Dynamic Problem of Coupled Thermoelasticity for a Thin Composite Structure

A dynamic problem of coupled thermoelasticity for a composite body is analyzed. The body is modeled as a composite beam in response to elastic deformation. It consists of two parts of different densities chosen such that the center of mass is the junction point. The beam is clamped at the center of mass and free at its ends. The body acts as a two-dimensional structure in response to the heat flow. The problem reduces to three partial integro-differential equations, and the two beam equations are coupled by the heat equation. A method based on integral transformations and expansion of the transforms of the beam deflection in terms of shape and vibration modes is proposed. Ultimately, it leads to an infinite system of linear algebraic equations with respect to the Laplace transforms of the vibration modes solved by the method of successive approximations. In addition, expressions for the generalized free energy and the dissipation function are derived. By employing the variational principle it is shown that this formulation deduces the governing differential equations of thermoelastic vibration of the thin structure that follow from dynamic thermoelasticity and beam theory.