CONSTITUTIVE MODELING AND OPTIMAL DESIGN OF POLYMERIC FOAMS FOR CRASHWORTHINESS Jun Zhang December 19, 1997 Computational Mechanics Laboratory Department of Mechanical Engineering and Applied Mechanics The University of Michigan
ACKNOWLEDGMENTS Computational Mechanics Lab American Automobile Manufacturer Association Chrysler Motor Company Livermore Software Technology Corporation The University of Michigan ACE-MRL Lab
Outline Introduction Experimental Investigation and Result Constitutive Modeling Numerical Implementation Procedures Image-based Fixed-grid Homogenization Method Foam Design Optimization Conclusion and Future Work
Introduction Background * In 1991, 56,000 people died in auto accident in the US (NHTSA); * New federal motor vehicle safety standards (FMVSS); * Usage of polymeric foam for cushion purpose; * Mathematical Modeling of Transportation Safety. Hybrid III Dummy and Honeycomb Padding Impact Biomechanics Computational Model
Objective and Tasks Tasks * Phenomenological modeling of PU, PS and PP foams; * Numerical implementation as user defined material subroutine in LS- DYNA3D; *Model validation: simple loading and structural test; *Microscopic constitutive modeling by image-based fixed-grid; representative volume element analysis using homogenization method *Optimization of polymeric foam structure. Foam specific cushion character * Limited compressive stress by long plateau regime * Compression and shear properties * Large deformation (80% volumetric strain) and low bulk modulus * Rate sensitive:high strain rate (35 mph) * Temperature sensitive:-20 o C to 80 o C
. Polymer Material Properties Types of polymer foams (at room temperature 20 o C) : Flexible(elastomeric) foam: Polyurethane foam Rigid (elastic-plastic) foam: Polystyrene foam Semi rigid foam: Polypropylene foam Time-Temperature Correspondence t E s ( t,t 0 )= E s,t 1 a t Material ( ) loga T = C 1 T 1 T g C 2 + T 1 T g Table 1.1 Properties of Solid Polymers (at 20 C) Densi ty (Mg/ m 3 ) Glass Temperatur e (K) Young s Modulus E s (GN/m 2 ) Yi eld S trength s ys (MN/ m 2 ) Fracture S trength (MN/ m 2 ) Fracture Toughness K IC (MN/m 1.5 ) Polyurethane 1.2-1.6 127 130 - Polystyrene Polypropylene 1.05 0.91 373 253 1.2-1.7 1.2-1.7 30-70 30-70 40-80 35-90 - 2 E Modulus Glass Regime 1.0 Glass Transition Rubber Regime Viscous Flow Normalized Temperature T/T g
Related Work Dimensional mechanism model Gibson and Ashby (1988); Gibson et al (1989) and Triantafillou et al (1989); Puso and Govindjee (1995). Simple loading phenomenological model Rush, 1969; Ramon et al, 1990; Sherwood and Frost, 1992. Continuum model Roscoe's critical state theory (Schofield and Worth, 1968); Krieg (1972); Neilsen et al (1995).
Experiment Program Foam type PP foam PS foam PU foam Test mode Density (pcf) 1.89 3.06 1.0 4.0 4.3 6.0 Strain rate (sec -1 ) 1.60 x 10-3 Uniaxial 8.00 x 10-1 Compressi 4.60 on 8.80 x10 1 4.00 x 10-3 Hydrostat 2.00 x 10-1 ic Compressi 1.15 x 10 1 on (a) (b) 1.60 x 10-3 Uniaxial 8.00 x 10-1 Tension 4.60 8.80 x10 1 1.60 x 10-3 Simple 8.00 x 10-1 Shear 4.60 8.80 x10 1 * ASTM Standard D1621 * 50 x 50 x 50 mm 3 for uniaxial and hydrostatic tests * 100 x 50 x 50 mm 3 for shear tests (c) (d)
Compressive Response of Polymeric Foam 0.60 0.50 Stress (MPa) 0.40 0.30 Stress Plateau Regime Densification Regime 0.20 0.10 Elasticity Regime 0.00 0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 Nominal Strain (%) * Negligible size effect * Uniform deformation * Near zero Poisson s ratio Quasi-static Response (BASF Polypropylene foam,1.89 pcf)
