Inhomogeneous Shells Research Articles

Algorithms of the Hybrid Projection Method Applied for Analyzing the Excitation of an Inhomogeneous Dielectric Body of Revolution by a Radial Magnetic Dipole

A revolution body consisting of a homogeneous dielectric sphere and an exterior inhomogeneous dielectric layer is considered. The problem of excitation of this body by a magnetic dipole located on the revolution axis is analyzed. Solution algorithms based on the hybrid projection method are developed. The method includes the projection of electromagnetic fields onto transverse spherical functions and application of the 1D finite element method for the solution of the obtained system of ordinary differential equations. The algorithms are also generalized to the case of the perfectly conducting interior sphere. The comparison results characterizing the efficiency of the method are presented. The calculation results are given for the pattern of the dipole in the presence of a conducting sphere surrounded by an inhomogeneous dielectric shell having the shape of a prolate spheroid in one case and a plasma shell having the same shape in the other case.

On an improved 3D stress analysis for elastic composite shells

In this paper inhomogeneous shells subjected to static loading are considered. The theory is based on a multi–field functional, where the associated Euler-Lagrange equations include besides the global shell equations formulated in stress resultants, the local equilibrium in terms of stresses. Within representative volume elements the displacement field is enriched with warping displacements and relative thickness displacements. Elimination of a set of parameters by static condensation leads to a material matrix for the stress resultants. It is used in displacement based elements or in mixed elements with the usual 5 or 6 nodal degrees of freedom. A second model, where decoupling of the local and global boundary value problems is assumed, proves to be computationally very effective.

Laminated cross-ply cylindrical shell due to transient heating

The stress-deformed state of an antisymmetric cross-ply laminated circular closed cylindrical shell subjected to initial local heating in initially is investigated. For the study, the mathematical model of the shear theory of inhomogeneous shells of the Tymoshenko type is used. The two-dimensional heat equation is deduced under the linear distribution of temperature over the thickness. By methods of Fourier and Laplace integral transformations the solution of the non-stationary heat conduction problem and the quasi-static thermoelasticity problem for a finite simply supported circular cylindrical shell is obtained. Numerical results are presented for four-layer orthotropic composite. Cite as: U. V. Zhydyk, “Laminated cross-ply cylindrical shell due to transient heating,” Prykl. Probl. Mekh. Mat. , Issue 17, 113–120 (2019) (in Ukrainian), https://doi.org/10.15407/apmm2019.17.113-120

Solution of the axisymmetric problem of thermoelasticity of a radially inhomogeneous cylindrical shell by numerical-analytical method and the finite element method

The aim of research is to compare two calculation methods using the example of solving the axisymmetric thermoelasticity problem. Methods. The calculation of a thick-walled cylindrical shell on the temperature effect was carried out by the numerical-analytical method and the finite element method, implemented in the LIRA-CAD software package. The shell consists of three layers: two layers of heat-resistant concrete and an outer steel layer. In the calculation, a piecewise linear inhomogeneity of the shell due to its three-layer structure and continuous inhomogeneity caused by the influence of a stationary temperature field is taken into account. The numerical-analytical method of calculation involves the derivation of a resolving differential equation, which is solved by the sweep method, it is possible to take into account the nonlinear nature of the deformation of the material using the method of successive approximations. To solve this problem by the finite element method, a similar computational model of the shell was constructed in the LIRA-CAD software package. The solution of the problem of thermoelasticity for an infinite cylinder (under conditions of a plane deformed state) and for a cylinder of finite length with free ends is given. Results . Comparison of the calculation results is carried out according to the obtained values of ring stresses σθ.

Control of elastic shear waves by periodic geometric transformation: cloaking, high reflectivity and anomalous resonances

A doubly periodic geometric transformation is applied to the problem of out of plane shear wave propagation in a doubly periodic perforated elastic medium. The technique leads to the design of a system of radially anisotropic and inhomogeneous shells surrounding the void inclusions, that can be tuned to give the desired filtering properties.For a regular transformation, the transformed elastic system displays the same dispersion properties than the original homogeneous one, but for overlapping and unfolding transformations new filtering properties can be obtained, which include anomalous resonances at zero and finite frequencies. Low-frequency homogenisation reveals how it is possible to tune the phase and group velocity in the long-wave limit at any value, or to obtain a zero frequency band gap for Neumann boundary conditions. The dispersion properties of the medium are studied both semi-analytically by the multipole expansion and numerically by the finite element methods.Several applications are shown, including the transmission problem throughout a grating of void inclusions and an interface in a waveguide, where the capability of the proposed model is quantitatively demonstrated by computing the transmitted power flow. Finally, we gave a demonstration of defect modes in a waveguide and of tuning the transformation for the research of Dirac points.

