Thermomechanical Characteristics of a Functionally Graded Mounted Uniform Disc with/without Rigid Casing

ABSTRACT

In the present paper, thermomechanical attributes of a power-law graded uniform mounted disc with or without rigid casing are investigated analytically under centrifugal and steady-state thermal loads within the framework of axisymmetric infinitesimal plane-stress elasticity theory. Young's modulus, material density, thermal expansion coefficient and thermal conductivity are all assumed to be constantly changed in the radial direction with different inhomogeneity indexes while Poisson's ratio is kept constant. Two parametric studies are conducted by both hypothetically and physically chosen metal-ceramic pairs namely nickel-silicon nitride (Ni-Si₃N₄), aluminum-aluminum oxide (Al-Al₂O₃), and stainless steel-zirconium oxide (SUS304-ZrO₂).

KEYWORDS:
Thermomechanical analysis; Exact solution; Functionally graded material; Circular annulus; Rotating disc

INTRODUCTION

After making the scene in Japan in the late 1980s, functionally graded materials (FGM), which have exceptional gradually changing mechanical and thermal properties along the favored directions of the structure, have gained great attention from scholars. The considerable number of studies has focused on the investigation of the elastic behavior of many types of FGM structures subjected to various loads and conditions. Those studies cover also disks as a basic structural element of diverse mechanical applications such as disk brakes, flywheels, gears, circular saws, hard disks, gas and steam turbine rotors, internal combustion engines, centrifugal compressors, and aerospace structural components. A disk or a circular annulus may simultaneously be subjected to centrifugal, pressure, thermal, magnetic, electrical and whatnot forces. The literature closely related to the theme of the present study is outlined forward.

Research related to only centrifugal force effects on elastic behavior of FGM discs is of the top priority in the open literature. Horgan and Chan (1999a) analytically examined the effects of material inhomogeneity on the response of power-law graded solid circular disks or cylinders, rotating at constant angular speeds. By employing a direct numerical integration and the finite element method, Durodola and Attia (2000) researched elastic fields in FG rotating disks. For exponentially graded uniform rotating annular disks under combinations of clamped and free boundary conditions, an elastic solution in terms of Whittaker's functions were presented by Zenkour (2005). Zenkour (2007) later handled a rotating FG annular disk with rigid casing. Eraslan and Akış (2006) exercised a parabolically varying thickness disk made of FGMs. Generalizing an available two-dimensional plane-stress solution to a three-dimensional one, Asghari and Ghafoori (2010) proposed a semi-analytical 3-D elasticity solution for rotating FG hollow and solid disks. Argeso (2012) considered analytically both a homogeneous non-uniform rotating disk, and a FG uniform disc by verifying the results with the nonlinear shooting method. Peng and Li (2012) utilized Fredholm integrals for elastic analysis of arbitrarily graded uniform solid rotating disks. Nejad et al. (2014) presented originally exact solutions for elasto-plastic deformations and stresses in a simple-power law graded rotating disk. They disclosed that the plasticity can occur in different regions of the disk. Dai and Dai (2015) investigated the angular speed variation of a FGM uniform rotating disc with the help of Runge-Kutta and shooting methods. Çallıoğlu et al. (2015) studied elastic-plastic stress analysis with non-work hardening case of power-law graded rotating discs; the results were verified by Ansys.

To consider variable-thickness FGM discs subjected to only centrifugal forces many methods such as semi-analytical methods in which the main domain is divided into uniform sub-domains (Bayat et al. 2008), the finite element method (Zafarmand and Hassani 2014), the finite difference method (Zheng et al. 2016), analytical (Yildirim 2016; 2018), the complementary functions method (Boğa and Yildirim 2017; Yildirim and Kacar 2017), and the transfer matrix method (Yildirim and Kacar 2017).

The issue of elastic analysis of an annulus or a disc subjected to only internal/external pressures has also been studied. Horgan and Chan (1999b) showed that the stress response of an inhomogeneous cylinder (or disk) subjected to pressure is significantly different from that of a homogeneous body. You et al. (2009) presented an analytical solution with hypergeometric functions for linearly graded uniform disks subjected to internal and/or external pressure. Tutuncu and Temel (2009) studied elastic analysis of FG hollow cylinders, disks and spheres subjected to only uniform internal pressure by the complementary functions method. Lotfian et al. (2011) presented a two-dimensional elasticity solution together with a finite element solution to study the elastic analysis of parabolically graded uniform disks subjected to both the internal and external pressures. Nejad et al. (2013) extended the previous study by Lotfian et al. (2011) to exponentially graded uniform discs subjected to internal and external pressures.

Yildirim (2016) proposed analytic solutions to power-law graded hyperbolic rotating discs subjected to both pressure and centrifugal loads under different boundary conditions. By exploiting Yildirim's (2016) closed-form formulas, Yildirim (2017a) conducted analytically a broad parametric study to investigate the effects of both the inhomogeneity parameter, and a profile index of Stodola's hyperbolic function on the static response of such structures subjected to both the inner and outer pressures. He observed that one of two materials in which Young's modulus is higher than the other is better to locate at the inner surface of the disc having divergent thickness profile to get reasonable maximum hoop stresses. However, much smaller radial displacements may be obtained by using positive inhomogeneity indexes for all discs in which inner surfaces host a material in which Young's modulus is smaller than the other.

Thermoelastic analyses of discs were also handled independently from the inertia and other forces. Noda (1991) has reviewed works in a wide range of topics conducted in 1980-1991. Tanigawa (1995) also performed a broad review on thermoelastic analysis of FGMs. Bakshi et al. (2006) worked on the coupled thermoelasticity of FG disks. Tokovy and Ma (2008) studied thermal stresses in anisotropic and radially inhomogeneous annular domains. Zenkour (2009) carried out an analytical study with Whittaker's functions to study a FG annular sandwich disk subjected to only steady-state thermal loads. After transforming the governing equation to a Fredholm integral equation, Peng and Li (2009) focused analytically and numerically on the thermoelastic analysis of either a power-law graded annulus or an arbitrarily graded annulus. Kurşun and Topcu (2013) achieved the elastic stress analysis of a power-law graded hyperbolic disk subjected to a linearly increasing temperature distribution under free-free and fixed-free boundary conditions. Based on the 2-D thermoelastic theories and finite difference method, Arnab et al. (2014) considered a thermoelastic analysis for a thin power-law and exponentially graded Al₂O₃/Al hollow disk subjected to only thermal loads. Under a logarithmic thermal gradient assumption, Aleksandrova (2016) investigated analytically elasto-plastic thermal stresses and deformations in a thin annular plate embedded into rigid container and made of a homogeneous and isotropic material.

Çallıoğlu et al. (2011) analytically handled the thermoelastic analysis of power-law graded stress-free annular discs subjected to both pressure and assumed temperature distributions. Kurşun et al. (2011) worked on the elastic stress analysis of power-law graded annular discs subjected to both uniform pressure on the inner surface and a linearly decreasing temperature distribution. Gönczi and Ecsedi (2015) studied analytically and numerically the thermo-mechanical behavior of a hollow power-law graded stress-free uniform circular disc under axisymmetric pressure and thermal loads.

Combined effects of magnetic and thermal loads on the elastic behavior of exponentially graded uniform annular discs were considered by Zenkour (2014).

Zenkour (2006) also proposed an analytical solution in terms of Whittaker's functions for exponentially graded uniform rotating annular disks under steady-state thermal and centrifugal loads. By dividing the radial domain into some virtual sub-domains, Kordkheili and Naghdabadi (2007) presented a semi-analytical solution for a thin axisymmetric uniform rotating traction-free disk made of FGM with power-law distribution of the volume fraction under centrifugal force and uniform thermal loads. By utilizing a finite element method, Go et al. (2010) demonstrated that a circular power-law graded free-free uniform cutter or grinding disk can be designed with better thermo-elastic characteristics if certain parameters are controlled properly. A finite element model based on the variational approach and Ritz method was developed by Afsar et al. (2010) to study the thermoelastic characteristics due to a thermal load and rotation of a thin uniform circular free-free rotating disk having an exponentially-graded coating at its outer surface. Based on the 2-D thermoelastic theories, Afsar and Go (2010) conducted a finite element analysis for the thermoelastic field in a thin circular exponentially graded Al₂O₃/Al disk subjected to a combined thermal and centrifugal loads. Afsar and Sohag (2010) considered the thermoelastic characteristics of a thin free-free disc having a FGM coating at the outer surface under thermal and centrifugal loads. Gong et al. (2014) used a finite volume method to study the 3-D thermoelastic analysis of FG uniform rotating discs. They showed that the least square method achieves better performances than the Gaussian method but least square method costs slightly more iteration and computer memory under different mesh types.

Thermomechanical behavior of variable-thickness FGM disks under thermal and centrifugal forces were investigated by using quite a few numerical and analytical methods: an analytical method with hypergeometric fuctions (Vivio and Vullo 2007; Vullo and Vivio 2008; Vivio et al. 2014), a semi-analytical method in which the radial domain is divided into sub-domains having uniform properties (Bayat et al. 2009a; 2009b; 2009c; 2010; Hassani et al. 2011; 2012a; Mahdavi et al. 2013; 2016; Jabbari et al. 2016; Dai and Dai 2016), He's variational iteration method (Hassani et al. 2011), Adomian's decomposition method (Hassani et al. 2011), Liao's homotopy analysis method (Hassani et al. 2011), Runge-Kutta's method (Hassani et al. 2011; 2012a; 2012b), finite elements (Hassani et al. 2012b; Mahdavi et al. 2013; 2016; Jabbari et al. 2016), the complementary functions method (Tutuncu and Temel 2013), dynamic relaxation method combined with the finite difference technique (Golmakani 2013), analytical methods (Çetin et al. 2014), finite difference method (Amin et al. 2015), and an approximate variational principle (Nayak and Saha 2016).

