MATHEMATICAL MODEL IN A THIN NON-HOMOGENEOUS ROTATING DISC FOR ISOTROPIC MATERIAL WITH RIGID SHAFT BY USING SETH ’ S TRANSITION THEORY

In this paper we investigate stresses and displacem ent in thin nonhomogeneous rotating disc has been derived by using Seth’s Transition theory and results have been discussed and depicted graphically. Non-h omogeneity is assumed due to the variation of modulus of rigidity. As a numerical ex ample, it has been seen that in the presence of non-homogeneity having values k > 0 at the bore, reduces stresses, displacement and the angular speed as compare to le sser value i.e. k <0 of nonhomogeneity. Radial stresses maximum at the interna l surface.


INTRODUCTION
Rotating discs are historically of interest to designers due to their vast spectrum of use in the aerospace industry.Gas turbine discs are an important example of such applications.In turbojet engines, rotating discs are simultaneously subjected to mechanical and thermal loads.A disc may be under internal pressure due to shrink fit on its mounting shaft; in addition an external tensile load may be applied to its outer edge resulting from the blade effects installed on its outer periphery.Rotating Discs form an essential part of the design of rotating machinery, namely rotors, turbines, compressors, flywheel and computer's disc drive etc. Helicopter rotor blades are typically built-up composite structure and made of material that may be anisotropic and non-homogeneous.For wide class of materials such as hot rolled copper, aluminum and magnesium alloys some degree of non-homogeneity is presents.OLSZAK et al. [1] solved the problems of thick walled cylinder, non-homogeneous both elastically and plastically subjected to internal and external pressures and showed that plastic flow may start from either surface depending on the character and intensity of the nonhomogeneity.GHOSH [2] on the problem involving the study of elastic-plastic stress in a spherically pressure vessel of non-homogeneous material and MUKHOPADHYAY [3 ] studied the effect of non-homogeneity on yields stress in a thick walled cylinder tube under pressure.Gupta et al. [4] solved the problems effect of non-homogeneity on elastic-plastic transition in a thin rotating disc by using Seth's transition theory and plane stress condition.This theory [5] does not required any assumptions like an yield condition, incompressibility condition and thus poses and solves a more general problem from which cases pertaining to the above assumptions can be worked out.It utilizes the concept of generalized strain measure and asymptotic solution at critical points or turning points of the differential equations defining the deformed field and has been successfully applied to a large number of problems [7][8][9][10][11][12].SETH [5] has defined the generalized principal strain measure as: where n is the measure and ii A e .is the Almansi finite strain components.For n = -2, -1,0, 1, 2, it gives Cauchy, Green Hencky, Swainger and Almansi measures, respectively.The rigidity modulus of a thin rotating disc is assumed to vary radially i.e. (2) where 0 µ and k are real constants.In this research paper, we discuss mathematical model in a thin non-homogeneous rotating disc for isotropic material with shaft by using Seth's transition theory.Result obtained have been numerically and depicted graphically.

MATHEMATICAL MODELS AND GOVERNING EQUATION
We consider an annular disc of inner radius a and outer radius b rotating with an angular speed ω about an axis perpendicular to its plane and passed through the center as shown in Figure 1.The annular disc is mounted on a rigid shaft.The disc is made of the material having constant density ρ but variable modulus of rigidity ( ) and thickness of disc is assumed sufficiently small so that it is effectively in a state of plane stress, that is, the axial stress zz T is zero.
The displacement components in cylindrical polar co-ordinate are given by [6]: where β is function of only and d is a constant.The finite strain components are given by Seth [6] as:

[ ]
n n e where .r P The stress -strain relations for isotropic material are given by [13]: where ij T and ij e are the stresses and strain components, λ are lame's constant, ( ) Equations (6) for this problem become:   4) in equation ( 6), the strain components in terms of stresses are obtained as [16]: where Substituting equation (5) in equation ( 7), we get the stress as: where ρ is the density of the material of the rotating disc.
Using equation ( 9) and ( 10), we get a non-linear differential equation in β as: ( P is function of β and β is function of r only).From equation ( 11), the transition points of β are 1 − = P and ∞ ± .

