The molecular chain network model for elastic deformation behavior and the reptation theory for viscoelastic deformation behavior are used to derive a constitutive equation for rubber. The new eight-chain-like model contains eight standard models consisting of Langevin springs and dashpot to account for the interaction of chains with their surroundings. Monotonic and cyclic deformation behavior of rubber with relaxation under different strain rates have been examined. The results reveal the roles of the individual springs and dashpot, and the strain rate dependence of materials in the monotonic and cyclic deformation behaviors, particularly softening and hysteresis loss, that is, the Mullins effect, occurring in stress-stretch curves under cyclic deformation processes. The validity of the results is checked through comparison with experimental results. The deformation behaviors of a plane strain rubber unit cell containing Carbon-Black (CB) under monotonic and cyclic straining are investigated by computational simulation using the proposed constitutive equation and homogenization method. The results reveal the AEPA2006 Special Issue Tomita et. al.-3-substantial enhancement of the resistance of CB-filled rubber to macroscopic deformation, which is caused by the marked orientation hardening due to the highly localized deformation of rubber. The role of strain rate sensitivity on such characteristic deformation behaviors as increases in the resistance to deformation, hysteresis loss, and the effects of the distribution morphology and the volume fraction of CB on the deformation behavior is clarified. The increases in the volume fraction and in the aggregation of the distribution of CB substantially raise the resistance to deformation and hysteresis loss. Volume fraction of CB 19% Volume fraction of CB 16% (a) (b) Volume fraction of CB 19% Volume fraction of CB 16%

We investigate the characteristic deformation behavior of rubber with carbon black (CB) filler. The deformation behaviors of a plane strain rubber unit cell containing CB fillers under monotonic and cyclic strain are investigated by computational simulation with a nonaffine molecular-chain network model. The results reveal the substantial enhancement of the resistance of the rubber to macroscopic deformation, which is caused by the marked orientation hardening due to the highly localized deformation in the rubber. The disentanglement of the molecular chain during the deformation of rubber results in the magnification of the hysteresis loss, i.e., the Mullins effect, occurring in stress-stretch curves under cyclic deformation processes. The increase in volume fraction and in aggregation of the distribution of CB substantially raises the resistance of the rubber to deformation and hysteresis loss. The effect of the heterogeneous distribution of the initial average number of segments of molecular chains on the hysteresis loss has been clarified. r

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We visualized the strain field of a polymer matrix using a finite element method (FEM) simulation based on knowledge of the three-dimensional (3D) structural configuration of silica particles in rubber. The 3D structural configuration was obtained using the Zernike-type phase contrast X-ray imaging method. Based on the structural information, the inhomogeneous local deformation of the rubber matrix was visualized using the FEM simulation.

The rubbers containing various kinds of fillers exhibit fabulous mechanical characteristic and therefore, they are widely used in various production. Owing to the wide range of controllability in mechanical characteristics by adding the coupling agent, silica-filled rubber draws attention for extensive usage. Here, to clarify the mechanism of the marked increase in deformation resistance in silica-filled rubber in detail, we will construct the finite element homogenization models of silica-filled rubber. These models can reflect various experimental observations that include changes in microscopic structural characteristics such as distribution morphology of silica particles, the thickness of the interfacial phase between silica and rubber, and the networklike gel structures developed from the interfacial phase. The obtained results clarified the essential physical enhancement mechanisms of deformation resistance and hysteresis loss, i.e., the Mullins effect, for rubber filled with silica. The volume fraction of the silica coupling agent essentially affects the deformation behavior of silica-filled rubber suggesting the high controllability of the material characteristics of silica-filled rubber compared with carbon-black-filled rubber. Although, the present model underestimates the hysteresis loss as compared with the experimental results, it has a capability to evaluate the effect of adding coupling agents on the fundamental deformation behavior of silica filled rubber.

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