Transient Finite-Speed Heat Transfer Influence on Deformation of a Nanoplate with Ultrafast Circular Ring Heating

Abstract

The present study provides a theoretical estimate for the thermal stress distribution and the displacement vector inside a nano-thick infinite plate due to an exponentially temporal decaying boundary heating on the front surface of the elastic plate. The surface heating is in the form of a circular ring; therefore, the axisymmetric formulation is adopted. Three different hyperbolic models of thermal transport are considered: the Maxwell-Cattaneo-Vernotte (MCV), hyperbolic Dual-Phase-Lag (HDPL) and modified hyperbolic Dual-Phase-Lag (MHDPL), which coincides with the two-step model under certain constraints. A focus is directed to the main features of the corresponding hyperbolic thermoelastic models, e.g., finite-speed thermal waves, singular surfaces (wave fronts) and wave reflection on the rear surface of the plate. Explicit expressions for the thermal and mechanical wave speeds are derived and discussed. Exact solution for the temperature in the short-time domain is derived when the thermalization time on the front surface is very long. The temperature, hydrostatic stress and displacement vector are represented in the space-time domain, with concentrating attention on the thermal reflection phenomenon on the thermally insulated rear surface. We find that the mechanical wave speeds are approximately equal for the considered models, while the thermal wave speeds are entirely different such that the modified hyperbolic dual-phase-lag thermoelasticity has the faster thermal wave speed and the Lord-Shulman thermoelasticity has the slower thermal wave speed.

1. Introduction

Thermal-induced stresses inherent inside a rigid heat conductor, of the Hookean type, are of a great importance to many physicists and engineers in understanding and predicting the elastic material responses to different thermal effects. The Biot assumption [1], in which both thermal-induced stresses and stress-induced heat are considered, presumes the Fourier heat transfer mechanism in which a definite thermal wave speed disappears, see recent illustrative numerical examples depicted through contour figures in [2,3]. Biot thermomechanical model, however, is still the most successful, rigorously established model, in particular its compatibility with irreversible thermodynamics [4,5,6].

In many experimental observations [7,8,9], Fourier’s law fails to describe the transmission of a heat wave with a finite velocity, in particular, when the short-time domain is considered and therefore; the need arose for an alternative equation to be capable capturing the thermal wave with a specific speed in the short-time domain, which quickly turns into a Fourier in the intermediate- and long-time domains, namely, transient heat transfer process [10]. The Maxwell-Cattaneo-Vernotte (MCV) with its familiar form can perfectly simulate the finite-speed thermal wave in the short-time response and its speed was experimentally verified [11,12]. The effect of finite-speed thermal transport on the internal stresses production was rigorously discussed and established by Lord and Shulman [13] (denoted in what follows as “LSTE”). In a few years later, the Lord-Shulman model was generalized to the case of anisotropic elastic solids [14].

Ultrafast heating process in metals distinguishes two main steps: Heating up the electron subsystem, which produces the diffusive hot electron; and Heating up the lattice through electron-phonon collisions, see [15,16]. The Maxwell-Cattaneo-Vernotte model was not capable to simulate such a behavior, see [17,18,19], rather, the two-temperature model with its two versions (parabolic and hyperbolic) shows the best fitting with experimental results, see also [20,21]; the recent investigation on the two-temperature model in heterogeneous materials [22], and the most recent numerical experiments [23,24]. When the material parameters are assumed temperature-independent, the microscopic parabolic and hyperbolic two-temperature model can be compared to the macroscopic approximations of the dual-phase-lag model, [25,26,27,28]. It is worth mentioning the criticism on the MCV and the parabolic DPL Equations [29,30,31,32], where both models may mispredict the thermal behavior on the infinite domains when higher dimensions are studied. Nevertheless, the recent investigation [33] has defined some sufficient conditions when using both models on the infinite domain for avoiding the possible negative distribution (temperature or concentration) values. On the other hand, the proposed parabolic dual-phase-lag model (or Jeffreys equation) has been derived from the continuous-time random walk model and has found many applications in anomalous diffusion [34]. In the case of the semi-infinite or bounded heat conductors, there is no a theoretical proof of the possibility of negative values for the temperature, except some numerical investigations depending on the finite difference and “Conservation Element and Solution Element” methods, see e.g., [35,36]. The singular surface, or the sharp wave front, which defines the exact value of the thermal wave velocity, disappear from these numerical techniques. In [37], a mathematical aspect on the hyperbolic dual-phase-lag equation was introduced and alleged that it refines its connection with the hyperbolic two-temperature model, provided that the material parameters are kept constant during heating. The corresponding thermomechanical theory that uses the parabolic and hyperbolic DPL heat conduction laws was developed by Tzou [28] and Chandrasekharaiah [38]. Some theoretical aspects on the DPL thermoelasticity were presented in [39,40], see also the recent application on thermoviscoelastic nanobeam [41]. We refer the reader to the review monographs on the hyperbolic theories of thermoelasticity [38,42,43]. On the other hand, the thermomechanics theory, which uses the modified hyperbolic DPL model, was developed in [37] with application on the Danilovskaya problem and on the Euler-Bernoulli beams [44].

The problem of axisymmetric stress distribution in an infinite plate, due to a circular continuously heating spot on the front surface, was firstly discussed in hyperbolic thermoelasticity of Lord-Shulman type by Sherief and Hamza [45]. In [46], the authors extended the axisymmetric temperature distribution in an infinite thick plate problem to the thermoelastic diffusion theory with considerations to reflecting boundary conditions for temperature and concentration on the rear surface. A version of the thick plate problem, subject to axisymmetric temperature distribution, was generalized to the hyperbolic dual-phase-lag thermoelasticity in [47]. Most recently, similar setting has been employed on a thermoviscoelastic medium [48]. It is noteworthy that counterparts in the heat conduction problems of such an axisymmetric formulation for finite plates was independently considered in [35] and has been applied to a circular cylindrical heat conductor with thermal boundary condition on one of its bases at the form of a circular ring [49], refer also to the thorough monograph [50]. In addition, the thermal wave reflection was referred to for MCV equation in [25] and for the DPL and modified DPL in [37], see also [51,52] for anomalous diffusion case.

In the present work, we revisit the axisymmetric temperature/stress problem considering the thermal reflection phenomenon on the rear surface and its effects on the stress components under three finite-speed models of thermoelasticity; Lord-Shulman thermoelasticity (LSTE), hyperbolic DPL thermoelasticity (HDPLTE), and modified hyperbolic DPL thermoelasticity (MHDPLTE). In Section 2, we present a review on the mathematical models of the considered thermomechanical theories and formulate the axisymmetric temperature/stress problem for an infinite elastic nanoplate. The method of solution, depending on the Laplace and Hankel transforms, is introduced in Section 3 and exact solutions for the stresses and temperature in the transformed domain are obtained. We use suitable numerical arguments to bring the solutions in the physical domain in Section 4. In addition, we give a discussion on the numerical results with concentrate on the finite-velocity thermal waves and the reflection phenomenon on the rear surface. Finally, we draw our remarks and future generalizations in Section 5.

2. Mathematical Model

According to the linear theory of thermoelasticity [4,5,6,43], the governing equations for an isotropic, homogeneous thermoelastic material consist of the conservation of momentum in the absence of body forces and in terms of the displacement vector $u$ and temperature $\theta$:

$$\mu \mathsf{\Delta }u+\left(\lambda +\mu \right)\nabla \left(\nabla \cdot u\right)-\left(3\lambda +2\mu \right){\alpha }_T\nabla \theta =\rho \ddot{u}$$

(1)

and the conservation of energy equation without internal heat source:

$$-\nabla \cdot q=\rho C_E\dot{\theta }+\left(3\lambda +2\mu \right){\alpha }_T{T}_0\nabla \cdot \dot{u},$$

(2)

where $q$ is the heat flux vector, $\lambda$ and $\mu$ are Lame constants, ${\alpha }_T$ is the coefficient of thermal expansion, $C_E$ is the specific heat capacity, $T_0$ is the reference temperature, $\rho$ is the density, $\nabla$ is the del operator, $\mathsf{\Delta }=\nabla \cdot \nabla$ is the Laplace operator and the dot over variable stands for time derivative. In (2), the temperature $\theta =T-T_0$, where $T$ is the absolute temperature. When the heat flux vector $q$ is given through the classical Fourier law:

$$q=-\kappa \nabla \theta ,$$

(3)

where $\kappa$ is the thermal conductivity, the coupled system (1) and (2) represents the governing equations of the Biot model of thermomechanics [1]. If we phenomenologically replace (3) with the generalized law [2]

$$\left[1+{\tau }_q\frac{\partial }{\partial t}+\left(n_0\frac{{\tau }_q^2}{2}+n_1{\tau }_m^2\right)\frac{{\partial }^2}{\partial t^2}\right]q=-\kappa \left(1+n_2{\tau }_{\theta }\frac{\partial }{\partial t}\right)\nabla \theta ,$$

(4)

where ${\tau }_q$ and ${\tau }_{\theta }$ are called phase lags of heat flux and temperature respectively, ${\tau }_m$ is a time constant for the modified DPL equation [37] and $n_0$, $n_1$ and $n_2$ are controlling non-negative integers take their values from the set $\left\{0,1\right\}$, we have a generalized constitutive law consisting of five heat conduction mechanisms. More specifically, Equation (4) consists of three hyperbolic and two parabolic heat conduction constitutive laws, combined in one unphysical equation for the sake of numerical comparisons. Since, we confine our study to the hyperbolic models, we disregard discussing the two parabolic models, Fourier and parabolic DPL and when using Equation (4) into the energy conservation (2), we get the governing equations of three hyperbolic thermoelastic models:

(i)

When $n_0=n_1=n_2=0$, Equation (4) reduces to the MCV constitutive law, namely

$$\left(1+{\tau }_q\frac{\partial }{\partial t}\right)q=-\kappa \nabla \theta ,$$

(5)

and therefore, Equations (1) and (2) with (5) result in the governing equations for the hyperbolic theory of thermoelasticity of Lord-Shulman type (LSTE) [13].

