# Effect of Imposing Linear Particle Gradient on Creep Behavior in Composite Disc Having Hyperbolic Thickness

**ABSTRACT**

Iii the present work, an effort has been made to study the effect of variation in linear particle gradient (PG) on the steady-state creep behavior in anisotropic disc made of functionally graded material disc (FGM) with hyperbolic thickness. In the anisotropic FGM disc, the content of silicon carbide particles decreases linearly from the inner radius to the outer radius of the disc. The creep response of the anisotropic disc under stresses developing due to rotation at 15,000 rpm has been determined by Sherby’s law. The creep parameters of the anisotropic FGM disc vary along the radial distance due to varying composition. The creep behavior of anisotropic disc is expressed by a thr eshold stress with the value of stress exponent as 8. The study reveals that in the FGM disc, the radial stress increases throughout the disc with an increase in PG, whereas the stresses (tangential and effective) increase near the inner radius but decrease near the outer radius. By employing higher PG in anisotropic FGM disc with hyperbolic thickness, the distribution of steady-state strain rates becomes more uniform compared to an anisotropic disc having uniform reinforcement distribution. Thus, the care to introduce PG in anisotropic FGM disc should be taken for an optimal design of composite disc.

## INTRODUCTION

Nowadays, advancement in technology has made it possible to synthesize materials for components that exhibit graded-variation in their properties. Under severe thermomechanical loadings, the conventional materials (metals or ceramics) may not survive alone. Thus, a new material concept of functionally graded material was introduced and led to the development of superior heat resistant materials. The idea of FGM was conceived to provide a material that may withstand severe thennomechanical loadings requiring heat-resistant ceramics on the high-temperature side and tough metals with high thermal conductivity on the lower temperature side. This variation in composition from ceramic to metal is made gradually in between the two sides are characterized in a way that the composition of each one and the volume fraction of materials are changed gradually. By gradually varying the volume fraction of reinforcement, their material properties exhibit a smooth and continuous change from one surface to another. The smooth variation of properties always offers many advantages such as the reduction of thermal stress, increased strength, etc. A major advantage of FGMs is the possibility of designing its gradation to optimize its performance. While designing the material properties, we need to determine the material phase volume fraction of each point of the structure. However, several authors have studied the stress and strain in the non-FGM/FGM discs made of isotropic/anisotropic material at elevated temperature using different yield criteria and creep law. Wahl et al. [29] were the first to investigate theoretically steady-state creep behavior by a power function in a rotating turbine disc made of 12% chromium steel using von Mises and Tresca yield criteria theoretically describing creep behavior and compared the results with experimental values. Ma [15] derived the formulas based on the maximum shear theory associated with the von Mises flow rule for calculating the results for the distributions of stresses and strain rates creep in variable thickness disc rotating at uniform temperature by using the power function for creep behavior. The results obtained by using the von Mises criterion are found to be in excellent agreement with the available experimental creep data

