Shell Edges Research Articles

Effect of porosity on the modal response of doubly-curved laminated shell structures made of functionally graded materials employing higher order theories

The manuscript investigates the modal response of laminated anisotropic doubly-curved shell structures of variable thickness made of Functionally Graded Materials (FGM), according to an efficient equivalent single layer strategy where the displacement field is described employing a condensed unified formulation, accounting for a higher order through-the-thickness expansion. A generalized power law distribution is adopted for the assessment of the FGM layers, whereas the presence of voids is considered starting from different homogenization assumptions. Unlike previous works which presented a variation of the homogenized material properties within the shell solid, in this paper the problem of material porosity is addressed, therefore a general distribution of the volume fraction of the constituent materials is considered, along with the presence of voids. To this end, both linear and trigonometric through-the-thickness distributions are here assumed. The fundamental governing equations are derived starting from a proper set of curvilinear principal coordinates, while a generalized set of blending functions accounts for arbitrarily shaped structures. Non-conventional boundary conditions are, here, modelled with a distribution of linear springs along the shell edges. The numerical implementation of the problem is performed with the generalized differential quadrature method. A systematic set of validating examples is presented, whose results are compared to predictions based on refined models, as well as some experimental evidence. After a validation step, an extensive parametric analysis points out the sensitivity of the porosity parameters on the modal response of structures with different curvatures and external constraints, with valid findings for engineering design purposes.

Choice of the shape imperfections model in dynamics problems of a long flexible cylindrical shell subjected to force couples

The issue of modeling geometrical imperfections in the dynamics problems of thin-walled shells was little researched. In cases when the natural modes of shell coincided with its buckling modes, the issue of choosing a dangerous imperfection model did not arise. When these shell modes did not coincide, it was important to investigate and compare the effect of different imperfections models on the static and dynamic characteristics of such shells. The choosing the shape imperfections model of a long flexible cylindrical shell subjected to force couples, the natural and buckling modes of which did not coincide, was studied using procedures of the finite element analysis software NASTRAN. The shell wall as a set of plat rectangular elements with six degrees of freedom at the node in the cylindrical coordinate system was modeled. The action of force couples as the concentrated forces were distributed at the nodes of the shell edges in accordance to the presentation of A.S. Volmir. The linear buckling problem and the geometrical nonlinear static analysis of the perfect shell by the Lanzosh method and the Newton-Raphson one were performed, respectively. The long half-waves buckling mode was taken as the first shell imperfections model. The modeling of the second shape imperfections as the first natural mode of the perfect shell using the natural vibration analysis by the Lanzosh method was performed. The different amplitudes of geometrical imperfections in proportion to the shell thickness using a program adapted to this software were set. The results of the geometrical nonlinear static analysis of the imperfect shell by the Newton-Raphson method showed that the shape imperfection model in the form of long half-waves more reduced the values of critical buckling loads. Investigations of natural shell vibrations by the Lanzosh method revealed the same influence of different imperfections models on the natural frequencies and natural forms. We think that the shape imperfections model in the form of long half-waves in studies of forced vibrations and dynamic stability of a long flexible cylindrical shell subjected to force couples will be more effective.

The signal crayfish Pacifastacus leniusculus (Dana, 1852) (Crustacea, Decapoda) is threatening the near future of Margaritifera margaritifera Linnaeus, 1758 (Bivalvia, Unionoida) in the Negro River (NW Zamora, Spain)

We studied the incidence of the signal crayfish on a population of Margaritifera margaritifera in the Negro River (Zamora, Spain) during the summers of 2019 to 2022. The incidence of predation was assessed as a mortality factor in relation to floods and the hydrology of the river. The crayfish were trapped in a small plot of high pearl mussels density monitored since 2002, and collected each summer since 2019, during which time their abundance increased (65 % in three years). Simultaneously, we counted the shells carried by the floods to the gravel riverbanks. The incidence of floods was stable between 2019 and 2022 (10.3 % to 19.4 % of mortality), while at the bottom of the river the mortality of pearl mussels increased due to predation from 2.7 % to 43.3 %. During the 2022 dry season, 29 pearl mussels that had recently died and whose shell edges were widely gnawed by crayfishes were collected from the plot. The shells appeared bitten only in the contour exposed above the gravel, ruling out the possibility that the marks could be the effect of the scavenging of dead specimens by the crayfish. Low intensity trapping barely affected the crayfish population, since the following year their abundance in the controlled section had recovered.

