Temperature Distribution In Thickness Direction Research Articles

Nonlinear axisymmetric response of temperature-dependent FGM conical shells under rapid heating

Considering the uncoupled thermoelasticity assumptions, a thermally induced vibration analysis for functionally graded material (FGM) conical shells is performed in this research. Thermo-mechanical properties of the conical shell are assumed to be temperature and position dependent. The conical shell is under rapid heating with various cases of thermal loads on the outer surface, whereas the opposite surface is kept at reference temperature or thermally insulated. Since the ratio of thickness to radius is much smaller than one, the transient heat conduction equation, for simplicity, may be established and solved for one-dimensional condition. Assuming temperature-dependent material properties, the heat conduction equation is nonlinear and should be solved using a numerical method. A hybrid generalized differential quadrature (GDQ) and Crank–Nicolson method is used to obtain the temperature distribution in thickness direction, respectively. Based on the first-order shear deformation theory and geometrically nonlinear assumptions, the equations of motion are obtained applying the Hamilton principle. Discretization of the equations of motion in the space domain and boundary conditions is performed by applying the GDQ method, and then, the system of highly nonlinear coupled ordinary differential equations is solved by the iterative Newmark time-marching scheme and well-known Newton–Raphson method. Since the thermally induced vibration of the conical shells is not reported in the literature, the results are compared with the case of a circular plate. Also, studies of the FGM conical shells for various types of boundary conditions, functionally graded patterns, and thermal loads are provided. The effects of temperature dependency, geometrical nonlinearity, semi-vertex angle, shell length, and shell thickness upon the deflections of the conical shells are investigated.

Free vibration of functionally graded sandwich shallow shells in thermal environments by a differential quadrature hierarchical finite element method

This paper presents a differential quadrature hierarchical finite element method (DQHFEM) for dynamic analyses of functionally graded material (FGM) sandwich shallow shells in thermal and non-thermal environments. A layer-wise theory based on the first-order shear deformation theory (FSDT) for each layer was adopted. Effective material properties of the FGM are estimated according to Voigt’s rule of mixture (ROM) and/or Mori–Tanaka (MT) scheme. For the shells in thermal environment, a nonlinear temperature distribution in thickness direction is considered and the elastic properties are assumed to be temperature dependent. The results obtained from the proposed formulation are validated with those available in literatures. Natural frequencies obtained from Sander’s, Love’s and Donnell’s shell theories for different geometric and boundary conditions are compared with each other first to assess the performance of different shell theories for FGM sandwich shells under non-thermal environment. Then the effects of volume fraction index, core thickness and temperature gradient on natural frequencies of FGM sandwich shells are investigated. The presented DQHFEM is much like the fixed interface mode synthesis method but does not need modal analysis and is of high accuracy. This work is the first application of the method to functionally graded sandwich shells in thermal environments.

Thermal effect on radially symmetric vibrations of temperature-dependent FGM circular plates with nonlinear thickness variation

Thermo-elastic analysis for the free axisymmetric vibration of functionally graded circular plates with quadratic thickness variation along radial direction has been presented on the basis of classical theory of plates. Following a power law model, the plate is assumed to be graded in thickness direction and mechanical properties of the material are temperature-dependent. The plate is subjected to a non-linear temperature distribution in thickness direction while the thermal environment over the top and bottom surfaces is uniform. The equations for thermo-elastic equilibrium and axisymmetric motion for such a plate model have been derived using Hamilton’s principle. Employing differential quadrature method, the numerical values of thermal displacements and frequencies for clamped and simply supported plates have been computed. The influence of thickness parameters, material graded index and temperature difference on the vibration characteristics for the first three modes of vibration has been analyzed. The benchmark results for uniform and linear temperature distributions have been computed. A study with the plate material having temperature-independent mechanical properties has also been performed. Because, functionally graded materials inherently withstand high temperature gradients due to low thermal conductivity, core ductility, low thermal expansion coefficient, etc, the performance of results can help for optimal designing of non-uniform circular plates by controlling one or both the taper parameters. In special cases, the results have been compared with those obtained by other methods.

Nonlinear Static and Dynamic Thermal Buckling Analysis of Spiral Stiffened Functionally Graded Cylindrical Shells with Elastic Foundation

In this paper, an analytical method is used to study the nonlinear static and dynamic thermal buckling analysis of imperfect spiral stiffened functionally graded (SSFG) cylindrical shells. The SSFG cylindrical shell is surrounded by a linear and nonlinear elastic foundation. The proposed linear model is based on the two-parameter elastic foundation (Winkler and Pasternak). A three-parameter elastic foundation with hardening/softening cubic nonlinearity is used for nonlinear model. The material properties are temperature dependent and assumed to be continuously graded in the thickness direction. Also, for thermal buckling analysis, the uniform and linear temperature distribution in thickness direction is considered. The SSFG cylindrical shells are considered with various angles for spiral stiffeners. The strain–displacement relations are obtained based on the von Kármán nonlinear equations and the classical plate theory of shells. The smeared stiffener technique and the Galerkin method are applied to solve the nonlinear problem. In order to find the nonlinear dynamic thermal buckling responses, the fourth-order Runge–Kutta method is used. To validate the results, comparisons are made with those available in literature and good agreements are shown. The effects of various geometrical and material parameters are investigated on the nonlinear static and dynamic thermal buckling response of SSFG cylindrical shells.

A layerwise finite element formulation for free vibration analysis of functionally graded sandwich shells

This paper presents a layerwise finite element formulation for dynamic analysis of functionally graded material (FGM) sandwich shell in thermal and non-thermal environments. The layerwise theory used in this work is based on the first-order shear deformation theory (FSDT) for each layer satisfying the displacement continuity at layer interface. Two configurations of FGM sandwich shells are considered, the first with FGM facesheet and homogenous core, and the second having homogenous facesheets and FGM core. Effective material properties of the FGM are estimated according to two micromechanical models, namely, Voigt’s rule of mixture (ROM) and Mori–Tanaka (MT) scheme. For the shells in thermal environment, a nonlinear temperature distribution in thickness direction is considered and the elastic properties are assumed to be temperature dependent. The results obtained from the proposed formulation are validated with those available in the literature. Natural frequencies obtained from Sanders’, Love’s and Donnell’s shell theories for different geometric and boundary conditions are compared to assess the performance of different shell theories for FGM sandwich shells under non-thermal environment. The effects of volume fraction index, core thickness and temperature gradient on natural frequencies of FGM sandwich shells are investigated. The present formulation is simple, accurate and computationally efficient.

EFFECT OF THE THROUGH-THE-THICKNESS TEMPERATURE DISTRIBUTION ON THE RESPONSE OF LAYERED AND COMPOSITE SHELLS

This paper considers the thermal stress problem of thick and thin multilayered cylindrical and spherical shells including carbon fiber reinforced layers and/or a central soft core. The following two cases are considered: (i) the temperature distribution in thickness direction is assumed linear; (ii) the temperature distribution in thickness direction is calculated via Fourier's heat conduction equation. Carrera's Unified Formulation and the Principle of Virtual Displacements are used to obtain the governing equations in the case of shells with constant radii of curvature subjected to established temperature conditions on their upper and lower surfaces. Both Equivalent Single Layer and Layer Wise models with an order of expansion in the thickness direction from linear to fourth order are considered. The importance of refined models for a correct evaluation of displacement and stress fields in multilayered shells can be noted. Furthermore, it has been shown that results obtained assuming a linear temperature profile in the thickness direction can be meaningless.