One-dimensional analysis for magneto-thermo-mechanical response in a functionally graded annular variable-thickness rotating disk

In this paper, the magneto-thermo-mechanical response of a functionally graded magneto-elastic material (FGMM) annular variable-thickness rotating disk is investigated. The material properties namely material stiffness, heat conduction coefficient, thermal expansion coefficient, mass density and magnetic permeability are assumed to vary continuously along the radial direction according to a power law. The thickness profile of the disk placed in a uniform magnetic field and subjected to the thermal load is assumed to be hyperbolic in nature. The effects of the magnetic field, grading index and geometric nonlinearity on the mechanical and thermal stresses of the disk are investigated. For a specific value of the grading index the maximum radial stress due to magneto-mechanical load in a mounted FGMM disk with hyperbolic convergent profile is found away from the center. This result is different from other thickness profile disks where the radial stresses are always at the center. It is observed that unlike radial stress in a mounted FGM disk subjected to mechanical load only where it is always tensile, the radial stress due to magneto-thermal load in a mounted FGMM disk can be both tensile and compressive type. It is seen that a decrease in the value of grading index invokes shifting of the location of the maximum temperature in FGMM disk with hyperbolic convergent profile towards the outer surface of the disk.     

Post buckling response of functionally graded materials plate subjected to mechanical and thermal loadings with random material properties

In this paper, the second order statistics of post buckling response of functionally graded materials plate (FGM) subjected to mechanical and thermal loading with nonuniform temperature changes subjected to temperature independent (TID) and dependent (TD) material properties is examined. Material properties such as material properties of each constituent’s materials, volume fraction index are taken as independent random input variables. The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear kinematic using modified C⁰ continuity. A direct iterative based C⁰ nonlinear finite element method (FEM) combined with mean centered first order perturbation technique (FOPT) proposed by last two authors for the composite plate is extended for Functionally Graded Materials (FGMs) plate with reasonable accuracy to compute the second order statistics (mean and coefficient of variation) of the post buckling load response of the FGM plates. The effect of random material properties with amplitude ratios, volume fraction index, plate thickness ratios, aspect ratios, boundary conditions and types of loadings subjected to TID and TD material properties are presented through numerical examples. The performance of outlined present approach is validated with the results available in literatures and independent Monte Carlo simulation (MCS).     

Analysis of functionally graded plates using higher order shear deformation theory

This work addresses a static analysis of functionally graded material (FGM) plates using higher order shear deformation theory. In the theory the transverse shear stresses are represented as quadratic through the thickness and hence it requires no shear correction factor. The material property gradient is assumed to vary in the thickness direction. Mori and Tanaka theory (1973) [1] is used to represent the material property of FGM plate at any point. The thermal gradient across the plate thickness is represented accurately by utilizing the thermal properties of the constituent materials. Results have been obtained by employing a C° continuous isoparametric Lagrangian finite element with seven degrees of freedom for each node. The convergence and comparison studies are presented and effects of the different material composition and the plate geometry (side-thickness, side–side) on deflection and temperature are investigated. Effect of skew angle on deflection and axial stress of the plate is also studied. Effects of material constant n on deflection and the temperature distribution are also discussed in detail.     

Nonlinear dynamic response for functionally graded shallow spherical shell under low velocity impact in thermal environment

Based on Giannakopoulos’s 2-D functionally graded material (FGM) contact model, a modified contact model is put forward to deal with impact problem of the functionally graded shallow spherical shell in thermal environment. The FGM shallow spherical shell, having temperature dependent material property, is subjected to a temperature field uniform over the shell surface but varying along the thickness direction due to steady-state heat conduction. The displacement field and geometrical relations of the FGM shallow spherical shell are established on the basis of Timoshenko–Midlin theory. And the nonlinear motion equations of the FGM shallow spherical shell under low velocity impact in thermal environment are founded in terms of displacement variable functions. Using the orthogonal collocation point method and the Newmark method to discretize the unknown variable functions in space and in time domain, the whole problem is solved by the iterative method. In numerical examples, the contact force and nonlinear dynamic response of the FGM shallow spherical shell under low velocity impact are investigated and effects of temperature field, material and geometrical parameters on contact force and dynamic response of the FGM shallow spherical shell are discussed.     

