Higher-order Shear Deformation Shell Theory Research Articles

Nonlinear vibration and dynamic buckling responses of stiffened functionally graded graphene‐reinforced cylindrical, parabolic, and sinusoid panels using the higher‐order shear deformation theory

AbstractThe nonlinear vibration and dynamic responses of functionally graded graphene‐reinforced composite (FG‐GRC) laminated cylindrical, parabolic, and sinusoid panels stiffened by FG‐GRC stiffeners in the uniformly distributed temperature variation are presented in this paper. An improved smeared stiffener technique is used to model the added stiffnesses of stiffeners to the total stiffnesses of panels. The higher‐order shear deformation shell theory (HSDT) with the geometrical nonlinearities of von Kármán is applied to establish the governing formulations. The stress function form is estimated using the approximated technique for complex curvature panels. Lagrange function and Euler‐Lagrange equations are applied, and the Rayleigh dissipation function is taken into account to obtain the nonlinear equation of motion. Numerical examples are investigated using the Runge‐Kutta method to obtain the dynamic responses of panels, and the critical dynamic buckling loads of panels are considered using the Budiansky‐Roth criterion. Some significant remarks on the nonlinear vibration and dynamic buckling responses of three types of stiffened panels can be recognized from the numerical examples.

Free-damped vibration analysis of viscoelastic foam-filled FGM anisogrid lattice cylindrical shells

Our aim with this paper is to model and investigate the vibration and damping of a new hybrid composite shell. The considered composite cylindrical shell includes an FGM anisogrid lattice shell perfectly filled with viscoelastic foams. The modeling of the lattice part composed of spiral and hoop ribs is accomplished according to a global continuous standard based on orthotropic deep shells. The distribution pattern of the metal and ceramic constituents along the lattice ribs is specified by a power law. The homogenizations between ceramic and metal phases within the ribs, as well as between the FGM lattice structure and foam, are governed by the rule of mixtures. Based on the transferred Kelvin–Voigt viscoelastic scheme, the dynamic moduli of the foam portion are acquired. Because viscoelastic foam is a soft material, the higher-order shear deformation shell theory is used to estimate the system's displacement components. After emanating the dynamic equations by Hamilton's principle, the Chebyshev collocation-based semi-numerical method is implemented to detect the system's frequencies and loss factors. The comprehensive results show the role of each composite characteristic in the vibration and damping behavior of the defined structure.

A new analytical approach to the nonlinear buckling and postbuckling behavior of functionally graded graphene reinforced composite laminated cylindrical, parabolic, and half‐sinusoid shallow imperfect panels

AbstractThis paper presents and analyzes the nonlinear buckling responses of three types of shallow imperfect panels (cylindrical panel, parabolic panel, and half‐sinusoid panel) made from functionally graded graphene reinforced composite (FG‐GRC) on nonlinear elastic foundations subjected to axial compressive load in thermal environments. The nonlinear governing equations are formulated on Reddy's higher‐order shear deformation shell theory (HSDST) and take into account von Karman‐type nonlinearity. A new approximation technique to determine the stress function in average sense is developed and Galerkin's method is used to obtain the algebraically nonlinear equation system. Then, the simple calculation process can be used to solve the obtained equation systems, and the formulations to calculate the critical buckling loads and postbuckling load‐deflection curves are expressed in explicit form. The results can be flexibly applied to FG‐GRC panels with different curvatures in engineering designs. The effects of panel types, geometrical parameters, temperature increase, initial imperfection, and nonlinear elastic foundations on the critical buckling loads and postbuckling curves of cylindrical, parabolic, and half‐sinusoid FG‐GRC panels are discussed in numerical results. Numerical results also show a small disadvantage in the load‐carrying capacity of cylindrical panels compared to parabolic panels and half‐sinusoid shallow panels.Highlights Postbuckling of cylindrical, parabolic, and half‐sinusoid panels are studied. The panels are made of functionally graded graphene reinforced composite. Reddy's higher‐order shear deformation shell theory is applied. A new approximation technique to determine the stress function is developed. Critical buckling loads and postbuckling expressions are explicitly obtained.

