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

Jean-Jacques Alibert 1 Alessandro Della Corte 2
1 IMATH - Institut de Mathématiques de Toulon - EA 2134
2 DIMA - Dipartimento di Ingegneria Meccanica e Aerospaziale
Abstract : 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 Gamma-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.
Keywords : Elasticity Microstructure Pantographic structures Gamma-convergence Homogenization Second-gradient continua Continuum mechanics
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Journal articles
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Contributor : Jean-Jacques Alibert <>
Submitted on : Friday, February 26, 2016 - 12:20:42 PM
Last modification on : Tuesday, June 19, 2018 - 3:50:01 PM

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Jean-Jacques Alibert, Alessandro Della Corte. Second-gradient continua as homogenized limit of pantographic microstructured plates: a rigorous proof. Zeitschrift für Angewandte Mathematik und Physik, Springer Verlag, 2015, 66 (5), pp 2855-2870. ⟨10.1007/s00033-015-0526-x⟩. ⟨hal-01279491⟩

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