Abstract
An extensive parametric study on the variation of the centrifugalforceinduced 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 powerlaw 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 powerlaw indexes
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/nonhomogeneous 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.
Güven (1995) Güven, U. (1995).Tresca's yield condition and the linear hardening rotating solid disk of variable thickness, Zeitschrift fur Angewandte Mathematik und Mechanik 75: 805 – 807. studied Tresca's yield condition and the linear hardening rotating solid disk of variable thickness. Eraslan (2003a) Eraslan, A.N. (2003a). Elastoplastic deformations of rotating parabolic solid disks using Tresca’s yield criterion, European Journal of Mechanics A/Solids 22: 861–874. obtained analytical solutions for the stress distribution in rotating parabolic solid disks made of an isotropic and homogeneous material based on Tresca’s yield criterion associated with the flow rule and linear strain hardening. Eraslan (2003a) Eraslan, A.N. (2003a). Elastoplastic deformations of rotating parabolic solid disks using Tresca’s yield criterion, European Journal of Mechanics A/Solids 22: 861–874. showed that the deformation behavior of the convex parabolic disk is similar to that of the uniform thickness disk, but in the case of concave parabolic solid disk, it is different. Eraslan (2003a) Eraslan, A.N. (2003a). Elastoplastic deformations of rotating parabolic solid disks using Tresca’s yield criterion, European Journal of Mechanics A/Solids 22: 861–874. also showed mathematically that in the limiting case the parabolic disk solution reduces to the solution of rotating uniform thickness solid disk. Based on Tresca’s yield criterion, its associated flow rule and linear strain hardening material behavior, Eraslan (2003b) Eraslan A.N. (2003b). Elastic–plastic deformations of rotating variable thickness annular disks with free, pressurized and radially constrained boundary conditions, International Journal of Mechanical Sciences 45: 643 – 667. offered analytical solutions for the elastic–plastic stress distribution in rotating parabolic disks with free, pressurized and radially constrained boundary conditions. In this study it was also shown mathematically that in the limiting case the parabolic disk solution reduces to the uniform disk solution. Apatay and Eraslan (2003) Apatay, T., and Eraslan, A.N. (2003). Elastic deformation of rotating parabolic discs: analytical solutions (in Turkish), Journal of the Faculty of Engineering and Architecture of Gazi University 18: 115135. achieved analytical solutions in terms of hypergeometric functions for the elastic deformation of rotating parabolic discs made of isotropic and homogeneous materials. Calderale et al. (2012) Calderale, P.M., Vivio, F., and Vullo, V. (2012). Thermal stresses of rotating hyperbolic disks as particular case of nonlinearly variable thickness disks, Journal of Thermal Stresses 35: 877891. studied theoretically a thermoelastic analysis of the Stodola's hyperbolic disk made of an isotropic and homogeneous material, axisymmetric and symmetric with respect to the midplane, and subjected to a radially polynomially varying thermal load. Vivio et al. (2014) Vivio, F., Vullo, V., and Cifani, P. (2014). Theoretical stress analysis of rotating hyperbolic disk without singularities subjected to thermal load, Journal of Thermal Stresses, 37: 117–136. introduced a theoretical method for the evaluation of elastic stresses and strains in rotating hyperbolic disks made of an isotropic and homogeneous material. For rotating discs made of an isotropic and homogeneous material, Eraslan and Ciftci (2015) Eraslan, A.N., and Ciftci, B. (2015). Analytical and numerical solutions to rotating variable thickness disks for a new thickness profile, Journal of Multidisciplinary Engineering Science and Technology (JMEST) 2/9: 23592364. used a disk profile changing with an exponential function. Yıldırım (2017) Yýldýrým, V., (2017). Heatinduced, pressureinduced and centrifugalforceinduced exact axisymmetric thermomechanical analyses in a thickwalled spherical vessel, an infinite cylindrical vessel, and a uniform disc made of an isotropic and homogeneous material, International Journal of Engineering & Applied Sciences (IJEAS), 9 (2): 6687. offered allinone formulas for uniform discs, cylinders and spheres subjected to the mechanical and thermal loads. In this Reference centrifugal forceinduced, heatinduced, and pressureinduced elastic responses have all been considered for those structures made of an isotropic and homogeneous material.
