Order Shear Deformation Shell Theory Research Articles

Vibrations of skew cylindrical shells made of FG-GPLRCs under rapid surface heating

An analysis is performed in this research to examine the thermally induced vibrations in skew cylindrical shells. It is assumed that shell is made from a composite laminated material where each layer is reinforced with graphene platelets (GPLs). The amount of GPLs in the layers may be different which results in a piecewise functionally graded media. The one-dimensional transient heat conduction equation is established through the thickness of the shell. Heat conduction equation is solved using the finite difference method for each layer and continuity of temperature and heat flux is applied between the layers. Crank Nicolson algorithm is implemented to obtain the temperature profile at each time step and after that thermally induced force and moment are evaluated. The kinematics of the shell are estimated following the first order shear deformation shell theory. Also it should be noted that an oblique coordinates system in addition to orthogonal system is defined which make it easier to apply the boundary conditions. The definition of strain and kinetic energies are constructed and Following the Ritz method whose shape functions are estimated by Chebyshev polynomials, energies are provided in discrete form of the governing equations. Results of this study are provided to explore the effects of thermal and mechanical boundary conditions, geometry of the shell, weight fraction and distribution patterns of GPLs. It is highlighted that thermally induced vibrations indeed exist especially in thin class of shells.

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.

Low-velocity impact resistance of all composite cylindrical shell panels with a foam filled honeycomb core: Theoretical and experimental investigation

The low-velocity impact resistance of all composite cylindrical shell panels (CCSPs) with a foam filled honeycomb core (FFCHC) is studied experimentally and theoretically. Initially, a dynamical model of the FFCHC-CCSPs is proposed, with the equivalent material properties of the core being determined by combining the Hamilton equivalence theory and the Gibson theory. By using the virtual work principle and quasi-static method, together with the von Karman's theory and the high order shear deformation shell theory, the constitutive relationship related to each damage case is derived. Also, by considering the influence of collapse of the FFCHC, the failure criteria and damage modes are modified. The impact displacement, load and impact energy absorption are solved by using the elastoplastic contact principle and energy principle. Several numerical results from the literature are employed for preliminary validation of the model. Furthermore, the FFCHC-CCSP specimens with and without polyurethane foam are fabricated and the detailed measurements associated with different impact energies are performed to provide a solid validation of the present model as well as to evaluate the structural impact resistance. Finally, the influences of the thickness of FFCHC, wall thickness and side length of honeycomb cell and density of foam on the impact resistance of the FFCHC-CCSP structures are discussed, with some useful conclusions being refined for better impact suppression design and manufacture of such advanced shells.

Application of GDQ method to large amplitude response of FGM joined spherical-conical shells under rapid surface heating

Geometrically non-linear thermally induced vibrations of functionally graded material (FGM) joined spherical-conical shells are analyzed in the current research. Thermo-mechanical properties of the shells are assumed to be temperature and position dependent. The system of joined spherical-conical shells is subjected to thermal shock on the ceramic-rich surface, whereas the metal-rich one is kept at reference temperature. The one-dimensional transient heat conduction equation is established and solved via the generalized differential quadrature (GDQ) and Crank-Nicolson methods. This equation is non-linear since the thermo-mechanical properties of the shells are temperature dependent. The total functional of the shells is obtained under the assumptions of uncoupled thermoelasticity laws, first order shear deformation shell theory, and the von Kármán type of geometrical non-linearity. Non-linear coupled equations of motion are solved via the iterative Picard method accompanied with the β-Newmark time approximation technique. Numerical results are well-validated with the available data for the case of single FGM spherical shell. Parametric studies are conducted to examine the influences of conical and spherical shell geometries, material composition, temperature dependence, in-plane and out-of-plane mechanical boundary conditions, various configuration of conical/spherical shell system, and thermal boundary conditions. It is highlighted that thermally induced vibrations indeed exists.

