First-order Shear Deformation Shell Theory Research Articles

Isogeometric analysis of functionally graded triply periodic minimal surface shells

Functionally graded triply periodic minimal surface (FG-TPMS) structures are known as bio-inspired structures. They possess some remarkable advantages such as porous structures with high inter-connectivity, mathematically controllable geometry features and smooth surfaces. As another advantage, FG-TPMS structures can be fast and numerously manufactured by 3D printing technology. Nevertheless, modeling these structures is a challenging task. This paper investigates static bending and free vibration behaviors of FG-TPMS shells. The proposed formulation is established upon isogeometric analysis (IGA) and first-order shear deformation shell theory (FSDT). The governing equations are discretized by a Galerkin weak form and numerically solved by using non-uniform rational B-Spline (NURBS) basis functions. Exact geometries of structures are described via NURBS basis functions. A fitting technique is used to compute the mechanical characteristics of FG-TPMS materials. We investigate behaviors of FG-TPMS shells considering three types of cell geometries which are primitive (P), gyroid (G), I-graph and wrapped package-graph (IWP), and six porosity distribution patterns. The present solutions are verified with the reference ones in the literature. Effects of boundary, type of cell geometry, porosity distribution pattern, length-to-radius and thickness ratios on behaviors of FG-TPMS shells are rigorously studied. Especially, many numerical results of FG-TPMS shells are first proposed in this paper.

Sound Insulation Characteristic Analysis of Corrugated Core Sandwich Panel by Using Differential Quadrature Finite Element Method

This paper proposes a universal vibroacoustic coupling theoretical analysis model for investigating the sound insulation characteristic of corrugated core sandwich panel (CCSP) under different boundary conditions. The vibroacoustic coupling model consists of CCSP (structural field) and two three-dimensional acoustic cavities which simulate the incident sound field and accept the sound field, respectively. Based on first-order shear deformation plate and shell theories and acoustic theory, the dynamic model of CCSP and three-dimensional acoustic cavities are established by employing the differential quadrature finite element method in conjunction with the energy principle. The coupling of CCSP and three-dimensional acoustic cavities is realized by the velocity continuity condition at the acoustic–structure interfaces. The validation of the vibroacoustic coupling model is verified by comparing the results calculated by the established model with the corresponding results calculated by ABAQUS. The influence of model parameters on the vibroacoustic characteristic of vibroacoustic coupling model is investigated systematically for determining the critical structure parameters which have influence on the vibroacoustic behaviors. The effect of critical structure parameters on the sound insulation characteristic of CCSP is analyzed systematically for the optimization of the sound insulation performance of CCSP and provides theoretical basis and technique guidance.

Time-frequency analysis of plate-shell coupled structures under moving stochastic load

Random vibration is of critical importance in understanding the dynamic behavior of engineering structures, especially the nonstationary responses for their coupled enclosers consisting of plates and shells subjected to moving stochastic loads. In this study, the pseudo excitation method (PEM) is integrated into the method of reverberation ray matrix (MRRM) to establish a full-domain stochastic dynamic system. With the help of rearrangement matrices and coupling spring connection technologies, the two plate-shell models of sequential and reinforced coupling are considered by employing the simple first-order shear deformation shell theory (S-FSDST) and Hamilton's principle. Then, the continuous movement of the excitation between each substructure is achieved through load coordinate normalization in nonhomogeneous equations, which are separated from the generalized solutions in the frequency domain. Therefore, it can be easily converted into a global pseudo-excitation and obtain the time-history responses relying on the inverse Laplace transform. After matching the present results with the finite element software and Monte Carlo simulation (MCS), the accuracy and efficiency of the proposed RRM-PEM are well-confirmed. Finally, by performing a series of numerical cases of power spectral density (PSD) and time-varying root mean square (RMS) under different loading velocities, frequency bands and system modal dampings, the time-frequency rules for different substructures of closed plate-shell coupled system under moving random loads are revealed for the first time.

