Boundary Element, Coupled Thermoelasticity
In this chapter, considering the Lord and Shulman’s theory, a Laplace-transform boundary element method is developed for the dynamic problems in coupled thermoelasticity with relaxation time involving a finite two dimensional domain. The boundary element
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Boundary Element, Coupled Thermoelasticity
In this chapter, considering the Lord and Shulman’s theory, a Laplace-transform boundary element method is developed for the dynamic problems in coupled thermoelasticity with relaxation time involving a finite two dimensional domain. The boundary element formulation is presented and a single heat excitation is used to drive the boundary element formulations. Aspect of numerical implementation are discussed. It is shown that the distributions of temperature, displacement, and stress show jumps at their wave fronts. The thermo-mechanical waves propagation in a finite domain and the influence of relaxation time on them are presented. The results of this section are compared with the classical coupled theory (CCT) and the GreenLindsay theory (GL). It is verified that the LS theory of the generalized thermoelasticity results into significant differences in the patterns and wave fronts of temperature, displacement, and stress compared to the CCT and GL theories, although the material and geometrical properties of the solution domain are identical. The details of these differences are given in the result section.
29.1 Governing Equations A homogeneous isotropic thermoelastic solid is considered. In the absence of body forces and heat flux, the governing equations for the dynamic coupled generalized thermoelasticity in the time domain based on the Lord and Shulman theory are written as 1 (λ + μ)uj,ij + μui,jj − γT0 T,i − ρ¨ui = 0 (29.1) kT,jj − ρce T˙ − ρce t0 T¨ − γ(t0 u¨ j,j + u˙ j,j ) = 0
(29.2)
1 Tehrani, P., Eslami, M.R., Boundary Element Analysis of Finite Domain Under Thermal and Mechanical Shock with the Lord-Shulman Theory, Proceedings of I.Mech.E, Journal of Strain, Analysis, 38(1), 53–64 (2003).
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_29, © Springer Science+Business Media Dordrecht 2013
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29 Boundary Element, Coupled Thermoelasticity
Throughout this section, the summation convention on repeated indices is used. A dot indicates time differentiation and the subscript i after a comma is partial differentiation with respect to xi (i = 1, 2). In these equations λ, μ, ui , ρ, T , T0 , k, γ, ce , and t0 are the Lame’s constants, the components of displacement vector, mass density, absolute temperature, reference temperature, coefficient of thermal conductivity, stress-temperature modulus, specific heat at constant strain, and relaxation times proposed by Lord and Shulman, respectively. When t0 vanish, Eq. (29.2) reduces to the classical coupled theory. In the Lord and Shulman’s theory, Fourier’s law of heat conduction is modified by introducing the relaxation time t0 . It is convenient to introduce the usual dimensionless variables as tCs t0 Cs x ; ˆt = ; tˆ0 = α α α σij (λ + 2μ)ui T − T0 σˆ ij = ; uˆ i = ; T= γT0 α · γ · T0 T0 xˆ =
(29.3)
where α = k/ρce Cs is the dimensionless characteristic length and Cs = √ (λ + 2μ)/ρ is the velocity of the longitudinal wave
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