^{1}

With regard to the transmission of a thermomechanical signal on extremely short temporal and spatial scales, which represents an issue particularly important dealing with micro- and nanosized electromechanical systems, it is well known that the so-called non-Fourier effects become not negligible. In addition, it has to be considered that the interaction among multiple energy carriers has, as a direct consequence, the involvement of high-order terms in the time-differential formulation of the dual-phase lag heat conduction constitutive equation linking the heat flux vector with the temperature variation gradient. Accepting that the deformations caused by the temperature variations are small enough to be modeled under the assumptions typical of the linear thermoelasticity, in the present article we take into account the highest Taylor expansion orders able to guarantee (under appropriate assumptions) stability conditions, thermodynamic consistency, and at the same time the existence of an influence domain of the external data linked to the energy transmission as thermal waves. To this aim, a cylindrical domain filled by an anisotropic and inhomogeneous thermoelastic material is investigated, although the results obtained will be independent from the considered geometry: for such a reason, we will be able to consider as illustrative examples some simulations referred to single-layer graphene and to show how the expansion orders selected strongly influence the domain of influence depth.

It is well known that the Fourier heat conduction equation leads to a diffusive model which predicts that a thermal signal can propagate infinitely fast through the medium under investigation. Likewise, it is now clear that it turns out to be not applicable to ultrafast and ultrasmall contexts: dealing, for example, with micro- and nanosized electromechanical systems the so-called non-Fourier effects involving the coupled diffusion and the wavelike heat propagation become not negligible [

In this regard, we are convinced that the dual-phase lag (DPL) model, together with its time-differential formulation, turned out to be particularly suited to respond to this kind of needs, having its features in fact already deepened in a very wide number of works, of which [

Considering in more detail the possible choices of the Taylor series expansion orders

Through this contribution we want to move forward and complete the study on the penetration depth of a thermomechanical signal in a micro-/nanosized deformable thermal conductor, considering the relationship between heat flux vector

Emphasizing once again that our results will be shown to be independent of the shape of the region considered, we take into consideration a right cylinder

For the investigation at issue, we consider it unnecessary to propose here a series of preliminary considerations already explained in [

By resorting to the linear thermoelastic theory under inhomogeneous and anisotropic assumptions, we recall that the convention for which a comma stands for partial differentiation with respect to the corresponding Cartesian coordinate will be employed, together with the following notations and properties:

The lateral boundary conditions, i.e., when

The upper end boundary conditions, i.e., when

The lower base boundary conditions assimilable to appropriate actions insisting on

where the functions

The initial-boundary value problem

The transformed equations of motion

The transformed equation of energy

The transformed constitutive equations

The transformed geometrical equations

The ordered array

Referring to the above defined three-dimensional region

We insert now relation (

Some important features of the above functional

Under the hypotheses

In order to prove that

The following theorem can then be proved, concerning the solution

Let

Avoiding, just for reasons of synthesis, reporting the explicit expression of

It is worth noting that the investigation techniques used in this article, although deriving from the classical linear thermoelasticity, turn out to be particularly suitable for the handling of innovative thermomechanical models such as the one here treated: in this regard it is not trivial to underline that, again in the linear thermoelastic field, there are numerous examples in the literature concerning the study of mixed initial-boundary value problems also involving very peculiar material structures (see, for instance, [

In order to provide useful elements for a complete understanding of the heat exchange mechanisms able to take into account also elastic deformations on the micro-/nanoscale, the influence of the DPL high-order effects up to

independent of the shape of the region

depending only on the features of the selected material, among which there are the constitutive coefficients and tensors and the relaxation times

Imagining the lower base of the deformable, anisotropic, and inhomogeneous cylinder

In the attempt to apply the result obtained to a concrete estimate, and in continuity with what was done in [

Mass density | |

| |

Thermal conductivity | |

| |

Specific heat capacity | |

| |

Elastic modulus | |

| |

Shear modulus | |

| |

Thermal expansion coefficient | |

The behavior of the parameter

In order to integrate and facilitate a direct comparison with the results shown in [

The behavior of the parameter

The behavior of the parameter

Again as highlighted in [

In conclusion, we have proved that the latter feature highlighted by Tzou in [

The main results shown in the manuscript are analytical. With regard to the proposed illustrative simulations, the values of the material parameters considered can be easily found in the literature. For this reason, no information seems to be necessary regarding the availability of specific data to support the results.

The author declares that there are no conflicts of interest regarding the publication of this article.

The author acknowledges the Italian National Group of Mathematical Physics (GNFM-INdAM) for supporting the research project