The effect of network chain distribution on mechanical behavior of elastomers is one of the long standing problems in rubber mechanics. The classical theory of rubber elasticity is built upon the assumption of entropic elasticity of networks whose constitutive strands are of uniform length. The kinetic theories for vulcanization, computer simulations, and indirect experimental measurements all indicate that the microstructure of vulcanizates is made of polymer strands with a random distribution of length. The polydispersity in strand length is expected to control the mechanical strength of rubber as the overloaded short strands break at small deformations and transfer the load to the longer strands. The purpose of this contribution is to present a simple theory of rubber mechanics which takes into account the length distribution of strands and its effect on the onset of bulk failure.

In this paper, a new hybrid unscented Kalman (UKF) and unscented H ∞ (U H ∞ F) filter is presented that can adaptively adjust its performance better than that of either UKF and/or U H ∞ F , accordingly. In this way, two Takagi-Sugeno-Kang (TSK) fuzzy logic systems are presented to adjust automatically some weights that combine those UK and U H ∞ filters, independent of the dynamics of the problem. Such adaptive fuzzy hybrid unscented Kalman/ H ∞ filter (AFUK H ∞ ) is based on the combination of gain, a priori state estimation, and a priori measurement estimation. The simulation results of an inverted pendulum and a re-entry vehicle tracking problem clearly demonstrate robust and better performance of this new AFUK H ∞ filter in comparison with those of both UKF and U H ∞ F, appropriately. It is shown that, therefore, the new presented AFUK H ∞ filter can simply eliminate the need for either UKF or U H ∞ F effectively in the presence of Gaussian and/or non-Gaussian noises.

An a priori assumption in micromechanical analysis of polymeric networks is that the constitutive polymer strands are of equal length. Monodisperse distribution of strands, however, is merely a simplifying assumption. This paper relaxes this assumption and considers a vulcanized network with a broad distribution of strand length. In the light of this model, this study predicts the damage initiation and stress–stretch dependency in filled polymer networks with random internal structures. The degradation of network mechanical behavior is assumed to be controlled by the adhesive failure of the strands adsorbed to the filler surface. This study shows that the short adsorbed strands are the culprits for damage initiation and their finite extensibility is a key determinant of the mechanical strength.

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