Predicting Material Properties of Flow Formed Work-piece Based on a Finite Deformation Method
Predicting Material Properties of Flow Formed Work-piece Based on
a Finite Deformation Method
Gyeong-Bok Lee1, Cheol-Soo Lee1*, Eun-Young Heo2, Dong-Won Kim3
1Mechanical Engineering
Sogang University
Seoul, Mapo- gu, 121-742, Korea
2Sogan Institute of Advanced
Technology, Sogang University
Seoul, South Korea
3Department of Industrial and
Information Systems Engineering,
Chonbuk National University Jeonju,
561-756 South Korea
ABSTRACT
Flow Forming Process is a manufacturing process to deform rotating axis symmetric work-piece with rotating
rollers. It is difficult to predict the behaviour of work-piece during the process because the shear and compression
deformation occurs simultaneously in the shape of helix along the axis. This study presents a model which can
predict the deformed geometric shape and material properties using geometric shape change and material
properties of work-piece. This model is based on finite deformation theory, and assumes that the structure of the
material is isotropic lattice structure. To simulate the flow forming process, firstly the axial displacement is
calculated using the stress-strain relation. And then the radial displacement is calculated using the volume
constancy theory. The material properties of flow formed material are easily calculated with the deformed
geometry. The presented model is verified by experimenting actually, and a tensile test demonstrates the predicted
material properties.
1. INTRODUCTION
Flow Forming Process is the application of plastic deformation. In a general procedure, axis symmetric work-piece
is installed on a mandrel, and as mandrel rotates, rollers pass the work-piece on the desired tool paths. Flow forming
process is similar to spinning process but with one main difference. In spinning process, the work-piece is a plate or
tube in general, and it would be shaped to a certain form without changing its thickness. Whereas in flow forming
processes, the work-piece is more likely to be a preformed shape which can change its thickness and form freely in the
process.
There are many studies concerning spinning process and flow forming process. It is hard to conduct the
experiments of several cases of 3D Model to this process: First of all, the angular velocity of a work-piece is very high,
so it is likely that the calculations will not be precise. Moreover, the process is complex, because Compression stress
and Tensile stress occur at same time, while the roller contacts the work-piece on a small area making ongoing
deformations at minimal intervals of time. For this reason, an enormous number of iterations are required for each
step, resulting to extended calculation time for each case. C. C. Wong et al. suggested the 3D FEM model for flow
forming process of cylindrical work-pieces at 1 roller [1]. F. A. Hua et al. suggested FEM Model of tube spinning at 3
rollers process [2]. M. Zhan et al. analyzed spinning process with cone shaped work-pieces with 3D FEA [3]. L. Wang
and H. Long investigated the spinning process which deformed cone shaped work-pieces with multi-pass, by
experiment and FEM [4]. O. Music and J. M. Allwood studied spinning process of plate deformed by rollers which
move along the curved tool path [5]. M. Hayama et al. studied the factors for deciding the path schedule of roller in
spinning process [6]. Y. Jianguo and M. Makoto presented an experimental study on spinning for tube diameter
reduction process [7]. S. H. Yeom et al. investigated the relationship of forming depth and lead angle with forming
process [8].
J. H. Kim et al. suggested the theoretical model of cone spinning [9].Upper bound method is a trend of theoretical
approaches with plastic deformation process. Upper bound method can explain the characteristics of the plastic
deformation processes relatively accurately, but that is hard to use because of the many differential equation. R.
Ebrahimi et al. applied the upper bound method to tube extrusion process [10].J. W. Park et al. analyzed the tube
spinning process with upper bound method [11].
* Corresponding author: Tel.: (+82) 70- 8668-9900; E-mail: cscam@sogang.ac.kr
Flexible Automation and Intelligent Manufacturing, FAIM2014
In this study, the presented prediction model is based on finite deformation theory. The model is designed to obtain
the material properties from geometric information and condition of the process directly. The model in this case is
considered as a static problem rather than a transient problem. In order to verify the model, experiments and finite
element analysis for the flow forming process of cup shaped work-pieces were conducted. First, FEA was conducted
and then it was compared with the results from the experiments. After FEA was validated, several cases of FEA were
conducted and compared with the model.
