Brauer diagram model of suppositional phantom network in double-network gel undergoing mechanochemical coupling

•A Brauer diagram model is formulated to describe the suppositional phantom double networks and explore their coupling effects.•A constitutive relationship among mechanochemical coupling, rubber elasticity, and stress-elongation ratio has been proposed.•Effectiveness of the Brauer diagram model was verified using analytical data from finite element analysis and molecular dynamics. Complex coupling of brittle and stretchable networks in double-network (DN) gel has been a hot research topic to understand working principle of ultra-high mechanical toughness, in comparison with the conventional gels. In this study, both the percolation theory and phantom network model were employed to formulate a Brauer diagram model, which was used to describe the suppositional phantom double networks and explore their coupling effects in DN gel. The mechanochemical coupling was investigated using the proposed Brauer diagram model, in which three critical parameters of coupling fraction, surviving probability and phantom degree, were proposed to formulate a constitutive relationship among rubber elasticity, mechanochemical coupling and stress-elongation ratio. Finally, effectiveness of the Brauer diagram model was verified using analytical data from the finite element analysis and molecular dynamics, as well as experimental results of DN gels reported in the literature. Schematic illustration of theoretical modelling process of networks in DN gel based on the percolation theory and phantom network model [Display omitted]