Free vibration analysis of FGM cylindrical shells surrounded by Pasternak elastic foundation in thermal environment considering fluid-structure interaction

This paper presents an investigation on partially fluid-filled cylindrical shells made of functionally graded materials (FGM) surrounded by elastic foundations (Pasternak elastic foundation) in thermal environment. Material properties are assumed to be temperature dependent and radially variable in terms of volume fraction of ceramic and metal according to a simple power law distribution. The shells are reinforced by stiffeners attached to their inside and outside in which the material properties of shell and the stiffeners are assumed to be continuously graded in the thickness direction. The formulations are derived based on smeared stiffeners technique and classical shell theory using higher-order shear deformation theory which accounts for shear flexibility through shell's thickness. Displacements and rotations of the shell middle surface are approximated by combining polynomial functions in the meridian direction and truncated Fourier series with an appropriate number of harmonic terms in the circumferential direction. The governing equations of liquid motion are derived using a finite strip element formulation of incompressible inviscid potential flow. The dynamic pressure of the fluid is expanded as a power series in the radial direction. Moreover, the quiescent liquid free surface is modeled by concentric annular rings. A detailed numerical study is carried out to investigate the effects of power-law index of functional graded material, fluid depth, stiffeners, boundary conditions, temperature and geometry of the shell on the natural frequency of eccentrically stiffened functionally graded shell surrounded by Pasternak foundations.     

Investigation on a rotating FGPM circular disk under a coupled hygrothermal field

In this paper, the distributions of the temperature, moisture, displacement and stress of a functionally graded piezoelectric material (FGPM) circular disk rotating around its axis at a constant angular velocity under a coupled hygrothermal field are presented by a numerical method. The material properties of the FGPM circular disk are assumed to vary along the radial coordinate exponentially. First, the coupled hygrothermal field along the radius of a rotating circular disk is achieved by solving the coupled hygrothermal equations, and then the dynamic equilibrium is solved by utilizing the finite difference method. Finally, numerical results show the effects of functionally graded index, inner radius, angular speed and hygrothermal index on the hygrothermal behaviors of the FGPM circular disk. The results can be useful for the optimal design of rotating FGPM circular disks under a coupled hygrothermal field.     

Thermoelastic instability of functionally graded coating with arbitrarily varying properties considering contact resistance and frictional heat

Using the homogeneous multi-layered model, this paper studies the thermoelastic instability (TEI) of the functionally graded material (FGM) coating with arbitrary varying properties considering the frictional heat and thermal contact resistance. A homogeneous half-plane slides on an FGM coated half-plane at the out-of-plane direction under a uniform pressure. The perturbation method and transfer matrix method are used to deduce the characteristic equation of the TEI problem, which is then solved to obtain the relationship between the critical sliding speed and critical heat flux. The effects of the gradient index and varying form of material properties of the FGM coating on the stability boundaries are examined. The results show that FGM coating can adjust the thermoelastic contact stability of sliding systems.     

3D dynamic coupled thermoelastic solution for constant thickness disks using refined 1D finite element models

This paper deals with the generalized coupled thermoelastic solution for disks with constant thickness. It is a sequel to the authors’s previous work in which refined 1D Galerkin finite element models with 3D-like accuracies are developed for theories of coupled thermoelasticity. Use of the reduced models with low computational costs may be of interest in a laborious time history analysis of the dynamic problems. In this paper, the developed models are applied and evaluated for a 3D solution of the dynamic generalized coupled thermoelasticity problem in the disk subjected to thermal shock loads. Comparison of the obtained result with the results available in the literature verified the proposed finite element models are quite efficient with very high rate of convergence and able to provide results with analytical accuracy. In addition, propagation of the thermoelastic waves, the wave reflection from the boundaries and the Poisson effect in an axisymmetric and asymmetric disk problem are represented as contour plots to demonstrate 3D capabilities of the models.     

Thermoelasticity of a Thermosensitive Hollow Anisotropic Sphere with a Thin Foreign Inclusion

By using generalized functions, we obtain the set of thermoelasticity equations for a hollow thermosensitive, transversally isotropic sphere with a thin foreign inclusion. By approximating the thermoelastic characteristics of the basic material and the material of the inclusion by piecewise-constant functions, we deduce a differential equation with coefficients of the type of impulse functions to determine a radial displacement. We carry out the numerical analysis of the temperature and level of thermal stresses in a system made of graphites.     

Approximation of functionally graded plates with non-conforming finite elements

In this paper rectangular plates made of functionally graded materials (FGMs) are studied. A two-constituent material distribution through the thickness is considered, varying with a simple power rule of mixture. The equations governing the FGM plates are determined using a variational formulation arising from the Reissner–Mindlin theory. To approximate the problem a simple locking-free Discontinuous Galerkin finite element of non-conforming type is used, choosing a piecewise linear non-conforming approximation for both rotations and transversal displacement. Several numerical simulations are carried out in order to show the capability of the proposed element to capture the properties of plates of various gradings, subjected to thermo-mechanical loads.     

