Shear Deformation Shell Theory Research Articles

Static stability analysis of multi-scale hybrid agglomerated nanocomposite shells

This paper investigates the buckling problem of a multi-scale hybrid nanocomposite shell for the first time while the cylinder is supposed to be rested on an elastic substrate. The constituent material is considered to be made from a polymeric matrix strengthened via both macro- and nano-scale reinforcements. The influence of nanofillers’ agglomeration, generated due to the high surface to volume ratio in nanostructures, is included by implementing the Eshelby-Mori-Tanaka homogenization scheme. Afterwards, the equivalent material properties of the carbon nanotube-reinforced (CNTR) nanocomposite are coupled with those of carbon fibers (CFs) within the framework of a modified rule of the mixture. A cylindrical coordinate system is chosen and mixed with the infinitesimal strains of first-order shear deformation theory of shells to obtain the motion equations on the basis of the static form of principle of virtual work. Next, the achieved governing equations will be solved by Galerkin’s method to obtain the buckling load of the structure for both simply supported and clamped boundary conditions. The results indicate on the fact that if the agglomeration effects are ignored, the designed nanocomposite system may fail under low static excitations. Also, it is better to avoid from the construction of a large number of small clusters to make the nanocomposite structure more powerful against buckling mode failure.

Isogeometric analysis for postbuckling of sandwich cylindrical shell panels with graphene platelet reinforced functionally graded porous core

The proposed work investigates for the first time the postbuckling behavior of sandwich cylindrical shell panels with two metallic face layers and a graphene platelet (GPL) reinforced functionally graded porous (FGP) core subjected to central point loads, uniform and non-uniform pressure loadings. Based on a general higher-order shear deformation shell theory and the von Kármán large deflection assumption, the discrete equilibrium equations of the sandwich cylindrical panels are established by the principle of virtual work. The NURBS (non-uniform rational B-spline) based isogeometric analysis (IGA) in conjunction with the modified arc-length method of Crisfield are utilized synthetically to acquire nonlinear load–deflection responses for the panels numerically and both snap-through and snap-back types of instability are captured. Several benchmarks are carried out to verify accuracy and reliability of IGA and arc-length method in present formulation. Parametric studies are performed to demonstrate the influences of weight fraction of the graphene platelet, porosity distribution, porosity coefficient and thickness of the sandwich core on the nonlinear load–deflection responses of the sandwich cylindrical panels. Some complex load–deflection curves of the FGP-GPL cylindrical panels may be useful for future references.

Mechanics of shear and normal deformable doubly-curved delaminated sandwich shells with soft core

In this work the first- and second-order shear and normal deformation as well as the third-order shear deformation theories are applied to delaminated sandwich shells having constant radii of curvatures. The normal deformation of the first- and second-order models was captured by quadratic functions in the core, while it was ignored in the facesheets. The third-order model was created using cubic displacement functions in the terms of through-thickness coordinate in the core, the facesheets were still captured by second-order shear deformable shell theory. The governing equations were deduced through the virtual work principle, eventually three possible scenarios were considered: core-core, face-core and face-face delamination, respectively. To prove the applicability of the developed shell models several examples were solved including the former three scenarios combined with elliptic and hyperbolic doubly-curved sandwich shell geometries with different facesheet/core material pairs. The traditional simply-supported boundary conditions were assumed involving the Lévy-type formulation. Moreover, the state space method was applied to solve the examples. The mechanical fields calculated were compared to results obtained through spatial finite element models. The J-integral was also determined and separated into mode-II and mode-III components making it possible to perform the comparison to energy release rates computed by the virtual crack closure method. The results indicate that the proposed models capture the mechanical fields with high accuracy in the vicinity of points lying along the delamination front. The J-integrals are also very well predicted by the proposed shell models.

Static and dynamic analysis of thin functionally graded shell with in-plane material inhomogeneity

The static and dynamic problems of thin functionally graded shells (FGSs) with in-plane material inhomogeneity are investigated in this work. Governing equations are derived based on the first-order shear deformation shell theory. A meshfree radial basis collocation method (RBCM) which employs infinitely continuous radial basis functions (RBFs) as the approximation and utilizes collocation method for the solution is introduced for the static and dynamic eigenvalue analysis. Comparison studies with the analytical solutions of the homogeneous shell problems demonstrate that this method can achieve high accuracy and spectral convergence. FGSs with three representative gradient distributions are studied. Numerical simulations reveal that material inhomogeneity can shift the dangerous positions of maximum displacements or stresses in statics, and has prominent influences on the natural frequencies and mode shapes in dynamics. These observations can be utilized for the design of material properties of shell structures in the engineering applications.

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.