Nonlocal strain gradient-based dynamic stability of functionally graded piezoelectric nanoshells considering thermo-electro-mechanical coupling effects

In this paper, a nonclassical shell model is utilized to investigate the dynamic stability of functionally graded piezoelectric (FGP) nanoshells considering the thermo-electro-mechanical coupling effects. Also, the FGP nanoshells on elastic foundations are subjected to periodic axial loading. The theoretical formulations are derived by implementing the first-order shear deformation (FSD) shell theory into the nonlocal strain gradient theory (NSGT). Hamilton’s principle is utilized to obtain the equations of motion and related boundary conditions of the present nanoshell model. Moreover, the Mathieu-Hill equations determining the instability regions are presented and analyzed based on Bolotin’s method. Results demonstrate that the external electric voltage, the elastic foundations, the temperature rise, the static load factor, the power-law index and the small-scale parameters have significant influences on the dynamic stability responses of FGP nanoshells.

Dynamic characteristics of novel damping sandwich composite open cylindrical shell: From theory to simulation

Based on the first-order shear deformation shell theory, the free vibration model of Multilayer Damping Sandwich Composite Open Cylindrical Shell (MDSCOCS) under different lay-up angles is established. The Navier double series and Rayleigh-Ritz method are used to solve the free vibration equation under four sided simply supported boundary conditions. The theoretical solutions are compared with the existing literature and finite element simulation results respectively to verify the validity of theoretical calculation model. The influences of the thickness, position, shell opening angle, composite lay-up angle and other factors on the natural frequency and loss factor of the structure under different number of core layers are investigated in detail. The analysis shows that the damping loss factor of the composite structure is greatly improved by the addition of multilayer damping sandwich, and the dynamic performance of the composite structure is optimized, compared with that of single-layer damping sandwich composite structure. It provides a theoretical basis for the design of multilayer damping sandwich composite open cylindrical shell.

Elasticity and stability of corrugated conical shells with diverse orthotropy

Curved thin-walled structures have attracted significant attention within engineering communities thanks to their remarkable properties, including high load-bearing capacity, tailorable directional stiffness, adaptability, and lightweight characteristics. The geometric intricacies of these structures not only influence their mechanical behavior but also provide a vast design space for exploring new frontiers in structural development. This study investigated a unique form of curved thin-walled structure, namely the corrugated annular conical shell, which exhibits diverse degrees of cylindrical orthotropy across various corrugation patterns. An analytical model was formulated based on the minimum total potential energy principle and first-order shear deformable shell theory to predict the non-monotonous force-deflection relationships and non-uniform stress distributions in corrugated conical shells when axially loaded under compression. Within this novel framework, the local undulation shape, corrugation pitch, and amplitude emerged as influential determinants of the responses of corrugated shells, intrinsically intertwined with the polar and shear orthotropy ratios. The accuracy of our analytical model was verified by comparing its results to those obtained using the finite element method and experimental measurements. A machine learning method was employed to classify the stability behaviors of corrugated annular conical shells and derive comprehensive design maps for the first time. Corrugated annular conical shells hold enticing potential for applications as adaptive spring units or building blocks for more sophisticated architectural metamaterials.

Free Vibration Analyses of Stiffened Functionally Graded Graphene-Reinforced Composite Multilayer Cylindrical Panel

In this paper, the free vibration response of a stiffened functionally graded graphene nanoplatelet (GPL)-reinforced composite multilayer cylindrical shell panel is studied for the first time. The shell is stiffened by both stringers and rings. Additionally, the effect of reinforcing the shell panel, ring and stinger with GPLs is independently studied. Halpin–Tsai relations are employed to evaluate the mechanical properties of the shell panel, rings and stringers. The first-order shear deformation shell theory, accompanied by the Lekhnitsky smeared stiffener model, using the numerical finite element method and Hamilton principle, is employed to develop the governing motion equations of the shell panel. Four different types of GPL patterns, including FG-A, FG-X, FG-O and UD, are assumed across the thickness of the shell panel, rings and stringers. The effects of different factors, including various weight fractions and patterns of GPLs nanofillers, the geometry of the shell panel and stiffeners and two displacement boundary conditions, on the natural frequencies of the shell panel, have been studied.

