Nonlinear thermal stability of eccentrically stiffened FGM double curved shallow shells

This article presents analytical solutions for the nonlinear static and dynamic stability of imperfect eccentrically stiffened functionally graded material (FGM) higher order shear deformable double curved shallow shell on elastic foundations in thermal environments. It is assumed that the shell’s properties depend on temperature and change according to the power functions of the shell thickness. The shell is reinforced by the eccentrically longitudinal and transversal stiffeners made of full metal. Equilibrium, motion, and compatibility equations are derived using Reddy’s higher order shear deformation shell theory and taking into account the effects of initial geometric imperfection and the thermal stress in both the shells and stiffeners. The Galerkin method is applied to determine load–deflection and deflection–time curves. For the dynamical response, motion equations are numerically solved using Runge–Kutta method. The nonlinear dynamic critical buckling loads are found according to the criterion suggested by Budiansky–Roth. The influences of inhomogeneous parameters, dimensional parameters, stiffeners, elastic foundations, initial imperfection, and temperature increment on the nonlinear static and dynamic stability of thick FGM double curved shallow shells are discussed in detail. Results for various problems are included to verify the accuracy and e?ciency of the approach.     

Effects of Thermal Loading on the Buckling and Vibration of Ring-Stiffened Functionally Graded Shell

A theoretical method is developed to investigate the effects of thermal load and ring stiffeners on buckling and vibration characteristics of the functionally graded cylindrical shells, based on the first-order shear deformation theory (FSDT) considering rotary inertia. Heat conduction equation across the shell thickness is used to determine the temperature distribution. Material properties are assumed to be graded across the shell wall thickness of according to a power-law, in terms of the volume fractions of the constituents. The Rayleigh–Ritz procedure is applied to obtain the frequency equation. The effects of stiffener's number and size on natural frequency of functionally graded cylindrical shells are investigated. Moreover, the influences of material composition, thermal loading and shell geometry parameters on buckling and vibration are studied. The obtained results have been compared with the analytical results of other researchers, which showed good agreement. The new features of thermal vibration and buckling of ring-stiffened functionally graded cylindrical shells and some meaningful and interesting results obtained in this article are helpful for the application and the design of functionally graded structures under thermal and mechanical loads.     

Thermal Buckling of Axially Functionally Graded Thin Cylindrical Shell

In the present research, thermal buckling of shell made of functionally graded material (FGM) under thermal loads is investigated. The material properties of functionally graded materials (FGMs) are assumed to be graded in the axial direction according to a simple power law distribution in terms of the volume fractions of the constituents. In the previous articles that published, these properties are assumed to be graded in the thickness direction. Nonlinear kinematic (strain-displacement) relations are considered based on the first order shear deformation shell theory. By substituting kinematic and stress-strain relations of functionally graded shell in the total potential energy equation and employing Euler equations, the equilibrium equations are obtained. Applying Euler equations to the second variation of total potential energy equation leads to the stability equations. Then, buckling analysis of functionally graded shell under three types of thermal loads is carried out resulting into closed-form solutions.     

Axisymmetric nonlinear rapid heating of FGM cylindrical shells

Large amplitude thermally induced vibrations of cylindrical shells made of a through-the-thickness functionally graded material (FGM) are investigated in the current research. All of the thermo-mechanical properties of the FGM shell are assumed to be functions of temperature and thickness coordinate. Shell is subjected to rapid surface heating on the ceramic-rich surface while the other surface of the shell is kept at reference temperature. One dimensional heat conduction equation is constructed and solved by means of a hybrid finite difference-Crank–Nicolson algorithm. The constructed heat conduction equation is nonlinear since the thermal conductivity is temperature dependent. With the aid of first-order shear deformation shell theory under the axisymmetric Donnell kinematic assumptions and von Kármán type of strain-displacement relations, the total energy of the shell is established. Implementing the conventional Ritz method, a set of nonlinear coupled algebraic equations are obtained which govern the dynamics of the shell under thermal shock. These equations are solved in time domain using the Newmark time marching scheme and the simple Picard successive method. Parametric studies are given to explore the dynamics of an FGM cylindrical shell under thermal shock.     

