Dynamic equations for fluid-loaded porous plates using approximate boundary conditions

Systematically derived equations for fluid-loaded thin poroelastic layers are presented for time-harmonic conditions. The layer is modeled according to Biot theory for both open and closed pores. Series expansion techniques in the thickness variable are used, resulting in separate symmetric and antisymmetric plate equations. These equations, which are believed to be asymptotically correct, are expressed in terms of approximate boundary conditions and can be truncated to arbitrary order. Analytical and numerical results are presented and compared to the exact three dimensional theory and a flexural plate theory. Numerical comparisons are made for two material configurations and two thicknesses. The results show that the presented theory predicts the plate behavior accurately

A hierarchy of dynamic equations for micropolar plates

AbstractThis work considers homogeneous isotropic micropolar plates adopting a power series expansion method in the thickness coordinate. Variationally consistent equations of motion and end boundary conditions are derived in a systematic fashion up to arbitrary order for extensional and flexural displacement cases. The plate equations are asymptotically correct to all studied orders. Numerical results are presented for various orders of the present method, other approximate theories as well as the exact three dimensional theory. The results illustrate that the present approach may render benchmark solutions provided higher order truncations are used, and act as engineering plate equations using low order truncation

Dynamic equations for a fully anisotropic piezoelectric rectangular plate

A hierarchy of dynamic plate equations based on the three dimensional piezoelectric theory is derived for a fully anisotropic piezoelectric rectangular plate. Using power series expansions results in sets of equations that may be truncated to arbitrary order, where each order set is hyperbolic, variationally consistent and asymptotically correct (to all studied orders). Numerical examples for eigenfrequencies and plots on mode shapes, electric potential and stress distributions curves are presented for orthotropic plate structures. The results illustrate that the present approach renders benchmark solutions provided higher order truncations are used, and act as engineering plate equations using low order truncation

Dispersion free wave splittings for structural elements

Wave splittings are derived for three types of structural elements: membranes, Timoshenko beams, and Mindlin plates. The Timoshenko beam equation and the Mindlin plate equation are inherently dispersive, as is each Fourier component of the membrane equation in an angular decomposition of the field. The distinctive feature of the wave splittings derived in the present paper is that, in homogeneous regions, they transform the dispersive wave equations into simple one-way wave equations without dispersion. Such splittings have uses both for radial scattering problems in the 2D cases and for scattering problems in dispersive media. As an example of how the splittings may be applied, a direct scattering problem is solved for a membrane with radially varying density. The imbedding method is utilized, and agreement is obtained with an FE simulation

Time domain Green functions for the homogeneous Timoshenko beam

In this paper, wave splitting technique is applied to a homogeneous Timoshenko beam. The purpose is to obtain a diagonal equation in terms of the split fields. These fields are calculated in the time domain from an appropriate set of boundary conditions. The fields along the beam are represented as a time convolution of Green functions with the excitation. The Green functions do not depend on the wave fields but only on the parameters of the beam. Green functions for a Timoshenko beam are derived, and the exponential behaviour of these functions as well as the split modes are discussed. A transformation that extracts the exponential part is performed. Some numerical examples for various loads are presented and compared with results appearing in the literature

Dynamic equations for a fully anisotropic elastic plate

A hierarchy of dynamic plate equations is derived for a fully anisotropic elastic plate. Using power series expansions in the thickness coordinate for the displacement components, recursion relations are obtained among the expansion functions. Adopting these in the boundary conditions on the plate surfaces and along the edges, a set of dynamic equations with pertinent edge boundary conditions are derived on implicit form. These can be truncated to any order and are believed to be asymptotically correct. For the special case of an orthotropic plate, explicit plate equations are presented and compared analytically and numerically to other approximate theories given in the literature. These results show that the present theory capture the plate behavior accurately concerning dispersion curves, eigenfrequencies as well as stress and displacement distributions

Approximate boundary conditions for a fluid-loaded elastic plate

Approximate boundary conditions for an infinite elastic layer immersed in a fluid are derived. By using series expansions in the thickness coordinate of the plate fields, the displacements fields are eliminated, adopting the three-dimensional equations of motion. The sums and differences of the boundary pressure fields and their normal derivatives are related through a set of approximate boundary conditions, one symmetric and one antisymmetric. These equations involve powers in the layer thickness together with partial derivatives with respect to time as well as the spatial variables in the plate plane. The approximate boundary conditions can be truncated to an arbitrary order, and explicit relations are presented including terms of order five. Comparisons are made with effective boundary conditions using classical plate theories. The numerical examples involve reflection and transmission of plane waves incident on the plate at different angles, as well as the pressure fields due to a line force. Three fluid-loading cases are studied: modest, heavy, and light loadings. The results using truncated approximate boundary conditions are compared to exact and classical plate solutions. The examples show that the accuracies of the power series approximations of order three and higher are very good in the frequency interval considered

