First-order Shear Deformation Shell Theory Research Articles

Nonlinear dynamic analysis of piezoelectric functionally graded porous truncated conical panel in thermal environments

Abstract In this article, the nonlinear dynamic response and free vibration of functionally graded porous (FGP) truncated conical panel with piezoelectric actuators in thermal environments are investigated by an analytical method. The panel resting on an elastic foundation which is modeled according to the Winkler–Pasternak theory. The material properties including Young's modulus, shear modulus, and density are assumed to smoothly through the shell thickness. Three types of porosity distribution across the thickness, namely, symmetric porosity distribution, non-symmetric porosity, and uniform porosity distribution, are considered. Theoretical formulations are presented based on the first-order shear deformation shell theory with a von Karman-Donnell type of kinematic nonlinearity. The non-linear motion equations and resulting equations are derived by using Hamilton's principle, Galerkin's method, and Runge-Kutta method. Lastly, some numerical results are presented to study the effects of shell characteristics, porosity distribution, porosity coefficient, applied actuator voltage, temperature increment and elastic foundations on the nonlinear dynamic response and the natural frequencies of the piezoelectric FGP truncated conical panel.

Bézier extraction based isogeometric analysis for bending and free vibration behavior of multilayered functionally graded composite cylindrical panels reinforced with graphene platelets

This study attempts to explore the bending and free vibration response of GPL (graphene platelet) reinforced composite cylindrical panels by using Bézier extraction based IGA (isogeometric analysis) in combination with first-order shear deformation shell theory. The cylindrical panel is composed of a finite number of layers in which the GPL concentration has stepped variation across the thickness to construct layerwise FG (functionally graded) structure. The dispersion patterns of GPLs have three FG forms such as FG-O, FG-X and FG-A distributions. Uniform arrangement of GPLs is also considered for comparison. The effective elastic modulus of the nanocompoiste cylindrical panel is determined from the Halpin-Tsai model, whilst the mass density and Poisson's ratio are estimated by the modified rule of mixture. The governing equations for the static and dynamic problems are derived, and Bézier extraction based isogeometric formulation is made to get the central deflections and natural frequencies of the multilayered FG GPL strengthened cylindrical panels. The validity of the present isogeometric approach incorporated with first-order shear deformation shell theory is first evidenced, followed by the illustrative parametric studies to further investigate the static and dynamic behavior of the nanocomposite cylindrical panels with various reinforcement schemes.

Buckling analysis of embedded graphene oxide powder-reinforced nanocomposite shells

In this study, the buckling analysis of a Graphene oxide powder reinforced (GOPR) nanocomposite shell is investigated. The effective material properties of the nanocomposite are estimated through Halpin-Tsai micromechanical scheme. Three distribution types of GOPs are considered, namely uniform, X and O. Also, a first-order shear deformation shell theory is incorporated with the principle of virtual work to derive the governing differential equations of the problem. The governing equations are solved via Galerkin’s method, which is a powerful analytical method for static and dynamic problems. Comparison study is performed to verify the present formulation with those of previous data. New results for the buckling load of GOPR nanocomposite shells are presented regarding for different values of circumferential wave number. Besides, the influences of weight fraction of nanofillers, length and radius to thickness ratios and elastic foundation on the critical buckling loads of GOP-reinforced nanocomposite shells are explored.

On the isogeometric analysis of geometrically nonlinear shell structures with the consideration of surface energies

Presented in this article is a size-dependent analysis which is aimed to discover the surface stress effects on the large deformation characteristics of small-scaled shell structures. Consideration of surface energies in the micro/nanodimensions, and also in the cases of structures with high ratios of surface to volume like shells, significantly affects the predicted responses. To take the surface stress effects into account, it is assumed that the body is surrounded by thin surface layers at the top and bottom. The first-order shear deformation shell theory with seven parameters is adopted in the Lagrangian configuration system for the bulk part of structure. Also, the effects of surface stresses are captured based on the surface elasticity theory. Because of its unique features, IGA (isogeometric analysis) solution methodology is implemented to solve the governing equations. In this regard, the matrix–vector form of nonlinear relations, including the constitutive equations and energy functionals, is presented to directly utilize IGA for the problem. As case studies and also to show the main contribution of present investigation, i.e., studying the effects of surface energies on the mechanical behavior of geometrically nonlinear shells, the well-known benchmarks in the literature are studied in micro/nanodimensions with the consideration of surface influences.