Temperature-dependent material properties were also included into thermoelastic analysis of FGM non-uniform rotating disks by Bayat et al. (2009c), Amin et al. (2015),Mahdavi et al. (2016), and Dai and Dai (2016).

Arani et al. (2010) and Bayat et al. (2014) studies cover combined effects of magnetic, thermal, and centrifugal loads on the elastic response of variable-thickness FG rotating discs based on a semi-analytical solution technique.

In the present study two types of disc are studied simultaneously: i) a disc mounted on a circular rigid shaft at its center having traction-free outer surface and subjected to both centrifugal and thermal loads (Fig. 1a), ii) a disc mounted on a rigid shaft at its center having a rigid casing at the outer surface under both centrifugal and thermal loads (Fig. 1b). Those discs are assumed to be made of both hypothetical and physical metal-ceramic pairs. A benchmark example, which was using only hypothetical inhomogeneity indexes for both infinite FGM cylinders (Jabbari et al. 2002) and FGM spheres (Eslami et al. 2005), is to be rehandled originally with those discs in Figs. 1 and 2 in the present study. Then another parametric study is to be conducted originally where the discs are assumed to be made of three types of physical metal-ceramic pairs. Separate and combined effects of both mechanical and thermal loads are to be investigated.

Figure 1
Mounted discs (a) without rigid casing (fi xed-free), and (b) with rigid casing (fixed-fixed).

DERIVATON OF NAVIER EQUATION

Consider a hollow disc of inside radius a, and outside radius b. Let's denote the radial and tangential coordinates by r, and θ, and use the prime symbol to indicate the derivatives with respect to the radial coordinate. Under axisymmetric plane-stress assumptions, the infinitesimal strain-displacement relationship for a disc is given by (Eq. 1)

(1) ${\mathit{\varepsilon }}_{\mathit{r}}\left(\mathit{r}\right)\mathbf{=}{{\mathit{u}}^{\prime }}_{\mathit{r}}\left(\mathit{r}\right)\mathbf{;}{\mathit{\varepsilon }}_{\mathit{\theta }}\left(\mathit{r}\right)\mathbf{=}\frac{{\mathit{u}}_{\mathit{r}}\left(\mathit{r}\right)}{\mathit{r}}$

where u_r is the radial displacement, ε_r is the unit radial strain and ε_θ is the unit tangential strain. Thermoelastic stress-strain constitutive relations for a FGM disc are used in the form of (Eq. 2)

(2) $\begin{array}{c}{\mathit{\sigma }}_{\mathit{r}}\left(\mathit{r}\right)\mathbf{=}{\mathit{C}}_{\mathbf{11}}\left(\mathit{r}\right){\mathit{\varepsilon }}_{\mathit{r}}\mathbf{+}{\mathit{C}}_{\mathbf{12}}\left(\mathit{r}\right){\mathit{\varepsilon }}_{\mathit{\theta }}\mathbf{-}\left({\mathit{C}}_{11}\left(\mathit{r}\right)+{\mathit{C}}_{12}\left(\mathit{r}\right)\right)\mathit{\alpha }\left(\mathit{r}\right)\mathit{T}\left(\mathit{r}\right)\\ {\mathit{\sigma }}_{\mathit{\theta }}\left(\mathit{r}\right)\mathbf{=}{\mathit{C}}_{\mathbf{12}}\left(\mathit{r}\right){\mathit{\varepsilon }}_{\mathit{r}}\mathbf{+}{\mathit{C}}_{\mathbf{11}}\left(\mathit{r}\right){\mathit{\varepsilon }}_{\mathit{\theta }}\mathbf{-}\left({\mathit{C}}_{11}\left(\mathit{r}\right)+{\mathit{C}}_{12}\left(\mathit{r}\right)\right)\mathit{\alpha }\left(\mathit{r}\right)\mathit{T}\left(\mathit{r}\right)\end{array}$

where σ_r is the radial stress, σ_θ is the hoop stress, α is the coefficient of thermal expansion, T(r) is the axisymmetric temperature distribution along the radial coordinate. The coefficients of unit strains in the above equations are defined as (Eq. 3)

(3) ${\mathit{C}}_{\mathbf{11}}\left(\mathit{r}\right)\mathbf{=}\frac{1}{1-{\mathit{v}}^2}\mathit{E}\left(\mathit{r}\right)\mathbf{;}{\mathit{C}}_{\mathbf{12}}\left(\mathit{r}\right)\mathbf{=}{\mathit{vC}}_{\mathbf{11}}\left(\mathit{r}\right)$

where E is elasticity modulus and v is Poisson's ratio. The arithmetic mean of Poisson's ratios of ceramic and material is used and kept constant along the radial coordinate in the present numerical calculations. The equilibrium equation in the radial coordinate of a rotating disc with a constant circular velocity, ω, is (Eq. 4)

(4) ${\left(\mathit{r}{\mathit{\sigma }}_{\mathit{r}}\left(\mathit{r}\right)\right)}^{\prime }\mathbf{-}{\mathit{\sigma }}_{\mathit{\theta }}\mathbf{=}{{\mathit{\sigma }}^{\prime }}_{\mathit{r}}\left(\mathit{r}\right)\mathbf{+}\frac{{\mathit{\sigma }}_{\mathit{r}}\left(\mathit{r}\right)-{\mathit{\sigma }}_{\mathit{\theta }}\left(\mathit{r}\right)}{\mathit{r}}\mathbf{=}\mathbf{-}\mathbf{\rho }\left(\mathit{r}\right){\mathit{\omega }}^{\mathbf{2}}\mathit{r}$

where ρ is the material density. After substitution of Eqs. 1 and 2 into Eq. 4, Navier equation in general form is obtained as follows (Eq. 5)

(5) $\begin{array}{c}{\mathit{u}}_{\mathit{r}}{\left(r\right)}^{″}+\left(\frac{1}{\mathit{r}}+\frac{{{\mathit{C}}^{\prime }}_{11}\left(\mathit{r}\right)}{{\mathit{C}}_{11}\left(\mathit{r}\right)}\right){{\mathit{u}}^{\prime }}_{\mathit{r}}\left(\mathit{r}\right)+\left(-\frac{1}{{\mathit{r}}^2}+\frac{\mathit{v}}{\mathit{r}}\frac{{{\mathit{C}}^{\prime }}_{11}\left(\mathit{r}\right)}{{\mathit{C}}_{11}\left(\mathit{r}\right)}\right){\mathit{u}}_{\mathit{r}}\left(\mathit{r}\right)\hfill \\ =-\frac{\mathit{\rho }\left(\mathit{r}\right){\mathit{\omega }}^2\mathit{r}}{{\mathit{C}}_{11}\left(\mathit{r}\right)}+\left(\frac{{{\mathit{C}}^{\prime }}_{11}\left(\mathit{r}\right)}{{\mathit{C}}_{11}\left(\mathit{r}\right)}\mathit{\alpha }\left(\mathit{r}\right)+{\mathit{\alpha }}^{\prime }\left(\mathit{r}\right)\right)\left(1+\mathit{v}\right)\mathit{T}\left(\mathit{r}\right)+\left(1+\mathit{v}\right)\mathit{\alpha }\left(\mathit{r}\right){\mathit{T}}^{\prime }\left(\mathit{r}\right)\hfill \end{array}$

Now, it is time to determine the temperature distribution along the radial coordinate in a uniform disc, T(r). After applying Fourier heat conduction equation with the first law of thermodynamics, the following differential equation governing the axisymmetric temperature distribution along the radial coordinate may be derived in cylindrical coordinates for discs made of arbitrarily graded FGMs (Eq. 6).

(6) $\frac{1}{\mathit{r}}{\left(\mathit{rk}\left(\mathit{r}\right){\mathit{T}}^{\prime }\left(\mathit{r}\right)\right)}^{\prime }={\mathit{T}}^{″}\left(\mathit{r}\right)+{\mathit{T}}^{\prime }\left(\mathit{r}\right)\left(\frac{1}{\mathit{r}}+\frac{{\mathit{k}}^{\prime }\left(\mathit{r}\right)}{\mathit{k}\left(\mathit{r}\right)}\right)=0$

where k(r) is the non-uniform thermal conduction coefficient. Yildirim (2017b) solved the above equation under steady-state Dirichlet's boundary conditions,$\mathit{T}\left(\mathit{a}\right)={\mathit{T}}_{\mathit{a}}\mathrm{and}\mathit{T}\left(\mathit{b}\right)={\mathit{T}}_{\mathit{b}}$, for a power-law graded disc,$\mathit{k}\left(\mathit{r}\right)={\mathit{k}}_{\mathit{a}}{\left(\mathit{r}/\mathit{a}\right)}^{\mathit{\mu }}$, as follows (Eq. 7)

(7) $\mathit{T}\left(\mathit{r}\right)=\frac{{\mathit{a}}^{\mathit{\mu }}{\mathit{T}}_{\mathit{a}}-{\mathit{b}}^{\mathit{\mu }}{\mathit{T}}_{\mathit{b}}}{{\mathit{a}}^{\mathit{\mu }}-{\mathit{b}}^{\mathit{\mu }}}+{\mathit{r}}^{-\mathit{\mu }}\frac{{\mathit{a}}^{\mathit{\mu }}{\mathit{b}}^{\mathit{\mu }}\left(-{\mathit{T}}_{\mathit{a}}+{\mathit{T}}_{\mathit{b}}\right)}{{\mathit{a}}^{\mathit{\mu }}-{\mathit{b}}^{\mathit{\mu }}}={\mathit{\Phi }}_1+{\mathit{r}}^{-\mathit{\mu }}{\mathit{\Phi }}_2$

The solution of Eq. 6 for isotropic and homogeneous materials, µ = 0, under the same boundary conditions takes the form of (Eq. 8)

(8) $\mathit{T}\left(\mathit{r}\right)=\mathit{ln}\mathit{r}\frac{{\mathit{T}}_{\mathit{a}}-{\mathit{T}}_{\mathit{b}}}{\mathit{ln}\left(\frac{\mathit{a}}{\mathit{b}}\right)}+\frac{\mathit{ln}{\mathit{aT}}_{\mathit{b}}-{\mathit{T}}_{\mathit{a}}\mathit{ln}\mathit{b}}{\mathit{ln}\left(\frac{\mathit{a}}{\mathit{b}}\right)}=\mathit{ln}\mathit{r}{\mathit{\Psi }}_2+{\mathit{\Psi }}_1$

Let's choose a simple power law grading rule with inhomogeneity indexes by β, q, and n (Eq. 9).

