Investigation of thermo-elastic characteristics in functionally graded rotating disk using finite element method

: In this research paper, displacement, stresses and strains are presented for rotating FGM disk with variable thickness by using finite element method (FEM). Thermo-elastic material properties and thickness of FGM diskcontinuouslyvaryasexponentialandpowerlawfunc-tion in radial direction along radius of disk. The value of Poisson’s ratio is taken as constant. The problem of thermo-elasticity is converted into second order governing differential equation in terms of radial coordinate. This conversion is based upon equilibrium equation for disk and stress-strain relationship. The influence of variable thickness, angular velocity and functionally graded materials is discussed on thermo-elastic characteristics of rotating disk for exponential variation of material properties. Further, these thermo-elastic characteristics of disk are plotted for various values of non-homogeneity parameter under power law distribution of material properties. Thus, the investigations done in this research paper may be useful for industrial area in construction an appropriate FGM disk by controlling above mentioned parameters.


Introduction
Functionally graded materials are a new class of composites that are also known as multifunctional materials due to their spatial variation composition. The positive impact of compressive residual surface stresses on wear resistance as well as on strength may be counted as a potential advantage of these multifunctional materials. So, it become a matter of concern to design an appropriate blueprint of the progressive compositional gradient for an optimal distribution of the residual stresses to correlate with mechanical properties [1]. Composite materials have found vast applications in distinguished elds of structural engineering such as aerospace, automobile, bio-medical, civil structures, etc. [2]. In the past few decades, many researchers have investigated the behavior of these multifunctional materials in the presence of di erent environmental conditions. Wang and Zu [3,4] published and concluded that the damping, translational speed, and the excitation amplitude signi cantly a ect the nonlinear dynamical responses of the translational FG plate using an analytical and numerical method approach. The nonlinear vibration analysis of rotating functionally graded cylindrical shell has been reported by Sheng and Wang [5]. Liew et al. [6] have studied the nonlinear vibration behaviour of a laminated FG cylindrical panel. Talha and Singh [7] have presented the nonlinear free exural vibration analysis of an FG plate using an improved higher-order theory. Akbari and Ghanbari [8] studied in uence of internal pressure, thermal load and rotation on functionally graded hollow discs from exact analytical solution under radially varying material properties. Kalali et al. [9] derived elasto-plastic stress solution numerically in axisymmetric functionally graded rotating disk, cylindrical and spherical vessel with help of Hencky's stress-strain relation. Salehi et al. [10] used linear plane elasticity theory to nd analytical solution for axisymmetric thick-walled FGM cylinders. Habib et al. [11] made stress and strain analysis in functionally graded cylinder under exponentially varying material properties by nite element method and ANASYS software. Yadav and Jiwari [12,13] used nite element to solve di erential equations in various mathematical models. Afsar and Go [14] shown e ect of radial thickness, angular speed and temperature pro le on thermoelastic eld in a thin circular FGM disk for exponentially varying material properties. Khorsand and Tang [15] used di erential quadrature method and co-evolutionary particle swarm optimization approach to minimize stress and displacement in functionally graded hollow circular disk under variable thickness. Go [16] derived second order di erential equation based upon two-dimensional thermal elastic theory to study stress, strain and temperature distribution in rotating circular disk. Thawati et al. [17] represented stress and deformation in functionally graded disk for three types of material properties named as: mori-tanaka scheme, power law distribution and exponential distribution under variable thickness. Zheng et al. [18] developed stress eld numerically by using nite di erence method in functionally graded rotating disk. To carried out stress eld material properties are assumed to followed power law distribution. Nejad et al. [19] obtained elasto-plastic deformation and stress from exact form of analytical solution in FGM rotating disk for constant Poisson's ration. Calliogue et al. [20] made