Numerical investigation on the buckling and vibration of cracked FG cylindrical panels based on the phase-field formulation

In this research, a numerical study is performed on the buckling and vibration of cracked functionally graded (FG) cylindrical panels under external pressure based on the phase-field formulation (PFF). For this purpose, the matrix form of energy functional is first derived on the basis of first-order shear deformation shell theory (FSDST) and phase-field model. To numerically handle the problem, the finite difference differential and integral operators are employed to directly discretize the energy functional. Then, by the use of Hamilton’s principle, the discretized governing equations are obtained. The accuracy and efficiency of the developed model are demonstrated through various comparative and convergence studies. In order to analyze the vibration and buckling behavior of cracked FG cylindrical panels, two different crack patterns are considered. In addition, a wide range of numerical results is provided to study the effects of crack’s shape, length and inclination angle, boundary conditions (BCs) and geometrical parameters on the buckling pressures and natural frequencies of FG cylindrical panels.

Torsional stability capacity of a nano-composite shell based on a nonlocal strain gradient shell model under a three-dimensional magnetic field

This paper considers a single-walled composite nano-shell (SWCNS) exposed in a torsional critical stability situation. As the magnetic field affects remarkably nanostructures in the small size, a three-dimensional magnetic field is assessed which contains magnetic effects along the circumferential, radial and axial coordinates system. Based on the results of the nonlocal model of strain gradient small-scale approach and the first-order shear deformation shell theory (FSDST), the problem is estimated. Afterward, the numerical results are taken analytically and compared with other existing literature. Hereafter, the influences of various factors, such as the magnetic field, are discussed deeply. It is observed that when the magnetic field is studied in three dimensions, the transverse magnetic effect is the most serious factor that affects fundamentally the torsional stability of the shell.

A novel data-driven nonlinear solver for solid mechanics using time series forecasting

In this paper, a novel data-driven nonlinear solver (DDNS) for solid mechanics using time series forecasting is first proposed. The key concept behind this work is to modify the starting point of iterations of the modified Riks method (M-R). The modified Riks method starts iterations at the previously converged solution point while the proposed method starts at a predicted point which is very close to the converged solution of the current step. In the prediction phase, the predicted starting point of the current step is simply determined only based on the previously converged solutions and the predictive networks built via group method of data handling (GMDH) known as a self-organizing deep learning method for time series forecasting problems. Then, the correction phase of the modified Riks method is used to obtain the converged solution via an iterative procedure starting at the predicted point. In this work, the training and applying processes of networks are continuously performed during the analysis to predict the starting point of each increment. It is interesting that the present deep learning networks are built with small data in very short time. Especially, the proposed method is not only simple in implementation but also reduces significantly number of iterations and computational cost compared with the conventional modified Riks method. Some benchmark problems on geometrically nonlinear analysis of shells are provided and solved by using isogeometric analysis (IGA) in conjunction with the first-order shear deformation shell theory (FSDT). The high accuracy, efficiency and stability of the proposed method are confirmed.

Effects of nanotube agglomeration on wave dynamics of carbon nanotube-reinforced piezocomposite cylindrical shells

Dynamics of wave propagation in carbon nanotube (CNT)-reinforced piezocomposite cylindrical shells affected by nanotube agglomeration is investigated in this study for the first time by developing an analytical approach incorporating existing theories and models. The Mori-Tanaka micromechanics model in combination of the first-order shear deformation shell theory and wave propagation solution are employed to determine wave propagation characteristics of piezocomposite cylindrical shells reinforced with agglomerated CNTs. The effects of both partial and complete nanotube agglomerations on the effective elastic properties and wave dynamics are examined within various axial and circumferential wave numbers for different wave modes by solving an eigenvalue problem. It is found that nanotube agglomeration leads to the reduction of wave phase velocity as a result of decrease in the effective elastic properties. The developed methodology in this study can be used for analysis of the data of structural health monitoring by the non-destructive testing (NDT) in estimating the degree of nanotube agglomeration in nanocomposites.