(ii)

When $n_1=0$ and $n_0=n_2=1$, Equation (4) reduces to the hyperbolic DPL equation [26]

$$\left(1+{\tau }_q\frac{\partial }{\partial t}+\frac{{\tau }_q^2}{2}\frac{{\partial }^2}{\partial t^2}\right)q=-\kappa \left(1+{\tau }_{\theta }\frac{\partial }{\partial t}\right)\nabla \theta .$$

(6)

Therefore, Equations (1) and (2) with (6) represent the governing equations of the hyperbolic DPL thermoelasticity (HDPLTE) [28,38].

(iii)

When $n_0=0$ and $n_1=n_2=1$, Equation (4) reduces to the modified hyperbolic DPL equation [37]

$$\left(1+{\tau }_q\frac{\partial }{\partial t}+{\tau }_m^2\frac{{\partial }^2}{\partial t^2}\right)q=-\kappa \left(1+{\tau }_{\theta }\frac{\partial }{\partial t}\right)\nabla \theta ,$$

(7)

where

$${\tau }_q={\tau }_{q1}+{\tau }_{q2},{\tau }_m=\sqrt{{\tau }_{q1}{\tau }_{q2}},$$

(8)

${\tau }_{q1}$ and ${\tau }_{q2}$ are successive lags, ${\tau }_{q1}$, ${\tau }_{q2}$ and ${\tau }_{\theta }$ are defined as

$${\tau }_{q1}=\frac{1}{G}{\left(\frac{1}{C_e}+\frac{1}{C_l}\right)}^{-1}, {\tau }_{q2}={\tau }_F, {\tau }_{\theta }=\frac{C_l}{G},$$

(9)

$C_l$ and $C_e$ are the heat capacities of the lattice and the electrons respectively, $G$ is the electron-phonon coupling factor, and ${\tau }_F$ is the electron relaxation at the Fermi level. Upon eliminating the heat flux vector between (7) and the conservation of energy equation without deformation effects, i.e., $-\nabla \cdot q=\rho C_E\dot{\theta }$, and utilizing Equations (8) and (9), we get

$$\left(1+\frac{C_l}{G}\frac{\partial }{\partial t}\right)\nabla \theta =\frac{{\tau }_F{C}_e{C}_l}{G{\kappa }_e}\frac{{\partial }^3\theta }{\partial t^3}+\left(\frac{{\tau }_F\left(C_e+C_l\right)}{{\kappa }_e}+\frac{C_e{C}_l}{G{\kappa }_e}\right)\frac{{\partial }^2\theta }{\partial t^2}+\frac{C_e+C_l}{{\kappa }_e}\frac{\partial \theta }{\partial t},$$

(10)

which coincides exactly the energy equation governing the lattice temperature in the hyperbolic two-temperature model [18,26] provided that $\rho C_E=C_e+C_l$, ${\kappa }_e$ is the electron thermal conductivity, the lattice thermal conduction is neglected, and all the thermophysical properties are kept constants during the temperature elevation. On the contrary, in the original hyperbolic DPL constitutive law (6), it requires to neglect the effects of ${\tau }_F^2$ and ${\tau }_T^2$ for satisfying this coincidence, refer to the paragraphs below relations (13d) and (13e) in [26]. In the case of variable coefficients, the activation of lattice thermal conduction, or even the presence of an internal source term, this matching between the hyperbolic DPL model and the hyperbolic two-temperature model is questionable. Therefore, upon adopting the heat flux described by (7), Equations (1) and (2) yield the governing equations of the modified hyperbolic thermoelasticity [37].

Now, we utilize the proposed generalized constitutive law (4) into (2) to get the generalized energy balance equation:

$$\left(1+n_2{\tau }_{\theta }\frac{\partial }{\partial t}\right)\mathsf{\Delta }\theta =\eta \left[1+{\tau }_q\frac{\partial }{\partial t}+\left(n_0\frac{{\tau }_q^2}{2}+n_1{\tau }_m^2\right)\frac{{\partial }^2}{\partial t^2}\right]\left(\dot{\theta }+\frac{\left(3\lambda +2\mu \right){\alpha }_T{T}_0}{\rho C_E}\nabla \cdot \dot{u}\right),$$

(11)

where $\eta =1/{\alpha }_T$, and ${\alpha }_T=\kappa /\left(\rho C_E\right)$ is the thermal diffusivity.

In what follows, we consider the problem of axisymmetric temperature/stress generated by thermal boundary conditions on the front and rear surfaces of a homogeneous isotropic elastic infinite plate with thickness $\ell$. We take the origin on the front surface and $z$-axis oriented to the plate body such that $z=0$ stands for the front surface and $z=\ell$ is the rear surface. In addition, we set the surface heating on the form of a circular ring, and it is exponentially decaying with time, see Figure 1. Further, we append a reflecting thermal boundary condition (thermal isolation) on the rear surface of the plate.

From the above settings, the axisymmetric formulation is suitable, in which the displacement vector is given by

$$u=\langle u_r,0,u_z\rangle ,$$

(12)

where the cylindrical coordinate system $\left(r,\psi ,z\right)$ is adopted, and thereby we can define the region $\mathsf{\Omega }=\left\{\left(r,\psi ,z\right), 0\le r<\infty , 0\le \psi \le 2\pi , and 0\le z\le \ell \right\}$ occupied by this plate. Moreover, the problem variables are functions of $r$, $z$ and $t$, i.e.,

$$u_r=u_r\left(r,z,t\right), u_z=u_z\left(r,z,t\right), \theta =\theta \left(r,z,t\right).$$

(13)

Hence, the governing Equations (1) and (11) reduce to

$$\left(\mu \mathsf{\Delta }-\rho \frac{{\partial }^2}{\partial t^2}\right)u_r=\left(3\lambda +2\mu \right){\alpha }_T\frac{\partial \theta }{\partial r}-\left(\lambda +\mu \right)\frac{\partial e}{\partial r},$$

(14)

$$\left(\mu \mathsf{\Delta }-\rho \frac{{\partial }^2}{\partial t^2}\right)u_z=\left(3\lambda +2\mu \right){\alpha }_T\frac{\partial \theta }{\partial z}-\left(\lambda +\mu \right)\frac{\partial e}{\partial z},$$

(15)

$$\left(1+n_2{\tau }_{\theta }\frac{\partial }{\partial t}\right)\mathsf{\Delta }\theta =\eta \frac{\partial }{\partial t}\left[1+{\tau }_q\frac{\partial }{\partial t}+\left(n_0\frac{{\tau }_q^2}{2}+n_1{\tau }_m^2\right)\frac{{\partial }^2}{\partial t^2}\right]\left(\theta +\frac{\left(3\lambda +2\mu \right){\alpha }_T{T}_0}{\rho C_E}e\right),$$

(16)

where $e$ is the cubical dilatation given by

$$e=\nabla \cdot u.$$

(17)

The generalized Hooke’s law for linear isotropic thermoelastic materials is given by [43]:

$${\sigma }_{ij}=2\mu e_{ij}+\lambda e{\delta }_{ij}-\left(3\lambda +2\mu \right){\alpha }_T\theta {\delta }_{ij},$$

(18)

where ${\sigma }_{ij}$ is the Cauchy stress tensor and $e_{ij}$ is the strain tensor defined as

$$e_{ij}=\frac{1}{2}\left(u_{i,j}+u_{j,i}\right),$$

(19)

and ${\delta }_{ij}$ is the Kronecker delta function. In this axisymmetric formulation, the non-vanishing strain tensor elements $e_{ij}$ are given by

$$e_{rr}=\frac{\partial u_r}{\partial r}, e_{\psi \psi }=\frac{u_r}{r},e_{zz}=\frac{\partial u_z}{\partial z}, e_{rz}=\frac{1}{2}\left(\frac{\partial u_r}{\partial z}+\frac{\partial u_z}{\partial r}\right),$$

(20)

while the stress tensor elements, determined through (18) and (20) are given by

$${\sigma }_{rr}=2\mu \frac{\partial u_r}{\partial r}+\lambda e-\left(3\lambda +2\mu \right){\alpha }_T\theta ,$$

(21)

$${\sigma }_{\psi \psi }=2\mu \frac{u_r}{r}+\lambda e-\left(3\lambda +2\mu \right){\alpha }_T\theta ,$$

(22)

$${\sigma }_{zz}=2\mu \frac{\partial u_z}{\partial z}+\lambda e-\left(3\lambda +2\mu \right){\alpha }_T\theta ,$$

(23)

$${\sigma }_{rz}=\mu \left(\frac{\partial u_r}{\partial z}+\frac{\partial u_z}{\partial r}\right),$$

(24)

where $e_{r\psi }=e_{\psi z}=0$ and ${\sigma }_{r\psi }={\sigma }_{\psi z}=0$. Moreover, let us append the following set of thermal and mechanical boundary conditions to the above governing equations:

(a)

Thermal boundary condition:

$$\theta \left(r,0,t\right)={\Theta }_0\left[H\left(r_2-r\right)-H\left(r_1-r\right)\right]\exp \left(-\frac{t}{t_p}\right), {\frac{\partial \theta \left(r,z,t\right)}{\partial z}|}_{z=\ell }=0,$$

(25)

where $H(\cdot )$ is the Heaviside unit step function, ${\Theta }_0$ is a constant to be determined later, $r_1$ and $r_2$ are the radii of the inner and outer circles of the thermal ring ($r_1<r_2$) and $t_p$ is the thermalization time. The second thermal boundary condition on the rear surface represents a thermal isolation, which paves the way for the thermal reflection phenomenon.

