A 3D rigid–viscoplastic FEM simulation of the isothermal precisionforging of a blade with a damper platform

He Yang*, Mei Zhan, Yuli LiuCollege of Materials Science and Engineering, Northwestern Polytechnical University, P.O. Box 542, Xi’an 710072, PR China

Received 18 January 2001

Abstract

A blade with a damper platform, with excellent anti-vibration characteristics and high efficiency, has become one of the most important new

types of blade being developed in the aeronautical engine. However, the blade is complicated in shape, and the material used for its

manufacture is difficult to deform. Therefore, it is important to undertake research on the blade-oriented precision forging process using three-

dimensional finite element method (3D FEM) method numerical simulation for the practice and the development of the process. However, up

to now, literature on such research has been scant. In this paper, based on the rigid–viscoplastic principle, three-dimensional finite element

simulation is reported for the isothermal precision forging of the blade using the penalty function, and eight-node hexahedral isoparameteric

elements for discretizing the deforming workpiece and triangular elements for discretizing the die cavity. The method of contracting from the

boundary to the interior, proposed by the authors, is used for remeshing a distorted mesh system, and the method of modifying the position of

nodes touching the die according to its original normal, also proposed by the authors, is used to avoid the ‘‘dead lock’’ problem due to the

normal uncontinuity of scatted die meshes, to enable the simulation to be successful. Friction is considered for the die–workpiece interface

boundary condition, and an arc is considered for the tenon–body joint, and a damper platform–body joint on the blade die cavity, respectively,

which make it possible for the simulation to approach the practical forging process of a blade with a damper platform. 3D FEM simulation

results have been obtained for the initial and deformed configurations, the deformed meshes of typical cross-sections, the distribution of

effective strain at the final stage, load–displacement curves, in this way the deformation law of the forging of a blade with a damper platform

being revealed. The achievements of this research serve as a significant guide to the optimization of design for the relevant process and dies.

The method used is also of general significance to the forging processes of other type of blades and other complicated massive deformation

processes. # 2002 Elsevier Science B.V. All rights reserved.

Keywords: Blade with a damper platform; 3D FEM; Rigid–viscoplastic; Isothermal forging; Precision forming

1. Introduction

Playing a role in energy transformation, an aeronautical

blade is one of the most important mechanical components

in the aviation industry. A blade manufactured by forging is

required to achieve high geometrical precision and to have

good mechanical properties in order to be able to be

employed under severe working conditions. A blade with

a damper platform, with excellent anti-vibration character-

istics and high efficiency, has become one of the most

important new types of blade being developed for the

aeronautical engine. However, because the blade with a

damper platform has a complicated shape, and the material

used is difficult to deform, its forging is a complex process of

three-dimensional nature. Therefore, how to design the

process and the dies of a blade with a damper platform

has become one of the important subjects in this field.

Until now, much of the work done has been based on the

assumption that blade forging is a two-dimensional plane-

strain problem [1–8]. Argyris et al. [9] and Yang et al. [10]

simulated the deformation of turbine blade forging by three-

dimensional finite element method (3D FEM), however in

the former, the absence of friction is assumed for the work-

piece–die interface boundary condition, which covers up the

phenomenon that friction leads to a distorted mesh. Also, in

the above two 3D simulations, a right angle is considered for

the tenon–body joint, and for the damper platform–body

joint on the die cavity, so that the two simulations are far

from the practical situation in hot forging of the blade. Yang

et al. [11] simulated the isothermal forging of a compressor

blade, but there is no damper platform on the body of this

compressor blade, so its shape is simpler than that of the

present blade with a damper platform.

Journal of Materials Processing Technology 122 (2002) 45–50

* Corresponding author. Tel.: þ86-29-849-5632; fax: þ86-29-849-1000.

E-mail address: yanghe@nwpu.edu.cn (H. Yang).

0924-0136/02/$ – see front matter # 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 9 2 4 - 0 1 3 6 ( 0 2 ) 0 0 0 3 2 - 8

In order to obtain more realistic deformation and more

precise information to help to improve the design precision of

the process and the die, it is necessary to simulate the blade

forging process by using a 3D FEM. However, up to now,

there has been scant literature reported on the 3D FEM

simulation of the forging of a blade with a damper platform.