Calculation of Equal Strength Thick-Walled Concrete Cylinder with Free Ends

The article discusses radially inhomogeneous thick-walled cylindrical shells in which inhomogeneity may be due to various reasons (structure, the effect of the temperature field, etc.).Thick-walled shells are called equal-stressed, if the principle stress σ1 in all points of the body is constant. In inhomogeneous body, in addition to the elastic modulus of a material, its strength can also variable. In accordance with the first theory of strength, if the stress σ1 is variable and at each point the strength of the material is equal, then the thick-walled shell is called equal-strength. The article discusses the modelling of equal strength concrete thick-walled shell with free ends.

On Existence of Solutions of Nonlinear Equilibrium Problems on Shallow Inhomogeneous Anisotropic Shells of the Timoshenko Type

We prove the existence of solutions to the geometrically nonlinear boundary value problem for elastic inhomogeneous anisotropic shallow shells with rigidly clamped edges in the framework of the S.P. Timoshenko shear model. The boundary value problem is reduced to one nonlinear operator equation, whose solvability is established using the principle of compressed maps. The investigation method is based on integral representations for generalized displacements containing holomorphic functions determined by boundary conditions involving the theory of Cauchy type integrals with real densities.

Modal analysis of thin parabolic shells

The modal analysis of parabolic shells of revolution is based on using the finite-element model of inhomogeneous shell. The shells can have complex-shaped midsurface, geometrical features throughout the thickness, or multilayer structure. To develop the finite-element shell model we approximate a thin shell by one spatial finite element throughout the thickness. The structural elements of an inhomogeneous shell require the finite element to be universal: it should be eccentrically arranged relative to the mid-surfaces of the casing, it should be possible to vary the thickness of the lateral edges of the finite element and ets. The universal finite element is based on an isoparametric spatial finite element with polylinear shape functions for coordinates and displacements. Additional variable parameters are introduced to enhance the capabilities of the modified finite element. Two hypotheses are used to describe the features of the stress–strain state of a thin inhomogeneous shell. The static hypothesis assumes that the compressive stresses in the fibers throughout the thickness are constant. The nonclassical kinematic hypothesis of deformed straight line is used: a straight segment along the thickness remains straight though stretched or shortened during deformation. This segment is not necessarily normal to the mid-surface of the shell.The stress–strain state of a shell and its structural elements is determined using the geometrically nonlinear equations of the three-dimensional theory of thermoelasticity. A linear elastic continuous medium with large displacements and small strains is used as a model whose properties correspond to the generalized Duhamel–Neumann law. To derive the governing finite-element equations for displacements the moment finite-element scheme is used. The moment finite-element scheme approximations of displacements and strains guarantee a correct description of the rigid-body displacements of finite elements, which enhances the convergence and accuracy of solutions on coarse meshes.The natural vibrations of parabolic shells with various heights have been investigated. The convergence of solutions has been studied and compared with the results obtained by other authors.During operation, realistic shell structures often undergo various changes in the temperature field. This can significantly affect their dynamic characteristics. Extension of this work to modal analysis of parabolic shells considering heating is currently being pursued.

Analysis of the prediction of a bilayered cylindrical shell’s reduced cutoff frequency with data-driven approaches

Abstract The reduced cutoff frequency is one of the most important properties of elastic circumferential waves of cylindrical shells. For homogeneous thin cylindrical shells, this frequency can be calculated based on the eigen modes theory. In contrast, this theory is not practical for inhomogeneous cylindrical shells. This paper presented the methods of reduced cutoff frequency prediction using data-driven approaches. Three widely known data-driven methods, the Artificial Neural Network (ANN), the Adaptive-Network-based Fuzzy Inference System (ANFIS), the Support Vector Machine (SVM), are employed to develop the prediction models. The training and testing datasets for these models are determined from the plane of modal identification. In this study, the considered bilayered cylindrical shells are made of stainless steel/polymer. Three prediction scenarios are assessed. The difference between these scenarios is the physical and geometrical parameters which considered as relevant entries of data-driven approaches. In the first and second scenarios, a comparative analysis indicates that the ANN models performed better than other models with highest prediction accuracy value of 0.9761 and the lowest mean relative error (MRE) of 0.4017%. In the third scenario the SVM model was the best model with highest accuracy of 0.9895 and lowest root mean square error (RMSE) of 0.0227.