(b)

Mechanical boundary condition: We assume that both the front and the rear surfaces are traction free, such that the normal stress component ${\sigma }_{zz}$ and the shear stress component ${\sigma }_{rz}$ are zero on both surfaces

$${\sigma }_{zz}\left(r,0,t\right)={\sigma }_{zz}\left(r,\ell ,t\right)=0, {\sigma }_{rz}\left(r,0,t\right)={\sigma }_{rz}\left(r,\ell ,t\right)=0.$$

(26)

Furthermore, we consider that the system begins the experiment from the rest with zero accelerations, namely,

$$\begin{gathered} {u_r\left(r,z,t\right)|}_{t=0}={\frac{\partial u_r\left(r,z,t\right)}{\partial t}|}_{t=0}=0, {u_z\left(r,z,t\right)|}_{t=0}={\frac{\partial u_z\left(r,z,t\right)}{\partial t}|}_{t=0}=0, \\ {\theta \left(r,z,t\right)|}_{t=0}={\frac{\partial \theta \left(r,z,t\right)}{\partial t}|}_{t=0}={\frac{{\partial }^2\theta \left(r,z,t\right)}{\partial t^2}|}_{t=0}=0, \end{gathered}$$

(27)

Let us introduce the following dimensionless transformations [45]:

$$\begin{gathered} r\to \frac{r}{c_0\eta }, r_1\to \frac{r_1}{c_0\eta }, r_2\to \frac{r_2}{c_0\eta }, z\to \frac{z}{c_0\eta }, u_r\to \frac{u_r}{c_0\eta }, u_z\to \frac{u_z}{c_0\eta }, \ell \to \frac{\ell }{c_0\eta }, \\ t\to \frac{t}{c_0^2\eta }, t_p\to \frac{t_p}{c_0^2\eta }, {\tau }_{\varsigma }\to \frac{{\tau }_{\varsigma }}{c_0^2\eta }, {\sigma }_{ij}\to \left(\lambda +2\mu \right){\sigma }_{ij}, \theta \to \frac{\lambda +2\mu }{\left(3\lambda +2\mu \right){\alpha }_T}\theta , \end{gathered}$$

(28)

where ${\tau }_{\varsigma }\in \left\{{\tau }_q,{\tau }_m,{\tau }_{\theta }\right\}$ and $c_0=\sqrt{\left(\lambda +2\mu \right)/\rho }$. Thus, the governing Equations (14)–(16) with the constitutive relations (17) and (21)–(24) subject to the boundary conditions (25), (26) and the initial conditions (27) can be written as

$$\left(\mathsf{\Delta }-{\xi }^2\frac{{\partial }^2}{\partial t^2}\right)u_r={\xi }^2\frac{\partial \theta }{\partial r}-\left({\xi }^2-1\right)\frac{\partial e}{\partial r},$$

(29)

$$\left(\mathsf{\Delta }-{\xi }^2\frac{{\partial }^2}{\partial t^2}\right)u_z={\xi }^2\frac{\partial \theta }{\partial z}-\left({\xi }^2-1\right)\frac{\partial e}{\partial z},$$

(30)

$$\left(1+n_2{\tau }_{\theta }\frac{\partial }{\partial t}\right)\mathsf{\Delta }\theta =\frac{\partial }{\partial t}\left[1+{\tau }_q\frac{\partial }{\partial t}+\left(n_0\frac{{\tau }_q^2}{2}+n_1{\tau }_m^2\right)\frac{{\partial }^2}{\partial t^2}\right]\left(\theta +\epsilon e\right),$$

(31)

$$e=\nabla \cdot u,$$

(32)

$${\sigma }_{rr}=\frac{2}{{\xi }^2}\frac{\partial u_r}{\partial r}+\left(1-\frac{2}{{\xi }^2}\right)e-\theta ,$$

(33)

$${\sigma }_{\psi \psi }=\frac{2}{{\xi }^2}\frac{u_r}{r}+\left(1-\frac{2}{{\xi }^2}\right)e-\theta ,$$

(34)

$${\sigma }_{zz}=\frac{2}{{\xi }^2}\frac{\partial u_z}{\partial z}+\left(1-\frac{2}{{\xi }^2}\right)e-\theta ,$$

(35)

$${\sigma }_{rz}=\frac{1}{{\xi }^2}\left(\frac{\partial u_r}{\partial z}+\frac{\partial u_z}{\partial r}\right),$$

(36)

subject to the dimensionless boundary conditions:

$$\begin{gathered} \theta \left(r,0,t\right)=\left[H\left(r_2-r\right)-H\left(r_1-r\right)\right]\exp \left(-\frac{t}{t_p}\right), {\frac{\partial \theta \left(r,z,t\right)}{\partial z}|}_{z=\ell }=0, \\ {\sigma }_{zz}\left(r,0,t\right)={\sigma }_{zz}\left(r,\ell ,t\right)=0, {\sigma }_{rz}\left(r,0,t\right)={\sigma }_{rz}\left(r,\ell ,t\right)=0, \end{gathered}$$

(37)

and the dimensionless initial conditions

$$\begin{gathered} {u_r\left(r,z,t\right)|}_{t=0}={\frac{\partial u_r\left(r,z,t\right)}{\partial t}|}_{t=0}=0, {u_z\left(r,z,t\right)|}_{t=0}={\frac{\partial u_z\left(r,z,t\right)}{\partial t}|}_{t=0}=0, \\ {\theta \left(r,z,t\right)|}_{t=0}={\frac{\partial \theta \left(r,z,t\right)}{\partial t}|}_{t=0}={\frac{{\partial }^2\theta \left(r,z,t\right)}{\partial t^2}|}_{t=0}=0, \end{gathered}$$

(38)

where ${\xi }^2=\left(\lambda +2\mu \right)/\mu$, $\epsilon ={\left(3\lambda +2\mu \right)}^2{\alpha }_T^2{T}_0/\left[\rho C_E\left(\lambda +2\mu \right)\right]$, and we have chosen ${\Theta }_0=\left(\lambda +2\mu \right)/\left[\left(3\lambda +2\mu \right){\alpha }_T\right]$.

3. Analytical Results in Laplace-Hankel Domain

In this section, we obtain an analytical solution for the temperature, displacement components and stresses in the transformed domain. First, for the temporal variable, we introduce the Laplace transform defined for any generic well-behaved function $f\left(r,z,t\right)$ as [53]

$$\tilde{f}\left(r,z,s\right)=ℒ\left\{f\left(r,z,t\right);t\right\}\left(r,z,s\right)={{\displaystyle \int }}_0^{\infty }f\left(r,z,t\right)e^{-st}dt,$$

(39)

where the tilde denotes to the function in Laplace domain, $s$ is the Laplace parameter and the inverse Laplace transform is given by

$${ℒ}^{-1}\left(\tilde{f}\left(r,z,s\right)\right)=\frac{1}{2\pi ı}\underset{\beta \to \infty }{\lim }\underset{\alpha -ı\beta }{\overset{\alpha +ı\beta }{{\displaystyle \int }}}\tilde{f}\left(r,z,s\right)e^{st}ds=f\left(r,z,t\right), ı=\sqrt{-1}.$$

(40)

On the other hand, we introduce the Hankel transform with respect to the radial distance $0\le r<\infty$, defined for any well-behaved function $f\left(r,z,t\right)$ as [54]

$${\widehat{f}}_{\nu }\left(q,z,t\right)={\mathscr{H}}_{\nu }\left[f\left(r,z,t\right)\right]=\underset{0}{\overset{\infty }{{\displaystyle \int }}}f\left(r,z,t\right)r{J}_{\nu }\left(qr\right)dr,$$

(41)

with its inversion formula

$$f\left(r,z,t\right)={\mathscr{H}}_{\nu }^{-1}\left[{\widehat{f}}_{\nu }\left(q,z,t\right)\right]=\underset{0}{\overset{\infty }{{\displaystyle \int }}}\widehat{f}\left(q,z,t\right) q{J}_{\nu }\left(qr\right)dq,$$

(42)

where the hat denotes to the function in Hankel domain, $q$ is the Hankel parameter, and $J_{\nu }(\cdot )$ is the Bessel function of the first kind and order $\nu$. For brevity in our analysis, we denote ${\mathscr{H}}_0\left[f\left(r,z,t\right)\right]=\widehat{f}\left(q,z,t\right)$, instead of ${\widehat{f}}_0\left(q,z,t\right)$, in the case of $\nu =0$.

