# On the Increase in Signal Depth due to High-Order Effects in Micro- and Nanosized Deformable Conductors

Authors: Vittorio Zampoli
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

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.


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## Body

## 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 [3–14] 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 T,i at a certain time t+τT with the heat flux vector qi at a different time t+τq. To the delay/relaxation times or phase lags involved τq (heat flux vector delay time) and τT (temperature variation gradient delay time), respectively, it is possible to give the following physical interpretations: τq is connected to the fast-transient effects of thermal inertia, while τT 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 n and m 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):(1)qix,t+τq1!∂qi∂tx,t+⋯+τqnn!∂nqi∂tnx,t=-kijxT,jx,t+τT1!∂T,j∂tx,t+⋯+τTmm!∂mT,j∂tmx,t,kij 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, 12–14] 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 ordersn and m, 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 n≥5 or m≥5 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 [12–14] the selection of the expansion orders has concerned the following choices: (a)n=4, m=3, (b)n=3, m=2, and (c)n=m=3. 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 a, b, and c, we notice that only the choices a and b lead to models that can express a wavelike behavior, whereas c depicts a diffusive behavior. However, all of these expansion orders are able to account for the high-order effects in τq and τT 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 vectorqi and temperature variation gradient T,i given by (1) considered under the assumption a. This actually represents an advancement, since the corresponding study related to the comparison between the orders selection b and its counterpart n=2, m=1 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 n=m+1, 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 ProblemP^

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 cylinderB of base D(0) and height L and choose the referring axes system Ox1x2x3 in such a way that x3∈(0,L) and that the base D(0) lies on the coordinate plane Ox1x2. We denote by D(x3) the plane cross section at distance x3 from the base of the cylinder, assuming it smooth enough in order to allow the application of the divergence theorem; moreover, B(x3) denotes the portion of B comprised between the parallel cross sections D(x3) and D(L). 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 D(0): 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 x3 from the perturbed base D(0).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 P^ (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 D(0) of the cylinder. The definitive formulation of our initial-boundary value problem is obtained through the hat integral operator defined, for any continuous function gt, as follows:(2)g^t=∫0t∫0s∫0z∫0rgξdξdrdzds+τq∫0t∫0s∫0zgrdrdzds+τq22∫0t∫0sgzdzds+τq36∫0tgsds+τq424gt,and that for the sake of brevity and with an obvious meaning of the Roman superscripts involved is shortened as(3)g^t=gIVt+τqgIIIt+τq22gIIt+τq36gIt+τq424gt.In 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 3Description 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 [15–17]).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:tji are the components of the stress tensor, ρ is the mass density, ui are the components of the displacement vector, T is the temperature variation from the constant ambient temperature T0 (with T0 strictly positive), η is the entropy per unit mass, qi are the components of the heat flux vector, εij are the components of the strain tensor, and Cijkl, βij, kij (kij, as already highlighted, represents the conductivity tensor), and a are constitutive tensors depending only on the spatial variable x and are characterized by the following symmetries:(4)Cijkl=Cjikl=Cklij,βij=βji,kij=kji.As anticipated, null initial conditions are assumed while, with regard to the boundary conditions, we set for any t≥0 the following:(i)
The lateral boundary conditions, i.e., when(x1,x2)∈∂D(x3) with x3∈(0,L):(5)tαix1,x2,x3,tnα=0oruix1,x2,x3,t=0,qαx1,x2,x3,tnα=0orTx1,x2,x3,t=0(ii)
The upper end boundary conditions, i.e., when(x1,x2)∈D(L):(6)t3ix1,x2,L,t=0oruix1,x2,L,t=0,q3x1,x2,L,t=0orTx1,x2,L,t=0(iii)
The lower base boundary conditions assimilable to appropriate actions insisting onD(0), i.e., when (x1,x2)∈D(0):(7)t3ix1,x2,0,t=tix1,x2,toruix1,x2,0,t=vix1,x2,t,(8)q3x1,x2,0,t=Q3x1,x2,torTx1,x2,0,t=Θx1,x2,twhere the functionsti(x1,x2,t), vi(x1,x2,t), Q3(x1,x2,t), and Θ(x1,x2,t) are assumed to be prescribed and sufficiently smooth on D(0).The initial-boundary value problemP^ (we recall, deduced indifferently from [12] or [13]) object of investigation is structured as follows:The transformed equations of motion(9)t^ji,jt=ρ∂2u^i∂t2tinB×0,+∞The transformed equation of energy(10)ρT0∂η^∂tt=-q^i,itinB×0,+∞The transformed constitutive equations(11)t^ijt=Cijklε^klt-βijT^tinB¯×0,+∞,(12)ρη^t=βijε^ijt+aT^tinB¯×0,+∞,(13)q^it=-kijT,jIVt+τTT,jIIIt+τT22T,jIIt+τT36T,jItinB¯×0,+∞The transformed geometrical equations(14)ε^ijt=12u^i,jt+u^j,itinB¯×0,+∞The ordered arrayS^x,t=u^i,T^,ε^ij,t^ij,η^,q^ix,t defined in B¯×0,+∞ and associated with the set of transformed and nontrivial given data D^=t^i,v^i,Q^3,Θ^ is identified as the (unique (see [12])) solution of the initial-boundary value problem P^.