[29]. Ma [15] extended liis work for rotating disc having a variable thickness, used in gas turbine and jet engine which are used in aircraft power plants and the nuclear space auxiliary power system. The study was based on the theoiy of the Tresca criterion. It is concluded that the distributions of stresses over the central portion of the variable thickness disc are quite different from the constant thickness disc. Bhatnagar et al. [2] investigated the creep response of orthotropic variable thickness discs by describing creep behavior with Norton’s power law. Thickness of rotating disc can be constant, linear, and hyperbolic. It is concluded that stresses and strain rates in composite disc can be reduced by selecting an optimum profile of thickness and a certain type of anisotropy. Mishra and Panday [16] proposed that creep response in rotating composite disc made by aluminum alloy can be described in a better way by Sherby’s constitutive creep model, as compared to Norton’s creep model. Pandey et al. [18] studied the steady-state creep response in aluminum-based composites consisting of silicon carbide particles under uniaxial loading conditions with variation in the temperature range between 623K and 723K for a different combination of particle sizes (1.7 /«», 14.5 *ftm* and 45.9 /////) with varying particle content (10 vo/%, 20 vo/% and 30 vo/%). It is concluded that the composite with finer particle size has better creep resistance than that containing coarser ones. Durodola and Attia [7] investigated the benefits of using different forms of fiber gradation in rotating hollow and solid FGM discs with constant thickness. It is noticed that the stress and deformation distribution can be modified by the different forms of property gradation with the same nominal volume fraction of reinforcement modify in the FGM discs compared with uniformly reinforced discs. Singh and Ray [26] studied the creep analysis in an isotropic FGM rotating disc at uniform elevated temperature by using Norton’s power law and concluded that the steady-state creep response in FGM disc is significantly superior compared to a non-FGM disc. Orcan and Eraslan [ 17] investigated that the stresses in composite discs with variable thickness are lower as compared to the results obtained for the disc having constant thickness than. Gupta et al. [12] have analyzed creep behavior of a rotating isotropic constant thickness disc made of FGM containing varying amounts of silicon carbide in the radial direction and operating in presence of radial thermal gradient and concluded that the steady-state strain rates in the rotating disc with the presence of thermal gradient and a linear particle gradient (PG) are significantly lower than that observed in an isotropic disc having uniform distribution of particle content and operating under isothermal condition. Jalied et al. [13] observed that the use of variable thickness disc helps in minimizing the weight of disc which helps to reduce the overall payload in the aerospace industry. This implies that a disc with variable thickness has no restrictions on the limiting value of maximum disc speed compared to a disc with constant thickness. Rattan et al. [22] have analyzed steady-state creep response of an isotropic FGM disc with constant thickness by using Sherby’s constitutive model. The results of isotropic disc having nonlinear variation of particle distribution along the radial distance are compared with the discs having a uniform and linear distribution of particles along the radial distance. Hasan Callioglu et al. [4] studied that stress analysis on functionally graded rotating annular discs subjected to temperature distributions parabolically decreasing with radius. He concluded that the tangential str ess by the increase in temperature decreases at the inner part of disc, but increases at the outer radius in disc whereas the radial stress reduces gradually for all distributions in the temperature. Garg et al. [9] studied the creep behavior analysis in variable thickness disc made of functionally gr aded material and noticed that the magnitudes of strain rates in functionally gr aded rotating discs are significantly lower than in a uniform composite disc. An FGM disc having reinforcement distributed in a nonlinear way possesses the lowest and relatively uniform distribution of strain rates. Deepak et al. [5] investigated the effect of thickness gradient on creep response in a linearly varying thickness disc made of functionally graded material containing silicon carbide particles in a matrix of pure aluminum. It can be concluded that the stresses and strain rates in both the radial as well as tangential directions reduce significantly with the increase in thickness gradient of the composite disc. Khanna et al. [14] analyzed creep behavior in a rotating isotropic disc with constant thickness by using the Tresca criterion. It is concluded that the gradient variation significantly effects the distribution of stresses and strain rates of the composite disc for purpose of designing a disc. Gupta and Singh [10, 27] studied an analytic framework for analyzing stresses and strain rates in isotropic rotating non-FGM/FGM disc with the thickness (constant, linear, and hyperbolic). The study revealed that the stresses and strain rates in the FGM hyperbolic thickness disc, are the lowest and more uniform compared to non-FGM/FGM discs with the thickness (constant and linear). Garg [8] studied the effect of varying thickness profile in the rotating FGM disc having linearly varying thickness. It is concluded that the strain rates are lower in the FGM disc having a higher thickness gradient along the radial direction. Bose and Rattan [3] have investigated the effect of the temperature gradient on anisotropic disc having a constant thickness. The results for the stress and strain rate distributions under a graded temperature field are expressed graphically. It is concluded that the effect of the parabolic temperature gradient should be considered while designing the anisotropic rotating disc due to its effect on the Disc’s creep behavior.

S. B. Singh and coworkers have also analyzed Elastic-Plastic and creep transition in the disc using transition theory where the assumptions: (i) incompressibility condition, (ii) creep-strain laws like Norton, (iii) yield condition like that of Tresca, (iv) associated flow rule were not considered. The necessity of the use of ad-hoc semi-empirical laws in the classical theory of elastic-plastic transition is based on the approach that the transition is a linear phenomenon which is not possible. Under the elastic-plastic and creep transition, the fundamental structure of the object undergoes a change and rearrange themselves to cause non-linear effects. Therefore, it suggests that at transition behavior, non-linear terms are significant and cannot be ignored. The generalized strain measures are useful to solve the various problems of elastic-plastic by solving the non-linear differential equations at the transition points. This concept of generalized strain measures and transition theory has been applied to find transitional stresses in various problems. Thakur [19] discusses the problems in creep transition stresses of a thick isotropic spherical shell by finitesirnal deformation under a steady-state of temperature by using Seth transition theory. All these problems based on the recognition of the transition state as a separate state necessitates showing the existence of the constitutive equation for that state. Thakur et al. [20, 21] further studied elastic-plastic and creep deformation in a rotating disc subjected to parameters such as variable thickness, variable density, etc.

The whole above literature study is based on pertaining to creep in rotating composite disc. With these forethoughts, it is decided to investigate the effect of imposing various kinds of linear PG on steady-state creep behavior for norr-FGM/FGM disc. The material of the anisotropic disc is made by aluminum alloy based metal matrix composites containing silicon carbide particulates due to the excellent mechanical properties like high specific strengtlr/stiffness and high-temperature stability. The thickness of the composite is assumed to be hyperbolic, because the stresses produced in the disc are due to rotary motion, can be minimized by varying the thickness of the disc. The analysis has been done by using Hill’s criterion for yielding. The creep behavior of the composite disc with stress exponent 8 has been described by Sherby’s constitutive law. The creep parameters in the law have been determined using the regr ession equations developed on the basis of available experimental results in the literature.