On the depth of a thermomechanical signal in a dual-phase lag deformable medium

In view of the continuous need for sensors miniaturization, also with reference to contexts such as the Internet of things or the monitoring of critical situations and to the related sensoristics, we believe that a deep understanding of the phenomena that concretely drive the heat exchange mechanisms in devices of characteristic dimensions of the order of the micro/nanometer can be objectively useful, especially if referred to an optimal design process of the most recent instrumentation used in monitoring systems. This contribution aims to provide, through the use of an appropriate mathematical model, a thorough analysis of the problem concerning the penetration depth of a thermomechanical signal in deformable, anisotropic and inhomogeneous thermal conductors. Imagining to refer to a micro/nanoscale framework, it is evident that the classical models theorized to describe the heat transfer are no longer applicable, while the high-order effects linked to the thermal lagging phenomena and closely related to the number of heat carriers involved become increasingly important; moreover it is reasonable to assume, dealing with such scales, that the deformations caused by the temperature variations are small enough to be modeled under the hypotheses typical of the linear thermoelasticity. With these assumptions, and although our results remain valid even for more general shapes, we consider here just for convenience a cylindrical domain filled by a thermoelastic material, and assume that its lower base is subjected to appropriate thermomechanical actions defined externally, investigating through a suitable initial-boundary value problem the spatial behavior of the transient solutions in terms of existence of an influence domain of the assigned data. The analysis is concluded taking into account, by way of example, single-layer graphene and showing some graphic comparisons related to different expansion orders, through which the influence on the signal's depth of the related high-order effects is evident.