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Closed-Form Formulas for Hyperbolically Tapered Rotating Disks Made of Traditional Materials under Combined Thermal and Mechanical Loads

Using the true temperature distribution along the radial coordinate, closed-form formulas are offered for readers to study the thermo-mechanical behavior of variable thickness disks having both convergent and divergent hyperbolic thickness profiles made of conventional materials. Internal and external pressures, centrifugal forces and thermal loads due to the differences in prescribed surface temperatures are all considered with three boundary conditions: free-free (a circular annulus), fixed-free (a disk mounted on a rotating shaft at the inner surface), and fixed-fixed (a disk mounted on a rotating shaft at the inner surface and cased at the outer surface) boundary conditions. A parametric study is also conducted in almost real working environment in which the outer surface of the disk has considerably higher temperature rather than the inner surface. The thermomechanical linear elastic response of a hyperbolic mounted rotating disk subjected to the external pressure induced by blades is originally handled by those proposed formulas.


Introduction
As a rotating machinery element, rotating disks may operate as a circular annulus or as a disk attaching a rotating shaft at its center. A rigid casing may exist at the outer surface of such a disk. Their thickness may vary along the radial direction linearly, hyperbolically, parabolically, exponentially or so on and so forth. It may be made of any kind of traditional or advanced materials. Rotating disks may act in high temperature environments like turbine rotors, flywheels and gears and may be subjected to simultaneously external pressures due to the existence of the blades. In such cases, the thermo-mechanical analyses come into prominence in the design of such structures. It is obvious that the existence of closed-form formulas that can be directly used in the design stage in relation to the subject matter is to be of great convenience for engineers.
Thermal-related analyses of uniform disks made of an isotropic and homogeneous materials have paid much attention than disks having varying section properties and made of advanced composite materials [1][2][3][4][5][6][7][8][9][10]. From those, Güven and Altay [1] investigated the elastic-plastic stress distribution of a solid disk due to nonuniform heat source under external pressure. Kulkarni and Deshmukh [4] addressed the thermal stresses in a thick annular disk under steady temperature field. Nejad and Afshin [6] offered an analytical solution of transient thermoelastic behaviors of rotating pressurized disks subjected to arbitrary boundary and initial conditions. Kaur et al. [8] observed that thermal effect in the disk increase the value of circumferential stress at the internal surface and radial stresses at the external surface for compressible as compare to incompressible material. Based on a variational principle considering the radial displacement field as unknown, Nayak and Saha [9] evaluated the influence of thermomechanical loading on stresses and deformation states in a rotating disk with varying thicknesses. They considered different disk geometries as well as temperature distribution profiles to calculate the limit angular speed of the disks under thermo-mechanical loading. Yıldırım [10] offered a consistent and an all-in-one analytical study for the determination of heat-induced, pressure-induced, and centrifugal force-induced axisymmetric exact deformation and stresses in a thick-walled spherical vessel, a cylindrical vessel, and a uniform disk at specified constant surface temperatures and at a constant angular velocity. Yıldırım [10] included both the inner and outer pressures in the formulation of annular structures made of an isotropic and homogeneous linear elastic material. For disks, three different boundary conditions were taken into account to consider mechanical engineering applications in the study [10].
Thermal-related analyses of uniform disks made of anisotropic materials [11][12][13][14][15][16][17][18][19] and functionally graded materials [20][21][22][23][24][25][26][27][28] are other investigation themes. Unfortunately, the number of studies on the thermal/thermal related analysis of variable thickness disks made of either functionally graded (FG) materials [29][30][31][32][33][34][35][36][37][38][39][40][41][42] or traditional materials [43][44][45][46] are not enough. In those studies, a variable-thickness disk is generally considered as a combination of multilayered uniform disks [29, 30-32, 34, 36, 40]. For example, Chiba [29] assumed the annular disk is to be a multilayered one with stepwise thickness variation, where each layer is assumed to have constant deterministic material properties. Bayat et al. [30] considered a rotating FG disk with either parabolic or hyperbolic thickness under a steady temperature field. The disk was assumed to