Nonlinear Shell Theory Research Articles

Large-amplitude vibrations of thin-walled rotating laminated composite cylindrical shell with arbitrary boundary conditions

The study of the large-amplitude vibrations of thin-walled rotating laminated composite cylindrical shell with arbitrary boundary conditions is presented in this paper, in which the artificial spring is used to simulate the arbitrary boundary conditions. The nonlinearity is introduced by using Donnell's nonlinear shell theory, and the orthogonal polynomials are used as the admissible displacement functions. By using the Lagrange equation based on the energy method, the governing equation of motion is obtained. Then, the Incremental Harmonic Balance Method (IHBM) and the arc-length method are used in the process of solving the governing equation. The influences of rotating speed, boundary spring stiffness, and geometric parameters on the nonlinear vibration characteristics of the shell are investigated. The results show that these parameters have a significant impact on the nonlinear vibration of the rotating laminated cylindrical shell.

On rotational instability within the nonlinear six-parameter shell theory

Within the six-parameter nonlinear shell theory we analyzed the in-plane rotational instability which occurs under in-plane tensile loading. For plane deformations the considered shell model coincides up to notations with the geometrically nonlinear Cosserat continuum under plane stress conditions. So we considered here both large translations and rotations. The constitutive relations contain some additional micropolar parameters with so-called coupling factor that relates Cosserat shear modulus with the Cauchy shear modulus. The discussed instability relates to the bifurcation from the static solution without rotations to solution with non-zero rotations. So we call it rotational instability. We present an elementary discrete model which captures the rotational instability phenomenon and the results of numerical analysis within the shell model. The dependence of the bifurcation condition on the micropolar material parameters is discussed.

Dynamic instability responses of the substructure living biological cells in the cytoplasm environment using stress-strain size-dependent theory

Over the last few years, some novel researches in the field of medical science made a tendency to have a therapy without any complications or side-effects of the disease with the aid of prognosis about the behaviors of the substructure living biological cell. Regarding this issue, nonlinear frequency characteristics of substructure living biological cell in axons with attention to different size effect parameters based on generalized differential quadrature method is presented. Supporting the effects of surrounding cytoplasm and MAP Tau proteins are considered as nonlinear elastic foundation. The Substructure living biological cell are modeled as a moderately thick curved cylindrical nanoshell. The displacement- strain of nonlinearity via Von Karman nonlinear shell theory is obtained. Extended Hamilton’s principle is used for obtaining nonlinear equations of the living biological cells and finally, GDQM and PA are presented to obtain large amplitude and nonlinear frequency information of the substructure living biological cell. Based on presented numerical results, increasing the nonlinear MAP tau protein parameter causes to improve the hardening behavior and increase the maximum amplitudes of resonant vibration of the microtubule. The crucial consequence is when the fixed boundary conditions in the microstructure switch to cantilevered, the living part of the cells could manage to have irrational feedback at the broad field of the excitation frequency. The current study has been made into the influences of the NSG parameters, geometrical and physical parameters on the instability of the curved microtubule employing continuum mechanics model. Communicated by Ramaswamy H. Sarma

A Shell Description of Beams Incorporating Transverse Thickness Strain Energies for Receding Contact

Abstract The geometrically exact nonlinear deflection of a beamshell is considered here as an extension of the formulation derived by Libai and Simmonds (1998, The Nonlinear Theory of Elastic Shells, Cambridge University Press, Cambridge, UK) to include deformation through the thickness of the beam, as might arise from transverse squeezing loads. In particular, this effect can lead to receding contact for a uniform beamshell resting on a smooth, flat, rigid surface; traditional shell theory cannot adequately such behavior. The formulation is developed from the weak form of the local equations for linear momentum balance, weighted by an appropriate tensor. Different choices for this tensor lead to both the traditional shell equations corresponding to linear and angular momentum balance, as well as the additional higher-order representation for the squeezing deformation. In addition, conjugate strains for the shell forces are derived from the deformation power, as presented by Libai and Simmonds. Finally, the predictions from this approach are compared against predictions from the finite element code abaqus for a uniform beam subject to transverse applied loads. The current geometrically exact shell model correctly predicts the transverse shell force through the thickness of the beamshell and is able to describe problems that admit receding contact.