Therefore, utilizing the Laplace transform, the governing Equations (29)–(31) are simplified to [45,46,48]

$$\left(\mathsf{\Delta }-{\xi }^2{s}^2\right){\tilde{u}}_r={\xi }^2\frac{\partial \tilde{\theta }}{\partial r}-\left({\xi }^2-1\right)\frac{\partial \tilde{e}}{\partial r},$$

(43)

$$\left(\mathsf{\Delta }-{\xi }^2{s}^2\right){\tilde{u}}_z={\xi }^2\frac{\partial \tilde{\theta }}{\partial z}-\left({\xi }^2-1\right)\frac{\partial \tilde{e}}{\partial z},$$

(44)

$$\left[\mathsf{\Delta }-s{𝒫}_0\left(s\right)\right]\tilde{\theta }=\epsilon s{𝒫}_0\left(s\right)\tilde{e},$$

(45)

where ${𝒫}_0\left(s\right)$ is given by

$${𝒫}_0\left(s\right)=\frac{1+{\tau }_{q}s+\left(n_0{\tau }_q^2/2+n_1{\tau }_m^2\right)s^2}{1+n_2{\tau }_{\theta }s}.$$

(46)

Using the operator $\left(1/r\right)\left(\partial /\partial r\right)\left(r\right)$ on both sides of (43) and the operator $\partial /\partial z$ on (44), then adding the results leads to

$$\left[\mathsf{\Delta }-s^2\right]\tilde{e}=\mathsf{\Delta }\tilde{\theta }.$$

(47)

Eliminating both the cubical dilatation $\tilde{e}$ and temperature $\tilde{\theta }$ between (45) and (47) one obtains

$$\left(\mathsf{\Delta } -k_1^2\right)\left(\mathsf{\Delta } -k_2^2\right)\left[\begin{array}{c}\tilde{\theta }\\ \tilde{e}\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right],$$

(48)

where $k_1^2$ and $k_2^2$ satisfy the relations

$$k_1^2{k}_2^2=s^3{𝒫}_0\left(s\right), k_1^2+k_2^2=s^2+\left(1+\epsilon \right)s{𝒫}_0\left(s\right).$$

(49)

Using the Hankel transform in (48), with its operational property [54] ${\mathscr{H}}_{\nu }\left[\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial f\left(r,z,t\right)}{\partial r}\right)-\frac{{\nu }^2}{r^2}f\left(r,z,t\right)\right]=-q^2{\widehat{f}}_{\nu }\left(q,z,t\right)$, we can assume the trial solutions for the temperature $\tilde{\theta }$ and cubical dilatation $\tilde{e}$

$$\begin{gathered} \widehat{\tilde{\theta }}\left(q,z,s\right)={\displaystyle \sum }_{i=1}^2\left[A_i\exp \left(m_{i}z\right)+B_i\exp \left(-m_{i}z\right)\right], \\ \widehat{\tilde{e}}\left(q,z,s\right)={\displaystyle \sum }_{i=1}^2\left[A_i^{\prime }\exp \left(m_{i}z\right)+B_i^{\prime }\exp \left(-m_{i}z\right)\right], \end{gathered}$$

(50)

where $A_i$, $B_i$, $A_i^{\prime }$ and $B_i^{\prime }$ are unknown functions of the Hankel and Laplace parameters, to be determined later, and $m_i=\sqrt{k_i^2+q^2}$. Reduction of the number of coefficients can be implemented by making a mathematical substitution from (50) into (47), we obtain

$$A_i^{\prime }=\frac{k_i^2}{k_i^2-s^2}{A}_i, B_i^{\prime }=\frac{k_i^2}{k_i^2-s^2}{B}_i, i=1,2.$$

(51)

As a crucial step which simplifies the final forms of temperature and stresses, thus simplify the numerical implementations, we use the following transform:

$$A_i\to \left(k_i^2-s^2\right)A_i, B_i\to -\left(k_i^2-s^2\right)B_i,$$

(52)

we get the solutions in the Laplace-Hankel domain

$$\widehat{\tilde{\theta }}\left(q,z,s\right)={\displaystyle \sum }_{i=1}^2\left(k_i^2-s^2\right)\left[A_i\exp \left(m_{i}z\right)-B_i\exp \left(-m_{i}z\right)\right],$$

(53)

$$\widehat{\tilde{e}}\left(q,z,s\right)={\displaystyle \sum }_{i=1}^2{k}_i^2\left[A_i\exp \left(m_{i}z\right)-B_i\exp \left(-m_{i}z\right)\right].$$

(54)

Inverting the Hankel transform of equations (53) and (54) we get the temperature and the cubical dilatation in the Laplace domain

$$\tilde{\theta }\left(r,z,s\right)={{\displaystyle \int }}_0^{\infty }{\displaystyle \sum }_{i=1}^2\left(k_i^2-s^2\right)\left[A_i\exp \left(m_{i}z\right)-B_i\exp \left(-m_{i}z\right)\right]q{J}_0\left(qr\right)dq,$$

(55)

$$\tilde{e}\left(r,z,s\right)={{\displaystyle \int }}_0^{\infty }{\displaystyle \sum }_{i=1}^2{k}_i^2\left[A_i\exp \left(m_{i}z\right)-B_i\exp \left(-m_{i}z\right)\right]q{J}_0\left(qr\right)dq.$$

(56)

The Hankel transform of (44), together with the temperature (53) and the cubical dilatation (54), yields the displacement component ${\widehat{\tilde{u}}}_z$

$${\widehat{\tilde{u}}}_z\left(q,z,s\right)={\displaystyle \sum }_{i=1}^2\left\{m_i\left[A_i\exp \left(m_{i}z\right)+B_i\exp \left(-m_{i}z\right)\right]+C_i\exp \left({\left(-1\right)}^{i+1}mz\right)\right\},$$

(57)

where $m=\sqrt{q^2+{\xi }^2{s}^2}$ and $C_i$, $i=1,2$, are unknown coefficients depending on $q$ and $s$. Upon inverting the Hankel transform in (57) we get the normal component of displacement in the Laplace domain:

$${\tilde{u}}_z\left(r,z,s\right)={{\displaystyle \int }}_0^{\infty }{\displaystyle \sum }_{i=1}^2\left\{m_i\left[A_i\exp \left(m_{i}z\right)+B_i\exp \left(-m_{i}z\right)\right]+C_i\exp \left({\left(-1\right)}^{i+1}mz\right)\right\}q{J}_0\left(qr\right)dq.$$

(58)

On the other hand, the combined Laplace-Hankel transform of (32) reads

$${\mathscr{H}}_0\left[\frac{1}{r}\frac{\partial }{\partial r}\left(r{\tilde{u}}_r\left(r,z,s\right)\right)\right]=-{\displaystyle \sum }_{i=1}^2\left\{q^2\left[A_i\exp \left(m_{i}z\right)-B_i\exp \left(-m_{i}z\right)\right]+{\left(-1\right)}^{i+1}m{C}_i\exp \left({\left(-1\right)}^{i+1}mz\right)\right\},$$

(59)

where Equations (54) and (58) have been used in deriving (59) Making use of the useful relation of Hankel transform [54], ${\mathscr{H}}_0\left[\frac{1}{r}\frac{d}{dr}\left(rf\left(r\right)\right)\right]=q{\mathscr{H}}_1\left[f\left(r\right)\right]$, we can obtain the radial component of displacement in the Laplace domain as, refer to Equation (42),

$${\tilde{u}}_r\left(r,z,s\right)=-{{\displaystyle \int }}_0^{\infty }{\displaystyle \sum }_{i=1}^2\left\{q^2\left[A_i\exp \left(m_{i}z\right)-B_i\exp \left(-m_{i}z\right)\right]+{\left(-1\right)}^{i+1}m{C}_i\exp \left({\left(-1\right)}^{i+1}mz\right)\right\} J_1\left(qr\right)dq.$$

(60)

Lastly, the stress components (33)–(36), with the help of the expressions (55), (56), (58) and (60) can be written in the Laplace domain as

$$\begin{array}{cc}\hfill {\tilde{\sigma }}_{rr}\left(r,z,s\right)& =\frac{-2}{{\xi }^2}{{\displaystyle \int }}_0^{\infty }{\displaystyle {\displaystyle \sum }_{i=1}^2}\left\{\left(m_i^2-\frac{{\xi }^2{s}^2}{2}\right)\left[A_i\exp \left(m_{i}z\right)-B_i\exp \left(-m_{i}z\right)\right]+{\left(-1\right)}^{i+1}m{C}_i\exp \left({\left(-1\right)}^{i+1}mz\right)\right\}q{J}_0\left(qr\right) dq\hfill \\ \hfill & +\frac{2}{{\xi }^{2}r}{{\displaystyle \int }}_0^{\infty }{\displaystyle {\displaystyle \sum }_{i=1}^2}\left\{q^2\left[A_i\exp \left(m_{i}z\right)-B_i\exp \left(-m_{i}z\right)\right]+{\left(-1\right)}^{i+1}m{C}_i\exp \left({\left(-1\right)}^{i+1}mz\right)\right\} J_1\left(qr\right) dq,\hfill \end{array}$$

