Abstract
Two models for evaluation of elastic and elastic–plastic stress and strain in non-linear variable thickness rotating disks, either solid or annular, subjected to such a rotational speed to determine stresses beyond yielding are presented. The first model regards the elastic field and the second concerns the elastic–plastic field, assuming that material hardening is isotropic and follows a most general hardening power-law. In the elastic region, a non-homogeneous hypergeometric differential equation is obtained. Such differential equation, which solves the elastic problem in closed form for non-linear variable thickness disks, with thickness given by the power of a linear function which may define a fourfold infinity of profiles, is integrated in closed form. As concerns the elastic–plastic analysis, first of all the introduction is made of a correlation between equivalent plastic strain and equivalent stress according to Von Mises, which is more general than those known from literature. In the case of isotropic hardening a second-order, non-homogeneous, non-linear differential equation is found, which governs the stress state in the plastic region of the disk. The procedure allows to calculate stress and displacement states in the two regions—plastic and elastic—of the disk subjected to prestressing by overspeeding. Several examples of disks, also prestressed by overspeeding, are considered (annular and solid, convex or concave, and linear tapered disks); the matching of results of the theoretical model and those obtained by means of FEA is very good. Lastly, the residual stress state can be found in a prestressed disk by overspeeding and the stress state in the actual operating condition in the same disk is evaluated.
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References
Abdul-Mihsein MJ, Bakr AA, Parker AP (1985) Stresses in axisymmetric rotating bodies determined by the boundary integral equation method. J Strain Anal 20(2):79–86
Abramowitz M, Stegun IH (1972) Handbook of mathematical functions. Dover Publications, New York
Banerjee PK, Butterfield R (1981) Boundary element methods in engineering science. McGraw-Hill Inc, New York
Burr AH (1982) Mechanical analysis and design. Elsevier, New York
Calderale PM (1960) Tavole per il calcolo delle sollecitazioni centrifughe e termiche in dischi a profilo iperbolico e a spessore costante, aventi peso specifico e temperatura variabili lungo il raggio. Ingegneria Meccanica IX(11):39–43
Eraslan AN (2002) Von Mises yield criterion and nonlinearly hardening variable thickness rotating annular disks with rigid inclusion. Mech Res Commun Sci 29(5):339–350
Eraslan AN (2003) Elastic-plastic deformations of rotating variable thickness annular disks with free, pressurized and radially constrained boundary conditions. Int J Mech Sci 45(4):643–667
Eraslan AN, Argeşo H (2002) Limit angular velocities of variable thickness rotating disks. Int J Solids Struct 39(12):3109–3130
Eraslan AN, Orçan Y (2002) Elastic-plastic deformation of a rotating solid disk of exponentially varying thickness. Mech Mater 34(7):423–432
Eraslan AN, Orçan Y (2002) On the rotating elastic-plastic solid disks of variable thickness having concave profiles. Int J Mech Sci 44(7):1445–1466
Eraslan AN, Orçan Y (2002) A parametric analysis of rotating variable thickness elastoplastic annular disks subjected to pressurized and radially constrained boundary conditions. Turk J Eng Environ Sci 28(6):381–395
Eraslan AN, Orçan Y, Güven U (2005) Elastoplastic analysis of nonlinearly hardening variable thickness annular disks under external pressure. Mech Res Commun 32(3):306–315
Faupel JH (1964) Engineering design: a synthesis of stress analysis and materials engineering. Wiley, New York
Fenner DN (1987) Engineering stress analysis a finite element approach with FORTRAN 77 software. E. Horwood, Halsted Press, Chichester
Forsyth AR (1996) A treatise of differential equations, 6th edn. Dover Publications, New York
Gamer U (1984) Elastic-plastic deformation of the rotating solid disk. Archiv Appl Mech (Ingenieur Archiv) 54(5):345–354
Gamer U, Mack W (1985) Thermal stress in an elastic–plastic disk exposed to a circular heat source. J Appl Math Phys 36:568–580