On the nonlinear dynamics of a multi-scale hybrid nanocomposite disk

This is the first research on the nonlinear frequency analysis of a multi-scale hybrid nanocomposite (MHC) disk (MHCD) resting on an elastic foundation subjected to nonlinear temperature gradient and mechanical loading is investigated. The matrix material is reinforced with carbon nanotubes (CNTs) or carbon fibers (CF) at the nano- or macroscale, respectively. We present a modified Halpin–Tsai model to predict the effective properties of the MHCD. The displacement–strain of nonlinear vibration of multi-scale laminated disk via third-order shear deformation theory (TSDT) and using Von Karman nonlinear shell theory is obtained. Hamilton’s principle is employed to establish the governing equations of motion, which is finally solved by generalized differential quadrature method (GDQM) and perturbation method (PM). Finally, the results show that FG patterns, different orientation angle of the fiber, the VF and WCNT parameters, axial load, nonlinear temperature gradient, and applied temperature of the top surface play an essential impact on the linear and nonlinear dynamic responses of the MHCD. The more significant outcome of this research is that the effects of the VF, WCNT, $$\theta$$, and β parameters on the nonlinear frequency of the MHCD can be considered at the higher value of the large deflection parameter and the effect of negative axial load on the dynamic responses of the structure is more intensive. As an applicable result show that the best functionally graded (FG) pattern for serving the highest nonlinear dynamic response of an MHC reinforced annular plate is FG-A.

Nonlinear vibrations of truncated conical shells considering multiple internal resonances

The geometrically nonlinear vibration response of truncated thin conical shells is studied for the first time considering the one-to-one internal resonance, a phenomenon typically observed in symmetric structures such as conical shells. The Novozhilov nonlinear shell theory, retaining all nonlinear terms in the in-plane strain–displacement relationships of the three mid-surface displacements, is applied to study nonlinear vibrations of truncated conical shells. In-plane inertia is also taken into account, and a relatively large number of generalized coordinates, associated with the global discretization of the shell, is considered. This gives very accurate numerical solutions for simply supported, truncated thin conical shells. The effect of an exact one-to-one internal resonance, due to the axial symmetry of conical shells, is fully considered and the results are presented for different excitation levels. The numerical results show that also an almost exact one-to-one internal resonance with a mode presenting a different number of circumferential waves can also arise, which further complicates the nonlinear vibrations and leads to 1:1:1:1 internal resonance. The numerical model was augmented with additional generalized coordinates to capture this phenomenon. Pitchfork, Neimark–Sacker and period-doubling bifurcations of the forced vibration responses arising from internal resonances are detected, followed and presented, showing complex nonlinear dynamics.

Construction of elastoplastic shallow panels manufactured by stretching from sheet metal

In the article, in a geometrically and physically nonlinear formulation, the problem of the stress-strain state of shallow shells on a rectangular plane under pressure loading is solved. The solving equations of the theory of flexible shallow shells are obtained on the basis of the Henki-Ilyushin deformation theory of plasticity and the equations of the geometrically non-linear theory of shells, which contain quadratic terms with respect to the angles of rotation of the normals to the median surface. The geometrical parameters of the median surface are taken in the initial undeformed state. The equations are written in a form that makes it possible to carry out numerical calculations of problems with allowance for geometric and physical nonlinearities or only with geometric nonlinearity, to take into account unloading, compressibility of the material and secondary plastic deformations. The results of numerical calculations of critical loads, unloading zones and secondary plastic deformations are presented. Recommendations are given on the choice of geometrical parameters when designing shallow shells.

