Continuum Mech. Thermodyn. DOI 10.1007/s00161-016-0540-z

O R I G I NA L A RT I C L E

Evgeniy Yu. Vitokhin · Mikhail B. Babenkov

Influence of boundary conditions on the solution of a hyperbolic thermoelasticity problem

Received: 23 May 2016 / Accepted: 10 November 2016 © Springer-Verlag Berlin Heidelberg 2016

Abstract We consider a series of problems with a short laser impact on a thin metal layer accounting various boundary conditions of the first and second kind. The behavior of the material is modeled by the hyperbolic thermoelasticity of Lord–Shulman type. We obtain analytical solutions of the problems in the semi-coupled formulation and numerical solutions in the coupled formulation. Numerical solutions are compared with the analytical ones. The analytical solutions of the semi-coupled problems and numerical solutions of the coupled problems show qualitative match. The solutions of hyperbolic thermoelasticity problems are compared with those obtained in the frame of the classical thermoelasticity. It was determined that the most prominent difference between the classical and hyperbolic solutions arises in the problem with fixed boundaries and constant temperature on them. The smallest differences were observed in the problem with unconstrained, thermally insulated edges. It was shown that a cooling zone is observed if the boundary conditions of the first kind are given for the temperature. Analytical expressions for the velocities of the quasiacoustic and quasithermal fronts as well as the critical value for the attenuation coefficient of the excitation impulse are verified numerically. Keywords Hyperbolic thermoelasticity · Maxwell–Cattaneo law · Heat flux relaxation · Finite difference method

1 Introduction In the modern theory of thermoelasticity, there are different approaches that take into account the finite speed of heat propagation. The need to describe the heat waves propagation process arises when considering small distances of the order of micrometers and ultra-fast thermal impulses. An example of such effects is a short laser impact on a thin film. One of the most well-known approaches describing the wave nature of the heat propagation processes in an elastic medium was proposed by Lord and Shulman [1]. This approach is based on the generalized Fourier law, taking into account the relaxation time of the heat flux τ : τ h˙ + h = −λ∇T, Communicated by Andreas Öchsner. E. Yu. Vitokhin (B) · M. B. Babenkov Department of Theoretical Mechanics, Peter the Great Saint-Petersburg Polytechnic University, Polytechnicheskaya, 29, St. Petersburg, Russia 195251 E-mail: [email protected] M. B. Babenkov Institute for Problems in Mechanical Engineering RAS, Bolshoy pr. V. O., 61, St. Petersburg, Russia 199178

(1)

E. Yu. Vitokhin, M. B. Babenkov

where h is the heat flux, λ is the coefficient of thermal conductivity, ∇ is the Hamiltonian operator, T is the temperature. The heat flux in the media does not appear or disappear instantly. The time lag it takes is characterized by the relaxation time of the heat flux. Problems considering the ultra-fast laser impact on a medium has been attracting significant interest caused by the development of microelectronics [2–4]. In addition, the internal heat sources at the nano scale level (the size of a modern transistor is of a few nanometers) cause a more intense temperature rise [5], than predicted by the classical Fourier theory of heat conduction, which increases the requirements for future cooling systems [6]. Temperature substantially affects the thermomechanical properties of layered nanomaterials, since the thinner the layer, the more it is sensitive to temperature effects [7]. Propagation of thermoelastic waves may be useful for determining the thermomechanical characteristics of micro- and nanoscale objects, such as thin plates and rods, which are used in micro- and nanoelectromechanical devices (MEMS and NEMS). A number of experimental [8] and theoretical papers [9] are dedicated to the thermoelastic wave propagation processes in nanometer-sized films. Some numerical studies of the hyperbolic thermoelasticity problems were done in [10–12]. An extensive review on the hyperbolic thermoelasticity is given in [13,14]. The second half of the century brought a surge of interest in the hyperbolic theory of thermal conductivity. The topic was thoroughly researched, and the numerous studies devoted to the theory were published. Over recent decades the problems of coupled hyperbolic thermoelasticity have drawn a certain attention. Regardless the large number of the research articles presented in the modern literature, see e.g., [13–19,19–21], the question remains investigated insufficiently. In particular, it would be of interest to know how the boundary conditions influence the contribution of the wave terms of the hyperbolic heat conduction equation to the overall solution of a coupled thermoelasticity problem. Besides, theoretical works [13,22] provide analytical expressions for velocities of the quasiacoustic and quasithermal wave fronts as well as a critical value for the attenuation coefficient that is responsible for the qualitative change in the behavior of a solution. Up to the date, no researches have yet validated these relations numerically. The thermoelasticity problem can be solved in the semi-coupled formulation. In this case, a solution of the heat equation with the given boundary conditions is substituted into the equation of motion. It is known [23] that the solution of the classical thermoelasticity problem in semi-coupled formulation gives a good approximation, at the same time, this solution is much easier to obtain than the solution of fully coupled thermoelasticity problems. Significance of the coupling effect for the solution of a hyperbolic thermoelasticity problem needs an additional examination. These questions are covered in the present research. Four combinations of the boundary conditions of the first and the second kind are compared in order to find the most prominent difference between the solutions of the problems of the classical and hyperbolic theories. Moreover, the velocities of the wave fronts as well as the critical attenuation coefficient are verified numerically. The problems of the hyperbolic thermoelasticity with an ultra-fast impact have certain peculiarities complicating its numerical solution. Impulse heating leads to the function discontinuity and erroneous high frequency oscillations in its neighborhood. Heat conduction equation contains a small multiplier of the higher derivative that results in appearing of fast oscillating functions regardless the impact type. Besides, the coefficient of thermal expansion, which provides the thermomechanical coupling, is also a small value that causes significant difference between the heat and acoustic peak values at the front. Obtaining of analytical solutions is also associated with certain difficulties. The solution of the semi-coupled hyperbolic thermoelasticity problem can be represented in the form of a series. Since this series converges slowly, one must sum up a large number of terms to get an accurate result. It can lead to a cumulative error and emergence of erroneous oscillations near the discontinuities. If one holds a small number of the series terms, then a rounding error may exceed the acceptable limit. Comparing the numerical and analytical solution, we can conclude which features of the obtained solutions reflect the actual behavior of the system. In order to validate the results obtained numerically, we calculate the error for the different step sizes.

2 The equations of the hyperbolic thermal conductivity and thermoelasticity Let us derive the heat equation using generalized Fourier’s law (1). To do this, we write the energy balance equation for the undeformable medium: ρ U˙ = −∇ · h + ρq, (2) where ρ is the density, U is the mass density of the internal energy, q is the mass density of the internal heat sources. Since the medium is undeformable, ρ = const.