be composed of sub-disks of uniform thicknesses. In another study, Bayat et al. [31] analyzed the thermoelastic bending of FGM rotating disks based on the first order shear deformation theory. Bayat et al. [32] studied on a thermo elastic analysis for axisymmetric rotating disks made of temperature-dependent power-law FG material with variable thickness. It was assumed in Ref. [32] that the temperature field is to be uniformly distributed over the disk surface and varied in the radial direction. Bayat et al. [32] presented semi-analytical solutions for the displacement field for solid/annular disks under free-free and fixed-free boundary conditions. Bayat et al. [33] also offered exact elastic solutions for axisymmetric variable-thickness hollow rotating disks with heat source made of functionally power-lawgraded (FG) materials under free-free and fixed-free boundary conditions. They showed that the temperature distribution in a hyperbolic disk is the smallest compared with other thickness profiles. A hyperbolic-convergent FG disk has smaller stresses because of thermal load compared with the uniform disk. Nie and Batra [35] studied axisymmetric stresses in a rotating disk made of a rubber-like material that was modeled as isotropic, linear thermoelastic, and incompressible. Damircheli and Azadi [36] carried out thermal and mechanical stress analyses of a rotating disk having either parabolically or hyperbolically varying thicknesses made of functionally graded material by using finite element method (FEM). By utilizing a 2D plane stress analysis and assuming a power form temperature distribution of over the disk with the higher temperature at the outer surface, Hassani et al. [37] obtained stress and strains of rotating disks with non-uniform thickness and material properties subjected to thermo-elastic loading under different boundary conditions. They used semi-exact methods of Liao's homotopy analysis method (HAM), Adomian's decomposition method and He's variational iteration method (VIM). Hassani et al. [37] compared the results of those three methods with Runge-Kutta's solutions. Tütüncü and Temel [38] studied numerically the thermoelastic analysis of FG rotating disks of variable thickness. Golmakani [39] scrutinized large deflection thermoelastic analysis of FG solid and hollow rotating axisymmetric disk with uniform and variable thickness subjected to thermo-mechanical loading. Kurşun and Topçu [40] handled the elastic stress analysis of a hollow disk with variable thickness made of FG materials under linearly increasing temperature distribution. Mahdavi et al. [41] worked on the thermoplastic analysis of FG rotating hyperbolic disks by dividing the domain into some finite sub-domains in the radial direction, in which the properties were assumed to be constant. Recently, Jabbari et. al. [42] presented a thermoelastic analysis of rotating disks with different thickness profiles made of power-graded and exponential-graded materials subjected to internal pressure. They verified the results with finite element method.
As stated above, there are also relatively less study on the thermal-related analyses of isotropic and homogeneous disks with varying thickness. By using hyper-geometric differential equation in terms of radial displacement, Vivio and Vullo [43] and Vullo and Vivio [44] introduced an analytical procedure for evaluation of elastic stresses and strains in rotating conical disks and in non-linear variable thickness rotating disks made of an isotropic and homogeneous material, either solid or annular, subjected to thermal loads. In the presence of the linear thermal gradient, Garg et al. [45] analyzed the steady state creep in a rotating disk having linearly varying thickness and made of composite containing silicon carbide particles (SiC) in a matrix of pure aluminium. Çetin et al. [46] studied analytically an elastic stress analysis of annular isotropic and homogeneous bi-material hyperbolic disks subjected to the mechanical and thermomechanical loads.
As can be seen from the literature survey mentioned above, the realm of the thermal-related analyses of such disks having varying thicknesses needs further works.
In the present study, the exact distribution of the temperature along the radial coordinate in a hyperbolic disk is, first, obtained analytically based on the solution of Fourier heat-conduction differential equation under thermal boundary conditions defined by specified surface temperatures. This closed-form solution for temperature distribution is, then, substitute in the Navier equation obtained for the elasto-static response of such disks. Finally, Navier equations containing thermal effects are solved by applying mechanical boundary conditions. In the solution process of both the heat conduction and Navier equations, both of which are in the form of a second order differential equation with constant coefficients, a well-known Euler-Cauchi technique is employed [47]. Those formulas are compared with those available in the open literature.

