Shear Deformation Shell Theory Research Articles

Nonlinear vibration analysis of CNT-reinforced functionally graded composite cylindrical shells resting on elastic foundations

Nonlinear vibration analysis of carbon-nanotube reinforced functionally graded composite (FG-CNTRC) cylindrical shells resting on elastic foundations is carried out. Four carbon nanotube distributions, namely UD, FGV, FGΛ and FGX, are considered and the elastic foundation is described using the Pasternak model. According to the first-order shear deformation shell theory, the nonlinear governing equations of FG-CNTRC cylindrical shells are derived from Lagrange equations. The forced nonlinear vibrations of FG-CNTRC cylindrical shells with/without surrounding elastic media are then investigated using the harmonic balance method combined with the arc length continuation technique, and chaotic vibrations due to the 1:1 internal resonance are further studied through direct integration. Finally, the effects of the distribution patterns and the stiffness of elastic foundations on the nonlinear behaviors of FG-CNTRC cylindrical shells are presented in detail.

Vibration analysis of functionally graded porous cylindrical shells filled with dense fluid using an energy method

In this paper, a novel energy approach is proposed for fully coupled fluid-structure problems of functionally graded porous (FGP) fluid-filled cylindrical shells under arbitrary boundary conditions. Kinds of continuous FG materials are considered including various patterns in which the voids distributed. A modified variational principle is developed in fluid domain as well as in structural domain, naturally reinforcing the restrains on the boundaries and interfaces between adjacent subdomains. Energy functions in structural and fluid domain are deduced based on the first-order shear deformable shell theory (FSDT) and Helmholtz equation, respectively. The fluid-structure interactions are successfully introduced with the work done by the sound pressure and displacement continuous conditions at fluid-structural interface. A semi-analytical solution is obtained by expanding the displacement and sound pressure components analytically in the circumferential direction and numerically in the axial direction. Good convergence, high accuracy, superior efficiency and flexibility in using admissible functions are demonstrated by comparing reasonably many proposed results with those in literatures or obtained by FEM/BEM. Numerous examples are developed to examine the effects of material properties and boundary conditions on free vibration of FGP cylindrical shells with and without water. With added mass factor introduced, the influences of the key parameters on the fluid-structure coupling are also presented. The slight dependences of the added mass factor on material and boundary conditions greatly broaden the applied scope of the numerical results.

Nonlinear vibrations and dynamic snap-through behaviors of four-corner simply supported bistable asymmetric laminated composite square shell

The nonlinear vibrations and dynamic snap-through behaviors are studied for a four-corner simply supported bistable asymmetric laminated composite square shell under the foundation excitation by using the theoretical, Abaqus finite element (FE) and experiment approaches. The first-order shear deformation shell theory and Hamilton principle are adopted to derive the partial differential governing equations of motion. A novel type of nonlinear strain-displacement relations is given by Sander’s strain. Galerkin approach is utilized to discretize the partial differential equations of motion into a three-degree-of-freedom system. The natural frequencies and vibration modes of the bistable shell are calculated by using Chebyshev polynomial method. The nonlinear vibrations and dynamic snap-through are investigated by using the maximum Lyapunov exponent, bifurcation diagrams, time histories, phase portraits, 3D phase portraits and Poincare maps. The theoretical results of the vibrations are well compared with those of the FE and experiment. The effects of the excitation and structural parameters on the dynamic snap-through and nonlinear vibrations are fully discussed. The influences of the initial radius of the curvature on the critical load of the dynamic snap-through behaviors and rule of the chaotic vibrations are analyzed for the bistable asymmetric laminated composite shell.