Boundary conditions in the hyperbolic thermoelasticity

If the temperature deviation from its initial value T ∗ is small, the mass density of the internal energy can be represented as follows: U = cv T˜ , T˜ = T − T ∗ , (3) where cv is the specific heat at a constant volume. After introducing (3) into (2) and applying generalized Fourier’s law (1) to eliminate the heat fluxes, we obtain the heat equation of the hyperbolic type, known as the Maxwell–Cattaneo equation:   ˙ (4) λT˜ − ρcv T˙˜ + τ T¨˜ = −ρ (q + τ q) Let us consider a deformable medium taking into account the thermal effects. We assume that the medium is elastic. The energy balance equation for the medium has the following form: ρ U˙ = σ · ·˙e − ∇ · h + ρq

(5)

There are various ways of introducing the temperature and the entropy [23,24]. In the case of an elastic medium they all lead to the same results. Let us use the method proposed by Zhilin [25,26]: ρT H˙ = −∇ · h + ρq,

(6)

where H is the mass density. Let us rewrite the energy balance Eq. (5) considering (6) ρ U˙ = σ · ·˙e + ρT H˙

(7)

Equation (7) is called the reduced energy balance equation. If the elastic medium considered, the internal energy depends only on the deformation and entropy. Then the time derivative of the internal energy is as follows: ρ U˙ =

∂ρU ∂ρU ˙ · ·˙e + H ∂e ∂H

(8)

Subtracting (7) and (8) we obtain     ∂ρU ˙ ∂ρU H · ·˙e + ρT − 0= σ− ∂e ∂H

(9)

A bracketed term in (9) does not depend on e˙ and H˙ . Thus one can obtain the Cauchy-Green relations from (9): ∂ρU ∂ρU , ρT = (10) σ = ∂e ∂H Expanding the function of the internal energy into the Taylor series and keeping only quadratic terms, since in the linear theory e and H are small, we obtain: 1 1 ρU = ρT ∗ H + e · ·4 C · ·e + e · ·C ∗ H + C˜ H 2 2 2

(11)

Substituting (11) in (10), we obtain the expression for stress tensor σ and temperature T˜ : σ = 4 C · ·e + C ∗ H, ρ T˜ = e · ·C ∗ + C˜ H,

(12) (13)

Next, we consider the isotropic material. In this case, all of the material tensors are isotropic and expressions (12), (13) can be are simplified to:   2 σ = K ad − G ε E + 2Ge + C∗ H E, (14) 3 (15) ρ T˜ = C∗ ε + C˜ H,

E. Yu. Vitokhin, M. B. Babenkov

where K ad is the adiabatic bulk modulus, G is the shear modulus, ε = tr e, C∗ = 1/3 tr C ∗ . An expression for the entropy can be obtained from (15): 1 C∗ H = ρ T˜ − ε (16) C˜ C˜ In this case, given that T˜ is a small quantity: ρT H˙ = ρT ∗



1 ˙˜ C∗ ρT − ε˙ C˜ C˜

 (17)

The divergence of (1) takes the form: τ (∇ · h)· + ∇ · h = −λT˜ .

(18)

Introducing (17) into (6) yields

ρ 2 T ∗ ˙˜ ρT ∗ C∗ T− ε˙ = −∇ · h + ρq. C˜ C˜ Calculating the time derivative of (19), multiplying the result by τ and adding it to (19), we arrive at:  ρT ∗ C   ρ 2 T ∗  ˙˜ ∗ T + τ T¨˜ − (˙ε + τ ε¨ ) = − τ (∇ · h)· + ∇ · h + τρ q˙ + ρq C˜ C˜

(19)

(20)

The heat flux (20) can be excluded from (20) taking into account (18). Thus we obtain the heat equation with regard to the elastic properties of the medium: −λT˜ +

 ρT ∗ C ρ 2 T ∗  ˙˜ ∗ T + τ T¨˜ − ˙ . (˙ε + τ ε¨ ) = ρ (q + τ q) ˜ ˜ C C

(21)

Constants C˜ and C∗ can be expressed in the terms of thermodynamic parameters: ρT ∗ α Ki z T ∗ C˜ = , C∗ = − , cv cv

(22)

where K i z is the isothermal bulk modulus, α is the volumetric coefficient of thermal expansion. Considering (22), Eq. (21) takes the following form:   λT˜ − ρcv T˙˜ + τ T¨˜ − α K i z T ∗ (˙ε + τ ε¨ ) = −ρ (q + τ q) ˙ (23) Now we derive the equations describing the dynamics of the thermoelastic medium. The continuum dynamics equation in the local form is as follows: ¨ ∇ · σ + ρf = ρ u,

(24)

where f is the body forces vector, u is the displacement vector. The relation between the stress, strain and temperature, known as the Duhamel–Neumann law, can be obtained by introducing the entropy (16) to (14) and taking into account the relation between the adiabatic and isothermal bulk moduli:   2 σ = K i z − G ε E + 2Ge − α K i z T˜ E. (25) 3 Introducing (25) to equation of motion (24) without taking into account bulk forces, we obtain:   2 ¨ K i z − G ∇ε + 2G∇ · e − α K i z ∇ T˜ = ρ u. 3

(26)

Calculating the divergence of (26), we obtain an equation describing three-dimensional deformation of a thermoelastic medium:   4 K i z + G ε − α K i z T˜ = ρ ε¨ (27) 3

Boundary conditions in the hyperbolic thermoelasticity

Calculating the curl of (24), we obtain an equation for the shear waves. The temperature terms are absent from this equation. However, the temperature change may affect the shear strain, because the shear and bulk strain are coupled by boundary conditions. Hereinafter, we consider such boundary conditions under which shear and bulk strain are independent. In order to obtain the analytical solution it’s convenient to operate the equation of motion in terms of stress. To do that, we calculate the trace of (25) and express the strain tensor in terms of the stress and temperature. By introducing it into (23) and (27) we obtain the equations of coupled hyperbolic thermoelasticity for bulk deformations:   4 K i z + G σ − α K i z ρ T¨˜ − ρ σ¨ = 0 3 (28)    ∗ α 2 K i2z T ∗ α K T i z ˙ ¨ λT˜ − T˜ + τ T˜ − + ρcv ˙ (σ˙ + τ σ¨ ) = −ρ (q + τ q), K i z + 43 G K i z + 43 G where σ = 1/3 trσ . If the heat flux relaxation time τ in (28) tends to zero, then the solution of this system will tend to the solution of the classical problem of the thermoelasticity of the parabolic type. 3 Thermoelastic waves in a layer with internal heat sources 3.1 Problem statement Let us consider an infinite layer, which is exposed to the laser radiation. If the layer is irradiated uniformly over the entire surface, waves will propagate only along the x axis. The interaction of a laser and a medium is modeled by the BeerLambert law (Fig. 1): J (x) = J0 e−γ x , where J0 is the laser intensity, γ is the attenuation coefficient of the medium. Eventually, the value of J0 depends not only on the properties of the laser, but also on the properties of a surface. A part of the incident radiation reflects from the surface, with this in mind, the value of J0 is considered to have the sense of the laser radiation intensity absorbed by the material. The dependence of the laser intensity on time is quite difficult and, therefore, J0 is assumed to be an average characteristic. Generally, there are two characteristics of the short laser impulse found in the literature: exposure duration δ0 and energy E 0 absorbed per unit area. Thus, the laser intensity is considered to be a function of the form

J0 e−γ x , 0 ≤ t ≤ δ0 ; J˜(x, t) = 0, t > δ0 . And



∞

E(x) ≡

J˜(x, t)dt = J0 δ0 e−γ x

⇒

E 0 ≡ E(0) = J0 δ0 .

