CONSTITUTIVE MODELING AND OPTIMAL DESIGN OF POLYMERIC FOAMS FOR CRASHWORTHINESS
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1 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
2 ACKNOWLEDGMENTS Computational Mechanics Lab American Automobile Manufacturer Association Chrysler Motor Company Livermore Software Technology Corporation The University of Michigan ACE-MRL Lab
3 Outline Introduction Experimental Investigation and Result Constitutive Modeling Numerical Implementation Procedures Image-based Fixed-grid Homogenization Method Foam Design Optimization Conclusion and Future Work
4 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
5 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
6 . 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 Polystyrene Polypropylene E Modulus Glass Regime 1.0 Glass Transition Rubber Regime Viscous Flow Normalized Temperature T/T g
7 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, Continuum model Roscoe's critical state theory (Schofield and Worth, 1968); Krieg (1972); Neilsen et al (1995).
8 Experiment Program Foam type PP foam PS foam PU foam Test mode Density (pcf) Strain rate (sec -1 ) 1.60 x 10-3 Uniaxial 8.00 x 10-1 Compressi 4.60 on 8.80 x 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 x x 10-3 Simple 8.00 x 10-1 Shear 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)
9 Compressive Response of Polymeric Foam Stress (MPa) Stress Plateau Regime Densification Regime Elasticity Regime Nominal Strain (%) * Negligible size effect * Uniform deformation * Near zero Poisson s ratio Quasi-static Response (BASF Polypropylene foam,1.89 pcf)
10 Compressive Responses of Polypropylene Foams Stress (MPa) m/s m/s m/s m/s rate sensitivity Stress (MPa) m/s m/s m/s m/s Strain Strain Polypropylene foam (1.89 pcf) Polypropylene foam (3.06 pcf)
11 Hydrostatic Compression Response of Polypropylene Foam Pressure (MPa) BASF Polypropylene Foam (Density 1.89 pcf) εý v = sec -1 εý v = sec -1 Hydrostatic Pressure (MPa) ε = sec -1 v ε = sec -1 v ε = sec -1 v v/v 0 Volumetric Strain -- dv / V 0 Polypropylene foam (3.06 pcf)
12 Shear Response of Rigid Polystyrene Foam τ m/s m/s m/s Shear strain (mm/mm) (MPa) Polystyrene Foam (1.0 pcf) under shear loading
13 Tensile Response of Polyurethane Foams m/s m/s m/s m/s m/s m/s Stress (MPa) Stress (MPa) Strain Strain PU Foam (4.3 pcf) PU Foam (6.0 pcf)
14 Temperature Effect on Polypropylene Foam Stress (MPa) o C 25 o C 80 o C Stress (MPa) o C 0 o C 25 o C 50 o C 80 o C Strain Strain rate /sec Strain Strain rate 4.6 1/sec Polypropylene Foam (3.06 pcf) under Uniaxial Compression
15 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 E 1 E 2 E 3 ν 12 1 ν E 1 E 2 E 3 ν 13 ν E S = 1 E 2 E G G G 12
16 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 σ 22 σ 33 k 22 k σ 33 σ 11 k 33 k σ 12 k σ 23 k σ 31 k 31 2 I ( s ) = σ 11 + σ 22 + σ 33 k 11 k 22 k 33
17 Temperature Sensitivity Nominal Stress (MPa) 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 Nominal Strain
18 Rate Dependency of PP foam (3.06 pcf) Effective Stress (MPa) % Strain 10% Strain 20% Strain 30% Strain 40% Strain 50% Strain 60% Strain 70% Strain 80% Strain Nagy et al, 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 Strain Rate
19 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 p σ ρ * ysł ρ sł 2 ø œ œ ß œ Buckling surface (Puso and Govindjee 1995) σ vm R ( 2 p2 - h 2 )= 0
20 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
21 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
22 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
23 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 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 σ vm
24 Model Validation under Uniaxial Compression Nominal Stress (MPa) 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) Nominal Strain PP Foam (3.06 pcf)
25 Model Validation under Simple Shear Shear Stress (MPa) Test Result ( /sec) Test Result (0.08 1/sec) Test Result (4.6 1/sec) Numerical Result ( /sec) Numerical Result (0.08 1/sec) Numerical Result (4.6 1/sec) Shear Strain Polystyrene Foam (1.0 pcf)
26 Hemispherical Free Drop Test Guide Rail Drop Head Experimental Result Numerical Result Foam Specimen Contact Force (N) Penetration (mm) Free Drop Machine Indenter:φ127 mm, 22.2 kgm, 4.5m/sec PP foam, 203x 203 x101 mm 3, 3.06 pcf
27 Hemispherical Free Drop Numerical Simulation Original Mesh Deformed Mesh at t=9.0 ms
28 Hemispherical Free Drop Numerical Simulation Deformed Mesh at t=16.0 ms Effective plastic strain t=16.0 ms
29 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;
30 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.
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