Identification of microstructured continua with Gedanken experiments

In this lecture I will analyze the role of the principle of virtual work in continuum mechanics and an introduction of first and second gradient continua. Characterization of first gradient elastic continua is standard in the literature. Recently, the same problem for second gradient materials has been investigated. Even though it is not complete, a dynamic investigation is done with an analysis of the dispersion relation. Another possibility is a static characterization, that is achieved through proper ideal (or gedanken) experiments. In particular, we will consider an isotropic second gradient elastic two-dimensional solid, for which the deformation (strain or internal) energy depends on 6 constitutive parameters: the 2 Lam\'e constants and 4 more coefficients. Therefore, gedanken experiments are designed to allow to write equations that have as unknowns the six constants that define the internal energy and as known terms the values of the experimental measurements of appropriately selected quantities. Besides, we relax the isotropic hypothesis and consider a D4 orthotropic material. The reason is that such anisotropy is the most general for pantographic structures. The corresponding strain energy depends on 9 constitutive parameters, that will be identified in terms of the material characteristics of the beams composing the pantographic structure.