Nonlinear vibrations and chaotic phenomena of functionally graded material truncated conical shell subject to aerodynamic and in-plane loads under 1:2 internal resonance relation

This paper focuses on the nonlinear dynamic responses of a functionally graded material (FGM) truncated conical shell under 1:2 internal resonance relation. The FGM truncated conical shell is subjected to the in-plane load and the aerodynamic load along the meridian direction. According to a power-law distribution, the material properties are assumed to be modified along the thickness direction smoothly and continuously and the material properties are temperature dependent. The aerodynamic load is obtained by the first-order piston theory with the curvature correction term. According to von Karman type nonlinear geometric relations, first-order shear deformation shell theory, Hamilton principle, the nonlinear equations of motion for the FGM truncated conical shell are established. Furthermore, the nonlinear equations of motion are reduced into a system of the ordinary differential equations by utilizing Galerkin procedure. The multiple scales method is used to obtain the averaged equations for the FGM truncated conical shell under the relations of 1:2 internal resonance and 1/2 subharmonic resonance. The frequency–response curves, time history diagrams, phase portraits, Poincare maps and bifurcation diagrams with different parameters are yielded by employing numerical calculations. The influences of exponent of volume fraction, Mach number, damping coefficient and in-plane load on the nonlinear resonance behaviors of the FGM truncated conical shell are investigated. The chaotic and periodic motions of the FGM truncated conical shell have been discussed in detail.

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.

Large-amplitude oscillations of composite conical nanoshells with in-plane heterogeneity including surface stress effect

In the current investigation, a new formulation for nonlinear vibration behaviors of functionally graded (FG) composite conical nanoshells are constructed using Gurtin-Murdoch elasticity theory based on higher-order shear deformation shell theory (HSDFST) framework. Both Voigt and Reuss homogenization procedures are considered for the estimation of the mechanical characteristics of FG materials. Using generalized differential quadrature method (GDQM) together with Galerkin technique, the surface elastic-based nonlinear frequency-responses of FG composite conical nanoshell are obtained. It has been illustrated that the decrease of material property gradient index or transformation of boundary condition from full simply supported to full clamped, surface stress effect on the nonlinear frequency of a FG composite conical nanoshell reduces. Also, decreasing semi-vertex angle increases the frequency ratio of ωNL/ωL which reveals higher geometrical nonlinearity. However, it is seen that surface elasticity effect on the nonlinear vibration behavior of FG composite conical nanoshells is not significant.

Free vibration analysis of FG-CNTRC conical shell panels using the kernel particle Ritz element-free method

The free vibration analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) conical shell is carried out by using element-free kp-Ritz method. Based on the first-order shear deformation shell theory, the approximate displacement field is expressed by the shape function of the nuclear particle, and the governing equation is established. The material properties of conical shell plates are determined by rule of mixtures and change with the thickness. In this work, the convergence is studied in terms of the number of nodes, and the accuracy of the proposed method is verified by comparing the current solution with the solution in the literature from different aspects. The effects of volume fraction, boundary conditions, distribution patterns of CNTs, half vertex angle and radius thickness ratio on their free vibration frequency characteristics are discussed in detail.

Analytical study of the impact response of shear deformable sandwich cylindrical shell with a functionally graded porous core

The impact response of shear deformable sandwich cylindrical shells composed of two face layers and a functionally graded (FG) porous core is investigated. The mass density and elastic moduli of FG porous core are assumed to vary along the thickness in terms of the coefficients of mass density and foam, whose relationships are determined by employing the typical mechanical characteristics of an open-cell metal foam. Governing equations are derived using the first-order shear deformation shell theory and Hamilton’s principle. The contact force is achieved based on linearized Hertzian contact law. The Duhamel integration is applied to find the transverse displacement of porous sandwich shell. Results show that the core thickness, impact velocity, foam type, foam coefficient, and masses of impactor and shell play important roles on the impact response of sandwich porous shells.

Thermoacoustic response of porous FGM cylindrical shell surround by elastic foundation subjected to nonlinear thermal loading

Due to defects in processing, there are some voids or pores inside the functionally graded material during the production progress. It has been proved that porosity has been shown to have a large influence on the mechanical response of the structure, but the effect of porosity on the acoustic performance is still worth studying. To reveal these effects, the thermoacoustic response of porous functionally graded material (PFGM) cylindrical shell on elastic foundation under nonlinear thermal environment are investigated analytically. In the analysis, three common types of PFGM cylindrical shell, namely, uniform, symmetric and asymmetric distribution are considered. Assuming that material properites are related to temperature, and the thermomechanical stresses are obtained according to the nonlinear heat conduction equation. The Pasternak model is used to describe elastic foundation, and the cylindrical shell dynamic equation based on the first-order shear deformation shell theory is established by Hamilton's principle. By using the velocity continuity condition at fluid-structure interfaces to consider the fluid-structure coupling. 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 porosity distribution type, gradient index, porosity volume fraction, material properties, elastic foundations and nonlinear temperature fields on the acoustic characteristics of PFGM cylindrical 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.