Nonlinear evolutionary problem for a laminated inhomogeneous spherical shell

In the present paper, the finite deformations of a laminated inhomogeneous spherical shell are studied. A laminated shell can be considered as a limit case for multilayered shells when the thickness of each layer tends to zero while their quantity tends to infinity. Such a limit might be useful in modeling of multilayered structures with large amount of layers, for example, produced by layer-by-layer additive manufacturing. It is easy to explain the nature of inhomogeneity for multilayered structures (discrete inhomogeneity). It is just the result of the fact that in the stress-free state the shapes of layers do not fit to each other. Thus, they cannot be assembled without gaps or overlaps. Proper assembly becomes possible only after individual deformations of layers that cause self-equilibrated stresses in their assembly. The explanation of inhomogeneity for laminated structures (continuous inhomogeneity) is slightly more complicated. It can be given upon the idea of a continuous family of reference shapes that are free from stresses only locally. In the present paper, this approach is discussed in detail. To define measures for stresses and strains on laminated structures, one has to determine corresponding fields in some specific way. Such definitions that are obtained by formalism, adopted in the theory of smooth manifolds with non-Euclidean connection, are also given. To compare a discrete inhomogeneity with its continuous counterpart, the stress–strain states for the sequence of multilayered structures have been examined. The common factor of these structures is that they have equal final volume. Meanwhile, the number of layers increases with their order in the sequence. The measure for inhomogeneity related with non-Euclidean connection is found from a nonlinear evolutionary problem. To support improved understanding of the interplay between multilayered and laminated structures, the stresses and strains exerted in them are studied in comparison. Considerations of the reverse situation, in which some multilayered structure with discrete inhomogeneity is defined upon a given laminated structure, are also carried out. The convergence with decreasing maximal thickness for layers to the original laminated structure is illustrated numerically. In this, one can see similarity with the partitioning procedure in the theory of integrals.

Method of Integral Equations for Studying the Solvability of Boundary Value Problems for the System of Nonlinear Differential Equations of the Theory of Timoshenko Type Shallow Inhomogeneous Shells

The solvability of the boundary value problem for a system of second-order nonlinear partial differential equations with given boundary conditions which describes the equilibrium of elastic inhomogeneous shallow shells with free edges in the framework of the Timoshenko shear model is considered. The boundary value problem is reduced to a single nonlinear equation whose solvability is established by using the contraction mapping principle.

О существовании решений нелинейных задач равновесия пологих неоднородных анизотропных оболочек типа Тимошенко

О существовании решений нелинейных задач равновесия пологих неоднородных анизотропных оболочек типа Тимошенко

Dynamics of Structurally Inhomogeneous Lamellar and Shell Mechanical Systems. Part 1

A vibrational statement, method and algorithm to assess the damping capacity of structurally inhomogeneous viscoelastic mechanical systems, consisting of a package of rectangular plates and shells with point relations and concentrated masses at different rheological properties of deformable elements, are proposed in the paper. To describe rheological properties of the material, the linear hereditary Boltzmann-Volter theory was used. To assess the damping capacity of the system, the problem in question in each case was reduced to solving the proper problems of algebraic equations with complex parameters solved by the Muller method. The accuracy of the methods was demonstrated by comparing the calculated results with known published data and a numerical experiment. Complex natural frequency of the system was used to assess the damping capacity of inhomogeneous viscoelastic systems. Various eigenvalue problems have been solved for structurally inhomogeneous mechanical systems consisting of a package of plate and shell systems with concentrated masses and shock absorbers. A number of new mechanical effects have been discovered, related to the manifestation of the damping capacity of mechanical systems under consideration.

Application of the method of spline-collocations to the solution of dynamic and static problems for structurally inhomogeneous cylindrical shells

Application of the method of spline-collocations to the solution of dynamic and static problems for structurally inhomogeneous cylindrical shells

Axisymmetric Thermal Stress in an Inhomogeneous Circular Cylindrical Shell Due to Cyclic Temperature Change

Axisymmetric Thermal Stress in an Inhomogeneous Circular Cylindrical Shell Due to Cyclic Temperature Change