In this paper, the isothermal forging of a blade with the

damper platform has been simulated by the 3D rigid–visco-

plastic FEM. In the present simulation, friction is considered

for the die–workpiece interface boundary condition, and an

arc is considered for the tenon–body joint, and the damper

platform–body joint on the blade die cavity, respectively,

which make it possible for the simulation to approach the

practical forging process of a blade with a damper platform.

The method of contracting from the boundary to the inside,

proposed by the present authors [12], is used for remeshing a

distorted mesh system, and the method of modifying the

position of nodes touching the die according to its original

normal, also proposed by the authors [13], is used to avoid the

‘‘dead lock’’ problem due to the normal uncontinuity of

scatted die meshes, to enable the simulation to be successful.

2. Rigid–viscoplastic FEM for simulation

2.1. Variational principle

The rigid–viscoplastic FEM is based on the rigid–visco-

plastic variational principle. The variational principle can be

expressed as: among admissible velocities that satisfy the

conditions of compatibility and incompressibility, as well as

the velocity boundary condition, the actual solution gives the

following functional a stationary value:

p ¼Z

V

Eð_eijÞ dV �Z

SF

Fiui dS (1)

where Eð_eijÞ is a work function:

Eð_eijÞ ¼Z _eij

0

s0ij d_eij ¼Z _e

0

s d_e (2)

when the incompressibility condition is handled by using the

penalty function method, the incompressibility constraint on

the admissible velocities in Eq. (1) is removed by introdu-

cing a penalty constant a, and then the first-order variational

of the functional is

dp ¼Z

V

sd_e dV þ aZ

V

_eVd_eV dV �Z

SF

Fidui dS (3)

where s, _e, _eV, Fi, and a are the effective stress, effective

strain rate, volume strain rate, surface traction, and penalty

constant.

2.2. Friction model

During the forging process of the blade with a damper

platform, the deformation and friction between the die and

the workpiece are very complicated due to the complicated

shape of the blade, and as friction has an important influence

on the deformation, it is necessary to introduce a friction

model into the FEM simulation. One feature of the com-

plicated deformation is in that there exists a point (or a

region) along the die–workpiece interface where the velocity

of the deforming material relative to the die becomes zero,

the location of this point (or region) depending on the

magnitude of the frictional stress itself. In order to deal

with this situation, a velocity-dependent frictional stress is

used as an approximation to the condition of constant

frictional stress. Along the interface the velocity boundary

condition is given in the direction normal to the interface by

the die velocity, and the friction boundary condition may be

expressed by

f ¼ � 2

pmk arctg

us

u0

� �t (4)

where m is the friction factor, k the yield shear stress, us the

velocity vector of the workpiece relative to the die, u0 is

a very small positive number as compared to us, and t the

unit vector in the direction of us.

2.3. Remeshing of the distorted mesh system

Because the practical forging process of the blade with a

damper platform is of a large deformation and of unsteady

three-dimensional nature, and the process simulation of

the blade forging is usually carried out by using the incre-

ment method, this causing the FEM initial mesh system to

distort to such a degree that a new FEM mesh system is

necessary for a further simulation. Therefore, the inter-

mediate remeshing stages play an important role in the 3D

FEM simulation of the forging process of the blade with a

damper platform.

In this paper, the remeshing method of contracting from

the boundary to the inside, proposed by the authors [12], has

been used. A three-dimensional mesh system with eight-

node hexahedral element can be obtained by using this

remeshing method. Its procedures are as follows:

(1) Project all the boundary or sensitive surfaces into

different surfaces, respectively.

(2) Divide the distorted deforming body into given cross-

sections along their length direction according to the

use.

(3) Generate a quadrilateral element mesh on every cross-

section, including all the boundary and sensitive

surfaces after projection.

(4) Generate an eight-node hexahedral element mesh by

linking the corresponding nodes of corresponding

quadrilateral elements on the adjacent cross-section.

