Analysis of a Hyperbolic System Modelling the Thermoelastic Impact of Two Rods

The problem on collinear collision of two thermoelastic rods possessing the same rheological parameters but of different length and temperature is considered, in so doing before the impact the rods move unidirectionally along a common longitudinal axis with distinct constant velocities. Lateral surfaces and free ends of both rods are heat insulated, and free heat exchange between the rods occurs within contacting ends. The rods' thermoelastic behavior is described by the Green–Naghdy theory. Two methods are used as the methods of solution, namely: D'Alembert's method and the ray method. D'Alembert's method is based on the analytical solution of equations of the hyperbolic type describing the dynamic behavior of the thermoelastic rods. This solution involves four arbitrary functions which are determined from the initial and boundary conditions and are piecewise constant functions. The ray method is based on the theory of discontinuities and also allows one to obtain the solution involving four arbitrary constants. Both solutions complement each other, since D'Alembert's solution enables one to analyze the influence of thermoelastic parameters on the values to be found, while the solution via the ray method is best suited for numerical investigation of the longitudinal coordinate dependence of the desired values at each fixed instant of the time beginning from the moment of rods' collision up to the moment of their rebound. To illustrate the both approaches, the problem of the rods' collision without account for the stress and temperature fields coupling is also considered.