Initial Postbuckling of Elastic Rods Subjected to Thermal Loads and Resting on an Elastic Foundation

In practical applications encountered in some mechanical systems initially straight slender elements laterally supported by elastic foundations may be prevented from thermal expansion. In such conditions a compressive force is created and a serious equilibrium instability phenomenon may take place when a critical temperature gradient is imparted to the structure. If the temperature is further increased the rod may experience relatively large deflections, and a geometrically non-linear two-point boundary value problem describes this behaviour. However, if finite but moderate deflections are admitted, an approximate solution may be obtained if a classical perturbation expansion method is applied to the non-linear ordinary differential governing equations. This approach renders a set of linear equations which can be sequentially solved. This work proposes an analytical solution for the assessment of the initial post-buckling configuration of slender elastic rods supported on linear elastic foundations and subjected to a uniform temperature gradient. The rod ends are assumed hinged and immovable and the thermal strain-temperature relationship is linear. The governing equations are derived and made non-dimensional so it is seen that two parameters rule the problem: the foundation stiffness modulus and the rod slenderness ratio. Results are presented and discussed for one slenderness ratio and a range of foundation stiffness that corresponds to the fourth buckling mode. Generally an increase in the applied temperature is associated with a decrease in the compressive force, but it is shown particular conditions where an opposite behavior is observed. The initial post-buckling results for unsupported rods are compared with existing full post-buckling solution and show excellent agreement, given that in practical situations the deflections are moderate.     

Thermal buckling and post-buckling of FGM Timoshenko beams on nonlinear elastic foundation

Buckling and post-buckling thermomechanical deformations of a functionally graded material (FGM) Timoshenko beam resting on a two-parameter non-linear elastic foundation and subjected to only a temperature rise have been numerically investigated with the shooting method. The material properties are assumed to vary only in the thickness direction according to a power law function. Through-the-thickness temperature distribution is determined by numerically solving the one-dimensional heat conduction equation. Geometric non-linearities in the strain-displacement relations and the non-linear traction-displacement relations at the interface between the beam and the foundation are considered. For clamped-clamped and immovable simply supported beams, critical values of the ratio of temperatures of the top and the bottom surfaces of the beam for transitions in buckling modes to occur are determined. Post-buckled equilibrium paths and configurations of the heated FGM beam are illustrated for different values of the elastic foundation stiffness parameters, exponent in the power law variation of material properties and the slenderness ratio. Results for the Timoshenko beam are compared with those of the corresponding homogeneous Euler–Bernoulli beam available in the literature.     

Thermal Post-Buckling of Functionally Graded Material Circular Plates Subjected to Transverse Point-Space Constraints

Axisymmetrical thermal post-buckling of functionally graded material (FGM) circular plates with immovably clamped boundary and a transversely central point-space constraint was studied. The material properties of the plate were assumed to vary as power law functions in the thickness direction and the temperature rise field to change only in the thickness direction. Based on von Karman's non-linear plate theory, governing equations in terms of the displacements of the middle plane were established. Temperature rise field was obtained by solving the one-dimensional heat conduction equation associated with specified boundary conditions at the top and bottom surface of the plate. By using the shooting method, thermal post-buckling deformation of the FGM circular plate was obtained before and after the plate contacting the point-space constraint. The changes in the characteristics of the deformation and the internal forces of FGM plates were discussed. The effects of gradients of material properties and non-uniform temperature rise parameters on the thermal post-buckling behaviors of FGM circular plates were also examined.     

Thermal post-buckling behavior of simply supported sandwich panels with truss cores

This article presents a thermal post-buckling solution for sandwich panels with truss cores under simply supported conditions, when subjected to uniform temperature rise. The Reissner assumptions are adopted and truss cores are assumed to be continuous and homogeneous. Differential governing equations are developed based on the variational principle. The perturbation technique is employed to determine the thermal post-buckling path of sandwich panels with truss cores. Based on the present method, influences of truss core configuration, relative density, aspect ratio, and initial imperfection on the thermal post buckling behavior are discussed.     

