Prismatic Shells Research Articles

The research of the connective tissue dysplasia effect on dental eruption and hard tissues mineralization

The purpose of this investigation was to explore the quality of 38, 48 teeth's hard tissues at different eruption stages and mandibular bone in different postpartum ontogenesis periods. The research involved 102 male patients divided into groups according to their age: 15-20, 21-30 years old, they were extracted one tooth 38, 48 without inflammatory process signs and a fragment of the mandibular bone alveolar part in the projection of the teeth 38, 48 for orthodontic indications. In the comparison group (49 observations without signs of connective tissue dysplasia), in the study group (53 observations with signs of connective tissue dysplasia) we extracted teeth 38, 48 which were in the bone tissue. We analyzed condition of crown and root systems of extracted 38, 48 teeth, densitometric density of mineral component, size of enamel prisms, size of prismatic shells of organic matrix, spatial organization of collagen fibers in bone tissue, size characteristics of bone plates and mineralization centers of bone tissue. Processing of the obtained data was performed by methods of variation statistics using standard packages Microsoft Excel 2008, Statistica 12.0. The paper reveals one of the surgical dentistry pressing issues related to the tething mechanism of lower human wisdom teeth, considered by the authors from the position of tissue disorders in collagen type 1 observed in connective tissue dysplasia. At age of 15-20 years in connective tissue dysplasia, the enamel prisms hypomineralized areas are generalized; at age of 21-30 years the hypomineralized areas are characterized by local changes. At the age of 15-20 years in connective tissue dysplasia the bone plates splitting is observed at the level of most fibrils, collagen fibers have insufficiently oriented direction in contrast to the age group of 21-30 years where bone plates splitting is characteristic for single fibrils and collagen fibrils are clearly oriented. Morphological and histological changes in teeth 38, 48 and in bone tissue prevent correct and timely teething at 15-20 years and create more favorable conditions for teething at 21-30 years with a slower rate in connective tissue dysplasia, but in both groups teeth eruption occurs under unfavorable anatomic conditions.

The stability of geometrically nonlinear plate systems under the action of dynamic loads

Relevance. Single-connected and multi-connected plate systems are widely used in construction, aircraft, shipbuilding, mechanical engineering, instrument making. As a result, the study of the stability of geometrically nonlinear plate systems is an urgent topic. But, despite significant achievements in this area, there are still many unsolved problems. Thus, the requests of the above-mentioned areas of application of thin-walled spatial systems require further study of the issue of static and dynamic stability. The aim of the work - development of a method of the dynamic stability analysis of geometrically nonlinear plate systems such as prismatic shells under the action of dynamic compression loads. Methods. A plate system, which is subject to dynamic compression loads in the longitudinal direction, is considered. Kirchhoff - Love hypotheses are taken into account. The material stress-deformation diagram is linear. The displacement of points in the normal direction to the median plane of the plates is determined in the form of the Vlasov expansion. To derive the basic differential equations of stability, the energy method and the variational Vlasov method are used. The extreme value of the total energy is determined using the Euler - Lagrange equation. As a result, a set of basic nonlinear differential equations for studying the buckling of the plate system under the action of dynamic compression loads is obtained. Results. The developed method is used to stability analysis of a geometrically nonlinear prismatic shell with a closed contour of the cross section, under central compression under the action of dynamic loading. The edges of the shell rest on the diaphragm. The buckling of the prismatic shell in the longitudinal direction along one and two half-waves of a sinusoid is studied. The numerical integration of nonlinear differential equations is performed by the Runge - Kutta method. Based on the calculation results, graphs of the dependence of the relative deflection on the dynamic coefficient are constructed. The influence of the rate of change of compression stress, the initial imperfection of the system, and other parameters on the criteria for the dynamic stability of the plate system is investigated.