## 3. The Domain of Influence Theorem

Referring to the above defined three-dimensional regionB(x3) and using (9)-(12) and (14) together with the divergence theorem and the null upper end and lateral boundary conditions, we get(15)ddt∫Bx312ρ∂u^i∂t∂u^i∂t+Cijklε^ijε^kl+aT^2dv=∫Dx31T0q^3T^-t^3i∂u^i∂tda+1T0∫Bx3q^iT^,idv,and multiplying (15) by e-δt we receive(16)ddt∫Bx3e-δt2ρ∂u^i∂t∂u^i∂t+Cijklε^ijε^kl+aT^2dv+δ∫Bx3e-δt2ρ∂u^i∂t∂u^i∂t+Cijklε^ijε^kl+aT^2dv=∫Dx3e-δt1T0q^3T^-t^3i∂u^i∂tda+1T0∫Bx3e-δtq^iT^,idv.The 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:(17)1T0∫Bx3e-δtq^iT^,idv=-τq448T0d4dt4∫Bx3e-δtkijT,jIVT,iIVdv-τq412T0δ+τT3+τq312T0d3dt3∫Bx3e-δtkijT,jIVT,iIVdv-τTτq448T0d3dt3∫Bx3e-δtkijT,jIIIT,iIIIdv-τq48T0δ2+τT3+τq34T0δ+τT2+τq24T0d2dt2∫Bx3e-δtkijT,jIVT,iIVdv-τTτq416T0δ+τTτq3+τT3τq12T0-τq412T0d2dt2∫Bx3e-δtkijT,jIIIT,iIIIdv-τT2τq496T0d2dt2∫Bx3e-δtkijT,jIIT,iIIdv-τq412T0δ3+τT3+τq34T0δ2+τT2+τq22T0δ+τT+τq2T0ddt∫Bx3e-δtkijT,jIVT,iIVdv-τTτq416T0δ2+τTτq3+τT3τq-τq46T0δ+τTτq2+τT2τq4T0-τT3+τq34T0ddt∫Bx3e-δtkijT,jIIIT,iIIIdv-τT2τq448T0δ+τT2τq3+τT3τq224T0-τTτq416T0ddt∫Bx3e-δtkijT,jIIT,iIIdv-τT3τq4288T0ddt∫Bx3e-δtkijT,jIT,iIdv-τq448T0δ4+τT3+τq312T0δ3+τT2+τq24T0δ2+τT+τq2T0δ+1T0∫Bx3e-δtkijT,jIVT,iIVdv-τTτq448T0δ3+τTτq3+τT3τq-τq412T0δ2+τTτq2+τT2τq-τT3-τq34T0δ+τTτqT0-τT2+τq22T0∫Bx3e-δtkijT,jIIIT,iIIIdv-τT2τq496T0δ2+2τT2τq3+2τT3τq2-3τTτq448T0δ+τT2τq24T0+τq424T0-τTτq3+τT3τq6T0∫Bx3e-δtkijT,jIIT,iIIdv-τT3τq4288T0δ+τT3τq336T0-τT2τq448T0∫Bx3e-δtkijT,jIT,iIdv.At this point, in order to proceed, we need to set a threshold value for the parameter δ, which we select as follows:(18)δ=2M=4max⁡τT+τqτTτq-2τT+τq,3τT.It is fundamental to underline that δ, 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 n=3, m=2, 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 n=4, m=3 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 get(19)∫0t∫0s∫0z∫Bx3e-δr2ρ∂u^i∂r∂u^i∂r+Cijklε^ijε^kl+aT^2dvdrdzds+∫0t∫0s∫0z∫0r∫Bx3δe-δξ2ρ∂u^i∂ξ∂u^i∂ξ+Cijklε^ijε^kl+aT^2dvdξdrdzds+τq448T0∫Bx3e-δtkijT,jIVT,iIVdv+τq412T0δ+τT3+τq312T0∫0t∫Bx3e-δskijT,jIVT,iIVdvds+τTτq448T0∫0t∫Bx3e-δskijT,jIIIT,iIIIdvds+τq48T0δ2+τT3+τq34T0δ+τT2+τq24T0∫0t∫0s∫Bx3e-δzkijT,jIVT,iIVdvdzds+τTτq416T0δ+τTτq3+τT3τq12T0-τq412T0∫0t∫0s∫Bx3e-δzkijT,jIIIT,iIIIdvdzds+τT2τq496T0∫0t∫0s∫Bx3e-δzkijT,jIIT,iIIdvdzds+τq412T0δ3+τT3+τq34T0δ2+τT2+τq22T0δ+τT+τq2T0∫0t∫0s∫0z∫Bx3e-δrkijT,jIVT,iIVdvdrdzds+