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.     

GEOMETRICALLY NONLINEAR ANALYSIS OF TIMOSHENKO BEAMS UNDER THERMOMECHANICAL LOADINGS

Considering the axial extension and the transversal shear deformation, geometrically nonlinear governing equations for static deformations of Timoshenko beams subjected to thermal as well as mechanical loadings are formulated. As an example, on the basis of the governing equations, thermal postbuckling response of an immovably pinned-fixed Timoshenko beam subjected to a static transversely nonuniform temperature rise is numerically analyzed by using a shooting method. Characteristic curves showing the relationships between the beam deformation and temperature rise are presented. The thermal postbuckled configurations and the equilibrium paths of the beam are presented. In particular, the effects of shear deformation on the buckling response are quantitatively investigated. The numerical results show, as we know, that shear deformation effects become significant with decrease of the slenderness and with increase of the shear flexibility.     

Hygrothermal buckling analysis of magnetically actuated embedded higher order functionally graded nanoscale beams considering the neutral surface position

In this article, the effects of humidity and thermal loads on buckling behavior of functionally graded (FG) nanobeams resting on elastic foundation and subjected to a unidirectional magnetic field is investigated. The nanobeam is modeled using different higher order refined beam theories which capture shear deformation influences needless of shear correction factors. The neutral axis position for all proposed beam models is determined. The material properties of FG nanobeam are temperature dependent and change gradually in spatial coordinate through the sigmoid and power-law models. Small-scale behavior of the nanobeam is described applying nonlocal elasticity theory of Eringen. Nonlocal governing equations for an embedded nanosize functionally graded material beam under hygrothermal loads obtained from Hamilton's principle are solved by an analytic method which satisfies various boundary conditions including S–S, C–S, and C–C. The validation of developed refined beam model has been proved with comparison to a previously published work on FG nanobeams. Numerical results are calculated for various beam theories to reveal the influences of moisture and temperature rise, elastic medium, nonlocality, volume fraction index, boundary conditions, and longitudinal magnetic field on the hygrothermal buckling responses of nanoscale P-FGM and S-FGM beams. The present study would be useful in the design of the nanoscale systems as one of the most demanded technologies in the near future.     

Hygrothermal effects on static stability of embedded single-layer graphene sheets based on nonlocal strain gradient elasticity theory

Abstract

This article develops a nonlocal strain gradient plate model for buckling analysis of graphene sheets under hygrothermal environments. For more accurate analysis of graphene sheets, the proposed theory contains two scale parameters related to the nonlocal and strain gradient effects. Graphene sheet is modeled via a two-variable shear deformation plate theory needless of shear correction factors. Governing equations of a nonlocal strain gradient graphene sheet on elastic substrate are derived via Hamilton’s principle. Galerkin’s method is implemented to solve the governing equations for different boundary conditions. Effects of different factors such as moisture concentration rise, temperature rise, nonlocal parameter, length scale parameter, elastic foundation and geometrical parameters on buckling characteristics a graphene sheets are examined.     

Thermal Buckling of Functionally Graded Plates Resting On Elastic Foundations Using the Trigonometric Theory

In this article, thermal buckling analysis of functionally graded material (FGM) plates resting on two-parameter Pasternak's foundations is investigated. Equilibrium and stability equations of FGM plates are derived based on the trigonometric shear deformation plate theory and includes the plate foundation interaction and thermal effects. The material properties vary according to a power law form through the thickness coordinate. The governing equations are solved analytically for a plate with simply supported boundary conditions and subjected to uniform temperature rise and gradient through the thickness. Resulting equations are employed to obtain the closed-form solution for the critical buckling load for each loading case. The influences of the plate aspect ratio, side-to-thickness ratio, gradient index, and elastic foundation stiffnesses on the buckling temperature difference are discussed.     