Nonlinear Resonance of Functionally Graded Porous Circular Cylindrical Shells Reinforced by Graphene Platelet with Initial Imperfections Using Higher-Order Shear Deformation Theory

This paper uses the higher-order shear deformation theory (HOSDT) to analyze the nonlinear resonance of functionally graded graphene platelet-reinforced porous (FG-GPL-RP) circular cylindrical shell with geometrical imperfections subjected to the harmonic transverse loading. The shell considered is surrounded by the elastic Winkler–Pasternak foundation. Based on Hamilton’s principle, the nonlinear governing equations of motion of the imperfect system considered are established using the first-order and various higher-order shear deformation shell theories that contain a unified higher-order displacement field. Four types of porosity and graphene nanoplatelet distribution patterns are considered. The modified Halpin-Tsai scheme and rule of mixture are utilized to calculate the effective properties of FG-GPL-RP materials. Explicit expressions of the nonlinear primary resonance of imperfect FG-GPL-RP cylindrical shells for simply-supported boundary conditions are achieved by the Galerkin technique and method of multiple scales. After verifying the accuracy of the model used and the results obtained, the influences of the geometric parameters, material properties and distribution and imperfection value on the nonlinear primary resonant response of the imperfect FG-GPL-RP circular cylindrical shells are examined in detail.

Vibration behavior analysis of novelty corrugated-core sandwich plate structure by using first-order shear deformation plate and shell theories

Vibration behavior analysis of novelty corrugated-core sandwich plate structure by using first-order shear deformation plate and shell theories

Wave Propagation Characteristics in Submerged Pipes Conveying Viscous Flowing Fluid Based on the Shear Deformation Shell Theory

Wave Propagation Characteristics in Submerged Pipes Conveying Viscous Flowing Fluid Based on the Shear Deformation Shell Theory

A couple-stress-based moving Kriging meshfree shell model for axial postbuckling analysis of random checkerboard composite cylindrical microshells

The current investigation intends to devise a couple stress continuum-based moving Kriging meshfree shell model for axial nonlinear buckling and postbuckling responses of random checkerboard composite cylindrical microshells. Accordingly, the graphene nanoplatelet reinforcements are supposed to be randomly dispersed based upon a checkerboard scheme within the resin matrix, the effective material properties of which are estimated based upon a probabilistic method. The modified couple stress continuum elasticity is implemented into the higher-order shear deformation shell theory incorporating geometrical nonlinearity. The established microstructural-dependent shell model is then numerically analyzed via the moving Kriging meshfree technique with the ability to take the essential boundary conditions into account accurately by employing proper moving Kriging shape function. By tracing the postbuckling paths, the snap-through phenomenon is observed. It is found that for a higher graphene nanoplatelet volume fraction, the significance of the stiffening character associated with the rotation gradient tensor increases. Furthermore, for a specific value of graphene nanoplatelet volume fraction, by reducing the length to width aspect ratio of graphene nanoplatelet reinforcements, the role of couple stress size dependency in the critical buckling load and critical shortening of an axially compressed cylindrical microshell becomes more important.

Free vibration analysis of coupled structures of laminated composite conical, cylindrical and spherical shells based on the spectral-Tchebychev technique

In this article, a numerical spectral-Tchebyshev (ST) technique is applied to solve the free vibration strong solution of the coupled structures of laminated composite conical, cylindrical and spherical shells under various boundary conditions. The artificial spring technique is introduced to realize the coupling connection of substructures and simulate the arbitrary boundary conditions. Considering the first-order shear deformation shell theory (FSDST), and using Hamilton variational analysis under its framework to derive the governing equations of motions of three coupled structures: coupled conical-cylindrical shells, coupled spherical-cylindrical shells and coupled spherical-cylindrical-conical shells. The solution of the governing equations of motions is obtained using the spectral-Tchebyshev technique. The variables along the generatrix direction are discretized based on the Gauss-Lobatto nodes, while the Tchebyshev polynomials are used for spectral approximation. The variables along the circumferential direction are approximated by Fourier series. The calculated free vibration characteristics of the coupled structures are compared with the numerical results based on previous references and the finite element method, which completely reflects the excellent convergence and calculation accuracy of the spectral-Tchebyshev technique. On this basis, the effect of the material properties and geometric dimensions of the substructures (conical, cylindrical and spherical shells) on the free vibration frequencies of the coupled structure is further studied.

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.