0

The Dirac delta function δ(t −0) is usually adopted to model the time dependence of the laser intensity, because such an approximation allows to obtain an exact analytical solution. The scale factor is selected so that full

Fig. 1 The hyperbolic thermoelasticity problem for an infinite layer under the influence of a laser impulse

E. Yu. Vitokhin, M. B. Babenkov

energy E 0 absorbed per unit area during the time of exposure, would be consistent with the experimental data. It is readily seen that function J (x, t) satisfying the above conditions has the form: Jˆ(x, t) = E 0 e−γ x δ(t − 0). Subsequently,

 E(x) ≡

∞

(29)

Jˆ(x, t)dt = E 0 e−γ x .

0

Hereinafter, formula (29) will be used for the laser intensity. The relation between heat supply rate per unit volume ρq(x, t) and the laser intensity is determined by the energy balance equation: Jˆ(x, t) − ρq(x, t)d x − Jˆ(x + d x, t) = 0. Expanding function Jˆ(x + d x, t) in the Taylor series and taking into account (29) we obtain E 0 e−γ x δ(t − 0) − ρq(x, t)d x − E 0 e−γ x δ(t − 0)(1 − γ d x) = 0. From the last equation it follows that q(x, t) =

E 0 γ −γ x δ(t − 0), e ρ

E 0 = J0 δ0 .

(30)

Internal heat sources density expression (30) is widely used in the literature, e.g., [27]. At the initial moment of time the layer is unperturbed, the thermal and mechanical loads are absent: u˙ t=0 = 0; u t=0 = 0; (31) T˙˜ t=0 = 0; T˜ t=0 = 0; (32) If the problem is formulated in terms of the stresses, then the initial conditions must be formulated for the stress instead of the strain: σ˙ t=0 = 0; σ t=0 = 0; (33) 3.2 A numerical method for solving the coupled problem If waves travel only along the x-axis, Eq. (28) has the form:   2 4 ∂ 2 T˜ ∂ 2σ ∂ σ Ki z + G − α Ki z ρ 2 − ρ 2 = 0 2 3 ∂x ∂t ∂t   α 2 K i2z T ∗ ρcv ∂ 2 T˜ ∂ T˜ ∂ 2 T˜ −  +τ 2  + ∂x2 λ ∂t ∂t K i z + 43 G λ α Ki z T ∗

−   λ K i z + 43 G



∂ 2σ ∂σ +τ 2 ∂t ∂t



(34)

  ∂q ρ q +τ =− λ ∂t

Poor convergence is caused by two factors: the second time derivatives are multiplied by a small parameter (34), and the source term includes the Dirac delta function. The heat equation in (34) has a small parameter τ , and on the right side of the equation there is the source term with the Dirac delta function, which degrades the convergence speed of a solution obtained in a series form. In order to find the numerical solution of system (34) let us represent it in a dimensionless form: ∂ 2η ∂ 2ζ ∂ 2η − − =0 ∂s 2 ∂θ 2 ∂θ 2       ∂ζ ∂q ∗ ∂ 2ζ ∂ 2η ∂η ∂ 2ζ ∗ − + d + d + d − c = − q ∂s 2 ∂θ ∂θ 2 ∂θ ∂θ 2 ∂θ

(35)

Boundary conditions in the hyperbolic thermoelasticity

Fig. 2 Modeling of the Dirac delta function

Non-dimensional variables are introduced as follows x = k1 s, t = k2 θ, σ = k3 η, T˜ = k4 ζ. The coefficients are given as:

  λ ρ K i z + 43 G λρ k1 =   , k2 =  , 4 2 2 2 ∗ 2 ∗ α K i z T + cv ρ K i z + 3 G α K i z T + cv ρ K i z + 43 G k3 = K i z , k4 = d=

1 , α

α 2 K i2z T ∗ ρk12 E 0 γ τ ∗ ∗ , q δ(θ − 0)e−γ s , , c= =   4 2 2 ∗ k2 λk4 k2 ρ α K i z T + cv ρ K i z + 3 G

γ ∗ = k1 γ . We adopt the finite differences method. The derivatives in Eq. (35) are approximated by the means of the explicit integration scheme f =

f i−1 − f i+1 , 2w

f  =

f i−1 − 2 f i + f i+1 , w2

(36)

where w is the integration step, f is the required function. The stability of scheme (36) is determined by the well-known Courant necessary criterion, which defines the ratio between the space and time integration steps. In the one-dimensional case for the problems of the hyperbolic type, the Courant criterion has the form: t

 x , Ca = (K i z + 4/3G)/ρ, Ca

(37)

where x and t are the space and time integration steps, Ca is the sound velocity. The Dirac delta function and its derivative are modeled on the first two time steps by the rectangular pulses with amplitudes 1/(2t) and ±1/(t)2 respectively (see Fig. 2). The properties of copper are taken as the thermomechanical parameters of the medium: Young’s modulus E = 127 GPa, Poisson’s ratio ν = 0.33, density ρ = 8960 kg/m3 , coefficient of thermal conductivity λ = 401 Wt/(m K), volumetric coefficient of thermal expansion α = 49.5 · 10−6 1/K, heat capacity at a constant pressure c p = 390 J/K, radiation absorption constant γ = 11.7 µm−1 , specimen width l = 1 µm. The heat flux relaxation constant ranges from 10−8 to 10−12 seconds in metals according to various works [8,10,27]. We assume the mean value of the interval as a heat flux relaxation constant for copper: τ = 0.1ns. Radiation energy E 0 is assumed to be equal to 0.15 J/m2 . In this case, the temperature deviation varies in the range from 0 to 10 ◦ C on the entire time interval. This allows to consider the propagation of thermoelastic waves within the linear theory. Since all the following results are obtained in the framework of the linear theory, radiation energy E 0 alteration will not lead to qualitative changes of the solution. To obtain a solution

E. Yu. Vitokhin, M. B. Babenkov

for the different values of the radiation energy, one needs to multiply the result of numerical calculations by a factor that shows how many times the given radiation energy value differs from the required one. The authors of experimental work [8] exposed gold films to the laser pulse with radiation energy E 0 = 0, 4 J/m2 , which roughly corresponds to the exposure level we use. However, in [27] the impact energy is about 10 J/m2 . Apparently, this is due to the fact that the authors were able to measure the temperature only at the back surface of the specimen, which is opposite to the irradiated side. If the radiation energy is of the order of 10 J/m2 , the temperature deviation on the back surface will range within of 10 ◦ C. As regards the temperature inside the layer, its value depends on the theory being used. The maximum temperatures calculated by the means of the classical theory are significantly less than the maximum temperature values calculated by the hyperbolic theory. The radiation energy value was chosen two orders of magnitude less than in work [27] so, that the temperature deviations evaluated within the hyperbolic theory were within the range of 10 ◦ C at all moments of time and at all sections of the layer. 3.3 Change of variables in the problem of the hyperbolic thermoelasticity The equation of motion in the system of coupled thermoelasticity (28) can be written as follows: ∂ 2σ 1 ∂ 2σ α Ki z ∂ 2 T˜ − 2 = , 4 2 2 ∂x Ca ∂t K i z + 3 G ∂t 2