TEMPERATURE AND STRESS FIELDS IN SILVER DUE TO LASER PICOSECOND HEATING PULSE

Laser short-pulse heating of metallic surfaces is involved with nonequilibrium energy transport in the region irradiated by a laser beam. In this case, the Fourier heating model fails to predict correct temperature rise in this region. Moreover, for completeness of analysis, the thermomechanical coupling needs to be incorporated in the governing equations. In the present study, electron kinetic theory approach is introduced to model the heating process and thermomechanical coupling is formulated and accommodated in the energy transport equation. Temperature and stress fields are computed numerically for silver. It is found that electron temperature well in excess of lattice site temperature occurs in the surface vicinity of the substrate material. Although lattice site temperature rise is low (~170°C), stress levels as high 3 2 10 8 Pa are computed in the region heated by a laser beam. The thermal expansion of the surface at the irradiated spot center reaches 0.5 nm after 4 ns of the heating period.     

Generalized differential quadrature nonlinear buckling analysis of smart SMA/FG laminated beam resting on nonlinear elastic medium under thermal loading

The nonlinear thermal buckling analysis of functionally graded (FG) beam integrated with shape memory alloy (SMA) layer(s), with different lay-up configurations and supported on a nonlinear elastic foundation, has been investigated. The FG layer is graded through the beam thickness direction and thermomechanical properties are assumed to be temperature dependent. The Brinson one-dimensional constitutive law are used to model the characteristics of SMA. The von Kármán strain–displacement fields with the Timoshenko beam theory are applied to the Hamilton’s principle to derive the set of nonlinear equilibrium equations. Generalized differential quadrature method along with direct iterative scheme is utilized to discretize and solve the nonlinear equilibrium equations. The accuracy of proposed model is compared and validated with previous research in literature. The detailed parametric study has been performed to investigate the influence of geometrical, material, and some other key parameters on the nonlinear thermal buckling solutions. The results show that selecting the proper lay-up is of great importance because the type of SMA/FG lay-up can considerably affect the nonlinear buckling solutions. Moreover, adequate application of SMA layers in a proper lay-up configuration significantly postpones the thermal buckling temperature of the beam.     

Thermal buckling and postbuckling responses of geometrically imperfect FG porous beams based on physical neutral plane

Abstract

Considering the third-order shear deformation and physical neutral plane theories, thermal postbuckling analysis for functionally graded (FG) porous beam are performed in this research. The cases of shear deformable functionally graded materials (FGM) beams with initial deflection and uniformly distributed porosity are considered. Geometrically imperfect FG porous beams with two different types of immovable boundary conditions as clamped–rolling and clamped–clamped are analyzed. Thermomechanical nonhomogeneous material properties of the FG porous beam are assumed to be temperature and position dependent. FG porous beams are subjected to different types of thermal loads as heat conduction and uniform temperature rise. Heat conduction equation is solved analytically using the polynomial series solution for the one-dimensional condition. The governing equilibrium equations are obtained by applying the virtual displacement principle. Assuming von Kármán type of geometrical nonlinearity, equilibrium equations are nonlinear and are solved using an analytical method. A two-step perturbation technique is used to obtain the thermal buckling and postbuckling responses of FG porous beams. The numerical results are compared with the case of perfect FGM Timoshenko beams without porosity distribution based on the midplane formulation. Parametric studies of the perfect/imperfect FG porous beams for two types of thermal loading and boundary conditions are provided.     

Effect of Initial Imperfections on Thermal Buckling of Functionally Graded Plates

ABSTRACT

Thermal buckling analysis of rectangular functionally graded plates with initial geometrical imperfections is presented in this article. The equilibrium, stability, and compatibility equations of an imperfect functionally graded plate are derived using the first-order shear deformation plate theory. It is assumed that the nonhomogeneous mechanical properties of the plate, graded through the thickness, are described by a power function of the thickness variable. The plate is assumed to be under three types of thermal loading, namely: uniform temperature rise, nonlinear temperature rise through the thickness, and axial temperature rise. Resulting equations are employed to obtain the closed-form solutions for the critical buckling temperature change of an imperfect functionally graded plate. The influence of transverse shear on thermal buckling load is discussed.     

