A phase transition model describing auxetic materials

In this paper we introduce a new model describing the behavior of auxetic materials in terms of a phase-field PDE system. More precisely, the evolution equations are recovered by a generalization of the principle of virtual power in which microscopic motions and forces, responsible for the phase transitions, are included. The momentum balance is written in the setting of a second gradient theory, and it presents nonlinear contributions depending on the phases. The evolution of the phases is governed by variational inclusions with non-linear coupling terms. By use of a fixed point theorem and monotonicity arguments, we are able to show that the resulting initial and boundary value problem admits a weak solution.