2. FINITE DEFORMATION OF FLOW FORMING PROCESS
Purpose of the prediction model is to get the material property from controllable variables. The variables are
geometric information of roller and work-pieces such as deformation thickness, federate, etc. Material properties
which are the results of model are elongation, yield stress, residual stress, etc.
To derive the material properties of the work-pieces, stress – strain relationship and volume constancy is
considered. The problem is simplified by ignoring the middle step of flow forming process. The boundary conditions
which occurred from the rotation of work-pieces and mandrel were ignored for the reason that the effect of the rotation
is too small to consider: The roller rotates in the opposite direction of the mandrel, as a result, when friction of
tangential direction occurs because of the contact between them, it cancels out. So the problem can be presumed to be
a 2-dimensional deformation problem.
2.1. THEORETICAL STUDY
Figure 1. Diagram of lattice structure of the work-pieces.
Work-pieces is assumed to be a lattice structure. Fig 1 shows the diagram of lattice structure. Based on finite
deformation theory, if the middle steps can be ignored, the final coordinates can be expressed by initial coordinates:
= +
(0 ≤ , 0 ≤ ) (1)
= + + (2)
, , , , are functions of input variables. and are the matrices of initial coordinates of assumed lattice
structure. and are matrices of final coordinates predicted. In this case, only thickness does change and the shape
doesn’t change, so
= 0 (3)
can be expressed by ratio of change of thickness to the deformation.
=
(4)
is the restoration of thickness by elastic force. It can be expressed as
= (/ )
(5)
Predicting Material Properties of Flow Formed Work-piece Based on a Finite Deformation Method
!"# is the initial thickness of an element of the lattice. dt& is the final thickness of an element and can be calculated
using volume constancy.
Figure 2. Section of cylindrical workpice.
Fig2 shows the section of work-pieces of radial direction. From the constraints, simultaneous equations can be
derived. The first equation is
'(() + "′) − ()
,!-
.
/# 0 = '(() + ") − ()
,!-#
.1
/# 0 (6)
!-# is the initial length of an element of the lattice. !- is the final length of an element in the end of the
process.2, 2is initial and final rotational angle of the element, relatively.2 is same as 2.
"′is the height of the mandrel from the i-th element in the axial direction. () is the radius of mandrel and " is the
initial height of each element.
'(() + "′) − (() + "′),!-
.
/# 0 = '(() + 2") − (() + "),!-#
.1
/# 0 (7)
Equation of i-th element from mandrel is expressed as:
'(() + "′) − (() + "′4),!-
.
/# 0 = '(() + 5") − (() + (5 − 1)"),!-#
.1
/# 0 (8)
Equation of the last elements which are furthest from mandrel is expressed as:
'(() + "′) − ()
,!-
.
/# 0 = '(() + 5") − ()
,!-#
.1
/# 0 (9)
"′ can be calculated as:
"′ =
"# +
(10)
Solving the equations (7) ~ (9), change of height of each element "′, "′,⋯, "′4, !- can be calculated.
is the ratio of change of length of work-pieces.
= 8
8
(11)
To calculate, sheer stress-strain relationship is used.
9 = :; = < = =>? (12)
> = @AB
1C
(13)
= "DE> = "DE @A
B
1C
(14)
Flexible Automation and Intelligent Manufacturing, FAIM2014
In the way of calculating the final formation of the structure, yield stress of the workpiece during the process and
elongation can be calculated.
F = FG "DEℎ @I
B (15)
Stress-strain equation which is suggested by Prager is used [12]. FG, J
can be calculated from tensile tests.
Figure 3. Stress-strain relationship of material.