(9) $\mathit{E}\left(\mathit{r}\right)={\mathit{E}}_{\mathit{a}}{\left(\frac{\mathit{r}}{\mathit{a}}\right)}^{\mathit{\beta }};\mathit{\rho }\left(\mathit{r}\right)={\mathit{\rho }}_{\mathit{a}}{\left(\frac{\mathit{r}}{\mathit{a}}\right)}^{\mathit{q}};\mathit{\alpha }\left(\mathit{r}\right)={\mathit{\alpha }}_{\mathit{a}}{\left(\frac{\mathit{r}}{\mathit{a}}\right)}^{\mathit{n}}$

Substitute Eq. 7 into Eq. 5. Then the governing equation called Navier equation, which is a second order non-homogeneous Euler-Cauchy differential equation with constant coefficients, governs the thermomechanical behavior of such discs turns to be (Eq. 10)

(10) $\frac{\left(-1+\mathit{\beta }\mathit{v}\right)}{{\mathit{r}}^2}{\mathit{u}}_{\mathit{r}}\left(\mathit{r}\right)\mathbf{+}\frac{\left(1+\mathit{\beta }\right)}{\mathit{r}}{{\mathit{u}}^{\prime }}_{\mathit{r}}\left(\mathit{r}\right)\mathbf{+}{{\mathit{u}}^{″}}_{\mathit{r}}\left(\mathit{r}\right){\mathbf{=}}^{.}\mathbf{-}\frac{{\mathit{a}}^{-\mathit{q}+\mathit{\beta }}{\mathit{r}}^{1+\mathit{q}-\mathit{\beta }}\left(1-{\mathit{v}}^2\right){\mathit{\rho }}_{\mathit{a}}{\mathit{\omega }}^2}{{\mathit{E}}_{\mathit{a}}}\mathbf{+}{\mathit{r}}^{\mathbf{-}\mathbf{1}\mathbf{+}\mathit{n}\mathbf{-}\mathit{\mu }}\left({\mathit{r}}^{\mathit{\mu }}{∆}_1+{∆}_2\right)$

where (Eq. 11)

(11) $\begin{array}{c}{\mathbf{∆}}_{\mathbf{1}}\mathbf{=}{\mathit{a}}^{\mathbf{-}\mathit{n}}{\mathit{\alpha }}_{\mathit{a}}\left(\mathit{n}+\mathit{\beta }\right)\left(1+\mathit{v}\right){\mathit{\Phi }}_{\mathbf{1}}\hfill \\ {\mathbf{∆}}_{\mathbf{2}}\mathbf{=}{\mathit{a}}^{\mathbf{-}\mathit{n}}{\mathit{\alpha }}_{\mathit{a}}\left(\mathit{n}+\mathit{\beta }-\mathit{\mu }\right)\left(1+\mathit{v}\right){\mathit{\Phi }}_{\mathbf{2}}\hfill \end{array}$

It is possible to find a closed form general solution (homogeneous + particular) to Navier equation with constant coefficients as follows (Eq. 12)

(12) $\begin{array}{c}{\mathit{u}}_{\mathit{r}}\left(\mathit{r}\right)={\mathit{r}}^{-\mathit{\beta }}\left({\mathit{r}}^{\frac{\mathit{\beta }-\mathit{\xi }}{2}}\left({\mathit{B}}_1+{\mathit{B}}_2{\mathit{r}}^{\mathit{\xi }}\right)+{\mathit{r}}^{3+\mathit{q}}\mathit{\Omega }\right)\hfill \\ +{\mathit{∆}}_3\left({\mathit{r}}^{1+\mathit{n}-\mathit{\mu }}{∆}_4+{\mathit{r}}^{1+\mathit{n}}{∆}_1{∆}_5+{\mathit{C}}_1{\mathit{r}}^{\frac{1}{2}\left(-\mathit{\beta }-\mathit{\xi }\right)}{∆}_5{∆}_6+{\mathit{C}}_2{\mathit{r}}^{\frac{1}{2}\left(-\mathit{\beta }+\mathit{\xi }\right)}{∆}_5{∆}_6\right)\hfill \\ {\mathit{\sigma }}_{\mathit{r}}\left(\mathit{r}\right)=\frac{1}{2}{\mathit{r}}^{-1-\mathit{\beta }-\frac{\mathit{\xi }}{2}}{\mathit{C}}_{11}\left(\mathit{r}\right)\left(\mathit{r}{}^{\mathit{\beta }/2}\left(-{\mathit{B}}_1\left(\mathit{\beta }-2\mathit{v}+\mathit{\xi }\right)+{\mathit{B}}_2{\mathit{r}}^{\mathit{\xi }}\left(-\mathit{\beta }+2\mathit{v}+\mathit{\xi }\right)\right)\right.\hfill \\ \left.+2{\mathit{r}}^{3+\mathit{q}+\frac{\mathit{\xi }}{2}}\left(3+\mathit{q}-\mathit{\beta }+\mathit{v}\right)\mathit{\Omega }\right)\hfill \\ +{\mathit{C}}_{11}\left(\mathit{r}\right){\mathit{r}}^{\mathit{n}}{\mathit{∆}}_3\left({∆}_1{∆}_5\left(1+\mathit{n}+\mathit{v}\right)+{\mathit{r}}^{-\mathit{\mu }}{∆}_4\left(1+\mathit{n}-\mathit{\mu }+\mathit{v}\right)\right)\hfill \\ +\frac{1}{2}{\mathit{C}}_{11}\left(\mathit{r}\right){\mathit{r}}^{\frac{1}{2}\left(-2-\mathit{\beta }-\mathit{\xi }\right)}{\mathit{∆}}_3{∆}_5{∆}_6\left(-{\mathit{C}}_1\left(\mathit{\beta }-2\mathit{v}+\mathit{\xi }\right)+{\mathit{C}}_2{\mathit{r}}^{\mathit{\xi }}\left(-\mathit{\beta }+2\mathit{v}+\mathit{\xi }\right)\right)\hfill \\ +\frac{{\mathit{a}}^{-\mathit{n}-\mathit{\beta }}{\mathit{E}}_{\mathit{a}}{\mathit{r}}^{\mathit{n}+\mathit{\beta }-\mathit{\mu }}{\mathit{\alpha }}_{\mathit{a}}\left({\mathit{r}}^{\mathit{\mu }}{\mathit{\Phi }}_1+{\mathit{\Phi }}_2\right)}{-1+\mathit{v}}\hfill \\ {\mathit{\sigma }}_{\mathit{\theta }}\left(\mathit{r}\right)=\frac{1}{2}{\mathit{r}}^{-1-\mathit{\beta }-\frac{\mathit{\xi }}{2}}{\mathit{C}}_{11}\left(\mathit{r}\right)\left(-{\mathit{r}}^{\mathit{\beta }/2}\left(-{\mathit{B}}_2{\mathit{r}}^{\mathit{\xi }}\left(2-\mathit{\beta }\mathit{v}+\mathit{v}\mathit{\xi }\right)+{\mathit{B}}_1\left(-2+\mathit{v}\left(\mathit{\beta }+\mathit{\xi }\right)\right)\right)\right.\hfill \\ \left.+2{\mathit{r}}^{3+\mathit{q}+\frac{\mathit{\xi }}{2}}\left(1+\left(3+\mathit{q}-\mathit{\beta }\right)\mathit{v}\right)\mathit{\Omega }\right)\hfill \\ +{\mathit{C}}_{11}\left(\mathit{r}\right){\mathit{r}}^{\mathit{n}}{\mathit{∆}}_3\left({∆}_1{∆}_5\left(1+\mathit{v}+\mathit{nv}\right)+{\mathit{r}}^{-\mathit{\mu }}\left({∆}_4+{∆}_4\left(1+\mathit{n}-\mathit{\mu }\right)\mathit{v}\right)\right)\hfill \\ +\frac{1}{2}{\mathit{C}}_{11}\left(\mathit{r}\right){\mathit{r}}^{\frac{1}{2}\left(-2-\mathit{\beta }-\mathit{\xi }\right)}{\mathit{∆}}_3{∆}_5{∆}_6\left({\mathit{C}}_2{\mathit{r}}^{\mathit{\xi }}\left(2-\mathit{\beta }\mathit{v}+\mathit{v}\mathit{\xi }\right)-{\mathit{C}}_1\left(-2+\mathit{v}\left(\mathit{\beta }+\mathit{\xi }\right)\right)\right)\hfill \\ +\frac{{\mathit{a}}^{-\mathit{n}-\mathit{\beta }}{\mathit{E}}_{\mathit{a}}{\mathit{r}}^{\mathit{n}+\mathit{\beta }-\mathit{\mu }}{\mathit{\alpha }}_{\mathit{a}}\left({\mathit{r}}^{\mathit{\mu }}{\mathit{\Phi }}_1+{\mathit{\Phi }}_2\right)}{-1+\mathit{v}}\hfill \end{array}$

where (Eq. 13)

