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Mechanical characteristics of polymers are markedly different depending on the temperature. At lower temperatures and up to a temperature known as the glass transition temperature Tg, polymers behave as glass-like materials with a high elasticity modulus and a low fracture strain. At higher temperatures than Tg, they behave as rubber-like materials with the ability to undergo large and reversible elastic deformations known as rubber elasticity. They are called hyperelastic materials and have a strain energy function W, a scalar function of the strain tensor, whose derivative with respect to the strain components determines the corresponding stress components (Fung, 1965). Many strain energy functions were proposed and were extensively used for the evaluation of the deformation behaviors of hyperelastic materials. The details of the concrete forms of strain energy functions and their experimental verifications were seen in Ogden (1977) and others. On the other hand, the constitutive equations of rubber elasticity using macromolecular network models were proposed based on the concept of a network of chains of randomly oriented rigid links that are connected at the chemical cross-links between macromolecules. The overall properties of the network are then obtained by summing the contributions of the individual chains. Non-Gaussian statistics were employed to single chain configurations taking into account the finite extendibility of molecular chains (Kuhn and Grun,1942, James and Guth, 1943). However, a rigorous treatment for summing the contributions of the individual chains, averaging process was not made available for arbitrary deformation until the development of computational resources. Many approximate methods have been proposed for the averaging process (James and Guth, 1943, Arruda and Boyce, 1993, Wu and van der Giessen, 1993) and have been employed toward the evaluation of the deformation behaviors of rubber. It is well known that the mechanical characteristics of rubbers are markedly improved by filling the hard particles. Therefore, their usage extends to an extensive range of engineering fields. Proper predictions of mechanical

Constitutive equations of rubber based on molecular chain network model and evaluation of deformation behaviors of particle-filled rubbers