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.

1. Introduction

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 [1]. In this connection the phonon-electron coupling factor and, by extension, the related phase lags can represent thermophysical properties able to take into account also the microstructural interaction effects (see [2], Chapters 1, 5). Therefore, a deep understanding of the heat exchange mechanisms at micro-/nanoscales is today more than ever a topic of great technological interest: it is sufficient to think about its impact on the correct design of MEMS (microelectromechanical systems) and NEMS (nanoelectromechanical systems) or even, in general, to the continuous need for sensors and actuators miniaturization and to the growing interest in controlling the related heat transfer processes.

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 [314] and the references therein just represent a selection also accounting for high expansion orders in thermoelasticity. The DPL constitutive equation puts in relation the temperature variation gradient at a certain time with the heat flux vector at a different time . To the delay/relaxation times or phase lags involved (heat flux vector delay time) and (temperature variation gradient delay time), respectively, it is possible to give the following physical interpretations: is connected to the fast-transient effects of thermal inertia, while is able to take account of microstructural interactions such as phonon scattering or phonon-electron interactions. Specifically within the context of the time-differential formulation for the DPL constitutive law, it is then possible to consider high-order effects by stopping at appropriate Taylor series expansion orders and the following relation between the heat flux vector and the temperature variation gradient (the reader can refer, for instance, to [2], eq. (2.115) page 103, or even to [12], page 930): being the conductivity tensor. It is also worth emphasizing that, dealing with micro-/nanosystems, it seems more than reasonable to assume that the deformations caused by the temperature variations are small enough to be modeled under the assumptions typical of the linear thermoelasticity: such an observation makes therefore particularly fitting to the framework above described the studies reported in [10, 1214] about the well-posedness question as well as the spatial behavior of the solutions for different linear thermoelastic models founded on constitutive equation (1).

Considering in more detail the possible choices of the Taylor series expansion orders and , we recall that Chiriţă et al. [11] showed very interesting results about some limitations in the application of constitutive equation (1), having proved in fact that for or the corresponding models unavoidably lead to instable mechanical systems; conversely, when the expansion orders are lower than or equal to four, the related models can be thermodynamically compatible, provided that suitable assumptions are made upon the delay times. This is the reason why in [1214] the selection of the expansion orders has concerned the following choices: , , , , and . Moreover, in order to capture the second sound effect for the heat propagation at the micro-/nanoscale, and with reference to the selections just cited , , and , we notice that only the choices and lead to models that can express a wavelike behavior, whereas depicts a diffusive behavior. However, all of these expansion orders are able to account for the high-order effects in and linked to the thermal lagging phenomena and are closely related to the number of heat carriers involved (see again [2], p. 442).

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 and temperature variation gradient given by (1) considered under the assumption . This actually represents an advancement, since the corresponding study related to the comparison between the orders selection and its counterpart , has already been investigated in [14], showing unequivocally the depth increase of the thermoelastic signal with the rise of the expansion orders (provided that the condition , able to guarantee the identification of an influence domain, is fulfilled). We take into account a cylindrical domain filled by an anisotropic and inhomogeneous thermoelastic material, although we will prove that the results obtained are valid regardless of the geometry considered: this feature will allow us to apply our analytical results to some simulations referred to single-layer graphene, showing that also in this case the influence of the expansion orders selected on the signal’s depth is evident. An external disturbance is simulated through the application on the lower base of the cylinder of specific thermal and mechanical actions defined from the outside, and a suitable initial-boundary value problem is implemented and investigated in order to show the spatial behavior of the solution in terms of existence and extension of an influence domain of the assigned data.

2. A Suitably Defined Initial-Boundary Value Problem

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 of base and height and choose the referring axes system in such a way that and that the base lies on the coordinate plane . We denote by the plane cross section at distance from the base of the cylinder, assuming it smooth enough in order to allow the application of the divergence theorem; moreover, denotes the portion of comprised between the parallel cross sections and . The cylinder is intended to be subjected to null initial conditions and null body force and heat supply, and also the boundary data are assumed null except for some prescribed nonzero actions insisting from the outside on the base : specifically, they are able to simulate the external thermomechanical perturbation of which the propagation depth will be estimated. Starting from an appropriately defined initial-boundary value problem, for which uniqueness and continuous dependence results from the given data have already been proved in [12, 13], we are able to show the spatial behavior of the solution with respect to the distance from the perturbed base .

