The nonlinear elastic response of filled elastomers: Experiments vs. theory for the basic case of particulate fillers of micrometer size

ABAQUS Version 6.14 Documentation, 2014. Dassault Systemes Simulia Corp.Providence, RI, USA. Benevides, 2015, Mechanical behavior of the alumina-filled silicone rubber under pure shear at finite strain, Mech. Mater., 85, 57, 10.1016/j.mechmat.2015.02.011 Bergström, 1999, Mechanical behavior of particle-filled elastomers, Rubber Chem. Technol., 69, 781 Berriot, 2003, Reinforcement of model filled elastomers: characterization of the crosslinking density at the filler-elastomer interface by 1h NMR measurements, Polymer, 44, 1437, 10.1016/S0032-3861(02)00882-0 Braides, 1985, Homogenization of some almost periodic coercive functionals, Rend. Accad. Naz. XL, 9, 313 Chi, 2016, A variational formulation with rigid-body constraints for finite elasticity: theory, finite element implementation, and applications, Comput. Mech., 57, 325, 10.1007/s00466-015-1234-2 Danas, 2012, Experiments and modeling of iron-particle-filled magnetorheological elastomers, J. Mech. Phys. Solids, 60, 120, 10.1016/j.jmps.2011.09.006 Diguet, 2010, Shape effect in the magnetostriction of ferromagnetic composite, J. Magn. Magn. Mater., 322, 3337, 10.1016/j.jmmm.2010.06.020 Einstein, 1906, Eine neue bestimmung der moleküldimensionen [A new determination of molecular dimensions], Ann. Phys., 324, 289, 10.1002/andp.19063240204 Frogley, 2003, Mechanical properties of carbon nanoparticlereinforced elastomers, Compos. Sci. Technol., 63, 1647, 10.1016/S0266-3538(03)00066-6 Fukahori, 2007, Generalized concept of the reinforcement of elastomers. part 1: carbon black reinforcement of rubbers, Rubber Chem. Technol., 80, 701, 10.5254/1.3548189 Goudarzi, 2015, Filled elastomers: A theory of filler reinforcement based on hydrodynamic and interphasial effects, J. Mech. Phys. Solids, 80, 37, 10.1016/j.jmps.2015.04.012 Govindjee, 1991, A micromechanically based continuum damage model for carbon black-filled rubbers incorporating mullins effect, J. Mech. Phys. Solids, 39, 87, 10.1016/0022-5096(91)90032-J Gusev, 1997, Representative volume element size for elastic composites: a numerical study, J. Mech. Phys. Solids, 45, 1449, 10.1016/S0022-5096(97)00016-1 Guth, 1938, On the hydrodynamical theory of the viscosity of suspensions, Phys. Rev., 53, 322 Heinrich, 2002, Reinforcement of elastomers, Curr. Opin. Solid State Mater. Sci., 6, 195, 10.1016/S1359-0286(02)00030-X Hill, 1972, On constitutive macro-variables for heterogeneous solids at finite strain, Proc. R. Soc. Lond. A, 326, 131, 10.1098/rspa.1972.0001 Johnston, 2014, Mechanical characterization of bulk sylgard 184 for microfluidics and microengineering, J. Micromech. Microeng., 24, 035017, 10.1088/0960-1317/24/3/035017 Leblanc, 2010 Lefèvre, 2017, A general result for the magnetoelastic response of isotropic suspensions of iron and ferrofluid particles in rubber, with applications to spherical and cylindrical specimens, J. Mech. Phys. Solids, 107, 343, 10.1016/j.jmps.2017.06.017 Lefévre, 2017, Nonlinear electroelastic deformations of dielectric elastomer composites: I — ideal elastic dielectrics, J. Mech. Phys. Solids, 99, 409, 10.1016/j.jmps.2016.07.004 Lefévre, 2017, Nonlinear electroelastic deformations of dielectric elastomer composites: II — non-gaussian elastic dielectrics, J. Mech. Phys. Solids, 99, 438, 10.1016/j.jmps.2016.07.005 Lopez-Pamies, 2010, A new I1-based model for rubber elastic materials, Comptes Rendus Mécanique, 338, 3, 10.1016/j.crme.2009.12.007 Lopez-Pamies, 2010, An exact result for the macroscopic response of particle-reinforced neohookean solids, J. Appl. Mech., 77, 021016, 10.1115/1.3197444 Lopez-Pamies, 2013, The nonlinear elastic response of suspensions of rigid inclusions in rubber: I — An exact result for dilute suspensions, J. Mech. Phys. Solids, 