Second-gradient continua as homogenized limit of pantographic microstructured plates: a rigorous proof

Since the works by Gabrio Piola, it has been debated the relevance of higher-gradient continuum models in mechanics. Some authors even questioned the logical consistency of higher-gradient theories, and the applicability of generalized continuum theories seems still open. The present paper considers a pantographic plate constituted by Euler beams suitably interconnected and proves that Piola's heuristic homogenization method does produce an approximating continuum in which deformation energy depends only on second gradients of displacements. The I-convergence argument presented herein shows indeed that Piola's conjecture can be rigorously proven in a Banach space whose norm is physically dictated by energetic considerations.