elastic-plastic stress analysis in functionally graded rotating disk from analytical and numerical solution for di erent values of angular velocities under radially varying material properties. Kordkheili and Livani [21] used material properties as function of temperature to study thermoelastic creep behaviour in functionally graded rotating disk with variable thickness. Arnab et al. [22] employed power and exponential function variation in radial direction to study thermoelastic eld in thin circular FGM disk. Allam et al. [23] used in nitesimal theory to drive accurate and e ective solutions for displacement and stress in rotating annular disk with variable thickness. Jalali and Shahriari [24] obtained stresses and deformation in rotating variable disk for three types of boundary conditions by nite di erence method. Torabnia et al. [25] shown e ect of young's modulus, density and yield stress on stress, strain and radial displacement in FGM hollow rotors from analytical solution of equilibrium equation. Garg [26] carried out elastic stress and strain in rotating FGM disk for di erent values of thickness gradation index under variable material properties in radial direction. Abdalla et al. [27] studied thermomechanical stress in functionally graded rotating disk with nite element method in two-dimensional model. Kursun et al. [28] made stress analysis in FGM discs under condition of uniform pressure on inner surface and linearly decreasing temperature distribution for radially varying material properties. Nkene et al. [29] considered second law of Newton, Hooke's law and stress-strain relationship to drive analytical and numerical solution in FGM rotating hollow cylinder for di erent values of inhomogeneity parameter. Saadtfar [30] analysed e ect of angular velocity, hygrothermal loading and moisture concentration on piezomagnetic rotating thick walled cylinder for hygrothermal boundary conditions. Abrinia et al. [31] assumed material properties varying along thickness as power law function to obtained radial and circumferential stress in FGM cylinder under internal pressure and temperature. Rahimi and Nejad [32] investigated thermal stresses from an exact solution in FGM rotating hollow thick-walled cylinder under internal and external pressure. Evci and Gulgec [33] derived analytical solution to represent stresses and displacement in FGM hollow cylinder by using airy stress function. Further, failure analysis done with coulomb-mohr theory and tresca yield criterion. Mantena et al. [34] obtained solution in form of bessels and trigonometric functions to study heat conduction and thermal stresses in hollow cylinder under non-homogeneous material properties. Yildirim [35] presented thermomechanical analysis analytically with Navier equations under assumption of spherically-symmetric plain strain for non-homogeneous isotropic spheres. Farhan et al. [36] proposed nite difference method to study thermoelasticity in isotropic and in nitely long circular cylinder under temperature dependent material properties. Hu et al. [37] investigated in uence of various parameters on natural vibration in rotating functionally graded ring plate. Go [38] made thermoelastic characteristics analysis numerically under variable contact force and homogeneous thickness. Thakare and Warbhe [39] studied temperature distribution and stress distribution in thick hollow cylinder for homogeneous and inhomogeneous material properties. Saeedi et al. [40] used successive approximation method to study stresses and strains in thick walled FGM cylindrical shells under internal pressure and temperature gradient. In the present work, rotating disks of functionally graded materials are constructed due to its utilization in di erent elds of Engineering and Science. A stress eld in FGM isotropic and anistropic cylinder under internal pressure and uniform heat generation for di erent values of gradient index, is represented. These disks are modelled under variety of thermo-mechanical properties to extend our published work [41][42][43][44][45].
Basic Equations of the problem: For rotating disk, the strain-displacement relationship irrespective of thickness can be written as: Where u is displacement, r is radial coordinate and (εr , ε θ ) are strains in radial and circumferential direction. For elastic deformation, by using Hooke's law relations between strains and stresses can be written as: where (σr , σ θ ) represents stresses in radial and circumferential direction, E(r) is young's modulus and ν is Poisson's ration of disk. Equilibrium equation for disk of variable thickness is given as: where ω is angular velocity, ρ(r) is density of rotating disk and h(r) is thickness of disk.

