Developing 3D Anisotropic Mechanics
Model of Powder Compaction
Wenhai Wang
Advisor: Dr. Antonios Zavaliangos
Department of Materials Science & Engineering
12-10-2004
1
Outline
1. INTRODUCTION
Powder compaction
Literature review
2. PHENOMENOLOGICAL MODELS AND VUMAT
Phenomenological models
Introduction of VUMAT
Results and discussion
3. ANISOTROPY IN POWDER COMPACTION
Anisotropy in powder compaction
Anisotropic models
4. CONCLUSIONS AND FUTURE WORK
2
Powder Compaction
Metal Industry
Pharmaceutical Industry
Chemical Industry
Ceramics Industry
Food Industry
3
Research Motivation
How do we get there?
How the product performs?
¾ To understand the physics of compaction mechanisms.
¾ To develop robust and rigorous mathematical models
of compaction.
¾ To Provide via models and FEM a design and
optimization tool for the engineers.
4
Length Scales & Models
Macroscopic
Meso-scopic
Microscopic
10 mm
Phenomenological Models
50 µm
Micromechanical
Models
Network
Models
MPFEM
Atomistic
Simulation
5
Past Work
Macroscopic
The powder is considered as a continuum.
References:
1-10
Meso-scopic
Study the particle collection. (statistics
information are inherently considered)
References:
11-18
Microscopic
Look into microscopic level, the local
anisotropy is considered and macrobehavior is deduced.
References:
19-20
Selected References:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
H.A. Kuhn, C.L. Downey, Int. J. Powder Metall. 7 (1) (1971) 15-25
R.J.Green, Int. J. Mech. Sci. 14 (1972) 215-224
S. Shima, M. Oyane, Int. J. Mech. Sci. 18 (1976)
D.C. Drucker, W. Prager Q. Appl. Math. 10 (1952) 157-175
A.N. Schofield, C.P. Wroth, McGrawHill, London, 1968
F.L. DiMaggio, I.S. Sandler, J. Eng. Mech. Div., Proc. – ASCE 96 (1971)
935-950
PM Modnet Computer Modelling Group, Powder Metall. 42 (1999) 301311
I.C. Sinka, J.C. Cunningham, A. Zavaliangos, Powder Tech. 133 (2003)
33-43
Sofronis P, Memeeking RM, Mechanics of Materials 18 (1): 55-68 May
1994
A, Zavaliangos L, Anand J. of the Mech. and Phy. Of solid 41 (6): 10871118 JUN 1993
N.A. Fleck, J. Mech. Phys. Solids 43 (1995) 1409-1431
12.
13.
14.
15.
16.
17.
18.
19.
20.
A.L. Gurson J. Eng. Mater. Tech. (Trans. ASME) (1977 January)
2-15
B. Storakersa, N.A. Fleck, R.M. McMeeking, J. Mech. Phy. of
Solids 47 (1999) 785-815
M.Kailasam, N. Aravas, P. Ponte Castaneda CMES, Vol. 1, pp.
105-118 2000
N. Aravas a, P. Ponte Castaneda, Comput. Methods Appl. Mech.
Engrg. 193 (2004) 3767–3805
P.R. Heyliger & R. M. McMeeking, J. Mech. Phy. Of Solid 49
(2001) 2031-2054
P. Redanz, N. A. Fleck, Acta mater. 49 (2001) 4325–4335
C.L. Martin, D. Bouvard, Acta Mate. 51 (2003) 373–386
Francisco X. –Castilloa S. and Anwarb J., Heyes D.M. J. of
Chem. PHy. Vol 18(10) Mar. 8 2003
A.T. Procopio and A. Zavaliangos, submitted to J. Mech. Phy.
of Solids
6
Phenomenological Models
σ
Ellipse Model
Relative density increase
tensile
compressive
p
Φ (σ , p, D ) = A( D)σ 2 + B ( D ) p 2 − 1 = 0
σ
P
D
™
™
™
™
equivalent stress
hydrostatic pressure
relative density
Yield is pressure dependant
Single state variable – Relative Density
Model parameters can be calibrated by experiments
They can be implemented in FEM to simulate complex
shape compaction operations.