(13) $\begin{array}{c}\mathit{\xi }=\sqrt{4+{\mathit{\beta }}^2-4\mathit{\beta }\mathit{v}}\\ \Omega =\frac{{\mathit{a}}^{-\mathit{q}+\mathit{\beta }}\left(-1+{\mathit{v}}^2\right){\mathit{\rho }}_{\mathit{a}}{\mathit{\omega }}^2}{{\mathit{E}}_{\mathit{a}}\left(8+\mathit{q}\left(6+\mathit{q}-\mathit{\beta }\right)-3\mathit{\beta }+\mathit{\beta }\mathit{v}\right)}\\ {\mathit{∆}}_3\frac{16}{\left(-2-2\mathit{n}-\mathit{\beta }+\mathit{\xi }\right)\left(2+2\mathit{n}+\mathit{\beta }+\mathit{\xi }\right)\left(2+2\mathit{n}+\mathit{\beta }-2\mathit{\mu }+\mathit{\xi }\right)\left(-2-2\mathit{n}-\mathit{\beta }+2\mathit{\mu }+\mathit{\xi }\right)}\\ {∆}_4={∆}_2\left(\mathit{\beta }+\mathit{n}\left(2+\mathit{n}+\mathit{\beta }\right)+\mathit{\beta }\mathit{v}\right)\\ {∆}_5={\mathit{n}}^2+\mathit{n}\left(2+\mathit{\beta }-2\mathit{\mu }\right)+\left(-2+\mathit{\mu }\right)\mathit{\mu }+\mathit{\beta }\left(1-\mathit{\mu }+\mathit{v}\right)\\ {∆}_6=\mathit{\beta }+\mathit{n}\left(2+\mathit{n}+\mathit{\beta }\right)+\mathit{\beta }\mathit{v}\end{array}$

Integration constants in the solution of Eq. 10 are determined under fixed-free boundary conditions (Fig. 1a), u_r (a) = 0 and σ_r (b) = 0, as (Eq. 14)

(14) $\begin{array}{c}{\mathit{B}}_1=\frac{\mathit{\Omega }{\mathit{a}}^{\frac{\mathit{\xi }-\mathit{\beta }}{2}}{\mathit{b}}^{\frac{\mathit{\xi }-\mathit{\beta }}{2}}\left({\mathit{a}}^{\mathit{q}+3}{\mathit{b}}^{\frac{\mathit{\beta }+\mathit{\xi }}{2}}\left(\mathit{\beta }-2\mathit{v}-\mathit{\xi }\right)-2{\mathit{b}}^{\mathit{q}+3}{\mathit{a}}^{\frac{\mathit{\beta }+\mathit{\xi }}{2}}\left(\mathit{\beta }-\mathit{v}-\mathit{q}-3\right)\right)}{{\mathit{a}}^{\mathit{\xi }}\left(\mathit{\beta }-2\mathit{v}+\mathit{\xi }\right)+{\mathit{b}}^{\mathit{\xi }}\left(-\mathit{\beta }+2\mathit{v}+\mathit{\xi }\right)}\\ {\mathit{B}}_2=-\frac{\mathit{\Omega }{\mathit{a}}^{-\mathit{\beta }/2}{\mathit{b}}^{-\mathit{\beta }/2}\left({\mathit{b}}^{\mathit{\beta }/2}{\mathit{a}}^{\frac{\mathit{\xi }}{2}+\mathit{q}+3}\left(\mathit{\beta }-2\mathit{v}+\mathit{\xi }\right)-2{\mathit{a}}^{\mathit{\beta }/2}{\mathit{b}}^{\frac{\mathit{\xi }}{2}+\mathit{q}+3}\left(\mathit{\beta }-\mathit{v}-\mathit{q}-3\right)\right)}{{\mathit{a}}^{\mathit{\xi }}\left(\mathit{\beta }-2\mathit{v}+\mathit{\xi }\right)+{\mathit{b}}^{\mathit{\xi }}\left(-\mathit{\beta }+2\mathit{v}+\mathit{\xi }\right)}\\ {\mathit{C}}_1=\frac{{\mathit{b}}^{\frac{1}{2}\left(\mathit{\xi }-2\mathit{\mu }\right)}{\mathit{a}}^{\frac{1}{2}\left(\mathit{\xi }-2\left(\mathit{\mu }+\mathit{n}\right)\right)}}{{\mathit{∆}}_3{∆}_5{∆}_6\left({\mathit{a}}^{\mathit{\xi }}\left(\mathit{\beta }-2\mathit{v}+\mathit{\xi }\right)+{\mathit{b}}^{\mathit{\xi }}\left(-\mathit{\beta }+2\mathit{v}+\mathit{\xi }\right)\right)}\left({\mathit{∆}}_3{\mathit{a}}^{\frac{\mathit{\beta }}{2}+2\mathit{n}+1}{\mathit{b}}^{\mathit{\mu }+\frac{\mathit{\xi }}{2}}\left(\mathit{\beta }-2\mathit{v}-\mathit{\xi }\right)\left({∆}_1{∆}_5{\mathit{a}}^{\mathit{\mu }}+{∆}_4\right)\right.\\ +2{\mathit{a}}^{\mathit{\mu }+\frac{\mathit{\xi }}{2}}{\mathit{b}}^{\frac{\mathit{\beta }}{2}+\mathit{n}+1}\left({\mathit{∆}}_3{\mathit{a}}^{\mathit{n}}\left({∆}_1{∆}_5{\mathit{b}}^{\mathit{\mu }}\left(\mathit{v}+\mathit{n}+1\right)+{∆}_4\left(-\mathit{\mu }+\mathit{v}+\mathit{n}+1\right)\right)\right.\\ \left.\left.-{\mathit{\alpha }}_{\mathit{a}}\left(\mathit{v}+1\right)\left({\mathit{\Phi }}_1{\mathit{b}}^{\mathit{\mu }}+{\mathit{\Phi }}_2\right)\right)\right)\\ {\mathit{C}}_2=\frac{{\mathit{b}}^{-\mathit{\mu }}{\mathit{a}}^{-\mathit{\mu }-\mathit{n}}}{{\mathit{∆}}_3{∆}_5{∆}_6\left({\mathit{a}}^{\mathit{\xi }}\left(\mathit{\beta }-2\mathit{v}+\mathit{\xi }\right)+{\mathit{b}}^{\mathit{\xi }}\left(-\mathit{\beta }+2\mathit{v}+\mathit{\xi }\right)\right)}\\ \left(-{∆}_3{\mathit{b}}^{\mathit{\mu }}\left(\mathit{\beta }-2\mathit{v}+\mathit{\xi }\right){\mathit{a}}^{\frac{1}{2}\left(\mathit{\beta }+4\mathit{n}+\mathit{\xi }+2\right)}\left({∆}_1{∆}_5{\mathit{a}}^{\mathit{\mu }}+{∆}_4\right)\right.\\ -2{\mathit{a}}^{\mathit{\mu }}{\mathit{b}}^{\frac{1}{2}\left(\mathit{\beta }+2\mathit{n}+\mathit{\xi }+2\right)}\left({\mathit{∆}}_3{\mathit{a}}^{\mathit{n}}\left({∆}_1{∆}_5{\mathit{b}}^{\mathit{\mu }}\left(\mathit{v}+\mathit{n}+1\right)\right.+{∆}_4\left(-\mathit{\mu }+\mathit{v}+\mathit{n}+1\right)\right)\\ \left.\left.-{\mathit{\alpha }}_{\mathit{a}}\left(\mathit{v}+1\right)\left({\mathit{\Phi }}_1{\mathit{b}}^{\mathit{\mu }}+{\mathit{\Phi }}_2\right)\right)\right)\end{array}$

and under fixed-fixed boundary conditions (Fig. 1b), u_r (a) = 0 and u_r (b) = 0 as (Eq. 15)

(15) $\begin{array}{c}{\mathit{B}}_1=-\frac{\mathit{\Omega }{\mathit{a}}^{\mathit{\xi }}{\mathit{b}}^{-\frac{\mathit{\beta }}{2}+\frac{\mathit{\xi }}{2}+\mathit{q}+3}-\mathit{\Omega }{\mathit{b}}^{\mathit{\xi }}{\mathit{a}}^{-\frac{\mathit{\beta }}{2}+\frac{\mathit{\xi }}{2}+\mathit{q}+3}}{{\mathit{a}}^{\mathit{\xi }}-{\mathit{b}}^{\mathit{\xi }}}\hfill \\ {\mathit{B}}_2=\frac{\mathit{\Omega }\left({\mathit{b}}^{-\frac{\mathit{\beta }}{2}+\frac{\mathit{\xi }}{2}+\mathit{q}+3}-{\mathit{a}}^{-\frac{\mathit{\beta }}{2}+\frac{\mathit{\xi }}{2}+\mathit{q}+3}\right)}{{\mathit{a}}^{\mathit{\xi }}-{\mathit{b}}^{\mathit{\xi }}}\hfill \\ {\mathit{C}}_1=-\frac{{\mathit{a}}^{\mathit{\xi }}\left({∆}_1{∆}_5{\mathit{b}}^{\mathit{\mu }}+{∆}_4\right){\mathit{b}}^{\frac{1}{2}\left(\mathit{\beta }-2\mathit{\mu }+2\mathit{n}+\mathit{\xi }+2\right)}-{\mathit{b}}^{\mathit{\xi }}\left({∆}_1{∆}_5{\mathit{a}}^{\mathit{\mu }}+{∆}_4\right){\mathit{a}}^{\frac{1}{2}\left(\mathit{\beta }-2\mathit{\mu }+2\mathit{n}+\mathit{\xi }+2\right)}}{{∆}_5{∆}_6\left({\mathit{a}}^{\mathit{\xi }}-{\mathit{b}}^{\mathit{\xi }}\right)}\hfill \\ {\mathit{C}}_2=-\frac{\left({∆}_1{∆}_5{\mathit{a}}^{\mathit{\mu }}+{∆}_4\right){\mathit{a}}^{\frac{1}{2}\left(\mathit{\beta }-2\mathit{\mu }+2\mathit{n}+\mathit{\xi }+2\right)}-\left({∆}_1{∆}_5{\mathit{b}}^{\mathit{\mu }}+{∆}_4\right){\mathit{b}}^{\frac{1}{2}\left(\mathit{\beta }-2\mathit{\mu }+2\mathit{n}+\mathit{\xi }+2\right)}}{{∆}_5{∆}_6\left({\mathit{a}}^{\mathit{\xi }}-{\mathit{b}}^{\mathit{\xi }}\right)}\hfill \end{array}$

