates that the velocity field stream function y/(x1, x2, x3), also has the form of a Laplace equation. When ^ is a constant c2, the equipotential lines of the scalar field ^are obtained, and it can be proven that yz are the stream lines of the velocity field[1].

In conclusion, if the space flow field is non-spinning, the velocity potential exists and the velocity field can be calculated[2]. Furthermore, if the velocity field is not divergent, it can also be derived from the stream function. It is important for the space stream field that the gradient field dot product of potential function, and the stream function should be zero:

grad grad + + = 0 (4)

ox1 ox1 ox2 ox2 ox3 ox3

which indicates that (¡> and y/ are conjugate functions, the equipotential lines (p( x1, x2, x3) = c1 and the stream

lines y/(x1, x2, x3) = c2 are two groups of mutually

orthogonal curves.

It is known from basic electrostatic fields that the potential value (p at a point in the electrostatic field satisfies the Poisson's equation:

+ d2y + d2p

dx2 dx2 dx2

= -p/s0

where p is the charge density, and s0 is the dielectric constant.

If there is no charge, the governing equation is represented by a Laplace equation:

8x1 dx2

According to the above analysis, the minimum deformation power path between the undeformed and the deformed shapes can be described by the equipotential lines generated between two conductors at different voltages. There is an infinite number of equipotential lines which are not overlapped. The fields have distinct features in shape between the two conductors[3], and the metal plastic-flow law obeys the minimum deformation power theory. Based on the comparability of the field equations and the minimum energy theory, the preform during the deformation can be described by the equipotential lines in an electrostatic field.

In this study, two isotropic dielectric electrodes are produced, whose shapes are identical with the initial blank and the final deformed component, respectively. Then, different voltages are applied to them. The potential field distribution of the above-cited physical model has the same controlling equation as the metal flow law during the bulging process. According to the electric field simulation theory, the equipotential line distribution in the electrostatic field can be used to simulate the different deformation stages of the blank.

The initial blank with dimension of 0130 mm x 100 mm and the ratio of height to diameter being 0.796 is amplified twice. Equipotential field between the initial blank and the final forging is obtained by endowing the final form boundary with 0 V and the initial blank with 1 V. Six equipotential lines close to the final forming shape (from 1 to 6) are selected to be the preforming die contours separately (see Fig.1).

Symmetry axis Fig.1 Equipotential lines close to final shape.

2.2. FEM model

Rigid-plastic finite element method (FEM) analysis

of the thermal dynamic couple is applied to simulate isothermal forging of the disk made from FGH96 P/M superalloy with the processing techniques[4-7]. Fig.2 shows the initial model of FEM simulation which includes 70 elements and 88 nodes at the initial stage of the deformation. In the simulation, the upper die moves to the fixed lower die at a speed of 1 mm/s during upsetting and 0.1 mm/s during preforming and final forming. Remeshing is taken into account in the simulation process because of over-deformation of the material. In the beginning, the die temperature is set to be 1 373 K in order to take account of the possible temperature drop of the blank in the transport process from heating furnace to forging die. The die temperature remains unchanged in the deforming process. The friction coefficient between the blank and the die surface is set to be 0.3. The constitutive relationship of the P/M superalloy adopted is shown as follows[8]:

cr = 113.25 arsinhp/A)1 n exp(Q/nRT)] (7)

where A is a constant, n a sensitivity exponent of strain rate, Q the deformation activation energy (J/mol), R the universal gas constant (R = 8.314 J-mol^K1), a the true stress (MPa), s the strain rate (s1), and T the deformation temperature (K). Furthermore, n = 3.909 4,

A = exp (74.860 O^"0 140 7), mol.

Q = 8.504 8 x 10V0 142 4 J/

1—Upper die for upsetting; 1'—Lower die for upsetting;

2—Upper die for performing; 2'—Lower die for performing;

3—Upper die for final forming; 3'—Lower die for final forming;

4—Symmetry axis; 5— Initial mesh of blank

Fig.2 Geometric model of FEM simulation.

3. Controlling Criteria for Selecting the Best Preforming Die Contour

On the basis of the above-introduced preforming design method, i.e. the equipotential field method, some preforming die contours can be acquired. In order to select the best one of them, the controlling criteria must be established at first, for which, in addition to the forging quality and the die life, the following factors must be taken into account: (1) Equivalent strain of forging It is well known that the metal flow will be more reasonable and the deformation distribution of forging more uniform, if the difference between the maximum and the minimum equivalent strains becomes smaller. Reasonable metal flow and uniform deformation dis-

tribution improve forging quality and productivity. Therefore, the equivalent strain of forgings can be used as a criterion to judge the forging quality.

(2) Deformation degree of forging

With comprehensive consideration of the quality of forging and the die life, the difference of deformation degrees between preforming and final forming should be as small as possible in the production processes.

(3) Stress distribution in die cavity

Larger shear stresses and normal stresses on the carrying surfaces make stress concentration easier between the billet and the die profile when the billet fills into the die cavity. However, from the view of ameliorating working conditions and extending die life, shear stresses and normal stresses must be ensured to be smaller.

Among the above-listed controlling criteria, the first one should be first taken into account when designing preforming dies.

4. Analysis of Simulation Results

4.1. Equivalent strain of forging

For the ease of analyzing simulation results, the disk is divided into three parts: wheel disk, wheel rim, and hub (see Fig.3).