Analysis of non-stationary reaction of elastic shell to impulse load

An effective numerical method for studying non-stationary vibrations of thin elastic shells is proposed. The method is based on the finite element model of a thin elastic inhomogeneous shell and the reduced model created on its basis for the dynamics problems.The finite-element shell model is based on the relations of the three-dimensional theory of thermo-elasticity and is developed with the use a tensor calculus apparatus, a geometrically nonlinear formulation of the problem in increments and the application of the moment finite-element scheme. To develop the finite-element shell model we approximate a thin shell by one spatial finite element throughout the thickness which is an efficient approach. The structural elements of an inhomogeneous shell require the finite element to be universal: it should be eccentrically arranged relative to the mid-surfaces of the casing (of the shell’s sections without stepwise-variable thickness), it should be possible to vary the thickness of the lateral edges of the finite element; the lateral edges of the neighboring finite elements should be in continuous contact; and it should be possible to model sharp bends and the multilayer structure of the shell. The universal finite element is based on an isoparametric spatial finite element with polylinear shape functions for coordinates and displacements. Additional variable parameters are introduced to enhance the capabilities of the modified finite element. Two hypotheses are used to describe the features of the stress–strain state of a thin inhomogeneous shell. The first static hypothesis assumes that the compressive stresses in the fibers throughout the thickness are constant. The next is the nonclassical kinematic hypothesis of deformed straight line: though stretched or shortened during deformation, a straight segment along the thickness remains straight. This segment is not necessarily normal to the mid-surface of the shell.The method for studying non-stationary vibrations of the shells under the action of short-term loads is based on the application of reduced models. The use of the basic nodes method allowed us to develop a simple and effective algorithm for solving this problem. We have transformed a system of coupled differential equations describing the motion of a shell to independent ones. The solution of obtained Cauchy problems is easily found by the well-developed Runge-Kutta numerical method.The possibility of applying the developed method to assess the effect of short-term load on the behavior of a thin-walled structure is shown on the test problems. Convergence of solutions is investigated and a comparison with theoretical data and results obtained with the help of the SCAD software is made.

Effect of Static Loads on the Natural Vibrations of Ribbed Shells

The article is devoted to a further analysis of the natural vibrations of inhomogeneous shells under the action of static loads. The method of investigation is based on a unified methodology that combines the problems of static stability and the vibrations of elastic shells. The problems of natural vibrations take into account the presence of a prestressed state of the shell structure from the action of static loads. The presence of a static load significantly affects the spectrum of the natural frequencies of the shell. This approach allows us to determine the critical load by the dynamic criterion.The method of investigating of inhomogeneous shells is based on the uniform methodological positions of the 3-d geometrically nonlinear theory of thermoelasticity and the finite-element method in the form of the moment finite-element scheme. So, a thin shell is considered by this method as a three-dimensional body which is modeled throughout the thickness by one isoparametric solid finite element with multilinear shape functions.Two nonclassical hypotheses are used to describe the stress–strain state of a thin inhomogeneous shell. The kinematic hypothesis of deformed straight line in the thickness direction: though stretched or shortened during deformation, a straight segment along the thickness remains straight. This segment is not necessarily normal to the mid-surface of the shell. The displacements are assumed distributed linearly along the thickness, which is conventional in the theory of thin shells. The static hypothesis compressive assumes that the stresses in the fibers are constant throughout the thickness of the shell.Modal analysis of a shallow ribbed panel demonstrates the effectiveness of the developed method. The natural frequencies and mode shapes are determined at each increment of static loading.

Analysis of the thermoelastic state of the inhomogeneous through the thickness isotropic cylindrical shell

The stress-deformed state of a functionally graded isotropic circular closed cylindrical shell subjected to local heating of the outer surface of the environment by means of convective heat transfer is investigated. For the study, the mathematical model of the shear theory of inhomogeneous shells of the Tymoshenko type is used. The two-dimensional heat equation is deduced under the linear distribution of temperature over the thickness. By methods of Fourier and Laplace integral transformations the solution of the non-stationary heat conduction problem and the quasi-static thermoelasticity problem for a finite simply supported circle cylindrical shell is obtained. Numerical results are presented for metal–ceramic composites. Cite as: U. V. Zhydyk, “Analysis of the thermoelastic state of the inhomogeneous through the thickness isotropic cylindrical shell,” Prykl. Probl. Mekh. Mat. , Issue 16, 119–125 (2018) (in Ukrainian), http://doi.org/10.15407/apmm2018.16.119-125

The radially inhomogeneous isotropic elastic equal thickness spherical shell

The axisymmetric elastostatic problem with axisymmetric geometry and loading is analyzed for a radially inhomogeneous equal thickness spherical shell, in linear elasticity. Considering that the stress field, the displacement field and the stiffness matrix have radial dependence, after a series of admissible functional manipulations, the general differential system which solves the isotropic linear elastic problem is developed in spherical coordinates. For the isotropic axisymmetric elastostatic problem, a differential equation in terms of the radial coordinate of the displacement field is derived. Exact analytical solution in spherical coordinates, is developed for a radially inhomogeneous equal thickness spherical shell of exponential varying Young's modulus and constant Poisson's ratio. In addition in the case that the Young's modulus is expressed by power law, the displacements and stress fields are deduced. An application has been made for a radially inhomogeneous equal thickness spherical shell where the stress and displacement fields are determined.

Application of Riemann–Hilbert Problem Solutions to a Study of Nonlinear Boundary Value Problems for Timoshenko Type Inhomogeneous Shells with Free Edges

The paper deals with the study of solvability of geometrically nonlinear boundary value problem for elastic shallow isotropic inhomogeneous shells with free edges within S. P. Timoshenko shear model. The problem is reduced to one nonlinear equation whose solvability is proved with the use of contracting mappings principle.