Simple first-order shear deformation theory for free vibration of FGP-GPLRC spherical shell segments

This study presents a free vibration analysis of spherical shell segments using simple first-order shear deformation shell theory (S-FSDT) for the first time. The shell structure is made of functionally graded porous graphene platelet reinforced composite (FGP-GPLRC) – one kind of porous material strengthened by graphene platelets (GPLs). Effective material properties of the FGP-GPLRC are determined by the modified Halpin-Tsai micromechanical model and the mixture rule. Four types of porosity distributions and GPL dispersions are considered fully in this study. The governing equations of the shell are derived based on the S-FSDT and classical shell theory, then solved by the well-known Rayleigh-Ritz method and the artificial spring technique. The results show that the S-FSDT can capture well the behaviors of the spherical shell segments. Besides, the best profile of novel FGP-GPLRC is not fixed; the physical parameters, geometric parameters, and material characteristics of the shells need to be investigated fully.

Critical velocities of a two-layer composite tube incorporating the effects of transverse shear, rotary inertia and material anisotropy

Critical velocities of a two-layer composite tube subjected to a uniform internal pressure moving at a constant velocity are analytically derived by using a first-order shear deformation shell theory incorporating the transverse shear, rotary inertia and material anisotropy. The composite tube consists of two perfectly bonded axisymmetric circular cylindrical layers of dissimilar materials, which can be orthotropic, transversely isotropic, cubic or isotropic. Closed-form expressions for four critical velocities are first derived for the general case by including the effects of transverse shear, rotary inertia, material orthotropy and radial stress. The formulas for composite tubes without the transverse shear, rotary inertia or radial stress effect and with simpler anisotropy are then obtained as special cases. In addition, it is shown that the model for a single-layer, homogeneous tube is included in the current model as a special case. To illustrate the newly derived closed-form formulas, a composite tube with an isotropic inner layer and an orthotropic outer layer is analyzed as an example. The numerical values of the lowest critical velocity of the two-layer composite tube predicted by the new formulas compare well with existing data.

Free Vibration Analysis of Functionally Graded Porous Cylindrical Panels Reinforced with Graphene Platelets.

The free vibration of functionally graded porous cylindrical shell panels reinforced with graphene platelets (GPLs) was numerically investigated. The free vibration problem was formulated using the first-order shear deformation shell theory in the framework of the 2-D natural element method (NEM). The effective material properties of the GPL-reinforced shell panel were evaluated by employing the Halpin-Tsai model and the rule of mixtures and were modified by considering the porosity distribution. The cylindrical shell surface was transformed into the 2-D planar NEM grid to avoid complex computation, and the concept of the MITC3+shell element was employed to suppress shear locking. The numerical method was validated through benchmark experiments, and the free vibration characteristics of FG-GPLRC porous cylindrical shell panels were investigated. The numerical results are presented for four GPL distribution patterns (FG-U, FG-X, FG-O, and FG-Λ) and three porosity distributions (center- and outer-biased and uniform). The effects of GPL weight, porosity amount, length-thickness and length-radius ratios, and the aspect ratio of the shell panel and boundary condition on the free vibration characteristics are discussed in detail. It is found from the numerical results that the proposed numerical method accurately predicts the natural frequencies of FG-GPLRC porous cylindrical shell panels. Moreover, the free vibration of FG-GPLRC porous cylindrical shell panels is significantly influenced by the distribution pattern as well as the amount of GPLs and the porosity.

Free Vibration Responses of Functionally Graded CNT-Reinforced Composite Conical Shell Panels.

Functionally graded CNT (carbon nanotube)-reinforced composites (FG-CNTRCs) are intensively studied because the mechanical behaviors of conventional composites can be dramatically improved. Only a small amount of CNTs are appropriately distributed through the thickness. However, the studies on conical shell panels have been poorly reported when compared with beams, plates and cylindrical shells, even though more parameters are associated with the mechanical behaviors of conical shell panels. In this context, this study intends to profoundly investigate the free vibration of FG-CNTRC conical shell panels by developing an effective and reliable 2-D (two-dimensional) numerical method. The displacement field is expressed using the first-order shear deformation shell theory, and it is approximated by the 2-D planar natural element method (NEM). The conical shell surface is transformed into the 2-D planar NEM grid, and the approach for MITC3+shell element is employed to suppress the shear locking. The developed numerical method is validated through the benchmark experiments, and the free vibration responses of FG-CNTRC conical shell panels are investigated with respect to all the associated parameters. It is found from the numerical results that the free vibration of FG-CNTRC conical shell panels is significantly influenced by the volume fraction and distribution pattern of CNTs, the geometry parameters of the conical shell, and the boundary condition.

