On the Axiom of the Second Law of Thermodynamics and the Modelling of Flexoelectricity

The present paper addresses models of non-local materials and, in particular, flexoelectric materials as dielectrics with higher-order gradients. While the literature develops the subject through variational formulations, here the modelling is developed within a thermodynamic framework with a general form of the Clausius–Duhem inequality, where the extra-entropy flux and the entropy production are given by constitutive functions. Furthermore, the polar character of the material, associated with flexoelectricity, is modelled by a body couple and no use is made of hyper-stresses. As a result of the thermodynamic analysis, it follows that the symmetric part of the stress and the electric polarization are given by a variational derivative of the electric enthalpy along with the dissipative parts that involve the selected function of entropy production. The skewed part of the stress is shown to be free from thermodynamic restrictions and is simply required to counterbalance the body couple.