Static analysis of anisotropic doubly-curved shells with arbitrary geometry and variable thickness resting on a Winkler-Pasternak support and subjected to general loads

Laminated doubly-curved shells constituted by innovative materials have become a standard application in several engineering fields. However, this requires a proper formulation of such structures, since several very complex issues affecting such applications must be kept in mind, among all, the curvature effect and the coupling issues between two adjacent layers. In the present work a generalized formulation based on Higher Order Theories is proposed for the linear static analysis of curved structures with completely anisotropic materials in each layer of the lamination scheme. The shell model is described according to the Equivalent Single Layer approach and a mapping procedure based on a Non-Uniform Rational Basis Spline (NURBS) description of the shell edges is applied for the geometric description. Different thickness variations have been considered in the analyses. The fundamental equations of the static problem are derived from the minimum potential energy principle directly in the strong form and the effect of the Winkler-Pasternak support has been accounted for. The external surface load has been applied in each principal direction and arbitrary actions have been enforced to the external edges of the structure. The proposed approach has been validated with respect to the outcomes of some 3D Finite Element models and very good agreement is found between such simulations. Shells characterized by very complex shapes have been accounted for and accurate results have been found with a reduced number of degrees of freedom.

Imperfection sensitivity of FGM cylindrical shells under compression

The behavior of FGM cylindrical shells with imperfections under compression was analyzed. Two cases of the ceramics-metal phase arrangement regarding the shell thickness, as well as a case where the coupling matrix determined according to the classical laminate plate theory equaled zero, owing to which the shell was homogenous transversely, was assumed in the analytical–numerical calculations. Cylindrical shell edges were simply supported. Thermal loads were neglected in this study. Within the first-order approximation of the non-linear Byskov-Hutchinson theory, a value of the ultimate load-carrying capacity corresponding to the critical load of the actual structure with imperfections was determined. This approach allows one to determine the imperfection sensitivity surface as well. It was assumed in the numerical calculations that the material of cylindrical shell gradients complied with Hooke’s law. The critical load of real shells (i.e., with imperfections) is treated as the failure load in standard provisions, design and verifications calculations. An analytical–numerical method (ANM) developed by the authors was applied in the calculations. In this approach, Lagrange’s description, precise strains for thin-walled panels and the second Piola-Kirchhoff’s stress tensor, the exact transition matrix method and the numerical method of the transition matrix using Godunov’s orthogonalization are used. All results were validated with the finite element method.

Static Analysis of Doubly-Curved Shell Structures of Smart Materials and Arbitrary Shape Subjected to General Loads Employing Higher Order Theories and Generalized Differential Quadrature Method

The article proposes an Equivalent Single Layer (ESL) formulation for the linear static analysis of arbitrarily-shaped shell structures subjected to general surface loads and boundary conditions. A parametrization of the physical domain is provided by employing a set of curvilinear principal coordinates. The generalized blending methodology accounts for a distortion of the structure so that disparate geometries can be considered. Each layer of the stacking sequence has an arbitrary orientation and is modelled as a generally anisotropic continuum. In addition, re-entrant auxetic three-dimensional honeycomb cells with soft-core behaviour are considered in the model. The unknown variables are described employing a generalized displacement field and pre-determined through-the-thickness functions assessed in a unified formulation. Then, a weak assessment of the structural problem accounts for shape functions defined with an isogeometric approach starting from the computational grid. A generalized methodology has been proposed to define two-dimensional distributions of static surface loads. In the same way, boundary conditions with three-dimensional features are implemented along the shell edges employing linear springs. The fundamental relations are obtained from the stationary configuration of the total potential energy, and they are numerically tackled by employing the Generalized Differential Quadrature (GDQ) method, accounting for nonuniform computational grids. In the post-processing stage, an equilibrium-based recovery procedure allows the determination of the three-dimensional dispersion of the kinematic and static quantities. Some case studies have been presented, and a successful benchmark of different structural responses has been performed with respect to various refined theories.

