1

Developing 3D Anisotropic Mechanics Developing 3D Anisotropic Mechanics Model of Powder CompactionModel of Powder Compaction

Wenhai WangAdvisor: Dr. Antonios Zavaliangos

Department of Materials Science & Engineering

12-10-2004

2

OutlineOutline

1. INTRODUCTIONPowder compaction Literature review

2. PHENOMENOLOGICAL MODELS AND VUMATPhenomenological modelsIntroduction of VUMATResults and discussion

3. ANISOTROPY IN POWDER COMPACTIONAnisotropy in powder compactionAnisotropic models

4. CONCLUSIONS AND FUTURE WORK

3

Powder CompactionPowder CompactionMetal IndustryMetal Industry Pharmaceutical IndustryPharmaceutical Industry

Food IndustryFood Industry

Chemical IndustryChemical Industry

Ceramics IndustryCeramics Industry

4

Research MotivationResearch Motivation

To understand the physics of compaction mechanisms.

To develop robust and rigorous mathematical models of compaction.

To Provide via models and FEM a design and optimization tool for the engineers.

How do we get there?

How the product performs?

5

Length Scales & ModelsLength Scales & Models

10 mm 50 µm

MesoMeso--scopicscopicMacroscopicMacroscopic MicroscopicMicroscopic

Network Network ModelsModels

Micromechanical Micromechanical ModelsModels

PhenomenologicalPhenomenological ModelsModels

MPFEMMPFEM

AtomisticAtomisticSimulationSimulation

6

Past WorkPast Work

References: 19-20

Look into microscopic level, the local anisotropy is considered and macro-behavior is deduced.

Microscopic

References: 11-18

Study the particle collection. (statistics information are inherently considered)

Meso-scopic

References: 1-10

The powder is considered as a continuum.Macroscopic

1. H.A. Kuhn, C.L. Downey, Int. J. Powder Metall. 7 (1) (1971) 15-252. R.J.Green, Int. J. Mech. Sci. 14 (1972) 215-2243. S. Shima, M. Oyane, Int. J. Mech. Sci. 18 (1976) 4. D.C. Drucker, W. Prager Q. Appl. Math. 10 (1952) 157-1755. A.N. Schofield, C.P. Wroth, McGrawHill, London, 19686. F.L. DiMaggio, I.S. Sandler, J. Eng. Mech. Div., Proc. – ASCE 96 (1971)

935-9507. PM Modnet Computer Modelling Group, Powder Metall. 42 (1999) 301-

3118. I.C. Sinka, J.C. Cunningham, A. Zavaliangos, Powder Tech. 133 (2003)

33-439. Sofronis P, Memeeking RM, Mechanics of Materials 18 (1): 55-68 May

199410. A, Zavaliangos L, Anand J. of the Mech. and Phy. Of solid 41 (6): 1087-

1118 JUN 1993 11. N.A. Fleck, J. Mech. Phys. Solids 43 (1995) 1409-1431

Selected References:12. A.L. Gurson J. Eng. Mater. Tech. (Trans. ASME) (1977 January)

2-1513. B. Storakersa, N.A. Fleck, R.M. McMeeking, J. Mech. Phy. of

Solids 47 (1999) 785-81514. M.Kailasam, N. Aravas, P. Ponte Castaneda CMES, Vol. 1, pp.

105-118 200015. N. Aravas a, P. Ponte Castaneda, Comput. Methods Appl. Mech.

Engrg. 193 (2004) 3767–380516. P.R. Heyliger & R. M. McMeeking, J. Mech. Phy. Of Solid 49

(2001) 2031-205417. P. Redanz, N. A. Fleck, Acta mater. 49 (2001) 4325–4335 18. C.L. Martin, D. Bouvard, Acta Mate. 51 (2003) 373–38619. Francisco X. –Castilloa S. and Anwarb J., Heyes D.M. J. of

Chem. PHy. Vol 18(10) Mar. 8 200320. A.T. Procopio and A. Zavaliangos, submitted to J. Mech. Phy.

of Solids

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Yield is pressure dependantSingle state variable – Relative DensityModel parameters can be calibrated by experimentsThey can be implemented in FEM to simulate complex shape compaction operations.

Phenomenological ModelsPhenomenological Models

01)()(),,( 22 =−+=Φ pDBDADp σσσ equivalent stressP hydrostatic pressureD relative density

Ellipse Model

tensile compressive p

σ

Relative density increase

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Examples of Phenomenological ModelsExamples of Phenomenological Models

= Experimental Measurements

Classical Classical elastoplasticityelastoplasticity

Soil mechanicsSoil mechanics

“Kuhn-Shima” model (1970’s)1

2 4

3

9

Which Phenomenological Model to Use?Which Phenomenological Model to Use?