If the disk material is homogeneous and isotropic, then Eq. 10 gives the elastic fields under fixed- free boundary conditions as (Eq. 16) (Yildirim 2017c)

(16) $\begin{array}{c}{\mathit{u}}_{\mathit{r}}\left(\mathit{r}\right)=\frac{\left({\mathit{v}}^2-1\right){\mathit{\rho }\mathit{\omega }}^2\left(\mathit{a}-\mathit{r}\right)\left(\mathit{a}+\mathit{r}\right)\left({\mathit{a}}^2\left({\mathit{b}}^2\left(\mathit{v}+1\right)-\left(\mathit{v}-1\right){\mathit{r}}^2\right)+{\mathit{b}}^2\left(\left(\mathit{v}+1\right){\mathit{r}}^2-{\mathit{b}}^2\left(\mathit{v}+3\right)\right)\right)}{8\mathit{Er}\left({\mathit{a}}^2\left(\mathit{v}-1\right)-{\mathit{b}}^2\left(\mathit{v}+1\right)\right)}\hfill \\ +\frac{1}{2{\mathit{a}}^2\left(\mathit{v}-1\right)\mathit{r}-2{\mathit{b}}^2\left(\mathit{v}+1\right)\mathit{r}}\left\{{\mathit{a}}^2\mathit{\kappa }\mathit{ln}\mathit{a}\left({\mathit{b}}^2\left(\mathit{v}+1\right)-\left(\mathit{v}-1\right){\mathit{r}}^2\right)-\mathit{\kappa }{\mathit{r}}^2\mathit{ln}\mathit{r}\right.\left({\mathit{b}}^2\left(\mathit{v}+1\right)-{\mathit{a}}^2\left(\mathit{v}-1\right)\right)\hfill \\ \left.-{\mathit{b}}^2\left(\mathit{a}-\mathit{r}\right)\left(\mathit{a}+\mathit{r}\right)\left(-2\mathit{\alpha }\left(\mathit{v}+1\right){\mathit{\Psi }}_1+\left(\mathit{v}+1\right)\mathit{ln}\mathit{b}\left(\mathit{\kappa }-2\mathit{\alpha }{\Psi }_2\right)+\mathit{\kappa }\right)\right\}\hfill \\ {\mathit{\sigma }}_{\mathit{r}}\left(\mathit{r}\right)=\frac{{\mathit{\rho }\mathit{\omega }}^2\left({\mathit{r}}^2-{\mathit{b}}^2\right)\left({\mathit{a}}^4\left({\mathit{v}}^2-1\right)-{\mathit{a}}^2\left(\mathit{v}-1\right)\left(\mathit{v}+3\right)\left({\mathit{b}}^2+{\mathit{r}}^2\right)+{\mathit{b}}^2\left(\mathit{v}+1\right)\left(\mathit{v}+3\right){\mathit{r}}^2\right)}{8{\mathit{r}}^2\left({\mathit{a}}^2\left(\mathit{v}-1\right)-{\mathit{b}}^2\left(\mathit{v}+1\right)\right)}\hfill \\ +\frac{1}{2\left({\mathit{v}}^2-1\right){\mathit{r}}^2\left({\mathit{a}}^2\left(\mathit{v}-1\right)-{\mathit{b}}^2\left(\mathit{v}+1\right)\right)}\left\{E\left(\left(\mathit{v}+1\right)\left(\mathit{\kappa }-2\mathit{\alpha }{\mathit{\Psi }}_2\right)\left({\mathit{b}}^2\mathit{ln}\mathit{b}\right.\left({\mathit{a}}^2\left(\mathit{v}-1\right)-\left(\mathit{v}+1\right){\mathit{r}}^2\right)\right.\right.+\hfill \\ \left.+{\mathit{r}}^2\mathit{ln}\mathit{r}\left({\mathit{b}}^2\left(\mathit{v}+1\right)-{\mathit{a}}^2\left(\mathit{v}-1\right)\right)\right)+{\mathit{a}}^2\left(\mathit{v}-1\right)\left(\mathit{b}-\mathit{r}\right)\left(\mathit{b}+\mathit{r}\right)\left(\mathit{\kappa }-2\mathit{\alpha }\left(\mathit{v}+1\right){\mathit{\Psi }}_1\right)-\hfill \\ \left.\left.-{\mathit{a}}^2\kappa \left({\mathit{v}}^2-1\right)\mathit{ln}\mathit{a}\left(\mathit{b}-\mathit{r}\right)\left(\mathit{b}+\mathit{r}\right)\right)\right\}\hfill \\ {\mathit{\sigma }}_{\mathit{\theta }}\left(\mathit{r}\right)=\frac{1}{8{\mathit{r}}^2\left({\mathit{a}}^2\left(\mathit{v}-1\right)-{\mathit{b}}^2\left(\mathit{v}+1\right)\right)}\left({\mathit{\rho }\mathit{\omega }}^2\left({\mathit{a}}^4\left({\mathit{v}}^2-1\right)\left({\mathit{b}}^2+{\mathit{r}}^2\right)-{\mathit{a}}^2\left(\mathit{v}-1\right)\left({\mathit{b}}^4\left(\mathit{v}+3\right)+\right.\right.\right.\hfill \\ \left.\left.\left.+\left(3\mathit{v}+1\right){\mathit{r}}^4\right)-{\mathit{b}}^2\left(\mathit{v}+1\right){\mathit{r}}^2\left({\mathit{b}}^2\left(\mathit{v}+3\right)-\left(3\mathit{v}+1\right){\mathit{r}}^2\right)\right)\right)\hfill \\ +\frac{1}{2\left({\mathit{v}}^2-1\right){\mathit{r}}^2\left({\mathit{a}}^2\left(\mathit{v}-1\right)-{\mathit{b}}^2\left(\mathit{v}+1\right)\right)}\left\{E\left({\mathit{a}}^2\left(\mathit{v}-1\right)\left(2\mathit{\alpha }\left(\mathit{v}+1\right){\mathit{\Psi }}_1\left(\mathit{b}{}^2+{\mathit{r}}^2\right)-\mathit{\kappa }\left({\mathit{b}}^2+{\mathit{vr}}^2\right)\right)\right.\right.\hfill \\ \left.\left.\left.-{\mathit{b}}^2\ln \mathit{b}\left({\mathit{a}}^2\left(\mathit{v}-1\right)+\left(\mathit{v}+1\right){\mathit{r}}^2\right)\right)+{\mathit{a}}^2\kappa \left({\mathit{v}}^2-1\right)\mathit{ln}\mathit{a}\left({\mathit{b}}^2+{\mathit{r}}^2\right)+{\mathit{b}}^2\kappa \left({\mathit{v}}^2-1\right){\mathit{r}}^2\right)\right\}\hfill \end{array}$

under fixed-fixed boundary conditions as (Eq. 17)

(17) $\begin{array}{c}{\mathit{u}}_{\mathit{r}}\left(\mathit{r}\right)=\frac{\left({\mathit{v}}^2-1\right){\mathit{\rho }\mathit{\omega }}^2\left({\mathit{r}}^2-{\mathit{a}}^2\right)\left({\mathit{r}}^2-{\mathit{b}}^2\right)}{8\mathit{Er}}\hfill \\ +\frac{\mathit{\kappa }\left({\mathit{b}}^2\left({\mathit{r}}^2-{\mathit{a}}^2\right)\mathit{ln}\mathit{b}+{\mathit{a}}^2\mathit{ln}\mathit{a}\left(\mathit{b}-\mathit{r}\right)\left(\mathit{b}+\mathit{r}\right)+{\mathit{r}}^2\left(\mathit{a}-\mathit{b}\right)\left(\mathit{a}+\mathit{b}\right)\mathit{ln}\mathit{r}\right)}{2\mathit{r}\left(\mathit{a}-\mathit{b}\right)\left(\mathit{a}+\mathit{b}\right)}\hfill \\ {\mathit{\sigma }}_{\mathit{r}}\left(\mathit{r}\right)=\frac{{\mathit{\rho }\mathit{\omega }}^2\left({\mathit{a}}^2\left(\left(\mathit{v}+1\right){\mathit{r}}^2-{\mathit{b}}^2\left(\mathit{v}-1\right)\right)+{\mathit{r}}^2\left({\mathit{b}}^2\left(\mathit{v}+1\right)-\left(\mathit{v}+3\right){\mathit{r}}^2\right)\right)}{8{\mathit{r}}^2}\hfill \\ +\frac{\mathit{E}}{2\left(\mathit{v}-1\right)\left(\mathit{v}+1\right){\mathit{r}}^2\left(\mathit{a}-\mathit{b}\right)\left(\mathit{a}+\mathit{b}\right)}\left\{\left({\mathit{a}}^2\mathit{\kappa }\mathit{ln}\mathit{a}\left(\left(\mathit{v}+1\right){\mathit{r}}^2-{\mathit{b}}^2\left(\mathit{v}-1\right)\right)\right.\right.\hfill \\ +{\mathit{b}}^2\mathit{\kappa }\mathit{ln}\mathit{b}\left({\mathit{a}}^2\left(\mathit{v}-1\right)-\left(\mathit{v}+1\right){\mathit{r}}^2\right)\hfill \\ \left.\left.-{\mathit{r}}^2\left(\mathit{a}-\mathit{b}\right)\left(\mathit{a}+\mathit{b}\right)\left(-2\mathit{\alpha }\left(\mathit{v}+1\right){\mathit{\Psi }}_1+\mathit{\kappa }+\left(\mathit{v}+1\right)\mathit{ln}\mathit{r}\left(\mathit{\kappa }-2\mathit{\alpha }{\mathit{\Psi }}_2\right)\right)\right)\right\}\hfill \\ {\mathit{\sigma }}_{\mathit{\theta }}\left(\mathit{r}\right)=\frac{{\mathit{\rho }\mathit{\omega }}^2\left({\mathit{a}}^2\left({\mathit{b}}^2\left(\mathit{v}-1\right)+\left(\mathit{v}+1\right){\mathit{r}}^2\right)+{\mathit{r}}^2\left({\mathit{b}}^2\left(\mathit{v}+1\right)-\left(3\mathit{v}+1\right){\mathit{r}}^2\right)\right)}{8{\mathit{r}}^2}\hfill \\ +\frac{\mathit{E}}{2\left(\mathit{v}-1\right)\left(\mathit{v}+1\right){\mathit{r}}^2\left(\mathit{a}-\mathit{b}\right)\left(\mathit{a}+\mathit{b}\right)}\left\{\left({\mathit{a}}^2\mathit{\kappa }\mathit{ln}\mathit{a}\left({\mathit{b}}^2\left(\mathit{v}-1\right)+\left(\mathit{v}+1\right){\mathit{r}}^2\right)\right.\right.\hfill \\ -{\mathit{b}}^2\mathit{\kappa }\mathit{ln}\mathit{b}\left({\mathit{a}}^2\left(\mathit{v}-1\right)+\left(\mathit{v}+1\right){\mathit{r}}^2\right)\hfill \\ \left.\left.-{\mathit{r}}^2\left(\mathit{a}-\mathit{b}\right)\left(\mathit{a}+\mathit{b}\right)\left(-2\mathit{\alpha }\left(\mathit{v}+1\right){\mathit{\Psi }}_1+\mathit{\kappa }\mathit{v}+\left(\mathit{v}+1\right)\mathit{ln}\mathit{r}\left(\mathit{\kappa }-2\mathit{\alpha }{\mathit{\Psi }}_2\right)\right)\right)\right\}\hfill \end{array}$