61, 1, 10.1016/j.jmps.2012.08.010 Lopez-Pamies, 2013, The nonlinear elastic response of suspensions of rigid inclusions in rubber: II — A simple explicit approximation for finite concentration suspensions, J. Mech. Phys. Solids, 61, 19, 10.1016/j.jmps.2012.08.013 Lopez-Pamies, 2006, On the overall behavior, microstructure evolution, and macroscopic stability in reinforced rubbers at large deformations: I — Theory, J. Mech. Phys. Solids, 54, 807, 10.1016/j.jmps.2005.10.006 Mark, 2007 Meddeb, 2019, Extreme enhancement of the nonlinear elastic response of elastomer nanoparticulate composites via interphases, Compos. Part B, 156, 166, 10.1016/j.compositesb.2018.08.064 Meinecke, 1988, Effect of carbon-black on the mechanical properties of elastomers, Rubber Chem. Technol., 61, 534, 10.5254/1.3536199 Michel, 1999, Effective properties of composite materials with periodic microstructure: a computational approach, Comput. Methods Appl. Mech. Eng., 172, 109, 10.1016/S0045-7825(98)00227-8 Mooney, 1940, A theory of large elastic deformation, J. Appl. Phys., 11, 582, 10.1063/1.1712836 Müller, 1987, Homogenization of nonconvex integral functionals and cellular elastic materials, Arch. Rat. Mech. Anal., 99, 189, 10.1007/BF00284506 Mullins, 1965, Stress softening in rubber vulcanizates. Part I. use of a strain amplification factor to describe the elastic behavior of filler-reinforced vulcanized rubber, J. Appl. Polym. Sci., 9, 2993, 10.1002/app.1965.070090906 Ogden, 1972, Large deformation isotropic elasticity — on the correlation of theory and experiment for incompressible rubberlike solids, Proc. R. Soc. Lond. A, 326, 565, 10.1098/rspa.1972.0026 Ogden, 2004, Fitting hyperelastic models to experimental data, Comput. Mech., 34, 484, 10.1007/s00466-004-0593-y Poulain, 2017, Damage in elastomers: nucleation and growth of cavities, micro-cracks, and macro-cracks, Int. J. Fract., 205, 1, 10.1007/s10704-016-0176-9 Poulain, 2018, Damage in elastomers: healing of internally nucleated cavities and micro-cracks, Soft Matter, 14, 4633, 10.1039/C8SM00238J Qu, 2011, Nanoscale visualization and multiscale mechanical implications of bound rubber interphases in rubber carbon black nanocomposites, Soft Matter, 7, 1066, 10.1039/C0SM00645A Roscoe, 1973, Isotropic composites with elastic or viscoelastic phases: general bounds for the moduli and solutions for special geometries, Rheol. Acta, 12, 404, 10.1007/BF01502992 Schöbel, 1997, Netgen an advancing front 2d/3d-mesh generator based on abstract rules, Comput. Visual. Sci., 1, 41, 10.1007/s007910050004 Segurado, 2002, A numerical approximation to the elastic properties of sphere-reinforced composites, J. Mech. Phys. Solids, 50, 2107, 10.1016/S0022-5096(02)00021-2 Smallwood, 1944, Limiting law of the reinforcement of rubber, J. Appl. Phys., 15, 758, 10.1063/1.1707385 Song, 2016, Concepts and conflicts in nanoparticles reinforcement to polymers beyond hydrodynamics, Prog. Mater. Sci., 84, 1, 10.1016/j.pmatsci.2016.09.002 Spinelli, 2015, Dielectric elastomer composites: a general closed-form solution in the small-deformation limit, J. Mech. Phys. Solids, 83, 263, 10.1016/j.jmps.2015.06.009 Tadiello, 2015, The filler-rubber interface in styrene butadiene nanocomposites with anisotropic silica particles: morphology and dynamic properties, Soft Matter, 11, 4022, 10.1039/C5SM00536A Valentin, 2008, Uncertainties in the determination of cross-link density by equilibrium swelling experiments in natural rubber, Macromolecules, 41, 10.1021/ma8005087 Wagner, 1976, Reinforcing silicas and silicates, Rubb. Chem. Technol., 49, 703, 10.5254/1.3534979 Wiegand, 1937, The carbon reinforcement of rubber, Rubb. Chem. Tech., 10, 395, 10.5254/1.3538994 Zou, 2008, Polymer/silica nanocomposites: preparation, characterization, properties, and applications, Chem. Rev., 108, 3893, 10.1021/cr068035q