Harmonic resonances of graphene-reinforced nonlinear cylindrical shells: effects of spinning motion and thermal environment

This work investigates nonlinear harmonic resonance behaviors of graded graphene-reinforced composite spinning thin cylindrical shells subjected to a thermal load and an external excitation. The volume fraction of graphene platelets varies continuously in the shell’s thickness direction, which generates position-dependent useful material properties. Natural frequencies of shell traveling waves are derived by considering influences of the initial hoop tension, centrifugal and Coriolis forces, thermal expansion deformation, and thermal conductivity. A new Airy stress function is introduced. Harmonic resonance behaviors and their stable solutions for the spinning cylindrical shell are analyzed based on an equation of motion which is established by adopting Donnell’s nonlinear shell theory. The necessary and sufficient conditions for the existence of the subharmonic resonance of the spinning composite cylindrical shell are given. Besides the shell’s intrinsic structural damping, the Coriolis effect due to the spinning motion has a contribution to the damping terms of the system as well. Comparisons between the present analytical results and those in other papers are made to validate the existing solutions. Influences of main factors on vibration characteristics, primary resonance, and subharmonic resonance behaviors of the novel composite cylindrical shell are discussed. Furthermore, the mechanism of how the spinning motion affects the amplitude–frequency curves of harmonic resonances of the cylindrical shell is analyzed.

Mechanics of the confined functionally graded porous arch reinforced by graphene platelets

This paper presents the stability mechanism of the confined functionally graded porous (FGP) arch reinforced by graphene platelets (GPLs). The Halpin-Tsai micromechanics theory is used to evaluate the distribution of Young’s modulus in the cross-section of the arch. The Gaussian random field is employed to describe the porosity coefficient and mass density. Both pores and GPLs are distributed symmetrically to the mid-surface of the arch. Theoretical predictions are obtained to express the buckling load (load bearing capacity) based on the nonlinear thin-walled shell theory. Excellent numerical verification is obtained by comparing the buckling load, as well as the equilibrium paths with the theoretical predictions. Moreover, a confinement factor is defined to quantize the confinement effect between the confined and unconfined arches. Finally, the buckling load may be influenced by the following parameters: thermal rise field, porosity coefficient, central angle of the arch, weight fraction and geometric parameters of GPLs, friction coefficient, deformability of the medium. These parameters are analyzed and discussed.

Nonlinear Thermo-Electro-Mechanical Vibration of Functionally Graded Piezoelectric Nanoshells on Winkler–Pasternak Foundations Via Nonlocal Donnell’s Nonlinear Shell Theory

The thermo-electro-mechanical nonlinear vibration of circular cylindrical nanoshells on the Winkler–Pasternak foundation is investigated. The nanoshell is made of functionally graded piezoelectric material (FGPM), which is simulated by the nonlocal elasticity theory and Donnell’s nonlinear shell theory. The Hamilton’s principle is employed to derive the nonlinear governing equations and corresponding boundary conditions. Then, the Galerkin’s method is used to obtain the nonlinear Duffing equation, to which an approximate analytical solution is obtained by the multiple scales method. The results reveal that the system exhibits hardening-spring behavior. External applied voltage and temperature change have significant effect on the nonlinear vibration of the FGPM nanoshells. Moreover, the effect of power-law index on the nonlinear vibration of the FGPM nanoshells depends on parameters such as the external applied voltage, temperature change and properties of the Winkler–Pasternak foundation.

Nonlinear vibration of magnetoelectroelastic nanoscale shells embedded in elastic media in thermoelectromagnetic fields

This study investigates the nonlinear vibration of magnetoelectroelastic composite cylindrical nanoshells embedded in elastic media for the first time. The small-size effect and thermoelectromagnetic loadings are considered. Based on the nonlocal elasticity theory and Donnell’s nonlinear shell theory, the nonlinear governing equations and the corresponding boundary conditions are derived using Hamilton’s principle. Then, the Galerkin method is utilized to transform the governing equations into a nonlinear ordinary differential equation and subsequently the method of multiple scales is employed to obtain an approximate analytical solution to nonlinear frequency response. The present results are verified by the comparison with the published ones in the literature. Finally, an extensive parametric study is conducted to examine the effects of the nonlocal parameter, the external magnetic potential, the external electric potential, the temperature change, and the elastic media on the nonlinear vibration characteristics of magnetoelectroelastic composite nanoshells.

