# Abstract

An extensive parametric study on the variation of the centrifugal-force-induced stress and displacements with the inhomogeneity indexes, profile parameters and boundary conditions is conducted based on the author’s recently published analytical formulas for radially functionally power-law graded rotating hyperbolic discs under axisymmetric conditions. The radial variation of the thickness of the disc is chosen to obey a hyperbolic function defined either convergent or divergent. In the present work, contrary to the published one, it is assumed that both Young’s modulus and density radially vary with the same inhomogeneity index to enable to conduct a parametric study. Under this additional assumption, for the values of the chosen power-law indexes β=5, 0, 5 for the material grading rule, and the chosen profile parameters m=1, 0.75, 0.5, 0.25, 0, 0.25, 0.5, 0.75, 1 for a hyperbolic disc; the variations of the radial stress, the hoop stress and the radial displacement are all illustrated graphically for a rotating disc whose both surfaces are stress-free, for a rotating disc mounted a rigid shaft at its center and its outer surface is stress-free, and finally for a rotating disc attached a rigid shaft at its center and guided at its outer surface (a rigid casing exists at the outer surface).

Keywords
Elasticity solution; rotating disc; functionally graded; axisymmetric; variable thickness disk

# 1 INTRODUCTION

Analytical and numerical studies on functionally graded discs have gained a momentum since 1990s. There are numerous studies on stationary/rotating discs with constant/variable thickness and made of an isotropic and homogeneous/non-homogeneous material in the available literature. Some of those studies performed analytically and almost directly relevant to this study are cited in the present paper. In the literature, especially analytical studies on such structures subjected to only the inner pressure are relatively large. In this section, especially, just discs rotating at a constant speed and mainly analytical studies about those are cited.

In this study, under the additional assumption that both Young’s modulus and material density have a variation with the same inhomogeneity index, closed-form formulas derived by Yıldırım (2016) are employed in the present work in a customized form to study the variation of centrifugal force-induced stress and displacements in power-law graded hyperbolic discs with inhomogeneity parameter, profile parameter and boundary conditions ( Fig. 1 ). As mentioned above, radial variation of Poisson’s ratio is neglected. As You et al. (2007) expressed “Doubling Poisson’s ratio, the radial and circumferential stresses and radial displacement have very few changes. However, doubling Young’s modulus, the radial stress is increased obviously, the circumferential stress is raised greatly, and the radial displacement is reduced noticeably. Therefore, compared to the effects of Young’s modulus, the variation of Poisson’s ratio can be omitted.”

Figure 1
3-D view of convergent/divergent hyperbolic and uniform disc profiles

# 2 EXPANDING YILDIRIM’S (2016) FORMULAS

The disk whose inner radius is denoted by a and outer radius is denoted by b is assumed to be symmetric with respect to the mid plane, and its profile vary radially continuously in an hyperbolical form

h ( r ) = h a ( r a ) m (1)

where ha is the thickness of the disc at the inner surface, r is the radial coordinate, and m is the disc profile parameter. In Eq. (1) , a uniform disc profile is obtained with m=0, a convergent hyperbolic dick profile is attained with m<0 and for m>0 a divergent hyperbolic disc profile is reached ( Fig. 1 ).

Yıldırım (2016) solved the following nonhomogeneous equation governing the elastostatic behavior of a rotating disc made of functionally graded materials for the hyperbolic discs rotating at a constant angular velocity, ω.

( 1 + m ν + β ν ) r 2 u r + ( 1 + m + β ) r u r ' + u r '' = r 1 + q β ( 1 ν 2 ) ρ a ω 2 E a (2)

In the above equation, the prime symbol, (‘), denotes the derivative with respect to the radial coordinate. Poisson’s ratio is indicated by v ; Ea and ρa are the inner surface values of elasticity modulus and the density, respectively. Representing Young’s modulus by E , and material density by ρ , in the above either