Nonlinear continuum thermo-mechanics of the composite thick panel used in concrete ceilings via general expression of the polynomial series theory

Proper concrete vibration is vital to the final quality and durability of concrete structures. There is a lack of objective methods to assess conformance to the requirements of vibration behavior of concrete systems used in various engineering structures. So, in the current work for the first time, nonlinear vibrations of a composite thick panel made of concrete materials are presented. In the current work, the GPLs are used to reinforce the thick concrete panel in the length direction. In the current work, according to the higher-order shear deformation shell theory the effects of the von Kármán strain-displacement kinematic nonlinearity are presented in the constitutive laws of the shell. The nonlinear governing equations for various nonlinear boundary edges are solved via discretization of equations on the space domain, derivation of Duffing-type equations, and Hadamard and Kronecker Products. The results are validated by comparing the current results with deep neural networks (DNN) and open-source results in the literature. For DNN, it is introduced a supervised neural network based on physical information to predict the vibrational behavior of the current system. In this context, data-driven solutions and data-driven discovery are presented to solve the problem of determining nonlinear frequency. Consequently, new results are investigated to show the effects of the Pasternak modulus, Winkler modulus, and the GPLs’ weight fraction on the nonlinear vibrations of concrete panels.

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.

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.

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.

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 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.

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.

Static bending and free vibration analysis of multilayered composite cylindrical and spherical panels reinforced with graphene platelets by using isogeometric analysis method

This study makes an attempt at analyzing numerically the bending and free vibration behavior of multilayered composite cylindrical and spherical panels reinforced with graphene platelets (GPLs). A numerical model for the static and dynamic analyses of the GPL reinforced composite curved panels based on isogeometric analysis (IGA) combined with higher-order shear deformation shell theory is developed for the first time. NURBS (non-uniform rational B-spline) based IGA considered in the present investigation is capable of constructing exactly the complex geometries of the shell-type structures. The cylindrical and spherical panels are fabricated by stacking several layers in which the GPL concentration alters in a layerwise manner to form functionally graded (FG) structure. Uniform, FG-O, FG-X and FG-A types of GPL dispersions are taken into account. The Halpin-Tsai model is adopted to determine the effective modulus of elasticity, whilst the modified rule of mixture is utilized to estimate Poisson’s ratio and the mass density of the nanocompoiste curved panels. The motion equations for the static and dynamic problems are derived from the higher-order shear deformation shell theory established based on the Reddy’s shell theory, and NURBS based isogeometric formulation is made to obtain the central deflections and natural frequencies of the multilayered FG GPL strengthened composite curved panels. The validity of the present IGA method is first evidenced, followed by the illustrative parametric studies to further scrutinize the static and dynamic responses of the nanocomposite cylindrical and spherical panels with the various reinforcement schemes.

Novel study on functionally graded anisotropic doubly curved nanoshells

This work aims at investigating the dispersion of elastic eaves in doubly curved nanoshells. The nonlocal strain gradient theory is adopted in conjunction with a higher-order shear deformation shell theory, to include the size-dependent effects. The nanoshells are made of functionally graded anisotropic materials, whose properties are changed exponentially through the thickness direction. Hamilton’s principle is employed to obtain the governing equations of wave motion which are solved analytically to compute the wave frequencies as well as phase velocities as a function of the wave number. The sensitivity of the wave response is analyzed for the exponential factor, small-scale parameters, geometrical conditions as well as wave number. In addition, the accuracy of modeling the nanoshells with less elastic coefficients compared to the anisotropic model is studied to disregard any kind of complex equations.

On pre-stressed functionally graded anisotropic nanoshell in magnetic field

Elastic bulk wave characteristics of doubly curved nanoshell made of functionally graded (FG) anisotropic material are studied. The effective properties of FG anisotropic material vary along the thickness direction. A higher-order shear deformation shell theory and the Bi-Helmholtz nonlocal strain gradient theory are, respectively, utilized to model the nanoshell as a continuum model and predicting the size-dependent behavior. The Hamiltonian principle is adopted to obtain the governing equations of wave motion. These equations are solved analytically to evaluate the wave characteristics of the nanoshell. Emphasizing the effect of parametric excitation, the influences of small-scale parameters, exponential factor, magnetic field intensity, initial stress, elastic foundation parameters, and wave number are assessed on the wave dispersion response of FG anisotropic nanoshell.