where (Eq. 18)

(18) $\mathit{\kappa }\mathbf{=}\mathit{\alpha }\left(1+\mathit{v}\right){\mathit{\Psi }}_{\mathbf{2}}$

EXAMPLES WITH HYPOTHETICAL FGMS

Let's begin with Jabbari et al. (2002) and Eslami et al. (2005) benchmark example data by adding ω.

They studied the thermomechanical analysis of both traction-free cylinders (Jabbari et al. 2002) and spheres (Eslami et al. 2005), which are subjected to both internal pressure and surface temperature differences. In the present study the same example is to be solved originally for mounted disks (Fig. 1). Let's assume that the homogeneity indexes are to be all the same, -3 ≤ β = q = µ = n ≤ 3. Based on von-Mises failure criterion, equivalent stress is defined by (Eq. 19)

(19) ${\mathit{\sigma }}_{\mathit{eq}}\left(\mathit{r}\right)\mathbf{=}\sqrt{{\mathit{\sigma }}_{\mathit{r}}^2-{\mathit{\sigma }}_{\mathit{r}}{\mathit{\sigma }}_{\mathit{\theta }}+{\mathit{\sigma }}_{\mathit{\theta }}^2}$

Some of the results are presented in Tables 1 and 2. It is worth noting that, for the disks made of isotropic and homogeneous materials, β = q = n = µ = 0, Eqs. 16 and 17 are used in all examples instead of Eq. 12 for FGM discs. Variation of the dimensionless elastic fields with the inhomogeneity index for fixed-free and fixed-fixed boundary conditions are presented in Figs. 2 and 3. Figure 2 implies that

Table 1
Variation of the dimensionless elastic fields with the inhomogeneity index under individual and combined loads for mounted rotating disks.

	ur /a		σr /pa		σθ /pa
r/a	β = -2	β = 2	β = -2	β = 2	β = -2	β = 2
Fixed-free
Thermal
1.00	0.00	0.00	0.00327254	0.00459112	-0.0470182	-0.0466227
1.04	0.0000005647	0.00000059	0.00166211	0.00276367	-0.0309148	-0.0389849
1.08	0.0000009674	0.00000105	0.00069278	0.00138161	-0.0186591	-0.0298611
1.12	0.0000012298	0.00000137	0.00017605	0.00045252	-0.00931328	-0.0191255
1.16	0.0000013704	0.00000155	-0.00002269	-0.00001265	-0.00218269	-0.00664791
1.2	0.0000014047	0.00000161	0.00	0.00	0.00325169	0.00770582
Centrifugal
1.00	0.00	0.00	0.300775	0.444601	0.0902324	0.13338
1.04	0.0000025014	0.00000354	0.232462	0.367487	0.0786334	0.124969
1.08	0.0000045049	0.00000614	0.168738	0.285286	0.064926	0.112092
1.12	0.0000059694	0.00000788	0.109049	0.197175	0.04971	0.0944611
1.16	0.0000068521	0.00000885	0.0529312	0.102346	0.0334387	0.0717666
1.2	0.0000071089	0.00000910	0.00	0.00	0.0164559	0.0436823
Combined (thermal + centrifugal)
1.	0.00	0.00	0.304047	0.449192	0.0432142	0.0867576
1.04	0.0000030660	0.00000413	0.234124	0.370251	0.0477187	0.0859837
1.08	0.0000054723	0.00000719	0.169431	0.286667	0.0462669	0.082231
1.12	0.0000071992	0.00000925	0.109225	0.197627	0.0403967	0.0753356
1.16	0.0000082224	0.00001040	0.0529085	0.102334	0.031256	0.0651187
1.2	0.0000085137	0.00001071	0.00	0.00	0.0197076	0.0513882
Fixed-fixed
Thermal
1.	0.00	0.00	-0.0245472	-0.0414957	-0.0553642	-0.0604487
1.04	0.0000003063	0.00000020	-0.0254226	-0.042107	-0.0399588	-0.0540915
1.08	0.0000004400	0.00000030	-0.0257345	-0.0424101	-0.0282619	-0.0462308
1.12	0.0000004220	0.00000031	-0.0256586	-0.0423822	-0.0193636	-0.0367431
1.16	0.0000002703	0.00000021	-0.0253195	-0.0419991	-0.0125908	-0.0255005
1.2	0.00	0.00	-0.0248054	-0.0412358	-0.00744163	-0.0123707
Centrifugal
1.	0.00	0.00	0.159987	0.183347	0.047996	0.0550041
1.04	0.0000011941	0.00000130	0.0953936	0.113127	0.0328641	0.0393328
1.08	0.0000018358	0.00000189	0.0349973	0.0370416	0.0163286	0.0192969
1.12	0.0000018813	0.00000185	-0.0216937	-0.0456441	-0.00115191	-0.00540828
1.16	0.0000012849	0.00000121	-0.0750889	-0.135664	-0.019234	-0.0351038
1.2	0.00	0.00	-0.125533	-0.233755	-0.03766	-0.0701265
Combined (thermal + centrifugal)
1.	0.00	0.00	0.135439	0.141851	-0.0073682	-0.00544463
1.04	0.0000015004	0.00000150	0.069971	0.0710199	-0.00709467	-0.0147588
1.08	0.0000022758	0.00000220	0.00926279	-0.00536849	-0.0119334	-0.0269339
1.12	0.0000023033	0.00000216	-0.0473523	-0.0880263	-0.0205155	-0.0421513
1.16	0.0000015552	0.00000141	-0.100408	-0.177663	-0.0318248	-0.0606043
1.2	0.00	0.00	-0.150339	-0.274991	-0.0451017	-0.0824972

Table 2
Variation of the dimensionless equivalent stress under individual and combined loads for mounted rotating disks and β = 2, - 2.

σeq /pa
r/a	β = -2	β = 2	β = -2	β = 2
	Fixed-free		Fixed-fixed
1.	0.284909	0.41271	0.13927	0.14465
1.04	0.214287	0.335624	0.0737747	0.0794344
1.08	0.151685	0.25567	0.0184049	0.0246914
1.12	0.0956535	0.172753	0.0411299	0.0762557
1.16	0.046071	0.089715	0.0888773	0.156429
1.2	0.0197076	0.0513882	0.133624	0.244417

Figure 2
Variation of the dimensionless elastic fields with the inhomogeneity index under individual and combined loads for fixed-free boundary conditions.

As the inhomogeneity indexes change from negative to positive, the elastic fields get higher.

• The maximum radial displacement is at the outer surface, and it increases with increasing r/a ratios under fixed-free boundary conditions.

• The maximum radial stress is at the inner surface, and it decreases with increasing r/a ratios.

• All the stresses, except the hoop stress due to thermal loads, are in tension.

• The maximum hoop stress is mostly at the inner surface.

• The radial thermal stresses for fixed-free boundary conditions are completely in tension.

• Effects of the thermal loads on the hoop stresses are significantly higher than those on the radial ones under fixed-free surfaces.

• The maximum equivalent stress is located at the inner surface and decreases with increasing r/a ratios.

Figure 3 suggests that

Figure 3
Variation of the dimensionless elastic fields with the inhomogeneity index under individual and combined loads for fixed-fixed boundary conditions.

• The maximum radial displacement is at the vicinity of the mid-surface.

• The maximum radial stress is at the outer surface under individual centrifugal and combined loads. Magnitudes of the radial stresses are much greater than the hoop stresses under rotation.

• Both individual radial and hoop stresses due to rotation are in tension-compression character.

• The maximum hoop stress is at the outer surface under individual thermal, individual centrifugal and combined loads.