(61)

$$\begin{array}{cc}\hfill {\tilde{\sigma }}_{\psi \psi }\left(r,z,s\right)& =\frac{-2}{{\xi }^{2}r}{{\displaystyle \int }}_0^{\infty }{\displaystyle {\displaystyle \sum }_{i=1}^2}\left\{q^2\left[A_i\exp \left(m_{i}z\right)-B_i\exp \left(-m_{i}z\right)\right]+{\left(-1\right)}^{i+1}m{C}_i\exp \left({\left(-1\right)}^{i+1}mz\right)\right\}{J}_1\left(qr\right) dq\hfill \\ \hfill & -\frac{2}{{\xi }^2}{{\displaystyle \int }}_0^{\infty }{\displaystyle {\displaystyle \sum }_{i=1}^2}\left(k_i^2-\frac{{\xi }^2{s}^2}{2}\right)\left[A_i\exp \left(m_{i}z\right)-B_i\exp \left(-m_{i}z\right)\right] q{J}_0\left(qr\right)dq,\hfill \end{array}$$

(62)

$${\tilde{\sigma }}_{zz}\left(r,z,s\right)=\frac{2}{{\xi }^2}{{\displaystyle \int }}_0^{\infty }{\displaystyle \sum }_{i=1}^2\left\{\left(q^2+\frac{{\xi }^2{s}^2}{2}\right)\left[A_i\exp \left(m_{i}z\right)-B_i\exp \left(-m_{i}z\right)\right]+{\left(-1\right)}^{i+1}m{C}_i\exp \left({\left(-1\right)}^{i+1}mz\right)\right\}q{J}_0\left(qr\right)dq,$$

(63)

$${\tilde{\sigma }}_{rz}\left(r,z,s\right)=\frac{-1}{{\xi }^2}{{\displaystyle \int }}_0^{\infty }{\displaystyle \sum }_{i=1}^2\left\{2q^2{m}_i\left[A_i\exp \left(m_{i}z\right)+B_i\exp \left(-m_{i}z\right)\right]+\left(m^2+q^2\right)C_i\exp \left({\left(-1\right)}^{i+1}mz\right)\right\}{J}_1\left(qr\right)dq,$$

(64)

where the following recurrence relations have been invoked during the derivation of stress components, refer to relations (9.8) and (9.9) in [54]:

$$\frac{d}{dz}\left[J_0\left(z\right)\right]=-J_1\left(z\right), \frac{d}{dz}\left[J_1\left(z\right)\right]=J_0\left(z\right)-\frac{J_1\left(z\right)}{z}.$$

(65)

To complete the above analytical expressions for temperature, cubical dilatation and stresses, we determine the six coefficients $A_i$, $B_i$, and $C_i$, $i=1$,$2$, through the set of boundary conditions (37), which can be written in the Laplace-Hankel domain as [54]

$$\begin{gathered} \widehat{\tilde{\theta }}\left(q,0,s\right)=\frac{t_p}{t_{p}s+1}\left[\frac{r_2{J}_1\left(r_{2}q\right)-r_1{J}_1\left(r_{1}q\right)}{q}\right], {\frac{\partial \widehat{\tilde{\theta }}\left(q,z,s\right)}{\partial z}|}_{z=\ell }=0, \\ {\widehat{\tilde{\sigma }}}_{zz}\left(q,0,s\right)={\widehat{\tilde{\sigma }}}_{zz}\left(q,\ell ,s\right)=0, \\ {\widehat{\tilde{\sigma }}}_{rz}\left(q,0,s\right)={\widehat{\tilde{\sigma }}}_{rz}\left(q,\ell ,s\right)=0. \end{gathered}$$

(66)

Therefore, we have the following system

$${\displaystyle \sum }_{i=1}^2\left(k_i^2-s^2\right)\left[A_i-B_i\right]=\frac{t_p}{t_{p}s+1}\left[\frac{r_2{J}_1\left(r_{2}q\right)-r_1{J}_1\left(r_{1}q\right)}{q}\right],$$

(67a)

$${\displaystyle \sum }_{i=1}^2\left(k_i^2-s^2\right)m_i\left[A_i\exp \left(m_i\ell \right)+B_i\exp \left(-m_i\ell \right)\right]=0,$$

(67b)

$${\displaystyle \sum }_{i=1}^2\left[\left(q^2+\frac{{\xi }^2{s}^2}{2}\right)\left[A_i-B_i\right]+{\left(-1\right)}^{i+1}m{C}_i\right]=0,$$

(67c)

$${\displaystyle \sum }_{i=1}^2\left[\left(q^2+\frac{{\xi }^2{s}^2}{2}\right)\left[A_i\exp \left(m_i\ell \right)-B_i\exp \left(-m_i\ell \right)\right]+{\left(-1\right)}^{i+1}m{C}_i\exp \left({\left(-1\right)}^{i+1}m\ell \right)\right]=0,$$

(67d)

$${\displaystyle \sum }_{i=1}^2\left\{2q^2{m}_i\left[A_i+B_i\right]+\left(m^2+q^2\right)C_i\right\}=0,$$

(67e)

$${\displaystyle \sum }_{i=1}^2\left\{2q^2{m}_i\left[A_i\exp \left(m_i\ell \right)+B_i\exp \left(-m_i\ell \right)\right]+\left(m^2+q^2\right)C_i\exp \left({\left(-1\right)}^{i+1}m\ell \right)\right\}=0.$$

(67f)

By solving the system (67a)–(67f) simultaneously, one can obtain the problem coefficients $A_i$, $B_i$ and $C_i$, $i=1$,$2$, thus, the solution of the problem in the Laplace domain is completed.

4. Numerical Results and Discussions

In this section, we recover numerically the temperature (55), displacement components (58) and (60) and stresses (61)–(64) to the physical domain. Inversion of the Laplace parameter $s$ to the physical time $t$ requires the implementation of Durbin formula [55,56]

$$\begin{gathered} f\left(r,z,t\right)=\frac{2\exp \left(\gamma \right)}{T_1}\{-\frac{1}{2}\tilde{f}\left(r,z,\frac{\gamma }{t}\right)+\Re \left[{\displaystyle \sum }_{j=0}^{NSum}\tilde{f}\left(r,z,\frac{\gamma }{t}+\frac{2\pi ıj}{T_1}\right)\cos \left(\frac{2\pi jt}{T_1}\right)\right] \\ -\Im \left[{\displaystyle \sum }_{j=0}^{NSum}\tilde{f}\left(r,z,\frac{\gamma }{t}+\frac{2\pi ıj}{T_1}\right)\sin \left(\frac{2\pi jt}{T_1}\right)\right]\}, \end{gathered}$$

(68)

where $\gamma \cong 4.7 ~ 10$ and $T_1$ is chosen so that $0<t\le 2T_1$. For accelerating the computations, we invoke a FORTRAN subroutine [57] for the series (68). The iteration $Nsum$ is taken ${10}^6$ and the maintained accuracy for the inverse Laplace implementation is five digits for all variables. The bounded integral over the Hankel parameter, $q$, in the temperature (55), displacement components (58) and (60) and stresses (61)–(64), will be numerically computed using successive Fortran subroutines that use the Bulirsch-Stoer step method, the rational function extrapolation and modified midpoint method [58]. Iteration of the sum, which approximating the integral over $q$, is set to ${10}^5$ and the error is approximated to ${10}^{-6}$ for temperature and stresses and to ${10}^{-7}$ or ${10}^{-9}$ for displacement. The program consisting of all subroutines in addition to the main part was built by Professor Hany Sherief, was successfully employed in [45,46,48,59] and has been developed here for reproducing the functions in the full space-time resolution. The numerical results are illustrated graphically using MATLAB robust graphical features (MATLAB R2019a). Most variables are represented within the meridian plane $rz$, on a spatial rectangular lattice consisting of $106\times 101$ points (i.e., $106$ for $r$ and $101$ for $z$) for stresses and temperature and on the square lattice $30\times 30$ for displacement vector.

In order to present the numerical simulations on a physical basis, pure Copper metal ($\mathrm{Cu}$) is chosen for this purpose, with its thermophysical parameters in SI units listed in

Table 1. Material parameters for copper at room temperature $T_0=293° \mathrm{K}$.

Property	$\mathit{\lambda }$	$\mathit{\mu }$	$\mathit{\rho }$	${\mathit{c}}_{\mathit{E}}$	${\mathit{\alpha }}_{\mathit{T}}$	$\mathit{k}$	${\mathit{\tau }}_{\mathit{q}}$	${\mathit{\tau }}_{\mathit{m}}$	${\mathit{\tau }}_{\mathit{\theta }}$
Value	$7.76\times {10}^{10}$	$3.86\times {10}^{10}$	8954	$383.1$	$1.78\times {10}^{-5}$	$386.0$	$0.4648$	0.1142	70.833
SI unit	$\mathrm{Kg}\cdot {\mathrm{m}}^{-1}\cdot {\sec }^{-2}$	$\mathrm{Kg}\cdot {\mathrm{m}}^{-1}\cdot {\sec }^{-2}$	$\mathrm{Kg}\cdot {\mathrm{m}}^{-3}$	$\mathrm{J} ·{ \mathrm{Kg}}^{-1}·{\mathrm{K}}^{-1}$	${\mathrm{K}}^{-1}$	$\mathrm{W}·{\mathrm{m}}^{-1}·{\mathrm{K}}^{-1}$	picoseconds	picoseconds	picoseconds

, the values are taken from [26,37,46]:

Based on the above thermophysical quantities, the constants of the problem are given as

$$\begin{gathered} \eta =8886.729, {\xi }^2=4.01036, \epsilon =0.0168, c_0\eta =3.695\times {10}^7, \\ {c}_0^2\eta =1.536\times {10}^{11}, {\tau }_q=0.0714, {\tau }_m=0.0175. \end{gathered}$$

(69)

The physical value of the dimensionless temperature-gradient phase lag ${\tau }_{\theta }$ is $10.883$. The thermal reflection phenomenon occurred inside the plate at this value of ${\tau }_{\theta }$ is not graphically clear as in the case of LSTE model, so we replace it with the arbitrary value ${\tau }_{\theta }=2{\tau }_q$ to show the main difference between HDPLTE and MHDPLTE. The other problem dimensionless parameters are chosen as

$$t_p=0.2, \ell =1.$$

(70)

In (70), the dimensionless thermalization time is clearly set to $0.2$ and the dimensionless plate thickness is set as one, which means upon referring to the dimensionless transformations (28) and the values (69) that the dimensional thermalization time is equal to $0.2/\left(c_0^2\eta \right)=1.302 \mathrm{picoseconds}$ and the dimensional thickness of the plate is equal to $1/(c_0\eta )=27.063 \mathrm{nanometer}$, accordingly, the current problem “ultrafast heating of a nanoplate”.