Fig 3 shows the stress- strain curve. The curve is shortened with strain direction and expanded with stress direction
when reduction ratio is increased. To calculate the stress-strain curve of the processed workpiece, the energy equation
is used:
K F# −
K F
= L
=
K F#
I1
#
+ LM + L (16)
σ# is the stress-strain curve of the initial workpiece, ε is the strain which caused by the flow forming process. σ
is the stress-strain curve of the final workpiece. L, LM, L are the total energy, redundant work and friction energy,
respectively. Friction energy can be ignored in this case because the friction of tangential direction is canceled by the
rotation of the roller, and friction of axial direction can be ignored because the axial direction speed of roller is very
smaller than the rotating speed. Energy from redundant work is ignored to simplify the problem. So the equation can
be arranged as:
K F#
=
K F#
I1
#
+ K F (17)
2.2.3D FINITE ELEMENT ANALYSIS
For the model established in this paper, FEM and experiment is conducted. By comparing the result of FEA with
the experiment, conditions which cannot be controlled in experiments can be simplified to small amounts of
constraints in FEA. And if the results of FEA are verified, model can be verified with FEA with the factors which
cannot be measured in experiments.
Table 1. Condition for FEA.
Character of Part Roller Rigid body
Mandrel Rigid body
Workpiece A1050
Feedrate 300 mm/min
Predicting Material Properties of Flow Formed Work-piece Based on a Finite Deformation Method
Figure 4. Axisymmetric mesh for FEM.
Fig 4 shows the 3D finite element method model and Table1 presents the boundary condition for the flow forming
process. ABAQUS/Explicit is used for the case. Mandrel and work-pieces rotate at 300 rpm, while the roller doesn’t
rotate. Friction effects exist only at axial direction. For the example Coulomb friction coefficient is used.
2.3. EXPERIMENT CONDITIONS
Experiments were conducted to verify the FEM Model. Fig 5 shows the experiment equipment of flow forming
process. Table 2 shows the specifications of the equipment.
Figure 5. Flow fomring machine for experiments.
Table 2. Condition of the experiments.
Feedrate 300 mm/min
Spindle speed 300 rpm
Temperature 17P
The work-pieces has a cup shape and a roller moves along a linear path to deform the thickness of the work-pieces.
A second roller is the idle roller, which helps the axis of the mandrel to maintain verticality.
To compare the result of the experiment with FEA and the established model in this paper, two different small
holes with different depth are machined on the work-pieces. The holes are filled with colored powder and sealed. By
cutting the work-pieces to axial direction, the aspect of the material’s motion, when the work-pieces is deformed can
be observed.
3. RESULT AND DISCUSSION
Precision of the shape is the most important factor of the flow forming process. In order to verify the accuracy of
the model, experiment and finite element analysis have to be conducted. First, two or three experiments are conducted
Flexible Automation and Intelligent Manufacturing, FAIM2014
and compared to the model and the result of analysis. Then, variable case of FEA is conducted to verify the model
established in this paper.
3.1. VERIFICATION WITH FEA AND EXPERIMENT
Figure 6. Result of the Experiment.
Figure 6 a) shows the final result of the experiment, and Figure 6 b) shows the flow forming process. The height of
the work-pieces is 100mm, and the thickness is 5mm. The thickness was reduced to 3mm, and the height of the
work-pieces was extended to 155.36mm.
Figure 7. a) UTS comparison between the model, experiments b) Yield stress comparison between the model, experiments c) Final
shape comparison between the model, experiments and FEA.
Figure 7 a) shows the aspect of the yield stress and figure 7 b) shows ultimate tensile stress according to reduction
ratio. 4 cases of the experiment is conducted. Each case is predicted by the model. The maximum error of UTS
between experiment and model is 13.08%, and the average value of the error of UTS is 4.29%. The maximum error of
yield stress between experiment and model is 7.85%, and the average value of the error of yield stress is 5.03%. Yield
stress and Ultimate tensile stress increase nonlinearly. Figure 7 c) shows the comparison of thickness at the case 4
which the thickness reduction amount is 1.8mm. The results of FEA are compared with model and experiment. The
maximum error between experiment and FEA is 9.41%, and the mean value of the error is 0.20%.
Predicting Material Properties of Flow Formed Work-piece Based on a Finite Deformation Method
Figure 8. Result of the FEA and Experiments.
Figure 8 shows the result of FEA and the Experiment. Slip in tangential direction can be observed. This is caused
by the flow stress and the friction that doesn’t get canceled by the rotation of the work-pieces. The result of the flow
forming process is presented. The edge of the right side of the work-pieces is moved upwards because of the elastic
force of the free end.
