A complete Galerkin’s type approach of finite element for the solution of a problem on modified Green–Lindsay thermoelasticity for a functionally graded hollow disk

Abstract

The present work is concerned with the modified Green–Lindsay thermoelasticity theory involving strain-rate. This theory has been proposed very recently to modify the Green–Lindsay (GL) model of thermoelasticity by introducing both temperature and strain-rate in the constitutive relations of coupled thermo-mechanics. We consider a problem involving coupled thermo-mechanical interactions inside a functionally graded hollow disk due to thermal shock applied at its stress-free inner and outer boundaries. The material properties of the disk are assumed to be non-homogeneous and changing along the radial direction according to a volume fraction rule with a power of non-homogeneity index term. We formulate the problem by considering the basic governing equations of GL and modified GL thermoelasticity theories in a unified form and derive a linear system of coupled partial differential equations with variable coefficients. The major aim of this work is to apply a complete finite element method to obtain the solution of the problem. We solve the system of equations by applying a Galerkin’s approach of FEM for the space domain and derive the time differential system of equations. To solve this time differential system of equations, we use two different approaches: (1) a Galerkin’s type FE approach and (2) the Newmark time integration scheme. We compare the results obtained from these two different approaches and find that the solution obtained by complete Galerkin’s approach of FEM perfectly matches with the corresponding solution obtained from Newmark time integration scheme. We further compare the CPU time taken by these two methods with the CPU time taken by the trans-finite element method and reveal the efficiency of the present method over trans-FE method. The variation of different physical field variables with space and time is discussed for various values of the non-homogeneity index by highlighting the difference in the results under the GL model and modified GL model.