Modelling the mechanical behaviors of double-network hydrogels

Abstract

Double-network hydrogels (DN gels), consisting of a rigid, highly cross-linked short-chain network and a flexible, loosely cross-linked long-chain network, are a kind of promising soft and tough material with superior mechanical properties. In order to capture the mechanical behaviors of DN gels more accurately and with more physical depth than some existing models, a constitutive formulation is developed in this paper. The Helmholtz free energy function is decomposed into three parts accounting for the free energies of each network and the mixing of all the components. The eight-chain worm-like chain (WLC) based model is then employed to describe the deformation of each network with two network structure parameters: the chain density and the average chain length. Based on the existing experimental investigations of the internal fracture process of DN gels, a detailed and clear physical picture is illustrated. According to this damage picture, the two network structure parameters of the short-chain network are regarded as two internal variables. Within the framework of general thermodynamics, the energy dissipation is defined through the variation of the internal variables, which allows one to obtain the evolutions of internal variables from analyzing the macroscopic dissipated behaviors of DN gels. For the cases without energy dissipation, the network structure parameters are constant and can be determined directly through the stress-strain relation at one typical point on the stress-strain curve and the stored Helmholtz free energy of each state. These determined values of the parameters are then used to validate the mass conservation. For the cases with energy dissipation, the evolution equations of the two internal variables are determined through the analysis of the energy dissipation and the mass conservation. As a result, not only the unloading stress-stain curves but also the primary loading curve can be theoretically predicted by the constitutive formulation. The mechanical behaviors of both t-DN gels and c-DN gels before necking are analyzed. The comparisons of the theoretical results with the experimental data show sufficiently good agreements, demonstrating the ability and efficiency of the formulation to capture the Mullins effect of DN gels.