Hygrothermal effect on vibroacoustic behaviour of higher-order sandwich panel structure with laminated composite face sheets

This article is a study of vibroacoustic behaviour of sandwich shell panels with laminated composite faces under concentrated harmonic loading in a hygrothermal environment by means of a higher-order finite-boundary element model. The system governing equations of the vibrating structure are derived using the higher-order shear deformation shell theory coupled with finite element and boundary element approach. The corrugated composite material properties due to temperature and moisture variation are included in the model macroscopically. The stiffness, mass tensors and modal values of the hygro-thermally stressed vibrating panel are first obtained and the acoustic radiation responses are then computed by solving the Helmholtz wave equation discretized on the structure boundary. The proposed scheme is implemented via a domestic MATLAB program to compute the desired responses. The results for natural frequencies, critical buckling moisture, critical buckling temperature and radiated sound power are found to be more accurate when matched with the available benchmark solutions. Additional numerical simulations are also performed with commercial software to authenticate the correctness of the present model. Finally, numerous numerical experimentations are carried out to reveal the versatility of the present scheme. The influence of hygrothermal conditions, geometrical parameters, support conditions and lamination scheme of the faces on the hygro-thermo-acoustic responses of laminated composite sandwich shell panels are probed and deliberated in detail.

Torsional postbuckling behavior of FG-GRC laminated cylindrical shells in thermal environments

This paper presents the buckling and postbuckling behaviors of graphene-reinforced composite (GRC) laminated cylindrical shells subjected to torsion in thermal environments. The GRC layers of the shell are arranged in a piece-wise functionally graded (FG) distribution pattern in the thickness direction and each layer of the shell contains different volume fraction of graphene reinforcement. The extended Halpin-Tsai micromechanical model is employed to determine the temperature dependent material properties of GRC layers. The governing equations of the GRC laminated cylindrical shells under torsion are derived based on a higher-order shear deformation shell theory with the geometric nonlinearity being defined by the von Kármán strain-displacement relationship. A singular perturbation technique along with a two-step perturbation approach is employed to determine the buckling torques and the torsional postbuckling equilibrium paths of the FG-GRC laminated cylindrical shells in thermal environments. The numerical results obtained reveal that the piece-wise FG distribution of graphene volume fraction can enhance the buckling torque and the torsional postbuckling strength while the rise of temperature may lead to the reduction of the torsional buckling torques and torsional postbuckling strength of the GRC laminated cylindrical shell.

Wave Propagation of Porous Nanoshells.

This study aims at investigating the wave propagation of porous nanoshells. The Bi-Helmholtz non-local strain gradient theory is employed in conjunction with a higher-order shear deformation shell theory, in order to include the size-dependent effects. The nanoshells are made of a porous functionally graded material (P-FGM), whose properties vary continuously along the thickness direction. A variational approach is here applied to handle the governing equations of the problem, which are solved analytically to compute the wave frequencies and phase velocities as function of the wave numbers. The sensitivity of the wave response is analyzed for a varying porosity volume fraction, material properties, non-local parameters, strain gradient length scales, temperature, humidity, and wave numbers. Based on the results, it is verified that the size-dependence of the response is almost the same to the one of plates, beams and tubes.

Vibroacoustic analysis of thermo-elastic laminated composite sandwich curved panel: a higher-order FEM–BEM approach