Thermoelastic analysis of laminated and functionally graded sandwich cylindrical shells with two refined higher order models

Analytical solutions for laminated and functionally graded sandwich open cylindrical shells under mechanical and thermal loads are presented using a refined higher order shear and normal deformation theory. Temperature variation through thickness is assumed as thickness coordinate polynomial. Present study also extends the classical thickness criteria with more reliable extension to moderately thick shells. Navier solution method is used to solve system of equations derived using principle of minimum potential energy for all edges diaphragm supported. Two kinds of sandwich panels with core or face sheets made of thickness graded material are studied. Several examples are numerically evaluated to establish the accuracy of present models.     

Nonlinear dynamic response of eccentrically stiffened FGM plate using Reddy’s TSDT in thermal environment

The nonlinear dynamics of an eccentrically stiffened functionally graded material (ES-FGM) plates resting on the elastic Pasternak foundations subjected to mechanical and thermal loads is considered in this article. The plates are reinforced by outside stiffeners with temperature-dependent material properties in two cases: uniform temperature rise and through the thickness temperature gradient. Both stiffeners and plate are deformed under temperature. Using Reddy’s third-order shear deformation plate theory, stress function, Galerkin and fourth-order Runge–Kutta methods, the effects of material and geometrical properties, temperature-dependent material properties, elastic foundations, and stiffeners on the nonlinear dynamic response of the ES-FGM plate in thermal environments are studied and discussed. Some obtained results are validated by comparing with those in the literature.     

Thermal buckling analysis of FGM sandwich truncated conical shells reinforced by FGM stiffeners resting on elastic foundations using FSDT

This work presents an analytical approach to investigate the mechanical and thermal buckling of functionally graded materials sandwich truncated conical shells resting on Pasternak elastic foundations, subjected to thermal load and axial compressive load. Shells are reinforced by closely spaced stringers and rings, in which the material properties of shells and stiffeners are graded in the thickness direction following a general sigmoid law distribution and a general power law distribution. Four models of coated shell-stiffener arrangements are investigated. The change of spacing between stringers in the meridional direction also is taken into account. Two cases on uniform temperature rise and linear temperature distribution through the thickness of shell are considered. Using the first-order shear deformation theory, Lekhnitskii smeared stiffener technique and the adjacent equilibrium criterion, the linearization stability equations have been established. Approximate solution satisfies simply supported boundary conditions and Galerkin method is applied to obtain closed-form expression for determining the critical compression buckling load and thermal buckling load in cases uniform temperature rise and linear temperature distribution across the shell thickness. The effects of temperature, foundation, core layer, coating layer, stiffeners, material properties, dimensional parameters and semi-vertex angle on buckling behaviors of shell are shown.     

Application of two-steps perturbation technique to geometrically nonlinear analysis of long FGM cylindrical panels on elastic foundation under thermal load

Present research deals with the geometrically nonlinear bending of a long cylindrical panel made of a through-the-thickness functionally graded material subjected to thermal load. A panel under the action of uniform temperature rise loading is considered. Formulation of the shell is based on the third-order shear deformation shell theory, where the first-order shear deformation and classical shell theory may be extracted as special cases. Thermomechanical properties of the shell are assumed to be temperature dependent and are estimated according to a power law function across the shell thickness. Also, it is assumed that shell is in contact with an elastic foundation which acts in tension as well as in compression. The nonlinear governing equations of the shell are obtained using the von Kármán type of geometrical nonlinearity. The obtained governing equations are solved for two cases, i.e., simply supported shells and clamped shells. The developed equations are solved using a two-step perturbation technique. Accurate closed-form expressions are provided to obtain the mid-span deflection of the shell as a function of temperature elevation. Numerical results are provided to analyze the effects of power law exponent, boundary conditions, temperature dependency, side to radius ratio, and side to thickness ratio.     

Nonlinear vibration and dynamic response of shear deformable imperfect functionally graded double-curved shallow shells resting on elastic foundations in thermal environments

In this article, nonlinear vibration and dynamic response of imperfect functionally graded materials (FGM) thick double-curved shallow shells resting on elastic foundations are investigated using Reddy's third-order shear deformation shell theory in thermal environments. Material properties are assumed to be temperature dependent and graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The FGM shells are subjected to mechanical, damping, and thermal loads. The Galerkin method and fourth-order \hboxRunge–Kutta method are used to calculate natural frequencies, nonlinear frequency–amplitude relation, and dynamic response of the shells. In numerical results, the effects of geometrical parameters, material properties, imperfections, shear deformation, the elastic foundations, mechanical, thermal and damping loads on the nonlinear dynamic response, and nonlinear vibration of FGM double-curved shallow shells are investigated. Accuracy of the present formulation is shown by comparing the results of numerical examples with the ones available in literature.     