Option Of The Method Of Successive Approximations In Calculating The Epicenters Of Extreme And Emergency Situations

The mechanics of aspiration flows formation is considered in the paper as an additional safety mechanism. A solution to nonlinear differential equation describing the equilibrium of a closed cylindrical shell subject to axisymmetric radial pressure is obtained in the framework of strain theory of plasticity and on the basis of an option of the method of successive approximations for calculating the epicenters of emergency phenomena. It is assumed that the nonlinear components related to the ring strength of plastic zones are located near the shell edges of the surrounding operative environment. The problem solutions are generalized for the case when the nonlinear components of the ring strength of the zone of emergency phenomena as a plastic zone are located near the shell of the operative headquarters created in an emergency. As a result of the studies, the plastic state of the shell between the linear state and the state of limit equilibrium of the zone of emergency is determined. This makes it possible to confine the center of extreme and emergency situations zone. The test results conducted for various building structures are given. It is proposed to use the knowledge of aspiration flows strength as an additional safety mechanism, as an option of additional fire safety mechanism.

P.744 Early response to trazodone in depression: clinical overview

Through biological evolution, bivalve mollusks developed shells to improve the utilization of metabolic energy and provide protection against external threats. In addition to the mechanical optimization of the microstructure, the design of the macroscopic shape of a bivalve shell naturally becomes a potential approach to achieving the aforementioned purposes. While the functions of some features of mollusk shells have been studied, the role of the suture-serrate margins, a common morphology of bivalve shell edges, in the global mechanical behaviors of bivalve shells requires further exploration. Here, we present how the serrate margins contribute to the global mechanical properties of bivalve shells. The results of the compression tests employed on a typical bivalve, M. mercenaria, showed that the complete bivalve shells with suture-serrate margins perform better in terms of strength and work to fracture than those without the margins under the same conditions (dry and wet). The primary failure types observed during compression reveal that the failure mechanisms of valve shells are dependent on the suture-serrate margin morphology and water content. Using numerical simulations, the mechanical functions of the suture-serrate margins were demonstrated. Specifically, serrate margins provide mutual resistance by “locking” complementary valves to redistribute and eliminate stress concentrations around pre-existing defects, thereby enhancing the mechanostability and strength of the entire structure.

Sunburst pattern by kinetic segregation in core-shell nanowires: A phase-field study

Compositional segregation is commonly observed during the growth of III-V alloys in core-shell nanowires. Nanometer-thin stripes, enriched in one of the alloy components, are observed along the six <1 1 2> directions perpendicular to the [1 1 1]-wire axis, departing from the core to the outer shell edges in between the {1 1 0} faceted sidewalls. While it has been well established that the phenomenon occurs because of the different mobility of the alloy components, the actual mechanisms by which it happens are yet unclear. A phase-field model, coupling deposition and surface diffusion dynamics, is here developed to inspect the simultaneous evolution of the shell morphology and composition during growth. Both surface energy anisotropy and orientation-dependent growth rates are taken into account to identify their different role. Simulations reveal that the observed segregation is mainly triggered by the enhanced growth rate at the facet edges, while surface anisotropy keeps the stripes thin. Polarity effects are also included to differentiate the behavior between <1 1 2>A and <1 1 2>B orientations, so to reproduce the experimental observation of a 3-fold symmetry.

