A recommendation of computation of normal streeses in single point incremental forming technology •

Single Point Incremental Forming (SPIF) has become popular for metal sheet forming technology in industry in many advanced countries. In the recent decade, there were lots of related studies that have concentrated on this new technology by Finite Element Method as well as by empirical practice. There have had very rare studies by pure analytical theory and almost all these researches were based on the formula of ISEKI. However, we consider that this formula does not reflect yet the mechanics of destruction of the sheet work piece as well as the behavior of the sheet in reality. The main aim of this paper is to examine ISEKI’s formula and to suggest a new analytical computation of three elements of stresses at any random point on the sheet work piece. The suggested formula is carefully verified by the results of Finite Element Method simulation.


A recommendation of computation of normal streeses in single point incremental forming technology
• Le Khanh Dien

ABSTRACT:
Single Point Incremental Forming (SPIF) has become popular for metal sheet forming technology in industry in many advanced countries.In the recent decade, there were lots of related studies that have concentrated on this new technology by Finite Element Method as well as by empirical practice.There have had very rare studies by pure analytical theory and almost all these researches were based on the formula of ISEKI.However, we consider that this formula does not reflect yet the mechanics of destruction of the sheet work piece as well as the behavior of the sheet in reality.

The main aim of this paper is to examine ISEKI's formula and to suggest a new analytical computation of three elements of stresses at any random point on the sheet work piece. The suggested formula is carefully verified by the results of Finite Element Method simulation.
Keywords: SPIF, Strains, Stresses, Computation, FEM Analysis.researchers have used as a basic theoretical analysis for their empirical researches.According to [3], [4] the basic normal stresses of Iseki's formula are displayed in (1):

AN OVERVIEW OF ISEKI'S FORMULA
Herein: y is the Yield stress of sheet workpiece, it is constant and depends on the characteristic of sheet material, rtool is radius of spherical tip of no cutting edges tool.
t is the thickness of the sheet workpiece.
In examination of Iseki's formula in (1) we could find out some important problems: The stresses at a random point in the sheet workpiece are always constant so they are independent to the position of the tool on the sheet workpiece that could not explain the reason of the worksheet.In the other hand, these stresses are equal the 3 principal stresses.
When calculating the partial differential of thickness t of 3 elements stresses of Iseki's formula in (1) we have result: That means that all 3 elements of stresses are inverse to the thickness t of the sheet workpiece.
So when the thickness of workpiece increases, all stresses as well as forming force and consuming power will decrease.This is the paradoxical result of the Iseki's formula to the empirical reality.
By the above reason, this paper attempts to recommend a new more accuracy calculating of stresses by pure analytics formula that is base on Ludwik 's formula [5] and then check the results to the one of a FEM software such as Abaqus and comparison with the empirical result.

A RECOMMENDED ANALYTICS FORMULA OF THE GENERATED STRESSES IN SPIF
Model of calculating stresses at a random point in contact area of tool and sheet workpiece are described in figure 1 The circumference of (H, r=MH) is also the initial length to p-direction: ). 2 The circumference of (H', r'=M'H') is also the deformed length to p-direction: Notice that r'=M'H' > r=MH so l'=2r' > l0=2r and l'/l0>1, so According to Ludwik's formula Calculating the differential of (3): σP is proportional to the thickness t.
-On -direction: The deformation increases from tip of tool to margin of the contact circle and M displaces to M'.Initial length: Ludwik's formula is applied for -direction: Calculating the differential of σt: Remained deformation on radial r-direction or In conclusion, referring to the result of ( 3), ( 4) and ( 5) we can see that in among 3 normal stresses at a random point: -p is proportional to the thickness t of the sheet workpiece, -t is inverse to the thickness t of the sheet workpiece, -r is independent to the thickness t of the sheet workpiece, ) cos .cos .

( ln
So the result of normal stresses is written in (6), these stresses have a complicated relation to the thickness t of the sheet, it could not be always inverse to the thickness of the sheet as in the result of Iseki's formula in (1).This result will be checked with Abaqus simulation

CHECKING FEM AND ABAQUS SIMULATION
In FEM simulation, we apply forming process      4) and (5).

CONCLUSIONS
In conclusion, the simulation in Abaqus proves that recommended formula in ( 6) is approval and more convincing then the Iseki's formula in (1). Figure 9 shows that the Iseki's formula is not true and could not explicable for the Đề xuất một phương pháp tính ứng suất pháp trong công nghệ SPIF

Figure 1 .
Figure 1.Model of calculating normal stresses in SPIF

Figure 2 .
Figure 2. Absolute deformation of sheet workpiece in 3 perpendicular directions p-plane: the plane that is perpendicular to OZ axis and parallel to OXY plane and passes After deformed, initial circle (H, r=MH) becomes (H', r'=M'H'): DEVELOPMENT, Vol 17, No.K2-2014 Trang 24


normal n-direction to the thickness of the sheet atTAÏ P CHÍ PHAÙ T TRIEÅ N KH&CN, TAÄ P 17, SOÁ K2-2014Trang 25 point M'.Sheet is extended to p-direction and tdirection is pressed in r-direction.According to[4] the relation of the initial thickness of sheet ti at M' and the deformed thickness t followed Cos law So stress of this direction is not depended on the thickness t model of SPIF in Abaqus software for stainless steel 304L sheet with different thickness 0,1mm and 0,4mm.The mechanical properties of empirical model sheets by documents and by testing are given in the following tabula and diagram: Model 1: thickness is 1mm

4. 1 .
Result of simulation of 0,1mm thickness model Shapes of 2 models of in Abaqus are circular conic lateral and tool material is HSS with haft spherical tip of 5mm diameter.The processes of simulation and the result of simulation of 0,1mm thickness model are displayed in figure 3 and figure 4:

Figure 3 .Figure 4 . 4 . 2 . 27 Figure 5 .
Figure 3. Stresses of simulated sheet 0,1mm thickness and red diameter band on the model is the position of measure stresses

Figure 6 .Figure 7 .
Figure 6.Diagram of stresses of 0,4mm thickness through the diameter band of the material The comparison of 2 diagrams of tresses of simulation in Abaqus is displayed in figure 9: result of the simulation by Abaqus software.ACKNOWLEDGMENTS: This research was supported by National Key Laboratory of Digital Control and System Engineering (DCSELAB), HCMUT, VNU-HCM.