theory of single point incremental forming

THEORY OF SINGLE POINT INCREMENTAL FORMINGFORMING

P. A. F. Martins(a), N. Bay(b) (1), M. B. Silva(a), M. Skjoedt(b), y ( ), , j(a) Instituto Superior Tecnico, TULisbon, Portugal

(b) Department of Mechanical Engineering, Technical University of Denmark, Denmark

IntroductionMotivation

Credits:Leszak (1967), Kitazawa et al. (1996),Jeswiet (2001), Filice et al. (2002) , among others

The incremental nature of SPIF raises important questions that need to be addressed:

Jeswiet (2001), Filice et al. (2002) , among others

(i) What is the state of strain and stress at the small plastic zone created by the forming tool?

(ii) What is the physics behind cracking at the transition between the inclined wall and the corner radius of the sheet being formed? Are the cracks formed by shear or by tensile stresses?

Theory of single point incremental forming 2

radius of the sheet being formed? Are the cracks formed by shear or by tensile stresses?

(iii) Is it possible to give a theoretical/quantitative explanation to the high formability in SPIF?

IntroductionState-of-the-art

Until very recently SPIF was one of the very few - if not the only - sheet metal forming process inUntil very recently SPIF was one of the very few if not the only sheet metal forming process inwhich the advantage of experimentation over theory was absolute even for solving the simplestpractical problem.

Disputes in explanations on SPIF mechanics and formability:

(i) Deformation by stretching or shearing

(ii) F bilit li it d b ki f t(ii) Formability limited by necking or fracture

(iii) Raising of FLC due to through thickness shear

(iv) Raising of FLC due to serrated strain paths arising from cyclic, local plastic deformation

Credits:Jeswiet et al. (2007), Allwood , Shouler and Tekkaya (2007), Jackson and Allwood (2007), Emmens and van den Boogaard (2007),

( ) g p g y p

Eyckens et al. (2007), Silva et al. (2008), among others.


IntroductionObjective

To embrace and extend the ongoing discussion on theTo embrace and extend the ongoing discussion on the mechanics of deformation and formability limits of SPIF with the aim of sharing ideas and providing contributions to the advance of knowledge in the field.

Methodology

(i) Development of a theoretical framework based on membrane equilibrium conditions under contact friction forces that can easily be applied to the different modes

y ppof deformation that are commonly found in single point incremental processes

(ii) Investigation of formability limits by means of ductile damage mechanics

(iii) Assessment of theoretical developments by means of FE analysis and experimentation.


Theoretical FrameworkAssumptions

(i) Material is rigid-perfectly plastic(ii) Material is isotropic(iii) Deformation by stretching and

b di (1)AB

CF

ED

bending(1)

(iv) Friction stress at the tool-sheet contact interface is small and assumed to be made of two in-plane

B

(a)

axis

(cut by a meridional plane)

cross section view

O

σσ +dφ φassumed to be made of two in plane components (circumferential and meridional)

(v) The sheet is considered as a shelldα

+dσ σsym

met

ry a

t+dt

r1

r2

φ φQ

α

dα/2tσ

σ t

σ

dα

θσ cos α

φ

dθ2

sinφ-μ

(vi) Circumferential σθ, meridional σø and thickness σt stresses may be considered as principal stresses and treated as membrane stresses

σ

dr

r

tφ

σθ sinαdθ2

+dσ σθ θ

sinθσ2

dθ

sin dθ2treated as membrane stresses

(1) Bending will be neglected in the equilibrium but can be added afterwards at specific

dθ

σθ+dσθtool rotation

dθ

detail view2dθ

-μθσt


but can be added afterwards at specific locations.

σθ top view

detail viewθσ

2

Theoretical FrameworkMembrane equilibrium

(1) The distributions of strain and stress are derived from the governing mode of deformation (extreme assumptions)

State of strain State of stress Hydrostatic stress

0>σY

SPIF(plane strain)

00

0

<ε

=ε

>ε−=ε

θ

φ

t

t

dd

dd ( )( )

( ) 0

01

3

3121

2

1

<σ−=σ=σ

σ+σ=σ=σ

>+

=σ=σ

θ

φ

t

rt

Yt

tool

Y

⎥⎦

⎤⎢⎣

⎡+−σ

=σtrtr

tool

toolYm 2

SPIF( l bi i l)

( ) 03 <+

σ=σ=σtrtool

Yt

( ) 0211 >

+σ

=σ=σ=σ θφ

trt tool

Y

⎥⎦

⎤⎢⎣

⎡ −σ=σ

ttrtoolY

m 2320>ε=ε θφ dd

(equal bi-axial)

Conventional 0>ε=ε θφ dd

( ) 02

23 <+

σ−=σ=σtr

t

toolYt

⎥⎦

⎢⎣ + trtool

m 230<εtd

( ) 011 >

+σ

=σ=σ=σ θφ rt punch

Y

⎥⎤

⎢⎡ −σ trpunchY 22


Stamping(equal bi-axial) 0

0

<ε

εε θφ

td

dd ( )

( ) 03 <+

σ−=σ=σtr

t

punchYt

punch

⎥⎥⎦⎢

⎢⎣ +

σ=σ

trpunch

punchYm 3

2


(2) Friction at the tool-sheet contact interface:

ϕμφφ σ=σ kB expφφ B

(3) Thinning at the corner radius:

tt

B0

φφ σ=σ

(4) Stresses at the inclined wall adjacent to the forming tool :

D

CCD r

rφφ σ=σ

0

0 1

=σ=σ

σ<σ=σ<

θ

φ

t

Y



(5) Dummy sheet may prevent galling in case tribologically difficult sheet material and avoid contact between the tool and the weld seam.


