A Semianalytical Model for the Simulation of Polymers

A Semianalytical Model for the Simulation of Polymers Paul Du Bois 3, Stefan Kolling 1, Markus Feucht 1 & André Haufe 2 1 DaimlerChrysler AG, Sindelfingen, Germany 2 Dynamore GmbH, Stuttgart, Germany 3 Consultant, Offenbach, Germany 4. LS DYNA FORUM 25, 2. 21. Oktober 25, Bamberg

Overview Introduction and motivation Discussion of appropriate p yield surfaces Details of a new material model A few examples Conclusions 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 2

Motivation More and more structural parts are made from plastics. At present important crashworthiness applications are: Pedestrian protection (bumper, fascia, hood, adhesives) Passenger protection (cockpit, internal structures) Their mechanical behaviour is strongly dependent on Underlaying chemical structure and production process Temperature, strain rate, humidity etc. Many thousand different blends are available today!! Shell-like but rib reinforced structures 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 3

V&V processes for complete FE models Semi Analytical Model for Polymers V&V process of each individual part of the numerical model necessary: Global solution strategy Constitutive model Element formulation, type & size Discretisation depth & modeling techniques more In engineering practise the numerical model shall correlate to experimental data close enough, to allow sound predictions in a predefined range of model variations. 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 4

Identification challenge: Structures made of plastics Plastics is a group name comprising many different materials Mechanical response at room temperature may be glassy or rubbery, brittle or viscous ε& 1 2 3 T st tress 4 5 thermoset plastic (Duroplast) thermoplastic elastomer strain 1 glasslike behaviour 2 plastic or viscous flow 3 low ductility 4 high ductility 5 rubbery crystalline thermoplastic amorphous thermoplastic Fiber reinforcement may causeanisotropic anisotropic response Rib reinforcements represent an important modeling effort 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 5

Mechanical behaviour of thermoplastics (uniaxial loading) True Stress Yield curve AND Young s Modulus are strain rate dependent d Non linear elasticity Uniform necking due to stabilisation True Stre ess True strain Visco elasto visco plasticity Different yielding under tension/compression (and shear) Plastic incompressibility for compression only (ν p 5).5) Under tension foam like (ν p ) True strain Tensile Test Compression Test No Von Mises type of plasticity! 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 6

Simulation of a tensile test: thermoplastic response Von Mises (piecewise linear) plasticity, linear elastic visco plastic, input of strain rate dependent test data (= state of the art: MAT24 in LS DYNA) Generally good representation of tensile responses 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 7

Simulation of a tensile test: thermoplastic response Positive curvature in the hardening curve causes stabilisation of the cross section after necking 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 8

Deficiencies of Von Mises plasticity to describe plastics Plastic incompressibility not correct for thermoplastics No Von Mises relationship between yield stress in tension, compression, shear σ c σ t σ b σ t σ Loss of isotropy during hardening due to reorientation of molecules (load induced anisotropy) Crazing ( volumetric plasticity and low biaxial strength) cannot be described Modulus is rate dependent (viscosity below the yield surface) Modulus of elasticity it depends d on plastic df deformation ( damage) s σ t 3 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 9

Major concerns for the simulation of plastic parts Bending stiffness is usually underestimated Under biaxial loads (punching) the plastic tends to develop crazes Important elastic rebound is currently the major stumbling block 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 1

SAMP 1 : goals and limitations Semi Analytical Model for Polymers Describe isotropic ductile thermoplastic materials Implement for solid and shell (plane stress) elements Fully tabulated input data Most general quadratic isotropic yield surface formulation, allows to fit 3 experiments exactly and 4 approximately (least squares) Damage model to simulate unloading response Tabulated visco plasticity Not suited for the simulation of fiber reinforced structures with high anisotropy and low rupture strain Load induced anisotropy neglected Large plastic deformation expected : rate dependency in elastic region is neglected No sophisticated failure model yet 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 11

SAMP A Semi Analytical Model for Polymers Hardening curves: tabulated data σ y ε& σ y σ c σ y σ s σ t d l l l l h tensile hardening curve from tensile test at different strain rates σt ε pt = ε t, ε t = E l ln l ε pt ε pc compressive hardening curve from compression test shear hardening curve from shear test ε ps c l ε = σ σ 1 x& s 1 d pc εc, εc = ln ε = = = E l ps εs, εs dt 2G 2 y 2 h difficult! 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 12

