Shell Theory Research Articles

Construction of homogeneous solutions of the torsion problem for a radially inhomogeneous transversely isotropic cylinder

The torsion problem for a radially inhomogeneous transversely isotropic cylinder of small thickness was investigated by the method of asymptotic integration of elasticity theory equations. It is assumed that the side part of the cylinder is stress-free, and boundary conditions are set at the ends of the cylinder, leaving the cylinder in equilibrium. The elastic moduli are thought to be arbitrary continuous functions of the variable along the cylinder radius. The formulated boundary value problem is reduced to a spectral problem containing a small parameter characterizing the thin-walledness of the cylinder. Homogeneous solutions are built, i.e. any solutions of the equilibrium equation satisfying the condition of no stresses on the side surfaces. It is shown that the solution of the torsion problem consists of a penetrating solution and a boundary layer character solution similar to Saint-Venant's edge effect in the theory of inhomogeneous plates. The penetrating solution determines the internal stress-strain state of a radially inhomogeneous cylinder. The stress state determined by the penetrating solution is equivalent to the torsional moments of stresses acting in the cross-section perpendicular to the cylinder axis. Solutions having the boundary layer character are localized at the ends of the cylinder and decrease exponentially with distance from the ends. These solutions are absent in applied shell theories. Asymptotic formulas for displacement and stresses are built, which make it possible to calculate the three-dimensional stress-strain state of a radially inhomogeneous transversely isotropic cylinder of small thickness. Based on the obtained asymptotic expansions, it is possible to assess the applicability of applied theories and build a refined applied theory for radially inhomogeneous cylindrical shells

СОБСТВЕННЫЕ ЧАСТОТЫ И ФОРМЫ КОЛЕБАНИЙ ОБОЛОЧЕК ВРАЩЕНИЯ.

A method is proposed for constructing a solution to the problem of natural oscillations of shells of revolution described by surfaces of rotation of the second order. The initial equations of shell theory are arranged in a sequence that allows one to calculate the unknowns in an iterative process. The system of the first two momentless equations and the first two equations of infinitesimal bending in the proposition that their inverse operators can be constructed, can be reduced to a single equation for the vector of tangential deformations. The possibility of such a representation is due to the research of V.Z. Vlasov shells of revolution, the meridians of which are described by second-order curves. It is shown that the results of calculating the natural frequencies of vibrations of loose shells given in the well-known reference book on strength, stability and vibrations are incorrect.

A unified framework for equivalent single layer theories of composite beams, plates, and shells: a review

This article presents a comprehensive review of the unified framework for the equivalent single layer theories of composite beams, plates and shells. First, the approach to unified displacement fields has two main categories: one is to start from the assumption of linear displacement field variables. And the another is to start from the assumption of transverse strains and transverse displacement variable. In terms of unified shape functions, the shape function is defined as the function that exist in the transverse strain rather than the in-plane displacement variables. Moreover, the basic requirements for the choose of shape functions are also given and analysed in this paper. The unified shape functions are introduced in terms of the several functions and the unified polynomial model. In addition, a construction method for new shear strain shape functions is described in this review. Finally, the current research progress and future prospects are also given.

A novel isogeometric model for dynamic buckling analysis of doubly curved two-directional functionally graded porous shallow microshells in thermal environments via variable length-scale parameters

In this study, the thermo-mechanical dynamic buckling analysis of two-directional functionally graded porous (2D-FGP) shallow microshells resting on elastic foundations with arbitrary boundary conditions is performed using the isogeometric approach. The isogeometric approach takes advantage of non-uniform rational B-spline basic functions to exactly represent the structure geometry models and the attainment of higher order approximation conditions. The characteristics of the material vary in both the thickness and axial x orientation. Especially, a new point in this material length-scale parameter of microshell is analyzed as a function of spatial coordinates and as a function of the material gradient parameters. Using Hamilton’s principle, the modified couple stress theory, and Kirchhoff-Love’s shell theory, the governing equations of a 2D-FGP microshell lying on a Pasternak medium are obtained. A computer program that employs an isogeometric analysis method to analyze the static and dynamic buckling of a 2D-FGP doubly curved shallow microshell with various boundary conditions. The numerical results for thin cylindrical, spherical, and hyperbolic paraboloidal shallow microshells with various planforms, such as rectangular and circular, are presented. Several special cases are used to demonstrate the accuracy and efficacy of the developed model by comparing the numerical results obtained by the proposed formulations with other published data. In addition, the effects of certain parameters, such as the length scale parameters, the power-law indexes, the thickness-to-sides ratio, and the radius ratio, on the dynamic thermo-mechanical buckling response of the 2D-FGP shallow microshells with double curvature are explored in depth.

