ABSTRACT
Introduction The system to be analyzed is an isotropic rod 0 ≤ x ≤ l that undergoes a motion in which the displacement vector is parallel to the x-axis. The displacement u(x, t) and the temperature θ(x, t) are functions of the coordinate x and the time t ≥ 0 only. Consider the equations of dynamic
linear thermoelasticity [5, 18]
κθxx = cθt + (3λ + 2µ)αθ0uxt,
(λ+ 2µ)uxx = (3λ + 2µ)αθx + utt,
where θ0 is the uniform reference temperature, is the mass density, λ and µ are the elastic moduli, c is the specific heat per unit mass, α is the coefficient of thermal expansion, and κ is the thermal conductivity. The subscripts containing t or x denote partial derivatives. Following the usual practice, we change variables to simplify the notation x → x l , t → ktcl2 , θ → θ−θ0θ0 , u → ul
, and get the system above in a nondimensional form: θxx = θt + auxt, butt − uxx = −aθx where a2 = θ0α
c(λ+2µ) , b = k2
c2(λ+2µ)l2 . These equations are considered in the
domain ΩT = (0, 1) × (0, T ). To establish the boundary conditions, we consider the case when the rod is situated between two walls that are kept at different temperatures. One end of the rod is permanently attached to a wall, while the other end is free to expand or contract. The expansion of the rod resulting from the evolution of the temperature and the stresses is limited by the existence of the other wall. For the displacement at the free edge, we choose the Signorini boundary conditions [7]: u(1, t) ≤ g, ux(1, t) ≤ aθ(1, t), [u(1, t) − g][ux(1, t) − aθ(1, t)] = 0 where the constant g is the nominal gap size between the wall and the free edge in the reference configuration. These conditions are completed by θ(0, t) = 0, −θx(1, t) = kθ(1, t), u(0, t) = 0, and all boundary conditions hold for 0 < t < T .