• Cap region is “OK” for these models• Shear ( ) region is not well captured• Drucker-Prager Cap model is the

best but needs more experiments

σ

0

20

40

60

80

100

120

140

160

180

0 20 40 60 80 100 120 140 160 180

Hydrostatic Stress, MPa

Effe

ctiv

e St

ress

, MPa

10

Phenomenological Models and FE Phenomenological Models and FE SimulationSimulation

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– 43PM Modnet Computer Modeling Group, Powder Metallurgy, Vol. 42, 1999, 301-311

Numerical implementation of phenomenological models in FE program to solve engineering problems.ABAQUS is one of the commercial finite element program software.A lot of applications can be found in literature.

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Phenomenological Model SuccessPhenomenological Model Success

Apply Drucker-Prager Cap model (DPC) into ABAQUS/Standard simulation (All parameters are taken as function of RD);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

UnUn--lubricated Dielubricated Die Lubricated DieLubricated Die

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Current DPC model in ABAQUS/Standard is OK but convergence is a problem.ABAQUS/Explicit does not have flexible enough DPC model but it can address more complex geometry problems.To this end, a versatile version DPC model (All parameters are taken as function of RD) was implemented in VUMAT of ABAQUS/Explicit.

Why Do We Need VUMAT?Why Do We Need VUMAT?ABAQUS

Integrating)(),(),( tFtVtX iii )( ttX i ∆+

iε∆

)( tti ∆+σ

VUMAT

Solving equations of mechanics

)(tiσ

)( ttFi ∆+

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Unit Cell Comparison Against ABAQUS/StandardUnit Cell Comparison Against ABAQUS/Standard

Simple compression

Constraint compression

Hydrostatic compression

Simple tensile

Constraint tensile

Hydrostatic tensile

Loading conditions:

Porosity

0.69

0.7

0.71

0.72

0.73

0.74

0.75

0 0.01 0.02 0.03 0.04 0.05 0.06

Time (s)

s11/s22

0

20000

40000

60000

80000

100000

120000

140000

0 0.01 0.02 0.03 0.04 0.05 0.06

Time (s)

Material: Avicel

ABAQUS/StandardVUMAT

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Simple compression

Constraint compression

Hydrostatic compression

Simple tensile

Constraint tensile

Hydrostatic tensile

Loading conditions:

S22

-120000

-100000

-80000

-60000

-40000

-20000

0

20000

0 0.05 0.1 0.15 0.2

Time (s)

Porosity

0.69

0.71

0.73

0.75

0 0.05 0.1 0.15 0.2

Time (s)

S11

0

10000

20000

30000

40000

50000

60000

70000

80000

90000

0 0.05 0.1 0.15 0.2

Time (s)

Unit Cell Comparison Against ABAQUS/StandardUnit Cell Comparison Against ABAQUS/Standard

Material: Avicel

ABAQUS/StandardVUMAT

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Unit Cell Comparison Against ABAQUS/StandardUnit Cell Comparison Against ABAQUS/Standard

Simple compression

Constraint compression

Hydrostatic compression

Simple tensile

Constraint tensile

Hydrostatic tensile

Loading conditions:

Porosity

0.62

0.64

0.66

0.68

0.7

-0.2 -0.15 -0.1 -0.05 0

Strain

S11

-2500000

-2000000

-1500000

-1000000

-500000

0-0.2 -0.15 -0.1 -0.05 0

Strain

S22

-800000

-700000

-600000

-500000

-400000

-300000

-200000

-100000

0-0.2 -0.15 -0.1 -0.05 0

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.

ABAQUS/StandardVUMAT

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Convex Tablet CompactionConvex Tablet Compaction

0 0.0125

Porosity

Unlubricated

Lubricated

ABAQUS/Explicit results (run with VUMAT) show good agreement with experimental results!

0.25

0.30

0.35

0.40

0.45

0.50

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

radius

poro

sity

--- EXPLICIT--- Experiment

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

radius

poro

sity

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Density distribution is well predicted!

DruckerDrucker--Prager Cap ModelPrager Cap Model

How about the strength & modes of fracture prediction?

σ

p

Shear failure Region

CapRegion

Non-associated plasticity

Associated plasticity

Failure+Dilation Densification

DPC model shows the different densification trend when the stress hit different yield surface regions. (Shear failure region v.s. Cap region)

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Tablet Diametrical CompactionTablet Diametrical CompactionDiametrical Compaction

Tablets compacted with different die lubrication show different fracture behaviors.

Die Compaction

Unlubricated Lubricated

Diametrical compression tests are carried out in the pharmaceutical industry to test the “hardness” of tablets.