Nonlinear static and dynamic stability of functionally graded toroidal shell segments under axial compression

This work presents an analytical approach to investigate nonlinear static and dynamic stability of toroidal shell segments resting on elastic foundation subjected to axial compression. The shells are made of functionally graded material (FGM) which created from metal and ceramic, and the volume fraction of constituents is supposed to gradually vary from one surface to another according to a power law function. Basic formulations are established based on Reddy's third-order shear deformation shell theory (TSDT) considering geometrical nonlinearity in von Kármán sense. Governing system of four-partial differential equations are converted into nonlinear differential equation using Galerkin method. Runge-Kutta method is used to solve nonlinear differential equation of motion and then nonlinear dynamic response of shell are examined. Budiansky-Roth criterion are used to obtain critical dynamic buckling load and then nonlinear dynamic stability of shells under axial compressive load linearly varying on time is analyzed. The influences of material and geometrical parameters, and elastic foundations on the static and dynamic stability of FGM toroidal sell segments are discussed in detail. The obtained results are validated by comparing with other results in the literature.

Nonlinear buckling and post-buckling behavior of shear deformable sandwich toroidal shell segments with functionally graded core subjected to axial compression and thermal loads

This paper presents an analytical investigation on buckling and post-buckling of shear deformable sandwich toroidal shell segments with functionally graded core and homogeneous face sheets. The effective material properties of core layer are graded in the thickness direction according to a simple power law distribution in terms of volume fraction index. The shells are surrounded by an elastic foundation and subjected to axial compressive or thermal loads. Reddy's third-order shear deformation shell theory (TSDT) is used to established formulations. A new solution using Galerkin method for solving a governing system of four-partial differential equations is carried out to obtain closed-form expressions of buckling stresses and post-buckling curves. Two cases of bowed-out and bowed-in sandwich toroidal shells are considered. Effects of material, geometric parameters and elastic foundation on stability behavior of the shells is analyzed in detail. The difference between bucking stresses basing on TSDT and those basing on classical shell theory (CST) is also clearly shown in this study.

Geometrically nonlinear postbuckling behavior of imperfect FG-CNTRC shells under axial compression using isogeometric analysis

The postbuckling and geometrically nonlinear behaviors of imperfect functionally graded carbon nanotube-reinforced composite (FG-CNTRC) shells under axial compression are investigated in this paper. For the first time, a new type of instability named “snap-backward” with the presence of three limit points on the equilibrium path is proposed. A novel formulation based on non-uniform rational B-Spline (NURBS) basis functions and the first-order shear deformation shell theory (FSDT) using the von Karman assumption and considering the initial deformation of the shells is presented. In addition, the advantage of NURBS in modeling exactly geometries of shells is exploited. In numerical implementation, the discrete nonlinear equation system is iteratively solved by a modified Riks method. The rule of mixture is used to estimate the effective material properties of FG-CNTRC shells. Some benchmark problems are solved to verify the high reliability of the proposed formulation. Effects of geometrical imperfection, CNTs distribution, volume fraction, CNTs orientation, thickness, radius of shell on the postbuckling and geometrically nonlinear behaviors of FG-CNTRC shells are rigorously investigated. Especially, some new and complex load-deflection curves of imperfect FG-CNTRC shells under axial compression are first provided that could be useful for future references.

Semi-analytical wave characteristics analysis of graphene-reinforced piezoelectric polymer nanocomposite cylindrical shells

Graphene platelets (GPLs) can be used to enhance mechanical and electric properties of the polymer polyvinylidene fluoride (PVDF), which efficiently controls and improves dispersion characteristics of wave motion in piezoelectric composite structures. Based on the first order shear deformation shell theory and the higher-order spectral elements, this paper proposes a semi-analytical formulation to investigate wave characteristics in the piezoelectric polymer nanocomposite cylindrical shell. The cylindrical shell is composed of PVDF reinforced with graphene platelets that are dispersed along the thickness direction, following various functionally graded patterns UD, FG-X and FG-O. Governing equation of waves in the piezoelectric composite shell is derived from Hamilton’s principle. By employing the Fourier transform, a second-order polynomial eigenvalue problem is obtained for wave analysis in frequency domain. Comparisons with other methods indicate that the present approach has the higher efficiency and accuracy on computation. Then, the study of significant parameters are conducted for analyzing the effects on wave propagation and the results show that dispersion behaviors of the piezoelectric nanocomposite shell are improved apparently by the addition of a small amount of GPLs. The increasing content of GPLs promotes the propagation of elastic waves and the richer concentration on the boundaries of the shells is beneficial to only the out-of-plane modes. Meanwhile, size parameters of GPLs provide an efficient means for the modulation of wave characteristics in nanocomposite shells. It is envisaged that the present work can provide guidelines to develop novel smart nanocomposites and structures.