Boundary condition:
The boundary conditions are given by: where rr T are the radial stresses and u are the displacement.

SOLUTION THROUGH THE PRINCIPAL STRESSES
For finding the plastic stress, the transition function is taken through the principal stress (see SETH [5,6], HULSURKAR [7], GUPTA etl.[4,[10][11][12], SHUKLA [8,9 ] , THAKUR [14 -42]) at the transition point P → ±∞ .We take the transition function T is defined as: where T is function of r only.Taking the logarithmic differentiation of equation ( 13) with respect to r, we get: Substituting the value of / dP dβ from equation (11) in equation ( 14) and taking the asymptotic value as ±∞ → P , we get after integration: ( ) where ( ) ( ) and A is a constant of integration and can be determined by the given boundary condition From equations ( 13) and ( 15), we have: ( ) Substituting equation ( 16) in equation ( 10) and integrating, we get: where B 1 is a constant of integration and can be determined by the given boundary condition and ( ) ( ) 16) and (17) in second equation of equation ( 8), we get: Substituting equation (18) in equation ( 3), we get ( ) ( ) where ( ) ( ) is the Young's modulus.Using boundary condition (12 in equations ( 17) and ( 19), we get the values of constant of integration: Substituting the values of constant of integration A and B from equations ( 16), (17), and ( 19) respectively, we get the transitional stresses and displacement as: Substituting eqn.(2) in equation ( 20), we get: Initial yielding.From equation ( 21), it is seen that θθ T T rr − is maximum at the internal surface (that is at r = a), therefore yielding will take place at the internal surface of the disc and equation ( 21) are becomes: ) angular velocity i ω required for initial yielding is given by: where The angular velocity f ω for fully-plastic state is given by: ( ) where .Stresses, displacement and angular speed for fully-plastic state ( ) λ → ∞ in non dimensional form are obtained from equations ( 21) and ( 24) become: )

NUMERICAL ILLUSTRATION AND DISCUSSION
For calculating the stresses, angular speed and displacement based on the above analysis, the following values have been taken as k = -1, 0.5, 2, E/Y = 1, 2 and 0 / 2 respectively.We considered two types of discs as shown in Table 1.
• Discs having less non-homogeneity at the bore than at the rim, i.e. k < 0.
• Discs having more non-homogeneity at the bore than at the rim, i.e. k > 0. It can be seen from Table 1, that disc having more non-homogeneity values k > 0 at the bore than at the rim required higher percentage increase in angular speed to become fully plastic from its initial yielding as compared to rotating disc having lesser value of non-homogeneity k < 0 at the bore.Curves have been drawn in fig. 1    It has been observed that discs having less non-homogeneity (i.e.k = -1) require higher angular speed to yield at the internal surface as compare to disc having more nonhomogeneity at the bore than at the rim, i.e. k = 2.In fig. 3 and 4, curves have been drawn for stresses and displacement with respect to radii ratio R = r/b for elastic-plastic transition and fully plastic state respectively.Discs having less non-homogeneity at the bore required maximum radial stresses and circumferential stresses as compared to disc have more nonhomogeneity at the bore than at the rim.It has been seen that radial stresses is maximum at the internal surface.Rotating disc likely fracture at the bore.

CONCLUSION
It has been observed that discs having less non-homogeneity ( k < 0 ) require higher angular speed to yield at the internal surface as compare to disc having more nonhomogeneity at the bore than at the rim ( k > 2).Discs having less non-homogeneity at the bore required maximum radial stresses and circumferential stresses as compared to disc have more non-homogeneity at the bore than at the rim.It has been seen that radial stresses is maximum at the internal surface.
in equation(1), the generalized components of strain are: of equilibrium are all satisfied except: the following non-dimensional components as R = r/b,

Figure 2 .Figure 3 .Figure 4 .
Figure 2. Angular speed required for initial yielding at the internal surface of the rotating disc with rigid inclusion along the radii ratio b a R / 0 = for k = -1, 0.5, 2.

Table 1 :
Angular speed required for initial yielding and fully plastic state