For the investigation at issue, we consider it unnecessary to propose here a series of preliminary considerations already explained in [12, 13], but rather we start directly from a suitably transformed initial-boundary value problem (inferable indifferently from the aforementioned articles) assuming for it null initial data, null body force and heat supply, and null boundary data, except for the abovementioned nonzero external actions insisting on the base of the cylinder. The definitive formulation of our initial-boundary value problem is obtained through the integral operator defined, for any continuous function , as follows:and that for the sake of brevity and with an obvious meaning of the Roman superscripts involved is shortened asIn order to clarify the origin and significance of (3) and of its previous extended form (2), we believe it is appropriate to direct the reader towards [12], Section 3 Description of the investigation strategy. We recall that this kind of integral operator has been employed to obtain a number of results in terms of well-posedness and spatial behavior of the solutions for models based on the dual-phase lag as well as on the three-phase lag constitutive law (see at this regard, in addition to the articles already cited, also [1517]).

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: are the components of the stress tensor, is the mass density, are the components of the displacement vector, is the temperature variation from the constant ambient temperature (with strictly positive), is the entropy per unit mass, are the components of the heat flux vector, are the components of the strain tensor, and , , (, as already highlighted, represents the conductivity tensor), and are constitutive tensors depending only on the spatial variable and are characterized by the following symmetries:As anticipated, null initial conditions are assumed while, with regard to the boundary conditions, we set for any the following:(i)The lateral boundary conditions, i.e., when with :(ii)The upper end boundary conditions, i.e., when :(iii)The lower base boundary conditions assimilable to appropriate actions insisting on , i.e., when :

where the functions , , , and are assumed to be prescribed and sufficiently smooth on .

The initial-boundary value problem (we recall, deduced indifferently from [12] or [13]) object of investigation is structured as follows:The transformed equations of motionThe transformed equation of energyThe transformed constitutive equationsThe transformed geometrical equations

The ordered array defined in and associated with the set of transformed and nontrivial given data is identified as the (unique (see [12])) solution of the initial-boundary value problem .

3. The Domain of Influence Theorem

Referring to the above defined three-dimensional region and using (9)-(12) and (14) together with the divergence theorem and the null upper end and lateral boundary conditions, we getand multiplying (15) by we receiveThe next step is to make explicit, with the aid of (3) and (13), the last integral term of (16) through suitable handling of the partial differentiations with respect to the time variable and an appropriate manipulation of the coefficients. The aim is to obtain an expression in which only products between temperature variation gradients characterized by common superscripts are present and, as highlighted in [13], this will result in the emergence of time derivatives up to the order four:At this point, in order to proceed, we need to set a threshold value for the parameter , which we select as follows:It is fundamental to underline that , chosen in this way, not only results to be strictly positive but also makes the right-hand side of (17) nonpositive. Then (as already mentioned in [14] for the corresponding case relative to the orders selection , , and confirming once again the validity of the model) we underline that the above threshold value (18) able to guarantee the nonpositivity of the RHS of (17) coincides with what was found for the present model , in [13], p. 837, investigating the corresponding continuous dependence issue.

We insert now relation (17) into (16) and then, in view of the null initial conditions, it is sufficient to integrate the expression obtained four times with respect to the time variable to getSubsequently, to the unique solution of the initial-boundary value problem we associate, for every possible value of the variable , the following functional :

Some important features of the above functional can then be easily shown.

Lemma 1. Under the hypotheses , positive definite tensors and selecting for the threshold value defined in (18), is not increasing with respect to the variable, or in other words . Moreover, can be identified as a measure for the (unique) solution of the initial-boundary value problem , in the sense that it turns out to be not negative for every possible value of the variables and .

Proof. In order to prove that , it is sufficient to remember how the volume is structured and evaluate, starting from (19), the partial derivative of with respect to the variable: it is therefore evident that, under the hypotheses at issue, is not positive. Consequently, in view of the null upper end boundary conditions and of definition (20) of we havefor every possible value of and . Specifically, it is easy to prove that the condition necessarily correlates to a null solution for the considered initial-boundary value problem.

The following theorem can then be proved, concerning the solution of the initial-boundary value problem (no ) from which the problem derives, in view of the considered hypothesis of linearity (see again [12] or [13] for details).

Theorem 2 (existence of a domain of influence). Let be the solution of the initial-boundary value problem . Let and be greater than zero, let and be positive definite tensors, and let be identified with the threshold value (18). Moreover, suppose that the cylinder is subjected on its lower base to the external actions described through (7) and (8). Then there exists a positive constant , dimensionally expressed by a speed, such thati.e., the whole activity in the cylinder due to the presence of the assigned data acting on its base vanishes at distances from greater than or equal to .