Finally, we introduce the hydrostatic stress as the mean value of the normal stresses ${\sigma }_{rr}$, ${\sigma }_{zz}$, and ${\sigma }_{\psi \psi }$, namely,

$${\sigma }_H\left(r,z,t\right)=\frac{{\sigma }_{rr}+{\sigma }_{zz}+{\sigma }_{\psi \psi }}{3},$$

(71)

as a representative stress carrying all features of other stress components. In view of Equations (33)–(35), the hydrostatic stress (71) can be written in terms of the temperature and the cubical dilatation, i.e.,

$${\sigma }_H\left(r,z,t\right)=\left(1-\frac{4}{3{\xi }^2}\right)e\left(r,z,t\right)-\theta \left(r,z,t\right),$$

(72)

where the temperature $\theta \left(r,z,t\right)$ and the cubical dilatation $e\left(r,z,t\right)$ are determined from the numerical inversion of the Laplace parameter $s$ and the numerical integration over $q$ in Equations (55) and (56).

The temporal evolution of the temperature and the hydrostatic stress at different “six” instants of time, in $rz$-plane, are respectively represented in Figure 2 and Figure 3. The effect of the thermalization time, $t_p=0.2$, incorporated by the exponent $\exp \left(-t/t_p\right)$ in the boundary conditions, is clearly notable on the plane $z=0$, where the height of the box varies with time progress, see the temperature distributions in Figure 2. Because of the thermal insulation of the rear surface, i.e., the reflection boundary condition ${\partial \theta \left(r,z,t\right)/\partial z|}_{z=1}=0$, the thermal reflection phenomenon on the rear surface $z=1$ is apparent in both the temperature and hydrostatic stress distributions, specifically, in the presence of the employed finite-speed models of thermoelasticity, LSTE, HDPLTE, and MHDPLTE. It is noteworthy that, in the parabolic models of thermomechanics, the reflection phenomenon is not observable due to the infinite thermal wave speed.

In the heat transfer problems [2,28,37], it is known that there is a single finite speed for the thermal wave and $v_{MCV}<v_{DPL}<v_{MDPL}$, where $v_{MCV}$ stands for the velocity of Maxwell-Cattaneo-Vernotte thermal wave speed, $v_{DPL}$ is the velocity of the hyperbolic DPL model and $v_{MDPL}$ is the velocity of the modified hyperbolic DPL model, or in other words, $v_{MDPL}$ represents the speed of a thermal wave governed by the hyperbolic two-step model (10). On the other hand, in the hyperbolic thermoelastic models, there are two propagating waves, where the thermal wave dominates the first and the mechanical wave dominates the second, refer to the hydrostatic distributions in Figure 3. In the next part of this discussion, we review the speeds of waves in the LSTE [45], and derive approximate formulas for the waves speed in both HDPLTE and MHDPLTE.

Firstly, we determine the exact solution for small values of time. With the aid of the initial value theorem of the Laplace transforms [60], this solution corresponds to large values of $s$ in the Laplace domain. We note that the characteristic roots $k_1^2$ and $k_2^2$ in Equation (49) can be written as

$$k_i^2=\frac{1}{2}\left\{s^2+\left(1+\epsilon \right)s{𝒫}_0\left(s\right)+{\left(-1\right)}^{i+1}\sqrt{{\left[s^2+\left(1+\epsilon \right)s{𝒫}_0\left(s\right)\right]}^2-4s^3{𝒫}_0\left(s\right)}\right\}, i=1,2.$$

(73)

Therefore, for small values of time, the roots $k_1^2$ and $k_2^2$ in (73) can be approximated in the form

$$k_i^2=s^2\left[{\zeta }_{i0}+\frac{{\zeta }_{i1}}{s}+O\left(\frac{1}{s^2}\right)\right], i=1,2,$$

(74)

where ${\zeta }_{i0}$ and ${\zeta }_{i1}$ are given by

$$\begin{gathered} {\zeta }_{i0}=\frac{1}{2}\left[1+\left(1+\epsilon \right){\delta }_1+{\left(-1\right)}^{i+1}\sqrt{1+2\left(\epsilon -1\right){\delta }_1+{\left(1+{\epsilon }^2\right)}^2{\delta }_1^2}\right], \\ {\zeta }_{i1}=\frac{{\delta }_2}{2}\left[1+\epsilon +\frac{{\left(-1\right)}^{i+1}\left[\epsilon -1+{\left(1+{\epsilon }^2\right)}^2{\delta }_1\right]}{\sqrt{1+2\left(\epsilon -1\right){\delta }_1+{\left(1+{\epsilon }^2\right)}^2{\delta }_1^2}}\right], \end{gathered}$$

(75)

${\delta }_1$ and ${\delta }_2$ depend on the relaxation times of the hyperbolic thermoelastic models, namely,

$${\delta }_1=\{\begin{array}{c}{\tau }_q, \mathrm{for} \mathrm{LSTE},\\ \frac{{\tau }_q^2}{2{\tau }_{\theta }}, \mathrm{for} \mathrm{HDPLTE},\\ \frac{{\tau }_m^2}{{\tau }_{\theta }}, \mathrm{for} \mathrm{MHDPLTE},\end{array}$$

(76)

and

$${\delta }_2=\{\begin{array}{c}1, \mathrm{for} \mathrm{LSTE},\\ \frac{{\tau }_q}{{\tau }_{\theta }}-\frac{{\tau }_q^2}{2{\tau }_{\theta }^2}, \mathrm{for} \mathrm{HDPLTE},\\ \frac{{\tau }_q}{{\tau }_{\theta }}-\frac{{\tau }_m^2}{{\tau }_{\theta }^2}, \mathrm{for} \mathrm{MHDPLTE}.\end{array}$$

(77)

In (75)–(77), we have employed the useful relation $\sqrt{x^2+ax+b}=\sqrt{b}+\frac{a}{2\sqrt{b}}x+O\left(x^2\right)\cong \sqrt{b}+\frac{a}{2\sqrt{b}}x$ as $x\to 0$. Furthermore, the parameters $m_i^2=q^2+k_i^2$, $i=1,2$, is approximated for large values of $s$ as

$$m_i^2=s^2\left[{\zeta }_{i0}+\frac{{\zeta }_{i1}}{s}+O\left(\frac{1}{s^2}\right)\right]\cong {\zeta }_{i0}{s}^2, i=1,2,$$

(78)

and the parameter $m\left(s\right)$, is $m^2=s^2\left[{\xi }^2+O\left(\frac{1}{s^2}\right)\right]\cong {\xi }^2{s}^2.$ Again, using the relation $\sqrt{x^2+ax+b}=\sqrt{b}+\frac{a}{2\sqrt{b}}x+O\left(x^2\right)$ as $x\to 0$, Equation (78) reduces to

$$m_i=s\left(\sqrt{{\zeta }_{i0}}+\frac{{\zeta }_{i1}}{2\sqrt{{\zeta }_{i0}}}\frac{1}{s}+O\left(\frac{1}{s^2}\right)\right)=\sqrt{{\zeta }_{i0}}s+\frac{{\zeta }_{i1}}{2\sqrt{{\zeta }_{i0}}}+O\left(\frac{1}{s}\right)=\sqrt{{\zeta }_{i0}}s+O\left(\frac{1}{s}\right),$$

(79)