Free vibration analysis of cylindrical honeycomb sandwich panels using state-space Levy method

In the current study, free vibrations of a honeycomb sandwich panel were investigated. The cylindrical sandwich panel consisted of a hexagonal honeycomb core layer and two face sheets. The hexagonal cell honeycomb core was modeled as a solid core made of aluminum according to Gibson’s hypothesis. Based on the first order shear deformation shell theory, the coupled governing equations of the panel were derived by using Hamilton’s principle. By using the state-space method, the equations were decoupled by shifting the equations from spatial space to modal space and were finally recast in the form of a system of equations. In a free vibration analysis, the roots of the coefficient matrix determinant return the eigenvalues. The validity and accuracy of the current technique were proved by comparing the obtained results with the results of previously published studies. Influences of various geometrical and material parameters and boundary conditions on natural frequencies of the panel were investigated. Numerical examples presented in this study showed that the geometrical parameters of the honeycomb core play a key role in the resulting natural frequencies of the sandwich panel. The results of this paper provide possible benchmark solutions for assessing future works on the free vibration of cylindrical honeycomb sandwich panels and can be relied on for the design of such structures in various engineering applications.

Geometrically nonlinear response of FGM joined conical–conical shells subjected to thermal shock

Geometrically nonlinear thermally induced vibrations of functionally graded material (FGM) joined conical–conical shells is analyzed in the current research. Thermo-mechanical properties of the shell are assumed to be temperature and position dependent. The system of joined conical–conical shells is subjected to thermal shock on the ceramic-rich surface, whereas the metal-rich one is kept at reference temperature. The one-dimensional transient heat conduction equation is established and solved via the generalized differential quadrature (GDQ) and the Crank–Nicolson methods. This equation is nonlinear since the thermo-mechanical properties of the shell are temperature dependent. The total functional of the shell is obtained under the assumptions of uncoupled thermoelasticity laws, first order shear deformation shell theory, and the von Kármán type of geometrical non-linearity. Non-linear coupled equations of motion are solved via the iterative Picard method accompanied with the β-Newmark time approximation technique. Numerical results are well validated with the available results for the case of FGM single conical shell. Parametric studies are conducted to examine the influences of conical geometry, material composition, temperature dependence, in-plane and out-of-plane mechanical, various configuration of joined shell system, and thermal boundary conditions.

Free vibration analysis of laminated cylindrical adhesive joints with conical composite shell adherends

Free vibration analysis of adhesive joins between two laminates circular cylindrical along with conical laminated composite shells considering various boundary conditions are examined. The numerical analysis are studied based on the first order shear deformation shell theory (FSDT) and considering compatibility conditions. Using Hamilton principle method, the governing equations of composite cylindrical shell lap adhesive joined to the composite conical shells are obtained. The analysis for carbon/epoxy, glass/epoxy and aramid/epoxy laminated composite materials and various type of adhesive materials are studied. For solving the equilibrium equations of lap joined the cylindrical to the conical shells with adhesive layer, the generalized differential quadrature method (GDQM) is used. The effect of circumferential wave number, length to radius ratio of the shell, length of overlap to length of shell ratio, thickness to radius ratio and thickness of adhesive to thickness of shell ratio are investigated. Furthermore, the various cone angles of conical shell, type of materials and adhesive layer on the natural frequency of laminated circular cylindrical connected to the conical shells are studied. In order to validate the numerical results of present analysis with previous research compared that the result shown that these comparisons show very good agreement. Numerical results shown that with increasing the value of cone angle of the conical shell and the ratio of overlap length to shell length, due to increasing the structural stiffness, the non-dimensional frequency of the structure with adhesive layer increased.