(38)

where Ca = (K i z + 43 G)/ρ is the propagation velocity of the acoustic waves in the medium. It is convenient to use the notation:  ρcv cv ρ ς=x ; ϑ = T˜ ; (39) λ E0 γ After the variable change (39) Eq. (38) takes the form: E0 γ α Ki z ρ ρcv  ¨ σ¨ = σ − ϑ, 4 λ Ki z + 3 G ρcv (K i z + 43 G)

(40)

where primes denote the derivatives with respect to the coordinate ς , points denote the time derivatives. Let us divide Eq. (40) by ρcv /λ and introduce a new notation for the stress: σ = σ∗

λE 0 γ α K i z 2 ρ cv 2 (K i z + 43 G)

(41)

After these substitutions Eq. (40) takes the form: ¨ σ∗ −  σ¨ ∗ = ϑ, where =

λ cv (K i z + 43 G)

(42)

(43)

√ If the heat waves velocity C h = λ/(ρcv τ ) is greater than the acoustic waves velocity Ca , it is equivalent to the statement  > τ . Conversely, if the velocity of the acoustic waves is greater than the velocity of heat waves, than  < τ . Let us find the relationship between displacements u and stress σ . The scalar equation of motion in the absence of the external forces, adopting notation (39), (41) can be written as:    ρλ K i z + 43 G ρcλv 3/2  σ∗ = u¨ (44) Ki z α E0 γ After introducing a new variable for the displacements u = u∗

Ki z α E0 γ   ρλ K i z + 43 G ρcλv 3/2 

(45)

Boundary conditions in the hyperbolic thermoelasticity

~

γ > γ0

γ < γ0

~

T

T

0

x 0

x

Fig. 3 The analytical solution of the heat conduction problem. The temperature at the boundaries is kept constant. Solid lines show the solution of the hyperbolic heat conduction problem, dashed lines show the solution of the classical heat conduction problem

and with regard to (45) Eq. (44) takes the form: σ∗ = u¨ ∗

(46)

Under the homogeneous initial conditions displacements u ∗ can calculated by the formula: t t u∗ = 0

σ∗ (ς, t)dtdt

(47)

0

3.4 Unconstrained boundaries of the layer are kept at a constant temperature The boundary conditions have the form: σ x=0 = 0; σ x=l = 0; T˜ x=0 = 0; T˜ x=l = 0;

(48)

Let us find an analytic solution for the semi-coupled thermoelasticity problem with the initial conditions (31)–(33), boundary conditions (48), and with heat source (30). The solution of the corresponding heat conduction problem (4) is given by [28]:   ∞ n (eμγ − (−1)n ) sin nπ ς  μ   ϑ(ς, t) = 2πe Sn n 2 π 2 + μ2 γ 2 n=1      1 1 H (t), × Sn cos t Sn + μ sin t Sn 2 2 −μγ − 2τt

(49)

 √ where Sn = 4n 2 π 2 τ − μ2 , μ = l ρcv /λ, H (t) is the Heaviside unit step function. Solution (49) is shown in Fig. 3, the curves are plotted for the moments of time √ τ , 3τ and 5τ . The left graphs are plotted for γ > γ0 , the right graphs are plotted for γ < γ0 , where γ0 = ρcv /(4λτ ). Jumps and discontinuities arise in the presented solutions because of the Dirac delta function is adopted to approximate a time dependence of the laser intensity while impacting the specimen and, as well, the form of the non-Fourier source term containing the first time derivative of the Dirac delta function. A cooling region can be observed under the boundary conditions of the first kind) if γ > γ0 [28] due to the amplitude of the wave reflected from the irradiated border with the opposite sign is sufficient to alter the sign of the original wave in the vicinity of the irradiated surface. Taking into account temperature distribution (49) and boundary conditions (48) we obtain the displacements in the layer: ∞  u ∗ (ς, t) = un , (50) n=1

E. Yu. Vitokhin, M. B. Babenkov

γ > γ0

u

u

γ < γ0

0 0

x

x

Fig. 4 The analytical solution of the semi-coupled thermoelasticity problem. The temperature at the unconstrained boundaries is kept constant. The solid lines show the solution of the hyperbolic thermoelasticity problem, the dashed lines show the solution of the classical thermoelasticity problem

where u n is given by: un = 

   t  2e−lγ − 2τ −(−1)n + elγ l cos nπl ς   n 2 π 2 + l 2 γ 2 l 2  + n 2 π 2 ( − τ )2 Q n       t  t Qn nπt Qn × (−l 2  − n 2 π 2 τ (−3 + τ )) sin + e 2τ l 2  + n 2 π 2 τ (− + τ ) cos √ 2τ l       2  t t Qn nπt 2 2 3/2 2τ Q n − e lnπ sin √ − l  + n π τ (− + τ ) cos (51) Q n H (t), 2τ l 

2 2 where Q n = 4n lπ2 τ − 1. In the classical limit of solution (51) and hereinafter, if τ comes closer to zero, the overall value of the expression still remains finite. The limiting values should be carefully evaluated taking into account the fact that each term with the relaxation time in its denominator, e.g., et/(2τ ) and sin[t Q n /(2τ )], is multiplied by an exponential function e−t/(2τ ) vanishing when τ tends to zero. Graphs in Fig. 4 show solution (50) for the layer exposed to the short laser pulse (30). The graphs are obtained for  = τ , which means that the velocities of thermal and acoustic waves are equal. The solution is plotted at the following times: τ , 3τ and 5τ . The left graphs are plotted for γ > γ0 , the right graphs are plotted for γ < γ0 . With the growth of the absorption constant γ , the difference between the displacements in classical and hyperbolic thermoelasticity problems increases. The graph in Fig. 4 illustrates the hyperbolic thermoelasticity problem solution. It has two salient points, while the graph plotted using the classical thermoelasticity is smooth. Under these boundary conditions, the most noticeable difference between the hyperbolic and classical uncoupled thermoelasticities is achieved. The velocities of the quasiacoustic and quasithermal fronts in a coupled problem are not equal to the sound and heat velocities in an uncoupled problem due to the influence of the coupling effects on the dispersion law [22]. If such corrections are significant for a problem under consideration, it is recommended to solve it in the most general coupled formulation. The smaller is the time scale, and the higher is the acoustic quality factor of the material (or the lower is the thermal expansion coefficient), the less is the difference between the solutions obtained for the coupled and semi-coupled formulation. The velocities of the quasiacoustic and quasithermal fronts in an uncoupled problem can be equal, while in the coupled formulation the distance between the fronts is constantly increasing. The velocities values of the quasiacoustic and quasithermal fronts in a coupled thermoelasticity problem [13,22] are as follows: √  √  C1 = 2/ A1 + A2 − a ∗ , C2 = 2/ A1 + A2 + a ∗ , (52)  (A1 − A2 )2 + 4 A1 A2 A4 ,  ρ 1 α2 Ki z T ∗ , ρcv + A1 = , A3 = λ K i z + 43 G 1 + 43 G K i−1 z 1 A2 = τ A3 , A4 = . 4 2 ∗ 1 + ρcv (1 + 3 G K i−1 z )/(α K i z T ) a∗ =

