Modelling the effects of various contents of fillers on the relaxation rate of filled rubbers
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1 Modelling the effects of various contents of fillers on the relaxation rate of filled rubbers L. Laiarinandrasana, A. Jean, D. Jeulin, S. Forest MINES ParisTech MAT Centre des Matériaux CNRS UMR 7633 BP 87 F-9003 Evry Cedex - France lucien.laiarinandrasana@mines-paristech.fr Tire Expo Conference Köln 9-/0/200 78/9
2 Motivations s/s max s/s max Unfilled SBR Filled SBR Time (s) Time (s) SBR with or without silica filler (Robisson, 2000) s/s max Unfilled NR with or without crystallization (Bennani, 2006) E+00.E+04 2.E+04 3.E+04 4.E+04 5.E+04 Relaxation time (s) 0% 2% 5% 5% 30% 45% NBR with various carbon black contents (Omnes, 2007) The higher the particles content, the higher the stress relaxation rate What about ~00% fillers? Tire Expo Conference Köln 9-/0/200 79/9
3 Motivations cont d ECCMR 2007 (numerical homogenization) A. Jean et al. Rubber with carbon black fillers: parameters identification of multiscale nanostructure model, pp. 4 V. Jha et al. Micro-structural finite element modelling of the stiffness of filled elastomers: the effect of filler number, shape and position, pp. 65 M. Naito et al. 3D modeling and simulation of micro to macroscopic deformation behavior of filled rubber, pp. 27 Applied to monotonous tensile loading (no viscous effect) Basic constitutive equations, with arbitrary materials coefficients Tire Expo Conference Köln 9-/0/200 80/9
4 Content Background FE computations on 2D axi-symmetrical unit cell Extension to 3D: digitized microstructure FE computations on periodic cell Summary Future works Tire Expo Conference Köln 9-/0/200 8/9
5 Small-strain linear viscoelasticity Material coefficients K=3000MPa G=MPa t = 0.0(s) g = 0.3 t 2 = 0.2(s) g 2 =0. Loading step: imposed strain rate 0.2s F/S 0 (MPa) Stress-strain curve l(-) Loading step Tire Expo Conference Köln 9-/0/200 82/ l(-) s/s max (-) l = 2 Stress relaxation curve l =.5 l =
6 Small-strain linear viscoelasticity Material coefficients K=3000MPa G=MPa 2.6 F/S 0 (MPa) Stress-strain curve t = 0.0(s) g = t 2 = 0.2(s) g 2 = Loading step: imposed time s l(-) Loading step l(-) s/s max (-) Stress relaxation curve Tire Expo Conference Köln 9-/0/200 83/
7 Summary Small strain viscoelasticity: Controlled loading strain rate effect on relaxation rate Controlled loading time (actual) no effect on relaxation rate Finite strain hyperviscoelasticity Controlled loading time effect on relaxation rate (non linearity) Tire Expo Conference Köln 9-/0/200 84/9
8 Unit cell 2D axisymmetrical simulation Meshes (axi-symmetrical) f =0% f =20% f =40% 3 f Particle radius: r Inter-particle distance : h-2r 3 2 Loading Imposed time to loading : second 20% (uniaxial tensile) Constitutive relations Matrix: visco-hyperelasticity Particle: elasticity Tire Expo Conference Köln 9-/0/200 85/9
9 Matrix: Finite-strain viscohyperelasticity Inclusion: elasticity Material coefficients Matrix Rivlin: C 0 =C 0 =0.46MPa C 20 =0.5MPa t = 0.0(s) g = 0.3 t 2 = 0.2(s) g 2 =0. Loading step to l=.2 : time s Relaxation = 0s l(-).3 Loading history Inclusion E=80000MPa n=0.3 Particles contents: 0%, 0%, 20%, 40% F/S.4 0 (MPa) Stress-strain curve 40% 20% 0% 0% l(-) s/s max (-) Stress relaxation curve Tire Expo Conference Köln 9-/0/200 86/ % 0% 20% 40%
10 Strain amplification Matrix: Finite-strain viscohyperelasticity Inclusion: elasticity Average stresses (MPa) 20% particles time (s) s =s 33 s Tire Expo Conference Köln 9-/0/200 87/ <l 22 >/l applied (-) % 0% Matrix average strain amplification 20% 40%.E-04.E-03.E-02.E-0.E+00 Matrix proportion Gage length effect: l 0 Dl/l 0 Particles contents(-)
