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Developing 3D Anisotropic Mechanics Developing 3D Anisotropic Mechanics Model of Powder Compaction Model of Powder Compaction Wenhai Wang Advisor: Dr. Antonios Zavaliangos Department of Materials Science & Engineering 12-10-2004 1

Outline 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 Powder Compaction Pharmaceutical Industry Metal Industry Metal Industry Pharmaceutical Industry Chemical Industry Chemical Industry Food Industry Food Industry Ceramics Industry Ceramics Industry 3

Research Motivation Research Motivation How do we get there? How the product performs? � 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. 4

Length Scales & Models Length Scales & Models Macroscopic Meso- -scopic scopic Microscopic Macroscopic Meso Microscopic 10 mm 50 µm Phenomenological Models Phenomenological Models Micromechanical Micromechanical MPFEM MPFEM Models Models Network Network Atomistic Atomistic Models Models Simulation Simulation 5

Past Work Past Work Macroscopic The powder is considered as a continuum. References: 1-10 Meso-scopic Study the particle collection. (statistics References: 11-18 information are inherently considered) Microscopic Look into microscopic level, the local References: 19-20 anisotropy is considered and macro- behavior is deduced. Selected References: 1. H.A. Kuhn, C.L. Downey, Int. J. Powder Metall. 7 (1) (1971) 15-25 12. A.L. Gurson J. Eng. Mater. Tech. (Trans. ASME) (1977 January) 2. R.J.Green, Int. J. Mech. Sci. 14 (1972) 215-224 2-15 3. S. Shima, M. Oyane, Int. J. Mech. Sci. 18 (1976) 13. B. Storakersa, N.A. Fleck, R.M. McMeeking, J. Mech. Phy. of 4. D.C. Drucker, W. Prager Q. Appl. Math. 10 (1952) 157-175 Solids 47 (1999) 785-815 5. A.N. Schofield, C.P. Wroth, McGrawHill, London, 1968 14. M.Kailasam, N. Aravas, P. Ponte Castaneda CMES, Vol. 1, pp. 6. F.L. DiMaggio, I.S. Sandler, J. Eng. Mech. Div., Proc. – ASCE 96 (1971) 105-118 2000 935-950 15. N. Aravas a, P. Ponte Castaneda, Comput. Methods Appl. Mech. 7. PM Modnet Computer Modelling Group, Powder Metall. 42 (1999) 301- Engrg. 193 (2004) 3767–3805 311 16. P.R. Heyliger & R. M. McMeeking, J. Mech. Phy. Of Solid 49 8. I.C. Sinka, J.C. Cunningham, A. Zavaliangos, Powder Tech. 133 (2003) (2001) 2031-2054 33-43 17. P. Redanz, N. A. Fleck, Acta mater. 49 (2001) 4325–4335 9. Sofronis P, Memeeking RM, Mechanics of Materials 18 (1): 55-68 May 18. C.L. Martin, D. Bouvard, Acta Mate. 51 (2003) 373–386 1994 19. Francisco X. –Castilloa S. and Anwarb J., Heyes D.M. J. of 10. A, Zavaliangos L, Anand J. of the Mech. and Phy. Of solid 41 (6): 1087- Chem. PHy. Vol 18(10) Mar. 8 2003 1118 JUN 1993 20. A.T. Procopio and A. Zavaliangos, submitted to J. Mech. Phy. 11. N.A. Fleck, J. Mech. Phys. Solids 43 (1995) 1409-1431 of Solids 6

Phenomenological Models Phenomenological Models σ Ellipse Model Relative density increase compressive p tensile Φ σ = σ + − = 2 2 ( , p , D ) A ( D ) B ( D ) p 1 0 σ equivalent stress P hydrostatic pressure D 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 Examples of Phenomenological Models Soil mechanics Soil mechanics Classical Classical elastoplasticity elastoplasticity 3 1 “Kuhn-Shima” model (1970’s) 2 4 = Experimental Measurements 8

Which Phenomenological Model to Use? Which Phenomenological Model to Use? 180 160 140 Effective Stress, MPa 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 σ • Shear ( ) region is not well captured • Drucker-Prager Cap model is the best but needs more experiments 9