Free vibration analysis of functionally graded cylindrical shells reinforced with graphene platelets

This research investigates the free vibrations of graphene platelet reinforced composite (GPLRC) cylindrical shells. Composite cylindrical shell is composed of a number of layers, each layer of composite shell has a different amount of graphene reinforcement, which leads to functionally graded (FG) distribution of the reinforcements. For this purpose, the first-order shear deformation theory of shells and Donnell kinematic relations have been used. To estimate the elastic properties of the composite material, Halpin-Tsai micro-mechanical relationship has been used where the effects of nanofillers is also included. On the other hand to obtain the mass density and Poisson's ratio of the media, the simple rule of mixtures approach is adopted. The Hamilton principle has been used to derive the governing equations for the free vibrations of shells as well as the boundary conditions. The obtained equations are five in number which are coupled in terms of displacements and rotations. Suitable for simply supported boundary conditions of the shell, Fourier expansions have been used as the conventional Navier solution. An analytical solution is provided to obtain the natural frequencies as well as the numbers of their modes. The results of this study at first are compared with the known data for isotropic homogeneous shells and after that novel results are provided for FG-GPLRC cylindrical shells. The results of the present study show that the weight fraction and the distribution of GPLs are two important factors on the natural frequencies of the shell, so that with increasing the weight fraction of GPLs, the natural frequencies of the shell are strongly increased. Also when the inner and outer layers of the shell have the maximum amount of reinforcements, maximum frequencies are achieved.

Nonlinear vibration of multilayer shell-type structural elements with double curvature consisting of CNT patterned layers within different theories

In this article, the nonlinear vibration of moderately thick multilayer shell-type structural elements with double curvature consisting of carbon nanotube (CNT) patterned layers is investigated within different shell theories. The first order shear deformation theory has been generalized on the motion for moderately thick multilayer shell-type structural elements with double curvature consisting of CNT patterned layers for the first time. Then, by applying Galerkin and semi-inverse perturbation methods to motion equations, and the frequency-amplitude relationship is obtained. From these formulas, the expressions for nonlinear frequencies of multilayer spherical and hyperbolic-paraboloid shells, rectangular plate and cylindrical panels patterned by CNTs within shear deformation and classical shell theories are obtained in special cases. The reliability of obtained results is verified by comparison with other results reported in the literature. The effects of transverse shear strains, volume fraction, sequence and number of nanocomposite layers on nonlinear frequency are discussed in detail.

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.

Modeling and Solution of Large Amplitude Vibration Problem of Construction Elements Made of Nanocomposites Using Shear Deformation Theory.

The main purpose of the study is to investigate the vibration behaviors of carbon nanotube (CNT) patterned double-curved construction elements using the shear deformation theory (SDT). After the visual and mathematical models of CNT patterned double-curved construction elements are created, the large amplitude stress–strain relationships and basic dynamic equations are derived using the first order shear deformation theory (FSDT). Then, using the Galerkin method, the problem is reduced to the nonlinear vibration of nanocomposite continuous systems with quadratic and cubic nonlinearities. Applying the Grigolyuk method to the obtained nonlinear differential equation, large-amplitude frequency-amplitude dependence is obtained. The expressions for nonlinear frequencies of homogenous and inhomogeneous nanocomposite construction members such as plates, panels, spherical and hyperbolic-paraboloid (hypar) shells in the framework of FSDT are found in special cases. The accuracy of the results of the current study has been confirmed by comparing them with the reliable results reported in the literature. Original analyses are carried out to examine the effects of nonlinearity, CNT patterns and volume fraction changes on frequencies in the framework of shear deformation and classical shell theories.

Nonlinear dynamic response of functionally graded cylindrical microshells conveying steady viscous fluid

Nonlinear free and forced vibrations for functionally graded (FG) cylindrical microshells conveying steady viscous fluid are investigated based on the shear deformation shell and potential flow theory. The size effects are considered by modified couple stress theory. Steady viscous force produced by fluid are added to the inviscid and incompressible microshell-fluid coupled system using time-mean Navier-Stokes equations. Using Hamilton’s principle, the nonlinear partial differential equations are derived for microshell-fluid system. The reduced nonlinear ordinary differential equations are obtained based on static condensation method and Galerkin’s method. The critical flow velocity and natural frequency are here discussed by changing material composition and scale parameters. Moreover, the effects of axial load, scale parameter, fluid flow velocity and viscosity on nonlinear dynamic response are also investigated. Comparisons with corresponding inviscid case and previously literatures are also here discussed

