A sigmoid function to characterise the mechanical behaviour of rubber materials

- A phenomenological hybrid chain network model for polymers is proposed.
- It captures the sigmoidal nature of stress-stretch behaviour and stiffness rise.
- Transition from 3-chain to 8-chain network happens earlier for biaxial deformation.
- Validity of the new model is verified for experimental data on spider silk & rubber.

A new phenomenological hybrid chain network model for polymer based materials has been developed. At the microscopic scale, the “beta function for growth rate”, as introduced by Yin et al. [30,29] for describing growth, has been taken to describe the sigmoidal shape of the stiffness rise to the breaking point at high strains of a single chain molecule. In this paper the model is redefined and used to model an entropy-elastic (rubber elastic) behaviour in relatively small strain regimes. In order to capture large strain regimes while maintaining the overall sigmoidal shape of the stiffness rise the model is extended by an energy-elastic term. On the macroscopic scale, the network of polymer chains are represented by isotropic three-chain and eight-chain network models that are linked together by a weighting function for describing the transition between them.The proposed model has been calibrated for different polymers simultaneously in the extensional range. It successfully predicted the experimental findings from literature, Treloar [21,22], other material data for Spider silk reported by Denny [4] and data of natural rubber compounds of different hardnesses reported by Koshal [12].

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