Primary resonance of double-curved nanocomposite shells using nonlinear theory and multi-scales method: Modeling and analytical solution

In this article, the forced vibration of double-curved nanocomposite shells under a time dependent excitation is studied using nonlinear shell theory and multi-scales method in primary resonance. The nanocomposite representative volume element consists of two phases, including carbon nanotube (CNT) and matrix. By generalizing the Ambartsumyan’s first order shear deformation shell theory (FSDT) to the heterogeneous nanocomposite shells, the nonlinear partial differential equations are derived. Then, the problem is reduced to the nonlinear forced vibration of damped nanocomposite shells with quadratic and cubic nonlinearities. For the occurrence of the primary resonance, the damping, nonlinearity, and excitation terms in the disturbance circuit are reduced to the same order. Applying the multi-scales method to nonlinear ordinary differential equation, nonlinear frequency–amplitude dependence in primary resonance is obtained.

A novel [formula omitted] continuity finite element based on Mindlin theory for doubly-curved laminated composite shells

As a widely used structural form, doubly-curved composite shells have been applied in aviation and other engineering fields. With a refined mechanical model, structural performances can be accurately predicted to help designers choosing best geometrical and material parameters. This work introduces a novel rectangular finite element for doubly-curved laminated composite shells based on a new set of strain-based shape functions. The governing equations are established on the Mindlin shell theory (a type of first order shear deformable shell theory), which incorporates a rectification of shear correction factor for laminates and von Kármán type nonlinearity. Based on strain approach, the shape functions for rectangular element are assumed as polynomial of 28 parameters to consider the influence of shear effect. Apart from 20 geometrical relations of four element nodes, shear force equilibrium equations are introduced to offer the additional eight equations to derive a new set of shape functions for finite element model. Using shape functions, a novel rectangular shell element is proposed for doubly curved laminated shell, which also maintains a compatibility with Kirchhoff shells. Numerical results for linear static bending, dynamic vibration and nonlinear bending cases of flat plate, cylindrical shell and doubly-curved laminated shell are compared with the available results in literatures and ABAQUS simulation for the sake of validating the present method.

Effect of negative Poisson’s ratio on the postbuckling behavior of pressure-loaded FG-GRMMC laminated cylindrical shells

Auxetic materials have emerged to be a new type of novel engineering materials with unique material properties. This paper reports the postbuckling behaviors of pressure loaded graphene-reinforced metal matrix composite (GRMMC) laminated cylindrical shells under the influence of in-plane negative Poisson’s ratio (NPR) in temperature environments. The GRMMCs have temperature-dependent material properties which can be determined using an extended micromechanical model of Halpin–Tsai type. A cylindrical shell is made of GRMMC layers of different graphene volume fractions to achieve a piece-wise functionally graded (FG) pattern. The postbuckling equations for the pressure-loaded GRMMC laminated cylindrical shells are derived using the Reddy’s third order shear deformation shell theory with the effects of von Kármán-type kinematic nonlinearity and temperature variation being included. Applying the singular perturbation technique in conjunction with a two-step perturbation approach, the governing equations for the shell postbuckling problem are solved. The results show that the postbuckling behaviors of pressure-loaded GRMMC laminated cylindrical shells are affected substantially by the in-plane NPR.

Assessment of negative poisson's ratio effect on thermal post-buckling of FG-GRMMC laminated cylindrical panels

This paper examines the thermal post-buckling behaviors of graphene-reinforced metal matrix composite (GRMMC) laminated cylindrical panels which possess in-plane negative Poisson's ratio (NPR) and rest on an elastic foundation. A panel consists of GRMMC layers of piece-wise varying graphene volume fractions to obtain functionally graded (FG) patterns. Based on the MD simulation results, the GRMMCs exhibit in-plane NPR as well as temperature-dependent material properties. The governing equations for the thermal post-buckling of panels are based on the Reddy's third order shear deformation shell theory. The von Karman nonlinear strain-displacement relationship and the elastic foundation are also included. The nonlinear partial differential equations for GRMMC laminated cylindrical panels are solved by means of a singular perturbation technique in associate with a two-step perturbation approach and in the solution process the boundary layer effect is considered. The results of numerical investigations reveal that the thermal post-buckling strength for (0/90)5T GRMMC laminated cylindrical panels can be enhanced with an FG-X pattern. The thermal post-buckling load-deflection curve of 6-layer (0/90/0)S and (0/90)3T panels of FG-X pattern are higher than those of 10-layer (0/90/0/90/0)S and (0/90)5T panels of FG-X pattern.