The present paper deals with the vibration induced acoustic responses of baffled sandwich curved shell panels constituted of laminated composite face sheets and subjected to harmonic point excitation in an elevated thermal environment by an effective computational approach. A coupled finite-boundary elements formulation based on the higher-order shear deformation shell theory has been proposed and implemented via a domestic MATLAB code for the computation of the responses. The eigen frequencies in conjunction with the conforming modes of the vibrating thermally pre-stressed composite sandwich panel are obtained using competent finite element steps engaging the Hamilton’s principle. The critical buckling temperature of the sandwich panel structure is assessed to prevent any excess thermal loading during analysis followed by the computation of acoustic responses by solving the Helmholtz wave equation using the boundary element steps. The performance and the versatility of the present scheme is established by the convergence study and the successive comparison of the present numerical responses, for instance the critical buckling temperature and the radiated sound power, with the available benchmark solutions. The sound power emitted by thermally stressed composite sandwich panel is also computed via simulation model developed using commercial software ANSYS and LMS Virtual.Lab and compared with the present values. Numerous numerical examples are solved to deeply discuss the influence of various design parameters on the vibroacoustic responses of laminated composite sandwich curved panels under high temperature environment.

Thermo-acoustic analysis of higher-order shear deformable laminated composite sandwich flat panel

The acoustic radiation responses of laminated sandwich baffled flat panels subjected to harmonic loading in an elevated thermal environment are investigated via a novel coupled finite and boundary elements formulation based on the higher-order shear deformation shell theory. The structural stiffness and mass tensors are obtained using competent finite element steps engaging the Hamilton’s principle followed by computation of acoustic responses by resolving the Helmholtz partial differential equation. An in-house MATLAB code is developed based on the present formulation for the computation of all the desired responses. The accuracy and robustness of the present scheme are recognized by the close conformance of the critical buckling temperature, natural frequencies and the sound power level values with the available benchmark solutions alongside the values obtained via a simulation model implemented using commercially available finite element (ANSYS) and boundary element (LMS Virtual.Lab) packages. Subsequently, the present model is employed to solve wide variety of numerical illustrations and the useful inferences related to the influence of elevated temperature, core-to-face thickness ratio, core-to face modular ratio and lay-up scheme on the sound emission characteristics of sandwich composite flat panels are deliberated in detail.

Numerical analysis of acoustic radiation responses of shear deformable laminated composite shell panel in hygrothermal environment

In this article, the vibroacoustic responses of laminated composite curved panels subjected to harmonic point excitation in a combined temperature and moisture environment are investigated numerically using a novel higher-order finite-boundary element model. The hygrothermal dependent composite material properties are incorporated macroscopically in the formulation. The governing equations are derived using the higher-order shear deformation shell theory coupled with finite and boundary element approach. First, the Hamilton's principle is employed to obtain the stiffness, mass tensors and modal values of the vibrating structure subjected to hygrothermal stresses. The acoustic radiation responses are then computed by solving the Helmholtz wave equation discretized on the structure boundary using boundary elements coupled with the structural finite elements. Compared to those reported in open literature, the results for natural frequencies, critical buckling temperature, critical buckling moisture and sound power level values computed using the present scheme are found to be more accurate. The sound power values are also acquired via a simulation model implemented using commercial tools ANSYS and LMS. Virtual Lab and compared with present numerical results. The scheme is further extended to solve numerous numerical examples highlighting the influence of hygrothermal loads, geometry, curvature ratio, modular ratio, support conditions and lamination scheme on the hygro-thermo-acoustic responses of laminated composite shell panels.

Moving load induced dynamic response of functionally graded-carbon nanotubes-reinforced pipes conveying fluid subjected to thermal load

Dynamic response of functionally graded Carbon nanotubes (FG-CNT) reinforced pipes conveying viscous fluid under accelerated moving load is presented. The mixture rule is used for obtaining the material properties of nano-composite pipe. The radial force induced by viscous fluid is calculated by Navier–Stokes equation. The material properties of pipe are considered temperature-dependent. The structure is simulated by Reddy higher-order shear deformation shell theory and the corresponding motion equations are derived by Hamilton\'s principal. Differential quadrature (DQ) method and the Integral Quadrature (IQ) are applied for analogizing the motion equations and then the Newmark time integration scheme is used for obtaining the dynamic response of structure. The effects of different parameters such as boundary conditions, geometrical parameters, velocity and acceleration of moving load, CNT volume percent and distribution type are shown on the dynamic response of pipe. Results indicate that increasing CNTs leads to decrease in transient deflection of structure. In accelerated motion of the moving load, the maximum displacement is occurred later with respect to decelerated motion of moving load.