<p>Microstructured mechanical metamaterials exhibit size-dependent behaviors that necessitate generalized continuum models beyond classical mechanics. This paper develops a novel variational asymptotic homogenization scheme to derive a second-gradient continuum model of a periodic lattice metamaterial. The theoretical framework relies on variational principles and asymptotic expansions to upscale micro-scale behavior, incorporating higher-order gradients to capture nonlocal effects. We derive effective stiffness and higher-order modulus tensors, formulate the governing higher-order partial differential equations, and analyze wave propagation to identify frequency band gaps. Analytical solutions validate the model, demonstrating its ability to describe size-dependent stiffness and dynamic phenomena. The derivations ensure thermodynamic consistency, aligning with the principles of continuum mechanics. This work advances the theoretical modeling of microstructured materials for the study of higher-order continuum mechanics.</p>