Nonlinear oscillations of composite conical microshells with in-plane heterogeneity based upon a couple stress-based shell model

Abstract The current investigation deals with the nonlinear oscillation response of conical microshells made of a composite material with functional graded (FG) in-plane heterogeneity is studied in the presence of the size dependency. To accomplish this purpose, various types of homogenization schemes including Voigt model, Reuss model, Mori-Tanaka model, and Hashin-Shtrikman bounds model are employed. The size-dependent characteristics are taken into consideration on the basis of the modified couple stress theory of elasticity within the framework of the higher-order shear deformation shell theory. The couple stress-based differential equations of motion are constructed via the Hamilton's principle. An efficient numerical solution methodology adopting generalized differential quadrature (GDQ) method together with the pseudo-arc technique is put to use to obtain the modified couple stress-based nonlinear frequency of homogenized FG composite conical microshells. It is demonstrated that by increasing the value of the maximum shell deflection, the couple stress type of size effect plays more important role in the nonlinear vibration response of FG composite conical microshells. Additionally, it is indicated by changing the boundary conditions from simply supported one to clamped one, the influence of couple stress size dependency decreases.

Free vibration of joined cylindrical–hemispherical FGM shells

Free vibration response of a joined shell system including cylindrical and spherical shells is analyzed in this research. It is assumed that the system of joined shell is made from a functionally graded material (FGM). Properties of the shells are assumed to be graded through the thickness. Both shells are unified in thickness. To capture the effects of through-the-thickness shear deformations and rotary inertias, first-order shear deformation theory of shells is used. The Donnell type of kinematic assumptions is adopted to establish the general equations of motion and the associated boundary and continuity conditions with the aid of Hamilton’s principle. The resulting system of equations is discretized using the semi-analytical generalized differential quadrature method. Considering the clamped and free boundary conditions for the end of the cylindrical shell and intersection continuity conditions, an eigenvalue problem is established to examine the vibration frequencies of the joined shell. After proving the efficiency and validity of the present method for the case of thin isotropic homogeneous joined shells, some parametric studies are carried out for the system of combined moderately thick cylindrical–spherical shell system. Novel results are provided for the case of FGM joined shells to explore the influence of power-law index and geometric properties.

Couple stress-based nonlinear buckling analysis of hydrostatic pressurized functionally graded composite conical microshells

The present study deals with the size-dependent nonlinear buckling characteristics of conical microshells made of functionally graded (FG) composite materials under uniform hydrostatic pressure based upon the modified couple stress theory of elasticity. Accordingly, a modified couple stress-based shell model within the framework of the higher-order shear deformation shell theory and von Karman geometrical nonlinearity is constructed. Using the virtual work's principle in conjunction with the adjacent equilibrium criterion, the non-classical governing differential equations are established. The material properties of FG composite conical microshells are estimated on the basis of different homogenization schemes. To solve the size-dependent nonlinear problem, the generalized differential quadrature discretization pattern together with the Galerkin technique is employed. It is seen that among various types of homogenization scheme, the Voigt and Reuss models represent, respectively, the overestimated and underestimated critical buckling pressures. Also, it is found that for a FG composite conical microshell with higher semi-vertex angle, the influence of the material property gradient index on the nonlinear critical buckling pressure diminishes. In addition, it is observed that the couple stress size dependency plays more important role in the nonlinear buckling behavior of FG composite conical microshells with lower ratio of R1/h. These patterns are the same for all types of boundary conditions.

Studying propagation of wave in metal foam cylindrical shells with graded porosities resting on variable elastic substrate

This investigation deals with wave propagation analysis of porous metal foam cylindrical shells resting on the variable elastic substrate within the framework of the first-order shear deformation shell theory. Magnesium, nickel, titanium and tungsten foams are considered as constitutive materials of cylindrical shell. The pores are distributed through the thickness uniformly, symmetrically and asymmetrically. The principle of Hamilton is employed in order to reach motion equations of porous metal foam cylindrical shell. Next, governing equations of porous metal foam are derived for a first-order shear deformation shell and then solved analytically. The effects of various parameters including porosity coefficient, various types of porosity distribution, length-to-thickness ratio, radius-to-thickness ratio, circumferential wave number, Pasternak coefficient and variable Winkler coefficient on the variation of wave frequency and phase velocity of metal foam cylindrical shells are covered and presented within the framework of a group of figures which can be seen in detail.