Nonlinear Vibration Analyses of Cylindrical Shells Composed of Hyperelastic Materials

The nonlinear vibration problem is studied for a thin-walled rubber cylindrical shell composed of the classical incompressible Mooney–Rivlin material and subjected to a radial harmonic excitation. With the Kirchhoff–Love hypothesis, Donnell’s nonlinear shallow shell theory, hyperelastic constitutive relation, Lagrange equations and small strain hypothesis, a system of nonlinear differential equations describing the large-deflection vibration of the shell is derived. First, the natural frequencies of radial, circumferential and axial vibrations are studied. Then, based on the bifurcation diagrams and the Poincare sections, the nonlinear behaviors describing the radial vibration of the shell are illustrated. Examining the influences of structural and material parameters on radial vibration of the shell shows that the vibration modes are highly sensitive to the thickness–radius ratio when the ratio is less than a certain critical value. Moreover, in terms of the results of multimodal expansion, it is found that the response of the shell to radial motion is more regular than that without considering the coupling between modes, while there are more phenomena for the uncoupled case.

Aerothermoelastic analysis of imperfect FG cylindrical shells in supersonic flow

The supersonic panel flutter of functionally graded cylindrical shells under thermal loadings is investigated by considering the effects of various axisymmetric and asymmetric geometric imperfections. The problem is formulated using a nonlinear first-order shear deformation theory of shells with the first-order aerodynamic piston theory. The temperature is assumed to vary in the thickness direction according to the steady-state heat conduction equation. The semi-analytical finite element method (FEM), based on the field consistency redistribution approach is used to obtain discretized nonlinear aeroelastic equations. In the process of FEM, the imperfection functions are also discretized and efficiently approximated by the Hermitian polynomials in order to facilitate the required integrations. For analysis purpose, the deformations and stresses induced by the temperature rise are first computed by solving the nonlinear static aerothermoelastic equations using the Newton-Raphson method. The pseudo-arclength continuation method is also employed to detect the possible snap-through of the shell. The linearized stability equation about the equilibrium state is then used for determining the flutter boundaries. Numerous parametric studies are conducted to examine the effects of temperature rise and imperfections on the flutter boundaries, which show considerable effects of imperfections on changing the trend of variation of the flutter pressure with temperature.

In-plane shear nonlinearity in failure behavior of angle-ply laminated shells

The paper concerns the progressive failure analysis of laminates with the in-plane shear nonlinearity accounted for. The nonlinear shear response of the layer is described by the constitutive relation treating the stresses as a function of strains. Thus it can be easily incorporated into the displacement-based FEM codes. The brittle failure mechanisms of the fibers and the matrix of the layer are recognized with the use of the Hashin criterion. The model is implemented into the non-commercial FEM program for the analysis of multifold shells which is based on the 6-parameter nonlinear shell theory of the Cosserat type.

Lagrangian Mixed Finite Element Methods for Nonlinear Thin Shell Problems

Abstract A class of Lagrangian mixed finite element methods is constructed for an approximate solution of a problem of nonlinear thin elastic shell theory, namely, the problem of finding critical points of the functional of potential energy according to the Budiansky–Sanders model. The proposed numerical method is based on the use of the second derivatives of the deflection as auxiliary variables. Sufficient conditions for the solvability of the corresponding discrete problem are obtained. Accuracy estimates for approximate solutions are established. Iterative methods for solving the corresponding systems of nonlinear equations are proposed and investigated.

Buckling and post-buckling responses of smart doubly curved composite shallow shells embedded in SMA fiber under hygro-thermal loading

In this research, the post-buckling behaviors of doubly curved composite shells in hygro-thermal environment are investigated by employing multiple scales perturbation method. Three-phase composites shells with polymer/Carbon nanotube/fiber and polymer/Graphene platelet/fiber (PGF) and Shape Memory Alloy (SMA)/matrix according to Halpin-Tsai model are taken into consideration. The displacement-strain of laminated doubly curved shells via third-order shear deformation theory (TSDT) and using von-Kármán nonlinear shell theory is obtained. The governing equations of shallow shell are derived by implementing Hamilton's principle. For investigating correctness and accuracy, this paper is validated with other previous researches. Finally, different parameters such as volume fraction of SMA, temperature rise, various distribution patterns, aspect and curvature ratios are considered in this article. It is found that these parameters have significant effect on the thermal buckling loading.