(5) Project all the boundary and sensitive surfaces after

remeshing back into their original boundary and

sensitive surfaces, respectively.

46 H. Yang et al. / Journal of Materials Processing Technology 122 (2002) 45–50

2.4. Position modifying of the node touching on the die

Since the forging process of the blade with a damper

platform is an unsteady and large deformation process, during

the forging process the workpiece geometry and the boundary

condition progressively change, i.e., some free nodes may be

in contact with the die patches or some nodes on the die

patches may be off the die. These constitute a dynamic

boundary condition. Three aspects in the handling of the

dynamic boundary condition are included: the determination

of a free node touching the die patch; the determination of a

node touching the die then separating from the die patch; and

the modification of the position of a node touching the die.

For the former two, there are some effective handling

methods, but for the third, the present modification methods

have some disadvantages, such as ‘‘dead lock’’, low preci-

sion, etc., so for nodes which have touched onto the die

patches and are without the trend to separate from the die, in

order to avoid these nodes invading the die or separating

from the die, some special handling is needed. In this paper,

because the die cavity is described by the finite element

mesh method, the die cavity, which is composed of scatted

meshes, is not smooth, and the normals of the die patches are

not continuous. Thus in order to avoid the ‘‘dead lock’’ due

to scattered die meshes and to improve modification preci-

sion, a new method of modifying the position of the node

touching the die, proposed by the authors [13], is used in this

paper, i.e., the position of a node touching the die is modified

according to its original normal. The procedure is as follows:

(1) Identify the state of the node touching the die.

(2) Determine the die mesh element on which the node has

touched.

(3) Determine the original normal of the node touching the

die.

(4) Modify the position of the node touching the die

according to its original normal.

2.5. 3D rigid–viscoplastic FEM simulation system

A 3D rigid–viscoplastic FEM simulation system consid-

ering friction and arc transition has been developed in the

authors laboratory [11] in terms of the structured program-

ming method. The system can be used for 3D FEM simula-

tion of not only precision forging processes of a blade with a

damper platform, but also for other complicated massive

deformation processes.

3. Results and discussion

The computation was carried out with one intermediate

remeshing stage and 1006 increment steps. At every incre-

ment step, the method of modifying the node touching the

die according to its original normal has been performed.

The material of the workpiece used in the analysis is

Ti–6Al–4V, its constitutive equation at 950 8C being as

follows [14]:

s ¼ 86:1ð_eÞ0:24 ðMPaÞ (5)

The upper die velocity is taken to be 20 mm/s, with the lower

die remaining stationary. The friction factor m is taken to be

0.20. The equivalent angle of the dies is chosen as 08450.The die cavity is discretized by the finite element mesh

method. The die meshes used in the simulation are shown in

Fig. 1, in which 2005 triangular elements and 1077 nodes are

included in the upper die mesh; and 2103 triangular elements

and 1129 nodes are included in the lower die mesh. In order

to easily inspect the damper platform on the lower die, a

rotation of 1808 around the z-axis has been carried out for the

lower die mesh.

The 3D FEM simulation results obtained are shown in

Fig. 2 for the deformed configuration at various deformation

stages of the blade with a damper platform.

Fig. 2(a) shows the initial finite element mesh system of a

billet with 1728 eight-node hexahedral isoparameteric ele-

ments and 2231 nodes totally. The billet satisfies a condition

under which the billet cannot only fill up the whole die

cavity, but can also achieve flashless forging. Fig. 2(d) shows

the deformed mesh at a top die travel of 7.0 mm. Because

of the friction and the large deformation difference between

the tenon and the body, and between the damper platform

and the body, the deformation on these arc transition zones

is very severe. Therefore, mesh distortion usually occurs in

Fig. 1. Showing: (a) the upper, and (b) the lower die meshes.

H. Yang et al. / Journal of Materials Processing Technology 122 (2002) 45–50 47

these zones, as shown in Fig. 2(d). At this stage, the

deformed mesh system just before arriving at the mesh

distortion is obtained. Remeshing was carried out by using

the remeshing method of contracting from the boundary to

the interior in order to obtain the new mesh system as shown

in Fig. 2(e) with a total of 3352 elements and 4331 nodes.

