Finite Element of Coupled Thermoelasticity
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 variationa
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Finite Element of Coupled Thermoelasticity
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 in V σi j, j + X i = ρu¨ i ˙ qi,i + ρcθ + βT0 ˙ii = R
(28.1) 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 = θs
on A
for t > t0
(28.3)
θ,n + aθ = b
on A
for t > t0
(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 and Its Applications 197, DOI: 10.1007/978-94-007-6356-2_28, © Springer Science+Business Media Dordrecht 2013
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28 Finite Element of Coupled Thermoelasticity
the boundary. The first condition is related to the specified temperature and the second condition describes the convection and radiation on the boundary. The mechanical boundary conditions are specified through the traction vector on the boundary. The traction components are related to the stress tensor through the Cauchy‘s formula given by tin = σij n j
on A
for t > t0
(28.5)
where tin is the prescribed traction component on the boundary surface whose outer unit normal vector is n. For displacement formulations, using the constitutive laws of linear thermoelasticity along with the strain-displacement relations, the traction components can be related to the displacements as tin = μ(u i, j + u j,i )n j + λu k,k n i − (3λ + 2μ)αθn i
(28.6)
where θ = T − T0 is the temperature change above the reference temperature T0 . It is further possible to have kinematical boundary conditions where the displacements are specified on the boundary as u i = u¯ i (s)
on A
for t > t0
(28.7)
The system of coupled Eqs. (28.1) and (28.2) does not have a general analytical solution. A finite element formulation may be developed based on the Galerkin method. The finite element model of the problem is obtained by discretizing the solution domain into a number of arbitrary elements. In each base element (e), the
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