Theoretical FrameworkAssessment by FE analysis (LS-DYNA)

Rigid perfectly plastic material (yield stress = 100 MPa)Rigid perfectly plastic material (yield stress = 100 MPa)Full-scale model discretized by means of shell elementsFull integration scheme with 5 integration points along the thicknessTool movement consists of rotation and helical tool pathFriction by means of Amonton-Coulomb μ=0.1

100Stress1 (FEM)

Stress1 (Theory)

60

80

s (%

)

Stress1 (Theory)

Stress2 (FEM)

Stress2 (Theory)

Stress3 (FEM)

Stress3 (Theory)

20

40

/ Yie

ld S

tres

s

-20

00.55 0.75 0.82St

ress


-40

Thickness (mm)

FormabilityyHydrostatic stress

(1) Hydrostatic stress in SPIF under bi-axial versus plane strain condition (proportional loading is assumed):

( )( )( ) 3

423

4≅⎥

⎦

⎤⎢⎣

⎡++

=σ

σ −

trtr

tool

tool

strainplanem

axialbim

(2) Triaxiality ratio in conventional stamping versus SPIF:

⎤⎡ t

⇔⎟⎟⎠

⎞⎜⎜⎝

⎛σσ

>⎟⎟⎠

⎞⎜⎜⎝

⎛σσ

>⎟⎟⎠

⎞⎜⎜⎝

⎛σσ

− strainplaneY

m

axialbiY

m

stampingY

m ⎥⎦

⎤⎢⎣

⎡+−

>⎥⎦

⎤⎢⎣

⎡+−

>

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

−

trtr

trtr

tr

tr

tool

tool

tool

tool

punch

punch

21

2322

32


FormabilityyDuctile damage

ε1fracture

1ε + ε =2q

(3) Ductile damage criterion (FLC vs FFL):

⎞⎛εf

local necking

1ε + ε =2 nq

FFL

ε⎟⎠

⎞⎜⎝

⎛σ

σ= ∫

ε

dfDf

m

0

FFL

pure shear

simple tension

FLC

plane strain

n

25

23

22

43

011

⎞⎛

−⎟⎟⎠

⎞⎜⎜⎝

⎛

−=⎤⎡

⎥⎦

⎤⎢⎣

⎡−+

−⎥⎦

⎤⎢⎣

⎡−

+

=ε−ε −

rt

r

ttrtr

trtr tool

tool

tool

tool

tool

axialbi

strainplaneaxialbi

ε0-n/2

1

1-1

11

-1/2

biaxial stretching6324302 +⎟⎟

⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡−

+−ε −

tr

trtr tool

tool

toolaxialbi

ε0-n/2 2


FormabilityyMorphology of the cracks

tool path

o

a

b

c

d

e

f

σφδ

φσ

o

g

c e

σφ φσ

σφt l th

o

σ

φtool path

ed


σφ

FormabilityyAssessment by means of experimentation

2.00

1.60

1.80Tensile Test

Elliptical Bulge Test

Circular Bulge Test

SPIF Cone

FFL (SPIF [1])

FFL

AA1050 – H111

1 mm

2 mm

1 20

1.40

1.60SPIF Pyramid

SPIF Cone & Pyramid (max. depth)

rain

FFL(Experimental)

Sheet blanks with 1, 1.5 and 2.0 mm initial thicknessFLC constructed by using

the strains at failure 1 mm

2 mm

1.00

1.20

ajor

True

Str

FFL (constant thickness strain)from grid-elements placed just outside the neck

FFL constructed by

2 mm

0.60

0.80

Ma FFL (constant thickness strain)

FLC FFL constructed by measuring the thickness at fracture in order to obtain the gauge-length strains.

0.20

0.40 Kink in the strain path

FLC(Experimental)


gauge length strains.0.00

-0.60 -0.40 -0.20 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40

MinorTrue Strain

Conclusions

The opening mode of cracks is similar toconventional stamping or deep drawingconventional stamping or deep drawing.

Cracks are opened by meridional tensile stresses(mode I of fracture mechanics) and not by shearstresses (mode II of fracture mechanics).

The governing mode of deformation is, therefore,consistent with stretching being the governing modeof deformation in SPIF.


Conclusions

Measurements of thickness reveals that plastic deformation takes place by uniform thinninguntil fracture without experimental evidence of localized necking before fracture.p g

0.9

1.0

Rolling direction

Perpendicular rolling direction

0.7

0.8

Perpendicular rolling direction

Sine law

0.5

0.6

knes

s (m

m)

0.3

0.4Thic

0 0

0.1

0.2

ψ = 65º ψ = 70º ψ = 75ºψ = 60º


0.00 5 10 15 20 25 30 35 40 45 50 55 60 65 70

Depth (mm)

Conclusions

Suppression of localized necking in SPIF is due to the inability of necks to grow.

Growth is inhibited by the bend path that circumvents the tool and by the surrounding materialhaving considerably lower stress level.

ε1fracture

1ε + ε =2 q

This implies that forming limit diagrams (FLD’s)of conventional sheet metal forming areinapplicable to describe failure in SPIF Instead local necking

1ε + ε =2 nq

FFL

inapplicable to describe failure in SPIF. Instead,fracture forming limit diagrams (FFLD’s)showing the fracture strains, placed well abovethe necking strains, should be employed.

pure shear

simple tension

FLCn

g y

ε0-n/2

1

1-1

11

-1/2

2

biaxial stretchingplane strain


Acknowledgementsg

The support provided by Prof. Anthony G. Akins during the investigation is greatlyacknowledged by the authors.g y


theory of single point incremental forming

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