Transcript Plane Strain Extrusion – Slip-line Field Solution vs. FEM

Plane Strain Extrusion –
Slip-line Field Solution vs.
FEM Solution
Nanshu Lu
ES 246 Plasticity Project
Jan. 11, 2006
Outline
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Metal Forming – Extrusion
Slip-line Field Solution
Unsymmetrical Extrusion
FEM Solution
Conclusions
Metal Forming – Extrusion

Extrusion: Metal
forming process
whereby the workpiece
is placed in a chamber
with an opening and is
forced to escape
through the opening,
usually being pushed
out by a mandrel.
Extrusion (a) and an assortment of extrudates (b)
Metal Forming, Betzalel Avitzur, Lehigh University
Slip-line Field Solution
CDE

k
 11   p1  k sin 30   p1  2

k

  15   22   p1  k sin 30   p1 
2


3k
 12  k cos 30 
2

Fan CAD
 11   p2  k sin 2

 22   p2  k sin 2
  k cos 2
 12
 
  
p0  2k     p1  2k     const
 4
 12 
 
 p1   k  1  
3

  22   p1 
 P  2 H  22
k
 1
 k  
2
 3 2
 1
 2k    H
 3 2
F
E
C
D
x1
ABC
 11   p0  k

  45   22   p0  k  0  p0   k
  0
 12
A
x2
Hill R. (1948)
B
G
H
Unsymmetrical Extrusion
Green A.P. (1955)
Fractional Reduction
 D  2d 
r
D
Eccentricity  
bc
bc
Unsymmetrical Extrusion

Rough container walls
p
1
 1 

 1.33log 

2k
2
 1 r 
r1 
r 1   
p 

1  r
r2 
r 1   
1  r
1
 p1  p2   r  p1  p2  
2
Smooth container walls
p
 1 
 0.13  r 1  r   1.33log 

2k
 1 r 
FEM Solution – Modeling Skills
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Stress-Strain Curve
Part:
Billet – 2D deformable shell
Ram – 2D analytical rigid body
Property
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Material – steel
450
400
350
Stress σ (MPa)
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 y  400MPa
300
250
200
150
100
50
0
0
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0.2
Elastic – 2000GPa, v=0.3
Plastic – Yield Stress = 400MPa, perfectly plastic
Material Orientation – local coordinate system
Assembly

0.1
Billet – punch contact
0.3
Strain ε
0.4
0.5
0.6
FEM Solution – Modeling Skills
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Step
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“Move ram” after initial step
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Maximum number of increments = 1000
Initial Increment = 0.0001
Minimum time increment = 1e-6
Nonlinear geometry
Output request
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History output – reaction force on the ram reference
point
DOF monitor – monitor the horizontal displacement of
the ram reference point
FEM Solution – Modeling Skills
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Interaction

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Frictionless surface-to-surface contact between
billet and the ram at initial step
Load

Boundary Condition
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Symmetric boundary condition
Displacement constraints
Imposed displacement to the ram (=0.05m)
FEM Solution – Modeling Skills
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Mesh
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Structured CPE4R elements
Global seed size 0.002m
Job

Monitor
FEM Solution – Deformed Shape
FEM Solution – Animation
FEM Solution – Reaction Force History
FEM Solution – P~H Relation
H (m)
0
0.2
0.4
0.6
0.8
1
P/k
0.0000
0.6187
1.2373
1.8560
2.4747
3.0933
Pmax (N)
0
75.6
154.2
217.9
308.4
377.5
Cal-P/k
0.0000
0.6547
1.3354
1.8871
2.6708
3.2692
Extrusion Force ~ Billet Width Relation
3.5000
3.0000

 1 
4H 0.13  r 1  r   1.33log 

 1  r 

2.5000
 1
2H   
 3 2
P/k
2.0000
1.5000
1.0000
0.5000
r=0.5,
 y  400MPa
0.0000
0
0.1
0.2
0.3
0.4
0.5
H (m)
0.6
0.7
0.8
0.9
1
FEM Solution – P~k Relation
σy (MPa)
360
380
400
420
440
P/(2H) (MPa)
321.6
339.4
357.3
375.2
393.0
Pmax (N)
68.8
72.2
75.6
78.9
82.2
Cal-P/(2H)
344
361
378
394.5
411
Extrusion Force ~ Yeild Stress Relation
420.0
410.0
400.0
P/(2H)(MPa)
390.0
380.0
370.0
360.0
 1
k  
 3 2
350.0
340.0
330.0
H=0.2m
320.0
360
370
380
410
400
390
Yield Stress (MPa)
420
430
440
FEM Solution – Unsymmetrical Extrusion
ε
0
0.2
0.4
0.6
Δp/k
0.0000
0.0300
0.1300
0.2950
P (N)
204.8
206.6
211.3
218.6
Cal- Δp/k
0.0000
0.0390
0.1407
0.2988
Increase in extrusion pressure due to eccentricity
0.4000
0.3500
0.3000
Δp/k
0.2500
0.2000
0.1500
0.1000
0.0500
r=0.8
0.0000
0
0.1
0.2
0.3
Eccentricity ε
0.4
0.5
0.6
Conclusion
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Slip-line field method gives good solutions for
metal-forming process
ABAQUS/CAE v6.5 is able to calculate perfectlyplastic material
Adaptive meshing is needed to simulate metalforming process accurately
Thermal contact interaction should be included
further
Acknowledgement
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Joost Vlassak
Jennifer Furstenau
Xuanhe Zhao