As to rotating discs made of functionally graded materials, material grading function was chosen as a simple power function in some References ( Horgan and Chan, 1999a Horgan, C., Chan, A. (1999a). The pressurized hollow cylinder or disk problem for functionally graded isotropic linearly elastic materials, Journal of Elasticity 55: 4359. , b Horgan, C., Chan A. (1999b). The stress response of functionally graded isotropic linearly elastic rotating disks, Journal of Elasticity 55: 219230. , You et al. (2007) You, L.H. You, X.Y. Zhang J.J. and Li J. (2007). On rotating circular disks with varying material properties, Z. Angew. Math. Phys., ZAMP 58: 1068–1084. ; Bayat et al., 2008 Bayat, M., Saleem M., Sahari B., Hamouda A. and Mahdi E. (2008). Analysis of functionally graded rotating disks with variable thickness, Mechanics Research Communications 35: 283309. ; Çallıoğlu et al., 2011; Yıldırım, 2016; Gang, 2017 Gang, M. (2017). Stress analysis of variable thickness rotating FG disc, International Journal of Pure and Applied Physics 13/1: 158161. ), and as an exponential function in some studies ( Zenkour, 2005 Zenkour, A.M. (2005). Analytical solutions for rotating exponentiallygraded annular disks with various boundary conditions, Int. Journal of Struct. Stability and Dynamics 5: 557577. , ^{2007} Zenkour, A.M. (2007). Elastic deformation of the rotating functionally graded annular disk with rigid casing, Journal of Materials Science 42: 97179724. ; Zenkour and Mashat, 2011 Zenkour, A.M., and Mashat, D.S. (2011). Stress function of a rotating variablethickness annular disk using exact and numerical methods, Engineering 3: 422430. ; Eraslan and Arslan, 2015 Eraslan, A.N., and Arslan, E., (2015). Analytical and numerical solutions to a rotating FGM disk, Journal of Multidisciplinary Engineering Science and Technology (JMEST) 2/10: 28432850. ) to get analytical solutions. From those, Horgan and Chan ( ^{1999a} Horgan, C., Chan, A. (1999a). The pressurized hollow cylinder or disk problem for functionally graded isotropic linearly elastic materials, Journal of Elasticity 55: 4359. , b Horgan, C., Chan A. (1999b). The stress response of functionally graded isotropic linearly elastic rotating disks, Journal of Elasticity 55: 219230. ) gave explicit solutions for rotating discs of constant density and thickness. Zenkour (2005) Zenkour, A.M. (2005). Analytical solutions for rotating exponentiallygraded annular disks with various boundary conditions, Int. Journal of Struct. Stability and Dynamics 5: 557577. studied analytically exponentially graded rotating annular discs with constant thickness. Eraslan and Akış (2006) Eraslan, A.N., and Akýþ, T. (2006). On the plane strain and plane stress solutions of functionally graded rotating solid shaft and solid disk problems, Acta Mechanica 181/(1–2): 43–63. used two variants of a parabolic function for disks made of functionally graded materials. Zenkour (2007) Zenkour, A.M. (2007). Elastic deformation of the rotating functionally graded annular disk with rigid casing, Journal of Materials Science 42: 97179724. extended his study ( Zenkour, 2005 Zenkour, A.M. (2005). Analytical solutions for rotating exponentiallygraded annular disks with various boundary conditions, Int. Journal of Struct. Stability and Dynamics 5: 557577. ) for such discs with rigid casing. Bayat et al. (2008) Bayat, M., Saleem M., Sahari B., Hamouda A. and Mahdi E. (2008). Analysis of functionally graded rotating disks with variable thickness, Mechanics Research Communications 35: 283309. , based on the powerlaw distribution, gave both analytical and semianalytical elastic solutions for axisymmetric rotating hollow discs with parabolic and hyperbolic thickness profiles. This semianalytic solution was obtained by dividing the disc with varying thickness into subdomains with uniform thickness. By taking Young’s modulus, thermal expansion coefficient and density to be functions of the radial coordinate, a closed form solution of rotating uniform circular disks made of powerlaw graded materials subjected to a constant angular velocity and a uniform temperature is proposed by You et al. (2007) You, L.H. You, X.Y. Zhang J.J. and Li J. (2007). On rotating circular disks with varying material properties, Z. Angew. Math. Phys., ZAMP 58: 1068–1084. . Vivio and Vullo (2007) Vivio, F., and Vullo, V. (2007). Elastic stress analysis of rotating converging conical disks subjected to thermal load and having variable density along the radius. International Journal of Solids and Structures 44: 7767–7784. presented an analytical procedure based on the hypergeometric differential equation for evaluation of elastic stresses and strains in rotating solid or annular conical disks subjected to thermal load, and having a fictitious density variation along the radius. Vivio and Vullo (2007) Vivio, F., and Vullo, V. (2007). Elastic stress analysis of rotating converging conical disks subjected to thermal load and having variable density along the radius. International Journal of Solids and Structures 44: 7767–7784. also verified their analytical results with finite element solutions. Vullo and Vivio (2008) Vullo, V., Vivio, F. (2008). Elastic stress analysis of nonlinear variable thickness rotating disks subjected to thermal load and having variable density along the radius, Inter. J. Solids Struct. 