Boundary conditions in the hyperbolic thermoelasticity

Condition C1 > C2 is always true. If τ > τ0 , where τ0 is given by [22]:     λρ ρcv K i z + 43 G − α 2 K i2z T ∗ τ0 =     , ρcv K i z + 43 G + α 2 K i2z T ∗ 2

(53)

propagation velocity C2 of a quasithermal component of the thermoelastic waves will be less than propagation velocity of a quasiacoustic component C1 . On the contrary, if τ < τ0 , than the propagation velocity of a quasithermal component will be given by C1 , wich is greater than C2 , as well. With the growth of the heat flux relaxation constant, the velocities of the both fronts C1 and C2 decrease [29]. In the classical limit, if the relaxation time approaches zero, it immediately follows that τ < τ0 , speed of the quasithermal front C1 tends to infinity, while speed of the quasiacoustic front C2 tends to the conventional speed of longitudinal waves in √ a media 1/ A1 . More details can be found in [22]. We find a numerical solution of the coupled hyperbolic thermoelasticity problem for a copper layer with the unconstrained boundaries kept at a constant temperature with the use of the numerical method and parameters set forth in Sect. 3.2. The radiation absorbtion constant of a medium is taken to be γ > γ0 = 4.61 µm−1 . Since the thermal expansion coefficient is a small quantity if γ > γ0 , we expect the behavior of the temperature front in the coupled problem of the hyperbolic thermoelasticity (Fig. 5, left) to be similar to the solution of the heat equation shown in Fig. 3, left. In this paper, we qualitatively compare analytical solutions of semi-coupled problems with numerical solutions of fully coupled problems. The analytical solution of semi-coupled problem (49)–(50) is in qualitative agreement with the analytical solution of the coupled problem given in [29]. It follows that the comparison of numerical and analytical solutions of the same problem solved both in the semi-coupled and coupled statements can be of a particular interest, especially in the case if the analytical solution of a coupled problem can not be obtained easily. The results of such a comparison will help us to find out if all the types of the boundary conditions considered in this work allow to obtain precize enough results in the semi-coupled formulation. Beforehand, we carry out a comparison of the analytical solution given in [29] with the numerical solution of the same problem to estimate a numerical error. A comparison of the temperature and displacement curves of the analytical and numerical solutions at t = 0.1 ns and τ = 0.1 ns is shown in Fig. 5. It can be easily seen from Fig. 5 that the numerical solution qualitatively agrees with the analytical. To determine the degree of quantitative agreement, we find the numerical error. The following formula gives the error value P ( f num , f an ) =

|| f an − f num || · 100%, || f an ||

(54)

where f an is a value obtained analytically, f num is a value obtained numerically. We define norm ||·|| of function f at a moment of time t as the Euclidean norm of a vector. The components of the vector are the function values at mesh nods:  || f || = (55) f i2 (x ∗ , t ∗ ). i

Figure 6 shows the error of the numerical solution at the various space steps h. The error decreases, when refining the finite difference mesh. The largest error is observed in the vicinity of the temperature jump, which is caused by the derivative discontinuity. p 0.6 0.2 -0.2 -0.6

Fig. 5 Comparison of the analytical and numerical solutions of the coupled thermoelasticity problem. A solid line represents the analytical solution

E. Yu. Vitokhin, M. B. Babenkov P,% 20 15

h =1 nm

10

h =0.1 nm

5 0

h =0.01 nm 0.2

0.6

1.0

x, μm

Fig. 6 The error of the numerical solution

3.5 The boundaries of the layer are constrained and kept at a constant temperature The boundary conditions have the form: u x=0 = 0; u x=l = 0; T˜ x=0 = 0; T˜ x=l = 0;

(56)

To find the stress distribution in the semi-coupled problem of thermoelasticity, expression for the temperature (49) is substituted in Eq. (42). This equation is homogeneous, and its solution can be conveniently obtained by the Grinberg method [30]. Then, using (47), we find the displacements in the layer:  n−1 ∞ 1  ∞ ∞     (57) u n,m + u n,m + u n,m u ∗ (ς, t) = n=1 m=2

n=2

m=1

m=n+1

Here, the value of u n,m is given by:     ς 4 (−1)m+n − 1 e−μγ ((−1)n − eμγ ) μn 2 sin mπ μ     u n,m =  2 2 m − n 2 π n 2 π 2 + μ2 γ 2 μ2 m 2  + π 2 Sn,m    t  mπt  2 e− 2τ 2 × m cos μ  − π τ Sn,m + 2 √ μ − 4n 2 π 2 τ μ       t mπt × e 2τ μn 2 π 3/2 −μ2 + 4n 2 π 2 τ sin √ μ       t Qn 2 2 2 2 2 +μ m μ  + π τ −3n  + m τ sin Qn 2τ     2   t Qn  2 2 2 2 −m μ − 4n π τ cos μ  − π τ Sn,m H (t), 2τ

(58)

where Sn,m = n 2  − m 2 τ . The solution for the classical thermoelasticity is obtained similarly. Figure 7 presents solution (57) at the following times: τ , 3τ and 5τ . The graphs are obtained for  = τ , which means that the velocities of thermal and acoustic waves are equal. Graphs on the left are plotted for γ > γ0 , graphs on the right are plotted for γ < γ0 . Figure 8 shows the numerical solution of the coupled thermoelasticity. The copper layer is exposed to a short laser pulse (30). At the initial moment time, the layer was in the unperturbed state. Graphs on the left are built at t = τ = 0.1 ns, graphics right at t = 2τ . In the hyperbolic thermoelasticity at the front of the quasithermal wave, the minimum displacements values are observed, the maximum displacements are observed at the quasiacoustic front. In the classical thermoelasticity, the displacements values on quasiacoustic and quasithermal fronts are lower than in the hyperbolic thermoelasticity. The maximum difference between the displacements in the classical and hyperbolic thermoelasticity is observed at the quasiacoustic front propagating with the speed given by (52). Graphs in Fig. 7 show that the negative displacement minimums are observed longer in the hyperbolic than in the classical thermoelasticity. Comparing the temperature distributions in the layer, obtained both

Boundary conditions in the hyperbolic thermoelasticity

γ < γ0

γ > γ0 u

u

0

x

0

x

Fig. 7 The analytical solution of the semi-coupled thermoelasticity problem with the constrained borders kept at a constant temperature. Solid lines represent curves of the hyperbolic thermoelasticity, dotted lines represent classical curves p

p

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

Fig. 8 The numerical solution of the coupled thermoelasticity problem with the constrained borders kept at a constant temperature. The solid lines represent curves of the hyperbolic thermoelasticity, the dotted lines represent classical curves Table 1 The numerical solution of the coupled thermoelasticity problem with the constrained borders kept at a constant temperature t (ns) 0.001 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200

