Size-Dependent Thermal Buckling and Postbuckling of Functionally Graded Annular Microplates Based on the Modified Strain Gradient Theory

This article reports on the thermal instability of functionally graded (FG) annular microplates with different boundary conditions. The modified strain gradient elasticity theory is employed to capture size effects. The non-linear governing equations and boundary conditions are derived based on the first-order shear deformation theory (FSDT) and virtual displacements principle. The generalized differential quadrature technique is implemented so as to discretize. To obtain the critical buckling temperature, the set of linear discretized governing equations is solved as an eigenvalue problem. Also, the non-linear problem of thermal postbuckling is solved by the pseudo arc-length continuation method. The effects of boundary conditions, length scale parameter, and the variation of material through the thickness and geometrical properties on both critical buckling temperature and thermal postbuckling behavior are studied.     

Geometrical nonlinear free vibration analysis of FGM spherical panel under nonlinear thermal loading with TD and TID properties

In this article, the nonlinear free vibration behavior of functionally graded (FG) spherical shell panel is examined under nonlinear temperature field. The functionally graded material (FGM) constituents are assumed to be the function of temperature and the thermal conductivity. The effective \hboxmaterial properties of the FGM are obtained using the Voigt micromechanical model through power-law distribution. The mathematical model of the shell panel is developed using Green–Lagrange nonlinear kinematics in the framework of the higher order shear deformation theory. The desired governing \hboxequation of the FG shell panel under thermal environment is obtained using the classical Hamilton's principle. The domain is discretized with the help of the \hboxisoparametric finite element steps and the responses are computed using the direct \hboxiterative method. The convergence behavior of the present nonlinear numerical model has been checked and compared with the previous reported results. Numerous examples have been demonstrated for the FG spherical panel to show the influence of different geometrical and material parameters and support conditions on the linear and nonlinear frequency parameters.     

Research on shakedown analysis for functionally graded material plate under thermomechanical loading

Shakedown is an important problem in the design and analysis of functionally graded structures subjected to cyclic, thermal, and mechanical loadings. Subjected to constant mechanical load and cyclic temperature change, static shakedown of a functionally graded material plate and its homogenous counterpart were analyzed in this article with the approach proposed by the authors previously. The functionally graded material plate is composed of an elastoplastic matrix Al and elastic particles SiC, and the particle volume fraction varies through the thickness. The distributions of the effective mechanical and thermal properties of the composites through the thickness are described graded continuously with an exponential law. The results show that a proper and continuously graded distribution of material properties can efficiently improve the shakedown capability of the functionally graded material plate and also show the significance of shakedown analysis and application for functionally graded material plates.     

Two-dimensional thermoelastic analysis of FG cylindrical shell resting on the Pasternak foundation subjected to mechanical and thermal loads based on FSDT formulation

Two-dimensional thermoelastic analysis of functionally graded thick-walled cylinder is investigated under thermal and mechanical loadings and based on the Pasternak foundation. The first-order shear deformation theory is used to describe the displacement field. Fundamental governing differential equations of system are obtained by energy method and Euler equations. Effect of gradation of material properties and Pasternak foundation parameters are considered as important results of this study. The obtained results indicate that the end supports have considerable effect on the longitudinal distribution of components. Furthermore, it can be concluded that with increasing the non-homogeneous index, both radial and axial displacements decreases.     

Nonlinear thermoelastic frequency analysis of functionally graded CNT-reinforced single/doubly curved shallow shell panels by FEM

In this article, the nonlinear vibration frequencies of functionally graded carbon nanotube-reinforced composite doubly curved shell panels under elevated thermal environment are numerically investigated using finite element method. The doubly curved carbon nanotube-reinforced shell panel has been modeled mathematically using higher-order kinematics theory and Green–Lagrange geometrical nonlinear strains. The properties of the individual constituents of the graded composite are assumed to be temperature dependent. In addition, the properties of the media are obtained based on the modified rule of mixture. The carbon nanotubes are dispersed nonuniformly through the thickness direction. The large deformation kinematic effects on the structural responses are counted by including all the nonlinear higher-order terms in the formulation. The desired nonlinear responses are computed numerically using our in-house computer code in conjunction with the direct iterative scheme. The convergence and the accuracy of the present numerical model have been checked by solving various numerical examples. Nonlinear mechanical responses were affected by several other design parameters and explored numerically for the thickness ratios, volume fractions, temperature loading, type of geometries, and type of grading under the uniform thermal environment.     

