Abstract
Disks of rotating machineries like gas turbine engines of aircraft are subjected to very high centrifugal stresses during extreme maneuvering conditions. These disks operate in nonlinear plastic region and may grow plastically during over-speed/over-load resulting in permanent deformation. As per certification criterion, disks should have acceptable permanent growth after over-speed. A closed-form solution is developed to predict permanent residual growth in rotating disk with variable thickness for linearly strain hardening material behavior using Tresca’s yield criteria and its associated flow rule. Results obtained using analytical solutions have been compared with finite element method (FEM) and experimental tests results for uniform thickness disks
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Abbreviations
- E :
-
Young’s modulus of elasticity of disk material
- \(f_{1} (\lambda ), f_{2} (\lambda )\) :
-
Function depending on disk material properties
- h :
-
Disk thickness at radial location r (non-dimensional form \(\bar{h} = h/h_{0}\)) \(\bar{h}^{{\prime }} = {\text{d}}\bar{h}/{\text{d}}\bar{r}\)
- \(h_{0}\) :
-
Disk thickness at the bore
- H :
-
Profile parameter of hyperbola disk
- \(H_{m}\) :
-
\(H_{m} = \eta \sigma_{0} /E\)
- \(K_{1} , K_{2} ,K_{3} ,K_{4}\) :
-
Integration constant
- p :
-
Profile parameter of hyperbola disk
- r :
-
Radial location from axis of rotation (non-dimensional form \(\bar{r} = r/r_{2}\))
- \(r_{1}\) :
-
Disk bore radius (non-dimensional form \(\bar{r}_{1} = r_{1} /r_{2}\))
- \(r_{2}\) :
-
Disk rim radius (non-dimensional form \(\bar{r}_{2} = r_{2} /r_{2}\))
- \(r_{\text{p}}\) :
-
Elastic plastic interface radius (non-dimensional form \(\bar{r}_{\text{p}} = r_{\text{p}} /r_{2}\))
- u :
-
Radial displacement at r (non-dimensional form \(\bar{u} = uE/r_{2} \sigma_{0}\))
- \(u^{\text{e}}\) :
-
Elastic displacement (non-dimensional \(\bar{u}^{\text{e}} = u^{\text{e}} E/r_{2} \sigma_{0}\))
- \(u^{\text{p}}\) :
-
Plastic displacement (non-dimensional \(\bar{u}^{\text{p}} = u^{\text{e}} E/r_{2} \sigma_{0}\))
- \(\varepsilon_{\theta } , \varepsilon_{\text{r}}\) :
-
Tangential and radial strain (non-dimensional form \(\bar{\varepsilon }_{\theta } = \varepsilon_{\theta } E/\sigma_{0} ,\bar{\varepsilon }_{\text{r}} = \varepsilon_{\text{r}} E/\sigma_{0}\))
- \(\varepsilon_{\text{eq}}\) :
-
Equivalent plastic strain (non-dimensional form \(\bar{\varepsilon }_{\text{eq}} = \varepsilon_{\text{eq}} E/\sigma_{0}\))
- \(\varepsilon_{\theta }^{\text{p}} ,\varepsilon_{\text{r}}^{\text{p}} ,\varepsilon_{\text{z}}^{\text{p}}\) :
-
Plastic tangential, radial and axial strain (non-dimensional form \(\bar{\varepsilon }_{\theta }^{\text{p}} = \varepsilon_{\theta }^{\text{p}} E/\sigma_{0} ,\bar{\varepsilon }_{\text{r}}^{\text{p}} = \varepsilon_{\text{r}}^{\text{p}} E/\sigma_{0} ,\bar{\varepsilon }_{\text{z}}^{\text{p}} = \varepsilon_{\text{z}}^{\text{p}} E/\sigma_{0}\))
- \(\eta\) :
-
Hardening parameter
- \(\lambda\) :
-
Constant depending on disk material properties
- \(\vartheta\) :
-
Poisson’s ratio
- \(\rho\) :
-
Density
- \(\sigma_{0}\) :
-
Initial yield stress
- \(\sigma_{\text{y}}\) :
-
Yield stress (non-dimensional form \(\bar{\sigma }_{\text{y}} = \sigma_{\text{y}} /\sigma_{0}\))
- \(\sigma_{\theta } , \sigma_{\text{r}}\) :
-
Tangential and radial stress (normalized form \(\bar{\sigma }_{\theta } = \sigma_{\theta } /\upsigma_{0 } ,\bar{\sigma }_{\text{r}} = \sigma_{\text{r}} /\upsigma_{0}\))
- \(\sigma_{\theta }^{\text{e}} , \sigma_{\text{r}}^{\text{e}}\) :
-
Tangential and radial stresses in elastic regime
- \(\sigma_{\theta }^{\text{p}} , \sigma_{\text{r}}^{\text{p}}\) :
-
Tangential and radial stresses in plastic regime
- \(\omega\) :
-
Angular velocity in radians per second (non-dimensional form \(\Omega = \sqrt {\rho \omega^{2} r_{2}^{2} /\sigma_{0} }\))
References
Timoshenko, S., Goodier, J.N.: Theory of Elasticity. McGraw-Hill, New York (1970)
Stodola, A.: Dampf und Gasturbinen, 6th edn. Julius Springer, Berlin (1924)