Thermal Buckling of Simply Supported Piezoelectric FGM Cylindrical Shells

A thermal buckling analysis is presented for functionally graded cylindrical shells that are integrated with surface-bonded piezoelectric actuators and are subjected to the combined action of thermal load and constant applied actuator voltage. The material properties are assumed to vary as a power form of the thickness coordinate. Derivation of the equations is based on the higher-order shear deformation shell theory using the Sanders nonlinear kinematic relations. Results for the buckling temperatures are obtained in the closed form solution. The effects of the applied actuator voltage, shell geometry, and volume fraction exponent of functionally graded material on the buckling temperature are investigated. The results for simpler states are validated with known data in the literature.     

Free vibration analysis of size-dependent functionally graded porous cylindrical microshells in thermal environment

In this article, the free vibration analysis of a functionally graded (FG) porous cylindrical microshell subjected to a thermal environment is investigated on the basis of the first-order shear deformation shells and the modified couple stress theories. The material properties are assumed to be temperature dependent and are graded in the thickness direction. The equations of motion and the related boundary conditions are derived using the principle of minimum potential energy and they are solved analytically. The model is validated by comparing the benchmark results with the obtained ones. The effects of material length scale parameter, temperature changes, volume fraction of the porosity, FG power index, axial and circumferential wave number, and length on the vibration behavior of the FG porous cylindrical microshell are studied. The results can have many applications such as in modeling of microrobots and biomedical microsystems.     

Thermal stability of eccentrically stiffened FGM plate on elastic foundation based on Reddy's third-order shear deformation plate theory

This article studies the nonlinear thermal buckling and postbuckling of eccentrically stiffened functionally graded plates on elastic foundation subjected to mechanical, thermal, and thermomechanical loads. The noticeable point of this study is using the Reddy's higher order shear deformation plate theory and a general formula for the forces and moments of eccentrically stiffened functionally graded material (FGM) plate, which takes into account the influence of temperature on both the FGM plate and stiffeners. The article used the Galerkin method, stress function, and iterative method to determine the thermal buckling loads and postbuckling response of the eccentrically stiffened FGM plates in three different cases of boundary conditions. The effects of material, temperature-dependent material properties, elastic foundations, boundary conditions, outside stiffeners, and temperature on the buckling and postbuckling loading capacity of the FGM plates in thermal environments are analyzed and discussed. A good agreement is obtained by comparing the present analysis with other available literature.     

Dynamic Thermal Postbuckling Analysis of Shear Deformable Piezoelectric-FGM Cylindrical Shells

Dynamic thermal postbuckling behavior of functionally graded cylindrical shells with surface-bonded piezoelectric actuators subjected to the combined action of thermal load and applied actuator voltage is studied. The shell material is graded across the thickness according to a power law. The material properties of the functionally graded cylindrical shells are considered to be temperature dependent. The theoretical formulations are based on the Sanders nonlinear kinematic relations, which account for the transverse shear strains, and the third-order shear deformation shell theory is employed. Hamilton's principle is used to derive the equations of motion governing piezoelectric FGM cylindrical shells. A finite difference approximation combined with the Runge-Kutta method is employed to predict the postbuckling equilibrium paths, and the dynamic buckling temperature difference is detected according to Budiansky's stability criterion. Numerical results are presented to demonstrate the effects of the applied actuator voltage, shell geometry, volume fraction exponent in the power-law variation of the FGM, and the temperature dependency of the material properties on the postbuckling behavior of the shell. The results for simpler states are validated with the known results in the literature.     

Three-dimensional thermo-elastic analysis of a functionally graded cylindrical shell with piezoelectric layers by differential quadrature method

Three-dimensional thermo-elastic analysis of a functionally graded cylindrical shell with piezoelectric layers under the effect of asymmetric thermo-electro-mechanical loads is carried out. Numerical results of displacement, stress and thermal fields are obtained using two versions of the differential quadrature methods, namely polynomial and Fourier quadrature methods. Material properties of the shell are assumed to be graded in the radial direction according to a power law but the Poisson’s ratio is assumed to be constant. Shells are considered to be under the effect of the pressure loading in the form of cosine and ring pressure loads, electric potentials and temperature fields. Numerical results for various boundary conditions are obtained and the effects of the thickness of piezoelectric layers, grading index of material properties and the ratio of the thickness to the radius of the shell on these results is presented.     