7
Examples of Phenomenological Models
Soil mechanics
Classical
elastoplasticity
1
2
“Kuhn-Shima” model (1970’s)
3
4
= Experimental Measurements
8
Which Phenomenological Model to Use?
180
160
Effective Stress, MPa
140
120
100
80
60
40
20
0
0
20
40
60
80
100
120
140
160
180
Hydrostatic Stress, MPa
• Cap region is “OK” for these models
• Shear (σ ) region is not well captured
• Drucker-Prager Cap model is the
best but needs more experiments
9
Phenomenological Models and FE
Simulation
¾Numerical implementation of phenomenological
models in FE program to solve engineering
problems.
¾ABAQUS is one of the commercial finite element
program software.
¾A lot of applications can be found in literature.
W. Wang, J. Cunningham and A. Zavaliangos, PM2Tec, Las Vegas, Nevada, June 8-12, 2003
I.C. Sinka, J.C. Cunningham and A. Zavaliangos Powder Technology 133 (2003) 33– 43
PM Modnet Computer Modeling Group, Powder Metallurgy, Vol. 42, 1999, 301-311
10
Phenomenological Model Success
Un-lubricated Die
Lubricated Die
¾ Apply Drucker-Prager Cap model (DPC) into ABAQUS/Standard
simulation (All parameters are taken as function of RD);
¾ Model predicts the inversion of radial variation of relative density and
hardness (lubricated V.S. unlubricated die).
I.C. Sinka, J.C. Cunningham and A. Zavaliangos Powder Technology 133 (2003) 33– 43
11
Why Do We Need VUMAT?
ABAQUS
X i (t ), Vi (t ), Fi (t )
Integrating
X i (t + ∆t )
σ i (t )
Fi (t + ∆t )
σ i (t + ∆t )
VUMAT
∆ε i
Solving equations
of mechanics
¾ Current DPC model in ABAQUS/Standard is OK but
convergence is a problem.
¾ ABAQUS/Explicit does not have flexible enough DPC model
but it can address more complex geometry problems.
¾ To this end, a versatile version DPC model (All parameters are
taken as function of RD) was implemented in VUMAT of
ABAQUS/Explicit.
12
Unit Cell Comparison Against ABAQUS/Standard
Loading conditions:
Hydrostatic
tensile
Constraint
tensile
Simple
tensile
Hydrostatic
compression
Porosity
Constraint
compression
Simple
compression
s11/s22
0.75
140000
ƒABAQUS
/Standard
ƒVUMAT
0.74
0.73
0.72
120000
100000
80000
60000
0.71
40000
0.7
20000
0
0.69
0
0.01
0.02
0.03
Tim e (s)
0.04
0.05
0.06
0
0.01
0.02
0.03
0.04
0.05
0.06
Tim e (s)
13
Material: Avicel
Unit Cell Comparison Against ABAQUS/Standard
Loading conditions:
Hydrostatic
tensile
Constraint
tensile
Simple
tensile
Hydrostatic
compression
Constraint
compression
S11
Porosity
Simple
compression
S22
20000
90000
0
80000
0.75
0
70000
-20000
60000
0.73
-40000
50000
40000
-60000
30000
0.71
0.05
0.1
0.15
0.2
ƒABAQUS
/Standard
ƒVUMAT
-80000
20000
10000
-100000
0
0.69
0
0.05
0.1
Tim e (s)
0.15
0.2
0
0.05
0.1
Tim e (s)
0.15
0.2
-120000
Tim e (s)
14
Material: Avicel
Unit Cell Comparison Against ABAQUS/Standard
Loading conditions:
Hydrostatic
tensile
Constraint
tensile
Simple
tensile
Porosity
Hydrostatic
compression
Constraint
compression
Simple
compression
S22
S11
0.7
0
0
-0.2
-0.15
-0.1
-0.05
0
-500000
0.68
-1000000
0.66
-0.2
ƒABAQUS
/Standard
ƒVUMAT
-0.15
-0.1
-0.05
-100000
0
-200000
-300000
-400000
-1500000
-500000
0.64
-600000
-2000000
-700000
0.62
-0.2
-0.15
-0.1
Strain
-0.05
0
-800000
-2500000
Strain
Strain
The origin of the difference is the Elastic modulus. It appears that ABAQUS/Standard does
not update the modulus. Simulations with higher modulus show no difference.