Compressive Responses of Polypropylene Foams Stress (MPa) 1.20 1.00 0.80 0.60 0.40 4.45 10 0 m/s 2.29 10-1 m/s 4.00 10-3 m/s 8.00 10-5 m/s rate sensitivity Stress (MPa) 2.00 1.50 1.00 0.50 4.45 10 0 m/s 2.29 10-1 m/s 4.00 10-3 m/s 8.00 10-5 m/s 0.20 0.00 0.0 0.2 0.4 0.6 0.8 Strain 0.00 0.0 0.2 0.4 0.6 0.8 Strain Polypropylene foam (1.89 pcf) Polypropylene foam (3.06 pcf)
Hydrostatic Compression Response of Polypropylene Foam Pressure (MPa) 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 BASF Polypropylene Foam (Density 1.89 pcf) εý v = 0.133 sec -1 εý v = 0.0027 sec -1 Hydrostatic Pressure (MPa) 1.20 1.00 0.80 0.60 0.40 0.20 ε = 1.15 10 1 sec -1 v ε = 2.00 10-1 sec -1 v ε = 4.00 10-3 sec -1 v 0.0 0 0.2 0.4 0.6 0.8 0.00 0.0 0.2 0.4 0.6 0.8 v/v 0 Volumetric Strain -- dv / V 0 Polypropylene foam (3.06 pcf)
Shear Response of Rigid Polystyrene Foam 0.12 0.10 τ 0.08 0.06 0.04 2.29 10-1 m/s 0.02 4.00 10-3 m/s 8.00 10-5 m/s 0.00 0.00 0.05 0.10 0.15 0.20 Shear strain (mm/mm) (MPa) Polystyrene Foam (1.0 pcf) under shear loading
Tensile Response of Polyurethane Foams 0.30 0.35 0.25 0.20 2.29 10-1 m/s 4.00 10-3 m/s 8.00 10-5 m/s 0.30 0.25 2.29 10-1 m/s 4.00 10-3 m/s 8.00 10-5 m/s Stress (MPa) 0.15 0.10 Stress (MPa) 0.20 0.15 0.10 0.05 0.05 0.00 0.0 0.1 0.2 0.3 0.4 Strain 0.00 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Strain PU Foam (4.3 pcf) PU Foam (6.0 pcf)
Temperature Effect on Polypropylene Foam 2.00 3.00 Stress (MPa) 1.50 1.00-20 o C 25 o C 80 o C Stress (MPa) 2.50 2.00 1.50 1.00-20 o C 0 o C 25 o C 50 o C 80 o C 0.50 0.50 0.00 0.0 0.2 0.4 0.6 0.8 Strain Strain rate 0.0016 1/sec 0.00 0.0 0.2 0.4 0.6 0.8 Strain Strain rate 4.6 1/sec Polypropylene Foam (3.06 pcf) under Uniaxial Compression
Rigid Polymeric Foam Elasticity Foam Elasticity Ý s = C: Ýe where objective stress rate s Ý J is the Jaumman stress rate in a corotational frame Isotropic Foam s Ý = 2G( e Ý d Ý ) K( ε Ý v Ý )I e dp ε vp Anisotropy Foam e Ý = S : s Ý 1 ν 21 ν 31 0 0 0 E 1 E 2 E 3 ν 12 1 ν 32 0 0 0 E 1 E 2 E 3 ν 13 ν 23 1 0 0 0 E S = 1 E 2 E 3 1 0 0 0 0 0 G 23 0 0 0 0 0 0 0 0 0 1 G 13 0 1 G 12
Yield Locus for Rigid Polymeric Foam Foam Yield Locus Isotropic elasto-plastic foam σ 1 σ 1 Dimensional Argument(Gibson and Ashby, 1988) M = σ ys bt2 4 P crit 1 σ a σ ys = n 2 π 2 E s I h 2 Proposed yield locus F( s ) F 0 = p x 0 ε vp a ε vp 2 [ ( )] 2 ( ) σ vm = κ 1 p β + σ vm 2 b( ε vp ) 1 = 0 2 σ a Μ t -σ ys c +σ ys σ Μ σ a Anisotropy elasto-plastic foam F( s ) = J + a I 1 J s ( ) = 1 2 σ 11 σ 22 k 11 k 22 2 + σ 22 σ 33 k 22 k 33 2 + σ 33 σ 11 k 33 k 11 2 + 3 σ 12 k 12 2 + σ 23 k 23 2 + σ 31 k 31 2 I ( s ) = σ 11 + σ 22 + σ 33 k 11 k 22 k 33
Temperature Sensitivity Nominal Stress (MPa) 3.00 2.50 2.00 1.50 1.00 0.50-20 C 0 C 25 C 50 C 80 C -20 C, test 0 C, test 25 C, test 50 C, test 80 C, test Williams-Landel-Ferry (WLF) Equation (Williams et al, 1955) L(T) = exp C 1 (T T r) C 2 +T T r PP foam (3.06 pcf) C1=6.52ÞC, C2=468.7ÞC 0.00 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Nominal Strain
Rate Dependency of PP foam (3.06 pcf) Effective Stress (MPa) 10 1 10 0 5% Strain 10% Strain 20% Strain 30% Strain 40% Strain 50% Strain 60% Strain 70% Strain 80% Strain Nagy et al,1964 1 n f 0 ε Ý p = D f σ( ε) = σ 0 ( ε)l(t) Ý ε Ý ε 0 n = a+ bε p Combined temperature and rate effect a+bε 10-1 10-3 10-2 10-1 10 0 10 1 Strain Rate
Comparison of Yield Criterion Plastic yield envelop (Gibson et al, 1989) σ vm σ ys = γ ρ* Ł ρ s ł Ø Œ Œ º Œ 3 2 1-3p σ ρ * ysł ρ sł 2 ø œ œ ß œ Buckling surface (Puso and Govindjee 1995) σ vm 2 + 1 R ( 2 p2 - h 2 )= 0