Formulation of the Thermal Behavior of the Disk
An accurate solution of the temperature field in the structure is a crucial first step to study the thermal-related analyses in an appropriate manner. The rate of the heat flux, q , in a solid object is proportional to the temperature gradient, . The Fourier law governing the heat transfer by conduction is where k is the thermal conductivity. Temperature gradient is given in cylindrical coordinates, where ( z r e e e , ,  ) are the unit vectors in cylindrical coordinates, r is the radial coordinate,  is the circumferential coordinate. By using the first law of thermodynamics, Fourier heat conduction equation may be written as follows.
where  is the density, and is the specific heat capacity, ̇ is the heat generation per unit volume. Using Eq. (1) the following may be written for the divergence of the heat flux as By assuming that there is no heat generation in the structure, ̇= 0, and the steady state case ( ⁄ = 0) exists, substitution of Eq. (4) into the heat conduction equation (3) gives the Laplacian of the temperature that is the divergence of the gradient of the temperature as follows This is a differential equation governing steady state 3-D temperature distribution for both cylinders and discs having uniform thickness and constant thermal conductivity without heat generation. As observed from this equation and Eq. (1), both the temperature field and heat flux distribution will be the same for both cylinders and disks with uniform thickness. However, a disk with variable thickness offers different temperature and heat flux profiles than hollow cylinders and uniform disks even for a constant thermal conductivity. For axisymmetric problems (derivatives with respect to and are all zero) of a variable thickness disk, Eq. (5) takes the form of It can be concluded from the above that the temperature field is to be the same for all materials exhibiting both isotropy and homogeneity properties as in the traditional materials although the material type is to be of important in the heat flux calculations. For any thickness profile, ℎ( ), defined by a differentiable function, Eq. (6) may also be written as Now, consider a hyperbolic disc profile defined by where denotes the inner radius of the disk, is the profile parameter, and ℎ is the reference thickness (Fig. 1). While > 0 defines divergent hyperbolic profiles, < 0 identifies convergent ones. Uniform thickness disks are characterized by = 0 (Fig. 1).
This is an Euler-Cauchy type differential equation with constant coefficients. Solution of Eq. (9) is given by [47] where 1  and 2  are the distinct characteristic real roots of the differential equation while 1 C and 2 C are integration constants. All types of thermal boundary conditions such as Dirichlet's, Neumann's, Robin's and mixed boundary conditions may be applied to determine the integration constants of the physical problem. In the present study the first kind boundary conditions (Dirichlet) are to be considered.
In the above, a T and b T are the inner surface and outer surface temperatures, respectively. Considering Eq. (10) together with Eq. (11), a closed form solution for the temperature distribution in a hyperbolic disk along the radial coordinate is obtained as follows As seen from Eq. (12), temperature distribution in a hyperbolic disk becomes indefinite when the uniform thickness is concerned with = 0. In this case either a numerical value very close to zero but not exactly equal to zero such as = 0.000000000001 may be used directly in Eq. (12) or Eq. (9) is resolved for = 0 under the same boundary conditions [11] to get the following

Formulation of the Thermo-mechanical Behavior of the Disk
Under small deformations and a state of axisymmetric plane stress assumptions for thin plates, field equations of a variable thickness rotating disk made of a traditional material in polar coordinates ( , ) are reduced to where ( ) is the radial displacement, ( ) and ( ) are the radial and circumferential strains, respectively; r ( ) is the radial stress, ( )is the hoop stress, is a constant angular velocity, is the material density, is Young's modulus, is is the coefficient of expansion of the disc material and is Poisson's ratio.

Closed-form Solutions of Navier Equations
The closed-form solutions of both Eqs. (16) and (18) under each mechanical boundary conditions presented in Table 1 are to be given in this section. Those formulas may be used by recalling that the superposition principle holds.
The equivalent von-Mises stresses by an axisymmetric plane stress assumption may be computed by

V. Yıldırım
Equation (21) is a special application of formulas derived for functionally graded disks in Ref. [48]. Homogeneous solutions of Eq. (18) for free-free disks are ( = 0, = 0) Closed-form solutions in Eq. (24) overlaps with Roark's formulas [49]. For a uniform mounted disk subjected to the only external pressure, solutions become To the best of the author's knowledge, Eqs. (22) and (25) are offered for the first time in the present study. These equations may be directly used to better simulate aero-disks subjected to blade pressures.