Exact thermoelastic analysis of a thick cylindrical functionally graded material shell under unsteady heating using first order shear deformation theory

In this article, a new analytical formulation is presented for an axisymmetric thick‐walled functionally graded material cylinder with power‐law variation in mechanical and thermal properties under transient heating using first order shear deformation theory. Equilibrium equations are derived by virtual work principles and the energy method. The unsteady heat conduction equation is solved using the method of separation of variables, generalized Bessel functions, and an Eigen‐function method. Validation of the analytical solutions is conducted with a finite element method. The effects of time on stress and displacement distribution are studied in detail. The numerical values used in this study are selected based on earlier studies. The influence of effect of transient heat transfer on heterogeneous thick‐walled cylinder elasticity is clearly demonstrated. In particular, the significant influence of time and the heterogenous constant on radial displacement, hoop stress, and temperature distributions is computed. The study is relevant to rocket chamber thermomechanics, propulsion duct thermophysical design, industrial thermal storage systems, and so forth.     

Thermoelastic vibration analysis of functionally graded skew plate using nonlinear finite element method

Numerical investigation of nonlinear free vibration of functionally graded skew (FGS) plate in the thermal environment is presented. The mathematical model is proposed for the first time based on higher order shear deformation theory in conjunction with Green–Lagrange-type geometric nonlinearity for the FGS plate subjected to a thermal load. The material properties are considered to be temperature dependent and are graded along the thickness direction as per simple power law of distribution in terms of volume fraction of the constituent phase. The governing algebraic equations are derived using Hamilton’s principle, and the solutions are obtained using the direct iterative method. The proposed finite element model has discretized into an eight-noded quadratic serendipity elements. To validate the model, the obtained results are compared with the available literature. The influence of volume fraction index, skew angle, temperature change, aspect ratio, side–thickness ratio, and boundary conditions on the linear and nonlinear frequency of skew functionally graded material plate is examined and discussed in detail.     

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.     

Transient thermal stress analysis of a functionally graded thick hollow cylinder with temperature-dependent material properties

This article deals with the study of temperature distribution and thermal stresses of a functionally graded thick hollow cylinder with temperature dependent material properties. All the material properties except Poisson’s ratio are assumed to be dependent on temperature and spatial coordinate z. The two-dimensional transient heat conduction equation is solved under convective heat transfer condition with varying point heat source. The influence of inhomogeneity parameters on the thermal and mechanical behavior is examined. Numerical computations are performed for ceramic-metal-based functionally graded material, in which alumina is selected as ceramic and nickel as metal.     

Elastoplastic analysis of FGM plate with a central cutout of various shapes under thermomechanical loading

The present work aims to study the elastoplastic buckling, postbuckling, and failure behavior of perforated Ni/Al₂O₃ functionally graded material (FGM) plate with various shaped cutouts (i.e., circular, square, diamond, and elliptical) of various sizes under thermomechanical loading conditions using finite element method (FEM). The nonlinear FEM formulation is based on the first-order shear deformation theory and von Kármán’s nonlinear kinematics in which the material nonlinearity is incorporated. The nonlinear temperature-dependent thermoelastic material properties of FGM plate are varied in the thickness direction by controlling the volume fraction of the constituent materials (i.e., ceramic and metal) as per a power law, and Mori–Tanaka homogenization scheme is applied to evaluate the properties at a particular thickness coordinate of FGM. In accordance with the Tamura–Tomota–Ozawa model (TTO model), the ceramic phase of FGM is considered to be elastic, whereas the metal phase is assumed to be elastoplastic. Further, the elastoplastic analysis of FGM is assumed to follow J₂ plasticity with isotropic hardening. After validating the present formulation with the results available in the literature, various numerical studies are conducted to examine the effects of material inhomogeneity, thermal loading, cutout shape, and size on the elastoplastic buckling, postbuckling, and failure behavior of perforated FGM plate. It is observed that for smaller cutout sizes, the FGM plate with square shape cutout possesses maximum value of ultimate failure load; however, for larger cutout size, the FGM plate with diamond cutout depicts highest ultimate failure load. Furthermore, for all cutout shapes, the ultimate failure load of FGM plate decreases with an increase in cutout size. It is also revealed that irrespective of shape and size of cutout, the material plastic flow has considerable effect on postbuckling path of FGM plate, and under thermomechanical loading conditions, the FGM plate shows destabilizing response after the point of maximum postbuckling strength.     

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.