$s\to \infty$ and $i=1,2$. Now, by solving the linear algebraic system (67), and expanding the coefficients $A_i\left(q,s\right), B_i\left(q,s\right), C_i\left(q,s\right)$ and $i=1,2$, into Maclaurin’s series for large value of $s$, using a suitable symbolic program, we obtain the following approximate expressions:

$$\begin{gathered} A_i\cong \frac{\left(r_2{J}_1\left(r_{2}q\right)-r_1{J}_1\left(r_{1}q\right)\right)\exp \left(-2m_i\ell \right)}{q{s}^3}\left[\frac{{\zeta }_{10}-\sqrt{{\zeta }_{10}{\zeta }_{20}}+{\zeta }_{20}-1}{\left({\zeta }_{10}-2\sqrt{{\zeta }_{10}{\zeta }_{20}}+{\zeta }_{20}\right)\left({\zeta }_{10}+\sqrt{{\zeta }_{10}{\zeta }_{20}}+{\zeta }_{20}-1\right)}\right], \\ {B}_i\cong \frac{\left(r_2{J}_1\left(r_{2}q\right)-r_1{J}_1\left(r_{1}q\right)\right)}{q{s}^3}\left[\frac{1}{{\zeta }_{10}-{\zeta }_{20}}\right], \\ {C}_1\cong -\frac{ 2q\left(r_2{J}_1\left(r_{2}q\right)-r_1{J}_1\left(r_{1}q\right)\right)\exp \left(-2m\ell \right)}{s^4}\left[\frac{1}{{\xi }^2\left(\sqrt{{\zeta }_{10}}+\sqrt{{\zeta }_{20}}\right)}\right], \end{gathered}$$

$$C_2\cong \frac{ 2q\left(r_2{J}_1\left(r_{2}q\right)-r_1{J}_1\left(r_{1}q\right)\right)}{s^4}\left[\frac{1}{{\xi }^2\left(\sqrt{{\zeta }_{10}}+\sqrt{{\zeta }_{20}}\right)}\right],$$

(80)

$i=1,2$. Hence, the problem coefficients $A_i, B_i$ are of order $O\left(\frac{1}{s^3}\right)$ and $C_i$ are of order $O\left(\frac{1}{s^4}\right).$ We shall now use the following Boley theorem [45,61,62], which is very effective in determining approximating values for the longitudinal wave fronts and speeds from the Laplace transform expressions in the small values of Laplace parameter $s$ when these expressions include exponential functions. These values of waves fronts and speeds are exactly representing the case of short-time domain, in which the second-sound phenomenon is significant.

Theorem 1. 

Let the inverse of the Laplace transform of a function$\tilde{f}\left(r,z,s\right)$, representing a longitudinal wave propagating along$z$-axis, be written at the form:

$$f\left(r,z,t\right)=\frac{1}{2\pi ı}\underset{\beta \to \infty }{\lim }\underset{\alpha -i\beta }{\overset{\alpha +i\beta }{{\displaystyle \int }}}\widetilde{𝒻}\left(r,z,s\right)\exp \left(g\left(z,s,t\right)\right) ds.$$

(81)

Further, if for large $\mathrm{s}$, $\widetilde{𝒻}\left(r,s\right)$ can be written as

$$\widetilde{𝒻}\left(r,z,s\right)=\frac{\mathcal{K}\left(r,z\right)}{s^n}\left[1+O\left(\frac{1}{s}\right)\right], n>0,$$

(82)

and if there exists a function $y\left(z,t\right)$ such that for large $s$

$$g\left(z,s,t\right)-sy\left(z,t\right)=O\left(\frac{1}{s}\right),$$

(83)

then, the magnitude (size) of discontinuity of the longitudinal wave $f\left(r,z_0,t_0\right)$, at the instant $t=t_0$ and the plane $z=z_0$, is denoted by $\mathcal{S}f\left(r,z,t_0\right)$ and determined through the relation [61]

$$\mathcal{S}f\left(r,z,t_0\right)=f\left(r,z,t_0+0\right)-f\left(r,z,t_0-0\right)=\{\begin{array}{c} 0 for y\ne 0,\\ =\left\{\begin{array}{c}0\\ K\left(r,z\right)\\ \infty \end{array} \begin{array}{c}if n>1,\\ if n=1,\\ if n<1,\end{array}\right\} for y=0.\end{array}$$

(84)

Clearly, $\mathcal{S}f\left(r,z,t_0\right)=0$ stands for the absence of discontinuities in the longitudinal wave propagation at the plane $z=z_0$ and $t=t_0$, while $\mathcal{S}f\left(r,z,t_0\right)=\mathcal{K}\left(r,z\right)$ means that there is a discontinuity at the instant $t_0$ and the position $z=z_0$, its magnitude is a function of $r$ and $z$. Employing Boley theorem to the temperature profile given in the Laplace domain by (55), we firstly get the temperature solution in the Laplace domain for large values of $s$ as

$$\tilde{\theta }\left(r,z,s\right)={\tilde{\theta }}_1\left(r,z,s\right)+{\tilde{\theta }}_2\left(r,z,s\right),$$

(85a)

$${\tilde{\theta }}_1\left(r,z,s\right)\cong \frac{H\left(r_2-r\right)-H\left(r_1-r\right)}{\left({\zeta }_{20}-{\zeta }_{10}\right)s}{\displaystyle \sum }_{i=1}^2\left({\zeta }_{i0}-1\right)\exp \left(-m_{i}z\right),$$

(85b)

$$\begin{gathered} {\tilde{\theta }}_2\left(r,z,s\right)\cong \frac{\left({\zeta }_{10}-\sqrt{{\zeta }_{10}{\zeta }_{20}}+{\zeta }_{20}-1\right)\left[H\left(r_2-r\right)-H\left(r_1-r\right)\right]}{\left({\zeta }_{10}-2\sqrt{{\zeta }_{10}{\zeta }_{20}}+{\zeta }_{20}\right)\left({\zeta }_{10}+\sqrt{{\zeta }_{10}{\zeta }_{20}}+{\zeta }_{20}-1\right)s} \\ \times {\displaystyle \sum }_{i=1}^2\left({\zeta }_{i0}-1\right)\exp \left(m_i\left(z-2\ell \right)\right). \end{gathered}$$

(85c)

The first and the second terms of ${\tilde{\theta }}_1\left(r,z,s\right)$ in (85b) represent respectively the two incident waves, orienting from the front surface $z=0$ to the rear surface $z=\ell$. On the other hand, the first and the second terms of ${\tilde{\theta }}_2\left(r,z,s\right)$ in (85c) represent respectively the two travelling waves, reflecting from the rear surface $z=\ell$ to the front surface $z=0$.

For the first incident wave from the front surface $z=0$ to the rear surface $z=\ell$, i.e., the first term of (85b), we choose $g\left(z,s,t\right)=-m_{1}z+st$ and $y\left(z,t\right)=-\sqrt{{\zeta }_{10}}z+t$, such that relation (83) is verified by the virtue of Equation (79). We define the first term of (85b) as

$${\tilde{\theta }}_{11}\left(r,z,s\right)\cong \frac{H\left(r_2-r\right)-H\left(r_1-r\right)}{\left({\zeta }_{20}-{\zeta }_{10}\right)s}\left({\zeta }_{10}-1\right)\exp \left(-m_{1}z\right),$$

and using the proposed function $g\left(z,s,t\right)$ in (81), (82), we get

$${\theta }_{11}\left(r,z,t\right)\cong \frac{1}{2\pi ı}\underset{\beta \to \infty }{\lim }\underset{\alpha -i\beta }{\overset{\alpha +i\beta }{{\displaystyle \int }}}\frac{{\mathcal{K}}_1^{Inc}\left(r,z\right)}{s}\exp \left(-m_{1}z\right)\exp \left(st\right) ds.$$

(86)

where

$${\mathcal{K}}_1^{Inc}\left(r,z\right)=\frac{\left({\zeta }_{10}-1\right)\left[H\left(r_2-r\right)-H\left(r_1-r\right)\right]}{\left({\zeta }_{20}-{\zeta }_{10}\right)}\exp \left(-\frac{{\zeta }_{11}}{2\sqrt{{\zeta }_{10}}}z\right).$$

(87)

In comparing (82) with (86), we find that $n=1$, therefore, if $y\left(z,t\right)=0$, namely, if the first incident wave travels with a finite velocity $1/\sqrt{{\zeta }_{10}}$, then the wave will suffer a cut at $z=t/\sqrt{{\zeta }_{10}}$. By similar arguments, if we define the second term of (85b) as

$${\tilde{\theta }}_{12}\left(r,z,s\right)\cong \frac{H\left(r_2-r\right)-H\left(r_1-r\right)}{\left({\zeta }_{20}-{\zeta }_{10}\right)s}\left({\zeta }_{20}-1\right)\exp \left(-m_{2}z\right),$$

and introduce suitable functions $g\left(z,s,t\right)$ and $y\left(z,t\right)$, we can define the velocity of the second incident wave as $1/\sqrt{{\zeta }_{20}}$ and its front at $z=t/\sqrt{{\zeta }_{20}}$. Furthermore, the magnitude of the discontinuity of the second wave is given by

$${\mathcal{K}}_2^{Inc}\left(r,z\right)=\frac{\left({\zeta }_{20}-1\right)\left[H\left(r_2-r\right)-H\left(r_1-r\right)\right]}{\left({\zeta }_{20}-{\zeta }_{10}\right)}\exp \left(-\frac{{\zeta }_{21}}{2\sqrt{{\zeta }_{20}}}z\right).$$

(88)

The magnitude of discontinuities in the incident waves (87) and (88) indicates that they occur within the interval $r_1\le r\le r_2$ and decay exponentially along $z$. From the above discussion, it is easy to bring the analytical approximated solution of the temperature when the dimensionless thermalization time is very large, i.e., $t_p\to \infty$, which is valid within the short-time domain as

$$\begin{gathered} \theta \left(r,z,t\right)={\mathcal{K}}_1^{Inc}\left(r,z\right)H\left(t-\sqrt{{\zeta }_{10}}z\right)+{\mathcal{K}}_2^{Inc}\left(r,z\right)H\left(t-\sqrt{{\zeta }_{20}}z\right)+{\mathcal{K}}_1^{Ref}\left(r,z\right)H\left(t+\sqrt{{\zeta }_{10}}\left(z-2\ell \right)\right) \\ +{\mathcal{K}}_2^{Ref}\left(r,z\right)H\left(t+\sqrt{{\zeta }_{20}}\left(z-2\ell \right)\right), \end{gathered}$$

