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We study the effect of a large obstacle on the so-called residence time, i.e., the time that a particle performing a symmetric random walk in a rectangular (two-dimensional, 2D) domain needs to cross the strip. We observe complex behavior: We find out that the residence time does not depend monotonically on the geometric properties of the obstacle, such as its width, length, and position. In some cases, due to the presence of the obstacle, the mean residence time is shorter with respect to the one measured for the obstacle-free strip. We explain the residence time behavior by developing a one-dimensional (1D) analog of the 2D model where the role of the obstacle is played by two defect sites having smaller probability to be crossed with respect to all the other regular sites. The 1D and 2D models behave similarly, but in the 1D case we are able to compute exactly the residence time, finding a perfect match with the Monte Carlo simulations.

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Pantographic fabrics are presented as a paradigmatic example to discuss the research perspective on multiphysics and multiscale materials. Reduced-order modeling, obtained by introducing higher-gradient or microstructured continua, shows much smaller computational needs than those required by full-scale 3D modeling calculations and by equivalent discrete spring systems. Researches already available in the literature compare theoretical predictions with results obtained in real experiments, analyzing both in-plane and outof-plane deformations possibly induced by local buckling phenomena. The goal is to achieve three major objectives: (1) formulate coarse-scale nonlinear higher-gradient continuum models describing a more general class of pantographic metamaterials, based on finer-scale descriptions, through homogenization techniques;(2) implement finite element analyses with shape functions possessing higher regularity or employing mixed formulations to perform simulations with the above-formulated models; and (3) validate and verify the derived models through the acquisition and analysis of experimental data. These goals would likely push toward the improvement of 3D printing protocols to enhance the quality of the pantographic prototypes and DIC as experiments measurement technique. It can be also conjectured that many macroscopic deformation energies can be synthesized by using as elementary elements, inside periodicity cells, some pantographic modules.