Stress Function of a Rotating Variable-Thickness Annular Disk Using Exact and Numerical Methods

Abstract

In this paper, the exact analytical and numerical solutions for rotating variable-thickness annular disk are presented. The inner and outer edges of the rotating variable-thickness annular disk are considered to have free boundary conditions. Two different annular disks for the radially varying thickness are given. The numerical Runge-Kutta solution as well as the exact analytical solution is available for the first disk while the exact analytical solution is not available for the second annular disk. Both exact and numerical results for stress function, stresses, strains and radial displacement will be investigated for the first annular disk of variable thickness. The accuracy of the present numerical solution is discussed and its ability of use for the second rotating variable-thickness annular disk is investigated. Finally, the distributions of stress function, displacement, strains, and stresses will be presented. The appropriate comparisons and discussions are made at the same angular velocity.

Analytic Solutions To Power-Law Graded Hyperbolic Rotating Discs Subjected To Different Boundary Conditions

“…There are numerous studies on stationary/rotating discs with constant/variable thickness and made of an isotropic and non-homogeneous material in the available literature. Some of those studies were performed analytically [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. For some types of those material grading patterns such as a simple power rule, or an exponential variation or a linear function it is possible to obtain a closed form solution to the problem.…”

“…Their analysis involving a Fredholm integral equation neither requires a special form of the gradient of material properties nor demands partitioning the entire structure into a multilayered homogeneous structure. Zenkour and Massat [8] used the modified Runge-Kutta algorithm in their numerical analysis while the hyper-geometric and Kummer's functions were employed in their analytical study. Çallıoğlu et al [10] 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.…”

“…For linearly varying properties, the solution is obtained in terms of hyper-geometric functions. If the material grading is performed by an exponential function [3,5,8,12] then Whittaker / Kumer functions or Frobenius series are employed in the solution. Apart from those, unlike general use of polar/cylindrical coordinates, elliptic cylindrical coordinate system may also be used to get closed form solutions in the formulation [14].…”

A Parametric Study on the Centrifugal Force-Induced Stress and Displacements in Power-Law Graded Hyperbolic Discs

“…As to rotating discs made of functionally graded materials, material grading function was chosen as a simple power function in some References Chan, 1999a,b, You et al (2007); Bayat et al, 2008;Çallıoğlu et al, 2011;Yıldırım, 2016;Gang, 2017), and as an exponential function in some studies (Zenkour, 2005(Zenkour, , 2007Zenkour and Mashat, 2011;Eraslan and Arslan, 2015) to get analytical solutions. From those, Horgan and Chan (1999a,b) gave explicit solutions for rotating discs of constant density and thickness.…”

“…Peng, and Li (2012) also studied effects of gradient on stress distribution in rotating functionally graded solid disks. Zenkour and Mashat (2011) used the modified Runga-Kutte 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.…”