Generalized continua. Foundations, material modeling, and uncertainty quantification.

This dissertation is devoted to the modeling of architectured metamaterials and particle-based materials. Such materials generally necessitate the use of sophisticated continuum models to properly characterize their mechanical behavior and make accurate numerical predictions. To this purpose, generalized continua, which improve the classical (or Cauchy) continuum, have been a topic of interest in the last decade. This dissertation begins with a uniform presentation of four different generalized continua, namely the micropolar continuum, the micromorphic continuum, the second-gradient continuum, and the second-gradient-micropolar continuum. Special attention is paid to the Euler-Lagrange equations that are derived using the least action principle and the Levi-Civita tensor calculus. Then, the dissertation focuses on pantographic structures that represent a paradigmatic case of architectured metamaterials described by generalized continua. A novel torsional energy for pantographic sheets is proposed and experimentally validated. A novel second-gradient continuum model for pantographic blocks is proposed and experimentally validated through digital volume correlation techniques. The effect of pivots-related local defects on the mechanical response of pantographic sheets is investigated via a noninformative prior probabilistic model. Finally, random generalized continuum models for particle-based materials with uncertain constitutive parameters and fields are analyzed: response and numerical identification of random Timoshenko-Ehrenfest beams are carried out via a noninformative prior probabilistic model, and sensitivity analysis of a second-gradient continuum model for particle-based materials is performed via an informative prior probabilistic model based on the maximum entropy principle. To make the work self-consistent, in Appendix, an overview of Levi-Civita tensor algebra is given.