15
Convex Tablet Compaction
Porosity
0.50
--- EXPLICIT
--- Experiment
Unlubricated
0
0.0125
porosity
0.45
0.40
0.35
0.30
0.25
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.01
0.012
0.014
radius
0.60
0.55
Lubricated
porosity
0.50
0.45
0.40
0.35
0.30
0.25
0
0.002
0.004
0.006
0.008
radius
ABAQUS/Explicit results (run with VUMAT) show good agreement
with experimental results!
16
Drucker-Prager Cap Model
Density distribution is well predicted!
How about the strength & modes of fracture prediction?
σ
Shear failure
Region
Cap
Region
p
Non-associated
plasticity
Failure+Dilation
Associated
plasticity
Densification
DPC model shows the different densification trend
when the stress hit different yield surface regions.
(Shear failure region v.s. Cap region)
17
Tablet Diametrical Compaction
Die Compaction
Diametrical Compaction
Diametrical compression tests are carried out in the pharmaceutical
industry to test the “hardness” of tablets.
Unlubricated
Lubricated
Tablets compacted with different die lubrication show different fracture
18
behaviors.
3-D FE Model of Tablet Diametrical
Compaction
Final Relative density distribution (2-D)
Die
Compaction
Mapping
Mapping
Initial Relative density distribution (3-D)
Diametrical
Compaction
Unlubricated
Lubricated
19
Tablet Diametrical Compression
- Unlubricated
Before failure
After failure
Low density in the middle somewhat indicates the initial fracture
development from the center.
20
Tablet Diametrical Compression
-Lubricated
Before failure
After failure
Convergence problems may happen when larger time step was
selected.
21
Force-displacement
Comparing with experimental data
Experiment
Simulation
unlubricated die
200
Force Displacement
180
120
Force (N)
100
80
0.612
0.559
140
0.59
0.59
60
0.560
0.510
120
0.505
100
80
0.472
0.433
0.464
60
40
40
20
20
0
0.00
0.590
160
Unlubricated
Lubricated
Force, N
140
lubricated die
0.380
0.374
0
0.20
0.40
0.60
0.80
Displacement (mm)
1.00
1.20
0.422
0.416
0
0.2
0.4
0.6
0.8
1
1.2
Distance, mm
Comparing with experiment results, Simulation
results show good trends.
22
Phenomenological Model Limits
Die
Isostatic
Compaction Compaction
σ
•
•
Stress path affect final
property.
Relative density is not
the only state variable.
Σ
Σ
Σ
p
Triaxial
Compaction
Σ=78 ;Τ∼0.5 Σ
RD 85%
σf 20 Ksi
Τ
Τ
Τ
Σ=Τ=60
85%
25 Ksi
Σ=80 ;Τ=12
85%
55 Ksi
Strength in Die ≠ Isostatic ≠ Triaxial Compaction
R.M. Koerner Ceramic Bulletin Vol. 52, No. 7 1973
23
Anisotropy In Powder Compaction
24
Anisotropy of Powder Compacts
- Path Dependence
Loading History
Triaxial Testing
Dibasic Calcium Phosphate (A-Tab) d = 180 µm
SR=Stress Ratio=
σ radial
σ axial
25
Data courtesy of Steve Galen
Strength Anisotropy of Powder Compacts
Transverse Strength ST
Normal Strength SN
The same sample after die
compaction shows the different
strength in transverse direction and
normal direction.
