The Effect of Membrane Load on the Usage of Berger’s Model in Electrostatically Actuated Pre-Stressed Circular Curved Micro Plates

The effect of membrane load on the behaviour of axisymmetric bistable circular curved microplates on Berger’s based axisymmetric reduced order (RO) model, incorporating radial prestress, is studied. The model is first validated for a “mechanical” load, against a Föppl-von-Kármán’s RO model with twenty degrees of freedom (DOF), a finite differences (FD) solution and a finite elements (FE) model, serving as the reference. All solutions implement the “Riks” method to track possible unstable branches, which can swerve in due to the presence of higher buckling modes. A convergence study is carried out for the snap-through location and load, as well as for the critical elevation and prestress required for bistability. Based on validated results of the analysis, the reliability of the model for predicting the effect of prestress on the plate behaviour under nonlinear electrostatic load is then investigated while using FD solutions as the reference. The study furnishes a reliable expended RO model, which includes prestress on the as-fabricated curved plate. The resulting model can further be used to estimate the value of residual prestress, present in an electrostatically actuated curved plate, based on its response.

Strain Gradient Theory Based Dynamic Mindlin-Reissner and Kirchhoff Micro-Plates with Microstructural and Micro-Inertial Effects

In this study, a dynamic Mindlin–Reissner-type plate is developed based on a simplified version of Mindlin’s form-II first-strain gradient elasticity theory. The governing equations of motion and the corresponding boundary conditions are derived using the general virtual work variational principle. The presented model contains, apart from the two classical Lame constants, one additional microstructure material parameter g for the static case and one micro-inertia parameter h for the dynamic case. The formal reduction of this model to a Kirchhoff-type plate model is also presented. Upon diminishing the microstructure parameters g and h, the classical Mindlin–Reissner and Kirchhoff plate theories are derived. Three points distinguish the present work from other similar published in the literature. First, the plane stress assumption, fundamental for the development of plate theories, is expressed by the vanishing of the z-component of the generalized true traction vector and not merely by the zz-component of the Cauchy stress tensor. Second, micro-inertia terms are included in the expression of the kinetic energy of the model. Finally, the detailed structure of classical and non-classical boundary conditions is presented for both Mindlin–Reissner and Kirchhoff micro-plates. An example of a simply supported rectangular plate is used to illustrate the proposed model and to compare it with results from the literature. The numerical results reveal the significance of the strain gradient effect on the bending and free vibration response of the micro-plate, when the plate thickness is at the micron-scale; in comparison to the classical theories for Mindlin–Reissner and Kirchhoff plates, the deflections, the rotations, and the shear-thickness frequencies are smaller, while the fundamental flexural frequency is higher. It is also observed that the micro-inertia effect should not be ignored in estimating the fundamental frequencies of micro-plates, primarily for thick plates, when plate thickness is at the micron scale (strain gradient effect).

Biogeography of Italy revisited: genetic lineages confirm major phylogeographic patterns and a pre-Pleistocene origin of its biota

Warm-adapted species survived the cold stages of the past glacial-interglacial cycles in southern European peninsulas and recolonized major parts of Central and Northern Europe in the wake of postglacial warming. However, many of the genetic lineages which differentiated within these refugia predate even the Pleistocene. One of these refugia is the Italian Peninsula with its surrounding islands. In this meta-analysis, we compiled phylogeographic patterns of multiple species across this region. We transformed existing geographic information on 78 animal and plant species (with a total of 471 genetic lineages) within 17 predefined areas into presence/absence matrices. We elaborated three matrices: (i) only old lineages exclusively distinguished by deep splits, (ii) only young lineages distinguished by shallow (i.e. recent) splits, and (iii) presence/absence of the respective species. To infer biogeographic relationships between the predefined areas, we performed bootstrapped neighbour joining cluster analyses on these three matrices. In addition, we reviewed the geological history of Italy to identify causes of the observed biogeographic patterns. We found Sardinia and Corsica to be biogeographically closely linked with each other, and that they diverge strongly from all other regions. Sicily also diverges strongly from all other regions, while the intra-island differentiation was comparatively low. On the Italian mainland, Calabria exhibited the most pronounced biogeographic differentiation, often with several lineages present, resulting from old vicariance events within the region. Furthermore, southern Apulia and the Po Plain with adjoining areas of northern peninsular Italy displayed considerable distinctiveness. Admixture prevailed in the areas between these three regions. The ancient isolation of Sicily, as well as Sardinia plus Corsica, resulted in endemic lineages with only moderate recent exchange with adjacent mainland regions. Pronounced diversification occurs within the Italian Peninsula. The complex tectonic activities, such as shifting (micro)plates, submergence of major parts of peninsular Italy with the genesis of numerous Pliocene islands, in combination with the climatic cycles during the Pleistocene have most likely generated the current biogeographic pattern of species. Immigrations from the Balkan Peninsula into northern Italy partly accounted for the distinctiveness of this region.