A finite element model for the analysis of flexible tube Coriolis mass flow meter using first order shear deformation shell theory

The numerical modelling of Coriolis Mass flow Meter (CFM) is essential for predicting its outcomes accurately in terms of sensitivity as well as exact mass flow rates. In the majority of mathematical and numerical modelling concerning the flexible structures, the authors neglect the dimensional and shape variation of the structure due to self-weight. The shell based on the First-order shear deformation shell theory (FSDST) is preferred in modelling shells compared to the beam model. The current work includes numerical modelling of CFM using eight noded isoparametric shell elements and twenty noded Acoustic fluid elements. The fluid energy describes as the potential, and the dynamic boundary condition is assumed utilising the displacement of structure and potential of the fluid. The fluid dynamic equation combining suitable numerical model, fluid-structure interaction module and cross-correlation technique helps to achieve the numerical modelling of CFM. The numerical model of CFM utilises the Newmark Beta method of numerical integration, and the response of two equidistant locations from the point of tube excitation is acquired. For the flexible tube conveying fluid, there exists sagging of tube due to the weight of tube and fluid. The Coriolis force and the external excitation force cause the fluid conveying tube to bend and twist, and as a result, the velocity responses picked from two equidistant points shows a difference in phase. The effect of sagging leads to a lower phase shift and time decay, and hence the sensitivity of the CFM is low for low pre-stretched flexible tubes. The pre-stretching of flexible tubes reduces the effect of sagging, facilitates to regain the cylindrical shape of the tube and increases the sensitivity of CFM. The result reveals that the shell element along with the three-dimensional acoustic fluid element provides the most accurate numerical model for the CFM and the change in sensitivity, as well as the change in mass flow measurements, can appropriately be analysed with the help of this numerical model. The amplitude of the velocity of the structure, measured from the two equidistant points, shows a difference. The severe variation in amplitude of velocity measured from two points is an implication of the out of plane deflection of the tube. For a CFM made up of metal tubes, the amplitude of velocity variation is minimal and ignored by the authors.

Nonlinear forced vibration of sandwich cylindrical panel with negative Poisson’s ratio auxetic honeycombs core and CNTRC face sheets

Recently, auxetic cellular solids in the forms of honeycombs and carbon nanotube reinforced composite (CNTRC) have great potential in a diverse range of applications. This paper studies the nonlinear free and forced vibration of sandwich cylindrical panel on visco-Pasternak foundations in thermal environment subjected to blast load The sandwich cylindrical panel consists of auxetic honeycombs core layer and two CNTRC face sheets. The Poisson’ ratio of the auxetic core is negative and the material properties of CNTRC face sheets are assumed to continuously vary in the thickness direction according to four different types of linear functions. The blast load is determined from the distance of the center of blast to center of the structure and the mass of explosive materials. The fundamental equations are established based on Reddy’s higher order shear deformation shell theory taking into account the effect of initial imperfection and visco-Pasternak foundations. The Galerkin and Runge–Kutta methods are used to obtain the nonlinear dynamic response and the natural frequency of the sandwich cylindrical panel. The numerical results show the effect of geometrical parameters, visco foundations, initial imperfection, temperature increment, nanotube volume fraction and blast load on the nonlinear vibration characteristics of the sandwich cylindrical panel. The accuracy of present approach and theoretical results is verified by some comparisons with the known data in the literature.