Calculation of Reinforced Concrete Prismatic Shells by the Finite Element Method Using Variable Elasticity Parameters

The paper considers the modification of the generally accepted formulation of the finite elements method by applying in the calculation I.Mileykovski’s refined technical theory of shells that takes into account the deformations of the transverse shear along the thickness of the model. When applying this solution path, it is possible to calculate thick and thin shells (plates) with equal efficiency, taking into account the complex strained state of an anisotropic material. It illustrates the inclusion in the computational algorithm of variable parameters of the elasticity of concrete, allowing more accurate evaluation of the stress-strain state in the finite element under complex (combined) loads. The presence of reinforcement in the material is modeled by dividing the structure into layers and sequentially reduction the elastic characteristics of the material based on the volume ratio of the components. The advantage of the algorithm is the ease of its integration with the conventional finite elements method. All transformations in this case consist in the modification of expressions for determining the elastic characteristics of the construction, calculating the gradient and stiffness matrices, while the sequence of further calculations does not change. This enables to use the proposed algorithm, including as a plug-in software module, expanding the capabilities of existing computing programs. The article demonstrates the application of the method in modeling a reinforced concrete slab made with the use of multi-component high-strength concrete of a heavy class having a prismatic strength under uniaxial compression of more than 110 MPa.

Interaction of thin-walled prismatic shell with elastic foundation

The problem of static analysis of prismatic shells consisting of rectangular plates rigidly interconnected along the longitudinal edges is considered. Shells freely lie on the surface of an elastic foundation or embedded in elastic medium. It is accepted that at the ends of the shell located diaphragm rigid in its plane and flexible out of the plane. According to applied bounding conditions the method of single trigonometric series is used which resulted in conversion of two-dimensional problem to one-dimensional. To solve one-dimensional problem uses the method of displacement, in accordance with which all the hard nodes of the system (the junction of the horizontal and vertical plates) are fixed against possible displacement. For ease of algorithmization of the calculation the plates are considered as generalized finite elements. To determine the nodal forces that correspond to the four degrees of freedom considered plane stress of the plate and her bending. The example of calculation of prismatic shell which is in two-parameter elastic medium and loaded on the upper plane by uniformly distributed load is given. It is shown that the account of friction forces leads to a significant reduction of the moments due to the fact that it reduced the total sediment of the shell.

Finite elements with continuous stress fields in calculation of folded prismatic thin-walled rods and shells

The article deals with methods of creating a rectangular wall-beam finite element with eight degrees of freedom per node and continuous stress fields along the boundaries. This effect is achieved by specifying displacement fields in the plane of the element in forms similar to those in finite elements of Bogner, Fox, and Schmitt plate. The article provides algebraic expressions for displacement forms; methods of forming reaction and stress matrices are also considered. Test calculations carried out with the help of “Computational mechanics” FEM complex have proved high efficiency of the finite element analysis performed. A rectangular shell finite element with twelve degrees of freedom per node was developed as a combination of membrane finite element and Bogner, Fox and Schmitt plate element.

Majorana states in prismatic core-shell nanowires

We consider core-shell nanowires with conductive shell and insulating core and with polygonal cross section. We investigate the implications of this geometry on Majorana states expected in the presence of proximity-induced superconductivity and an external magnetic field. A typical prismatic nanowire has a hexagonal profile, but square and triangular shapes can also be obtained. The low-energy states are localized at the corners of the cross section, i.e., along the prism edges, and are separated by a gap from higher energy states localized on the sides. The corner localization depends on the details of the shell geometry, i.e., thickness, diameter, and sharpness of the corners. We study systematically the low-energy spectrum of prismatic shells using numerical methods and derive the topological phase diagram as a function of magnetic field and chemical potential for triangular, square, and hexagonal geometries. A strong corner localization enhances the stability of Majorana modes to various perturbations, including the orbital effect of the magnetic field, whereas a weaker localization favorizes orbital effects and reduces the critical magnetic field. The prismatic geometry allows the Majorana zero-energy modes to be accompanied by low-energy states, which we call pseudo Majorana, and which converge to real Majoranas in the limit of small shell thickness. We include the Rashba spin-orbit coupling in a phenomenological manner, assuming a radial electric field across the shell.

DYNAMIC CALCULATION OF PRISMATIC SYSTEMS TAKING INTO ACCOUNT INTERNAL FRICTION

The paper shows a possibility of including a frequency-independent viscoelastic design resistance in calculation models representing composite structures with distributed parameters. Diff erential equations of the system motion include parameters of complex rigidity. The research puts forward a solution which helps solve stationary and non-stationary dynamic problems taking internal friction into account. This solution is found by using the calculation algorithm for prismatic shells involving the use of topological matrices. In the process of separation of variables, this approach yields a diff erential equation of linear oscillator motion with frequency-independent viscoelastic resistance.