(89)

where ${\mathcal{K}}_1^{Inc}$ and ${\mathcal{K}}_2^{Inc}$ are given by (87) and (88), while ${\mathcal{K}}_1^{Ref}$ and ${\mathcal{K}}_2^{Ref}$ are the magnitude of discontinuities of the reflected waves and given by

$$\begin{gathered} {\mathcal{K}}_1^{Ref}\left(r,z\right)=\frac{\left({\zeta }_{10}-\sqrt{{\zeta }_{10}{\zeta }_{20}}+{\zeta }_{20}-1\right)\left[H\left(r_2-r\right)-H\left(r_1-r\right)\right]\left({\zeta }_{10}-1\right)}{\left({\zeta }_{10}-2\sqrt{{\zeta }_{10}{\zeta }_{20}}+{\zeta }_{20}\right)\left({\zeta }_{10}+\sqrt{{\zeta }_{10}{\zeta }_{20}}+{\zeta }_{20}-1\right)}\exp \left(\frac{{\zeta }_{11}}{2\sqrt{{\zeta }_{10}}}\left(z-2\ell \right)\right) \\ {\mathcal{K}}_2^{Ref}\left(r,z\right)=\frac{\left({\zeta }_{10}-\sqrt{{\zeta }_{10}{\zeta }_{20}}+{\zeta }_{20}-1\right)\left[H\left(r_2-r\right)-H\left(r_1-r\right)\right]\left({\zeta }_{20}-1\right)}{\left({\zeta }_{10}-2\sqrt{{\zeta }_{10}{\zeta }_{20}}+{\zeta }_{20}\right)\left({\zeta }_{10}+\sqrt{{\zeta }_{10}{\zeta }_{20}}+{\zeta }_{20}-1\right)}\exp \left(\frac{{\zeta }_{21}}{2\sqrt{{\zeta }_{20}}}\left(z-2\ell \right)\right) \end{gathered}$$

(90)

It is clear that the first wave with velocity $1/\sqrt{{\zeta }_{10}}$ is of a mechanical nature, while the second wave with velocity $1/\sqrt{{\zeta }_{20}}$ is of a thermal nature [45]. Indeed, if we consider the pure thermal propagation in which $\epsilon =0$, refer to (31), then we have from Equation (75) that ${\zeta }_{10}=1$, ${\zeta }_{20}={\delta }_1$, ${\zeta }_{11}=1$ and ${\zeta }_{21}={\delta }_2$. Thereby, ${\mathcal{K}}_1^{Inc}\left(r,z\right)={\mathcal{K}}_1^{Ref}\left(r,z\right)=0$, while ${\mathcal{K}}_2^{Inc}\left(r,z\right)\ne 0$ and ${\mathcal{K}}_2^{Ref}\left(r,z\right)\ne 0$, which indicate that the mechanical behavior dominates the first incident and reflected waves, such that their fronts disappear when the thermomechanical coupling parameter vanishes. Meanwhile, the second incident and reflected waves fronts are still existing when $\epsilon =0$ and their dimensionless speed is $1/\sqrt{{\delta }_1}$, i.e., $1/\sqrt{{\tau }_q}$ for the MCV model, $\sqrt{2{\tau }_{\theta }}/{\tau }_q$ for HDPL model and $\sqrt{{\tau }_{\theta }}/{\tau }_m$ for the MHDPL model, refer to (76), which perfectly coincide with their corresponding values in heat conduction problem [2] when we set $c_0^2\eta ={\tau }_q$ in (28). Figuratively speaking, if we call the first wave as a mechanical wave and the second wave as a thermal wave, then we have

$$v_{Mech}=\frac{1}{\sqrt{{\zeta }_{10}}}, v_{Therm}=\frac{1}{\sqrt{{\zeta }_{20}}},$$

(91)

where ${\zeta }_{i0}$, $i=1,2$, are given by (75) and ${\delta }_1$, which depends on the type of thermoelastic model, is given by (76). Upon considering the proposed numerical values (69) and (70), the wave speeds (91) result in the following numerical values for the wave speeds and first arrival times to the rear surface under the three hyperbolic models of thermoelasticity, see

Table 2. Velocities and arrival/passage times for mechanical and thermal waves to the rear surface of the plate, $z=1$, for different finite speed models.

Model	Speed	First Arrival Times to the Rear Surface $\mathit{z}=1$	Speed	First Arrival Times to the Rear Surface $\mathit{z}=1$
	Mechanical Wave		Thermal Wave
LSTE	$0.9993546797$	$1.000645737$	$3.744547879$	$0.2670549375$
HDPLTE	$0.9998473403$	$1.000152683$	$7.485405609$	$0.1335932950$
MHDPLTE	$0.9999818549$	$1.000018145$	$21.53981318$	$0.04642565799$

,

Therefore, we can conclude that

$$v_{Mech}^{\mathrm{LSTE}}<v_{Mech}^{\mathrm{HDPLTE}}<v_{Mech}^{\mathrm{MHDPLTE}}, v_{Therm}^{\mathrm{LSTE}}<v_{Therm}^{\mathrm{HDPLTE}}<v_{Therm}^{\mathrm{MHDPLTE}}.$$

(92)

Basically, Equation (92) and Table 2 introduce a reasoning interpretation for the substantial differences among the waves of the three hyperbolic models of thermoelasticity, refer to Figure 2 and Figure 3.

In addition, the stress components have three types of waves originated on the front surface and reflected on the rear surface, more specifically: the mechanical, thermal waves and the shear transverse wave. From the above discussion for the temperature distribution and in view of the stresses components (61)–(64), one can easily see that every stress component has a finite discontinuity, corresponding to the temperature discontinuity, across the mechanical and thermal longitudinal waves at the wave fronts $\left(z_{Mech}, z_{Therm}\right)$. However, the shear waves, represented by the coefficients $C_1$ and $C_2$, are continuous at the shear wave front ${\mathrm{z}}_{Trans }=t/\xi ,$ with non-dimensional finite speed $v_{Trans}=1/\xi$. Since the coefficients of the displacement components, refer to (58) and (60), are of the orders $O\left(1/s^3\right)$ for $u_r$, and $O\left(1/s^2\right)$ and $O\left(1/s^4\right)$ for $u_z$, thus Equation (84) yields that $\mathcal{K}=0$, i.e., the disappearance of the wave front in both displacement components. In Figure 4 and Figure 5, we compare among the profiles of temperature and hydrostatic stress under the three considered theories of hyperbolic thermoelasticity at $r=2$ and different values of time. The main difference is the discrepancies among the thermal waves resulting from these models. However, the mechanical wave speed is approximately the same, refer to Table 2.

The displacement vector $u=\langle u_r,u_z,0\rangle$ is graphically represented in Figure 6. We have picked out four different instants: $t=0.1$, $0.15$, $0.3$ and $0.7$. At $t=0.1$, $0.15$, and $0.3$, the displacement effects on the rear surface are very apparent when the MHDPLTE, HDPLTE and LSTE models are adopted respectively. The last instant, $t=0.7$, is relatively long so that there is no substantial difference among the three models. In view of Table 2, we note that the first arrival times for the thermal wave are $t=0.046$, $0.134$, and $t=0.267$ for the MHDPLTE, HDPLTE, and LSTE respectively. Thereafter, these thermal waves are reflected on the rear surface returning to the front surface, which amplify the displacement influences on the rear surface.

5. Summary

In this article, we have revisited the infinite plate problem with axisymmetric temperature distribution on its front surface as the solely cause of the disturbances. The surface temperature distribution has been taken as a hot circular ring decaying exponentially with time, according to an incorporated controlling thermalization time $t_p$. The axisymmetric plane deformation has been thereby presented with dependence on the variables $\left(r,z,t\right)$. The Laplace and Hankel transforms have been used to obtain analytical expression for the temperature, cubical dilatation, displacement components and stress components in the Laplace-Hankel domain, then, suitable numerical techniques, based on accelerating sums of two infinite series, have been employed to recover the solutions in the real domain. Explicit expressions for the thermal and mechanical wave speeds are derived for the three models of thermoelasticity and we compare about their values for copper plate. Velocity of the thermal wave governed by the MHDPLTE (corresponding exactly to the two-step model) is the faster wave compared with the HDPLTE, which is faster than LSTE. The mechanical wave speed is also compared. The differences in the mechanical wave speeds come from the coupling with thermal wave. When the thermomechanical coupling is neglected, all the thermal wave speeds reduce to their correspondences in the heat transfer situation.

We have studied the thermal reflection phenomenon by setting a thermal insulation boundary condition on the rear surface. The numerical results are brought and represented graphically in the space-time domain for the temperature and hydrostatic stress. Exact solution for the temperature in the short time domain, when the thermalization time is very long, has been derived, which contains exactly the same wave front positions of the problem thermal wave.

The vector plot for displacement has been represented for the three models of hyperbolic thermoelasticity. We clearly see the effects of the thermal wave arrival and reflection on the rear surface. The distinct physical basis of the ultrafast models of thermoelasticity, i.e., the DPL and two-step models of heat transfer, apparently affects the deformation of the plate.