# Stresses and Deformation in Rotating Disk During Over-Speed

• Rajesh Kumar
• Rajeev Jain
Conference paper
Part of the Lecture Notes on Multidisciplinary Industrial Engineering book series (LNMUINEN)

## 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

## Keywords

Rotating disk Permanent deformation Residual growth Tresca’s Spin test ABAQUS Finite element method (FEM)

## Nomenclatures

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 $$\Omega = \sqrt {\rho \omega^{2} r_{2}^{2} /\sigma_{0} }$$)

## Notes

### Acknowledgements

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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