Nonlinear Temperature Rise Research Articles

Impact of the homogenization models on the thermoelastic response of FG plates on variable elastic foundation

This paper presents a theoretical investigation on the response of the thermo-mechanical bending of FG plate on variable elastic foundation. A quasi-3D higher shear deformation theory is used that contains undetermined integral forms and involves only four unknowns to derive. The FG plates are supposed simply supported with temperature-dependent material properties and subjected to nonlinear temperature rise. Various homogenization models are used to estimate the effective material properties such as temperature-dependent thermoelastic properties. Equations of motion are derived from the principle of virtual displacements and Navier\'s solution is used to solve the problem of simply supported plates. Numerical results for deflections and stresses of FG plate with temperature-dependent material properties are investigated. It can be concluded that the proposed theory is accurate and simple in solving the thermoelastic bending behavior of FG thick plates.

On radially symmetric vibrations of functionally graded non-uniform circular plate including non-linear temperature rise

In this study, the free axisymmetric vibrations of functionally graded circular plate of linearly varying thickness controlled by a taper parameter in radial direction and non-linear temperature rise along the thickness have been investigated on the basis of classical plate theory. The plate material is graded in transverse direction and its mechanical properties are temperature-dependent. The temperature over the top and bottom surfaces is assumed to be uniformly distributed. Hamilton's principle has been used to derive the governing equations for thermo-elastic equilibrium and axisymmetric motion of such a plate model. The generalized differential quadrature method has been employed to obtain the thermal displacements and characteristic equations, for clamped and simply supported plates. The lowest three roots of these equations have been computed and reported as the values of frequency parameter for the first three modes of vibration. Effect of thickness parameter, volume fraction index and temperature difference has been analyzed on the vibration characteristics of the plate. A study with temperature-independent material properties has also been performed. Results have been compared with the published work.

Isogeometric analysis for size-dependent nonlinear thermal stability of porous FG microplates

In this article, we present for the first time a research analysis for the size-dependent effects on thermal buckling and post-buckling behaviors of functionally graded material micro-plates with porosities (imperfect FGM) using isogeometric analysis. A seventh-order shear deformation plate theory associated with the modified couple stress theory (MCST) is particularly imposed to capture the size-dependent phenomenon within imperfect FGM micro-plates. The material properties of imperfect FGM micro-plates with three different distributions of porosities including even, uneven and logarithmic-uneven varying across the plate thickness are derived from the modified rule-of-mixture assumption. The nonlinear governing equation for size-dependent imperfect FGM micro-plate under uniform, linear and nonlinear temperature rise is derived using the Von-Kármán assumption and Hamilton’s principle. Through numerical example, the effect of temperature rise, boundary conditions, power index, porosity volume fraction, porosity distribution pattern and material length scale parameter on thermal buckling and post-buckling behaviors of FGP micro-plates are investigated.

Identifying the temperature effect on the vibrations of functionally graded cylindrical shells with porosities

The free thermal vibration of functionally graded material (FGM) cylindrical shells containing porosities is investigated. Both even distribution and uneven distribution are taken into account. In addition, three thermal load types, i.e., uniform temperature rise (UTR), nonlinear temperature rise (NLTR), and linear temperature rise (LTR), are researched to explore their effects on the vibration characteristics of porous FGM cylindrical shells. A modified power-law formulation is used to describe the material properties of FGM shells in the thickness direction. Love’s shell theory is used to formulate the strain-displacement equations, and the Rayleigh-Ritz method is utilized to calculate the natural frequencies of the system. The results show that the natural frequencies are affected by the porosity volume fraction, constituent volume fraction, and thermal load. Moreover, the natural frequencies obtained from the LTR have insignificant differences compared with those from the NLTR. Due to the calculation complexity of the NLTR, we propose that it is reasonable to replace it by its linear counterpart for the analysis of thin porous FGM cylindrical shells. The present results are verified in comparison with the published ones in the literature.

Nonlinear thermal vibration analysis of refined shear deformable FG nanoplates: two semi-analytical solutions

This paper deals with the semi-analytical nonlinear thermal vibration analysis of functionally graded (FG) nanoplates modeled by four-variable refined plate theory. The nanoplate is resting on a nonlinear hardening elastic medium and subjected to uniform and nonlinear temperature rises. Temperature-dependent gradient material properties of the nanoplate are described via simple power-law model. A closed-form expression for nonlinear frequency is presented using a novel Hamiltonian approach as well as He’s variational method. To the best of the author’s knowledge, the nonlinear governing equations and their solution for a refined four-variable nanoplate are very rare in the literature. It is seen that the nonlinear vibration frequency is affected by the uniform temperature rise, nonlinear temperature rise, scale parameter, maximum amplitude, material gradation, foundation parameters and temperature-dependency. The presented formulation and solution approaches can be used in future investigations on macro to nanoscale plates.

