International Journal of Mathematics and Computational Science

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Numerical Elasticity Solution for Continuously Tapered and Arbitrarily Functionally Graded (FG) Rotating Disks via the Transfer Matrix Approach

Pages: 48-73 Views: 395 Downloads: 217

[01]
Vebil Yıldırım, Mechanical Engineering Department, University of Çukurova, Adana, Turkey.

The main purpose of the present study is to introduce the transfer matrix method, which is an efficacious and accurate analytical/numerical method developed based on the initial value problem (IVP), to numerically study the elastostatic response of variable-thickness rotating thin disks made of an isotropic but non-homogeneous material which is composed of a metal and a ceramic constituents under mechanical pressure and centrifugal forces. The governing equation called Navier equation of such disks having any arbitrary thickness profile is a second order non-homogeneous differential equation with variable coefficients. It is possible to achieve an analytical solution of Navier equation by using some certain material grading rules and certain disk profiles. Those certain conditions are out of the scope of the present study. The present study deals with the numerical solution of Navier equation developed for both arbitrarily functionally graded metal-ceramic pairs and arbitrarily continuously varying disk profiles. To this end, several conventional material grading rules such as a simple power rule (P-FGM), an exponential function (E-FGM), a linear function (L-FGM), a Voigt mixture rule with power of volume fractions of constituents (V-FGM), a Mori-Tanaka scheme (MT-FGM), and a sigmoid function (S-FGM) are all considered with several parabolically/linearly/hyperbolically tapered disk profiles including uniform ones. Three boundary conditions namely free-free, fixed-free, and fixed-fixed are examined. Some numerical results are also presented to serve benchmark solutions for future advanced studies for an aluminum/aluminum oxide (Al/Al_{2}O_{3}) FG material.

Transfer Matrix Approach, Complementary Functions Method, Initial Value Problem, Numerical Analysis, Axisymmetric Elasticity Solution, Rotating Variable-thickness Disk, Functionally Graded, Inhomogeneous Material