11 Investigation on the matrix under visco-hyperelasticity Material coefficients Rivlin: C 0 =C 0 =0.46MPa C 20 =0.5MPa t = 0.0(s) g = 0.3 t 2 = 0.2(s) g 2 = F/S 0 (MPa) Stress-strain curve Loading step: imposed time s l(-) Loading step Tire Expo Conference Köln 9-/0/200 88/ l(-) s/s max (-) Stress relaxation curve l =.2 l =.5 l =
12 Results Experimental trend captured if: Matrix: visco-hyperelastic/ particle: elastic Imposed time to loading Gage length effect The higher the filler content, the smaller the inter-particle distance, the smaller the local initial gage length. Hyper-viscoelastic response The higher the particles content, the higher the stress level in the matrix, the higher the stress relaxation rate (due to non linear elasticity) Tire Expo Conference Köln 9-/0/200 89/9
13 Extension to 3D : digitized microstructure (Numerical homogenization) Distribution of the inter-particle distance (dispersion) Effect of transverse inter-particle distance Cluster effects A. Jean et al. «Rubber with carbon black fillers: parameters identification of multiscale nanostructure model» in ECCMR 2007 pp. 4 High amount of d.o.f. Filler content: 20% Parallel simulation under shear stress Time consuming Altering to periodic cell Tire Expo Conference Köln 9-/0/200 90/9
14 Periodic cell simulation (effect of inter-particle distance) Meshes (tetrakaidecahedron) f 3 2 d R 3 Loading Imposed time to loading : second 20% Boundary conditions at infinity (uniaxial tension or extension) Constitutive relations Matrix: visco-hyperelasticity Particle: elasticity Tire Expo Conference Köln 9-/0/200 9/9
15 Matrix: Finite-strain viscohyperelasticity Inclusion: elasticity Material coefficients Matrix Rivlin: C 0 =C 0 =0.46MPa C 20 =0.5MPa t = 0.0(s) g = 0.3 t 2 = 0.2(s) g 2 =0. Inclusion E=80000MPa n=0.3 Loading step to l=.2 : time s Relaxation = 0s Particles contents: 0%, 0%, 20%, 40% s/s max (-) s/s max (-) Stress relaxation curve 2D unit cell 0% 0% 20% 40% Tire Expo Conference Köln 9-/0/200 92/ Stress relaxation curve 3D Periodic cells 0% 40% 20%
16 Strain amplification Matrix: Finite-strain viscohyperelasticity Inclusion: elasticity Average stresses in the matrix s = s 22 = s 33 = 446MPa Transverse displacement blocked at infinity! <l 22 >/l applied (-) Matrix average strain amplification % 40% 20% 0%.E-05.E-04.E-03.E-02.E-0.E+00 Matrix proportion Particles contents(-) Tire Expo Conference Köln 9-/0/200 93/9
17 Periodic uniaxial tension vs extension Small strain visco-elasticity Boundary conditions at infinity No effect on the stress relaxation trend Effect on the mean stress (triaxiality) Tire Expo Conference Köln 9-/0/200 94/9
18 Summary Periodic cell vs unit cell, conclusion in agreement for Both global and local strain amplification(gage length effect) Stress relaxation rate Tire Expo Conference Köln 9-/0/200 95/9
19 Meshing: inter-particle distance distribution FE computation on digitized microstructure Third phase (occluded gum, oriented amorphous interphase, crystallization) Can be meshed on periodic cell Thickness/Consitutive equation of the interphase? Constitutive models Self consistent model (Omnes, 2007) Equivalent Homogenized Medium Statistical approach (Boukamel et al., ECCMR 2009, Dresden) Asymptotic filler content Outlooks Saturation at 40% : electrical, mechanical percolation Short inter-particle distance cavitation (tension), incompressibility (compression) Tire Expo Conference Köln 9-/0/200 96/9
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