Reddy, T.Y., Srinath, H.: Elastic stresses in a rotating anisotropic annular disk of variable thickness and variable density. Int. J. Mech. Sci. 16, 85–89 (1974)
Jain, R., Ramachandra, K., Simha, K.R.Y.: Rotating anisotropic disk of uniform strength. Int. J. Mech. Sci. 41, 639–648 (1999)
Military Handbook. Department of Defence, USA: Mil-HDBK-5H (1998)
Chakrabarty, J.: Theory of Plasticity. McGraw-Hill, New York (1987)
Johnson, W., Mellor, P.B.: Engineering Plasticity. Ellis Horwood, Chichester, UK (1983)
Gamer, U.: Tresca’s yield condition and the rotating disk. J. Appl. Mech. 50:676–678 (1983)
Eraslan, A.N., Argeso, H.: Limit angular velocities of variable thickness rotating disks. Int. J. Solids Struct. 39, 3109–3130 (2002)
Ma, G., Hao, H., Miyamoto, Y.: Limiting angular velocity disc with unified yield criterion. Int. J. Mech. Sci. 43, 1137–1153 (2001)
Güven, U.: Elastic-plastic stresses in a rotating annular disk of variable thickness and variable density. Int. J. Mech. Sci. 34, 133–138 (1992)
Eraslan, A.N.: Elastic–plastic deformations of rotating variable thickness annular disks with free, pressurized and radially constrained boundary conditions. Int. J. Mech. Sci. 45, 643–667 (2003)
Eraslan, A.N.: Elastoplastic deformations of rotating parabolic solid disks using Tresca’s yield criterion. Eur. J. Mech. A Solids 22, 861–874 (2003)
Rees, D.W.A.: The Mechanics of Solids and Structures, 1st edn. McGraw-Hill, New York (1990)
You, L.H., Long, S.Y., Zhang, J.J.: Perturbation solution of rotating solid disks with non-linear strain hardening. Mech. Res. Commun. 24, 649–658 (1997)
You, L.H., Zhang, J.J.: Elastic-plastic stresses in a rotating solid disk. Int. J. Mech. Sci. 41, 269–282 (1999)
You, L.H., Tang, Y.Y., Zhang, J.J., Zhen, C.Y.: Numerical analysis of elastic-plastic rotating disks with arbitrary variable thickness and density. Int. J. Solids Struct. 37, 7809–7820 (2000)
Bhowmick, S., Misra, D., Nath, K.: Variational formulation based analysis on growth of yield front in high speed rotating solid disks. Int. J. Eng. Sci. Technol. 2, 200–219 (2010)
Wilterdink, P.I., Holms, A.G., Manson, S.S.: A theoretical and experimental investigation of the influence of temperature gradient on the deformation and burst speeds of rotating disks. Lewis Flight Propulsion Laboratory Cleveland, Ohio, Technical Note 2803 (1952)
Zienkiewicz, O.C.: The Finite Element Method in Engineering Science. McGraw-Hill, London (1971)
Ayyappan, C., Rajesh, K., Ramesh, P., Jain, R.: Experimental and numerical studies to predict residual growth in an aero-engine compressor disk after over-speed. In: Procedia Engineering, 6th International Conference on Creep Fatigue and Creep-Fatigue Interaction, vol. 55, pp. 625–30 (2013)
Karlsson, H.: ABAQUS/Standard User’s Manual, vol. I & II, Version 5.4. SorensenInc. Pawtucket, Rhode Island, USA (1994)
Hsu, Y.C., Forman, R.G.: Elastic-plastic analysis of an infinite sheet having a circular hole under pressure. ASME J. Appl. Mech. 42, 347–352 (1975)
Wanlin, G.: Elastic-plastic analysis of a finite sheet with a cold-worked hole. Eng. Fract. Mech. 46, 465–472 (1993)
Callioğlu, H., Topcu, M., Tarakcilar, A.R.: Elastic-plastic stress analysis of an orthotropic rotating disc. Int. J. Mech. Sci. 48, 985–990 (2006)
Robinson, E.L., Schenectady, N.Y.: Bursting tests of steam turbine disk wheels. Trans. ASME 66, 373–386 (1944)
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Authors are thankful to Director, GTRE Dr. C. P. Ramanarayan, Outstanding Scientist for allowing this paper to publish in international referred journal.
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Kumar, R., Jain, R. (2019). Stresses and Deformation in Rotating Disk During Over-Speed. In: Sahoo, P., Davim, J. (eds) Advances in Materials, Mechanical and Industrial Engineering. INCOM 2018. Lecture Notes on Multidisciplinary Industrial Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-96968-8_28
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