Thermal buckling analysis of functionally graded sandwich plates

ABSTRACTThermal buckling of functionally graded sandwich plates are presented in this article. Two common types of FGM sandwich plates, namely, homogeneous face layers with FGM core and FGM face layers with homogeneous core are considered. Material properties and thermal expansion coefficient of FGM layers are assumed to vary continuously through-the-thickness according to a simple power-law distribution in terms of the volume fractions of the constituents. Equilibrium and stability equations of FGM sandwich plate with simply supported boundary conditions are derived using the higher-order shear deformation plate theory. The influence of the plate aspect ratio, the relative thickness, the gradient index, and the thermal loading conditions on the critical buckling temperature of FGM sandwich plates are investigated. The thermal loads are assumed to be uniform, linear, and nonlinear distribution through-the-thickness. A new simple solution for thermal buckling of FGM sandwich plates under nonlinear temperature rise is presented.

Thermal pre- and post-snap-through buckling of a geometrically imperfect doubly-clamped microbeam made of temperature-dependent functionally graded materials

Owing to inappropriate fabrication processing it is likely to fabricate a geometrically imperfect functionally graded (FG) microbeam. In addition, the experimental tests show that the classical continuum theory is incapable of interpreting the mechanical behavior of microstructures when the role of size-dependency is significant. Therefore, this investigation aims to examine the influences of geometric imperfection on the nonlinear stability behavior as a prominent characteristic of microstructures for a thermally loaded doubly-clamped microbeam made of temperature-dependent FGMs by taking into account the size effect phenomenon. The geometrically nonlinear size-dependent governing equation of system is derived in the framework of modified couple stress theory in conjunction with Euler-Bernoulli beam theory and the classical rule of mixture. Based on the static form of governing equation, a closed-form solution for the nonlinear critical snap-through buckling temperature rise as well as the nonlinear thermal stability behavior of system in pre- and post-snap-through buckling domains is proposed and analytical study is then carried out by consideration of different effective parameters i.e., dimensionless imperfection amplitude, size-dependency, temperature-dependency, and power index. To verify the closed-form solution, the dynamic response of system is numerically evaluated by implementation of Galerkin scheme in conjunction with Runge-Kutta finite difference method.

Nonlinear Finite Element Solution of Post-buckling Responses of FGM Panel Structure under Elevated Thermal Load and TD and TID Properties

The nonlinear finite element solutions for the buckling and post-buckling responses of the functionally graded shell panel subjected to the non-uniform thermal environment have been presented in this article. The thermal fields are assumed as uniform, linear and nonlinear temperature rise across the thickness of shell panel and the properties of each constituent are considered to be temperature dependent. The effective material properties of the graded structure are evaluated using the Voigt’s micromechanical rule in conjunction with power-law distribution. For the analysis purpose, a general nonlinear mathematical model of the functionally graded shell panel has been developed based on the higher order shear deformation theory and Green-Lagrange type geometrical nonlinear strains. The system governing equation of the panel structure is derived using the variational principle. Further, suitable nonlinear finite element steps have been adopted to discretize the model for the computation of the desired responses in association with the direct iterative method. The convergence and the validation behavior of the present numerical model are initially tested to demonstrate its efficacy and significance. Finally, the effects of curvature, power law index and different support conditions on the buckling and post-buckling responses of the functionally graded shell panels are investigated and discussed in details.

An analytical solution for thermal vibration of compositionally graded nanoplates with arbitrary boundary conditions based on physical neutral surface position

ABSTRACTIn this article, an analytical method is presented for thermo-mechanical vibration analysis of functionally graded (FG) nanoplates with different boundary conditions under various thermal loadings including uniform, linear, and nonlinear temperature rise via a four-variable plate theory considering neutral surface position. The temperature-dependent material properties of FG nanoplate vary gradually along the thickness according to the Mori-Tanaka homogenization scheme. The exactness of solution is confirmed by comparing obtained results with those provided in the literature. A parametric study is performed investigating the effects of nonlocal parameter, temperature fields, gradient index, and boundary conditions on vibration behavior of FG nanoplates.

Dynamic Behavior of Functionally Graded Beams in Thermal Environment due to a Moving Harmonic Load

This paper investigates the dynamic behavior of functionally graded beam in the thermal environment due to a moving harmonic load. The material properties are assumed to be graded in the thickness direction by a power-law function, and they are considered to be temperature dependent. Two types of temperature distribution, namely uniform and nonlinear temperature rises, are considered. Equations of motion based on Euler-Bernoulli beam theoryare derived from Hamilton’s principle and they are solved by a simple finite element formulation in combination with Newmark time-integration procedure. Numerical results show that the dynamic deflection and dynamic amplification factor is decreased with increasing the temperature rise, and the decreasein the uniform temperature rise is more significant that by the nonlinear temperature rise. The excitation frequency plays an important role in the dynamic behavior of the beams, and the frequency at which resonant phenomenon can occur depends on the temperature. A parametric study is carried out to highlight the effect of the temperature rise and moving load parameters on the dynamic behavior of the beams.