Vh (m/s) 1500 1141 1105 1096 1108 1093 1093 1085 1080

Va (m/s) 4614 4670 4668 4643 4584 4637 4636 4635 4611

Umax (pm) 0.000878 0.401 0.704 0.844 0.901 0.920 0.927 0.926 0.886

Umin (pm) −0.0147 −0.0804 −0.0388 −0.0285 −0.0320 −0.0371 −0.0304 −0.0241 −0.0106

Tmax (◦ C) 0.514 0.389 0.317 0.271 0.237 0.211 0.190 0.170 0.153

Tmin (◦ C) −0.109 −0.189 −0.193 −0.179 −0.159 −0.138 −0.119 −0.102 −0.086

t is a moment of time; Vh is the quasithermal front speed; Va is the quasiacoustic front speed; Umax , Umin are the maximum and minimum displacements, respectively; Tmax , Tmin are the maximum and minimum temperatures

numerically and analytically (Figs. 3, 8), we can conclude that the local cooling effect observed near the irradiated boundary of the layer is not a numerical error and is not a summation error of the analytical solution. The presence of the cooling zone is explained by the fact that at the initial moments of time when the material is struck by the laser pulse and the thermoelastic wave is excited in the layer, it begins to propagate both to the left and to the right directions starting from its origin. Since the heat source is close to the left boundary of the layer, the wave propagating to the left reflects with the opposite sign due to the given boundary conditions. The areas of the solution, where the numerical methods give strong oscillations (Fig. 8) correspond to a smooth analytical curve (Fig. 3). This indicates that the oscillations are an artifact of the numerical calculation. Moreover, such oscillations are observed only if the boundaries of the layer are kept at a constant temperature. The displacement graph is a smooth function, because the displacements are obtained by integrating the strains. Table 1 presents the results of the numerical calculations. Under the considered parameters, the displacements are evaluated to be of the picometer order of magnitude. These values are beyond the sensitivity of the existing experimental equipment. Therefore, the displacements themselves bear no practical interest. However,

E. Yu. Vitokhin, M. B. Babenkov

from a theoretical point of view, its analysis allows the better understanding of the energy transfer processes between the thermal and mechanical degrees of freedom. During the period of the first 0.15 ns the maximum displacement value corresponding to the quasiacoustic front increases, and it starts to decrease only at the last 0.05 ns. Thus, the effect of thermal-elastic damping is observed starting at 0.15 ns after the exposure. Increasing of the displacements amplitude in the first 0.15 ns can be attributed to the transition of the heat energy into the mechanical energy due to the coupling effect. The minimum displacements value is observed at the quasiathermal front and has a negative sign, decreasing by three times for a time period of 0.1 ns. During this time, the maximum and minimum temperatures lower by 38%. 3.6 The boundaries of the layer are constrained and insulated The boundary conditions have the form: u x=0 = 0; u x=l = 0; h x=0 = 0; h x=l = 0;

(59)

A solution of the hyperbolic heat conduction problem of the Maxwell–Cattaneo type with the homogeneous initial conditions and the insulation at the boundaries under the laser impact of the form (30) is given by [28]:   nπ ς ∞ cos (eμγ − (−1)n )  μ −μγ −μγ − 21 t/τ   H (t) − 2μγ e ϑ(ς, t) = e Sn n 2 π 2 + μ2 γ 2 n=1       t Sn t Sn t − μ sin H (t) (60) × Sn e 2τ − cos 2μτ 2μτ Solution (60) at the moments of time τ , 2τ and 5τ is presented in Fig. 9. The graphs on the left are plotted for γ > γ0 , the graphs on the right are plotted for γ < γ0 . If γ > γ0 , the temperature maximum is located at the wave front, coinciding with the salient point (the observation time is of the order of τ ). If γ < γ0 than the temperature maximum is located at the irradiated boundary (x = 0). Displacements in a thermally insulated layer with the constrained boundaries and the temperature distribution given by (60) can be evaluated as: ∞  u ∗ (ς, t) = un , (61) n=1

where u n has the form:    t  2e−lγ − 2τ −(−1)n + elγ l 2 γ sin nπl ς    un = nπ n 2 π 2 + l 2 γ 2 l 2  + n 2 π 2 ( − τ )2    2  t Qn 2 2 × l  + n π τ (− + τ ) cos 2τ

~

γ > γ0

~

0

γ < γ0

T

T

x

0

x

Fig. 9 The analytical solution of the thermal conductivity problem with the thermally insulated boundaries. Solid lines represent curves for the hyperbolic thermal conductivity, the dotted lines represent curves for the Fourier thermal conductivity

Boundary conditions in the hyperbolic thermoelasticity

γ > γ0

u

0

u

x

γ < γ0

0

x

Fig. 10 The analytical solution of the semi-coupled thermoelasticity problem with the thermally insulated and constrained boundaries. Solid lines represent curves of the hyperbolic thermal conductivity, dotted lines represent curves for the Fourier thermal conductivity

Table 2 The numerical solution of the semi-coupled thermoelasticity problem with the thermally insulated and constrained boundaries t (ns) 0.001 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200

Vh (m/s) 909 1056 1056 1069 1070 1072 1076 1074 1070

Va (m/s) 3714 4640 4620 4613 4586 4617 4607 4619 4595

Umax (pm) 0.0023 0.448 0.925 1.181 1.324 1.397 1.437 1.458 1.467

Tmax (◦ C) 0.519 0.397 0.324 0.277 0.243 0.217 0.195 0.175 0.157

t is a moment of time; Vh is the quasithermal front speed; Va is the quasiacoustic front speed; Umax is the maximum displacement; Tmax is the maximum temperature

     nπt nπt −l 2  − n 2 π 2 τ (− + τ ) cos √ + lnπ 3/2 sin √ l  l      Qn Qn  l 2 l 2  + n 2 π 2 τ (−3 + τ ) sin t 2τ H (t) − l 2 − 4n 2 π 2 τ t

+e 2τ

 

(62)

Figure 10 illustrates solution (61) at the following times: τ , 2τ and 5τ . The graphs are obtained for  = τ , which means that the velocities of thermal and acoustic waves are equal. The graphs on the left are plotted for γ > γ0 , the graphs on the right are plotted for γ < γ0 . Analysis of the graphs in Fig. 10 allows to conclude that the solution obtained by the hyperbolic heat conduction equation contains a salient point, while the solution obtained via the classical heat conduction equation is smooth. If the attenuation coefficient of the medium increases, the difference between the peak temperature values in both cases also increases. Starting from a certain moment of time if γ < γ0 , one can observe a displacements extremum located to the left of the wavefront. In the semi-coupled thermoelasticity problem if γ > γ0 , the displacements extremum is always located on the wave front. The numerical solution of the coupled thermoelasticity problem for the copper layer is shown in Fig. 11. The graphs on the left are built at t = τ = 0.1 ns, the graphs on the right are built at t = 2τ . The most noticeable difference between the displacements of the classical and hyperbolic thermoelasticities is achieved at the quasithermal front, moving with the speed given by formula (52). From the numerical data (Table 2), it follows that the displacements maximum increases by 15% within 1 ns, and the temperature maximum is reduces by 38% at the same time. The graphs in Figs. 9 and 11 show that over the time the temperature of the irradiated border in the hyperbolic problem of thermoelasticity can be either larger or smaller than in the classical one. However, the temperature difference is small in these cases. Also, the displacement obtained via the hyperbolic and classical thermoelasticity do not have qualitative differences and the observed quantitative differences are neglectable.