Proof. Avoiding, just for reasons of synthesis, reporting the explicit expression of , anyway easily obtainable from (19), let us estimate instead the partial derivative of with respect to the time variable . To do this, we recall thatand, thus, from the definition (20) we get In order to estimate the right-hand side of (24) (in particular, the terms containing and ) we first take into account constitutive equation (11) and denote by , positive and constant, the greatest elastic modulus of the constitutive tensor such that for every it is . In view of the arithmetic-geometric mean inequality, the following relation is valid:On the other side, recalling constitutive equation (13), distributing with respect to the terms of the sum, and then applying the Cauchy-Schwarz inequality we can writeand then through squaring we getIn addition, identifying with the smallest (and strictly positive) eigenvalue of the conductivity tensor , the following relations are valid:and so from (24) it follows thatAssuming again for the threshold value (18) and reconsidering the partial derivative , but retaining this time only the terms with a number of integrals equal to that characterizing (29), we reach the following estimate:At this point it is sufficient to defineand to impose the equality between the following coefficients, which is a consequence of the comparison between the estimates involving the partial derivatives of with respect to and . We denote them by (strictly positive) and setIt follows that being the positive root of the algebraic equationWith these choices, comparing relations (29) and (30) we reach the following differential inequality, valid for all :Such a differential inequality can be integrated, for example, as in [17] in order to easily deduce thata condition that evidently completes the proof of the theorem. The existence of an influence domain of the external data with respect to the coordinate remains therefore proved.

4. Final Remarks and Illustrative Simulations

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, [18, 19]). We also report that, with reference to the investigation of the spatial behavior of the solutions for time-differential DPL deformable thermal conductors, the analysis proposed in this article has been further deepened in [20], currently in press.

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 , (directly related to the number of heat carriers) on the depth of the influence domain has been taken into account with the aid of a properly formulated initial-boundary value problem. To this aim a cylindrical domain has been considered, although the resulting parameter (to be understood as the signal transmission speed) able to completely describe the abovementioned domain depth results indeed to be(i)independent of the shape of the region (ii)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 excited by an appropriate thermomechanical perturbation, the spatial behavior of the solution in terms of existence and extension of an influence domain has been investigated.

In the attempt to apply the result obtained to a concrete estimate, and in continuity with what was done in [14], a homogeneous and isotropic single-layer graphene sheet at the temperature of has been considered, showing a first graphic simulation in which the signal transmission speed is given with respect to both the relaxation times and (Figure 1). The material features of the considered single-layer graphene are summarized in Table 1 (in this regard, the reader can refer to [21] and to the references therein).

In order to integrate and facilitate a direct comparison with the results shown in [14], the same significant combination of intervals has been considered for and for : , . Subsequently, in Figures 2 and 3 the value of is given separately with respect to and , showing in parallel also the homologous behaviors obtained in [14] relative to the expansion orders , and , . The value has been selected to fix, alternately, and in Figures 2 and 3, being possible to appreciate how the depth of the signal is further increased if , are selected as done in the present article.

Again as highlighted in [14] for the model , , but in this case even more markedly, we can observe in Figure 1 that, for increasing values of , a significantly shorter relaxation time makes the penetration depth of the signal extremely wide: also in this case, in fact, in view of the expansion orders involved, a suitable combination of values for the phase lags makes the model tend to assume almost a diffusive behavior. For the fixed value of (Figure 2) a sudden enlargement of the amplitude of the influence domain for decreasing values of is detectable. We recall that this is due to the fact that the transmission speed becomes extremely high, bringing the behavior of the model very close to the classical thermoelasticity (let us recall the paradox of the infinite propagation speed). Also in Figure 3, after having selected , the direct proportionality between and is preserved.

In conclusion, we have proved that the latter feature highlighted by Tzou in [2] (Chapter 12) up to the orders selection , As a general trend, the high-order waves and diffusion (corresponding to higher-order effects in and ) gradually increase the temperature level in the heat-affected zone. When thermal waves are activated [as in the case here considered] … the physical domain of the heat-affected zone gradually increases as well” is preserved also in the context of the linear thermoelasticity if we cross the sequence of expansion orders , ; , ; , . We also specify that all the proposed simulations, performed through Wolfram Mathematica 9, were conducted under the assumption of flux-precedence.

Data Availability

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.

Conflicts of Interest

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 Progetto Giovani 2018 – Heat-pulse propagation in FGMs.