General isotropic quadratic yield surface: SAMP 1 f = tension q shear compression Yield surface rface(modified Schlimmer): (,, pl 2 2 f p σ ε ) = σ A A p A p vm vm 1 2 3 1 p To achieve a fast and easy calibration i process, test results from tension, compression and shear tests can be fd fed directly into SAMP 1 via table definitions: σt σc σs Tension test data Compression test data Shear test data ε t ε c ε s Therefore the yield surface coefficients A,1,2 are calculated directly from stresses gained from a table lookup. A 2 c t t c s = σ A = 9( σ ) A = 9( ) 2 3 s 1 s σ σ σ σ c t 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 13 2 2 σ σ 3σ σ σ t c

General isotropic quadratic yield surface: special cases Von Mises q Drucker Prager q f = f = 3 tension tension 1 3 compression 1 p p σ t σ t σ c Tension test data Tension test data Compression test data ε t εt εc alternatively υ p.5 σt σs Tension test data Shear test data ε t ε s 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 14

General isotropic quadratic yield surface: SAMP 1 f = q tension q shear compression Yield surface: (,, pl 2 2 f p σ ε ) = σ A A p A p vm vm 1 2 Biaxial tension 3 1 Plastic potential: g = σ α p 2 2 vm p To achieve a fast and easy calibration process, test results from tension, compression and shear tests can be fed directly into SAMP 1 via table definitions: σ t σt σc σs Tension test data Biaxial test data Compression test data Shear test data ε t ε t ε c ε s Therefore the yield surface coefficients A 12,1,2 are calculated each time step by a least squares approach fitting the yield stress values obtained by a table lookup 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 15

General isotropic quadratic yield surface: anything goes! f = q Drucker Prager q tension q shear compression f = Biaxial tension 3 tension 1 biaxial 3 1 p p σ t σ t σ c σ t σ t Tension test data Biaxial test data Compression test data Tension test data Biaxial test data ε ε t ε t ε c ε t ε t σ t σ t σ s Tension test data Biaxial test data Shear test data ε t εt εs 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 16

General isotropic quadratic yield surface: convexity f = q compression tension q shear A 2 Yield surface: (,, pl 2 2 f p σ ε ) = σ A A p A p vm vm 1 2 Biaxial tension 3 1 A 2 2 Condition for convexity : A 2 σ s σ σ t 3 c p In the numerical algorithm, a unique solution is guaranteed only for convex yield surfaces In order to avoid numerical problems convexity can be enforced in SAMP 1 by a user controlled flag The hardening curve in shear is then scaled as necessary every timestep This allows to achieve robust numerical behaviour under most loading conditions 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 17

SAMP 1: Flow rule & g g = & λ F = σ = q vm σ c ε p σ plastic potential: ti g=const. σ t σ s m = g σσ g = 2 2 q A A1 p A2 p associated 2 2 q + αpα p non associated σ t 3 σ c 3 p flow parameter correlates to plastic ν p Poisson s ratio: l 9 2α α υ p =.5 18 + 2α l l 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 18 ε pt

SAMP 1: Flow rule & g g = & λ F = σ = q vm σ c ε p σ plastic potential: ti g=const. σ t σ s m = g σσ g = 2 2 q A A1 p A2 p associated 2 2 q + αpα p non associated σ t 3 σ c 3 p flow parameter correlates to plastic ν p Poisson s ratio: l 9 2α α υ p =.5 18 + 2α l l 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 19 ε pt

SAMP1: Visco plasticity Elastoplastic consistency condition f ( ) 2 2 λ, & λ = = σ A A p A p vm 1 2 σ y ε& Viscoplastic constitutive law: & λ = f Φ( f ) 2η ( ) 1 λ, & λ Φ ( 2ηλ& ) = σ t l ε pt l 1 Φ ( 2ηλ& ) = f ( λ, & λ) f ( λ,) 2 = A ( λ, & λ) + A ( λ, & λ) p + A ( λ, & λ) p A ( λ, ) A ( λ, ) p A ( λ, ) p 2 1 2 1 2 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 2

25 2 15 1 5,, 1, 2, 3, 4, 5 Semi Analytical Model for Polymers Unloading test: (1 d) damage damage model: Basics E E eff1 E eff2 Schädigung d 1 Tabulated damage curve: d(ε pl ) =1 - E eff / E wahre True Spann St nung tress [MPa] d,8,6,4,2 Test Versuch Input Simulation data 125 Test % Streckgrenze 8% Approximated Versagendehnung effective modulus of elasticity 75% Streckgrenze,,1,2,3,4,5 Plastic strain ε pl plastische Dehnung True strain wahre Dehnung [-] Damage parameter d : E eff = E (1 d) d = 1 E eff / E σ eff y σ y = 1 d 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 21

