Self-consistent Sierpinski iteration of toughening mechanism in elastomer undergoing scaled segment-chain-network

Understanding working principles and toughening mechanisms of soft elastomers has been a huge challenge due to their significant scaling effects from molecules to bulk polymer. In this study, by combining Sierpinski fractal and scaling theory, an extended Kelvin model is developed to investigate the mechanical behaviour of the soft elastomer undergoing scaled segment-chain-network. According to the scaling theory, a radial distribution function was initially introduced to explain the diffusion and relaxation behaviours of molecular segments with the Sierpinski fractal features. The self-consistent iteration of the Sierpinski fractal is then used to describe the scaling effects of segments and networks. Rubber elasticity of the polymer network is further formulated based on the self-consistent iteration equation and scaling theory. A constitutive stress–strain relationship is also derived to explore the toughness mechanism and working principle in the polymer elastomer. Finally, the effectiveness of the proposed model is verified using finite-element analysis and experimental results reported in the literature, to explore a scaling insight into toughening mechanisms of elastomers governed by the self-consistent Sierpinski iteration.