Vibrations of FG-CNT reinforced composite cylindrical panels with cutout

An investigation is performed in this research to analyze the free vibration characteristics of carbon nanotube reinforced composite (CNTRC) cylindrical panels with cutout. Distribution of CNTs in the matrix of the composite panel may be uniform or functionally graded. The cylindrical panel has a cutout at the center in a rectangular shape. Following the first-order shear deformation shell theory with the Donnell type of kinematic assumptions, the total kinetic and strain energies of the cylindrical panel are obtained. The eigenvalues problem governing the natural frequencies of the panel is established using the generalized Ritz method where the shape functions are constructed by means of the Chebyshev polynomials. Different boundary conditions such as free, simply supported, shear diaphragm, and clamped may be assumed for the edge of the panel, however, the perforation part of the panel is considered to be free all around. The obtained frequencies are first validated with the simple cases such as CNTRC plates with cutout and CNTRC panels without cutout. Afterwards novel numerical results are provided to analyze the effects of volume fraction of CNT, graded pattern of CNT, boundary conditions, panel geometry, and the hole size. Results of this study show that as the volume fraction of CNTs as reinforcement increases, the natural frequencies of the plate with/without cutout enhance. Also through a proper functionally graded distribution of CNTs as reinforcement the natural frequencies of the panel may be decreased/increased.

Wave based method for free vibration characteristics of functionally graded cylindrical shells with arbitrary boundary conditions

Abstract The wave based method (WBM) is used to analyze the free vibration characteristics of functionally graded material (FGM) cylindrical shell with arbitrary boundary conditions. The motion relationship is described by the first-order shear deformation shell theory (FSDST). The displacement components and transverse rotations are expressed as wave function expansions. In accordance with the dynamic relationship, the final governing equation and global matrix are assembled by incorporating the boundary matrices. The natural frequency of the system is obtained by solving the determinant of the global matrix. By comparing the results with those in the literature, the validity of the proposed method is verified. In addition, the influences of power-law exponents and boundary conditions on natural frequencies are analyzed. The effects of geometric parameters including the ratio of thickness to radius and the ratio of length to the radius on natural frequencies are discussed. The purpose of this paper is to demonstrate the ease of application of the WBM for the free vibration of FGM cylindrical shells with arbitrary boundary conditions. Furthermore, the advantage of the WBM are: (1) the global matrix is easy to construct; (2) different boundary conditions can be conveniently adjusted; (3) it is with high computational efficiency and precision.

Free vibration analysis of smart laminated carbon nanotube-reinforced composite cylindrical shells with various boundary conditions in hygrothermal environments

Abstract Free vibration analysis is carried out for piezoelectric coupled carbon nanotube (CNT)-reinforced composite cylindrical shells with the influences of various boundary conditions and hygrothermal environmental conditions for the first time. A simple and effective non-iterative mathematical method is used to calculate the natural frequencies. The equilibrium equations of motion are obtained based on the first-order shear deformation shell theory with the coupling effects of piezoelectricity, temperature, and moisture, respectively based on the Maxwell equation, the Fourier heat conduction equation, and the Fickian moisture diffusion equation. The Mori-Tanaka micromechanics model is used to estimate the resulting material properties for a composite shell reinforced with CNTs. Presented methodology and attained results are validated with the existing results in the literature. The effects of the boundary conditions, lamination stacking sequence, volume fraction and orientation of CNTs, piezoelectricity, and geometry of the shell on the natural frequencies of the shell are investigated. A moderate effect of temperature/moisture variation on the natural frequencies is also observed. It is found that the influence of structural boundary conditions is more significant at higher CNT volume fractions and for thicker and shorter shells, and the piezoelectricity effect is more obvious at higher circumferential mode. The model and results presented in this study can be utilized to determine vibration characteristics of smart CNT-reinforced composites subjected to hygrothermal loading as well as mechanical loading.

Wave Based Method for Free Vibration Analysis of Cross-Ply Composite Laminated Shallow Shells with General Boundary Conditions.