Phenomenological Models and FE Phenomenological Models and FE Simulation Simulation � 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. W. Wang, J. Cunningham and A. Zavaliangos, PM2Tec, Las Vegas, Nevada, June 8-12, 2003 10 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

Phenomenological Model Success Phenomenological Model Success Un- -lubricated Die lubricated Die Lubricated Die Un Lubricated Die � 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). 11 I.C. Sinka, J.C. Cunningham and A. Zavaliangos Powder Technology 133 (2003) 33– 43

Why Do We Need VUMAT? Why Do We Need VUMAT? ABAQUS X ( t ), V ( t ), F ( t ) + ∆ X i ( t t ) Integrating i i i σ ( t ) i ∆ ε + ∆ VUMAT F i ( t t ) i Solving equations σ + ∆ ( t t ) of mechanics i � 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. 12

Unit Cell Comparison Against ABAQUS/Standard Unit Cell Comparison Against ABAQUS/Standard Loading conditions: Hydrostatic Constraint Simple Hydrostatic Constraint Simple tensile tensile tensile compression compression compression Porosity s11/s22 0.75 140000 � ABAQUS 120000 0.74 /Standard 100000 0.73 � VUMAT 80000 0.72 60000 0.71 40000 0.7 20000 0 0.69 0 0.01 0.02 0.03 0.04 0.05 0.06 0 0.01 0.02 0.03 0.04 0.05 0.06 Time (s) Time (s) 13 Material: Avicel

Unit Cell Comparison Against ABAQUS/Standard Unit Cell Comparison Against ABAQUS/Standard Loading conditions: Hydrostatic Constraint Simple Hydrostatic Constraint Simple tensile tensile tensile compression compression compression Porosity S11 S22 20000 90000 0 80000 0.75 0 0.05 0.1 0.15 0.2 70000 -20000 � ABAQUS 60000 -40000 /Standard 0.73 50000 40000 � VUMAT -60000 30000 0.71 -80000 20000 10000 -100000 0 0.69 0 0.05 0.1 0.15 0.2 -120000 0 0.05 0.1 0.15 0.2 Time (s) Time (s) Time (s) 14 Material: Avicel

Unit Cell Comparison Against ABAQUS/Standard Unit Cell Comparison Against ABAQUS/Standard Loading conditions: Hydrostatic Constraint Simple Hydrostatic Constraint Simple tensile tensile tensile compression compression compression Porosity S11 S22 0.7 0 0 -0.2 � ABAQUS -0.15 -0.1 -0.05 0 -0.2 -0.15 -0.1 -0.05 0 -100000 0.68 -500000 /Standard -200000 � VUMAT -300000 -1000000 0.66 -400000 -1500000 -500000 0.64 -600000 -2000000 -700000 0.62 -0.2 -0.15 -0.1 -0.05 0 -800000 -2500000 Strain Strain Strain The origin of the difference is the Elastic modulus. It appears that ABAQUS/Standard does 15 not update the modulus. Simulations with higher modulus show no difference.

Convex Tablet Compaction Convex Tablet Compaction Porosity 0.50 --- EXPLICIT 0.45 --- Experiment Unlubricated 0.40 porosity 0.35 0 0.0125 0.30 0.25 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 radius 0.60 0.55 0.50 porosity 0.45 Lubricated 0.40 0.35 0.30 0.25 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 radius ABAQUS/Explicit results (run with VUMAT) show good agreement 16 with experimental results!

Drucker- -Prager Cap Model Prager Cap Model Drucker Density distribution is well predicted! How about the strength & modes of fracture prediction? σ Cap Shear failure Region Region p Associated Non-associated plasticity 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) 17

Tablet Diametrical Compaction Tablet Diametrical Compaction Diametrical Compaction Die 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 D FE Model of Tablet Diametrical 3 Compaction Compaction Final Relative density distribution (2-D) Die Compaction Mapping Mapping Initial Relative density distribution (3-D) Diametrical Compaction 19 Unlubricated Lubricated

Tablet Diametrical Compression Tablet Diametrical Compression - Unlubricated Unlubricated - Before failure After failure Low density in the middle somewhat indicates the initial fracture development from the center. 20