Free vibration behaviour of composite laminated skew cylindrical shells reinforced with graphene platelets

Current Investigation deals with the free vibration response of composite laminated shells which are reinforced with graphene platelet. Shell is in the form of skew cylinder where rectangular and skew plates and cylindrical shell are considered as especial cases. Each layer of the shell may be enriched with different amount of graphene platelets which results in a functionally graded pattern of reinforcements. The basic governing equations of the shell are obtained by means of the first order shear deformation shell theory. To implement the boundary conditions, it is more preferred to use an oblique coordinate system. Consequently, definition of strains and stresses are provided in an oblique coordinate system. After that, total kinetic and strain energies of the shell are established. Following the idea of Ritz method where the basic shape functions are Chebyshev polynomials, the motion equations of the shell associated to free vibrations are established. Results of this research are compared with the available data in the open literature and after that novel numerical results from the present study are provided to explore the effects of skew angle, opening angle, shallowness, side to thickness ratio, boundary conditions, graphene platelet weight fraction and profile. It is shown that all of these factors are important for the free vibration response of such structure.

Systematic failure mechanism of an FGMs polyhedral arched liner under a fire disaster environment

This paper investigated the failure performance of an arched polyhedral liner yielding a fire disaster environment. The functionally graded materials (FGMs) are used in the liner to increase the ability of fire resistance. A concentrated load may occur at the invert of the arched liner due to the debris, which is considered in the study. The polyhedral arched liner deforms in a single-lobe shape, and this deformation is expressed by a trigonometric function. The nonlinear equilibrium equations are presented by applying the thin-walled shell theory and the principle of minimum potential energy, respectively. Based on the above investigations, two innovative contributions of the present study are addressed: (1) a novel FGMs polyhedral liner is introduced to rehabilitate the cracked non-circular tunnel in the combination of the mechanical loading and fire environment; and (2) the critical buckling load, as well as the load-displacement nonlinear equilibrium curves, is derived explicitly by solving the nonlinear equilibrium equations. The effect of the fire disaster on the buckling load is quantized to describe the destabilization behavior of the arched liner in a fire environment, respectively. Then, the present study is compared efficiently with other closed-form formulations when the polyhedral FGM arched liner simplifies to a circular homogeneous liner. It is shown that a smaller number of polyhedral edges and a higher content of ceramic are beneficial to resist external loadings and fire disasters in engineering practices. Finally, the effects of geometric parameters, volume fraction exponents, and temperature variations on the nonlinear buckling behavior of the polyhedral arched liner are evaluated.

On the wave propagation characteristics of functionally graded porous shells

This paper concentrates on the wave propagation characteristics of functionally graded (FG) porous shells. The transversal shear deformation of the shell is taken into consideration by employing a simple higher-ordered shear deformation shell theory. The equations of motion are derived for the proposed model based on Hamilton’s principle. An eigenvalue problem that relates the wave propagation elements is formulated and solved to present the various dispersion relations of FG cylindrical and spherical shells. The effects of porosities, shells’ geometrical parameters, FG material exponent, and wavenumbers on the principal wave propagation frequency and the associated phase velocity are investigated in detail.

AXISYMMETRIC DYNAMIC BEHAVIOUR OF THIN OBLATE SHELLS

This paper presents a theoretical investigation of the axisymmetric free vibrations of an isotropic thin oblate spheroid shell. The analysis depends on the Rayliegh – Ritz's method. The non – shallow shell theory is used for the analysis. The analysis based on considering the oblate spheroid as a continuous system constructed from two spherical shell elements matched at the continuous boundaries.Throughout the results, it is shown that when the eccentricity reaches zero, an exact thin sphere solution is emerged and when the eccentricity equals one an exact thin circular plate solution is emerged. Therefore, the eccentricity of an oblate shell at medium value lies between these two values.It was found that the Rayleigh – Ritz's method is suitable for all eccentricities, while the literature showed that the Rayleigh's method is suitable for eccentricities less than 0.6.