Thermoelastic buckling of FGM conical shells under non-linear temperature rise in the framework of the shear deformation theory

Abstract In this study, the thermoelastic buckling of functionally graded material (FGM) conical shells under non-linear temperature rise across the thickness in the framework of the shear deformation theory (SDT) is investigated. To the derivation of basic equations is used modified Donnell-type shell theory. The Galerkin method is used to obtain the formula for non-linear buckling temperature difference of freely supported FGM truncated conical shell in the framework of the SDT. By changing the properties of FGMs and volume fraction index, the effect of transverse shear deformations on the non-linear buckling temperature difference is evaluated by comparing with the results of the classical shell theory (CST). Meanwhile, the effect of geometric parameters on the non-linear buckling temperature difference of the shear deformable FGM conical shells is discussed in detail.

Thermal buckling analysis of nanoplates based on nonlocal elasticity theory with four-unknown shear deformation theory resting on Winkler–Pasternak elastic foundation

ABSTRACTThe thermal buckling analysis of nanoplates is based on nonlocal elasticity theory with four-unknown shear deformation theory resting on Winkler–Pasternak elastic foundation. The nanoplate is assumed to be under three types of thermal loadings, namely uniform temperature rise, linear temperature rise, and nonlinear temperature rise through the thickness. The theory involves four unknown variables with small-scale effects, as against five in the case of other higher-order theories and first-order shear deformation theory. Closed-form solution for theory was also presented. Results are presented to discuss the influences of the nonlocal parameter, aspect ratio, side-to-thickness ratio, and elastic foundation parameters on the thermal buckling characteristics of analytical rectangular nanoplates.

Vibration analysis of nonlocal beams made of functionally graded material in thermal environment

In this paper, thermal vibration behavior of functionally graded (FG) nanobeams exposed to various kinds of thermo-mechanical loading including uniform, linear and non-linear temperature rise embedded in a two-parameter elastic foundation are investigated based on third-order shear deformation beam theory which considers the influence of shear deformation without the need to shear correction factors. Material properties of FG nanobeam are supposed to be temperature-dependent and vary gradually along the thickness according to the Mori-Tanaka homogenization scheme. The influence of small scale is captured based on nonlocal elasticity theory of Eringen. The nonlocal equations of motion are derived through Hamilton’s principle and they are solved applying analytical solution. The comparison of the obtained results is conducted with those of nonlocal Euler-Bernoulli beam theory and it is demonstrated that the proposed modeling predicts correctly the vibration responses of FG nanobeams. The influences of some parameters including gradient index, nonlocal parameter, mode number, foundation parameters and thermal loading on the thermo-mechanical vibration characteristics of the FG nanobeams are presented.

Nonlocal thermo-mechanical vibration analysis of functionally graded nanobeams in thermal environment

In this paper, the thermal effect on free vibration characteristics of functionally graded (FG) size-dependent nanobeams subjected to various types of thermal loading is investigated by presenting a Navier type solution and employing a semi analytical differential transform method (DTM) for the first time. Two kinds of thermal loading, namely, linear temperature rise and nonlinear temperature rise are studied. Material properties of FG nanobeam are supposed to vary continuously along the thickness according to the power-law form and the material properties are assumed to be temperature-dependent. The small scale effect is taken into consideration based on nonlocal elasticity theory of Eringen. The nonlocal equations of motion are derived through Hamilton׳s principle and they are solved applying DTM. According to the numerical results, it is revealed that the proposed modeling and semi analytical approach can provide accurate frequency results of the FG nanobeams as compared to analytical results and also some cases in the literature. The detailed mathematical derivations are presented and numerical investigations are performed while the emphasis is placed on investigating the effect of the several parameters such as thermal effect, material distribution profile, small scale effects, mode number and boundary conditions on the normalized natural frequencies of the temperature-dependent FG nanobeams in detail. It is explicitly shown that the vibration behaviour of a FG nanobeams is significantly influenced by these effects. Numerical results are presented to serve as benchmarks for future analyses of FG nanobeams.

Suppression of nonlinear phonon relaxation in Yb:YAG thin disk via zero phonon line pumping.