• The radial thermal stresses are in compression under fixed-fixed boundary conditions. As stated before, for fixed-free boundary conditions they turn completely into tension.

• The maximum equivalent stress is located at the outer surface. Positive inhomogeneity indexes offer the highest magnitudes.

EXAMPLES WITH PHYSICAL FGMS

In the present work three types of functionally graded materials (FGMs) are considered to study the thermomechanical behavior of such discs under centrifugal and steady-state thermal loads. Constituent material properties have been given in Table 3 for nickel-silicon nitride (Ni-Si₃N₄), aluminum-aluminum oxide (Al-Al₂O₃), and stainless steel-zirconium oxide (SUS304-ZrO₂). The other properties are as follows: ; T_a = 373 K; T_b = 273 K; a = 0.5 m; b = 1.0 m. In this section, as opposed to the previous section, the inhomogeneity indexes are all to be determined under the assumption that the inner surface is to be full ceramic, and the outer is full metal (Eq. 20).

Table 3
Constituent material properties.

Materials		E (GPa)	ρ (kg/m3)	ν	k (W/mK)	α (1/K)
Metals	Nickel (Ni)	199.5	8900	0.3	90.7	13.3 10-6
	Aluminum (Al)	70	2700	0.3	204	23. 10-6
	SUS304 (Stainless Steel)	201.04	7800	0.3262	15.379	12.33 10-6
Ceramics	Silicon Nitride (Si3N4)	348.43	4429	0.24	1.209	5.8723 10-6
	Aluminum Oxide (Al2O3)	393	3970	0.3	30.1	8.8 10-6
	Zirconium oxide (ZrO2)	116.4	3657	0.3	1.78	8.7 10-6

$\begin{array}{c}\mathit{a}=1.0\mathit{m};\mathit{b}=1.2\mathit{m};\mathit{\omega }=100\mathit{rad}/\mathit{s}\\ \mathit{v}=0.3;{\mathit{E}}_{\mathit{a}}=200\mathit{GPa};{\mathit{\rho }}_{\mathit{a}}=7800\frac{\mathit{kg}}{{\mathit{m}}^3}\\ {\mathit{\alpha }}_{\mathit{a}}=1.2{10}^{-6}\frac{1}{°C};{\mathit{k}}_{\mathit{a}}=15.379\frac{\mathit{W}}{\mathit{m}°C}\\ {\mathit{T}}_{\mathit{a}}=10°\mathit{C};{\mathit{T}}_{\mathit{b}}=0°C\end{array}$

As seen from Eq. 20, the numerical values of inhomogeneity indexes depend on the disc aspect ratio, a/b, as well as the metal and ceramic properties located individually at both surfaces. If the full metal is placed at the inner surface, the sign of the inhomogeneity indexes will all change. The positive inhomogeneity index means an increase in the related property from the inner surface towards the outer.

Tables 4 and 5 include some numerical results for the elastic fields of the example with physical FGMs under fixed-free and fixed-fixed boundary conditions.

Table 4
Elastic fields for physical FGMs under fixed-free boundary conditions.

r/b	Thermal			Centrifugal			Thermal + centrifugal
	Si3N4	Ni-Si3N4	Ni	Si3N4	Ni-Si3N4	Ni	Si3N4	Ni-Si3N4	Si3N4
ur (mm)
0.5	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
0.6	0.3227	0.3542	0.74496	0.0046	0.008080	0.0154	0.3273	0.3623	0.7604
0.7	0.5871	0.6918	1.35457	0.0080	0.015174	0.0266	0.5951	0.7070	1.3811
0.8	0.8132	1.0405	1.87476	0.0103	0.020926	0.0341	0.8234	1.0614	1.9088
0.9	1.0121	1.4112	2.33162	0.0115	0.024729	0.0381	1.0236	1.4359	2.3697
1.0	1.1908	1.8090	2.74121	0.0118	0.025754	0.0387	1.2026	1.8347	2.7799
σr (GPa)
0.5	0.3346	0.3895	0.42067	0.0199	0.032058	0.0393	0.3545	0.4215	0.4600
0.6	0.1835	0.2383	0.230133	0.0153	0.025922	0.0305	0.1988	0.2642	0.2606
0.7	0.0989	0.1495	0.123707	0.0114	0.020327	0.0228	0.1103	0.1698	0.1465
0.8	0.0490	0.0874	0.061041	0.0077	0.014430	0.0154	0.0567	0.1019	0.0765
0.9	0.0186	0.0395	0.023099	0.0039	0.007758	0.0079	0.0225	0.0473	0.0310
1.0	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
σθ (GPa)
0.5	-0.683	-0.658	-0.86350	0.0048	0.008656	0.0118	-0.678	-0.649	-0.852
0.6	-0.478	-0.425	-0.60317	0.0064	0.011051	0.0143	-0.472	-0.413	-0.589
0.7	-0.348	-0.357	-0.43774	0.0067	0.011250	0.0144	-0.341	-0.345	-0.423
0.8	-0.259	-0.342	-0.32395	0.0063	0.010141	0.0131	-0.252	-0.332	-0.311
0.9	-0.193	-0.348	-0.24092	0.0054	0.008061	0.0108	-0.188	-0.340	-0.230
1.0	-0.144	-0.363	-0.17749	0.0041	0.005138	0.0077	-0.140	-0.358	-0.170
r/b	Al2O3	Al-Al2O3	Al	Al2O3	Al-Al2O3	Al	Al2O3	Al-Al2O3	Al
ur (mm)
0.5	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
0.6	0.4929	0.5326	1.28828	0.0035	0.003615	0.0133	0.4964	0.5362	1.3016
0.7	0.8963	1.0872	2.34249	0.0060	0.007549	0.0230	0.9023	1.0947	2.3655
0.8	1.2404	1.6815	3.24206	0.0077	0.011338	0.0295	1.2482	1.6928	3.2715
0.9	1.5427	2.3249	4.03212	0.0086	0.014237	0.0330	1.5514	2.3392	4.0651
1.0	1.8137	3.0226	4.74044	0.0088	0.015190	0.0335	1.8225	3.0377	4.7739
σr (GPa)
0.5	0.5483	0.4507	0.255254	0.0175	0.014537	0.0119	0.5658	0.4653	0.2672
0.6	0.3000	0.2337	0.13964	0.0136	0.011001	0.0092	0.3136	0.2447	0.1489
0.7	0.1612	0.1273	0.075063	0.0102	0.008010	0.0069	0.1714	0.1353	0.0820
0.8	0.0796	0.0662	0.037039	0.0069	0.005253	0.0047	0.0864	0.0714	0.0417
0.9	0.0301	0.0270	0.014016	0.0035	0.002604	0.0024	0.0336	0.0296	0.0164
1.0	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
σθ (GPa)
0.5	-1.125	-1.155	-0.52395	0.0053	0.004361	0.0036	-1.120	-1.150	-0.520
0.6	-0.786	-0.632	-0.36599	0.0064	0.004804	0.0043	-0.780	-0.627	-0.362
0.7	-0.571	-0.418	-0.26561	0.0064	0.004237	0.0044	-0.564	-0.414	-0.261
0.8	-0.422	-0.316	-0.19657	0.0059	0.003305	0.0040	-0.416	-0.313	-0.193
0.9	-0.314	-0.261	-0.14619	0.0048	0.002221	0.0033	-0.309	-0.259	-0.143
1.0	-0.231	-0.228	-0.10770	0.0034	0.001063	0.0023	-0.228	-0.227	-0.105
r/b	ZrO2	SUS304-ZrO2	SUS304	ZrO2	SUS304-ZrO2	SUS304	ZrO2	SUS304-ZrO2	SUS304
ur (mm)
0.5	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
0.6	0.4873	0.5370	0.696216	0.0109	0.016374	0.0131	0.4982	0.5533	0.7093
0.7	0.8861	0.9831	1.2655	0.0187	0.027204	0.0225	0.9048	1.0104	1.2880
0.8	1.2263	1.3864	1.7509	0.0240	0.033985	0.0288	1.2503	1.4204	1.7797
0.9	1.5252	1.7698	2.17686	0.0268	0.037395	0.0322	1.5520	1.8072	2.2090
1.0	1.7931	2.1458	2.55842	0.0273	0.037721	0.0326	1.8204	2.1835	2.5910
σr (GPa)
0.5	0.1606	0.2378	0.387834	0.0162	0.025836	0.0342	0.1767	0.2636	0.4220
0.6	0.0878	0.1529	0.211934	0.0125	0.021015	0.0266	0.1004	0.1739	0.2385
0.7	0.0472	0.0976	0.113778	0.0094	0.016592	0.0200	0.0566	0.1142	0.1337
0.8	0.0233	0.0577	0.056058	0.0063	0.011878	0.0135	0.0296	0.0696	0.0696
0.9	0.0088	0.0263	0.021176	0.0032	0.006448	0.0069	0.0121	0.0328	0.0281
1.0	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
σθ (GPa)
0.5	-0.330	-0.303	-0.79809	0.0048	0.008089	0.0112	-0.325	-0.295	-0.787
0.6	-0.230	-0.247	-0.55699	0.0059	0.010247	0.0131	-0.224	-0.237	-0.544
0.7	-0.167	-0.225	-0.40371	0.0059	0.011093	0.0130	-0.161	-0.214	-0.391
0.8	-0.124	-0.221	-0.29823	0.0054	0.010882	0.0117	-0.118	-0.211	-0.287
0.9	-0.092	-0.230	-0.22123	0.0044	0.009706	0.0095	-0.088	-0.220	-0.212
1.0	-0.068	-0.245	-0.16237	0.0032	0.007583	0.0066	-0.065	-0.238	-0.156

Table 5
Elastic fields for physical FGMs under fixed-fixed boundary conditions.