Incompatible Deformations in Hyperelastic Plates

The design of thin-walled structures is commonly based on the solutions of linear boundary-value problems, formulated within well-developed theories for elastic plates and shells. However, in modern appliances, especially in MEMS design, it is necessary to take into account non-linear mechanical effects that become decisive for flexible elements. Among the substantial non-linear effects that significantly change the deformation properties of thin plates are the effects of residual stresses caused by the incompatibility of deformations, which inevitably arise during the manufacture of ultrathin elements. The development of new methods of mathematical modeling of residual stresses and incompatible finite deformations in plates is the subject of this paper. To this end, the local unloading hypothesis is used. This makes it possible to define smooth fields of local deformations (inverse implant field) for the mathematical formalization of incompatibility. The main outcomes are field equations, natural boundary conditions and conservation laws, derived from the least action principle and variational symmetries taking account of the implant field. The derivations are carried out in the framework of elasticity theory for simple materials and, in addition, within Cosserat’s theory of a two-dimensional continuum. As illustrative examples, the distributions of incompatible deformations in a circular plate are considered.

Does the openness of the Boundary Shell system influence the sustainable development of the high-tech industry?

High-technology industries have gained substantial recognition as pivotal drivers of economic growth and technological advancement in modern society. The imperative of sustainable development in high-tech industries cannot be overemphasized, as it plays a crucial role in enabling long-term growth, fostering innovation, and assuming environmental responsibility. This article presents a study on sustainable development in high-tech industries using Boundary Shell theory. The study investigates the role of the stable and sustainable entropy criterion for the Boundary Shell system of high-tech industries from an entropy balance perspective. It analyzes the upper and lower limits of the Boundary Shell support force. Additionally, it improves the traditional boundary system ratio model to comprehensively and objectively evaluate the sustainable development of high-tech industries. The results illustrate that the Boundary Shell of industrial innovation is stronger than that of external dependency, with a reversed ranking of internal evaluation factor strengths compared to the traditional model. This research integrates reaction-diffusion equations theory with entropy balance equations theory to address sustainability issues in the high-tech industry. We further analyze the sustainable development of the high-tech industry through a Boundary Shell theory perspective to advance sustainability in high-tech industries. Moreover, it provides useful insights into the sustainable development of high-tech industries.

Dynamic characteristics of novel damping sandwich composite open cylindrical shell: From theory to simulation

Based on the first-order shear deformation shell theory, the free vibration model of Multilayer Damping Sandwich Composite Open Cylindrical Shell (MDSCOCS) under different lay-up angles is established. The Navier double series and Rayleigh-Ritz method are used to solve the free vibration equation under four sided simply supported boundary conditions. The theoretical solutions are compared with the existing literature and finite element simulation results respectively to verify the validity of theoretical calculation model. The influences of the thickness, position, shell opening angle, composite lay-up angle and other factors on the natural frequency and loss factor of the structure under different number of core layers are investigated in detail. The analysis shows that the damping loss factor of the composite structure is greatly improved by the addition of multilayer damping sandwich, and the dynamic performance of the composite structure is optimized, compared with that of single-layer damping sandwich composite structure. It provides a theoretical basis for the design of multilayer damping sandwich composite open cylindrical shell.

Free vibration of rotating bifunctional gradient CNTs-reinforced ceramic matrix composites pre-twisted blade in thermal environment

Considering the thermal environment’s effect, we built a dynamics model of rotating pre-twisted ceramic matrix composites blades with elastic boundary constraints based on shell and first-order shear deformation theories. The model considers the effect of temperature and simplifies the blade to a twisted cylindrical plate. Based on the Lagrange equation, the differential equation of the vibration of a rotating pre-twisted blade of CNTs-reinforced ceramic matrix composites in the thermal environment was derived by using the orthogonal polynomial as the acceptable function. The natural frequencies and vibration modes of a rotating pre-twisted blade take into account the effects of Coriolis and centrifugal forces. The accuracy of the model is verified by comparing with the literature and Ansys results. The effects of temperature, the volume fraction of functional gradient, pre-twisted angle, and rotational speed on the model were studied in detail. The results show that the addition of carbon nanotubes can greatly enhance the ceramic-based blade. And both temperature and pre-twisted angle significantly affect the carbon nanotube enhanced functional gradient blades.

Curved tempered vacuum glazing: Mechanical model, preparation and wind load performance

This study aimed to establish a mechanical model for curved tempered vacuum glazing (CTVG) based on thin shells theory. By utilizing the windshield of a specific vehicle as a research object, the ANSYS mechanical simulation model for CTVG was developed. Additionally, CTVG samples were prepared and subjected to wind load performance testing. The simulation results demonstrated that the maximum support stress sustained by the CTVG reached 55.807 MPa, accompanied by a maximum glass deformation of 27.756 μm, both of which satisfied the safety requirements. Furthermore, the analysis of the mechanical model revealed that the maximum surface stress of the CTVG under wind load was directly proportional to the square of the wind speed. The CTVG was capable of withstanding a maximum wind pressure of 1.99 kPa and a maximum wind speed of 56.427 m·s−1, and would be safely deployed at a vehicle speed of 203 km·h−1. The wind load performance test results indicated that the observed trend in the maximum surface stress of the CTVG closely aligned with the theoretically predicted values, deviating within a range of 2.7 MPa. This correspondence signified the reliability and validation of our developed mechanical model. Consequently, this research served as a foundational reference for the advancement and utilization of CTVG products, which would play a beneficial role in reducing vehicle energy consumption when used as automotive window glass in the future.