A quantitative comparison of conventional absorption line (940 nm) pumping and zero phonon line (ZPL) (969 nm) pumping of a Yb:YAG thin disk laser is reported. Characteristics of an output beam profile, surface temperature, and deformation of a thin disk under the different pump wavelengths are evaluated. We found that a nonlinear phonon relaxation (NPR) of the excited state in Yb:YAG, which induces nonlinear temperature rise and large aspheric deformation, did not appear in the case of a ZPL pumped Yb:YAG thin disk. This means that the advantage of ZPL pumping is not only the reduction of quantum defect but also the suppression of NPR. The latter effect is more important for high power lasers.

Nonlocal elasticity theory for thermal buckling of nanoplates lying on Winkler–Pasternak elastic substrate medium

In the present work, thermal buckling of single-layered graphene sheets lying on an elastic medium is analyzed. For this purpose, the sinusoidal shear deformation plate theory in tandem with the nonlocal continuum theory, which takes the small scale effects into account, is employed. The non-linear stability equations, which contain the reaction of Winkler–Pasternak elastic substrate medium, are derived and then solved analytically for a plate with various boundary conditions and based on various plate theories. Closed form solutions are formulated for three types of thermal loadings as uniform, linear and nonlinear temperature rise through the thickness of the plate. A number of examples are presented to illustrate the numerical results concerned with the buckling temperature response of nanoplates resting on two-parameter elastic foundations. The influences played by transversal shear deformation, plate aspect ratio, side-to-thickness ratio, nonlocal parameter, and elastic foundation parameters are all investigated.

Thermal buckling of functionally graded skew and trapezoidal plates with different boundary conditions using the element-free Galerkin method

Abstract In this paper, thermal buckling of functionally graded skew and trapezoidal plates is investigated using the element-free Galerkin method. The shape functions are constructed using the moving least squares (MLS) approximation and the essential boundary conditions are introduced into the formulation through the use of the Lagrange multiplier method and the orthogonal transformation techniques. The material properties are assumed to vary as a power form of the thickness coordinate. Uniform, Linear and nonlinear temperature rise across the thickness are considered. The effects of aspect ratio, thickness ratios, gradient index and skew angle on the critical buckling temperature difference are examined.

Thermal Buckling of Imperfect Deep Functionally Graded Spherical Shell

Thermal buckling analysis of deep imperfect functionally graded (FGM) spherical shell is considered in this paper. A mixture of ceramic and metal is considered for the FGM shell and the material properties, such as the modulus of elasticity and coefficient of thermal expansion, vary by a power law function through the thickness. Employing the Sanders non-linear kinematic relations, total potential energy function is derived and the equilibrium and stability equations are obtained for the imperfect shell. Approximate solutions satisfying the simply supported boundary condition are assumed and using the Galerkin method the error due to the approximation is minimized. The geometrically imperfect shell is considered and three types of thermal loadings, such as the uniform temperature rise (UTR), linear temperature rise through the thickness (LTR), and non-linear temperature rise through the thickness (NLTR) are considered and their associated buckling temperatures are obtained. The effects of different temperature functions and the magnitude of initial geometric imperfection are examined on the thermal buckling loads of the shell.

Thermoelastic Stability Analysis of Imperfect Functionally Graded Shallow Spherical Shells

Thermal buckling analysis of functionally graded spherical shells with geometrical imperfection is presented in this article. It is assumed that the material properties of functionally graded spherical shell are changed by a power law function through the thickness of shell. The equilibrium and stability equations of an imperfect functionally graded spherical shell are derived using the Donnell-Mushtari-Velasov theory (DMV). Approximate solutions considered for displacement components are based on the assumption that they satisfy the boundary conditions. The Galerkin method is used to minimize the errors caused by using the approximation. Three various types of thermal loadings including the uniform temperature rise (UTR), linear radial temperature rise (LTR), and the nonlinear radial temperature rise (NTR) are assumed for analytical solution. The results are validated with the known data in literature.

Thermal Free Vibration Analysis of Temperature-Dependent Functionally Graded Plates Using Second Order Shear Deformation

This research has been conducted to approach second-order shear deformation theory (SSDT) to analysis vibration characteristics of Functionally Graded Plates (FGP’s). Material properties in FGP's were assumed to be temperature dependent and graded along the thickness using a simple power law distribution in term of the volume fractions of the constituents. FGP was subjected to a linear and nonlinear temperature rise. The energy method was chosen to derive the equilibrium equations. The solution was based on the Fourier series that satisfy the simply supported boundary condition (Navier's method). Numerical results indicated the effect of material composition, plate geometry, and temperature fields on the vibration characteristics and mode shapes. The results revealed that, the temperature field and volume fraction distribution had significant effect on the vibration of FGPs. It was observed the second order theory was very close to the other shear deformation theorem as reported in the literature.