Research Projects

In the real world, mathematics is about solving unsolved problems, analyzing unfamiliar situations, inventing innovative ideas and tools, exploring new landscapes, and much more, while all the reasons are clear and require proper logic. Nonlinear differential equations are often used to describe various physical systems in applied mathematics. Many structural systems include rotating discs and are subjected to thermal-mechanical loading. Thus, the proper material selection and the estimation of thermal stresses are essential parts of the design process because of their capacity to withstand the combined effects of high pressures, radial loads, and radial temperature gradients. Now-a-days, functionally graded materials (FGMs), a special class of composite materials, are often used as structural material for high temperature applications. However, due to the continuous variation of thermal and mechanical properties from one surface to the other of the FGM disc, the mathematical solution for heat transfer and, subsequently, the thermal stresses, is a great challenge. In this research proposal, a convective-conductive-radiative rotating disc made of functionally graded materials (FGM) is considered. The parameters in describing the thermo-mechanical behavior will be taken either functions of temperature or local coordinate systems. The governing nonlinear differential equation for heat transfer, involving all the modes of heat transfer, such as conduction, convection and radiation, will be constructed and then solved using the homotopy perturbation method (HPM). The stress fields of the proposed disc subjected to thermo-mechanical loading will be derived from the HPM based approximate closed-form solution for a steady-state non-homogeneous temperature field coupled with the solution of the classical theory of elasticity. Both the Dirichlet and Neumann boundary conditions will be taken into account in the current study. The effect of various thermo-mechanical parameters on the local temperature distribution and, subsequently, on the stress fields will be thoroughly investigated. For the sake of confidence and correctness, the approximate closed-form solutions for the temperature field and stress fields in the rotating disc will be validated with the numerical solution based on finite difference and finite element methods. An inverse study for estimating the thermo-mechanical parameters for a reference temperature field and stress fields will also be part of the present proposal.