Vibration characteristics of smart laminated carbon nanotube-reinforced composite cylindrical shells resting on elastic foundations with open circuit

This study analyzes the vibration of smart laminated carbon nanotube (CNT)-reinforced composite cylindrical shells integrated with piezoelectric materials and resting on elastic foundations. The effect of open circuit electrical boundary condition on vibration characteristics is investigated when a quadratic variation is considered for the electrical potential. Furthermore, the elastic foundations are included in the mathematical modeling to see their effects when the open circuit electrical boundary condition is applied. The first-order shear deformation shell theory and Maxwell’s static electricity equation are employed to derive the governing equations, and natural frequencies are calculated by solving an eigenvalue problem. Vibration characteristics with the open circuit electrical boundary condition are compared with those of closed (short) circuit electrical boundary condition for various mechanical boundary conditions and shell and piezoelectric geometrical parameters. The results demonstrate that the open circuit electrical boundary condition leads to higher frequencies than the closed circuit condition and it must be noticed in the design of smart composite structures with embedded or surface-bonded piezoelectric materials.

Acoustic Radiation From Stiffened Double Concentric Large Cylindrical Shells: Part I Circumferential Harmonic Waves

Abstract Acoustic radiation from stiffened double concentric large cylindrical shells with periodic cavities is analytically investigated via circumferential harmonic waves driven by a point force. The vibration of double isotropic circular cylindrical shells is described by the first-order shear deformation shell theory. One set of uniformly spaced annular bulkheads connects the inner and outer cylindrical shells. In-plane motion equations of the annular bulkheads are expressed by two displacement potential functions. Sound pressure loadings of periodic cavities exerting on the inner and outer cylindrical shells are derived according to the Fourier transform and Poisson summation formula. Far-field sound pressure of the stiffened double cylindrical shells is obtained using the stationary phase method and acoustic radiation features of stiffened double concentric large cylindrical shells with periodic cavities are analyzed in terms of sound pressure power spectra and sound pressure level. Acoustic propagation features of stiffened double cylindrical shells with or without acoustic cavities are shown.

Wave propagation analysis of cylindrical sandwich shell with auxetic core utilizing first-order shear deformable theory (FSDT)

The high mechanical properties of auxetic materials with negative Poisson’s ratio make them suitable for a variety of applications. This article investigated how the auxetic layer affects wave propagation and its effective characteristics in cylindrical sandwich shells. An analytical solution is established to describe wave propagation through wave frequency and wave velocity according to first-order shear deformable shell theory. Moreover, the Hamilton principle was applied. Then, the governing equation was solved by utilizing an analytical method. Different parameters such as auxetic cell angle inclined and dimensions on wave behavior were investigated. The final step is to conduct a numerical investigation. The results are reported in the form of several figures and analyze how each parameter influences wave propagation.

Thermal strain energy induced wave propagation for imperfect FGM sandwich cylindrical shells

An investigation is developed to analyze the heat-induced wave propagation characteristics of imperfect functionally graded material (FGM) sandwich cylindrical shells composed of a metal core layer and two functionally graded materials (FGMs) surface layers in a thermal environment by using the first-order shear deformation (FSD) shell theory. The temperature-dependent material properties are assumed to vary continuously along the thickness direction. Two types of porosity in FGMs layers of sandwich cylindrical shells are taken into account. To describe the porosity effects, the new porosity function composed of the porosity distribution function, core-to-thickness ratio, and porosity volume fraction is proposed. A novel thermal strain energy approach is established for cylindrical shell structures in a high-temperature environment by introducing Green’s nonlinear strains. The Hamilton’s principle is applied to derive the wave motion equations that govern the wave propagation behaviors. The analytical solutions of dispersion relations are determined by solving a generalized eigenvalue problem. In addition, a parametric investigation is conducted to study the effects of axial and circumferential wave number, porosity type and volume fraction, core-to-thickness ratio, power-law exponent, temperature changes, and radius-to-thickness ratio on the wave propagation characteristics of imperfect FGM sandwich cylindrical shells in high-temperature environments.

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

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

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.