Case I: Exponential variation of material properties
In this section, we assumed that material properties of rotating disk are varying exponentially along radial direction. Therefore, E(r), ρ(r) and h(r) vary as: Where r is radial coordinate, E , ρ and h are constants of young's modulus, mass density and thickness respectively at inner surface of FGM disk. The index m and n are material parameters, the k is the geometric parameter.
After substituting values from (5-7) into equation (4), governing di erential equation of problem in terms of radial displacement become:

Case II: Power law variation of material properties
In this case, to study thermoelastic characteristics in FGM disk material properties of disk are consider to follow power law variation along radius as follows: ρ(r) = ρ r n (10) h(r) = h r n By substituting values of various material parameters from Eqs. (9)(10)(11) into Eq. (4), the di erential equation which represents problem of thermoelasticity is obtained, as: After simpli cation equation (15), we get: Where P = n +n + , Q = v(n +n )− and R = ρ E ( −v )ω

Numerical results and discussions
In this section, numerical results for thermo-elastic characteristics have been presented graphically for rotating FGM disk under exponential and power-law distribution of material properties. The value of Poisson's ratio is taken as 0.3 constant throughout material.

Case I:
In this section, an e ect of angular velocity, thickness and functionally graded materials study on stresses, strains and displacement under exponentially varying material properties.         Figure 5 reveals that the behaviour of circumferential strain and displacement curves are same for di erent values of ω. The maximum and minimum numerical value of circumferential strain exists corresponding maximum and minimum value of angular velocity respectively. The displacement and circumferential strain curves are tensile for inner surface of disk. Figure 6 exhibits that, radial strain and angular velocity are opposite in nature for di erent values of r i.e εr decreases as value of r increases. Figure 7, illustrate circumferential stress against radial coordinate for di erent values of ω. The behaviour of circumferential stress curves are changes at r = . . At inner surface of disk maximum value of circumferential stress is obtained for maximum value of angular velocity but at outer surface of disk maximum value is achieved for minimum value for angular velocity. Figure 8 represents, e ect of angular ve-locity on radial stress. For ω = radial stress curve is very uctuating in nature but for ω = curve shows steady behaviour. Figures 9 -13 show the e ect of thickness on thermoelastic characteristics by taking di erent values of thickness parameter k. As per Figure 9, for inner surface of disk maximum value of displacement is obtained corresponding to k = . and for outer surface maximum value it exists for k = . . Figure 10 shows that, circumferential strain curves are more variable for inner surface of disk. All circumferential curves are decreasing in nature i.e. as r increases ε θ decreases. But from Figure 11, one can see that behaviour of radial strain curve is increasing and maximum value of radial strain is achieved corresponding to maximum value of thickness parameter. From Figures 12 and 13 it is noted that behaviour of circumferential and radial stress curves are di erent in nature. Maximum value of circumferential stress and minimum value of radial stress are obtained for k = . . Also, circumferential stress is maximum at inner surface of disk but radial stress is minimum at this surface. Figure 14 -18 show distribution of thermo-elastic characteristics of rotating disk that disk is constructed from di erent FGM's. The thermo-mechanical properties of FGM's are presented in Table 1.
From Figure 14, it is observed that displacement decreases as value of radial coordinate increases. The maximum value of displacement is obtained for stainless steel and minimum value is attained for silicon nitride. Figure 15 displays that circumferential strain converges to zero as radial coordinate moves toward outer surface of disk. Radial strain curves are increasing in nature, according to Figure 16. The behaviour of all radial strain curves are linear when . ≤ r ≤ . and for remaining region behaviour is non-linear in nature. The circumferential stress is maximum for silicon nitride and least for aluminimum. The behaviour of displacement and circumferential strain curves are same in nature. As we can see from Figure 18, radial stress curve is rstly linearly decreasing in nature for . ≤ r ≤ . and it is decreasing for remaining values of radial coordinate.

Case II:
In this section, graphical representation of thermoelastic characteristics of rotating disk is done under exponential material properties. The e ect of non-homogeneity index (n = n = n = n) is shown on stress, strain and displacement by taking its various values. Figures 19 -26 show thermo-mechanical properties and thermo-elastic characteristics for power law distribution of material properties. The e ect of non-homogeneity   Figures 19 -21, one can see that maximum value of material properties obtained for n = and minimum values are obtained for n = . Modulus of elasticity, density and thickness are constant when non homogeneity parameter is absent from material properties. As Figure 22 shows, displacement curve is decreasing for n = and n = but for n = curve is strictly increasing. Maximum displacement is obtained when material properties follow quadratic variation along radius of disk. For n = circum-  ferential strain Curve shows increasing behaviour in nature for inner surface of rotating disk but decreasing for outer surface of disk. For n = , n = and n = behaviour of circumferential strain linear and stable. Radial strain is uctuating for n = and n = but this behaviour is linear for n = and very less elastic for n = . From Figures  25 -26, it is noted that both stress curves are increasing in nature for n = but shows decreasing behaviour for remaining values of n. Maximum variation in stresses exits for outer surface of disk.

Conclusion
In presented research, thermo-elastic characteristics analysis is made for di erent material properties pro les in rotating disk which is constructed from FGM material. Finite element method is employed to nd displacement, strains and stresses numerically from equilibrium equation. The following conclusion can be made: 1) In exponential distribution of material properties, the value of thermo-mechanical properties increases as material parameter increases but for power law distribution this behaviour of material properties is opposite in nature. 2) Maximum variation in displacement, strains and stresses exist for inner surface of rotating disk when thickness is taken as variable parameter in case-I. 3) The behaviour of thermo-elastic characteristics is same in nature but di er numerically for di erent types of functionally graded materials. 4) In power law distribution of material properties maximum value of thermo-elastic characteristics is obtained for n = . 5) For n = and n = thermo-elastic curves are more stable in nature in power law distribution of material properties.
Finally, these ndings may be helpful in construction/designing of FGM rotating disks, which have a numerous application in di erent elds of engineering.

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The authors state no funding involved.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

Con ict of interest:
The authors state no con ict of interest.