Influence of three-parameter viscoelastic medium on vibration behavior of a cylindrical nonhomogeneous microshell in thermal environment: An exact solution

In this article, the free vibration behavior of a functionally graded (FG) size-dependent microshell surrounded by viscoelastic foundation subjected to various thermal loading conditions is analytically studied. The material properties of the cylindrical FG microshell are supposed to be temperature dependent and vary continuously along the thickness direction according to the modified rule of mixtures. The size-dependent FG microshell is analyzed based on the modified couple stress theory. The analytical modeling is developed using the first-order shear deformation theory and the equations of motion are derived by the principle of minimum total potential energy. Then the governing equations for the free vibration behavior of a simply supported FG cylindrical microshell subjected to thermal loading are solved using the Navier procedure. The effects of some important parameters, such as material length scale parameter, stiffness and damping of the visco-Pasternak foundation, temperature changes, axial and circumferential wave number, and length of the microshell on the natural frequency are investigated and discussed.     

Thermal Effects on Stress Analysis in Large Amplitude Vibration of a Thick Annular Functionally Graded Plate

This paper deals with the nonlinear free and forced vibration of thick annular functionally graded material plates. The temperature field considered is assumed to be a uniform distribution over the plate surface and varied in the thickness direction only. Material properties are assumed to be temperature-dependent, and graded in the thickness direction. The formulations are based on the first-order shear deformation plate theory and von Kármán-type equation. The numerical illustrations concern with nonlinear vibration characteristics of functional graded plates with two constituent materials in thermal environments. Effects of material compositions and thermal loads on the vibration characteristics and stresses are examined.     

On the Behavior of a Rotating Functionally Graded Hybrid Cylindrical Shell with Imperfect Bonding Subjected to Hygrothermal Condition

The static behavior of a rotating cylindrical shell with surface bounded sensor and actuator in an axisymmetric hygrothermal condition is analyzed. The shell is simply supported and could be rested on an elastic foundation. The material properties of the shell and piezoelectric sensor and actuator are assumed to be functionally graded in the radial direction. Using the Fourier series expansion method through the longitudinal direction and the differential quadrature method (DQM) across the radial direction, and governing differential equations are solved. The validity of the present work was verified by comparisons with other published works. Numerical results are presented to illuminate the effects of key parameters on the responses of the hybrid shell.     

Variable kinematic shell elements for composite laminates accounting for hygrothermal effects

This article presents advanced shell models for the steady-state hygrothermal analysis of composite laminates. The Carrera Unified Formulation is used to derive refined models that include both layer-wise and equivalent single layer models. The governing equations are derived from the principle of virtual displacement taking into account thermal and hygroscopic effects. The geometrical relations for the exact cylindrical geometry are here considered. Through-the-thickness variations of temperature and moisture concentration are calculated by solving the Fourier equation and the Fick law, respectively. The mixed interpolation of tensorial component method is applied to a nine-node shell element to contrast the membrane and shear locking phenomena. Simply supported cross-ply cylindrical shells with antisymmetrical lamination subjected to bisinusoidal thermal/hygroscopic loads are analyzed considering various thickness/curvature ratios. Results obtained with assumed linear and calculated temperature/hygroscopic profiles are presented. Variable kinematics are compared regarding both accuracy and computational costs. The results show that all the kinematics can approximate the transverse shear stress distribution through the thickness with satisfactory accuracy when su?cient expansion terms are adopted. In some cases, miscellaneous expansions can lead to significant reductions in computational costs. The results presented here can be used as benchmark solutions for future works.     

Thermal Response of Symmetric Cross-Ply Laminated Plates Subjected to Linear and Non-Linear Thermo-Mechanical Loads

The flexural response of symmetric cross-ply laminated plates subjected to uniformly distributed linear and non-linear thermo-mechanical loads is presented using trigonometric shear deformation theory. The in-plane displacement field uses sinusoidal function in terms of thickness coordinate to include the shear deformation effect. The theory satisfies the shear stress-free boundary conditions on the top and bottom surfaces of the plate. The present theory obviates the need of shear correction factor. Governing equations and boundary conditions of the theory are obtained using the principle of virtual work. Thermal stresses and displacements for three-layer symmetric square cross-ply laminated plates subjected to uniform linear and nonlinear and thermo-mechanical loads are obtained. The results of present theory are compared with those of classical plate theory, first-order shear deformation theory and higher-order shear deformation theory.