Modeling of the Effect of Rigid Fillers on the Stiffness of Rubbers Vineet Jha, Amir A. Hon, * Alan G. Thomas, James J. C. Busfield Department of Materials, Queen Mary, University of London, London E1 4NS, United Kingdom Received 3 May 2007; accepted 10 August 2007 DOI 10.1002/app.27324 Published online 13 November 2007 in Wiley InterScience (www.interscience.wiley.com). ABSTRACT: The theories that predict the increase in the modulus of elastomers resulting from the presence of a rigid filler are typically derived from Einstein’s viscosity law. For example, Guth and Gold used this approach to predict how the Young’s modulus of an elastomer is related to the filler volume fraction. Hon et al. have shown using finite element microstructural models that stiffness predictions at small strains were also possible. Here, microstructural finite element models have been used to investigate the modulus of filled elastomer over a wider range of strains than has been possible previously. At larger strains, finite extensibility effects are significant and here an appropriate stored energy function proposed by Gent was adopted. In this work, the effect of spherical MT-type carbon-black filler behavior was considered. Dif- ferent models were made and the results were then com- pared to experimental measurement of the stiffness taken from the literature. It is shown that the boundary condi- tions of the microstructural unit cell lie between the two extremes of free surfaces and planar surfaces. Also as the filler volume fraction increases then the number of filler particles required in the representative volume to predict the actual stiffness behavior also increases. Ó 2007 Wiley Periodicals, Inc. J Appl Polym Sci 107: 2572–2577, 2008 Key words: elastomers; fillers; modeling; modulus; rubber INTRODUCTION Elastomers deform to large strains under load and recover to their original shape upon unloading. Rein- forcing rigid fillers such as carbon black are added to elastomers to increase the mechanical properties such as the modulus, strength, and wear resistance. The addition of these rigid fillers therefore increases the range of properties available for using elastomers in industrial applications. A theory for the stiffening of elastomers by car- bon-black fillers is based on Einstein’s theory 1,2 for the increase in viscosity of a suspension due to the presence of spherical colloidal particles. The Einstein equation is given as h ¼ h 0 ð1 þ 2:5/Þ (1) where h is the viscosity of suspension, h 0 is the vis- cosity of the incompressible fluid, and / is the vol- ume fraction of the spherical particles. Guth and Gold 3 and Smallwood 4 adapted the vis- cosity law given in Eq. (1) to predict the modulus of an elastomer filled with rigid spherical particles and they included an additional term to account for the interaction of rigid fillers at larger filler volume frac- tions. They proposed that the increase in the modu- lus due to the incorporation of spherical rigid fillers was given by E ¼ E 0 ð1 þ 2:5/ þ 14:1/ 2 Þ; (2) where E is the modulus of the filled rubber, E 0 is the modulus of the unfilled rubber, and / is the filler volume fraction. This relation also assumes that the carbon-black filler particles are spherical, well dis- persed throughout the matrix, and that each is per- fectly bonded to the rubber. The prediction of the stiffness of filled rubber is well described at small strains below about 10% by the relationship derived by Guth and Gold 3 and Smallwood. 4 Hon et al. 5 have shown that these stiff- ness predictions at small strains were well repre- sented using finite element microstructural models. For moderate strain of less than 100%, Kashani and Padovan 6 have shown that the mechanical properties of filled rubbers follow a spring in series model, sim- ilar to the one shown in Eq. (2). Further, the rubber matrix and rigid fillers are in state of uniform stress rather than state of uniform strain. However, at larger strains these existing theories cannot predict the stiffness accurately and the microstructural finite element approach has not been tried. Hence, the present work uses micro mechanical modeling to predict large strain behavior including a considera- *Present address: Universiti Teknologi PETRONAS, Ban- dar Seri Iskandar, 31750 Tronoh, Perak, Malaysia. Correspondence to: James J. C. Busfield (j.busfield@qmul. ac.uk). Journal of Applied Polymer Science, Vol. 107, 2572–2577 (2008) V V C 2007 Wiley Periodicals, Inc.