Heat transfer in a nonuniform rod of variable cross section

An integral formula is used to average a coupled problem of thermoelasticity for a nonuniform rod of variable cross section. Effective characteristics are found. It is shown that, in addition to the expected effective constants, there appear five independent constants characterizing the temperature rate change on the stresses in the rod, on the longitudinal heat flux, and on the entropy distribution along the length of the rod. A feature of these new constants is that they become equal to zero in the case of a homogeneous material. The homogenization of the thermoelasticity equations for nonuniform rods allows one to propose a new theory of thermal conductivity in rods. This new theory differs from the classical one by the fact that some new terms are added to the Duhamel–Neumann law, to the Fourier thermal conductivity law, and to the entropy expression. These new terms are proportional to the temperature rate change with time. It is also shown that, in the new theory of thermal conductivity, the propagation velocity of harmonic thermal perturbations is dependent on the oscillation frequency and is finite when the frequency tends to infinity.