Kinematic Hardening of Polypropylene Foam (3.06 pcf) Kinematic hardening F = F(σ,ε vp, Ý ε ) g = g(σ, ε vp, Ý ε ) Evolution of Yield Ellipse with Plastic Volumetric Strain
Stress Integration Procedure for Elastic-plastic Materials Deformation decomposition Plasticity consistency condition de = de e + de p de = D 1 ds + g s dλ F ds - Adλ = 0 s g de {} D-1 0 = s ds T F { dλ} A s ds = D * ep de * = D D g F s s D ep T D A + F s T D g s 1 * if F g D ep is a non-symmetric matrix no solution σ vm Non-unique solution for non-associative plastic flow The stress return is not radial g(s ) F(s ) p
Non-smooth Multisurface Plasticity Plastic potential variation (assuming associative plastic potential) Ý F i = F i s : C : e Ý λ F i s : C : s = 0 (i=1,2,...) If plastic yield and loading condition active F i ( s )= 0 and F i s : C : e > 0 σ 2 Plasticity consistency condition (1) Ý λ i = 0 λ Ý = s F i : C : e s F i : C : s (i=1,2,...), loading is not active; F 1 dε ij p1 = λ 1 F 1 σ ij (2) Ý λ i > 0 λ Ý i > 0 Ý λ i = s F i :C:e s F i :C:s (3) for multiple surfaces Closest-point-projection (Simo et al, 1988) In summary λ Ý = max( s F i :C:e s F i :C:s,i = 1,2,... ) F 2 dε p2 F ij = λ 2 2 σ ij σ 1 C C ep = C [ C : s ] C : s F i s F i : C : s [ ] if λ Ý = 0 ( i = 1,2,... ) if λ Ý > 0 σ 3 Illustration of a sigular point in yield Surface
Polymeric Rigid Foam Plastic Flow Law Non-associative plastic potential Plasticity consistency condition Plastic Poisson s Ratio Under uniaxial compression g( s ˆ,φ) = αp 2 2 + σ vm ( ) e Ý ˆ p = λ Ý g s ˆ,φ ˆ s ε Ý xxp = Ý ε yyp = ν p ε Ý zzp ε Ý vp = ( 1 2ν p ) Ý ε zzp P e Ý ˆ p = λ Ý 1 2g 2σ vm σ vm ˆ s + 2αp p ˆ s or e Ý ˆ p = λ Ý 3 s 2g ˆ 2α pˆ I 9 ε Ý zzp = λ Ý 3 2g s zz 2α 9 p ε Ý vp = λ Ý αp g ( ) ( ) α = 9 1 2ν p 2 1+ ν p zero plastic Poisson s ratio g = 9 2 p 2 2 + σ vm
Model Validation under Uniaxial Compression Nominal Stress (MPa) 2.00 1.50 1.00 0.50 Numerical Result (1.6E-3 1/sec) Experimental Result (1.6E-3 1/sec) Numerical Result (8.0E-2 1/sec) Experimental Result (8.0E-2 1/sec) Numerical Result (4.6 1/sec) Experimental Result (4.6 1/sec) 0.00 0.0 0.2 0.4 0.6 0.8 Nominal Strain PP Foam (3.06 pcf)
Model Validation under Simple Shear 0.12 0.10 Shear Stress (MPa) 0.08 0.06 0.04 0.02 Test Result (0.0016 1/sec) Test Result (0.08 1/sec) Test Result (4.6 1/sec) Numerical Result (0.0016 1/sec) Numerical Result (0.08 1/sec) Numerical Result (4.6 1/sec) 0.00 0.00 0.05 0.10 0.15 0.20 0.25 Shear Strain Polystyrene Foam (1.0 pcf)
Hemispherical Free Drop Test Guide Rail Drop Head 16 10 3 14 10 3 12 10 3 Experimental Result Numerical Result Foam Specimen Contact Force (N) 10 10 3 8 10 3 6 10 3 4 10 3 2 10 3 0 10 0 0 20 40 60 80 100 Penetration (mm) Free Drop Machine Indenter:φ127 mm, 22.2 kgm, 4.5m/sec PP foam, 203x 203 x101 mm 3, 3.06 pcf
Hemispherical Free Drop Numerical Simulation Original Mesh Deformed Mesh at t=9.0 ms
Hemispherical Free Drop Numerical Simulation Deformed Mesh at t=16.0 ms Effective plastic strain t=16.0 ms
Conclusion PU foams are flexible while PS and PP foams are rigid at 20 o C; Yield stress of polymeric foams are sensitive to strain rate, temperature and pressure; A phenomenological rate dependent single surface elasto-plastic yield criterion is developed and implemented in LS-DYNA3D program;
Future Work Experiment on polymeric foams under multiaxial loading; Constitutive modeling considering different failure mechanism; Validate rigid foam model under multiaxial loading; Couple homogenization constitutive modeling and LS-DYNA3D; Three-dimentional RVE modeling and analysis by using more powerful CT scanner. Three-dimentional foam design optimization.