45: 5337–5355. presented an analytical procedure for evaluation of elastic stresses and strains in nonlinear variable thickness rotating disks, either solid or annular, subjected to thermal load, and having a fictitious density variation along the radius. Thickness variation of disks was described by means of a power of linear function by Vullo and Vivio (2008) Vullo, V., Vivio, F. (2008). Elastic stress analysis of nonlinear variable thickness rotating disks subjected to thermal load and having variable density along the radius, Inter. J. Solids Struct. 45: 5337–5355. . Peng and Li (2009) Peng, X.L., and Li, X.F. (2009). Thermoelastic analysis of a functionally graded annulus with an arbitrary gradient, Appl Math Mech. 30: 1211–1220. studied a thermoelastic problem of a circular annulus made of functionally graded materials with an arbitrary gradient. Peng, and Li (2012) Peng, X.L. and Li, X.F. (2012). Effects of gradient on stress distribution in rotating functionally graded solid disks, J. Mech. Sci. Technol. 26: 14831492. also studied effects of gradient on stress distribution in rotating functionally graded solid disks. Zenkour and Mashat (2011) Zenkour, A.M., and Mashat, D.S. (2011). Stress function of a rotating variablethickness annular disk using exact and numerical methods, Engineering 3: 422430. used the modified RungaKutte algorithm in their numerical analysis. Çallıoğlu et al. (2011) performed an exact stress analysis of annular rotating discs made of functionally graded materials by assuming that both elasticity modulus and material density vary radially as a function of a simple power rule with the same inhomogeneity parameter. Hassani et al. (2011) Hassani, A., Hojjati, M.H., Farrahi, G., and Alashti, R.A. (2011). Semiexact elastic solutions for thermomechanical analysis of functionally graded rotating disks, Composite Structures 93: 32393251. obtained distributions of stress and strain components of rotating hyperbolic disks with nonuniform material properties subjected to a power form thermoelastic loading under different boundary conditions by semiexact methods of Liao’s homotopy analysis method. Argeso (2012) Argeso, H. (2012). Analytical solutions to variable thickness and variable material property rotating disks for a new threeparameter variation function, Mechanics Based Design of Structures and Machines 40: 133152. presented analytical solutions for two different annular rotating disk problems for the elastic stress state: The first problem involves an exponentially variable profile rotating disk made of an isotropic and homogeneous material, the second is a uniform disc made of exponentially functionally graded materials. Nejad et al. (2013 Nejad, M.Z., Abedi, M., Lotfian, M.H., and Ghannad, M. (2013) Elastic analysis of exponential FGM disks subjected to internal and external pressure, Central European Journal of Engineering 3: 459465. , ^{2014} Nejad, M.Z., Rastgoo, A. and Hadi, A. (2014). Exact elastoplastic analysis of rotating disks made of functionally graded materials, International Journal of Engineering Science 85: 4757. ) gave a closedform analytical solution in terms of hyper geometric functions to elastic analysis of exponentially functionally graded stationary discs subjected to internal and external pressures. Eraslan and Arslan (2015) Eraslan, A.N., and Arslan, E., (2015). Analytical and numerical solutions to a rotating FGM disk, Journal of Multidisciplinary Engineering Science and Technology (JMEST) 2/10: 28432850. developed analytical and numerical solutions to a rotating uniform thickness exponentially functionally graded (FGM) solid and annular disks. Yıldırım (2016) studied the exact elastic response of a rotating disk having a continuously varying hyperbolic thickness profile under different boundary conditions. Both convergenthyperbolic and divergenthyperbolic disk profiles together with uniform profile are all studied. Powerlaw grading is used for material gradation pattern. Yıldırım’s (2016) formulation comprises both continuously variations of elasticity modulus and material density including continuously variation of the thickness of the disc except variation of Poisson’s ratios. Contrary to the literature all effects affecting the elastic behavior of the disk with varying thickness such as internal and external pressures including rotation at a constant angular velocity are all studied under four physical boundary conditions and presented in compact forms in Yıldırım’s study (Yıldırım, 2016). Recently, Gang (2017) Gang, M. (2017). Stress analysis of variable thickness rotating FG disc, International Journal of Pure and Applied Physics 13/1: 158161. analytically studied the stress analysis of hyperbolic simplepower law graded rotating discs under stressfree conditions for four convergent disc profiles and negative inhomogeneity indexes.