E ( r ) = E a ( r a ) β ρ ( r ) = ρ a ( r a ) q (3a)
E ( r ) = E b ( r b ) β ρ ( r ) = ρ b ( r b ) q (3b)

may be applied as a material grading rule. In Eq. (3)β and q are called inhomogeneity parameters for both elasticity modulus and density, respectively. If one suppose that Material-a is located at the inner surface and Material-b is located at the outer surface, inhomogeneity parameters in these equations are defined as follows

β = ln ( E a E b ) ln ( a b ) = ln ( E b E a ) ln ( b a ) (4a)
q = In ( ρ a ρ b ) In ( a b ) = In ( ρ b ρ a ) In ( b a ) (4b)

Derivation of Eq. (2) is presented in Appendix A . The general solution of Eq. (2) is written in terms of unknown coefficients C1 and C2 as follows (Yıldırım, 2016)

u r = r 1 2 ( m β ξ ) ( C 1 + C 2 r ξ ) + r 3 + q β ( 1 + ν 2 ) ρ a ω 2 E a ( 8 + q ( 6 + q β ) 3 β + β ν + m ( 3 + q β + ν ) ) (5)

Where

ξ = ( 4 + ( m + β ) ( m + β 4 ν ) ) (6)

Yıldırım (2016) presented explicit definitions of unknown coefficients, C1 and C2 , for all possible boundary conditions. Although Yıldırım’s (2016) formulas are valid for the different inhomogeneity parameters for both elasticity modulus and material density, in the present study, those indexes are assumed to be equal to each other, that is q=β is to be used. That is those formulas will be customized for q=β to allow a parametric study.

Let’s do this. Under this assumption, q=β , the solution in Eq. (5) together with Eq. (3a) turns into the following

u r = r 3 ( 1 + ν 2 ) ρ a ω 2 E a ( 8 + m ( 3 + ν ) + β ( 3 + ν ) ) + r 1 2 ( m β ξ ) ( C 1 + r ξ C 2 ) (7)

Boundary conditions considered in the present study are presented in Fig. 2 . For those boundary conditions and q=β , the closed-form expressions of the radial displacement, radial and hoop stresses are presented in Tables 1 - 3 .

Figure 2
Boundary conditions considered in the present study
Table 1
Table 2
Closed-form formulas for fixed-guided boundary conditions
Table 3
Closed-form formulas for fixed-free boundary conditions

Çallýoðlu et al. (2011) Çallýoðlu, H., Bektaþ, N.B., and Sayer, M. (2011). Stress analysis of functionally graded rotating discs: analytical and numerical solutions, Acta Mechanica Sinica 27: 950-955. studied the elastic response of power-graded uniform stress-free rotating disks with boundary conditions: ur(a)=0 and σr(b)=0 ( Fig. 2 ). They assumed that both the Young’s modulus and the material density change with the same inhomogeneity index, q=β , in their formulation as in the present work. However, they further assumed that the thickness of the disc remains constant along the radial coordinate, that is m=0 . Yýldýrým (2016) Yýldýrým, V. (2016). Analytic solutions to power-law graded hyperbolic rotating discs subjected to different boundary conditions, International Journal of Engineering & Applied Sciences (IJEAS) 8/1:38-52. also showed that her formulas coincides with Çallýoðlu et al.’s (2011) Çallýoðlu, H., Bektaþ, N.B., and Sayer, M. (2011). Stress analysis of functionally graded rotating discs: analytical and numerical solutions, Acta Mechanica Sinica 27: 950-955. study under those assumptions stated in this paragraph.

# 3 A NUMERICAL STUDY

The following geometrical properties are used in the parametric study: a=0.02 m;b=0.1 m. Poisson’s ratio is assumed to be constant along the radial coordinate as ν=0.3. Dimensionless elastic stress and displacements are defined as

σ¯r=σrρaω2b2 ; σ¯θ=σθρaω2b2 ; u¯r=Eaurρaω2b3 (8)

For the values of the chosen simple power-law indexes β=5, 0, 5 for the material grading rule ( Fig. 3 ), and the chosen profile parameters m=1, 0.75, 0.5, 0.25, 0, 0.25, 0.5, 0.75, 1 for a hyperbolic disc; the variations of the radial stress, the hoop stress and the radial displacement are all illustrated graphically for a rotating disc whose both surfaces are stress-free, for a rotating disc mounted a rigid shaft at its center and its outer surface is stress-free, and finally for a rotating disc attached a rigid shaft at its center and guided at its outer surface (a rigid casing exists at the outer surface) in Figs. 3 - 5 . Some numerical results are also presented in Tables 4 - 6 to serve as a numerical data for investigators.