Creep of a closed cylindrical tank at hydrostatic pressure

Objective. When constructing the resolving relations of the theory of shells, the validity of the basic assumptions about the material of the structure under consideration is assumed, which is considered homogeneous, isotropic and viscoelastic, i.e. obeying the Maxwell-Gurevich law. The subject to study is a polymer cylindrical shell, rigidly clamped at the base and subject to internal hydrostatic pressure. Method. The problem is reduced to an inhomogeneous differential equation of the fourth degree with respect to the displacement of the middle surface w along the z axis. Since the closed form representation of the solution to this equation is extremely difficult, the search for it is presented numerically, in particular, using the grid method. The creep strain components ε*x, ε*θ, γ*xθ were determined as a linear approximation of the velocity by the Runge-Kutta method. Result. In the process of calculating the shell using moment theory, it was found that as a result of shell creep, tangential stresses increased by more than 12 percent. Conclusion. The proposed technique makes it possible to simulate changes in the mechanical properties of the shell (for example, indirect heterogeneity) caused by the influence of physical fields.

Sub-harmonic and super-harmonic resonances of spiral stiffened FG cylindrical panels of variable thickness embedded within a nonlinear elastic foundation

This research devotes to study the behaviors of sub- and super-harmonic resonances of spiral stiffened functionally graded (SSFG) variable thickness cylindrical panels (VTCP) exposed to external loading. The panel considered is rested on a nonlinear elastic foundation (NEF), which is composed of two linear foundations namely Pasternak and Winkler foundations, and a nonlinear foundation with cubic stiffness. The cylindrical panels of variable thickness are reinforced with internal spiral stiffeners. Both the cylindrical panels and spiral stiffeners are considered to be continuously graded along their thickness direction. To model the spiral stiffeners, the smeared stiffeners technique is utilized, and for deriving the discretized nonlinear governing equation of the SSFG-VTCP, the improved Donnell shell theory, von-Kármán equation, and Galerkin’s method are applied. In continuing, to examine the sub- and super-harmonic resonances, the method of multiple scales is implemented. Also, to validate the present results, in addition to comparing the present results with that of the previous research, the response of super-harmonic resonance, which is obtained theoretically, is compared with the 4th-order P-T method.

A machine learning approach to evaluate the influence of higher-order generalized variables on shell free vibrations

This work focuses on deriving guidelines for choosing structural theories for composite shells using Convolutional Neural Networks (CNN). The Axiomatic/Asymptotic Method (AAM) is used to evaluate higher-order structural theories’ accuracy and computational efficiency based on polynomial expansions. AAM exploits the Carrera Unified Formulation to derive the finite element matrices and obtain natural frequencies. The outcomes of AAM concerning the accuracy and computational cost are used to train CNN for various composite shell configurations. The trained network can then be used as a substitute for finite element models to estimate the accuracy of a given structural theory. The results are provided via Best Theory Diagrams (BTD), in which the set of generalized displacement variables to retain the best accuracy can be read for a given amount of nodal degrees of freedom. Verification is carried out using results from FEM. The results proved the computational efficiency of CNN and highlighted the influence of the shell thickness for the proper choice of the structural theory. Third-order terms and transverse stretching are often necessary to obtain acceptable accuracy.

Nonlinear combined harmonic resonances of composite cylindrical shells operating in hygro-thermo-electro-magneto-mechanical fields

By considering the effects of the hygro-thermo-electro-magnetic environments, von Karman nonlinear terms, and multi-harmonic excitations, a coupled nonlinear vibration modeling of composite cylindrical shells comprising a carbon nanotube-reinforced composite (CNTRC) core and two piezoelectric/magnetic composite (PEMC) skins is developed, and the nonlinear dynamic behaviors of such cylindrical shells under primary, super/sub-harmonic, and combined resonance states are investigated. In the theoretical modeling process, the effective mechanical properties of the CNTRC core are first determined using the mixing and Schapery laws, and the hygro-thermo-electro-magneto-mechanical constitutive relations of the PEMC skins are then formulated. Within the framework of Reissner-Mindlin shell theory, the Lagrangian of the system containing Green-Lagrange and von Karman nonlinear terms is derived, and solution techniques based on the multiscale method are provided to obtain nonlinear frequencies, dynamic responses, and phases of CNTRC-PEMC cylindrical shells under multi-physics fields. Consequently, comparison studies are conducted to validate the correctness of the proposed model from different aspects. Based on this, various resonance and chaos behaviors of such structures subjected to multiple load components are revealed and compared, and the influences of environmental factors and structure composition on the amplitude-frequency curves, time-history responses, and phase planes are explored, with several recommendations and findings being drawn.

