Modeling the mechanics of covalently adaptable polymer networks with temperature-dependent bond exchange reactions

We study a class of polymers with bond exchange reactions that can rearrange the polymer network topology while maintaining the network integrity. Examples include the recently developed epoxy-acid network with transesterification reaction and polybutadiene polymer with olefin metathesis reaction. This class of polymers, although covalently crosslinked and thus insoluble, are rendered malleable through viscoelastic flow caused by the bond exchange reactions. The reaction also facilitates unique and technologically important behavior such as self-healing, thermoforming, and recyclability. Here we develop a thermodynamically consistent continuum model to describe the coupling between the macroscopic thermo-mechanics and the microscopic network-altering chemical reactions. Our basic approach is to decompose the network into groups of chains that are reformed at different times during the loading history; at any instant the network strain energy is obtained by summing the contributions from all chains. Evolution of the chain composition, determined by the temperature-dependent chemical reaction kinetics, changes the total strain energy and leads to macroscopic relaxation. Our model is applicable to general three-dimensional finite deformation with arbitrary thermal and mechanical boundary conditions. In the case of isothermal processes, only four material parameters are required: two describing the mechanical behavior and two describing the chemical kinetics. Using the epoxy-acid network as an example, we show that our model captures the key features of recently reported experiments. An important feature of these materials is that they can be processed like a thermoplastic, while retaining the appealing properties of a thermoset. Specifically, one can deform the material and then heat it to relax stresses, after which it will maintain its deformed shape, although there may be some elastic springback. To this end, we use our model to derive simple analytical estimates for a fixity parameter, defined as the ratio of the permanent deformation after the thermomechanical process to the prescribed deformation. We also implement our model in a 3D nonlinear finite element code and use it to study two representative thermoforming examples: twisting of a strip and indentation of a substrate. We extract the fixity parameter from the finite element results and show that our analytical estimates provide a reasonable approximation.