Figure 3
Variation of the displacement, the radial and hoop stresses with the boundary conditions and profile indexes for β=5
Figure 4
Variation of the displacement, the radial and hoop stresses with the boundary conditions and profile indexes for β=0
Figure 5
Variation of the displacement, the radial and hoop stresses with the boundary conditions and profile indexes for β=5
Table 4
Some numerical results for β=5
Table 5
Some numerical results for β=0
Table 6
Some numerical results for β=5

According to Eq. (3) the positive inhomogeneity indexes suggest that the outer surface and its vicinity is highly stiffer than the middle and the inner surfaces. However, the inner surface is stiffer than the middle and outer surfaces for the negative inhomogeneity indexes. The results obtained from a parametric study of the present work may be outlined concisely as follows (see Figs 3 - 5 ):

Convergent hyperbolic dick profiles, m<0 , offer smaller elastic field than divergent ones for negative inhomogeneity indexes including isotropic and homogeneous materials with β=0 . However, for the positive inhomogeneity parameters some differences in the behavior may be observed. For instance, while the hoop stresses are smaller for fixed-free and free-free ends of convergent disc profiles, the radial stresses behave contrarily to this for all boundary conditions. Similar to this, for fixed-free and free-free conditions, convergent profiles having positive inhomogeneity parameter present much smaller radial displacement values.

As expected, fixed-guided discs have higher elastic field than fixed-free and free-free boundary conditions.

For fixed-guided disc and positive inhomogeneity index, hoop stresses are in tension-compression. For other boundary conditions they are in tension. However, negative inhomogeneity parameters offer hoop stresses in tension for all boundary conditions and for both convergent and divergent disc profiles.

While positive inhomogeneity indexes present the maximum hoop stress at the outer surface of the disc, their locations are at the inner surface of the disc for negative inhomogeneity indexes.

It may be noted that Gang (2017) Gang, M. (2017). Stress analysis of variable thickness rotating FG disc, International Journal of Pure and Applied Physics 13/1: 158-161. analytically studied convergent hyperbolic discs with m=1, 0.75, 0.5, 0.25 , and negative inhomogeneity indexes which may be defined approximately as β=0.00265 and q=0.019 under stress-free conditions (free-free). He concluded that radial and tangential stresses in convergent FGM hyperbolic disc with negative inhomogeneity indexes is significantly reduced as compared to FGM uniform disc. This comment is in agreement with the first item of the conclusions given above of Figs. 3 - 5 and obtained from a widespread search.

# 4 CONCLUSIONS

After they are customized, in this study, the closed-form formulas derived by Yýldýrým (2016) Yýldýrým, V. (2016). Analytic solutions to power-law graded hyperbolic rotating discs subjected to different boundary conditions, International Journal of Engineering & Applied Sciences (IJEAS) 8/1:38-52. are employed to study the variation of the centrifugal-force-induced stress and displacements in power-law graded hyperbolic discs with inhomogeneity parameter, profile parameter, and boundary conditions. Contrary to Yýldýrým’s (2016) Yýldýrým, V. (2016). Analytic solutions to power-law graded hyperbolic rotating discs subjected to different boundary conditions, International Journal of Engineering & Applied Sciences (IJEAS) 8/1:38-52. study, it is assumed that both Young’s modulus and material density change with the same inhomogeneity parameter. If one suppose that Material-a is located at the inner surface and Material-b is located at the outer surface, inhomogeneity indexes should be defined by Eq. (4) . Under this assumption, in practice, it is hardly confronted to get a physical metal-ceramic pair to satisfy that condition which may be defined by the following derived from Eq. (4) .