Magneto-thermoelastic coupled resonance characteristics of a rotating functionally graded ferromagnetic cylindrical shell under double harmonic line loads

The magneto-thermoelastic coupled resonance of a rotating functionally graded (FG) ferromagnetic cylindrical shell subjected to double harmonic line loads in magnetic and temperature fields is investigated. In accordance with Donnell's thin shell theory on the physical neutral surface, the nonlinear geometric equations are determined. Utilizing the thermoelastic constitutive relationships, combining with the gradient characteristics and temperature dependence of physical parameters, the governing equations of kinetic energy and strain energy are given. Based on the magnetoelasticity theory, and considering the nonlinear magnetization of ferromagnetic metal, the calculation formulas of Lorentz force and magnetizing force acting on the shell are deduced. The magneto-thermoelastic coupled dynamic equations are derived by applying Hamilton principle and discretized by Galerkin method. Subsequently, considering the different resonance forms, the multiple scales method and Lyapunov stability theory are employed to obtain the theoretical solutions and stability discriminants. Numerical examples are carried out to plot the characteristic diagrams with different parameters, which are used to analyze the characteristics of nonlinear coupled resonance. Furthermore, the primary resonance and combination resonance are selected to analyze the bifurcation and chaotic behavior. Results show that the variation of different parameters affects obviously the resonance amplitude and stability. The changes of amplitude and position of hard excitation can lead to resonance region drift. Amplitude, frequency, and position of excitation have significant effects on bifurcation and chaos. The motion state of cylindrical shell is transmitted from periodic response to chaotic response through the accumulation of period-doubling bifurcation.

Free vibration analysis of functionally graded shallow shells with variable thickness and physical neutral surface effects

The study and manufacture of functionally graded (FG) shallow shells with the variable thickness are essential for the actual application of these types of structures. Previously, many researchers have developed different approaches to analyzing the mechanical behaviors of the FG structures, in which the thickness parameter has been assumed to be constant. Moreover, studies on these types of structures with the variable thickness are still quite limited. For this reason, this paper provides an effective and simple approach via mathematical formulations for analyzing the free vibrational responses of FG shallow shells with the variable thickness. Also, the presence of porosities that can occur during the fabrication of FG materials is considered. One of the highlights here is to consider the influence of the physical neutral surface. Before conducting the analysis of these FG structures, firstly, it is important to establish governing equations based on the classical shell theory. These essential equations are used for the FG shallow shells. Then, by setting the Navier solution with the help of simply supported boundary conditions, the natural frequencies are obtained. Finally, the numerical examples will be given to illustrate the effects of porosity coefficient, material and geometry parameters as well as the physical neutral surface on the natural frequencies. In addition, two numerical examples for verification will also be performed and thereby show the accuracy of the current method and model. It is hoped that the main results of this study can be useful not only to designers and researchers but also to be extended to more complex problems.

Geometrically nonlinear bending of stiffened composite skewed cylindrical shells under transverse pressure

The literature review on relative performances of laminated composite skewed shells confirms that research reports on bending performances of moderately thin, stiffened, laminated composite skewed cylindrical panels, using the geometrically nonlinear approach, are not available. This paper aims to fill that deficiency and proposes a finite element code combining eight-noded, doubly curved elements with modified Sanders’ first approximation theory for thin shells and von Kármán-type nonlinear strains. Correctness of the proposed geometrically nonlinear bending formulation for skewed shells are verified through solutions of benchmark problems. The deflections, force, and moment resultants are reported for different skew angles, laminations, stacking sequences, radius of curvature, plan dimension ratios, and stiffener properties like orientations, numbers, and eccentric positions. The results are discussed critically which reveals that shells having curved edges free and straight edges clamped fabricated using 0°/90°/0° laminate offer the best performances. The biaxial stiffeners, nx = 7, ny = 7, show the minimum deflections and stress resultants. The skewed shells offer greater deflections and hence, must be avoided in industrial practices.