Thermodinamically-consistent dynamic continualization of block-lattice materials

Lattice-like materials featuring periodic planar tessellation of regular rigid blocks connected by linear elastic interfaces and chiral or achiral properties are considered. The chirality results from a uniform rotation of the blocks with respect to their centroidal joining line and leads to interesting auxetic and dispersive acoustic behaviors. The governing equations of the discrete Lagrangian model are properly manipulated via the novel enhanced continualization scheme in such a way to obtain equivalent non-local integral and gradient-type higher-order continua. Based on the formal Taylor series expansion of the integral kernels or the corresponding pseudo-differential functions accounting for shift operators and proper pseudo-differential downscaling laws, the proposed enhanced continualization technique allows formulating homogeneous non-local continuum models of increasing orders, analytically featured by characteristic non-local constitutive and inertial terms. The enhanced continualization shows thermodynamic consistency in the definition of the overall non-local constitutive tensors, as well as qualitative agreement and quantitative convergent matching of the actual complex frequency band structure. The theoretical findings are successfully verified though the study of wave dispersion and attenuation properties as referred to a representative tetrachiral geometry.