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The nonlinear buckling and postbuckling of functionally graded carbon nanotube reinforcement composite (FG-CNTRC) plates resting on the nonlinear effect of elastic foundation under combined load including external pressure and axial compression loads with uniformly distributed temperature rises are fully investigated in this paper. The FG-CNTRC plates are reinforced by FG-CNTRC stiffeners and the plate-foundation interaction is modeled using a nonlinear model. The higher-order shear deformation plate theory with the geometrical nonlinearities of von Kármán is applied to establish the basic formulations. Additionally, by using the anisotropic higher-order shear deformation beam theory, a new smeared stiffener technique is successfully developed for FG-CNTRC stiffeners. Galerkin's method is used to achieve the equilibrium equation system in the nonlinear algebraic forms, and the critical load and postbuckling load–deflection curve expression are explicitly determined. The numerical investigations discuss the influences of FG-CNTRC stiffeners, hardening/softening nonlinear elastic foundation, uniformly distributed temperature rises, geometrical and material features on nonlinear stability of stiffened FG-CNTRC plates. [ABSTRACT FROM AUTHOR]

An analytical investigation on buckling and postbuckling behavior of sandwich cylindrical shells comprising functionally graded material layers and subjected to uniform temperature rise and external pressure in thermal environments is presented in this paper. Two sandwich models corresponding to functionally graded material face sheets and core layer are considered. The properties of constitutive materials are assumed to be temperature dependent and effective properties of functionally graded material are determined according to linear rule of mixture. Formulations are based on the classical shell theory taking Von Karman–Donnell nonlinearity and elastic constraint at edges into consideration. Multi-term solutions of deflection and stress function are assumed to satisfy simply supported boundary conditions and Galerkin method is applied to derive nonlinear load–deflection relations from which buckling loads and postbuckling paths are determined. The effects of sandwich configurations, material gradation, tangential constraints of edges, and temperature dependence of material properties on buckling loads and postbuckling curves are analyzed through numerical examples. [ABSTRACT FROM AUTHOR]

Buckling and postbuckling behavior of thin composite cylindrical shells reinforced by carbon nanotubes (CNTs), surrounded by elastic media and exposed to uniform temperature rise, are investigated in this article. CNTs are reinforced into isotropic matrix phase through uniform distribution or functionally graded distributions across the thickness direction. Material properties are assumed to be temperature dependent, and effective elastic moduli of CNT-reinforced composite are determined according to extended rule of mixture. Formulations are based on the classical thin shell theory taking Von Karman–Donnell nonlinearity, surrounding elastic media and elastic constraints of boundary edges into consideration. Multi-term solutions of deflection and stress function are assumed to satisfy simply supported boundary condition, and Galerkin method is applied to obtain nonlinear relation of thermal load and deflection. An iteration algorithm is used to determine buckling temperatures and postbuckling paths. Numerical examples are given to analyze the effects of volume fraction and distribution type of CNTs, geometrical parameters, degree of tangential edge constraint, buckling mode, and surrounding elastic media on the buckling temperatures and postbuckling strength of thermally loaded nanocomposite cylindrical shells. [ABSTRACT FROM AUTHOR]

Buckling and postbuckling behavior of carbon nanotube‐reinforced composite (CNTRC) cylindrical shells with tangentially restrained edges exposed to preexisting temperature conditions and subjected to uniform external pressure are presented in this analytical study. Three temperature conditions considered are that uniform temperature rise, through‐the‐thickness temperature gradient, and in‐plane linear temperature distribution. Carbon nanotubes (CNTs) are reinforced into matrix phase through uniform or functionally graded distributions. The properties of CNTs and matrix are assumed to be temperature‐dependent and effective moduli of CNTRC are determined according to extended rule of mixture. Governing equations are based on the classical shell theory taking into account Von Karman‐Donnell nonlinearity and elasticity of tangential constraints of edges. Multi‐term solutions of deflection and stress function are assumed to satisfy simply supported boundary conditions and Galerkin method is applied to derive closed‐form expression of nonlinear pressure‐deflection relation from which critical buckling pressures and postbuckling paths are determined. A variety of numerical examples are given and interesting remarks are achieved. Due to practical situations of boundary edges and various temperature conditions, this paper aims to analyze separate and combined influences of tangential edge constraints and preexisting temperatures on thermomechanical postbuckling behavior of pressure‐loaded nanocomposite cylindrical shells. [ABSTRACT FROM AUTHOR]