In this study, under the additional assumption that both Young’s modulus and material density have a variation with the same inhomogeneity index, closedform formulas derived by Yıldırım (2016) are employed in the present work in a customized form to study the variation of centrifugal forceinduced stress and displacements in powerlaw 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.”
2 EXPANDING YILDIRIM’S (2016) FORMULAS
The disk whose inner radius is denoted by
where
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.
In the above equation, the prime symbol, (‘), denotes the derivative with respect to the radial coordinate. Poisson’s ratio is indicated by
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 Materiala is located at the inner surface and Materialb is located at the outer surface, inhomogeneity parameters in these equations are defined as follows
Derivation of Eq. (2) is presented in Appendix A . The general solution of Eq. (2) is written in terms of unknown coefficients
Where
Yıldırım (2016) presented explicit definitions of unknown coefficients,
Let’s do this. Under this assumption,
Boundary conditions considered in the present study are presented in Fig. 2 . For those boundary conditions and
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: 950955. studied the elastic response of powergraded uniform stressfree rotating disks with boundary conditions:
3 A NUMERICAL STUDY
The following geometrical properties are used in the parametric study:
For the values of the chosen simple powerlaw indexes
Variation of the displacement, the radial and hoop stresses with the boundary conditions and profile indexes for
Variation of the displacement, the radial and hoop stresses with the boundary conditions and profile indexes for
Variation of the displacement, the radial and hoop stresses with the boundary conditions and profile indexes for
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,
As expected, fixedguided discs have higher elastic field than fixedfree and freefree boundary conditions.
For fixedguided disc and positive inhomogeneity index, hoop stresses are in tensioncompression. 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: 158161. analytically studied convergent hyperbolic discs with
4 CONCLUSIONS
After they are customized, in this study, the closedform formulas derived by Yýldýrým (2016) Yýldýrým, V. (2016). Analytic solutions to powerlaw graded hyperbolic rotating discs subjected to different boundary conditions, International Journal of Engineering & Applied Sciences (IJEAS) 8/1:3852. are employed to study the variation of the centrifugalforceinduced stress and displacements in powerlaw graded hyperbolic discs with inhomogeneity parameter, profile parameter, and boundary conditions. Contrary to Yýldýrým’s (2016) Yýldýrým, V. (2016). Analytic solutions to powerlaw graded hyperbolic rotating discs subjected to different boundary conditions, International Journal of Engineering & Applied Sciences (IJEAS) 8/1:3852. study, it is assumed that both Young’s modulus and material density change with the same inhomogeneity parameter. If one suppose that Materiala is located at the inner surface and Materialb 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 metalceramic pair to satisfy that condition which may be defined by the following derived from Eq. (4) .
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 Materiala is located at the inner surface, the mixture of two materials which is multiples of
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 powerlaw graded hyperbolic rotating discs subjected to different boundary conditions, International Journal of Engineering & Applied Sciences (IJEAS) 8/1:3852. should be employed to get accurate results for hyperbolic discs made of physically exist materialceramic pairs in the last decision stage of a design process.
Taking into consideration the above ballpark 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,
Where
Where
Where
In the above equation called Navier equation, after choosing either
ACKNOWLEDGEMENTS
The author is grateful to the anonymous Referees for their valuable suggestions, the effort and time spent.
References
 Apatay, T., and Eraslan, A.N. (2003). Elastic deformation of rotating parabolic discs: analytical solutions (in Turkish), Journal of the Faculty of Engineering and Architecture of Gazi University 18: 115135.
 Argeso, H. (2012). Analytical solutions to variable thickness and variable material property rotating disks for a new threeparameter variation function, Mechanics Based Design of Structures and Machines 40: 133152.
 Bayat, M., Saleem M., Sahari B., Hamouda A. and Mahdi E. (2008). Analysis of functionally graded rotating disks with variable thickness, Mechanics Research Communications 35: 283309.
 Çallýoðlu, H., Bektaþ, N.B., and Sayer, M. (2011). Stress analysis of functionally graded rotating discs: analytical and numerical solutions, Acta Mechanica Sinica 27: 950955.
 Calderale, P.M., Vivio, F., and Vullo, V. (2012). Thermal stresses of rotating hyperbolic disks as particular case of nonlinearly variable thickness disks, Journal of Thermal Stresses 35: 877891.
 Eraslan, A.N. (2003a). Elastoplastic deformations of rotating parabolic solid disks using Tresca’s yield criterion, European Journal of Mechanics A/Solids 22: 861–874.