τTτq416T0δ2+τTτq3+τT3τq-τq46T0δ+τTτq2+τT2τq4T0-τT3+τq34T0∫0t∫0s∫0z∫Bx3e-δrkijT,jIIIT,iIIIdvdrdzds+τT2τq448T0δ+τT2τq3+τT3τq224T0-τTτq416T0∫0t∫0s∫0z∫Bx3e-δrkijT,jIIT,iIIdvdrdzds+τT3τq4288T0∫0t∫0s∫0z∫Bx3e-δrkijT,jIT,iIdvdrdzds+τq448T0δ4+τT3+τq312T0δ3+τT2+τq24T0δ2+τT+τq2T0δ+1T0∫0t∫0s∫0z∫0r∫Bx3e-δξkijT,jIVT,iIVdvdξdrdzds+τTτq448T0δ3+τTτq3+τT3τq-τq412T0δ2+τTτq2+τT2τq-τT3-τq34T0δ+τTτqT0-τT2+τq22T0×∫0t∫0s∫0z∫0r∫Bx3e-δξkijT,jIIIT,iIIIdvdξdrdzds+τT2τq496T0δ2+2τT2τq3+2τT3τq2-3τTτq448T0δ+τq424T0+τT2τq24T0-τTτq3+τT3τq6T0×∫0t∫0s∫0z∫0r∫Bx3e-δξkijT,jIIT,iIIdvdξdrdzds+τT3τq4288T0δ+τT3τq336T0-τT2τq448T0∫0t∫0s∫0z∫0r∫Bx3e-δξkijT,jIT,iIdvdξdrdzds=∫0t∫0s∫0z∫0r∫Dx3e-δξ1T0q^3T^-t^3i∂u^i∂ξdadξdrdzds.Subsequently, to the unique solution S^x,t of the initial-boundary value problem P^ we associate, for every possible value of the variable x3, the following functional Jδ(x3,t):(20)Jδx3,t=-∫0t∫0s∫0z∫0r∫Dx3e-δξt^3i∂u^i∂ξ-1T0q^3T^dadξdrdzds∀t≥0.Some important features of the above functionalJδ(x3,t) can then be easily shown.Lemma 1.
Under the hypothesesρ,a>0, Cijkl,kij positive definite tensors and selecting for δ the threshold value defined in (18), Jδ(x3,t) is not increasing with respect to the x3 variable, or in other words ∂Jδ/∂x3≤0. Moreover, Jδ(x3,t) can be identified as a measure for the (unique) solution S^x,t of the initial-boundary value problem P^, in the sense that it turns out to be not negative for every possible value of the variables x3 and t.Proof.
In order to prove that∂Jδ/∂x3≤0, it is sufficient to remember how the volume Bx3 is structured and evaluate, starting from (19), the partial derivative of Jδ(x3,t) with respect to the x3 variable: it is therefore evident that, under the hypotheses at issue, ∂Jδ/∂x3 is not positive. Consequently, in view of the null upper end boundary conditions and of definition (20) of Jδ(x3,t) we have(21)Jδx3,t≥JδL,t=0,for every possible value of x3 and t. Specifically, it is easy to prove that the condition Jδ(x3,t)=0 necessarily correlates to a null solution for the considered initial-boundary value problem.The following theorem can then be proved, concerning the solutionSx,t of the initial-boundary value problem P (no hat) from which the problem P^ derives, in view of the considered hypothesis of linearity (see again [12] or [13] for details).Theorem 2 (existence of a domain of influence).