功能梯度梁在热-机械荷载作用下的几何非线性分析

基于经典梁理论，运用虚功原理和变分法推导了均匀变温场与横向均布荷载联合作用的功能梯度梁的几何非线性控制方程.考虑端部不可移夹紧边界条件，运用打靶法求解了该两点边值问题.当横向均布荷载为0时，考察了功能梯度梁的热屈曲临界升温和屈曲平衡路径.当均匀变温与横向均布荷载都不为0时，考察了功能梯度梁的荷载 挠度曲线.数值结果表明:随材料体积分数指数增加，梁的有量纲热屈曲临界升温显著减小，后屈曲变形显著增加；变温对功能梯度梁的荷载 挠度曲线影响非常显著.发现了功能梯度梁的双稳态构形及其转换现象，梁的最终平衡位形不但与变温及荷载参数有关，还与加载历程有关.     

功能梯度材料杆的热后屈曲分析

对两端不可移简支陶瓷-金属功能梯度材料(FGM)杆建立了在热载荷作用下的非线性控制微分方程,采用打靶法分析了由二氧化锆和Ti-6Al-4V两种材料组成的FGM杆的热后屈曲行为．首先给出了在均匀温度场中不同梯度指标的FGM杆的热后屈曲平衡路径,并与二氧化锆和Ti-6Al-4V两种均质材料杆的相应特性进行了比较,同时讨论了不同端部转角下梯度指标对FGM杆稳定性的影响;然后分别研究了在温差一定、下表面温度变化时和在下表面温度一定、温差变化时FGM杆的热后屈曲特性,也与两种均质材料杆的后屈曲特性进行了比较．     

Temperature and displacement fields in brake and clutch systems with thermoelastic instabilities

In brake and clutch systems kinetic energy is converted into thermal energy. Experiments show that the corresponding temperature field can develop unstable periodic structures. The temperature field couples to the displacement field by thermal expansion. Local pressure maxima in the frictional plane and the corresponding maxima in heat generated cause thermoelastic instabilities (TEI). A model describing both effects covers layers of thermoelastic materials for all necessary mechanical components of the system. The set of field equations of each layer can analytically be solved by separation of constants. These solutions must fulfill the boundary conditions e.g. in the sliding plane. A stability discussion yields whether TEI appear or not. As a study a brake system is analyzed comprising two pads pressed against a rotating disk with cooling channels inside. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Thermal buckling and post-buckling analysis of geometrically imperfect FGM clamped tubes on nonlinear elastic foundation

Present research deals with the thermal buckling and post-buckling analysis of the geometrically imperfect functionally graded tubes on nonlinear elastic foundation. Imperfect FGM tube with immovable clamped–clamped end conditions is subjected to thermal environments. Tube under different types of thermal loads, such as heat conduction, linear temperature change, and uniform temperature rise is analyzed. Material properties of the FGM tube are assumed to be temperature dependent and are distributed through the radial direction. Displacement field satisfies the tangential traction free boundary conditions on the inner and outer surfaces of the FGM tube. The nonlinear governing equations of the FGM tube are obtained by means of the virtual displacement principle. The equilibrium equations are based on the nonlinear von Kármán assumption and higher order shear deformation circular tube theory. These coupled differential equations are solved using the two-step perturbation method. Approximate solutions are provided to estimate the thermal post-buckling response of the perfect/imperfect FGM tube as explicit functions of the various thermal loads. Numerical results are provided to explore the effects of different geometrical parameters of the FGM tube subjected to different types of thermal loads. The effects of power law index, springs stiffness of elastic foundation, and geometrical imperfection parameter of tube are also included.     

Transient piezothermoelastic analysis of a cross-ply laminated cylindrical panel bonded to a piezoelectric actuator

In this study, the theoretical treatment of transient piezothermoelastic problem is developed for a cross-ply laminated cylindrical panel bonded to a piezoelectric actuator due to nonuniform heat supply. By using the exact solutions for cross-ply laminate and piezoelectric layer of crystal class mm2, the theoretical analysis of a transient piezothermoelasticity is developed for a simple supported cylindrical composite panel under the state of plane strain. Analysis of a piezothermoelastic problem leads to an appropriate electric potential applied to the piezoelectric layer which suppresses the induced thermoelastic displacement in the radial direction at the midpoint on the free surface of the cross-ply laminate. Some numerical results for the temperature change, the displacement, the stress in a transient state when the transient thermoelastic displacement is controlled are shown in figures.     

