Finite Element of Coupled Thermoelasticity

Abstract

Due to the mathematical complexities encountered in analytical treatment of the coupled thermoelasticity problems, the finite element method is often preferred. The finite element method itself is based on two entirely different approaches, the variational approach based on the Ritz method, and the weighted residual methods. The variational approach, which for elastic continuum is based on the extremum of the total potential and kinetic energies has deficiencies in handling the coupled thermoelasticity problems due to the controversial functional relation of the first law of thermodynamics. On the other hand, the weighted residual method based on the Galerkin technique, which is directly applied to the governing equations, is quite efficient and has a very high rate of convergence. 28.1 Galerkin Finite Element The general governing equations of the classical coupled thermoelasticity are the equation of motion and the first law of thermodynamics as σ i j, j + X i = ρ ¨ u i in V (28.1) q i,i + ρc ˙ θ + βT 0 ˙ ii = R in V (28.2) These equations must be simultaneously solved for the displacement components u i and temperature change θ. The thermal boundary conditions are satisfied by either of the equations θ = θ s on A for t > t 0 (28.3) θ ,n + aθ = b on A for t > t 0 (28.4) where θ ,n is the gradient of temperature change along the normal to the surface boundary A, and a and b are either constants or given functions of temperature on M. Reza Eslami et al., Theory of Elasticity and Thermal Stresses, Solid Mechanics 727 and Its Applications 197,