SAMP 1: Damage and failure Semi Analytical Model for Polymers Failure onset defined by the parameter further fading of the element defined by d c p Δ ε rupt d 1. d c Δ ε p rupt ε p 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 22

Yield surfaces: experiment vs. SAMP Polyvinyl Chloride (PVC) 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 23

Experimental data vs. SAMP Polystyrene (PS) 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 24

Experimental data vs. SAMP Polycarbonate (PC) 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 25

Experimental data vs. SAMP Polypropylene (PP) 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 26

Experimental data vs. SAMP Polyethylene (PE) 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 27

Experimental datavs vs. SAMP Acrylonitrile Butadiene Styrene (ABS) 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 28

Verification and validation of PP EPDM Quasi static tensile tests with unloading 5 45 4 tensile tests with unloading experiment unloading simulation with damage simulation no damage 35 8x2 mm 48.6 mm [N] force 3 25 2 15 1 5.5 1 1.5 2 displacement [mm] 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 29

Verification and validation of PP EPDM Compression test F 3 2.5 DKI PP_EPDM compression tests : 2.5mm mesh experiment simulation 2 force [N N] 1.5 4x6 mm 1.5 F 2 4 6 8 1 12 14 16 displacement [mm] 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 3

Verification and validation of PP EPDM Shear test F 5 45 PP_EPDM - Shear Test experiment simulation 4 35 force N 3 25 2 F 15 1 5.5 1 1.5 2 2.5 3 3.5 4 4.5 5 displacement mm 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 31

Verification and validation of PP EPDM SAMP 1: experimental results (tension and compression test) has been used directly as input data for this three point bending test. Tensile Test Compression Test Specimen size 1 x 133 x 3.8 mm Downward prescribed displacement (quasi static) 3 integration points across thickness direction 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 32

Unloading test: (1 d) damage damage model: Results Flow curve definition and failure parameters for a simple dogbone tension test at different loading velocities including proper unloading behaviour 6 5 u 4 u 3 F-x_dyn (test data) F-x_static (test data) 2 F-x_dyn_dam (SAMP-) F-x_static_dam (SAMP-) t 1 5 1 15 2 25 3 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 33

Other applications Material model: SAMP not only valid for thermoplastics It covers metals as well Also suitable for Adhesives (if you have a clue how to model) Structural foams Crushable foams Example: validation of a high strength, low density, expandable epoxy polymer (CORE Products) using a single material input card of SAMP 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 34

Material laws for crushable foams in LS DYNA No. keyword formulation input 5,14 MAT_SOIL_AND_FOAM isotropic, el pl parameter 26,126 MAT_HONEYCOMB anisotropic, el pl LC 53 MAT_CLOSED_CELL_FOAM CELL isotropic, el pl LC 63,163 MAT_CRUSHABLE_FOAM isotropic, el pl LC / table υ variable 75 MAT_BILKHU/DUBOIS_FOAM _ isotropic, el pl LC strain rate parameter 142 MAT_TRANSVERSELY_ANISO TROPIC_CRUSHABLE_FOAM anisotropic el pl LC 144 MAT_PITZER_CRUSHABLE isotropic, el pl LC + strain rate _FOAM υ variable parameter user MAT_SAMP isotropic, el pl variable LC / table υ 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 35

Validation of structural high density foam (CORE) 45 4 FOAM compression experiment simulation 6 FOAM tension experiment simulation 35 5 Fad force [kn] 3 25 2 15 1 force [kn] 4 3 2 ading out via damage formulation on 5 1-5 -3-2.5-2 -1.5-1 -.5.5.5 1 1.5 2 2.5 displacement [mm] FOAM shear displacement [mm] experiment simulation -1 force [kn] -2-3 -4-5 -6 -.8 -.7 -.6 -.5 -.4 -.3 -.2 -.1 displacement [mm] 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 36

Conclusions and outlook Semi Analytical Model for Polymers Most critical issues have been addressed with SAMP 1 Damage and strain rate dependency Failure and unloading General quadratic yield condition Pressure dependent plastic potential First shot at crazing Features still missing: Transition to anisotropic behaviour Rate dependency in elastic region Future work: Further testing of material ilmodel dl Calibration and validation of more polymers Simulation of more component tests SAMP 1 will be available in the next release of LS DYNA (971) 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 37

Thanks for your attention! Semi Analytical Model for Polymers This presentation was pedagogical valuable or somehow 4. LS DYNA FORUM, 2. 21. Oktober 25, Bamberg S. 38