In this paper, a semi-analytical method is adopted to analyze the free vibration characteristics of composite laminated shallow shells under general boundary conditions. Combining two kinds of shell theory, that is, first-order shear deformation shell theory (FSDT) and classical shell theory (CST), to describe the dynamic relationship between the displacement resultants and force vectors, the theoretical formulations are established. According to the presented work, the displacement and transverse rotational variables are transformed into wave function forms to satisfy the theoretical formulation. Related to diverse boundary conditions, the total matrix of the composite shallow shell can be established. Searching the determinant of the total matrix using the dichotomy method, the natural frequency of composite laminated shallow shells is obtained. Through several classical numerical examples, it is proven that the results calculated by the presented method are more accurate and reliable. Furthermore, to discuss the effect of geometric parameters and material constants on the natural frequencies of composite laminated shallow shells, some numerical examples are calculated to analyze. Also, the influence of boundary elastic restrained stiffness is discussed.

Static analysis of two-directional functionally graded cylindrical panels under the effect of symmetric loads using finite element method (FEM)

This paper offers the linear analysis of the static behavior of two directional functionally graded(2D-FG) cylindrical panels under the effect of internal symmetric loads. The mechanicalproperties of the cylindrical panel are given to be changed simultaneously through the thicknessand longitudinal directions as a function to the volume fraction of the constituents by a simplepower-law distribution. Based on Sander’s first order shear deformation shell theory (FSDT), theequations of motion for (2D-FG) panels are derived using the principle of minimum totalpotential energy (MPE). The finite element method (FEM) as an effective numerical tool isutilized to solve the equations of motion. The model has been compared with those available inthe literature and it observed good correspondence. The influences of the material variationalong the thickness and longitudinal directions, geometrical parameters, boundary conditionsand load parameters on the panel deformation are studied in detail.

Nonlinear Dynamical Analysis of a Circular Truss Antenna in Thermal Environments

This paper investigates the nonlinear dynamics of the circular truss antenna with considering the thermal effect. The first-order shear deformation shell theory and the Hamilton’s principle are used to establish the mechanical model for the circular truss antenna system. The motion governing equation is derived for the equivalent circular cylindrical shell. For interest in 1:2:3 internal resonances, the continuous system of nonlinear partial governing differential equation of motion is truncated into a three-degree-of-freedom system of ordinary differential equation including quadratic and cubic nonlinearities. From the averaged equations obtained by the method of multiple scales, numerical simulations are presented to investigate the effects of the thermal excitation on the nonlinear responses of the truss antenna system. Bifurcation diagram, phase portraits, time histories, power spectra and max Lyapunov exponents are used to get the numerically results. It is found that there exist period doubling bifurcations, periodic motions and chaotic motions in the system. The numerical results show that the thermal excitation has significant influence on the nonlinear dynamical behaviors of the circular truss antenna system. The findings in this paper is also useful for the nonlinear vibration control of the shell structure.

Semi-analytical postbuckling analysis of polymer nanocomposite cylindrical shells reinforced with functionally graded graphene platelets

In this study, the postbuckling analysis of functionally graded graphene platelets reinforced composite (FG-GPLRC) cylindrical shells is presented. The continuous uniform and functionally graded distribution of GPLs is considered through the thickness direction of the shell. In order to present the effective mechanical properties of GPL-reinforced composites the modified Halpin-Tsai micromechanical model is taken into account. On the basis of first-order shear deformation shell theory and von-Karman geometrically nonlinear relations, the nonlinear governing equations are present in the context of the variational formulation. Employing the Fourier series and variational differential quadrature (VDQ) numerical approach, the semi-analytical solution methodology is further presented. The pseud arc-length continuation method in conjunction with a load disturbance approach was utilized to solve the nonlinear governing equations and trace the postbuckling path. Note that based on the proposed formulation, the mode changes and secondary buckling can be considered through the postbuckling path. The results indicate that in addition to geometrical parameters, distribution patterns and weight fractions of GPLs have significant effects on the buckling and postbuckling characteristics of FG-GPLRC cylindrical shells.