Effects of nonlinear hygrothermomechanical loading on bending of FGM rectangular plates resting on two-parameter elastic foundation using four-unknown plate theory

In the present study, a simple four-unknown exponential shear deformation theory is developed for the bending of functionally graded material (FGM) rectangular plates resting on two-parameter elastic foundation and subjected to nonlinear hygrothermomechanical loading. The elastic properties, coefficient of thermal expansion, and coefficient of moisture expansion of the plate are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of volume fractions of material constituents. Unlike first-order and other higher-order plate theories, the present theory has four independent unknowns. The in-plane displacement field of the present theory uses exponential functions in terms of thickness co-ordinate for calculating out-of-plane shearing strains. The transverse displacement includes bending and shear components. The principle of virtual displacement is employed to derive the governing equations and associated boundary conditions. A Navier solution technique is employed to obtained an analytical solutions. The elastic foundation is modelled as two-parameter Winkler–Pasternak foundation. The numerical results obtained are compared with previously published results wherever possible to prove the efficacy and accuracy of the present theory. The effects of stiffness and gradient index of the foundation on the hygrothermomechanical responses of the plates are discussed.     

Thermal Buckling of Functionally Graded Plates According to a Four-Variable Refined Plate Theory

In this article, a four-variable refined plate theory is presented for buckling analysis of functionally graded plates. The theory, which has strong similarity with classical plate theory in many aspects, accounts for a quadratic variation of the transverse shear strains across the thickness and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. A power law distribution is used to describe the variation of volume fraction of material compositions. Equilibrium and stability equations are derived based on the present theory. The non-linear governing equations are solved for plates subjected to simply supported boundary conditions. The thermal loads are assumed to be uniform, linear and non-linear distribution through-the-thickness. The influences of many plate parameters on buckling temperature difference will be investigated. It is noticed that the present refined plate theory can predict accurately the critical temperatures of simply supported functionally graded plates.     

Exact Solution for Thermal Buckling of Functionally Graded Plates Resting on Elastic Foundations with Various Boundary Conditions

In this article, we investigate the buckling analysis of plates that are made of functionally graded materials (FGMs) resting on two-parameter Pasternak's foundations under thermal loads. Three different thermal loads were considered, i.e., uniform temperature rise (UTR), linear and non-linear temperature distributions (LTD and NTD) through the thickness. The mechanical and thermal properties of functionally graded material (FGM) vary continuously along the plate thickness according to a simple power law distribution. Employing an analytical approach, the five coupled governing stability equations, which are derived based on first-order shear deformation plate theory, are converted into two uncoupled partial differential equations (PDEs). Considering the Levy-type solution, these two PDEs are reduced to two ordinary differential equations (ODEs) with variable coefficients. Then, the ODEs are solved using an exact analytical solution, which is called the power series Frobenius method. The appropriate convergence study and comparison with previously published related articles was employed to verify the accuracy of the proposed method. After such verifications, the effects of parameters such as the plate aspect ratio, side-to-thickness ratio, gradient index, and elastic foundation stiffnesses on the critical buckling temperature difference are illustrated and explained. The critical buckling temperatures of functionally graded rectangular plates with six various boundary conditions are reported for the first time and can serve as benchmark results for researchers to validate their numerical and analytical methods in the future.     

Analytical Solution for Thermomechanical Vibration of Double-Viscoelastic Nanoplate-Systems Made of Functionally Graded Materials