r/b	Thermal			Centrifugal			Thermal + centrifugal
	Si3N4	Ni-Si3N4	Ni	Si3N4	Ni-Si3N4	Ni	Si3N4	Ni-Si3N4	Si3N4
ur (mm)
0.5	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
0.6	0.0315	0.0038	0.074886	0.0018	0.003091	0.0060	0.0333	0.0069	0.0808
0.7	0.0427	-0.009	0.101444	0.0026	0.005198	0.0089	0.0453	-0.004	0.1103
0.8	0.0392	-0.018	0.092971	0.0026	0.005859	0.0089	0.0418	-0.012	0.1019
0.9	0.0242	-0.016	0.057431	0.0018	0.004413	0.0060	0.0260	-0.011	0.0634
1.0	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
σr (GPa)
0.5	-0.839	-0.943	-1.18188	0.0083	0.013091	0.0167	-0.831	-0.930	-1.165
0.6	-0.854	-0.945	-1.20104	0.0050	0.009080	0.0103	-0.849	-0.936	-1.191
0.7	-0.857	-0.940	-1.20412	0.0019	0.004812	0.0041	-0.855	-0.936	-1.200
0.8	-0.853	-0.938	-1.19972	-0.001	-0.00018	-0.002	-0.854	-0.939	-1.202
0.9	-0.847	-0.939	-1.19168	-0.005	-0.00618	-0.009	-0.852	-0.945	-1.201
1.0	-0.839	-0.942	-1.18188	-0.008	-0.01341	-0.017	-0.848	-0.956	-1.199
σθ (GPa)
0.5	-0.965	-1.018	-1.34426	0.0020	0.003535	0.0050	-0.963	-1.014	-1.339
0.6	-0.896	-0.920	-1.25532	0.0022	0.004002	0.0051	-0.894	-0.916	-1.250
0.7	-0.848	-0.917	-1.19322	0.0018	0.003273	0.0037	-0.846	-0.913	-1.189
0.8	-0.812	-0.935	-1.14651	0.0008	0.001701	0.0015	-0.811	-0.933	-1.145
0.9	-0.784	-0.957	-1.10947	-0.0004	-0.00060	-0.001	-0.784	-0.958	-1.111
1.0	-0.760	-0.979	-1.07893	-0.002	-0.00362	-0.005	-0.762	-0.982	-1.084
r/b	Al2O3	Al-Al2O3	Al	Al2O3	Al-Al2O3	Al	Al2O3	Al-Al2O3	Al
ur (mm)
0.5	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
0.6	0.0495	0.1991	0.129503	0.0013	0.001939	0.0051	0.0509	0.2010	0.1347
0.7	0.0671	0.3086	0.175429	0.0020	0.003636	0.0077	0.0691	0.3122	0.1831
0.8	0.0615	0.3219	0.160776	0.0020	0.004505	0.0077	0.0635	0.3264	0.1685
0.9	0.0380	0.2250	0.099317	0.0014	0.003684	0.0052	0.0394	0.2287	0.1045
1.0	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
σr (GPa)
0.5	-1.540	-0.786	-0.71714	0.0074	0.008323	0.0051	-1.533	-0.777	-0.712
0.6	-1.565	-0.868	-0.72877	0.0046	0.005462	0.0031	-1.561	-0.863	-0.726
0.7	-1.569	-0.887	-0.73064	0.0018	0.002914	0.0012	-1.568	-0.884	-0.729
0.8	-1.564	-0.883	-0.72797	-0.002	0.000484	-0.0007	-1.565	-0.882	-0.729
0.9	-1.553	-0.871	-0.72309	-0.004	-0.00191	-0.003	-1.557	-0.872	-0.726
1.0	-1.540	-0.855	-0.71714	-0.007	-0.00430	-0.005	-1.548	-0.860	-0.722
σθ (GPa)
0.5	-1.752	-1.526	-0.81567	0.0022	0.002500	0.0015	-1.750	-1.523	-0.814
0.6	-1.636	-1.101	-0.76170	0.0023	0.002445	0.0015	-1.634	-1.099	-0.760
0.7	-1.555	-0.912	-0.72403	0.0017	0.001758	0.0011	-1.554	-0.910	-0.723
0.8	-1.494	-0.808	-0.69568	0.0007	0.000832	0.0005	-1.494	-0.807	-0.695
0.9	-1.446	-0.743	-0.67320	-0.0006	-0.00020	-0.0004	-1.447	-0.743	-0.674
1.0	-1.406	-0.696	-0.65467	-0.002	-0.00129	-0.002	-1.409	-0.697	-0.656
r/b	ZrO2	SUS304-ZrO2	SUS304	ZrO2	SUS304-ZrO2	SUS304	ZrO2	SUS304-ZrO2	SUS304
ur (mm)
0.5	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
0.6	0.0490	-0.109	0.070824	0.0042	0.005022	0.0051	0.0532	-0.104	0.0759
0.7	0.0664	-0.152	0.095941	0.0062	0.007250	0.0076	0.0726	-0.145	0.1035
0.8	0.0608	-0.142	0.087927	0.0063	0.007110	0.0076	0.0671	-0.135	0.0955
0.9	0.0376	-0.090	0.054315	0.0042	0.004710	0.0051	0.0418	-0.085	0.0594
1.0	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
σr (GPa)
0.5	-0.451	-0.740	-1.14708	0.0069	0.008655	0.0146	-0.444	-0.731	-1.132
0.6	-0.458	-0.723	-1.16497	0.0042	0.005626	0.0091	-0.454	-0.717	-1.156
0.7	-0.460	-0.719	-1.16785	0.0017	0.002241	0.0036	-0.458	-0.717	-1.164
0.8	-0.458	-0.724	-1.16374	-0.001	-0.00186	-0.002	-0.459	-0.725	-1.166
0.9	-0.455	-0.734	-1.15623	-0.004	-0.00692	-0.008	-0.459	-0.741	-1.164
1.0	-0.451	-0.749	-1.14708	-0.007	-0.01316	-0.015	-0.458	-0.762	-1.162
σθ (GPa)
0.5	-0.513	-0.609	-1.29878	0.0021	0.002710	0.0048	-0.511	-0.607	-1.294
0.6	-0.479	-0.666	-1.21568	0.0021	0.002886	0.0047	-0.477	-0.663	-1.211
0.7	-0.455	-0.727	-1.15767	0.0015	0.002273	0.0034	-0.454	-0.724	-1.154
0.8	-0.438	-0.788	-1.11403	0.0006	0.000917	0.0013	-0.437	-0.787	-1.113
0.9	-0.423	-0.850	-1.07943	-0.0006	-0.00120	-0.001	-0.424	-0.851	-1.081
1.0	-0.412	-0.911	-1.0509	-0.002	-0.00412	-0.005	-0.414	-0.915	-1.056

Variation of the radial displacement, radial and hoop stresses with homogeeous/inhomogeneous material types and boundary conditions along the radial coordinate are presented in Figs. 4 to 6.

Figure 4
Variation of the radial displacements.

Figure 5
Variation of the radial stresses.

Figure 6
Variation of the hoop stresses.

Let's consider Fig. 4.

• The characteristics of three types FGMs are close to the characteristics of individual ceramic constituent under all types of loads and fixed-free boundary conditions. This observation is valid only for the radial displacements of the mounted disc with rigid casing made of Ni/Si₃N₄.

• The radial displacements build up as r/b increase under all load and material types for mounted discs with traction-free outer surface.

• In these examples, the individual thermal effects become more significant than the centrifugal ones since the surface temperature difference is assumed to be much higher.

• Ni/Si₃N₄ has the smallest radial displacements due to thermal loads, compared to the other two FGMs, while Al/Al₂O₃ has the highest ones.

• The maximum radial displacement due to the rotation takes the smallest value for Al/Al₂O₃ under two boundary conditions.

• SUS304-ZrO₂ seem to be the best for the radial displacements due to combined loads under two boundary conditions.

If the radial stresses in Fig. 5 are considered, it is observed that the characteristic of thermal and thermomechanical behaviors of Ni/Si₃N₄ is very close to the characteristics of individual Ni under fixed-free boundary conditions. This is observed between the radial centrifugal stresses of the discs made of either Ni/Si₃N₄ or individual Ni under fixed-fixed boundaries. Maximum individual thermal, individual centrifugal, and combined radial stress are observed at the inner surface for all FGM discs under fixed-free boundary conditions. However, the individual thermal and combined radial stress distribution is almost uniform for FGM discs under fixed-fixed boundaries.

From Fig. 6 it is observed that, for all FGM discs and under two boundary conditions, thermal and combined hoop stresses are completely in compression. Under individual centrifugal loads, hoop stresses in fixed-fixed FGM discs may be either compression or tension while those stresses are totally in tension for fixed-free boundary conditions. The variation of the combined hoop stresses in fixed-free disc made of SUS304/ZrO₂ material is almost uniform. This may be observed for discs made of Ni/Si₃N₄ under fixed-fixed boundary conditions.

CONCLUSIONS

The present study is bound up with an analytical thermomechanical analysis of mounted rotating uniform power-law graded discs with or without rigid casing at the outer surface. It presents closed form solutions together with two parametric studies under individual and combined mechanical and thermal loads. The first parametric study, in which centrifugal force effects are circumstantial when comparing to the thermal load effects, deals with an investigation of the effects of inhomogeneity indexes on the elastic fields of such discs. The second one considers three FGM discs made of physical metal and ceramic pairs. In this example, the thermal loads are essential compared to inertia forces. Some remarkable conclusions of the two parametric studies are achieved.

Generally speaking, in some applications, although the effect of thermal loads is negligible compared to inertia forces, it may be of equal or more importance in others. Closed-form formulas proposed in the present study may be helpful to get a concrete idea for specific applications.

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Edited by

Section Editor: Marcia Mantelli

Publication Dates

• Publication in this collection
  13 June 2019
• Date of issue
  2019

History

• Received
  01 Jan 2018
• Accepted
  04 May 2018