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33--D FE Model of Tablet Diametrical D FE Model of Tablet Diametrical CompactionCompaction

Final Relative density distribution (2-D)

Initial Relative density distribution (3-D)

Die Compaction

Diametrical Compaction

Mapping

Mapping

Unlubricated Lubricated

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Tablet Diametrical CompressionTablet Diametrical Compression-- UnlubricatedUnlubricated

Low density in the middle somewhat indicates the initial fracture development from the center.

Before failure After failure

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Tablet Diametrical CompressionTablet Diametrical Compression--LubricatedLubricated

Convergence problems may happen when larger time step was selected.

Before failure After failure

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ForceForce--displacement displacement Comparing with experimental dataComparing with experimental data

Comparing with experiment results, Simulation results show good trends.

Force Displacement

0

20

40

60

80

100

120

140

0.00 0.20 0.40 0.60 0.80 1.00 1.20

Displacement (mm)

Forc

e (N

)

UnlubricatedLubricated

020406080

100120140160180200

0 0.2 0.4 0.6 0.8 1 1.2

Distance, mmFo

rce,

N

lubricated die unlubricated die

0.3740.3800.416

0.464

0.505

0.560

0.590

0.422

0.433

0.472

0.510

0.5590.612

Simulation Experiment

0.59

0.59

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Phenomenological Model LimitsPhenomenological Model LimitsDie Compaction

Isostatic Compaction

Triaxial Compaction

Σ

Τ

Σ

Τ

Σ

Τ

Σ=78 ;Τ∼0.5 Σ Σ=Τ=60 ;Τ=12Σ=80

RDσf

85% 85% 85%20 Ksi 25 Ksi 55 Ksi

R.M. Koerner Ceramic Bulletin Vol. 52, No. 7 1973

• Stress path affect final property.

• Relative density is not the only state variable.

Strength in Die Strength in Die ≠≠ Isostatic Isostatic ≠≠ Triaxial CompactionTriaxial Compaction

σ

p

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Anisotropy In Powder CompactionAnisotropy In Powder Compaction

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Anisotropy of Powder CompactsAnisotropy of Powder Compacts-- Path DependencePath Dependence

Data courtesy of Steve Galen

Loading History

Triaxial Testing

SR=Stress Ratio= axial

radial

σσDibasic Calcium Phosphate (A-Tab) d = 180 µm

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Strength Anisotropy of Powder CompactsStrength Anisotropy of Powder Compacts

Normal Strength SN

Transverse Strength ST

SN ST

Data courtesy of Steve Galen

The same sample after die compaction shows the different strength in transverse direction and normal direction.

Anisotropy!

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State VariablesState Variables

p

σ Drucker-Prager Cap

...),,( RDPσΦState variables:

Relative densityRelative density

““BB”” tensortensor““ss”” tensortensor

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Anisotropic Mechanics Models

M. KailasamN. Aravas

P.Ponte Castaneda

N. A. Fleck

Continuum model with isolated pores;Takes into account the evolution of the porosity and the development of anisotropy due to change in the shape and the orientation of the voids during deformation.

Micromechanics model with discrete particles;An internal state variable (B tensor) is used to describe the evolution of anisotropy under general loading.

Anisotropic constitutive modelAnisotropic constitutive modelMicromechanics model Micromechanics model

N.A. Fleck, J. Mech. Phys. Solids 43 (1995) 1409-1431M.Kailasam, N. Aravas, P. Ponte Castaneda CMES, Vol. 1, pp. 105-118 2000N. Aravas a, P. Ponte Castaneda, Comput. Methods Appl. Mech. Engrg. 193 (2004) 3767–3805

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FleckFleck’’s Models Model

N.A. Fleck, J. Mech. Phys. Solids 43 (1995) 1409-1431B. Storakersa, N.A. Fleck, R.M. McMeeking, J. Mech. Phy. of Solids 47 (1999) 785-815

Assumes affine motion.Macro plastic strain Micro velocity field

ijE& ijV

Goal: Find the yield locus in macroscopic level.

jiji nERv &02=

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FleckFleck’’s Model (cont.)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.

]cos)sin(sin)cos(sin

[)1(4

1 222

0

0

zzyyxx

zzyyxxjiij EEE

EEEDDD

nnB&&&

&&&

++

++

−−

=φθφθφ

Hydrostatic Compaction:

)1(121

0

0

DDDnnB jiij −

−= Constant!