In order to easily examine and analyze the forging defor-

mation process of the blade with a damper platform, four

typical cross-sections are selected, as shown in Fig. 2(a). The

3D FEM simulation results obtained are shown in Figs. 3–6

for the deformed meshes of four typical cross-sections at the

top die travel of dh ¼ 0:0, 4.0 and 8.0 mm. Fig. 7 shows the

Fig. 2. The deformed configurations at various stages, where dh is: (a) 0; (b) 2.0; (c) 4.0; (d) 7.0 mm (before remeshing); (e) 7.0 mm (after remeshing); (f)

8.0 mm.

Fig. 3. Deformed meshes of section A–A, where dh is: (a) 0; (b) 4.0; (c) 8.0 mm.

Fig. 4. Deformed meshes of section B–B, where dh is: (a) 0; (b) 4.0; (c) 8.0 mm.

Fig. 5. Deformed meshes of section C–C, where dh is: (a) 0; (b) 4.0; (c) 8.0 mm.

48 H. Yang et al. / Journal of Materials Processing Technology 122 (2002) 45–50

distribution of effective strain of the four typical cross-

sections at the final stage.

Form Figs. 3–7, it can be seen that: (1) the deformation of

the tenon and the damper platform is smaller than that of the

blade body; (2) the deformation of the center is smaller than

that of two laterals on the tenon and the damper platform; (3)

the deformation of the center is larger than that of two

laterals on the blade body, which is entirely different from

that of the tenon and the damper platform. Because of this

big difference between them, deformation coordination is

necessary, and the arc transition zones between the tenon

and the body, and between the damper platform and the

body, exactly play this coordinating role.

The 3D FEM simulation results obtained are shown in

Fig. 8 for the load–displacement curves. The forging load

versus the top die travel curve is shown in Fig. 8(a). A steep

increase in the forging load takes place at the final forging

stage, which is in accordance with the practical forging

process of the blade with the damper platform. Fig. 8(b) and

(c) show the horizontal forces acting on the upper and lower

dies versus the top die travel curves, respectively. These two

figures demonstrate that during the deformation, the hor-

izontal forces acting on the upper and lower dies are small,

and their wave range is also small, especially for the

horizontal force acting on the upper die. This shows that

the chosen equivalent angle of the dies is reasonable.

4. Conclusions

Using the 3D rigid–viscoplastic FEM and allowing for the

frictional boundary condition and arc transition, process

simulation has been carried out for the isothermal precision

forging of a blade with a damper platform The remeshing

scheme of contracting from the boundary to the interior and

the method of modifying the node touching the die accord-

ing to its original normal at every increment step have been

performed successfully. The 3D FEM simulation results

have been obtained for the deformed configuration at various

stages, for the deformed meshes of four typical cross-sec-

tions, for the contours of the effective strain on four typical

cross-section at final stage, and for the forging load curves.

The conclusion can be drawn as follows:

(1) The deformation law of forging the blade with a

damper platform is that: the deformation of the tenon

Fig. 6. Deformed meshes of section D–D, where dh is: (a) 0; (b) 4.0; (c) 8.0 mm.

Fig. 7. Distribution of effective strain of four typical cross-sections at the final stage: (a) section A–A; (b) section B–B; (c) section C–C; (d) section D–D.

Fig. 8. Load–displacement curves of precision forging of the blade with the damper platform showing: (a) the vertical load–displacement curve; (b) the

horizontal load acting on the upper die–displacement curve; (c) the horizontal load acting on the lower die–displacement curve.

H. Yang et al. / Journal of Materials Processing Technology 122 (2002) 45–50 49

and the damper platform is smaller than that of the

blade body; the deformation of the center is smaller

than that of two laterals on the tenon and the damper

platform; the deformation of the center is larger than

that of two laterals on the blade body, which is entirely

different from that for the tenon and the damper

platform.