Nonlinear transient thermal stress and elastic wave propagation analyses of thick temperature-dependent FGM cylinders,using a second-order point-collocation method

Nonlinear transient thermal stress and elastic wave propagation analyses are developed for hollow thick temperature-dependent FGM cylinders subjected to dynamic thermomechanical loads. Stress wave propagation, wave shape distortion, and speed variation under impulsive mechanical loads in thermal environments are also investigated. In contrast to researches accomplished so far, a second-order formulation rather than a first-order one is employed to improve the accuracy. The FDM method (as a point-collocation FEM method) is used. It is known that other FEM methods cannot show the actual trend jumps due to distributing the abrupt changes in the quantities as the numerical errors and the residuals of the governing equations among the nodal results. Furthermore, the required computational time and allocated computer memory are much reduced by the present solution algorithm. The cylinder is not divided into isotropic sub-cylinders. Therefore, artificial wave reflections from the hard interfaces are avoided. Time variations of the temperatures, displacements, and stresses due to the dynamic or impulsive loads are determined by solving the resulted highly nonlinear governing equations using an iterative updating solution scheme. A sensitivity analysis includes effects of the volume fraction indices, dimensions, and temperature-dependency of the material properties is performed. Results reveal the significant effect of the temperature-dependency of the material properties on the thermoelastic stresses and present some interesting characteristics of the thermoelastic and wave propagation behaviors.     

The dynamic analysis of the functionally graded piezoelectric (FGP) shell panel based on three-dimensional elasticity theory

In this paper, three-dimensional elasticity solution is extended to investigate a FGPM finite length, simply supported shell panel under dynamic pressure excitation. The host panel is assumed to be of some functionally graded piezoelectric material (FGPM). The ordinary differential equations (o.d.e.) are derived from the highly coupled partial differential equations (p.d.e.) using series expansions of mechanical and electrical displacements. The resulting system of ordinary differential equations is solved by means of Galerkin finite element method. At last, numerical examples are presented for a FGPM shell panel. To verify the validity of code and formulation, the results of a FGM panel and a FGM plate are compared with the published results.     

Static response and free vibration analysis of FGM plates using higher order shear deformation theory

Free vibration and static analysis of functionally graded material (FGM) plates are studied using higher order shear deformation theory with a special modification in the transverse displacement in conjunction with finite element models. The mechanical properties of the plate are assumed to vary continuously in the thickness direction by a simple power-law distribution in terms of the volume fractions of the constituents. The fundamental equations for FGM plates are derived using variational approach by considering traction free boundary conditions on the top and bottom faces of the plate. Results have been obtained by employing a continuous isoparametric Lagrangian finite element with 13 degrees of freedom per node. Convergence tests and comparison studies have been carried out to demonstrate the efficiency of the present model. Numerical results for different thickness ratios, aspect ratios and volume fraction index with different boundary conditions have been presented. It is observed that the natural frequency parameter increases for plate aspect ratio, lower volume fraction index n and smaller thickness ratios. It is also observed that the effect of thickness ratio on the frequency of a plate is independent of the volume fraction index. For a given thickness ratio non-dimensional deflection increases as the volume fraction index increases. It is concluded that the gradient in the material properties plays a vital role in determining the response of the FGM plates.     

Nonlinear thermoelastic analysis of layered structures

The nonlinear thermoelastic behavior of orthotropic layered slabs and cylinders including radiation boundaries, temperature-dependent material properties, and stress-dependent layer interface conditions is investigated. A one-dimensional finite element formulation employing quadratic layer and linear interface elements is used to perform the analyses. The transient heat conduction portion of the program is temporally discretized via an implicit linear time interpolation algorithm which includes Crank-Nicolson, Galerkin, and Euler backward differencing. The nonlinear heat conduction equations are iteratively evaluated using a modified Newton-Raphson scheme. Direct iteration between heat conduction and stress analysis is employed when stress-dependent interface thermal resistance coefficients are utilized. Verification problems are presented to demonstrate the accuracy of the finite element code.     

陶瓷/金属功能梯度材料圆筒的热应力分析

对金属-功能梯度材料-陶瓷的三层组合圆筒进行了热应力分析,导出了定常温度分布及热应力分布的计算表达式,并就ZrO₂/Ti-6Al-4V梯度材料的热应力进行了计算和讨论.