Giovannozzi R (1956) Calculation, tabulation and uses of some functions occurring in the theory of the conical disc subjected to centrifugal and thermic stress. In: Proceedings of II Congresso Aeronautico Europeo, Scheveningen, 30.1-30.12
Giovannozzi R (1965) Costruzione di Macchine, vol. II, 4th ed., Pàtron, Bologna
Güven U (1998) Elastic-plastic stress distribution in a rotating hyperbolic disk with rigid inclusion. Int J Mech Sci 40(1):97–109
Güven U (1998) Stress distribution in a linear hardening annular disk of variable thickness subjected to external pressure. Int J Mech Sci 40(6):589–601
Honegger von E (1927) Festigkeitsberechnung von Rotierenden Konischen Scheiben. Zeitschrift für Angewandte Mathematik und Mechanik. pp (120–128)
Klein F (1933) Vorlesungen über die hypergeometrische Funktion. J. Springer, Berlin
László F (1925) Geschleuderte Umdrehungskörper im Gebiet bleibender Deformation. Zietschrift für Angewandte Mathematik and Mechanik ZAMM 5:281–293
Lenard J, Haddow JB (1972) Plastic collapse speeds for rotating cylinders. Int J Mech Sci 14(5):285–292
Love AEH (1944) A treatise on the mathematical theory of elasticity. Dover Publication, New York
Ludwik P (1909) Elemente der Technologischen Mechanik. Springer, Berlin
Ma G, Hao H, Miyamoto Y (2001) Limit angular velocity of rotating disc with unified yield criterion. Int J Mech Sci 43(5):1137–1153
Manson SS (1947) Determination of elastic stresses in gas-turbine disks. NACA Report 871:241–251
Millenson MB, Manson SS (1948) Determination of stresses in gas-turbine disks subjected to plastic flow and creep. NACA Report 906:277–292
Orçan Y, Eraslan AN (2002) Elastic-plastic stresses in linearly hardening rotating solid disks of variable thickness. Mech Res Commun 29(4):269–281
Ramberg W, Osgood WR (1943) Description of stress-strain curves by three parameters. NACA Technical Note 902
Rees DWA (1999) Elastic–plastic stresses in rotating disks by Von Mises and Tresca. Zeitschrift fur Angewandte Mathematik und Mechanik ZAMM 79(4):281–288
Reid SR (1972) On the influence of acceleration stresses on the yielding of disks of uniform thickness. Int J Mech Sci 14(11):755–763
Saada AS (1974) Elasticity: theory and applications. Pergamon Press, New York
Smirnov V (1972) Cours de Mathématiques Supérieures. Editions MIR, Moscow
Stanley Thompson A, Lester PA (1946) Stresses in rotating disks at high temperatures. J Appl Mech 13(1):A45–A52
Sterner SC, Saigal S, Kistler W, Dietrich DE (1994) A unified numerical approach for the analysis of rotating disks including turbine rotors. Int J Solids Struct 31(2):269–277
Swift HW (1952) Plastic instability under plane stress. J Mech Phys Solids 1:1–18
Timoshenko SP, Goodier JN (1970) Theory of easticity, 3rd edn. McGraw Hill, New York
Tricomi FG (1967) Equazioni differenziali, 4th edn. Boringhieri, Turin
Ugural SC, Fenster SK (1987) Advanced strength and applied elasticity. Elsevier, New York
Vivio F, Vullo V (2007) Elastic stress analysis of rotating converging conical disks subjected to thermal load and having variable density along the radius. Int J Solids Struct 44:7767–7784
Vullo V, Vivio F (2008) Elastic stress analysis of non-linear variable thickness rotating disks subjected to thermal load and having variable density along the radius. Int J Solids Struct 45:5337–5355
You LH, Zhang JJ (1999) Elastic-plastic stresses in a rotating solid disk. Int J Mech Sci 41(3):269–282
You LH, Long SY, Zang JJ (1997) Perturbation solution of rotating solid disks with nonlinear strain–hardening. Mech Res Commun 24(6):649–658
You LH, Tang YY, Zhang JJ, Zheng CY (2000) Numerical analysis of elastic-plastic rotating disks with arbitrary variable thickness and density. Int J Solids Struct 37(52):7809–7820
Zimmermann RW, Lutz MP (1999) Thermal stresses and thermal expansion in a uniformly heated functionally graded cylinder. J Therm Stress 22:177–188
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Vivio, F., Vullo, L. Elastic–plastic analysis of rotating disks having non-linearly variable thickness: residual stresses by overspeeding and service stress state reduction. Ann. Solid Struct. Mech. 1, 87–102 (2010). https://doi.org/10.1007/s12356-010-0007-z
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DOI: https://doi.org/10.1007/s12356-010-0007-z