3.2.
VERIFICATION WITH FEA FOR MORE CASES
To verify the model and observe the aspect of the flow forming process in more cases, parametric study with FEA
is conducted. The factors are thickness of work-pieces and thickness reduction ratio. The condition of FEA is
presented at Table 3.
Table 3. Factors and condition of FEA.
Factor
Thickness of work-pieces[mm] Thickness reduction[%]
No. 1 5 40
No.2 10 10
No. 3 10 20
Figure 9. Error of the Process in FEA and model.
Error of the flow forming processes is presented in Figure 9. The mean value of the error is 2.77%, 3.49%, 4.03%,
relatively. The maximum error between the model and FEA is almost same for all cases. The deformation thickness
Flexible Automation and Intelligent Manufacturing, FAIM2014
becomes greater and excessive deformation occurs so the error factors are more evident. Therefore, the mean value of
the error is proportional to the initial thickness of the work-piece, as well as the thickness decrement of.
4.CONCLUSION
A model is used to analyze the flow forming process and the following results are obtained:
1. Prediction model for flow forming process is established, and verified by Experiments and FEA.
2. Through the case study, reliability of the model is approved.
3. Error of the model increases when the initial thickness of the work-piece increases, and the thickness decrement
of the work-piece becomes higher.
4. Slips which cannot be predicted in model are observed during the process.
ACKNOWLEDGEMENTS
This research was supported by the Converging Research Center Program through the Korean Ministry of
Education, Science and Technology (Grant Number: 2013K001051).
REFERENCES
[1] C. C. Wong, T. A. Dean and J. Lin: "Incremental forming of solid cylindrical components using flow forming principles",
Journal of Materials Processing Technology, pp.60-66, 2004.
[2] F. A. Hua, Y. S. Yang, Y. N. Zhang, M. H. Guo, D. Y. Guo, W. H. Tong
and
Z. Q. Hu: “Three-dimensional finite element
analysis of tube spinning”, Journal of Material Processing Technology, pp.68-74, 2005.
[3] M. Zhan, H. Yang, J. H. Zhang, Y. L. Xu and F. Ma, “3D FEM analysis of influence of roller feed rate on forming force and
quality of cone spinning”, Journal of Materials Processing Technology, pp.486-491, 2007.
[4] L. Wang and H. Long: "Investigation of material deformation in multi-pass conventional metal spinning", Materials and
Design, Vol.32, pp.2891-2899, 2011.
[5] O. Music and J. M. Allwood: “Flexible asymmetric spinning”, CIRP Annals – Manufacturing Technology, Vol.60,
pp.319-322, 2011.
[6] M. Hayama, H. Kudo and T. Shinokura: "Study of the pass schedule in conventional simple spinning", The Bulletin of JSME,
pp.1358-1365, 1969.
[7] Y. Jianguo and M. Makoto: “An experimental study in paraxial spinning of one tube end”, Journal of Materials Processing
Technology, Vol.128, pp.324-329, 2002.
[8] S. H. Yeom, K. O. Nam, H. J. Park and S. I. Hong: "The effects of forming depth and lead angle on forming force of shear
spinning", KSPE, Vol.11, No.2, pp.27-33, 2007.
[9] J. H. Kim, J. H. Park and C. Kim, “A study on the mechanics of sheer spinning of cones”, Journal of Material Science and
Technology, Vol.20, No.6, pp.806-818, 2006.
[10] R. Ebrahimi, M. Reihanian, M. Kanaani and M. M. Moshksar: “An upper-bound analysis of the tube extrusion process”,
Journals of Materials Processing Technology, Vol.199, pp.214-220, 2008.
[11] J. W. Park, Y. H. Kim and Y. B. Bae: "Analysis of tube-spinning processes by the upper-bound stream-function method",
Journal of Materials Processing Technology, Vol.66, pp.195-203, 1997.
[12] Prager. W. Proc: 5th Int Congr Appl Mech, Cambridge, 1938, S.234.