Free vibration analysis of composite conical shells using Walsh series method

As a typical structure of underwater vehicles, the vibration characteristics of the conical shell are of great significance to the selection of structural design parameters. This paper aims at the free vibration of composite conical shells with general boundary conditions by presenting a simple yet efficient solution based on the Walsh series (WS). The theoretical model is formulated on the basis of the first-order shear deformation shell theory. Elastic boundary conditions are equivalent by introducing weight parameters at boundary positions. The governing differential equation and boundary equation are obtained directly by the Hamilton principle, and decomposed by separating variables. Then Walsh series (WS) is applied for the axial direction and the Fourier series is assumed with respect to the circumferential direction. The unknown constants generated during integration can be determined by boundary conditions, and thus the partial differential equations are transformed into algebraic equations. By solving these algebraic equations, then natural frequencies and mode shapes of the composite conical shells are obtained. Compared with the existing solutions, the accuracy and reliability of the proposed method are verified. The effects of geometric and material parameters on the natural frequencies of composite conical shells are studied.

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.

A study on the effect of electric potential on vibration of smart nanocomposite cylindrical shells with closed circuit

A vibration analysis of smart laminated carbon nanotube-reinforced composite cylindrical shells bonded by a piezoelectric layer with closed circuit is presented. A solution for different distributions of electric potential along the thickness direction of the piezoelectric layer satisfying the closed circuit electrical boundary condition is developed for the first time. The micromechanics and macromechanics models and solution for the vibration analysis of piezoelectric coupled laminated carbon nanotube-reinforced composite cylindrical shells are then established based on different electric potential distributions (i.e. linear, quadratic, and cosine), the Mori–Tanaka model, the first-order shear deformation shell theory, and the Maxwell’s static electricity equation. Natural frequencies for various vibration modes are computed with different electric potential distributions and the effects of mechanical boundary conditions, piezoelectric thickness, nanoparticles, and shell geometry. The numerical results indicate that linear variation of the electric potential provides higher estimate of natural frequencies and quadratic and cosine variations lead to similar vibration trends and results.

Large deflection of functionally graded carbon nanotube reinforced composite cylindrical shell exposed to internal pressure and thermal gradient

In this paper, the nonlinear bending of functionally graded carbon nanotube‐reinforced composite (FG‐CNTRC) shell exposed to thermomechanical loading is perused. It is assumed that the composite shell is reinforced in the longitudinal axis and is also made from a polymeric matrix. Mechanical features of the constituents are obtained based on the modified rule of mixture, and they are considered to be temperature dependent (TD). Using the first‐order shear deformation shell theory (FSDT) as well as von Kármán type of geometrical nonlinearity, the equilibrium mathematical relations are derived. Utilizing the dynamic relaxation (DR) procedure combined with the central finite difference method, these mathematical relations are solved in diverse boundary conditions. Finally, roles of carbon nanotube (CNT) distributions, boundary conditions, shell radius, thickness‐to‐radius ratios, volume fraction of CNTs, mechanical loads, thermal gradient, and temperature dependency are examined on the results. From the numerical results, it can be inferred that in the shell with the CC boundary condition, the FG‐O distribution of nanotubes has the maximum deflection, and the lowest deflection belongs to the uniform distribution. However, in the SS boundary condition, the highest and lowest values of deflections are related to V and uniform distributions, respectively.

Analysis of vibration characteristics of FGM sandwich joined conical–conical shells surrounded by elastic foundations

The main objective of this paper is focused on the vibration behavior analysis of functionally graded material (FGM) sandwich joined conical–conical shell surround by elastic foundations. 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 governing equation of conical shell surround by elastic foundations is established by the first-order shear deformation shell theory and Hamilton’s principle. Then generalized differential quadrature (GDQ) method is used to discretize the obtained equations. By using the continuity conditions of displacement, rotation, force and moment between two adjacent shells and the boundary conditions at the end of the connected shell, a set of homogeneous equations are obtained, which control the free vibration of the two joined shell. 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 boundary conditions on the vibration behavior of FGM sandwich joined conical–conical shell have been investigated. It has found that for different sandwich distribution types, the influences of gradient index on the natural frequency are obvious when the gradient index is minor, but when high values of the gradient index are taken, the gradient index has no significant effect on the natural frequency.

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.