Nonlinear free vibration of piezoelectric cylindrical nanoshells

The nonlinear vibration characteristics of the piezoelectric circular cylindrical nanoshells resting on a viscoelastic foundation are analyzed. The small scale effect and thermo-electro-mechanical loading are taken into account. Based on the nonlocal elasticity theory and Donnell’s nonlinear shell theory, the nonlinear governing equations and the corresponding boundary conditions are derived by employing Hamilton’s principle. Then, the Galerkin method is used to transform the governing equations into a set of ordinary differential equations, and subsequently, the multiple-scale method is used to obtain an approximate analytical solution. Finally, an extensive parametric study is conducted to examine the effects of the nonlocal parameter, the external electric potential, the temper-ature rise, and the Winkler-Pasternak foundation parameters on the nonlinear vibration characteristics of circular cylindrical piezoelectric nanoshells.

Robust four-node elements based on Hu\u2013Washizu principle for nonlinear analysis of Cosserat shells

Mixed 4-node shell elements with the drilling rotation and Cosserat-type strain measures based on the three-field Hu–Washizu principle are proposed. In the formulation, apart from displacement and rotation fields, both strain and stress resultant fields are treated as independent. The elements are derived in the framework of a general nonlinear 6-parameter shell theory dedicated to the analysis of multifold irregular shells. The novelty of the developed elements stems from the fact that the measures of assumed strains and stress resultants are asymmetric. The original interpolation of drilling and bending components of strains and stress resultants is proposed. In the formulation of new mixed elements, two variants of the interpolation of membrane components are used and interpolation of the independent fields is defined in the natural or skew coordinates. Accuracy and efficiency of the developed elements are tested in several linear and nonlinear numerical examples. It is shown that smaller number of independent parameters in the interpolation of membrane components gives more accurate results. The proposed mixed 4-node elements enable the use of large load steps in nonlinear computations. Moreover, they require significantly less equilibrium iterations than other shell elements formulated in the 6-parameter shell theory.

An efficient solver for fully coupled solution of interaction between incompressible fluid flow and nanocomposite truncated conical shells

In this research, the dynamic instabilities of nanocomposite truncated conical shells containing a quiescent or a flowing inviscid fluid are scrutinized. Nonlinear dynamic equations are established according to the Novozhilov nonlinear shell theory along with Green’s strains and Hamilton principle. The velocity potential and Bernoulli’s equations are adopted to calculate fluid pressure acting on the conical shell. The nonlinear governing equations are discretized using trigonometric expansion through the circumferential direction and generalized differential quadrature method (GDQM) through the meridional direction. A detailed parametric study is directed to provide an insight into the influence of volume fraction of carbon nanotubes (CNTs), CNT dispersion, geometrical parameters, and boundary conditions on the divergence and flutter instabilities of nanocomposite truncated conical shells. This study shows the superb efficiency of the outlined solution procedure in reducing computational costs and virtual storage. The simulation indicates that the beginning of divergence and flutter instabilities can be significantly postponed by selecting an appropriate dispersion of CNTs through the thickness of the conical shell. Furthermore, the onset of flutter and divergence instabilities are found to be very sensitive to the semi-vertex angle and thickness-to-radius ratio. The results of this research shed light into using ultra-high-strength and low-weight nanocomposite for pressure vessels applications.

Chaotic dynamics of a non-autonomous nonlinear system for a smart composite shell subjected to the hygro-thermal environment

In this research, nonlinear dynamic behaviors of multiscale composites doubly curved shells have been investigated by employing multiple scales Perturbation Method. Three-phase composites shells with polymer/Carbon nanotube/fiber (PCF) according to Halpin–Tsai model have been assumed. The displacement- strain of nonlinear vibration of multiscale laminated doubly curved shells via higher order shear deformation (HSDT) theory and using Green–Lagrange nonlinear shell theory is obtained. The governing equations of composite doubly curved shell have been derived by implementing Hamilton’s principle and shell considered to be simply supported. For investigating correctness and accuracy, this paper is validated by other previous researches. Finally, bifurcation diagram, phase portraits and Poincare maps are investigated. The results indicate different dimensionless force; curvature ratio and kind of distribution pattern have strong influence on nonlinear vibration control of the composite multiscale doubly curved shell.