LetSx,t=ui,T,εij,tij,η,qix,t be the solution of the initial-boundary value problem P. Let ρ and a be greater than zero, let Cijkl and kij be positive definite tensors, and let δ be identified with the threshold value (18). Moreover, suppose that the cylinder B is subjected on its lower base D(0) to the external actions described through (7) and (8). Then there exists a positive constant σ, dimensionally expressed by a speed, such that(22)Sx,t=ui,T,εij,tij,η,qix,t=0for  everyx∈Bsuch  thatx3≥σt;i.e., the whole activity in the cylinder B due to the presence of the assigned data acting on its base D(0) vanishes at distances from D(0) greater than or equal to σt.Proof.
Avoiding, just for reasons of synthesis, reporting the explicit expression of∂Jδ/∂x3, anyway easily obtainable from (19), let us estimate instead the partial derivative of Jδ(x3,t) with respect to the time variable t. To do this, we recall that(23)2ab≤εa2+1εb2∀a,b∈R,∀ε>0,and, thus, from the definition (20) we get (24)∂∂tJδx3,t≤12∫0t∫0s∫0z∫Dx3e-δrε1ρt^3it^3i+ρε1∂u^i∂r∂u^i∂r+ε2aT02q^32+aε2T^2dadrdzds∀ε1,ε2>0.In order to estimate the right-hand side of (24) (in particular, the terms containing t^3it^3i and q^32) we first take into account constitutive equation (11) and denote by μM, positive and constant, the greatest elastic modulus of the constitutive tensor Cijkl such that for every ξij it is Cijklξijξkl≤μMξijξij. In view of the arithmetic-geometric mean inequality, the following relation is valid:(25)t^3it^3i≤t^ijt^ij≤1+εμMCijklε^ijε^kl+1+1εB2T^2,∀ε>0and  whereB2=βijβij>0.On the other side, recalling constitutive equation (13), distributing (-kijq^i) with respect to the terms of the sum, and then applying the Cauchy-Schwarz inequality we can write(26)q^iq^i=-kijq^iT,jIVt+τTT,jIIIt+τT22T,jIIt+τT36T,jIt≤krskrs1/2T,jIVq^iT,jIVq^i1/2+τT2krskrs1/2T,jIIIq^iT,jIIIq^i1/2+τT44krskrs1/2T,jIIq^iT,jIIq^i1/2+τT636krskrs1/2T,jIq^iT,jIq^i1/2=q^iq^i1/2krskrs1/2T,jIVT,jIV1/2+τTkrskrs1/2T,jIIIT,jIII1/2+τT22krskrs1/2T,jIIT,jII1/2+τT36krskrs1/2T,jIT,jI1/2,and then through squaring we get(27)q^32≤q^iq^i≤4krskrsT,jIVT,jIV+τT2krskrsT,jIIIT,jIII+τT44krskrsT,jIIT,jII+τT636krskrsT,jIT,jI.In addition, identifying with km the smallest (and strictly positive) eigenvalue of the conductivity tensor kij, the following relations are valid:(28)T,jIVT,jIV≤1kmkijT,jIVT,iIV,T,jIIIT,jIII≤1kmkijT,jIIIT,iIII,T,jIIT,jII≤1kmkijT,jIIT,iII,T,jIT,jI≤1kmkijT,jIT,iI,and so from (24) it follows that(29)∂∂tJδx3,t≤12∫0t∫0s∫0z∫Dx3e-δrρε1∂u^i∂r∂u^i∂r+ε11+εμMρCijklε^ijε^kl+1ε2+B2ε1aρ1+1εaT^2+4ε2T0krskrsakm1T0kijT,jIVT,iIV+τT2T0kijT,jIIIT,iIII+τT44T0kijT,jIIT,iII+τT636T0kijT,jIT,iIdadrdzds∀ε,ε1,ε2>0.Assuming again for δ the threshold value (18) and reconsidering the partial derivative ∂Jδ/∂x3, but retaining this time only the terms with a number of integrals equal to that characterizing (29), we reach the following estimate:(30)-∂∂x3Jδx3,t≥12∫0t∫0s∫0z∫Dx3e-δrρ∂u^i∂r∂u^i∂r+Cijklε^ijε^kl+aT^2+τq46δ3+τT3+τq32δ2+τT2+τq2δ+τT+τq1T0kijT,jIVT,iIV+τTτq48δ2+τTτq3+τT3τq-τq43δ+τTτq2+τT2τq2-τT3+τq321T0kijT,jIIIT,iIII+τT2τq424δ+τT2τq3+τT3τq212-τTτq481T0kijT,jIIT,iII+τT3τq41441T0kijT,jIT,iIdadrdzds≥0.At this point it is sufficient to define(31)Ψ=max⁡122τq4δ3+6τT3+τq3δ2+12τT2+τq2δ+12τT+τq,24τT23τTτq4δ2+8τTτq3+τT3τq-τq4δ+12τTτq2+τT2τq-τT3-τq3,6τT4τT2τq4δ+2τT2τq3+τT3τq2-3τTτq4,4τT3τq4supDx3⁡krskrsakmand to impose the equality between the following coefficients, which is a consequence of the comparison between the estimates involving the partial derivatives of Jδ(x3,t) with respect to x3 and t. We denote them by σ (strictly positive) and set(32)1ε1=ε11+εμMρ=1ε2+B2ε1aρ1+1ε=4ε2T0Ψ=σ.It follows that(33)ε1=ρ1+εμM,ε2=T04Ψ1+εμMρ,ε being the positive root of the