Three dimensional frequency analysis of bidirectional functionally graded thick cylindrical shells using a radial point interpolation method (RPIM)

This paper considers a functionally graded (FG) shell using a meshless radial point interpolation method (RPIM). The material is assumed to be bidirectional FG, where the variation is present in both the radial and the axial directions. Based on the three-dimensional equations of motion, the frequency equations are stated using RPIM. Numerical results are presented for a thick shell for various boundary conditions. These results illustrate the influence from the material variation concerning eigenfrequencies and eigenmodes. In addition, the study shows that the RPIM is an efficient method to solve dynamical shell problems

Marine mammal hotspots across the circumpolar Arctic

Aim: Identify hotspots and areas of high species richness for Arctic marine mammals. Location: Circumpolar Arctic. Methods: A total of 2115 biologging devices were deployed on marine mammals from 13 species in the Arctic from 2005 to 2019. Getis-Ord Gi* hotspots were calculated based on the number of individuals in grid cells for each species and for phylogenetic groups (nine pinnipeds, three cetaceans, all species) and areas with high species richness were identified for summer (Jun-Nov), winter (Dec-May) and the entire year. Seasonal habitat differences among species’ hotspots were investigated using Principal Component Analysis. Results: Hotspots and areas with high species richness occurred within the Arctic continental-shelf seas and within the marginal ice zone, particularly in the “Arctic gateways” of the north Atlantic and Pacific oceans. Summer hotspots were generally found further north than winter hotspots, but there were exceptions to this pattern, including bowhead whales in the Greenland-Barents Seas and species with coastal distributions in Svalbard, Norway and East Greenland. Areas with high species richness generally overlapped high-density hotspots. Large regional and seasonal differences in habitat features of hotspots were found among species but also within species from different regions. Gap analysis (discrepancy between hotspots and IUCN ranges) identified species and regions where more research is required. Main conclusions: This study identified important areas (and habitat types) for Arctic marine mammals using available biotelemetry data. The results herein serve as a benchmark to measure future distributional shifts. Expanded monitoring and telemetry studies are needed on Arctic species to understand the impacts of climate change and concomitant ecosystem changes (synergistic effects of multiple stressors). While efforts should be made to fill knowledge gaps, including regional gaps and more complete sex and age coverage, hotspots identified herein can inform management efforts to mitigate the impacts of human activities and ecological changes, including creation of protected areas

Marine mammal hotspots across the circumpolar Arctic

Aim: Identify hotspots and areas of high species richness for Arctic marine mammals. Location: Circumpolar Arctic. Methods: A total of 2115 biologging devices were deployed on marine mammals from 13 species in the Arctic from 2005 to 2019. Getis-Ord Gi* hotspots were calculated based on the number of individuals in grid cells for each species and for phyloge-netic groups (nine pinnipeds, three cetaceans, all species) and areas with high spe-cies richness were identified for summer (Jun-Nov), winter (Dec-May) and the entire year. Seasonal habitat differences among species’ hotspots were investigated using Principal Component Analysis. Results: Hotspots and areas with high species richness occurred within the Arctic continental-shelf seas and within the marginal ice zone, particularly in the “Arctic gateways” of the north Atlantic and Pacific oceans. Summer hotspots were generally found further north than winter hotspots, but there were exceptions to this pattern, including bowhead whales in the Greenland-Barents Seas and species with coastal distributions in Svalbard, Norway and East Greenland. Areas with high species rich-ness generally overlapped high-density hotspots. Large regional and seasonal dif-ferences in habitat features of hotspots were found among species but also within species from different regions. Gap analysis (discrepancy between hotspots and IUCN ranges) identified species and regions where more research is required. Main conclusions: This study identified important areas (and habitat types) for Arctic marine mammals using available biotelemetry data. The results herein serve as a benchmark to measure future distributional shifts. Expanded monitoring and teleme-try studies are needed on Arctic species to understand the impacts of climate change and concomitant ecosystem changes (synergistic effects of multiple stressors). While efforts should be made to fill knowledge gaps, including regional gaps and more com-plete sex and age coverage, hotspots identified herein can inform management ef-forts to mitigate the impacts of human activities and ecological changes, including creation of protected areas