In this article, based on the nonlocal elasticity theory of Eringen, dynamic characteristics of a double-FGM viscoelastic nanoplates-system subjected to temperature change with considering surface effects (surface elasticity, tension and density) is studied. Two Kirchhoff nanoplates are coupled by an internal Kelvin–Voigt viscoelastic medium and also are limited to the external Pasternak elastic foundation. The material properties of the simply supported functionally graded nanoplates are assumed to follow power law distribution in the thickness direction. The governing equations of motion for three cases (out-of-phase vibration, in-phase vibration and one nanoplate fixed) are derived from Hamilton's principle. The analytical approach is employed to determine explicit closed-form expression for complex natural frequencies of the system. Numerical results are presented to show variations of the frequency of double-FGM viscoelastic nanoplates corresponding to various values of the nonlocal parameter, temperature change, power law index, aspect ratio and transverse and shear stiffness coefficients of the Pasternak elastic foundation. Moreover, influence of higher order modes, viscoelastic structural damping and damping coefficient of the viscoelastic medium on vibration characteristics are investigated. Numerical results show that natural frequency is greatly influenced by surface elastic modulus and residual surface stress.     

Vibration Control of Composite Spacecraft Booms Subjected to Solar Heating

In this paper, piezoelectric feedback control of vibration and instability of spacecraft booms modeled as circular thin-walled cross-section beams and subjected to solar radiant heating is investigated. Having in view that composite material systems are likely to play a great role in the design of these devices, the beam constituent materials encompass non-classical effects such as anisotropy and transverse shear. In addition, in order to induce beneficial elastic couplings, a special ply-angle distribution achieved via the usual helically wounding fiber-reinforced technology, the so called filament winding, is implemented. The dynamic governing equations including the temperature effects and the related boundary conditions are obtained via the application of Hamilton's principle. Toward the end of controlling the oscillations and prevent the occurrence of the thermal dynamic instability, a feedback control capability based on the use of the piezoelectric induced strain actuation is implemented. The performance of its implementation considered in conjunction with that of the structural tailoring are highlighted and pertinent conclusions are derived.     

Asymmetric thermal buckling of annular plates on a partial elastic foundation

In this investigation, the asymmetrical buckling behavior of isotropic homogeneous annular plates resting on a partial Winkler-type elastic foundation under uniform temperature elevation is investigated. First-order shear deformation plate theory is used to obtain the governing equations and the associated boundary conditions. Prebuckling deformations and stresses of the plate are obtained under the solution of a plane stress formulation, neglecting the rotations and lateral deflection. Applying the adjacent equilibrium criterion, the linearized stability equations are obtained. The governing equations are divided into two sets. The first set, which is associated with the in-contact region, and the second set, which is related to contact-less region. The resulting equations are solved using a hybrid method, including the analytical trigonometric functions through the circumferential direction and generalized differential quadratures method through the radial direction. The resulting system of eigenvalue problem is solved to obtain the critical conditions of the plate and the associated circumferential mode number. Benchmark results are given in tabular and graphical presentations for combinations of simply supported and clamped types of boundary conditions. Numerical results are given to explore the effects of elastic foundation, foundation radius, plate thickness, plate hole size, and the boundary conditions.     

Thermomechanical postbuckling behavior of CNT-reinforced composite sandwich plate models resting on elastic foundations with elastically restrained unloaded edges

Buckling and postbuckling behaviors of two models of sandwich plate reinforced by carbon nanotubes (CNTs) resting on elastic foundations and subjected to uniaxial compressive and thermomechanical loads are investigated in this paper. Material properties of all constituents are assumed to be temperature dependent and effective properties of CNT-reinforced composite layer are determined according to extended rule of mixture. Governing equations are established within the framework of first-order shear deformation theory taking into account von Kármán nonlinearity, initial geometrical imperfection, plate-foundation interaction and tangential elastic constraints of unloaded edges. Three types of loading are considered including uniaxial compression, preexisting thermal load combined with uniaxial compression and preexisting mechanical load combined with thermal load. Approximate analytical solutions are assumed to satisfy simply supported boundary conditions and the Galerkin method is used to derive nonlinear load-deflection relations from which buckling loads and postbuckling equilibrium paths are determined. The most important findings are that tangential constraints of unloaded edges significantly lowers buckling loads and postbuckling load capacity of sandwich plates and, in contrast, buckling loads and postbuckling strength are considerably enhanced as sandwich plate is constructed from CNT-reinforced composite core layer and homogeneous face sheets.