Free-edge stress fields in cylindrically curved cross-ply laminated shells

This paper discusses a semi-analytical approach to the determination of three-dimensional stress fields in cylindrically curved orthotropic symmetrically or unsymmetrically laminated composite shells under bending load under consideration of free-edge effects. The analysis method incorporates a plane-strain analysis in the innermost regions of the shell (the so-called inner solution) in conjunction with a free-edge solution which enables the accurate determination of the interlaminar stress field at the free shell edges. The free-edge solution employs a discretization of the laminated shell into a number of mathematical layers with respect to the thickness direction wherein in each layer interface approximate and a priori unknown displacement functions are introduced. Lagrangian shape functions are used to interpolate between the displacement functions in the interfaces, and various orders of shape functions are investigated in this paper. The differential equations and boundary conditions that describe the current stress problem are derived from the principle of minimum elastic potential of the laminated shell. The resultant differential equations allow for an exact solution for the displacement functions whereas the boundary conditions at the free shell edges are satisfied in an integral sense. The results of the developed analysis method are verified by comparison with detailed finite element computations, and it is found that the semi-analytical approach works with comparable accuracy, however only at a fraction of the required computational effort.

Designed for the enhancement of structure mechanostability and strength: Suture-serrate margins of bivalve shells

Through biological evolution, bivalve mollusks developed shells to improve the utilization of metabolic energy and provide protection against external threats. In addition to the mechanical optimization of the microstructure, the design of the macroscopic shape of a bivalve shell naturally becomes a potential approach to achieving the aforementioned purposes. While the functions of some features of mollusk shells have been studied, the role of the suture-serrate margins, a common morphology of bivalve shell edges, in the global mechanical behaviors of bivalve shells requires further exploration. Here, we present how the serrate margins contribute to the global mechanical properties of bivalve shells. The results of the compression tests employed on a typical bivalve, M. mercenaria, showed that the complete bivalve shells with suture-serrate margins perform better in terms of strength and work to fracture than those without the margins under the same conditions (dry and wet). The primary failure types observed during compression reveal that the failure mechanisms of valve shells are dependent on the suture-serrate margin morphology and water content. Using numerical simulations, the mechanical functions of the suture-serrate margins were demonstrated. Specifically, serrate margins provide mutual resistance by “locking” complementary valves to redistribute and eliminate stress concentrations around pre-existing defects, thereby enhancing the mechanostability and strength of the entire structure.

Local buckling of axially compressed cylindrical shells with different boundary conditions

The concept of local buckling of compressed isotropic cylindrical shells is developed in this paper. Buckling of axially compressed cylindrical shells under different types of local perturbations has been studied in many theoretical and experimental researches. Different methodologies, based on these studies, were suggested for design buckling load estimation. However, in most calculations and experiments only two classical boundary conditions were considered: fixed displacement of the shell edges in the axial direction or fixed and uniformly distributed compressive load. Both of them are hard or even impossible to realize in practical applications. In the present paper we studied the shell behaviour with 6 different types of boundary conditions, which included the above mentioned two types. We considered shell loading through rigid plates with different types of possible displacements (degrees of freedom) and through stiffening ribs at the edges of the shell. Two theoretical methods were applied: numerical (finite element method) and analytical (Pogorelov's geometrical method). Lateral force was applied as a perturbation and stability of the shell under load combination was studied. The results of calculations were compared with obtained experimental data for validation of numerical solutions. Then metastability and post-buckling behaviour of the structure was studied using theoretical and experimental methods. In particular the interval of existence and stability of post-buckling equilibrium states of the shells with one or several buckles was studied systematically. The post-buckling equilibrium paths and corresponding energy barriers for all mentioned types of boundary conditions were analyzed. Parametric analysis of boundary value problem allowed to establish the main (Batdorf) structure parameter. Formula for design buckling load was suggested and discussed.