Geometric nonlinear analysis of prismatic shells using the semi-analytical finite strip method

The present study sheds light on geometric nonlinear static analysis of prismatic shells using the semi-analytical finite strip method. A new computational model, which includes a fully nonlinear compound strip with a longitudinal and transverse stiffener, has been presented. Furthermore, strips with non-uniform characteristics in the longitudinal direction have been used in nonlinear analysis. This has, to the best knowledge of authors, never been reported. Also, this paper describes the design and implementation of eighteen ideal boundary conditions using three different longitudinal and six well-known transverse displacement interpolation functions.The results of the presented study were obtained using an open-source software and multi-purpose software Abaqus. Moreover, the accuracy of the applied computational approach has been verified by comparison with results from the literature. An excellent agreement of displacement fields is achieved for large deflection analyses of plates with a hole and stiffeners as well as for shells with a stepped thickness in the longitudinal direction. Additionally, results from post-buckling analyses of thin-walled structures, a snap-through and snap-back behavior of shallow shells, are matched. The work presented here has profound implications for future studies of the finite strip deployment.

On micropolar elastic cusped prismatic shells

A huge literature is devoted to the study of cusped prismatic shells on the basis of the classical theory of elasticity. It was stimulated by the works of I. Vekua. I. Vekua considered very important to carry out investigations of boundary value and initial boundary value problems for such bodies, since they are connected with degenerate partial differential equations and, therefore, are not classical, in general. The present paper is devoted to cusped prismatic shells on the basis of the theory of micropolar elasticity. Namely, on the basis of the N=0 approximation of hierarchical models for micropolar elastic cusped prismatic shells constructed by the I. Vekua dimension reduction method.

Buckling of uniaxially compressed composite anisogrid lattice plate with clamped edges

An analytical solution of the buckling problem for a uniaxially compressed composite lattice plate with the clamped edges is presented in this paper. The compressive load acting in-plane and applied to the two opposite sides of the plate induces compression in the orthogonal direction due to Poisson effect. The lattice plate composed of the diagonal and transverse ribs is modelled as an equivalent orthotropic plate with effective stiffness parameters. The deflection of buckled plate is approximated using the mode shape functions of a clamped-clamped beam. A formula providing fast and reliable way of calculation of the critical buckling load is derived and applied to the analyses of the plates with various parameters of lattice structures. The results are verified using a finite-element method. Based on these calculations, the significance of the allowance for Poisson effect is demonstrated. An estimation of the mass efficiency of the lattice plates designed for a required critical load is presented. The solution obtained in this work provides an efficient analytical tool for the design and analysis of prismatic shells composed of composite lattice panels and subjected to axial compressive loading.

Vekua type hierarchical models for prismatic shells with mixed conditions on face surfaces

I. Vekua constructed hierarchical models for elastic prismatic shells, in particular, plates of variable thickness, when on the face surfaces either stresses or displacements are known. In the present paper other hierarchical models for cusped, in general, elastic isotropic and anisotropic prismatic shells are constructed and analyzed, namely, when on the face surfaces (i) a normal to the projection of the prismatic shell component of a stress vector and parallel to the projection of the prismatic shell components of a displacement vector, (ii) a normal to the projection of the prismatic shell component of the displacement vector and parallel to the projection of the prismatic shell components of the stress vector are prescribed. We construct also hierarchical models, when other mixed conditions are given on face surfaces. In the zero approximations of the models under consideration peculiarities (depending on sharpening geometry of the cusped edge) of correct setting boundary conditions at edges are investigated. In concrete cases some boundary value problems are solved in an explicit form. As an example of application of the constructed Vekua-type models to composite structures, an unidirectional lamina with fibers parallel to x2-axis under shear strain is considered. Tension–compression is treated as well.

On one problem of a cusped elastic prismatic shells in case of the third model of Vekua s hierarchical model

On one problem of a cusped elastic prismatic shells in case of the third model of Vekua s hierarchical model

On basic problems for elastic prismatic shells with microtemperatures

In the present paper on the basis of the linear theory of thermoelasticity of homogeneous isotropic bodies with microtemperatures the zeroth order approximation of hierarchical models of elastic prismatic shells with microtemperatures in the case of constant thickness (but, in general, with bent face surfaces) is considered. The existence and uniqueness of solutions of basic boundary value problems when the projections of the bodies under consideration are bounded and unbounded domains with closed contours are established. The ways of solving boundary value problems in explicit forms and of their numerical solution are indicated.

