Second Law of Thermodynamics and Strain Gradient Theories of Elasticity

The paper addresses the second law of thermodynamics through the Clausius-Duhem inequality in its general form, with entropy flux and entropy production given by suitable constitutive functions. For definiteness the paper investigates possible models of elastic solids where, to account for non-local properties, the stress depends on strain gradients up to second order. While previous approaches are developed through variational formulations or by applying the virtual power method, here it is shown that no change in the energy balance or the form of kinetic energy is necessary; it is sufficient that the entropy flux be given by a suitable constitutive function. The paper also emphasizes that non-local constitutive properties arise from the Clausius-Duhem thermodynamic inequality, while variational formulations and the virtual power method are in fact limited to the purely mechanical context, as they involve only the equation of motion.