Expansion patterns and parallaxes for planetary nebulae

Aims.We aim to determine individual distances to a small number of rather round, quite regularly shaped planetary nebulae by combining their angular expansion in the plane of the sky with a spectroscopically measured expansion along the line of sight.Methods.We combined up to three epochs ofHubbleSpace Telescope imaging data and determined the angular proper motions of rim and shell edges and of other features. These results are combined with measured expansion speeds to determine individual distances by assuming that line of sight and sky-plane expansions are equal. We employed 1D radiation-hydrodynamics simulations of nebular evolution to correct for the difference between the spectroscopically measured expansion velocities of rim and shell and of their respective shock fronts.Results.Rim and shell are two independently expanding entities, driven by different physical mechanisms, although their model-based expansion timescales are quite similar. We derive good individual distances for 15 objects, and the main results are as follows: (i) distances derived from rim and shell agree well; (ii) comparison with the statistical distances in the literature gives reasonable agreement; (iii) our distances disagree with those derived by spectroscopic methods; (iv) central-star “plateau” luminosities range from about 2000L⊙to well below 10 000L⊙, with a mean value at about 5000L⊙, in excellent agreement with other samples of known distance (Galactic bulge, Magellanic Clouds, and K648 in the globular cluster M 15); (v) the central-star mass range is rather restricted: from about 0.53 to about 0.56M⊙, with a mean value of 0.55M⊙.Conclusions.The expansion measurements of nebular rim and shell edges confirm the predictions of radiation-hydrodynamics simulations and offer a reliable method for the evaluation of distances to suited objects.

Mixed-Form Equations for Stiffened Orthotropic Shells of Arbitrary Canonical Shape with Static Loading

AbstractThin-walled orthotropic shells of arbitrary form reinforced from the concave side by a cross-sectional stiffening system oriented in parallel to coordinate lines are examined. Geometrical nonlinearity and transverse shears are taken into account, but it is presumed that a shell is shallow.Mixed-form equations are more simplified equations of a shell theory as compared to displacement equations, but they are more convenient for some types of fixing of the shell edges (for example, for movable pin fixing).Forces are expressed using a stress function in a middle surface of a shell in such a way that the first two equilibrium equations are satisfied identically. Shell deformation is also expressed using this function.The third equation of strain compatibility is used to form one of the mixed-form equations. Curvature and torsion change functions for this equation are written in the same way as for the Kirchhoff–Love model, though also taking into account transverse shears.

Free vibration analysis of moderately thick isotropic homogeneous open cylindrical shells using improved Fourier series method

In this paper an Improved Fourier series method has been employed to study the free vibrations of isotropic homogeneous moderately thick open cylindrical shells with arbitrary subtended angle and general elastic restraints. In this method, regardless of the boundary conditions, each of the displacement components of open shell is invariably expressed as a simple trigonometric series with accelerated and uniform convergence over the solution domain. Distributed elastic restraints are used to specify the elastic boundary conditions along the shell edges and therefore, arbitrary boundary restraints can be achieved by varying the values of spring’s stiffness. All the unknown expansion coefficients are treated as the generalized coordinates and solved using the Rayleigh-Ritz technique. A considerable number of new vibration results for isotropic open cylindrical shells with various geometric parameters and boundary conditions are presented. The effects of boundary stiffness, thickness to radius ratio and subtended angle on the vibration characteristics are also discussed in detail.

High resolution microscopy reveals significant impacts of ocean acidification and warming on larval shell development in Laternula elliptica.

Environmental stressors impact marine larval growth rates, quality and sizes. Larvae of the Antarctic bivalve, Laternula elliptica, were raised to the D-larvae stage under temperature and pH conditions representing ambient and end of century projections (-1.6°C to +0.4°C and pH 7.98 to 7.65). Previous observations using light microscopy suggested pH had no influence on larval abnormalities in this species. Detailed analysis of the shell using SEM showed that reduced pH is in fact a major stressor during development for this species, producing D-larvae with abnormal shapes, deformed shell edges and irregular hinges, cracked shell surfaces and even uncalcified larvae. Additionally, reduced pH increased pitting and cracking on shell surfaces. Thus, apparently normal larvae may be compromised at the ultrastructural level and these larvae would be in poor condition at settlement, reducing juvenile recruitment and overall survival. Elevated temperatures increased prodissoconch II sizes. However, the overall impacts on larval shell quality and integrity with concurrent ocean acidification would likely overshadow any beneficial results from warmer temperatures, limiting populations of this prevalent Antarctic species.