algebraic equation(34)ε2+1-T0B2+4ρaΨaT0μMε-B2aμM=0.With these choices, comparing relations (29) and (30) we reach the following differential inequality, valid for all x3,t∈0,L×0,∞:(35)∂∂tJδx3,t+σ∂∂x3Jδx3,t≤0.Such a differential inequality can be integrated, for example, as in [17] in order to easily deduce that(36)Jδx3,t=0for  everyx3,t∈σt,L×0,∞,a condition that evidently completes the proof of the theorem. The existence of an influence domain of the external data with respect to the x3 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 ton=4, m=3 (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 B has been considered, although the resulting parameter σ (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 regionB(ii)
depending only on the features of the selected material, among which there are the constitutive coefficients and tensors and the relaxation timesImagining the lower base of the deformable, anisotropic, and inhomogeneous cylinderB 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 300K has been considered, showing a first graphic simulation in which the signal transmission speed σ is given with respect to both the relaxation times τq and τT (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).Table 1 Mass density 2.35 × 1 0 3  k g   m - 3  Thermal conductivity 3.10 × 1 0 3  W   m - 1 K - 1  Specific heat capacity 7.70 × 1 0 2  J   k g - 1 K - 1  Elastic modulus 1.05 × 1 0 12  N   m - 2  Shear modulus 4.45 × 1 0 11  N   m - 2  Thermal expansion coefficient - 4.00 × 1 0 - 6  K - 1Figure 1
The behavior of the parameterσ with respect to both the relaxation times τT and τq.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 τq and for τT: 1.0×10-12s≤τq≤1.0×10-11s, 1.0×10-10s≤τT≤1.0×10-9s. Subsequently, in Figures 2 and 3 the value of σ is given separately with respect to τq and τT, showing in parallel also the homologous behaviors obtained in [14] relative to the expansion orders n=3, m=2 and n=2, m=1. The value 2×10-11s has been selected to fix, alternately, τT and τq in Figures 2 and 3, being possible to appreciate how the depth of the signal is further increased if n=4, m=3 are selected as done in the present article.Figure 2
The behavior of the parameterσ for the expansion orders 4,3 (continuous red line) with respect to the relaxation time τq, having selected τT=2×10-11s, in comparison with the expansion orders 3,2 (dash-dot green line) and 2,1 (dashed blue line).Figure 3
The behavior of the parameterσ for the expansion orders 4,3 (continuous red line) with respect to the relaxation time τT, having selected τq=2×10-11s, in comparison with the expansion orders 3,2 (dash-dot green line) and 2,1 (dashed blue line).Again as highlighted in [14] for the model n=3, m=2, but in this case even more markedly, we can observe in Figure 1 that, for increasing values of τT, a significantly shorter relaxation time τq 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 τT (Figure 2) a sudden enlargement of the amplitude of the influence domain for decreasing values of τq 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 τq=2×10-11s, the direct proportionality between σ and τT is preserved.In conclusion, we have proved that the latter feature highlighted by Tzou in [2] (Chapter 12) up to the orders selection n=2, m=1 “As a general trend, the high-order waves and diffusion (corresponding to higher-order effects inτTandτq) 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 n=2, m=1; n=3, m=2; n=4, m=3. We also specify that all the proposed simulations, performed throughWolfram Mathematica 9, were conducted under the assumption of flux-precedence.



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*Source: 2629012-2019-01-31.xml*