Assessment of negative Poisson’s ratio effect on the postbuckling of pressure-loaded FG-CNTRC laminated cylindrical shells

This paper examines the postbuckling behaviors of pressure-loaded laminated cylindrical shells made of carbon nanotube reinforced composite (CNTRC) under thermal environmental conditions. The shell has negative in-plane or out-of-plane effective Poisson’s ratio (EPR). The thermo-mechanical properties of CNTRCs are temperature dependent. The CNT volume fraction of the shell is in a piece-wise functionally graded (FG) distribution across the shell thickness. The extended rule of mixture (EROM) model is applied to determine the thermo-mechanical properties of CNTRCs. Based on the framework of the Reddy's third order shear deformation shell theory and the von Kármán-type of kinematic assumptions, the governing partial differential equations for pressure-loaded CNTRC laminated cylindrical shells are formulated. The thermal environmental condition effect is also included. The postbuckling solution for pressure-loaded FG-CNTRC laminated cylindrical shell is obtained in the asymptotic sense by means of a singular perturbation technique in associate with a two-step perturbation approach. Numerical investigations are carried out for the (10/60/10)S shell including in-plane NPR effect, and (20/-20/20)S and (70/-70/70)S shells including out-of-plane NPR effect. Assessment is presented through numerical comparisons.

Application of Haar wavelet method for free vibration of laminated composite conical–cylindrical coupled shells with elastic boundary condition

In this paper, the Haar wavelet discretization method (HWDM) is proposed for the calculation of natural frequency of the laminated composite conical-cylindrical coupled shells. The displacement components of the system are set by the first order shear deformation shell theory (FSDST), and all the displacement functions of conical and cylindrical shells including boundary conditions are obtained by the Haar wavelet and its integrals. The Hamilton’s principle is applied to the constitution equation and the artificial spring technique is introduced to generalize the boundary and continuity conditions of the coupled structure. The convergence and accuracy of the proposed method are validated by comparing with the results obtained from the previous works and finite element analysis, and the proposed method shows high accuracy, reliability and good convergence in both individual structure and coupled structure. Using the proposed method, in the laminated composite conical–cylindrical coupled shell with any boundary condition, new numerical results on the natural frequency are obtained along with the parameter studies, and these results can provide the referential data for the following research in this field.

Nonlinear vibration and dynamic instability analyses of laminated doubly curved panels in thermal environments

Nonlinear dynamic analysis for a shear deformable anisotropic laminated doubly curved panel of rectangular planform resting on elastic foundations is presented. A new panel model of arbitrary curvature and arbitrary fiber stacking sequences but constant thickness is developed and established. The governing equations are based on an extended higher order shear deformation shell theory with von Kármán-type of kinematic nonlinearity and including the anisotropic couplings. The nonlinear deformations and initial deflections of the panel are both taken into account. A two-step perturbation technique combining with the Galerkin method is employed to determine the linear and nonlinear vibration frequencies and forced responses. Nonlinear forced harmonic vibration in a Duffing oscillator for anisotropic laminated doubly curved panel is also investigated and discussed together with the effects of damping and excitation by using multiscale and Melnikov methods. The numerical results are validated for various lamination schemes through a comparison with solutions for free vibration frequencies of saddle (hyperbolic), flat plate, cylindrical and spherical shells available in the literature or obtained by the commercial finite element software. The numerical illustrations concern nonlinear vibration and dynamic response of moderately thick, anisotropic laminated doubly curved panels under different values of panel geometric parameters and mechanical excitation.

Study on dynamic instability characteristics of functionally graded material sandwich conical shells with arbitrary boundary conditions

This paper presents analytical studies on the dynamic instability of functionally graded material (FGM) sandwich conical shell subjected to time dependent periodic parametric axial and lateral load. In the analysis, four kinds of sandwich distributions, including FGM core and isotropic skins, and isotropic core and FGM skins are considered and the power law distribution is employed to estimate the volume of the constituents. The arbitrary boundary conditions of the FGM sandwich conical shell are achieved by using the artificial spring boundary. The governing equations of conical shell subjected to parametric excitation are established by the Hamilton’s principle considering first order shear deformation shell theory. Then the Mathieu-Hill equations describing the parametric stability of conical shell are obtained by generalized differential quadrature (GDQ) method, and the Bolotin’s method is utilized to obtain the first-order approximations of principal instability regions of shell structure. By comparing the numerical results with the existing solutions in open literature, the validity of the proposed theoretical model is verified. Finally, the influences of sandwich distribution types, gradient indexes, skin-core-skin ratio and load forms on the dynamic stability of FGM sandwich conical shell have been investigated.