(2) Because of the above large difference between the

deformation on the tenon, the damper platform and the

body of the blade, deformation coordination is

necessary, the arc transition zones between the tenon

and the body, and between the damper platform and the

body, exactly playing this coordination role.

(3) At the beginning, the vertical forging load increases

slowly, but at the final stage, the forging load increases

sharply, which is in agreement with the practical

forging process of a blade with a damper platform.

During the deformation, the horizontal forces acting on

the upper and lower dies is small, and their variational

range is also small. This shows that the chosen

equivalent angle of the dies is reasonable.

The achievements of this research can serve as a sig-

nificant guide to the optimization design for the relevant

process and dies. The method used is also of general

significance to the forging processes of other type blades

and other complicated massive deformation process.

Acknowledgements

The authors would like to express their appreciation for

the financial support of the Project Foundation of National

Key Laboratory of Precision Hot Processing of Metals of

China and the Aeronautical Science Foundation of China for

the present research work.

References

[1] J. Zhu, Z.W. Zhang, Simulation of blade forging, Forging Stamping

Technol. (1) (1991) 2–21 (in Chinese).

[2] L.B. Aksenov, N.R. Chitkata, W. Johnson, Pressure and deformation

in the plane strain pressing of circular section bar to form compressor

blades, Int. J. Mech. Sci. 17 (1975) 681–691.

[3] N. Akgerman, T. Altan, Application of CAD/CAM in forging

compressor and compressor blades, Eng. Power Trans. ASME 4

(1976) 290–296.

[4] A. Morita, S. Hattori, K. Tani, A. Takemura, Y. Ashida, Near net

shape forging of titanium alloy turbine blade, ISIJ Int. 31 (8) (1991)

827–833.

[5] N.L. Dung, O. Mahrenholtz, Progress in the analysis of unsteady

metal-forming processes using the FEM, Num. Meth. Ind. Form.

Proc., Pineridge, Swansea, 1982, pp. 187–196.

[6] B.S. Kang, N.S. Kim, S. Kobayashi, Computer aided preform design

in forging of an airfoil section blade, Int. J. Mach. Tools Manuf. 30

(1) (1990) 43–52.

[7] Z. Wang, K.M. Xue, Y.W. Liu, Backward UBET simulation of the

forging of a blade, J. Mater. Process. Technol. 65 (1997) 18–21.

[8] B. Soltsni, K. Mattiasson, A. Samuelsson, Implicit and dynamic

explicit solutions of blade forging using the finite element method, J.

Mater. Process. Technol. 45 (1994) 69–74.

[9] J.H. Argyris, J.S. Doltsinis, J. Luginsland, Three-dimension thermo-

mechanical analysis of metal forming processes, in: Proceedings of the

International Workshop Simulation of Metal Forming Processes by the

Finite Element Method (SIMOP-I), Stuttgart, 1985, pp. 125–160.

[10] D.Y. Yang, N.K. Lee, J.H. Yoon, A three-dimensional simulation of

isothermal compressor blade forging by the rigid–viscoplastic finite-

element method, J. Mater. Eng. Perform. 2 (1) (1993) 119–124.

[11] H. Yang, M. Zhan, Y.L. Liu, 3-D FEM simulation for precision

forging of blade, Mater. Sci. Technol. 7 (Suppl.) (1999) 87–90 (in

Chinese).

[12] M. Zhan, Y.L. Liu, H. Yang, Research on a new remeshing for the 3-

D FEM simulation of blade forging, J. Mater. Process. Technol. 94

(2–3) (1999) 231–234.

[13] M. Zhan, Y. Liu, H. Yang, A new method of modifying the position

of the node touching on the die for 3-D FEM simulation of metal

forming, J. Plast. Eng. 7 (2) (2000) 62–65 (in Chinese).

[14] T. Altan, et al., in: S. Lu, Modern Forging-equipment, Materials and

Processes, Defense Industrial Press, Beijing, 1982, p. 150 (in Chinese).

50 H. Yang et al. / Journal of Materials Processing Technology 122 (2002) 45–50