E. Yu. Vitokhin, M. B. Babenkov

p

1.4

1

1

0.6

0.6

0.2

0.2

p

Fig. 11 The numerical solution of the semi-coupled thermoelasticity problem with the thermally insulated and constrained boundaries. Solid lines represent curves of the hyperbolic thermal conductivity, dotted lines represent curves for the Fourier thermal conductivity

3.7 The boundaries of the layer are thermally insulated and unconstrained The boundary conditions have the form: σ x=0 = 0; σ x=l = 0; h x=0 = 0; h x=l = 0;

(63)

Knowing the temperature distribution for an insulated layer (60), we obtain the displacements for a layer with the unconstrained boundaries:  n−1 ∞ 1  ∞ ∞ ∞      u ∗ (ς, t) = u n,m + u n,m + u n,m + um (64) n=1 m=2

n=2

m=1

m=n+1

m=1

where u n,m and u m are given by:      t  4 −1 + (−1)m+n e−lγ − 2τ −(−1)n + elγ l 2 γ cos mπl ς u n,m = −      2 2  2 m 2 − n 2 π 2 n 2 π 2 + l 2 γ 2 −1 + 4n lπ2 τ l 2 m 2  + π 2 Sn,m          t Qn t Qn + Q n m 2 cos × −m 2 l 2  + π 2 τ −3n 2  + m 2 τ sin 2τ 2τ        √ t mπt × −l 2  + π 2 τ Sn,m + e 2τ lm lm − n 2 π  sin √ l      mπt + Sn,m +π 2 Sn,m −n 2  cos √ H (t) l         √t 2e−lγ −(−1)m + elγ l cos mπl ς −1 + cos mπ l  um = H (t) 2 2 2 2 m π +l γ Figure 12 illustrates solution (64) at the following times: τ , 2τ and 5τ . The graphs are obtained for  = τ , which means that the velocities of thermal and acoustic waves are equal. The graphs on the left are plotted for γ > γ0 , the graphs on the right are plotted for γ < γ0 . Figure 12 shows that the difference between the displacements calculated via the classical and hyperbolic theories increases with attenuation coefficient γ . The temperature extremums in Fig. 9 do not qualitatively affect the displacements shown in Fig. 12. Numerical solution for the copper layer is shown in Fig. 13. The graphs on the left are plotted for t = τ = 0.1 ns, and the graphs on the right are plotted for t = 2τ ns. As in the previous case, the most noticeable difference between the displacements calculated via classical and hyperbolic theories is achieved at the quasithermal front, moving with the speed given by (52). As seen from a comparison of Figs. 11 and 13, the mechanical

Boundary conditions in the hyperbolic thermoelasticity

γ > γ0

γ < γ0 u

u

0

0

x

x

Fig. 12 The analytical solution of the semi-coupled thermoelasticity problem with the thermally insulated and unconstrained boundaries. Solid lines represent curves of the hyperbolic thermal conductivity, dotted lines represent curves for the Fourier thermal conductivity

Fig. 13 The numerical solution of the coupled thermoelasticity problem with the thermally insulated and unconstrained boundaries. Solid lines represent curves for the hyperbolic thermal conductivity, dotted lines represent curves for the Fourier thermal conductivity Table 3 The numerical solution of the coupled thermoelasticity problem with the thermally insulated and unconstrained boundaries t (ns) 0.001 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200

Vh (m/s) 918 1096 1076 1075 1071 1080 1075 1074 1071

Va (m/s) 4613 4668 4620 4640 4604 4640 4633 4637 4610

Umax (pm) 0.00235 0.410 0.641 0.712 0.748 0.747 0.749 0.759 0.922

Umin (pm) −0.078 −1.03 −1.27 −1.31 −1.32 −1.31 −1.31 −1.31 −1.30

Tmax (◦ C) 0.512 0.396 0.325 0.278 0.244 0.218 0.196 0.176 0.158

Tmin (◦ C) 0 0 0 0 −0.003 −0.006 −0.007 −0.007 −0.008

t is a moment of time; Vh is the quasithermal front speed; Va is the quasiacoustic front speed; Umax , Umin are the maximum and minimum displacements, respectively; Tmax , Tmin are the maximum and minimum temperatures

boundary conditions have little effect on the temperature distribution in the layer, due to the smallness of the thermal expansion coefficient. As can be seen in Fig. 12, the difference between the classical and hyperbolic solutions under the chosen boundary conditions is less noticeable, than in the previously considered cases. The numerical solution of the coupled problem and the analytical solution of the semi-coupled problem demonstrate negative displacements extremum, which disappears over a short period of time. This means that at the initial time period the points of the layer are moving toward the radiation source. The same effect is observed in the solution of the classical problem. The displacements peak values in the semi-coupled hyperbolic thermoelasticity (Fig. 12, solid lines) are greater than in the classical thermoelasticity problem (Fig. 12, dashed lines). On the contrary, the displacements

E. Yu. Vitokhin, M. B. Babenkov

a

b

c

d

Fig. 14 Numerically evaluated displacements in the layer subjected to the laser pulse under the various boundary conditions at t = τ = 0, 1 ns. Unconstrained boundaries of the layer are kept at a constant temperature (a); the boundaries of the layer are constrained and kept at a constant temperature (b); the boundaries of the layer are constrained and insulated (c); the boundaries of the layer are thermally insulated and unconstrained (d)

peak values in the coupled hyperbolic thermoelasticity (Fig. 13) are less than in classical one. Such effect is observed because the solution presented in Fig. 12 is obtained if quasithermal and quasielastic wave components propagate with an equal speed. The resulting thermoelastic wave is formed of these components by adding up its amplitudes. However, in the coupled statement the component’s velocities can not be equal. In this case, the numerical solution reflects the actual behavior of the material better. Numerical results (Table 3) show that the minimum value of the displacements is stabilized after 0.1 ns and reaches a value of −1.3 pm. At the same time, the maximum and minimum temperature decreases. 4 Analysis of the stress distribution Despite the fact that the equations system of thermoelasticity is formulated in terms of stresses, only displacements were studied in details. However, without the stress study the survey can not be considered complete. Setting aside the boundary behavior, hereafter, we consider a region of the layer sufficiently distant from its ends. If the boundaries are unconstrained, a jump is observed at the acoustic front (Fig. 14a, d). The compression zone is observed before the front, and the extension zone is observed after the front. If the boundaries are constrained, one can observe a salient point at the acoustic front, and a compression zone in its neighborhood (Fig. 14b, c). If the constrained boundaries are kept at a constant temperature, it result into an extra maximum of the tensile stresses observed in the vicinity of the acoustic front. A jump at the thermal front is observed if the boundary temperature is kept constant. The stress values at the heat front are substantially smaller than at the acoustic one. Further conclusions are made under an assumption that the both fronts are sufficiently distant from each other. If the boundaries are thermally insulated, the salient point is observed at the thermal front (Fig. 14c, d). Stress values at the thermal front change its sign only in the problem with the unconstrained boundaries kept at a constant temperature (Fig. 14a). The tensile stresses are observed in the neighborhood of the thermal front in all other problems discussed above. 5 Conclusion The hyperbolic effects have introduced the qualitative and quantitative changes in the state of stress and strain of the layer in all of the considered problems. The most significant differences between the classical and hyperbolic thermoelasticity theories arise in the problem with the constrained boundaries and a constant temperature maintained on them. On the other hand, the problem with unconstrained and thermally insulation boundaries shows the least differences. The temperature at the irradiated thermally insulated border evaluated via the hyperbolic thermoelasticity theory can be either larger or smaller than in the classical thermoelasticity. The most prominent difference between the solutions of the classical and hyperbolic equations of motion is observed in the interval between the acoustic and thermal front. The less the distance between the fronts, the