 Eraslan A.N. (2003b). Elastic–plastic deformations of rotating variable thickness annular disks with free, pressurized and radially constrained boundary conditions, International Journal of Mechanical Sciences 45: 643 – 667.
 Eraslan, A.N., and Akýþ, T. (2006). On the plane strain and plane stress solutions of functionally graded rotating solid shaft and solid disk problems, Acta Mechanica 181/(1–2): 43–63.
 Eraslan, A.N., and Arslan, E., (2015). Analytical and numerical solutions to a rotating FGM disk, Journal of Multidisciplinary Engineering Science and Technology (JMEST) 2/10: 28432850.
 Eraslan, A.N., and Ciftci, B. (2015). Analytical and numerical solutions to rotating variable thickness disks for a new thickness profile, Journal of Multidisciplinary Engineering Science and Technology (JMEST) 2/9: 23592364.
 Gang, M. (2017). Stress analysis of variable thickness rotating FG disc, International Journal of Pure and Applied Physics 13/1: 158161.
 Güven, U. (1995).Tresca's yield condition and the linear hardening rotating solid disk of variable thickness, Zeitschrift fur Angewandte Mathematik und Mechanik 75: 805 – 807.
 Hassani, A., Hojjati, M.H., Farrahi, G., and Alashti, R.A. (2011). Semiexact elastic solutions for thermomechanical analysis of functionally graded rotating disks, Composite Structures 93: 32393251.
 Horgan, C., Chan, A. (1999a). The pressurized hollow cylinder or disk problem for functionally graded isotropic linearly elastic materials, Journal of Elasticity 55: 4359.
 Horgan, C., Chan A. (1999b). The stress response of functionally graded isotropic linearly elastic rotating disks, Journal of Elasticity 55: 219230.
 Nejad, M.Z., Abedi, M., Lotfian, M.H., and Ghannad, M. (2013) Elastic analysis of exponential FGM disks subjected to internal and external pressure, Central European Journal of Engineering 3: 459465.
 Nejad, M.Z., Rastgoo, A. and Hadi, A. (2014). Exact elastoplastic analysis of rotating disks made of functionally graded materials, International Journal of Engineering Science 85: 4757.
 Peng, X.L., and Li, X.F. (2009). Thermoelastic analysis of a functionally graded annulus with an arbitrary gradient, Appl Math Mech. 30: 1211–1220.
 Peng, X.L. and Li, X.F. (2012). Effects of gradient on stress distribution in rotating functionally graded solid disks, J. Mech. Sci. Technol. 26: 14831492.
 Vivio, F., and Vullo, V. (2007). Elastic stress analysis of rotating converging conical disks subjected to thermal load and having variable density along the radius. International Journal of Solids and Structures 44: 7767–7784.
 Vivio, F., Vullo, V., and Cifani, P. (2014). Theoretical stress analysis of rotating hyperbolic disk without singularities subjected to thermal load, Journal of Thermal Stresses, 37: 117–136.
 Vullo, V., Vivio, F. (2008). Elastic stress analysis of nonlinear variable thickness rotating disks subjected to thermal load and having variable density along the radius, Inter. J. Solids Struct. 45: 5337–5355.
 Yýldýrým, V. (2016). Analytic solutions to powerlaw graded hyperbolic rotating discs subjected to different boundary conditions, International Journal of Engineering & Applied Sciences (IJEAS) 8/1:3852.
 Yýldýrým, V., (2017). Heatinduced, pressureinduced and centrifugalforceinduced exact axisymmetric thermomechanical analyses in a thickwalled spherical vessel, an infinite cylindrical vessel, and a uniform disc made of an isotropic and homogeneous material, International Journal of Engineering & Applied Sciences (IJEAS), 9 (2): 6687.
 You, L.H. You, X.Y. Zhang J.J. and Li J. (2007). On rotating circular disks with varying material properties, Z. Angew. Math. Phys., ZAMP 58: 1068–1084.
 Zenkour, A.M. (2005). Analytical solutions for rotating exponentiallygraded annular disks with various boundary conditions, Int. Journal of Struct. Stability and Dynamics 5: 557577.
 Zenkour, A.M. (2007). Elastic deformation of the rotating functionally graded annular disk with rigid casing, Journal of Materials Science 42: 97179724.
 Zenkour, A.M., and Mashat, D.S. (2011). Stress function of a rotating variablethickness annular disk using exact and numerical methods, Engineering 3: 422430.
Publication Dates

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

Received
09 July 2017 
Reviewed
07 Oct 2017 
Accepted
11 Oct 2017