Spatial behavior of the steady state vibrations in a dual-phase-lag rigid conductor

This paper studies the spatial behavior of the steady state vibrations in a cylinder made of a dual-phase-lag anisotropic rigid conductor material. We analyze the influence of the lagging model upon the spatial behavior of the amplitude of vibration along the axis of the cylinder, providing the explicit expressions of the decay rate and of the corresponding critical frequency in terms of the coefficients of the considered constitutive equation (or delay times). In fact, for the amplitude of the harmonic vibrations we obtain some exponential decay estimates of Saint-Venant type, provided the frequency of vibration is lower than a critical value. This gives information on the thermal penetration depth of the steady state vibrations describing the heat affected zone. Illustrative examples are given for the class of lagging behavior models that are thermodynamically compatible.