Boundary conditions in the hyperbolic thermoelasticity

more is the difference between solutions of the classical and hyperbolic thermoelasticity problem. As it was shown by numerical analyses, the grows of the fronts’ velocities is observed for the first 0,025 ns in all of the considered problems if the relaxation time is taken to be 0.1 ns. If the relaxation time differs from this value, fronts’ growing time may also take different values. Since the end of this period, the velocity values deviate from its mean value by no more than 5%, which is within the error margin due to the calculation method. The comparison of the analytical solutions of the semi-coupled problem and the numerical solutions of the coupled problems shows the good agreement. This means that the solution of the semi-coupled problem can be used to study the waves propagation processes in the hyperbolic thermoelasticity problems. Acknowledgements The authors are grateful to E. A. Ivanova, A. K. Belyaev, V. A. Polyansky and V. M. Pestrenin for valuable discussions.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

Lord, H.W., Shulman, Y.A.: Generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids 15(5), 299–309 (1967) Dahotre, N.B.: Laser surface engineering. Adv. Mater. Process. 160(7), 35–39 (2002) Miyazaki, K.: Generation and control of hight intensity ultrashort laser pulses. Laser Res. 27(5), 319–323 (1999) Yabe, A.: Advanced material processing using hight-intensity laser pulse. Laser Res. 27(5), 336–340 (1999) Pop, E., Sinha, S., Goodson, K.E.: Heat generation and transport in nanometer scale transistors. In: Proceedings of the IEEE, 2006, vol. 94, pp. 1587–1601 Tong, X.C.: Development and Application of Advanced Thermal Management Materials. Springer, New York (2011) Haque, M.A., Saif, M.T.A.: Thermo-mechanical properties of nano-scale freestanding aluminum films. Thin Solid Films 484(1), 364–368 (2005) Poletkin, K.V., Gurzadyan, G.G., Shang, J., Kulish, V.: Ultrafast heat transfer on nanoscale in thin gold films. Appl. Phys. B 107(1), 137–143 (2012) Wang, H.-D., Ma, W.-G., Zhang, X., Wang, W., Guo, Z.-Y.: Theoretical and experimental study on the heat transport in metallic nanofilms heated by ultra-short pulsed laser. Int. J. Heat Mass Transf. 54(4), 967–974 (2011) Ning, Y., Imatani, S., Inoue, T.: Hyperbolic thermoelastic analysis due to pulsed heat input by numerical simulation. JSME Int. J. Ser. A Solid Mech. Mater. Eng. 49(2), 180–187 (2006) Melnik, R.V.: Discrete models of coupled dynamic thermoelasticity for stresstemperature formulations. Appl. Math. Comput. 122(1), 107–132 (2001) Youssef, H.M.: State-space approach on generalized thermoelasticity for an infinite material with a spherical cavity and variable thermal conductivity subjected to ramp-type heating. Can. Appl. Math. Q. 13, 4 (2005) Ignaczak, J., Ostoja-Starzewski, M.: Thermoelasticity with Finite Wave Speeds. Oxford University Press, Oxford (2010) Chandrasekharaiah, D.S.: Hyperbolic thermoelasticity: a review of recent literature. Appl. Mech. Rev. 51(12), 705–729 (1998). doi:10.1115/1.3098984 Engelbrecht, J., Maugin, G.A.: Deformation waves in thermoelastic media and the concept of internal variables. Arch. Appl. Mech. 66(3), 200–207 (1996) Jou, D., Casas-Vazquez, J., Lebon, G.: Extended Irreversible Thermodynamics. Springer, Berlin (1996) Wang, C.C.: The principle of fading memory. Arch. Ration. Mech. Anal. 18(5), 343–366 (1965) Tzou, D.Y.: Macro-to-Microscale Heat Transfer. The Lagging Behaviour. Taylor and Francis, New York (1997) Ivanova, E.A.: Derivation of theory of thermoviscoelasticity by means of two-component Cosserat continuum. Tech. Mech. 32(2–5), 273–286 (2012) Ivanova, E.A.: On one model of generalized continuum and its thermodynamical interpretation. In: Altenbach, H., Maugin, G.A., Erofeev, V. (eds.) Mechanics of Generalized Continua, pp. 151–174. Springer, Berlin (2011) Ivanova, E.A.: Description of mechanism of thermal conduction and internal damping by means of two component Cosserat continuum. Acta Mech. 225(3), 757–795 (2014) Babenkov, M.B.: Propagation of harmonic perturbations in a thermoelastic medium with heat relaxation. J. Appl. Mech. Tech. Phys. 54(2), 277–286 (2013) Nowacki, W.: Dynamic Problems of Thermoelasticity. Springer, New York (1975) Palmov, V.A.: The Fundamental Laws of Nature in the Non-linear Continuum Thermomechanics. Polytechnic University Publishing House, St. Petersburg (2008). (in Russian) Zhilin, P.A.: Rational Continuum Mechanics. Polytechnic University Publishing House, St. Petersburg (2012). (In Russian) Altenbach, H., Naumenko, K., Zhilin, P.A.: A micro-polar theory for binary media with application to phase-transitional flow of fiber suspensions. Contin. Mech. Thermodyn. 15(6), 539–570 (2003) Vovnenko, N.V., Zimin, B.A., Sud’enkov, Y.V.: Nonequilibrium motion of a metal surface exposed to sub-microsecond laser pulses. Zhurnal tekhnicheskoi fiziki 80(7), 41–45 (2010). (in Russian) Babenkov, M.B., Ivanova, E.A.: Analysis of the wave propagation processes in heat transfer problems of the hyperbolic type. Contin. Mech. Thermodyn. 26(4), 483–502 (2014) Vitokhin, E.Y., Babenkov, M.B.: Numerical and analytical studies of the propagation of thermoelastic waves in a medium taking into account heat flow relaxation. J. Appl. Mech. Tech. Phys. 57(3), 537–549 (2016) Lebedev, N.N., Skalskaya, I.P., Uflyand, I.P.: Worked Problems in Applied Mathematics. Dover, New York (1965)