Anisotropy!
SN
ST
26
Data courtesy of Steve Galen
State Variables
σ
Drucker-Prager Cap
Φ(σ , P, RD...)
p
State variables:
¾Relative density
¾“B” tensor
¾“s” tensor
27
Anisotropic Mechanics Models
Micromechanics model
Anisotropic constitutive model
N. A. Fleck
M. Kailasam
N. Aravas
P.Ponte Castaneda
¾Micromechanics model with ¾Continuum model with isolated
discrete particles;
pores;
¾An internal state variable
¾Takes into account the
(B tensor) is used to
evolution of the porosity and
describe the evolution of
the development of anisotropy
anisotropy under general
due to change in the shape and
loading.
the orientation of the voids
during deformation.
N.A. Fleck, J. Mech. Phys. Solids 43 (1995) 1409-1431
M.Kailasam, N. Aravas, P. Ponte Castaneda CMES, Vol. 1, pp. 105-118 2000
N. Aravas a, P. Ponte Castaneda, Comput. Methods Appl. Mech. Engrg. 193 (2004) 3767–3805
28
Fleck’s Model
Goal: Find the yield locus in
macroscopic level.
Assumes affine motion.
Macro plastic strain Ù Micro velocity field
E& ij
Vij
N.A. Fleck, J. Mech. Phys. Solids 43 (1995) 1409-1431
B. Storakersa, N.A. Fleck, R.M. McMeeking, J. Mech. Phy. of Solids 47 (1999) 785-815
vi = 2 R0 E& ij n j
29
Fleck’s Model (cont.)
Anisotropic factor ----“B” Tensor :
(i) The distribution of contact area;
(ii) The number of contacts per unit
surface area of particle;
(iii)The hardness of each contact.
2
2
2
1 D − D0 E& xx (sin φ cos θ ) + E& yy (sin φ sin θ ) + E& zz cos φ
Bij ni n j =
[
]
4 (1 − D0 )
E& xx + E& yy + E& zz
Hydrostatic Compaction:
Bij ni n j =
1 D − D0
12 (1 − D0 )
Constant!
Die Compaction:
Bij ni n j =
1 D − D0
(
) cos 2 φ
4 1 − D0
30
Obtaining The Yield Surface
Macroscopic stress Σ ij may be
calculated by differentiation of plastic
dissipation W& with respect to plastic
strain rate E& ij .
∂W&
Σ ij =
∂E&
Hydrostatic
Compaction
2
1.5
1
0.5
∑
-1.5
0
-1
-0.5
0
0.5
1
1.5
-0.5
ij
-1
Put a perturbation, calculate the
differential
-1.5
∑m
-2
∆W&
∑=
∆E&
Die
Compaction
Initial loading point
2-D Axisymmetric
Plastic strain rate
H& = E& ZZ + 2 E XX
2
E& = ( E& ZZ − E& XX )
3
Macroscopic stress
∂W&
∑m =
∂H&
∂W&
∑=
∂E&
31
Critique of Fleck’s Model
¾Predicts path dependence but exaggerates the
effect
¾Major “problems” :
--Affine motion assumption shown to be incorrect by DEM,
leads to overestimate of loads;
--Cannot address triaxialities less than die compaction (can
not be implemented into FEM)
¾Predicts “wrong” anisotropy in diametrical
compression
σ Transverse
ratio =
<1
σ Normal
which is constant ratio and does not vary with relative density.
However, experiments show opposite trend and vary with
relative density.
32
Anisotropic constitutive model
(P&A Model)
N. Aravas a, P. Ponte Castaneda, Comput. Methods Appl. Mech. Engrg. 193 (2004) 3767–3805
33
Representative Volume Elements
E3
Inclusions
Voids
P
Magnified
Cracks
Grain
boundaries
E2
E1
P is a material point
surrounded by a
material neighborhood.