Die Compaction:

φ2

0

0 cos)1

(41

DDDnnB jiij −

−=

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Obtaining The Yield SurfaceObtaining The Yield Surface

2-D Axisymmetric

Put a perturbation, calculate the differential

EW&

&

∆∆

=∑

Macroscopic stress may be calculated by differentiation of plastic dissipation with respect to plastic strain rate .

ijΣ

W&

ijE&

ijij E

W&

&

∂∂

=Σ

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-1.5 -1 -0.5 0 0.5 1 1.5

Initial loading pointm∑

∑

XXZZ EEH 2+= &&

)(32

XXZZ EEE &&& −=

Plastic strain rate

HW

m &

&

∂∂

=∑

EW&

&

∂∂

=∑

Macroscopic stress

Die Compaction

Hydrostatic Compaction

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Critique of FleckCritique of Fleck’’s Models ModelPredicts path dependence but exaggerates the effectMajor “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)Predicts “wrong” anisotropy in diametrical compression

which is constant ratio and does not vary with relative density.However, experiments show opposite trend and vary with relative density.

1<=Normal

Transverseratioσ

σ

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Anisotropic constitutive modelAnisotropic constitutive model(P&A Model)(P&A Model)

N. Aravas a, P. Ponte Castaneda, Comput. Methods Appl. Mech. Engrg. 193 (2004) 3767–3805

34

Representative Volume ElementsRepresentative Volume Elements

P

P is a material point surrounded by a material neighborhood.(macro element)

E1

E2

E3

Cracks Grain boundaries

Voids

Inclusions

Magnified

Possible microstate of an RVEfor material neighborhood of P

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Porous Material RepresentationPorous Material Representation

The local state is represented by the average shapes and orientations of voidsAll the voids initially have the same shape and orientation and distributed randomly in a elastic-plastic matrixUnder finite plastic deformation, the voids remain ellipsoidal but change their volume, shape and orientation with the “local” macroscopic deformationThe size of the voids is assumed to be much smaller than the scale of variation of the macroscopic fields

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Description of the Constitutive ModelDescription of the Constitutive Model

1. Average rate-of-deformation tensor

2. Elastic part

pe DDD +=

0~ σMDe =is the effective elastic compliance tensorM~

is the spin of the voids (antisymmetric tensor) ωf is porosity and Q is a microstructure tensor )( sILQ −=

Q /s depend on the shape and orientation of the ellipsoidal voids.

),,,,,( )3()2()1(21 nnnwwfs =

X1

X2

X3

ab

c

bcwacw /;/ 11 ==

ωσσωσσ +−= &0

1

1~ −

−+= Q

ffMM

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The plastic behavior described by the macroscopic potential is fully compressible.Φ~

Yield Condition and Plastic FlowYield Condition and Plastic Flow

2)(1

))(~(),(~yf

sms σσσσ −−

⋅=Φ

The effective yield function can be written:

yσ is the yield strength in tension of the matrix material.

m~ corresponds to an appropriately normalized effective viscous compliance tensor.

3. Plastic partND p Λ= &

σ∂Φ∂

=N

Λ& is the plastic multiplier, larger than zero.

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Evolution of the MicrostructureEvolution of the Microstructure

When the porous material deforms, the state variables evolve and, in turn, influence the response of the material.

Porosity),()1()1( shNfDff kk

pkk σΛ≡−Λ=−= &&&

Shape),()( 1

'11

'3311 shDDww pp σΛ=−= &&

),()( 2'

22'

3322 shDDww pp σΛ=−= &&

Orientation)3,2,1()()( == inn ii ω&

M.Kailasam, N. Aravas, P. Ponte Castaneda CMES, Vol. 1, pp. 105-118 2000N. Aravas a, P. Ponte Castaneda, Comput. Methods Appl. Mech. Engrg. 193 (2004) 3767–3805

39

Models ComparisonModels Comparison

Limitations:No contact area and coordination number evolutionSymmetric yield surface, not appropriate for “powder”

Limitations:Affine motion

Stage II Compaction (RD>0.9)Stage I Compaction (RD<0.9)

Ellipsoid voids with shape and orientation

Contact area and coordination number

“s” tensor“B” tensorMatrix with voids insideInteraction of particles

P&A ModelP&A ModelFleckFleck’’s Models Model

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ConclusionsConclusionsA versatile version of the Drucker-Prager model was

implemented in VUMAT of ABAQUS/Explicit.State of the art models of compaction predict densification but not post processing properties because of

- Path dependence- Anisotropy- Brittle behavior of compacts

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 problemsAffine motion assumption; Cannot address low triaxialites cases; Predicts wrong anisotropy…

- P&A model is not appropriateNo contact area and coordination number evolution; Symmetric yield surface, notappropriate for “powder”

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Framework of Future WorkFramework of Future WorkFE ModelFE Model Micromechanical ModelMicromechanical Model

Continuum mechanics ModelContinuum mechanics Model

Stage I compaction

Take into account of the anisotropy in microscopic level (“B” Tensor)

Modify the assumption of “affine motion”

Combine micromechanical model and continuum mechanics model

Develop new model and implement it to VUMAT

Study the modes of fracture during diametrical compaction of tablet with new model