Numerical Examples
where = is used for both free-free and fixed-fixed boundary conditions while = for fixed-free surfaces.
As a first example, the radial temperature distribution in a hyperbolic disk is investigated regarding different profile parameters and aspect ratios defined by ( / ). Solutions are illustrated by Fig. 2. As can be seen from Fig. 2 that the differences in the temperature distributions in a hyperbolic disk become much obvious as the aspect ratios get smaller. A convergent disk profile having = −1 exhibits a linear temperature distribution while the other profile parameters offer different curves. The temperatures at the intermediate surfaces of a disk having divergent profiles decrease faster than convergent ones towards the inner surface and increase slowly towards the outer. It is not suitable the use of a linear temperature profile instead of the true one, except = −1, for even uniform disks as seen from Fig. 2. Fig. 2. Variation of the radial temperature distribution with the aspect ratios and profile parameters As a second example, the radial variations of the elastic fields in a hyperbolic disk with the profile parameters are to be examined. Solutions for three boundary conditions are demonstrated by Fig. 3 in a comparative manner. The numerical values of some equivalent stresses are presented in Table 2. Fig. 3. The radial variations of the elastic fields in a hyperbolic disk with the profile parameters As can be observed from Fig. 3 that, the curves are in accordance with the related boundary conditions. Observations from this figure are outlined below: i. The maximum radial displacement is located at the outer surface for free-free and fixedfree boundary conditions while it is in the vicinity of the mid-surface for fixed-fixed hyperbolic disks. ii. The convergent profiles offer smaller radial displacements than divergent and uniform ones under free-free and fixed-free boundary conditions. The radial displacements in divergent hyperbolic disks are higher than even uniform profiles under free-free and fixed-free boundary conditions.
iii. The radial stress in free-free divergent hyperbolic disks is in compression at the vicinity of both the inner and outer surfaces while it is in tension at the vicinity of mid-surface. However, the radial stresses are all in compression in both uniform and convergent disks under free-free conditions. iv.
The maximum radial stress is at the inner surface under both free-free and fixed-free conditions except for = −0.4 in free-free and fixed-fixed disks. v. Convergent disk profiles exhibit better response to the radial stress variation than both uniform and divergent ones for fixed-free disks. vi.
Fixed-fixed disks have radial and hoop stresses which are completely in compression. Convergent disk profiles exhibit better response to the combined thermal and centrifugal loads than uniform and divergent ones under fixed-fixed boundary conditions. vii.
When the variation of the hoop stresses are concerned, convergent profiles seem to be better than divergent ones under free-free and fixed-fixed boundaries. The converse is true for fixed-free hyperbolic disks. viii. For fixed-free disks, divergent profiles having higher parameter values in absolute present better hoop stress distribution.  ix. For free-free disks, convergent profiles having smaller parameter values in absolute offer better hoop stress variations than divergent ones. For this boundary condition, the uniform disks exhibit the best response to the circumferential stresses together with = −0.2. x. As to the equivalent von-Misses stresses, fixed-free and fixed-fixed boundaries need convergent disks having higher parameter values in absolute while free-free disks require convergent disk having smaller parameter values. xi. Convergent profiles may exhibit almost uniform equivalent stresses under free-free boundary conditions. According to Fig. 3, = −0.2 is the best for the distribution of equivalent von-Misses stresses under free-free conditions.

Conclusions
Closed-form formulas were proposed for hyperbolic disks made of traditional materials under free-free, fixed-free, and fixed-fixed boundary conditions to obtain the exact thermomechanical fields. Combined thermal loads, internal and external pressures, and centrifugal forces were considered as thermo-mechanical loads. Those formulas are to be very helpful tools in the pre-design stage of such disks.
To the best of the author's knowledge, formulas for the hyperbolic fixed-free disks subjected to the external pressure induced by the existence of the blades have been originally offered in the present study.