(macro element)
Possible microstate of an RVE
for material neighborhood of P
34
Porous Material Representation
¾ The local state is represented by the
average shapes and orientations of
voids
¾ All the voids initially have the same
shape and orientation and distributed
randomly in a elastic-plastic matrix
¾ Under finite plastic deformation, the
voids remain ellipsoidal but change
their volume, shape and orientation
with the “local” macroscopic
deformation
¾ The size of the voids is assumed to be
much smaller than the scale of
variation of the macroscopic fields
35
Description of the Constitutive Model
X3
1. Average rate-of-deformation tensor
D = De + D p
2. Elastic part
~ 0
D =Mσ
e
~
M
b a
0
σ = σ& − ωσ + σ ω
~
M =M +
f
Q −1
1− f
X2
c
X1
is the effective elastic compliance tensor
ω is the spin of the voids (antisymmetric tensor)
f is porosity and Q is a microstructure tensor
Q = L( I − s )
Q /s depend on the shape and orientation of the ellipsoidal voids.
s = ( f , w1 , w2 , n (1) , n ( 2 ) , n (3) )
w1 = c / a ; w1 = c / b
36
Yield Condition and Plastic Flow
The effective yield function can be written:
~ (s)σ )
σ ⋅ (m
~
Φ(σ , s) =
− (σ y ) 2
1− f
σ y is the yield strength in tension of the matrix material.
~ corresponds to an appropriately normalized effective
m
viscous compliance tensor.
The plastic behavior described by the macroscopic potential
~
Φ is fully compressible.
3. Plastic part
&N
D =Λ
p
N=
∂Φ
∂σ
& is the plastic multiplier, larger than zero.
Λ
37
Evolution of the Microstructure
When the porous material deforms, the state variables
evolve and, in turn, influence the response of the
material.
Porosity
& (1 − f ) N ≡ Λ
& h(σ , s)
f& = (1 − f ) Dkkp = Λ
kk
Shape
'
'
& h (σ , s)
w& 1 = w1 ( D33p − D11p ) = Λ
1
'
'
& h (σ , s )
w& = w ( D p − D p ) = Λ
2
2
33
22
2
Orientation
n& (i ) = ωn (i ) (i = 1,2,3)
M.Kailasam, N. Aravas, P. Ponte Castaneda CMES, Vol. 1, pp. 105-118 2000
N. Aravas a, P. Ponte Castaneda, Comput. Methods Appl. Mech. Engrg. 193 (2004) 3767–3805
38
Models Comparison
Fleck’s Model
P&A Model
Interaction of particles
Matrix with voids inside
“B” tensor
“s” tensor
Contact area and coordination
number
Ellipsoid voids with shape and
orientation
Stage I Compaction (RD<0.9)
Stage II Compaction (RD>0.9)
Limitations:
Affine motion
Limitations:
No contact area and
coordination number evolution
Symmetric yield surface, not
appropriate for “powder”
39
Conclusions
¾ A versatile version of the Drucker-Prager model was
implemented in VUMAT of ABAQUS/Explicit.
¾ State of the art models of compaction predict densification
but not post processing properties because of
- Path dependence
- Anisotropy
- Brittle behavior of compacts
¾ Fleck’s and P&A models were reviewed to check if they can
address the weakness of Drucker-Prager model
- Fleck’s model has major problems
Affine motion assumption; Cannot address low triaxialites cases; Predicts wrong
anisotropy…
- P&A model is not appropriate
No contact area and coordination number evolution; Symmetric yield surface, not
appropriate for “powder”
40
Framework of Future Work
FE Model
Micromechanical Model
¾Stage I compaction
¾Take into account of the anisotropy
in microscopic level (“B” Tensor)
¾Modify the assumption of “affine
motion”
Continuum mechanics Model
¾Combine micromechanical model
and continuum mechanics model
¾Develop new model and implement it
to VUMAT
¾Study the modes of fracture during
diametrical compaction of tablet with
new model
41