E a / E b = ρ a / ρ b (9)

On the other hand, in the present parametric study, Eq. (4) is not used to define the inhomogeneity indexes. Equation (3) is used by attributing hypothetically chosen values to the inhomogeneity indexes instead. For positive inhomogeneity indexes this means that while Material-a is located at the inner surface, the mixture of two materials which is multiples of Ea exist at the other surfaces including the outer surface. The change of the properties of the mixture is defined by Eq. (3) .

Taking the same inhomogeneity index for both Young’s modulus and density helps to conduct a parametric study by eliminating subordinate changes in some variables (See Eqn. (2) ). That is, by doing so, we may acquire, at least, a ballpark estimate about the variation of the elastic response of hyperbolic rotating discs made of functionally graded materials. It may be noted that the formulas derived in the present study may be used for both arbitrarily chosen inhomogeneity indexes as in the parametric study and inhomogeneity indexes computed by Eq. (4) .

It is obvious that analytical formulas offered by Yýldýrým (2016) Yýldýrým, V. (2016). Analytic solutions to power-law graded hyperbolic rotating discs subjected to different boundary conditions, International Journal of Engineering & Applied Sciences (IJEAS) 8/1:38-52. should be employed to get accurate results for hyperbolic discs made of physically exist material-ceramic pairs in the last decision stage of a design process.

Taking into consideration the above ball-park estimations, the true material tailoring may be done without consuming much time in the design process of such rotating hyperbolic discs made of functionally graded materials.

# APPENDIX A

Under axisymmetric plane stress and small deformation assumptions, in a polar coordinate system, (r,θ) , the relations between the strain and displacement components are as follows

ε r ( r ) = d d r u r ( r ) ε θ ( r ) = u r ( r ) r (A.1)

Where ur is the radial displacement; εr and εθ are the radial and tangential strain components, respectively. For an isotropic and non-homogeneous material, the stress-strain relations (Hooke’s law) is

σ r ( r ) = E ( r ) 1 ν 2 ( ε r ( r ) + ν ε θ ( r ) ) σ θ ( r ) = E ( r ) 1 ν 2 ( ε θ ( r ) + ν ε r ( r ) ) (A.2)

Where E(r) is the elasticity modulus; v is Poisson’s ratio; σr and σθ are radial stress and circumferential stress (hoop stress), respectively. The equilibrium equation of a variable thickness disk rotating at a constant angular velocity, ω , is

d d r ( h ( r ) r σ r ( r ) ) h ( r ) σ θ ( r ) + h ( r ) ρ ( r ) ω 2 r 2 = 0 (A.3)

Where ρ(r) is the material density, and h(r) defines the profile of the disc, h(r)=ha(r/a)m . Substituting Eq. (A.1) in Eq. (A.2) , and then Eq. (A.2) in Eq. (A.3) yields the following non-homogeneous governing equation in terms of radial displacement and its derivatives.

d 2 d r 2 u r ( r ) + d d r u r ( r ) ( d d r E ( r ) E ( r ) + d d r h ( r ) h ( r ) + 1 r ) + u r ( r ) ( ν d d r E ( r ) r E ( r ) 1 r 2 + ν d d r h ( r ) r h ( r ) ) = ω 2 r ρ ( r ) ( 1 ν 2 ) E ( r ) (A.4)

In the above equation called Navier equation, after choosing either E(r)=Ea(r/a)β with ρ(r)=ρa(r/a)q or E(r)=Eb(r/b)β with ρ(r)=ρb(r/b)q as a material grading rule ((dE(r)/dr/E(r)=;(dh(r)/dr)/h(r)=m/r) then one may reach the following.

( 1 + m ν + β ν ) r 2 u r ( r ) + ( 1 + m + β ) r d d r u r ( r ) + d 2 d r 2 u r ( r ) = r 1 + q β ( 1 ν 2 ) ρ a ω 2 E a (A.5)

# ACKNOWLEDGEMENTS

The author is grateful to the anonymous Referees for their valuable suggestions, the effort and time spent.

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# Publication Dates

• Publication in this collection
15 May 2018
• Date of issue
2018