Cylindrical shells are usually buckled under complex and combined loading conditions. This article presents an analytical approach to investigate the buckling and postbuckling behaviors of cylindrical shells reinforced by single-walled carbon nanotubes, surrounded by an elastic medium, exposed to thermal environments, and subjected to combined axial compression and lateral pressure loads. Carbon nanotubes (CNTs) are imbedded into matrix phase by uniform distribution or functionally graded distribution along the thickness direction. The properties of constituents are assumed to be temperature dependent, and effective properties of CNT-reinforced composite (CNTRC) are determined by an extended rule of mixture. Governing equations are based on the classical shell theory (CST) taking von Karman–Donnell nonlinearity and surrounding elastic foundations into consideration. Three-term form of deflection is assumed to satisfy simply supported boundary conditions, and Galerkin method is applied to obtain nonlinear load–deflection relations from which buckling loads and postbuckling equilibrium paths are determined. Numerical examples are carried out to show the effects of CNT volume fraction, distribution types, thermal environments, preexisting nondestabilizing lateral pressure and axial compression loads, and elastic medium on the buckling and postbuckling behaviors of CNTRC cylindrical shells. [ABSTRACT FROM AUTHOR]

Buckling and postbuckling behaviors of toroidal shell segment reinforced by single-walled carbon nanotubes, surrounded by an elastic medium, exposed to a thermal environment and subjected to uniform external pressure are investigated in this paper. Carbon nanotubes are reinforced into matrix phase by uniform distribution or functionally graded distribution along the thickness direction. Material properties of constituents are assumed to be temperature dependent, and the effective properties of carbon nanotube reinforced composite are estimated by extended mixture rule through a micromechanical model. Governing equations for toroidal shell segments are based on the classical thin shell theory taking into account geometrical nonlinearity, surrounding elastic medium, and varying degree of tangential constraints of edges. Three-term solution of deflection and stress function are assumed to satisfy simply supported boundary condition, and Galerkin method is applied to derive nonlinear load–deflection relation from which buckling loads and postbuckling equilibrium paths are determined. Analysis shows that tangential edge restraints have significant effects on nonlinear buckling of carbon nanotube reinforced composite toroidal shell segments. In addition, the effects of carbon nanotube volume fraction, distribution types, geometrical ratios, elastic foundation, and thermal environments on the buckling and postbuckling behaviors of carbon nanotube reinforced composite toroidal shell segments are analyzed and discussed. [ABSTRACT FROM AUTHOR]

A new nonlinear approach on the buckling and postbuckling of functionally graded orthogonal and/or spiral-stiffened circular cylindrical shells subjected to torsional loads is proposed in this paper. The shells skin are stiffened by eccentrically rings, stringers, and/or spiral stiffeners at the surface of shells assuming that the material distribution laws of shell skin and stiffeners are graded by two distribution models. Lekhnitskii's smeared stiffeners technique is improved for spiral stiffeners with effect of thermal terms. This is the significant novelty and scientific contribution of this paper. Theoretical formulations were established by using the Donnell shell theory taking into account the geometrical nonlinearity of von Kármán. The obtained results investigated in numerical forms show effects of volume fraction exponent of shell skin and stiffeners, geometrical parameter and stiffeners on the torsional buckling, and postbuckling behavior of functionally graded cylindrical shells. Especially, very large effects of spiral stiffeners on torsional stability behavior are obtained in comparison with same quantity material of orthogonal stiffeners. [ABSTRACT FROM AUTHOR]

The problem of finding a solution of a variational inequality over the set of common fixed points of a nonexpansive semigroup is considered in a real and uniformly convex Banach space without imposing the sequential weak continuity of the normalized duality mapping. Two new explicit iterative methods are introduced based on the steepest-descent method, and conditions are given to obtain their strong convergence. A numerical example is showed to illustrate the convergence analysis of the proposed methods. [ABSTRACT FROM AUTHOR]