Thermo-Mechanical Instability Characteristics of Laminated Composite Cylindrical Shells

Even with small infinitesimal increase in load, change of equilibrium configuration of completely different character is highly possible for thin shell structures and that could eventually result the whole structure become unstable. This article presents numerical investigation on buckling and post-buckling characteristics of laminated composite circular cylindrical/conical shell subjected to torsional, thermal and combined loads. Using commercial finite element software ABAQUS, both linear and non-linear buckling response of composite shells are simulated. Linear critical buckling load due to uncoupled torsional and thermal loads is determined using eigen value buckling analysis. Non-linear post-buckling response of shell is simulated using Riks arc length method. Un-coupled and coupled influence of thermal-torsional influence to critical buckling load is estimated. Mode shape associated with linear Eigen value analysis are used as initial geometric imperfection for non-linear analysis. Thermo-mechanical coupling found reduces the buckling load resistance of cylindrical shell by maximum of 47%. It was observed that the generation of additional axial compressive reaction force due to thermal expansion restrain on axial constrained shell edges could be the major influential factor. Except a small radial expansion of cylindrical surface, deformation shape predicted from non-linear post buckling analysis resembles rather similar irrespective of any proportion of thermal-torsional coupling.

Structural, magnetic and electronic properties of Cu-Fe nanoclusters by density functional theory calculations

We present results from density functional theory calculations referring to the magnetic properties of 13, 55, 147 and 309 atoms Cu-Fe icosahedral nanoclusters. Aiming in finding the nanocluster with the optimum magnetic moment (μΒ) we explored the various sizes considering several compositions and atomic conformations. It came out that configurations with agglomerated Fe atoms inside the Cu-Fe nanoclusters and pure Cu surface shell are energetically favoured as demonstrated e.g. for the Cu49Fe6 with 2.3μΒ compared to 2.1μΒ of the Fe bcc. The highest magnetic moment, 3.6μΒ, was found in the Cu12Fe case with the Fe atom located at the surface cell, while 3.18μΒ was found for the Cu297Fe12 in a similar configuration having Fe atoms surrounded by Cu that occupy the surface shell's edges. The magnetic moment is mainly due to Fe's spin up - down electronic density of states difference close to the Fermi level (EF). In particular, the Spin-up Fe d electronic density of states are fully occupied yielding wavefunctions with homogeneous change distribution while the Spin-down is almost unoccupied exhibiting dangling bonding states close to EF. These results could be used for the design of environmental sustainable smart alloys with superior magnetic properties e.g. by depositing Fe or FeCu on Cu nanoclusters or including new elements that provide the possibility of keeping the Fe Spin up-down electronic occupation difference close to EF.

Expansion parallaxes and intrinsic stellar properties for 15 simple planetary nebulae

AbstractWe determined individual distances to a small number of rather round, quite regularly shaped planetary nebulae by combining their angular expansion in the plane of the sky with a spectroscopically measured expansion along the line of sight. For this goal, we combined up to three epochs of Hubble Space Telescope imaging data and determined the respective proper motions of rim and shell edges, and of other features as well. Ground-based radial velocities are assigned separately to the nebular rims and shells and used to determine individual distances, thereby assuming that the expansions in the line-of-sight and in the plane of sky are equal. We employed 1D radiation-hydrodynamics simulations of planetary nebulae evolution to correct for the difference between the spectroscopically measured expansion velocities of rim and shell and the expansion speeds of their respective shock fronts.

Vibration analysis of high-speed rotating conical shell with arbitrary boundary conditions

Based on the energy method, an analytical model for the vibration characteristics of high speed rotating conical shell with arbitrary boundary is constructed. Effects of centrifugal and Coriolis accelerations as well as initial hoop tension due to rotating are all taken into account. A set of translational and rotational artificial springs to simulate the elastic constraints imposed on the conical shell edges. Under the framework, each of the displacement components of the conical shell is invariantly described by a modified Fourier series, which is composed of a standard Fourier series and closed-form auxiliary functions. All the expansion coefficients are determined by using the familiar Rayleigh-Ritz procedure as the generalized coordinates. To validate the present analysis, a series of comparisons between the present results and those available in the open literature are carried out and very good agreements are obtained. On this basis, some numerical results are given to illustrate the effects of rotating speed and geometric parameters on the frequencies of rotating conical shell. Further, the influence of the boundary springs’stiffness are also discussed.