The remarkable bending properties of perforated plates

Motivated by recent experiments on the bending of porous strips towed through a fluid bath, a combined theoretical and experimental study is made of the bending response of perforated plates. The focus is on the practically relevant class of thin plates (with thickness h) made of a homogeneous isotropic material that is perforated with periodic distributions (with unit-cell size ε) of monodisperse holes spanning a large range of porosities, from the dilute limit to nearly the percolation threshold. From the theoretical point of view, with the objective of quantifying the roles that the various constitutive and geometric inputs play on the their bending response, the perforated plates are modeled by means of three different approaches: (i) as 3D structures made of a perforated nonlinear elastic material and as 2D structures made of homogeneous linear elastic materials whose effective properties result from (ii) first taking the limit of dimension reduction (h↘0) and then that of homogenization (ε↘0) and, vice versa, (iii) first taking the limit of homogenization (ε↘0) and then that of dimension reduction (h↘0). From the experimental point of view, laser-engraved molds are utilized to fabricate perforated elastomeric plates with hexagonal distributions of elliptical holes. The resulting perforated plates are then subject to cantilever-bending due to their self weight and their pointwise deformation measured by means of X-ray tomography. Remarkably, counter to the general expectation from available mathematical results for heterogeneous plates at large, the results indicate that the bending response of perforated plates is fairly insensitive to whether the holes are smaller or larger than the plate thickness. Instead, it is dominated by their porosity, while the spatial distribution and shape of the underlying holes only have relatively marginal effects.