[01]
Güven U. 1995. Tresca's yield condition and the linear hardening rotating solid disk of variable thickness, Zeitschrift fur Angewandte Mathematik und Mechanik, 75, pp. 805-807.
[02]
Eraslan A. N. 2003. Elastoplastic deformations of rotating parabolic solid disks using Tresca’s yield criterion, European Journal of Mechanics A/Solids, 22, pp. 861-874.
[03]
Eraslan A. N. 2003. Elastic–plastic deformations of rotating variable thickness annular disks with free, pressurized and radially constrained boundary conditions, Int J Mech Sci, 45, pp. 643-667.
[04]
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, pp. 115-135.
[05]
Calderale P. M., Vivio F., and Vullo V. 2012. Thermal stresses of rotating hyperbolic disks as particular case of non-linearly variable thickness disks, Journal of Thermal Stresses, 35, pp. 877-891.
[06]
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, pp. 117-136.
[07]
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), pp. 2359-2364.
[08]
Argeso H. 2012. Analytical solutions to variable thickness and variable material property rotating disks for a new three-parameter variation function, Mechanics Based Design of Structures and Machines, 40, pp. 133-152.
[09]
Murthy D., and Sherbourne A. 1970. Elastic stresses in anisotropic discs of variable thickness. Int J Mech Sci, 12, pp. 627-640.
[10]
Reddy T. Y., and Srinath H. 1974. Elastic stresses in a rotating anisotropic annular disc of variable thickness and variable density, Int J Mech Sci, 16 (2), pp. 85-89.
[11]
Gurushankar G. V. 1975. Thermal stresses in a rotating nonhomogeneous, anisotropic disc of varying thickness and density, The Journal of Strain Analysis for Engineering Design, 10, pp. 137-142.
[12]
Zenkour A. M., and Allam N. M. N. 2006. On the rotating fiber-reinforced viscoelastic composite solid and annular disks of variable thickness, International Journal of Computational Methods in Engineering Science and Mechanics, 7, pp. 21-31.
[13]
Eraslan A. N., Kaya Y., and Varlı E. 2016. Analytical solutions to orthotropic variable thickness disk problems, Pamukkale University Journal of Engineering Sciences, 22 (1), pp. 24-30.
[14]
Kacar I., Yıldırım V. 2017. Effect of the anisotropy ratios on the exact elastic behavior of functionally power-graded polar orthotropic rotating uniform discs under various boundary conditions, Digital Proceeding of ICOCEE – Cappadocia 2017, Nevsehir, Turkey, pp. 1743-1752.
[15]
Essa S., and Argeso H. 2017. Elastic analysis of variable proﬁle and polar orthotropic FGM rotating disks for a variation function with three parameters, Acta Mechanica, 228, pp. 3877–3899.
[16]
Zheng Y., Bahaloo H., Mousanezhad D., Vaziri A., and Nayeb-Hashemi H. 2017. Displacement and stress fields in a functionally graded fiber-reinforced rotating disk with nonuniform thickness and variable angular velocity, Journal of Engineering Materials and Technology, 39, 031010-1-9.
[17]
Yıldırım V. 2018. Unified exact solutions to the hyperbolically tapered pressurized/rotating disks made of nonhomogeneous isotropic/orthotropic materials, International Journal of Advanced Materials Research (to be published)
[18]
Eraslan, A. N. and Akiş, 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), pp. 43–63.
[19]
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, pp. 7767–7784.
[20]
Vullo V., and Vivio F. 2008. Elastic stress analysis of non-linear variable thickness rotating disks subjected to thermal load and having variable density along the radius, International Journal of Solids and Structures, 45, pp. 5337–5355.
[21]
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, pp. 283-309.
[22]
Zenkour A. M., and Mashat D. S. 2011. Stress function of a rotating variable-thickness annular disk using exact and numerical methods, Engineering, 3, pp. 422-430.
[23]
Tütüncü N., and Temel B. 2011. An efficient unified method for thermoelastic analysis of functionally graded rotating disks of variable thickness, Mechanics of Advanced Materials and Structures, 20 (1), pp. 38-46.
[24]
Hassani, A., Hojjati, M. H., Farrahi, G. and Alashti, R. A. 2011. Semi-exact elastic solutions for thermomechanical analysis of functionally graded rotating disks, Composite Structures, 93, pp. 3239-3251.
[25]
Bayat M., Sahari B. B., Saleem M., Dezvareh E., and Mohazzab A. H. 2011. Analysis of functionally graded rotating disks with parabolic concave thickness applying an exponential function and the Mori-Tanaka scheme, IOP Conf. Series: Materials Science and Engineering, 17 012005.
[26]
Bayat M., Sahari B. B., and Saleem M. 2012. The effect of ceramic in combination of two sigmoid functionally graded rotating disks with variable thickness, International Journal of Computational Methods, 9 (2), 1240029 (22 pages).
[27]
Ghorbani M. T. 2012. A semi-analytical solution for time-variant thermoelastic creep analysis of functionally graded rotating disks with variable thickness and properties, International Journal of Advanced Design and Manufacturing Technology, 5, pp. 41-50.
[28]
Amin H., Saber E., and Khourshid A. M. 2015. Performance of functionally graded rotating disk with variable thickness, International Journal of Engineering Research & Technology, 4 (3), pp. 556- 564.
[29]
Yıldırım 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), pp. 38-52.
[30]
Zheng Y., Bahaloo H., Mousanezhad D., Mahdi E., Vaziri A., and Nayeb-Hashemi H. 2016. Stress analysis in functionally graded rotating disks with non-uniform thickness and variable angular velocity, Int J Mech Sci, 119, pp. 283–293.
[31]
Gang M. 2017. Stress analysis of variable thickness rotating FG disc, International Journal of Pure and Applied Physics, 13 (1), pp. 158-161.
[32]
Yıldırım V. 2017. Effects of inhomogeneity and thickness parameters on the elastic response of a pressurized hyperbolic annulus/disc made of functionally graded material, International Journal of Engineering & Applied Sciences, 9 (3), pp. 36-50.
[33]
Yıldırım V. 2018. A parametric study on the centrifugal force-induced stress and displacements in power-law graded hyperbolic discs, Latin American Journal of Solids and Structures, LAJSS, 15 (4), pp. 1-16.
[34]
Young W. C., and Budynas R. G. 2002. Roark’s Formulas for Stress and Strain, McGraw-Hill, Seventh Edition, New York.
[35]
İnan, M., 1968. The Method of Initial Values and the Carry-Over Matrix in Elastomechanics. ODTU Publication, Ankara, No: 20.
[36]
Haktanır V., and Kiral E. 1993. Statical analysis of elastically and continuously supported helicoidal structures by the transfer and stiffness matrix methods, Computers and Structures, 49 (4), pp. 663-677.
[37]
Yıldırım V. 1999. In-plane free vibration of symmetric cross-ply laminated circular bars, Journal of Engineering Mechanics-ASCE, 125, pp. 630-636.
[38]
Aktas Z. 1972. Numerical Solutions of Two-Point Boundary Value Problems. Ankara, Turkey, METU, Dept of Computer Eng.
[39]
Roberts S., and Shipman J. 1979. Fundamental matrix and two-point boundary-value problems, Journal of Optimization Theory and Applications, 28 (1), pp. 77-88.
[40]
Haktanır V. 1995. The complementary functions method for the element stiffness matrix of arbitrary spatial bars of helicoidal axes, International Journal for Numerical Methods in Engineering, 38 (6), pp. 1031–1056.
[41]
Yıldırım V. 2018. The complementary functions method (CFM) solution to the elastic analysis of polar orthotropic rotating discs, Journal of Applied and Computational Mechanics (JACM) (in press).
[42]
Yıldırım V., 2018. Numerical/analytical solutions to the elastic response of arbitrarily functionally graded polar orthotropic rotating discs, Journal of the Brazilian Society of Mechanical Sciences and Engineering (accepted for publication).
[43]
Chung Y. L., and Chi S. H. 2001. The residual stress of functionally graded materials, Journal of the Chinese Institute of Civil and Hydraulic Engineering, 3, pp. 1–9.

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