Homogenization of Thermal Properties in Metaplates

Three-dimensional metamaterials endowed with two-dimensional in-plane periodicity exhibit peculiar thermoelastic behaviour when heated or cooled. By proper design of the unit cell, the equivalent thermal expansion coefficient can be programmed and can also reach negative values. The heterogeneity in the third direction of such metamaterials also causes, in general, a thermal-induced deflection. The prediction of the equivalent thermal properties is important to design the metamaterial suitable for a specific application. Under the hypothesis of small thickness with respect to the global in-plane dimensions, we make use of asymptotic homogenization to describe the thermoelastic behaviour of these metamaterials as that of an equivalent homogenous plate. The method provides explicit expressions for the effective thermal properties, which allow for a cost-effective prediction of the thermoelastic response of these metaplates.