Effect of negative poisson's ratio on the axially compressed postbuckling behavior of FG-GRMMC laminated cylindrical panels on elastic foundations

Auxetic materials are one of the metamaterials that have potential applications in many scientific and engineering fields. In this paper, an investigation is presented on the postbuckling responses of axially loaded graphenereinforced metal matrix composite (GRMMC) laminated cylindrical panels with in-plane negative Poisson's ratio (NPR). The panels are made of GRMMC laminates and rest on an elastic foundation in thermal environments. The graphene volume fraction in GRMMC layer may vary across the panel thickness in order to achieve a piece-wise functionally graded (FG) pattern. The GRMMC layers have in-plane NPR and temperature dependent material properties. The governing differential equations for the GRMMC laminated cylindrical panels are formulated based on the Reddy's third order shear deformation shell theory and are solved by using a singular perturbation technique along with a two-step perturbation approach. Numerical investigation is carried out to study the influence of the in-plane NPR, the FG patterns, the thermal environmental conditions and the foundation stiffness on the postbuckling responses of the GRMMC laminated cylindrical panels under axial compressive load. The results reveal that the in-plane NPR can lead to a substantial change of the postbuckling behaviors of the GRMMC laminated cylindrical panels.

Dynamic instability analysis of FG-CNTRC laminated conical shells surrounded by elastic foundations within FSDT

A comprehensive understanding of the dynamic instability of shell structure is critical to avoid resonance damage. On the basis of that, an accurate and analytical method for investigation the dynamic instability of laminated functionally graded carbon nanotube reinforced composite (FG-CNTRC) conical shell surrounded by the elastic foundations is presented in this work based on the first-order shear deformation theory. In the analysis, uniform or functionally graded distributions of reinforcements across the shell thickness are considered and the extended Voigt model is employed to estimate the CNTRC material properties. The governing equations of conical shell subjected to parametric excitation are established by the Hamilton's principle considering first order shear deformation shell theory. Then the Mathieu-Hill equations describing the parametric stability of conical shell are obtained by generalized differential quadrature (GDQ) method, and the Bolotin's method is utilized to obtain the first-order approximations of principal instability regions of shell structure. By comparing the numerical results with the existing solutions in open literature, the validity of the proposed theoretical model is verified. Finally, the influences of volume fractions and types of CNTs, lamination angle, elastic foundations stiffness and lamination angle on the dynamic stability of laminated FG-CNTRC conical shell have been investigated.

Effect of external pressure on the vibration analysis of higher order shear deformable FG-CNTRC spherical panels

The numerical investigation is performed on the vibration of the higher order shear deformable carbon nanotube reinforced composite (CNTRC) spherical panels subjected to the initial external pressure. The functionally graded nanocomposite is reinforced with the non-uniform distribution of CNTs. The problem is formulated following the higher order shear deformation shell theory (HSDT) within the variational differential quadrature numerical approach. For this purpose, the direct discretization of Hamilton’s principle is carried out with the aid of differential quadrature operators. Derivation of the variational formulation of nanocomposite spherical panels based on the HSDT and studying the effects of external pressure on the vibration behavior are the main novel aspects of this research. Several numerical examples are provided to survey the impacts of geometrical and material factors on the vibration of pressurized functionally graded CNTRC spherical panels. It is shown that the internal pressure has the most influence on the vibrational behavior of the thicker panel.