On a Mathematical Problem of Cusped Double‐Layered Plates

AbstractThe present paper is devoted to the system of degenerate partial differential equations arise from the investigation of elastic two layered prismatic shells. Harmonic vibration of cusped double‐layered plates is consider. The weak setting of the BVPs in the case of the zero approximation of hierarchical models is considered. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A model of layered prismatic shells

The present paper is devoted to a model for elastic layered prismatic shells which is constructed by means of a suggested in the paper approach which essentially differs from the known approaches for constructing models of laminated structures. Using Vekua’s dimension reduction method after appropriate modifications, hierarchical models for elastic layered prismatic shells are constructed. We get coupled governing systems for the whole structure in the projection of the structure. The advantage of this model consists in the fact that we solve boundary value problems separately for each ply. In addition, beginning with the second ply, we use a solution of a boundary value problem of the preceding ply. We indicate ways of investigating boundary value problems for the governing systems. For the sake of simplicity, we consider the case of two plies, in the zeroth approximation. However, we also make remarks concerning the cases when either the number of plies is more than two or higher-order approximations (hierarchical models) should be applied. As an example, we consider a special case of deformation and solve the corresponding boundary value problem in the explicit form.

Сomputation of prismatic shells in elastic medium

Сomputation of prismatic shells in elastic medium

On Some Analytic Methods for Calculating of Cusped Prismatic Shells

AbstractCusped plates, on the one hand, are very important details from the practical point of view, such plates and beams are often encountered in spatial structures with partly fixed edges, e.g., stadium ceilings, aircraft wings, submarine wings etc., in machine‐tool design, as in cutting‐machines, planning‐machines, in astronautics, turbines, and in many other areas of engineering (e.g., dams); on the other hand, their theoretical analysis and calculation are mathematically connected with the study of the problems for degenerate partial differential equations which are not covered by the general theory for degenerate partial differential equations (see, e.g., [3]). Some satisfactory results are achieved in this direction in the case of Lipschitz domains but in the case of non‐Lipschitz domains there are a lot of open problems. To investigate such problems are the main part of the the present paper. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Investigation of hyperbolic systems with order degeneration arising in I. Vekua's hierarchical models

We study the well-posedness of an initial-boundary value problem corresponding to the zeroth approximation of I. Vekua's hierarchical models for elastic cusped prismatic shells. The mathematical model is described by a two-dimensional order-degenerating hyperbolic system. We formulate the problem in the weak setting and prove the uniqueness and existence theorems. We show that the sequence of corresponding explicit Galerkin approximations converges to the exact solution in an appropriate weighted Lebesgue space.

Harmonic vibration of prismatic shells in zero approximation of Vekua's hierarchical models

In this article elastic cusped symmetric prismatic shells (i.e., plates of variable thickness with cusped edges) in the zero approximation of I.Vekua's hierarchical models is considered. The well-posedness of the boundary value problems (BVPs) under the reasonable boundary conditions at the cusped edge and given displacements at the non-cusped edge is studied in the case of harmonic vibration. The approach works also for non-symmetric prismatic shells word for word. The classical and weak setting of the BVPs in the case of the zero approximation of hierarchical models is considered. Appropriate weighted functional spaces are introduced. Uniqueness and existence results for the variational problem are proved. The structure of the constructed weighted space is described and its connection with weighted Sobolev spaces is established. Moreover, some sufficient conditions for a linear functional arising in the right-hand side of the variational equation to be bounded are given.

Harmonic vibration of cusped plates in the N-th approximation of Vekua's hierarchical models

In this paper elastic cusped symmetric prismatic shells (i.e., plates of variable thickness with cusped edges) in the N -th approximation of Vekua’s hierarchical models are considered. The well-posedness of the boundary value problems (BVPs) under the reasonable boundary conditions at the cusped edge and given displacements at the non-cusped edge is studied in the case of harmonic vibration. The classical and weak setting of the BVPs in the case of the N -th approximation of hierarchical models is considered. Appropriate weighted functional spaces are introduced. Uniqueness and existence results for the variational problem are proved. The structure of the constructed weighted space is described and its connection with weighted Sobolev spaces is established.