Analytical solutions to orthotropic variable thickness disk problems

Abstract

An analytical model is developed to estimate the mechanical response of nonisothermal, orthotropic, variable thickness disks under a variety of boundary conditions. Combining basic mechanical equations of disk geometry with the equations of orthotropic material, the elastic equation of the disk is obtained. This equation is transformed into a standard hypergeometric differential equation by means of a suitable transformation. An analytical solution is then obtained in terms of hypergeometric functions. The boundary conditions used to complete the solutions simulate rotating annular disks with two free surfaces, stationary annular disks with pressurized inner and free outer surfaces, and free inner and pressurized outer surfaces. The results of the solutions to each of these cases are presented in graphical forms. It is observed that, for the three cases investigated the elastic orthotropy parameter turns out to be an important parameter affecting the elastic behavior

Keywords: Orthotropic disk, Variable thickness, Thermoelasticity, Hypergeometric equation

Keywords

• -

Ortotropik değişken kalınlıklı disk problemlerinin analitik çözümleri

Öz

Bu çalışmada eş ısıl olmayan, ortotropik, değişken kesitli disklerin, farklı sınır koşulları altında mekanik davranışlarını tahmin edebilmek için analitik bir model geliştirilmiştir. Disk geometrisi için temel mekanik denklemleri, ortotropik malzeme denklemleri ile birleştirilerek elastik denklem elde edilmiştir. Bu denklem uygun bir dönüşüm tekniği ile standart hipergeometrik diferansiyel denkleme dönüştürülmüş ve bunun analitik çözümü hipergeometrik fonksiyonlar cinsinden bulunmuştur. Çözümü tamamlayan sınır koşulları, iki ucu serbest dönen, iç veya dış yüzeyden basınçlandırılmış durağan değişken kesitli disklerin benzetişimini sağlayacak şekilde seçilmiştir. Elde edilen sonuçlar grafiksel olarak sunulmuştur. Sonuçlar göstermiştir ki makalede incelenen her üç problem için de elastik ortotropi parametresi diskin elastik davranışını etkileyen en önemli parametre olarak ortaya çıkmıştır.

Anahtar Kelimeler: Ortotropik diskler, Değişken kesit, Termo elastisite, Hipergeometrik denklem

Anahtar Kelimeler

• Ortotropik diskler, Değişken kesit, Termo elastisite, Hipergeometrik denklem

Kaynakça

1. Timoshenko S, Goodier JN. Theory of elasticity. 3rd ed. New York, McGraw-Hill, 1970.
2. Rees DWA. The Mechanics of solids and structures. New York, McGraw-Hill, 1990.
3. Boresi AP, Schmidt RJ, Sidebottom OM. Advanced mechanics of materials. 5th ed. New York, Wiley, 1993.
4. Ugural AC, Fenster SK. Advanced strength and applied elasticity. 3rd ed. London, Prentice-Hall, 1995.
5. Eraslan AN, Argeso H. "Limit angular velocities of variable thickness rotating disks". International Journal of Solids and Structures, 39(12), 3109-3130, 2002.
6. Eraslan AN, Orcan Y. "Elastic-plastic deformation of a rotating solid disk of exponentially varying thickness". Mechanics of Materials, 34(7), 423-432, 2002.
7. Eraslan, AN, Orçan, Y. "A parametric analysis of rotating variable thickness elastoplastic annular disks subjected to pressurized and radially constrained boundary conditions". Turkish Journal of Engineering and Environmental Sciences, 28, 381-395, 2004.
8. Eraslan AN. "Stress distributions in elastic-plastic rotating disks with elliptical thickness profiles using tresca and vonmises criteria". ZAMM Journal of Applied Mathematics and Mechanics, 85(4), 252-266, 2005.

9. Vullo V, Vivio F. "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(20), 5337-5355, 2008.
10. You LH, Wang JX, Tang, BP. "Deformations and stresses in annular disks made of functionally graded materials subjected to internal and/or external pressure". Meccanica, 44(3), 283-292, 2009.
11. Zenkour AM. "Stress distribution in rotating composite structures of functionally graded solid disks". Journal of Materials Processing Technology, 209(7), 3511-3517, 2009.
12. Bayat M, Sahari BB, Saleem M, Hamouda AMS, Reddy JN. "Thermoelastic analysis of functionally graded rotating disks with temperature-dependent material properties: uniform and variable thickness". International Journal of Mechanics and Materials in Design, 5(3), 263-279, 2009.
13. Argeso H. "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(2), 133-152, 2012.
14. Tutuncu N, Temel B. "An efficient unified method for thermoelastic analysis of functionally graded rotating disks of variable thickness". Mechanics of Advanced Materials and Structures, 20(1), 38-46, 2013.
15. Dumir PC, Mehta AK. "Rectilinearly orthotropic annular disc under uniform pressure". Technical Note: Applied Mathematical Modelling, 11(5), 397-399, 1987.
16. Gunay E, Sonmez M. “Mechanical behavior of wood under torsional and tensile loadings”. Gazi University Journal of Science, 16(4), 733-749, 2003.
17. Shipsha A, Berglund LA. “Shear coupling effects on stress and strain distributions in wood subjected to transverse compression”. Composites Science and Technology, 67(7-8), 1362-1369, 2007.
18. Tutuncu N. "Effect of Anisotropy on stresses in rotating discs". International Journal Mechanical Sciences, 37(8), 873-881, 1995.
19. Jain R, Ramachandra K, Simha KRY. "Rotating anisotropic disc of uniform strength". International Journal Mechanical Sciences, 41(6), 639-648, 1999.
20. Guven U, Celik A, Baykara C. “On transverse vibrations of functionally graded polar orthotropic rotating solid disk with variable thickness and constant radial stress”. Journal of Reinforced Plastics and Composites, 23(12), 1279-1284, 2004.
21. Callioglu H. “Stress analysis of an orthotropic rotating disc under thermal loading”. Journal of Reinforced Plastics and Composites, 23(17), 1859-1867, 2004.
22. Callioglu H, Topcu M, Tarakcilar AR. "Elastic-plastic stress analysis of an orthotropic rotating disc". International Journal of Mechanical Sciences, 48(9), 985-990, 2006.
23. Nie GJ, Zhong, Z, Batra RC. “Material tailoring for orthotropic elastic rotating disks”. Composites Science and Technology, 71(3), 406-414, 2011.
24. Gupta V, Singh SB. “Creep analysis in anisotropic composite rotating disc with hyperbolically varying thickness”. Applied Mechanics and Materials, 116, 4171-4178, 2012.
25. Lubarda VA. "On pressurized curvilinearly orthotropic circular disk, cylinder and sphere made of radially nonuniform material". Journal of Elasticity, 109(2), 103-133, 2012.
26. Eraslan AN, Kaya Y, Ciftci B. “A computational model for partially-plastic stress analysis of orthotropic variable thickness disks subjected to exterternal pressure”. Mathematical Sciences and Applications E-Notes, 2(1), 1-13, 2014.
27. Hill R. “User-friendly theory of orthotropic plasticity in sheet metals”. International Journal of Mechanical Sciences, 35(1), 19-25, 1993.