(Area I)

Fig.3 Integral profile of a new-type P/M superalloy disk.

Tables 1-3 show the equivalent strains in the three parts of the disk. In area I, among the six preforming contours, the differences between the maximum and the minimum equivalent strains of No.1, No.3, and No.6 preforming contours are relatively small. In area II, the differences of No.1, No.2, and No.3 preforming contours are relatively small. In area III, the difference of the No.3 preforming die contour is the smallest.

Table 1 Equivalent strains in area I

Number of

preforming die contour Max Min Difference

1 3.22 0.26 2.96

2 3.46 0.38 3.08

3 3.34 0.42 2.92

4 3.94 0.51 3.43

5 4.51 0.54 3.97

6 3.50 0.59 2.91

Table 2 Equivalent strains in area II

Number of preforming die contour Max Min Difference

1 3.90 1.37 2.53

2 4.24 2.25 1.99

3 4.19 2.23 1.96

4 4.73 2.26 2.47

5 4.39 2.21 2.18

6 4.81 2.23 2.58

Table 3 Equivalent strains in area III

Number of preforming die contour Max Min Difference

1 3.82 1.29 2.03

2 4.73 1.72 3.01

3 4.08 1.96 2.12

4 4.32 1.51 2.81

5 4.28 1.65 2.63

6 4.65 1.51 3.14

According to the first controlling criterion, the preforming contour No.3 is reasonably the best choice. Fig.4 shows the distribution isolines of the equivalent strain in superalloy disk simulated in the preforming die contour No.3.

Fig.4 Distribution isolines of the equivalent strain in super-

alloy disk simulated in the preforming die contour No.3.

4.2. Deformation degree

Generally speaking, the differences of deformation degree in preforming and final forming should be small in production processes, but excessively small or large ones will also bring ill influences to bear on the die life.

Fig.5 shows the variation of the deformation degree of blank in the preforming and final forming processes.

It can be found form the figure that the difference of deformation degrees in preforming and final forming in die contour No.3 is 2.4% with the degree during final forming being little less than in preforming, which renders the die contour No.3 more reasonable.

Fig.5 Comparison of deformation degrees of preforming and final forming.

4.3. Die stresses

(1) Normal stresses

Fig.6 shows the maximum normal stresses in preforming and the final forming die contours in the areas

50 -'-■-1-■-'-'-1-■-1-■-"

1 2 3 4 5 6

Number of the preforming die (c) Aera III

Fig.6 Maximum normal stresses in preforming and final forming die cavities in areas I, II, and III.

I, II, and III. It can be seen from the figures that the maximum normal stresses in the preforming and the final forming die contour No.3 in all three areas reach the smallest. Larger normal stresses on the carrying surfaces cause stress concentration easier between the billet and the die profile when the blank fills into them, which is to blame for accelerating die wear and shortening die life. As a result, normal stresses in die cavities must be decreased to improve work conditions and prolong die life. This makes the preforming die contour No.3 the most favorable.

(2) Shear stresses

Fig.7 shows maximum shear stresses in the preforming and the final forming die cavities in areas I, II, and III. It can be seen from the figures that the maximum shear stresses in the preforming and the final forming die cavities No.3 in all three areas reach the smallest. Larger shear stresses on the carrying surfaces cause stress concentration easier between the billet and the die profile when the blank fills into them, which is to blame for accelerating die wear and shortening die life. Consequently, shear stresses in die cavities must be decreased to improve work conditions and prolong die life. This allows the preforming die contour No.3 to be the most preferable.

0 -1-'-1-1-1-1-1-'-1-1-1

1 2 3 4 5 6

Number of the preforming die (a) Aera I

0 _i_i_i_i_i_i_i_i_i_i_i

1 2 3 4 5 6

Number of the preforming die (b) Aera II

1 2 3 4 5 6

Number of the preforming die (c) Aera III

Fig.7 Maximum shear stresses in preforming and final forming die cavities in areas I, II, and III.

5. Conclusions

This article demonstrates the comparability of the flow law of metal during plastic deformation and the equipotential line distribution in an electro-static field. The deformation stages of the P/M Superalloy disk, i.e., upsetting, performing, and final forming, can be simulated with the equipotential lines in an electrostatic field. On the basis of the equivalent strain distribution in the final shape of the superalloy disk, the deformation in the preforming and final forming stages and the stress distribution in the die cavities, it can be concluded that among six preforming die contour variants, No.3 will be the best choice.

References

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[3] Li C, Li F G, Zhang Y. Reverse educing method of bugle-blank based on equipotential field. Chinese Journal of Mechanical Engineering 2005; 41(11): 127-133. [in Chinese]

[4] Shen X H, An T, Yan J. Finite element analysis of preforming for 840 railway wheel. Journal of Iron Steel Research 2005; 17(1): 30-33. [in Chinese]

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Biographies:

Wang Xiaona Born in 1978, she is a Ph.D. candidate in School of Materials Science and Engineering at Northwestern Polytechnical University, Xi'an, China. Her major research fields are materials processing engineering, simulation and control of material deformation. E-mail: wangxiaona78@163.com

Li Fuguo Born in 1965, he is a professor and doctorial tutor in School of Materials Science and Engineering at Northwestern Polytechnical University, Xi'an, China. His major research fields are materials processing engineering, numerical simulation and date integration. E-mail: fuguolx@nwpu.edu.cn