A Unified Approach of Free Vibration Analysis for Stiffened Cylindrical Shell with General Boundary Conditions

A unified approach of free vibration analysis for stiffened cylindrical shell with general boundary conditions is presented in this paper. The vibration of stiffened cylindrical shell is modeled mathematically involving the first-order shear deformation shell theory. The improved Fourier series is selected as the admissible displacement function while the arbitrary boundary conditions are simulated by adjusting the equivalent spring stiffness. The natural frequencies and modal shapes of the stiffened shell are obtained by solving the dynamic model with the Rayleigh-Ritz procedure. Various numerical results of free vibration analysis for stiffened cylindrical shell are obtained, including natural frequencies and modes under simply supported, free, and clamped boundary conditions. Moreover, the effects of stiffener on natural frequencies are discussed. Compared with several state-of-the-art methods, the feasibility and validity of the proposed method are verified.

A novel analysis-prediction approach for geometrically nonlinear problems using group method of data handling

A novel analysis-prediction (ANP) approach for geometrically nonlinear problems of solid mechanics is for the first time proposed in this paper. The key concept of this approach is: (1) A part of equilibrium path is traced by numerical analysis; (2) Data of this part is then used for training predictive network; (3) Applying the trained network, the rest of the equilibrium path is simply traced by pure prediction without using any analysis. As an illustration for ANP approach, the analysis package is in this study established based on isogeometric shell analysis using the first-order shear deformation shell theory (FSDT). As the main advantage of the proposed approach, computational cost is significantly lower than that of the conventional approach based on pure numerical analysis. In addition, the predictive networks are built via group method of data handling (GMDH) known as a self-organizing deep learning method for time series forecasting problems without requirement of big data. Some numerical examples are provided to confirm the high accuracy and efficiency of the proposed approach. The approach not only could be applied to a wide range of computational mechanics problems in which nonlinear response occurs but also to other computational engineering fields.

Vibration analysis of porous metal foam shells rested on an elastic substrate

In this article, the vibration problem of an embedded cylindrical shell consisted of porous metal foam is solved via an analytical method with respect to the influences of various porosity distributions. Three types of porosity distribution across the thickness are covered, namely, uniform, symmetric, and asymmetric. The strain–displacement relations of the shell are assumed to be derived on the basis of the first-order shear deformation shell theory. Then, the achieved relations will be incorporated with the Hamilton’s principle in order to reach the Navier equations of the cylindrical shell. Next, the well-known Galerkin’s method is utilized to calculate the natural frequencies of the system. The influences of both simply supported and clamped boundary conditions are included. In order to show the accuracy of the presented method, the results of the present research are compared with those reported by former published papers. The reported results show that an increase in the porosity coefficient can decrease the frequency of the shell. Also, the stiffness of the system can be lesser decreased while symmetric porosity distribution is chosen.

Vibration and Buckling of Shear Deformable Functionally Graded Nanoporous Metal Foam Nanoshells

This article aims to investigate free vibration and buckling of functionally graded (FG) nanoporous metal foam (NPMF) nanoshells. The first-order shear deformation (FSD) shell theory is adopted and the theoretical model is formulated by using Mindlin’s most general strain gradient theory, which can derive several well-known simplified models. The symmetric and unsymmetric nanoporosity distributions are considered for the structural composition. Hamilton’s principle is employed to deduce the governing equations as well as the boundary conditions. Then, via the Navier solution technique, an analytical solution for the free vibration and buckling of FG NPMF nanoshells is presented. Afterwards, a detailed parametric analysis is conducted to highlight the effects of the nanoporosity coefficient, nanoporosity distribution, length scale parameter, and geometrical parameters on the mechanical behaviors of FG NPMF nanoshells.

NURBS-based postbuckling analysis of functionally graded carbon nanotube-reinforced composite shells

An investigation into the postbuckling and geometrically nonlinear behaviors of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) shells is carried out in this study. The discrete nonlinear equation system is established based on non-uniform rational B-Spline (NURBS) basis functions and the first-order shear deformation shell theory (FSDT). The nonlinearity of shells is formed in the Total Lagrangian approach considering the von Karman assumption. The incremental solutions are obtained by using a modified Riks method. In the present formulation, the rule of mixture is used to estimate the effective material properties of FG-CNTRC shells. Effects of CNTs distribution, volume fraction and CNTs orientation on the postbuckling behavior of FG-CNTRC shells are particularly investigated. Exact geometries of shells are modeled by using NURBS interpolation. Several verifications are given to show the high reliability of the proposed formulation. Especially, some complex postbuckling curves of FG-CNTRC panels and cylinders are first provided that could be useful for future references.